tag:blogger.com,1999:blog-72516868255609413612014-09-13T23:59:50.103-04:00Launchings 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.comBlogger38125tag:blogger.com,1999:blog-7251686825560941361.post-1128075813547451872014-09-01T05:00:00.000-04:002014-09-01T05:00:02.147-04:00Beyond the Limit, III<div class="MsoNormal"><span style="font-family: "Times New Roman","serif"; font-size: 11.0pt;">In my last two columns, <i><a href="http://launchings.blogspot.com/2014/06/beyond-limit-i.html" target="_blank">Beyond the Limit, I</a></i>and <i><a href="http://launchings.blogspot.com/2014/08/beyond-limit-ii.html" target="_blank">Beyond the Limit, II</a></i>, I looked at common student difficulties with the concept of limit and explained Michael Oehrtman’s investigations into the metaphors that students use when they try to apply the concept of limit to problems of first-year calculus. The point of this exploration is to identify the most productive and useful ways of thinking about limits so that we can channel calculus instruction toward these understandings. In this month’s column, I will describe Oehrtman’s suggestions for how to accomplish this.<o:p></o:p></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-family: "Times New Roman","serif"; font-size: 11.0pt;">In the MAA <i>Notes</i> volume <i>Making the Connection</i> (Carlson and Rasmussen 2008), Oehrtman focuses on the last of the strong metaphors described in his 2009 paper, that of limit as approximation. The point of building instruction around this approach is that it arises spontaneously from the students themselves, providing what Tall refers to as a <i>cognitive root</i>: <o:p></o:p></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal" style="margin-left: .5in;"><span style="font-family: "Times New Roman","serif"; font-size: 11.0pt;">Rather than deal initially with formal definitions that contain elements unfamiliar to the learner, it is preferable to attempt to find an approach that builds on concepts that have the dual role of being familiar to the students and providing the basis for later mathematical development. Such a concept I call a <i>cognitive root</i>. (Tall 1992, p. 497)<o:p></o:p></span></div><div class="MsoNormal" style="margin-left: .5in;"><br /></div><div class="MsoNormal"><span style="font-family: "Times New Roman","serif"; font-size: 11.0pt;">While limit as approximation was not always the most commonly employed way of understanding limits—that honor frequently went to limit as collapse—it has several major advantages. First, because it does arise spontaneously from many of the students, we know that it is easily accessible to many of them. Second, it is the metaphor that comes closest to the mathematically correct definition of limit. This is important. Because it comes so close to the formal understanding of limits, it provides a means for students to reason consistently throughout the course, providing coherence and making it easier to transfer this understanding to novel situations. Finally, explorations of how to approximate quantities such as instantaneous velocity or the force on a dam provide direct connections between the concepts of calculus and the modeling situations students will encounter in engineering or the sciences.<o:p></o:p></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-family: "Times New Roman","serif"; font-size: 11.0pt;">In student responses to the eleven problems that Oehrtman posed to the 120 students in his study (reproduced at the end of this column), Oehrtman found that the approximation metaphor was employed by 11% in answering questions #1 and #2, 26% for #6, 35% for #4, 70% for #3, and 74% for #8. This last asked to explain in what sense \( \displaystyle \sin x = x - \frac{1}{3!} x^3 + \frac{1}{5!} x^5 - \frac{1}{7!}x^7 + \cdots . \) Oehrtman quotes at length one of these explanations that I wish to reproduce here because it amply demonstrates how, without any explicit instruction in ideas of approximation, it was the metaphor instinctively seized by a student trying to verbalize what she knew about Taylor series.<o:p></o:p></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal" style="margin-left: .5in;"><span style="font-family: "Times New Roman","serif"; font-size: 11.0pt;">When calculating a Taylor polynomial, the accuracy of the approximation becomes greater with each successive term. This can be illustrated by graphing a function such as sin(<i>x</i>) and its various polynomial approximations. If one such polynomial with a finite number of terms is centered around some origin, the difference in <i>y</i>-values between the points along the polynomial and the points along the original curve (sin <i>x</i>) will be greater the further the <i>x</i>-values are from the origin. If more terms are added to the polynomial, it will hug the curves of the sin function more closely, and this error will decrease. As one continues to add more and more terms, the polynomial becomes a very good approximation of the curve. Locally, at the origin, it will be very difficult to tell the difference between sin(<i>x</i>) and its polynomial approximation. If you were to travel out away from the origin however, you would find that the polynomial becomes more and more loosely fitted around the curve, until at some point it goes off in its own direction and you would have to deal once again with a substantial error the further you went in that direction. Adding more terms to the polynomial in his case increases the distance that you have to travel before it veers away from the approximated function, and decreases the error at any one <i>x</i>-value. Eventually, if an infinite number of terms could be calculated, the error would decrease to zero, the distance you would have to travel to see the polynomial veer away would become infinite, and the two functions would become equal. This is a very important and useful characteristic, as it allows for the approximation of complicated functions. By using polynomials with an appropriate number of terms, one can find approximations with reasonable accuracy. (Oehrtman 2008, pp. 72–3)<o:p></o:p></span></div><div class="MsoNormal" style="margin-left: .5in;"><br /></div><div class="MsoNormal"><span style="font-family: "Times New Roman","serif"; font-size: 11.0pt;">I am certain that this explanation echoes much of what this student heard and saw in the classroom. I read in it much of what I say when I explain Taylor series. Yet this observation is useful because it demonstrates which images and explanations have resonated with this student.<o:p></o:p></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-family: "Times New Roman","serif"; font-size: 11.0pt;">For Oehrtman, this explanation is classified as <i>approximation</i>not just because that word appears frequently in the student’s explanation, but because the student combines it with a sense of the numerical size of the error. This is very different from the metaphor of limit as proximity. For this student, what is important is not just that the graphs of the polynomials are spatially close but that she can control the size of the error.<o:p></o:p></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-family: "Times New Roman","serif"; font-size: 11.0pt;">Given this observation, Oehrtman has built a series of activities designed to encourage and strengthen student reliance on the metaphor of approximation, several of which are described in his 2008 article, many more of which are coming available on his new website, </span><a href="http://clearclaculus.okstate.edu/" target="_blank"><span style="font-family: "Times New Roman","serif"; font-size: 11.0pt;">CLEAR Calculus</span></a><span style="font-family: "Times New Roman","serif"; font-size: 11.0pt;"> (contact Mike for access to the posted materials by writing to michael.oehrtman@gmail.com). Thus, instead of introducing the slope of the tangent as the limit of the slopes of secant lines, he chooses a particular point on a particular curve, in this case <i>x</i> = –1 on <i>y </i>=<i></i>2<i><sup>x</sup></i>, and introduces the secants as lines whose slopes approximate the slope of the tangent line. He gets students to identify those secant lines that provide an upper bound on the slope of the tangent and those that provide a lower bound and then has them explore secant lines that tighten these bounds until they can approximate the slope of the tangent to within an error of 0.0001. The word <i>limit</i> need never arise.<o:p></o:p></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-family: "Times New Roman","serif"; font-size: 11.0pt;">In a similar vein, integrals are approached via Riemann sums, but not as the limit of these sums. Rather, these sums provide approximations to the desired quantity. One can identify those approximations that overshoot and those that undershoot the true value and adjust the partitioning of the interval to make the error as small as one wishes.<o:p></o:p></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-family: "Times New Roman","serif"; font-size: 11.0pt;">Each of Oehrtman’s activities is built around five questions:</span></div><ol><li>What are you approximating?</li><li>What are the approximations?</li><li>What are the errors?</li><li>What are the bounds on the size of the errors?</li><li>How can the error be made smaller than any predetermined bound?</li></ol><div class="MsoNormal"><span style="font-family: "Times New Roman","serif"; font-size: 11.0pt;">As Oehrtman explains, the last two are intentionally reciprocal: Given a choice of approximation, what are the bounds on the error? Given a bound on the error, what approximation will achieve it?<o:p></o:p></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-family: "Times New Roman","serif"; font-size: 11.0pt;">While this approach provides a route through calculus that does not require the use of the word “limit,” Oehrtman does not avoid it. For those students who will pursue mathematics, it is a term that will come up in other contexts. For the serious student of mathematics, it is absolutely essential. What Oehrtman does recognize is that performing a series of exercises in which one finds limits or explains why they do not exist has little or no bearing on the development of a robust personal understanding of derivatives and integrals. <o:p></o:p></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-family: "Times New Roman","serif"; font-size: 11.0pt;">Not surprisingly, students whose understanding of limits is deeply rooted in the concept of approximations, including the reciprocal processes of determining the bound from the approximation and finding an approximation that will satisfy a particular bound, find it much easier to grasp the formal epsilon-delta definition of limit. In fact, Oehrtman, Swinyard, and Martin (2014) have documented the relative ease with which students schooled in this approach are able to rediscover the mathematically correct definition of limit for themselves.<o:p></o:p></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-family: "Times New Roman","serif"; font-size: 11.0pt;">This is not a new insight. In Emil Artin’s <i>A Freshman Honors Course in Calculus and Analytic Geometry Taught at Princeton University</i> (published by MAA in 1958), he talks about approximations to the slope of the tangent line before introducing <i>limit</i> as “the number approached by the approximations to the slope” (page 23). The Five Colleges Calculus Project, <i>Calculus in Context</i>, also begins with approximations, as do <i>Calculus with Applications</i> by Peter Lax and Maria Terrell and <i>The Sensible Calculus Program</i> by Martin Flashman.</span></div><div class="MsoNormal"><br /><hr /><br />The following are abbreviated statements of the problems posed by Michael Oehrtman (2009) to 120 students in first-year calculus via pre-course and post-course surveys, quizzes, and other writing assignments as well as two hour-long clinical interviews with twenty of the students.<br /><div><ol><li>Explain the meaning of \( \displaystyle \lim_{x\to 1} \frac{x^3-1}{x-1} = 3. \) </li><li>Let \( f(x) = x^2 + 1.\) Explain the meaning of \( \displaystyle \lim_{h\to 0} \frac{f(3+h)-f(3)}{h}.\) </li><li>Explain why \( 0.\overline{9} = 1.\) </li><li>Explain why the derivative \( \displaystyle f’(x) = \lim_{h\to 0} \frac{f(x+h)-f(x)}{h}\) gives the instantaneous rate of change of <i>f</i> at <i>x</i>. </li><li>Explain why L’Hôpital’s rule works. </li><li>Explain how the solid obtained by revolving the graph of <i>y</i> = 1/<i>x</i> around the <i>x</i>-axis can have finite volume but infinite surface area. </li><li>Explain why the limit comparison test works. </li><li>Explain in what sense \( \displaystyle \sin x = x - \frac{1}{3!} x^3 + \frac{1}{5!} x^5 - \frac{1}{7!}x^7 + \cdots . \) </li><li>Explain how the length of each jagged line shown below can be \( \sqrt{2} \) while the limit has length 1. <div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-bj4CKnNxNbs/U6mYLanFSYI/AAAAAAAAJ1o/D7pvrgYOeiE/s1600/14-7-1+(1).jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://1.bp.blogspot.com/-bj4CKnNxNbs/U6mYLanFSYI/AAAAAAAAJ1o/D7pvrgYOeiE/s1600/14-7-1+(1).jpg" width="500" /></a></div></li><li>Explain what it means for a function of two variables to be continuous.</li><li>Explain why the derivative of the formula for the volume of a sphere, \( V = (4/3)\pi r^3 \), is the surface area of the sphere, \( dV/dr = 4\pi r^2 = A. \) </li></ol></div><hr /><br /><span style="font-family: "Times New Roman","serif"; font-size: 11pt;">Artin, E. (1958). </span><i><span style="font-family: "Times New Roman","serif"; font-size: 11.0pt;">A Freshman Honors Course in Calculus and Analytic Geometry Taught at Princeton University</span></i><span style="font-family: "Times New Roman","serif"; font-size: 11pt;">. Buffalo, NY: Committee on the Undergraduate Program of the Mathematical Association of America</span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-family: "Times New Roman","serif"; font-size: 11.0pt;">Callahan, J., Hoffman, K., Cox, D., O’Shea, D., Pollatsek, H., Senechal, L. (2008). <i>Calculus in Context: The Five College Calculus Project</i>. Accessed August 11, 2014. </span><a href="http://www.math.smith.edu/Local/cicintro/book.pdf" target="_blank"><span style="font-family: "Times New Roman","serif"; font-size: 11.0pt;">www.math.smith.edu/Local/cicintro/book.pdf</span></a><span style="font-family: "Times New Roman","serif"; font-size: 11.0pt;"><o:p></o:p></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-family: "Times New Roman","serif"; font-size: 11.0pt; mso-fareast-font-family: "Times New Roman";">Carlson, M.P. & Rasmussen, C. (Eds.). (2008). <i>Making the Connection: Research and Teaching in Undergraduate Mathematics Education</i>. <i>MAA Notes </i>#73. Washington, DC: Mathematical Association of America.<o:p></o:p></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-family: "Times New Roman","serif"; font-size: 11.0pt; mso-fareast-font-family: "Times New Roman";">Flashman, M. (2014). <i>The Sensible Calculus Program</i>. Accessed August 11, 2014. </span><a href="http://users.humboldt.edu/flashman/senscalc.Core.html" target="_blank"><span style="font-family: "Times New Roman","serif"; font-size: 11.0pt; mso-fareast-font-family: "Times New Roman";">users.humboldt.edu/flashman/senscalc.Core.html</span></a><span style="font-family: "Times New Roman","serif"; font-size: 11.0pt; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-family: "Times New Roman","serif"; font-size: 11.0pt; mso-fareast-font-family: "Times New Roman";">Lax, P. & Terrell, M.S. (2014). <i>Calculus with Applications</i>, Second Edition. New York, NY: Springer.<o:p></o:p></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-family: "Times New Roman","serif"; font-size: 11.0pt; mso-fareast-font-family: "Times New Roman";">Oehrtman, M. (2008). Layers of abstraction: theory and design for the instruction of limit concepts. Pages 65–80 in Carlson & Rasmussen (Eds.), <i>Making the Connection: Research and Teaching in Undergraduate Mathematics Education</i>. <i>MAA Notes </i>#73. Washington, DC: Mathematical Association of America.<o:p></o:p></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-family: "Times New Roman","serif"; font-size: 11.0pt; mso-fareast-font-family: "Times New Roman";">Oehrtman M. (2009). Collapsing Dimensions, Physical Limitation, and Other Student Metaphors for Limit Concepts. <i>Journal For Research In Mathematics Education</i>, <b>40</b>(4), 396–426.</span><span style="font-family: "Times New Roman","serif"; font-size: 11.0pt;"><o:p></o:p></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-family: "Times New Roman","serif"; font-size: 11.0pt; mso-fareast-font-family: "Times New Roman";">Oehrtman, M., Swinyard, C., Martin, J. (2014). Problems and solutions in students’ reinvention of a definition for sequence convergence. <i>The Journal of Mathematical Behavior</i>, <b>33</b>, 131–148.</span><span style="font-family: "Times New Roman","serif"; font-size: 11.0pt;"><o:p></o:p></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-family: "Times New Roman","serif"; font-size: 11.0pt; mso-fareast-font-family: "Times New Roman";">Tall, D. (1992). The transition to advanced mathematical thinking: functions, limits, infinity, and proof. Chapter 20 in Grouws D.A. (ed.) <i>Handbook of Research on Mathematics Teaching and Learning</i>, Macmillan, New York, 495-511</span>.<span style="font-family: "Times New Roman","serif"; font-size: 11.0pt;"><o:p></o:p></span></div>Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com1tag:blogger.com,1999:blog-7251686825560941361.post-24481820520888592222014-08-01T06:00:00.000-04:002014-08-01T08:15:49.465-04:00Beyond the Limit, IILast month, in "<a href="http://launchings.blogspot.com/2014/06/beyond-limit-i.html" target="_blank">Beyond the Limit, I</a>," I discussed some of the difficulties and misconceptions surrounding student understanding of limits in first-year calculus. I also raised the question of how serious these misconceptions actually are and introduced the work of Michael Oehrtman (2009, also see the preprint). This month I wish to explain what Oehrtman found as he analyzed student responses to the eleven problems that he posed and that are reproduced at the end of this column.<br /><br />Rather than focusing on student mistakes and misconceptions, Oehrtman was interested in how students can use what they know or think they know about limits to reason through and explain some of the central ideas of calculus. He chose to focus on student metaphors, nontechnical ways of thinking and talking about limits that reflect individual experiences and that enable students to relate unfamiliar mathematical ideas to familiar concepts. Mathematics and science are full of useful metaphors. Oehrtman cites Max Black’s 1962 <i>Models and Metaphors</i> as the first serious study of how scientists use metaphors. Black paid particular attention to James Clerk Maxwell’s use of the "heuristic fiction" of thinking of electrical fields as incompressible fluids. In 2000 Lakoff and Núñez explained one route to an understanding of limit via a metaphorical map that takes the experience of an iterative process that terminates and maps it onto an iterative process that does not terminate, using the final stage of the terminating iteration as a metaphor for the limit of the infinite iterative process.<br /><br />Oehrtman investigated the metaphors or nontechnical experiences on which students rely as they grapple with the notion of limit. His study involved 120 students who he followed throughout a yearlong single variable calculus sequence, observing the class throughout the year and gathering information from the students via pre- and post-course surveys, quizzes, and writing assignments, followed up with clinical interviews with 20 of the students. As he suspected and as his study confirmed, students employed a wide variety of metaphors for limit, many of which were reflected in significant idiosyncrasies in their understanding of limits.<br /><br />He found eight distinct metaphors or ways of relating limits to previous experience, three of which he classified as weak because they were not used very often. Five were classified as strong because they were commonly employed. He then looked at how helpful these metaphors were in reasoning through to an explanation. An important point in mathematics is that, as with Maxwell, metaphors can be useful and even insightful. Thus, thinking of the derivative as a ratio of very small changes in the dependent and independent variables is a metaphor that is certainly not correct but that is very powerful in understanding many of the results of differential calculus.<br /><br />It is also important to keep in mind that the metaphors for limits are fluid and often context specific. Oehrtman found that students used different metaphors in different situations and that if one metaphor was failing, many students would begin to transition to another that seemed more promising. Nevertheless, he did find that, across the board, some metaphors were more useful than others. In describing the metaphors that Oehrtman identified, I am going to expand beyond his work and incorporate into his framework some of the insights into student thinking that have been identified by other researchers.<br /><br /><h3>The Weak Metaphors</h3><div><br /></div><b>Lim</b><b>it as Motion.</b> One surprising finding was that the metaphor of limit as motion was very weak. This was initially unexpected because students consistently spoke of "approaching a limit," and the previous research literature had emphasized a dynamic interpretation of limit as the most common student misconception. Nevertheless, as he explored student use of the idea of "approaching," Oehrtman discovered that students were thinking of a sequence of discrete steps toward the limiting value rather than a continuous motion. The only question for which continuous motion did play a role was #10, explaining what is meant by continuity of a function of two variables. Here several students did speak of moving over the surface defined by this function without falling into holes or encountering cliffs, explaining continuity in terms of the physical topography of the surface.<br /><br /><b>Limit as Zooming.</b> In the early days of graphing calculators, one frequently touted advantage was the ability to zoom in on the graph of a differentiable function to reveal its local linearity. This approach was emphasized by the instructor of the course whose students Oehrtman observed, but the students did not invoke this image on their own and, in fact, frequently misinterpreted its significance.<br /><br /><b>Limit as Informal Version of Correct Definition.</b> While many students used the phrases "arbitrarily precise" and "sufficiently close" that they had heard in class, when pressed they defined these as meaning "very" or "very, very" or "very, very, very" close. None of the students used these phrases in the sense implied by the informal version of the correct definition: The limit is that value <i>L</i> to which the function is forced to be arbitrarily close by taking <i>x</i> sufficiently close but not equal to <i>c</i>.<br /><br /><h3>The Strong Metaphors, in rough order of increasing productivity</h3><div><br /></div><b>Limit as Physical Limitation.</b> The research literature is replete with evidence that a small but significant number of students never move beyond the colloquial definition of limit as a boundary that cannot be surpassed. While this view of limit is commonly held, none of the students in the study employed it directly in trying to answer the questions, yet it did surface in an interesting way. What Oehrtman found was a number of students who attempted to explain #6 and #9 by invoking a physical limit on how small a quantity could be. Thus, Torricelli’s trumpet has a finite volume because eventually the tail is too small to accommodate any matter, and the limit of the jagged line may look like a straight line, but it is really not. There is a smallest positive distance, perhaps the width of an atom, below which it cannot move. Thus, while the limit may look like a straight line, it is really microscopically jagged and so still has length \( \sqrt{2} \). This was, by far, the most counterproductive of all student metaphors for limit.<br /><br /><b>Limit with Infinity as Number.</b> A number of students spoke of the last term of an infinite sequence or an infinitesimal as the smallest distance between a converging sequence and its limit. Sierpińska (1987) and Cornu (1991) refer to these students as "unconscious infinitists" who say "infinite" but think "very big." Infinitesimals can be useful metaphors as witnessed in the work of Leibniz, the Bernoullis, and Euler. Even Cauchy, while introducing the formal language of epsilons and deltas, explained continuity in the language of infinitesimals:<br /><blockquote class="tr_bq">The function <i>f</i>(<i>x</i>) is continuous within given limits if between these limits an infinitely small increment <i>i</i> in the variable <i>x</i> produces always an infinitely small increment, <i>f</i>(<i>x</i>+<i>i</i>) – <i>f</i>(<i>x</i>), in the function itself. (as translated in Boyer, 1949, p. 277)</blockquote>This metaphor was particularly strong in the students’ attempts to explain #5 through #8: L’Hôpital’s rule, Torricelli’s trumpet, the limit comparison test, and Taylor series. There are significant drawbacks. An obvious problem is the creation of a clear distinction between the number represented by \( 0.\overline{9} \) and 1. Many of these students, asked to explain why they are equal, argued instead that they are not. A less obvious but more insidious difficulty is that understanding infinity as very large number encourages belief in the generic limit property as described by Tall (1992), the assumption that any property held by all terms of the sequence must also be held by the limit. Question #9 is extremely problematic for students who attempt to employ this metaphor.<br /><br /><b>Limit as Proximity.</b> The next two metaphors elaborate on a dynamic interpretation of limit in the sense actually employed by students, a description of closeness to the limit value. In exploring how students use this metaphor to explain the limit of a function or the meaning of continuity, we see a phenomenon—recorded by a number of other researchers—of the tendency to focus on the spatial proximity, one manifestation of which is to identify the limit as a point in the Cartesian plane located on or touched by the graph. In explaining #8, the Taylor series, students employing this metaphor talked of the graphs of the Taylor polynomials and emphasized their closeness to the graph of the sine. As Oehrtman explains, they essentially reinvented the L1 norm as a measure of closeness. <br /><br />An additional problem with this metaphor was evident in explanations of the derivative as a limit. These students focused on the distance between the secant lines and the position of the tangent line, sometimes measuring the distance between two lines as the difference of their slopes, but sometimes referring to their physical separation. Students who relied on this metaphor had difficulty making the transition from spatial proximity to quantitative difference.<br /><br /><b>Limit as Collapse. </b>This was the most interesting metaphor that Oehrtman encountered. While incorrect, it could be productive and insightful. It was a common student response to the inherent contradiction of an infinite sequence as an unending process of coming ever closer to the limit against the assertion that in some sense this limit value is equated with the sequence. The image is of a sequence that comes closer and closer until at some point it "collapses" onto the limit value.<br /><br />Oehrtman’s choice of the term "collapse" arises from student use of this metaphor for question #11, why the derivative of the formula for volume yields the formula for surface area. Students spoke of small changes in the volume represented by thin outer shells that became thinner and thinner until they became the surface, collapsing down from three to two dimensions. <br /><br />This phenomena had been observed earlier by Thompson (1994) as he explored student understanding of the fundamental theorem of calculus. In explaining the antiderivative part (finding the derivative of a function defined in terms of a definite integral), students may begin with the limit definition,<br /><br />\[ \lim_{\Delta x \to 0} \frac{ \int_a^{x+\Delta x} f(t)\,dt-\int_a^x f(t)\,dt}{\Delta x}, \]<br /><br />but they then ignore the denominator and view the difference as a thin rectangle of width \(\Delta x\) and height <i>f</i>(<i>x</i>) that collapses down to the one-dimensional height in the limit. This is very close to the way in which Newton and Leibniz first explained this result. In the other direction, they see an area as built up from one-dimensional lines, a metaphor that Finney, Demana, Waits, Kennedy, and I use in preparing students for the fundamental theorem of calculus (Exploration 2 on page 293 of Finney et al. 2012).<br /><br />The collapse metaphor was also important for understanding of the derivative as a limit. Some selected passages from the transcript of a student working through the application of this metaphor to answer #2 are instructive.<br /><blockquote class="tr_bq">You take your values and you squish them really small until … you can go no more, and magically that’s the limit. … As this gets smaller and this gets smaller [points at the vertical and horizontal changes], … you’re getting really, really close to the rise over the run of <b>THIS</b> [points at (3, <i>f</i>(3))]. And when you reach your limit, that’s what the rise over the run of this is [points at (3,<i>f</i>(3))], so I guess that’s the tangent, which is the derivative. Yeah. That does make sense. Because that’s what happens on a limit. (Oehrtman 2009, p. 411)</blockquote>It is worth mentioning that this student made four cycles of attempts to answer question #2 before grasping at the collapse metaphor and finding in it an explanation that she found satisfying. Once she discovered this metaphor, Oehrtman reports that she applied it repeatedly and consistently across multiple representations and contexts. Furthermore, this was the most popular metaphor used by students in answering questions #2 and #4, the two that deal with the definition of the derivative.<br /><br />We see in the collapse metaphor an attempt by students to connect their understanding of limit as part of an unending iterative process with the recognition that this process must be equated with a single value. One of the potential idiosyncrasies of this metaphor is the phenomenon, mentioned in last month’s column, of accepting both \(0.\overline{9} \) and 1 as limits of the sequence \( (0.9, 0.99, 0.999, \ldots) \) while still arguing that they are distinct.<br /><br /><b>Limit as Approximation.</b> The last of the strong metaphors was both the most productive and the one that comes closest to the mathematically correct definition of the limit. This metaphor differs from mere proximity in its focus on numerical difference, rather than geometric distance, and the recognition of the need for a bound, either explicitly or implicitly described, on the difference between the limit and the terms that are approaching it. It is usually accompanied by recognition that this error can be made as small as one wishes. One student wrote,<br /><blockquote class="tr_bq">In fact the power series for sin <i>x</i> will approximate a value infinitely close to the value of sin <i>x</i> and even a remainder can be calculated … The power series of sin <i>x</i> continues forever depending on how close you want your value to come to the value of sin <i>x</i> … The remainder is designed to show how much a power series deviates from the value of a function at a particular point … the power series or polynomial for sin <i>x</i> is an approximation of its value that can be as close of value as you want it to be. (Oehrtman 2009, p. 415)</blockquote>Strictly speaking, approximation is not a metaphor for limit. It is an essential component of what a mathematician means by a limit. What is interesting is that although classroom instruction on limits had not focused on the notion of approximation as a way of understanding limits, many students instinctively drew on it as something from their experience that helped them to understand this concept.<br /><br />Most students used the language of approximation to answer questions #3 and #8, and it was also popular, though not as common as the collapse metaphor, in explaining the meaning of the derivative, #2 and #4.<br /><br /><b>Conclusion. </b>The point of this exploration of student metaphors for limit has not been to illustrate student errors and misconceptions, but rather to illuminate legitimate student attempts to build an understanding of the limit concepts that undergird calculus and to help instructors recognize the source of many of the idiosyncrasies they might encounter in student responses. The question before us as teachers is how to channel these attempts so that our students can build robust and productive ways of thinking about the fundamental ideas of calculus. In my next and final column on "Beyond the Limit," I will look at some of the approaches to teaching that have arisen from Oehrtman’s work.<br /><br /><hr /><br />The following are abbreviated statements of the problems posed by Michael Oehrtman (2009) to 120 students in first-year calculus via pre-course and post-course surveys, quizzes, and other writing assignments as well as two hour-long clinical interviews with twenty of the students.<br /><div><ol><li>Explain the meaning of \( \displaystyle \lim_{x\to 1} \frac{x^3-1}{x-1} = 3. \) </li><li>Let \( f(x) = x^2 + 1.\) Explain the meaning of \( \displaystyle \lim_{h\to 0} \frac{f(3+h)-f(3)}{h}.\) </li><li>Explain why \( 0.\overline{9} = 1.\) </li><li>Explain why the derivative \( \displaystyle f’(x) = \lim_{h\to 0} \frac{f(x+h)-f(x)}{h}\) gives the instantaneous rate of change of <i>f</i> at <i>x</i>. </li><li>Explain why L’Hôpital’s rule works. </li><li>Explain how the solid obtained by revolving the graph of <i>y</i> = 1/<i>x</i> around the <i>x</i>-axis can have finite volume but infinite surface area. </li><li>Explain why the limit comparison test works. </li><li>Explain in what sense \( \displaystyle \sin x = x - \frac{1}{3!} x^3 + \frac{1}{5!} x^5 - \frac{1}{7!}x^7 + \cdots . \) </li><li>Explain how the length of each jagged line shown below can be \( \sqrt{2} \) while the limit has length 1. <div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-bj4CKnNxNbs/U6mYLanFSYI/AAAAAAAAJ1o/D7pvrgYOeiE/s1600/14-7-1+(1).jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://1.bp.blogspot.com/-bj4CKnNxNbs/U6mYLanFSYI/AAAAAAAAJ1o/D7pvrgYOeiE/s1600/14-7-1+(1).jpg" width="500" /></a></div></li><li>Explain what it means for a function of two variables to be continuous.</li><li>Explain why the derivative of the formula for the volume of a sphere, \( V = (4/3)\pi r^3 \), is the surface area of the sphere, \( dV/dr = 4\pi r^2 = A. \) </li></ol></div><hr /><br />Black, M. (1962). Models and Metaphors: Studies in Language and Philosophy. Cornell, NY: Cornell University Press.<br /><br />Boyer, C.B. (1949). The History of the Calculus and Its Conceptual Development. Reprinted 1959. New York, NY: Dover Publications.<br /><br />Cornu, B. (1991). Limits. In D. Tall (Ed.) Advanced Mathematical Thinking. (pp. 153–166). Dordrecht, The Netherlands: KluwerAcademic.<br /><br />Finney, R.L., Demana, F.D., Waits, B.K., Kennedy, D. (2012). Calculus: Graphical, Numerical, Algebraic, 4th ed. Boston, MA: Pearson.<br /><br />Lakoff, G. & Núñez, R. (2000) Where mathematics comes from: How the embodied mind brings mathematics into being. New York, NY: Basic Books.<br /><br />Oehrtman M. (2009). Collapsing Dimensions, Physical Limitation, and Other Student Metaphors for Limit Concepts. Journal For Research In Mathematics Education, <b>40</b>(4), 396–426.<br /><br />Oehrtman, M. (Preprint). Students’ Metaphors for Limit Concepts in Introductory Calculus. To appear in Lessons Learned from Research: Volume 2 Useful Research on Teaching Important Mathematics to All Students. NCTM<br /><br />Sierpińska, A. (1987). Humanities students and epistemological obstacles related to limits. Educational Studies in Mathematics, 18(4), 371-397.<br /><br />Tall, D. (1992). The transition to advanced mathematical thinking: functions, limits, infinity, and proof. Chapter 20 in Grouws D.A. (ed.) Handbook of Research on Mathematics Teaching and Learning, Macmillan, New York, 495-511.<br /><br />Thompson, P.W. (1994). Images of rate and operational understanding of the fundamental theorem of calculus. Educational Studies in Mathematics. <b>26</b>(2), 229–274.Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com1tag:blogger.com,1999:blog-7251686825560941361.post-27251866910063106802014-06-30T13:00:00.000-04:002014-06-30T13:51:56.710-04:00Beyond the Limit, IIn my May column, <a href="http://launchings.blogspot.com/2014_05_01_archive.html" target="_blank">FDWK+B</a>, I said that I would love to ignore limits until we get to infinite series. One of my readers called me out on this, asking how I would motivate the definition of the derivative. Beginning this month and continuing through September, I would like to use my postings to give a brief overview of some of the problems with limit as an organizing principle for first-year calculus and to describe research that supports a better approach. <br /><br />To a mathematician, the limit of <i>f</i>(<i>x</i>) as <i>x</i> approaches <i>c</i> is informally defined as that value <i>L</i> to which the function is forced to be arbitrarily close by taking <i>x</i> sufficiently close (but not equal) to <i>c</i>. In most calculus texts, this provides the foundation for the definition of the derivative: The derivative of <i>f</i> at <i>c</i> is the limit as <i>x</i> approaches <i>c</i> of the average rate of change of <i>f</i> over the interval from <i>x</i> to <i>c</i>. Most calculus texts also invoke the concept of limit in defining the definite integral, though here its application is much more sophisticated.<br /><br />There are many pedagogical problems with this approach. The very first is that any definition of limit that is mathematically correct makes little sense to most students. Starting with a highly abstract definition and then moving toward instances of its application is exactly the opposite of how we know people learn. This problem is compounded by the fact that first-year calculus does not really use the limit definitions of derivative or integral. Students develop many ways of understanding derivatives and integrals, but limits, especially as correctly defined, are almost never employed as a tool with which first-year calculus students tackle the problems they need to solve in either differential or integral calculus. The chapter on limits, with its attendant and rather idiosyncratic problems, is viewed as an isolated set of procedures to be mastered.<br /><br />This student perception of the material on limits as purely procedural was illustrated in a Canadian study (Hardy 2009) of students who had just been through a lesson in which they were shown how to find limits of rational functions at a value of <i>x</i> at which both numerator and denominator were zero. Hardy ran individual observations of 28 students as they worked through a set of problems that were superficially similar to what they had seen in class, but in fact should have been simpler. Students were asked to find \(\lim_{x\to 2} (x+3)/(x^2-9)\). This was solved correctly by all but one of the students, although most them first performed the unnecessary step of factoring <i>x</i>+3 out of both numerator and denominator. When faced with \( \lim_{x\to 1} (x-1)/(x^2+x) \), the fraction of students who could solve this fell to 82%. Many were confused by the fact that <i>x</i>–1 is not a factor of the denominator. The problem \( \lim_{x \to 5} (x^2-4)/(x^2-25) \) evoked an even stronger expectation that <i>x</i>–5 must be a factor of both numerator and denominator. It was correctly solved by only 43% of the students. <br /><br />The Canadian study hints at what forty years of investigations of student understandings and misunderstandings of limits have confirmed: Student understanding of limit is tied up with the process of finding limits. Even when students are able to transcend the mere mastery of a set of procedures, almost all get caught in the language of “approaching” a limit, what many researchers have referred to as a dynamic interpretation of limit, and are unable to get beyond the idea of a limit as something to which you simply come closer and closer.<br /><br />Many studies have explored common misconceptions that arise from this dynamic interpretation. One is that each term of a convergent sequence must be closer to the limit than the previous term. Another is that no term of the convergent sequence can equal the limit. A third, and even more problematic interpretation, is to understand the word “limit” as a reference to the entire process of moving a point along the graph of a function or listing the terms of a sequence, a misconception that, unfortunately, may be reinforced by dynamic software. This plays out in one particularly interesting error that was observed by Tall and Vinner (1981): They encountered students who would agree that the sequence 0.9, 0.99, 0.999, … converges to \(0.\overline{9} \) and that this sequence also converges to 1, but they would still hold to the belief that these two limits are not equal. In drilling into student beliefs, it was discovered that \(0.\overline{9} \) is often understood not as a number, but as a process. As such it may be approaching 1, but it never equals 1. Tied up in this is student understanding of the term “converge” as describing some sort of equivalence.<br /><br />Words that we assume have clear meanings are often interpreted in surprising ways by our students. As David Tall has repeatedly shown (for example, see Tall & Vinner, 1981), a student’s concept image or understanding of what a term means will always trump the concept definition, the actual definition of that term. Thus, Oehrtman (2009) has found that when faced with a mathematically correct definition of limit—that value <i>L</i> to which the function is forced to be arbitrarily close by taking <i>x</i> sufficiently close but not equal to <i>c</i>—most students read the definition through the lens of their understanding that limit means that as <i>x</i> gets closer to <i>c</i>, <i>f</i>(<i>x</i>) gets closer to <i>L</i>. “Sufficiently close” is understood to mean “very close” and “arbitrarily close” becomes “very, very close,” and the definition is transformed in the student’s mind to the statement that the function is very, very close to <i>L</i> when <i>x</i> is very close to <i>c</i>.<br /><br />That raises an interesting and inadequately explored question: Is this so bad? When we use the terminology of limits to define derivatives and definite integrals, is it sufficient if students understand the derivative as that value to which the average rates are getting closer or the definite integral as that value to which Riemann sums get progressively closer? There can be some rough edges that may need to be dealt with individually such as the belief that the limit definition of the derivative does not apply to linear functions and Riemann sums cannot be used to define the integral of a constant function (since they give the exact value, not something that is getting closer), but it may well be that students with this understanding of limits do okay and get what they need from the course.<br /><br />There has been one very thorough study that directly addresses this question, published by Michael Oehrtman in 2009. This involved 120 students in first-year calculus at “a major southwestern university,” over half of whom had also completed a course of calculus in high school. Oehrtman chose eleven questions, described below, that would force a student to draw on her or his understanding of limit. Through pre-course and post-course surveys, quizzes, and other writing assignments as well as clinical interviews with twenty of the students chosen because they had given interesting answers, he probed the metaphors they were using to think through and explain fundamental aspects of calculus.<br /><br />The following are abbreviated statements of the problems he posed, all of which ask for explanations of ideas that I think most mathematicians would agree are central to understanding calculus:<br /><div><ol><li>Explain the meaning of \( \displaystyle \lim_{x\to 1} \frac{x^3-1}{x-1} = 3. \) </li><li>Let \( f(x) = x^2 + 1.\) Explain the meaning of \( \displaystyle \lim_{h\to 0} \frac{f(3+h)-f(3)}{h}.\) </li><li>Explain why \( 0.\overline{9} = 1.\) </li><li>Explain why the derivative \( \displaystyle f’(x) = \lim_{h\to 0} \frac{f(x+h)-f(x)}{h}\) gives the instantaneous rate of change of <i>f</i> at <i>x</i>. </li><li>Explain why L’Hôpital’s rule works. </li><li>Explain how the solid obtained by revolving the graph of <i>y</i> = 1/<i>x</i> around the <i>x</i>-axis can have finite volume but infinite surface area. </li><li>Explain why the limit comparison test works. </li><li>Explain in what sense \( \displaystyle \sin x = x - \frac{1}{3!} x^3 + \frac{1}{5!} x^5 - \frac{1}{7!}x^7 + \cdots . \) </li><li>Explain how the length of each jagged line shown below can be \( \sqrt{2} \) while the limit has length 1. <div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-bj4CKnNxNbs/U6mYLanFSYI/AAAAAAAAJ1o/D7pvrgYOeiE/s1600/14-7-1+(1).jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://1.bp.blogspot.com/-bj4CKnNxNbs/U6mYLanFSYI/AAAAAAAAJ1o/D7pvrgYOeiE/s1600/14-7-1+(1).jpg" width="500" /></a></div></li><li>Explain what it means for a function of two variables to be continuous.</li><li>Explain why the derivative of the formula for the volume of a sphere, \( V = (4/3)\pi r^3 \), is the surface 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Name="Bibliography"/> <w:LsdException Locked="false" Priority="39" QFormat="true" Name="TOC Heading"/> </w:LatentStyles></xml><![endif]--> <!--[if gte mso 10]><style> /* Style Definitions */ table.MsoNormalTable {mso-style-name:"Table Normal"; mso-tstyle-rowband-size:0; mso-tstyle-colband-size:0; mso-style-noshow:yes; mso-style-priority:99; mso-style-parent:""; mso-padding-alt:0in 5.4pt 0in 5.4pt; mso-para-margin:0in; mso-para-margin-bottom:.0001pt; mso-pagination:widow-orphan; font-size:12.0pt; font-family:Cambria; mso-ascii-font-family:Cambria; mso-ascii-theme-font:minor-latin; mso-hansi-font-family:Cambria; mso-hansi-theme-font:minor-latin;} </style><![endif]--> <!--StartFragment--> <!--EndFragment--><br />In next month’s column, I will summarize Oehrtman’s findings. I then will show how they have led to a fresh approach to the teaching of calculus that avoids many of the pitfalls surrounding limits.<br /><hr />Hardy, N. (2009). Students' Perceptions of Institutional Practices: The Case of Limits of Functions in College Level Calculus Courses. <i>Educational Studies In Mathematics</i>, <b>72</b>(3), 341–358.<br /><br />Oehrtman M. (2009). Collapsing Dimensions, Physical Limitation, and Other Student Metaphors for Limit Concepts. <i>Journal For Research In Mathematics Education</i>, <b>40</b>(4), 396–426.<br /><br />Tall, D., & Vinner, S. (1981). Concept image and concept definition in mathematics with particular reference to limits and continuity. <i>Educational Studies in Mathematics</i>, <b>12</b>(2), 151–169.</div>Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com7tag:blogger.com,1999:blog-7251686825560941361.post-86157886628559907822014-06-01T05:00:00.000-04:002014-06-01T05:00:05.083-04:00AP Calculus and the Common CoreIn February, 2013, Trevor Packer, Senior Vice-President for the Advanced Placement Program and Instruction at The College Board, appeared before the American Association of School Administrators (AASA), the professional society for school superintendents, to provide information about the Advanced Placement Program. Following that session, he had a short video interview in which he was asked to comment on the relationship between the Common Core State Standards (CCSS) and the College Board’s Advanced Placement Program. What he said about CCSS and AP Calculus has, unfortunately, been misreported. With Trevor Packer’s encouragement, I would like to attempt a clarification: There is no conflict between the Common Core and AP Calculus. In fact, it is just the opposite. If faithfully implemented, the Common Core can improve the preparation of students for AP Calculus or any college-level calculus.<br /><br />In the AASA summary of the conference proceedings, “<a href="http://www.aasa.org/content.aspx?id=27296">College Board: Reconciling AP Exams With Common Core</a>,” Packer’s comment on AP Calculus was reported as follows:<br /><blockquote class="tr_bq">“Despite these measures, there are still difficulties in reconciling many AP courses with the Common Core. In particular, AP Calculus is in conflict with the Common Core, Packer said, and it lies outside the sequence of the Common Core because of the fear that it may unnecessarily rush students into advanced math classes for which they are not prepared. </blockquote><blockquote class="tr_bq">“The College Board suggests a solution to the problem of AP Calculus ‘If you’re worried about AP Calculus and fidelity to the Common Core, we recommend AP Statistics and AP Computer Science,’ he told conference attendees.”</blockquote>This article, written by a high school junior serving as an intern at the meeting, is not in line with the video of Packer’s remarks, “<a href="https://www.youtube.com/watch?v=BbgEo52DEqs">College Board’s Trevor Packer on Common Core and AP Curriculum</a>,” where he says that<br /><blockquote class="tr_bq">“AP Calculus sits outside of the Common Core. The Calculus is not part of the Common Core sequence, and in fact the Common Core asks that educators slow down the progressions for math so that students learn college-ready math very, very well. So that can involve a sequence that does not culminate in AP Calculus. There may still be a track toward AP Calculus for students who are interested in majoring in Engineering or other STEM disciplines, but by and large, the Common Core math sequence is best suited to prepare students for AP Statistics or AP Computer Science, which have dependencies on the math requirements of the Common Core.”</blockquote>The assertion of a conflict between the Common Core and AP Calculus was a misinterpretation on the part of the student. Nevertheless, this lack of clear articulation between Common Core and AP Calculus is easy to misinterpret. <br /><br />Packer’s remarks arose from concerns that I and others have expressed about the headlong rush to calculus in high school (see, in particular, <a href="http://launchings.blogspot.com/2012/04/maanctm-joint-position-on-calculus.html" target="_blank">MAA/NCTM Joint Position on Calculus</a>). As I pointed out in last month’s column (<a href="http://launchings.blogspot.com/2014_05_01_archive.html" target="_blank">FDWK+B, May, 2014</a>), almost 700,000 students begin the study of calculus while in high school each year. Not all of them are in AP programs. Not all in an AP program take or even intend to take an AP Calculus exam. But we are now closing in on 400,000 students who take either the AB or BC Calculus exam each year, a number that is still growing at roughly 6% per year with no sign that we have reached an inflection point. Over half the students in Calculus I in our colleges and universities have already completed a calculus course while in high school. At our leading universities, the fraction is over three-quarters. Unfortunately, merely studying calculus in high school does not mean that these students are ready for college-level calculus and the subsequent mathematics courses required for engineering or the mathematical or physical sciences.<br /><br />The problem for many students who enter with the aspiration of a STEM degree is inadequate proficiency at the level of precalculus: facility with algebra; understanding of trigonometric, exponential, and logarithmic functions; and comprehension of the varied and interconnected ways of viewing functions. Packer speaks of slowing down the progressions through mathematics. This is in response to a shared concern that the rush to get to calculus while in high school can interfere with the development of a solid foundation on which to build mathematical proficiency. Much of the impetus for the Common Core State Standards in Mathematics comes from the recognition that there are clear benchmarks consisting of skills and understandings that must be mastered before students are ready to move on to the next level of abstraction and sophistication. Failure to achieve those benchmarks at the appropriate point in a student’s mathematical development risks seriously handicapping future mathematical achievement.<br /><br />The Common Core was designed as a common core, a set of expectations we intend for all students. There is an intentional gap between where the Common Core in Mathematics ends and where mathematics at the level of calculus begins. This gap is partially filled with the additional topics marked with a “+” in the Common Core State Standards in Mathematics, topics that usually get the required level of attention in a course called Precalculus. As the name suggests, Precalculus is the course that prepares students for calculus. This is the articulation problem to which Packer alludes. Completing the Common Core does not mean one is ready for the study of AP Calculus or any other calculus. It means one is ready for a number of options that include AP Statistics, AP Computer Science, or a Precalculus class.<br /><br />There is no conflict between AP Calculus and the Common Core. Rather, there is an expectation that if the Common Core is faithfully implemented, then students will be better prepared when they get to AP Calculus and the courses that follow it.Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com5tag:blogger.com,1999:blog-7251686825560941361.post-78184286661776366932014-05-01T01:00:00.000-04:002014-05-01T01:00:04.059-04:00FDWK+B<div class="MsoNormal"><span style="font-size: 11pt;"><span style="font-family: inherit;">I am very pleased to announce that I will be joining the team of authors for the AP Calculus text <i>Calculus: Graphical, Numerical, Algebraic</i>by Finney, Demana, Waits, and Kennedy (commonly known as FDWK). Ross Finney has not been an active member of the team for some years (he died in 2000), and Frank Demana and Bert Waits are easing out of their roles, but their names reflect the incredible pedigree of this text. It began with George Thomas in 1951 and has variously been known as Thomas; Thomas & Finney; Finney & Thomas; Finney, Thomas, Demana, Waits; and Finney, Demana, Waits, Kennedy.<o:p></o:p></span></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-size: 11pt;"><span style="font-family: inherit;">I was fortunate to be able to get to know George Thomas after he retired from MIT and moved to State College, Pennsylvania. I knew him as an extremely modest and gentle person with a continuing fascination with mathematics. I have long admired Frank Demana and Bert Waits for their pioneering work in the Calculus Reform efforts. Dan Kennedy and I have known each other for many years through the AP Program, and it is a particular delight for me now to be collaborating with him.<o:p></o:p></span></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-size: 11pt;"><span style="font-family: inherit;">I also am very happy to be joining an effort aimed at high school calculus. Roughly one million U.S. students begin the study of calculus each year, and close to 700,000 of them, at least two-thirds of the total, start this journey in high school. This is the place where one can have the greatest impact in shaping students’ understanding of calculus.<o:p></o:p></span></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-size: 11pt;"><span style="font-family: inherit;">There are limitations that I, as an author of “niche textbooks” for which I can take whatever approach I wish, find constraining. First of all, the text has to be closely tied to the AP Calculus syllabus and exams, which, in their turn, are closely tied to the curricula as enacted at the major universities, the big consumers of AP Calculus results. The emphasis on limits is one of those limitations. I would love to ignore them until we get to infinite series, but that really is not an option.<o:p></o:p></span></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-size: 11pt;"><span style="font-family: inherit;">Second, the books I write for my own pleasure can assume whatever level of sophistication on the part of the reader I choose to impose. I recognize that this text will be used by teachers and students for whom digressions and elaborations may be more confusing than helpful. That said, I do hope to push both teachers and students a little and to open more perspectives, especially historical perspectives, on this subject.<o:p></o:p></span></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-size: 11pt;"><span style="font-family: inherit;">Third, I am now working for the behemoth that is Pearson. I’ve worked with Pearson people on several projects and have always found them to be intelligent, conscientious, and seriously concerned with producing quality products. Nevertheless, this is a mass-market endeavor that travels with its own peculiar baggage of demands and constraints. I am pleased that in the face of so much pressure to bulk up with every tidbit relevant to Calculus, FDWK has managed to maintain a lean profile of only 717 pages (16 fewer than the first edition of Thomas).<o:p></o:p></span></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-family: inherit;"><span style="font-size: 11pt;">Also on the plus side is the large and talented staff that will be working with us to produce the next edition of this text. As I observed in my contribution to “</span><a href="http://www.ams.org/notices/201401/rnoti-p69.pdf" target="_blank"><span style="font-size: 11pt;">Musing on MOOCS</span></a><span style="font-size: 11pt;">,” which appeared in the <i style="mso-bidi-font-style: normal;">Notices of the AMS</i> this past January, the real revolution in education created by the online world is not the disappearance of the live instructor but the richness of supporting resources that instructors can now draw upon. Robert Ghrist argued that the ease with which individuals can produce their own online materials will eliminate the need for big publishers. I argued that the situation is exactly the opposite: “The problem is that few of us will have the time to develop our own materials, and anyone who searches for such resources online is quickly inundated with options. In an era of overwhelming choices, it is the reputable bundlers who will dominate.” MAA is one reputable supplier, as evidenced by WeBWorK (see my column from </span><a href="http://www.maa.org/external_archive/columns/launchings/launchings_04_09.html" target="_blank"><span style="font-size: 11pt;">April, 2009</span></a><span style="font-size: 11pt;">). Pearson is well aware of this need and is actively building these supports.</span></span></div><div class="MsoNormal"><span style="font-family: inherit;"><span style="font-size: 11pt;"><br /></span></span></div><div class="MsoNormal"><span style="font-family: inherit;"><span style="font-size: 11pt;">By an opportune coincidence, I also am working with Karen Marrongelle and Karen Graham on the calculus chapter for the next version of the </span><i style="mso-bidi-font-style: normal;"><span style="font-size: 11pt;">NCTM Handbook of Research on Learning and Teaching Mathematics. </span></i><span style="font-size: 11pt;"> This means that I am currently steeped in the accumulated research on how students understand and misunderstand the key concepts of calculus. I expect to translate some of this knowledge into the shaping of future editions of FDWK, and I also hope to share some of what I’m learning in future <i>Launchings</i> columns.</span></span><span style="font-family: "Times New Roman"; font-size: 11.0pt; mso-bidi-font-family: "Times New Roman";"><o:p></o:p></span></div>Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com2tag:blogger.com,1999:blog-7251686825560941361.post-9200655971647460982014-04-01T01:00:00.000-04:002014-04-01T01:00:05.570-04:00Age Is Not the Problem<div class="MsoNormal"><span style="font-family: "Times New Roman","serif"; font-size: 11.0pt;">Edward Frenkel recently resurrected an old complaint in his <i>Los Angeles Times</i> op-ed, “</span><span style="font-family: "Times New Roman","serif"; font-size: 11.0pt;"><a href="http://www.latimes.com/opinion/commentary/la-oe-adv-frenkel-why-study-math-20140302,0,5177338.story#axzz2vPNhS0zO" target="_blank">How our 1,000-year–old math curriculum cheats America’s kids</a>.</span><span style="font-family: "Times New Roman","serif"; font-size: 11.0pt;">” He observes that no one would exclude an appreciation for the beauty of art or music from the need to build technique. Why do we do that in mathematics? As I said, this is an old complaint. Possibly no one has voiced it more eloquently than Paul Lockhart in </span><a href="http://www.maa.org/sites/default/files/pdf/devlin/LockhartsLament.pdf" target="_blank"><i><span style="font-family: "Times New Roman","serif"; font-size: 11.0pt;">A Mathematician’s Lament</span></i></a><span style="font-family: "Times New Roman","serif"; font-size: 11.0pt;">, the theme of Keith Devlin’s 2008 MAA column, “</span><a href="https://www.maa.org/external_archive/devlin/devlin_03_08.html" target="_blank"><span style="font-family: "Times New Roman","serif"; font-size: 11.0pt;">Lockhart’s Lament</span></a><span style="font-family: "Times New Roman","serif"; font-size: 11.0pt;">.” Enough time has passed that it is worth my while to bring this lament back to the attention of the readers of MAA columns. I also want to respond to Frenkel’s post. I have two problems with what he writes.<o:p></o:p></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-family: "Times New Roman","serif"; font-size: 11.0pt;">The first is the suggestion that we spend too much time on “old” mathematics and not enough on what is “new.” I share Frenkel’s disappointment that too few have any appreciation of mathematics as a fresh, creative, and self-renewing field of study. Frenkel himself has made a significant contribution toward correcting this. In his recent book, </span><a href="http://www.amazon.com/dp/0465050743/?tag=googhydr-20&hvadid=43868416769&hvpos=1t1&hvexid=&hvnetw=g&hvrand=16378492594770513284&hvpone=&hvptwo=&hvqmt=e&hvdev=c&ref=pd_sl_9hkrjlnrtf_e" target="_blank"><i><span style="font-family: "Times New Roman","serif"; font-size: 11.0pt;">Love and Math</span></i></a><span style="font-family: "Times New Roman","serif"; font-size: 11.0pt;">, he has opened a window for the educated layperson to glimpse the fascination of the Langland’s program. But I disagree with Frenkel’s solution of devoting “just 20% of class time [to] opening students’ eyes to the power and exquisite harmony of modern math.” There is power and exquisite harmony in everything from early Babylonian and Egyptian discoveries through Euclid’s <i>Elements</i> to the <i>Arithmetica</i> of Diophantus and the development of trigonometry in the astronomical centers of Alexandria and India, all of which were accomplished more than a millennium ago and are still capable of inspiring awe. </span></div><div class="MsoNormal"><span style="font-family: "Times New Roman","serif"; font-size: 11.0pt;"><br /></span></div><div class="MsoNormal"><span style="font-family: "Times New Roman","serif"; font-size: 11.0pt;">In fact, I believe that one of the worst things we could do is to create a dichotomy in students’ minds between beautiful modern math and ugly old math. We must communicate the timeless beauty of all real mathematics. The challenge of the educator is to engage students in rediscovering this beauty for themselves, not outside of the standard curriculum, but embedded within it. The question of how to accomplish this leads to my second problem with Frenkel.<o:p></o:p></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-family: "Times New Roman","serif"; font-size: 11.0pt;">Frenkel makes the implicit assumption that what we need is a wake-up call, that it is time to recognize that mathematics education must do more than create procedural facility. In fact, the need to combine the development of technical ability with an appreciation for the ideas that motivate and justify the mathematics that we teach goes back at least a century to Felix Klein and his </span><a href="http://www.amazon.com/Elementary-Mathematics-Advanced-Standpoint-Arithmetic/dp/048643480X/ref=sr_1_2?s=books&ie=UTF8&qid=1395576510&sr=1-2&keywords=elementary+mathematics+from+an+advanced+standpoint" target="_blank"><i><span style="font-family: "Times New Roman","serif"; font-size: 11.0pt;">Elementary Mathematics from an Advanced Standpoint</span></i></a><span style="font-family: "Times New Roman","serif"; font-size: 11.0pt;">. It is front and center in the </span><a href="http://www.corestandards.org/Math/Practice/" target="_blank"><span style="font-family: "Times New Roman","serif"; font-size: 11.0pt;">Practice Standards</span></a><span style="font-family: "Times New Roman","serif"; font-size: 11.0pt;"> of the </span><a href="http://www.corestandards.org/math" target="_blank"><span style="font-family: "Times New Roman","serif"; font-size: 11.0pt;">Common Core State Standards in Mathematics</span></a><span style="font-family: "Times New Roman","serif"; font-size: 11.0pt;">. It was a driving concern of Paul Sally at the University of Chicago, who we so recently and unfortunately lost. It continues to motivate Al Cuoco and his staff engaged in the development of the materials of the </span><a href="https://mpi.edc.org/" target="_blank"><span style="font-family: "Times New Roman","serif"; font-size: 11.0pt;">Mathematical Practice Institute</span></a><span style="font-family: "Times New Roman","serif"; font-size: 11.0pt;">. It lies at the root of Richard Rusczyk’s creation of the </span><a href="http://www.artofproblemsolving.com/" target="_blank"><i><span style="font-family: "Times New Roman","serif"; font-size: 11.0pt;">Art of Problem Solving</span></i></a><span style="font-family: "Times New Roman","serif"; font-size: 11.0pt;">. It permeates the efforts of literally thousands of us who are struggling to enable each of our students to encounter the thrill of mathematical exploration and discovery.<o:p></o:p></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-family: "Times New Roman","serif"; font-size: 11.0pt;">As we know, it takes more than good curricular materials and good intentions to accomplish this. It requires educators who understand mathematics both broadly and deeply and can bring this expertise to their teaching. Many are working to spread this knowledge among all who would teach mathematics to our children. This is the inspiration behind the reports of the Conference Board of the Mathematical Sciences on </span><a href="http://cbmsweb.org/MET2/" target="_blank"><span style="font-family: "Times New Roman","serif"; font-size: 11.0pt;">The Mathematical Education of Teachers</span></a><span style="font-family: "Times New Roman","serif"; font-size: 11.0pt;">. It is a goal of the </span><a href="https://www.mathcircles.org/" target="_blank"><span style="font-family: "Times New Roman","serif"; font-size: 11.0pt;">Math Circles</span></a><span style="font-family: "Times New Roman","serif"; font-size: 11.0pt;">, in particular the </span><a href="http://www.mathteacherscircle.org/" target="_blank"><span style="font-family: "Times New Roman","serif"; font-size: 11.0pt;">Math Teachers’ Circles</span></a><span style="font-family: "Times New Roman","serif"; font-size: 11.0pt;"> that reach those who too often are unaware of the exciting opportunities for exploration and discovery within the curricula they teach.</span></div><div class="MsoNormal"><span style="font-family: 'Times New Roman', serif; font-size: 11pt;"><br /></span></div><div class="MsoNormal"><span style="font-family: 'Times New Roman', serif; font-size: 11pt;">The mathematician’s lament is still all too relevant, but it is neither unheard nor unheeded. I am encouraged by the many talented and dedicated individuals and organizations working to meet its challenge.</span></div>Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com0tag:blogger.com,1999:blog-7251686825560941361.post-55154997901709670832014-03-01T07:00:00.000-05:002014-03-01T07:00:08.944-05:00Collective Action by STEM Disciplinary Societies<div class="MsoNormal"><span style="font-family: "Times New Roman","serif"; font-size: 11.0pt;">At the end of January, it was my great pleasure to be part of the leadership for a meeting at the MAA Carriage House of representatives of a collection of STEM disciplinary societies [1] and concerned educational associations [2] to consider ways that these societies can coordinate efforts to increase their collective impact on undergraduate education. Across academia, but especially at research universities, most faculty identify first with their discipline and department and only secondarily with their university. Disciplinary societies therefore have the potential to impact how faculty think about their teaching and how willing they are to reach outside their own department in seeking ideas and support for improving undergraduate education.<o:p></o:p></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-family: "Times New Roman","serif"; font-size: 11.0pt;">Many disciplinary societies are actively promoting effective methods for engaging students to improve both what they learn and their desire to persist. The American Physical Society and the American Association of Physics Teachers have been particularly effective in this regard. See, for example, the Physics Education Research User’s Guide, </span><a href="http://perusersguide.org/" target="_blank"><span style="font-family: "Times New Roman","serif"; font-size: 11.0pt;">perusersguide.org</span></a><span style="font-family: "Times New Roman","serif"; font-size: 11.0pt;">, described in my column </span><a href="http://launchings.blogspot.com/2012/07/learning-from-physicists.html" target="_blank"><span style="font-family: "Times New Roman","serif"; font-size: 11.0pt;">“Learning from the Physicists,” July, 2012</span></a><span style="font-family: "Times New Roman","serif"; font-size: 11.0pt;">. Over the past several years, the life sciences community, scattered over some 147 disciplinary societies, has come together to produce a joint report, </span><a href="http://visionandchange.org/files/2013/11/aaas-VISchange-web1113.pdf" target="_blank"><i><span style="font-family: "Times New Roman","serif"; font-size: 11.0pt;">Vision and Change in Undergraduate Biology Education: A Call to Action</span></i></a><i><span style="font-family: "Times New Roman","serif"; font-size: 11.0pt;"> </span></i><span style="font-family: "Times New Roman","serif"; font-size: 11.0pt;">[3]. Recognizing that it is not sufficient to issue a report, </span><a href="file:///C:/Users/kmerow/Desktop/visionandchange.org" target="_blank"><i><span style="font-family: "Times New Roman","serif"; font-size: 11.0pt;">Vision and Change</span></i></a><span style="font-family: "Times New Roman","serif"; font-size: 11.0pt;">continues to seek ways to implement the changes it champions. One outgrowth has been </span><a href="http://www.pulsecommunity.org/"><span style="font-family: "Times New Roman","serif"; font-size: 11.0pt;">PULSE</span></a><span style="font-family: "Times New Roman","serif"; font-size: 11.0pt;">, the Partnership for Undergraduate Life Sciences Education, which is building communities that share experiences of department-level implementation of the <i>Vision and Change</i> recommendations. Inspired by the example of PULSE, the mathematics community began last summer to build a comparable effort, </span><a href="http://www.ingeniousmathstat.org/" target="_blank"><span style="font-family: "Times New Roman","serif"; font-size: 11.0pt;">INGenIOuS</span></a><span style="font-family: "Times New Roman","serif"; font-size: 11.0pt;">, </span><span style="font-size: 11.0pt; mso-fareast-font-family: "Times New Roman";">Investing in the Next Generation through Innovative and Outstanding Strategies.</span><span style="font-size: 11pt;"><o:p></o:p></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-size: 11.0pt; mso-fareast-font-family: "Times New Roman";">We have much to learn from each other. Beyond just sharing information, an ability to offer comparable statements of vision and comparable programs to promulgate effective practices would increase their collective impact. This would be especially true if the disciplinary societies were to establish and promote linkages that enable individuals to connect with others at their university who are working toward the same ends but within other departments.</span><span style="font-size: 11pt;"><o:p></o:p></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-size: 11.0pt; mso-fareast-font-family: "Times New Roman";">With these goals in mind, 28 representatives of disciplinary societies and educational associations met at the MAA Carriage House in Washington, DC on January 30–31 for an NSF-sponsored workshop [4] entitled </span><a href="http://serc.carleton.edu/issues/index.html" target="_blank"><span style="font-family: "Times New Roman","serif"; font-size: 11.0pt; mso-fareast-font-family: "Times New Roman";">ISSUES</span></a><span style="font-size: 11.0pt; mso-fareast-font-family: "Times New Roman";">, Integration of Strategies that Support Undergraduate Education in STEM, to look for opportunities to work collectively. As preparation, most of the societies provided a summary of their current activities directed toward faculty development and the improvement of undergraduate education. These </span><a href="http://serc.carleton.edu/issues/profiles.html" target="_blank"><span style="font-family: "Times New Roman","serif"; font-size: 11.0pt; mso-fareast-font-family: "Times New Roman";">Profiles</span></a><span style="font-size: 11.0pt; mso-fareast-font-family: "Times New Roman";"> can be found within the ISSUES website at </span><a href="http://serc.carleton.edu/issues" target="_blank"><span style="font-family: "Times New Roman","serif"; font-size: 11.0pt; mso-fareast-font-family: "Times New Roman";">serc.carleton.edu/issues</span></a><span style="font-size: 11.0pt; mso-fareast-font-family: "Times New Roman";">. A summary of the workshop is available at </span><a href="http://serc.carleton.edu/issues/workshop14" target="_blank"><span style="font-family: "Times New Roman","serif"; font-size: 11.0pt; mso-fareast-font-family: "Times New Roman";">serc.carleton.edu/issues/workshop14</span></a><span style="font-size: 11.0pt; mso-fareast-font-family: "Times New Roman";">.</span><span style="font-size: 11pt;"><o:p></o:p></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-size: 11pt;">The workshop identified five concrete areas in which disciplinary societies could increase their effectiveness by sharing and coordinating their efforts:</span><strong><span style="font-size: 11.0pt; font-weight: normal; mso-bidi-font-weight: bold; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></strong></div><div class="MsoNormal"></div><ol><li><b style="text-indent: -0.25in;"><span style="font-family: "Times New Roman","serif"; font-size: 11.0pt;">Supporting Early Career Faculty. </span></b><span style="font-family: 'Times New Roman', serif; font-size: 11pt; text-indent: -0.25in;">Within the disciplinary societies, the task is to develop workshops for and build communities of early career faculty, as well as partnering with the Discipline-Based Educational Research community to assess the long-term effectiveness of this work. On individual campuses, the task is to work with deans and chairs to build cross-disciplinary networks of faculty who have been through these experiences, supported by networks of mentors both from the individual’s profession and from within the individual’s home institution.</span></li><li><b style="text-indent: -0.25in;"><span style="font-family: "Times New Roman","serif"; font-size: 11.0pt;">Strengthening Departments. </span></b><span style="font-family: 'Times New Roman', serif; font-size: 11pt; text-indent: -0.25in;">There is a need to increase the value placed on the department chair and to provide support for the chair by supplying tools for departmental self-assessment of teaching effectiveness together with practical suggestions that chairs and departmental leaders can implement to improve teaching effectiveness.</span></li><li><b style="text-indent: -0.25in;"><span style="font-family: "Times New Roman","serif"; font-size: 11.0pt;">Communicating Career Pathways.</span></b><span style="font-family: 'Times New Roman', serif; font-size: 11pt; text-indent: -0.25in;">We need to increase the diversity of students within our disciplines by increasing student awareness of the variety of pathways that are available to them, actively recruiting students to these pathways, preparing them for a variety of careers, and introducing them to a network of potential employers.</span></li><li><b style="text-indent: -0.25in;"><span style="font-family: "Times New Roman","serif"; font-size: 11.0pt;">Shifting Cultural Norms. </span></b><span style="font-family: 'Times New Roman', serif; font-size: 11pt; text-indent: -0.25in;">Disciplinary societies should strive to move their members toward embracing teaching practices that align with what educational research has shown to be most effective and toward a mindset of continual efforts to improve undergraduate teaching and learning. This can be accomplished through policy statements, rubrics for assessing effective educational processes, and active promotion of these practices. Part of our collective goal should be the adoption of consistent language that reinforces this message across disciplinary boundaries.</span></li><li><b style="text-indent: -0.25in;"><span style="font-family: "Times New Roman","serif"; font-size: 11.0pt;">Measuring the Impact of Our Own Programs for Improving Undergraduate Education.</span></b><span style="font-family: 'Times New Roman', serif; font-size: 11pt; text-indent: -0.25in;"> The disciplinary societies can benefit from developing common rubrics for assessing the effectiveness of their own programs and using these to help frame discussion and dialog across the societies.</span></li></ol><br /><div class="MsoNormal"><span style="font-family: "Times New Roman","serif"; font-size: 11.0pt;">On point <b>1</b>, we are already working with the Association of American Universities (AAU) to put together a pilot project on AAU campuses that will build local networks of faculty from multiple disciplines who have each been through an early career professional development program run by their disciplinary society. On point <b>5</b>, we are beginning the task of gathering information from the disciplinary societies about their experiences with assessment of their own programs. 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class="MsoNormal"><span style="font-family: "Times New Roman","serif"; font-size: 11.0pt;">Footnotes and References: <o:p></o:p></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-family: "Times New Roman","serif"; font-size: 11.0pt;">[1] The disciplinary societies that were represented were the American Association of Physics Teachers, American Chemical Society, American Geophysical Union, American Institute of Biological Sciences, American Institute of Physics, American Mathematical Society, American Physical Society, American Psychological Association, American Society for Engineering Education, American Society for Microbiology, American Statistical Association, Mathematical Association of America, National Association of Biology Teachers, National Association of Geoscience Teachers, and the Society for Industrial and Applied Mathematics.<o:p></o:p></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-family: "Times New Roman","serif"; font-size: 11.0pt;">[2] The educational associations that were represented included the American Association for the Advancement of Science, Association of American Universities, Association of Public Land-Grant Universities, Howard Hughes Medical Institute, National Academy of Sciences, National Science Foundation, and Project Kaleidoscope of the Association of American Colleges and Universities.<o:p></o:p></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-family: "Times New Roman","serif"; font-size: 11.0pt;">[3] Brewer, C.A., and Smith, D. (eds.). 2011. <i>Vision and Change in Undergraduate Biology Education: A Call to Action</i>. Washington, DC: American Association for the Advancement of Science. Available at </span><span style="font-family: "Times New Roman","serif"; font-size: 11.0pt;"><a href="http://visionandchange.org/files/2013/11/aaas-VISchange-web1113.pdf" target="_blank">visionandchange.org/files/2013/11/aaas-VISchange-web1113.pdf</a></span></div><div class="MsoNormal"><span style="font-family: 'Times New Roman', serif; font-size: 11pt;"><br /></span></div><div class="MsoNormal"><span style="font-family: 'Times New Roman', serif; font-size: 11pt;">[4] The workshop was made possible by a grant from the National Science Foundation, #1344418. The opinions expressed here do not necessarily reflect those of NSF.</span></div>Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com0tag:blogger.com,1999:blog-7251686825560941361.post-17056220484883200122014-02-01T06:00:00.000-05:002014-02-01T06:00:00.545-05:00Mathematics for the Biological Sciences<div class="MsoNormal" style="text-align: left;">MAA has just published a Notes volume, <a href="http://www.maa.org/publications/ebooks/undergraduate-mathematics-for-the-life-sciences"><i>Undergraduate Mathematics for the Life Sciences: Models, Processes, and Directions</i></a> [1] that provides examples and advice for mathematics departments that want to reach out to the growing population of biological science majors.</div><div class="MsoNormal"><o:p></o:p></div><div class="MsoNormal"><br /></div><div class="MsoNormal">Biological science majors have replaced prospective engineers as the largest group of students taking regular Calculus I. From the MAA’s Calculus Survey [2], just over 28% of all students in mainstream Calculus I intend to pursue a major in the biological sciences, the largest single group of majors in this course. It is larger than engineers (just under 28%) or the combined physical science (7%), computer science (7%), and mathematical science majors (1%). For women in mainstream Calculus I, 42% intend a biological science major. For Black or Hispanic students, 34% are going into biological sciences. This dominance is certain to only increase. As the graph in Figure 1 illustrates, the growth in science, engineering, and mathematical sciences majors is occurring almost exclusively in the biological sciences.<o:p></o:p></div><div class="MsoNormal"><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="http://2.bp.blogspot.com/-YUZjkNMDQns/UuZhxZ0JNvI/AAAAAAAAJT0/vobKaNca1rM/s1600/IntendedMajors.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto; text-align: center;"><img border="0" src="http://2.bp.blogspot.com/-YUZjkNMDQns/UuZhxZ0JNvI/AAAAAAAAJT0/vobKaNca1rM/s1600/IntendedMajors.jpg" width="600" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><span style="font-family: 'Times New Roman', serif;"><i>Figure 1: Number of full-time entering freshmen who identified a STEM field as their most likely major. Data from The American Freshman surveys. [2]</i></span></td></tr></tbody></table><div class="MsoNormal">Mathematics has done well by encouraging students who have to study mathematics to continue its study. Mathematics departments actually graduate more majors than the number of students who enter with the intention of pursuing a math major. In 2012, 18,842 students graduated with a Bachelor’s degree in mathematics.[3] Four years earlier, in 2008, only 11,583 entered a full-time program with the intention of majoring in mathematics.[4] Even after subtracting the roughly 5,000 students per year who are heading into K-12 mathematics teaching and who get a degree in mathematics but identify education as their intended field when they enter, we see that mathematics—uniquely among the major STEM disciplines—still has a net gain in majors. If we are to maintain this happy state of affairs, then we need to convince our audience that mathematics is relevant to its interests.<o:p></o:p></div><div class="MsoNormal"><br /></div><div class="MsoNormal">Mathematics departments are recognizing this fact. The 2010 CBMS report revealed that 41% of those at research universities had added interdisciplinary courses in mathematics and biology within the past five years. As the new MAA Notes volume illustrates, there is a tremendous amount of experimentation under way.<o:p></o:p></div><div class="MsoNormal"><br /></div><div class="MsoNormal">This volume begins with descriptions of thirteen programs that range from calculus for biology majors, to programs that draw calculus and statistics together into a year-long course, to bioinformatics, to research programs for biology majors that incorporate significant quantitative analysis. The institutions include large universities: Illinois at Urbana-Champaign, Ohio State, and the Universities of Minnesota, Nebraska-Lincoln, and Utah. There also are smaller places: Benedictine University, Macalester College, University of Richmond, Chicago State, Sweet Briar College, University of Wisconsin-Stout, and East Tennessee State.<o:p></o:p></div><div class="MsoNormal"><br /></div><div class="MsoNormal">The volume continues with a collection of essays on “Processes.” These are nine accounts of the trials and tribulations of getting such a program started and keeping it going. This is particularly useful because these essays describe both programs that have survived, moving beyond the small group of individuals who initiated them, and programs that have failed or are failing, the ones that have not managed to establish themselves as a permanent feature of the local curriculum. The final four essays, labeled “Directions,” speak to opportunities and needs.<o:p></o:p></div><div class="MsoNormal"><br /></div><div class="MsoNormal">Unfortunately, this book is only available as a pdf file ($25) or as Print-on-Demand ($43), so you probably will not see a display copy at MAA meetings. But it is well worth checking out.<o:p></o:p></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><b>References<o:p></o:p></b></div><div class="MsoNormal"><br /></div><div class="MsoNormal">[1] Ledder, G., J.P. Carpenter, and T.D. Comar (Eds.) (2013). <i>Undergraduate Mathematics for the Life Sciences: Models, Processes, and Directions</i>. MAA Notes #81. Washington, DC: Mathematical Association of America. <a href="http://www.maa.org/publications/ebooks/undergraduate-mathematics-for-the-life-sciences">www.maa.org/publications/ebooks/undergraduate-mathematics-for-the-life-sciences</a><o:p></o:p></div><div class="MsoNormal"><br /></div><div class="MsoNormal">[2] Higher Education Research Institute. (Multiple years). <i>The American Freshman</i>. <a href="http://www.heri.ucla.edu/tfsPublications.php">www.heri.ucla.edu/tfsPublications.php</a><o:p></o:p></div><div class="MsoNormal"><br /></div><div class="MsoNormal">[3] NCES. (2013). <i>Digest of Education Statistics</i>. Table 322.10. <a href="http://nces.ed.gov/programs/digest/d13/tables/dt13_322.10.asp">nces.ed.gov/programs/digest/d13/tables/dt13_322.10.asp</a><o:p></o:p></div><div class="MsoNormal"><br /></div><div class="MsoNormal">[4] Higher Education Research Institute. (2008). <i>The American Freshman: National norms for fall 2008</i>. <a href="http://www.heri.ucla.edu/tfsPublications.php">www.heri.ucla.edu/tfsPublications.php</a></div>Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com0tag:blogger.com,1999:blog-7251686825560941361.post-46699673393902998262014-01-01T05:00:00.000-05:002014-01-01T05:00:00.922-05:00MAA Calculus Study: Seven Characteristics of Successful Calculus Programs<div class="MsoNormal"><span style="font-family: 'Times New Roman', serif; font-size: 11pt;"><i>By David Bressoud and Chris Rasmussen</i></span></div><div align="center" class="MsoNormal" style="text-align: center;"><br /></div><div class="MsoNormal"><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><span style="font-family: inherit;"><span style="font-size: 11pt;">In these days of tight budgets and pressure to improve retention rates for science and engineering majors, many mathematics departments want to know what works, what are the most effective means of improving the effectiveness of calculus instruction. This was the impetus behind the study of </span><a href="http://www.maa.org/cspcc"><i><span style="font-size: 11pt;">Characteristics of Successful Programs in College Calculus</span></i></a><span style="font-size: 11pt;"> undertaken by the MAA. The study consisted of a national survey in fall 2010, followed by case study visits to 17 institutions that were identified as “successful” because of their success in retention and the maintenance of “productive disposition,” defined in [</span><a href="http://www.nap.edu/catalog.php?record_id=9822"><span style="font-size: 11pt;">NRC 2001</span></a><span style="font-size: 11pt;">] as “habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy.”<o:p></o:p></span></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-size: 11pt;"><span style="font-family: inherit;">Our survey revealed that Calculus I, as taught in our colleges and universities, is extremely efficient at lowering student confidence, enjoyment of mathematics, and desire to continue in a field that requires further mathematics. The institutions we selected bucked this trend. This report draws on our experiences at all 17 colleges and universities, but focuses on the insights drawn from those universities that offer a Ph.D. in mathematics, the universities that both produce the largest numbers of science and engineering majors and that often struggle with how to balance the maintenance of high quality research with attention to undergraduate education. </span><span style="font-family: Times New Roman, serif;"><o:p></o:p></span></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-family: inherit; font-size: 11.0pt;">Case studies were conducted in the fall of 2012 at five of these universities: two large public research universities, one large private research university, one public technical institute, and one private technical institute. We shall refer to these as</span><br /><br /><ul><li><span style="font-family: inherit;"><b>ETI</b>: Eastern Technological Institute. Private university. Data from nine sections of calculus with an average enrollment of 33.</span></li><li><span style="font-family: inherit;"><b>MTI</b>: Midwestern Technological Institute. Public university. Data from seven sections with an average enrollment of 38 and one with an enrollment of 110.</span></li><li><span style="font-family: inherit;"><b>MPU</b>: Large Midwestern Public University. Data from 41 sections with an average enrollment of 27.</span></li><li><span style="font-family: inherit;"><b>WPU</b>: Large Western Public University. Data from four sections with an average enrollment of 200.</span></li><li><span style="font-family: inherit;"><b>WPR</b>: Large Western Private University. Data from three sections with an average enrollment of 196 and one section with 32 students.</span></li></ul></div><div><div class="MsoNormal"><span style="font-family: inherit; font-size: 11.0pt;">In addition to productive disposition and improved retention rates, the five also had noticeably higher grades (see Figure 1), cutting the DFW rate from 25% across all doctoral universities to only 15% at the case study sites. The difference was in B’s and C’s. The five case study universities actually gave out a slightly lower percentage of A’s than the overall average.</span></div><div class="MsoNormal"><br /></div><div class="separator" style="clear: both; text-align: center;"><b><a href="http://2.bp.blogspot.com/-wC_vlwoYMQ4/UrL1rCgPhXI/AAAAAAAAJEQ/pUna-tvMqi0/s1600/fig1.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://2.bp.blogspot.com/-wC_vlwoYMQ4/UrL1rCgPhXI/AAAAAAAAJEQ/pUna-tvMqi0/s1600/fig1.jpg" width="550" /></a></b></div><br /><div class="MsoNormal"><span style="font-family: inherit; font-size: 11.0pt;">We identified seven characteristics of the calculus programs at these five universities, characteristics that, as applicable, were also found at the other twelve:<o:p></o:p></span></div><div class="MsoNormal"><span style="font-family: inherit;"><br /></span></div><ol><li><b>Regular use of local data to guide curricular and structural modifications.</b> In his description of the MAA study of <i>Models that Work</i> [<a href="http://serc.carleton.edu/resources/2605.html">Tucker, 1995</a>], Alan Tucker wrote, “No matter how successful their current programs are, faculty members in the visited departments are not yet satisfied with the programs. Experimentation is continuous.” [<a href="http://www.ams.org/notices/199611/comm-tucker.pdf">Tucker, 1996</a>] We found that not only was this true of the successful programs we studied, these universities used the annual gathering and sharing of data on retention and grade distributions to guide this continuous experimentation. A bad semester was not dismissed as an anomaly, but was viewed as an opportunity to understand what went wrong and what could be done to avoid a similar occurrence.</li><li><b>Attention to the effectiveness of placement procedures.</b> Though this could be considered part of the first characteristic of successful programs, it received so much attention from all of the universities that we have elevated it to the level of a separate point. These universities evaluate and adjust their placement procedures on an annual basis. We also found a great deal of attention paid to those students near the cut-off, paying particular attention to programs in support of those allowed into Calculus I but most at risk and working with those who did not quite make the cut so that they were placed in programs that addressed their actual needs.</li><li><b>Coordination of instruction, including the building of communities of practice.</b> As Tucker reported in 1996, “There is a great diversity of instructional and curricular approaches, varying from one visited department to another, and even varying within a single department.” We found this, but we also found that those teaching Calculus were in regular communication with the other instructors of this class. Of course, where classes were taught by graduate teaching assistants, there was much tighter coordination of instruction. In all cases, we found that common exams were used. The simple act of creation of such exams fostered communication among those teaching the course. In some cases, communication about teaching was much more intentional, sharing innovative pedagogies, assignments, and approaches to particular aspects of the curriculum. In all cases there was also a course coordinator, a position that was not rotating but a more or less permanent position with commensurate reduction in teaching load.</li><li><b>Construction of challenging and engaging courses.</b> This is reflected in an observation that Tucker made in 1996: “Faculty members communicate explicitly and implicitly that the material studied by their students is important and that they expect their students to be successful in mathematical studies.” It also is the first example of effective educational practice in <i>Student Success in College</i> [<a href="http://nsse.iub.edu/_/?cid=185">Kuh et al, 2010, p 11</a>]: “Challenging intellectual and creative work is central to student learning and collegiate quality.” None of the successful programs we studied believed that one could improve retention by making the course easier. Instructors used textbooks and selected problems that required students to delve into concepts and to work on modeling-type problems, or even problems involving proofs. Interviews with students—most of whom had taken calculus in high school—revealed that they felt academically challenged in ways that went far beyond their high school courses. </li><li><b>Use of student-centered pedagogies and active-learning strategies.</b> This is the second example of effective educational practice in [<a href="http://nsse.iub.edu/_/?cid=185">Kuh et al, 2010</a>]: “Students learn more when they are intensely involved in their education and have opportunities to think about and apply what they are learning in different settings.” As the first author learned twenty years ago when he surveyed Calculus I students at Penn State [<a href="http://www.macalester.edu/~bressoud/pub/StudentAttitudes/StudentAttitudes.pdf">Bressoud, 1994</a>], few students know how to study or what it means to engage the mathematics, and most take a very passive role when attending a lecture. Active-learning strategies force students to engage the mathematical ideas and confront their own misconceptions. The successful programs we studied made much greater use of group projects and student presentations. </li><li><b>Effective training of graduate teaching assistants.</b> Graduate students play an important role in calculus instruction at all universities with doctoral programs, whether as teaching assistants in the breakout sections for large lectures or as the instructors of their own classes. The most successful universities have developed extensive programs for training, monitoring, and supporting these instructors. Running a successful training program is not a task that can be handed off to a single person. While there is always one coordinator, their effectiveness requires a core of faculty who are willing to participate in the graduate students’ training that takes place before the start of the fall term and to assist in visiting classes and providing feedback.</li><li><b>Proactive student support services, including the fostering of student academic and social integration. </b>This is a broad category that ranges from the building of a student-faculty community within the mathematics department to the specifics of support mechanisms for at-risk students. These are addressed in three of the effective practices identified in [<a href="http://nsse.iub.edu/_/?cid=185">Kuh et al, 2010</a>]: “Student Interactions with Faculty Members,” “Enriching Educational Experiences,” and “Supportive Campus Environment.” The first is mentioned in [<a href="http://www.ams.org/notices/199611/comm-tucker.pdf">Tucker, 1996</a>]: “Extensive student-faculty interaction characterizes both the teaching and learning of mathematics, both inside and outside of the classroom.” The universities we visited had rich programs of extra-curricular activities within the Mathematics Department. They also had a variety of responses to supporting at-risk students. These included stretching Calculus I over two terms to allow for supplemental instruction in precalculus topics, providing “fallback” courses for students who discovered after the first exam that they were in trouble in Calculus, and working with student support services to ensure that students who were struggling got the help they needed. There also were heavily utilized learning centers that attracted all students as places to gather, work on assignments, and get help as needed. Often, these were centers dedicated solely to helping students in Calculus. What was common among all of the successful calculus programs was attention to the support of all students and a willingness to monitor and adjust the programs designed to help them.</li></ol><div class="MsoNormal"><span style="font-family: inherit; font-size: 11.0pt;">There were some dramatic differences between instruction at the doctoral universities that were selected for the case study visits and instruction at all doctoral universities (see Table 1). Where the section size facilitated this—at ETI, MTI, MPU, and one section of WPR—instructors made much less use of lecture and much more use of students working together, holding discussions, and making presentations. Three of the five have almost universal use of online homework, and a fourth uses it for half of the sections. Graphing calculators were allowed on exams in two of the five universities, though use was not consistent across sections. The most striking difference between these five universities and the overall survey was the number of instructors who ask students to explain their thinking.<o:p></o:p></span></div><div class="MsoNormal"><span style="font-family: inherit;"><br /></span></div><div class="MsoNormal"><span style="font-size: 11.0pt;"><span style="font-family: inherit;">Instructors at the case study sites still consider themselves to be fairly traditional (see Figure 2), though slightly less so than the national average. They also tend to agree with the statement, “Calculus students learn best from lectures, provided they are clear and well organized” (see Figure 3). Interestingly, not a single instructor at any of the case study sites <i>strongly</i> agreed with this statement. On the other hand, the instructors at the case study sites were slightly less likely to disagree with it. They clumped heavily toward mild agreement, suggesting an attitude of keeping an open mind and a willingness to try an approach that might be more productive.</span><span style="font-family: Times New Roman, serif;"><o:p></o:p></span></span></div><br /><div style="text-align: center;"><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-_G4633WE1F8/UrMLX8_zbwI/AAAAAAAAJE8/k05LEMJOPFE/s1600/Table1.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://2.bp.blogspot.com/-_G4633WE1F8/UrMLX8_zbwI/AAAAAAAAJE8/k05LEMJOPFE/s1600/Table1.jpg" width="550" /></a></div><br /></div></div><div align="center" class="MsoNormal" style="text-align: center;"><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-G5uMWcNjLSA/UrL10D8-gJI/AAAAAAAAJEc/HSKZvUYJqUA/s1600/fig2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://1.bp.blogspot.com/-G5uMWcNjLSA/UrL10D8-gJI/AAAAAAAAJEc/HSKZvUYJqUA/s1600/fig2.jpg" width="550" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div></div><div class="MsoNormal"><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-ZHGgMsGjfO0/UrL2SJ_zWPI/AAAAAAAAJEg/jei884buLYw/s1600/fig3.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://1.bp.blogspot.com/-ZHGgMsGjfO0/UrL2SJ_zWPI/AAAAAAAAJEg/jei884buLYw/s1600/fig3.jpg" width="550" /></a></div></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><b><span style="font-family: inherit;">References<o:p></o:p></span></b></div><div class="MsoNormal"><span style="font-family: inherit;"><br /></span></div><div class="MsoNormal"><span style="font-family: inherit;"><span style="font-size: 11.0pt; mso-fareast-font-family: "Times New Roman";">Bressoud, D. 1994. Student attitudes in first semester calculus. <i>MAA Focus</i>, vol 14, pages 6–7. </span><a href="http://www.macalester.edu/~bressoud/pub/StudentAttitudes/StudentAttitudes.pdf"><span style="font-size: 11.0pt; mso-fareast-font-family: "Times New Roman";">http://www.macalester.edu/~bressoud/pub/StudentAttitudes/StudentAttitudes.pdf</span></a><span style="font-size: 11.0pt; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></span></div><div class="MsoNormal"><span style="font-family: inherit;"><br /></span></div><div class="MsoNormal"><span style="font-family: inherit; font-size: 11.0pt; mso-fareast-font-family: "Times New Roman";">Kuh, G.D., J. Kinzie, J.H. Schuh, E.J. Whitt. 2010. <i>Student Success in College: Creating Conditions that Matter</i>. Jossey-Bass. <o:p></o:p></span></div><div class="MsoNormal"><span style="font-family: inherit;"><br /></span></div><div class="MsoNormal"><span style="font-family: inherit;"><span style="font-size: 11.0pt; mso-fareast-font-family: "Times New Roman";">National Research Council. 2001. <i>Adding It Up: Helping Children Learn Mathematics</i>. Kilpatrick, Swafford, and Findell (Eds.). National Academy Press. </span><a href="http://www.nap.edu/catalog.php?record_id=9822"><span style="font-size: 11.0pt; mso-fareast-font-family: "Times New Roman";">http://www.nap.edu/catalog.php?record_id=9822</span></a><span style="font-size: 11.0pt; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></span></div><div class="MsoNormal"><span style="font-family: inherit;"><br /></span></div><div class="MsoNormal"><span style="font-family: inherit; font-size: 11.0pt; mso-fareast-font-family: "Times New Roman";">Tucker, A. 1995. <i>Models that Work: Case Studies in Effective Undergraduate Mathematics Programs</i>. <i>MAA Notes </i>#38. Mathematical Association of America. <o:p></o:p></span></div><div class="MsoNormal"><span style="font-family: inherit;"><br /></span></div><div class="MsoNormal"><span style="font-family: inherit;"><span style="font-size: 11.0pt; mso-fareast-font-family: "Times New Roman";">Tucker, A. 1996. Models that Work: Case Studies in Effective Undergraduate Mathematics Programs. <i>Notices of the AMS</i>, vol 43, pages 1356–1358. </span><a href="http://www.ams.org/notices/199611/comm-tucker.pdf"><span style="font-size: 11.0pt; mso-fareast-font-family: "Times New Roman";">http://www.ams.org/notices/199611/comm-tucker.pdf</span></a><span style="font-size: 11.0pt; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></span></div><div class="MsoNormal" style="text-align: justify; text-justify: inter-ideograph;"><!--[if gte vml 1]><v:line id="Straight_x0020_Connector_x0020_8" o:spid="_x0000_s1026" style='position:absolute; left:0;text-align:left;z-index:251659264;visibility:visible;mso-wrap-style:square; mso-wrap-distance-left:9pt;mso-wrap-distance-top:0;mso-wrap-distance-right:9pt; mso-wrap-distance-bottom:0;mso-position-horizontal:absolute; mso-position-horizontal-relative:text;mso-position-vertical:absolute; mso-position-vertical-relative:text' from="0,10.1pt" 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style="height: 17px; left: -2px; mso-ignore: vglayout; position: relative; top: 12px; width: 539px; z-index: 251659264;"><img height="5" src="file:///C:/Users/kmerow/AppData/Local/Temp/msohtmlclip1/01/clip_image007.png" v:shapes="Straight_x0020_Connector_x0020_8" width="539" /></span><!--[endif]--><span style="font-size: 11.0pt;"> </span></span></div><span style="font-family: inherit;"><br clear="ALL" /></span><span style="font-family: inherit;"><br /></span><br /><div class="MsoNormal"><span style="font-family: inherit;"><i><span style="font-size: 11.0pt;">Characteristics of Successful Programs in College Calculus</span></i><span style="font-size: 11.0pt;"> is supported by NSF #0910240. The opinions expressed in this article do not necessarily reflect those of the National Science Foundation.</span></span></div>Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com0tag:blogger.com,1999:blog-7251686825560941361.post-92006023189205769122013-12-01T01:00:00.000-05:002013-12-01T01:00:05.395-05:00MAA Calculus Study: Persistence through CalculusA successful Calculus program must do more than simply ensure that students who pass are ready for the next course. It also needs to support as many students as possible to attain this readiness. And it must encourage those students to continue on with their mathematics. As I wrote in my January 2010 column, "<span style="color: blue;"><span lang="zxx"><u><a href="http://www.maa.org/external_archive/columns/launchings/launchings_01_10.html">The Problem of Persistence</a></u></span></span>," just because a student needs further mathematics for the intended career and has done well in the last mathematics course is no guarantee that he or she will decide to continue the study of mathematics. This loss between courses is a significant contributor to the disappearance from STEM fields of at least half of the students who enter college with the intention of pursuing a degree in science, technology, engineering, or mathematics. Chris Rasmussen and Jess Ellis, drawing on data from MAA’s Calculus Study, have now shed further light on this problem. This column draws on some of the results they have gleaned from our data.<br /><div style="margin-bottom: 0in;"><br /></div><div style="margin-bottom: 0in;">For the MAA Calculus Study, students were surveyed both at the start and end of the fall term in mainstream Calculus I. A student was classified as a <i>persister</i>if she or he indicated at the start of the term an intention to continue on to Calculus II and still held that intention at the end of the term. A student was classified as a <i>switcher</i> if she or he intended at the start of the term to continue on to Calculus II, but changed his or her mind by the end of the term.</div><div style="margin-bottom: 0in;"><br /></div><div style="margin-bottom: 0in;">Not all students completed both the start and end of term surveys. While 50% of all Calculus I students received an A or B in the course, A or B students accounted for 80% of those who completed both surveys. Almost all of the remainder received a C. This implies that our data reflect what happened to the students who were doing well in the class. Of the students who started the term with the intention of taking Calculus II (74% of the students who answered both surveys), 15% turned out to be switchers. Less than 2% of all Calculus I students started with the expectation that they would not continue on to Calculus II but changed their minds by the end of the course.</div><div style="margin-bottom: 0in;"><br /></div><div style="margin-bottom: 0in;">The rates of switchers varied considerably. Women were far more likely to switch (20%) than men (11%). Those at large research universities were also more likely to switch (16%), particularly if they were taught by a graduate teaching assistant (19%). Rates varied by intended major, from a low of 6% switchers for those headed into engineering to 23% for pre-med majors and 27% for business majors taking mainstream calculus.</div><div style="margin-bottom: 0in;"><br /></div><div style="margin-bottom: 0in;">Classroom instruction had a significant effect on switcher rates (see Figure 1). "Good Teaching" reflects the collection of highly correlated observations described in this column in March 2013, "<span style="color: blue;"><span lang="zxx"><u><a href="http://launchings.blogspot.com/2013_03_01_archive.html">MAA Calculus Study: Good Teaching</a></u></span></span>." "Progressive Teaching" refers to those practices described in the following column from April, "<span style="color: blue;"><span lang="zxx"><u><a href="http://launchings.blogspot.com/2013/04/maa-calculus-study-progressive-teaching.html">MAA Calculus Study: Progressive Teaching</a></u></span></span>." Good Teaching is most important. In combination, Good and Progressive Teaching can significantly lower switcher rates.</div><div style="margin-bottom: 0in;"><br /></div><div align="CENTER" style="margin-bottom: 0in;"></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="http://4.bp.blogspot.com/-1JhS4AOBi4g/Uo6Wduf1LrI/AAAAAAAAI5M/-2ByNI4cH60/s1600/Figure1.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" src="http://4.bp.blogspot.com/-1JhS4AOBi4g/Uo6Wduf1LrI/AAAAAAAAI5M/-2ByNI4cH60/s1600/Figure1.jpg" height="243" width="400" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Figure 1.</td></tr></tbody></table><div style="margin-bottom: 0in;"><br /></div><div style="margin-bottom: 0in;">Our study offered students who had chosen to switch out a variety of reasons from which they could select any with which they agreed. Just over half reported that they had changed their major to a field that did not require Calculus II. A third of these students, as well as a third of all switchers, identified their experience in Calculus I as responsible for their decision. It also was a third of all switchers who reported that the reason for switching was that they found calculus to require too much time and effort.</div><div style="margin-bottom: 0in;"><br /></div><div style="margin-bottom: 0in;">This observation was supported by other data from our study that showed that switchers visit their instructors and tutors more often than persisters and spend more time studying calculus. As stated before, these are students who are doing well, but have decided that continuing would require more effort than they can afford.</div><div style="margin-bottom: 0in;"><br /></div><div style="margin-bottom: 0in;">I am concerned by these good students who find calculus simply too hard. As I documented in my column from May 2011, "<span style="color: blue;"><span lang="zxx"><u><a href="http://www.maa.org/external_archive/columns/launchings/launchings_05_11.html">The Calculus I Student</a></u></span></span>," these students experienced success in high school, and an overwhelming majority had studied calculus in high school. They entered college with high levels of confidence and strong motivation. Their experience of Calculus I in college has had a profound effect on both confidence and motivation. </div><div style="margin-bottom: 0in;"><br /></div><div style="margin-bottom: 0in;">The solution should not be to make college calculus easier. However, we do need to find ways of mitigating the shock that hits so many students when they transition from high school to college. We need to do a better job of preparing students for the demands of college, working on both sides of the transition to equip them with the skills they need to make effective use of their time and effort.</div><div style="margin-bottom: 0in;"><br /></div><div style="margin-bottom: 0in;">Twenty years ago, I surveyed Calculus I students at Penn State and learned that most had no idea what it means to study mathematics. Their efforts seldom extended beyond trying to match the problems at the back of the section to the templates in the book or the examples that had been explained that day. The result was that studying mathematics had been reduced to the memorization of a large body of specific and seemingly unrelated techniques for solving a vast assortment of problems. No wonder students found it so difficult. I fear that this has not changed.</div>Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com0tag:blogger.com,1999:blog-7251686825560941361.post-66708383631440575692013-11-01T01:00:00.000-04:002013-11-01T01:00:01.097-04:00An International Comparison of Adult Numeracy<div class="MsoNormal"><span style="font-family: inherit;">This past October, the Organization for Economic Cooperation and Development (OECD) released the first results from its survey of adult skills, <a href="http://www.oecd-ilibrary.org/education/oecd-skills-outlook-2013_9789264204256-en"><i>OECD Skills Outlook 2013</i></a><i> </i>[1]. It presents more evidence that the United States is lagging behind other economically developed nations in building a quantitatively literate workforce. A rich source of data, the report is unusual in its focus on the numerical skills of adults, covering ages 16 through 65, and on its parallel investigations of literacy and "problem solving in technology-rich environments." Intriguingly, its data suggest that—although their numerical skills rank near the bottom—U.S. workers consider the numerical demands of their work and their ability to handle those demands to be greater than do workers in most other developed countries.<o:p></o:p></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-family: inherit;">The OECD measured numerical proficiency at five levels:<o:p></o:p></span></div><div class="MsoListParagraphCxSpFirst" style="margin-left: .25in; mso-add-space: auto; mso-list: l0 level1 lfo1; text-indent: -.25in;"><!--[if !supportLists]--><span style="font-family: inherit;">1.<span style="font-size: 7pt;"> </span><!--[endif]-->Able to perform basic calculations in common, concrete situations.<o:p></o:p></span></div><div class="MsoListParagraphCxSpMiddle" style="margin-left: .25in; mso-add-space: auto; mso-list: l0 level1 lfo1; text-indent: -.25in;"><!--[if !supportLists]--><span style="font-family: inherit;">2.<span style="font-size: 7pt;"> </span><!--[endif]-->Can identify and act on mathematical information in a common context.<o:p></o:p></span></div><div class="MsoListParagraphCxSpMiddle" style="margin-left: .25in; mso-add-space: auto; mso-list: l0 level1 lfo1; text-indent: -.25in;"><!--[if !supportLists]--><span style="font-family: inherit;">3.<span style="font-size: 7pt;"> </span><!--[endif]-->Can identify and act on mathematical information in an unfamiliar or complex context.<o:p></o:p></span></div><div class="MsoListParagraphCxSpMiddle" style="margin-left: .25in; mso-add-space: auto; mso-list: l0 level1 lfo1; text-indent: -.25in;"><!--[if !supportLists]--><span style="font-family: inherit;">4.<span style="font-size: 7pt;"> </span><!--[endif]-->Can perform multi-step tasks and work with a broad range of mathematical information in unfamiliar or complex contexts.<o:p></o:p></span></div><div class="MsoListParagraphCxSpLast" style="margin-left: .25in; mso-add-space: auto; mso-list: l0 level1 lfo1; text-indent: -.25in;"><!--[if !supportLists]--><span style="font-family: inherit;">5.<span style="font-size: 7pt;"> </span><!--[endif]-->Can understand complex mathematical or statistical ideas and integrate multiple types of mathematical information where interpretation is required.<o:p></o:p></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-family: inherit;">As an illustration of a task at level 3 (from the <a href="http://www.oecd-ilibrary.org/education/the-survey-of-adult-skills_9789264204027-en"><i>Reader’s Companion</i></a> to the report [2, p. 30]): In 2005, the Swedish government closed its Barsebäck nuclear power plant, which was generating 3,572 GWh (Gigawatt hours) of power per year. Given that a wind power station generates about 6,000 MWh (Megawatt hours) of power per year, that 1 MWh = 1,000,000 Wh (Watt hours), and 1 GWh = 1,000,000,000 Wh, how many wind power stations would be needed to replace the Barsebäck plant?<o:p></o:p></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-family: inherit;">Now the discouraging news. Only just over a third, 34.4%, of U.S. adults were capable of solving such a problem. In many OECD countries, over half the working age population was numerate at level 3 or above, including Austria (50.8%), the Czech Republic (51.9%), Finland (57.8%), Japan (62.5%), Norway (54.8%), the Slovak Republic (53.7%), and Sweden (56.6%). Germany came in just under at 49.1%. South Korea, at 41.4%, suffered from the fact that many of its older workers, especially those over 45, have skills that are far below those of younger Koreans. Other countries in which less than 40% of the population reached level 3 include Poland (38.9%), France (37.3%), and Ireland (36.4%). Only Italy (28.9%) and Spain (28.6%) came in lower than the United States. [1, Table A2.5, p. 262]<o:p></o:p></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-family: inherit;">While the top 5% of U.S. adults are capable of working at level 4, the scores at the 95<sup>th</sup> percentile in the United States were well below those in most other OECD countries. The exceptions were France, Ireland, Italy, South Korea (again the unequal opportunity effect for older workers), Poland, and Spain. Only Finland had more than 2% of the adult population capable of working at level 5. In the United States, 0.7% of the adult population was capable of answering questions at level 5. [1, Table A2.8, p. 266]<o:p></o:p></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-family: inherit;">The OECD data also reveal that the weakness of U.S. adults is not a recent phenomenon. The report separates numeracy skill levels by age decade: 16–24, 25–34, 35–44, 45–54, and 55–65. The United States is near the bottom of every age cohort, though it stayed above Italy and Spain and managed to climb above France and Ireland for adults 45 and older and above Poland and South Korea for adults 55 and older. [1, Table A3.2 (N), p. 272]<o:p></o:p></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-family: inherit;">Given the low marks on numerical ability, it is interesting that when U.S. workers were asked whether they need to use their numeracy skills at work, the percentages were near the top of the OECD list. All of the following comparisons are for workers in the top 25% in terms of numeracy level. In the United States, 28.8% of these workers said that they need to use their numeracy skills frequently, as opposed to 28.0% in Finland, 26.7% in Germany, and only 17.7% in Japan. Only the Czech Republic at 30.0% and the Slovak Republic at 29.4% reported higher rates of frequent use of numerical skills. [1, Table A4.3, p. 303]<o:p></o:p></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-family: inherit;">In addition, U.S. workers are more inclined to consider their numeracy skills to over qualify them for the requirements of their job. In the United States, 9.4% of workers considered their numeracy skills greater than the requirements of their job. In Italy, it was 12.6%; in Spain, 15.8%. In contrast, only 7.9% of the workers in Japan and 7.0% in Finland considered their numeracy skills to be greater than the demands of their job. [1, Table A4.25, p. 358] Across the OECD countries, there is a strong negative correlation between numerical ability and the perception of how well one has mastered the numerical skills required for one’s work.<o:p></o:p></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-family: inherit;">That should be the most troubling aspect of this study.<o:p></o:p></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><!--[if gte vml 1]><v:line id="Straight_x0020_Connector_x0020_1" o:spid="_x0000_s1026" style='position:absolute;z-index:251659264;visibility:visible; mso-wrap-style:square;mso-wrap-distance-left:9pt;mso-wrap-distance-top:0; mso-wrap-distance-right:9pt;mso-wrap-distance-bottom:0; mso-position-horizontal:absolute;mso-position-horizontal-relative:text; mso-position-vertical:absolute;mso-position-vertical-relative:text' from="0,.45pt" to="414pt,.45pt" 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style="mso-ignore: vglayout; position: relative; z-index: 251659264;"><span style="height: 6px; left: -2px; position: absolute; top: -1px; width: 556px;"><img height="6" src="file:///C:/Users/kmerow/AppData/Local/Temp/msohtmlclip1/01/clip_image001.png" v:shapes="Straight_x0020_Connector_x0020_1" width="556" /></span></span><!--[endif]--><o:p> </o:p></span></div><span style="font-family: inherit;"><br clear="ALL" /> </span><br /><div class="MsoNormal"><span style="font-family: inherit;">[1] OECD (2013), <i>OECD Skills Outlook 2013: First Results from the Survey of Adult Skills</i>, OECD Publishing. <a href="http://dx.doi.org/10.1787/9789264204256-en">http://dx.doi.org/10.1787/9789264204256-en</a><o:p></o:p></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-family: inherit;">[2] OECD (2013), <i>The Survey of Adult Skills: Reader’s Companion</i>, OECD Publishing.<o:p></o:p></span></div><div class="MsoNormal"><span style="font-family: inherit;"><a href="http://dx.doi.org/10.1787/9789264204027-en">http://dx.doi.org/10.1787/9789264204027-en</a><o:p></o:p></span></div><div class="MsoNormal"><br /></div><br /><div class="MsoNormal"><span style="font-family: inherit;">[3] The OECD countries in the survey were Australia, Austria, Czech Republic, Denmark, Estonia, Finland, France, Germany, Ireland, Italy, Japan, Korea, Netherlands, Norway, Poland, Slovak Republic, Spain, Sweden, United States, and three subnational entities: Flanders (Belgium), England (UK), and Northern Ireland (UK). Some data are also presented for Cyprus and the Russian Federation.</span><o:p></o:p></div>Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com0tag:blogger.com,1999:blog-7251686825560941361.post-81978821403202545332013-10-01T02:00:00.000-04:002013-10-01T08:25:42.355-04:00Evidence of Improved Teaching<div class="MsoNormal"><span style="font-family: inherit;">Last December I discussed the NRC report, <a href="http://www.nap.edu/catalog.php?record_id=13362"><i>Discipline-Based Education Research: Understanding and Improving Learning in Undergraduate Science and Engineering</i></a>. One of its themes is the importance of the adoption of “evidence-based teaching strategies.” It is hard to find carefully collected quantitative evidence that certain instructional strategies for undergraduate mathematics really are better. I was pleased to see two articles over the past month that present such evidence for active learning strategies.<o:p></o:p></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-family: inherit;">One of the articles is the long-anticipated piece by Jerry Epstein, "<a href="http://www.ams.org/notices/201308/rnoti-p1018.pdf">The Calculus Concept Inventory—Measurement of the Effect of Teaching Methodology in Mathematics</a>" which appeared in the September 2013 <i>Notices of the AMS</i> [1]. Because this article is so readily available to all mathematicians, I will not say much about it. Epstein’s Calculus Concept Inventory (CCI) represents a notable advancement in our ability to assess the effectiveness of different pedagogical approaches to basic calculus instruction. He presents strong evidence for the benefits of Interactive Engagement (IE) over more traditional approaches. As with the older Force Concept Inventory developed by Hestenes <i>et al.</i> [2], CCI has a great deal of surface validity. It measures the kinds of understandings we implicitly assume our students pick up in studying the first semester of calculus, and it clarifies how little basic conceptual understanding is absorbed under traditional pedagogical approaches. Epstein claims statistically significant improvements in conceptual understanding from the use of Interactive Engagement, stronger gains than those seen from other types of interventions including plugging the best instructors into a traditional lecture format. Because CCI is so easily implemented and scored, it should spur greater study of what is most effective in improving undergraduate learning of calculus.<o:p></o:p></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-family: inherit;">The second paper is "<a href="http://link.springer.com/article/10.1007/s10755-013-9269-9">Assessing Long-Term Effects of Inquiry-Based Learning: A Case Study from College Mathematics</a>" by Marina Kogan and Sandra Laursen [3]. This was a carefully controlled study of the effects of Inquiry-Based Learning (IBL) on persistence in mathematics courses and performance in subsequent courses. They were able to compare IBL and non-IBL sections taught at the same universities during the same terms. <o:p></o:p></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-family: inherit;">IE and IBL describe comparable pedagogical approaches. Richard Hake defined IE as<o:p></o:p></span></div><div class="MsoNormal" style="margin-left: .5in;"><span style="font-family: inherit;">“… those [methods] designed at least in part to promote conceptual understanding through interactive engagement of students in heads-on (always) and hands-on (usually) activities which yield immediate feedback through discussion with peers and/or instructors.” [4]<o:p></o:p></span></div><div class="MsoNormal"><span style="font-family: inherit;">IBL does this and also is expected to incorporate a structured curriculum that builds toward the big ideas, a component that may or may not be present in IE. For the Kogan and Laursen study, IBL was a label that the universities chose to apply to certain sections. The trained observers in the Kogan and Laursen study found significant differences between IBL and non-IBL sections. They rated IBL sections “higher for creating a supportive classroom atmosphere, eliciting student intellectual input, and providing feedback to students on their work” than non-IBL sections. IBL sections spent an average of 60% of the time on student-centered activities; in non-IBL sections the instructor talked at least 85% of the time.<o:p></o:p></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-family: inherit;">Kogan and Laursen compared IBL and non-IBL sections for three courses:</span></div><div class="MsoNormal"></div><ul><li><span style="font-family: inherit; text-indent: -0.25in;">G1, the first term of a three-term sequence covering multivariable calculus, linear algebra, and differential equations, taken either in the freshman or sophomore year;</span></li><li><span style="font-family: inherit; text-indent: -0.25in;">L1, a sophomore/junior-level introduction to proof course; and</span></li><li><span style="font-family: inherit; text-indent: -0.25in;">L2, an advanced junior/senior-level mathematics course with an emphasis on proofs.</span></li></ul><br /><div class="MsoNormal"><span style="font-family: inherit;">For L1 and L2, students did not know in advance whether they were enrolling in IBL or non-IBL sections. The IBL section of G1 was labeled as such. In all cases, the authors took care to control for discrepancies in student preparation and ability.<o:p></o:p></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-family: inherit;">IBL had the least impact on the students in the advanced course, L2. IBL students had slightly higher grades in subsequent mathematics courses (2.6 for non-IBL, 2.8 for IBL) and took slightly fewer subsequent mathematics courses (1.5 for non-IBL, 1.4 for IBL).<o:p></o:p></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-family: inherit;">For the introduction to proof course, L1, IBL students again had slightly higher grades in the following term (2.8 for non-IBL, 3.0 for IBL). There were statistically significant gains (<i>p</i> < 0.05) from IBL in the number of subsequent courses that students took and that were required for a mathematics major, both for the overall population (0.5 for non-IBL, 0.6 for IBL) and, especially, for women (0.6 for non-IBL, 0.8 for IBL).<o:p></o:p></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-family: inherit;">For L1, the sample size was large enough (1077 non-IBL, 204 IBL over seven years) to investigate persistence and subsequent performance broken down by student overall GPA, recorded as low (< 2.5), medium (2.5 to 3.4), or high (> 3.4). For the non-IBL students, differences in overall GPA were reflected in dramatic differences in their grades in subsequent mathematics courses required for the major, all statistically significant at <i>p</i> < 0.001. Low GPA students averaged 1.96, medium GPA students averaged 2.58, and high GPA students averaged 3.36. All three categories of IBL students performed better in subsequent required courses, but the greatest improvement was seen with the weakest students. Taking this course as IBL wiped out much of the difference between low GPA students and medium GPA students. It also decreased the difference between medium and high GPA students in subsequent required courses. For IBL students, low GPA students averaged 2.43, medium GPA students averaged 2.75, and high GPA students averaged 3.38 in subsequent required courses. See Figure 1.</span></div><div class="MsoNormal"><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="http://4.bp.blogspot.com/-_pLOLEPE3aY/UkBIQN7PKyI/AAAAAAAAIQ4/L1kMSDaWFyI/s1600/fig1.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><span style="font-family: inherit;"><img border="0" src="http://4.bp.blogspot.com/-_pLOLEPE3aY/UkBIQN7PKyI/AAAAAAAAIQ4/L1kMSDaWFyI/s1600/fig1.jpg" height="245" width="400" /></span></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><i><span style="font-family: inherit; font-size: small;">Figure 1: Average grade in subsequent courses required for the major following introduction to proof class taught either as non-IBL or IBL.</span></i></td></tr></tbody></table></div><div class="MsoNormal"><span style="font-family: inherit;">While the number of subsequent courses satisfying the requirements for a mathematics major was higher for all students taking the IBL section of L1, here the greatest gain was among those with the highest GPA. For low GPA students, the number of courses was 0.50 for non-IBL and 0.51 for IBL; for medium GPA the number was 0.53 for non-IBL, 0.62 for IBL; and for high GPA the number was 0.49 for non-IBL, 0.65 for IBL. See Figure 2.</span></div><div class="MsoNormal"><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="http://2.bp.blogspot.com/--ymiQ44BX5I/UkBIi2_C2VI/AAAAAAAAIRA/56HmCO_pPvA/s1600/fig2.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><span style="font-family: inherit;"><img border="0" src="http://2.bp.blogspot.com/--ymiQ44BX5I/UkBIi2_C2VI/AAAAAAAAIRA/56HmCO_pPvA/s1600/fig2.jpg" height="245" width="400" /></span></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><i><span style="font-family: inherit; font-size: small;">Figure 2: Average number of subsequent courses taken and required for the major following introduction to proof class taught either as non-IBL or IBL.</span></i></td></tr></tbody></table></div><div class="MsoNormal"><span style="font-family: inherit;">For the first course in the sophomore sequence, G1, IBL did have a statistically significant effect on grades in the next course in the sequence (<i>p</i> < 0.05). The average grade in the second course was 3.0 for non-IBL students, 3.4 for IBL students. There also was a modest gain in the number of subsequent mathematics courses that students took and that were required for the students’ majors: 1.96 courses for non-IBL students, 2.09 for IBL students.<o:p></o:p></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-family: inherit;">These have been the highlights of the Kogan and Laursen paper. Most striking is the very clear evidence that IBL does no harm, despite the fact that spending more time on interactive activities inevitably cuts into the amount of material that can be “covered.” In fact, it was the course with the densest required syllabus, G1, where IBL showed the clearest gains in terms of preparation of students for the next course. <o:p></o:p></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-family: inherit;">IBL is often viewed as a luxury in which we might indulge our best students. 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</span><br /><div class="MsoNormal"><span style="font-family: inherit;">[1] J. Epstein. 2013. The Calculus Concept Inventory—Measurement of the Effect of Teaching Methodology in Mathematics. <i>Notices of the AMS</i> <b>60</b> (8), 1018–1026. <a href="http://www.ams.org/notices/201308/rnoti-p1018.pdf">http://www.ams.org/notices/201308/rnoti-p1018.pdf</a><cite><span style="font-style: normal; mso-bidi-font-style: italic; mso-fareast-font-family: "Times New Roman";"><o:p></o:p></span></cite></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-family: inherit;">[2] D. Hestenes, M. Wells, and G. Swackhamer. 1992. Force concept inventory. <i>Physics Teacher</i> <b>30</b>, 141–158. <a href="http://modelinginstruction.org/wp-content/uploads/2012/08/FCI-TPT.pdf">http://modelinginstruction.org/wp-content/uploads/2012/08/FCI-TPT.pdf</a><o:p></o:p></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-family: inherit;">[3] M. Kogan and S. Laursen. 2013. Assessing Long-Term Effects of Inquiry-Based Learning: A Case Study from College Mathematics. <i>Innovative Higher Education</i> <b>39 </b>(3). <a href="http://link.springer.com/article/10.1007/s10755-013-9269-9" target="_blank">http://link.springer.com/article/10.1007/s10755-013-9269-9</a><o:p></o:p></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-family: inherit;">[4] R.R. Hake. 1998. Interactive engagement versus traditional methods: A six-thousand student survey of mechanics test data for physics courses. <i>American J. Physics</i> <b>66</b>(1), 64–74. </span><a href="http://www.physics.indiana.edu/~sdi/ajpv3i.pdf">http://www.physics.indiana.edu/~sdi/ajpv3i.pdf</a></div>Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com0tag:blogger.com,1999:blog-7251686825560941361.post-68837736905548761142013-09-01T01:00:00.000-04:002013-09-03T08:12:55.496-04:00JPBM Presentation to PCAST<span style="font-family: inherit;">On July 18, 2013, I had the pleasure of being part of a presentation from the Joint Policy Board for Mathematics (JPBM, the umbrella organization for AMS, ASA, MAA, and SIAM) to a joint meeting of the <span style="color: blue;"><span lang="zxx"><u><a href="http://www.whitehouse.gov/administration/eop/ostp/pcast">President’s Council of Advisors for Science and Technology</a></u></span></span>(PCAST) and the British Prime Minister’s <span style="color: blue;"><span lang="zxx"><u><a href="http://www.bis.gov.uk/cst">Council for Science and Technology</a></u></span></span>. The title of the presentation was <i>Mathematics Education: Toward 2025</i>, and the focus was on the recent NRC report, <span style="color: blue;"><span lang="zxx"><u><a href="http://www.nap.edu/catalog.php?record_id=15269"><i>The Mathematical Sciences in 2025</i></a></u></span></span>(see my column on this report from <span style="color: blue;"><span lang="zxx"><u><a href="http://launchings.blogspot.com/2013/02/mathematics-in-2025.html">February 1, 2013</a></u></span></span>). I was one of four presenters. The others were <span style="color: blue;"><span lang="zxx"><u><a href="http://www8.nationalacademies.org/cp/committeeview.aspx?key=49237">Mark Green</a></u></span></span>, vice-chair of the committee that produced the report; <span style="color: blue;"><span lang="zxx"><u><a href="http://www-bcf.usc.edu/~ericmf/">Eric Friedlander</a></u></span></span>, Past-President of AMS; and <span style="color: blue;"><span lang="zxx"><u><a href="http://www.statslab.cam.ac.uk/~frank/">Frank Kelly</a></u></span></span>, Chair of the British <span style="color: blue;"><span lang="zxx"><u><a href="http://www.cms.ac.uk/">Council for the Mathematical Sciences</a></u></span></span> (the British equivalent of JPBM). A webcast of the presentations and copies of the slides are available on the <span style="color: blue;"><span lang="zxx"><u><a href="http://www.whitehouse.gov/administration/eop/ostp/pcast/meetings/past">White House PCAST website</a></u></span></span>.</span><br /><div style="margin-bottom: 0in;"><span style="font-family: inherit;"><br /></span></div><div style="margin-bottom: 0in;"><span style="font-family: inherit;">The impetus for JPBM’s request to make this presentation was PCAST’s <span style="color: blue;"><span lang="zxx"><u><a href="http://www.whitehouse.gov/sites/default/files/microsites/ostp/pcast-engage-to-excel-final_2-25-12.pdf"><i>Engage to Excel</i></a></u></span></span>report (see my column from <span style="color: blue;"><span lang="zxx"><u><a href="http://launchings.blogspot.com/2012/03/on-engaging-to-excel.html">March 1, 2012</a></u></span></span>). While there is much in this report with which the mathematical community disagrees, especially the implication that mathematicians are not engaged in trying to improve undergraduate education, it was quickly decided that a positive message would be most productive. We told the Council that we appreciate the attention they have drawn to undergraduate mathematics education, we assured them that our community is actively seeking ways to improve the teaching and learning of post-secondary mathematics, and we offered to work with PCAST as we move forward.</span></div><div style="margin-bottom: 0in;"><span style="font-family: inherit;"><br /></span></div><div style="margin-bottom: 0in;"><span style="font-family: inherit;">There was a great deal of preparation in the months leading up to the presentation. It would be impossible to overstate the importance of <span style="color: blue;"><span lang="zxx"><u><a href="http://umdphysics.umd.edu/people/faculty/135-gates.html">Jim Gates</a></u></span></span>’ role in making this happen. He has been a strong friend of the mathematical community, helping to ensure that our voice is heard. It was through his efforts that the July meeting was made possible. I also must emphasize the role that <span style="color: blue;"><span lang="zxx"><u><a href="http://www.terpconnect.umd.edu/~lvrmr/">David Levermore</a></u></span></span>played in helping to refine our message and coordinate the preparation of our presentations. I had hoped and expected that he would be included in those making the presentation to PCAST. Unfortunately, he was cut from the list of proposed speakers. </span></div><div style="margin-bottom: 0in;"><span style="font-family: inherit;"><br /></span></div><div style="margin-bottom: 0in;"><span style="font-family: inherit;">Leaders of all four mathematical societies helped to develop our position statement, which was distributed to PCAST in advance of the meeting and is available on the web as <i style="background-color: white;"><a href="http://www.maa.org/sites/default/files/pdf/MathReport2PCAST.pdf">Meeting the Challenges of Improved Post-Secondary Education in the Mathematical Sciences</a></i><i>.</i>It includes a substantial appendix describing many of the activities of the JPBM societies that are directed toward the improvement of undergraduate mathematics education, the provision of evidence of what works, and the encouragement of widespread adoption of approaches to teaching and learning that are known to improve student outcomes. Following is the one-page opening statement from this document, written by Eric Friedlander, David Levermore, and myself and created with extensive feedback from and ultimate endorsement by the leadership of all four societies.</span></div><div style="margin-bottom: 0in;"><br /></div><div align="CENTER" style="margin-bottom: 0in;"><span style="font-family: Times New Roman, serif;"><b>MEETING THE CHALLENGES OF IMPROVED POST-SECONDARY</b></span></div><div align="CENTER" style="margin-bottom: 0in;"><span style="font-family: Times New Roman, serif;"><b>EDUCATION IN THE MATHEMATICAL SCIENCES</b></span></div><div align="CENTER" style="margin-bottom: 0in;"><span style="font-family: Times New Roman, serif;"><span style="font-size: x-small;">DAVID M. BRESSOUD, ERIC M. FRIEDLANDER, C. DAVID LEVERMORE</span></span></div><div style="margin-bottom: 0in;"><br /></div><span style="font-family: inherit;">The mathematical sciences play a foundational and crosscutting role in enabling substantial advances across a broad array of fields: medicine, engineering, technology, biology, chemistry, computer science, social sciences, and others. Due to this foundational role, the delivery of excellent post-secondary mathematics education is essential to the present and future well being of our nation and its citizens.</span><br /><span style="font-family: inherit;"><br /></span><span style="font-family: inherit;">We greatly appreciate the engagement of PCAST in the challenges of post-secondary mathematics education. A key finding of the 2012 PCAST <i>Engage to Excel</i> report is that mathematics education is a critical component of all undergraduate STEM degrees. We share this perspective of mathematics education as an enabler of STEM careers, provider of broad mathematics literacy, and shaper of the next generation of leaders in our increasingly technological, data-driven, and scientific society.</span><br /><span style="font-family: inherit;"><br /></span><span style="font-family: inherit;">The report also found that current deficiencies in mathematics learning are partly driving the loss of STEM majors in the early college years. We acknowledge many of the shortcomings highlighted by the report. The wake-up call delivered by PCAST has sharpened the awareness of the mathematical sciences community of the need for intensive, broad-scale efforts to address these problems. We emphasize that efforts by a great many in the mathematical sciences community predated PCAST's report, that progress is being made, and that plans are in place to broaden these to a community-wide effort. </span><br /><span style="font-family: inherit;"><br /></span><span style="font-family: inherit;">Our task is to encourage and help lead constructive actions that will address the difficult and varied challenges facing post-secondary education in the mathematical sciences. How should mathematics educators improve developmental education in order to enable students to aspire to STEM careers? How should mathematical scientists in colleges and universities augment their cooperative efforts with “partner disciplines” to best serve the needs of students needing basic university mathematics? How should mathematical sciences departments reshape their curricula to suit the needs of a well-educated workforce in the 21st century? How can technology be best used to serve educational needs? </span><br /><span style="font-family: inherit; font-size: small;"><br /></span><span style="font-family: inherit;"><span style="font-size: small;">These questions must be answered in the context of a changing landscape. </span><span style="font-size: small;">There are growing disparities in the preparation of incoming students. </span><span style="font-size: small;">A third of all undergraduate mathematics students are enrolled in precollege level mathematics. At the other extreme, almost 700,000 high school students in the US completed a course of calculus this past year. The mathematical sciences themselves are changing as the needs of big data and the challenges of modeling complex systems reveal the limits of traditional curricula.</span></span><br /><span style="font-family: inherit; font-size: small;"><br /></span> <br /><span style="font-family: inherit;"><span style="font-size: small;">The NRC report </span><span style="font-size: small;"><i>The Mathematical Sciences in 2025</i></span><span style="font-size: small;">eloquently describes the opportunities and challenges of this shifting landscape. </span><span style="font-size: small;">This report should serve as a springboard for initiatives in mathematics education that more closely intertwine the learning of mathematics with the appreciation of its applications. However, t</span><span style="font-size: small;">he mathematical community alone cannot bring about the scale of changes called for in </span><span style="font-size: small;"><i>Engage to Excel</i></span><span style="font-size: small;">. Building on all the activities in mathematics education underway or that have arisen as a result of the PCAST report, we ask for PCAST’s help in promoting greater awareness, collaboration, and cooperation among all of the scientific disciplines who are working to prepare the STEM workforce of the future.</span></span>Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com1tag:blogger.com,1999:blog-7251686825560941361.post-5950243490371015312013-08-01T06:00:00.000-04:002013-08-01T15:01:36.655-04:00MAA Calculus Study: Effects of Calculus in High School<div class="MsoNormal"><span style="font-family: inherit;">This month I return to the MAA Calculus Study, <a href="http://www.maa.org/programs/faculty-and-departments/curriculum-development-resources/characteristics-of-successful-programs-in-college-calculus"><i>Characteristicsof Successful Programs in College Calculus</i></a><i>, </i>with a report on what we learned about the effects of taking calculus in high school. Because this study only looked at students in Calculus I, we can say nothing about how many of these students never take another calculus class or how many start their college mathematics with Calculus II or higher. Since the survey was conducted in the fall term, we cannot even say anything about students who might postpone taking Calculus I until later in the academic year. But, thanks to this survey, we <i>can</i> say a lot about the students who study calculus in high school and then begin with Calculus I in their first term at college. <o:p></o:p></span></div><div class="MsoNormal"><span style="font-family: inherit;"><br /></span></div><div class="MsoNormal"><span style="font-family: inherit;">First of all, we have a good idea of how many students this involves. Of the 300,000 who enrolled in Calculus I in fall 2010, just over half had studied calculus in high school. That represents just about one quarter of the 600,000 or so students who had studied calculus in high school the previous year. Of those who had taken the AP Calculus exam that spring, just over a quarter were in Calculus I that fall. From the latest CBMS report [1], about 55,000 incoming freshman arrived with credit for Calculus I. This leaves almost two-thirds of those who studied calculus in high school neither acquiring college credit for their high school work nor enrolling in Calculus I in a fall term.<o:p></o:p></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-family: inherit;">We can break these data down further by score on the AP exam:<o:p></o:p></span></div><div class="MsoNormal"></div><ul><li><span style="background-color: white; color: #222222; font-family: inherit;">Half of those who scored a 1 or 2 on the BC exam or a 3 on the AB exam enrolled in Calculus I in the fall.</span></li><li><span style="background-color: white; color: #222222; font-family: inherit;">A third of those who scored a 3 on the BC exam enrolled in Calculus I.</span></li><li><span style="background-color: white; color: #222222; font-family: inherit;">A quarter of those who scored 1, 2, or 4 on the AB exam enrolled in Calculus I.</span></li><li><span style="background-color: white; color: #222222; font-family: inherit;">1 in 8 of those who scored a 5 on AB or a 4 on BC enrolled in Calculus I.</span></li><li><span style="background-color: white; color: #222222; font-family: inherit;">1 in 20 of those who scored a 5 on BC enrolled in Calculus I.</span></li><li><span style="background-color: white; color: #222222; font-family: inherit;">A quarter of those who scored a 3 or higher on the AB or BC exam received college credit for Calculus I.</span></li></ul><div><div class="MsoNormal"><span style="font-family: inherit;"><span style="text-indent: -0.25in;"><span style="color: #222222;">Of </span></span><span style="text-indent: -0.25in;">those who arrived with credit for Calculus I, we do not know how many used it to place into a higher calculus class and how many never studied any further calculus in college.</span><!--[if !supportLists]--></span></div><div class="MsoNormal"><span style="font-family: inherit;"><br /></span></div><div class="MsoNormal"><span style="font-family: inherit;">At the research universities, those characterized by offering a doctorate in mathematics and dominated by the flagship state universities, over 70% of the students in Calculus I had completed a course of calculus in high school. We have data for about 5,000 of these students and so can report fairly accurately on the effect of studying calculus in high school on student performance in Calculus I in college. As shown in Figure 1, less than a third studied no calculus in high school, less than a third studied calculus but did not take the AP Calculus exam, a third took the AB exam, and a small but significant percentage (8%) took the BC exam.</span><span style="font-family: Times New Roman, serif;"><o:p></o:p></span></div><div class="MsoNormal"><span style="font-family: "Times New Roman","serif";"> <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="http://4.bp.blogspot.com/-4FZ5nkdca84/UcHvde_DJsI/AAAAAAAAHac/qAgCChbOHB4/s1600/figure1.jpeg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" src="http://4.bp.blogspot.com/-4FZ5nkdca84/UcHvde_DJsI/AAAAAAAAHac/qAgCChbOHB4/s1600/figure1.jpeg" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><span style="font-family: 'Times New Roman', serif; font-size: small;"><i>Figure 1: Distribution of high school calculus experience among Calculus I students at research universities.</i></span></td></tr></tbody></table></span></div><div class="MsoNormal" style="text-align: left;"><span style="font-family: inherit;">There is a common perception among students that having studied calculus in high school gives students a significant advantage in Calculus I when they get to college. Our data tend to confirm that. Figure 2 compares the final grades of the total student population at research universities with three subsets: those who did not study calculus in high school, those who did and did not take the AP Calculus exam, and those who did and did take the AP Calculus exam. We see a much lower percentage of A’s and a much higher percentage of DFW’s (grades of D or F or a withdrawal from the course) among students who did not study calculus in high school as opposed to those who did. There is little difference in Calculus I grades between those who did and those did not take the AP Calculus exam.</span><br /><span style="font-family: inherit;"><br /></span></div><div class="MsoNormal"><span style="font-family: "Times New Roman","serif";"><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="http://2.bp.blogspot.com/-4YAhtln1kgI/UcHvq062QdI/AAAAAAAAHak/gUrsidoLgrA/s1600/figure2.jpeg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" src="http://2.bp.blogspot.com/-4YAhtln1kgI/UcHvq062QdI/AAAAAAAAHak/gUrsidoLgrA/s1600/figure2.jpeg" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><span style="font-family: 'Times New Roman', serif; font-size: small;"><i>Figure 2: Final grades of Calculus I students in research universities by experience with high school calculus.</i></span><span style="font-family: 'Times New Roman', serif;"> </span></td></tr></tbody></table></span></div><div class="MsoNormal"><span style="font-family: inherit;">If we separate the grades of those who took the AP Calculus exam by their performance on this exam, we see that it makes a large difference in their final grade (Figure 3). Students who score a 1 or 2 on the AP exam are comparable to students who did not study calculus in high school, although there is a slightly lower probability of receiving a D or F or withdrawing from the course. Students with a 3 on the AB exam are comparable to the average student who took calculus in high school and did not take the AP Calculus exam, but with a somewhat higher probability of earning a B. Not surprisingly, students who earned a 4 or higher on the AB exam or a 3 or higher on the BC exam did very well in Calculus I: 45% received an A, over two thirds at least a B. It is interesting that even among these students, roughly a quarter received a D or F or withdrew from the course. In fact, the rate of DFW is remarkably consistent across all levels of preparation, suggesting that the decision to stop working or to withdraw from the course is one aspect of course performance that has little to do with high school preparation.</span><br /><span style="font-family: inherit;"><br /></span></div><div class="MsoNormal"><span style="font-family: "Times New Roman","serif";"><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="http://3.bp.blogspot.com/-FQmly-cAfsA/UcHwa1FlLAI/AAAAAAAAHas/8_tu-mGtR0c/s1600/figure3.jpeg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" src="http://3.bp.blogspot.com/-FQmly-cAfsA/UcHwa1FlLAI/AAAAAAAAHas/8_tu-mGtR0c/s1600/figure3.jpeg" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><span style="font-family: 'Times New Roman', serif; font-size: small;"><i>Figure 3: Final grades of Calculus I students in research universities by performance on AP Calculus exam.</i></span></td></tr></tbody></table></span></div><div class="MsoNormal"><br /></div><br /><div class="MsoNormal"><i><span style="background: white; color: #222222; mso-bidi-font-family: "Times New Roman"; mso-fareast-font-family: "Times New Roman";">The MAA national study of calculus, Characteristics of Successful Programs in College Calculus, is funded by NSF grant no. 0910240. The opinions expressed in this column do not necessarily reflect those of the National Science Foundation.</span></i><span style="font-family: "Times New Roman","serif";"><o:p></o:p></span></div></div>Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com0tag:blogger.com,1999:blog-7251686825560941361.post-33324096614131146732013-07-01T06:00:00.000-04:002013-07-01T06:00:03.962-04:00Measuring Teacher Quality<div class="MsoNormal"><span style="font-family: inherit;">A perennial issue at every college and university is how to measure teacher quality. It is important because it directly influences decisions about retention, tenure, and promotion. Everyone complains about basing such decisions on end-of-course evaluations. This column will explore a recent study by Scott Carrell and James West [1], undertaken at the United States Air Force Academy (USAFA), that strongly suggests that such evaluations are even less useful than commonly believed and that the greatest long-term learning does not come from those instructors who receive the strongest evaluations at the end of the class.<o:p></o:p></span></div><div class="MsoNormal"><span style="font-family: inherit;"><br /></span></div><div class="MsoNormal"><span style="font-family: inherit;">The study authors chose to measure teacher effectiveness in Calculus I by examining value added in both Calculus I and Calculus II: comparing student performance on course assessments for each instructor after controlling for variables in student preparation and background that included academic background, SAT verbal and math scores, sex, and race and ethnicity. It is generally acknowledged that better teachers produce better results in their students, but this has only been extensively studied in elementary students, and even there it is not without its problems. The authors reference a 2010 study by Rothstein [2] that shows a strong positive correlation between the quality of fifth grade teachers and student performance on assessments taken in fourth grade, suggesting a significant selection bias: The best students seek out the best teachers. This may be even truer at the university level where students have much more control over who they take a class with. For this reason, Carrell and West were very careful to measure the comparability of the classes. At USAFA, everyone takes Calculus I, and there is little personal choice in which section to take, so such selection bias is less likely to occur. The authors also tested for and found no evidence of backward correlation, that the best Calculus II instructors were correlated with higher grades in Calculus I.<o:p></o:p></span></div><div class="MsoNormal"><span style="font-family: inherit;"><br /></span></div><div class="MsoNormal"><span style="font-family: inherit;">The authors had a large sample size with which to work, all of the students who took Calculus I from fall 2000 through spring 2007, over 10,000 students and 91 instructors. The faculty make-up at USAFA is unusual among post-secondary institutions. There is a small core of permanent faculty. Only 15% of Calculus I instructors held the rank of Associate or Full Professor, and only 31% held a doctorate in mathematics or a mathematical science. Most of the teaching is done by officers who hold a master’s degree and are doing a rotation through USAFA. The average number of years of teaching experience among all Calculus I instructors was less than four years. Because of this, there is tight control on these courses, which facilitates a careful statistical study. There are common syllabi and examinations. All instructors get to see the examinations before they are given so that there is opportunity, if an instructor so wishes, to “teach to the test,” emphasizing those parts of the curriculum that are known to be important for the assessment.<o:p></o:p></span></div><div class="MsoNormal"><span style="font-family: inherit;"><br /></span></div><div class="MsoNormal"><span style="font-family: inherit;">Positive responses to the following prompts all had positive influence on student performance in Calculus I, significant at the 0.05 level:<o:p></o:p></span></div><div class="MsoListParagraphCxSpFirst" style="mso-list: l0 level1 lfo1; text-indent: -.25in;"><br /><ol><li><span style="background-color: white; color: #222222;"><span style="font-family: inherit;">Instructor’s ability to provide clear, well-organized instruction.</span></span></li><li><span style="background-color: white; color: #222222;"><span style="font-family: inherit;">Value of questions and problems raised by instructor.</span></span></li><li><span style="background-color: white; color: #222222;"><span style="font-family: inherit;"> Instructor’s knowledge of course material.</span></span></li><li><span style="background-color: white; color: #222222;"><span style="font-family: inherit;">The course as a whole.</span></span></li><li><span style="background-color: white; color: #222222;"><span style="font-family: inherit;"> Amount you learned in the course.</span></span></li><li><span style="background-color: white; color: #222222;"><span style="font-family: inherit;">The instructor’s effectiveness in facilitating my learning in the course.</span></span></li></ol><span style="font-family: inherit;"><span style="background-color: white; color: #222222;">Surprisingly, these evaluations of the Calculus I instructor all had negative impact on student performance in Calculus II, with responses 1, 2, and 6 significant at the 0.05 level.</span><br /><br />On the other hand, faculty rank, highest degree, and years of teaching experience were negatively correlated with examination performance in Calculus I, but positively correlated with performance in Calculus II, with statistical significance for years of teaching experience for both the negative impact in Calculus I and the positive impact in Calculus II.</span></div><div class="MsoNormal"><span style="font-family: inherit;"><br /></span></div><div class="MsoNormal"><span style="font-family: inherit;">The suggested implication is that less experienced instructors tend to focus on the particular skills and abilities needed to succeed in the next assessment and that students like that approach. Experienced instructors may pay more attention to the foundational knowledge that will serve the student in subsequent courses, and students appear to be less immediately appreciative of what these instructors are able to bring to the class.<o:p></o:p></span></div><div class="MsoNormal"><span style="font-family: inherit;"><br /></span></div><div class="MsoNormal"><span style="font-family: inherit;">This study strongly suggests that end of course student evaluations are, at best, an incomplete measure of an instructor’s effectiveness. It also suggests a long-term weakness of simply preparing students for their next assessment, though it should be emphasized that this represents merely a guess as to why less experienced instructors appear to get better performance from their students in Calculus I assessments.<o:p></o:p></span></div><div class="MsoNormal"><span style="font-family: inherit;"><br /></span></div><div class="MsoNormal"><span style="font-family: inherit;">At Macalester College, we recognize the importance of student reflection on what they learned months or years earlier. When a promotion or tenure case comes to the Personnel Committee, we collect online student evaluations of that faculty member from all of the students who have taken a course with him or her over roughly the past five years, combining both recent and current appraisals with longer term assessments of the effect that instructor has had.<o:p></o:p></span></div><div class="MsoNormal"><span style="font-family: inherit;"><br /></span></div><div class="MsoNormal"><span style="font-family: inherit;">References:<o:p></o:p></span></div><div class="MsoNormal"><span style="font-family: inherit;"><br /></span></div><div class="MsoNormal"><span style="font-family: inherit;">[1] Scott E. Carrell & James E. West, 2010. "Does Professor Quality Matter? Evidence from Random Assignment of Students to Professors," Journal of Political Economy, University of Chicago Press, vol. 118(3), pages 409-432, 06. Available at <a href="http://www.nber.org/papers/w14081">http://www.nber.org/papers/w14081</a><o:p></o:p></span></div><div class="MsoNormal"><span style="font-family: inherit;"><br /></span></div><span style="font-family: inherit;"><br /></span><div class="MsoNormal"><span style="font-family: inherit;">[2] Jesse Rothstein, 2010. “Teacher Quality in Educational Production: Tracking, Decay, and Student Achievement.” Quarterly Journal of Economics 125 (1): 175–214. Available at <a href="http://gsppi.berkeley.edu/faculty/jrothstein/published/rothstein_vam_may152009.pdf">http://gsppi.berkeley.edu/faculty/jrothstein/published/rothstein_vam_may152009.pdf</a></span><cite><span style="font-family: "Cambria","serif"; font-style: normal; mso-ascii-theme-font: minor-latin; mso-bidi-font-style: italic; mso-fareast-font-family: "Times New Roman"; mso-hansi-theme-font: minor-latin;"><o:p></o:p></span></cite></div>Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com0tag:blogger.com,1999:blog-7251686825560941361.post-75456236556913120842013-06-01T06:00:00.000-04:002013-06-01T10:06:03.230-04:00Who Needs Algebra II?<div class="MsoNormal"></div><div class="MsoNormal"><a href="https://www.blogger.com/blogger.g?blogID=7251686825560941361" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"></a><span style="font-family: 'Times New Roman', serif;">In May this year, the National Center on Education and the Economy (NCEE) released its report, "</span><span style="font-family: "Times New Roman","serif";"><a href="http://www.ncee.org/college-and-work-ready/">What Does It Really Mean to Be College and Work Ready?</a>" </span><span style="font-family: 'Times New Roman', serif;">[1]. The report is in two parts: Mathematics and English Literacy. It is based on a national study of the proficiencies required and actually used for the most popular associate’s degree programs at two-year colleges. Just a few weeks earlier, Jordan Weissman published a piece in <i>The Atlantic</i>, "</span><span style="font-family: "Times New Roman","serif";"><a href="http://www.theatlantic.com/business/archive/2013/04/heres-how-little-math-americans-actually-use-at-work/275260/">Here’s How Little Math Americans Actually Use at Work</a>"</span><span style="font-family: 'Times New Roman', serif;">[2]. That was based on a 2010 report written by Michael Handel at Northeastern University, "</span><span style="font-family: "Times New Roman","serif";"><a href="http://www.northeastern.edu/socant/?page_id=366">What Do People Do at Work? A Profile of U.S. Jobs from the Survey of Workplace Skills, Technology, and Management Practices</a>"</span><span style="font-family: 'Times New Roman', serif;"> [3]. This large-scale survey includes an assessment of what mathematics is actually used in the workplace.</span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-family: "Times New Roman","serif";">I hope that it will come as a surprise to no one that not everyone actually uses the contents of Algebra II in their work or that College Algebra taught in two-year colleges is essentially high school Algebra II. Weissman highlights Handel’s data that less than a quarter of all workers use any mathematics that is more advanced than fractions, ratios, and percentages. He raises the question whether requiring Algebra II for high school graduation is placing an unnecessary roadblock in the way of too many students. NCEE poses more nuanced questions. Why are so many students being hurried through the critical early mathematics that they will need to be work and college ready, especially fractions, ratios, and percentages, just so that they can get to Algebra II? Isn’t there a better way to prepare them for what they will need?<o:p></o:p></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-family: "Times New Roman","serif";">I will begin with the data. Handel divided the workforce into five categories:</span><br /><br /><ul><li><span style="font-family: 'Times New Roman', serif; text-indent: -0.25in;">Upper White Collar (management, professional, technical occupations)</span></li><li><span style="font-family: 'Times New Roman', serif; text-indent: -0.25in;">Lower White Collar (clerical, sales)</span></li><li><span style="font-family: Symbol; text-indent: -0.25in;"><span style="font-family: 'Times New Roman';"> </span></span><span style="font-family: 'Times New Roman', serif; text-indent: -0.25in;">Upper Blue Collar (craft and repair workers, construction trades, mechanics)</span></li><li><span style="font-family: 'Times New Roman', serif; text-indent: -0.25in;">Lower Blue Collar (factory workers, truck drivers)</span></li><li><span style="font-family: 'Times New Roman', serif; text-indent: -0.25in;">Service (food service, home health care, child care, janitors)</span></li></ul></div><div class="MsoNormal"><span style="font-family: "Times New Roman","serif";">Generally, the best paying and most desirable jobs are Upper White Collar (UWC) and Upper Blue Collar (UBC). We should be equipping our students so that they can aspire to such jobs. It’s still not true that everyone needs Algebra II, but 35% of UWC workers reported using basic algebra, geometry, and/or statistics in their work. This level of mathematics is even more important for UBC workers, with 41% reporting using mathematics at the level of basic algebra, geometry, and/or statistics. This is still not Algebra II (which Handel lists as “complex Algebra” as opposed to “basic Algebra”), which was reported being used by 14% of UWC workers and 16% of UBC workers. Much less is it Calculus, which was reported by 8% of both UWC and UBC workers. But about 40% of those working in UWC or UBC jobs need a working knowledge of some high school mathematics, a higher bar than simply having passed the relevant courses. It is interesting to observe that UBC workers are more likely to use mathematics than UWC workers. <o:p></o:p></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-family: "Times New Roman","serif";">The NCEE report looked at the mathematics required for the nine most popular associate’s degree programs at two-year colleges: Accounting, Automotive Technology, Biotech/Electrical Technology, Business, Computer Programming, Criminal Justice, Early Childhood Education, Information Technology, and Nursing, as well as the General Track. This ties nicely to the Handel study because the nine are generally seen as preparation for UWC or UBC careers. NCEE selected seven two-year colleges in seven states and examined the texts, assignments, and exams in the introductory courses for these disciplines as well as for the mathematics courses required for these fields. There were three notable insights:<o:p></o:p></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><b><span style="font-family: "Times New Roman","serif";">First</span></b><span style="font-family: "Times New Roman","serif";">, except for some work on geometric visualization, NCEE found no content in either College Algebra or Statistics, two college-credit bearing courses, that goes beyond the high school curriculum described in the Common Core State Standards in Mathematics (CCSS-M). They found that College Algebra had a large component of middle school topics, especially CCSS-M for grades 6–8 in Expressions and Equations, Functions, Number Systems, Geometry, and Ratios and Proportions. Statistics was a mix of CCSS-M middle and high school level statistics, with a significant component of grades 6–8 Ratios and Proportions and Expressions and Equations. <o:p></o:p></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><b><span style="font-family: "Times New Roman","serif";">Second,</span></b><span style="font-family: "Times New Roman","serif";"> the introductory textbooks in the disciplinary fields used nothing beyond Algebra I. Ratios and proportions are important as well as interpreting quantitative relationship expressed in tables, graphs, and formulae, but, as the report says,</span></div><div class="MsoNormal" style="margin-left: .5in;"><blockquote class="tr_bq"><span style="font-family: "Times New Roman","serif";">When mathematics is present in the texts, equations are not solved, quadratics are absent, and functions are present but not named or analyzed, just treated as formulae. […] Students do not have to perform algebraic manipulations nor construct graphs or tables. […] The area of high school content with the highest representation in the texts, Number Systems, is found in six percent of the text chapters. [p. 16]</span></blockquote></div><div class="MsoNormal"><b><span style="font-family: "Times New Roman","serif";">Third,</span></b><span style="font-family: "Times New Roman","serif";"> the mathematical knowledge that was tested in these introductory courses in the disciplinary fields was far lower than what was in the textbooks. Not only was there nothing requiring Algebra II on the exams, the NCEE team could find nothing, or almost nothing, that reflected knowledge of Algebra I. Furthermore, the questions that were asked on examinations were of low difficulty. The NCEE team used the PISA (Program for International Student Assessment) Item-Difficulty Coding Framework with four levels. Examples of what is expected at each level include<o:p></o:p></span></div><div class="MsoListParagraphCxSpFirst" style="mso-list: l1 level1 lfo2; text-indent: -.25in;"><br /><ul><li><span style="font-family: 'Times New Roman', serif;">Level 0: perform simple calculations and make direct inferences;</span></li><li><span style="font-family: 'Times New Roman', serif;">Level 1: use simple functional relationships and formal mathematical symbols, interpret models;</span></li><li><span style="font-family: 'Times New Roman', serif;">Level 2: use multiple relationships, manipulate mathematical symbols, modify existing models; and</span></li><li><span style="font-family: Symbol; mso-bidi-font-family: Symbol; mso-fareast-font-family: Symbol;"><span style="font-family: 'Times New Roman';"> </span></span><span style="font-family: "Times New Roman","serif";">Level 3: solve multi-step application of formal procedures, evaluate arguments, create models.</span></li></ul></div><div class="MsoListParagraphCxSpLast" style="margin-left: 0in; mso-add-space: auto;"><span style="font-family: "Times New Roman","serif";">The team found that over 60% of the mathematical questions on the examinations given in introductory courses were at Level 0. Few rose to Level 2, much less Level 3. (This was not the case in College Algebra and Statistics where most of the examination items were at Level 1 or 2 and some attained Level 3. This suggests that even though the material of College Algebra and Statistics does not go beyond topics covered in CCSS-M, the level of expected proficiency may be higher than what is typically encountered in high school.)<o:p></o:p></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-family: "Times New Roman","serif";">NCEE did find three mathematical topics required for the introductory courses that are not covered in CCSS-M nor in the College Algebra or Statistics classes: complex applications of measurements, schematic diagrams (2-D schematics of 3-D objects and flow charts), and geometric visualization. They also found a much greater demand for knowledge of statistics, probability, and modeling (“how to frame a real-world problem in mathematical terms”) than is commonly taught in most mainstream high school mathematics programs today.<o:p></o:p></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-family: "Times New Roman","serif";">What makes the NCEE report even more depressing is that it restricted its attention to college-credit bearing courses. Most of the mathematics taught at two-year colleges is below the level of College Algebra (see Figure 1). The mathematical requirements for UWC and UBC jobs may not be high, but we do not seem to be doing a very good job of preparing students even for what they will need.<o:p></o:p></span></div><div class="MsoNormal"><br /></div><div align="center" class="MsoNormal" style="text-align: center;"><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="http://1.bp.blogspot.com/-a7M18h3r6nM/UaTLnLINZ2I/AAAAAAAAHWw/6UE5eSW5OrU/s1600/Figure1+(3).tiff" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" src="http://1.bp.blogspot.com/-a7M18h3r6nM/UaTLnLINZ2I/AAAAAAAAHWw/6UE5eSW5OrU/s1600/Figure1+(3).tiff" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><div align="center" class="MsoNormal"><span style="font-family: "Times New Roman","serif"; font-size: small;">Figure 1. <i>Fall term Mathematics course enrollments (thousands). “Introductory” includes College Algebra, Trigonometry, and Precalculus.</i><o:p></o:p></span></div><div align="center" class="MsoNormal"><div class="separator" style="clear: both;"><span style="font-size: small;"><a href="https://www.blogger.com/blogger.g?blogID=7251686825560941361" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"></a></span></div><span style="font-family: "Times New Roman","serif"; font-size: small;">Source: CBMS.</span></div></td></tr></tbody></table></div><div align="center" class="MsoNormal" style="text-align: center;"><br /></div><div class="MsoNormal"><span style="font-family: "Times New Roman","serif";">All of this raises serious questions about whether Algebra II should be expected of all graduating high school students. This parallels the situation that has been my primary concern: Should Calculus be expected of all graduating high school students who are going directly into a four-year undergraduate program, especially those who may need to take Calculus in college? I would far prefer a student who can operate at PISA Level 3 in Algebra I over a student who cannot handle problems above Level 1 in Algebra II. I would prefer Level 3 in Precalculus over Level 1 in Calculus. When students are short-changed in their mathematical preparation simply so that Algebra II or Calculus appears on the high school transcript, with little regard to what that actually means, then neither they nor society as a whole are well served.<o:p></o:p></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-family: "Times New Roman","serif";">It also raises questions about what mathematics should be required for an associate’s degree. College Algebra constitutes a significant hurdle for most two-year college students. Should there be alternatives? In this case, I believe that most two-year college students would be better served with a program that combines demanding use of the topics of Algebra I with a college-level introduction to Statistics.<o:p></o:p></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-family: "Times New Roman","serif";">We are not at the point where we can demand Algebra II for high school graduation. To do so would either create unacceptable rates of high school failure or force us to change what we mean by “understanding Algebra II.” But I worry that if we simply lower our sights and decide that, since few of our students actually will use anything from Algebra II once they have graduated, it should not be expected for graduation, then that will actually weaken the preparation that occurs in the earlier grades. Elementary and middle school mathematics should be laying the foundation for a student to succeed in Algebra II. If we want our students to have a strong working knowledge of the high school mathematics that <i>is</i> needed for 40% of the UWC and UBC jobs, then we want them to have the mathematical preparation that would enable them to succeed in Algebra II. <o:p></o:p></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-family: "Times New Roman","serif";">References:<o:p></o:p></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-family: "Times New Roman","serif";">[1] National Center on Education and the Economy. 2013. What does it really mean to be college and work ready? The mathematics requirements of first year community college students. Washington, DC. Available at <a href="http://www.ncee.org/college-and-work-ready/">http://www.ncee.org/college-and-work-ready/</a><o:p></o:p></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-family: "Times New Roman","serif";">[2] Jordan Weissman. April 24, 2013. Here’s how little Math Americans actually use at work. <i>The Atlantic</i>. Available at </span><a href="http://www.theatlantic.com/business/archive/2013/04/heres-how-little-math-americans-actually-use-at-work/275260/"><span style="font-family: "Times New Roman","serif";">http://www.theatlantic.com/business/archive/2013/04/heres-how-little-math-americans-actually-use-at-work/275260/</span></a><span style="font-family: "Times New Roman","serif";"><o:p></o:p></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-family: "Times New Roman","serif";">[3] Michael Handel. 2010. What do people do at work? A Profile of U.S. jobs from the Survey of Workplace Skills, Technology, and Management Practices. OECD (forthcoming). Available at <a href="http://www.northeastern.edu/socant/?page_id=366">http://www.northeastern.edu/socant/?page_id=366</a></span></div><div class="MsoNormal"><br /></div>Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com3tag:blogger.com,1999:blog-7251686825560941361.post-35840961580805713882013-05-01T06:00:00.000-04:002013-06-04T08:49:33.815-04:00MAA Calculus Study: Graphing Calculators and CAS<br /><span style="font-family: Times New Roman, serif;">This column continues my report on results of the MAA National Study of Calculus I, </span><span style="color: blue;"><span lang="zxx"><u><a href="http://www.maa.org/cspcc/"><span style="font-family: Times New Roman, serif;"><i>Characteristics of Successful Programs in College Calculus</i></span></a></u></span></span><span style="font-family: Times New Roman, serif;"><i>. </i></span><span style="font-family: Times New Roman, serif;">This month I am sharing what we learned about the use of graphing calculators (with or without computer algebra systems) and computer software such as </span><span style="font-family: Times New Roman, serif;"><i>Maple</i></span><span style="font-family: Times New Roman, serif;">or</span><span style="font-family: Times New Roman, serif;"><i> Mathematica</i></span><span style="font-family: Times New Roman, serif;">. Our results draw on three of the surveys:</span><br /><ul><li><div style="margin-bottom: 0in;"><span style="font-family: Times New Roman, serif;">Student survey at start of term: We asked students how calculators and/or computer algebra systems (CAS) were used in their last high school mathematics class and how comfortable they are in using these technologies.</span></div></li><li><div style="margin-bottom: 0in;"><span style="font-family: Times New Roman, serif;">Student survey at end of term: We asked students how calculators or CAS had been used both in class and for out of class assignments.</span></div></li><li><div style="margin-bottom: 0in;"><span style="font-family: Times New Roman, serif;">Instructor survey at start of term: We asked instructors what technologies would be allowed on examinations and which would be required on examinations.</span></div></li></ul><div style="margin-bottom: 0in;"><span style="font-family: Times New Roman, serif;">Our first question asked students how calculators were used on exams in their last high school mathematics class (see Figure 1). As in previous columns, “research” refers to the responses of students taking Calculus I at research universities (highest degree in mathematics is doctorate), “undergrad” refers to undergraduate colleges (highest degree is bachelor’s), “masters” to masters universities (highest degree is masters), and “two-year” to two-year colleges (highest degree is associate’s).</span></div><div style="margin-bottom: 0in;"><span style="font-family: Times New Roman, serif;"><br /></span></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="http://3.bp.blogspot.com/-6GYHjA0ZmDs/UX1-V7kVyHI/AAAAAAAAHGY/xxlG6CTLd7E/s1600/figure1.tiff" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" src="http://3.bp.blogspot.com/-6GYHjA0ZmDs/UX1-V7kVyHI/AAAAAAAAHGY/xxlG6CTLd7E/s1600/figure1.tiff" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"> <i> </i><span style="font-size: small;"><span style="font-family: Times New Roman, serif; font-style: italic;">Figure 1. GC = graphing calculator. CAS = graphing calculator with computer algebra system capabilities (</span><span style="font-family: Times New Roman, serif;">e.g</span><span style="font-family: Times New Roman, serif;">.<i>TI-89 or TI-92).</i></span></span></td></tr></tbody></table><div style="margin-bottom: 0in;"><span style="font-family: 'Times New Roman', serif;">There are several interesting observations to be made from this graph. First, not surprisingly, almost all Calculus I students reported having used graphing calculators on their exams at least some of the time (“always” and “sometimes” were mutually exclusive options). Second, there is a difference by type of institution. Students at undergraduate colleges were most likely to have used graphing calculators on high school exams (94%), then those at research universities (91%), then masters universities (86%), and finally two-year colleges (77%). The differences are small but statistically significant. My best guess is that these are reflections of the economic background of these students. A second observation is that for most students, access to a graphing calculator was not always allowed. However, it is still common practice in high schools (roughly one-third of all students) to always allow students to use graphing calculators on mathematics exams.</span></div><div style="margin-bottom: 0in;"></div><div style="margin-bottom: 0in;"><br /></div><div style="margin-bottom: 0in;"><span style="font-family: Times New Roman, serif;">Another striking observation from Figure 1 is that the percentage of students who were always allowed to use graphing calculators on exams is almost identical to the percentage of students who were always allowed to use graphing calculators with CAS capabilities on exams. For all categories of students, over half of them were allowed to use graphing calculators with CAS capabilities at least some of the time, which suggests that over half of the students in college Calculus I own or have had access to such calculators.</span></div><div style="margin-bottom: 0in;"><br /></div><div style="margin-bottom: 0in;"><span style="font-family: Times New Roman, serif;">The next graph (Figure 2) shows how students at the start of the term reported their comfort level with using graphing calculators or computer algebra systems (</span><span style="font-family: Times New Roman, serif;"><i>Maple</i></span><span style="font-family: Times New Roman, serif;">and </span><span style="font-family: Times New Roman, serif;"><i>Mathematica</i></span><span style="font-family: Times New Roman, serif;">were provided as examples of what we meant). The most interesting feature of this graph is that students at two-year colleges are much more likely to be comfortable with </span><span style="font-family: Times New Roman, serif;"><i>Maple</i></span><span style="font-family: Times New Roman, serif;">or </span><span style="font-family: Times New Roman, serif;"><i>Mathematica</i></span><span style="font-family: Times New Roman, serif;">than those at four-year programs. I suspect that the reason behind this is that most Calculus I students at two-year colleges are sophomores who took pre-calculus at that college the year before. This gave them more opportunity to experience these computer algebra systems.</span></div><div style="margin-bottom: 0in;"><span style="font-family: Times New Roman, serif;"><br /></span></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="http://4.bp.blogspot.com/-Hw-tG8bmOJE/UX1-eeWULyI/AAAAAAAAHGg/OAxSKMRvBu0/s1600/figure2.tiff" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" src="http://4.bp.blogspot.com/-Hw-tG8bmOJE/UX1-eeWULyI/AAAAAAAAHGg/OAxSKMRvBu0/s1600/figure2.tiff" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><span style="font-size: small;"><span style="font-family: Times New Roman, serif; font-style: italic;">Figure 2. Student attitude toward use of graphing calculator or CAS on a computer such as </span><span style="font-family: Times New Roman, serif;">Maple</span><span style="font-family: Times New Roman, serif; font-style: italic;">or </span><span style="font-family: Times New Roman, serif;">Mathematica</span><span style="font-family: Times New Roman, serif; font-style: italic;">.</span></span></td></tr></tbody></table><div style="margin-bottom: 0in;"><span style="font-family: Times New Roman, serif;">The graphs in Figures 3–5 show what students reported at the end of the term about use of technology. For the graph in Figure 3, students were asked how frequently each of these occurred in class. Percentage shows the fraction of students who responded “about half the class sessions,” “most class sessions,” or “every class session.” We note large differences in instructor use of technology generally (for this question, “technology” was not defined), and especially sharp differences for instructor use of graphing calculators or CAS (with </span><span style="font-family: Times New Roman, serif;"><i>Maple </i>and<i>Mathematica</i></span><span style="font-family: Times New Roman, serif;"> given as examples). It is interesting that students are most likely to encounter computer algebra systems in undergraduate and two-year colleges, much less likely in masters and research universities.</span></div><div style="margin-bottom: 0in;"></div><div style="margin-bottom: 0in;"><span style="font-family: Times New Roman, serif;"><br /></span></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="http://4.bp.blogspot.com/-1kh37Ccbw-w/UX1-kuD9swI/AAAAAAAAHGo/cVDOv8gWKQ8/s1600/figure3.tiff" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" src="http://4.bp.blogspot.com/-1kh37Ccbw-w/UX1-kuD9swI/AAAAAAAAHGo/cVDOv8gWKQ8/s1600/figure3.tiff" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><span style="font-size: small;"><i> </i><span style="font-family: Times New Roman, serif; font-style: italic;">Figure 3. End of term student reports on frequency of use of technology (at least once/month). For this question, CAS refers to a computer algebra system on a computer, such as </span><span style="font-family: Times New Roman, serif;">Maple</span><span style="font-family: Times New Roman, serif; font-style: italic;">or </span><span style="font-family: Times New Roman, serif;">Mathematica</span><span style="font-family: Times New Roman, serif; font-style: italic;">.</span></span></td></tr></tbody></table><div style="margin-bottom: 0in;"><span style="font-family: 'Times New Roman', serif;">The first two sets of bars in Figure 4 show student responses to “Does your calculator find the symbolic derivative of a function?” The first set gives the percentage responding “N/A, I do not use a calculator.” The second set displays the percentage responding “yes.” Looking at the complement of these two responses, we see that across all types of institutions, roughly 50% of students taking Calculus I own a graphing calculator without CAS capabilities. The third set records the percentage responding “yes” to the question, “Were you allowed to use a graphing calculator during your exams?” Note that there are some discrepancies between what students and instructors report about allowing graphing calculators on exams (Figures 4 and 6), but the basic pattern that graphing calculators are allowed far less frequently at research universities than at other types of institutions is consistently demonstrated.</span></div><div style="margin-bottom: 0in;"></div><div style="margin-bottom: 0in;"><span style="font-family: Times New Roman, serif;"><br /></span></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="http://1.bp.blogspot.com/-oh0RM5h1mcU/UX1-poT5hkI/AAAAAAAAHGw/u4cjzFBk7WY/s1600/figure4.tiff" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" src="http://1.bp.blogspot.com/-oh0RM5h1mcU/UX1-poT5hkI/AAAAAAAAHGw/u4cjzFBk7WY/s1600/figure4.tiff" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><i><span style="font-size: x-small;"> </span><span style="font-family: 'Times New Roman', serif;"><span style="font-size: small;">Figure 4. End of term student reports on calculator use. No calculator = do not use a calculator. Calculator with CAS = use a calculator with CAS capabilities. Calc allowed on exams = graphing calculators were allowed on exams.</span></span></i></td></tr></tbody></table><div style="margin-bottom: 0in;"><span style="font-family: 'Times New Roman', serif;">We also asked how often “The assignments completed outside of class time required that I use technology to understand ideas.” Again, we see much less use of technology at research universities, the greatest use at undergraduate and two-year colleges.</span></div><div style="margin-bottom: 0in;"><span style="font-family: Times New Roman, serif;"><br /></span></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="http://4.bp.blogspot.com/-qjz741EoovI/UX1-029kUNI/AAAAAAAAHG4/rOIbf4eUTLQ/s1600/figure5.tiff" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" src="http://4.bp.blogspot.com/-qjz741EoovI/UX1-029kUNI/AAAAAAAAHG4/rOIbf4eUTLQ/s1600/figure5.tiff" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><i><span style="font-size: x-small;"> </span><span style="font-family: 'Times New Roman', serif;"><span style="font-size: small;">Figure 5. Frequency with which technology (either graphing calculators or computers) was used for out of class assignments. Almost never = less than once per month (includes never). Sometimes = at least once per month but less than once per week. Often = at least once per week.</span></span></i></td></tr></tbody></table><div style="margin-bottom: 0in;"><span style="font-family: 'Times New Roman', serif;">The last two graphs (Figures 6 and 7) are taken from the instructor responses at the start of the term: what technology they would allow on their exams and what technology they would require on their exams. Again, we see a clear indication that technology, especially the use of graphing calculators without CAS capabilities, is much less common at research universities than other types of institutions.</span></div><div style="margin-bottom: 0in;"></div><div style="margin-bottom: 0in;"><br /></div><div style="margin-bottom: 0in;"><span style="font-family: Times New Roman, serif;">It is interesting to observe that there are large numbers of instructors who allow but do not require technology on the exams. At research universities, 26% require the use of some kind of technology, and a further 25% allow but do not require the use of some sort of technology. For undergraduate colleges, 38% of instructors require technology, an additional 42% allow it. At masters universities, 42% require, and a further 33% allow. At two-year colleges, 52% require, and an additional 36% allow.</span></div><div style="margin-bottom: 0in;"><span style="font-family: Times New Roman, serif;"><br /></span></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="http://2.bp.blogspot.com/-ZvyVfeU8f3g/UX1--rf9r6I/AAAAAAAAHHE/rRWRliTwvCg/s1600/figure6.tiff" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" src="http://2.bp.blogspot.com/-ZvyVfeU8f3g/UX1--rf9r6I/AAAAAAAAHHE/rRWRliTwvCg/s1600/figure6.tiff" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><i><span style="font-size: x-small;"> </span><span style="font-family: 'Times New Roman', serif;"><span style="font-size: small;">Figure 6. Start of term report by instructor of intended use of technology on exams. GC = graphing calculator. Most of those who checked “other” reported that they allowed graphing calculators on some but not all parts of the exam. Some reported allowing only scientific calculators.</span></span></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="http://3.bp.blogspot.com/-8bxho8dcABs/UX1--g82TfI/AAAAAAAAHHA/D3sS4oBPTA0/s1600/figure7.tiff" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" src="http://3.bp.blogspot.com/-8bxho8dcABs/UX1--g82TfI/AAAAAAAAHHA/D3sS4oBPTA0/s1600/figure7.tiff" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><span style="font-family: 'Times New Roman', serif;"><span style="font-size: small;"><i>Figure 7. Start of term report by instructor of intended use of technology on exams. GC = graphing calculator. Most of those who checked “other” reported that they required graphing calculators on some but not all parts of the exam. Some reported requiring only scientific calculators.</i></span></span></td></tr></tbody></table><div style="margin-bottom: 0in;"><span style="font-family: 'Times New Roman', serif;">We see a pattern of very heavy use of graphing calculators in high schools, driven, no doubt, by the fact that students are expected to use them for certain sections of the Advanced Placement Calculus exams. They are still the dominant technology at colleges and universities, but there the use is as likely to be voluntary as required. This implies that in many colleges and universities questions and assignments are posed in such a way that graphing calculators confer little or no advantage. The use of graphing calculators at the post-secondary level varies tremendously by type of institution. Yet even at the research universities, over half the instructors allow the use of graphing calculators for at least some portions of their exams. </span><br /><span style="font-family: 'Times New Roman', serif;"><br /></span><i style="background-color: white; color: #222222;">The MAA national study of calculus, Characteristics of Successful Programs in College Calculus, is funded by NSF grant no. 0910240. The opinions expressed in this column do not necessarily reflect those of the National Science Foundation.</i></div><br />Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com0tag:blogger.com,1999:blog-7251686825560941361.post-62433726465962587912013-04-01T04:00:00.000-04:002013-06-04T08:53:14.837-04:00MAA Calculus Study: Progressive Teaching<br /><div class="MsoNormal">Last month (<a href="http://launchings.blogspot.com/2013/03/maa-calculus-study-good-teaching.html" target="_blank">MAA Calculus Study: Good Teaching</a>) I discussed the student-described attributes of instructors that were highly correlated with improvements in student confidence, enjoyment of mathematics, and desire to continue to study mathematics. This month I will discuss a second set of instructor attributes that we are labeling "Progressive Teaching" because they are generally associated with approaches to teaching and learning that focus on active engagement of the students. <o:p></o:p></div><div class="MsoNormal"><br /></div><div class="MsoNormal">Here the evidence for improved results is less clear. In particular, Sadler and Sonnert discovered a strong interaction with the attributes we are calling "Good Teaching": teachers who rated high on Good Teaching improved student outcomes if they also rated high on Progressive Teaching. But if they rated low on Good Teaching, then a high rating on Progressive Teaching had a strongly negative effect on student confidence. This might have been expected. Good Teaching describes student-teacher interactions, including the degree to which students feel encouraged to participate in class and supported by the instructor. It is not surprising that students who are encountering unfamiliar approaches to classroom learning react negatively if they believe that that the instructor is not encouraging or supportive. <o:p></o:p></div><div class="MsoNormal"><br /></div><div class="MsoNormal">We also have evidence of some consistently positive effects from Progressive Teaching. Even with a low score on Good Teaching, Progressive Teaching was seen to be helpful in convincing students to continue the study of mathematics. Our conclusions are that:<o:p></o:p></div><div class="MsoNormal"><br /></div><div class="MsoListParagraphCxSpFirst" style="margin-left: .25in; mso-add-space: auto; mso-list: l0 level1 lfo2; text-indent: -.25in;"><!--[if !supportLists]-->a.<span style="font-size: 7pt;"> </span><!--[endif]-->Good Teaching and Progressive Teaching are independent clusters of student perceptions of instructor behaviors, <o:p></o:p></div><div class="MsoListParagraphCxSpMiddle" style="margin-left: .25in; mso-add-space: auto; mso-list: l0 level1 lfo2; text-indent: -.25in;"><!--[if !supportLists]-->b.<span style="font-size: 7pt;"> </span><!--[endif]-->Good Teaching is more important to student persistence than Progressive Teaching, <o:p></o:p></div><div class="MsoListParagraphCxSpMiddle" style="margin-left: .25in; mso-add-space: auto; mso-list: l0 level1 lfo2; text-indent: -.25in;"><!--[if !supportLists]-->c.<span style="font-size: 7pt;"> </span>both can serve to improve student outcomes, and <o:p></o:p></div><div class="MsoListParagraphCxSpLast" style="margin-left: .25in; mso-add-space: auto; mso-list: l0 level1 lfo2; text-indent: -.25in;"><!--[if !supportLists]-->d.<span style="font-size: 7pt;"> </span>teaching is most effective when instructors rate high on both measures.<o:p></o:p></div><div class="MsoNormal"><br /></div><div class="MsoNormal">There were 12 student responses that clustered into what we are calling Progressive Teaching:<o:p></o:p></div><div class="MsoNormal"><br />My calculus instructor frequently<br /><span style="text-align: center; text-indent: -0.25in;">1.</span><span style="font-size: 7pt; text-align: center; text-indent: -0.25in;"> </span><span style="text-align: center; text-indent: -0.25in;">Assigned sections of the textbook to read before coming to class.</span></div><div class="MsoNormal"><span style="text-indent: -0.25in;">2.</span><span style="font-size: 7pt; text-indent: -0.25in;"> </span><span style="text-indent: -0.25in;">Had students work with one another.</span><br /><span style="text-indent: -0.25in;">3.</span><span style="font-size: 7pt; text-indent: -0.25in;"> </span><span style="text-indent: -0.25in;">Had students give presentations.</span><br /><span style="text-indent: -0.25in;">4.</span><span style="font-size: 7pt; text-indent: -0.25in;"> </span><span style="text-indent: -0.25in;">Asked students to explain their thinking in class.</span><br /><span style="text-indent: -0.25in;">5.</span><span style="font-size: 7pt; text-indent: -0.25in;"> </span><span style="text-indent: -0.25in;">Required students to explain their thinking on homework assignments.</span><br /><span style="text-indent: -0.25in;">6.</span><span style="font-size: 7pt; text-indent: -0.25in;"> </span><span style="text-indent: -0.25in;">Required students to explain their thinking on exams.</span><br /><span style="text-indent: -0.25in;">7.</span><span style="font-size: 7pt; text-indent: -0.25in;"> </span><span style="text-indent: -0.25in;">Held whole class discussions.</span></div><div class="MsoNormal"><span style="text-indent: -0.25in;"><br /></span></div><div class="MsoNormal">My calculus instructor did <i>not</i> frequently<br /><span style="text-indent: -0.25in;">8.</span><span style="font-size: 7pt; text-indent: -0.25in;"> </span><span style="text-indent: -0.25in;">Lecture.</span></div><div class="MsoNormal"></div><div style="text-indent: -24px;"><br /></div>Assignments completed outside of class<br /><span style="text-indent: -0.25in;">9.</span><span style="font-size: 7pt; text-indent: -0.25in;"> </span><span style="text-indent: -0.25in;">Required that I solve word problems.</span><br /><span style="text-indent: -0.25in;">10.</span><span style="font-size: 7pt; text-indent: -0.25in;"> </span><span style="text-indent: -0.25in;">Were problems unlike those done in class or in the book.</span><br /><span style="text-indent: -0.25in;">11.</span><span style="font-size: 7pt; text-indent: -0.25in;"> </span><span style="text-indent: -0.25in;">Were often submitted as a group project.</span><br /><span style="text-indent: -0.25in;">12.</span><span style="font-size: 7pt; text-indent: -0.25in;"> </span><span style="text-indent: -0.25in;">Were returned with helpful feedback and comments.</span><o:p></o:p><br /><div class="MsoListParagraphCxSpFirst" style="mso-list: l1 level1 lfo1; text-indent: -.25in;"><o:p></o:p></div><div class="MsoListParagraphCxSpMiddle" style="mso-list: l1 level1 lfo1; text-indent: -.25in;"><o:p></o:p></div><div class="MsoListParagraphCxSpMiddle" style="mso-list: l1 level1 lfo1; text-indent: -.25in;"><o:p></o:p></div><div class="MsoListParagraphCxSpMiddle" style="mso-list: l1 level1 lfo1; text-indent: -.25in;"><o:p></o:p></div><div class="MsoListParagraphCxSpMiddle" style="mso-list: l1 level1 lfo1; text-indent: -.25in;"><o:p></o:p></div><div class="MsoListParagraphCxSpMiddle" style="mso-list: l1 level1 lfo1; text-indent: -.25in;"><o:p></o:p></div><div class="MsoListParagraphCxSpLast" style="mso-list: l1 level1 lfo1; text-indent: -.25in;"><o:p></o:p></div><div class="MsoNormal"><o:p></o:p></div><div class="MsoListParagraph" style="mso-list: l1 level1 lfo1; text-indent: -.25in;"><o:p></o:p></div><div class="MsoNormal"><o:p></o:p></div><div class="MsoListParagraphCxSpFirst" style="mso-list: l1 level1 lfo1; text-indent: -.25in;"><o:p></o:p></div><div class="MsoListParagraphCxSpMiddle" style="mso-list: l1 level1 lfo1; text-indent: -.25in;"><o:p></o:p></div><div class="MsoListParagraphCxSpMiddle" style="mso-list: l1 level1 lfo1; text-indent: -.25in;"><o:p></o:p></div><div class="MsoListParagraphCxSpLast" style="mso-list: l1 level1 lfo1; text-indent: -.25in;"><o:p></o:p></div><div class="MsoNormal"><br /></div><div class="MsoNormal">With one exception, the following graphs show the percentage of students who reported that their instructors employed each of these practices often or very often (a 5 or 6 on a Likert scale from 1 = not at all to 6 = very often). The exception is practice #8. Here we record the percentage of students who responded 1, 2, or 3 on the same scale to the question, "During class time, how frequently did your instructor lecture?". <o:p></o:p></div><div class="MsoNormal"><br /></div><div class="MsoNormal">We see that for most of the instructor behaviors (practices 1 through 8), the undergraduate colleges and two-year colleges are where these are most likely to be employed. The relatively large percentage of instructors at masters universities who had students give presentations in class (13% as opposed to 6% at all other types of institutions) is still small and may be an artifact of the relatively small number of responses from students at masters universities (305 students at 18 institutions). The research universities are where we find the most challenging problems being posed on assignments, either word problems or those unlike those done in class or in the book. Instructors at two-year colleges provide the most helpful feedback on assignments, instructors at research universities the least helpful feedback.<o:p></o:p><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="http://2.bp.blogspot.com/-Br5ZGw-UoI4/UVXjxO-BpuI/AAAAAAAAG4o/bIcFJfx0c0w/s1600/figure1+(2)-001.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" src="http://2.bp.blogspot.com/-Br5ZGw-UoI4/UVXjxO-BpuI/AAAAAAAAG4o/bIcFJfx0c0w/s1600/figure1+(2)-001.jpg" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><span style="font-size: small;"><i>Figure 1: Instructor practices 1 through 3 and 8</i></span></td></tr></tbody></table><br /><table 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="http://4.bp.blogspot.com/-YrE4hSEXEDY/UVXjxIiK_JI/AAAAAAAAG4k/nbfVvkXm6po/s1600/figure2+(1).jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" src="http://4.bp.blogspot.com/-YrE4hSEXEDY/UVXjxIiK_JI/AAAAAAAAG4k/nbfVvkXm6po/s1600/figure2+(1).jpg" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><span style="font-size: small;"><i>Figure 2: Instructor practices 4 through 7</i></span></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="http://1.bp.blogspot.com/-Q3gpvwmwzzI/UVXjxOKk-2I/AAAAAAAAG4g/qzKxba_CMuI/s1600/figure3+(1).jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" src="http://1.bp.blogspot.com/-Q3gpvwmwzzI/UVXjxOKk-2I/AAAAAAAAG4g/qzKxba_CMuI/s1600/figure3+(1).jpg" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><i><span style="font-size: small;">Figure 3: Instructor practices 9 through 12</span></i></td></tr></tbody></table></div><div class="MsoNormal"><i style="background-color: white; color: #222222;"><br /></i><i style="background-color: white; color: #222222;">The MAA national study of calculus, Characteristics of Successful Programs in College Calculus, is funded by NSF grant no. 0910240. The opinions expressed in this column do not necessarily reflect those of the National Science Foundation.</i></div><div class="separator" style="clear: both; text-align: center;"></div><br /><div class="separator" style="clear: both; text-align: center;"></div><div class="MsoNormal"><br /></div>Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com0tag:blogger.com,1999:blog-7251686825560941361.post-12241645743431072362013-03-01T08:00:00.000-05:002013-06-04T08:49:04.064-04:00MAA Calculus Study: Good Teaching<br /><div class="MsoNormal"><span style="font-family: inherit;">One of the primary goals of the MAA Calculus Study, <i><a href="http://www.maa.org/cspcc/index.html" target="_blank">Characteristics of Successful Programs in College Calculus</a></i> (NSF #0910240), has been to identify the factors that are highly correlated with an improvement in student attitudes from the start to the end of the calculus course: confidence in mathematical ability, enjoyment of mathematics, and desire to continue the study of mathematics. To this end, Phil Sadler and Gerhard Sonnert of the Science Education Department within the Harvard-Smithsonian Center for Astrophysics constructed a hierarchical linear model from our survey responses to identify these factors. The factors reside at three levels: institutional, classroom, and individual student. Not surprisingly, most of the variation in student attitudes can be explained by student background, but there are influences at the institutional and classroom level. We have been particularly interested in what happens at the classroom level where there is the greatest opportunity for improvement.<o:p></o:p></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-family: inherit;">Sadler and Sonnert ran a factor analysis of the classroom-level variables, clumping those responses that were highly correlated. They discovered that the responses broke into three distinct clusters, which we are labeling “technology,” “progressive teaching,” and “good teaching” because these seem to describe the characteristics of the instruction. By far, the most important of these in terms of high correlation with improved attitudes is “good teaching.” Listed below are the 21 student-reported characteristics of instruction that are highly correlated with each other and highly correlated with improvements in student attitudes, characteristics that collectively we are calling “good teaching”:<o:p></o:p></span></div><div class="MsoNormal"><span style="font-family: inherit;"><br /></span></div><div class="MsoNormal"></div><div class="MsoNormal"><span style="font-family: inherit;">My calculus instructor:<o:p></o:p></span></div><div class="MsoNormal"></div><ol><li><span style="font-family: inherit;">Asked questions to determine if I understood what was being discussed.</span></li><li><span style="font-family: inherit;">Listened carefully to my questions and comments.</span></li><li><span style="font-family: inherit;">Discussed applications of calculus.</span></li><li><span style="font-family: inherit;">Allowed time for me to understand difficult ideas.</span></li><li><span style="font-family: inherit;">Helped me become a better problem solver.</span></li><li><span style="font-family: inherit;">Encouraged students to enroll in Calculus II.</span></li><li><span style="font-family: inherit;">Acted as if I was capable of understanding the key ideas of calculus.</span></li><li><span style="font-family: inherit;">Made me feel comfortable asking questions during class.</span></li><li><span style="font-family: inherit;">Encouraged students to seek help during office hours.</span></li><li><span style="font-family: inherit;">Presented more than one method for solving problems.</span></li><li><span style="font-family: inherit;">Made class interesting.</span></li><li><span style="font-family: inherit;">Provided explanations that were understandable.</span></li><li><span style="font-family: inherit;">Was available to make appointments outside of office hours, if needed.</span></li></ol>My calculus instructor did not:<br /><ol start="14"><li><span style="font-family: inherit;">Discourage me from wanting to continue taking calculus.</span></li><li><span style="font-family: inherit;">Make students feel nervous during class.</span></li></ol><div>My instructor often or very often:</div><ol start="16"><li><span style="font-family: inherit;">Showed how to work specific problems.</span></li><li><span style="font-family: inherit;">Asked questions.</span></li><li><span style="font-family: inherit;">Prepared extra material to help students understand calculus concepts or procedures.</span></li></ol><div>In addition:</div><ol start="19"><li><span style="font-family: inherit;">My calculus exams were a good assessment of what I learned.</span></li><li><span style="font-family: inherit;">My exams were fairly graded.</span></li><li><span style="font-family: inherit;">My homework was fairly graded.</span></li></ol><div><div><div class="MsoNormal"><span style="font-family: inherit;">The good news is that most calculus instructors rated highly on most of these characteristics. This good news needs to be tempered by two facts: Instructors could and in many cases did elect not to participate even though other instructors at their institution were involved in the study, and these responses were all collected at the end of the term. They reflect the opinions of the students who had successfully navigated this course, predominantly students who were earning an A or a B in the course (roughly 40% A, 40% B, 20% C). <o:p></o:p></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-family: inherit;">It is interesting and informative to see how students at different types of institutions rated their instructors on these criteria. We followed CBMS in categorizing post-secondary institutions by the highest mathematics degree offered at that institution. I am using “research” to designate universities that offer a PhD in Mathematics (predominantly large state flagship universities), “masters” if the highest degree is a master’s (predominantly public comprehensive universities), “undergrad” if it is a bachelor’s degree (predominantly private liberal arts colleges), and “two-year” if it is an associate’s degree (predominantly community and technical colleges). As shown in the graphs at the end of this article, instructors at research universities got the lowest ratings on every characteristic except “showed how to work specific problems.” For most of these characteristics, instructors at undergraduate colleges were the next lowest, then masters universities, and most of the time instructors at two-year colleges received the highest ratings. </span><o:p></o:p></div><div class="MsoNormal"><span style="font-family: inherit;"><br /></span></div><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"></div><div class="MsoNormal"></div><div class="MsoNormal">There were a few notable exceptions. Instructors at undergraduate colleges received the highest ratings in some of the areas where one would expect them to be strong:<o:p></o:p></div><div class="MsoNormal"></div><ul><li>Acted as if I was capable of understanding the key ideas of calculus.</li><li>Encouraged students to seek help during office hours.</li><li>Was available to make appointments outside of office hours, if needed.</li><li>Did not make students feel nervous during class.</li></ul><br /><div class="MsoNormal">Masters universities scored highest in often or very often showing how to work specific problems, and just barely edged out two-year colleges in “listened carefully” and “my exams were fairly graded.”<o:p></o:p></div><div class="MsoNormal"><br /></div><div class="MsoNormal">There are a number of possible explanations for the weaknesses of research universities and the strengths of two-year colleges. One is class size. The largest classes are found at the research universities where average class size is 53, the smallest at two-year colleges where the average is 21. However, average class size at masters universities is larger than at undergraduate colleges, so class size cannot be the only explanatory variable. Some of the discrepancies between institution types may be explained by student expectations. This is because SAT scores and high school mathematics GPA are highest for research universities, then undergraduate colleges, then masters universities, and lowest for two-year colleges. Better students may have higher expectations of their instructors, or they may be more discouraged by encountering difficulties in this course. The differences may also have something to do with age and thus maturity of the students. The youngest students are at research universities, the oldest at two-year colleges. They also may be related to the relatively large number of instructors at research universities who teach calculus but have little or no interest in teaching this course, as opposed to two-year colleges where the interest is very high (see my November column, <a href="http://launchings.blogspot.com/2012/10/maa-calculus-study-instructors.html" target="_blank">MAA Calculus Study: The Instructors</a>). Nevertheless, it is discouraging that students at research universities seem to be getting calculus instruction that has a worse effect on student attitudes than instruction at other types of institutions.<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="http://3.bp.blogspot.com/-n39ff43HLfQ/USzs0tZyscI/AAAAAAAAGto/0vG-SzN4qgo/s1600/Figure1.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" src="http://3.bp.blogspot.com/-n39ff43HLfQ/USzs0tZyscI/AAAAAAAAGto/0vG-SzN4qgo/s1600/Figure1.jpg" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><span style="font-family: inherit;"><i>Figure 1: Instructor Characteristics 1–5.</i></span></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="http://2.bp.blogspot.com/-WbGv1Y0e8Vk/USzs0ht-HsI/AAAAAAAAGts/HAUU2YtzCo4/s1600/Figure3.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" src="http://2.bp.blogspot.com/-WbGv1Y0e8Vk/USzs0ht-HsI/AAAAAAAAGts/HAUU2YtzCo4/s1600/Figure3.jpg" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><span style="font-family: inherit;"><i>Figure 2: Instructor Characteristics 6–10.</i></span></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="http://3.bp.blogspot.com/-WWyGnuw_JZo/USzs0sosQSI/AAAAAAAAGtw/HKe67SIMf0A/s1600/Figure2.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" src="http://3.bp.blogspot.com/-WWyGnuw_JZo/USzs0sosQSI/AAAAAAAAGtw/HKe67SIMf0A/s1600/Figure2.jpg" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><div align="center" class="MsoNormal"><i>Figure 3: Instructor Characteristics 11–15.</i><o:p></o:p></div></td></tr></tbody></table><br /><div class="separator" style="clear: both; text-align: center;"></div><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="http://1.bp.blogspot.com/-yWHPlgj1ohQ/USzs0_uBmLI/AAAAAAAAGt4/N6Jmj0SP8Z8/s1600/Figure4.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" src="http://1.bp.blogspot.com/-yWHPlgj1ohQ/USzs0_uBmLI/AAAAAAAAGt4/N6Jmj0SP8Z8/s400/Figure4.jpg" height="340" width="400" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><div align="center" class="MsoNormal"><i>Figure 4: Instructor Characteristics 16–18.</i><o:p></o:p></div><div align="center" class="MsoNormal"><br /></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="http://2.bp.blogspot.com/--RSY7a_0UI4/USzs1Cr4MdI/AAAAAAAAGt0/6TwsyuTGp2U/s1600/Figure5.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" src="http://2.bp.blogspot.com/--RSY7a_0UI4/USzs1Cr4MdI/AAAAAAAAGt0/6TwsyuTGp2U/s400/Figure5.jpg" height="330" width="400" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><div align="center" class="MsoNormal"><i>Figure 5: Instructor Characteristics 19–21.</i><o:p></o:p></div></td></tr></tbody></table><i style="background-color: white; color: #222222;"><br /></i></div><div class="MsoNormal"><i style="background-color: white; color: #222222;">The MAA national study of calculus, Characteristics of Successful Programs in College Calculus, is funded by NSF grant no. 0910240. The opinions expressed in this column do not necessarily reflect those of the National Science Foundation.</i></div></div><o:p></o:p></div><o:p></o:p>Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com0tag:blogger.com,1999:blog-7251686825560941361.post-46123804901975223592013-02-01T09:00:00.000-05:002013-02-01T09:00:02.557-05:00Mathematics in 2025The National Research Council of the National Academies has just released the preliminary version of its report, <a href="http://www.nap.edu/catalog.php?record_id=15269"><i>The Mathematical Sciences in 2025</i></a><i> </i>[1]. This was produced in response to a request from the National Science Foundation. It comes as the latest in a series of glimpses into the future of mathematics that go back to the “David reports” of 1984 and 1990 [2,3] and the “Odom study” of 1998 [4]. This report is important because it will influence the direction NSF takes as it plans for the future.<br /><div class="MsoNormal"><o:p></o:p></div><div class="MsoNormal"><br /></div><div class="MsoNormal">The emphasis of the report is on the central role that the mathematical sciences are taking within research in areas as diverse as biology, finance, and climate science. Traditional disciplinary boundaries are blurring. There is an increasing need for scientists who are well grounded in mathematical sciences, especially the statistical and computational sciences, as well as other disciplines. This goes two ways. It means opening courses and programs in the mathematical sciences, especially at the graduate level, to those in other fields of study, and it means ensuring that students graduating in the mathematical sciences are prepared to work in this interdisciplinary world.<o:p></o:p></div><div class="MsoNormal"><br /></div><div class="MsoNormal">This has implications right down the line of mathematics education. The authors of the report question whether, in a scientific world that is dominated by big data and the challenges of large-scale computation, the traditional calculus-focused curriculum is the most appropriate for all students. As they say, “Different pathways are needed for students who may go on to work in bioinformatics, ecology, medicine, computing, and so on. It is not enough to rearrange existing courses to create alternative curricula; <i>a redesigned offering of courses and majors is needed</i> [my emphasis].” (NRC 2013, p. S-9)<o:p></o:p></div><div class="MsoNormal"><br /></div><div class="MsoNormal">The report also stresses the importance of attracting more women and students from traditionally underrepresented minorities to the mathematical societies. This is the one place where I disagree with the report, for it asserts that, “While there has been progress in the last 10–20 years, the fraction of women and minorities in the mathematical sciences drops with each step up the career ladder.” (NRC 2013, p. S-10). I don’t question the drop. I question whether there has been progress over the last 10–20 years.<o:p></o:p></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-ZbCVxE05hyI/UQltz8EyJII/AAAAAAAAGa4/LTVh1KxD0ek/s1600/Jan13-1.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="240" src="http://1.bp.blogspot.com/-ZbCVxE05hyI/UQltz8EyJII/AAAAAAAAGa4/LTVh1KxD0ek/s400/Jan13-1.jpg" width="400" /></a></div><div align="center" class="MsoNormal" style="text-align: center;"><br /></div><div class="MsoNormal">If we look at mathematics majors (bachelor’s degrees) by gender, we see that over the period 1990 to 2011 the number of men majoring in mathematics grew by 25% while the number of women grew by only 10% (Figure 1). As a result, the percentage of bachelor’s degrees in the mathematical sciences going to women has dropped to 43.1%, the first time it has been this low since 1981. This is having knock-on effects for graduate programs. The percentage of bachelor’s degrees in mathematics that went to women peaked in 1999 at 47.8%. The percentage of master’s degrees in mathematics that went to women peaked in 2004 at 45.1% and has since dropped back to 40.9%. The percentage of doctoral degrees in mathematics that went to women peaked in 2008 and ’09 at 31.0%. It has since dropped back to 28.6%. The good news is that the past decade has seen strong growth in the number of mathematics majors, but two-thirds of the growth since 2001 has been in the number of men. <o:p></o:p></div><div class="MsoNormal"><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-7SGX2OD7Bc8/UQlt-Jd2UuI/AAAAAAAAGbA/OGHRKQHlTy4/s1600/Jan13-2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="257" src="http://3.bp.blogspot.com/-7SGX2OD7Bc8/UQlt-Jd2UuI/AAAAAAAAGbA/OGHRKQHlTy4/s400/Jan13-2.jpg" width="400" /></a></div></div><div align="center" class="MsoNormal" style="text-align: center;"><br /></div><div class="MsoNormal">We see an even more discouraging pattern among Black students (Figure 2). The number of Black mathematics majors is essentially back to where it was twenty years ago despite the number of bachelor’s degrees earned by Black students almost tripling over this period. The number of Black mathematics majors peaked in 1997 at 1,089. It was back down to only 840 in 2011. The number of ethnically Asian mathematics majors has been growing strongly over the past decade. Even so, the number earning undergraduate degrees in the mathematical sciences has only doubled since 1990, while the number earning bachelor’s degrees has tripled. The growth in the number of Hispanic mathematics majors looks good, having slightly more than tripled in twenty years, until you realize that the number of Hispanic students graduating from college is almost five times what it was in 1990 (154,000 versus 33,000). Where we do see strong growth, especially since 2007, is in the number of non-resident aliens majoring in mathematics, which now stands at 7% of all US mathematics majors.<o:p></o:p></div><div class="MsoNormal"><br /></div><div class="MsoNormal">I must emphasize that the NRC report does highlight the importance of increasing the participation of women and members of underrepresented groups. It includes the following specific recommendation:<o:p></o:p><br /><br /></div><div style="margin-left: .5in;"><div style="text-align: justify;"><b><span style="font-family: "TimesNewRomanPS","serif"; font-size: 12.0pt;">Recommendation 5-4:</span></b><span style="font-family: "TimesNewRomanPS","serif"; font-size: 12.0pt;">Every academic department in the mathematical sciences should explicitly incorporate recruitment and retention of women and underrepresented groups into the responsibilities of the faculty members in charge of the undergraduate program, graduate program, and faculty hiring and promotion. Resources need to be provided to enable departments to adopt, monitor and adapt successful recruiting and mentoring programs that have been pioneered at other schools and to find and correct any disincentives that may exist in the department. (NRC 2013, p. 5-18)</span><o:p></o:p></div><span style="font-family: "TimesNewRomanPS","serif"; font-size: 12.0pt;"><br /></span></div><div class="MsoNormal">I have only touched on a few of the topics covered in the NRC report. It also discusses the increasingly important role of the mathematical sciences institutes, the issue of maintaining online repositories of mathematical research such as <a href="http://arxiv.org/">arXive</a>, and the threats to mathematics departments as more instruction—especially for the service courses that often provide the justification for a large mathematics faculty—is moved online. This is a report well worth reading and pondering.<span style="text-align: center;"> </span><br /><span style="text-align: center;"><br /></span></div><hr /><br />[1] National Research Council. 2013. <a href="http://www.nap.edu/catalog.php?record_id=15269">The MathematicalSciences in 2025</a>. Washington, DC. The National Academies Press.<br /><div class="MsoNormal"><o:p></o:p></div><div class="MsoNormal"><br /></div><div class="MsoNormal">[2] NRC. 1984, Renewing US Mathematics: Critical Resource for the Future. Washington, DC. The National Academies Press.<o:p></o:p></div><div class="MsoNormal"><br /></div><div class="MsoNormal">[3] NRC. 1990. Renewing US Mathematics: A Plan for the 1990s. Washington, DC. The National Academies Press.<o:p></o:p></div><div class="MsoNormal"><br /></div><div class="MsoNormal">[4] NSF. 1998. Report of the Senior Assessment Panel for the International Assessment of the US Mathematical Sciences. Arlington, VA. National Science Foundation.<o:p></o:p></div>Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com0tag:blogger.com,1999:blog-7251686825560941361.post-89123651461866971262013-01-01T09:00:00.000-05:002013-01-02T13:06:49.270-05:00The Red Herring of Grade Inflation <span style="font-family: inherit;">Two things happened in the week before Christmas that got me thinking about grade inflation. The first was that I graded the final exams for my multivariable calculus class. I have never before seen my students do so well. Out of 33 students in the class, 22 received an A. For my class, an A requires earning more than 92% of the total possible grade. The last time I graded on a curve was over 20 years ago.</span><br /><div class="MsoNormal"><span style="font-family: inherit;"><br /></span></div><div class="MsoNormal"><span style="font-family: inherit;">This past semester I had worked these students hard. They were responsible for and graded on:</span></div><div class="MsoListParagraphCxSpFirst" style="margin-left: .25in; mso-add-space: auto; mso-list: l1 level1 lfo1; text-indent: -.25in;"><br /><blockquote class="tr_bq"><ul><li><span style="text-indent: -0.25in;"> Reading Reflections (three times per week, reading the section and answering questions about the material before we discussed it in class).</span><span style="text-indent: -0.25in;"> </span></li></ul></blockquote><blockquote class="tr_bq"><ul><li><span style="text-indent: -0.25in;"> Two sets of homework each week (about 12 fairly straightforward questions on WeBWorK due on Thursdays and three challenging multi-part problems due on Mondays). </span></li></ul></blockquote><blockquote class="tr_bq"><ul><li><span style="text-indent: -0.25in;"> </span><span style="text-indent: -0.25in;">Seven short projects developed by Tevian Dray and Corinne Manogue as part of their Bridge Project (see </span><a href="http://www.math.oregonstate.edu/bridge/" style="text-indent: -0.25in;">http://www.math.oregonstate.edu/bridge/</a><span style="text-indent: -0.25in;">). These were started in groups of three or four, but each student was responsible for writing his or her own three to five page report of the solution. For the first report, I required a first draft that was critiqued and returned for revision and resubmission.</span></li></ul></blockquote><blockquote class="tr_bq"><ul><li><span style="text-indent: -0.25in;"> </span><span style="text-indent: -0.25in;">A major project based on the Hydro-Turbine Optimization chapter in </span><i style="text-indent: -0.25in;">Applications of Calculus</i><span style="text-indent: -0.25in;">[1]. The project was started in groups. Each student was responsible for an 8–12 page paper explaining the solution. The papers were turned in, critiqued, and returned for revision and resubmission. LaTeX and pdf files of my version of this project are available </span><a href="http://www.macalester.edu/~bressoud/courses/HTP.html" style="text-indent: -0.25in;">here</a><span style="text-indent: -0.25in;">.</span></li></ul></blockquote><blockquote class="tr_bq"><ul><li><span style="text-indent: -0.25in;"> </span><span style="text-indent: -0.25in;">Two examinations during the semester and a final exam. After each exam during the semester, students were required to write about the problems they had missed points on, explain what they did wrong, and explain how to do it correctly. They could earn back half the points they had lost. For the final exam, they had to explain what they were doing to solve the problems, not just give an answer.</span></li></ul></blockquote></div><div class="MsoNormal"><span style="font-family: inherit;">I was available to my students every afternoon, and I also had a great undergraduate preceptor (teaching assistant) who held help sessions Sunday and Thursday evenings, before the homework assignments were due. By the end of the semester, over half the class was coming to each of these, and so she organized them into groups working with each other on the homework while she circulated to help the groups that were stuck.<o:p></o:p></span></div><div class="MsoNormal"><span style="font-family: inherit;"><br /></span></div><div class="MsoNormal"><span style="font-family: inherit;">Not surprisingly, in the end of semester course evaluations my students wrote about how much work they had done for this course. And yet, when asked specifically whether or not they agreed with the statement, “The general workload was appropriate for this level course,” only five of my 33 students disagreed. One student comment that summarized the tenor of the end of course evaluations stated, “I would say that the course is difficult and a lot of work, but very rewarding, because if you put in a lot of time and effort then you can see yourself understand the material and do well. Although the course can be really hard at times, there is always somewhere to go for help.” <o:p></o:p></span></div><div class="MsoNormal"><span style="font-family: inherit;"><br /></span></div><div class="MsoNormal"><span style="font-family: inherit;">The second thing that happened this past week was my discovery of <i>How Learning Works: 7 Research-Based Principles for Smart Teaching</i> [2]. This collaborative effort, published in 2010, translates what has been learned by those engaged in research in undergraduate education into practical guidance for those of us in the classroom. What the authors call principles, I see more as facets of teaching to which I need to pay attention. This is my own paraphrasing of these principles or facets:</span></div><div class="MsoListParagraphCxSpFirst" style="mso-list: l0 level1 lfo2; text-indent: -.25in;"><br /><blockquote class="tr_bq"><ol><li><span style="text-indent: -0.25in;"> The need to understand the variety of prior knowledge that my students bring to my class and how it helps or hinders them. </span></li><li><span style="text-indent: -0.25in;"> The importance of how students organize the knowledge they are acquiring and the need for me to understand common misalignments and to help them make the necessary connections.</span></li><li><span style="text-indent: -0.25in;"> The critical role of student motivation and my responsibility to strengthen it.</span></li><li><span style="text-indent: -0.25in;"> The need to develop automaticity in basic skills and the fact that learning how to integrate and apply these skills requires guidance and directed practice from me.</span></li><li><span style="text-indent: -0.25in;"> How important it is that I provide useful feedback that is targeted at improving performance.</span></li><li><span style="text-indent: -0.25in;"> The role of the social, emotional, and intellectual climate in my classroom.</span></li><li><span style="text-indent: -0.25in;"> The need for me to guide students in practicing metacognition, monitoring what they are doing and why.</span></li></ol></blockquote></div><div class="MsoNormal"><span style="font-family: inherit;">The book discusses the relevant research, but is also full of examples of traps we can fall into and strategies for dealing with these principles or facets in order to improve our teaching.<o:p></o:p></span></div><div class="MsoNormal"><span style="font-family: inherit;"><br /></span></div><div class="MsoNormal"><span style="font-family: inherit;">One trap discussed under #3 describes the teacher who, with the intent of spurring his students to work hard, warned them at the start of the course that they could expect that a third of them would not pass. This had exactly the opposite effect. With the expectation that they would not do well regardless of how much effort they put into the course, a large proportion of the students directed their time and energy to other courses. <o:p></o:p></span></div><div class="MsoNormal"><span style="font-family: inherit;"><br /></span></div><div class="MsoNormal"><span style="font-family: inherit;">The issue here is motivation, getting the students to put in the effort needed to learn the material. I believe that I did succeed particularly well this past semester in motivating most of my multivariable calculus students. <i>How Learning Works</i> identifies three levers that motivate students to work hard. The first is <b>value</b>. They have to believe that what I want them to learn will be of value to them. Personal enthusiasm on my part goes a long way toward building this sense of value. The second is a <b>supportive environment</b>. They have to believe that the course is structured in such a way as to help them be successful, rather than throwing up obstacles to their success. Starting the projects and encouraging them to share their understanding of homework problems within groups, providing feedback and multiple opportunities to demonstrate understanding (as with WeBWorK and the chance to earn back points lost on exams), and the availability of myself and my preceptor build the sense of support. The third is <b>self-efficacy</b>, belief that one is capable of achieving success.<o:p></o:p></span></div><div class="MsoNormal"><span style="font-family: inherit;"><br /></span></div><div class="MsoNormal"><span style="font-family: inherit;">This last is the main reason I will never again grade on a curve. The message sent by grading on a curve is that the proportion of failures has been determined in advance, regardless of how much work students are prepared to invest in the course. It is also why I am disturbed that in our national survey of calculus, faculty at the start of the term were able to predict, almost perfectly, what their grade distributions would be at the end of the term (see the last bullet under Instructor Attitudes in <a href="http://maa.org/columns/launchings/launchings_06_11.html">The Calculus I Instructor, <i>Launchings</i>, June 2011</a><i>)</i>. Going into this course, I would never have predicted 67% A’s. I am delighted that what I did worked so well with so many of my students. [3]<o:p></o:p></span></div><div class="MsoNormal"><span style="font-family: inherit;"><br /></span></div><div class="MsoNormal"><span style="font-family: inherit;">Which brings me back to the issue of grade inflation. Grade inflation is a red herring because it misdirects our attention from what should be our true concerns: What do our grades mean in terms of expectation of student achievement and understanding? And how can we support as many students as possible to meet our highest expectations?</span></div><div class="MsoNormal"><span style="font-family: inherit;"><br /></span></div><br /><hr /><span style="font-family: inherit;">[1] Straffin, P. D., Jr. 1996. Hydro-Turbine Optimization. Pages 240–250 in <i><a href="http://maa-store.hostedbywebstore.com/APPLICATIONS-OF-CALCULUS-Philip/dp/B007BDPE14">Applications of Calculus</a></i>. P.D. Straffin, Jr., editor. Classroom Resource Materials. MAA. </span><br /><span style="height: 6px; margin-left: -2px; margin-top: 2px; mso-ignore: vglayout; position: absolute; width: 544px; z-index: 251659264;"></span><span style="font-family: inherit;"><span style="font-weight: normal;"> [2] Ambrose, S. A., M. W. Bridges, M. DiPietro, M. C. Lovett, M. K. Norman. 2010. <i><a href="http://www.wiley.com/WileyCDA/WileyTitle/productCd-0470484101.html">How Learning Works: 7 Research-Based Principles for Smart Teaching</a></i>. Jossey-Bass. </span><br /><span style="font-weight: normal;">[3] Not all my students did well. The class GPA was 3.5. What was important was that I had explicit expectations of what would constitute A work, that I clearly communicated what was required to meet those expectations, that students saw them as challenging but achievable, and that my students really were graded according to these expectations.</span></span><br /><br /><div class="MsoNormal"></div><br /><div class="MsoNormal"><br /></div>Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com0tag:blogger.com,1999:blog-7251686825560941361.post-39610866610302910522012-12-01T08:00:00.000-05:002012-12-01T08:00:05.980-05:00Mathematics and the NRC Discipline-Based Education Research Report <br /><div class="MsoNormal"><span style="font-family: inherit;">This past spring, the National Research Council of the National Academies released its report, <a href="http://www.nap.edu/catalog.php?record_id=13362"><i>Discipline-Based Education Research: Understanding and Improving Learning in Undergraduate Science and Engineering</i></a><i> </i>[1]<i>.</i> The charge to the committee writing this report was to synthesize existing research on teaching and learning in the sciences, to report on the effect of this research, and to identify future directions for this research. The project has its roots in two 2008 workshops on promising practices in undergraduate science, technology, engineering, and mathematics education.</span></div><div class="MsoNormal"><span style="font-family: inherit;"><o:p></o:p></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-family: inherit;">Unfortunately, between 2008 and 2012 undergraduate mathematics education dropped out of the picture. The resulting report discusses undergraduate education research only for physics, chemistry, engineering, biology, the geosciences, and astronomy. Nevertheless, it is an interesting report with useful information—especially the instructional strategies that have been shown to be effective—that is relevant for those of us who teach undergraduate mathematics.<o:p></o:p></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-family: inherit;">The studies that are described are founded on the assumption that students must build their own understanding of the discipline by applying its methods and principles, and this is best accomplished within a student-centered approach that puts less emphasis on simple transmission of factual information and more on student engagement with conceptual understanding, including active learning in the classroom.<o:p></o:p></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-family: inherit;">The great strength of this report is the wealth of resources that it references and the common themes that emerge across all of the scientific disciplines. A lot of attention is paid to the power of interactive lectures. Given that most science and mathematics instruction is still given in traditional lecture settings, finding ways of engaging students and getting them to <i>think</i> about the mathematics while they are in class is essential for increasing student understanding.<o:p></o:p></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-family: inherit;">The recommendations of effective practice range from simple techniques, such as starting each class with a challenging question for students to keep in mind, to transformative practices such as collaborative learning. A common intermediate practice involves student engagement by posing a challenging question, having students interact with their peers to think through the answer, and then testing the answer. In some respects, this is more easily done in the sciences where student predictions can be verified or falsified experimentally. Yet it is also a very effective tool in mathematics education where a well-chosen example can falsify an invalid expectation and careful analysis can support correct understanding. But most important is that it forces to try to use what they have been learning.<o:p></o:p></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-family: inherit;">In large classes, this type of peer instruction can be facilitated by the use of clickers. The report does include the caveat, with supporting research, that merely using clickers without attention to how they are used is of no measurable benefit.<o:p></o:p></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-family: inherit;">The greatest learning gains that have been documented occur when collaborative research is incorporated into the classroom. The NRC report includes many descriptions of how this can be accomplished in a variety of scientific disciplines. It also references the research that has established its effectiveness. Again, attention to how it is done is an important component of effective practice.<o:p></o:p></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-family: inherit;">Two of the areas that are identified as needing more research are issues of transference (see my September column on <a href="http://launchings.blogspot.com/2012_09_01_archive.html">Teaching and Learning for Transference</a>) and metacognition. Usefully, the authors point out that there are two sides to transference: the ability to draw on prior knowledge and the ability to carry what is currently being learned to future situations. Metacognition is an important issue in research in undergraduate mathematics education, especially for those studying the difference between experts and novices engaged in activities such as constructing proofs. Experts monitor their assumptions and progress and are prepared to change track when a particular approach is not fruitful. Novices are more likely to choose what to them seems the likeliest approach and then ignore alternatives.<o:p></o:p></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-family: inherit;">In sum, this is a useful and thought-provoking report. I wish that it had included undergraduate mathematics education research, but perhaps that omission can be corrected as we move forward.</span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-family: inherit;">[1] National Research Council. 2012. <i><a href="http://www.nap.edu/catalog.php?record_id=13362">Discipline-Based Education Research: Understanding and ImprovingLearning in Undergraduate Science and Engineering</a></i>. S.R. Singer, N.R. Nielsen, and H.A. Schweingruber, eds. Washington, DC. The National Academies Press. <o:p></o:p></span></div><div class="MsoNormal"><br /></div>Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com0tag:blogger.com,1999:blog-7251686825560941361.post-47249968214099119712012-11-01T08:00:00.000-04:002013-06-04T08:48:40.932-04:00MAA Calculus Study: The Instructors<br /><div class="MsoNormal"><span style="font-family: inherit;">One of the goals of the MAA Calculus Study, <a href="http://www.maa.org/cspcc"><i>Characteristics of Successful Programs in College Calculus</i></a><i>,</i> was to gather information about the instructors of mainstream Calculus I. Here, stratified by type of institution, is some of what we have learned, refining some of the data presented in <a href="http://maa.org/columns/launchings/launchings_06_11.html">“The Calculus I Instructor” (<i>Launchings</i>, June 2011)</a>. Again, I am using Research University as code for institutions for which the highest mathematics degree that is offered is the PhD, Masters University if the highest degree is a Master’s, Undergraduate College if it is a Bachelor’s, and Two Year College if it is an Associate’s degree. These surveys were completed by 360 instructors at research universities, 73 at masters universities, 118 at undergraduate colleges, and 112 at two year colleges.<o:p></o:p></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-family: inherit;">Calculus I instructors are predominantly white and male. Masters universities have the largest percentage of Black instructors, research universities of Asian instructors, and two-year colleges of Hispanic instructors. By and large, undergraduate colleges do not do well in representing any of these groups.<o:p></o:p></span></div><div align="center" class="MsoNormal" style="text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-Sp8OPpmM1fk/UJFzxwylsYI/AAAAAAAAFkY/9rg-6aoyAJw/s1600/bressoud1.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://2.bp.blogspot.com/-Sp8OPpmM1fk/UJFzxwylsYI/AAAAAAAAFkY/9rg-6aoyAJw/s400/bressoud1.jpg" height="232" width="400" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div align="center" class="MsoNormal" style="text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-QUOMkd3QNAI/UJFz1genPkI/AAAAAAAAFkg/sgqcsn2vVf0/s1600/bressoud2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://2.bp.blogspot.com/-QUOMkd3QNAI/UJFz1genPkI/AAAAAAAAFkg/sgqcsn2vVf0/s400/bressoud2.jpg" height="276" width="400" /></a></div><div align="center" class="MsoNormal" style="text-align: center;"><span style="font-family: inherit;"> </span><span style="font-family: inherit;"> </span></div><div class="MsoNormal"><span style="font-family: inherit;">There is a dramatic difference between the status and highest degree of Calculus I instructors at research universities and those at other types of colleges and universities. At research universities, instructors are less likely to be tenured or on tenure track, or to hold a PhD. They are also less likely to want to teach calculus: One in five has no interest or only a mild interest in teaching calculus. The high number of part-time faculty at masters universities and two year colleges is troubling because of the evidence that such instructors tend to be less effective in the classroom and much less accessible to their students [1]. Not surprisingly, less than a quarter of the Calculus I instructors at two-year colleges hold a PhD.<o:p></o:p></span></div><div align="center" class="MsoNormal" style="text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-9YxF6PwhBEw/UJF0QCzyAwI/AAAAAAAAFkw/WVG0eEXkERs/s1600/bressoud3.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://1.bp.blogspot.com/-9YxF6PwhBEw/UJF0QCzyAwI/AAAAAAAAFkw/WVG0eEXkERs/s400/bressoud3.jpg" height="286" width="400" /></a></div><div align="center" class="MsoNormal" style="text-align: center;"><br /></div><div align="center" class="MsoNormal" style="text-align: center;"><o:p><span style="font-family: inherit;"><br /></span></o:p></div><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-YsbDyoOrMJI/UJF0UKEo4wI/AAAAAAAAFk4/6-EVmBq38Cg/s1600/bressoud4.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://3.bp.blogspot.com/-YsbDyoOrMJI/UJF0UKEo4wI/AAAAAAAAFk4/6-EVmBq38Cg/s400/bressoud4.jpg" height="237" width="400" /></a></div><div align="center" class="MsoNormal" style="text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-bPF1zM8RXIM/UJF0Y7Xnv4I/AAAAAAAAFlA/3WRQX1w8lBQ/s1600/bressoud5.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://3.bp.blogspot.com/-bPF1zM8RXIM/UJF0Y7Xnv4I/AAAAAAAAFlA/3WRQX1w8lBQ/s400/bressoud5.jpg" height="251" width="400" /></a></div><div align="center" class="MsoNormal" style="text-align: center;"><span style="font-family: inherit;"> </span><span style="font-family: inherit;"> </span></div><div class="MsoNormal"><span style="font-family: inherit;">Generally, calculus instructors consider themselves to be somewhat traditional in their instructional approaches, and they believe that students learn best from lectures. The greatest divergence from these views is at undergraduate colleges where almost half consider themselves to be innovative and 45% disagree that lectures are the best way to teach. The greatest variation among faculty at different types of institutions is over the use of calculators on exams. Close to half of the instructors at research universities do not allow them; 71% of the instructors at two year colleges do.<o:p></o:p></span></div><div class="MsoNormal"><o:p><span style="font-family: inherit;"> </span></o:p><span style="font-family: inherit;"> </span></div><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-W3TkY7_BhwQ/UJF0z384wBI/AAAAAAAAFlY/VRfoYOj-CLI/s1600/bressoud6.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://1.bp.blogspot.com/-W3TkY7_BhwQ/UJF0z384wBI/AAAAAAAAFlY/VRfoYOj-CLI/s400/bressoud6.jpg" height="227" width="400" /></a></div><div align="center" class="MsoNormal" style="text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-jeZvWafZWpQ/UJF04CSuSjI/AAAAAAAAFlg/mZhpcy0bXKI/s1600/bressoud7.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://1.bp.blogspot.com/-jeZvWafZWpQ/UJF04CSuSjI/AAAAAAAAFlg/mZhpcy0bXKI/s400/bressoud7.jpg" height="305" width="400" /></a></div><div align="center" class="MsoNormal" style="text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-oGj3FucN9HI/UJF1CPmPTEI/AAAAAAAAFlo/eGgY_FFsqpY/s1600/bressoud8.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://4.bp.blogspot.com/-oGj3FucN9HI/UJF1CPmPTEI/AAAAAAAAFlo/eGgY_FFsqpY/s400/bressoud8.jpg" height="278" width="400" /></a></div><div align="center" class="MsoNormal" style="text-align: center;"><br /></div><div class="MsoNormal"><span style="font-family: inherit;">There also are institutional differences in beliefs about whether all of the students who enter Calculus I are capable of learning this material.<o:p></o:p></span></div><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-EhgNrYr7uZg/UJF1Hnilk2I/AAAAAAAAFlw/SBhDd0aXvNo/s1600/bressoud9.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://2.bp.blogspot.com/-EhgNrYr7uZg/UJF1Hnilk2I/AAAAAAAAFlw/SBhDd0aXvNo/s400/bressoud9.jpg" height="291" width="400" /></a></div><div align="center" class="MsoNormal" style="text-align: center;"><br /></div><div class="MsoNormal"><span style="font-family: inherit;">Finally, we look at the grade distributions by type of institution.<o:p></o:p></span></div><div align="center" class="MsoNormal" style="text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-Lh5JVTnbUrQ/UJF1N_Dc0KI/AAAAAAAAFl4/yZ37Bah7BHc/s1600/bressoud10.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://2.bp.blogspot.com/-Lh5JVTnbUrQ/UJF1N_Dc0KI/AAAAAAAAFl4/yZ37Bah7BHc/s400/bressoud10.jpg" height="280" width="400" /></a></div><div align="center" class="MsoNormal" style="text-align: center;"><o:p><span style="font-family: inherit;"> </span></o:p><span style="font-family: inherit;"> </span></div><div class="MsoNormal"><span style="font-family: inherit;">[1] Schmidt, P. <a href="http://chronicle.com/article/Conditions-Imposed-on/125573/">Conditions Imposed on Part-Time Adjuncts Threaten Quality ofTeaching, Researchers Say</a>. <i>Chronicle of Higher Education</i>. <b>Nov 30, 2010.</b></span><o:p></o:p><br /><span style="font-family: inherit;"><b><br /></b></span><i style="background-color: white; color: #222222;">The MAA national study of calculus, Characteristics of Successful Programs in College Calculus, is funded by NSF grant no. 0910240. The opinions expressed in this column do not necessarily reflect those of the National Science Foundation.</i></div>Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com0tag:blogger.com,1999:blog-7251686825560941361.post-9909484210090040452012-10-01T09:00:00.000-04:002013-06-04T08:48:16.879-04:00MAA Calculus Study: Intended Careers<br /><div class="MsoNormal">This is the first of what I intend to be a series of reports that delve into the data on Calculus I instruction that were collected in fall 2010 as part of MAA’s study <i>Characteristics of Successful Programs in College Calculus</i> (NSF #0910240). Some of the raw summative data was reported in earlier <i>Launchings</i>columns [1,2], but we have now separated the data by type of institution, characterized by highest degree offered in mathematics. Combined with knowledge of the number of students who took Calculus I in fall 2010 at each of the types of institutions—thanks to the Conference Board of the Mathematical Sciences (CBMS) 2010 survey—it is now possible to appropriately weight the data that has been collected. A preprint of an article that summarizes the methods and instruments used for the surveys with selected results [3] has been posted on the website <a href="http://www.maa.org/cspcc">maa.org/cspcc</a>.</div><div class="MsoNormal"><o:p></o:p></div><div class="MsoNormal"><br /></div><div class="MsoNormal">For this month, I want to focus on the intended careers of students as they begin mainstream Calculus I. This is a course that took on its modern form after the Second World War, and in most places it still follows the curriculum as laid out in George Thomas’s <i>Calculus and Analytic Geometry</i> of 1951 [4]. The course was designed to meet the needs of engineers and those in the physical sciences. However, as illustrated in Figure 1, just under 35% of those taking Calculus I today are on one of these tracks. Today’s Calculus I student is more likely to be pursuing a career in the biological or life sciences than in engineering.<o:p></o:p></div><div class="MsoNormal"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-XmneocUtKlk/UGX2ikq4qRI/AAAAAAAAFNI/qa_BNCk8gsc/s1600/bressoudOCT1.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://1.bp.blogspot.com/-XmneocUtKlk/UGX2ikq4qRI/AAAAAAAAFNI/qa_BNCk8gsc/s400/bressoudOCT1.jpg" height="243" width="400" /></a></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-family: "Times New Roman","serif";">These percentages were calculated by determining the percentages for each of the four types of postsecondary institutions as classified by CBMS. We refer to universities that offer a doctorate in mathematics as research universities, those for which the highest degree in mathematics is the master’s as masters universities. If bachelor’s is the highest degree we call it an undergraduate college, and if associate’s is the highest degree we call it a two-year college. The distributions were then weighted according to the number of students who were enrolled in Calculus I that fall: 110,000 at research universities, 41,000 at masters universities, 82,000 at undergraduate colleges, and 65,000 at two-year colleges.<o:p></o:p></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-family: "Times New Roman","serif";">It is interesting to look at the data by type of institution. Only at research universities and two-year colleges do those heading into engineering outnumber those with an intended major in the biological or life sciences. Even there, the majors that normally require a full year of single variable calculus account for less than half of the Calculus I students. See Figures 2 and 3. <o:p></o:p></span></div><div class="MsoNormal"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-or5LQjloDyo/UGX2nsfprXI/AAAAAAAAFNQ/itSv26_jqUM/s1600/bressoudOCT2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://3.bp.blogspot.com/-or5LQjloDyo/UGX2nsfprXI/AAAAAAAAFNQ/itSv26_jqUM/s400/bressoudOCT2.jpg" height="295" width="400" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-qxe63y4Nbe0/UGX2rj6GbZI/AAAAAAAAFNY/5aLLb6NwCCE/s1600/bressoudOCT3.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://4.bp.blogspot.com/-qxe63y4Nbe0/UGX2rj6GbZI/AAAAAAAAFNY/5aLLb6NwCCE/s400/bressoudOCT3.jpg" height="267" width="400" /></a></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-family: "Times New Roman","serif";">These are also the only types of institution where students going into science, technology, engineering, or mathematics (STEM) fields constitute over 75% of all Calculus I students. The biological sciences are much more dominant, and students are significantly less likely to be intending a STEM major, at masters universities and undergraduate colleges. See Figures 4 and 5.<o:p></o:p></span></div><div class="MsoNormal"><span style="font-family: "Times New Roman","serif";"><br /></span></div><div class="MsoNormal"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-R7KqRcsz5G8/UGX2yEND4qI/AAAAAAAAFNg/ddp4YwIDXfQ/s1600/bressoudOCT4.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://4.bp.blogspot.com/-R7KqRcsz5G8/UGX2yEND4qI/AAAAAAAAFNg/ddp4YwIDXfQ/s400/bressoudOCT4.jpg" height="276" width="400" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="MsoNormal"><b><span style="font-family: "Times New Roman","serif";"><br /></span></b></div><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-pjfB4bhgYA8/UGX22X1UJsI/AAAAAAAAFNo/rvIIicsNYRM/s1600/bressoudOCT5.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://4.bp.blogspot.com/-pjfB4bhgYA8/UGX22X1UJsI/AAAAAAAAFNo/rvIIicsNYRM/s400/bressoudOCT5.jpg" height="276" width="400" /></a></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-family: "Times New Roman","serif";">The following graphs show the distribution of intended careers for women, Asian-American students, Black students, and Hispanic students. See Figures 6–9.<o:p></o:p></span></div><div class="MsoNormal"><span style="font-family: "Times New Roman","serif";"><br /></span></div><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-myJz1npN4Sc/UGX28LdEA5I/AAAAAAAAFNw/j0-CilEI87A/s1600/bressoudOCT6.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://2.bp.blogspot.com/-myJz1npN4Sc/UGX28LdEA5I/AAAAAAAAFNw/j0-CilEI87A/s400/bressoudOCT6.jpg" height="250" width="400" /></a></div><div class="MsoNormal"><span style="font-family: "Times New Roman","serif";"><br /></span></div><div class="MsoNormal"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-6l45-K7i5SI/UGX3BSXnAyI/AAAAAAAAFN4/x71nW5JveXw/s1600/bressoudOCT7.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://3.bp.blogspot.com/-6l45-K7i5SI/UGX3BSXnAyI/AAAAAAAAFN4/x71nW5JveXw/s400/bressoudOCT7.jpg" height="255" width="400" /></a></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-A2zhJjSoPow/UGX3G8PHAII/AAAAAAAAFOA/kHSz8AjpVY4/s1600/bressoudOCT8.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://3.bp.blogspot.com/-A2zhJjSoPow/UGX3G8PHAII/AAAAAAAAFOA/kHSz8AjpVY4/s400/bressoudOCT8.jpg" height="240" width="400" /></a></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-28U7PHnEjVU/UGX3NQfIIOI/AAAAAAAAFOI/XHaSOFZNaoE/s1600/bressoudOCT9.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://1.bp.blogspot.com/-28U7PHnEjVU/UGX3NQfIIOI/AAAAAAAAFOI/XHaSOFZNaoE/s400/bressoudOCT9.jpg" height="242" width="400" /></a></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-family: "Times New Roman","serif";">The variation in distributions is most dramatic for women. A women in Calculus I is three times as likely to be headed into the biological sciences as into engineering. It may come as a surprise to some that Asian-American students in Calculus I are almost twice as likely to be majoring in the biological or life sciences as engineering, but this follows a general trend of Asian-American students out of engineering and into biology. While Asian-Americans are still very well represented in engineering, making up over 12% of the bachelor’s degrees in engineering while they are only 7% of all bachelor’s degrees earned in the United States, Asian-Americans constitute almost 18% of the bachelor’s degrees in the biological sciences. See Figure 10.</span></div><div class="MsoNormal"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-wZDtdWnWB3U/UGX3Q5iQ0ZI/AAAAAAAAFOQ/2GCGtFt9LxQ/s1600/bressoudOCT10.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://1.bp.blogspot.com/-wZDtdWnWB3U/UGX3Q5iQ0ZI/AAAAAAAAFOQ/2GCGtFt9LxQ/s400/bressoudOCT10.jpg" height="272" width="400" /></a></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-family: "Times New Roman","serif";">Of course, these data are skewed by the fact that many students, especially many of those going into engineering or the physical or the mathematical sciences, never take Calculus I in college. They begin their college mathematics at the level of Calculus II or higher. Unfortunately, there are no good estimates for the size of this population. But that still leaves the question that we have addressed at Macalester: Why teach Calculus I as if it is the first half of a year-long course when—for most of the students who take it—the next calculus course is not required or even expected?</span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-family: "Times New Roman","serif";">[1]</span><span style="font-family: "Times New Roman","serif"; font-size: 11.0pt;"> </span><span style="font-family: "Times New Roman","serif";">Bressoud, D.M. 2011. The Calculus I Student. <i>Launchings</i>. <o:p></o:p></span></div><div class="MsoNormal"><a href="http://www.maa.org/columns/launchings/launchings_05_11.html"><span style="font-family: "Times New Roman","serif";">http://www.maa.org/columns/launchings/launchings_05_11.html</span></a><span style="font-family: "Times New Roman","serif";"><o:p></o:p></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-family: "Times New Roman","serif";">[2] Bressoud, D.M. 2011. The Calculus I Instructor. <i>Launchings</i>. <o:p></o:p></span></div><div class="MsoNormal"><a href="http://www.maa.org/columns/launchings/launchings_06_11.html"><span style="font-family: "Times New Roman","serif";">http://www.maa.org/columns/launchings/launchings_06_11.html</span></a><span style="font-family: "Times New Roman","serif";"><o:p></o:p></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-family: "Times New Roman","serif";">[3] </span>D. Bressoud, M. Carlson, V. Mesa, C. Rasmussen. 2012. <a href="http://www.maa.org/cspcc/CSPCC4IJMEST-12-09-18.pdf">Description of and Selected Results from the MAA National Study of Calculus</a>(pdf). Submitted to<em><span style="font-family: "Cambria","serif"; mso-ascii-theme-font: minor-latin; mso-hansi-theme-font: minor-latin;"> International Journal of Mathematical Education in Science and Technology</span></em>. <span style="font-family: "Times New Roman","serif";"><o:p></o:p></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-family: "Times New Roman","serif";">[4] G.B. Thomas. 1951. <i>Calculus and Analytic Geometry</i>. Addison-Wesley. Reading, MA.<o:p></o:p></span></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><span style="font-family: "Times New Roman","serif";">[5] National Center for Education Statistics (NCES). 2011. </span><a href="http://nces.ed.gov/programs/digest/"><i><span style="font-family: "Times New Roman","serif";">Digest of Education Statistics</span></i></a><span style="font-family: "Times New Roman","serif";">. US Department of Education. Washington, DC.<o:p></o:p></span><br /><span style="font-family: "Times New Roman","serif";"><br /></span><span style="background-color: white; color: #222222;"><span style="font-family: inherit;"><i>The MAA national study of calculus, Characteristics of Successful Programs in College Calculus, is funded by NSF grant no. 0910240. The opinions expressed in this column do not necessarily reflect those of the National Science Foundation.</i></span></span></div>Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com0tag:blogger.com,1999:blog-7251686825560941361.post-81072385108877203922012-09-01T09:00:00.000-04:002012-09-04T14:57:30.605-04:00Teaching and Learning for Transference<b id="internal-source-marker_0.12387427757494152" style="font-weight: normal;"><span style="font-family: inherit;"><span style="vertical-align: baseline; white-space: pre-wrap;">The question whether algebra deserves its prominent role in the high school curriculum was raised once again on July 28 by Andrew Hacker in a </span><span style="font-style: italic; vertical-align: baseline; white-space: pre-wrap;">New York Times</span><span style="vertical-align: baseline; white-space: pre-wrap;"> opinion column, </span><a href="http://www.nytimes.com/2012/07/29/opinion/sunday/is-algebra-necessary.html?_r=1&nl=todaysheadlines&emc=edit_th_20120729"><span style="color: blue; vertical-align: baseline; white-space: pre-wrap;">“Is Algebra Necessary?’’</span></a><span style="vertical-align: baseline; white-space: pre-wrap;"> [1]. His piece echoes the argument made a year ago by Sol Garfunkel and David Mumford in the opinion pages of the </span><span style="font-style: italic; vertical-align: baseline; white-space: pre-wrap;">New York Times</span><span style="vertical-align: baseline; white-space: pre-wrap;">, “</span><a href="http://www.nytimes.com/2011/08/25/opinion/how-to-fix-our-math-education.html?scp=1&sq=fix%20our%20math%20education&st=cse"><span style="color: blue; vertical-align: baseline; white-space: pre-wrap;">How to Fix our Math Education</span></a><span style="vertical-align: baseline; white-space: pre-wrap;">” [2]. It also resonates with Mike Shaughnessy’s comments in his </span><span style="font-style: italic; vertical-align: baseline; white-space: pre-wrap;">President’s Column</span><span style="vertical-align: baseline; white-space: pre-wrap;"> for the NCTM newsletter, “</span><a href="http://www.nctm.org/about/content.aspx?id=182"><span style="color: blue; vertical-align: baseline; white-space: pre-wrap;">Endless Algebra—the Deadly Pathway from High School Mathematics to College Mathematics</span></a><span style="vertical-align: baseline; white-space: pre-wrap;">’’ [3].</span><br /><span style="vertical-align: baseline; white-space: pre-wrap;"></span><br /><span style="vertical-align: baseline; white-space: pre-wrap;">There have been many thoughtful responses to the Hacker article. For a clear explanation why algebra is important, the best is still Zal Usiskin’s piece from 1995, “Why is Algebra Important to Learn?” [4]. Daniel Willingham posted a good reply to Hacker on his blog, “</span><a href="http://www.danielwillingham.com/1/post/2012/07/yes-algebra-is-necessary.html"><span style="color: blue; vertical-align: baseline; white-space: pre-wrap;">Yes, algebra is necessary</span></a><span style="vertical-align: baseline; white-space: pre-wrap;">” [5]. In a </span><a href="http://www.macalester.edu/~bressoud/misc/Hacker%25Reply.pdf"><span style="color: blue; vertical-align: baseline; white-space: pre-wrap;">private communication</span></a><span style="vertical-align: baseline; white-space: pre-wrap;"> [6], Dan Kennedy of the Baylor School in Chattanooga describes the beauty of algebra and laments the fact that we are doing such a very poor job of communicating that beauty.</span><br /><span style="vertical-align: baseline; white-space: pre-wrap;"></span><br /><span style="vertical-align: baseline; white-space: pre-wrap;">I want to focus this column on a theme raised in Lynn Steen’s response, “</span><a href="http://maa.org/pubs/FOCUSoct-nov12_Steen.html"><span style="color: blue; vertical-align: baseline; white-space: pre-wrap;">Reflections on Mathematics and Democracy</span></a><span style="vertical-align: baseline; white-space: pre-wrap;">” [7]. His article was conceived as a reaction to the Garfunkel and Mumford editorial, but he also discusses Hacker. As Steen points out, Hacker’s argument is not that algebra is not important; it is that algebra is not working in the curriculum. The problem—as Steen distills it—is the fact that many if not most students never learn how to transfer the knowledge and skills that are taught in algebra. They are trapped within a perception of algebra as a system of arcane manipulations with no relevance to anything outside the mathematics classroom.</span><br /><span style="vertical-align: baseline; white-space: pre-wrap;"></span><br /><span style="vertical-align: baseline; white-space: pre-wrap;">Steen’s solution focuses on the curriculum: embedding applications into mathematics courses, team-teaching cross-disciplinary courses, and employing project- or career-focused curricula. I fully agree that the curriculum can be and has been an obstacle to learning how to transfer one’s knowledge and skills. I also recognize that the problem is greater than just the curriculum. </span><br /><span style="vertical-align: baseline; white-space: pre-wrap;"></span><br /><span style="vertical-align: baseline; white-space: pre-wrap;">Teaching for transference is one of the most difficult tasks we face as educators. I would like to share two personal stories that serve as touchstones for me.</span><br /><span style="vertical-align: baseline; white-space: pre-wrap;"></span><br /><span style="vertical-align: baseline; white-space: pre-wrap;">In my last year at Penn State, 1993–94, I taught a yearlong honors course in calculus using the </span><span style="font-style: italic; vertical-align: baseline; white-space: pre-wrap;">Project CALC</span><span style="vertical-align: baseline; white-space: pre-wrap;"> materials developed by David Smith and Lang Moore. CALC stands for </span><span style="font-style: italic; vertical-align: baseline; white-space: pre-wrap;">Calculus As a Laboratory Course</span><span style="vertical-align: baseline; white-space: pre-wrap;">. The course met five days a week, two of those days in computer labs where the students explored and applied the ideas of calculus. The classroom was used both to prepare students for the laboratories and to reflect on and distill what had happened there. Some students put up initial resistance to such an unconventional approach, and a few switched to a regular section at winter break, but most came to enjoy learning this way. They believed they were getting much more than they would have from traditional instruction.</span><br /><span style="vertical-align: baseline; white-space: pre-wrap;"></span><br /><span style="vertical-align: baseline; white-space: pre-wrap;">Midway through spring semester, one of my students came breathlessly to my office. She had just completed an engineering exam where, as she told me, she had forgotten the formula needed to solve one of the problems. But then she remembered what she had learned in my course, and she figured out how to solve the problem with what she knew from calculus. </span><br /><span style="vertical-align: baseline; white-space: pre-wrap;"></span><br /><span style="vertical-align: baseline; white-space: pre-wrap;">I do not know what the problem was or what tools she used, but she made it clear that she was relying on the ideas, on the conceptual knowledge she had acquired. I knew then that my job with her was done. Whatever specific information she might still learn from me, nothing would equal the power that came from recognizing that she did not have to rely on memorized procedures, that she was capable of applying the principles of calculus to derive solutions to problems that mattered to her. Not that she always would do this, but now she knew that she could.</span><br /><span style="vertical-align: baseline; white-space: pre-wrap;"></span><br /><span style="vertical-align: baseline; white-space: pre-wrap;">The second story is about myself. During the spring of 7</span><span style="vertical-align: super; white-space: pre-wrap;">th</span><span style="vertical-align: baseline; white-space: pre-wrap;"> grade, I met periodically after school with my math teacher, Mr. Checkley. He presented and challenged me with bits of mathematics. One of these consisted of the rules for determining divisibility by 3, 9, or 11 by considering the digits of the integer. He asked me to find an explanation why these rules work. I struggled, convincing myself that they are always valid but unable to frame a proof. </span><br /><span style="vertical-align: baseline; white-space: pre-wrap;"></span><br /><span style="vertical-align: baseline; white-space: pre-wrap;">The following year I took Algebra I. I had some difficulty at first with this strange kind of mathematics, as everyone did, until I realized that what I was learning in this class was exactly the language that I needed to provide Mr. Checkley with his proof. Once I knew that algebra is simply a language for exploring and explaining mathematical patterns, a language that I could use to answer mathematical questions of interest to me, it became easy.</span><br /><span style="vertical-align: baseline; white-space: pre-wrap;"></span><br /><span style="vertical-align: baseline; white-space: pre-wrap;">There are several lessons that I take from these experiences. First, transference need not be to a real-world application. Nor is it about the need to use one’s mathematical knowledge in a career. It is important because of the power that comes from discovering that one can rely on one’s own reasoning to recover a forgotten formula or uncover the logic behind an unexpected pattern. </span><br /><span style="vertical-align: baseline; white-space: pre-wrap;"></span><br /><span style="vertical-align: baseline; white-space: pre-wrap;">Second, I have learned that what works in a particular setting with one instructor and a particular group of students will not necessarily work when these parameters are changed. My Penn State class consisted of University Scholars, selected from the top 3% of the student body. I was highly motivated to make it work. </span><span style="font-style: italic; vertical-align: baseline; white-space: pre-wrap;">Project Calc</span><span style="vertical-align: baseline; white-space: pre-wrap;"> was used at Duke with mixed results. No curriculum by itself will ever be sufficient.</span><br /><span style="vertical-align: baseline; white-space: pre-wrap;"></span><br /><span style="vertical-align: baseline; white-space: pre-wrap;">Third and finally, learning for transference is a process that takes time. A student begins with new knowledge, either acquired through personal discovery or introduced and explained by a teacher. This is followed by an opportunity to apply this knowledge in a fresh context. The next steps are critical and far too often neglected. The student must now reflect on what worked and what difficulties were encountered, then distill the significant features that made it work. Even now the process is not done. There must be a fresh attempt at application, followed once again by reflection and distillation. I have found that most students need to progress through this cycle several times before they have real control of a new piece of knowledge. I still struggle with the difficulties of engaging my students in this process while balancing the demands of the course. A continuing challenge for me is to decide which concepts deserve this much attention.</span><br /><span style="vertical-align: baseline; white-space: pre-wrap;"></span><br /><span style="vertical-align: baseline; white-space: pre-wrap;">I work with privileged and highly motivated students. The difficulties inherent in accomplishing learning for transference in our public schools are far greater, but the goal should be the same. Lynn Steen has asked us to “organiz[e] the curriculum to pay greater attention to the goal of transferable knowledge and skills.” I would go beyond this. We need to organize the very way we teach so that we keep this end in mind.</span><br /><span style="vertical-align: baseline; white-space: pre-wrap;"></span><br /><span style="vertical-align: baseline; white-space: pre-wrap;">[1] Hacker, A. 2012. Is algebra necessary? </span><span style="font-style: italic; vertical-align: baseline; white-space: pre-wrap;">New York Times</span><span style="vertical-align: baseline; white-space: pre-wrap;">. July 28. </span><a href="http://www.nytimes.com/2012/07/29/opinion/sunday/is-algebra-necessary.html?_r=1&smid=fb-share"><span style="color: blue; vertical-align: baseline; white-space: pre-wrap;">http://www.nytimes.com/2012/07/29/opinion/sunday/is-algebra-necessary.html?_r=1&smid=fb-share</span></a><br /><a href="http://www.nytimes.com/2012/07/29/opinion/sunday/is-algebra-necessary.html?_r=1&smid=fb-share"><span style="color: blue; vertical-align: baseline; white-space: pre-wrap;"></span></a><br /><span style="vertical-align: baseline; white-space: pre-wrap;">[2] Garfunkel, S. and Mumford, D. 2011. How to fix our math education. </span><span style="font-style: italic; vertical-align: baseline; white-space: pre-wrap;">New York Times</span><span style="vertical-align: baseline; white-space: pre-wrap;">. August 24. </span><a href="http://www.nytimes.com/2011/08/25/opinion/how-to-fix-our-math-education.html?scp=1&sq=fix%20our%20math%20education&st=cse"><span style="color: blue; vertical-align: baseline; white-space: pre-wrap;">http://www.nytimes.com/2011/08/25/opinion/how-to-fix-our-math-education.html?scp=1&sq=fix%20our%20math%20education&st=cse</span></a><span style="font-style: italic; vertical-align: baseline; white-space: pre-wrap;"> </span><br /><span style="font-style: italic; vertical-align: baseline; white-space: pre-wrap;"></span><br /><span style="vertical-align: baseline; white-space: pre-wrap;">[3] Shaughnessy, J.M. 2011</span><span style="font-style: italic; vertical-align: baseline; white-space: pre-wrap;">. </span><span style="vertical-align: baseline; white-space: pre-wrap;">Endless Algebra—the Deadly Pathway from High School Mathematics to College Mathematics. </span><span style="font-style: italic; vertical-align: baseline; white-space: pre-wrap;">Summing Up</span><span style="vertical-align: baseline; white-space: pre-wrap;">. February 2. National Council of Teachers of Mathematics. Reston, VA. </span><a href="http://www.nctm.org/about/content.aspx?id=182"><span style="color: blue; vertical-align: baseline; white-space: pre-wrap;">www.nctm.org/about/content.aspx?id=182</span></a><span style="vertical-align: baseline; white-space: pre-wrap;"> </span><br /><span style="vertical-align: baseline; white-space: pre-wrap;"></span><br /><span style="vertical-align: baseline; white-space: pre-wrap;">[4] Usiskin, Z. 1995. Why is algebra important to learn? </span><span style="font-style: italic; vertical-align: baseline; white-space: pre-wrap;">American Educator</span><span style="vertical-align: baseline; white-space: pre-wrap;">. Vol. 19. pp 30–37.</span><br /><span style="vertical-align: baseline; white-space: pre-wrap;"></span><br /><span style="vertical-align: baseline; white-space: pre-wrap;">[5] Willingham, D. 2012. Yes, algebra is necessary. 7/30/2012. </span><a href="http://www.danielwillingham.com/1/post/2012/07/yes-algebra-is-necessary.html"><span style="color: blue; vertical-align: baseline; white-space: pre-wrap;">http://www.danielwillingham.com/1/post/2012/07/yes-algebra-is-necessary.html</span></a><br /><a href="http://www.danielwillingham.com/1/post/2012/07/yes-algebra-is-necessary.html"><span style="color: blue; vertical-align: baseline; white-space: pre-wrap;"></span></a><br /><span style="vertical-align: baseline; white-space: pre-wrap;">[6] Kennedy, D. 2012. Response to Hacker’s editorial. (private communication posted with permission of the author) </span><a href="http://www.macalester.edu/~bressoud/misc/Hacker%25Reply.pdf"><span style="color: blue; vertical-align: baseline; white-space: pre-wrap;">http://www.macalester.edu/~bressoud/misc/Hacker%Reply.pdf</span></a><br /><a href="http://www.macalester.edu/~bressoud/misc/Hacker%25Reply.pdf"><span style="color: blue; vertical-align: baseline; white-space: pre-wrap;"></span></a><br /><span style="vertical-align: baseline; white-space: pre-wrap;">[7] Steen, L.A. 2012. Reflections on mathematics and democracy. To appear in </span><span style="font-style: italic; vertical-align: baseline; white-space: pre-wrap;">MAA Focus</span><span style="vertical-align: baseline; white-space: pre-wrap;">. </span><a href="http://maa.org/pubs/FOCUSoct-nov12_Steen.html"><span style="color: blue; vertical-align: baseline; white-space: pre-wrap;">http://maa.org/pubs/FOCUSoct-nov12_Steen.html</span></a></span></b>Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com0