tag:blogger.com,1999:blog-72516868255609413612018-05-26T03:42:58.325-04:00Launchings by David BressoudDavid Bressoud is DeWitt Wallace Professor of Mathematics at Macalester College in St. Paul, Minnesota, and former president of the Mathematical Association of America.
Launchings is a monthly column sponsored by the Mathematical Association of America.Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.comBlogger83125tag:blogger.com,1999:blog-7251686825560941361.post-9316475793104760592018-05-01T08:00:00.000-04:002018-05-01T13:32:06.705-04:00Trends in Mathematics Majors<span style="font-family: "cambria" , serif; font-size: 11pt; margin: 0px;">By David Bressoud </span><br /><br /><span style="margin: 0px;"><b style="mso-bidi-font-weight: normal;"><span style="font-family: "cambria" , serif; font-size: 11pt; margin: 0px;">You can now follow me on Twitter <a href="https://twitter.com/dbressoud">@dbressoud</a></span></b></span><br /><div style="margin: 0px;"><span style="margin: 0px;"><b style="mso-bidi-font-weight: normal;"><span style="font-family: "cambria" , serif; font-size: 11pt; margin: 0px;"><br /></span></b></span></div><b></b><i></i><u></u><sub></sub><sup></sup><strike></strike>By the time this column appears, the full CBMS 2015 survey of math departments should be available at <a href="http://www.ams.org/profession/data/cbms-survey/cbms2015">www.ams.org/profession/data/cbms-survey/cbms2015</a>. I reported some of the data on faculty demographics in my <a href="http://launchings.blogspot.com/2017/10/">October</a> and <a href="http://launchings.blogspot.com/2017/11/women-in-profession.html">November</a> Launchings columns. This month I want to report on what is happening to undergraduate mathematics majors.<br /><br />From 2010 to 2015, the number of bachelor’s degrees in the mathematical sciences grew by just over 3,000, from 19,242 to 22,265, almost a 16% increase (Figure 1). However, most of the growth was in Actuarial Science (from 849 to 2354), Statistics (from 858 to 1509), joint majors (e.g. biomath, the total rising from 1222 to 1821), and “other” (including Operations Research, from 231 to 907). Degrees in Mathematics Education fell from 3,614 to 2,880. Traditional mathematics and applied mathematics degrees only rose by 326, from 12,468 to 12,794.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-9eVyfGKkn7Q/WuOGwLt1j0I/AAAAAAAALG0/CTwwxTdjJZYLR68_KBJMbvgplCb2yJ5rgCLcBGAs/s1600/Figure1.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="416" data-original-width="603" src="https://3.bp.blogspot.com/-9eVyfGKkn7Q/WuOGwLt1j0I/AAAAAAAALG0/CTwwxTdjJZYLR68_KBJMbvgplCb2yJ5rgCLcBGAs/s1600/Figure1.jpg" /></a></div><div style="text-align: center;">Figure 1. Bachelor’s degrees awarded by departments of Mathematics or Statistics. </div><div style="text-align: center;">Source: CBMS Surveys.</div><div style="text-align: center;"><br /></div>For comparison, the total number of Bachelor’s degrees over the years 2010 to 2015 increased by 15%, and the number of degrees in STEM fields (specifically bioscience, computer science, engineering, mathematical sciences, or physical sciences) rose by 34%, from 238,000 to 319,000.<br /><br />The period 2010 to 2015 saw a decrease in the percentage of Bachelor’s degrees in Mathematics or Statistics earned by women, dropping from 42.4% to 40.8% (Figure 2). This does not include degrees in Mathematics Education awarded by Math departments. If we include them, then women earned 43.3% of the Bachelor’s degrees in 2015.<br /><br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-fkXi_pXsBJ4/WuOHLyBgf1I/AAAAAAAALG8/2V0e8nczLKwM64t-wtpDxrQ35pm1D-o7wCLcBGAs/s1600/Figure2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="399" data-original-width="535" src="https://2.bp.blogspot.com/-fkXi_pXsBJ4/WuOHLyBgf1I/AAAAAAAALG8/2V0e8nczLKwM64t-wtpDxrQ35pm1D-o7wCLcBGAs/s1600/Figure2.jpg" /></a></div><div class="separator" style="clear: both; text-align: center;"><span style="background-color: transparent; color: black; display: inline; float: none; font-family: "times new roman"; font-size: 16px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; text-align: left; text-decoration: none; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px;"> Figure 2. Women as % of Mathematics or Statistics Bachelor’s degrees, organized by highest degree offered by the mathematics department. Source: CBMS Surveys.</span></div><b></b><i></i><u></u><sub></sub><sup></sup><strike></strike><br /><div style="text-align: left;">Figure 3 shows the representation of African-Americans, Hispanic-American, Asian-Americans (including Pacific Islanders), and nonresident aliens. Here we are drawing on data from the National Center for Education Statistics (NCES), which is collected annually. Two trends are particularly interesting: the number of African-Americans has remained pretty much unchanged since the mid-1990s, and the number of nonresident aliens has exploded since 2007. It should be noted that NCES began allowing the designation “two or more races” in 2011. In 2011, 216 Mathematics or Statistics majors chose this designation, growing to 684 in 2016. These numbers are not reflected in Figure 3.</div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-LGcJZZ0krZ4/WuOHcJ7UzCI/AAAAAAAALHE/jVizGM13uZIEo-bWtAEY7O0wyuNGCkpYgCLcBGAs/s1600/Figure3.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="373" data-original-width="601" src="https://3.bp.blogspot.com/-LGcJZZ0krZ4/WuOHcJ7UzCI/AAAAAAAALHE/jVizGM13uZIEo-bWtAEY7O0wyuNGCkpYgCLcBGAs/s1600/Figure3.jpg" /></a></div><div style="text-align: center;"><span style="background-color: transparent; color: black; display: inline; float: none; font-family: "times new roman"; font-size: 16px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; text-align: left; text-decoration: none; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px;"> </span><span style="background-color: transparent; color: black; display: inline; float: none; font-family: "times new roman"; font-size: 16px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; text-align: left; text-decoration: none; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px;">Figure 3. Number of Mathematics or Statistics majors by race, ethnicity, or resident status.</span><br /><span style="background-color: transparent; color: black; display: inline; float: none; font-family: "times new roman"; font-size: 16px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; text-align: left; text-decoration: none; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px;">Source: NCES.</span></div><div style="text-align: center;"><span style="background-color: transparent; color: black; display: inline; float: none; font-family: "times new roman"; font-size: 16px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; text-align: left; text-decoration: none; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px;"></span><b></b><i></i><u></u><sub></sub><sup></sup><strike></strike><br /></div>The following graphs, Figures 4–7, look closer at each of these four groups, comparing their percentage of all Bachelor’s degrees, of Bachelor’s degrees in Mathematics or Statistics, and of Bachelor’s degrees in Engineering. Again, these do not include students who designated as two or more races after 2010. We see that until 2000, African Americans were well represented among Mathematics majors in the sense that their representation was comparable to their representation among all undergraduates, but since then their percentage has noticeably dropped off. Hispanic Americans are underrepresented, but the trend is promising. Not surprisingly, Asian Americans are well represented among Mathematics and Engineering majors. Non-resident aliens are growing as a percentage of all Bachelor’s degrees and all Engineering degrees, but their growth among Mathematics majors is remarkable. This attests to the importance of student visas in maintaining our mathematical workforce, but it also suggests that more could be done to attract U.S. citizens to the pursuit of Mathematics, especially African Americans.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-9DZ3-G9sulU/WuOH5QucS9I/AAAAAAAALHM/SuIb71CyT-8z3vaohMpwGrx2nFq2BtWoQCLcBGAs/s1600/Figure4.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="288" data-original-width="468" src="https://1.bp.blogspot.com/-9DZ3-G9sulU/WuOH5QucS9I/AAAAAAAALHM/SuIb71CyT-8z3vaohMpwGrx2nFq2BtWoQCLcBGAs/s1600/Figure4.jpg" /></a></div><div style="text-align: center;">Figure 4. African Americans as percentage of all bachelor’s degrees and of bachelor’s degrees in Mathematics or Statistics and in Engineering. Source: NCES.</div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/--sGDMA0zJx0/WuOIClsqyTI/AAAAAAAALHQ/WEYpqul3qighIu2rWnFFOI9LLBmMJOYrQCLcBGAs/s1600/Figure5.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="288" data-original-width="469" src="https://4.bp.blogspot.com/--sGDMA0zJx0/WuOIClsqyTI/AAAAAAAALHQ/WEYpqul3qighIu2rWnFFOI9LLBmMJOYrQCLcBGAs/s1600/Figure5.jpg" /></a></div><div style="text-align: center;">Figure 5. Hispanic Americans as percentage of all bachelor’s degrees and of bachelor’s degrees in Mathematics or Statistics and in Engineering. Source: NCES.</div><div style="text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-ipDs9U8kB_k/WuOINis5-kI/AAAAAAAALHY/OMBu8wZLZlwX3Gx4fPsTVozBITmzdXV7QCLcBGAs/s1600/Figure6.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="288" data-original-width="468" src="https://3.bp.blogspot.com/-ipDs9U8kB_k/WuOINis5-kI/AAAAAAAALHY/OMBu8wZLZlwX3Gx4fPsTVozBITmzdXV7QCLcBGAs/s1600/Figure6.jpg" /></a></div><div style="text-align: center;">Figure 6. Asian Americans and Pacific Islanders as percentage of all bachelor’s degrees and of bachelor’s degrees in Mathematics or Statistics and in Engineering. Source: NCES.</div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-gMSi8pUJVZw/WuOIdsdrVnI/AAAAAAAALHk/0q7kuWXv_zo4Q5Q_nu1NrhZjZH8pkqvAQCLcBGAs/s1600/Figure7.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="289" data-original-width="470" src="https://1.bp.blogspot.com/-gMSi8pUJVZw/WuOIdsdrVnI/AAAAAAAALHk/0q7kuWXv_zo4Q5Q_nu1NrhZjZH8pkqvAQCLcBGAs/s1600/Figure7.jpg" /></a></div><div style="text-align: center;"> Figure 7. Non-resident aliens as percentage of all bachelor’s degrees and of bachelor’s degrees in Mathematics or Statistics and in Engineering. Source: NCES.</div>Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com0tag:blogger.com,1999:blog-7251686825560941361.post-77494117930733926332018-04-01T08:00:00.000-04:002018-04-02T08:48:10.395-04:00Gaps in Student Understanding of the Fundamental Theorem of Integral CalculusBy David Bressoud<br /><br />You can now follow me on Twitter <a href="https://twitter.com/dbressoud" target="_blank"><span style="color: #0b5394;">@dbressoud</span></a><br /><span style="color: #0b5394;"></span><span style="color: #0b5394;"></span><br />I have long held the belief (<a href="http://www.jstor.org/stable/10.4169/amer.math.monthly.118.02.099?seq=1%20-%20page_scan_tab_contents" target="_blank"><span style="color: #0b5394;">Bressoud, 2011</span></a>) that we should revert to the original name, the Fundamental Theorem of Integral Calculus (FTIC), for what in the 1960s came to be known as the Fundamental Theorem of Calculus (FTC). The reason is that the real importance of this theorem is not that integration and differentiation are inverse processes—for most students that is the working definition of integration—but that we have two very distinct ways of viewing integration, as limits of Riemann sums and in terms of anti-differentiation, and that for all practical purposes they are equivalent.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-ng6K6G8uc1c/Wr0NGfKUhsI/AAAAAAAALD4/NAdvGET_3LMDJ4Ge_PD-Va4WCSGjLj40ACLcBGAs/s1600/123.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="475" data-original-width="702" height="270" src="https://2.bp.blogspot.com/-ng6K6G8uc1c/Wr0NGfKUhsI/AAAAAAAALD4/NAdvGET_3LMDJ4Ge_PD-Va4WCSGjLj40ACLcBGAs/s400/123.gif" width="400" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div style="text-align: center;">Figure 1. Students working on integral as accumulator, reproduced from the homepage of Coherent Labs to Enhance Accessible and Rigorous Calculus Instruction (<a href="http://clearcalculus.okstate.edu/" target="_blank"><span style="color: #0b5394;">CLEAR Calculus</span></a>) </div><br />A recent paper by Joseph Wagner (<a href="https://link.springer.com/article/10.1007%2Fs40753-017-0060-7" target="_blank"><span style="color: #0b5394;">2017</span></a>) is an insightful study of the confusion experienced by most students about the nature of integration. As he points out, this is not about student deficits, but about common misconceptions that can be traced to the way we teach integration.<br /><br /><a href="https://3.bp.blogspot.com/-h0D-K4LzzeI/Wr0a3t425yI/AAAAAAAALEo/vNKLeAfm_zEDe9kTuY-mfHeAuyxk-pNAwCLcBGAs/s1600/2.png" imageanchor="1" style="-webkit-text-stroke-width: 0px; background-color: transparent; clear: right; color: #0066cc; float: right; font-family: Times New Roman; font-size: 16px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; margin-bottom: 1em; margin-left: 1em; orphans: 2; text-align: center; text-decoration: underline; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px;"></a><a href="https://3.bp.blogspot.com/-h0D-K4LzzeI/Wr0a3t425yI/AAAAAAAALEo/vNKLeAfm_zEDe9kTuY-mfHeAuyxk-pNAwCLcBGAs/s1600/2.png" imageanchor="1" style="-webkit-text-stroke-width: 0px; background-color: transparent; clear: right; color: #0066cc; float: right; font-family: Times New Roman; font-size: 16px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; margin-bottom: 1em; margin-left: 1em; orphans: 2; text-align: center; text-decoration: underline; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px;"></a>Previous work by Sealey (<a href="http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.500.3209&rep=rep1&type=pdf" target="_blank"><span style="color: #0b5394;">2006</span></a>, <a href="https://www.sciencedirect.com/science/article/pii/S0732312313001065" target="_blank"><span style="color: #0b5394;">2014</span></a>) and Jones (<a href="https://www.sciencedirect.com/science/article/pii/S0732312312000612" target="_blank"><span style="color: #0b5394;">2013</span></a>, <a href="https://www.sciencedirect.com/science/article/pii/S0732312315000024" target="_blank"><span style="color: #0b5394;">2015a</span></a>, <a href="https://www.tandfonline.com/doi/abs/10.1080/0020739X.2014.1001454" target="_blank"><span style="color: #0b5394;">2015b</span></a>) has shown that<br />there are three ways in which students describe the meaning of the definite integral, <br /><b></b><i></i><u></u><sub></sub><sup></sup><strike></strike><b></b><i></i><u></u><sub></sub><sup></sup><strike></strike><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-h0D-K4LzzeI/Wr0a3t425yI/AAAAAAAALEo/vNKLeAfm_zEDe9kTuY-mfHeAuyxk-pNAwCLcBGAs/s1600/2.png" imageanchor="1" style="-webkit-text-stroke-width: 0px; background-color: transparent; color: #0066cc; font-family: Times New Roman; font-size: 16px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; margin-left: 1em; margin-right: 1em; orphans: 2; text-align: center; text-decoration: underline; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px;"><img border="0" data-original-height="29" data-original-width="77" src="https://3.bp.blogspot.com/-h0D-K4LzzeI/Wr0a3t425yI/AAAAAAAALEo/vNKLeAfm_zEDe9kTuY-mfHeAuyxk-pNAwCLcBGAs/s1600/2.png" /></a></div><ul><a href="https://3.bp.blogspot.com/-h0D-K4LzzeI/Wr0a3t425yI/AAAAAAAALEo/vNKLeAfm_zEDe9kTuY-mfHeAuyxk-pNAwCLcBGAs/s1600/2.png" imageanchor="1" style="-webkit-text-stroke-width: 0px; background-color: transparent; clear: right; color: #0066cc; float: right; font-family: Times New Roman; font-size: 16px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; margin-bottom: 1em; margin-left: 1em; orphans: 2; text-align: center; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px;"></a><li> as an area,</li><li> in terms of an antiderivative, or</li><li> in terms of a summation.</li></ul>Overwhelmingly, students employ the first, the second is common, the third is rare.<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-dokMx0qGMlI/Wr0dNxaJLKI/AAAAAAAALFA/5yaWinlxswwJrfTelYl5gF7JiSCeB448ACLcBGAs/s1600/3.png" imageanchor="1" style="-webkit-text-stroke-width: 0px; background-color: transparent; color: #0066cc; font-family: Times New Roman; font-size: 16px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; margin-left: 1em; margin-right: 1em; orphans: 2; text-align: center; text-decoration: underline; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px;"></a></div>Nevertheless, when confronted with a problem in physics that requires integration, the interpretation in terms of a summation is more common. Jones (<a href="https://www.tandfonline.com/doi/abs/10.1080/0020739X.2014.1001454" target="_blank"><span style="color: #0b5394;">2015b</span></a>), after reminding second term calculus students that force is pressure times area, asked why<a href="https://1.bp.blogspot.com/-dokMx0qGMlI/Wr0dNxaJLKI/AAAAAAAALFA/5yaWinlxswwJrfTelYl5gF7JiSCeB448ACLcBGAs/s1600/3.png" imageanchor="1" style="-webkit-text-stroke-width: 0px; background-color: transparent; color: #0066cc; font-family: Times New Roman; font-size: 16px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; margin-left: 16px; margin-right: 16px; orphans: 2; text-align: center; text-decoration: underline; text-indent: 0px; text-transform: none; white-space: normal; word-spacing: 0px;"><img border="0" data-original-height="21" data-original-width="75" src="https://1.bp.blogspot.com/-dokMx0qGMlI/Wr0dNxaJLKI/AAAAAAAALFA/5yaWinlxswwJrfTelYl5gF7JiSCeB448ACLcBGAs/s1600/3.png" /></a>calculates the total force. Of 150 students, 61 (41%) produced an argument that involved summation, although only 25 of them (17%) indicated that any product was involved.<br /><br />Following up on this insight, Wagner explored the understanding of definite integrals by physics students. He interviewed eight students in an introductory calculus-based physics course focused on classical mechanics and seven third-year physics majors. Of the students in the introductory course, five had completed both single and multi-variable calculus, two were currently enrolled in multi-variable calculus, and one was still in single variable calculus. All were in majors that required this physics course.<br /><br />When students in the introductory course were asked what Riemann sums have to do with definite integrals, they split evenly between two types of answers: either as something that accomplishes the same task as an integral (usually finding areas) or as a means of approximating definite integrals. As we shall see, the connection between integration as a limit of Riemann sums and in terms of antiderivatives was hazy at best and not recognized as significant. As Wagner reports, several were mystified why they had to learn about Riemann sums, “Because like when they were teaching this, they were kind of like oh, like you’ll do this for the first test, and then you get rid of it and never have to do it again.”<br /><br />On the other hand, the third-year physics students were much more inclined to explain the meaning of the definite integral in terms of a summation. They were conversant with how to convert an accumulation problem into a definite integral. As Wagner suggested privately, this appears to be the result of repeated exposure to problems from physics in which definite integrals arise from “slice and add” procedures.<br /><br />But Wagner uncovered an intriguing gap in their understanding. All fifteen students were asked to make up a simple area problem and then solve it. All of them did so correctly, using a polynomial function and antidifferentiation. As an example the area under the graph of y=x^3 from 0 to 2 was calculated as follows,<br /><br /><div align="center" class="MsoNormal" style="mso-layout-grid-align: none; mso-pagination: none; text-align: center; text-autospace: none;"><!--[if gte msEquation 12]><m:oMathPara><m:oMath><m:nary><m:naryPr><m:limLoc m:val="subSup"/><span style='font-size:11.0pt;mso-ansi-font-size:11.0pt; mso-bidi-font-size:11.0pt;font-family:"Cambria Math",serif;mso-ascii-font-family: "Cambria Math";mso-fareast-font-family:"Times New Roman";mso-hansi-font-family: "Cambria Math";font-style:italic;mso-bidi-font-style:normal'><m:ctrlPr></m:ctrlPr></span></m:naryPr><m:sub><span class=gi><i style='mso-bidi-font-style:normal'><span style='font-size:11.0pt; font-family:"Cambria Math",serif;mso-fareast-font-family:"Times New Roman"'><m:r>0</m:r></span></i></span></m:sub><m:sup><span class=gi><i style='mso-bidi-font-style:normal'><span style='font-size:11.0pt; font-family:"Cambria Math",serif;mso-fareast-font-family:"Times New Roman"'><m:r>2</m:r></span></i></span></m:sup><m:e><m:sSup><m:sSupPr><span style='font-size:11.0pt;mso-ansi-font-size:11.0pt;mso-bidi-font-size: 11.0pt;font-family:"Cambria Math",serif;mso-ascii-font-family:"Cambria Math"; mso-fareast-font-family:"Times New Roman";mso-hansi-font-family:"Cambria Math"; font-style:italic;mso-bidi-font-style:normal'><m:ctrlPr></m:ctrlPr></span></m:sSupPr><m:e><span class=gi><i style='mso-bidi-font-style:normal'><span style='font-size: 11.0pt;font-family:"Cambria Math",serif;mso-fareast-font-family:"Times New Roman"'><m:r>x</m:r></span></i></span></m:e><m:sup><span class=gi><i style='mso-bidi-font-style:normal'><span style='font-size: 11.0pt;font-family:"Cambria Math",serif;mso-fareast-font-family:"Times New Roman"'><m:r>3</m:r></span></i></span></m:sup></m:sSup><span class=gi><i style='mso-bidi-font-style:normal'><span style='font-size:11.0pt; font-family:"Cambria Math",serif;mso-fareast-font-family:"Times New Roman"'><m:r> </m:r><m:r>dx</m:r></span></i></span></m:e></m:nary><span class=gi><i style='mso-bidi-font-style:normal'><span style='font-size:11.0pt;font-family: "Cambria Math",serif;mso-fareast-font-family:"Times New Roman"'><m:r>=</m:r></span></i></span><m:sSubSup><m:sSubSupPr><span style='font-size:11.0pt;mso-ansi-font-size:11.0pt;mso-bidi-font-size:11.0pt; font-family:"Cambria Math",serif;mso-ascii-font-family:"Cambria Math"; mso-fareast-font-family:"Times New Roman";mso-hansi-font-family:"Cambria Math"; font-style:italic;mso-bidi-font-style:normal'><m:ctrlPr></m:ctrlPr></span></m:sSubSupPr><m:e><m:d><m:dPr><m:begChr m:val=""/><m:endChr m:val="|"/><span style='font-size:11.0pt;mso-ansi-font-size: 11.0pt;mso-bidi-font-size:11.0pt;font-family:"Cambria Math",serif; mso-ascii-font-family:"Cambria Math";mso-fareast-font-family:"Times New Roman"; mso-hansi-font-family:"Cambria Math";font-style:italic;mso-bidi-font-style: normal'><m:ctrlPr></m:ctrlPr></span></m:dPr><m:e><m:f><m:fPr><span style='font-size:11.0pt;mso-ansi-font-size:11.0pt;mso-bidi-font-size: 11.0pt;font-family:"Cambria Math",serif;mso-ascii-font-family:"Cambria Math"; mso-fareast-font-family:"Times New Roman";mso-hansi-font-family:"Cambria Math"; font-style:italic;mso-bidi-font-style:normal'><m:ctrlPr></m:ctrlPr></span></m:fPr><m:num><span class=gi><i style='mso-bidi-font-style:normal'><span style='font-size: 11.0pt;font-family:"Cambria Math",serif;mso-fareast-font-family:"Times New Roman"'><m:r>1</m:r></span></i></span></m:num><m:den><span class=gi><i style='mso-bidi-font-style:normal'><span style='font-size: 11.0pt;font-family:"Cambria Math",serif;mso-fareast-font-family:"Times New Roman"'><m:r>4</m:r></span></i></span></m:den></m:f><m:sSup><m:sSupPr><span style='font-size:11.0pt;mso-ansi-font-size:11.0pt;mso-bidi-font-size: 11.0pt;font-family:"Cambria Math",serif;mso-ascii-font-family:"Cambria Math"; mso-fareast-font-family:"Times New Roman";mso-hansi-font-family:"Cambria Math"; font-style:italic;mso-bidi-font-style:normal'><m:ctrlPr></m:ctrlPr></span></m:sSupPr><m:e><span class=gi><i style='mso-bidi-font-style:normal'><span style='font-size: 11.0pt;font-family:"Cambria Math",serif;mso-fareast-font-family:"Times New Roman"'><m:r>x</m:r></span></i></span></m:e><m:sup><span class=gi><i style='mso-bidi-font-style:normal'><span style='font-size: 11.0pt;font-family:"Cambria Math",serif;mso-fareast-font-family:"Times New Roman"'><m:r>4</m:r></span></i></span></m:sup></m:sSup></m:e></m:d></m:e><m:sub><span class=gi><i style='mso-bidi-font-style:normal'><span style='font-size:11.0pt; font-family:"Cambria Math",serif;mso-fareast-font-family:"Times New Roman"'><m:r>0</m:r></span></i></span></m:sub><m:sup><span class=gi><i style='mso-bidi-font-style:normal'><span style='font-size:11.0pt; font-family:"Cambria Math",serif;mso-fareast-font-family:"Times New Roman"'><m:r>2</m:r></span></i></span></m:sup></m:sSubSup><span class=gi><i style='mso-bidi-font-style:normal'><span style='font-size:11.0pt; font-family:"Cambria Math",serif;mso-fareast-font-family:"Times New Roman"'><m:r>=</m:r></span></i></span><m:f><m:fPr><span style='font-size:11.0pt;mso-ansi-font-size:11.0pt;mso-bidi-font-size:11.0pt; font-family:"Cambria Math",serif;mso-ascii-font-family:"Cambria Math"; mso-fareast-font-family:"Times New Roman";mso-hansi-font-family:"Cambria Math"; font-style:italic;mso-bidi-font-style:normal'><m:ctrlPr></m:ctrlPr></span></m:fPr><m:num><span class=gi><i style='mso-bidi-font-style:normal'><span style='font-size:11.0pt; font-family:"Cambria Math",serif;mso-fareast-font-family:"Times New Roman"'><m:r>16</m:r></span></i></span></m:num><m:den><span class=gi><i style='mso-bidi-font-style:normal'><span style='font-size:11.0pt; font-family:"Cambria Math",serif;mso-fareast-font-family:"Times New Roman"'><m:r>4</m:r></span></i></span></m:den></m:f><span class=gi><i style='mso-bidi-font-style:normal'><span style='font-size:11.0pt; font-family:"Cambria Math",serif;mso-fareast-font-family:"Times New Roman"'><m:r>-0=4.</m:r></span></i></span></m:oMath></m:oMathPara><![endif]--><!--[if !msEquation]--><span style="font-family: "times new roman" , serif; font-size: 12.0pt;"><!--[if gte vml 1]><v:shapetype id="_x0000_t75" coordsize="21600,21600" o:spt="75" o:preferrelative="t" path="m@4@5l@4@11@9@11@9@5xe" filled="f" stroked="f"> <v:stroke joinstyle="miter"/> <v:formulas> <v:f eqn="if lineDrawn pixelLineWidth 0"/> <v:f eqn="sum @0 1 0"/> <v:f eqn="sum 0 0 @1"/> <v:f eqn="prod @2 1 2"/> <v:f eqn="prod @3 21600 pixelWidth"/> <v:f eqn="prod @3 21600 pixelHeight"/> <v:f eqn="sum @0 0 1"/> <v:f eqn="prod @6 1 2"/> <v:f eqn="prod @7 21600 pixelWidth"/> <v:f eqn="sum @8 21600 0"/> <v:f eqn="prod @7 21600 pixelHeight"/> <v:f eqn="sum @10 21600 0"/> </v:formulas> <v:path o:extrusionok="f" gradientshapeok="t" o:connecttype="rect"/> <o:lock v:ext="edit" aspectratio="t"/></v:shapetype><v:shape id="_x0000_i1025" type="#_x0000_t75" style='width:150.6pt; height:30.6pt'> <v:imagedata src="file:///C:\Users\AMARTI~1\AppData\Local\Temp\msohtmlclip1\01\clip_image001.png" o:title="" chromakey="white"/></v:shape><![endif]--><!--[if !vml]--><!--[endif]--></span><!--[endif]--><span class="gi"><span style="font-family: "cambria" , serif; font-size: 11.0pt;"><o:p></o:p></span></span></div><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-4hOLaeVV7cY/Wr0TbVnxKpI/AAAAAAAALEM/OAgulx_GFOQqBaWS8_yfm9qkEncvfPiRQCLcBGAs/s1600/1.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="159" data-original-width="560" height="89" src="https://2.bp.blogspot.com/-4hOLaeVV7cY/Wr0TbVnxKpI/AAAAAAAALEM/OAgulx_GFOQqBaWS8_yfm9qkEncvfPiRQCLcBGAs/s320/1.png" width="320" /></a></div><br />He then pushed each of these students to explain why this sequence of calculations produced the area. Only one of the fifteen, a third-year physics student, indicated that this was a consequence of FTC. Several of the others struggled to make sense of how the symbols in the definite integral led to the functional transformation implied by the first equality. Wagner argues that many students are looking for algebraic sense-making in that first equality. With two of the third-year students, he documented their growing sense of frustration as they realized that they could not explain why it works. Quoting the first student:<br /><br />"Yeah, I do it. I don’t–. I’m not proud of it, but I hope there is some way to justify it. […] When I think about integration as a sum of differentials, quantities–. When I think about that, I go, OK, that makes intuitive sense, and it works. Great. But then I wonder, you know, what is, in terms of more modernized math that I’m doing. Because I usually feel like what I’m doing is kind of a trick. And it works. I don’t feel great about doing this, like, intuitively I feel fine."<br /><br />From the second student:<br /><br />"So math gives us these sort of weird tools, and they behave differently than any, like, the physical tools we know of, and it doesn’t really make sense to ask why they work or how they work, because they work mathematically, not physically. So this mathematical tool called the integral allows us to change functions, to apply this operation that changes functions into other functions."<br /><br />Wagner concludes this article with a thoughtful discussion of the distinction between the algebraic equivalence of two expressions, a notion of equivalence with which students are familiar, and the transformational equivalence that is enabled by FTC. As he laments, “Nothing, however, in the standard calculus curriculum prepares students for the sudden transition from making sense of the symbolic processes of algebra to making sense of the symbolic processes of calculus.” He points out that a great deal of attention has been devoted to a Riemann-sum based understanding of the definite integral, but virtually none to helping students understand the transformational aspects of calculus that are so central.<br /><br />I believe that a shift from FTC to FTIC can help. As Thompson with others (<a href="https://www.cambridge.org/core/books/making-the-connection/1136E373DB89B6BFC7C7E4B23F074303" target="_blank"><span style="color: #0b5394;">2008</span></a>, <a href="https://www.researchgate.net/publication/306108323_A_Coherent_Approach_to_the_Fundamental_Theorem_of_Calculus_Using_Differentials" target="_blank"><span style="color: #0b5394;">2013</span></a>,<span style="color: #0b5394;"> </span><a href="https://www.researchgate.net/publication/306108323_A_Coherent_Approach_to_the_Fundamental_Theorem_of_Calculus_Using_Differentials" target="_blank"><span style="color: #0b5394;">2016</span></a>) has shown, and I have discussed in earlier columns (<a href="http://launchings.blogspot.com/2017/05/re-imagining-calculus-curriculum-i.html" target="_blank"><span style="color: #0b5394;">Re-imagining the Calculus Curriculum, I</span></a>, and<span style="color: blue;"> <a href="http://launchings.blogspot.com/2017/06/re-imagining-calculus-curriculum-ii.html" target="_blank"><span style="color: #0b5394;">Re-</span><span style="color: #0b5394;">imagining the Calculus Curriculum, II</span></a></span>), it makes sense to first develop the definite integral as an accumulator, making it very clear that Riemann sums are neither an introduction to a subject that eventually will be about antide<span style="color: #0b5394;"></span>rivatives nor just a tool for finding approximations, but the very essence of what a definite integral is<span style="color: #0b5394;"></span> and how it is used. Then, we bring in FTIC to show that there is another—entirely distinct because it is transformational—expression for this same integral and that this equivalent expression facilitates calculation. Wagner’s third-year physics students were struggling because they failed to realize that integration has these two very different manifestations. It is a very big deal that it does.<span style="color: #0b5394;"></span><br /><b><br /></b><b>References</b><br /><br />Bressoud, D. (2011). Historical reflections on teaching the Fundamental Theorem of Integral Calculus. American Mathematical Monthly. 118:99–115. <a href="http://www.jstor.org/stable/10.4169/amer.math.monthly.118.02.099?seq=1%20-%20page_scan_tab_contents" target="_blank"><span style="color: #0b5394;">http://www.jstor.org/stable/10.4169/amer.math.monthly.118.02.099?seq=1 - page_scan_tab_contents</span></a><br /><span style="color: #0b5394;"></span><br />Jones, S. R. (2013). Understanding the integral: Students’ symbolic forms. The Journal of Mathematical Behavior, 32(2), 122–141. <a href="https://www.sciencedirect.com/science/article/pii/S0732312312000612"><span style="color: #0b5394;">https://www.sciencedirect.com/science/article/pii/S0732312312000612</span></a><br /><span style="color: #0b5394;"></span><br />Jones, S. R. (2015a). Areas, anti-derivatives, and adding up pieces: Integrals in pure mathematics and applied contexts. The Journal of Mathematical Behavior, 38(1), 9–28. <a href="https://www.sciencedirect.com/science/article/pii/S0732312315000024"><span style="color: #0b5394;">https://www.sciencedirect.com/science/article/pii/S0732312315000024</span></a><br /><span style="color: #0b5394;"></span><br />Jones, S. R. (2015b). The prevalence of area-under-a-curve and anti-derivative conceptions over Riemann sum-based conceptions in students’ explanations of definite integrals. International Journal of Mathematics Education in Science and Technology, 46(5), 721–736. <a href="http://www.tandfonline.com/doi/abs/10.1080/0020739X.2014.1001454"><span style="color: #0b5394;">http://www.tandfonline.com/doi/abs/10.1080/0020739X.2014.1001454</span></a><br /><span style="color: #0b5394;"></span><br />Sealey, V. (2006). Definite integrals, Riemann sums, and area under a curve: What is necessary and sufficient. In S. Alatorre, J. L. Cortina, M. Sáiz & A. Méndez (Eds.) Proceedings of the 28th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 46-53). Mérida: Universidad Pedagógica Nacional. <a href="http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.500.3209&rep=rep1&type=pdf"><span style="color: #0b5394;">http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.500.3209&rep=rep1&type=pdf</span></a><br /><span style="color: #0b5394;"></span><br />Sealey, V. (2014). A framework for characterizing student understanding of Riemann sums and definite integrals. The Journal of Mathematical Behavior, 33, 230–245. <a href="https://www.sciencedirect.com/science/article/pii/S0732312313001065"><span style="color: #0b5394;">https://www.sciencedirect.com/science/article/pii/S0732312313001065</span></a><br /><span style="color: #0b5394;"></span><br />Thompson, P.W., and Silverman, J. (2008). The concept of accumulation in calculus. In M.P. Carlson & C. Rasmussen (Eds.), Making the connection: Research and teaching in undergraduate mathematics (MAA Notes Vol. 73, pp. 43–52). Washington, DC: Mathematical Association of America. <a href="https://doi.org/10.5948/UPO9780883859759.005"><span style="color: #0b5394;">https://doi.org/10.5948/UPO9780883859759.005</span></a><br /><span style="color: #3d85c6;"></span><span style="color: #0b5394;"></span><br />Thompson, P.W., Byerley, C. and Hatfield, N. (2013). A conceptual approach to calculus made possible by technology. Computers in the Schools. 30:124–147. <a href="http://pat-thompson.net/PDFversions/2013CalcTech.pdf"><span style="color: #0b5394;">http://pat-thompson.net/PDFversions/2013CalcTech.pdf</span></a><br /><span style="color: #0b5394;"></span><span style="color: #0b5394;"></span><br />Thompson, P.W., and Dreyfus, T. (2016). A coherent approach to the Fundamental Theorem of Calculus using differentials. In R. Göller. R. Biehler & R. Hochsmuth (Eds.), Proceedings of the Conference on Didactics of Mathematics in Higher Education as a Scientific Discipline (pp. 355–359 ) Hannover, Germany: KHDM. <a href="https://www.researchgate.net/publication/306108323_A_Coherent_Approach_to_the_Fundamental_Theorem_of_Calculus_Using_Differentials"><span style="color: #0b5394;">https://www.researchgate.net/publication/306108323_A_Coherent_Approach_to_the_Fundamental_Theorem_of_Calculus_Using_Differentials</span></a><br /><span style="color: #0b5394;"></span><br />Wagner, J.F. (2017). Students’ obstacles to using Riemann sum interpretations of the definite integral. International Journal of Research in Undergraduate Mathematics Education. <a href="https://doi.org/10.1007/s40753-017-0060-7"><span style="color: #0b5394;">https://doi.org/10.1007/s40753-017-0060-7</span></a><br /><div><span style="color: #073763;"></span><span style="color: #0b5394;"></span><br /></div>Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com0tag:blogger.com,1999:blog-7251686825560941361.post-39834933171877910112018-03-01T07:00:00.000-05:002018-03-05T08:56:49.906-05:00A False Dichotomy: Lecture vs. Active Learning<span style="font-family: "times" , "times new roman" , serif;">By David Bressoud</span><br /><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><div class="separator" style="clear: both; text-align: center;"><a href="https://www.blogger.com/blogger.g?blogID=7251686825560941361" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"></a></div><div class="separator" style="clear: both; text-align: center;"><a href="https://www.blogger.com/blogger.g?blogID=7251686825560941361" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"></a></div><span style="font-family: "times" , "times new roman" , serif;"><br /></span></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><span style="font-family: "times" , "times new roman" , serif;"><b><span style="color: black; font-family: "times new roman" , serif; mso-fareast-font-family: "Times New Roman";">You can now follow me on Twitter @dbressoud</span></b><o:p></o:p></span></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><span style="color: black; font-family: "times" , "times new roman" , serif; mso-fareast-font-family: "Times New Roman";"><br /></span><br /><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"></div><span style="font-family: "times" , "times new roman" , serif;"><span style="color: black; font-family: "times new roman" , serif; mso-fareast-font-family: "Times New Roman";">On January 31, I published a piece in </span><a href="https://theconversation.com/us/topics/mathematics-98"><i><span style="color: blue;">The Conversation</span></i></a><span style="color: black; font-family: "times new roman" , serif; mso-fareast-font-family: "Times New Roman";">, “</span><a href="https://theconversation.com/why-colleges-must-change-how-they-teach-calculus-90679"><span style="color: blue;">Why Colleges Must Change How They Teach Calculus</span></a><span style="color: black; font-family: "times new roman" , serif; mso-fareast-font-family: "Times New Roman";">.” The following is one of the statements that I made in this article:</span><o:p></o:p></span></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in; margin-left: .5in; margin-right: 0in; margin-top: 0in;"><span style="color: black; font-family: "times" , "times new roman" , serif; mso-fareast-font-family: "Times New Roman";"><br /></span><span style="font-family: "times" , "times new roman" , serif;"><span style="color: black; font-family: "times new roman" , serif; mso-fareast-font-family: "Times New Roman";">Active learning does not mean ban all lectures. A lecture is still the most effective means for conveying a great deal of information in a short amount of time. But the most useful lectures come in short bursts when students are primed with a need and desire to know the information. </span><o:p></o:p></span><br /><span style="color: black; font-family: "times" , "times new roman" , serif; mso-fareast-font-family: "Times New Roman";"><br /></span></div><div align="center" class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in; text-align: center;"><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-WD4VfJtXXB8/WpchT6CL0NI/AAAAAAAALCw/0JCWkx8HIfoBj8cCKM3009k4Jg8jgevWQCLcBGAs/s1600/David%2527s%2BPic.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><span style="font-family: "times" , "times new roman" , serif;"><img border="0" data-original-height="400" data-original-width="1180" height="216" src="https://1.bp.blogspot.com/-WD4VfJtXXB8/WpchT6CL0NI/AAAAAAAALCw/0JCWkx8HIfoBj8cCKM3009k4Jg8jgevWQCLcBGAs/s640/David%2527s%2BPic.jpg" width="640" /></span></a></div><span style="font-family: "times" , "times new roman" , serif;"><br /></span></div><div align="center" class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in; text-align: center;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: black; font-family: "times new roman" , serif; mso-fareast-font-family: "Times New Roman";">Figure: Image from the </span><a href="https://www.aau.edu/education-service/undergraduate-education/undergraduate-stem-education-initiative"><span style="color: blue;">AAU Undergraduate STEM Initiative homepage</span></a><span style="color: black; font-family: "times new roman" , serif; mso-fareast-font-family: "Times New Roman";">.</span><o:p></o:p></span></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: black; font-family: "times new roman" , serif; mso-fareast-font-family: "Times New Roman";">There is no simple binary choice between an active learning classroom and straight lecture. Furthermore, making a class an effective locus for student learning requires more than just active learning. </span></span></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: black; font-family: "times new roman" , serif; mso-fareast-font-family: "Times New Roman";">An article by </span><a href="https://link.springer.com/article/10.1007/s11162-016-9440-0"><span style="color: blue;">Campbell et al. (2017)</span></a><span style="color: black; font-family: "times new roman" , serif; mso-fareast-font-family: "Times New Roman";">, “From Comprehensive to Singular: A latent class analysis of college teaching practices,” reports on an interesting study of what happens in college classes (not just STEM classes), adding a few layers of complexity that are useful for anyone thinking about how to be a more effective teacher. The authors observed 587 courses in nine colleges and universities, ranging from Research 1 (public and private) to comprehensive state schools to liberal arts colleges at a range of levels of selectivity. They looked for seven types of activities in the classroom.</span><o:p></o:p></span></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><a href="https://www.blogger.com/blogger.g?blogID=7251686825560941361" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"></a><a href="https://www.blogger.com/blogger.g?blogID=7251686825560941361" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"></a><span style="font-family: "times" , "times new roman" , serif;"><br /></span></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: black; font-family: "times new roman" , serif; mso-fareast-font-family: "Times New Roman";">One of these is <b>lecture</b>, defined as “A presentation or recitation of course content by the faculty member to all students in the class.”</span><o:p></o:p></span></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: black; font-family: "times new roman" , serif; mso-fareast-font-family: "Times New Roman";">They split active learning into three sub-categories:</span><o:p></o:p></span><br /><span style="font-family: "times" , "times new roman" , serif;"><span style="color: black; font-family: "times new roman" , serif; mso-fareast-font-family: "Times New Roman";"><br /></span></span></div><ul style="margin-top: 0in;" type="disc"><li class="MsoNormal" style="color: black; line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in; mso-list: l0 level1 lfo1; tab-stops: list .5in; vertical-align: baseline;"><span style="font-family: "times" , "times new roman" , serif;"><b><span style="font-family: "times new roman" , serif; mso-fareast-font-family: "Times New Roman";">Class discussion.</span></b><span style="font-family: "times new roman" , serif; mso-fareast-font-family: "Times New Roman";"> Back and forth conversation between instructor and students or among students about the course content.</span></span></li></ul><ul style="margin-top: 0in;" type="disc"><li class="MsoNormal" style="color: black; line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in; mso-list: l0 level1 lfo1; tab-stops: list .5in; vertical-align: baseline;"><span style="font-family: "times" , "times new roman" , serif;"><b><span style="font-family: "times new roman" , serif; mso-fareast-font-family: "Times New Roman";">Class activities. </span></b><span style="font-family: "times new roman" , serif; mso-fareast-font-family: "Times New Roman";">A structured activity where students engaged with the course content (e.g., case studies, clickers, group work).</span></span></li></ul><ul style="margin-top: 0in;" type="disc"><li class="MsoNormal" style="color: black; line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in; mso-list: l0 level1 lfo1; tab-stops: list .5in; vertical-align: baseline;"><span style="font-family: "times" , "times new roman" , serif;"><b><span style="font-family: "times new roman" , serif; mso-fareast-font-family: "Times New Roman";">Student questions.</span></b><span style="font-family: "times new roman" , serif; mso-fareast-font-family: "Times New Roman";"> Students asking individual questions of the instructor about the course content.</span></span></li></ul><span style="font-family: "times new roman" , serif;"><br /></span><ul style="margin-top: 0in;" type="disc"></ul><ul style="margin-top: 0in;" type="disc"></ul><span style="color: black; font-family: "times new roman" , serif; mso-fareast-font-family: "Times New Roman";">They also picked up the three practices laid out in </span><a href="http://www.ashe.ws/files/Past%20Presidents/37.2.neumann.pdf" style="font-family: times, "times new roman", serif;"><span style="color: blue;">Neumann’s (2014) description</span></a><span style="color: black; font-family: "times new roman" , serif; mso-fareast-font-family: "Times New Roman";"> of cognitively response teaching. Active teaching should be cognitively responsive. Unfortunately, as their observations showed, it often is not. These three practices are:</span><br /><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span><br /><ul style="margin-top: 0in;" type="disc"><li class="MsoNormal" style="line-height: normal; margin-bottom: 0in; vertical-align: baseline;"><span style="font-family: "times" , "times new roman" , serif;"><b><span style="font-family: "times new roman" , serif; mso-fareast-font-family: "Times New Roman";">Core subject matter ideas. </span></b><span style="font-family: "times new roman" , serif; mso-fareast-font-family: "Times New Roman";">The instructor introduced in depth one or more concepts that are central to the subject matter of the course, the instructor created multiple representations of “core ideas,” or the instructor introduced students to how ideas play out in the field.</span></span></li></ul><ul style="margin-top: 0in;" type="disc"><li class="MsoNormal" style="line-height: normal; margin-bottom: 0in; vertical-align: baseline;"><span style="font-family: "times" , "times new roman" , serif;"><b><span style="font-family: "times new roman" , serif; mso-fareast-font-family: "Times New Roman";">Connections to prior knowledge. </span></b><span style="font-family: "times new roman" , serif; mso-fareast-font-family: "Times New Roman";">The instructor surfaced students’ prior knowledge about the subject “core ideas,” or the instructor worked to understand students’ prior knowledge about the subject matter “core ideas.”</span></span></li></ul><ul style="margin-top: 0in;" type="disc"><a href="https://www.blogger.com/blogger.g?blogID=7251686825560941361" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><a href="https://www.blogger.com/blogger.g?blogID=7251686825560941361" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><li class="MsoNormal" style="line-height: normal; margin-bottom: 0in; vertical-align: baseline;"><span style="font-family: "times" , "times new roman" , serif;"><b><span style="font-family: "times new roman" , serif; mso-fareast-font-family: "Times New Roman";">Support of changing views. </span></b><span style="font-family: "times new roman" , serif; mso-fareast-font-family: "Times New Roman";">The instructor provided a space for students to encounter dissonance between prior knowledge and new course material, or the instructor helped students to realize the difference similarities and sometimes conflict between prior knowledge and new subject matter ideas.</span></span></li></ul><span style="font-family: "times new roman" , serif;"><br /></span><span style="color: black; font-family: "times new roman" , serif; mso-fareast-font-family: "Times New Roman";">Developing over the past few decades and now accelerating thanks to the work of the community engaged in research in undergraduate mathematics education, there have been remarkable strides in understanding the misconceptions that are barriers to student learning. To cite just two examples that I have discussed elsewhere, students often have difficulty making the transition from trigonometric functions in terms of triangles to the circle definition, and they tend to interpret functions as static objects, impeding an understanding of them as descriptions of the linkage between variables that vary. I discussed this issue of the disconnection between what we say and what students hear in two columns in 2016: </span><a href="http://launchings.blogspot.com/2016/02/" style="font-family: times, "times new roman", serif;"><span style="color: blue;">What we say/what they hear</span></a><span style="color: black; font-family: "times new roman" , serif; mso-fareast-font-family: "Times New Roman";">and </span><a href="http://launchings.blogspot.com/2016/03/" style="font-family: times, "times new roman", serif;"><span style="color: blue;">What we say/what they hear II</span></a><span style="color: black; font-family: "times new roman" , serif; mso-fareast-font-family: "Times New Roman";">. The instructor who does not try to understand the prior conceptions and knowledge that students bring into the classroom is setting a large proportion of the students up for failure.</span></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: black; font-family: "times new roman" , serif; mso-fareast-font-family: "Times New Roman";">For the last practice, support of changing views, the physics education community knows how important this is. With their Force Concept Inventory (FCI), Halloun, Hestenes, and Wells (see </span><a href="about:blank"><span style="color: blue;">Hestenes et al., 1992</span></a><span style="color: black; font-family: "times new roman" , serif; mso-fareast-font-family: "Times New Roman";">) demonstrated that prior concepts are powerful. Students are reluctant to release them, even in the face of what instructors consider to be clear exposition of the actual state of affairs. Getting students to recognize cognitive dissonance requires skill.</span><o:p></o:p></span></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: black; font-family: "times new roman" , serif; mso-fareast-font-family: "Times New Roman";">Campbell et al. observed that traditional lecture—what the Progress through Calculus study (</span><a href="https://www.maa.org/sites/default/files/PtC%20Technical%20Report_Final.pdf"><span style="color: blue;">Apkarian and Kirin, 2017</span></a><span style="color: black; font-family: "times new roman" , serif; mso-fareast-font-family: "Times New Roman";">) has revealed to be standard practice in 72% of all Calculus I classes in university mathematics departments with PhD programs—did a pretty good job on <b>core subject matter ideas</b>, but almost nothing with <b>connections to prior knowledge</b> or <b>support of changing views</b>. And, of course, traditional lecture involved none of the first two active learning sub-categories. Less obvious but not surprising, <b>student questions</b> were seldom observed in traditional lecture.</span><o:p></o:p></span></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: black; font-family: "times new roman" , serif; mso-fareast-font-family: "Times New Roman";">Active lecture is the second most common form of calculus instruction, found in about 14% of the PhD-granting mathematics departments we surveyed in progress through Calculus (3% of departments relied mainly on active learning practices in the classroom and the remaining departments reported too much variation by instructor to classify their course as one type). These introduced <b>class activities</b> and did not decrease <b>core subject matter ideas</b>. Campbell et al. found that they noticeably increase <b>student questions</b>, but do nothing in and of themselves to improve <b>connections to prior knowledge</b> or <b>support of changing views</b>. </span><o:p></o:p></span></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: black; font-family: "times new roman" , serif; mso-fareast-font-family: "Times New Roman";">These last two practices were almost never observed in either traditional or active lecture classes. The only classes that were observed to improve these aspects of cognitively responsive teaching were those that made a point of employing all seven behaviors, including lecture. In other words, <b>connections to prior knowledge</b> and <b>support of changing views</b> do not come for free once one is using active learning. They have to be intentionally incorporated, and they rely heavily on carefully guided <b>class discussion</b>.</span><o:p></o:p></span></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: black; font-family: "times new roman" , serif; mso-fareast-font-family: "Times New Roman";">The lesson is that lecture has its place, and active learning is only one piece of what is needed for a truly effective class. </span><a href="http://aapt.scitation.org/doi/abs/10.1119/1.18898"><span style="color: blue;">David Hestenes (1998)</span></a><span style="color: black; font-family: "times new roman" , serif; mso-fareast-font-family: "Times New Roman";"> summed it up nicely in “Who needs physics education research!?”:</span><o:p></o:p></span></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></div><div class="MsoNormal" style="line-height: normal; margin-bottom: 12.0pt; margin-left: .5in; margin-right: 0in; margin-top: 0in;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: black; font-family: "times new roman" , serif; mso-fareast-font-family: "Times New Roman";">Managing <i>the quality of classroom discourse </i>is the single most important factor in teaching with interactive engagement methods. This factor accounts for wide differences in class FCI score among teachers using the same curriculum materials and purportedly the same teaching methods. Effective discourse management requires careful planning and preparation as well as skill and experience … <i>Effective teaching requires complex skills </i>which take years to develop.<sup> </sup>Technical knowledge about teaching and learning is as essential as subject content knowledge. </span><o:p></o:p></span></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><span style="font-family: "times" , "times new roman" , serif;"><b><span style="color: black;">References</span></b><o:p></o:p></span></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in; mso-outline-level: 3; text-indent: -.2in;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: black; font-family: "times new roman" , serif; mso-fareast-font-family: "Times New Roman";"> Apkarian, N. and Kirin, D. 2017. <i>Progress through Calculus: Census Survey Report</i>. </span><b><a href="https://www.maa.org/sites/default/files/PtC%20Technical%20Report_Final.pdf"><span style="color: blue; font-weight: normal;">https://www.maa.org/sites/default/files/PtC Technical Report_Final.pdf</span></a><o:p></o:p></b></span></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in; mso-outline-level: 3; text-indent: -.2in;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: black; font-family: "times new roman" , serif; mso-fareast-font-family: "Times New Roman";">Bressoud, D. 2016. What we say/What they hear. <i>Launchings</i>. </span><b><a href="http://launchings.blogspot.com/2016/02/"><span style="color: blue; font-weight: normal;">http://launchings.blogspot.com/2016/02/</span></a><o:p></o:p></b></span></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in; mso-outline-level: 3; text-indent: -.2in;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: black; font-family: "times new roman" , serif; mso-fareast-font-family: "Times New Roman";">Bressoud, D. 2016. What we say/What they hear. II. <i>Launchings</i>. </span><b><a href="http://launchings.blogspot.com/2016/03/"><span style="color: blue; font-weight: normal;">http://launchings.blogspot.com/2016/03/</span></a><o:p></o:p></b></span></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in; text-indent: -.2in;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: black; font-family: "times new roman" , serif; mso-fareast-font-family: "Times New Roman";">Bressoud, D. 2018. Why colleges must change how they teach calculus. <i>TheConversation</i>. January 31, 2018. </span><a href="https://theconversation.com/why-colleges-must-change-how-they-teach-calculus-90679"><span style="color: blue;">https://theconversation.com/why-colleges-must-change-how-they-teach-calculus-90679</span></a><o:p></o:p></span></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in; text-indent: -.2in;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: black; font-family: "times new roman" , serif; mso-fareast-font-family: "Times New Roman";">Campbell, C.M., Cabrera, A.F., Michel, J.O., and Patel, S. 2017. From Comprehensive to Singular: A Latent Class Analysis of College Teaching Practices. <i>Research in Higher Education</i>. <b>58</b>: 581–604.</span><span style="color: black; font-family: "ms mincho" , serif; mso-bidi-font-family: "Times New Roman"; mso-hansi-font-family: "Times New Roman";"> </span><a href="https://link.springer.com/article/10.1007/s11162-016-9440-0"><span style="color: blue;">https://link.springer.com/article/10.1007/s11162-016-9440-0</span></a><o:p></o:p></span></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in; text-indent: -.2in;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: black; font-family: "times new roman" , serif; mso-fareast-font-family: "Times New Roman";">Hestenes D., Wells M., Swackhamer G. 1992. Force concept inventory. <i>The Physics Teacher</i> <b>30</b>: 141-166. </span><a href="http://aapt.scitation.org/doi/10.1119/1.2343497"><span style="color: blue;">http://aapt.scitation.org/doi/10.1119/1.2343497</span></a><o:p></o:p></span></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in; text-indent: -.2in;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: black; font-family: "times new roman" , serif; mso-fareast-font-family: "Times New Roman";">Hestenes D. 1998. Who needs physics education research!?. <i>Am. J. Phys</i>. <b>66</b>:46.5. </span><a href="http://aapt.scitation.org/doi/abs/10.1119/1.18898"><span style="color: blue;">http://aapt.scitation.org/doi/abs/10.1119/1.18898</span></a><o:p></o:p></span></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in; text-indent: -.2in;"><span style="font-family: "times" , "times new roman" , serif;"><span style="color: black; font-family: "times new roman" , serif; mso-fareast-font-family: "Times New Roman";">Mathematical Association of America. 2017. <i>Instructional Practices Guide</i>. </span><a href="https://www.maa.org/programs-and-communities/curriculum%20resources/instructional-practices-guide"><span style="color: blue;">https://www.maa.org/programs-and-communities/curriculum resources/instructional-practices-guide</span></a><o:p></o:p></span></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in;"><span style="font-family: "times" , "times new roman" , serif;"><br /></span></div><div class="MsoNormal" style="line-height: normal; margin-bottom: .0001pt; margin-bottom: 0in; text-indent: -.2in;"><span style="color: black; font-family: "times" , "times new roman" , serif;">Neumann, A. 2014. Staking a claim on learning: What we should know about learning in higher education and why. <i>The Review of higher Education</i>. <b>37</b>:249–267. </span><a href="http://www.ashe.ws/files/Past%20Presidents/37.2.neumann.pdf"><span style="color: blue;"><span style="font-family: "times" , "times new roman" , serif;">http://www.ashe.ws/files/Past Presidents/37.2.neumann.</span><span style="font-family: "times new roman" , serif; font-size: x-small;">pdf</span></span></a><span style="font-family: "times new roman" , serif; font-size: x-small;"><o:p></o:p></span></div><div class="MsoNormal"><br /></div><br />Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com0tag:blogger.com,1999:blog-7251686825560941361.post-70795759343297317012018-02-01T07:00:00.000-05:002018-02-01T07:00:05.098-05:00Getting to Know the IP Guide<b><i>You can follow me on Twitter <a href="https://twitter.com/dbressoud" target="_blank">@dbressoud</a>.</i></b><br /><br />In 2015, the MAA’s Committee on the Undergraduate Program in Mathematics (CUPM) produced its latest <i><a href="http://maa.org/cupm" target="_blank">Curriculum Guide</a></i>. Extensive as this was, including specific recommendations for most courses and programs offered in departments of mathematics, the steering committee that it left out a big part of what is needed for effective teaching. Spurred by the <i><a href="https://www.maa.org/programs-and-communities/curriculum%20resources/common-vision" target="_blank">Common Vision</a></i> report that outlined what we know about effective teaching and called for their implementation, CUPM set out to describe in detail examples of instructional practices that can greatly improve teaching and learning. The result is the <a href="http://www.maa.org/node/789682" target="_blank"><i>Instructional Practices Guide</i></a> (IP Guide), now available for <a href="http://www.maa.org/node/789682" target="_blank">free download</a> from the MAA.<br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://3.bp.blogspot.com/-zJfUGMFsJCM/WmjTkBrSAsI/AAAAAAAALBY/NgKYC-qHj8Y9UD4p89IqgyFqfy7WbS9ZgCLcBGAs/s1600/Launchings_effectiveteaching_images.JPG" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="353" data-original-width="903" height="156" src="https://3.bp.blogspot.com/-zJfUGMFsJCM/WmjTkBrSAsI/AAAAAAAALBY/NgKYC-qHj8Y9UD4p89IqgyFqfy7WbS9ZgCLcBGAs/s400/Launchings_effectiveteaching_images.JPG" width="400" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">From the description of paired board work, page 20 of the IP Guide.</td></tr></tbody></table>The core of the IP Guide message is that<br /><blockquote class="tr_bq"><b>Effective teaching and deep learning require student engagement with content both inside and outside the classroom.</b></blockquote>This puts the emphasis within the phrase “active learning” where its advocates have always intended it to be, on <i>learning</i>, employing those practices that foster higher order thinking skills. <br /><br />The report is usefully divided into three sections: <b>Classroom Practices</b>, activities that can be used in the classroom to promote engagement with the material; <b>Assessment Practices</b>, how assessment can be used formatively and to probe student understanding; and <b>Design Practices</b>, which get to the broader questions of how to design courses that incorporate the classroom and assessment practices in ways that are most effective. It concludes with two short sections, one on the use of technology and one on equity issues. <br /><br />Classroom Practices constitutes the longest section, describing how to build a classroom community, use wait time, respond to students, and promote persistence. This section includes explanations and examples of some of the standard techniques of active learning: one-minute papers, think-pair-share, just-in-time teaching, and peer instruction. <br /><br />Almost as long as the section on Classroom Practices, Assessment Practices goes into detail on what effective, meaningful, and helpful assessment looks like and how it can be accomplished without overwhelming the instructor, even in large classes. <br /><br />We now have overwhelming evidence of the importance of active cognitive engagement with the mathematics we want our students to learn. Those of us who have succeeded in mathematics have known how to do this outside of the classroom. Most students do not. Most students still approach mathematics as a sequence of templates to be learned for solving specific sets of problems. If we want them to learn anything that will stay with them beyond the term, any knowledge that is transferable, then we must structure our classes so that students are forced to wrestle with the material. The IP Guide should prove to be a useful resource as we reconfigure our courses to meet these goals. <br /><br /><b>References </b><br />Karen Saxe and Linda Braddy. 2015. <i>A Common Vision for Undergraduate Mathematical Sciences Programs in 2025</i>. Washington DC: MAA Press. <a href="https://www.maa.org/sites/default/files/pdf/CommonVisionFinal.pdf" target="_blank">https://www.maa.org/sites/default/files/pdf/CommonVisionFinal.pdf</a><br /><br />Carol S. Schumacher and Martha J. Siegel, co-Chairs, Paul Zorn, editor. 2015. <i>2015 CUPM Curriculum Guide to Majors in the Mathematical Sciences</i>. <a href="https://www.maa.org/sites/default/files/CUPM%20Guide.pdf" target="_blank">https://www.maa.org/sites/default/files/CUPM Guide.pdf</a><br /><br />MAA. 2017. <i>Instructional Practices Guide</i>. <a href="https://www.dropbox.com/s/xpvkni52tkf0wgt/MAA_IP_Guide_V1-1.pdf?dl=0" target="_blank">https://www.dropbox.com/s/xpvkni52tkf0wgt/MAA_IP_Guide_V1-1.pdf?dl=0</a><br /><br />Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com0tag:blogger.com,1999:blog-7251686825560941361.post-88749109153315855842018-01-01T06:55:00.000-05:002018-01-02T10:18:55.474-05:00Indicators for STEM Education<b><i>You can follow me on Twitter <a href="https://twitter.com/dbressoud" target="_blank">@dbressoud</a>.</i></b><br /><br />The National Academies have just released the Board on Science Education report on Indicators for Monitoring Undergraduate STEM Education (available at <a href="http://sites.nationalacademies.org/DBASSE/BOSE">http://sites.nationalacademies.org/DBASSE/BOSE</a>).<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-H2C4nAP7Rnc/WjqxBk_FA5I/AAAAAAAALAM/tfwKqjps9no2551RxqBEKnm1pFHB9BMMACLcBGAs/s1600/Launchings_Stem_gears.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="263" data-original-width="200" src="https://2.bp.blogspot.com/-H2C4nAP7Rnc/WjqxBk_FA5I/AAAAAAAALAM/tfwKqjps9no2551RxqBEKnm1pFHB9BMMACLcBGAs/s1600/Launchings_Stem_gears.png" /></a></div><br />This report is a response to the concern raised by the President’s Council of Advisors in Science and Technology that despite the many initiatives that are seeking to improve the teaching and learning of STEM subjects, we do not have effective national-scale measures of their success. The core of the charge to the committee that produced this report was to identify objectives for the improvement of STEM education, describe indicators that would inform whether or not we are making progress, and catalog what currently exists or could be developed by way of research and data collection to track progress. This extensive report provides this information. <br /><br />The committee identified eleven objectives, organized into three general goals: <br /><br /><b>Goal 1: Increase students’ mastery of STEM concepts and skills by engaging them in evidence-based STEM practices and programs.</b><br /><blockquote class="tr_bq">1.1 Use of evidence-based stem educational practices both in and outside of classrooms<br />1.2 Existence and use of supports that help instructors use evidence-based STEM educational practices<br />1.3 An institutional culture that values undergraduate STEM education<br />1.4 Continuous improvement in STEM teaching and learning </blockquote><b>Goal 2: Strive for equity, diversity, and inclusion of STEM students and instructors by providing equitable opportunities for access and success.</b><br /><blockquote class="tr_bq">2.1 Equity of access to high-quality undergraduate STEM educational programs and experiences<br />2.2 Representational diversity among STEM credential earners<br />2.3 Representational diversity among STEM instructors<br />2.4 Inclusive environments in institutions and STEM departments</blockquote><b>Goal 3: Ensure adequate numbers of STEM professionals.</b><br /><blockquote class="tr_bq">3.1 Foundational preparation for STEM for all students<br />3.2 Successful navigation into and through STEM programs of study<br />3.3 STEM credential attainment</blockquote>Each of these objectives is explained in detail, together with indicators of success and suggestions for how these might be measured. To give an indication of the breadth of this report, I’ll summarize some of what it says about the first and third objective, “Use of evidence-based stem educational practices both in and outside of classroom” and “An institutional culture that values undergraduate STEM education.”<br /><br />The report first explains what evidence-based stem educational practices entail. For in-class practices, the report includes active learning and formative assessments. Acknowledging the lack of a common definition of active learning, this report uses it “to refer to that class of pedagogical practices that <i>cognitively</i> engage students in building understanding at the highest levels of Bloom’s taxonomy,” and then elaborates with examples that include “collaborative classroom activities, fast feedback using classroom response systems (e.g., clickers), problem-based learning, and peer instruction.”<br /><br />This resonates with the CBMS definition, “classroom practices that engage students in activities, such as reading, writing, discussion, or problem solving, that promote higher-order thinking” (<a href="https://www.cbmsweb.org/2016/07/active-learning-in-post-secondary-mathematics-education/" target="_blank">https://www.cbmsweb.org/2016/07/active-learning-in-post-secondary-mathematics-education/</a>). The point being to engage students in wrestling with the critical concepts while in class. Thus the emphasis is not on activity as such, but on the promotion of cognitive engagement in higher order thinking.<br /><br />I appreciate the emphasis on formative assessment: frequent, low-stakes, and varied assessments that clarify for students what they actually do and do not know. I also have found these helpful in informing me where student difficulties lie. The Indicators report references a 1998 review of formative assessment literature by Black and Wiliam, “Inside the Black Box: Raising Standards through Classroom Assessment,” that presents this as the single most effective means of raising student performance and describes how it needs to be done if it is to have these positive benefits. (Black and Wiliam article available at <a href="https://www.rdc.udel.edu/wp-content/uploads/2015/04/InsideBlackBox.pdf">https://www.rdc.udel.edu/wp-content/uploads/2015/04/InsideBlackBox.pdf</a>.)<br /><br />Another important insight from this report, also identified in the MAA’s calculus studies, is the importance of course coordination. If a department is to improve instruction, it is essential that its members share a common understanding of the goals of the course. These shape pedagogical and curricular decisions as well as how student accomplishment is to be measured. The degree of coordination is one of the aspects of objective 1.3: <b>An institutional culture that values undergraduate STEM education. </b>As the report states on page 3-12,<br /><blockquote class="tr_bq">A growing body of research indicates that many dimensions of current departmental and institutional cultures in higher education pose barriers to educators’ adoption of evidence- based educational practices (e.g., Dolan et al., 2016; Elrod and Kezar, 2015, 2016a, 2016b). For example, allowing each individual instructor full control over his or her course, including learning outcomes, a well-established norm in some STEM departments, can cause instructors to resist working with colleagues to establish shared learning goals for core courses, a process that is essential for improving teaching and learning.</blockquote>As I reported last February in "<a href="http://launchings.blogspot.com/2017/02/" target="_blank">MAA Calculus Study: PtC Survey Results</a>," there is very little departmental coordination around homework, exams, grades, or instructional approaches. <br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://1.bp.blogspot.com/-aAG8DTT1Uw0/WjqxFAOZwlI/AAAAAAAALAQ/5pteQ4mwO5wkXKEzowO-gWCNkK5oEm-PQCLcBGAs/s1600/STEM_education_table.JPG" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="345" data-original-width="575" height="240" src="https://1.bp.blogspot.com/-aAG8DTT1Uw0/WjqxFAOZwlI/AAAAAAAALAQ/5pteQ4mwO5wkXKEzowO-gWCNkK5oEm-PQCLcBGAs/s400/STEM_education_table.JPG" width="400" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Table. Of the 207 mainstream Calculus I courses with multiple sections taught in 121 PhD- granting departments and 103 such courses in 76 Masters-granting departments, the percentage of courses that have each feature in common across all sections. Source: PtC Census Survey Technical Report, available at<br /> <a href="https://www.maa.org/sites/default/files/PtC%20Technical%20Report_Final.pdf" target="_blank">https://www.maa.org/sites/default/files/PtC Technical Report_Final.pdf</a>.</td></tr></tbody></table>Of course, the big issue for an institutional culture that values undergraduate STEM education is how teaching is evaluated and role it plays in decisions of promotion and tenure. What is deeply discouraging is how poorly most departments do with just questions of coordination. <br /><br />Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com0tag:blogger.com,1999:blog-7251686825560941361.post-84544052187263609902017-12-01T07:36:00.000-05:002017-12-01T07:36:08.591-05:00Essential Questions<b><i>You can follow me on Twitter <a href="https://twitter.com/dbressoud" target="_blank">@dbressoud</a>.</i></b><br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-Glmxr5Ye65U/WhyEQTfVUrI/AAAAAAAAK-g/zw35eAJyP1k50ZtPzjblisUvWaXmRGG4wCLcBGAs/s1600/AAUSTEM.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="394" data-original-width="1175" height="214" src="https://4.bp.blogspot.com/-Glmxr5Ye65U/WhyEQTfVUrI/AAAAAAAAK-g/zw35eAJyP1k50ZtPzjblisUvWaXmRGG4wCLcBGAs/s640/AAUSTEM.jpg" width="640" /></a></div><br /><br />For over five years, the Association of American Universities (AAU), representing the 62 leading research universities in the United States and Canada, has been engaged in <br /><br /><blockquote class="tr_bq">an initiative to improve the quality of undergraduate teaching and learning in science, technology, engineering, and mathematics (STEM) fields at its member institutions. The overall objective is to influence the culture of STEM departments at AAU universities so that faculty members are encouraged to use teaching practices proven to be effective in engaging students in STEM education and in helping students learn. (See <a href="https://www.aau.edu/education-service/undergraduate-education/undergraduate-%20stem-education-%20initiative" target="_blank">https://www.aau.edu/education-service/undergraduate-education/undergraduate- stem-education- initiative</a>.)</blockquote><br />Products from this initiative that should be of help to every mathematics department seeking to improve instructional practice are now available online. These include a framework for improving undergraduate STEM education with examples of programs at AAU universities that address each of the elements of <a href="https://www.aau.edu/education-service/undergraduate-education/undergraduate-stem-education-initiative/stem-framework" target="_blank">the framework</a>.<br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://3.bp.blogspot.com/-bpvw_zzQguQ/WhyFEYqdC9I/AAAAAAAAK-o/PM4dIzGwPRcR4CdZ1GXETkiq_IBi4zVbACLcBGAs/s1600/AAUCover.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="1600" data-original-width="1237" height="320" src="https://3.bp.blogspot.com/-bpvw_zzQguQ/WhyFEYqdC9I/AAAAAAAAK-o/PM4dIzGwPRcR4CdZ1GXETkiq_IBi4zVbACLcBGAs/s320/AAUCover.png" width="247" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Figure: Cover of the AAU <i>Essential Questions & Data Sources </i>Report.</td></tr></tbody></table>This past summer, they released their report on <a href="https://www.aau.edu/essential-questions-data-sources-continuous-improvement-undergraduate-stem-teaching-and-learning" target="_blank"><i>Essential Questions & Data Sources for Continuous Improvement of Undergraduate Teaching and Learning</i></a>. Data sources include institutional data and tools for its visualization, observation protocols, rubrics, frameworks, student learning assessments, and surveys. The essential questions are separated into questions for institutional leadership as well as at the college, departmental, and instructor levels. <br /><br />Because I believe that departmental leadership is the critical juncture for effective improvement, I will focus the remainder of this column on the questions addressed to departmental leaders and comment on what we have learned from the MAA’s studies of calculus instruction. By departmental leadership, I mean not just chairs and associate chairs, but all of those who shape the department’s direction. Change does not happen without a chair who is committed to improving the teaching and learning within the department, but it cannot be maintained without the support of a core of senior faculty. <br /><br /><i>Do all of the courses in the department have articulated learning goals, and are these made clear to students? What process exists to ensure that individual course learning goals connect to learning goals for the program, major, and department?</i><br /><br />One of the clearest findings from the MAA calculus studies is that coordination of multiple section classes is essential. A prerequisite for effective coordination is a shared sense of what each course is seeking to accomplish. <br /><i><br /></i><i>What are the demographics of students in the department? What are the progression/retention/completion rates for students in the department or major broken out by relevant demographic categories? How do these compare with other departments and what steps are being taken to improve these rates? </i><br /><br />Most departments I have visited have a sense that they are not doing as much as they could or should for students from traditionally underrepresented groups. This is not just a question of race, ethnicity, or gender, but also for students who are first generation, of lower socio-economic status, or from under-resourced schools whether they be inner city or rural. A department cannot know what is working for which populations if it is not tracking success rates by student demographics. <br /><br /><i>What actions has the department chair taken to encourage instructors to take advantage of both on-campus and off-campus (e.g., through relevant disciplinary societies) resources and professional development related to pedagogy? How many instructors have taken advantage of these resources and what notable improvements have occurred as the result?</i><br /><br />The CBMS 2015 survey and other sources have documented that improvements in instructional pedagogy, support services, and course options almost always result from efforts initiated by individual faculty members. This question probes what the department is doing to nurture these faculty. <br /><br /><i>What resources are available to instructors in the department for encouraging all students to succeed, and what steps have been taken to ensure all instructors take advantage of these resources?</i><br /><br />We know that faculty expectations of student ability play a huge role in how well students do, and faculty attitudes toward support services shape how students think about using these resources. The department as a whole must work to ensure the effectiveness of these services and then actively support their use, not as remediation but as a source of support and enrichment. <br /><br /><i>To what extent do departmental instructors have access to learning spaces that support evidence-based pedagogy? What training in the use of those facilities is available to instructors in the department?</i><br /><br />The physical layout of classrooms and access to appropriate technology is critical for implementing effective pedagogies. This means tables where students can work together; sufficient space for instructors to walk around, answer questions, and observe how students are progressing; and sufficient board space for student groups to share their work. It does not have to be high tech classroom, but computer projection that is easily visible by all students is essential. <br /><br /><i>What is the department chair’s and distinguished faculty members’ support of evidence- based pedagogy? How well-known is this support to instructors and students?</i><br /><br />This returns to the issue of nurturing those faculty who are positioned to initiate effective improvements. They need to know that if they are going to sink time and energy into improving teaching and learning within the department, then they will have the support not just of the chair whose term is limited but also of a core group of senior faculty who can ensure that support continues. <br /><br /><i>What are the biggest barriers to evidence-based pedagogy for instructors in the department and how is the chair working to address them? How often does the chair discuss these issues with the dean or other institutional leaders?</i><br /><br />This addresses the chair’s critical role as the bridge between enthusiastic faculty, eager with ideas, and the college or university administrators with concerns to improve instruction and with access to resources that can support change. It is a position that requires insight and discernment on the part of the chair: to understand the priorities of the dean or provost and to comprehend the nature and potential of the initiative that faculty members are proposing. What will it take to implement a particular change? How can it be sold to the dean? What worries of the dean can be matched to ideas from the faculty? <br /><br /><i>How are all faculty who participate in annual/merit, promotion, and tenure evaluations educated about the meaningful inclusion of measures of teaching excellence in those processes? How closely does the chair review the outcomes of those processes to ensure teaching is indeed meaningfully included?</i><br /><br />Finally, there is this elephant standing in the background of every effort to improve teaching and learning: How will it effect promotion and tenure? In my early years at Penn State, I was told that the dean of science was concerned about any faculty member that received high praise for teaching, because that might be a sign that they were neglecting their research. Even in my later years there, I found it necessary to discourage untenured faculty from sinking too much time into educational efforts. Unfortunately, the bifurcation of the faculty that I wrote about in <a href="http://launchings.blogspot.com/2017/10/the-loss-of-tenure-positions-threats-to.html">October</a>, separating tenure line faculty from contract faculty, only exacerbates this problem. With the option to “drop down” to a non-tenure line, the pressure to publish and receive research grants is all the greater. <br /><br /><br /><br />Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com0tag:blogger.com,1999:blog-7251686825560941361.post-51375649911638477982017-11-02T11:15:00.000-04:002017-11-02T11:15:38.412-04:00Women in the Profession<b><i>You can follow me on Twitter <a href="https://twitter.com/dbressoud" target="_blank">@dbressoud</a>.</i></b><br /><b><i><br /></i></b>In last month’s column, I described the loss of tenure positions and their replacement with other full-time faculty appointments. This month, I will focus on how this has affected women earning PhDs in the mathematical sciences, also drawing on the Annual Survey of new PhDs, made available through AMS. <br /><br />The first observation is that, while the number of tenured and tenure-eligible female faculty has increased by a third since 1995, most of the employment gains have been in other-full-time positions, which have more than tripled (Figure 1). <br /><br />The first observation is that, while the number of tenured and tenure-eligible female faculty has increased by a third since 1995, most of the employment gains have been in other-full-time positions, which have more than tripled (Figure 1). <br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://1.bp.blogspot.com/-Ip0EuyW-7GI/Wfs0h7FlZ1I/AAAAAAAAK80/AOdLJc7-TK8yCExL-xqZxuQdZvCFh0YkgCLcBGAs/s1600/FemaleFaculty_fig1.jpeg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="275" data-original-width="484" src="https://1.bp.blogspot.com/-Ip0EuyW-7GI/Wfs0h7FlZ1I/AAAAAAAAK80/AOdLJc7-TK8yCExL-xqZxuQdZvCFh0YkgCLcBGAs/s1600/FemaleFaculty_fig1.jpeg" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><b>Figure 1.</b> The number of women employed in U.S. departments of mathematics,<br />applied mathematics, or statistics. T & TE = tenure or tenure-eligible. Other full-time includes post-docs.<br />Source: CBMS Surveys for 1995, 200, 2005, 2010, 2015.</td></tr></tbody></table><br />This is particularly noticeable in PhD-granting mathematics departments, where a woman employed full-time is far less likely than a man to be in a tenure or tenure-eligible position (Figures 2 & 3). In 2015, 80% of the men employed full-time in a PhD-granting department were in tenure or tenure-eligible positions, this fraction having dropped from 91% in 1995. For women, the percentage fell from 65% in 2015 to 44% in 2015.<br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://3.bp.blogspot.com/-VuanN54AcYM/Wfs08tZnN_I/AAAAAAAAK9E/WpCvtSj_5TcLBVw3a8G0hANjJXaGapqhQCEwYBhgL/s1600/WomenatPhDuniversities.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="248" data-original-width="446" src="https://3.bp.blogspot.com/-VuanN54AcYM/Wfs08tZnN_I/AAAAAAAAK9E/WpCvtSj_5TcLBVw3a8G0hANjJXaGapqhQCEwYBhgL/s1600/WomenatPhDuniversities.jpg" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><b>Figure 2. </b> The number of women employed in PhD-granting U.S. departments of mathematics, applied mathematics, or statistics.<br />T & TE = tenure or tenure-eligible. Other full-time includes post-docs.<br />Source: CBMS Surveys for 1995, 200, 2005, 2010, 2015.</td></tr></tbody></table><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://1.bp.blogspot.com/-raUTgCYsWQw/Wfs03P10zHI/AAAAAAAAK9E/vQFHiCz48WMy_xcD_xJBxqHyMpg_WW93QCEwYBhgL/s1600/MenatPhDuniversities.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="250" data-original-width="448" src="https://1.bp.blogspot.com/-raUTgCYsWQw/Wfs03P10zHI/AAAAAAAAK9E/vQFHiCz48WMy_xcD_xJBxqHyMpg_WW93QCEwYBhgL/s1600/MenatPhDuniversities.jpg" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><b>Figure 3.</b> The number of men employed in PhD-granting U.S. departments of mathematics, applied mathematics, or statistics. T & TE = tenure or tenure-eligible. Other full-time includes post-docs.<br />Source: CBMS Surveys for 1995, 200, 2005, 2010, 2015.</td></tr></tbody></table>Despite the appearance that women are making substantial gains in tenure and tenure-eligible positions in PhD-granting departments, the fact is that they have only grown from 9% of those faculty in 1995 to 16% in 2015. In comparison, in Masters-granting departments the percentage of women in tenure and tenure-eligible positions rose from 18% in 1995 to 29% in 2015. At undergraduate colleges, it rose from 26% in 1995 to 32% in 2015. Over the same two decades, women rose from 22% of the PhDs awarded by mathematics departments to 26%.<br /><br />If we look at all PhDs awarded to women in the mathematical sciences, now including departments of statistics or applied mathematics, the situation looks better, rising to 31% in 2015 (Figure 4), with women earning 33% of the PhDs in applied mathematics and 46% of those degrees in statistics. <br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://1.bp.blogspot.com/-Fwvpcx72tDk/Wfs0veR5e6I/AAAAAAAAK9E/7fNlUcxq6mg0hsC-v4KCa82rAnNvWBCKwCEwYBhgL/s1600/WomenaspercentofnewPhDs.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="251" data-original-width="447" src="https://1.bp.blogspot.com/-Fwvpcx72tDk/Wfs0veR5e6I/AAAAAAAAK9E/7fNlUcxq6mg0hsC-v4KCa82rAnNvWBCKwCEwYBhgL/s1600/WomenaspercentofnewPhDs.jpg" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><b>Figure 4. </b>Women as a percentage of new PhDs in the mathematical sciences in the U.S. by type of department.<br /> Source: The Joint Data Committee’s Annual Survey available at AMS.org, 1995 through 2015.</td></tr></tbody></table>CBMS does not collect the data that would enable us to make comparable statements about the type of employment gained by mathematicians from other underrepresented groups and the numbers are so small it is not clear how meaningful they would be, but it does appear that efforts to broaden the diversity of mathematics departments is being stymied by the trend to replace tenure-line positions with contract positions. At least for women, their expanding representation in mathematics faculty is happening primarily in those contract positions. <br /><br /><br /><br /><br /><br /><br /><br /><br />Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com0tag:blogger.com,1999:blog-7251686825560941361.post-40225990428760628442017-10-02T10:47:00.000-04:002017-10-27T17:31:43.957-04:00The Loss of Tenure Positions: Threats to the Profession<b><i>You can follow me on Twitter <a href="https://twitter.com/dbressoud" target="_blank">@dbressoud</a>.</i></b><br /><br />The preliminary tables from the CBMS 2015 surveys of U.S. departments of mathematics or statistics are now available from the CBMS homepage at <a href="http://cbmsweb.org/">CBMSweb.org</a> or by clicking <a href="http://www.ams.org/profession/data/cbms-survey/cbms2015-work" target="_blank">HERE</a>. I am using this month’s column to highlight one of the most dramatic developments: the loss of tenured and tenure-eligible faculty (Figure 1). At the end of this article, I reflect on the implications for our profession. <br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://3.bp.blogspot.com/-Xc0prV3hBnE/WfOQaHdf7MI/AAAAAAAAK8U/oi68_5VIxWcrNLfLfbruv1ANXOT8pOvywCLcBGAs/s1600/fig1.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="294" data-original-width="482" src="https://3.bp.blogspot.com/-Xc0prV3hBnE/WfOQaHdf7MI/AAAAAAAAK8U/oi68_5VIxWcrNLfLfbruv1ANXOT8pOvywCLcBGAs/s1600/fig1.jpg" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Figure 1. Number of faculty in mathematics departments.<br />T & TE = tenured or tenure-eligible, other full-time includes post-docs.</td></tr></tbody></table><br />The year 2015 saw the fewest tenured or tenure-eligible faculty, 15,270, since 1995, a drop of two thousand positions since 2005. Where they have gone is no mystery. The number of other full-time faculty, including post-docs, has tripled over the past two decades, from 2140 in 1995 to 6427 in 2015. <br /><br />The break-down by type of institution—according to the highest degree offered by the mathematics department: PhD, Master’s, or Bachelor’s—is interesting. PhD-granting universities have seen remarkably constant numbers of tenure positions, Master’s universities have seen the greatest loss, and undergraduate colleges saw a spike around 2005 and have now returned to the number of positions in 1995. The growth in other full-time positions has been most dramatic at the PhD-granting universities, from 758 in 1995 to 2336 in 2015 (Figures 2–4). <br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://1.bp.blogspot.com/-VrGx1fPOxP8/WfOQvfebNWI/AAAAAAAAK8Y/RbSAjOrmDIYDRdkJT3ElA4FiWjGngbjcgCLcBGAs/s1600/fig2.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="269" data-original-width="447" src="https://1.bp.blogspot.com/-VrGx1fPOxP8/WfOQvfebNWI/AAAAAAAAK8Y/RbSAjOrmDIYDRdkJT3ElA4FiWjGngbjcgCLcBGAs/s1600/fig2.jpg" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Figure 2. Distribution of faculty in PhD-granting mathematics departments.</td></tr></tbody></table><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://3.bp.blogspot.com/-xwAoLzkk24w/WfOQ6FMdjtI/AAAAAAAAK8g/1iA4f7ya_cU7KWJ8Z2MUMXPxEPxf-On6wCLcBGAs/s1600/fig3.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="267" data-original-width="448" src="https://3.bp.blogspot.com/-xwAoLzkk24w/WfOQ6FMdjtI/AAAAAAAAK8g/1iA4f7ya_cU7KWJ8Z2MUMXPxEPxf-On6wCLcBGAs/s1600/fig3.jpg" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Figure 3. Distribution of faculty in Master’s-granting mathematics departments.</td></tr></tbody></table><br /><div class="separator" style="clear: both; text-align: center;"></div><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://4.bp.blogspot.com/-zjBGw_J5GFo/WfORC5mr86I/AAAAAAAAK8k/FrXuybzaRn82V7Ox0USS56eLhYrCwqA0ACLcBGAs/s1600/fig4.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="267" data-original-width="447" src="https://4.bp.blogspot.com/-zjBGw_J5GFo/WfORC5mr86I/AAAAAAAAK8k/FrXuybzaRn82V7Ox0USS56eLhYrCwqA0ACLcBGAs/s1600/fig4.jpg" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Figure 4. Distribution of faculty in Bachelor’s-granting mathematics departments.</td></tr></tbody></table><br />It is not that we now have fewer students to teach. Since 2005, the number of students studying mathematics in four-year under undergraduate programs has grown from 1.6 to over 2.2 million, an increase of 38% (Figure 5). If we add in the statistics courses taught within mathematics departments, the number of students enrolled each fall has jumped from 1.79 to 2.53 million, almost three-quarters of a million additional students. This dramatic growth holds even when we restrict to students at the level of calculus instruction and above, where the past decade has seen an increase of 262,000 students (Figure 6). To meet this increased demand while dropping two thousand tenure positions, we have added over three thousand other full-time faculty and one thousand part-time faculty. <br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://4.bp.blogspot.com/-rQiro2X_OTM/WdJM7I7JS3I/AAAAAAAAK6s/uAb28guILlootEywcUTmzk8AT5deyYNeACLcBGAs/s1600/fig5.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="273" data-original-width="513" src="https://4.bp.blogspot.com/-rQiro2X_OTM/WdJM7I7JS3I/AAAAAAAAK6s/uAb28guILlootEywcUTmzk8AT5deyYNeACLcBGAs/s1600/fig5.jpg" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Table 5. Undergraduate enrollment in mathematics in four-year programs. Calculus level includes sophomore-level differential equations, linear algebra, and discrete mathematics. Advanced is any math course beyond calculus level. These do not include statistics.</td><td class="tr-caption"><br /></td><td class="tr-caption"><br /></td></tr></tbody></table><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://4.bp.blogspot.com/-lD8Q5nHzW7Y/WdJNXz1gqGI/AAAAAAAAK6w/rWLI-egp0TE86C5Qbicj2MaxlfbrStQ_gCLcBGAs/s1600/fig6.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="267" data-original-width="450" src="https://4.bp.blogspot.com/-lD8Q5nHzW7Y/WdJNXz1gqGI/AAAAAAAAK6w/rWLI-egp0TE86C5Qbicj2MaxlfbrStQ_gCLcBGAs/s1600/fig6.jpg" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Table 6. Undergraduate enrollment at calculus level and above.</td></tr></tbody></table><br />Not surprisingly, this means that undergraduate courses are now much less likely to be taught by a tenured or tenure-eligible faculty member. Figures 7 and 8 show what has happened at the PhD- granting universities. The 2015 survey was the first time that mainstream Calculus I and Calculus II were less likely to be taught by tenure line faculty than by other full-time faculty.<br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://1.bp.blogspot.com/-uAiInrHxZKc/WdJNlANkUcI/AAAAAAAAK60/drcCl1HLOQ0ZLXlc0eh_GV3TYJQt2sOFACLcBGAs/s1600/fig7.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="267" data-original-width="450" src="https://1.bp.blogspot.com/-uAiInrHxZKc/WdJNlANkUcI/AAAAAAAAK60/drcCl1HLOQ0ZLXlc0eh_GV3TYJQt2sOFACLcBGAs/s1600/fig7.jpg" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Table 7. T &TE = tenured or tenure-eligible, other full-time includes post-docs.<br />For 1995 and 2000, % is percentage of total students taking Calculus I.<br />After 2000, it is the percentage of sections.</td></tr></tbody></table><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://1.bp.blogspot.com/-bQsh1qjVZr8/WdJOdNq83PI/AAAAAAAAK68/cTx8fd2JfYEqBTfuICJNncljiI6dPJfdQCLcBGAs/s1600/fig8.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="268" data-original-width="448" src="https://1.bp.blogspot.com/-bQsh1qjVZr8/WdJOdNq83PI/AAAAAAAAK68/cTx8fd2JfYEqBTfuICJNncljiI6dPJfdQCLcBGAs/s1600/fig8.jpg" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Table 8. T & TE = tenured or tenure-eligible, other full-time includes post-docs.<br />For 1995 and 2000, % is percentage of total students taking Calculus II.<br />After 2000, it is the percentage of sections.</td></tr></tbody></table><br />The trends are similar at Master’s universities and Bachelor’s colleges, though not as dramatic (Figures 9–12, following the <b>Reflection</b>). <br /><br /><b>Reflection.</b> The CBMS data confirm what I have seen in departments across the country, especially in PhD- and Masters-granting departments. More and more of the undergraduate instruction is now the responsibility of contract faculty. In our research universities, it is becoming unusual for a tenured faculty member to teach any undergraduate courses. The unfortunate consequence is that the teacher-scholar, the ideal when I entered the profession, is fast disappearing. Those who are most active in mathematical research receive few teaching responsibilities. The remainder are saddled with heavy teaching loads that leave little time for research. <br /><br />The reality of this bifurcation of the profession hit home in a recent network analysis of faculty interaction around issues of teaching, undertaken by the MAA’s <i>Progress through Calculus</i>project at a large public university. We found that tenure line faculty only interact with other tenure line faculty, contract faculty only with other contract faculty, with just a few individuals to provide a bridge. In effect, it has become two departments, one for undergraduate teaching and the other for research and the preparation of graduate students. <br /><br />The teaching faculty are now manifestly second-class members of the profession: earning less money and receiving fewer benefits, carrying heavier prescribed duties, often lacking input in departmental decision-making, and living with the reality that, even with a renewable contract, long-term prospects are uncertain. It is no wonder so many of them have chosen to unionize. <br /><br />There also are disturbing implications for the research faculty. Unlike Engineering or many of the other sciences, tenured mathematics faculty members seldom receive research grants that cover the full cost of their employment. Our public research universities have justified the size of their departments of mathematics by the large load of service teaching these departments must provide. Administrators are already questioning the wisdom of supporting a large corps of mathematics researchers who contribute ever less to the activities that pay the university’s bills. <br /><br />We cannot turn back the clock, but there are mechanisms that can mitigate the dangers: involving contract faculty in departmental committees and decision making, involving tenure line faculty in observing and supporting those who carry the brunt of the teaching responsibilities, and ensuring that everyone is respected. There was one simple action that I observed at the Colorado School of Mines, a PhD-granting department. On the bulletin board that posts pictures of the faculty, contract faculty were not segregated from tenure line faculty. All members of the faculty were together in alphabetical order. What a radical idea. <br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://3.bp.blogspot.com/-Pg9bET-z_BU/WdJO5fgH8DI/AAAAAAAAK7M/c79vXdRCSwwYVhX9fngVELvbrcOLER0gACLcBGAs/s1600/fig9.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="269" data-original-width="449" src="https://3.bp.blogspot.com/-Pg9bET-z_BU/WdJO5fgH8DI/AAAAAAAAK7M/c79vXdRCSwwYVhX9fngVELvbrcOLER0gACLcBGAs/s1600/fig9.jpg" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Table 9. T & TE = tenured or tenure-eligible, other full-time includes post-docs.<br />For 1995 and 2000, % is percentage of total students taking Calculus I.<br />After 2000, it is the percentage of sections.</td></tr></tbody></table><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://2.bp.blogspot.com/-zC_2Na8m92g/WdJO4tiPdMI/AAAAAAAAK7A/A_LZ2W8WN_I2vbLWJRUHOv4SMM1SSvo8gCLcBGAs/s1600/fig10.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="268" data-original-width="448" src="https://2.bp.blogspot.com/-zC_2Na8m92g/WdJO4tiPdMI/AAAAAAAAK7A/A_LZ2W8WN_I2vbLWJRUHOv4SMM1SSvo8gCLcBGAs/s1600/fig10.jpg" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Table 10. T & TE = tenured or tenure-eligible, other full-time includes post-docs.<br />For 1995 and 2000, % is percentage of total students taking Calculus II.<br />After 2000, it is the percentage of sections.</td></tr></tbody></table><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://3.bp.blogspot.com/-YzJWfbPOUaI/WdJO4kEYZkI/AAAAAAAAK7I/1rbF-BvzC7MjGZF4biTs2pDGcCB7X7XcgCLcBGAs/s1600/fig11.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="271" data-original-width="450" src="https://3.bp.blogspot.com/-YzJWfbPOUaI/WdJO4kEYZkI/AAAAAAAAK7I/1rbF-BvzC7MjGZF4biTs2pDGcCB7X7XcgCLcBGAs/s1600/fig11.jpg" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Table 11. T & TE = tenured or tenure-eligible, other full-time includes post-docs.<br />For 1995 and 2000, % is percentage of total students taking Calculus I.<br />After 2000, it is the percentage of sections.<br /><br /></td></tr></tbody></table><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://3.bp.blogspot.com/-2-W5BDqb5NQ/WdJO4vBNwwI/AAAAAAAAK7E/8hnpWLZge2kB-Hq6tJ6Xjp7TvYDPonr8wCLcBGAs/s1600/fig12.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="268" data-original-width="451" src="https://3.bp.blogspot.com/-2-W5BDqb5NQ/WdJO4vBNwwI/AAAAAAAAK7E/8hnpWLZge2kB-Hq6tJ6Xjp7TvYDPonr8wCLcBGAs/s1600/fig12.jpg" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Table 12. T & TE = tenured or tenure-eligible, other full-time includes post-docs.<br />For 1995 and 2000, % is percentage of total students taking Calculus II.<br />After 2000, it is the percentage of sections.</td></tr></tbody></table><i><br /></i><i>**Editorial note: Figures 1-4 were updated on October 27, 2017.</i>Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com0tag:blogger.com,1999:blog-7251686825560941361.post-90207288729441419402017-09-01T06:30:00.000-04:002017-09-01T06:30:10.529-04:00Mathematics as Peacock Feathers<b><i>You can follow me on Twitter <a href="https://twitter.com/dbressoud" target="_blank">@dbressoud</a>.</i></b><br /><br />Mathematics occupies a privileged position in our educational system, generally equated with English language facility—reading and writing—for emphasis within the K-12 curriculum, in curriculum reform efforts such as Common Core, in admissions testing with SAT and ACT, and in college graduation requirements. Why? An important recent article by Daniel Douglas and Paul Attewell, “School Mathematics as Gatekeeper,”[1] draws on data from the Education Longitudinal Study of 2002 (ELS:2002) [2] to explore this question. <br /><br />A common response is that in today’s technologically driven society, mathematical knowledge is more essential than ever. Yet, as the authors document, the fact is that few workers, even in those jobs that require a bachelor’s degree, use mathematics at or above the level of Algebra II on a regular basis. <br /><br />Of course, no one argues that actually factoring a quadratic or finding a derivative are essential skills for today’s workplace. Instead it is the habits of mind that learning mathematics instills that are considered so important. Douglas and Attewell look at the other side of this connection. It has been extremely difficult to demonstrate that mathematics instruction does lead to the development of logical thinking and effective problem solving, but society does recognize those who are successful in mathematics as talented individuals who are primed for success. The authors explore the role of mathematical achievement as a signal that a prospective student or employee is going to succeed, just as a peacock’s feathers signal a male capable of fathering strong offspring (Figure 1). <br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://1.bp.blogspot.com/-geNy7_q-Y7c/Wag9FifqOgI/AAAAAAAAK5c/zA3l00ho9wkKBTN8wbxUwMZ-tksCWCUPgCLcBGAs/s1600/lauchings_peacock_calculus.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="550" data-original-width="768" height="229" src="https://1.bp.blogspot.com/-geNy7_q-Y7c/Wag9FifqOgI/AAAAAAAAK5c/zA3l00ho9wkKBTN8wbxUwMZ-tksCWCUPgCLcBGAs/s320/lauchings_peacock_calculus.jpg" width="320" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><span id="docs-internal-guid-e6fd4f04-392e-dea8-8d5b-e809076a4e26"><div dir="ltr" style="line-height: 1.2; margin-bottom: 0pt; margin-top: 0pt;"><span style="font-size: 11pt; vertical-align: baseline; white-space: pre-wrap;">Figure 1. From Bob Orlin’s “The Peacock Tail Theory of AP</span><span style="font-size: 6.6pt; vertical-align: super; white-space: pre-wrap;">®</span><span style="font-size: 11pt; vertical-align: baseline; white-space: pre-wrap;"> Calculus.”</span></div><div dir="ltr" style="line-height: 1.2; margin-bottom: 0pt; margin-top: 0pt;"><span style="color: blue; font-size: 11pt; text-decoration-line: underline; vertical-align: baseline; white-space: pre-wrap;"><a href="https://mathwithbaddrawings.com/2016/07/20/the-peacock-tail-theory-of-ap-calculus/" style="text-decoration-line: none;">mathwithbaddrawings.com/2016/07/20/the-peacock-tail-theory-of-ap-calculus/</a></span></div></span></td></tr></tbody></table><br />Signals are important. Those who are believed capable of succeeding are more likely to get the support and encouragement they need to succeed. Douglas and Attewell were able to draw on ELS:2002, a ten-year longitudinal study of survey data and transcripts of 15,000 U.S. students, to test whether mathematical achievement in high school has such a signaling effect. Able to control for the common variables associated with success: general academic performance in high school, motivation, effort, academic involvement, gender, race/ethnicity, socio-economic status (SES), and parental education, they took as their null hypothesis that mathematical achievement—especially having studied precalculus or calculus in high school—would add nothing to the chances of being admitted to and graduating from a four-year college program. <br /><br />That null hypothesis was firmly rejected with a p-value less than 0.001. Controlling for all of those other factors, taking trigonometry or precalculus as the last high school math class was associated with increased odds of attending a four-year college, close to two times those of students whose last mathematics class was Algebra II. The odds of attending a selective college were doubled. Calculus in high school is an even stronger signal, associated with the increased odds of attending a four-year college by a factor of two and a half, and attending a selective college by a factor of three. Again controlling for all of these other variables, completing any of these courses nearly doubled the odds of earning a bachelor’s degree. <br /><br />In the other direction and still controlling for all other factors, terminating high school mathematics at Algebra I was associated with far lower odds of attending a four-year college—by a factor of one half. The odds of earning a bachelor’s degree among students completing only Algebra I were about a quarter of that for students for whom Algebra II was the highest mathematics course taken in high school. <br /><br />Reporting marginal effects, the authors note that students taking precalculus as the last high school mathematics course were 12 percentage points more likely to attend a four-year college than those for whom Algebra II was the last class. A precalculus class also raised the likelihood of attending a selective college by 12 percent, and of earning a bachelor’s degree by nine percent. Similarly, taking calculus in high school boosted the likelihood even further: 16 percent for four-year colleges, 18 percent for a selective college, and 10 percent for earning a bachelor's degree. <br /><br />Perhaps surprising is the fact that this signaling effect is strongest for students of high SES. Using a composite score of mathematical ability as measured by the ELS:2002 standardized test in mathematics and the highest mathematics course taken in high school, students scoring one standard deviation above the mean increased their likelihood of attending a selective college by 12 percent. For students with high SES, it increased by 25 percent. It is important to note that while these findings are statistically significant associations, they should not be interpreted as statements of causality. <br /><br /><b>Conclusions</b><br /><br />The authors emphasize the irony of the very strong signal sent by advanced work in high school mathematics given how small a role it plays in actual workforce needs. It is my personal belief that the strong signaling effect of mathematical achievement points to something real, an analytic ability that goes beyond the other talents for which this study controlled: general academic performance in high school, motivation, effort, and academic involvement, but that the signal has been amplified beyond reason. This has important implications. <br /><br />The common perception that calculus on a high school transcript helps a student get into a selective college is supported by these data. It also appears to improve the chances of completing a bachelor’s degree. Given that this effect is strongest for students of high SES, those with parents who are best positioned to push to accelerate their sons and daughters, the trend to bring ever more students into calculus at an ever earlier point in their high school careers is rational. Rational does not mean desirable, or even necessarily appropriate, but it does mean that trying to counter the growth of high school calculus will require more than recommendations and policy statements. If misapplied acceleration can do harm, as many of us believe, we need convincing evidence of this. <br /><br />The work of Douglas and Attewell should also inform the debate over requiring Algebra II in high school. Those who oppose this as a requirement for all students point out that few will need the skills taught in this course; this perspective is highlighted by the study authors, though they do not believe mathematics requirements should be summarily dismissed. The problem is the self-reinforcing signal sent by not having Algebra II on one’s transcript. Their work also points to the importance of making precalculus and calculus available to all students who are prepared to study them. Lack of access in high school does more than postpone the opportunity for their study; the evidence suggests that it actually damages chances of post-secondary success. <br /><br /><b>References</b><br /><br />[1] Douglas, D. and Attewell, P. (2017). School mathematics as gatekeeper. <i>The Sociological Quarterly</i>. <a href="http://www.tandfonline.com/doi/abs/10.1080/00380253.2017.1354733" target="_blank">www.tandfonline.com/doi/abs/10.1080/00380253.2017.1354733</a><br /><br />[2] National Center for Education Statistics (NCES). <a href="http://nces.ed.gov/surveys/els2002/">nces.ed.gov/surveys/els2002/</a>Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com0tag:blogger.com,1999:blog-7251686825560941361.post-84612262767462956482017-08-01T16:01:00.000-04:002017-08-01T16:01:08.232-04:00Changing Demographics<b><i>You can follow me on Twitter <a href="https://twitter.com/dbressoud" target="_blank">@dbressoud</a>.</i></b> <br /><br />In the United States, the mathematically intensive disciplines—engineering, the mathematical sciences, and the physical sciences—have traditionally been dominated by White males. It is common knowledge that the U.S. population is changing. Data from the National Center for Education Statistics (NCES) of the Department of Education show that while 73% of high school graduates in 1995 were White, by 2015 that had decreased to 55%, on track to drop below 50% by 2025, in just eight years (Figure 0.1). <i>These and all data in this article are taken from the NCES Digests of Education Statistics, 1990 through 2017, available at <a href="http://nces.ed.gov/programs/digest/">nces.ed.gov/programs/digest/</a>. </i><br /><br />It has become a truism that if the United States is to maintain its pre-eminence in science and technology, we must ensure that traditionally underrepresented minorities share in this preparation for mathematically intensive careers. Groups like the <a href="http://maa.org/?utm_source=Launchings&utm_medium=blog&utm_campaign=MAA" target="_blank">Mathematical Association of America</a> have programs such as the <a href="https://www.maa.org/programs/maa-grants/women-and-mathematics-grants?utm_source=Launchings&utm_medium=blog&utm_campaign=Programs" target="_blank">Tensor grants</a> that encourage students from <a href="https://www.maa.org/programs/underrepresented-groups?utm_source=Launchings&utm_medium=blog&utm_campaign=Programs" target="_blank">underrepresented groups</a> to succeed in mathematics.<br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://2.bp.blogspot.com/-gQdWmBRAxnI/WYDLP9ozTgI/AAAAAAAAK20/oPnDZA1_YPkbWj5oIYOwiDeUTKNxu29tQCLcBGAs/s1600/launchings%2Bgraph%2B1.JPG" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="399" data-original-width="727" height="348" src="https://2.bp.blogspot.com/-gQdWmBRAxnI/WYDLP9ozTgI/AAAAAAAAK20/oPnDZA1_YPkbWj5oIYOwiDeUTKNxu29tQCLcBGAs/s640/launchings%2Bgraph%2B1.JPG" width="640" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><b>Figure 0.1.<i> </i></b><i>White non-Hispanic students as percentage of all high school graduates. Percentages after 2012 are estimates based on the number of students already in the K-12 pipeline.</i><b> Note: scale starts at 40%.</b><span id="docs-internal-guid-6837ae5a-9f1a-7684-668c-c3377e1e2fab"><span style="font-size: 11pt; font-style: italic; vertical-align: baseline; white-space: pre-wrap;"> </span></span></td></tr></tbody></table><br />While we have seen and should continue to see a modest increase in the percentage of Asian and Black students, most of the changing demographics are shaped by the dramatic growth in the number of Hispanic students, which grew from 9% of high school graduates in 1995 to 21% in 2015, projected to reach 27% by 2025 (Figure 0.2).<br /><div><br /></div><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://3.bp.blogspot.com/-133miWeFqFk/WYDLp4SaR8I/AAAAAAAAK24/tjEPrUWGXZ0Hi20IUQ7P83KjJLXc7cyRwCLcBGAs/s1600/launchings%2Bgraph%2B2.JPG" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="444" data-original-width="730" height="388" src="https://3.bp.blogspot.com/-133miWeFqFk/WYDLp4SaR8I/AAAAAAAAK24/tjEPrUWGXZ0Hi20IUQ7P83KjJLXc7cyRwCLcBGAs/s640/launchings%2Bgraph%2B2.JPG" width="640" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><b>Figure 0.02 </b> <span id="docs-internal-guid-6837ae5a-9f1b-ce02-febf-47204f12d8ab"><i>Black non-Hispanic, Hispanic, and Asian/Pacific Islander students as a percentage of all high school graduates. Percentages after 2012 are estimates based on the number of students already in the K-12.</i></span></td></tr></tbody></table><br />The intent of this article is simply to exhibit the data showing how well we are including various racial, ethnic, and gender groups among the recipients of bachelor’s degrees in engineering, the mathematical sciences, and the physical sciences. In future articles, I will address some of the ways in which MAA and other organizations are addressing the issues raised by these trends. <br /><br />The bulk of this paper is taken with three appendices that show the graphs of the percentage of bachelor’s degrees earned in these three areas by each of the following demographic groups:<br /><br /><ul><li>Figure x.1. Women</li><li>Figure x.2. White students, also reported by gender</li><li>Figure x.3. Black students, also reported by gender Figure x.4.</li><li>Hispanic students, also reported by gender</li><li>Figure x.5. Asian students, also reported by gender</li><li>Figure x.6. Non-resident alien students, also reported by gender where x is 1 for engineering, 2 for the mathematical sciences, and 3 for the physical sciences.</li></ul>Native Americans/Alaskan Natives account for 1.1% of high school graduates, 0.3% to 0.5% in engineering and mathematics, and 0.5% to 0.8% in the physical sciences. These numbers are so small that there is tremendous year-to-year variation, and the graphs do not exhibit meaningful trends. Only in 2011 did NCES begin separating Asian from Pacific Islander and begin to allow students to self-identify as two or more races. For the sake of consistency, all of the data reported for Asian students include Pacific Islander students. In the disciplines of engineering, mathematical sciences, and physical sciences, Pacific Islanders make up between 0.1% and 0.2% of the total majors. In the first year that the choice of two or more races was allowed, about 1% of the students in each of the three disciplines so identified. This had risen to 3% by 2015. From 2011 through 2015, about 2% of high school graduates identified as two or more races.<br /><br /><b>Observations</b><br /><br />Probably the most striking graph in this entire collection is Figure 2.6, showing the proportion of mathematics degrees going to non-resident aliens. Historically, this has been around 4%. It began to take off in 2008. By 2015, 13% of the bachelor’s degrees in the mathematical sciences were awarded to non-resident aliens. While we welcome these visitors and hope that many of them will stay, it is disturbing that so much of our mathematical talent must be imported. As shown in Figures 1.6 and 3.6, there have also been increases in the fraction of engineering and physical science degrees earned by non-resident aliens, but here the growth has not been nearly as dramatic.<br /><br />A very disturbing set of graphs are given in Figures 1.3, 2.3, and 3.3, showing the proportion of degrees earned by Black students. In all three disciplines, we see a pattern of substantial growth during the 1990s, followed by a period of leveling off, followed by substantial decline. In engineering and the physical sciences, the percentage of degrees earned by Black students has dropped to levels not seen since 1993. In mathematics, the percentage of degrees awarded to Black students in 2015, 4.6%, is below that of 1990, when it was 5.0%.<br /><div><br /></div>The graphs showing the percentage of women in these fields, Figures 1.1, 2.1, and 3.1, are also discouraging. Engineering has always had a difficult time attracting and retaining women. By 2000, they had managed to get the proportion of degrees going to women over 20%, but it then slipped back to 18%. The good news is that the fraction of engineering degrees to women began growing again in 2010 and is now back to the 20% mark, far too low, but headed in the right direction.<br /><div><br /></div>In the physical sciences, there was dramatic growth in the participation of women, from 31% in 1990 to over 42% in 2002. It has been slipping since then, now back almost to 38%.<br />Compared to engineering and the physical sciences, the mathematical sciences have done very well, but we were at 46% in 1990 and achieved 48% in 1998. We have since slipped back to 43%. The recent trend line looks decidedly flat.<br /><div><br /></div>The brightest spot in these data is the substantial increase in the proportion of Hispanic students among these mathematically intensive majors (Figures 1.4, 2.4, and 3.4). Given the dramatic increase in the percentage of students of traditional college age who are Hispanic, an increase of 125% from 1995 to 2015, engineering and mathematics—with only 110% increases in the proportion of majors who are Hispanic—are not doing as well as they should. The physical sciences have seen the most dramatic increase, but starting from an extremely low base. Nevertheless, today around 9% of the majors in all three disciplinary areas are Hispanic, and strong growth continues.<br /><div><br /></div>Asian students have always been well represented in engineering, the mathematical sciences, and the physical sciences, currently at or above 10% of those degrees (Figures 1.5, 2.5, and 3.5).<br /><b><br /></b><b><br /></b><b><br /></b><b>Appendix I. Bachelor’s degrees in Engineering</b><br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://3.bp.blogspot.com/-xj18BRomCxo/WYDNE_QypRI/AAAAAAAAK3E/NWq8mYv0CI44mwA5cPijTNQgL_-Nl3HTACLcBGAs/s1600/Launchings%2Bappendix%2B1.JPG" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="395" data-original-width="732" height="345" src="https://3.bp.blogspot.com/-xj18BRomCxo/WYDNE_QypRI/AAAAAAAAK3E/NWq8mYv0CI44mwA5cPijTNQgL_-Nl3HTACLcBGAs/s640/Launchings%2Bappendix%2B1.JPG" width="640" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><b>Figure 1.1. </b><span id="docs-internal-guid-6837ae5a-9f21-4ff4-0aa6-2f052dfdbc6b">Women as a percentage of all bachelor’s degrees in engineering.<span style="font-size: 11pt; font-weight: 700; vertical-align: baseline; white-space: pre-wrap;"> </span><b>Note: scale starts at 10%.</b><div><span style="font-size: 11pt; font-weight: 700; vertical-align: baseline; white-space: pre-wrap;"><br /></span></div></span></td></tr></tbody></table><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://1.bp.blogspot.com/-usXc4Gn7QJc/WYDNbjzKjbI/AAAAAAAAK3I/dbYO-LzehaQByTl-K5MpjqzbUJ-HGI9ZgCLcBGAs/s1600/Launchings%2Bappendix%2B2.JPG" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="436" data-original-width="731" height="380" src="https://1.bp.blogspot.com/-usXc4Gn7QJc/WYDNbjzKjbI/AAAAAAAAK3I/dbYO-LzehaQByTl-K5MpjqzbUJ-HGI9ZgCLcBGAs/s640/Launchings%2Bappendix%2B2.JPG" width="640" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><b>Figure 1.2. </b><i>White non-Hispanic students as percentage of all bachelor's degrees in engineering. </i></td></tr></tbody></table><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://1.bp.blogspot.com/-IoeKfXO_XeI/WYDSbOX81JI/AAAAAAAAK3g/nwzmu85rdzYKg2QHEivAu--r0MwCZHYqQCLcBGAs/s1600/Launchings%2Bappendix%2B3.JPG" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="440" data-original-width="733" height="384" src="https://1.bp.blogspot.com/-IoeKfXO_XeI/WYDSbOX81JI/AAAAAAAAK3g/nwzmu85rdzYKg2QHEivAu--r0MwCZHYqQCLcBGAs/s640/Launchings%2Bappendix%2B3.JPG" width="640" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><b>Figure 1.3.<i> </i></b><i>Black non-Hispanic students as a percentage of all bachelor's degrees in engineering.</i></td></tr></tbody></table><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><img border="0" data-original-height="438" data-original-width="733" height="382" src="https://1.bp.blogspot.com/-m7rSaubceds/WYDSbBe1NCI/AAAAAAAAK3U/c0IOJ42s8cYFwZzbh085rvD0JVMG-hhqACLcBGAs/s640/Lauchings%2Bappendix%2B5.JPG" style="margin-left: auto; margin-right: auto;" width="640" /></td></tr><tr><td class="tr-caption" style="text-align: center;"><b>Figure 1.4.</b> <i>Hispanic students as a percentage of all bachelor's degrees in engineering.</i></td></tr></tbody></table><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-m7rSaubceds/WYDSbBe1NCI/AAAAAAAAK3U/c0IOJ42s8cYFwZzbh085rvD0JVMG-hhqACLcBGAs/s1600/Lauchings%2Bappendix%2B5.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"></a><br /><a href="https://1.bp.blogspot.com/-m7rSaubceds/WYDSbBe1NCI/AAAAAAAAK3U/c0IOJ42s8cYFwZzbh085rvD0JVMG-hhqACLcBGAs/s1600/Lauchings%2Bappendix%2B5.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://1.bp.blogspot.com/-AE6lg_CpAtA/WYDSbPhBStI/AAAAAAAAK3Y/SX10P0eg3JIX-EEWa0K6CaH154Oj1dFrQCLcBGAs/s1600/Launchings%2Bappendix%2B4.JPG" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="438" data-original-width="734" height="381" src="https://1.bp.blogspot.com/-AE6lg_CpAtA/WYDSbPhBStI/AAAAAAAAK3Y/SX10P0eg3JIX-EEWa0K6CaH154Oj1dFrQCLcBGAs/s640/Launchings%2Bappendix%2B4.JPG" width="640" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><b>Figure 1.5.</b> <i>Asian students as a percentage of all bachelor's degrees in engineering. </i></td></tr></tbody></table><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://2.bp.blogspot.com/-LGNDjPITxVA/WYDSbWj9BwI/AAAAAAAAK3c/fP9nKIkyfAoneGg-MG7rkuQBr2mKDspCgCLcBGAs/s1600/Launchings%2Bappendix%2B6.JPG" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="441" data-original-width="736" height="380" src="https://2.bp.blogspot.com/-LGNDjPITxVA/WYDSbWj9BwI/AAAAAAAAK3c/fP9nKIkyfAoneGg-MG7rkuQBr2mKDspCgCLcBGAs/s640/Launchings%2Bappendix%2B6.JPG" width="640" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><b>Figure 1.6.</b> <i>Non-Resident Alien students as a percentage of all bachelor's degrees in engineering.</i></td></tr></tbody></table><b><br /></b><b><br /></b><b>Appendix II. Bachelor’s degrees in the Mathematical Sciences</b><br /><br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://3.bp.blogspot.com/-JmghIsRMoHk/WYDUchDx39I/AAAAAAAAK30/sxiFull7TFIgtaWIsyMvL9KCYQxRyd16wCLcBGAs/s1600/Launchings%2Bappendix%2B2_1.JPG" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="418" data-original-width="741" height="360" src="https://3.bp.blogspot.com/-JmghIsRMoHk/WYDUchDx39I/AAAAAAAAK30/sxiFull7TFIgtaWIsyMvL9KCYQxRyd16wCLcBGAs/s640/Launchings%2Bappendix%2B2_1.JPG" width="640" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><b>Figure 2.1.</b> <i>Women as a percentage of all bachelor's degrees in the mathematical sciences.</i> <b>Note: Scale starts at 40%.</b></td></tr></tbody></table><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://1.bp.blogspot.com/-tWgDSVq8Qow/WYDUcgkkZKI/AAAAAAAAK3s/6eEsKgsLVdAf74k98tduiTSsRQCXymC0wCLcBGAs/s1600/Launchings%2Bappendix%2B2_2.JPG" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="443" data-original-width="737" height="384" src="https://1.bp.blogspot.com/-tWgDSVq8Qow/WYDUcgkkZKI/AAAAAAAAK3s/6eEsKgsLVdAf74k98tduiTSsRQCXymC0wCLcBGAs/s640/Launchings%2Bappendix%2B2_2.JPG" width="640" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><b>Figure 2.2.</b> <i>White non-Hispanic students as a percentage of all bachelor's degrees in the mathematical sciences.</i></td></tr></tbody></table><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://4.bp.blogspot.com/-eaRMWgFvWQo/WYDUdJG4fNI/AAAAAAAAK3w/_S_Yb7QlXQUnq8iOf-FB5SF7KkQ-78w-QCLcBGAs/s1600/Launchings%2Bappendix%2B2_3.JPG" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="441" data-original-width="739" height="380" src="https://4.bp.blogspot.com/-eaRMWgFvWQo/WYDUdJG4fNI/AAAAAAAAK3w/_S_Yb7QlXQUnq8iOf-FB5SF7KkQ-78w-QCLcBGAs/s640/Launchings%2Bappendix%2B2_3.JPG" width="640" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><b>Figure 2.3. </b><i>Black non-Hispanic students as a percentage of all bachelor's degrees in the mathematical sciences.</i></td></tr></tbody></table><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://2.bp.blogspot.com/-IR_UqIKvjH4/WYDUdq3mT0I/AAAAAAAAK34/-dxQd93tps8Vj3NQBd6CSrN1jV9GCEE-wCLcBGAs/s1600/Launchings%2Bappendix%2B2_4.JPG" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="436" data-original-width="732" height="380" src="https://2.bp.blogspot.com/-IR_UqIKvjH4/WYDUdq3mT0I/AAAAAAAAK34/-dxQd93tps8Vj3NQBd6CSrN1jV9GCEE-wCLcBGAs/s640/Launchings%2Bappendix%2B2_4.JPG" width="640" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><b>Figure 2.4.</b> <i>Hispanic students as a percentage of all bachelor's degrees in the mathematical sciences. </i></td></tr></tbody></table><br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://3.bp.blogspot.com/-d0kfUV910O0/WYDUcomCBTI/AAAAAAAAK3o/4Xd-WWx4H4wJqRGhWWOuPM7IKMgabxzLQCLcBGAs/s1600/Launchings%2B2_5.JPG" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="440" data-original-width="735" height="382" src="https://3.bp.blogspot.com/-d0kfUV910O0/WYDUcomCBTI/AAAAAAAAK3o/4Xd-WWx4H4wJqRGhWWOuPM7IKMgabxzLQCLcBGAs/s640/Launchings%2B2_5.JPG" width="640" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><b>Figure 2.5. </b><i>Asian students as a percentage of all bachelor's degrees in the mathematical sciences.</i></td></tr></tbody></table><br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://1.bp.blogspot.com/-K2qAiq2XskM/WYDUd-mtpkI/AAAAAAAAK38/Sog4LRJIk3AGHQg8wsWQ00HxdQSug7uqQCLcBGAs/s1600/Launchings%2Bappendix%2B2_6.JPG" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="439" data-original-width="740" height="378" src="https://1.bp.blogspot.com/-K2qAiq2XskM/WYDUd-mtpkI/AAAAAAAAK38/Sog4LRJIk3AGHQg8wsWQ00HxdQSug7uqQCLcBGAs/s640/Launchings%2Bappendix%2B2_6.JPG" width="640" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><b>Figure 2.6. </b><i>Non-Resident Alien students as a percentage of all bachelor's degrees in the mathematical sciences.</i> </td></tr></tbody></table><b><br /></b><b><br /></b><b>Appendix III. Bachelor’s degrees in the Physical Sciences</b><br /><b><br /></b><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://4.bp.blogspot.com/-3jzYRVtQ300/WYDbYuyCeTI/AAAAAAAAK4g/u0kkBSgAomULfUZ2f66VweFBzoOa4dICACLcBGAs/s1600/Launchings%2Bappendix%2B3_1.JPG" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="448" data-original-width="742" height="386" src="https://4.bp.blogspot.com/-3jzYRVtQ300/WYDbYuyCeTI/AAAAAAAAK4g/u0kkBSgAomULfUZ2f66VweFBzoOa4dICACLcBGAs/s640/Launchings%2Bappendix%2B3_1.JPG" width="640" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><b>Figure 3.1.</b> <i>Women as a percentage of all bachelor's degrees in the physical sciences.</i> <b>Note: scale starts at 30%</b></td></tr></tbody></table><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://4.bp.blogspot.com/-yvP2iid8XNU/WYDbYlHxoII/AAAAAAAAK4U/PFAw-M67Zb81KU7vfTYHamqEu9gV9uuHQCLcBGAs/s1600/Launchings%2B3_2.JPG" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="438" data-original-width="735" height="380" src="https://4.bp.blogspot.com/-yvP2iid8XNU/WYDbYlHxoII/AAAAAAAAK4U/PFAw-M67Zb81KU7vfTYHamqEu9gV9uuHQCLcBGAs/s640/Launchings%2B3_2.JPG" width="640" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><b>Figure 3.2.</b> <i>White non-Hispanic students as a percentage of all bachelor's degrees in the physical sciences.</i></td></tr></tbody></table><br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://3.bp.blogspot.com/-jqWxaFwqLkY/WYDbYr-E20I/AAAAAAAAK4Y/4Xt6j71P8ss9ie_2UDt7EHLAZD3MaBRNACLcBGAs/s1600/Launchings%2B3_3.JPG" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="436" data-original-width="733" height="380" src="https://3.bp.blogspot.com/-jqWxaFwqLkY/WYDbYr-E20I/AAAAAAAAK4Y/4Xt6j71P8ss9ie_2UDt7EHLAZD3MaBRNACLcBGAs/s640/Launchings%2B3_3.JPG" width="640" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><b>Figure 3.3. </b><i>Black non-Hispanic students as a percentage of all bachelor's degrees in the physical sciences.</i></td></tr></tbody></table><br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://1.bp.blogspot.com/-efEBMLOUR6M/WYDbYxRCrUI/AAAAAAAAK4c/49-sujvNzEMtlT4A-dSSf6orJ-WXnH8jgCLcBGAs/s1600/Launchings%2Bappendix%2B3_4.JPG" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="440" data-original-width="734" height="382" src="https://1.bp.blogspot.com/-efEBMLOUR6M/WYDbYxRCrUI/AAAAAAAAK4c/49-sujvNzEMtlT4A-dSSf6orJ-WXnH8jgCLcBGAs/s640/Launchings%2Bappendix%2B3_4.JPG" width="640" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><b>Figure 3.4. </b><i>Hispanic students as a percentage of all bachelor's degrees in the physical sciences.</i></td></tr></tbody></table><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://3.bp.blogspot.com/-eehSuojEl3Q/WYDbainvEhI/AAAAAAAAK4o/Rmu_W9ItYZ4w0Xw1JIjOPkKM63YfWfXJQCLcBGAs/s1600/Launchings%2Bappendix%2B3_5.JPG" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="441" data-original-width="741" height="380" src="https://3.bp.blogspot.com/-eehSuojEl3Q/WYDbainvEhI/AAAAAAAAK4o/Rmu_W9ItYZ4w0Xw1JIjOPkKM63YfWfXJQCLcBGAs/s640/Launchings%2Bappendix%2B3_5.JPG" width="640" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><b>Figure 3.5.</b> <i>Asian students as a percentage of all Bachelor's degrees in the physical sciences. </i></td></tr></tbody></table><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://4.bp.blogspot.com/-L_v8b6HYi3I/WYDbZXTCP_I/AAAAAAAAK4k/zicIyJRBXnkVJ-LJdZKWeq1GxYDYS5MgQCLcBGAs/s1600/Launchings%2Bappendix%2B3_6.JPG" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="443" data-original-width="738" height="384" src="https://4.bp.blogspot.com/-L_v8b6HYi3I/WYDbZXTCP_I/AAAAAAAAK4k/zicIyJRBXnkVJ-LJdZKWeq1GxYDYS5MgQCLcBGAs/s640/Launchings%2Bappendix%2B3_6.JPG" width="640" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><b>Figure 3.6. </b><i>Non-Resident Alien students as a percentage of all bachelor's degrees in the physical sciences.</i></td></tr></tbody></table><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br /><br />Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com0tag:blogger.com,1999:blog-7251686825560941361.post-53802103807547776132017-07-05T12:44:00.000-04:002017-07-05T12:44:42.850-04:00The 2015 NAEP<b><i>You can follow me on Twitter <a href="https://twitter.com/dbressoud" target="_blank">@dbressoud</a>.</i></b><br /><br />On May 5, 2017, the presidents and executive directors of the member societies of CBMS received a report from Samantha Burg and Stephen Provasnik at the U.S. Department of Education’s National Center for Education Statistics (NCES) on the results in mathematics from the 2015 studies by the National Assessment of Educational Progress (NAEP) and Trends in International Mathematics and Science Study (TIMSS). The full PowerPoint of their presentation, covering both the 2015 NAEP and 2015 TIMSS, can be accessed at www.cbmsweb.org/2017/05/presentations. <br /><br />Both assessments are conducted for students at grades 4, 8 and 12. NAEP is a federally mandated assessment of student achievement in the U.S. and is conducted every other year. TIMSS provides an international comparison and is run every four years for ages equivalent to grades 4 and 8. The 12th grade TIMSS is restricted to advanced mathematics students (in the U.S. those who have taken a course like AP Calculus). It was administered in 2015 for the first time in the U.S. since 1995. <br /><br />The scores since 1990 for the 4 th and 8 th grade NAEP and since 2005 for the 12th grade are shown in Figures 1, 2, and 3. The distinguishing features for grades 4 and 8 are the strong growth from 1990 until 2007 and relative stagnation since then, with a small but statistically significant drop (except for the 90th percentile in grade 4) between 2013 and 2015. The 12th grade scores also show a drop since 2013 that is statistically significant at and below the 50th percentile. <br /><br />This drop is a cause for concern, but not yet alarm. NCES is eagerly anticipating the 2017 NAEP results to see whether the downturn was simply a blip in what is essentially a stable state or the start of something more troubling. <br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://2.bp.blogspot.com/-IBQmxGuzdBk/WV0W3ZlXL4I/AAAAAAAAK2E/N2O0x4cnlawgDpeZmctHVupUNKIQ96FKgCLcBGAs/s1600/Figure-1.tiff" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="646" data-original-width="1146" height="360" src="https://2.bp.blogspot.com/-IBQmxGuzdBk/WV0W3ZlXL4I/AAAAAAAAK2E/N2O0x4cnlawgDpeZmctHVupUNKIQ96FKgCLcBGAs/s640/Figure-1.tiff" width="640" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Figure 1: NAEP scores for grade 4. <br />Source: Burg & Provasnik, 2017.</td></tr></tbody></table><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://4.bp.blogspot.com/-AvlPYRYTntw/WV0W3VIp66I/AAAAAAAAK2A/A17eLRyZXV8RdtiGXy4RFpvKJoWlEOVzQCLcBGAs/s1600/Figure-2.tiff" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="645" data-original-width="1148" height="356" src="https://4.bp.blogspot.com/-AvlPYRYTntw/WV0W3VIp66I/AAAAAAAAK2A/A17eLRyZXV8RdtiGXy4RFpvKJoWlEOVzQCLcBGAs/s640/Figure-2.tiff" width="640" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Figure 2: NAEP scores for grade 8. Source: Burg & Provasnik, 2017.</td></tr></tbody></table><div class="separator" style="clear: both; text-align: center;"><br /></div><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://4.bp.blogspot.com/-w-ZraHzySf8/WV0WXjRzhTI/AAAAAAAAK10/LNsY0lJA1gsEaQbSwmeVezlpVpIwdWSgACLcBGAs/s1600/Figure-3.tiff" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="642" data-original-width="1147" height="356" src="https://4.bp.blogspot.com/-w-ZraHzySf8/WV0WXjRzhTI/AAAAAAAAK10/LNsY0lJA1gsEaQbSwmeVezlpVpIwdWSgACLcBGAs/s640/Figure-3.tiff" width="640" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Figure 3: NAEP scores for grade 12.<br />Source: Burg & Provasnik, 2017.</td></tr></tbody></table><br />An obvious question is whether the Common Core State Standards in Mathematics (CCSS-M) have had any effect on student scores. One hypothesis is that changing the curriculum has introduced enough confusion and uncertainty among teachers that it is having a visibly negative effect. Another hypothesis, which has some supporting evidence, is that the choices of topics for assessment may no longer be completely aligned with what is being taught. <br /><br />The largest drops at Grade 4 were in the subject areas of Geometry and Data Analysis (Table 1). The NAEP Validity Studies (NVS) panel (Daro, Hughes, & Stancavage, 2015) found some misalignment between the NAEP questions and the CCSS-M curriculum. They found that 32% of the Data Analysis questions were either not covered in CCSS-M or were covered after grade 4. In Geometry, 18%, of the NAEP questions were covered after grade 4 in CCSS-M. In the other direction, only 57% of CCSS-M standards for Operations and Algebraic Thinking by grade 4 were covered by NAEP questions. <br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://3.bp.blogspot.com/-aauYHK9VdIg/WV0U5mPclxI/AAAAAAAAK1w/MZqsUUjJ1EoYHFDv-jvX-6pabE7HJCN2wCLcBGAs/s1600/Table-1.tiff" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="468" data-original-width="840" height="355" src="https://3.bp.blogspot.com/-aauYHK9VdIg/WV0U5mPclxI/AAAAAAAAK1w/MZqsUUjJ1EoYHFDv-jvX-6pabE7HJCN2wCLcBGAs/s640/Table-1.tiff" width="640" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Table 1: Changes in NAEP scores, 2013 to 2015, by subscale topics.<br />Source: Burg & Provasnik, 2017</td></tr></tbody></table><br />For grade 8, the misalignment occurs in both directions within Data Analysis. In the 8 th grade NAEP, 17% of the Data Analysis questions had not yet been covered in CCSS-M, and 59% of what is specified for statistics and probability by grade 8 in CCSS-M was not assessed by NAEP. For grade 12, there was a uniform 2-point drop across all subscales. <br /><br />These observations raise interesting questions about the construction of future NAEP instruments. Because of the need for comparability from one test administration to the next, the distribution of topics has not changed. While CCSS-M is not the national curriculum that was once envisioned, the fact is that almost all states have aligned their standards with its expectations. NAEP may need to change to reflect the reality of what is taught by grades 4 and 8. <br /><br />The breakdowns by race/ethnicity and gender for the overall mathematics scores in grades 4 and 8 (Table 2) show comparable increases from 1990 to 2015, and comparable declines since 2013. Black students in grade 4 saw the greatest gains since 1990, but at a score of 224 they are still well below the national average. <br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://2.bp.blogspot.com/-a4E8QoBPbVo/WV0T-ZcMWkI/AAAAAAAAK1s/R6aRIbi46TM6kBCZ5Rm_LjZxmWe0duiKQCLcBGAs/s1600/Table-2.tiff" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="407" data-original-width="672" height="387" src="https://2.bp.blogspot.com/-a4E8QoBPbVo/WV0T-ZcMWkI/AAAAAAAAK1s/R6aRIbi46TM6kBCZ5Rm_LjZxmWe0duiKQCLcBGAs/s640/Table-2.tiff" width="640" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Table 2: Changes in NAEP Math scores for grades 4 and 8 by race/ethnicity and gender.<br /> Source: Burg & Provasnik, 2017.</td></tr></tbody></table><br />At grade 12, the strongest gains since 2005 have been for Asian and Hispanic students (Table 3, Pacific Islanders are such a small proportion of Asian/Pacific Islander that it is not clear how their scores have changed, and the doubling of the percentage identifying as Two or More Races makes it difficult to compare the 2005 and 2015 scores). An interesting insight lies in the shift in the demographics of 12 th grade students. In ten years, the percentage of White students dropped from 66% to 55%, while the percentage of Hispanic 12 th graders rose from 13% to 22%. <br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://1.bp.blogspot.com/-exeKowP20Ak/WV0TaqbVv2I/AAAAAAAAK1o/lzkHmDNB07sAfzuwREaDw24AT7BY0puvQCLcBGAs/s1600/Table-3.tiff" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="378" data-original-width="640" height="377" src="https://1.bp.blogspot.com/-exeKowP20Ak/WV0TaqbVv2I/AAAAAAAAK1o/lzkHmDNB07sAfzuwREaDw24AT7BY0puvQCLcBGAs/s640/Table-3.tiff" width="640" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Table 3: Changes in NAEP Math scores for grade 12 by race/ethnicity and gender.<br /> Source: Burg & Provasnik, 2017.</td></tr></tbody></table><br />Next month I will be looking at the changing demographics of bachelor’s degrees earned in engineering, the mathematical sciences, and the physical sciences. In mathematics, the decline in the percentage of degrees in mathematics going to White students has been in line with the decline in their overall percentage at that age group, from 72.4% in 2005 to 59.6% in 2015 (NCES, 2005–2015). Some of this has been made up by a significant increase in mathematics degrees going to Hispanic students, from 5.7% to 8.9%, but the percentage of bachelor’s degrees in mathematics earned by Black students decreased from 6.1% to 4.7% over this decade, while Asian students remained essentially stable, 10.2% to 10.6%. Most of the shift has gone to non- resident aliens who accounted for 5.0% of the mathematics degrees in 2005, but 12.9% in 2015. <br /><b><br /></b><b>References</b><br />Burg, S. & Provasnik, S. (2017). NAEP and TIMSS Mathematics 2015. Presentation to the Conference Board of the Mathematical Sciences, May 5, 2017. Available at <a href="http://www.cbmsweb.org/2017/05/presentations/" target="_blank">www.cbmsweb.org/2017/05/presentations/</a><br /><br />Daro, P., Hughes, G.B., & Stancavage, F. (2015). Study of the alignment of the 2015 NAEP mathematics items at grades 4 and 8 to the Common Core State Standards (CCSS) for Mathematics. NAEP Validity Studies Panel report. Washington, DC: American Institutes for Research. Available at <a href="http://www.air.org/sites/default/files/downloads/report/Study-of-Alignment-NAEP-Mathematics-Items-common-core-Nov-2015.pdf" target="_blank">www.air.org/sites/default/files/downloads/report/Study-of- Alignment-NAEP-Mathematics- Items-common- core-Nov- 2015.pdf</a><br /><br />National Center for Education Statistics (NCES). (2005–2015). <i>Digest of Education Statistics</i>. Available at <a href="http://nces.ed.gov/programs/digest/" target="_blank">nces.ed.gov/programs/digest/</a><br /><br /><b>Note:</b><br />In compliance with new standards from the U.S. Office of Management and Budget for collecting and reporting data on race/ethnicity, additional information was collected beginning in 2011 so that results could be reported separately for Asian students, Native Hawaiian/Other Pacific Islander students, and students identifying with two or more races. In earlier assessment years, results for Asian and Native Hawaiian/Other Pacific Islander students were combined into a single Asian/Pacific Islander category. <br /><br />As of 2011, all of the students participating in NAEP are identified as one of the following seven racial/ethnic categories:<br /><ul><li>White </li><li>Black (includes African American) </li><li>Hispanic (includes Latino) </li><li>Asian </li><li>Native Hawaiian/Other Pacific Islander </li><li>American Indian/Alaska Native </li><li>Two or more races</li></ul>When comparing the results for racial/ethnic groups from 2013 to earlier assessment years, results for Asian and Native Hawaiian/Other Pacific Islander students were combined into a single Asian/Pacific Islander category for all previous assessment years. <br /><br /><br /><br />Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com0tag:blogger.com,1999:blog-7251686825560941361.post-30502704421560068612017-06-01T07:00:00.000-04:002017-06-01T07:00:28.066-04:00Re-imagining the Calculus Curriculum, II<b><i>You can follow me on Twitter <a href="https://twitter.com/dbressoud" target="_blank">@dbressoud</a>.</i></b><br /><br />Last month, in "<a href="http://launchings.blogspot.com/2017/05/re-imagining-calculus-curriculum-i.html" target="_blank">Re-imagining the Calculus Curriculum</a>," I, I introduced <a href="http://patthompson.net/ThompsonCalc/index.html" target="_blank">Project DIRACC</a> (<i>Developing and Investigating a Rigorous Approach to Conceptual Calculus</i>), developed by Pat Thompson, Mark Ashbrook, and Fabio Milner at Arizona State University. References to the theory underpinning this approach are given at the end of this column. This month’s column will expand on some details of this curriculum. <br /><br />One of the first common student misconceptions that Project DIRACC tackles is that variables are simply stand-ins for unknown quantities. The authors begin the meat of his course in Chapter 3 with an explanation of the distinction between <i>variable</i>,<i> constant</i>, and <i>parameter</i>, pointing out how context-specific the designations as either variable or parameter can be. One of the distinctive features of this project is the thoughtful use of technology, in this case enabling students to play with the effect of varying a variable with a variety of choices of parameter (see <a href="http://patthompson.net/ThompsonCalc/section_3_1.html">patthompson.net/ThompsonCalc/section_3_1.html</a>). <br /><br />This leads to relationships between variables (how volume varies with height), and then functions as a special class of relationships between variables, one in which “<i><b>any value of one variable determines exactly one value of the other</b></i>.” The point is that the <i>f</i> in <i>f</i> (<i>x</i>) has meaning. It is the name of the relationship. This enables the authors to tackle the misconception that <i>f</i> (<i>x</i>) is simply a lengthy way of expressing the variable <i>y</i>. <br /><br />While acknowledging that <i>f</i>(<i>x</i>) can represent a second variable, they emphasize that it is shorthand for “the value of the relationship f when applied to a value of <i>x</i>.” This point is driven home by an example of the usefulness of functional notation. If <i>d</i>(<i>x</i>) relates a moment in time, <i>x</i> measured in years, to the distance between the Earth and the Moon at that time, then <i>d</i>(<i>x</i>) – <i>d</i>(<i>x</i>–5) enables us to express the change in distance over the five years before time x, while <i>d</i>(<i>x</i>+5) – <i>d</i>(<i>x</i>) expresses the change in distance over the succeeding five years. <br /><br />The authors also make the important distinction between functions defined conceptually—the distance between Earth and Moon at a given time—and those defined computationally, such as <i>V</i>(<i>u</i>) = <i>u</i>(13.76 – 2<i>u</i>)(16.42 – 2<i>u</i>). They then proceed to devote considerable effort to describing the structure of functions as they are built from sums, products, quotients, compositions, and inverses. This includes clarifying the distinction between the independent variable and the argument of a function. Thus for f (<i>x</i>/3 + 5) the independent variable is <i>x</i>, but the function argument is<i> x</i>/3 + 5, an important step toward understanding composition of functions. <br /><br />While function structure <i>should</i> be part of precalculus, the importance of including this material has been revealed in exploring student difficulties with differentiation. Given a complicated computational rule that defines a function, students often have difficulty parsing this rule and thus determining the choice and order of the techniques of differentiation they need to use. <br /><br />Rates of change are now introduced in Chapter 4. The authors distinguish between ∆x, the parameter that describes the length of a small subinterval of the domain, and the changes in x and y represented by the differentials dx and dy. These are variables that within the given subinterval are always connected by a linear relationship. <br /><br />A nice illustration of how this works is given with a photograph of a truck traveling through an intersection (Figure 1). <br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://4.bp.blogspot.com/-UG6YvHpIQrA/WS83mi8QEsI/AAAAAAAAK1Y/kgOGfpWSIa0ZMorjShP1QjVeSxvFDadqwCLcB/s1600/truck%2Bmoving%2Blaunchings.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="700" data-original-width="1050" height="213" src="https://4.bp.blogspot.com/-UG6YvHpIQrA/WS83mi8QEsI/AAAAAAAAK1Y/kgOGfpWSIa0ZMorjShP1QjVeSxvFDadqwCLcB/s320/truck%2Bmoving%2Blaunchings.jpg" width="320" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><b>Figure 1.</b> A photo of truck taken with a shutter setting of 1/1000 sec.</td></tr></tbody></table>Taken at a shutter speed of 1/1000th of a second, it appears to freeze the truck. But if you zoom in on the tail light (Figure 2, see <a href="http://patthompson.net/ThompsonCalc/section_4_3.html" target="_blank">Section 4.3</a> for a video of the zoom), the streaks reveal that the truck was moving.<br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://1.bp.blogspot.com/-HSbEuBPmB1U/WS83Vnr_M4I/AAAAAAAAK1U/rUo9PzVaq7U6DULmUiTeyKLX4iwXJwwUgCLcB/s1600/blurred%2Btruck%2Blaunchings.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="243" data-original-width="263" src="https://1.bp.blogspot.com/-HSbEuBPmB1U/WS83Vnr_M4I/AAAAAAAAK1U/rUo9PzVaq7U6DULmUiTeyKLX4iwXJwwUgCLcB/s1600/blurred%2Btruck%2Blaunchings.png" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><b>Figure 2.</b> A closer look at the truck's tail light shows small streaks. <br />The truck moved slightly while the camera's shutter was open.</td></tr></tbody></table><br /><br />One can even estimate the length of the streaks to approximate the velocity of the truck. Over 1/1000th of a second, it is doubtful that the truck’s velocity changed very much. The picture of the truck was taken at a “moment” in time, but that moment stretched over 0.001 seconds. The point is that this period of time is short enough that the truck’s velocity measured as change in distance over change in time is “essentially constant.” If <i>y</i> is position and <i>x</i> is time, then over this interval of length <i>∆x</i> = 0.001 seconds, we can treat the variable <i>dy</i> as a constant times <i>dx</i>. It is this constant that is used to define the rate of change at a moment, <br /><br /><blockquote>We say that <b>a function has a rate of change at the moment x<sub>0</sub> if, over a suitably small interval of its independent variable containing x<sub>0</sub>, the function’s value changes at essentially a constant rate with respect to its independent variable.</b></blockquote><br />Significantly, even as the authors are defining the rate of change at a moment, they emphasize that “all motion, and hence all variation, is blurry.” <br /><br />Note that there is no mention of limits, a means of defining the derivative that is often more confusing than enlightening (see the 2014 <i>Launchings</i> columns from <a href="http://launchings.blogspot.com/2014/06/beyond-limit-i.html">July</a>, <a href="http://launchings.blogspot.com/2014/08/beyond-limit-ii.html">August</a>, and <a href="http://launchings.blogspot.com/2014/09/beyond-limit-iii.html">September</a>). <br /><br />After further discussion and exploration of rate of change functions, the authors now move in Chapter 5 to Accumulation Functions, building up total changes from rates of change that are essentially constant on very small intervals. These give rise to what are anachronistically referred to as left-hand Riemann sums. Students use technology to explore the increasing accuracy as ∆x gets smaller. The effect of the choice of starting value is noted, and the definite integral with a variable upper limit now appears. It is important that the first time students see a definite integral it has a variable upper limit. <br /><br />In Chapter 6, the inverse problem, going from knowledge of an exact expression of the accumulation function to the discovery of the corresponding rate of change function, is now explored, leading to the Fundamental Theorem of Integral Calculus in the form: The derivative with respect to x of the definite integral from a to x of a rate of change function is equal to that rate of change function evaluated at x. Techniques and applications of differentiation follow as the semester concludes. <br /><br />The great strength and promise of this approach is that the traditional content of the first semester of calculus is only slightly tweaked, especially since it is increasingly common for university Calculus I courses to avoid or significantly downplay limits. But the curriculum has been totally reshaped to address common student difficulties and misconceptions. This route into calculus has the added advantage—though perhaps a disadvantage in the eyes of some students—that those who have been through a procedurally oriented course are unlikely to recognize this as an accelerated repetition of what they have already studied. It will challenge them to rethink what they believe calculus to be. <br /><br />References <br /><br />Thompson, P.W. and Silverman, J. (2008). The concept of accumulation in calculus. In M.P. Carlson & C. Rasmussen (Eds.), <i>Making the connection: Research and teaching in undergraduate mathematics</i> (<i>MAA Notes </i>Vol. 73, pp. 43–52). Washington, DC: Mathematical Association of America. <br /><br />Thompson, P.W., Byerley, C. and Hatfield, N. (2013). A conceptual approach to calculus made possible by technology. <i>Computers in the Schools</i>. 30:124–147. <br /><br />Thompson, P.W. and Dreyfus, T. (2016). A coherent approach to the Fundamental Theorem of Calculus using differentials. In R. Göller. R. Biehler & R. Hochsmuth (Eds.), <i>Proceedings of the Conference on Didactics of Mathematics in Higher Education as a Scientific Discipline</i> (pp. 355–359 ) Hannover, Germany: KHDM. <br /><br /><br /><br /><br /><br /><br />Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com0tag:blogger.com,1999:blog-7251686825560941361.post-64015698074392957852017-05-01T08:00:00.000-04:002017-05-01T08:00:05.679-04:00Re-imagining the Calculus Curriculum, I<b><i>You can now follow me on Twitter <a href="https://twitter.com/dbressoud" target="_blank">@dbressoud</a>.</i></b><br /><br />I was recently asked about calculus instruction: Which is easier, reforming pedagogy or curriculum? The answer is easy: pedagogy. This is not to say that it is easy to change how we teach this course, but it is far easier than trying to change what we teach. The order and emphasis of topics that emerged in the 1950s has proven extremely hard to shift. <br /><br />Not that we have not tried. During the Calculus Reform movement of the late 1980s and early 1990s, NSF encouraged curricular innovation. Several of these efforts adopted an emphasis on modeling dynamical systems, introducing calculus via differential equations and developing the tools of calculus in service to this vision. This provides wonderful motivation, and this approach survives in a few pockets. It is how we teach calculus at Macalester, and the U.S. Military Academy at West Point has successfully used this route into calculus for over a quarter century. But despite its appeal, this curriculum necessitates modifying the entire year of single variable calculus, raising problems for institutions that must accommodate students who are transferring in or out. It also is a difficult sell to those who worry about “coverage” since it requires devoting considerable time to topics—modeling with differential equations, functions of several variables, and partial differential equations—that receive little or no attention in the traditional course. <br /><br />This month, I want to talk about a promising curricular innovation that Pat Thompson at Arizona State University has been developing in collaboration with Fabio Milner and Mark Ashbrook, Project DIRACC (<i>Developing and Investigating a Rigorous Approach to Conceptual Calculus</i>). It has the advantage that it fits more easily into what is expected from each semester. It has been under development since 2010 and is slated to be the curriculum used for all Calculus I sections for mathematics or science majors at ASU beginning in fall, 2018. <br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-vWQ0GY9ER08/WQOmpcsjZdI/AAAAAAAAK1E/iDVvhbOe9s4GwhGk_bavZC-Xxg-AKIpaQCLcB/s1600/launchings%2Bfaces.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="161" src="https://2.bp.blogspot.com/-vWQ0GY9ER08/WQOmpcsjZdI/AAAAAAAAK1E/iDVvhbOe9s4GwhGk_bavZC-Xxg-AKIpaQCLcB/s400/launchings%2Bfaces.JPG" width="400" /></a></div><br />Thompson began with research into the misconceptions that students carry into calculus and that impede their ability to understand it. I quote these common misconceptions from his website (<a href="http://patthompson.net/ThompsonCalc/About.html">patthompson.net/ThompsonCalc/About.html</a>).<br /><ul><li>Calculus, like the school mathematics, is about rules and procedures. Students think that calculus is difficult primarily because there are so many rules and procedures. </li><li>Variables do not vary. Therefore rate of change is not about change. </li><li>Integrals are areas under a curve. Students wonder, "How can an area represent a distance or an amount of work?" </li><li>Average rate of change has little to do with rate of change. It is about the direction of a line that passes through two points on a graph. </li><li>A tangent is a line that "just touches" a curve. </li><li>Derivative is a slope of a tangent. The net result is that, in students' understandings, derivatives are not about rates of change.</li></ul>The second bullet point is particularly common and problematic. In an expression such as <br /><div style="text-align: center;"><i style="text-align: start;">f</i><span style="text-align: start;">(</span><i style="text-align: start;">x</i><span style="text-align: start;">)</span> = <i>x</i><sup>3</sup> – 3<i>x</i> + 2, </div><br />many students see the expression <i>f</i>(<i>x</i>) as nothing more than a lengthy way of writing the dependent variable, and functions are seen as static objects that prescribe how to turn the input x into the output <i>f</i>(<i>x</i>). With this mindset, differentiation and integration are nothing more than arcane rules for turning one static object into another. <br /><br />Choosing to define integrals as areas and derivatives as slopes, as is common in the standard curriculum, is equally problematic. It reinforces the notion that calculus is about computing values associated with geometric objects. To complicate matters, while area is a familiar concept, slope is far less real or meaningful to our students. Too many students never come to the realization that the real power of differentiation and integration arises from their interpretation as rate of change and as accumulation. <br /><br />While the earliest uses of accumulation were for determining areas, those Hellenistic philosophers who mastered it also recognized its equal applicability to questions of volumes and moments. By the 14th century, European philosophers were applying techniques of accumulation to the problem of determining distance from knowledge of instantaneous velocity. None of these come easily to students who are fixated on integrals as areas. <br /><br />Seeing the derivative as a slope is even more problematic, a static value of an obscure parameter. Differentiation arose from problems of interpolation for the purpose of approximating values of trigonometric functions in first millenium India, in understanding the sensitivity of one variable to changes in another in the work of Fermat and Descartes, and in relating rates of change as in Napier’s analysis of the logarithm and Newton’s <i>Principia</i>. Derivative as slope came quite late in the historical development of calculus precisely because its application to interesting questions is not intuitive. <br /><br />These insights provide the starting point for Thompson’s reformation of Calculus I. His textbook, which is still a work in progress, can be accessed at <a href="http://patthompson.net/ThompsonCalc">patthompson.net/ThompsonCalc</a>. See Thompson & Silverman (2008), Thompson et al. (2013), and Thompson & Dreyfus (2016) for additional background. I find it deliciously ironic that one of the first topics he tackles is the distinction among constants, parameters, and variables. If you look at the calculus textbooks of the late 18 th century through the middle of the 19 th century, this is exactly where they started. Somehow, we lost recognition of the importance of elevating this distinction for our students. Thompson goes on to spend considerable effort to clarify the role of a function as a bridge between two co-varying quantities. And then, he really breaks with tradition by first tackling integration, which he enters via problems in accumulation.<br /><br />This accomplishes several desiderata. First, it ensures that students do not begin with an understanding of the integral as area, but as an accumulator. Second, it makes it much easier to recognize this accumulator as a function in its own right. Students struggle with recognizing the definite integral from a to the variable x as a function of x (see the section of last month’s column, <a href="http://launchings.blogspot.com/2017/04/" target="_blank">Conceptual Understanding</a>, that addresses Integration as Accumulation). Thompson begins by viewing the integrand as a rate of change function. The variable upper limit arises naturally. Third, and perhaps most important, it gives meaning to the Fundamental Theorem of Integral Calculus, that the derivative of an accumulator function is the rate of change function. <br /><br />Differentiation can then be introduced in precisely the way Newton first understood it: Given a closed expression for the accumulator function, how can we find the corresponding rate of change function? <br /><br />Next month, I will expound on exactly how Thompson introduces these steps, but for now I would like to conclude with a comparison of the two curricular innovations, that of Thompson and the approach described at the start of this column that emphasizes calculus as a tool for modeling dynamical systems. The latter does overcome the problem of student belief that the derivative is to be understood as the slope of the tangent. It brings to the fore the derivative as describing a rate of change. The problem is that it does nothing to clarify the role of the integral as an accumulator. In some sense, it makes it more difficult. As we teach calculus at Macalester, the integral is introduced purely as an anti-derivative, making it extremely difficult to give meaning to the Fundamental Theorem of Integral Calculus. I have to work very hard in the second semester to help students understand the integral as an accumulator and so justify that this theorem has meaning. In a very real sense, Thompson approach <i>begins </i>with this fundamental theorem. <br /><br />Thompson, P.W. and Silverman, J. (2008). The concept of accumulation in calculus. In M.P. Carlson & C. Rasmussen (Eds.), <i>Making the connection: Research and teaching in undergraduate mathematics</i> (MAA Notes Vol. 73, pp. 43–52). Washington, DC: Mathematical Association of America. <br /><br />Thompson, P.W., Byerley, C. and Hatfield, N. (2013). A conceptual approach to calculus made possible by technology. <i>Computers in the Schools</i>. 30:124–147. <br /><br />Thompson, P.W. and Dreyfus, T. (2016). A coherent approach to the Fundamental Theorem of Calculus using differentials. In R. Göller. R. Biehler & R. Hochsmuth (Eds.), <i>Proceedings of the Conference on Didactics of Mathematics in Higher Education as a Scientitific Discipline</i> (pp. 355–359 ) Hannover, Germany: KHDM. Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com0tag:blogger.com,1999:blog-7251686825560941361.post-3069777966555504532017-04-01T06:49:00.000-04:002017-04-01T06:49:08.064-04:00Conceptual Understanding<b><i>You can now follow me on Twitter <a href="https://twitter.com/dbressoud" target="_blank">@dbressoud</a>.</i></b><br /><br />Continuing my series of summaries of articles that have appeared in the <i>International Journal of Research in Undergraduate Mathematics Education </i>(IJRUME), this month I want to briefly describe three studies that address issues of conceptual understanding. The first is a study out of Israel that probed student difficulties in understanding integration as accumulation (Swidan and Yerushalmy, 2016). The second is from France, exploring student difficulties with understanding the real number line as a continuum (Durand-Guerrier, 2016). The final paper, from England, explores a method of measuring conceptual understanding (Bisson, Gilmore, Inglis, and Jones, 2016). <br /><br /><b>Integration as Accumulation</b><br />To use the definite integral, students need to understand it as accumulation. In particular, the Fundamental Theorem of Integral Calculus rests on the recognition that the definite integral of a function <i>f</i>, when given a variable upper limit, is an accumulation function of a quantity for which <i>f </i>describes the rate of change. Pat Thompson (2013) has described the course he developed for Arizona State University that places this realization at the heart of the calculus curriculum. <br /><br />We know that students have a difficult time understanding and working with a definite integral with a variable upper limit. The authors of the IJRUME paper suggest that much of the problem lies in the fact that when students are introduced to the definite integral as a limit of Riemann sums, they only consider the case when the upper and lower limits on the Riemann sum are fixed. The limit is thus a number, usually thought of as the area under a curve. Making the transition to the case where the upper limit is variable is thus non-intuitive. <br /><br />The authors used software to explore student recognition of accumulation functions based on right-hand Riemann sums. They investigated student recognition of how the properties of these functions are shaped by the rate of change function. The experiment involved a graphing tool, Calculus UnLimited (CUL), in which students input a function and the software provides values of the corresponding accumulation function given by a right-hand sum with Δx = 0.5 (see Figure 1). Students could adjust the upper and lower limits, in jumps of 0.5. The software displays the rectangles corresponding to a right-hand sum. Students were not told that these were points on an accumulation function, merely that this was a function related to the initial function. They were encouraged to start with a lower limit of –3 and to explore the functions <i>x<sup>2</sup></i>, <i>x<sup>2</sup></i> – 9, and then cubic polynomials, and to discover what they could about this second function. Students received no further prompts. <br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://1.bp.blogspot.com/-GKuwCq75MKE/WN1EMekOm8I/AAAAAAAAKzs/Mj1PLQg-E3M9ibeXfim_tI2nVqkhaceJwCLcB/s1600/accumulation.tiff" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="305" src="https://1.bp.blogspot.com/-GKuwCq75MKE/WN1EMekOm8I/AAAAAAAAKzs/Mj1PLQg-E3M9ibeXfim_tI2nVqkhaceJwCLcB/s400/accumulation.tiff" width="400" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><b>Figure 1: </b>The CUL interface. Taken from Swidan and Yerushalmy (2016), page 33.</td></tr></tbody></table><br />Thirteen pairs of Israeli 17-year olds participated in the study. They all had been studying derivatives and indefinite integrals, but none had yet encountered definite integrals. Each pair spent about an hour exploring this software. Their actions and remarks were video-taped and then analyzed.<br /><br />One of the interesting observations was that the key to recognizing that the second function accumulates areas came from playing with the lower limit. Adjusting the upper limit simply adds or removes points, but adjusting the lower limit moves the plotted points up or down. Once students realized that the point corresponding to the lower limit is always zero, they were able to deduce that the y-value of the next point is the area of the first rectangle, and that succeeding points reflect values obtained by adding up the areas of the rectangles. Rectangles below the <i>x</i>-axis were shaded in a darker color, and students quickly picked up that they were subtracting values. Seven of the thirteen pairs of students went as far as remarking on how the concavity of the accumulation function is related to the behavior of the original function. <br /><br />This work suggests that a Riemann sum with a variable upper limit is more intuitive than a definite integral with a variable upper limit. In addition, it appears that students can discover many of the essential properties of a discrete accumulation function if allowed the opportunity to experiment with it. <br /><br /><b>Understanding the Continuum</b><br />The second paper explores student difficulties with the properties of the real number line and describes an intervention that appears to have been useful in helping students understand the structure of the continuum. Mathematicians of the nineteenth century struggled to understand the essential differences between the continuum of all real numbers and dense subsets such as the set of rational numbers. It comes as no surprise that our students also struggle with these distinctions. <br /><br />The author analyzes the transcripts from an intervention described by Pontille et al. (1996). It began with the following question: Given an increasing function, f, ( <i>x < y</i> implies <i>f</i>(<i>x</i>) ≤ <i>f</i>(<i>y</i>) ) from an ordered set S into itself, can we conclude that there will always exist an element s in S for which <i>f</i>(<i>s</i>) = <i>s</i>? The answer, of course, depends on the set. The intervention asks students to answer this question for four sets: a finite set of positive integers, the set of numbers with finite decimal expansions in [0,1], the set of rational numbers in [0,1], and the entire set [0,1]. In the original work, this question was posed to a class of lycée students in a scientific track. Over the course of an academic year, they periodically returned to this question, gradually building a refined understanding of the structure of the continuum. The author’s analysis of the transcripts from these classroom discussions is fascinating. <br /><br />Durand-Guerrier then posed this same question to a group of students in a graduate teacher- training program. In both cases, students were able to answer the question in the affirmative for the finite set, using an inductive proof or <i>reductio ad absurdum</i>. Almost all then tried to apply this proof to the dense countable sets. Here they ran into the realization that there is no “next” number. The graduate students, given only an hour to work on this, did not get much further. The lycée students did come to doubt that it was always true for these sets. As they began to think about the “holes” these sets left, they were able to construct counter-examples. <br /><br />The continuum provides the most difficulty. The lycée students were eventually able to prove that it is true in this instance, but only after being given the hint to consider the set of <i>x </i>in [0,1] for which <i>f</i>(<i>x</i>) > <i>x</i> and to draw on the property of the continuum that every bounded set has a least upper bound. <br /><br /><b>Measuring Conceptual Understanding</b><br />The last paper in this set addresses the problem of measuring conceptual understanding. We know that students can be proficient in answering procedural questions without the least understanding of what they are doing or why they are doing it. But measuring conceptual understanding is difficult. A meaningful assessment with limited possible answers, such as a concept inventory, requires a great deal of work to develop and validate. Open-ended questions can provide a better window into student thinking and understanding, but consistent application of scoring rubrics across multiple evaluators is hard to achieve. <br /><br />The authors build a solution from the observation that it is far easier to compare the quality of the responses from two students than it is to compare one student’s response against a rubric. They therefore suggest asking a simple, very open-ended question, scored by ranking student responses, which is achieved by pairwise comparisons. As an example, to evaluate student understanding of the derivative, they provided the prompt,<br /><blockquote>Explain what a derivative is to someone who hasn’t encountered it before. Use diagrams, examples and writing to include everything you know about derivatives.</blockquote>The 42 students in this study first read several examples of situations involving velocity and acceleration (presumably to prompt them to think of derivatives as rates of change rather than a collection of procedures) and were then given 20 minutes to write their responses to the prompt. <br /><br />Afterwards, 30 graduate students each judged 42 pairings. The authors found very high inter-rater reliability (r = .826 to .907). In fact, they found that comparative judgments appeared to do a better job of evaluating conceptual understanding than did Epstein’s Calculus Concept Inventory (Epstein, 2013). <br /><br />Similar studies were undertaken to evaluate student understanding of <i>p</i>-values and 11- to 12-year- olds understanding of the use of letters in algebra. Again, there was very high inter-rater reliability, and in these cases there were high levels of agreement with established instruments. <br /><br />This approach constitutes a very broad method of assessment, but it does enable the instructor to get some idea of what students are thinking and how they understand the concept at hand. It can be used even with large classes because it is not necessary to look at all possible pairs to get a meaningful ranking. <br /><br /><b>Conclusion</b><br />The three papers referenced here are very different in focus and goal, but I do see the common thread of searching for ways to encourage and assess student understanding. After all, that is what teaching and learning is really about. <br /><br /><b>References</b><br />Bisson, M.-J., Gilmore, C., Inglis, M., and Jones, I. (2016). Measuring conceptual understanding using comparative judgement. <i>IJRUME</i>. 2:141–164. <br /><br />Durand-Guerrier, V. (2016). Conceptualization of the continuum, an educational challenge for undergraduate students. <i>IJRUME</i>. 2:338–361. <br /><br />Epstein, J. (2013). The Calculus Concept Inventory - measurement of the effect of teaching methodology in mathematics. <i>Notices of the American Mathematical Society</i>, 60, 1018–27. <br /><br />Pontille, M. C., Feurly-Reynaud, J., & Tisseron, C. (1996). Et pourtant, ils trouvent. Repères IREM, 24, 10–34. <br /><br />Swidan, O. and Yerushalmy, M. (2016). Conceptual structure of the accumulation function in an interactive and multiple-linked representational environment. <i>IJRUME</i>. 2:30–58. <br /><br />Thompson, P.W., Byerley, C., and Hatfiled, N. (2013). A Conceptual approach to calculus made possible by technology. <i>Computers in the Schools</i>. 30:124–147. Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com0tag:blogger.com,1999:blog-7251686825560941361.post-74438220249816511302017-03-01T07:30:00.000-05:002017-03-01T07:30:07.391-05:00MAA Calculus Studies: Use of Local Data<b><i>You can now follow me on Twitter <a href="https://twitter.com/dbressoud" target="_blank">@dbressoud</a>.</i></b><br /><br />From our 2012 study, <i>Characteristics of Successful Programs in College Calculus</i> (NSF #0910240), the most successful departments had a practice of monitoring and reflecting on data from their courses. When we surveyed all departments with graduate programs in 2015 as part of <i>Progress through Calculus</i> (NSF #1430540), we asked about their access to and use of these data, what we are referring to as “local data.” <br /><br />The first thing we learned is that a few departments report no access to data about their courses or what happens to their students. For almost half, access is not readily available (see Table 1). When we asked, “Which types of data does your department review on a regular basis to inform decisions about your undergraduate program?”, most departments review grade distributions and pay attention to end of term student course evaluations (Table 2). Between 40% and 50% of the surveyed departments correlate performance in subsequent courses with the grades they received in previous courses and look at how well placement procedures are being followed. Given how important it is to track persistence rates (see <a href="http://www.maa.org/external_archive/columns/launchings/launchings_01_10.html">The Problem of Persistence</a>, <i>Launchings</i>, January 2010), it is disappointing to see that only 41% of departments track these data. Regular communication with client disciplines is almost non-existent. <br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://2.bp.blogspot.com/-HUrm4JawL9M/WLWkvd2yOOI/AAAAAAAAKyE/UryNQzbO1EMDiBG1NNzyWujHZ0QPVYKeQCLcB/s1600/launchings%2Btable%2B1responses.JPG" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="118" src="https://2.bp.blogspot.com/-HUrm4JawL9M/WLWkvd2yOOI/AAAAAAAAKyE/UryNQzbO1EMDiBG1NNzyWujHZ0QPVYKeQCLcB/s400/launchings%2Btable%2B1responses.JPG" width="400" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><div style="text-align: center;">Table 1. Responses to the question, “Does your department have access to data </div><div style="text-align: center;">to help inform decisions about your undergraduate program? PhD indicates </div><div style="text-align: center;">departments that offer a PhD in Mathematics. MA indicates departments for </div><div style="text-align: center;">which the highest degree offered in Mathematics is a Master’s.</div></td></tr></tbody></table><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://3.bp.blogspot.com/-1cmyZ8aL3HA/WLWkvdKE_9I/AAAAAAAAKyI/yDcWnnIio88Zu7UGQBLnzYQrl99UaDJ-ACLcB/s1600/launchings%2Btable%2B2%2Bresponses.JPG" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="188" src="https://3.bp.blogspot.com/-1cmyZ8aL3HA/WLWkvdKE_9I/AAAAAAAAKyI/yDcWnnIio88Zu7UGQBLnzYQrl99UaDJ-ACLcB/s400/launchings%2Btable%2B2%2Bresponses.JPG" width="400" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Table 2. Responses to the question, “Which types of data does your department<br /> review on a regular basis to inform decisions about your undergraduate program?”</td></tr></tbody></table><br />We also asked departments to describe the kinds of data they collect and regularly review. Several reported combining placement scores, persistence, and grades in subsequent courses to better understand the success of their program. Some of the other interesting uses of data included universities that<br /><ul><li>Built a model of “at-risk” students in Calculus I using admissions data from the past seven years. Using it, they report “developing a program to assist these students right at the beginning of Fall quarter, rather than target them after they start to perform poorly.” </li><li>Surveyed calculus students to get a better understanding of their backgrounds and attitudes toward studying in groups. </li><li>Collected regular information from business and industry employers of their majors. </li><li>Measured correlation of grade in Calculus I with transfer status, year in college, gender, whether repeating Calculus I, and GPA. </li><li>Used data from the university’s Core Learning Objectives and a uniform final exam to inform decisions about the course (including the ordering of topics, emphasis on material and time devoted to mastery of certain concepts, particularly in Calculus II). </li><li>Reviewed the performance on exam problems to decide if a problem type is too hard, a problem type needs to be rephrased, or an idea needs to be revisited on a future exam.</li></ul>The intelligent use of data to shape and monitor interventions is a central feature of the large- scale initiatives that are now underway. To mention just one, the AAU STEM Initiative (Association of American Universities, a consortium of 62 of the most prominent research universities in the U.S. and Canada) has established a Framework for sustainable institutional change. It can be found at <a href="https://stemedhub.org/groups/aau/framework">https://stemedhub.org/groups/aau/framework</a> (Figure 1). <br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://1.bp.blogspot.com/-UYZTXLKvuk4/WLWl4m4DV1I/AAAAAAAAKyM/17MV2vqJe4wgs8ucR1Drou4FYlMo7_xkwCLcB/s1600/Launchings%2Bfigure%2B1.JPG" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" src="https://1.bp.blogspot.com/-UYZTXLKvuk4/WLWl4m4DV1I/AAAAAAAAKyM/17MV2vqJe4wgs8ucR1Drou4FYlMo7_xkwCLcB/s1600/Launchings%2Bfigure%2B1.JPG" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Figure 1. AAU STEM Initiative Framework</td></tr></tbody></table><br />The three levels of change are subdivided into topics, each of which links to programs at member universities that illustrate work on this aspect of the framework. <br /><br />Cultural change encompasses<br /><ol><li> Aligning incentives with expectations of teaching excellence. </li><li> Establishing strong measures of teaching excellence. </li><li> Leadership commitment.</li></ol>Scaffolding includes<br /><ol><li> Facilities. </li><li> Technology. </li><li> Data. </li><li>Faculty professional development.</li></ol>Pedagogy is comprised of<br /><ol><li> Access. </li><li>Articulated learning goals. </li><li>Assessments. </li><li>Educational practices.</li></ol>In addition, AAU is now finalizing a list of “Essential Questions” to ask about the institution, the college, the department, and the course, illustrating the types of data and information that should be collected and pointing to helpful resources. This report, which should be published by the time this column appears, will be accessible through the AAU STEM Initiative homepage at <a href="https://stemedhub.org/groups/aau">https://stemedhub.org/groups/aau</a>. <br /><br /><br />Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com0tag:blogger.com,1999:blog-7251686825560941361.post-17551048236997298062017-02-01T06:54:00.000-05:002017-02-28T13:01:52.433-05:00MAA Calculus Study: PtC Survey Results <b><i>You can now follow me on Twitter <a href="https://twitter.com/dbressoud" target="_blank">@dbressoud</a>.</i></b><br /><br /><br />In spring 2015 the MAA’s <i>Progress through Calculus</i> (PtC) grant (NSF#1430540) surveyed all U.S. Departments of Mathematics that offer a graduate degree in Mathematics to learn about departmental practices, priorities, and concerns with respect to their mainstream courses in precalculus through single variable calculus. I have reported on some of the results from this study in <a href="http://launchings.blogspot.com/2015_11_01_archive.html" target="_blank">November, 2015</a>. This month’s column describes a variety of data relative to mainstream Calculus I that were collected in that survey. The full report can be found under <a href="mailto:http://www.maa.org/programs/faculty-and-departments/curriculum-development-resources/national-studies-college-calculus/ptc-publications" target="_blank">PtC Reports</a> (link from <a href="http://maa.org/cspcc">maa.org/cspcc</a>). <br /><br />The survey was sent to the chairs of all departments of mathematics in the United States that offer a graduate degree in Mathematics (PhD or Master’s). We received responses from 134 of the 178 PhD-granting universities (75%) and 89 of the 152 Master’s-granting universities (59%). <br /><br />Given how ineffective the standard precalculus course is known to be (see my <i>Launchings</i> column from <a href="http://launchings.blogspot.com/2014_10_01_archive.html" target="_blank">October, 2014</a>), we were particularly interested in efforts to teach precalculus topics concurrently with calculus. Accomplishing this through a stretched-out Calculus I is now fairly common (20 of 222 respondents use this approach to incorporate precalculus topics into Calculus I). Eleven universities have courses or options with extra hours to allow time on precalculus, and three offer precalculus courses designed to be taken concurrently with Calculus I. We also found 14 universities with an accelerated calculus specifically designed to meet the needs of students entering with AP® Calculus credit. Three universities have special lower credit courses that enable students who begin in a non-mainstream Calculus I to transition to mainstream calculus. <br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://2.bp.blogspot.com/-oxFy3DUgM4s/WJDl6Y8pzmI/AAAAAAAAKvw/U7Yn30lSR_wDCJ5j6mmI-Xh9lpsvM7LyQCLcB/s1600/launchings%2BPtC%2Btable%2B1.JPG" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="188" src="https://2.bp.blogspot.com/-oxFy3DUgM4s/WJDl6Y8pzmI/AAAAAAAAKvw/U7Yn30lSR_wDCJ5j6mmI-Xh9lpsvM7LyQCLcB/s400/launchings%2BPtC%2Btable%2B1.JPG" width="400" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Table 1: Number of surveyed universities that reported using each <br />of the listed variations in single variable calculus classes.</td></tr></tbody></table><br />Every five years, CBMS surveys departments of mathematics in the U.S. to get enrollment numbers, but those are only gathered for the fall term. In this survey, we were particularly interested in how these numbers vary over the full year, both academic and summer terms. While we only have results for a sample of universities, and no undergraduate colleges, the numbers are large enough, 150,000 in Precalculus, 200,000 in mainstream Calculus I, and 160,000 in subsequent mainstream single variable classes, to get a good idea of how these enrollments distribute over the year. For Precalculus, 57% of the enrollment occurs in the fall term. Fall term accounts for 60% of the Calculus I students. Not surprisingly, Calculus II is predominantly a second-term course (47%), but 40% of the students who take Calculus II do so in the fall. The distribution among the terms is complicated by the fact that some universities are on a quarter system, others on semesters. What I have labeled <i>2nd Term</i>, is either spring semester or winter quarter. The <i>3rd Term</i> refers to the spring quarter for those on a quarter system. <i>Summer</i> aggregates all summer terms. Figure 1 shows actual numbers from the universities that responded to give an idea of how enrollments drop off. For the purposes of the survey, “Precalculus” was defined as the last course before mainstream Calculus I. It is variously called Precalculus, College Algebra, College Algebra with Trigonometry, or Preparation for Calculus. Calculus II includes all mainstream single variable calculus courses that follow Calculus I. On a semester system, there is usually just one. On a quarter system, there usually are two such courses. <br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://2.bp.blogspot.com/-KEMpu4OSI5s/WJDzIcPU7KI/AAAAAAAAKwo/RwKsg18xBuw2tQWObZYKtmIjAwqpHDe2wCLcB/s1600/Fig1.tiff" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="263" src="https://2.bp.blogspot.com/-KEMpu4OSI5s/WJDzIcPU7KI/AAAAAAAAKwo/RwKsg18xBuw2tQWObZYKtmIjAwqpHDe2wCLcB/s400/Fig1.tiff" width="400" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Figure 1: Distribution of enrollments by term among the 205 universities that respond to this question. <br />2nd term = spring semester or winter quarter. 3rd term = spring quarter. <br />Calculus II includes all mainstream single variable calculus classes that follow Calculus I.</td></tr></tbody></table><br />The number of contact hours (including recitation sections) in Calculus I averaged 4.17 (SD = 0.77) at PhD-granting universities and 4.25 (SD = 0.64) at Masters-granting universities. The DFW rate in mainstream Calculus I was 21% (SD =12.2), at PhD-granting universities and 25% (SD = 13.7) at Masters-granting universities. <br /><br />The next table (Table 2) reports the fraction of universities in which Calculus I is frequently taught by each type of instructor. For each category of instructor, the options were “Never,” “Rarely,” or “Frequently.” <br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://4.bp.blogspot.com/-ZehM_eiA-ak/WJDnHaB-tPI/AAAAAAAAKv4/W7q5tR8VNPosni29iXaW5AKY6lUvZc_mgCLcB/s1600/launchings%2BPtC%2Btable%2B3.JPG" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="137" src="https://4.bp.blogspot.com/-ZehM_eiA-ak/WJDnHaB-tPI/AAAAAAAAKv4/W7q5tR8VNPosni29iXaW5AKY6lUvZc_mgCLcB/s400/launchings%2BPtC%2Btable%2B3.JPG" width="400" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Table 2: Percentage of universities for which each category of <br />instructor frequently teaches mainstream Calculus I.</td></tr></tbody></table><br />Recitation sections were far more common at PhD-granting universities. All classes have recitation sections for 49% of the institutions, some classes at 6%, and there are no recitation sections at 45% of the universities. For Masters-granting universities, the percentages were 18% for all classes, 6% for some classes, and 76% for no classes. <br /><br />We also found that active learning was much more common at Masters-granting universities than PhD-granting universities. Figures 2 and 3 record primary instructional format for mainstream Calculus I. “Some active learning” includes techniques such as use of clickers or think-pair-share. “Minimal lecture” includes Inquiry Based Learning and flipped classes. “Other” usually means too much variation to be able to identify a primary instructional format. We did find that 35% of the PhD-granting universities did report having at least some sections that were using active learning approaches. <br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://3.bp.blogspot.com/-dVOwQSNy5HU/WJDpXAvYGYI/AAAAAAAAKwE/hK0m7Mc0m3gmrA-O8HpCQ3hkeVLv3wEyQCLcB/s1600/launchings%2BPtC%2Btable%2B4.JPG" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="199" src="https://3.bp.blogspot.com/-dVOwQSNy5HU/WJDpXAvYGYI/AAAAAAAAKwE/hK0m7Mc0m3gmrA-O8HpCQ3hkeVLv3wEyQCLcB/s320/launchings%2BPtC%2Btable%2B4.JPG" width="320" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Figure 2. Primary instructional format for regular classes <br />(not recitation sections) at 214 PhD-granting universities.</td></tr></tbody></table><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://1.bp.blogspot.com/-mWN3-fSc3cQ/WJDqJUwfoPI/AAAAAAAAKwI/qsTcMuuz1wk3TXfEsTrlImp112vsx7uAACLcB/s1600/launchings%2BPtC%2Btable%2B5.JPG" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="194" src="https://1.bp.blogspot.com/-mWN3-fSc3cQ/WJDqJUwfoPI/AAAAAAAAKwI/qsTcMuuz1wk3TXfEsTrlImp112vsx7uAACLcB/s320/launchings%2BPtC%2Btable%2B5.JPG" width="320" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Figure 3. Primary instructional format for regular classes <br />(not recitation sections) at 109 Masters-granting universities.</td></tr></tbody></table><br />At 73% of the PhD-granting universities and 74% of the Masters-granting universities that offer recitation sections, they are simply homework help, Q&A, and review. Recitation sections are built around active learning approaches 21% of the time at PhD-granting universities, 4% of the time at Masters-granting universities. <br /><br />Table 3 reports which elements of mainstream Calculus I are common across all sections. We see much more uniformity at PhD-granting universities. In view of our findings from the earlier <i>Characteristics of Successful Programs in College Calculus</i> that coordination of course elements was one of the significant factors of successful calculus programs (see my <i>Launchings</i> column from <a href="http://launchings.blogspot.com/2014_01_01_archive.html" target="_blank">January 2014</a>), the results of this study suggest a great deal of room for improvement. <br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://3.bp.blogspot.com/-aGAXIAfvO98/WJDqqV5l_dI/AAAAAAAAKwQ/hJ77lqUXTJUQQ5IrSTI5ImsO8i5lEiD9QCLcB/s1600/launchings%2BPtC%2Btable%2B6.JPG" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="253" src="https://3.bp.blogspot.com/-aGAXIAfvO98/WJDqqV5l_dI/AAAAAAAAKwQ/hJ77lqUXTJUQQ5IrSTI5ImsO8i5lEiD9QCLcB/s400/launchings%2BPtC%2Btable%2B6.JPG" width="400" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Table 3: Percentage of reporting universities that have these elements across all sections of mainstream Calculus I.</td></tr></tbody></table><br />Another aspect of coordination that was characteristic of the most successful programs was the practice of regular meetings of the course instructors. As shown in Table 4, there is also a great deal of room for improvement here. <br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://4.bp.blogspot.com/-JdFC_iZHlZg/WJDrLqDpwPI/AAAAAAAAKwY/YGZ1HztWB3YwXdW97fXdpPXT_6hjI4kcACLcB/s1600/launchings%2BPtC%2Btable%2B7.JPG" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" src="https://4.bp.blogspot.com/-JdFC_iZHlZg/WJDrLqDpwPI/AAAAAAAAKwY/YGZ1HztWB3YwXdW97fXdpPXT_6hjI4kcACLcB/s1600/launchings%2BPtC%2Btable%2B7.JPG" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Table 4: Response to "When several instructors are teaching in the same term, <br />how often do they typically meet as a group to discuss the course?"</td></tr></tbody></table><br />The situation at PhD-granting universities is disappointing. The primary means of instruction is still large lecture with few or no structured opportunities for students to reflect on what is being presented to them, supplemented by recitation sections in which graduate students simply go over homework and answer student questions. At the Masters-granting universities, where classes are smaller and there is more emphasis on teaching, there is little coordination, often resulting in highly variable instruction. But there is room for hope. While there is no previous study with comparable data, there appears to be good deal of experimentation. My own experience in visiting these predominantly large public universities is that they are aware that what they are doing is not working, and they are looking for ways to improve what happens in this critical sequence. <br /><br /><br /><br /><br /><br /><br />Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com0tag:blogger.com,1999:blog-7251686825560941361.post-57055565074220777922017-01-01T07:00:00.000-05:002017-01-01T07:00:08.913-05:00IJRUME: Approximation in Calculus<b><i>You can now follow me on Twitter <a href="https://twitter.com/dbressoud" target="_blank">@dbressoud</a>.</i></b><br /><br />In an earlier column, "<a href="http://launchings.blogspot.com/2014/09/beyond-limit-iii.html" target="_blank">Beyond the Limit, III</a>," I talked about how Michael Oehrtman and colleagues have been able to use approximation as a unifying theme for single variable calculus that helps students avoid many of the confusing aspects of the language of limits. I also pointed out that this is hardly a new idea, having been used by many textbook authors including Emil Artin in <i>A Freshman Honors Course in Calculus and Analytic Geometry</i> and Peter Lax and Maria Terrell in <i>Calculus with Applications</i>. The IJRUME research paper I wish to highlight this month, “A study of calculus instructors’ perceptions of approximation as a unifying thread of the first-year calculus” by Sofronas et al., looks at how common this approach actually is.<br /><br />The authors address four research questions:<br /><br /><ol><li>Do calculus instructors perceive approximation to be important to student understanding of first-year calculus? </li><li>Do calculus instructors report emphasizing approximation as a central concept and-or unifying thread in the first-year calculus? </li><li>Which approximation ideas do calculus instructors believe are “worthwhile” to address in first-year calculus? </li><li>Are there any differences between demographic groups with respect to the approximation ideas they teach in first-year calculus courses? </li></ol>They surveyed calculus instructors at 182 colleges and universities, collecting 279 responses.<br /><br /><br />To the first two questions, 89% agreed that approximation is important, but only 51% considered it a central concept, and only 40% found that it provides a unifying thread (see Figure 1). For those who did consider it central and-or unifying, the reasons that they gave included: (a) it illuminates reasons for studying calculus, (b) most functions are not elementary and approximation is helpful in dealing with such functions, (c) approximation facilitates the understanding of fundamental concepts including limit, derivative, integral, and series, (d) linear approximations lie at the foundation of differential calculus, and (e) an emphasis on approximation resonates with the instructors personal interests in applied mathematics or numerical analysis.<br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://1.bp.blogspot.com/-PZ1XLkIML6g/WF1PH4Lhg5I/AAAAAAAAKuk/uQyPAwsf1G4t4SnApWh8NDyD-OLITocTQCLcB/s1600/IJRUME_Approx_in_Calc.tiff" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="335" src="https://1.bp.blogspot.com/-PZ1XLkIML6g/WF1PH4Lhg5I/AAAAAAAAKuk/uQyPAwsf1G4t4SnApWh8NDyD-OLITocTQCLcB/s400/IJRUME_Approx_in_Calc.tiff" width="400" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Figure 1: Graph depicting participants’ perceptions of approximation (N=214).<br /> Source: Sofronas et al. 2015.</td></tr></tbody></table><br />For those who did<i> not </i>consider approximation to be central or unifying, many stated that it is not sufficiently universal, only important in a few contexts such as motivating the definition of the derivative at a point or the value of a definite integral. Many stated other unifying threads such as limit or the study of change. Some objected to an emphasis on approximation because of its inevitable ties to the use of technology. There were also a large number of obstacles to the use of approximation that instructors identified. These included: (a) an overcrowded syllabus that left no room for the instructor to develop a unifying thread, (b) required adherence to a curriculum emphasizing procedural facility, (c) students with weak preparation who are not prepared to understand the subtleties of approximation arguments, (d) lack of access to technology, (e) lack of familiarity with how to use approximation ideas in developing calculus. I personally find these obstacles to be very sad, in particular the assumption on the part of many instructors that the only way to get through the required syllabus or to enable students to pass the course is to focus exclusively on memorizing procedures. <br /><br />Jumping ahead to the fourth question, the authors found that the single factor that most highly correlated with emphasizing approximation as a central concept and-or unifying thread was having served on either a local or national calculus committee. Not surprisingly, this factor was also highly correlated with number of years teaching calculus, rank, being the recipient of a teaching award, and having published or presented on a calculus topic. <br /><br />To the third research question, the combined list of topics gleaned from all of the responses truly spans first-year calculus: numerical limits, definition of limit, definition of the derivative, derivative values, tangent line approximations, differentials, error estimation, function change, functions roots and Newton’s method, linearization, integration, Riemann sums, Taylor polynomials and Taylor series, Newton’s second law, Einstein’s equation for force, L’Hospital’s rule, Euler’s method, and the approximation of irrational numbers. One unexpected outcome of the survey is that several of the respondents commented that answering this survey about their use of approximation in first-year calculus opened their eyes to the opportunity to use it as a unifying theme. As one respondent wrote,<br /><blockquote class="tr_bq">I agree that approximation is an important concept AND after taking this survey I can see teaching calculus using approximation as the main theme. The rate of change theme offers many opportunities for real-life applications but I can see how using approximations from the beginning would offer other opportunities. It is an interesting idea, and I would love to incorporate more of this theme into my lessons.</blockquote>For those who are interested in following up on the use of approximation as a unifying thread, this article also supplies a wealth of background information that includes a discussion of the different ways in which approximation can be used and the research evidence for its effectiveness as a guiding theme in developing student understanding of limits, derivatives, integrals, and series. <br /><br /><b>References </b><br /><br />Artin, E. (1958). <i>A Freshman Honors Course in Calculus and Analytic Geometry Taught at Princeton University</i>. Buffalo, NY: Committee on the Undergraduate Program of the Mathematical Association of America <br /><br />Lax, P. & Terrell, M.S. (2014). <i>Calculus with Applications</i>, Second Edition. New York, NY: Springer. <br /><br />Sofronas, K.S., DeFranco, T.C., Swaminathan, H., Gorgievski, N., Vinsonhaler, C., Wiseman, B., Escolas, S. (2015). A study of calculus instructors’ perceptions of approximation as a unifying thread of the first-year calculus. <i>Int. J. Res. Undergrad. Math. Ed</i>. 1:386–412 DOI 10.1007/s40753-015- 0019-5 <br /><br />Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com0tag:blogger.com,1999:blog-7251686825560941361.post-66675784552381122052016-12-01T07:31:00.000-05:002016-12-01T07:31:02.403-05:00IJRUME: Peer-Assisted Reflection<b><i>You can now follow me on Twitter <a href="https://twitter.com/dbressoud" target="_blank">@dbressoud</a>.</i></b><br /><br />The second paper I want to discuss from the <i>International Journal of Research in Undergraduate Mathematics Education</i> is a description of part of the doctoral work done by Daniel Reinholz, who earned his PhD at Berkeley in 2014 under the direction of Alan Schoenfeld. It consists of an investigation of the use of Peer-Assisted Reflection (PAR) in calculus [1]. <br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><img alt="Macintosh HD:Users:macalester:Dropbox:Personal:Launchings:2016:2016-12:21565092511_2e41878c9c_b.jpg" height="186" src="https://lh3.googleusercontent.com/y7Io02wwuQrFwnsPxOpi1E_ZEdmD8oneyGEiM4Amtu9uLSkn6wwhX8Q9lcT-wL02oNZVmL4jJlPA30Wo72hJVmENVkAgyMGmPCdnbwCYe933zbVzxNtFzqhspmqsPHkSS6X1xhp1RehWZzTyPA" style="border: none; margin-left: auto; margin-right: auto; transform: rotate(0rad);" width="279" /></td></tr><tr><td class="tr-caption" style="text-align: center;">Daniel Reinholz. <i>Photo Credit: David Bressoud</i></td></tr></tbody></table>PAR addresses an aspect of learning to do mathematics that Schoenfeld refers to as “self-reflection or monitoring and control” in his chapter on “Learning to Think Mathematically” [4]. As he observed in his problem-solving course at Berkeley, most students have been conditioned to assume that when presented with a mathematical problem, they should be able to identify immediately which tool to use. Among the possible activities that students might engage in while solving a problem—read, analyze, explore, plan, implement, and verify—most students quickly chose one approach to explore and then “pursue that direction come hell or high water” (Figure 1). <br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><img alt="Macintosh HD:Users:macalester:Dropbox:Personal:Launchings:2016:2016-12:Schoenfeld fig3.tiff" height="158" src="https://lh4.googleusercontent.com/FjKzvc4M1NZwaUE4FaMV09La3YM7GThgPTRu4ve-cJi-YaBe6RiXExHe_gkFodNk2gGe3t7f9F0RFbZOoCNQw3tEYmpB9gb0EX17PXE5x7iq98vag2PWoXa5M-NHOSsLzzXLTVsq83uYmeoKnA" style="border: none; margin-left: auto; margin-right: auto; transform: rotate(0rad);" width="300" /></td></tr><tr><td class="tr-caption" style="text-align: center;">Figure 1: Time-line graph of a typical student attempt to solve a non-standard problem. <br />Source: [4, p.356, Figure 15-3]</td></tr></tbody></table><div class="separator" style="clear: both; text-align: center;"><span style="font-size: 14.6667px; margin-left: 1em; margin-right: 1em; vertical-align: baseline; white-space: pre-wrap;"></span></div>In contrast, when he observed a mathematician working on an unfamiliar problem, he observed all of these strategies coming into play, a constant appraisal of whether the approach being used was likely to succeed and a readiness to try different ways of approaching the problem. He also found that mathematicians would verbalize the difficulties they were encountering, something seldom encountered among students (Figure 2). Note that over half the time was spent making sense of the problem rather than committing to a particular direction. Triangles represent moments when explicit comments were made such as “Hmm, I don’t know exactly where to start here.”<span id="docs-internal-guid-9f019c3f-b67c-fbc5-040d-19be399af837"></span><br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><img alt="Macintosh HD:Users:macalester:Dropbox:Personal:Launchings:2016:2016-12:Schoenfeld fig4.tiff" height="206" src="https://lh3.googleusercontent.com/uFcFd4bZWj65BeCoVvQ6-N47BWgbpLGL0tkTX0duxiIV9awBaP-su1Kn9K0xLQrkLEZmnVq4CWiILEfop02uVF7jpl0ZRkX_3dZfavs5bTQA3CZ4HtKhReTjl3CnIsVACCaVohwCxK0jBhtqLQ" style="border: none; margin-left: auto; margin-right: auto; transform: rotate(0rad);" width="374" /></td></tr><tr><td class="tr-caption" style="text-align: center;">Figure 2: Time-line graph of a mathematician working on a difficult problem.<br />Source: [4, p.356, Figure 15-4]</td></tr></tbody></table><div class="separator" style="clear: both; text-align: center;"><span style="font-size: 14.6667px; margin-left: 1em; margin-right: 1em; vertical-align: baseline; white-space: pre-wrap;"></span></div>In the conclusion to this section of his chapter, Schoenfeld wrote, “Developing self-regulatory skills in complex subject-matter domains is difficult.” In reference to two of the studies that had attempted to foster these skills, he concluded that, “Making the move from such ‘existence proofs’ (problematic as they are) to standard classrooms will require a substantial amount of conceptualizing and pedagogical engineering.”<span id="docs-internal-guid-9f019c3f-b67e-413c-aaa5-e34230d919ba"></span><br /><br />One of the problems with the early attempts at instilling self-reflection was the tremendous amount of work required of the instructor. Reinholz implemented PAR in Calculus I, greatly simplifying the role of the instructor by using students as partners in analyzing each other’s work. The study was conducted in two phases over two separate semesters in studies that each semester included one experimental section and eight to ten control sections, all of whom used the same examinations that were blind-graded. There were no significant differences between sections in either student ability on entering the class or in student demographics. The measure of success was an increase in the percentage of students earning a grade of C or higher. In the first phase of the study, the experimental section had a success rate of 82%, as opposed to the control sections where success was 69%. In the second phase, success rose from 56% in the control sections to 79% in the experimental section. <br /><br />Reinholz observed a noticeable improvement in student solutions to the PAR problems after they had received peer feedback. From student interviews, he found that many students in the PAR section had learned the importance of iteration, that homework is not just something to be turned in and then forgotten, but that getting it wrong the first time was okay as long as they were learning from their mistakes. Students were learning the importance of explaining how they arrived at their solutions. And they appreciated the chance to see the different approaches that other students in the class might take. <br /><br />What is most impressive about this intervention is how relatively easy it is to implement. Each week, the students would be given one “PAR problem” as part of their homework assignment. They were required to work on the problem outside of class, reflect on their work, exchange their solution with another student and provide feedback on the other student’s work in class, and then finalize the solution for submission. The time in class in which students read each other’s work and exchanged feedback took only ten minutes per week: five minutes for reading the other’s work (to ensure they really were focusing on reasoning, not just the solution) and five minutes for discussion. <br /><br />The difficulty, of course, lies in ensuring that the feedback provided by peers is useful. Reinholz identifies what he learned from several iterations of PAR instruction. In particular, he found that it is essential for the students to be explicitly taught how to provide useful feedback. By the time he got to Phase II, Reinholz was giving the students three sample solutions to that week’s PAR problem, allowing two to three minutes to read and reflect on the reasoning in each, and then engaging in a whole class discussion for about five minutes before pairing up to analyze and reflect on each other’s work. <br /><br />Further details can be found in [2] and [3]. For anyone interested in using Peer-Assisted Reflection, this is a useful body of work with a wealth of details on how it can be implemented and strong evidence for its effectiveness. <br /><br /><b>References </b><br /><br />[1] Reinholz, D.L. (2015). Peer-Assisted Reflection: A design-based intervention for improving success in calculus. <i>International Journal of Research in Undergraduate Mathematics Education</i>. <b>1</b>:234–267. <br /><br />[2] Reinholz, D. (2015). Peer conferences in calculus: the impact of systematic training. <i>Assessment & Evaluation in Higher Education</i>, DOI: 10.1080/02602938.2015.1077197 <br /><br />[3] Reinholz, D.L. (2016). Improving calculus explanations through peer review. <i>The Journal of Mathematical Behavior</i>. <b>44</b>: 34–49. <br /><br />[4] Schoenfeld, A.H. (1992). Learning to think mathematically: problem-solving, metacognition, and sense-making in mathematics. Pp. 334–370 in <i>Handbook for Research in Mathematics Teaching and Learning</i>. D. Grouws (Ed.). New York: Macmillan. Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com0tag:blogger.com,1999:blog-7251686825560941361.post-5249667226711665412016-11-01T07:41:00.000-04:002016-11-01T07:41:03.361-04:00IJRUME: Measuring Readiness for Calculus<b><i>You can now follow me on Twitter <a href="https://twitter.com/dbressoud" target="_blank">@dbressoud</a>.</i></b><br /><br />In 2015, the <i>International Journal of Research in Undergraduate Mathematics Education</i> (IJRUME) was launched by Springer with editors-in- chief Karen Marrongelle and Chris Rasmussen from the U.S. and Mike Thomas from New Zealand. It was established to “become the central, premier international journal dedicated to university mathematics education research.” While this is a journal by mathematics education researchers for mathematics education researchers, many of the articles are directly relevant to those of us engaged in the teaching of post-secondary mathematics. This then is the first of what I anticipate will be a series of columns abstracting some of the insights that I gather from this journal. <br /><br />I have chosen for the first of these columns the paper by Marilyn Carlson, Bernie Madison, and Richard West, “A study of students’ readiness to learn calculus.” [1] It is common to point to students’ lack of procedural fluency as the culprit behind their difficulties when they get to post- secondary calculus. Certainly, this is a problem, but not the whole story. Work over the past quarter century by Tall, Vinner, Dubinsky, Monk, Harel, Zandieh, Thompson, Carlson and many others have led the authors to identify major reasoning abilities and understandings that students need for success in calculus. This paper describes a validated diagnostic test that measures foundational reasoning abilities and understandings for learning calculus, the Calculus Concept Readiness (CCR) instrument. <br /><br />The reasoning abilities and conceptual understandings assessed by CCR require students to move beyond a procedural or action-oriented understanding of mathematics. Whether it is an equation such as 2 + 3 = 5 or a function definition, <i>f</i>(x) = <i>x</i><sup>2</sup> + 3<i>x</i> + 6, students are introduced to these as describing an action to be taken, adding 2 to 3 or plugging in various values for <i>x</i>. To make sense of and use the ideas of calculus, students need to view a function as a process (defined by a function formula, graph, or word description) that characterizes how the values of two varying quantities change together. Listed below are four of the reasoning abilities and understandings assessed by CCR and which the authors highlight in their article.<br /><div class="separator" style="clear: both; text-align: center;"><br /></div><ol><li><b>Covariational Reasoning.</b> When two variables are linked by an equation or a functional relationship, students need to understand how changes in one variable are reflected in changes in the other variable. The classic example considers how the rates of change of height and volume are related when water is poured into a non-cylindrical container such as a cone. At an even more basic level, students need to be able to interpret information on the velocities of two runners to an understanding of which is ahead at what times. Another example, which involves covariational reasoning as well as understanding rate as a ratio, considers the height of a ladder and its distance from a wall (Figure 1). When the authors administered their instrument to 631 students who were starting Calculus I, only 27% were able to select the correct answer (<b>c</b>) to the ladder problem.<br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://4.bp.blogspot.com/-grCVaQLEKVQ/WA_Bv_uBkaI/AAAAAAAAKrc/MNEja7EpMvA3dRDK-zmrh-zpbFzGyyR5gCEw/s1600/2016-11-a.tiff" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="376" src="https://4.bp.blogspot.com/-grCVaQLEKVQ/WA_Bv_uBkaI/AAAAAAAAKrc/MNEja7EpMvA3dRDK-zmrh-zpbFzGyyR5gCEw/s640/2016-11-a.tiff" width="640" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><b>Figure 1. The ladder problem.</b></td></tr></tbody></table></li><br /><li><b>Understanding the Function Concept.</b> Too many students interpret f(<i>x</i>) as an unnecessarily long-winded way of saying <i>y</i>. They see a function definition such as f(<i>x</i>) = <i>x</i><sup>2</sup>+ 3<i>x</i> + 6 as simply a prescription for how to take an input x and turn it into an output f(<i>x</i>). Such a limited view makes it difficult for students to manipulate functional relationships or to compose function formulas. Carlson et al. asked their 631 students for the formula for the area of a circle in terms of its circumference and offered the following list of possible answers:<br /><b> a.</b> <i>A</i> = <i>C</i><sup>2</sup>/4π<br /><b> b.</b> <i>A</i> = <i>C</i><sup>2</sup>/2<br /><b> c.</b> <i>A</i> = (2π<i>r</i>)<sup>2a</sup><b> <br /> d.</b> <i>A</i> = π<i>r</i><sup>2</sup><b> <br /> e.</b> <i>A</i> = π(<i>C</i><sup>2</sup>/4)<br />Only 28% chose the correct answer (<b>a</b>). As the authors learned from interviewing a sample of these students, those who answered correctly were the students who could see the equation C = 2πr as a process relating C and r which could be inverted and then composed with the familiar functional relationship between the area and radius.<br /></li><li><b>Proportional Relationships.</b> Too many students do not understand proportional reasoning. When Carlson et al. in an earlier study [2] administered the rain-gauge problem of Piaget et al. (Figure 2) to 1205 students who were finishing a precalculus course, only 43% identified the correct answer (as presented in Figure 2, it is 4⅔). Many students preserve the difference rather than the ratio, giving 5 as the answer. Difficulties with proportional reasoning are known to impede student understanding of constant rate of change, which in turn underpins average rate of change, which is fundamental to understanding the meaning of the derivative.<br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://4.bp.blogspot.com/--wQfot4FriY/WA_CYB6RyLI/AAAAAAAAKrM/l7HxlrvQE30fZj2OyMjkVEl5q8b1ytazwCLcB/s1600/2016-11-b.tiff" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="168" src="https://4.bp.blogspot.com/--wQfot4FriY/WA_CYB6RyLI/AAAAAAAAKrM/l7HxlrvQE30fZj2OyMjkVEl5q8b1ytazwCLcB/s640/2016-11-b.tiff" width="640" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><b>Figure 2. The rain gauge problem (taken from [3], [4])</b></td></tr></tbody></table></li><br /><li><b>Angle Measure and Sine Function.</b> As I described some years ago in an article for The Mathematics Teacher [5], the emphasis in high school trigonometry on the sine as a ratio of the lengths of sides of a triangle—often leading to the misconception that the sine is a function of a triangle rather than an angle—can lead to difficulties when encountering the sine in calculus, where it must be understood as a periodic function expressible in terms of arc length. An example is given in Figure 3, a problem for which only 21% of the Calculus I students chose the correct answer (<b>e</b>). Student interviews revealed that difficulties with this problem most often arose because students did not understand how to represent an angle measure using the length of the arc cut off by the angle’s rays.</li></ol><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-6u7_4whwHRE/WA_Bv3FW6AI/AAAAAAAAKrE/iyBEQ4f11kMThbr9pS--700pQOUw8QWUgCEw/s1600/2016-11-c.tiff" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="262" src="https://4.bp.blogspot.com/-6u7_4whwHRE/WA_Bv3FW6AI/AAAAAAAAKrE/iyBEQ4f11kMThbr9pS--700pQOUw8QWUgCEw/s640/2016-11-c.tiff" width="640" /></a></div><br /><br />What lessons are we to take away from this for our own classes? Last spring, in <a href="http://launchings.blogspot.com/2016/02/what-we-saywhat-they-hear.html">What we say/What they hear</a> and <a href="http://launchings.blogspot.com/2016/03/what-we-saywhat-they-hear-ii.html">What we say/What they hear II</a>, I discussed problems of communication between instructors and students. The work of Carlson, Madison, and West illustrates some of the fundamental levels at which miscommunication can occur and identifies the productive ways of thinking that students need to develop. <br /><br /><b>References</b><br /><br />[1] Carlson, M.P., Madison, B., & West, R.D. (2015). A study of students’ readiness to learn Calculus. <i>Int. J. Res. Undergrad. Math. Ed. </i>1:209–233. DOI 10.1007/s40753-015- 0013-y.<br /><br />[2] Carlson, M., Oehrtman, M., & Engelke, N. (2010). The precalculus concept assessment (PCA) instrument: a tool for assessing reasoning patterns, understandings and knowledge of precalculus level students. <i>Cognition and Instruction</i>, 28(2):113–145.<br /><br />[3] Piaget, J., Blaise-Grize, J., Szeminska, A., & Bang, V. (1977). <i>Epistemology and psychology of functions</i>. Dordrecht: Reidel.<br /><br />[4] Lawson, A.E. (1978). The development and validation of a classroom test of formal reasoning. <i>Journal of Research in Science Teaching</i>, 15, 11–24. doi:10.1002/tea.3660150103.<br /><br />[5] Bressoud, D.M. (2010). Historical reflections on teaching trigonometry. <i>The Mathematics Teacher</i>. 104(2):106–112. <br /><br /><br /><br />Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com1tag:blogger.com,1999:blog-7251686825560941361.post-7952692303864974242016-10-01T07:31:00.000-04:002017-03-30T13:52:19.121-04:00MAA Calculus Study: Women in STEMIt is nice to see that the national media has picked up one of the publications arising from the MAA’s national study, <i>Characteristics of Successful Programs in College Calculus</i> (NSF #0910240). It is the article by Ellis, Fosdick, and Rasmussen, “<a href="http://journals.plos.org/plosone/article?id=10.1371%2Fjournal.pone.0157447" target="_blank">Women 1.5 times more likely toleave STEM pipeline</a>,” that was published in PLoS ONE on July 13 of this year. The media coverage includes:<br /><ul><li>Rachel Feltman at the <i>Washington Post</i>, "<a href="https://www.washingtonpost.com/news/speaking-of-science/wp/2016/07/14/calculus-apprehensions-may-steer-women-away-from-science-careers/" target="_blank">Calculus apprehensions may steer women away from science careers</a>," who began her piece with her own experience, “Calculus II was one of the most demoralizing experiences of my college career.” </li><li>Dominique Mosbergen at the <i>Huffington Post</i>, "<a href="http://www.huffingtonpost.com/entry/calculus-stem-gender-gap_us_57a1b9eee4b0e2e15eb7df83?section" target="_blank">This Popular Math Class Is At The Heart Of The STEM Gender Gap, Study Suggests</a>"</li><li>Lauren Camera at <i>U.S. News</i>, "<a href="http://www.usnews.com/news/articles/2016-07-21/calculus-steers-women-away-from-stem" target="_blank">Calculus Steers Women Away From STEM</a>"</li><li>Maggie Kuo at <i>Science</i>, "<a href="http://www.sciencemag.org/careers/2016/07/low-math-confidence-discourages-female-students-pursuing-stem-disciplines" target="_blank">Low math confidence discourages female students from pursuing STEM disciplines</a>" </li></ul>...as well as a host of blogs and regional news sources.<br /><br />The article was an outgrowth of the “switcher” analysis that Jess Ellis and Chris Rasmussen had begun, using data from our 2010 national survey to study who came into Calculus I with the intention of staying on to Calculus II but then changed their minds by the end of the course. You can find a preliminary report on the Ellis and Rasmussen switcher analysis in my column for December 2013, <a href="http://launchings.blogspot.com/2013/12/maa-calculus-study-persistence-through.html" target="_blank">MAA Calculus Study: Persistence through Calculus</a> and a further analysis of the differences between men and women in the November, 2014 column, <a href="http://launchings.blogspot.com/2014/11/maa-calculus-study-women-are-different.html" target="_blank">MAA Calculus Study: Women are Different</a>. See also Rasmussen and Ellis (2013).<br /><br />The 2013 column reported that women were about twice as likely as men to switch out of the calculus sequence, but those data were compromised by several lurking variables, most significantly intended major. Women are heavily represented in the biological sciences, much less so in engineering and the physical sciences. Since the biological sciences are less likely to require a second semester of calculus, some of the effect was almost certainly due to different requirements.<br /><br />The study published in PLOS One controlled for student preparedness for Calculus I, intended career goals, institutional environment, and student perceptions of instructor quality and use of student-centered practices. They found that even with these controls, women were 50% more likely to switch out than men. As I discussed in my 2014 column, while Calculus I is very efficient at destroying the mathematical confidence of most of the students who take it, it is particularly effective for women (see Figure 1). As Ellis et al. report, 35% of the STEM-intending women who switched out chose as one of their reasons, “I do not believe I understand the ideas of Calculus I well enough to take Calculus II.” Only 14% of the men chose this reason.<br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://4.bp.blogspot.com/-Y2GaoYLT4P4/V-0_GAUwiCI/AAAAAAAAKoQ/goi7ieOWHO0dE1g5qQh5f1dFkVeWV7TnQCLcB/s1600/Women%2Bin%2BSTEM%2B1.tiff" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="331" src="https://4.bp.blogspot.com/-Y2GaoYLT4P4/V-0_GAUwiCI/AAAAAAAAKoQ/goi7ieOWHO0dE1g5qQh5f1dFkVeWV7TnQCLcB/s640/Women%2Bin%2BSTEM%2B1.tiff" width="640" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Figure 1: Change in standard mathematical confidence at the beginning of the Calculus I semester (pre- survey) and at the end of the semester (post-survey) separated by career intentions, gender and persistence status, [N = 1524] doi:10.1371/journal.pone.0157447.g004</td></tr></tbody></table><br />The last figure in the Ellis et al. article is enlightening (see Figure 2). If we could just raise the persistence rates of women once they choose enter Calculus I to match that of men, we could get a 50% increase in the percentage of women who enter the STEM workforce each year. <br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://4.bp.blogspot.com/-QuzGa9qfays/V-0_Y8irEJI/AAAAAAAAKoU/dsiixo7yp74ffdbxy6USgdi1JLucirGggCLcB/s1600/Womein%2Bin%2BSTEM%2B2.tiff" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="528" src="https://4.bp.blogspot.com/-QuzGa9qfays/V-0_Y8irEJI/AAAAAAAAKoU/dsiixo7yp74ffdbxy6USgdi1JLucirGggCLcB/s640/Womein%2Bin%2BSTEM%2B2.tiff" width="640" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Figure 2: Projected participation of STEM if women and men persisted at equal rates after Calculus I. The dotted line represents the projected participation of women. doi:10.1371/journal.pone.0157447.g005</td></tr></tbody></table><br />I believe that this issue of women’s confidence is cultural, not biological. It fits in with all we know about stereotype threat. When the message is that women are not expected to do as well as men in mathematics, negative signals loom very large. Calculus—as taught in most of our colleges and universities—is filled with negative signals. <br /><br /><b>Reference </b><br /><br />Ellis, J., Fosdick, B.K., and Rasmussen, C. (2016). Women 1.5 times more likely to leave STEM pipeline after calculus compared to men: Lack of mathematical confidence a potential culprit. PLoS ONE 11(7): e0157447. doi10.1371/journal.pone.0157447 <br /><br />Rasmussen, C., & Ellis, J. (2013). Who is switching out of calculus and why? In Lindmeier, A. M. & Heinze, A. (Eds.). <i>Proceedings of the 37th Conference of the International Group for the Psychology of Mathematics Education</i>, Vol. 4 (pp. 73-80). Kiel, Germany: PME.<br /><br />Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com0tag:blogger.com,1999:blog-7251686825560941361.post-49857314522695986432016-09-01T07:00:00.000-04:002016-09-02T16:29:20.555-04:00CBMS and Active Learning<div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-oZNNF_6dU1k/V8b6iB5dSjI/AAAAAAAAKlo/v8PN2JXAI2o3hs23baXJI71Z4JxAw51LQCLcB/s1600/CBMS%2Blogo.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="139" src="https://2.bp.blogspot.com/-oZNNF_6dU1k/V8b6iB5dSjI/AAAAAAAAKlo/v8PN2JXAI2o3hs23baXJI71Z4JxAw51LQCLcB/s320/CBMS%2Blogo.jpg" width="320" /></a></div><br /><br />I have just accepted the position of Director of the <a href="http://www.cbmsweb.org/" target="_blank">Conference Board of the Mathematical Sciences (CBMS)</a> and will be taking over from Ron Rosier at the end of this year. Most mathematicians, if they have heard of it at all, know of CBMS for its national survey of the mathematical sciences conducted every five years or for its regional research conferences. A few may know of CBMS through its forums on educational issues, its series on <i>Issues in Mathematics Education</i>, or the <i>Mathematical Education of Teachers</i> (MET II) report. <br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-wxbV5GyRmfY/V8b62QQZCxI/AAAAAAAAKls/5pZI-uQluyYLza0uh0Y6TKI8wiNV4bgJQCLcB/s1600/bressoud_CBMS.PNG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://3.bp.blogspot.com/-wxbV5GyRmfY/V8b62QQZCxI/AAAAAAAAKls/5pZI-uQluyYLza0uh0Y6TKI8wiNV4bgJQCLcB/s1600/bressoud_CBMS.PNG" /></a></div><br />These have emerged from the core mission of CBMS, which is to provide a structure within which the presidents of the societies that represent the mathematical sciences [1] can identify issues of common concern and coordinate efforts to address them. This is exemplified in the joint statement on <i><a href="http://www.cbmsweb.org/Statements/Active_Learning_Statement.pdf" target="_blank">Active Learning in Postsecondary Mathematics</a></i> [2] that was released this past July. This statement explains what is meant by active learning, presents the case for its importance, points to some of the published evidence of its effectiveness, lists society reports that have encouraged its use, and urges the following recommendation:<br /><br /><blockquote class="tr_bq"><i><b>We call on institutions of higher education, mathematics departments and the mathematics faculty, public policy-makers, and funding agencies to invest time and resources to ensure that effective active learning is incorporated into postsecondary mathematics classrooms. </b></i></blockquote><br />Ben Braun led the society representatives who drafted this position paper [2]. The presidents of all of the member societies with strong interest in mathematics education have signed onto it [3]. <br /><br />I see this statement as an example of what can be accomplished when the mathematical societies look to issues of common interest, and I am looking forward to working with them to coordinate efforts that will help colleges and universities identify and implement locally appropriate strategies for active learning. <br /><br />I also hope to use my position to assist these societies in addressing the issues of articulation that so plague mathematics education. These include the transitions from two-year to four-year institutions, from undergraduate to either graduate school or the workforce, and from graduate school to either academic or non-academic employment. But the transition on which I am currently focusing my attention is from secondary to postsecondary education. This point of discontinuity is rife with difficulties for many of our students who would seek STEM careers as well those who have struggled with mathematics. It is especially problematic for students from underrepresented groups: racially, ethnically, by socio-economic status, by gender, and by family experience with postsecondary education. <br /><br />The solutions—for there will be many pieces to be addressed if we are to succeed in ameliorating the problems—will require strong and coordinated efforts from both sides of the transition from high school to college. I am very encouraged by the clear messages of support for this work that I have received from NCTM, NCSM, and ASSM on the secondary side of the divide as well as AMS, MAA, AMATYC, ASA, and SIAM from the postsecondary side. CBMS is uniquely situated to bridge their work. <br /><br />While I expect my primary focus to be on educational concerns, CBMS has and must continue to work on all matters of common interest including public awareness of the role and importance of mathematics, advocacy for programs that improve opportunities for underrepresented minorities, and issues of employment in the mathematical sciences. <br /><br />I want to conclude by acknowledging the tremendous debt that the mathematical community owes to Ron Rosier and Lisa Kolbe who have been the entire staff of CBMS for roughly three decades. They have made this an effective organization. Under their direction, it has run smoothly and accomplished a great deal. They have left me with a very strong base on which to continue to build. <br /><br /><b>Endnotes </b><br /><br />[1] The seventeen professional societies that belong to CBMS can be grouped into those that are primarily focused at the postsecondary level:<br /><br /><ul><li><a href="http://www.cbmsweb.org/Members/amatyc.htm" target="_blank">American Mathematical Association of Two-Year Colleges (AMATYC)</a></li><li><a href="http://www.cbmsweb.org/Members/ams.htm" target="_blank">American Mathematical Society (AMS)</a></li><li><a href="http://www.cbmsweb.org/Members/asa.htm" target="_blank">American Statistical Association (ASA)</a></li><li><a href="http://www.cbmsweb.org/Members/asl.htm" target="_blank">Association for Symbolic Logic (ASL)</a> </li><li><a href="http://www.cbmsweb.org/Members/awm.htm" target="_blank">Association for Women in Mathematics (AWM)</a> </li><li><a href="http://www.cbmsweb.org/Members/amte.htm" target="_blank">Association of Mathematics Teacher Educators (AMTE)</a> </li><li><a href="http://www.cbmsweb.org/Members/ims.htm" target="_blank">Institute of Mathematical Statistics (IMS)</a> </li><li><a href="http://www.cbmsweb.org/Members/maa.htm" target="_blank">Mathematical Association of America (MAA)</a> </li><li><a href="http://www.cbmsweb.org/Members/nam.htm" target="_blank">National Association of Mathematicians (NAM)</a> </li><li><a href="http://www.cbmsweb.org/Members/siam.htm" target="_blank">Society for Industrial and Applied Mathematics (SIAM)</a><br /><br />Those that are primarily focused at preK–12 mathematics: </li><li><a href="http://www.cbmsweb.org/Members/assm.htm" target="_blank">Association of State Supervisors of Mathematics (ASSM)</a></li><li><a href="http://www.cbmsweb.org/Members/bba.htm" target="_blank">Benjamin Banneker Association (BBA)</a> </li><li><a href="http://www.cbmsweb.org/Members/ncsm.htm" target="_blank">National Council of Supervisors of Mathematics (NCSM)</a> </li><li><a href="http://www.cbmsweb.org/Members/nctm.htm" target="_blank">National Council of Teachers of Mathematics (NCTM)</a> </li><li><a href="http://www.cbmsweb.org/Members/todos.htm" target="_blank">TODOS: Mathematics for ALL (TODOS) </a><br /><br />And those that are primarily non-academic: </li><li><a href="http://www.cbmsweb.org/Members/informs.htm" target="_blank">Institute for Operations Research and the Management Sciences (INFORMS)</a></li><li><a href="http://www.cbmsweb.org/Members/soa.htm" target="_blank">Society of Actuaries (SOA)</a></li></ul><br /><br />[2] <i>Active Learning in Postsecondary Mathematics</i>, available at <a href="http://www.cbmsweb.org/Statements/Active_Learning_Statement.pdf" target="_blank">http://www.cbmsweb.org/Statements/Active_Learning_Statement.pdf</a><br />The writing team was led by Ben Braun and included myself as well as Diane Briars, Ted Coe, Jim Crowley, Jackie Dewar, Edray Herber Goins, Tara Holm, Pao-Sheng Hsu, Ken Krehbiel, Donna LaLonde, Matt Larson, Jacqueline Leonard, Rachel Levy, Doug Mupasiri, Brea Ratliff, Francis Su, Jane Tanner, Christine Thomas, Margaret Walker, and Mark Daniel Ward. The presidents of the member societies undertook the final wordsmithing. <br /><br />[3] The presidents of INFORMS and SOA were the only ones who were not engaged in the formulation or signing of this position paper. Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com0tag:blogger.com,1999:blog-7251686825560941361.post-13706480178912393302016-08-01T07:30:00.000-04:002016-08-01T07:30:00.156-04:00MAA Calculus Study: PlacementIn November’s column, <a href="http://launchings.blogspot.com/2015_11_01_archive.html" target="_blank">MAA Calculus Study: A New Initiative</a>, I described a survey that MAA has conducted of practices for and concerns about the precalculus through calculus sequence at departments of mathematics that have graduate programs. The initial summary of the survey results is now available as<i><a href="http://www.maa.org/sites/default/files/pdf/PtC%20Survey%20Report.pdf" target="_blank"> Progress through Calculus: National Survey Summary</a></i>, which can also be accessed through the Publications & Reports under Progress through Calculus on the web page <a href="http://maa.org/cspcc">maa.org/cspcc</a>. Universities were distinguished by whether the highest degree offered in mathematics was a Masters or a PhD.<br /><br />As I reported in November, placement was the number one issue among mathematics departments when comparing self-evaluation of importance to the program with confidence that the department is doing it well. Figure 1 shows that most PhD-granting departments rely on internally constructed instruments for placement.<br /><br /><div class="separator" style="clear: both; text-align: center;"></div><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://2.bp.blogspot.com/-acQc_8vF4mA/V45QLjmAkZI/AAAAAAAAKic/-_MixTGrTc0E3rz2MPGEt6nvwrLK8eorwCEw/s1600/bressoud_calculus_study.PNG" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="273" src="https://2.bp.blogspot.com/-acQc_8vF4mA/V45QLjmAkZI/AAAAAAAAKic/-_MixTGrTc0E3rz2MPGEt6nvwrLK8eorwCEw/s400/bressoud_calculus_study.PNG" width="400" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Figure 1. Percentage of respondents using specific placement tools for precalculus/calculus.<br />Respondents could select more than one.</td></tr></tbody></table>It is discouraging that a majority of Masters-granting departments and almost half of the PhD- granting departments use ACT or SAT scores for placement, instruments that are particularly ill suited to this purpose, even when only used to distinguish between placement into precalculus versus a previous course. It is also discouraging that so few PhD-granting universities use high school grades in determining placement. While not sufficient on their own, the study of <i>Characteristics of Successful Program in College Calculus</i> did reveal that including these grades improved departmental satisfaction with its placement decisions (see [1]). One of the striking results of the survey is that the number of PhD-granting departments using ALEKS increased from 10% in our 2010 survey to 28% in 2015. This may be somewhat misleading because the 2010 question only asked about placement into Calculus I, while the 2015 question asked about placement into precalculus or calculus, but from my own experience, the past several years have seen strong growing interest in and adoption of ALEKS.<br /><br />Figure 2 shows the overall degree of satisfaction of the department with their placement procedures. Note that the bars above the placement tools represent degree of satisfaction with the entire placement procedure among those institutions that include this particular tool. Thus it does not necessarily reflect the degree of satisfaction with that particular instrument. Nevertheless, this does indicate that there is no single instrument that guarantees satisfaction.<br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://2.bp.blogspot.com/-_x3J5I2ocJk/V45RORPs_sI/AAAAAAAAKik/l0BEVOvS0uE9FLT0OYfbaeVI-ZWMaljIgCLcB/s1600/bressoud_calculus_study2.PNG" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="248" src="https://2.bp.blogspot.com/-_x3J5I2ocJk/V45RORPs_sI/AAAAAAAAKik/l0BEVOvS0uE9FLT0OYfbaeVI-ZWMaljIgCLcB/s400/bressoud_calculus_study2.PNG" width="400" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Figure 2. Number of universities (out of 223) using each placement, with degree of<br />overall satisfaction with placement procedures.</td></tr></tbody></table>Among all of the surveyed universities, 9% were not satisfied with their placement procedures, and 39% considered them adequate but could be improved. Even though 52% were generally satisfied, we found that there is a lot of churn in placement procedures: 30% of the universities had recently replaced or were currently replacing their placement procedures, and an additional 29% were considering replacing these procedures.<br /><br />Perhaps the most interesting and potentially alarming result is that only 43% of respondents (45% of PhD-granting departments and 41% of Masters-granting departments) reported that they regularly review adherence to placement recommendations. It is hard to know how well your placement is working if you do not monitor it.<br /><br /><b>Reference</b><br /> [1] Hsu, E. and Bressoud, D. 2015. Placement and Student Performance in Calculus I. pages 59–67 in <i>Insights and Recommendations from the MAA National Study of College Calculus</i>, Bressoud, Mesa, and Rasmussen, editors. MAA Notes #84. Washington, DC: MAA Press. <a href="http://www.maa.org/sites/default/files/pdf/cspcc/InsightsandRecommendations.pdf">www.maa.org/sites/default/files/pdf/cspcc/InsightsandRecommendations.pdf</a>Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com0tag:blogger.com,1999:blog-7251686825560941361.post-15425189889976628502016-07-01T08:12:00.000-04:002016-07-19T11:56:56.942-04:00MAA and Active LearningThere is a general perception among both research mathematicians and those working in our partner disciplines that—with a few exceptional pockets such as the community of those promoting Inquiry Based Learning (IBL)—the mathematical community is only now beginning to wake up to the importance of active learning. In fact, the MAA’s Committee on the Undergraduate Program in Mathematics (CUPM) began to promote the use of active learning in 1981 and has never ceased. It is a cry to which many have responded, but which has recently been rediscovered and promoted with urgency as chairs, deans, provosts, and presidents have come to realize that the way mathematics instruction has traditionally been organized cannot meet our present needs, much less those of the future. I was reminded of the origins of MAA’s support for active learning after encountering a particular piece of misleading data in Andrew Hacker’s <i>The Math Myth</i>, an unpleasant little book filled with half-truths, deceptive innuendo, and misleading statistics. <br /><br />Hacker argues that the “math mandarins” cannot even attract students to major in mathematics and supports his argument with the fact that the number of Bachelor’s degrees in mathematics earned by US citizens dropped from 27,135 in 1970 to 17,408 in 2013. As a percentage of the total number of Bachelor’s degrees, the drop is even more impressive: from 3.4% to 1.0%. These numbers are symptomatic of Hacker’s deceptive use of data.<br /><br />The first thing that is deceptive about these numbers is that they suggest a steady erosion of interest in mathematics. In fact, as Figures 1 and 2 show, the drop was precipitous during the 1970s, with the total number of Bachelor’s degrees in mathematics bottoming out in 1981 at 11,078, showing some recovery in the ‘80’s, followed by a steady decline until 2001 when it dipped back below 12,000, only 0.94% of Bachelor’s degrees. Since then, the growth has been reasonably strong, rising to 20,980 (of whom 2,438 were non-resident aliens) in 2014. That was back up to 1.12%.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-bVZXtkAdUc0/V3LaKEy6WJI/AAAAAAAAKhs/I9dI7ZgBKfczRdtOwl6fU7noKEJVs71pgCLcB/s1600/MAA%2Bactive%2Blearning%2B1.PNG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://3.bp.blogspot.com/-bVZXtkAdUc0/V3LaKEy6WJI/AAAAAAAAKhs/I9dI7ZgBKfczRdtOwl6fU7noKEJVs71pgCLcB/s1600/MAA%2Bactive%2Blearning%2B1.PNG" /></a></div><br />(Note: The sharp increase in the early 1980s is almost certainly due to the high unemployment the United States was then suffering. Similarly, the noticeable increase in slope around 2010 is most probably a product of the unemployment rate that peaked in 2009.)<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-OYI2WtPWjRc/V3LaOc0h1oI/AAAAAAAAKh0/BplEXYCdWhcKLIpuiLILCsLea39KuONFQCLcB/s1600/MAA%2Bactive%2Blearning%2B2.PNG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://3.bp.blogspot.com/-OYI2WtPWjRc/V3LaOc0h1oI/AAAAAAAAKh0/BplEXYCdWhcKLIpuiLILCsLea39KuONFQCLcB/s1600/MAA%2Bactive%2Blearning%2B2.PNG" /></a></div><br />The other thing that is deceptive is the choice of when to start. The year 1970 came at the end of a strong national push for young people to enter mathematics and science. We had begun that decade in 1960 with only 11,399 mathematics degrees, though admittedly that was 2.9% of the total. Much of the loss during the ’70’s may be attributed to the creation of computer science majors. Bachelor’s degrees in computer science rose from 2388 in 1971 to 15,121 in 1981. Much, but not all. In fact, many members of the mathematical community were alarmed by this drop. Therein begins the story that is far more important than Hacker’s data.<br /><br />CUPM was established in the early 1950’s to bring order to the chaotic assortment of courses that constituted mathematics majors across the country. In 1965 this committee of leading mathematicians published <i><a href="http://www.maa.org/sites/default/files/pdf/CUPM/pdf/CUPM_Report_1965.pdf">A General Curriculum in Mathematics for Colleges</a><a href="http://www.maa.org/sites/default/files/pdf/CUPM/pdf/CUPM_Report_1965.pdf"></a></i>, which codified what by then was becoming the standard undergraduate major, beginning with three semesters of calculus and one semester of linear algebra. CUPM’s concern was almost entirely what to teach, not how to teach it. That changed in 1981 when Alan Tucker’s CUPM panel published <i><a href="http://www.maa.org/sites/default/files/pdf/CUPM/pdf/CUPM_Report_1981.pdf">Recommendations for a General Mathematical Sciences Program</a><a href="http://www.maa.org/sites/default/files/pdf/CUPM/pdf/CUPM_Report_1981.pdf"></a></i>.<br /><br />Concerned about the precipitous fall in the number of majors as well as enrollments in upper division courses, the attention in this report was focused on the goals of an undergraduate major and how they could be achieved. It laid out a five-point program philosophy that included an appeal to use active learning: <br /><ol><li>“The curriculum should have a primary goal of developing attitudes of mind and analytical skills required for efficient use and understanding of mathematics … </li><li>“The mathematical sciences curriculum should be designed around the abilities and academic needs of the average mathematical sciences student … </li><li>“<i>A mathematical sciences program should use interactive classroom teaching to involve students actively in the development of new material. Whenever possible, the teacher should guide students to discover new mathematics for themselves rather than present students with concisely sculptured theories.</i> (My italics.) </li><li>“Applications should be used to illustrate and motivate material in abstract and applied courses… </li><li>“First courses in a subject should be designed to appeal to as broad an audience as is academically reasonable …” </li></ol> In the 1991 CUPM report, <i>The Undergraduate Major in the Mathematical Sciences<a href="http://www.maa.org/sites/default/files/pdf/CUPM/pdf/CUPM_Report_1991.pdf"></a></i>, chaired by Lynn Steen, the third point from 1981 was expanded to a clearer articulation of active learning.<br /><blockquote><b>III. Interaction. </b>Since active participation is essential to learning mathematics, instruction in mathematics should be an interactive process in which students participate in the development of new concepts, questions, and answers. Students should be asked to explain their ideas both by writing and by speaking, and should be given experience working on team projects. In consequence, curriculum planners must act to assure appropriate sizes of various classes. Moreover, as new information about learning styles among mathematics students emerges, care should be taken to respond by suitably altering teaching styles.</blockquote>The next report, <a href="http://www.maa.org/programs/faculty-and-departments/curriculum-department-guidelines-recommendations/cupm/cupm-guide-2004"><i>CUPM Curriculum Guide 2004</i></a>, chaired by Harriet Pollatsek, continued to build on the theme of how we teach. In this iteration, CUPM expanded its vision to all of the courses taught by departments of mathematics, insisting that “Every course should incorporate activities that will help all students progress in developing analytical, critical reasoning, problem-solving, and communication skills and acquiring mathematical habits of mind.”<br /><br />This emphasis continues in the most recent CUPM guide, <a href="http://www.maa.org/sites/default/files/pdf/CUPM/pdf/CUPMguide_print.pdf">2015 CUPM Curriculum Guide toMajors in the Mathematical Sciences</a>, co-chaired by Carol Schumacher and Martha Siegel and edited by Paul Zorn. It begins with four “Cognitive Recommendations:"<br /><br /><ol><li>Students should develop effective thinking and communication skills. </li><li>Students should learn to link applications and theory. </li><li>Students should learn to use technological tools. </li><li>Students should develop mathematical independence and experience open-ended inquiry.</li></ol><br />Throughout these decades, MAA has done more than issue recommendations. All of these reports have been backed up by <i>MAA Notes</i> volumes that have pointed to successful programs and explained how such instruction can be implemented within specific courses. (For a list of all Notes volumes, click <a href="http://www.macalester.edu/~bressoud/maa/MAA_Notes.pdf" target="_blank">here</a>.) The Notes began in 1983 with <i><a href="http://files.eric.ed.gov/fulltext/ED229248.pdf">Problem Solving in the Mathematical Sciences</a></i>, edited by Alan Schoenfeld. MAA has run workshops as well as focused sessions and presentations at both national and regional meetings. Project NExT, MAA’s program for new faculty now in its third decade, has always had an emphasis on introducing newly minted PhDs to the use of active learning strategies.<br /><br />It is hard to say whether these measures have been responsible for arresting and reversing the slide in the number of majors. Economic factors have certainly played a role. But MAA publications and activities have established a depth of experience and expertise within the mathematical community. Now that there is broad recognition of the importance of active learning strategies in the teaching and learning of undergraduate mathematics, we are fortunate to have this foundation on which to build.<br /><br /><b>References</b><br /><br />Duren, W.L. Jr., Chair. 1965. <i><a href="http://www.maa.org/sites/default/files/pdf/CUPM/pdf/CUPM_Report_1965.pdf">A General Curriculum in Mathematics for Colleges</a></i>. Berkeley, CA: CUPM. www.maa.org/sites/default/files/pdf/CUPM/pdf/CUPM_Report_1965.pdf<br /><br />Hacker, A.<i> The Math Myth: and other STEM Delusions</i>. New York, NY: The New Press.<br /><br />Pollatsek, H., Chair. 2004. <i>CUPM Curriculum Guide</i> 2004. Washington, DC: MAA. <a href="http://www.maa.org/programs/faculty-and-departments/curriculum-department-guidelines-recommendations/cupm/cupm-guide-2004">www.maa.org/programs/faculty-and- departments/curriculum-department- guidelines-recommendations/cupm/cupm-guide- 2004</a><br /><br />Schoenfeld, A.H., Editor. 1983. Problem Solving in the Mathematical Sciences. Washington, DC: MAA. <a href="http://files.eric.ed.gov/fulltext/ED229248.pdf">files.eric.ed.gov/fulltext/ED229248.pdf</a><br /><br />Schumacher, C.S. and Siegel, M.J., Co-Chairs, and Zorn, P., Editor. 2015. 2015 <i>CUPM Curriculum Guide to Majors in the Mathematical Sciences</i>. Washington, DC: MAA. <a href="http://www.maa.org/programs/faculty-and-departments/curriculum-department-guidelines-recommendations/cupm">www.maa.org/programs/faculty-and- departments/curriculum-department- guidelines-recommendations/cupm</a><br /><br />Steen, L.A., Chair. 1991. <i>The Undergraduate Major in the Mathematical Sciences</i>. Washington, DC: MAA. <a href="http://www.maa.org/sites/default/files/pdf/CUPM/pdf/CUPM_Report_1991.pdf">www.maa.org/sites/default/files/pdf/CUPM/pdf/CUPM_Report_1991.pdf</a><br /><br />Tucker, A., Chair. 1981. <i>Recommendations for a General Mathematical Sciences Program</i>. Washington, DC: MAA. Reprinted on pages 1–59 in <i>Reshaping College Mathematics</i>, L.A. Steen, editor. Washington, DC: MAA, <a href="http://www.maa.org/sites/default/files/pdf/CUPM/pdf/CUPM_Report_1981.pdf">www.maa.org/sites/default/files/pdf/CUPM/pdf/CUPM_Report_1981.pdf</a>Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com0tag:blogger.com,1999:blog-7251686825560941361.post-29949409139426505142016-06-01T07:00:00.000-04:002016-06-03T09:13:34.655-04:00The Role of Calculus in the Transition from High School to College MathematicsThis past March I ran a small workshop in Washington, DC to look at “The Role of Calculus in the Transition from High School to College Mathematics,” funded by the National Science Foundation (#1550484). As the introduction to the workshop, cited below, makes clear, this is a critically important issue. Regular readers of my column will recognize that I return to it often. Part of the problem is that we don’t even know what we know and what we need to know (echoes of the “unknown unknowns”). The stated purpose was to bring together stakeholders to<br /><ol><li>Clarify the current state of knowledge of the effects on college performance and retention in STEM disciplines of acceleration into calculus at or before 12 th grade, as well as the effects of lack of access to calculus while in high school.</li><li>Identify the most pressing research questions.</li><li>Suggest strategies for answering these questions and identify the appropriate researchers or organizations to tackle them.</li></ol>As it turned out, my hopes for the workshop were too ambitious. We were not able to leave with a research agenda, but we did make progress toward identifying what we know about the problems surrounding this transition and in clarifying the complex issues surrounding the movement of calculus into the high school curriculum.<br /><br />The two days of the workshop provided an opportunity for a rich exchange. Participants included post-secondary mathematics faculty, high school calculus teachers, state and district supervisors of mathematics, researchers in both K-12 and post-secondary mathematics education, as well as representatives of The College Board, the National Council of Teachers of Mathematics, the National Research Council’s Board on Science Education, the National Math and Science Initiative, and Achieve. What is most gratifying is that all of these players recognize that there are serious problems around issues of preparation for post-secondary mathematics as well as equity and access that are being exasperated by the tremendous pressures to bring students into calculus ever earlier in their high school careers. My intention is that over the coming months it will be possible to build on the foundation laid by this workshop. <br /><br />Before concluding with the preamble that I wrote for this workshop, I would like to send out an appeal for all readers who would like to contribute to this conversation to send me their thoughts at <a href="mailto:bressoud@macalester.edu">bressoud@macalester.edu</a>, subject line: Role of Calculus.<br /><br /><blockquote class="tr_bq" style="text-align: left;">Last year, at least three quarters of a million U.S. high school students were enrolled in a calculus class.[1] This was three times the number of U.S. students who took their first calculus class in college. High school calculus enrollments are still growing at roughly 6% per year,[2] with increasing pressure on the most advantaged students to take calculus ever earlier. In 2015, over 120,000 students took the AP Calculus exam by the end of grade 11, and these numbers are growing by 9% per year. [3]</blockquote><blockquote class="tr_bq" style="text-align: left;">If we make the assumption that most of the students who enroll in calculus in high school will go on to matriculate as full-time students in a four-year undergraduate program—of whom there are 1.5 million each year[4]—then roughly half of these full-time, first-year students enter having already studied calculus. Calculus in high school is now commonly perceived as a prerequisite for college admission, with the result that high schools must start students with Algebra I in 8th grade high school if they are to be on track for calculus by grade 12, and calculus teachers find themselves under increasing pressure from parents and administrators to admit into their classes students they know are not adequately prepared. How do we provide support and alternatives for those students who cannot handle this accelerated progression?</blockquote><blockquote class="tr_bq" style="text-align: left;">We have learned from our study of Characteristics of Successful Programs in College Calculus (NSF DRL 0910240) that a quarter million of the students who study calculus in high school will retake mainstream Calculus I at the post-secondary level, and 40% of these quarter million will fail to get the A or B that signals they are prepared to continue in mathematics.[5] College faculty are very much aware that many of the students who enter with calculus on their high school transcript are, in fact, not ready for college-level mathematics. Of particular concern is that we know almost nothing about what happens to the remaining half million students who have studied calculus in high school: How many take advantage of advanced placement, and how well prepared are they? What are the mathematical trajectories of those who do not take calculus in college? How has the experience of calculus in high school shaped the aspirations and attitudes of these students and their ability to continue on toward mathematically demanding careers? </blockquote><blockquote class="tr_bq" style="text-align: left;">At the same time, half of all U.S. high schools do not even offer calculus. [6] Students from underrepresented groups, even when they are in a high school that offers calculus, are often discouraged from enrolling in this course. Those who have not taken calculus in high school find themselves in competition against students with a much richer preparation. What does this do to their chances for admission to college or the pursuit of a STEM major? </blockquote><blockquote class="tr_bq" style="text-align: left;">The questions are manifold. The goal of this workshop is to better understand these questions and to begin to develop strategies for answering them. Specifically, we seek to<br /><ol><li style="text-align: left;">Clarify the current state of knowledge of the effects on college performance and retention in STEM disciplines of acceleration into calculus at or before 12th grade, as well as the effects of lack of access to calculus while in high school.</li></ol><ol><li style="text-align: left;">Identify the most pressing research questions.</li></ol><ol><li style="text-align: left;">Suggest strategies for answering these questions and identify the appropriate researchers or organizations to tackle them.</li></ol></blockquote><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-0jwzHFVBUn0/V0dGaJmCAFI/AAAAAAAAKhE/GX5-h9EZQjkyakvXUB8SzXXpcn0lJJWDQCLcB/s1600/RoC.tiff" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="400" src="https://4.bp.blogspot.com/-0jwzHFVBUn0/V0dGaJmCAFI/AAAAAAAAKhE/GX5-h9EZQjkyakvXUB8SzXXpcn0lJJWDQCLcB/s400/RoC.tiff" width="318" /></a></div><br /><br /><b>Footnotes</b><br /><br />[1] Based on the NCES longitudinal study (HSLS:09) reporting that 19% of the four million students who were in 9th grade in 2009 had taken a calculus course in high school by the time they graduated.<br /><br />[2] From the College Board’s <i>AP Program Summary Reports</i> from 2002 to 2015.<br /><br />[3] Ibid.<br /><br />[4] Higher Education Research Institute.<i>The American Freshman: National Norms Fall 2015</i>.<br /><br /> [5] Data from <i>Characteristics of Successful Programs in College Calculus</i> project’s maalongdatafile. See David Bressoud. 2015. Insights from the MAA national study of college calculus. <i>The Mathematics Teacher</i>. 109 3:179–185.<br /><br />[6] U.S. Department of Education Office of Civil rights. Issue Brief No. 3 (March, 2014).<br /><a href="http://ocrdata.ed.gov/Downloads/CRDC-College-and-Career-Readiness-Snapshot.pdf">http://ocrdata.ed.gov/Downloads/CRDC-College- and-Career- Readiness-Snapshot.pdf</a><br /><br />Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com0tag:blogger.com,1999:blog-7251686825560941361.post-32443699703476837682016-05-01T09:06:00.000-04:002016-05-01T09:06:00.482-04:00Reflections on a Career in TeachingMonday, May 2 is my last day of teaching. Not that I will never step into a classroom again, but it marks the end of my full-time employment by Macalester College. My responsibilities for the future will not require any teaching. I am looking forward to the freedom this phased retirement will bring to focus on my writing and other educational activities, but I also approach this date with some sense of loss. For forty years, I have defined myself as a teacher first. I am taking advantage of this column to reflect on a few of the lessons I have learned over these four decades in the classroom.<br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: right;"><tbody><tr><td style="text-align: center;"><a href="https://1.bp.blogspot.com/-PSi1vvP1DlU/VyJkP12uThI/AAAAAAAAKgA/mk0P9Hwlxy8clih8wx5_xQkrkpNyWnYCgCLcB/s1600/Bressoud.PNG" imageanchor="1" style="clear: right; margin-bottom: 1em; margin-left: auto; margin-right: auto;"><img border="0" height="293" src="https://1.bp.blogspot.com/-PSi1vvP1DlU/VyJkP12uThI/AAAAAAAAKgA/mk0P9Hwlxy8clih8wx5_xQkrkpNyWnYCgCLcB/s400/Bressoud.PNG" width="400" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Class on Revolutions, taught at Penn State, Spring 1993, with John Harwood and Phil Jenkins (whose daughter Catherine is also in the picture). Taken on a field trip to the Joseph Priestley House, Northumberland, PA.</td></tr></tbody></table><b><br /></b><b>Students absorb far less than we think they do. As teachers, we must identify what is truly important and do all in our power to ensure that students are internalizing this knowledge.</b><br /><br />The first time this reality hit home for me was a differential equations class I taught at the University of Wisconsin in 1980–81, before the days of computers in the classroom. As taught back then, it was all about finding exact solutions to certain limited classes of differential equations. I thought I did a pretty good job of explaining the motivation and usefulness of these techniques, but homework and midterm exams were focused on finding these exact solutions. For the final exam, I wanted to make it interesting by presenting situations that my students would have to model using one of the types of differential equations we had studied, and then they would solve it. Not only did the students totally bomb this exam, they were angry that I was asking them to do something—modeling real world situations with these differential equations—that they had never been asked to do during the semester.<br /><br />They were absolutely right. Students learn by doing, not by listening. Knowing how to solve a differential equation is very different from knowing how to use a differential equation. It is far too easy to focus our instruction on the easily tested mechanical skills and assume that understanding and an appreciation for context will come as fortuitous byproducts.<br /><br />My second major experience of this truth occurred in a minicourse run by Kathy Heid and Joan Ferrini-Mundy at the 1994 Joint Math Meetings (in frigid Cincinnati). The theme was the use of student interviews as a tool for deep assessment of student learning. I practiced on a volunteer, a student from a local university who had completed four semesters of calculus and was now a junior. Eight months after the last of these classes, I wanted to find out what he had retained. I was horrified. As much as I pressed, differentiation for him carried no connotations beyond a method for turning functions into simpler functions (in the sense that 3x<sup>2</sup> is “simpler” than x<sup>3</sup>), and integration reverses that process. Nothing about tangents or rates or areas or accumulations had stuck. It drove home to me how little it is possible to learn while still passing a mathematics class.<br /><br />These experiences explain why I am such a fan of Inquiry Based Learning (IBL). It uses class time to focus on what we really want students to learn and provides a means of constantly probing student understanding. In my columns from February and March of this year, <a href="http://launchings.blogspot.com/2016/02/what-we-saywhat-they-hear.html">What We Say/What They Hear</a> and <a href="http://launchings.blogspot.com/2016_03_01_archive.html">What We Say/What They Hear II</a>, I described work in mathematics education that has validated my observation that what we think we are communicating is not what most students hear. I appreciate Stan Yoshinobu’s recent blog, <a href="http://theiblblog.blogspot.com/2016/04/a-practical-solution-to-what-we-saywhat.html">A Practical Solution to “What We Say/What They Hear,”</a> that illustrates how an IBL approach can address this problem of conveying meaning.<br /><br /><b>Very little of what is learned at the post-secondary level happens inside the classroom. As teachers, we have a responsibility to structure how students interact with the mathematics beyond the classroom walls.</b><br /><br />This point really struck home in 1990–91, the year I taught AP Calculus at the State College Area High School. The leisurely pace through the AB curriculum meant that I could actually watch light bulbs coming on in my class, something that I had never witnessed in the large lecture calculus classes at Penn State. This has always been for me the greatest distinction between high school and college mathematics. For the latter, students have to know how to learn outside of class from notes and textbook and in exchanges with other students.<br /><br />My last year at Penn State, 1993–94, I surveyed students in one of these large lecture sections to try to understand their experience. I found that most of them were very conscientious about studying, usually spending about two hours each evening that followed class going over that day’s lesson. But I also found that most of them had no idea how to study. They would read through their notes, paying particular attention to the problems that had been worked out in class, and then they would tackle problems from the end of that section, practicing the techniques that they had seen demonstrated. Anything that strayed too far from what had been worked by the instructor was considered irrelevant.<br /><br />Uri Treisman has shown the importance of students working together to clarify understandings (see [1]). I have found that if I want students to think more deeply about the mathematics introduced in class and to share those understandings with others, I have to structure out-of-class assignments designed to accomplish this. <br /><br /><b>Final exams carry far too much weight. As teachers, we need to administer frequent and varied assessments that truly measure what our students are learning and that provide opportunities for students to learn.</b> <br /><br />I hate final exams. I have had colleagues who promise students that if they do really well on the final, any poor test results earlier in the term will be forgiven. I have even seen students who manage to pull off a superior performance on the final despite a record leading up to it that would not have predicted this. Over forty years, I can count them on one hand. On the other side, I have seen many of my students who were steady and successful during term completely fall apart at the final. It is a stressful time, not just in my class but in almost every class a student is taking. It is a time of late nights and cramming and incredible pressure. The final straw for me was about fifteen years ago when there was a major incident of cheating on my final exam. This was by good students who had been doing well and who I knew did not need to cheat, but they were overwhelmed by the fear of doing poorly on this major component of their final grade for which there would be no opportunity to overcome a poor result.<br /><br />My policy now is—with rare exceptions—to never count a final exam for more than 15% of the total grade. As a complement to this, I test early, trying to get the first major assessment in by the end of the fourth week of class; I test often; and I assess student performance using a wide variety of measures: in class tests and quizzes that focus on procedural knowledge, take-home tests with problems that challenge students to apply their knowledge in unfamiliar situations requiring multiple steps, in multi-week projects that will be critiqued and returned for revision, in Reading Reflections—short answers to questions about the material read<i> before </i>class that help inform me before class begins of what students do and do not yet understand, and in written questions collected at the end of each class. Many of these have been inspired by Angelo and Cross’s <i>Classroom Assessment Techniques</i>, but there is nothing I have picked up from that book that I have not reshaped to fit my own style and needs.<br /><br />I also believe in looking for ways to enable students to learn from the assessments I use. Every major project is turned in for feedback before the final submission. For each exam during the semester, students are allowed to earn back some of the lost points by explaining where they went wrong and how to do the problem correctly. By giving myself some flexibility on how much of the grade students can earn back, I find I that I can give very challenging exams without needing to grade on a curve. Grading on a curve is a practice I consider to be particularly pernicious because it communicates to students that they are competing against each other, that what matters is less how much you have learned than how much better you can perform than your neighbor.<br /><br />I especially value group projects as a tool for teaching as well as assessing student learning. Most of my classes include several major projects. It was my last year at Penn State when I had the great good fortune to be able to teach the early <i>Project CALC</i> materials with David Smith present on campus (on sabbatical from Duke). I was able to meet with him weekly over lunch to talk about how the course was going. I have been able to watch how effective students are at teaching each other. It still never ceases to amaze me that I can say something in class without it registering with some student until the person next to them restates it as his or her own insight, though often verbatim, in a private conversation around solving a problem they are working on. I have found that group grades are problematic, and have experimented with a variety of techniques over the years to make them fairer. Now, as much as possible and for at least one project per class, everyone in the group is required to write up his or her own report of what the group has found. Clarity of exposition is every bit as important as the correctness of the results, a requirement that usually catches out those who simply attempt to reproduce the work done by other members of the group.<br /><br />With many writing assignments, projects, and exams, my students always complain about how much work they have to do for my classes. But no single assessment counts for very much, and my students have many opportunities to learn and recover from a bad performance. I am proud of the fact that it has been many years since a student complained that my grading was not fair. I am also proud that, despite complaining about the amount of work, my students also note how much they have learned.<br /><br /><b>Most students get too few opportunities to appreciate the culture of mathematics. As teachers, our instruction should communicate the true nature of mathematics. </b> <br /><br />I find it deeply discouraging that so many students graduate from college without any appreciation for mathematics as a rich venue for discovery and innovation. Burger and Starbird have done an excellent job of communicating this side of mathematics in <i>The Heart of Mathematics</i>. I have tried to do it through the history of the subject. This is reflected in all of the textbooks that I have written, and my students will attest that I am constantly interjecting the history of the subject into my classes.<br /><br />This spring, as a swan song, I am teaching a 100-level class on the history of mathematics for the first and last time. Fifty students are enrolled, most of whom are taking it to satisfy Macalester’s quantitative reasoning requirement. I am using Berlinghoff and Gouvêa’s <i>Math through the Ages</i>, a perfect book for my vision of this course. Their text consists of a stripped down history of mathematics, complemented by thirty short vignettes that survey topics from the development of negative numbers to non-Euclidean geometries to the rise of the computer. I use class time to tell the stories I love and introduce my students to the people who have been instrumental in the development of this vast subject. The real learning takes place in the writing students are required to do: short questions that must be submitted at the end of each class and from which I pick a few to answer at the start of the next, Reading Reflections in which students must tie what they have read to their own experiences, and many short papers in which they must pursue some of the many references provided by Berlinghoff and Gouvêa and explain to a younger version of themselves something about mathematics that they wish they had known earlier in their mathematical career. These have been especially insightful. I have seen so many of my students who were interested in, even excited about mathematical ideas early in their schooling, but had had that interest pounded out of them. It is emotional for me to see them, jaded as they now are about mathematics, reaching back to that younger self, trying to blow that ember back to life.<br /><br />None of the insights I have presented here are particularly original. Others have described them much more eloquently. But for me they are hard-won truths achieved through years of constantly striving to be a better teacher. After all, that is how we learn, not by listening to someone else expound or by reading a book or column, but by observantly striving to master our chosen profession. What we hear or read can suggest fruitful directions in which to explore and grow. Ultimately, this is our challenge, to constantly seek to improve how we teach.<br /><br />[1] Eric Hsu, Teri J. Murphy, Uri Treisman. 2008. Supporting high achievement in introductory mathematics courses: What we have learned from 30 years of the Emerging Scholars Program. Pages 205–220 in M.P. Carlson & C. Rasmussen (eds.), <i>Making the Connection: Research and Teaching in Undergraduate Mathematics Education</i>. MAA Notes #73. Washington, DC: Mathematical Association of America.Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com3