tag:blogger.com,1999:blog-72516868255609413612017-06-24T02:29:03.357-04:00Launchings 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.comBlogger72125tag:blogger.com,1999:blog-7251686825560941361.post-30502704421560068612017-06-01T07:00:00.000-04:002017-06-01T07:00:28.066-04:00Re-imagining the Calculus Curriculum, II<b><i>You can follow me on Twitter <a href="https://twitter.com/dbressoud" target="_blank">@dbressoud</a>.</i></b><br /><br />Last month, in "<a href="http://launchings.blogspot.com/2017/05/re-imagining-calculus-curriculum-i.html" target="_blank">Re-imagining the Calculus Curriculum</a>," I, I introduced <a href="http://patthompson.net/ThompsonCalc/index.html" target="_blank">Project DIRACC</a> (<i>Developing and Investigating a Rigorous Approach to Conceptual Calculus</i>), developed by Pat Thompson, Mark Ashbrook, and Fabio Milner at Arizona State University. References to the theory underpinning this approach are given at the end of this column. This month’s column will expand on some details of this curriculum. <br /><br />One of the first common student misconceptions that Project DIRACC tackles is that variables are simply stand-ins for unknown quantities. The authors begin the meat of his course in Chapter 3 with an explanation of the distinction between <i>variable</i>,<i> constant</i>, and <i>parameter</i>, pointing out how context-specific the designations as either variable or parameter can be. One of the distinctive features of this project is the thoughtful use of technology, in this case enabling students to play with the effect of varying a variable with a variety of choices of parameter (see <a href="http://patthompson.net/ThompsonCalc/section_3_1.html">patthompson.net/ThompsonCalc/section_3_1.html</a>). <br /><br />This leads to relationships between variables (how volume varies with height), and then functions as a special class of relationships between variables, one in which “<i><b>any value of one variable determines exactly one value of the other</b></i>.” The point is that the <i>f</i> in <i>f</i> (<i>x</i>) has meaning. It is the name of the relationship. This enables the authors to tackle the misconception that <i>f</i> (<i>x</i>) is simply a lengthy way of expressing the variable <i>y</i>. <br /><br />While acknowledging that <i>f</i>(<i>x</i>) can represent a second variable, they emphasize that it is shorthand for “the value of the relationship f when applied to a value of <i>x</i>.” This point is driven home by an example of the usefulness of functional notation. If <i>d</i>(<i>x</i>) relates a moment in time, <i>x</i> measured in years, to the distance between the Earth and the Moon at that time, then <i>d</i>(<i>x</i>) – <i>d</i>(<i>x</i>–5) enables us to express the change in distance over the five years before time x, while <i>d</i>(<i>x</i>+5) – <i>d</i>(<i>x</i>) expresses the change in distance over the succeeding five years. <br /><br />The authors also make the important distinction between functions defined conceptually—the distance between Earth and Moon at a given time—and those defined computationally, such as <i>V</i>(<i>u</i>) = <i>u</i>(13.76 – 2<i>u</i>)(16.42 – 2<i>u</i>). They then proceed to devote considerable effort to describing the structure of functions as they are built from sums, products, quotients, compositions, and inverses. This includes clarifying the distinction between the independent variable and the argument of a function. Thus for f (<i>x</i>/3 + 5) the independent variable is <i>x</i>, but the function argument is<i> x</i>/3 + 5, an important step toward understanding composition of functions. <br /><br />While function structure <i>should</i> be part of precalculus, the importance of including this material has been revealed in exploring student difficulties with differentiation. Given a complicated computational rule that defines a function, students often have difficulty parsing this rule and thus determining the choice and order of the techniques of differentiation they need to use. <br /><br />Rates of change are now introduced in Chapter 4. The authors distinguish between ∆x, the parameter that describes the length of a small subinterval of the domain, and the changes in x and y represented by the differentials dx and dy. These are variables that within the given subinterval are always connected by a linear relationship. <br /><br />A nice illustration of how this works is given with a photograph of a truck traveling through an intersection (Figure 1). <br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://4.bp.blogspot.com/-UG6YvHpIQrA/WS83mi8QEsI/AAAAAAAAK1Y/kgOGfpWSIa0ZMorjShP1QjVeSxvFDadqwCLcB/s1600/truck%2Bmoving%2Blaunchings.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="700" data-original-width="1050" height="213" src="https://4.bp.blogspot.com/-UG6YvHpIQrA/WS83mi8QEsI/AAAAAAAAK1Y/kgOGfpWSIa0ZMorjShP1QjVeSxvFDadqwCLcB/s320/truck%2Bmoving%2Blaunchings.jpg" width="320" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><b>Figure 1.</b> A photo of truck taken with a shutter setting of 1/1000 sec.</td></tr></tbody></table>Taken at a shutter speed of 1/1000th of a second, it appears to freeze the truck. But if you zoom in on the tail light (Figure 2, see <a href="http://patthompson.net/ThompsonCalc/section_4_3.html" target="_blank">Section 4.3</a> for a video of the zoom), the streaks reveal that the truck was moving.<br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://1.bp.blogspot.com/-HSbEuBPmB1U/WS83Vnr_M4I/AAAAAAAAK1U/rUo9PzVaq7U6DULmUiTeyKLX4iwXJwwUgCLcB/s1600/blurred%2Btruck%2Blaunchings.png" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" data-original-height="243" data-original-width="263" src="https://1.bp.blogspot.com/-HSbEuBPmB1U/WS83Vnr_M4I/AAAAAAAAK1U/rUo9PzVaq7U6DULmUiTeyKLX4iwXJwwUgCLcB/s1600/blurred%2Btruck%2Blaunchings.png" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><b>Figure 2.</b> A closer look at the truck's tail light shows small streaks. <br />The truck moved slightly while the camera's shutter was open.</td></tr></tbody></table><br /><br />One can even estimate the length of the streaks to approximate the velocity of the truck. Over 1/1000th of a second, it is doubtful that the truck’s velocity changed very much. The picture of the truck was taken at a “moment” in time, but that moment stretched over 0.001 seconds. The point is that this period of time is short enough that the truck’s velocity measured as change in distance over change in time is “essentially constant.” If <i>y</i> is position and <i>x</i> is time, then over this interval of length <i>∆x</i> = 0.001 seconds, we can treat the variable <i>dy</i> as a constant times <i>dx</i>. It is this constant that is used to define the rate of change at a moment, <br /><br /><blockquote>We say that <b>a function has a rate of change at the moment x<sub>0</sub> if, over a suitably small interval of its independent variable containing x<sub>0</sub>, the function’s value changes at essentially a constant rate with respect to its independent variable.</b></blockquote><br />Significantly, even as the authors are defining the rate of change at a moment, they emphasize that “all motion, and hence all variation, is blurry.” <br /><br />Note that there is no mention of limits, a means of defining the derivative that is often more confusing than enlightening (see the 2014 <i>Launchings</i> columns from <a href="http://launchings.blogspot.com/2014/06/beyond-limit-i.html">July</a>, <a href="http://launchings.blogspot.com/2014/08/beyond-limit-ii.html">August</a>, and <a href="http://launchings.blogspot.com/2014/09/beyond-limit-iii.html">September</a>). <br /><br />After further discussion and exploration of rate of change functions, the authors now move in Chapter 5 to Accumulation Functions, building up total changes from rates of change that are essentially constant on very small intervals. These give rise to what are anachronistically referred to as left-hand Riemann sums. Students use technology to explore the increasing accuracy as ∆x gets smaller. The effect of the choice of starting value is noted, and the definite integral with a variable upper limit now appears. It is important that the first time students see a definite integral it has a variable upper limit. <br /><br />In Chapter 6, the inverse problem, going from knowledge of an exact expression of the accumulation function to the discovery of the corresponding rate of change function, is now explored, leading to the Fundamental Theorem of Integral Calculus in the form: The derivative with respect to x of the definite integral from a to x of a rate of change function is equal to that rate of change function evaluated at x. Techniques and applications of differentiation follow as the semester concludes. <br /><br />The great strength and promise of this approach is that the traditional content of the first semester of calculus is only slightly tweaked, especially since it is increasingly common for university Calculus I courses to avoid or significantly downplay limits. But the curriculum has been totally reshaped to address common student difficulties and misconceptions. This route into calculus has the added advantage—though perhaps a disadvantage in the eyes of some students—that those who have been through a procedurally oriented course are unlikely to recognize this as an accelerated repetition of what they have already studied. It will challenge them to rethink what they believe calculus to be. <br /><br />References <br /><br />Thompson, P.W. and Silverman, J. (2008). The concept of accumulation in calculus. In M.P. Carlson & C. Rasmussen (Eds.), <i>Making the connection: Research and teaching in undergraduate mathematics</i> (<i>MAA Notes </i>Vol. 73, pp. 43–52). Washington, DC: Mathematical Association of America. <br /><br />Thompson, P.W., Byerley, C. and Hatfield, N. (2013). A conceptual approach to calculus made possible by technology. <i>Computers in the Schools</i>. 30:124–147. <br /><br />Thompson, P.W. and Dreyfus, T. (2016). A coherent approach to the Fundamental Theorem of Calculus using differentials. In R. Göller. R. Biehler & R. Hochsmuth (Eds.), <i>Proceedings of the Conference on Didactics of Mathematics in Higher Education as a Scientific Discipline</i> (pp. 355–359 ) Hannover, Germany: KHDM. <br /><br /><br /><br /><br /><br /><br />Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com0tag:blogger.com,1999:blog-7251686825560941361.post-64015698074392957852017-05-01T08:00:00.000-04:002017-05-01T08:00:05.679-04:00Re-imagining the Calculus Curriculum, I<b><i>You can now follow me on Twitter <a href="https://twitter.com/dbressoud" target="_blank">@dbressoud</a>.</i></b><br /><br />I was recently asked about calculus instruction: Which is easier, reforming pedagogy or curriculum? The answer is easy: pedagogy. This is not to say that it is easy to change how we teach this course, but it is far easier than trying to change what we teach. The order and emphasis of topics that emerged in the 1950s has proven extremely hard to shift. <br /><br />Not that we have not tried. During the Calculus Reform movement of the late 1980s and early 1990s, NSF encouraged curricular innovation. Several of these efforts adopted an emphasis on modeling dynamical systems, introducing calculus via differential equations and developing the tools of calculus in service to this vision. This provides wonderful motivation, and this approach survives in a few pockets. It is how we teach calculus at Macalester, and the U.S. Military Academy at West Point has successfully used this route into calculus for over a quarter century. But despite its appeal, this curriculum necessitates modifying the entire year of single variable calculus, raising problems for institutions that must accommodate students who are transferring in or out. It also is a difficult sell to those who worry about “coverage” since it requires devoting considerable time to topics—modeling with differential equations, functions of several variables, and partial differential equations—that receive little or no attention in the traditional course. <br /><br />This month, I want to talk about a promising curricular innovation that Pat Thompson at Arizona State University has been developing in collaboration with Fabio Milner and Mark Ashbrook, Project DIRACC (<i>Developing and Investigating a Rigorous Approach to Conceptual Calculus</i>). It has the advantage that it fits more easily into what is expected from each semester. It has been under development since 2010 and is slated to be the curriculum used for all Calculus I sections for mathematics or science majors at ASU beginning in fall, 2018. <br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-vWQ0GY9ER08/WQOmpcsjZdI/AAAAAAAAK1E/iDVvhbOe9s4GwhGk_bavZC-Xxg-AKIpaQCLcB/s1600/launchings%2Bfaces.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="161" src="https://2.bp.blogspot.com/-vWQ0GY9ER08/WQOmpcsjZdI/AAAAAAAAK1E/iDVvhbOe9s4GwhGk_bavZC-Xxg-AKIpaQCLcB/s400/launchings%2Bfaces.JPG" width="400" /></a></div><br />Thompson began with research into the misconceptions that students carry into calculus and that impede their ability to understand it. I quote these common misconceptions from his website (<a href="http://patthompson.net/ThompsonCalc/About.html">patthompson.net/ThompsonCalc/About.html</a>).<br /><ul><li>Calculus, like the school mathematics, is about rules and procedures. Students think that calculus is difficult primarily because there are so many rules and procedures. </li><li>Variables do not vary. Therefore rate of change is not about change. </li><li>Integrals are areas under a curve. Students wonder, "How can an area represent a distance or an amount of work?" </li><li>Average rate of change has little to do with rate of change. It is about the direction of a line that passes through two points on a graph. </li><li>A tangent is a line that "just touches" a curve. </li><li>Derivative is a slope of a tangent. The net result is that, in students' understandings, derivatives are not about rates of change.</li></ul>The second bullet point is particularly common and problematic. In an expression such as <br /><div style="text-align: center;"><i style="text-align: start;">f</i><span style="text-align: start;">(</span><i style="text-align: start;">x</i><span style="text-align: start;">)</span> = <i>x</i><sup>3</sup> – 3<i>x</i> + 2, </div><br />many students see the expression <i>f</i>(<i>x</i>) as nothing more than a lengthy way of writing the dependent variable, and functions are seen as static objects that prescribe how to turn the input x into the output <i>f</i>(<i>x</i>). With this mindset, differentiation and integration are nothing more than arcane rules for turning one static object into another. <br /><br />Choosing to define integrals as areas and derivatives as slopes, as is common in the standard curriculum, is equally problematic. It reinforces the notion that calculus is about computing values associated with geometric objects. To complicate matters, while area is a familiar concept, slope is far less real or meaningful to our students. Too many students never come to the realization that the real power of differentiation and integration arises from their interpretation as rate of change and as accumulation. <br /><br />While the earliest uses of accumulation were for determining areas, those Hellenistic philosophers who mastered it also recognized its equal applicability to questions of volumes and moments. By the 14th century, European philosophers were applying techniques of accumulation to the problem of determining distance from knowledge of instantaneous velocity. None of these come easily to students who are fixated on integrals as areas. <br /><br />Seeing the derivative as a slope is even more problematic, a static value of an obscure parameter. Differentiation arose from problems of interpolation for the purpose of approximating values of trigonometric functions in first millenium India, in understanding the sensitivity of one variable to changes in another in the work of Fermat and Descartes, and in relating rates of change as in Napier’s analysis of the logarithm and Newton’s <i>Principia</i>. Derivative as slope came quite late in the historical development of calculus precisely because its application to interesting questions is not intuitive. <br /><br />These insights provide the starting point for Thompson’s reformation of Calculus I. His textbook, which is still a work in progress, can be accessed at <a href="http://patthompson.net/ThompsonCalc">patthompson.net/ThompsonCalc</a>. See Thompson & Silverman (2008), Thompson et al. (2013), and Thompson & Dreyfus (2016) for additional background. I find it deliciously ironic that one of the first topics he tackles is the distinction among constants, parameters, and variables. If you look at the calculus textbooks of the late 18 th century through the middle of the 19 th century, this is exactly where they started. Somehow, we lost recognition of the importance of elevating this distinction for our students. Thompson goes on to spend considerable effort to clarify the role of a function as a bridge between two co-varying quantities. And then, he really breaks with tradition by first tackling integration, which he enters via problems in accumulation.<br /><br />This accomplishes several desiderata. First, it ensures that students do not begin with an understanding of the integral as area, but as an accumulator. Second, it makes it much easier to recognize this accumulator as a function in its own right. Students struggle with recognizing the definite integral from a to the variable x as a function of x (see the section of last month’s column, <a href="http://launchings.blogspot.com/2017/04/" target="_blank">Conceptual Understanding</a>, that addresses Integration as Accumulation). Thompson begins by viewing the integrand as a rate of change function. The variable upper limit arises naturally. Third, and perhaps most important, it gives meaning to the Fundamental Theorem of Integral Calculus, that the derivative of an accumulator function is the rate of change function. <br /><br />Differentiation can then be introduced in precisely the way Newton first understood it: Given a closed expression for the accumulator function, how can we find the corresponding rate of change function? <br /><br />Next month, I will expound on exactly how Thompson introduces these steps, but for now I would like to conclude with a comparison of the two curricular innovations, that of Thompson and the approach described at the start of this column that emphasizes calculus as a tool for modeling dynamical systems. The latter does overcome the problem of student belief that the derivative is to be understood as the slope of the tangent. It brings to the fore the derivative as describing a rate of change. The problem is that it does nothing to clarify the role of the integral as an accumulator. In some sense, it makes it more difficult. As we teach calculus at Macalester, the integral is introduced purely as an anti-derivative, making it extremely difficult to give meaning to the Fundamental Theorem of Integral Calculus. I have to work very hard in the second semester to help students understand the integral as an accumulator and so justify that this theorem has meaning. In a very real sense, Thompson approach <i>begins </i>with this fundamental theorem. <br /><br />Thompson, P.W. and Silverman, J. (2008). The concept of accumulation in calculus. In M.P. Carlson & C. Rasmussen (Eds.), <i>Making the connection: Research and teaching in undergraduate mathematics</i> (MAA Notes Vol. 73, pp. 43–52). Washington, DC: Mathematical Association of America. <br /><br />Thompson, P.W., Byerley, C. and Hatfield, N. (2013). A conceptual approach to calculus made possible by technology. <i>Computers in the Schools</i>. 30:124–147. <br /><br />Thompson, P.W. and Dreyfus, T. (2016). A coherent approach to the Fundamental Theorem of Calculus using differentials. In R. Göller. R. Biehler & R. Hochsmuth (Eds.), <i>Proceedings of the Conference on Didactics of Mathematics in Higher Education as a Scientitific Discipline</i> (pp. 355–359 ) Hannover, Germany: KHDM. Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com0tag:blogger.com,1999:blog-7251686825560941361.post-3069777966555504532017-04-01T06:49:00.000-04:002017-04-01T06:49:08.064-04:00Conceptual Understanding<b><i>You can now follow me on Twitter <a href="https://twitter.com/dbressoud" target="_blank">@dbressoud</a>.</i></b><br /><br />Continuing my series of summaries of articles that have appeared in the <i>International Journal of Research in Undergraduate Mathematics Education </i>(IJRUME), this month I want to briefly describe three studies that address issues of conceptual understanding. The first is a study out of Israel that probed student difficulties in understanding integration as accumulation (Swidan and Yerushalmy, 2016). The second is from France, exploring student difficulties with understanding the real number line as a continuum (Durand-Guerrier, 2016). The final paper, from England, explores a method of measuring conceptual understanding (Bisson, Gilmore, Inglis, and Jones, 2016). <br /><br /><b>Integration as Accumulation</b><br />To use the definite integral, students need to understand it as accumulation. In particular, the Fundamental Theorem of Integral Calculus rests on the recognition that the definite integral of a function <i>f</i>, when given a variable upper limit, is an accumulation function of a quantity for which <i>f </i>describes the rate of change. Pat Thompson (2013) has described the course he developed for Arizona State University that places this realization at the heart of the calculus curriculum. <br /><br />We know that students have a difficult time understanding and working with a definite integral with a variable upper limit. The authors of the IJRUME paper suggest that much of the problem lies in the fact that when students are introduced to the definite integral as a limit of Riemann sums, they only consider the case when the upper and lower limits on the Riemann sum are fixed. The limit is thus a number, usually thought of as the area under a curve. Making the transition to the case where the upper limit is variable is thus non-intuitive. <br /><br />The authors used software to explore student recognition of accumulation functions based on right-hand Riemann sums. They investigated student recognition of how the properties of these functions are shaped by the rate of change function. The experiment involved a graphing tool, Calculus UnLimited (CUL), in which students input a function and the software provides values of the corresponding accumulation function given by a right-hand sum with Δx = 0.5 (see Figure 1). Students could adjust the upper and lower limits, in jumps of 0.5. The software displays the rectangles corresponding to a right-hand sum. Students were not told that these were points on an accumulation function, merely that this was a function related to the initial function. They were encouraged to start with a lower limit of –3 and to explore the functions <i>x<sup>2</sup></i>, <i>x<sup>2</sup></i> – 9, and then cubic polynomials, and to discover what they could about this second function. Students received no further prompts. <br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://1.bp.blogspot.com/-GKuwCq75MKE/WN1EMekOm8I/AAAAAAAAKzs/Mj1PLQg-E3M9ibeXfim_tI2nVqkhaceJwCLcB/s1600/accumulation.tiff" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="305" src="https://1.bp.blogspot.com/-GKuwCq75MKE/WN1EMekOm8I/AAAAAAAAKzs/Mj1PLQg-E3M9ibeXfim_tI2nVqkhaceJwCLcB/s400/accumulation.tiff" width="400" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><b>Figure 1: </b>The CUL interface. Taken from Swidan and Yerushalmy (2016), page 33.</td></tr></tbody></table><br />Thirteen pairs of Israeli 17-year olds participated in the study. They all had been studying derivatives and indefinite integrals, but none had yet encountered definite integrals. Each pair spent about an hour exploring this software. Their actions and remarks were video-taped and then analyzed.<br /><br />One of the interesting observations was that the key to recognizing that the second function accumulates areas came from playing with the lower limit. Adjusting the upper limit simply adds or removes points, but adjusting the lower limit moves the plotted points up or down. Once students realized that the point corresponding to the lower limit is always zero, they were able to deduce that the y-value of the next point is the area of the first rectangle, and that succeeding points reflect values obtained by adding up the areas of the rectangles. Rectangles below the <i>x</i>-axis were shaded in a darker color, and students quickly picked up that they were subtracting values. Seven of the thirteen pairs of students went as far as remarking on how the concavity of the accumulation function is related to the behavior of the original function. <br /><br />This work suggests that a Riemann sum with a variable upper limit is more intuitive than a definite integral with a variable upper limit. In addition, it appears that students can discover many of the essential properties of a discrete accumulation function if allowed the opportunity to experiment with it. <br /><br /><b>Understanding the Continuum</b><br />The second paper explores student difficulties with the properties of the real number line and describes an intervention that appears to have been useful in helping students understand the structure of the continuum. Mathematicians of the nineteenth century struggled to understand the essential differences between the continuum of all real numbers and dense subsets such as the set of rational numbers. It comes as no surprise that our students also struggle with these distinctions. <br /><br />The author analyzes the transcripts from an intervention described by Pontille et al. (1996). It began with the following question: Given an increasing function, f, ( <i>x < y</i> implies <i>f</i>(<i>x</i>) ≤ <i>f</i>(<i>y</i>) ) from an ordered set S into itself, can we conclude that there will always exist an element s in S for which <i>f</i>(<i>s</i>) = <i>s</i>? The answer, of course, depends on the set. The intervention asks students to answer this question for four sets: a finite set of positive integers, the set of numbers with finite decimal expansions in [0,1], the set of rational numbers in [0,1], and the entire set [0,1]. In the original work, this question was posed to a class of lycée students in a scientific track. Over the course of an academic year, they periodically returned to this question, gradually building a refined understanding of the structure of the continuum. The author’s analysis of the transcripts from these classroom discussions is fascinating. <br /><br />Durand-Guerrier then posed this same question to a group of students in a graduate teacher- training program. In both cases, students were able to answer the question in the affirmative for the finite set, using an inductive proof or <i>reductio ad absurdum</i>. Almost all then tried to apply this proof to the dense countable sets. Here they ran into the realization that there is no “next” number. The graduate students, given only an hour to work on this, did not get much further. The lycée students did come to doubt that it was always true for these sets. As they began to think about the “holes” these sets left, they were able to construct counter-examples. <br /><br />The continuum provides the most difficulty. The lycée students were eventually able to prove that it is true in this instance, but only after being given the hint to consider the set of <i>x </i>in [0,1] for which <i>f</i>(<i>x</i>) > <i>x</i> and to draw on the property of the continuum that every bounded set has a least upper bound. <br /><br /><b>Measuring Conceptual Understanding</b><br />The last paper in this set addresses the problem of measuring conceptual understanding. We know that students can be proficient in answering procedural questions without the least understanding of what they are doing or why they are doing it. But measuring conceptual understanding is difficult. A meaningful assessment with limited possible answers, such as a concept inventory, requires a great deal of work to develop and validate. Open-ended questions can provide a better window into student thinking and understanding, but consistent application of scoring rubrics across multiple evaluators is hard to achieve. <br /><br />The authors build a solution from the observation that it is far easier to compare the quality of the responses from two students than it is to compare one student’s response against a rubric. They therefore suggest asking a simple, very open-ended question, scored by ranking student responses, which is achieved by pairwise comparisons. As an example, to evaluate student understanding of the derivative, they provided the prompt,<br /><blockquote>Explain what a derivative is to someone who hasn’t encountered it before. Use diagrams, examples and writing to include everything you know about derivatives.</blockquote>The 42 students in this study first read several examples of situations involving velocity and acceleration (presumably to prompt them to think of derivatives as rates of change rather than a collection of procedures) and were then given 20 minutes to write their responses to the prompt. <br /><br />Afterwards, 30 graduate students each judged 42 pairings. The authors found very high inter-rater reliability (r = .826 to .907). In fact, they found that comparative judgments appeared to do a better job of evaluating conceptual understanding than did Epstein’s Calculus Concept Inventory (Epstein, 2013). <br /><br />Similar studies were undertaken to evaluate student understanding of <i>p</i>-values and 11- to 12-year- olds understanding of the use of letters in algebra. Again, there was very high inter-rater reliability, and in these cases there were high levels of agreement with established instruments. <br /><br />This approach constitutes a very broad method of assessment, but it does enable the instructor to get some idea of what students are thinking and how they understand the concept at hand. It can be used even with large classes because it is not necessary to look at all possible pairs to get a meaningful ranking. <br /><br /><b>Conclusion</b><br />The three papers referenced here are very different in focus and goal, but I do see the common thread of searching for ways to encourage and assess student understanding. After all, that is what teaching and learning is really about. <br /><br /><b>References</b><br />Bisson, M.-J., Gilmore, C., Inglis, M., and Jones, I. (2016). Measuring conceptual understanding using comparative judgement. <i>IJRUME</i>. 2:141–164. <br /><br />Durand-Guerrier, V. (2016). Conceptualization of the continuum, an educational challenge for undergraduate students. <i>IJRUME</i>. 2:338–361. <br /><br />Epstein, J. (2013). The Calculus Concept Inventory - measurement of the effect of teaching methodology in mathematics. <i>Notices of the American Mathematical Society</i>, 60, 1018–27. <br /><br />Pontille, M. C., Feurly-Reynaud, J., & Tisseron, C. (1996). Et pourtant, ils trouvent. Repères IREM, 24, 10–34. <br /><br />Swidan, O. and Yerushalmy, M. (2016). Conceptual structure of the accumulation function in an interactive and multiple-linked representational environment. <i>IJRUME</i>. 2:30–58. <br /><br />Thompson, P.W., Byerley, C., and Hatfiled, N. (2013). A Conceptual approach to calculus made possible by technology. <i>Computers in the Schools</i>. 30:124–147. Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com0tag:blogger.com,1999:blog-7251686825560941361.post-74438220249816511302017-03-01T07:30:00.000-05:002017-03-01T07:30:07.391-05:00MAA Calculus Studies: Use of Local Data<b><i>You can now follow me on Twitter <a href="https://twitter.com/dbressoud" target="_blank">@dbressoud</a>.</i></b><br /><br />From our 2012 study, <i>Characteristics of Successful Programs in College Calculus</i> (NSF #0910240), the most successful departments had a practice of monitoring and reflecting on data from their courses. When we surveyed all departments with graduate programs in 2015 as part of <i>Progress through Calculus</i> (NSF #1430540), we asked about their access to and use of these data, what we are referring to as “local data.” <br /><br />The first thing we learned is that a few departments report no access to data about their courses or what happens to their students. For almost half, access is not readily available (see Table 1). When we asked, “Which types of data does your department review on a regular basis to inform decisions about your undergraduate program?”, most departments review grade distributions and pay attention to end of term student course evaluations (Table 2). Between 40% and 50% of the surveyed departments correlate performance in subsequent courses with the grades they received in previous courses and look at how well placement procedures are being followed. Given how important it is to track persistence rates (see <a href="http://www.maa.org/external_archive/columns/launchings/launchings_01_10.html">The Problem of Persistence</a>, <i>Launchings</i>, January 2010), it is disappointing to see that only 41% of departments track these data. Regular communication with client disciplines is almost non-existent. <br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://2.bp.blogspot.com/-HUrm4JawL9M/WLWkvd2yOOI/AAAAAAAAKyE/UryNQzbO1EMDiBG1NNzyWujHZ0QPVYKeQCLcB/s1600/launchings%2Btable%2B1responses.JPG" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="118" src="https://2.bp.blogspot.com/-HUrm4JawL9M/WLWkvd2yOOI/AAAAAAAAKyE/UryNQzbO1EMDiBG1NNzyWujHZ0QPVYKeQCLcB/s400/launchings%2Btable%2B1responses.JPG" width="400" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><div style="text-align: center;">Table 1. Responses to the question, “Does your department have access to data </div><div style="text-align: center;">to help inform decisions about your undergraduate program? PhD indicates </div><div style="text-align: center;">departments that offer a PhD in Mathematics. MA indicates departments for </div><div style="text-align: center;">which the highest degree offered in Mathematics is a Master’s.</div></td></tr></tbody></table><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://3.bp.blogspot.com/-1cmyZ8aL3HA/WLWkvdKE_9I/AAAAAAAAKyI/yDcWnnIio88Zu7UGQBLnzYQrl99UaDJ-ACLcB/s1600/launchings%2Btable%2B2%2Bresponses.JPG" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="188" src="https://3.bp.blogspot.com/-1cmyZ8aL3HA/WLWkvdKE_9I/AAAAAAAAKyI/yDcWnnIio88Zu7UGQBLnzYQrl99UaDJ-ACLcB/s400/launchings%2Btable%2B2%2Bresponses.JPG" width="400" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Table 2. Responses to the question, “Which types of data does your department<br /> review on a regular basis to inform decisions about your undergraduate program?”</td></tr></tbody></table><br />We also asked departments to describe the kinds of data they collect and regularly review. Several reported combining placement scores, persistence, and grades in subsequent courses to better understand the success of their program. Some of the other interesting uses of data included universities that<br /><ul><li>Built a model of “at-risk” students in Calculus I using admissions data from the past seven years. Using it, they report “developing a program to assist these students right at the beginning of Fall quarter, rather than target them after they start to perform poorly.” </li><li>Surveyed calculus students to get a better understanding of their backgrounds and attitudes toward studying in groups. </li><li>Collected regular information from business and industry employers of their majors. </li><li>Measured correlation of grade in Calculus I with transfer status, year in college, gender, whether repeating Calculus I, and GPA. </li><li>Used data from the university’s Core Learning Objectives and a uniform final exam to inform decisions about the course (including the ordering of topics, emphasis on material and time devoted to mastery of certain concepts, particularly in Calculus II). </li><li>Reviewed the performance on exam problems to decide if a problem type is too hard, a problem type needs to be rephrased, or an idea needs to be revisited on a future exam.</li></ul>The intelligent use of data to shape and monitor interventions is a central feature of the large- scale initiatives that are now underway. To mention just one, the AAU STEM Initiative (Association of American Universities, a consortium of 62 of the most prominent research universities in the U.S. and Canada) has established a Framework for sustainable institutional change. It can be found at <a href="https://stemedhub.org/groups/aau/framework">https://stemedhub.org/groups/aau/framework</a> (Figure 1). <br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://1.bp.blogspot.com/-UYZTXLKvuk4/WLWl4m4DV1I/AAAAAAAAKyM/17MV2vqJe4wgs8ucR1Drou4FYlMo7_xkwCLcB/s1600/Launchings%2Bfigure%2B1.JPG" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" src="https://1.bp.blogspot.com/-UYZTXLKvuk4/WLWl4m4DV1I/AAAAAAAAKyM/17MV2vqJe4wgs8ucR1Drou4FYlMo7_xkwCLcB/s1600/Launchings%2Bfigure%2B1.JPG" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Figure 1. AAU STEM Initiative Framework</td></tr></tbody></table><br />The three levels of change are subdivided into topics, each of which links to programs at member universities that illustrate work on this aspect of the framework. <br /><br />Cultural change encompasses<br /><ol><li> Aligning incentives with expectations of teaching excellence. </li><li> Establishing strong measures of teaching excellence. </li><li> Leadership commitment.</li></ol>Scaffolding includes<br /><ol><li> Facilities. </li><li> Technology. </li><li> Data. </li><li>Faculty professional development.</li></ol>Pedagogy is comprised of<br /><ol><li> Access. </li><li>Articulated learning goals. </li><li>Assessments. </li><li>Educational practices.</li></ol>In addition, AAU is now finalizing a list of “Essential Questions” to ask about the institution, the college, the department, and the course, illustrating the types of data and information that should be collected and pointing to helpful resources. This report, which should be published by the time this column appears, will be accessible through the AAU STEM Initiative homepage at <a href="https://stemedhub.org/groups/aau">https://stemedhub.org/groups/aau</a>. <br /><br /><br />Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com0tag:blogger.com,1999:blog-7251686825560941361.post-17551048236997298062017-02-01T06:54:00.000-05:002017-02-28T13:01:52.433-05:00MAA Calculus Study: PtC Survey Results <b><i>You can now follow me on Twitter <a href="https://twitter.com/dbressoud" target="_blank">@dbressoud</a>.</i></b><br /><br /><br />In spring 2015 the MAA’s <i>Progress through Calculus</i> (PtC) grant (NSF#1430540) surveyed all U.S. Departments of Mathematics that offer a graduate degree in Mathematics to learn about departmental practices, priorities, and concerns with respect to their mainstream courses in precalculus through single variable calculus. I have reported on some of the results from this study in <a href="http://launchings.blogspot.com/2015_11_01_archive.html" target="_blank">November, 2015</a>. This month’s column describes a variety of data relative to mainstream Calculus I that were collected in that survey. The full report can be found under <a href="mailto:http://www.maa.org/programs/faculty-and-departments/curriculum-development-resources/national-studies-college-calculus/ptc-publications" target="_blank">PtC Reports</a> (link from <a href="http://maa.org/cspcc">maa.org/cspcc</a>). <br /><br />The survey was sent to the chairs of all departments of mathematics in the United States that offer a graduate degree in Mathematics (PhD or Master’s). We received responses from 134 of the 178 PhD-granting universities (75%) and 89 of the 152 Master’s-granting universities (59%). <br /><br />Given how ineffective the standard precalculus course is known to be (see my <i>Launchings</i> column from <a href="http://launchings.blogspot.com/2014_10_01_archive.html" target="_blank">October, 2014</a>), we were particularly interested in efforts to teach precalculus topics concurrently with calculus. Accomplishing this through a stretched-out Calculus I is now fairly common (20 of 222 respondents use this approach to incorporate precalculus topics into Calculus I). Eleven universities have courses or options with extra hours to allow time on precalculus, and three offer precalculus courses designed to be taken concurrently with Calculus I. We also found 14 universities with an accelerated calculus specifically designed to meet the needs of students entering with AP® Calculus credit. Three universities have special lower credit courses that enable students who begin in a non-mainstream Calculus I to transition to mainstream calculus. <br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://2.bp.blogspot.com/-oxFy3DUgM4s/WJDl6Y8pzmI/AAAAAAAAKvw/U7Yn30lSR_wDCJ5j6mmI-Xh9lpsvM7LyQCLcB/s1600/launchings%2BPtC%2Btable%2B1.JPG" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="188" src="https://2.bp.blogspot.com/-oxFy3DUgM4s/WJDl6Y8pzmI/AAAAAAAAKvw/U7Yn30lSR_wDCJ5j6mmI-Xh9lpsvM7LyQCLcB/s400/launchings%2BPtC%2Btable%2B1.JPG" width="400" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Table 1: Number of surveyed universities that reported using each <br />of the listed variations in single variable calculus classes.</td></tr></tbody></table><br />Every five years, CBMS surveys departments of mathematics in the U.S. to get enrollment numbers, but those are only gathered for the fall term. In this survey, we were particularly interested in how these numbers vary over the full year, both academic and summer terms. While we only have results for a sample of universities, and no undergraduate colleges, the numbers are large enough, 150,000 in Precalculus, 200,000 in mainstream Calculus I, and 160,000 in subsequent mainstream single variable classes, to get a good idea of how these enrollments distribute over the year. For Precalculus, 57% of the enrollment occurs in the fall term. Fall term accounts for 60% of the Calculus I students. Not surprisingly, Calculus II is predominantly a second-term course (47%), but 40% of the students who take Calculus II do so in the fall. The distribution among the terms is complicated by the fact that some universities are on a quarter system, others on semesters. What I have labeled <i>2nd Term</i>, is either spring semester or winter quarter. The <i>3rd Term</i> refers to the spring quarter for those on a quarter system. <i>Summer</i> aggregates all summer terms. Figure 1 shows actual numbers from the universities that responded to give an idea of how enrollments drop off. For the purposes of the survey, “Precalculus” was defined as the last course before mainstream Calculus I. It is variously called Precalculus, College Algebra, College Algebra with Trigonometry, or Preparation for Calculus. Calculus II includes all mainstream single variable calculus courses that follow Calculus I. On a semester system, there is usually just one. On a quarter system, there usually are two such courses. <br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://2.bp.blogspot.com/-KEMpu4OSI5s/WJDzIcPU7KI/AAAAAAAAKwo/RwKsg18xBuw2tQWObZYKtmIjAwqpHDe2wCLcB/s1600/Fig1.tiff" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="263" src="https://2.bp.blogspot.com/-KEMpu4OSI5s/WJDzIcPU7KI/AAAAAAAAKwo/RwKsg18xBuw2tQWObZYKtmIjAwqpHDe2wCLcB/s400/Fig1.tiff" width="400" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Figure 1: Distribution of enrollments by term among the 205 universities that respond to this question. <br />2nd term = spring semester or winter quarter. 3rd term = spring quarter. <br />Calculus II includes all mainstream single variable calculus classes that follow Calculus I.</td></tr></tbody></table><br />The number of contact hours (including recitation sections) in Calculus I averaged 4.17 (SD = 0.77) at PhD-granting universities and 4.25 (SD = 0.64) at Masters-granting universities. The DFW rate in mainstream Calculus I was 21% (SD =12.2), at PhD-granting universities and 25% (SD = 13.7) at Masters-granting universities. <br /><br />The next table (Table 2) reports the fraction of universities in which Calculus I is frequently taught by each type of instructor. For each category of instructor, the options were “Never,” “Rarely,” or “Frequently.” <br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://4.bp.blogspot.com/-ZehM_eiA-ak/WJDnHaB-tPI/AAAAAAAAKv4/W7q5tR8VNPosni29iXaW5AKY6lUvZc_mgCLcB/s1600/launchings%2BPtC%2Btable%2B3.JPG" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="137" src="https://4.bp.blogspot.com/-ZehM_eiA-ak/WJDnHaB-tPI/AAAAAAAAKv4/W7q5tR8VNPosni29iXaW5AKY6lUvZc_mgCLcB/s400/launchings%2BPtC%2Btable%2B3.JPG" width="400" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Table 2: Percentage of universities for which each category of <br />instructor frequently teaches mainstream Calculus I.</td></tr></tbody></table><br />Recitation sections were far more common at PhD-granting universities. All classes have recitation sections for 49% of the institutions, some classes at 6%, and there are no recitation sections at 45% of the universities. For Masters-granting universities, the percentages were 18% for all classes, 6% for some classes, and 76% for no classes. <br /><br />We also found that active learning was much more common at Masters-granting universities than PhD-granting universities. Figures 2 and 3 record primary instructional format for mainstream Calculus I. “Some active learning” includes techniques such as use of clickers or think-pair-share. “Minimal lecture” includes Inquiry Based Learning and flipped classes. “Other” usually means too much variation to be able to identify a primary instructional format. We did find that 35% of the PhD-granting universities did report having at least some sections that were using active learning approaches. <br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://3.bp.blogspot.com/-dVOwQSNy5HU/WJDpXAvYGYI/AAAAAAAAKwE/hK0m7Mc0m3gmrA-O8HpCQ3hkeVLv3wEyQCLcB/s1600/launchings%2BPtC%2Btable%2B4.JPG" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="199" src="https://3.bp.blogspot.com/-dVOwQSNy5HU/WJDpXAvYGYI/AAAAAAAAKwE/hK0m7Mc0m3gmrA-O8HpCQ3hkeVLv3wEyQCLcB/s320/launchings%2BPtC%2Btable%2B4.JPG" width="320" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Figure 2. Primary instructional format for regular classes <br />(not recitation sections) at 214 PhD-granting universities.</td></tr></tbody></table><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://1.bp.blogspot.com/-mWN3-fSc3cQ/WJDqJUwfoPI/AAAAAAAAKwI/qsTcMuuz1wk3TXfEsTrlImp112vsx7uAACLcB/s1600/launchings%2BPtC%2Btable%2B5.JPG" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="194" src="https://1.bp.blogspot.com/-mWN3-fSc3cQ/WJDqJUwfoPI/AAAAAAAAKwI/qsTcMuuz1wk3TXfEsTrlImp112vsx7uAACLcB/s320/launchings%2BPtC%2Btable%2B5.JPG" width="320" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Figure 3. Primary instructional format for regular classes <br />(not recitation sections) at 109 Masters-granting universities.</td></tr></tbody></table><br />At 73% of the PhD-granting universities and 74% of the Masters-granting universities that offer recitation sections, they are simply homework help, Q&A, and review. Recitation sections are built around active learning approaches 21% of the time at PhD-granting universities, 4% of the time at Masters-granting universities. <br /><br />Table 3 reports which elements of mainstream Calculus I are common across all sections. We see much more uniformity at PhD-granting universities. In view of our findings from the earlier <i>Characteristics of Successful Programs in College Calculus</i> that coordination of course elements was one of the significant factors of successful calculus programs (see my <i>Launchings</i> column from <a href="http://launchings.blogspot.com/2014_01_01_archive.html" target="_blank">January 2014</a>), the results of this study suggest a great deal of room for improvement. <br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://3.bp.blogspot.com/-aGAXIAfvO98/WJDqqV5l_dI/AAAAAAAAKwQ/hJ77lqUXTJUQQ5IrSTI5ImsO8i5lEiD9QCLcB/s1600/launchings%2BPtC%2Btable%2B6.JPG" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="253" src="https://3.bp.blogspot.com/-aGAXIAfvO98/WJDqqV5l_dI/AAAAAAAAKwQ/hJ77lqUXTJUQQ5IrSTI5ImsO8i5lEiD9QCLcB/s400/launchings%2BPtC%2Btable%2B6.JPG" width="400" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Table 3: Percentage of reporting universities that have these elements across all sections of mainstream Calculus I.</td></tr></tbody></table><br />Another aspect of coordination that was characteristic of the most successful programs was the practice of regular meetings of the course instructors. As shown in Table 4, there is also a great deal of room for improvement here. <br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://4.bp.blogspot.com/-JdFC_iZHlZg/WJDrLqDpwPI/AAAAAAAAKwY/YGZ1HztWB3YwXdW97fXdpPXT_6hjI4kcACLcB/s1600/launchings%2BPtC%2Btable%2B7.JPG" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" src="https://4.bp.blogspot.com/-JdFC_iZHlZg/WJDrLqDpwPI/AAAAAAAAKwY/YGZ1HztWB3YwXdW97fXdpPXT_6hjI4kcACLcB/s1600/launchings%2BPtC%2Btable%2B7.JPG" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Table 4: Response to "When several instructors are teaching in the same term, <br />how often do they typically meet as a group to discuss the course?"</td></tr></tbody></table><br />The situation at PhD-granting universities is disappointing. The primary means of instruction is still large lecture with few or no structured opportunities for students to reflect on what is being presented to them, supplemented by recitation sections in which graduate students simply go over homework and answer student questions. At the Masters-granting universities, where classes are smaller and there is more emphasis on teaching, there is little coordination, often resulting in highly variable instruction. But there is room for hope. While there is no previous study with comparable data, there appears to be good deal of experimentation. My own experience in visiting these predominantly large public universities is that they are aware that what they are doing is not working, and they are looking for ways to improve what happens in this critical sequence. <br /><br /><br /><br /><br /><br /><br />Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com0tag:blogger.com,1999:blog-7251686825560941361.post-57055565074220777922017-01-01T07:00:00.000-05:002017-01-01T07:00:08.913-05:00IJRUME: Approximation in Calculus<b><i>You can now follow me on Twitter <a href="https://twitter.com/dbressoud" target="_blank">@dbressoud</a>.</i></b><br /><br />In an earlier column, "<a href="http://launchings.blogspot.com/2014/09/beyond-limit-iii.html" target="_blank">Beyond the Limit, III</a>," I talked about how Michael Oehrtman and colleagues have been able to use approximation as a unifying theme for single variable calculus that helps students avoid many of the confusing aspects of the language of limits. I also pointed out that this is hardly a new idea, having been used by many textbook authors including Emil Artin in <i>A Freshman Honors Course in Calculus and Analytic Geometry</i> and Peter Lax and Maria Terrell in <i>Calculus with Applications</i>. The IJRUME research paper I wish to highlight this month, “A study of calculus instructors’ perceptions of approximation as a unifying thread of the first-year calculus” by Sofronas et al., looks at how common this approach actually is.<br /><br />The authors address four research questions:<br /><br /><ol><li>Do calculus instructors perceive approximation to be important to student understanding of first-year calculus? </li><li>Do calculus instructors report emphasizing approximation as a central concept and-or unifying thread in the first-year calculus? </li><li>Which approximation ideas do calculus instructors believe are “worthwhile” to address in first-year calculus? </li><li>Are there any differences between demographic groups with respect to the approximation ideas they teach in first-year calculus courses? </li></ol>They surveyed calculus instructors at 182 colleges and universities, collecting 279 responses.<br /><br /><br />To the first two questions, 89% agreed that approximation is important, but only 51% considered it a central concept, and only 40% found that it provides a unifying thread (see Figure 1). For those who did consider it central and-or unifying, the reasons that they gave included: (a) it illuminates reasons for studying calculus, (b) most functions are not elementary and approximation is helpful in dealing with such functions, (c) approximation facilitates the understanding of fundamental concepts including limit, derivative, integral, and series, (d) linear approximations lie at the foundation of differential calculus, and (e) an emphasis on approximation resonates with the instructors personal interests in applied mathematics or numerical analysis.<br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://1.bp.blogspot.com/-PZ1XLkIML6g/WF1PH4Lhg5I/AAAAAAAAKuk/uQyPAwsf1G4t4SnApWh8NDyD-OLITocTQCLcB/s1600/IJRUME_Approx_in_Calc.tiff" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="335" src="https://1.bp.blogspot.com/-PZ1XLkIML6g/WF1PH4Lhg5I/AAAAAAAAKuk/uQyPAwsf1G4t4SnApWh8NDyD-OLITocTQCLcB/s400/IJRUME_Approx_in_Calc.tiff" width="400" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Figure 1: Graph depicting participants’ perceptions of approximation (N=214).<br /> Source: Sofronas et al. 2015.</td></tr></tbody></table><br />For those who did<i> not </i>consider approximation to be central or unifying, many stated that it is not sufficiently universal, only important in a few contexts such as motivating the definition of the derivative at a point or the value of a definite integral. Many stated other unifying threads such as limit or the study of change. Some objected to an emphasis on approximation because of its inevitable ties to the use of technology. There were also a large number of obstacles to the use of approximation that instructors identified. These included: (a) an overcrowded syllabus that left no room for the instructor to develop a unifying thread, (b) required adherence to a curriculum emphasizing procedural facility, (c) students with weak preparation who are not prepared to understand the subtleties of approximation arguments, (d) lack of access to technology, (e) lack of familiarity with how to use approximation ideas in developing calculus. I personally find these obstacles to be very sad, in particular the assumption on the part of many instructors that the only way to get through the required syllabus or to enable students to pass the course is to focus exclusively on memorizing procedures. <br /><br />Jumping ahead to the fourth question, the authors found that the single factor that most highly correlated with emphasizing approximation as a central concept and-or unifying thread was having served on either a local or national calculus committee. Not surprisingly, this factor was also highly correlated with number of years teaching calculus, rank, being the recipient of a teaching award, and having published or presented on a calculus topic. <br /><br />To the third research question, the combined list of topics gleaned from all of the responses truly spans first-year calculus: numerical limits, definition of limit, definition of the derivative, derivative values, tangent line approximations, differentials, error estimation, function change, functions roots and Newton’s method, linearization, integration, Riemann sums, Taylor polynomials and Taylor series, Newton’s second law, Einstein’s equation for force, L’Hospital’s rule, Euler’s method, and the approximation of irrational numbers. One unexpected outcome of the survey is that several of the respondents commented that answering this survey about their use of approximation in first-year calculus opened their eyes to the opportunity to use it as a unifying theme. As one respondent wrote,<br /><blockquote class="tr_bq">I agree that approximation is an important concept AND after taking this survey I can see teaching calculus using approximation as the main theme. The rate of change theme offers many opportunities for real-life applications but I can see how using approximations from the beginning would offer other opportunities. It is an interesting idea, and I would love to incorporate more of this theme into my lessons.</blockquote>For those who are interested in following up on the use of approximation as a unifying thread, this article also supplies a wealth of background information that includes a discussion of the different ways in which approximation can be used and the research evidence for its effectiveness as a guiding theme in developing student understanding of limits, derivatives, integrals, and series. <br /><br /><b>References </b><br /><br />Artin, E. (1958). <i>A Freshman Honors Course in Calculus and Analytic Geometry Taught at Princeton University</i>. Buffalo, NY: Committee on the Undergraduate Program of the Mathematical Association of America <br /><br />Lax, P. & Terrell, M.S. (2014). <i>Calculus with Applications</i>, Second Edition. New York, NY: Springer. <br /><br />Sofronas, K.S., DeFranco, T.C., Swaminathan, H., Gorgievski, N., Vinsonhaler, C., Wiseman, B., Escolas, S. (2015). A study of calculus instructors’ perceptions of approximation as a unifying thread of the first-year calculus. <i>Int. J. Res. Undergrad. Math. Ed</i>. 1:386–412 DOI 10.1007/s40753-015- 0019-5 <br /><br />Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com0tag:blogger.com,1999:blog-7251686825560941361.post-66675784552381122052016-12-01T07:31:00.000-05:002016-12-01T07:31:02.403-05:00IJRUME: Peer-Assisted Reflection<b><i>You can now follow me on Twitter <a href="https://twitter.com/dbressoud" target="_blank">@dbressoud</a>.</i></b><br /><br />The second paper I want to discuss from the <i>International Journal of Research in Undergraduate Mathematics Education</i> is a description of part of the doctoral work done by Daniel Reinholz, who earned his PhD at Berkeley in 2014 under the direction of Alan Schoenfeld. It consists of an investigation of the use of Peer-Assisted Reflection (PAR) in calculus [1]. <br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><img alt="Macintosh HD:Users:macalester:Dropbox:Personal:Launchings:2016:2016-12:21565092511_2e41878c9c_b.jpg" height="186" src="https://lh3.googleusercontent.com/y7Io02wwuQrFwnsPxOpi1E_ZEdmD8oneyGEiM4Amtu9uLSkn6wwhX8Q9lcT-wL02oNZVmL4jJlPA30Wo72hJVmENVkAgyMGmPCdnbwCYe933zbVzxNtFzqhspmqsPHkSS6X1xhp1RehWZzTyPA" style="border: none; margin-left: auto; margin-right: auto; transform: rotate(0rad);" width="279" /></td></tr><tr><td class="tr-caption" style="text-align: center;">Daniel Reinholz. <i>Photo Credit: David Bressoud</i></td></tr></tbody></table>PAR addresses an aspect of learning to do mathematics that Schoenfeld refers to as “self-reflection or monitoring and control” in his chapter on “Learning to Think Mathematically” [4]. As he observed in his problem-solving course at Berkeley, most students have been conditioned to assume that when presented with a mathematical problem, they should be able to identify immediately which tool to use. Among the possible activities that students might engage in while solving a problem—read, analyze, explore, plan, implement, and verify—most students quickly chose one approach to explore and then “pursue that direction come hell or high water” (Figure 1). <br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><img alt="Macintosh HD:Users:macalester:Dropbox:Personal:Launchings:2016:2016-12:Schoenfeld fig3.tiff" height="158" src="https://lh4.googleusercontent.com/FjKzvc4M1NZwaUE4FaMV09La3YM7GThgPTRu4ve-cJi-YaBe6RiXExHe_gkFodNk2gGe3t7f9F0RFbZOoCNQw3tEYmpB9gb0EX17PXE5x7iq98vag2PWoXa5M-NHOSsLzzXLTVsq83uYmeoKnA" style="border: none; margin-left: auto; margin-right: auto; transform: rotate(0rad);" width="300" /></td></tr><tr><td class="tr-caption" style="text-align: center;">Figure 1: Time-line graph of a typical student attempt to solve a non-standard problem. <br />Source: [4, p.356, Figure 15-3]</td></tr></tbody></table><div class="separator" style="clear: both; text-align: center;"><span style="font-size: 14.6667px; margin-left: 1em; margin-right: 1em; vertical-align: baseline; white-space: pre-wrap;"></span></div>In contrast, when he observed a mathematician working on an unfamiliar problem, he observed all of these strategies coming into play, a constant appraisal of whether the approach being used was likely to succeed and a readiness to try different ways of approaching the problem. He also found that mathematicians would verbalize the difficulties they were encountering, something seldom encountered among students (Figure 2). Note that over half the time was spent making sense of the problem rather than committing to a particular direction. Triangles represent moments when explicit comments were made such as “Hmm, I don’t know exactly where to start here.”<span id="docs-internal-guid-9f019c3f-b67c-fbc5-040d-19be399af837"></span><br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><img alt="Macintosh HD:Users:macalester:Dropbox:Personal:Launchings:2016:2016-12:Schoenfeld fig4.tiff" height="206" src="https://lh3.googleusercontent.com/uFcFd4bZWj65BeCoVvQ6-N47BWgbpLGL0tkTX0duxiIV9awBaP-su1Kn9K0xLQrkLEZmnVq4CWiILEfop02uVF7jpl0ZRkX_3dZfavs5bTQA3CZ4HtKhReTjl3CnIsVACCaVohwCxK0jBhtqLQ" style="border: none; margin-left: auto; margin-right: auto; transform: rotate(0rad);" width="374" /></td></tr><tr><td class="tr-caption" style="text-align: center;">Figure 2: Time-line graph of a mathematician working on a difficult problem.<br />Source: [4, p.356, Figure 15-4]</td></tr></tbody></table><div class="separator" style="clear: both; text-align: center;"><span style="font-size: 14.6667px; margin-left: 1em; margin-right: 1em; vertical-align: baseline; white-space: pre-wrap;"></span></div>In the conclusion to this section of his chapter, Schoenfeld wrote, “Developing self-regulatory skills in complex subject-matter domains is difficult.” In reference to two of the studies that had attempted to foster these skills, he concluded that, “Making the move from such ‘existence proofs’ (problematic as they are) to standard classrooms will require a substantial amount of conceptualizing and pedagogical engineering.”<span id="docs-internal-guid-9f019c3f-b67e-413c-aaa5-e34230d919ba"></span><br /><br />One of the problems with the early attempts at instilling self-reflection was the tremendous amount of work required of the instructor. Reinholz implemented PAR in Calculus I, greatly simplifying the role of the instructor by using students as partners in analyzing each other’s work. The study was conducted in two phases over two separate semesters in studies that each semester included one experimental section and eight to ten control sections, all of whom used the same examinations that were blind-graded. There were no significant differences between sections in either student ability on entering the class or in student demographics. The measure of success was an increase in the percentage of students earning a grade of C or higher. In the first phase of the study, the experimental section had a success rate of 82%, as opposed to the control sections where success was 69%. In the second phase, success rose from 56% in the control sections to 79% in the experimental section. <br /><br />Reinholz observed a noticeable improvement in student solutions to the PAR problems after they had received peer feedback. From student interviews, he found that many students in the PAR section had learned the importance of iteration, that homework is not just something to be turned in and then forgotten, but that getting it wrong the first time was okay as long as they were learning from their mistakes. Students were learning the importance of explaining how they arrived at their solutions. And they appreciated the chance to see the different approaches that other students in the class might take. <br /><br />What is most impressive about this intervention is how relatively easy it is to implement. Each week, the students would be given one “PAR problem” as part of their homework assignment. They were required to work on the problem outside of class, reflect on their work, exchange their solution with another student and provide feedback on the other student’s work in class, and then finalize the solution for submission. The time in class in which students read each other’s work and exchanged feedback took only ten minutes per week: five minutes for reading the other’s work (to ensure they really were focusing on reasoning, not just the solution) and five minutes for discussion. <br /><br />The difficulty, of course, lies in ensuring that the feedback provided by peers is useful. Reinholz identifies what he learned from several iterations of PAR instruction. In particular, he found that it is essential for the students to be explicitly taught how to provide useful feedback. By the time he got to Phase II, Reinholz was giving the students three sample solutions to that week’s PAR problem, allowing two to three minutes to read and reflect on the reasoning in each, and then engaging in a whole class discussion for about five minutes before pairing up to analyze and reflect on each other’s work. <br /><br />Further details can be found in [2] and [3]. For anyone interested in using Peer-Assisted Reflection, this is a useful body of work with a wealth of details on how it can be implemented and strong evidence for its effectiveness. <br /><br /><b>References </b><br /><br />[1] Reinholz, D.L. (2015). Peer-Assisted Reflection: A design-based intervention for improving success in calculus. <i>International Journal of Research in Undergraduate Mathematics Education</i>. <b>1</b>:234–267. <br /><br />[2] Reinholz, D. (2015). Peer conferences in calculus: the impact of systematic training. <i>Assessment & Evaluation in Higher Education</i>, DOI: 10.1080/02602938.2015.1077197 <br /><br />[3] Reinholz, D.L. (2016). Improving calculus explanations through peer review. <i>The Journal of Mathematical Behavior</i>. <b>44</b>: 34–49. <br /><br />[4] Schoenfeld, A.H. (1992). Learning to think mathematically: problem-solving, metacognition, and sense-making in mathematics. Pp. 334–370 in <i>Handbook for Research in Mathematics Teaching and Learning</i>. D. Grouws (Ed.). New York: Macmillan. Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com0tag:blogger.com,1999:blog-7251686825560941361.post-5249667226711665412016-11-01T07:41:00.000-04:002016-11-01T07:41:03.361-04:00IJRUME: Measuring Readiness for Calculus<b><i>You can now follow me on Twitter <a href="https://twitter.com/dbressoud" target="_blank">@dbressoud</a>.</i></b><br /><br />In 2015, the <i>International Journal of Research in Undergraduate Mathematics Education</i> (IJRUME) was launched by Springer with editors-in- chief Karen Marrongelle and Chris Rasmussen from the U.S. and Mike Thomas from New Zealand. It was established to “become the central, premier international journal dedicated to university mathematics education research.” While this is a journal by mathematics education researchers for mathematics education researchers, many of the articles are directly relevant to those of us engaged in the teaching of post-secondary mathematics. This then is the first of what I anticipate will be a series of columns abstracting some of the insights that I gather from this journal. <br /><br />I have chosen for the first of these columns the paper by Marilyn Carlson, Bernie Madison, and Richard West, “A study of students’ readiness to learn calculus.” [1] It is common to point to students’ lack of procedural fluency as the culprit behind their difficulties when they get to post- secondary calculus. Certainly, this is a problem, but not the whole story. Work over the past quarter century by Tall, Vinner, Dubinsky, Monk, Harel, Zandieh, Thompson, Carlson and many others have led the authors to identify major reasoning abilities and understandings that students need for success in calculus. This paper describes a validated diagnostic test that measures foundational reasoning abilities and understandings for learning calculus, the Calculus Concept Readiness (CCR) instrument. <br /><br />The reasoning abilities and conceptual understandings assessed by CCR require students to move beyond a procedural or action-oriented understanding of mathematics. Whether it is an equation such as 2 + 3 = 5 or a function definition, <i>f</i>(x) = <i>x</i><sup>2</sup> + 3<i>x</i> + 6, students are introduced to these as describing an action to be taken, adding 2 to 3 or plugging in various values for <i>x</i>. To make sense of and use the ideas of calculus, students need to view a function as a process (defined by a function formula, graph, or word description) that characterizes how the values of two varying quantities change together. Listed below are four of the reasoning abilities and understandings assessed by CCR and which the authors highlight in their article.<br /><div class="separator" style="clear: both; text-align: center;"><br /></div><ol><li><b>Covariational Reasoning.</b> When two variables are linked by an equation or a functional relationship, students need to understand how changes in one variable are reflected in changes in the other variable. The classic example considers how the rates of change of height and volume are related when water is poured into a non-cylindrical container such as a cone. At an even more basic level, students need to be able to interpret information on the velocities of two runners to an understanding of which is ahead at what times. Another example, which involves covariational reasoning as well as understanding rate as a ratio, considers the height of a ladder and its distance from a wall (Figure 1). When the authors administered their instrument to 631 students who were starting Calculus I, only 27% were able to select the correct answer (<b>c</b>) to the ladder problem.<br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://4.bp.blogspot.com/-grCVaQLEKVQ/WA_Bv_uBkaI/AAAAAAAAKrc/MNEja7EpMvA3dRDK-zmrh-zpbFzGyyR5gCEw/s1600/2016-11-a.tiff" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="376" src="https://4.bp.blogspot.com/-grCVaQLEKVQ/WA_Bv_uBkaI/AAAAAAAAKrc/MNEja7EpMvA3dRDK-zmrh-zpbFzGyyR5gCEw/s640/2016-11-a.tiff" width="640" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><b>Figure 1. The ladder problem.</b></td></tr></tbody></table></li><br /><li><b>Understanding the Function Concept.</b> Too many students interpret f(<i>x</i>) as an unnecessarily long-winded way of saying <i>y</i>. They see a function definition such as f(<i>x</i>) = <i>x</i><sup>2</sup>+ 3<i>x</i> + 6 as simply a prescription for how to take an input x and turn it into an output f(<i>x</i>). Such a limited view makes it difficult for students to manipulate functional relationships or to compose function formulas. Carlson et al. asked their 631 students for the formula for the area of a circle in terms of its circumference and offered the following list of possible answers:<br /><b> a.</b> <i>A</i> = <i>C</i><sup>2</sup>/4π<br /><b> b.</b> <i>A</i> = <i>C</i><sup>2</sup>/2<br /><b> c.</b> <i>A</i> = (2π<i>r</i>)<sup>2a</sup><b> <br /> d.</b> <i>A</i> = π<i>r</i><sup>2</sup><b> <br /> e.</b> <i>A</i> = π(<i>C</i><sup>2</sup>/4)<br />Only 28% chose the correct answer (<b>a</b>). As the authors learned from interviewing a sample of these students, those who answered correctly were the students who could see the equation C = 2πr as a process relating C and r which could be inverted and then composed with the familiar functional relationship between the area and radius.<br /></li><li><b>Proportional Relationships.</b> Too many students do not understand proportional reasoning. When Carlson et al. in an earlier study [2] administered the rain-gauge problem of Piaget et al. (Figure 2) to 1205 students who were finishing a precalculus course, only 43% identified the correct answer (as presented in Figure 2, it is 4⅔). Many students preserve the difference rather than the ratio, giving 5 as the answer. Difficulties with proportional reasoning are known to impede student understanding of constant rate of change, which in turn underpins average rate of change, which is fundamental to understanding the meaning of the derivative.<br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://4.bp.blogspot.com/--wQfot4FriY/WA_CYB6RyLI/AAAAAAAAKrM/l7HxlrvQE30fZj2OyMjkVEl5q8b1ytazwCLcB/s1600/2016-11-b.tiff" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="168" src="https://4.bp.blogspot.com/--wQfot4FriY/WA_CYB6RyLI/AAAAAAAAKrM/l7HxlrvQE30fZj2OyMjkVEl5q8b1ytazwCLcB/s640/2016-11-b.tiff" width="640" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><b>Figure 2. The rain gauge problem (taken from [3], [4])</b></td></tr></tbody></table></li><br /><li><b>Angle Measure and Sine Function.</b> As I described some years ago in an article for The Mathematics Teacher [5], the emphasis in high school trigonometry on the sine as a ratio of the lengths of sides of a triangle—often leading to the misconception that the sine is a function of a triangle rather than an angle—can lead to difficulties when encountering the sine in calculus, where it must be understood as a periodic function expressible in terms of arc length. An example is given in Figure 3, a problem for which only 21% of the Calculus I students chose the correct answer (<b>e</b>). Student interviews revealed that difficulties with this problem most often arose because students did not understand how to represent an angle measure using the length of the arc cut off by the angle’s rays.</li></ol><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-6u7_4whwHRE/WA_Bv3FW6AI/AAAAAAAAKrE/iyBEQ4f11kMThbr9pS--700pQOUw8QWUgCEw/s1600/2016-11-c.tiff" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="262" src="https://4.bp.blogspot.com/-6u7_4whwHRE/WA_Bv3FW6AI/AAAAAAAAKrE/iyBEQ4f11kMThbr9pS--700pQOUw8QWUgCEw/s640/2016-11-c.tiff" width="640" /></a></div><br /><br />What lessons are we to take away from this for our own classes? Last spring, in <a href="http://launchings.blogspot.com/2016/02/what-we-saywhat-they-hear.html">What we say/What they hear</a> and <a href="http://launchings.blogspot.com/2016/03/what-we-saywhat-they-hear-ii.html">What we say/What they hear II</a>, I discussed problems of communication between instructors and students. The work of Carlson, Madison, and West illustrates some of the fundamental levels at which miscommunication can occur and identifies the productive ways of thinking that students need to develop. <br /><br /><b>References</b><br /><br />[1] Carlson, M.P., Madison, B., & West, R.D. (2015). A study of students’ readiness to learn Calculus. <i>Int. J. Res. Undergrad. Math. Ed. </i>1:209–233. DOI 10.1007/s40753-015- 0013-y.<br /><br />[2] Carlson, M., Oehrtman, M., & Engelke, N. (2010). The precalculus concept assessment (PCA) instrument: a tool for assessing reasoning patterns, understandings and knowledge of precalculus level students. <i>Cognition and Instruction</i>, 28(2):113–145.<br /><br />[3] Piaget, J., Blaise-Grize, J., Szeminska, A., & Bang, V. (1977). <i>Epistemology and psychology of functions</i>. Dordrecht: Reidel.<br /><br />[4] Lawson, A.E. (1978). The development and validation of a classroom test of formal reasoning. <i>Journal of Research in Science Teaching</i>, 15, 11–24. doi:10.1002/tea.3660150103.<br /><br />[5] Bressoud, D.M. (2010). Historical reflections on teaching trigonometry. <i>The Mathematics Teacher</i>. 104(2):106–112. <br /><br /><br /><br />Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com1tag:blogger.com,1999:blog-7251686825560941361.post-7952692303864974242016-10-01T07:31:00.000-04:002017-03-30T13:52:19.121-04:00MAA Calculus Study: Women in STEMIt is nice to see that the national media has picked up one of the publications arising from the MAA’s national study, <i>Characteristics of Successful Programs in College Calculus</i> (NSF #0910240). It is the article by Ellis, Fosdick, and Rasmussen, “<a href="http://journals.plos.org/plosone/article?id=10.1371%2Fjournal.pone.0157447" target="_blank">Women 1.5 times more likely toleave STEM pipeline</a>,” that was published in PLoS ONE on July 13 of this year. The media coverage includes:<br /><ul><li>Rachel Feltman at the <i>Washington Post</i>, "<a href="https://www.washingtonpost.com/news/speaking-of-science/wp/2016/07/14/calculus-apprehensions-may-steer-women-away-from-science-careers/" target="_blank">Calculus apprehensions may steer women away from science careers</a>," who began her piece with her own experience, “Calculus II was one of the most demoralizing experiences of my college career.” </li><li>Dominique Mosbergen at the <i>Huffington Post</i>, "<a href="http://www.huffingtonpost.com/entry/calculus-stem-gender-gap_us_57a1b9eee4b0e2e15eb7df83?section" target="_blank">This Popular Math Class Is At The Heart Of The STEM Gender Gap, Study Suggests</a>"</li><li>Lauren Camera at <i>U.S. News</i>, "<a href="http://www.usnews.com/news/articles/2016-07-21/calculus-steers-women-away-from-stem" target="_blank">Calculus Steers Women Away From STEM</a>"</li><li>Maggie Kuo at <i>Science</i>, "<a href="http://www.sciencemag.org/careers/2016/07/low-math-confidence-discourages-female-students-pursuing-stem-disciplines" target="_blank">Low math confidence discourages female students from pursuing STEM disciplines</a>" </li></ul>...as well as a host of blogs and regional news sources.<br /><br />The article was an outgrowth of the “switcher” analysis that Jess Ellis and Chris Rasmussen had begun, using data from our 2010 national survey to study who came into Calculus I with the intention of staying on to Calculus II but then changed their minds by the end of the course. You can find a preliminary report on the Ellis and Rasmussen switcher analysis in my column for December 2013, <a href="http://launchings.blogspot.com/2013/12/maa-calculus-study-persistence-through.html" target="_blank">MAA Calculus Study: Persistence through Calculus</a> and a further analysis of the differences between men and women in the November, 2014 column, <a href="http://launchings.blogspot.com/2014/11/maa-calculus-study-women-are-different.html" target="_blank">MAA Calculus Study: Women are Different</a>. See also Rasmussen and Ellis (2013).<br /><br />The 2013 column reported that women were about twice as likely as men to switch out of the calculus sequence, but those data were compromised by several lurking variables, most significantly intended major. Women are heavily represented in the biological sciences, much less so in engineering and the physical sciences. Since the biological sciences are less likely to require a second semester of calculus, some of the effect was almost certainly due to different requirements.<br /><br />The study published in PLOS One controlled for student preparedness for Calculus I, intended career goals, institutional environment, and student perceptions of instructor quality and use of student-centered practices. They found that even with these controls, women were 50% more likely to switch out than men. As I discussed in my 2014 column, while Calculus I is very efficient at destroying the mathematical confidence of most of the students who take it, it is particularly effective for women (see Figure 1). As Ellis et al. report, 35% of the STEM-intending women who switched out chose as one of their reasons, “I do not believe I understand the ideas of Calculus I well enough to take Calculus II.” Only 14% of the men chose this reason.<br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://4.bp.blogspot.com/-Y2GaoYLT4P4/V-0_GAUwiCI/AAAAAAAAKoQ/goi7ieOWHO0dE1g5qQh5f1dFkVeWV7TnQCLcB/s1600/Women%2Bin%2BSTEM%2B1.tiff" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="331" src="https://4.bp.blogspot.com/-Y2GaoYLT4P4/V-0_GAUwiCI/AAAAAAAAKoQ/goi7ieOWHO0dE1g5qQh5f1dFkVeWV7TnQCLcB/s640/Women%2Bin%2BSTEM%2B1.tiff" width="640" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Figure 1: Change in standard mathematical confidence at the beginning of the Calculus I semester (pre- survey) and at the end of the semester (post-survey) separated by career intentions, gender and persistence status, [N = 1524] doi:10.1371/journal.pone.0157447.g004</td></tr></tbody></table><br />The last figure in the Ellis et al. article is enlightening (see Figure 2). If we could just raise the persistence rates of women once they choose enter Calculus I to match that of men, we could get a 50% increase in the percentage of women who enter the STEM workforce each year. <br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://4.bp.blogspot.com/-QuzGa9qfays/V-0_Y8irEJI/AAAAAAAAKoU/dsiixo7yp74ffdbxy6USgdi1JLucirGggCLcB/s1600/Womein%2Bin%2BSTEM%2B2.tiff" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="528" src="https://4.bp.blogspot.com/-QuzGa9qfays/V-0_Y8irEJI/AAAAAAAAKoU/dsiixo7yp74ffdbxy6USgdi1JLucirGggCLcB/s640/Womein%2Bin%2BSTEM%2B2.tiff" width="640" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Figure 2: Projected participation of STEM if women and men persisted at equal rates after Calculus I. The dotted line represents the projected participation of women. doi:10.1371/journal.pone.0157447.g005</td></tr></tbody></table><br />I believe that this issue of women’s confidence is cultural, not biological. It fits in with all we know about stereotype threat. When the message is that women are not expected to do as well as men in mathematics, negative signals loom very large. Calculus—as taught in most of our colleges and universities—is filled with negative signals. <br /><br /><b>Reference </b><br /><br />Ellis, J., Fosdick, B.K., and Rasmussen, C. (2016). Women 1.5 times more likely to leave STEM pipeline after calculus compared to men: Lack of mathematical confidence a potential culprit. PLoS ONE 11(7): e0157447. doi10.1371/journal.pone.0157447 <br /><br />Rasmussen, C., & Ellis, J. (2013). Who is switching out of calculus and why? In Lindmeier, A. M. & Heinze, A. (Eds.). <i>Proceedings of the 37th Conference of the International Group for the Psychology of Mathematics Education</i>, Vol. 4 (pp. 73-80). Kiel, Germany: PME.<br /><br />Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com0tag:blogger.com,1999:blog-7251686825560941361.post-49857314522695986432016-09-01T07:00:00.000-04:002016-09-02T16:29:20.555-04:00CBMS and Active Learning<div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-oZNNF_6dU1k/V8b6iB5dSjI/AAAAAAAAKlo/v8PN2JXAI2o3hs23baXJI71Z4JxAw51LQCLcB/s1600/CBMS%2Blogo.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="139" src="https://2.bp.blogspot.com/-oZNNF_6dU1k/V8b6iB5dSjI/AAAAAAAAKlo/v8PN2JXAI2o3hs23baXJI71Z4JxAw51LQCLcB/s320/CBMS%2Blogo.jpg" width="320" /></a></div><br /><br />I have just accepted the position of Director of the <a href="http://www.cbmsweb.org/" target="_blank">Conference Board of the Mathematical Sciences (CBMS)</a> and will be taking over from Ron Rosier at the end of this year. Most mathematicians, if they have heard of it at all, know of CBMS for its national survey of the mathematical sciences conducted every five years or for its regional research conferences. A few may know of CBMS through its forums on educational issues, its series on <i>Issues in Mathematics Education</i>, or the <i>Mathematical Education of Teachers</i> (MET II) report. <br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-wxbV5GyRmfY/V8b62QQZCxI/AAAAAAAAKls/5pZI-uQluyYLza0uh0Y6TKI8wiNV4bgJQCLcB/s1600/bressoud_CBMS.PNG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://3.bp.blogspot.com/-wxbV5GyRmfY/V8b62QQZCxI/AAAAAAAAKls/5pZI-uQluyYLza0uh0Y6TKI8wiNV4bgJQCLcB/s1600/bressoud_CBMS.PNG" /></a></div><br />These have emerged from the core mission of CBMS, which is to provide a structure within which the presidents of the societies that represent the mathematical sciences [1] can identify issues of common concern and coordinate efforts to address them. This is exemplified in the joint statement on <i><a href="http://www.cbmsweb.org/Statements/Active_Learning_Statement.pdf" target="_blank">Active Learning in Postsecondary Mathematics</a></i> [2] that was released this past July. This statement explains what is meant by active learning, presents the case for its importance, points to some of the published evidence of its effectiveness, lists society reports that have encouraged its use, and urges the following recommendation:<br /><br /><blockquote class="tr_bq"><i><b>We call on institutions of higher education, mathematics departments and the mathematics faculty, public policy-makers, and funding agencies to invest time and resources to ensure that effective active learning is incorporated into postsecondary mathematics classrooms. </b></i></blockquote><br />Ben Braun led the society representatives who drafted this position paper [2]. The presidents of all of the member societies with strong interest in mathematics education have signed onto it [3]. <br /><br />I see this statement as an example of what can be accomplished when the mathematical societies look to issues of common interest, and I am looking forward to working with them to coordinate efforts that will help colleges and universities identify and implement locally appropriate strategies for active learning. <br /><br />I also hope to use my position to assist these societies in addressing the issues of articulation that so plague mathematics education. These include the transitions from two-year to four-year institutions, from undergraduate to either graduate school or the workforce, and from graduate school to either academic or non-academic employment. But the transition on which I am currently focusing my attention is from secondary to postsecondary education. This point of discontinuity is rife with difficulties for many of our students who would seek STEM careers as well those who have struggled with mathematics. It is especially problematic for students from underrepresented groups: racially, ethnically, by socio-economic status, by gender, and by family experience with postsecondary education. <br /><br />The solutions—for there will be many pieces to be addressed if we are to succeed in ameliorating the problems—will require strong and coordinated efforts from both sides of the transition from high school to college. I am very encouraged by the clear messages of support for this work that I have received from NCTM, NCSM, and ASSM on the secondary side of the divide as well as AMS, MAA, AMATYC, ASA, and SIAM from the postsecondary side. CBMS is uniquely situated to bridge their work. <br /><br />While I expect my primary focus to be on educational concerns, CBMS has and must continue to work on all matters of common interest including public awareness of the role and importance of mathematics, advocacy for programs that improve opportunities for underrepresented minorities, and issues of employment in the mathematical sciences. <br /><br />I want to conclude by acknowledging the tremendous debt that the mathematical community owes to Ron Rosier and Lisa Kolbe who have been the entire staff of CBMS for roughly three decades. They have made this an effective organization. Under their direction, it has run smoothly and accomplished a great deal. They have left me with a very strong base on which to continue to build. <br /><br /><b>Endnotes </b><br /><br />[1] The seventeen professional societies that belong to CBMS can be grouped into those that are primarily focused at the postsecondary level:<br /><br /><ul><li><a href="http://www.cbmsweb.org/Members/amatyc.htm" target="_blank">American Mathematical Association of Two-Year Colleges (AMATYC)</a></li><li><a href="http://www.cbmsweb.org/Members/ams.htm" target="_blank">American Mathematical Society (AMS)</a></li><li><a href="http://www.cbmsweb.org/Members/asa.htm" target="_blank">American Statistical Association (ASA)</a></li><li><a href="http://www.cbmsweb.org/Members/asl.htm" target="_blank">Association for Symbolic Logic (ASL)</a> </li><li><a href="http://www.cbmsweb.org/Members/awm.htm" target="_blank">Association for Women in Mathematics (AWM)</a> </li><li><a href="http://www.cbmsweb.org/Members/amte.htm" target="_blank">Association of Mathematics Teacher Educators (AMTE)</a> </li><li><a href="http://www.cbmsweb.org/Members/ims.htm" target="_blank">Institute of Mathematical Statistics (IMS)</a> </li><li><a href="http://www.cbmsweb.org/Members/maa.htm" target="_blank">Mathematical Association of America (MAA)</a> </li><li><a href="http://www.cbmsweb.org/Members/nam.htm" target="_blank">National Association of Mathematicians (NAM)</a> </li><li><a href="http://www.cbmsweb.org/Members/siam.htm" target="_blank">Society for Industrial and Applied Mathematics (SIAM)</a><br /><br />Those that are primarily focused at preK–12 mathematics: </li><li><a href="http://www.cbmsweb.org/Members/assm.htm" target="_blank">Association of State Supervisors of Mathematics (ASSM)</a></li><li><a href="http://www.cbmsweb.org/Members/bba.htm" target="_blank">Benjamin Banneker Association (BBA)</a> </li><li><a href="http://www.cbmsweb.org/Members/ncsm.htm" target="_blank">National Council of Supervisors of Mathematics (NCSM)</a> </li><li><a href="http://www.cbmsweb.org/Members/nctm.htm" target="_blank">National Council of Teachers of Mathematics (NCTM)</a> </li><li><a href="http://www.cbmsweb.org/Members/todos.htm" target="_blank">TODOS: Mathematics for ALL (TODOS) </a><br /><br />And those that are primarily non-academic: </li><li><a href="http://www.cbmsweb.org/Members/informs.htm" target="_blank">Institute for Operations Research and the Management Sciences (INFORMS)</a></li><li><a href="http://www.cbmsweb.org/Members/soa.htm" target="_blank">Society of Actuaries (SOA)</a></li></ul><br /><br />[2] <i>Active Learning in Postsecondary Mathematics</i>, available at <a href="http://www.cbmsweb.org/Statements/Active_Learning_Statement.pdf" target="_blank">http://www.cbmsweb.org/Statements/Active_Learning_Statement.pdf</a><br />The writing team was led by Ben Braun and included myself as well as Diane Briars, Ted Coe, Jim Crowley, Jackie Dewar, Edray Herber Goins, Tara Holm, Pao-Sheng Hsu, Ken Krehbiel, Donna LaLonde, Matt Larson, Jacqueline Leonard, Rachel Levy, Doug Mupasiri, Brea Ratliff, Francis Su, Jane Tanner, Christine Thomas, Margaret Walker, and Mark Daniel Ward. The presidents of the member societies undertook the final wordsmithing. <br /><br />[3] The presidents of INFORMS and SOA were the only ones who were not engaged in the formulation or signing of this position paper. Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com0tag:blogger.com,1999:blog-7251686825560941361.post-13706480178912393302016-08-01T07:30:00.000-04:002016-08-01T07:30:00.156-04:00MAA Calculus Study: PlacementIn November’s column, <a href="http://launchings.blogspot.com/2015_11_01_archive.html" target="_blank">MAA Calculus Study: A New Initiative</a>, I described a survey that MAA has conducted of practices for and concerns about the precalculus through calculus sequence at departments of mathematics that have graduate programs. The initial summary of the survey results is now available as<i><a href="http://www.maa.org/sites/default/files/pdf/PtC%20Survey%20Report.pdf" target="_blank"> Progress through Calculus: National Survey Summary</a></i>, which can also be accessed through the Publications & Reports under Progress through Calculus on the web page <a href="http://maa.org/cspcc">maa.org/cspcc</a>. Universities were distinguished by whether the highest degree offered in mathematics was a Masters or a PhD.<br /><br />As I reported in November, placement was the number one issue among mathematics departments when comparing self-evaluation of importance to the program with confidence that the department is doing it well. Figure 1 shows that most PhD-granting departments rely on internally constructed instruments for placement.<br /><br /><div class="separator" style="clear: both; text-align: center;"></div><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://2.bp.blogspot.com/-acQc_8vF4mA/V45QLjmAkZI/AAAAAAAAKic/-_MixTGrTc0E3rz2MPGEt6nvwrLK8eorwCEw/s1600/bressoud_calculus_study.PNG" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="273" src="https://2.bp.blogspot.com/-acQc_8vF4mA/V45QLjmAkZI/AAAAAAAAKic/-_MixTGrTc0E3rz2MPGEt6nvwrLK8eorwCEw/s400/bressoud_calculus_study.PNG" width="400" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Figure 1. Percentage of respondents using specific placement tools for precalculus/calculus.<br />Respondents could select more than one.</td></tr></tbody></table>It is discouraging that a majority of Masters-granting departments and almost half of the PhD- granting departments use ACT or SAT scores for placement, instruments that are particularly ill suited to this purpose, even when only used to distinguish between placement into precalculus versus a previous course. It is also discouraging that so few PhD-granting universities use high school grades in determining placement. While not sufficient on their own, the study of <i>Characteristics of Successful Program in College Calculus</i> did reveal that including these grades improved departmental satisfaction with its placement decisions (see [1]). One of the striking results of the survey is that the number of PhD-granting departments using ALEKS increased from 10% in our 2010 survey to 28% in 2015. This may be somewhat misleading because the 2010 question only asked about placement into Calculus I, while the 2015 question asked about placement into precalculus or calculus, but from my own experience, the past several years have seen strong growing interest in and adoption of ALEKS.<br /><br />Figure 2 shows the overall degree of satisfaction of the department with their placement procedures. Note that the bars above the placement tools represent degree of satisfaction with the entire placement procedure among those institutions that include this particular tool. Thus it does not necessarily reflect the degree of satisfaction with that particular instrument. Nevertheless, this does indicate that there is no single instrument that guarantees satisfaction.<br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="https://2.bp.blogspot.com/-_x3J5I2ocJk/V45RORPs_sI/AAAAAAAAKik/l0BEVOvS0uE9FLT0OYfbaeVI-ZWMaljIgCLcB/s1600/bressoud_calculus_study2.PNG" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="248" src="https://2.bp.blogspot.com/-_x3J5I2ocJk/V45RORPs_sI/AAAAAAAAKik/l0BEVOvS0uE9FLT0OYfbaeVI-ZWMaljIgCLcB/s400/bressoud_calculus_study2.PNG" width="400" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Figure 2. Number of universities (out of 223) using each placement, with degree of<br />overall satisfaction with placement procedures.</td></tr></tbody></table>Among all of the surveyed universities, 9% were not satisfied with their placement procedures, and 39% considered them adequate but could be improved. Even though 52% were generally satisfied, we found that there is a lot of churn in placement procedures: 30% of the universities had recently replaced or were currently replacing their placement procedures, and an additional 29% were considering replacing these procedures.<br /><br />Perhaps the most interesting and potentially alarming result is that only 43% of respondents (45% of PhD-granting departments and 41% of Masters-granting departments) reported that they regularly review adherence to placement recommendations. It is hard to know how well your placement is working if you do not monitor it.<br /><br /><b>Reference</b><br /> [1] Hsu, E. and Bressoud, D. 2015. Placement and Student Performance in Calculus I. pages 59–67 in <i>Insights and Recommendations from the MAA National Study of College Calculus</i>, Bressoud, Mesa, and Rasmussen, editors. MAA Notes #84. Washington, DC: MAA Press. <a href="http://www.maa.org/sites/default/files/pdf/cspcc/InsightsandRecommendations.pdf">www.maa.org/sites/default/files/pdf/cspcc/InsightsandRecommendations.pdf</a>Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com0tag:blogger.com,1999:blog-7251686825560941361.post-15425189889976628502016-07-01T08:12:00.000-04:002016-07-19T11:56:56.942-04:00MAA and Active LearningThere is a general perception among both research mathematicians and those working in our partner disciplines that—with a few exceptional pockets such as the community of those promoting Inquiry Based Learning (IBL)—the mathematical community is only now beginning to wake up to the importance of active learning. In fact, the MAA’s Committee on the Undergraduate Program in Mathematics (CUPM) began to promote the use of active learning in 1981 and has never ceased. It is a cry to which many have responded, but which has recently been rediscovered and promoted with urgency as chairs, deans, provosts, and presidents have come to realize that the way mathematics instruction has traditionally been organized cannot meet our present needs, much less those of the future. I was reminded of the origins of MAA’s support for active learning after encountering a particular piece of misleading data in Andrew Hacker’s <i>The Math Myth</i>, an unpleasant little book filled with half-truths, deceptive innuendo, and misleading statistics. <br /><br />Hacker argues that the “math mandarins” cannot even attract students to major in mathematics and supports his argument with the fact that the number of Bachelor’s degrees in mathematics earned by US citizens dropped from 27,135 in 1970 to 17,408 in 2013. As a percentage of the total number of Bachelor’s degrees, the drop is even more impressive: from 3.4% to 1.0%. These numbers are symptomatic of Hacker’s deceptive use of data.<br /><br />The first thing that is deceptive about these numbers is that they suggest a steady erosion of interest in mathematics. In fact, as Figures 1 and 2 show, the drop was precipitous during the 1970s, with the total number of Bachelor’s degrees in mathematics bottoming out in 1981 at 11,078, showing some recovery in the ‘80’s, followed by a steady decline until 2001 when it dipped back below 12,000, only 0.94% of Bachelor’s degrees. Since then, the growth has been reasonably strong, rising to 20,980 (of whom 2,438 were non-resident aliens) in 2014. That was back up to 1.12%.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-bVZXtkAdUc0/V3LaKEy6WJI/AAAAAAAAKhs/I9dI7ZgBKfczRdtOwl6fU7noKEJVs71pgCLcB/s1600/MAA%2Bactive%2Blearning%2B1.PNG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://3.bp.blogspot.com/-bVZXtkAdUc0/V3LaKEy6WJI/AAAAAAAAKhs/I9dI7ZgBKfczRdtOwl6fU7noKEJVs71pgCLcB/s1600/MAA%2Bactive%2Blearning%2B1.PNG" /></a></div><br />(Note: The sharp increase in the early 1980s is almost certainly due to the high unemployment the United States was then suffering. Similarly, the noticeable increase in slope around 2010 is most probably a product of the unemployment rate that peaked in 2009.)<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-OYI2WtPWjRc/V3LaOc0h1oI/AAAAAAAAKh0/BplEXYCdWhcKLIpuiLILCsLea39KuONFQCLcB/s1600/MAA%2Bactive%2Blearning%2B2.PNG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://3.bp.blogspot.com/-OYI2WtPWjRc/V3LaOc0h1oI/AAAAAAAAKh0/BplEXYCdWhcKLIpuiLILCsLea39KuONFQCLcB/s1600/MAA%2Bactive%2Blearning%2B2.PNG" /></a></div><br />The other thing that is deceptive is the choice of when to start. The year 1970 came at the end of a strong national push for young people to enter mathematics and science. We had begun that decade in 1960 with only 11,399 mathematics degrees, though admittedly that was 2.9% of the total. Much of the loss during the ’70’s may be attributed to the creation of computer science majors. Bachelor’s degrees in computer science rose from 2388 in 1971 to 15,121 in 1981. Much, but not all. In fact, many members of the mathematical community were alarmed by this drop. Therein begins the story that is far more important than Hacker’s data.<br /><br />CUPM was established in the early 1950’s to bring order to the chaotic assortment of courses that constituted mathematics majors across the country. In 1965 this committee of leading mathematicians published <i><a href="http://www.maa.org/sites/default/files/pdf/CUPM/pdf/CUPM_Report_1965.pdf">A General Curriculum in Mathematics for Colleges</a><a href="http://www.maa.org/sites/default/files/pdf/CUPM/pdf/CUPM_Report_1965.pdf"></a></i>, which codified what by then was becoming the standard undergraduate major, beginning with three semesters of calculus and one semester of linear algebra. CUPM’s concern was almost entirely what to teach, not how to teach it. That changed in 1981 when Alan Tucker’s CUPM panel published <i><a href="http://www.maa.org/sites/default/files/pdf/CUPM/pdf/CUPM_Report_1981.pdf">Recommendations for a General Mathematical Sciences Program</a><a href="http://www.maa.org/sites/default/files/pdf/CUPM/pdf/CUPM_Report_1981.pdf"></a></i>.<br /><br />Concerned about the precipitous fall in the number of majors as well as enrollments in upper division courses, the attention in this report was focused on the goals of an undergraduate major and how they could be achieved. It laid out a five-point program philosophy that included an appeal to use active learning: <br /><ol><li>“The curriculum should have a primary goal of developing attitudes of mind and analytical skills required for efficient use and understanding of mathematics … </li><li>“The mathematical sciences curriculum should be designed around the abilities and academic needs of the average mathematical sciences student … </li><li>“<i>A mathematical sciences program should use interactive classroom teaching to involve students actively in the development of new material. Whenever possible, the teacher should guide students to discover new mathematics for themselves rather than present students with concisely sculptured theories.</i> (My italics.) </li><li>“Applications should be used to illustrate and motivate material in abstract and applied courses… </li><li>“First courses in a subject should be designed to appeal to as broad an audience as is academically reasonable …” </li></ol> In the 1991 CUPM report, <i>The Undergraduate Major in the Mathematical Sciences<a href="http://www.maa.org/sites/default/files/pdf/CUPM/pdf/CUPM_Report_1991.pdf"></a></i>, chaired by Lynn Steen, the third point from 1981 was expanded to a clearer articulation of active learning.<br /><blockquote><b>III. Interaction. </b>Since active participation is essential to learning mathematics, instruction in mathematics should be an interactive process in which students participate in the development of new concepts, questions, and answers. Students should be asked to explain their ideas both by writing and by speaking, and should be given experience working on team projects. In consequence, curriculum planners must act to assure appropriate sizes of various classes. Moreover, as new information about learning styles among mathematics students emerges, care should be taken to respond by suitably altering teaching styles.</blockquote>The next report, <a href="http://www.maa.org/programs/faculty-and-departments/curriculum-department-guidelines-recommendations/cupm/cupm-guide-2004"><i>CUPM Curriculum Guide 2004</i></a>, chaired by Harriet Pollatsek, continued to build on the theme of how we teach. In this iteration, CUPM expanded its vision to all of the courses taught by departments of mathematics, insisting that “Every course should incorporate activities that will help all students progress in developing analytical, critical reasoning, problem-solving, and communication skills and acquiring mathematical habits of mind.”<br /><br />This emphasis continues in the most recent CUPM guide, <a href="http://www.maa.org/sites/default/files/pdf/CUPM/pdf/CUPMguide_print.pdf">2015 CUPM Curriculum Guide toMajors in the Mathematical Sciences</a>, co-chaired by Carol Schumacher and Martha Siegel and edited by Paul Zorn. It begins with four “Cognitive Recommendations:"<br /><br /><ol><li>Students should develop effective thinking and communication skills. </li><li>Students should learn to link applications and theory. </li><li>Students should learn to use technological tools. </li><li>Students should develop mathematical independence and experience open-ended inquiry.</li></ol><br />Throughout these decades, MAA has done more than issue recommendations. All of these reports have been backed up by <i>MAA Notes</i> volumes that have pointed to successful programs and explained how such instruction can be implemented within specific courses. (For a list of all Notes volumes, click <a href="http://www.macalester.edu/~bressoud/maa/MAA_Notes.pdf" target="_blank">here</a>.) The Notes began in 1983 with <i><a href="http://files.eric.ed.gov/fulltext/ED229248.pdf">Problem Solving in the Mathematical Sciences</a></i>, edited by Alan Schoenfeld. MAA has run workshops as well as focused sessions and presentations at both national and regional meetings. Project NExT, MAA’s program for new faculty now in its third decade, has always had an emphasis on introducing newly minted PhDs to the use of active learning strategies.<br /><br />It is hard to say whether these measures have been responsible for arresting and reversing the slide in the number of majors. Economic factors have certainly played a role. But MAA publications and activities have established a depth of experience and expertise within the mathematical community. Now that there is broad recognition of the importance of active learning strategies in the teaching and learning of undergraduate mathematics, we are fortunate to have this foundation on which to build.<br /><br /><b>References</b><br /><br />Duren, W.L. Jr., Chair. 1965. <i><a href="http://www.maa.org/sites/default/files/pdf/CUPM/pdf/CUPM_Report_1965.pdf">A General Curriculum in Mathematics for Colleges</a></i>. Berkeley, CA: CUPM. www.maa.org/sites/default/files/pdf/CUPM/pdf/CUPM_Report_1965.pdf<br /><br />Hacker, A.<i> The Math Myth: and other STEM Delusions</i>. New York, NY: The New Press.<br /><br />Pollatsek, H., Chair. 2004. <i>CUPM Curriculum Guide</i> 2004. Washington, DC: MAA. <a href="http://www.maa.org/programs/faculty-and-departments/curriculum-department-guidelines-recommendations/cupm/cupm-guide-2004">www.maa.org/programs/faculty-and- departments/curriculum-department- guidelines-recommendations/cupm/cupm-guide- 2004</a><br /><br />Schoenfeld, A.H., Editor. 1983. Problem Solving in the Mathematical Sciences. Washington, DC: MAA. <a href="http://files.eric.ed.gov/fulltext/ED229248.pdf">files.eric.ed.gov/fulltext/ED229248.pdf</a><br /><br />Schumacher, C.S. and Siegel, M.J., Co-Chairs, and Zorn, P., Editor. 2015. 2015 <i>CUPM Curriculum Guide to Majors in the Mathematical Sciences</i>. Washington, DC: MAA. <a href="http://www.maa.org/programs/faculty-and-departments/curriculum-department-guidelines-recommendations/cupm">www.maa.org/programs/faculty-and- departments/curriculum-department- guidelines-recommendations/cupm</a><br /><br />Steen, L.A., Chair. 1991. <i>The Undergraduate Major in the Mathematical Sciences</i>. Washington, DC: MAA. <a href="http://www.maa.org/sites/default/files/pdf/CUPM/pdf/CUPM_Report_1991.pdf">www.maa.org/sites/default/files/pdf/CUPM/pdf/CUPM_Report_1991.pdf</a><br /><br />Tucker, A., Chair. 1981. <i>Recommendations for a General Mathematical Sciences Program</i>. Washington, DC: MAA. Reprinted on pages 1–59 in <i>Reshaping College Mathematics</i>, L.A. Steen, editor. Washington, DC: MAA, <a href="http://www.maa.org/sites/default/files/pdf/CUPM/pdf/CUPM_Report_1981.pdf">www.maa.org/sites/default/files/pdf/CUPM/pdf/CUPM_Report_1981.pdf</a>Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com0tag:blogger.com,1999:blog-7251686825560941361.post-29949409139426505142016-06-01T07:00:00.000-04:002016-06-03T09:13:34.655-04:00The Role of Calculus in the Transition from High School to College MathematicsThis past March I ran a small workshop in Washington, DC to look at “The Role of Calculus in the Transition from High School to College Mathematics,” funded by the National Science Foundation (#1550484). As the introduction to the workshop, cited below, makes clear, this is a critically important issue. Regular readers of my column will recognize that I return to it often. Part of the problem is that we don’t even know what we know and what we need to know (echoes of the “unknown unknowns”). The stated purpose was to bring together stakeholders to<br /><ol><li>Clarify the current state of knowledge of the effects on college performance and retention in STEM disciplines of acceleration into calculus at or before 12 th grade, as well as the effects of lack of access to calculus while in high school.</li><li>Identify the most pressing research questions.</li><li>Suggest strategies for answering these questions and identify the appropriate researchers or organizations to tackle them.</li></ol>As it turned out, my hopes for the workshop were too ambitious. We were not able to leave with a research agenda, but we did make progress toward identifying what we know about the problems surrounding this transition and in clarifying the complex issues surrounding the movement of calculus into the high school curriculum.<br /><br />The two days of the workshop provided an opportunity for a rich exchange. Participants included post-secondary mathematics faculty, high school calculus teachers, state and district supervisors of mathematics, researchers in both K-12 and post-secondary mathematics education, as well as representatives of The College Board, the National Council of Teachers of Mathematics, the National Research Council’s Board on Science Education, the National Math and Science Initiative, and Achieve. What is most gratifying is that all of these players recognize that there are serious problems around issues of preparation for post-secondary mathematics as well as equity and access that are being exasperated by the tremendous pressures to bring students into calculus ever earlier in their high school careers. My intention is that over the coming months it will be possible to build on the foundation laid by this workshop. <br /><br />Before concluding with the preamble that I wrote for this workshop, I would like to send out an appeal for all readers who would like to contribute to this conversation to send me their thoughts at <a href="mailto:bressoud@macalester.edu">bressoud@macalester.edu</a>, subject line: Role of Calculus.<br /><br /><blockquote class="tr_bq" style="text-align: left;">Last year, at least three quarters of a million U.S. high school students were enrolled in a calculus class.[1] This was three times the number of U.S. students who took their first calculus class in college. High school calculus enrollments are still growing at roughly 6% per year,[2] with increasing pressure on the most advantaged students to take calculus ever earlier. In 2015, over 120,000 students took the AP Calculus exam by the end of grade 11, and these numbers are growing by 9% per year. [3]</blockquote><blockquote class="tr_bq" style="text-align: left;">If we make the assumption that most of the students who enroll in calculus in high school will go on to matriculate as full-time students in a four-year undergraduate program—of whom there are 1.5 million each year[4]—then roughly half of these full-time, first-year students enter having already studied calculus. Calculus in high school is now commonly perceived as a prerequisite for college admission, with the result that high schools must start students with Algebra I in 8th grade high school if they are to be on track for calculus by grade 12, and calculus teachers find themselves under increasing pressure from parents and administrators to admit into their classes students they know are not adequately prepared. How do we provide support and alternatives for those students who cannot handle this accelerated progression?</blockquote><blockquote class="tr_bq" style="text-align: left;">We have learned from our study of Characteristics of Successful Programs in College Calculus (NSF DRL 0910240) that a quarter million of the students who study calculus in high school will retake mainstream Calculus I at the post-secondary level, and 40% of these quarter million will fail to get the A or B that signals they are prepared to continue in mathematics.[5] College faculty are very much aware that many of the students who enter with calculus on their high school transcript are, in fact, not ready for college-level mathematics. Of particular concern is that we know almost nothing about what happens to the remaining half million students who have studied calculus in high school: How many take advantage of advanced placement, and how well prepared are they? What are the mathematical trajectories of those who do not take calculus in college? How has the experience of calculus in high school shaped the aspirations and attitudes of these students and their ability to continue on toward mathematically demanding careers? </blockquote><blockquote class="tr_bq" style="text-align: left;">At the same time, half of all U.S. high schools do not even offer calculus. [6] Students from underrepresented groups, even when they are in a high school that offers calculus, are often discouraged from enrolling in this course. Those who have not taken calculus in high school find themselves in competition against students with a much richer preparation. What does this do to their chances for admission to college or the pursuit of a STEM major? </blockquote><blockquote class="tr_bq" style="text-align: left;">The questions are manifold. The goal of this workshop is to better understand these questions and to begin to develop strategies for answering them. Specifically, we seek to<br /><ol><li style="text-align: left;">Clarify the current state of knowledge of the effects on college performance and retention in STEM disciplines of acceleration into calculus at or before 12th grade, as well as the effects of lack of access to calculus while in high school.</li></ol><ol><li style="text-align: left;">Identify the most pressing research questions.</li></ol><ol><li style="text-align: left;">Suggest strategies for answering these questions and identify the appropriate researchers or organizations to tackle them.</li></ol></blockquote><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-0jwzHFVBUn0/V0dGaJmCAFI/AAAAAAAAKhE/GX5-h9EZQjkyakvXUB8SzXXpcn0lJJWDQCLcB/s1600/RoC.tiff" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="400" src="https://4.bp.blogspot.com/-0jwzHFVBUn0/V0dGaJmCAFI/AAAAAAAAKhE/GX5-h9EZQjkyakvXUB8SzXXpcn0lJJWDQCLcB/s400/RoC.tiff" width="318" /></a></div><br /><br /><b>Footnotes</b><br /><br />[1] Based on the NCES longitudinal study (HSLS:09) reporting that 19% of the four million students who were in 9th grade in 2009 had taken a calculus course in high school by the time they graduated.<br /><br />[2] From the College Board’s <i>AP Program Summary Reports</i> from 2002 to 2015.<br /><br />[3] Ibid.<br /><br />[4] Higher Education Research Institute.<i>The American Freshman: National Norms Fall 2015</i>.<br /><br /> [5] Data from <i>Characteristics of Successful Programs in College Calculus</i> project’s maalongdatafile. See David Bressoud. 2015. Insights from the MAA national study of college calculus. <i>The Mathematics Teacher</i>. 109 3:179–185.<br /><br />[6] U.S. Department of Education Office of Civil rights. Issue Brief No. 3 (March, 2014).<br /><a href="http://ocrdata.ed.gov/Downloads/CRDC-College-and-Career-Readiness-Snapshot.pdf">http://ocrdata.ed.gov/Downloads/CRDC-College- and-Career- Readiness-Snapshot.pdf</a><br /><br />Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com0tag:blogger.com,1999:blog-7251686825560941361.post-32443699703476837682016-05-01T09:06:00.000-04:002016-05-01T09:06:00.482-04:00Reflections on a Career in TeachingMonday, May 2 is my last day of teaching. Not that I will never step into a classroom again, but it marks the end of my full-time employment by Macalester College. My responsibilities for the future will not require any teaching. I am looking forward to the freedom this phased retirement will bring to focus on my writing and other educational activities, but I also approach this date with some sense of loss. For forty years, I have defined myself as a teacher first. I am taking advantage of this column to reflect on a few of the lessons I have learned over these four decades in the classroom.<br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: right;"><tbody><tr><td style="text-align: center;"><a href="https://1.bp.blogspot.com/-PSi1vvP1DlU/VyJkP12uThI/AAAAAAAAKgA/mk0P9Hwlxy8clih8wx5_xQkrkpNyWnYCgCLcB/s1600/Bressoud.PNG" imageanchor="1" style="clear: right; margin-bottom: 1em; margin-left: auto; margin-right: auto;"><img border="0" height="293" src="https://1.bp.blogspot.com/-PSi1vvP1DlU/VyJkP12uThI/AAAAAAAAKgA/mk0P9Hwlxy8clih8wx5_xQkrkpNyWnYCgCLcB/s400/Bressoud.PNG" width="400" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Class on Revolutions, taught at Penn State, Spring 1993, with John Harwood and Phil Jenkins (whose daughter Catherine is also in the picture). Taken on a field trip to the Joseph Priestley House, Northumberland, PA.</td></tr></tbody></table><b><br /></b><b>Students absorb far less than we think they do. As teachers, we must identify what is truly important and do all in our power to ensure that students are internalizing this knowledge.</b><br /><br />The first time this reality hit home for me was a differential equations class I taught at the University of Wisconsin in 1980–81, before the days of computers in the classroom. As taught back then, it was all about finding exact solutions to certain limited classes of differential equations. I thought I did a pretty good job of explaining the motivation and usefulness of these techniques, but homework and midterm exams were focused on finding these exact solutions. For the final exam, I wanted to make it interesting by presenting situations that my students would have to model using one of the types of differential equations we had studied, and then they would solve it. Not only did the students totally bomb this exam, they were angry that I was asking them to do something—modeling real world situations with these differential equations—that they had never been asked to do during the semester.<br /><br />They were absolutely right. Students learn by doing, not by listening. Knowing how to solve a differential equation is very different from knowing how to use a differential equation. It is far too easy to focus our instruction on the easily tested mechanical skills and assume that understanding and an appreciation for context will come as fortuitous byproducts.<br /><br />My second major experience of this truth occurred in a minicourse run by Kathy Heid and Joan Ferrini-Mundy at the 1994 Joint Math Meetings (in frigid Cincinnati). The theme was the use of student interviews as a tool for deep assessment of student learning. I practiced on a volunteer, a student from a local university who had completed four semesters of calculus and was now a junior. Eight months after the last of these classes, I wanted to find out what he had retained. I was horrified. As much as I pressed, differentiation for him carried no connotations beyond a method for turning functions into simpler functions (in the sense that 3x<sup>2</sup> is “simpler” than x<sup>3</sup>), and integration reverses that process. Nothing about tangents or rates or areas or accumulations had stuck. It drove home to me how little it is possible to learn while still passing a mathematics class.<br /><br />These experiences explain why I am such a fan of Inquiry Based Learning (IBL). It uses class time to focus on what we really want students to learn and provides a means of constantly probing student understanding. In my columns from February and March of this year, <a href="http://launchings.blogspot.com/2016/02/what-we-saywhat-they-hear.html">What We Say/What They Hear</a> and <a href="http://launchings.blogspot.com/2016_03_01_archive.html">What We Say/What They Hear II</a>, I described work in mathematics education that has validated my observation that what we think we are communicating is not what most students hear. I appreciate Stan Yoshinobu’s recent blog, <a href="http://theiblblog.blogspot.com/2016/04/a-practical-solution-to-what-we-saywhat.html">A Practical Solution to “What We Say/What They Hear,”</a> that illustrates how an IBL approach can address this problem of conveying meaning.<br /><br /><b>Very little of what is learned at the post-secondary level happens inside the classroom. As teachers, we have a responsibility to structure how students interact with the mathematics beyond the classroom walls.</b><br /><br />This point really struck home in 1990–91, the year I taught AP Calculus at the State College Area High School. The leisurely pace through the AB curriculum meant that I could actually watch light bulbs coming on in my class, something that I had never witnessed in the large lecture calculus classes at Penn State. This has always been for me the greatest distinction between high school and college mathematics. For the latter, students have to know how to learn outside of class from notes and textbook and in exchanges with other students.<br /><br />My last year at Penn State, 1993–94, I surveyed students in one of these large lecture sections to try to understand their experience. I found that most of them were very conscientious about studying, usually spending about two hours each evening that followed class going over that day’s lesson. But I also found that most of them had no idea how to study. They would read through their notes, paying particular attention to the problems that had been worked out in class, and then they would tackle problems from the end of that section, practicing the techniques that they had seen demonstrated. Anything that strayed too far from what had been worked by the instructor was considered irrelevant.<br /><br />Uri Treisman has shown the importance of students working together to clarify understandings (see [1]). I have found that if I want students to think more deeply about the mathematics introduced in class and to share those understandings with others, I have to structure out-of-class assignments designed to accomplish this. <br /><br /><b>Final exams carry far too much weight. As teachers, we need to administer frequent and varied assessments that truly measure what our students are learning and that provide opportunities for students to learn.</b> <br /><br />I hate final exams. I have had colleagues who promise students that if they do really well on the final, any poor test results earlier in the term will be forgiven. I have even seen students who manage to pull off a superior performance on the final despite a record leading up to it that would not have predicted this. Over forty years, I can count them on one hand. On the other side, I have seen many of my students who were steady and successful during term completely fall apart at the final. It is a stressful time, not just in my class but in almost every class a student is taking. It is a time of late nights and cramming and incredible pressure. The final straw for me was about fifteen years ago when there was a major incident of cheating on my final exam. This was by good students who had been doing well and who I knew did not need to cheat, but they were overwhelmed by the fear of doing poorly on this major component of their final grade for which there would be no opportunity to overcome a poor result.<br /><br />My policy now is—with rare exceptions—to never count a final exam for more than 15% of the total grade. As a complement to this, I test early, trying to get the first major assessment in by the end of the fourth week of class; I test often; and I assess student performance using a wide variety of measures: in class tests and quizzes that focus on procedural knowledge, take-home tests with problems that challenge students to apply their knowledge in unfamiliar situations requiring multiple steps, in multi-week projects that will be critiqued and returned for revision, in Reading Reflections—short answers to questions about the material read<i> before </i>class that help inform me before class begins of what students do and do not yet understand, and in written questions collected at the end of each class. Many of these have been inspired by Angelo and Cross’s <i>Classroom Assessment Techniques</i>, but there is nothing I have picked up from that book that I have not reshaped to fit my own style and needs.<br /><br />I also believe in looking for ways to enable students to learn from the assessments I use. Every major project is turned in for feedback before the final submission. For each exam during the semester, students are allowed to earn back some of the lost points by explaining where they went wrong and how to do the problem correctly. By giving myself some flexibility on how much of the grade students can earn back, I find I that I can give very challenging exams without needing to grade on a curve. Grading on a curve is a practice I consider to be particularly pernicious because it communicates to students that they are competing against each other, that what matters is less how much you have learned than how much better you can perform than your neighbor.<br /><br />I especially value group projects as a tool for teaching as well as assessing student learning. Most of my classes include several major projects. It was my last year at Penn State when I had the great good fortune to be able to teach the early <i>Project CALC</i> materials with David Smith present on campus (on sabbatical from Duke). I was able to meet with him weekly over lunch to talk about how the course was going. I have been able to watch how effective students are at teaching each other. It still never ceases to amaze me that I can say something in class without it registering with some student until the person next to them restates it as his or her own insight, though often verbatim, in a private conversation around solving a problem they are working on. I have found that group grades are problematic, and have experimented with a variety of techniques over the years to make them fairer. Now, as much as possible and for at least one project per class, everyone in the group is required to write up his or her own report of what the group has found. Clarity of exposition is every bit as important as the correctness of the results, a requirement that usually catches out those who simply attempt to reproduce the work done by other members of the group.<br /><br />With many writing assignments, projects, and exams, my students always complain about how much work they have to do for my classes. But no single assessment counts for very much, and my students have many opportunities to learn and recover from a bad performance. I am proud of the fact that it has been many years since a student complained that my grading was not fair. I am also proud that, despite complaining about the amount of work, my students also note how much they have learned.<br /><br /><b>Most students get too few opportunities to appreciate the culture of mathematics. As teachers, our instruction should communicate the true nature of mathematics. </b> <br /><br />I find it deeply discouraging that so many students graduate from college without any appreciation for mathematics as a rich venue for discovery and innovation. Burger and Starbird have done an excellent job of communicating this side of mathematics in <i>The Heart of Mathematics</i>. I have tried to do it through the history of the subject. This is reflected in all of the textbooks that I have written, and my students will attest that I am constantly interjecting the history of the subject into my classes.<br /><br />This spring, as a swan song, I am teaching a 100-level class on the history of mathematics for the first and last time. Fifty students are enrolled, most of whom are taking it to satisfy Macalester’s quantitative reasoning requirement. I am using Berlinghoff and Gouvêa’s <i>Math through the Ages</i>, a perfect book for my vision of this course. Their text consists of a stripped down history of mathematics, complemented by thirty short vignettes that survey topics from the development of negative numbers to non-Euclidean geometries to the rise of the computer. I use class time to tell the stories I love and introduce my students to the people who have been instrumental in the development of this vast subject. The real learning takes place in the writing students are required to do: short questions that must be submitted at the end of each class and from which I pick a few to answer at the start of the next, Reading Reflections in which students must tie what they have read to their own experiences, and many short papers in which they must pursue some of the many references provided by Berlinghoff and Gouvêa and explain to a younger version of themselves something about mathematics that they wish they had known earlier in their mathematical career. These have been especially insightful. I have seen so many of my students who were interested in, even excited about mathematical ideas early in their schooling, but had had that interest pounded out of them. It is emotional for me to see them, jaded as they now are about mathematics, reaching back to that younger self, trying to blow that ember back to life.<br /><br />None of the insights I have presented here are particularly original. Others have described them much more eloquently. But for me they are hard-won truths achieved through years of constantly striving to be a better teacher. After all, that is how we learn, not by listening to someone else expound or by reading a book or column, but by observantly striving to master our chosen profession. What we hear or read can suggest fruitful directions in which to explore and grow. Ultimately, this is our challenge, to constantly seek to improve how we teach.<br /><br />[1] Eric Hsu, Teri J. Murphy, Uri Treisman. 2008. Supporting high achievement in introductory mathematics courses: What we have learned from 30 years of the Emerging Scholars Program. Pages 205–220 in M.P. Carlson & C. Rasmussen (eds.), <i>Making the Connection: Research and Teaching in Undergraduate Mathematics Education</i>. MAA Notes #73. Washington, DC: Mathematical Association of America.Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com3tag:blogger.com,1999:blog-7251686825560941361.post-78920602829108638102016-04-01T08:00:00.000-04:002016-04-04T14:48:37.420-04:00A Common VisionFive major mathematical societies—AMATYC, AMS, ASA, MAA, and SIAM—have just released a joint report, <i><a href="http://www.maa.org/sites/default/files/pdf/CommonVisionFinal.pdf">A Common Vision for Undergraduate Mathematical Sciences Programs in 2025</a></i>, authored by Karen Saxe and Linda Braddy and distributed by the MAA. This is a coordinated call for “modernizing undergraduate programs in the mathematical sciences.” For many years, all five of the societies have been proclaiming the need to improve the teaching and learning of undergraduate mathematics and statistics. For the first time, they have come together to identify their common concerns and to support shared recommendations. What follows is my own précis of the contents of this report. I strongly encourage you to read it for yourself. A full reference, with the address of the PDF file of the report, is at the end of this column.<br /><br />The central message, repeated throughout this document, is that “The status quo is unacceptable.” Specifically, A Common Vision issues a joint appeal to<br /><ol><li>Update curricula, </li><li>Articulate curricula across the critical divide between high school and college mathematics,</li><li>Scale-up the use of evidence-based pedagogical methods, </li><li>Remove barriers at critical transition points, and </li><li>Establish stronger connections to other disciplines.</li></ol>These are accompanied by a call to support those faculty engaged in these efforts. After explaining the need for these changes and summarizing the reports that have been issued by the five societies, A Common Vision describes in further detail the common themes that have emerged:<br /><br /><b>Curricula.</b> The greatest concentration of themes reported in A Common Vision circles around curricular issues. These include calls for presenting key ideas from a variety of perspectives and motivating them through the use of applications to contemporary topics. The cited reports recognize the importance of providing multiple pathways into and through undergraduate mathematics, with particular concern that departments attend to the barriers that students often confront. Solutions should include entry points that emphasize modeling, statistics, and applications as well as programs that focus on the development of computational and statistical skills.<br /><br />There is a recognized need for more statistics, computation, and modeling for all students within the first two years of undergraduate mathematics. And there is recognition of the need for closer cooperation with and awareness of the needs of other disciplines. The report includes a call for more attention to the development of the skills needed for effective mathematical communication, both orally and in writing. And, finally, this report highlights the common awareness among the five societies of the need to address issues of transition: from high school to college, in transfer between institutions, in issues of placement, and at critical juncture points such as the start of proof-based courses.<br /><br /><b>Course Structure.</b> There is a consensus among all of the societies that instruction needs to move beyond simple lecture and embrace a variety of active learning approaches that engage students in grappling with the difficulties of mathematics. These include providing opportunities for collaboration and communication. In addition, all five societies advocate the use of technology in those situations where it can enhance student learning.<br /><b><br /></b><b>Workforce Preparation. </b>Mathematics departments need to work with those in client departments within their own institutions as well as with the consumers of our graduates in business, industry, and government to understand the workforce skills that graduates will need. This must be done not as narrow technical training but in the recognition that our task is to equip students with a broad base of skills that will serve them in our rapidly evolving economy.<br /><b><br /></b><b>Faculty Development and Support. </b>All of the five societies recognize the need to provide training opportunities for faculty to broaden their expertise in areas of the mathematical sciences where great needs have not been met. These include data analytics and computational science. We also must foster an institutional culture that encourages and values work on the issues raised in these reports.<br /><br /><b>Other Issues.</b> In addition, other issues have arisen in one or more of the society reports. These include the need<br /><ol><li>To attend to issues of student diversity, particularly the retention of students in at- risk groups, </li><li>To ease difficulties as students move between institutions, </li><li>To recognize the special needs of contingent faculty, </li><li>To devote energy toward the preparation of K-12 teachers, </li><li>To properly prepare graduate students for their contributions to the teaching mission of the department, </li><li>To recognize and address the issues that lead to high failure rates, </li><li>To look for ways of improving courses in developmental mathematics so that they retain and adequately prepare students, </li><li>To shape calculus instruction so that it responds to the reality that most students studying calculus in college have already experienced it in high school, </li><li>To be aware of technology-enabled models of delivery of course content and to critically consider when and where they might be beneficial, </li><li>To gather and use empirical data to refine programs and improve student learning, </li><li>To scale successful efforts by involving more faculty within each department, by increasing communication about these efforts within the mathematical sciences community, and by understanding the obstacles to effective transfer of successful programs.</li></ol><br /><br />This report, which is only intended to be a summary of the common themes of these five societies, is nevertheless an important first step in recognizing the commonalities in the messages they are all sending and in working toward coordinated efforts to improve undergraduate education in the mathematical sciences.<br /><br />———————————————————————————————————<br /><br />Karen Saxe & Linda Braddy. 2016. <i>A Common Vision for Mathematical Sciences Programs in 2025</i>. Forward by William “Brit” Kirwan. Washington, DC: Mathematical Association of America. <a href="http://www.maa.org/sites/default/files/pdf/CommonVisionFinal.pdf">www.maa.org/sites/default/files/pdf/CommonVisionFinal.pdf</a>Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com0tag:blogger.com,1999:blog-7251686825560941361.post-29923173871795555802016-03-01T08:00:00.000-05:002016-03-01T08:00:11.013-05:00What we say/What they hear. IIIn last month’s column, I introduced recent research by Kristen Lew, Tim Fukawa- Connelly, Juan Pablo Mejia-Ramos, and Keith Weber on the difficulties students encountered in picking out the points that the instructor wanted to emphasize. One of the lessons, of course, is that if you want to ensure that students note and remember a particular message that you, the instructor, wish to make, it is not enough to say it. You also need to write it. But something deeper is also at work. I appreciate that in response to my column Pat Thompson sent me copies of two of his articles on issues of meaning when teaching mathematics (see references).<br /><br />Pat begins by describing the work of Dewey, Piaget and others who explained that the communication of meaning lies at the heart of effective teaching. However, communicating meaning is extremely difficult. As Pat says, <br /><br /><blockquote class="tr_bq">Figure [1] shows Persons A and B attempting to have a meaningful conversation. Person A intends to convey something to Person B. The intention is constituted by a thought that A holds that he wishes B to hold as well. The figure shows A not just considering how to express his thought, but considering how B might interpret A’s utterances and actions. It is worthwhile noting that A’s action towards B is not really towards B. A’s action towards B is towards A’s image of B. In a sophisticated conversation A’s action towards B is not just towards B, but it’s towards B with some understanding of how B might hear A. Likewise, B is doing the same thing. He assimilates A’s utterances, imbuing them with meanings that he would have were he to say the same thing. But B then colors those understandings with what he knows about A’s meanings and according to the extent to which A said something differently than B would have said it to mean what B thinks A means. B then formulates a response to A with the intent of conveying to A what B now has in mind, but B colors his intention with his model of how he thinks A might hear him, where the model is updated by anything he has just learned from attempting to understand A’s utterance. And so on. (Thompson, 2013, p. 63)</blockquote><br /><br /><div style="text-align: center;"><a href="https://4.bp.blogspot.com/-dCVx6nusavs/VtBxqW8sGrI/AAAAAAAAKe0/1FoSMfmJQmw/s1600/Figure1.tiff" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" src="https://4.bp.blogspot.com/-dCVx6nusavs/VtBxqW8sGrI/AAAAAAAAKe0/1FoSMfmJQmw/s1600/Figure1.tiff" /></a></div><div style="text-align: center;"><span style="font-size: 12.8px;">Figure 1. Summary of intersubjective operations involved in the communication of </span><span style="font-size: 12.8px;">meaning. (Thompson, 2013, p. 64)</span></div><br /><br />The problem is that just because each party has a mental image of the other as understanding their meaning is no guarantee that there is such a mutual understanding:<br /><br /><blockquote class="tr_bq">In Piaget’s and Glasersfeld’s usage, A’s and B’s conversation enters a state of intersubjectivity when neither of them has a reason to believe that he has misunderstood the other. They may in fact have completely misunderstood each other, but they have not discerned any evidence of such. (Thompson, 2013, p. 64)</blockquote><br />I’d like to offer my own interpretation of what was happening in the class that Lew et al. observed. This is pure speculation, but it is based on more than forty years of teaching. I believe that the instructor and the students had attributed very specific and very different meanings to the proof that was presented in class. <br /><br />To the instructor, this proof was an opportunity to showcase general approaches. The fact is that the theorem that was proven, “If a sequence {<i>x<sub>n</sub></i>} has the property that there exists a constant <i>r</i> with 0<<i>r</i><1 such that |<i>x<sub>n</sub>–x</i><sub><i>n</i>–1</sub>| < <i>r<sup>n</sup></i> for any two consecutive terms in the sequence, then {<i>x<sub>n</sub></i>} is convergent,” is not particularly important to the study of convergence. What is clear from the five points that instructor believed he had made was that this provided an opportunity to showcase the usefulness of the Cauchy criterion, the triangle inequality, and the geometric series. This was his meaning. The ease with which peers identified the majority of these messages signifies that they shared his image of the meaning of this example.<br /><br />The student inability to recognize the points that the instructor had intended to convey suggests that their meaning for this proof was very different. They probably understood the instructor’s intention as one of communicating that this is a valid result worthy of being noted and remembered. Just laying out a formal proof immediately communicates this message to most students in real analysis. The fact that the only things written on the board were the steps in the proof of this result almost certainly reinforced their belief that it was the validity and significance of this statement that was the instructor’s meaning.<br /><br />I suspect that, had the instructor written his five points on the board, that might have succeeded in shifting the understanding by some of the students of the instructor’s meaning for this proof. But I would be willing to wager that not all of them, probably not even a majority of them, would have seen these as being as important as the actual statement of the theorem. Their reluctance to even recognize that the instructor had made particular points when these were singled out from the lecture suggests that just writing them on the board would not have been sufficient.<br /><br />Anyone who has probed student understanding has seen this miscommunication. This raises the obvious question: How do we manage to establish a shared understanding? Certainly, a lecture format with only occasional interaction between instructor and students is fertile ground for intersubjectivity that has nothing to do with mutually shared meanings. This is where clickers can help, especially in large format classes. But their effective use relies on a thorough understanding of the range of possible student understandings of the meanings of the lesson. And this understanding must be accompanied by careful construction of questions that can both identify miscommunication and create the cognitive dissonance that moves students toward understanding the instructor’s meaning.<br /><br />Flipped classes can be even more effective in establishing common meanings, but they also are not easy to run effectively. The work that is done in class must be carefully tailored to identify student misinterpretations of the intent of the lesson, complete with leverage points for addressing these misunderstandings. It is too easy for a flipped class to degenerate into supervised practice. How much more instructive it would have been for the instructor to ask the students to work on a proof of the stated result, emphasizing the usefulness of each of the tools needed for the proof as students discovered—or were led to discover—them. And, of course, you do not just do this once. You need the students to encounter multiple instances where these tools are useful before they fully grasp their versatility and importance. This approach is not easy. It requires an instructor who is finely attuned to the knowledge and the ability to draw on that knowledge of each of the students. And it requires a considerable investment of time. Such an approach takes far more than the ten minutes the instructor actually spent on this theorem.<br /><br />However, as the work in this study revealed, that ten minutes was largely wasted. The intended messages were never heard. If these were important messages, and I think that most of us who teach real analysis would acknowledge that they are, then they are worth the effort to communicate this importance. Inevitably, that will require “covering” less material. It forces the instructor not only to prioritize the understandings she intends that students carry away from this course, but also to prioritize her efforts to determine what students think she is saying.<br /><br /><b>References</b><br /><br />Lew, K., Fukawa-Connelly, T., Mejia-Ramos, J.P., and Weber, K. 2016. <a href="http://pcrg.gse.rutgers.edu/">Lectures in advanced mathematics: Why students might not understand what the mathematics professor is trying to convey</a>. <i>Journal for Research in Mathematics Education</i>. Preprint retrieved from pcrg.gse.rutgers.edu on January 24, 2016.<br /><r an="" and="" any="" believed="" br="" cauchy="" clear="" consecutive="" convergence.="" convergent="" criterion="" ease="" example.="" five="" for="" from="" geometric="" had="" he="" his="" identified="" image="" important="" in="" inequality="" instructor="" is="" made="" majority="" meaning.="" meaning="" messages="" not="" of="" opportunity="" particularly="" peers="" points="" provided="" rn="" sequence="" series.="" shared="" showcase="" signifies="" study="" such="" terms="" that="" the="" then="" these="" they="" this="" to="" triangle="" two="" usefulness="" was="" what="" which="" with="" xn=""><br /></r><r an="" and="" any="" believed="" br="" cauchy="" clear="" consecutive="" convergence.="" convergent="" criterion="" ease="" example.="" five="" for="" from="" geometric="" had="" he="" his="" identified="" image="" important="" in="" inequality="" instructor="" is="" made="" majority="" meaning.="" meaning="" messages="" not="" of="" opportunity="" particularly="" peers="" points="" provided="" rn="" sequence="" series.="" shared="" showcase="" signifies="" study="" such="" terms="" that="" the="" then="" these="" they="" this="" to="" triangle="" two="" usefulness="" was="" what="" which="" with="" xn=""> Thompson, P. W. (2013). In the absence of meaning… . In Leatham, K. (Ed.), <i>Vital directions for research in mathematics education</i> (pp. 57-93). New York, NY: Springer.<br /><br /> Thompson, P. W. (2015). Researching mathematical meanings for teaching. In English, L., & Kirshner, D. (Eds.), <i>Third Handbook of International Research in Mathematics Education </i>(pp. 435-461). London: Taylor and Francis.<br /><br /></r>Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com0tag:blogger.com,1999:blog-7251686825560941361.post-89507024724807880562016-02-01T07:00:00.000-05:002016-02-01T07:00:11.227-05:00What we say/What they hearAn important paper is about to appear in the <i>Journal for Research in Mathematics Education</i>, exploring why lecture is so ineffective for so many students: “<a href="http://pcrg.gse.rutgers.edu/">Lectures in advanced mathematics: Why students might not understand what the mathematics professor is trying to convey</a>” by Kristen Lew, Tim Fukawa-Connelly, Juan Pablo Mejia-Ramos, and Keith Weber. The authors video-taped a portion of a lecture given in a junior-level real analysis course and performed a detailed analysis of the differences between what both the professor and his peers thought had been conveyed and what the students were able to take from it.<br /><br />The study used a class by a professor at a large public university who is widely recognized as an excellent lecturer. It focused on a 10-minute stretch in which a proof was presented. The theorem in question is, “If a sequence {x<sub>n</sub>} has the property that there exists a constant <i>r</i> with 0 < <i>r </i>< 1 such that |x<sub>n</sub>–x<sub>n–1</sub>| < r<sup>n</sup> for any two consecutive terms in the sequence, then {x<sub>n</sub>}is convergent.” The four authors of this paper and an additional instructor who teaches real analysis each observed the video and noted the messages that they saw the professor conveying. They then interviewed the professor who identified five messages that he was trying to convey during this lecture. These are listed below. All except the first had been noted by all of the other peer observers. A full transcript of what transpired during these 10 minutes is included in the appendix to the paper. You may want to check whether you can see these points.<br /><br /><ol><li>Cauchy sequences can be thought of as sequences that “bunch up”</li><li>One can prove a sequence with an unknown limit converges by showing it is Cauchy</li><li>This shows how one sets up a proof that a sequence is Cauchy</li><li>The triangle inequality is useful in proving series in absolute value formulae are small</li><li>The geometric series formula is part of the mathematical toolbox that can be used to keep some desired quantities small</li></ol><br />Six students from this class agreed to participate in the extensive interviews required for the study. They were put into three pairs in order to encourage discussion that would help draw out and verbalize what they remembered. <br /><br />About two or three weeks after the class in question, students were asked to review their notes about this proof and identify the points that the professor had made. These were compared with the professor’s five points. None of the pairs brought up any of the instructors messages. This is not particularly surprising. Students tend to restrict what they write in their notes to what is being written on the board, and all five of the professor’s points had only been made orally.<br /><br />As a second pass, each of the students was given a transcript of all that had been written on the blackboard during this proof and then watched the 10-minute lecture, with the hope that they could now focus on what was being said rather than what had been written. They were again asked to identify the points that had been made. One pair did note the emphasis on the importance of the triangle inequality. Another pair noted the third point, that this was about how to set up a proof that a sequence is Cauchy. Nothing else from the list was mentioned.<br /><br />At a third pass, the students were shown just the five short clips where these five points had been made. Two of the pairs now picked up the first message, two picked up the second, and two picked up the fourth. No one new picked up the third point, that the professor had been illustrating a general approach to proving that a particular sequence is Cauchy.<br /><br />Finally, the students were told that these five messages might have been contained in the lecture and were asked whether, in fact, these points had been made. Now most of the students were able to see most of these messages, but one pair never acknowledged the second point, that one way to prove that a sequence converges is to show that it is Cauchy, and, even after seeing the clip in which this point was made, none of them acknowledged that the professor had made the fifth point: that the geometric series is part of the toolbox for approaching such proofs. <br /><br />What I find particularly interesting is the sharp distinction between what was seen in this lecture by those who are familiar with the material and what was seen by those who are still struggling to build an understanding. This echoes much of the work of John and Annie Selden who have shown how difficult it is for undergraduate students to extract the significant features of a proof. This paper shows that it is not enough to accompany what is written on the board with oral indications of what is important and how to think about it. It is not even enough when these indications are repeatedly emphasized.<br /><br />In the introduction, this paper presents the example of the Feynman Lectures, widely considered to be some of the finest scientific expositions ever made. Yet, the fact is that when they were given at Cal Tech, “Many of the students dreaded the course, and as the course wore on, attendance by the registered students dropped alarmingly.” (Goodstein and Negebauer, 1995, p. xxii–xxiii). There is no doubt that lectures have an important role to play in conveying information for which the recipients have a well-structured understanding in which to place it. However, as this study strongly suggests, lectures are not very helpful for students who are trying to find their way into a new area of mathematics and who still need to build such a structure of understanding.<br /><br /><b>References</b><br /><br />Goodstein, G. & Negebauer, G. 1995. Preface to R. Feynman’s <i>Six Easy Pieces</i>. pp. xix–xxii. New York: Basic Books.<br /><br />Lew, K., Fukawa-Connelly, T., Mejia-Ramos, J.P., and Weber, K. 2016. <a href="http://pcrg.gse.rutgers.edu/">Lectures in advanced mathematics: Why students might not understand what the mathematics professor is trying to convey</a>. <i>Journal for Research in Mathematics Education</i>. Preprint retrieved from <a href="http://pcrg.gse.rutgers.edu/">pcrg.gse.rutgers.edu</a> on January 24, 2016.Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com1tag:blogger.com,1999:blog-7251686825560941361.post-44975179900570340372016-01-01T15:46:00.000-05:002016-01-01T15:46:32.163-05:00MAA Calculus Study: Building NetworksI am beginning this month’s column with the announcement of a conference and workshop that should interest readers of this blog. A discussion of the background and context for the conference will follow the announcement.<br /><br />Announcing an NSF-sponsored MAA Conference on<br /><div style="text-align: center;"><b>Precalculus to Calculus: Insights & Innovations</b> </div><div style="text-align: center;">June 16–19, 2016 </div><div style="text-align: center;">University of Saint Thomas, Saint Paul, Minnesota </div><br />To be followed immediately by a workshop on<br /><div style="text-align: center;"> <b>Curriculum, Instruction, and Placement in Algebra and Precalculus </b></div><div style="text-align: center;">June 19–20, 2016</div><div style="text-align: center;"></div><div style="text-align: center;">Same location</div><div style="text-align: center;"><b><br /></b></div>The conference will provide opportunities to learn from the MAA’s studies of precalculus and calculus, to hear what is happening at peer institutions, and to build networks of shared experience and practice. The two and a half days will be built around four themes:<br /><br /><div style="text-align: left;"><b>Focus on Curriculum</b>. Content of and alternative approaches to precalculus, articulation issues, preparation for downstream courses</div><div style="text-align: left;"><br /></div><div style="text-align: left;"><b>Focus on Students</b>. Placement, early warning systems and support services, formative and summative assessment, supporting students from underrepresented groups</div><div style="text-align: left;"><br /></div><div style="text-align: left;"><b>Focus on Pedagogy</b>. Active learning strategies, making the most of large lectures, use of Learning Assistants, assessing effectiveness of innovations</div><div style="text-align: left;"><br /></div><div style="text-align: left;"><b>Focus on Instructors</b>. Building communities of practice, training of graduate teaching assistants, working with adjuncts, getting faculty buy-in for innovative practices</div><br />The workshop will be an opportunity to learn from the work of Marilyn Carlson, Bernie Madison, and Michael Tallman on <i>Using Research to Shape Instruction and Placement in Algebra and Precalculus </i>(NSF #1122965).<br /><br />There is no registration fee. Housing and meals are included at no cost to participants. Participants are responsible for their own transportation. Housing will be in the air- conditioned apartments in Flynn Hall. Each apartment consists of four single bedrooms, two bathrooms, and a kitchen and living room. The University of Saint Thomas sits on a bluff above the Mississippi River, six miles from the Minneapolis/St. Paul airport and midway between the downtowns of Minneapolis and Saint Paul.<br /><br />The number of participants accepted to the conference and workshop will be limited. A link to the application to attend the conference and workshop is at <a href="http://www.maa.org/cspcc">www.maa.org/cspcc</a>.<br /><br />Review of applications will begin March 15. Those accepted will be notified by April 1.<br /><br />This conference combines the efforts of the two studies on which I have been PI: <i>Characteristics of Successful Programs in College Calculus</i> (CSPCC, NSF #0910240) and Progress through Calculus (PtC, NSF #1430540). Part of its role is to disseminate results from CSPCC, many of which can also be found in the Notes volume <i><a href="http://www.maa.org/programs/faculty-and-departments/curriculum-development-resources/national-studies-college-calculus">Insights and Recommendations</a></i>. But the more important task is to foster the building of networks of peer colleges and universities who are seeking to improve the effectiveness of their precalculus through calculus sequences. The four themes reflect the four areas of concern and ongoing work that have emerged from our surveys and from the meeting held in Washington, DC over the October 31 to November 1 weekend.<br /><br />The DC workshop brought together representatives from 27 universities that either are now engaged in initiatives to improve this sequence or are seriously concerned about lack of student success in these courses and are looking to improve what they do. There were many common interests and concerns that emerged. I want to acknowledge the role of Naneh Apkarian, assisted by the other graduate students, who monitored the discussions and summarized the issues. These included:<br /><br /><ul><li>Aligning precalculus/calculus courses to create more coherent programs based on student and client discipline needs (with an emphasis on the transition from precalculus to calculus)</li><ul><li>What is “precalculus?” (content, purpose, function)</li><li>Aligning precalculus so that it is truly a preparation for calculus</li><li>Dealing with the multiple purposes for a variety of students (e.g., preparation, gen. ed., STEM, business)</li></ul><li>Encouraging/Supporting/Implementing Active Learning</li><ul><li>Especially when the institution insists on large classes </li></ul><li>Information about flexible and/or non-standard models for the precalculus/calculus </li><li>GTA Training Programs </li><ul><li>Specifically with regards to issues surrounding active learning</li></ul><li>Student skill retention within and across courses </li><li>Making calculus accessible for students from varying backgrounds</li><ul><li>Can it be done in one classroom, or are “flavors” needed? </li></ul><li>Placing students into appropriate courses and then supporting them</li><ul><li>Establishing what various high school calculus courses really are</li><li>Early warning systems</li><li>Various pathways through calculus</li></ul><li>Professional development/Increasing faculty buy-in</li><ul><li>With respect to active learning</li><li>Identifying ways of supporting faculty interested in using active learning strategies</li><li>With respect to utilizing technology to support student learning</li></ul><li>Strategies for increasing administrative support/handling administrative pressures </li><li>Collecting and managing data </li></ul>The Saint Paul conference in June will be an opportunity to learn what is known about these issues, with examples of successful or promising interventions. In response to the request for networking opportunities to share information about materials, case studies, guidelines, and the experiences of peer institutions, we have established a website, the PtC Discussion Group, on a new platform, Trellis, managed by AAAS. To join this discussion, go to <a href="http://www.trelliscience.com/">www.trelliscience.com</a>, register, then search for the PtC Discussion Group and request to join.Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com1tag:blogger.com,1999:blog-7251686825560941361.post-11625395203037742072015-12-01T07:30:00.000-05:002015-12-02T12:38:57.210-05:00Strategies for ChangeOne of the most striking findings from the MAA’s survey of university mathematics departments undertaken this past spring (see<a href="http://launchings.blogspot.com/2015/11/maa-calculus-study-new-initiative.html" target="_blank"> last month’s column</a>) is the almost universal recognition that current practice in the precalculus through single variable calculus sequence needs to be improved. Many such efforts are now underway, but many of them lack understanding of how institutional change occurs as well as recognition of the importance of this understanding.<br /><br />Much of the literature on institutional change lies too far from the contexts or concerns of mathematics departments to be easily translatable, but an important paper appeared a little over a year ago in the Journal of Engineering Education that provides an insightful framework for understanding change in the context of undergraduate STEM education: “<a href="http://onlinelibrary.wiley.com/doi/10.1002/jee.20040/abstract" target="_blank">Increasing the Use of Evidence-Based Teaching in STEM Higher Education</a>: A Comparison of Eight Change Strategies” by Borrego and Henderson (2014). This paper takes the framework distilled by Henderson, Beach, and Finkelstein in 2010 and 2011 from their literature review of change strategies and applies it to eight different approaches to bringing evidence-based teaching into the undergraduate STEM classroom. This short column cannot do justice to their extensive discussion, but it can perhaps whet interest in reading their paper.<br /><br />Henderson, Beach, and Finkelstein have identified two axes along which change strategies occur (Table 1): those whose focus is on changing individuals versus those that focus on changing environments and structures, and those that they describe as <i>prescribed</i>, meaning that they try to implement specific solutions, versus those they describe as <i>emergent</i>, meaning that they attempt to foster conditions that support local actors in finding their own solutions. This results in the four categories shown in Table 1.<br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="http://3.bp.blogspot.com/-eimSCpXcN8I/VlSbY0Me10I/AAAAAAAAKbE/cezmV-sgBXU/s1600/Launchings%2BDec%2B2015.PNG" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" src="http://3.bp.blogspot.com/-eimSCpXcN8I/VlSbY0Me10I/AAAAAAAAKbE/cezmV-sgBXU/s1600/Launchings%2BDec%2B2015.PNG" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><span style="font-size: 12.8px;">Table I: Change theories mapped to the four categories of change strategies. The italicized text lists two specific change strategies for each of the four categories. Reproduced from Borrego and Hnderson (2014).</span></td></tr></tbody></table><br /><div class="separator" style="clear: both; text-align: center;"></div>Within each of the four categories, they identify two strategies that have been used. For example, under a prescribed outcome focused on individuals, Category I, they identify <i>Diffusion</i> and <i>Implementation</i> as two change strategies. Diffusion describes the common practice of developing an innovation at a single location and then publicizing it in the hope that others will pick it up. <i>Implementation</i> involves the development of a curriculum or specified set of practices that are intended to be implemented at other institutions. For each of the eight change strategies, they describe the underlying logic of how it could effect change, describe what it looks like in practice, and give an example of how it has been used, accompanied by some assessment of its potential strengths and weaknesses. Diffusion, in particular, is very common and is known to be capable of raising awareness of what can be done, but it often runs into challenges of incompatibility together with a lack of support for those who would attempt to implement it.<br /><br />At the opposite corner are the emergent strategies that focus on environments and structures. Here Borrego and Henderson consider <i>Learning Organizations </i>and <i>Complexity Leadership Theory</i>. Learning organizations have emerged from management theory as a means of facilitating improvements. They involve informal communities of practice that share their insights into what is and is not working, embedded within a formal structure that facilitates the implementation of the best ideas that emerge from these communities. In management-speak, it is the middle-line managers who are the key to the success of this approach. In the context of higher education, these middle-line managers are the department chairs and the senior, most highly respected faculty. <br /><br />The effectiveness of Learning Organizations resonates with what I have seen of effective departments. They require an upper administration that recognizes there are problems in undergraduate mathematics education and are willing to invest resources in practical and cost- effective means of improving this education, together with faculty in the trenches who are passionate about finding ways of improving the teaching and learning that takes place at their institution. The faculty need to be encouraged to form such communities of practice, sharing their understanding and envisioning what changes would improve teaching and learning. Some of the best undergraduate teaching we have seen has been built on the practice of regular meetings of the instructors for a particular class. The role of the chair and senior faculty is one of encouraging the generation of these ideas, providing feedback and guidance in refining them, and then selling the result to the upper administration, conscious of how it fits into the concerns and priorities of deans and provosts. Throughout this process, it is critical to have access to robust and timely data on student performance for this class as well as for the downstream courses both within and beyond the mathematics department.<br /><br />Complexity Leadership Theory is based on recognition of the difficulties inherent in trying to change any complex institution and calls on the leadership to do three things: to disrupt existing patterns, to encourage novelty, and to make sense of the responses that emerge. Borrego and Henderson could not find any examples of Complexity Leadership Theory within higher education, but, as I interpret this approach as it might appear within a mathematics department, it speaks to the responsibility of the chair and leading faculty to draw attention to what is not working, to encourage faculty to seek creative solutions to these problems, and then to shape what emerges in a way that can be implemented. In many respects, it is not so different from Learning Organizations. The strategies of Category IV highlight the key role of the departmental leadership, which must involve more than just the chair or head of the department.<br /><br />In their discussion, Borrego and Henderson emphasize that they are not suggesting a preference for any of these categories, although they do note that Category I is the most common within higher education and Category IV the least. My own experience suggests that the strategies of Category IV have the greatest chance of making a lasting improvement. Nevertheless, anyone seeking systemic change will need to employ a variety of strategies that span all of these approaches. Their point is that anyone seeking change must be aware of the nature of what they seek to accomplish and must recognize which strategies are best suited to their desired goals.<br /><br /><b>Bibliography</b><br /><br />M. Borrego and C. Henderson. 2014. Increasing the use of evidence-based teaching in STEM higher education: A comparison of eight change strategies. <i>Journal of Engineering Education</i>. 103 (2): 220–252.<br /><br />C. Henderson, A. Beach, N. Finkelstein. 2011. Facilitating change in undergraduate STEM instructional practices: An analytic review of the literature. <i>Journal of Research in Science Teaching</i>. 48 (8): 952–984.<br /><br />C. Henderson, N. Finkelstein, A. Beach. 2010. Beyond dissemination in college science teaching: An introduction to four core change strategies. <i>Journal of College Science Teaching</i>. 39 (5): 18–25.<br /><br />Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com0tag:blogger.com,1999:blog-7251686825560941361.post-8793833164090389502015-11-01T10:30:00.000-05:002015-11-01T10:30:00.278-05:00MAA Calculus Study: A New Initiative<div class="separator" style="clear: both; text-align: center;"></div><br />With the publication of <i><a href="http://www.maa.org/sites/default/files/pdf/cspcc/InsightsandRecommendations.pdf" target="_blank">Insights and Recommendations from the MAA National Study of College Calculus</a></i>, <a href="http://www.maa.org/programs/faculty-and-departments/curriculum-development-resources/national-studies-college-calculus" target="_blank">we are wrapping up the original MAA calculus study, Characteristics of Successful Programs in College Calculus</a> (CSPCC, NSF #0910240). This past January, MAA began a new large-scale program, <i><a href="http://maa.org/cspcc" target="_blank">Progress through Calculus</a></i> (PtC, NSF #1430540), that is designed to build on the lessons of CSPCC. I am continuing as PI of the new project. Co-PIs Chris Rasmussen at San Diego State, Sean Larsen at Portland State, Jess Ellis at Colorado State, and senior researcher Estrella Johnson at Virginia Tech are leading local teams of post-docs, graduate students, and undergraduates who will be working on this effort.<br /><br />CSPCC sought to identify what made certain calculus programs more successful than others but was limited in its measures of success to what could be learned about changes in student attitudes between the start and end of Calculus I and to what could be observed from a single three-day visit to a select group of 20 colleges and universities. PtC is extending its purview to the entire sequence of precalculus through single variable calculus, and it will take broader measures of success, including performance on a standardized assessment instrument, persistence into subsequent mathematics courses, and performance in subsequent courses. It also is shifting emphasis from description of the attributes of successful programs to analysis of the process of change: What obstacles do departments encounter as they attempt to improve the success of their students? What accounts for the difference between departments that are successful in institutionalizing improvements and those that are not? <br /><br />We began this past spring with a survey of all mathematics departments offering a graduate degree in Mathematics, either MA/MS or PhD. This is a manageable number of institutions: 178 PhD and 152 Masters universities. These are the places that most often struggle with large classes and with the trade-off between teaching and research. We had an excellent participation rate: 75% of PhD and 59% of Masters universities filled out the survey. <br /><br />Data from this survey will appear in future papers and articles, but for this column I want to focus on the most important information we learned: what these departments see as critical to offering successful classes and how that compares to how well they consider themselves to be doing on these measures.<br /><br />CSPCC identified eight practices of successful programs. These are listed here in the order implied by the number of doctoral departments in the PtC survey that identified each as “very important to a successful precalculus/calculus sequence.” <br /><br /><ol><li>Student placement into the appropriate initial course </li><li>GTA teaching preparation and development </li><li>Student support programs (e.g. tutoring center) </li><li>Uniform course components (e.g. textbook, schedule, homework) </li><li>Courses that challenge students </li><li>Active learning strategies </li><li>Monitoring of the precalculus/calculus sequence through the collection of local data </li><li>Regular instructor meetings about course delivery.</li></ol><br />The graphs in Figures 1 and 2 show the percentage of respondents who identified each as “very important” (as opposed to “somewhat important” or “not important”), as well as the percentage of respondents who considered themselves to be “very successful” with each (opposed to “somewhat successful” or “not successful”).<br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><img height="407px;" src="https://lh4.googleusercontent.com/W3K7X1h0C4IDqCWJt2XEGyPJ0KstnJPiwcn2E0bxH6yb1xqo_9T0QTwOZrlENDEg0OBs0Rdz8LP5zzsZPQmDSVHZH36IrgoJ4_UmKoGWkdhQLn2dNQLBOXDY3xOcysq7GvquLHiQrRIFwkdK" style="border: none; margin-left: auto; margin-right: auto; transform: rotate(0rad);" width="576px;" /></td></tr><tr><td class="tr-caption" style="text-align: center;"><span id="docs-internal-guid-a9e8461f-a583-6a09-9804-a5751f40186b"><span style="font-weight: normal; vertical-align: baseline; white-space: pre-wrap;"><span style="font-size: x-small;">Figure 1. PhD universities. What they consider to be important versus how successful they consider themselves to be.</span></span></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;"><img height="414px;" src="https://lh3.googleusercontent.com/NI8rKI62xVQ4-Xf6iThe_e-PZYt7OgcC-u7GSjyUkusK6a7zeNOQbYwM-MFaw2ODEytW0zqKV9l1EbI7chMCPGckPYo9UjhkMxRNauMAm3GiV_54DFXOAMGmKUDyZh0drNHXaY7z6CAu0kr7" style="border: none; margin-left: auto; margin-right: auto; transform: rotate(0rad);" width="576px;" /></td></tr><tr><td class="tr-caption" style="text-align: center;">Figure 2. Masters universities. What they consider to be important versus how successful they consider themselves to be.</td></tr></tbody></table><div style="text-align: center;"><span id="docs-internal-guid-a9e8461f-a992-7611-7687-131ce9fea4aa"><span style="font-size: 16px; vertical-align: baseline; white-space: pre-wrap;"></span></span></div><br />What is most interesting for our purposes is where departments see a substantial gap between what they consider to be very important and where they see themselves as very successful. These are the areas where departments are going to be most receptive to change. If we look for large absolute or relative gaps, five of the eight practices show up as areas of concern (Table 1). The biggest absolute gap is for placement; approximately half of all universities consider placement to be very important but do not rate themselves as very successful. The largest relative gap is for active learning, where only 27% of doctoral universities and 36% of masters universities that consider this to be very important also consider themselves to be very successful at it.<br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><img src="http://2.bp.blogspot.com/-bWooxlprX3M/Vi55d9HiG_I/AAAAAAAAKaA/wYtrf5Y0reU/s1600/Nov%2BLaunchings%2BTable.PNG" style="margin-left: auto; margin-right: auto;" /></td></tr><tr><td class="tr-caption" style="text-align: center;">Table 1. Departments that consider themselves to be very successful as percentage of those that consider the practice to be very important.</td></tr></tbody></table>The next stage of this project will be the building of networks of universities with common concerns and the identification of twelve universities for intense study over a three-year period. This stage has begun with a small workshop for representatives of 27 universities, a workshop that will begin building these networks and is ending as this column goes live on November 1. It will be continuing with a larger conference in Saint Paul, MN, June 16–19, 2016. Watch this space for more information about that conference.Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com3tag:blogger.com,1999:blog-7251686825560941361.post-77108471036950705382015-10-01T08:30:00.000-04:002015-10-01T08:30:02.985-04:00Evidence for IBL<b>Special Note: The AMS Blog <a href="http://blogs.ams.org/matheducation/2015/09/10/active-learning-in-mathematics-part-i-the-challenge-of-defining-active-learning/" target="_blank">On Teaching and Learning Mathematics</a> has started a six-part series on active learning.</b><br /><br />Over the past decade, the Educational Advancement Foundation has supported programs to promote Inquiry-Based Learning (IBL) in mathematics at four major universities. IBL is not a curriculum. Rather, it is a guiding philosophy for instruction that takes a structured approach to active learning, directing student activities and projects toward building a fluent and comprehensive understanding of the central concepts of the course. Ethnography & Evaluation Research (E&ER) at the University of Colorado, Boulder has studied the effectiveness of these implementations. Several research papers have resulted, of which the paper by Kogan and Laursen (2014), discussed in my column <a href="http://launchings.blogspot.com/2013/10/evidence-of-improved-teaching.html" target="_blank">Evidence of Improved Teaching</a> (October 2013), presented very clear evidence that IBL prepares students for subsequent courses better than standard instruction and that IBL can result in students taking more mathematics courses, especially when offered early enough in the curriculum. Two recent papers document the benefits of IBL in preparing future teachers and in building personal empowerment.<br /><br />In <i>Implementation and outcomes of inquiry-based learning in mathematics content courses for pre-service teachers</i> (Laursen, Hassi, and Hough, 2015), the authors focused on the development of Mathematical Knowledge for Teaching (MKT), a term coined by Deborah Ball to describe the kind of knowledge that teachers must draw upon to teach mathematics well and that reflects understanding of how ideas and concepts relate to one another as well as the common difficulties and misunderstandings that students are likely to encounter. Being prepared for teaching requires more than being able to find solutions to particular problems. A good teacher must have at her or his disposal a variety of approaches to a solution and the ability to take a student’s incorrect attempt at an answer, recognize where the misunderstanding lies, and build on what the student does understand.<br /><br />In theory, IBL should help develop MKT because it focuses on precisely those characteristics of practicing mathematicians that teachers most need, the habits of mind than include sense-making, conjecture, experimentation, creation, and communication.<br /><br />E&ER studied students in thirteen sections of seven courses for pre-service teachers at two of the four universities, courses that collectively spanned preparation for primary, middle school, and secondary teaching. They used an instrument developed by Ball and colleagues, Learning Mathematics for Teaching (LMT) that has been validated as an effective measure of MKT for practicing teachers. The results were impressive. The students had begun the term with LMT scores that averaged at the mean for in-service teachers across the country. Each of the IBL classes saw mean LMT scores rise by 0.67 to 0.90 standard deviations. In line with the results of the 2014 report, all students experienced gains from IBL, but the weakest students saw the greatest gains.<br /><br />The second recent article is <i>Transforming learning: Personal empowerment in learning mathematics </i>(Hassi and Laursen, 2015). In <i>Adding It Up</i> (NRC 2001), mathematical proficiency is recognized as consisting of five strands: conceptual understanding, procedural fluency, strategic competence, adaptive reasoning, and productive disposition. They are each critically important. This paper investigates the effect of IBL on both strategic competence, what the authors term <i>cognitive empowerment</i>, and productive disposition, which they separate into <i>self-empowerment</i> and <i>social empowerment</i>, the last of which also incorporates effective communication.<br /><br />The study was conducted through interviews with students who had taken a class at one of the four universities using IBL. An overwhelming majority of students reported gains in each of the three areas of personal empowerment. Among women 77% and among men 69% reported an increase in self-esteem, sense of self-efficacy, and confidence from their IBL experience. For general thinking skills, deep thinking and learning, flexibility, and creativity, 77% of the women and 90% of the men described improvements. For ability to explain and discuss mathematics as well as skills in writing and presenting mathematics, 79% of the women and 76% of the men saw gains.<br /><br />When pressed for what made the IBL experience special, students identified their own role in influencing the course pace and direction, the importance of combining both individual and collaborative work, and the fact that they were faced with problems that were both challenging and meaningful. They appreciated that they were given responsibility to think on their own. Such experiences were especially important for women and for first-year students.<br /><br />In the very discouraging reports on the effects of Calculus I instruction in most US universities (Sonnert and Sadler 2015), we see courses that accomplish exactly the opposite of personal empowerment, courses that sharply decrease student confidence and sense of self-efficacy. It does not have to be this way.<br /><br />References<br /><br />Hassi, M.-L., and Laursen, S.L. 2015. Transformative learning: Personal empowerment in learning mathematics. <i>Journal of Transformative Education</i>. Published online before print May 24, 2015, doi: 10.1177/1541344615587111.<br /><br />M. Kogan and S. Laursen. 2014. Assessing long-term effects of inquiry-based learning: A case study from college mathematics. <i>Innovative Higher Education</i> <b>39</b> (3), 183–199. http://link.springer.com/article/10.1007/s10755-013-9269-9<br /><br />Laursen, S.L., Hassi, M.-L., and Hough, S. 2015. Implementation and outcomes of inquiry-based learning in mathematics content courses for pre-service teachers. <i>International Journal of Mathematical Education in Science and Technology</i>. Published online before print July 25, 2015, doi: 0.1080/0020739X.2015.1068390<br /><br />National Research Council (NRC). 2001 <i>Adding it up: Helping children learn mathematics</i>. J. Kilpatrick, J. Swafford, and B. Findell (Eds.). Washington, DC: National Academy Press.<br /><br />Sonnert, G. and Sadler, P. 2015. The impact of instructor and institutional factors on students’ attitudes. Pages 17–29 in <i>Insights and Recommendations from the MAA National Study of College Calculus</i>, D. Bressoud, V. Mesa, and C. Rasmussen (Eds.). Washington, DC: Mathematical Association of America Press.Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com2tag:blogger.com,1999:blog-7251686825560941361.post-41994610888156964412015-09-01T09:24:00.000-04:002015-09-01T10:00:14.934-04:00Calculus at Crisis V: Networks of Support<b>Special Notice: The MAA Notes volume summarizing the results of Characteristics of Successful programs in College Calculus (NSF #0910240), <i>Insights and Recommendations from the MAA National Study of College Calculus</i>, is now available for free download as a PDF file at <a href="http://www.maa.org/cspcc">www.maa.org/cspcc</a>.</b><br /><br />This is the last of my columns on <i>Calculus at Crisis</i>. In the first three, from <a href="http://launchings.blogspot.com/2015_05_01_archive.html" target="_blank">May</a>, <a href="http://launchings.blogspot.com/2015_06_01_archive.html" target="_blank">June</a>, and <a href="http://launchings.blogspot.com/2015_07_01_archive.html" target="_blank">July</a>, I explained why we can no longer afford to continue doing what we have always done. <a href="http://launchings.blogspot.com/2015_08_01_archive.html" target="_blank">Last month</a> I described some of the lessons that have been learned in recent years about best practices with regard to placement, student support, curriculum, and pedagogy. Unfortunately, as those who seek to improve the teaching and learning of introductory mathematics and science have come to realize, knowing what works is not enough.<br /><br />There are many barriers to change, both individual and institutional. Lack of awareness of what can be done is seldom one of them. In recent years, leaders in physics and chemistry education research, especially Melissa Dancy, Noah Finkelstein, and Charles Henderson have studied these barriers and begun to translate insights from the study of how institutional change comes about in order to assist those who seek to improve post- secondary science, mathematics, and engineering education.<br /><br />One of the best short summaries describing specific steps toward achieving long-term change is <i><a href="https://www.aacu.org/pkal/sourcebook" target="_blank">Achieving Systemic Change</a></i>, a report issued jointly by the American Association for the Advancement of Science (AAAS), the American Association of Colleges and Universities (AAC&U), the Association of American Universities (AAU), and the Association of Public and Land-grant Universities (APLU) that I discussed this past <a href="http://launchings.blogspot.com/2014/12/reforming-undergraduate-math-and.html" target="_blank">December</a>. Its emphasis on creating supportive networks within and across institutions is reflected in our own findings in the MAA’s calculus study.<br /><br />There has always been lively interest from individual faculty members in improving mathematics education. Heroic efforts have often succeeded in moving the dial, but without strong departmental support they are not sustainable. As I have explained over the past months, deans, provosts, and even presidents now realize that something must be done. I have yet to meet a dean of science who is not willing—usually even eager—to fund a proposal from the mathematics department for improving student outcomes provided it is concrete, workable, and cost-effective. (Just hiring more mathematicians does not cut it.) The key link between eager faculty and concerned administrators is the department chair, together with the senior, most highly respected faculty. Without their support and cooperation, no lasting improvements are possible.<br /><br />The department chair is essential. This is the person who can take an enthusiastic proposal and massage it into a workable plan whose benefits are understandable to the upper administration. This is the person who can take a request from the dean, understand the resources that will be required, and find the right people to work on it. Unfortunately, appointment as chair does not automatically confer such wisdom. Part of what is needed is an understanding of what is being done at comparable institutions, how it is being implemented, what is working or failing and why. This is where the mathematical societies have an important role to play. AMS does this through its <a href="http://www.ams.org/profession/leaders/leaders" target="_blank">Information for Department Leaders</a>, the work of the <a href="http://www.ams.org/about-us/governance/committees/coe-home" target="_blank">Committee on Education</a>, and its blog <a href="http://blogs.ams.org/matheducation/" target="_blank">On the Teaching and Learning of Mathematics</a>. The MAA’s <a href="http://www.maa.org/cupm" target="_blank">CUPM</a>, <a href="http://www.maa.org/programs/faculty-and-departments/curriculum-department-guidelines-recommendations/guidelines-for-undergrad-programs" target="_blank">CTUM</a>, and <a href="http://www.maa.org/cupm/crafty" target="_blank">CRAFTY</a> committees provide this information through publications, panels, and contributed paper sessions. SIAM, ASA, and AMATYC also embrace this mission. <a href="http://www.maa.org/programs/faculty-and-departments/common-vision" target="_blank">Common Vision</a> began this year as an effort to coordinate these activities across the five societies.<br /><br />But a supportive department chair is not enough. The lasting power center in any department consists of senior faculty who are highly respected for their research visibility. The most successful calculus programs we have seen in the MAA study <i>Characteristics of Successful Programs of College Calculus</i> involved some of these senior faculty in an advisory capacity: monitoring the annual data on student performance, observing occasional classes, mentoring graduate students not just for research but also for the development of teaching expertise, and providing encouragement and a sounding board to those—usually younger faculty—engaged in trying new methods in the classroom. It will be the chair’s responsibility to identify the right people for this advisory group, but once it is in existence it can help ensure that future chairs are sympathetic to these efforts.<br /><br />Finally, any mathematics department seeking to improve undergraduate education must remember that it is not alone within its institution. Similar efforts are underway in each of the sciences as well as engineering. Deans and provosts can help by formally recognizing those who serve in these senior roles across all STEM departments and encouraging links between these groups of faculty. They can draw on support and advice from consortia of colleges and universities such as <a href="https://stemedhub.org/groups/aau" target="_blank">AAU</a>, <a href="http://www.aplu.org/projects-and-initiatives/stem-education/science-and-mathematics-teaching-imperative/" target="_blank">APLU</a>, and <a href="https://www.aacu.org/pkal" target="_blank">AAC&U</a>, as well as multidisciplinary societies and consortia such as <a href="http://www.aaas.org/enhancing-education" target="_blank">AAAS</a> and the <a href="http://www.pulsecommunity.org/" target="_blank">Partnership for Undergraduate Life Science Education (PULSE)</a>, all of whom are working to promote networks of educational innovation that cross STEM disciplines. Joining with other departments within the institution can dispel the perception of mathematics as insular and unconcerned with the needs of others as it strengthens individual departmental efforts. All STEM departments are facing similar difficulties. This crisis presents us with an exceptional opportunity to work across traditional boundaries.<br /><br />Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com0tag:blogger.com,1999:blog-7251686825560941361.post-18359665409655065002015-08-01T00:05:00.000-04:002015-08-01T00:05:00.612-04:00Calculus at Crisis IV: Best PracticesIn my last three columns I explained the reasons that college calculus instruction is now at crisis:<br /><br /><ol><li>The need to teach ever more students, who often bring weaker preparation, using fewer resources.</li><li>The fact that most Calculus I students have already studied calculus in high school (this past spring 424,000 students took an AP Calculus exam, an increase of 100,000 over the past five years).</li><li>The pressures from the client disciplines to equip their students with the mathematical knowledge and habits of mind that they actually will need.</li></ol><br /><br />As I have traveled this country to meet with mathematics departments, I have seen that there is a general recognition on the part of chairs, deans, provosts, and occasionally even presidents that the past solutions for calculus instruction are no longer adequate. I am encouraged by the fact that the mainstream calculus sequence is so central to all of the STEM disciplines that, even in these tight budget times, many deans and provosts can find the resources to support innovative programs if they can be convinced these efforts are sustainable, cost-effective, and will actually make a difference. <br /><br />There are four basic leverage points for improving the calculus sequence so that it better meets at least some of these pressures: placement, student support, curriculum, and pedagogy. We know a lot about what does work for each of these. Much of this knowledge—relevant to the teaching of calculus—is contained in the new MAA publication <i>Insights and Recommendations from the MAA National Study of College Calculus</i>, the report on a five-year study of <i>Characteristics of Successful Programs in College Calculus</i> undertaken by the MAA with support from NSF (#0910240). I briefly summarize some of the insights.<br /><br /><b>Placement.</b> Placement can have a huge impact on student success rates. However, given the demands of the client disciplines and the fact that remediation is usually of doubtful value (see <a href="http://launchings.blogspot.com/2014/10/the-pitfalls-of-precalculus.html" target="_blank">The Pitfalls of Precalculus</a>), just tightening up the requirements for access to calculus is unlikely to make a dean or provost happy. We do have evidence of the effectiveness of adaptive online exams such as ALEKS that probe student understanding to reveal individual strengths and weaknesses, especially when combined with tools that can help students address specific topics on which they need refreshing. But there is no one placement exam or means of implementation that will work for all institutions. Further elaboration on what we have learned about placement exams can be found in Chapter 5, Placement and Student Performance in Calculus I, of <i>Insights and Recommendations</i>.<br /><br /><b>Student Support.</b> Programs modeled on the Emerging Scholars Programs can be very effective for supporting at-risk students (see Hsu, Murphy, Treisman, 2008). Tutoring centers are virtually universal, but not always as useful as they could be. The best we have seen put thought into the training of the tutors, require classroom instructors to hold some of their office hours in the center, and are located conveniently with a congenial atmosphere that encourages students to drop in to study or work on group projects even if they do not need the assistance of a tutor. In addition, quick identification and effective guidance of students who are struggling with the course is essential. More on these points can be found in Chapter 6, Academic and Social Supports, of <i>Insights and Recommendations</i>.<br /><br /><b>Curriculum.</b> This is the toughest place at which to apply leverage. Most faculty are fine with changes to placement procedures and support services but are appalled at the very thought of touching the curriculum. The pushback against the Calculus Reform movement of the early 1990s was strongest where curricular changes were suggested. Yet this is where we are most likely to be successful in meeting the needs of students who studied calculus in high school, and it must be part of any strategy for meeting the needs of the client disciplines. Research coming out of Arizona State University and other centers of research in undergraduate mathematics education has revealed the basic wisdom of many of the Calculus Reform curricula that approached calculus as a study of dynamical systems. Curricular materials are now being developed that have a much firmer basis in an understanding of student difficulties with the concepts of calculus (for an example, see <a href="http://launchings.blogspot.com/2014/06/beyond-limit-i.html" target="_blank">Beyond the Limit</a>).<br /><br /><b>Pedagogy.</b> Another aspect of the Calculus Reform movement that was poorly received was the emphasis on active learning. The evidence is now overwhelming that active learning is critical, especially important for at-risk students and essential for meeting the needs of the client disciplines. We have learned a lot in the intervening quarter century about how to do it well and cost-effectively, and this is one of the places where new technologies can be particularly helpful. There are now many models for implementation of active learning strategies, spanning classrooms of all sizes, student audiences at varied levels of expertise, and faculty with different levels of commitment to changing how they teach (see <a href="http://launchings.blogspot.com/2015/04/reaching-students.html" target="_blank">Reaching Students</a>). Evidence for the effectiveness of active learning and recommendations of strategies for implementing it can be found in Donovan & Bransford, 2005; Freeman et al., 2014; Fry, 2014; Kober, 2015; and Kogan & Laursen, 2014.<br /><br />The bottom line is that we do have knowledge that can help us face this crisis. There is no universal solution. Each department will have to find its own way toward its own solutions. But it need not stumble alone. As I will explain next month in the fifth and final column in this series, making meaningful and lasting change requires networks of support both within and beyond the individual department. Here also our knowledge base of what works and why has expanded in recent years.<br /><br />References<br /><br />Bressoud, D., Mesa, V., Rasmussen, C. (eds.) (2015). <i>Insights and Recommendations from the MAA National Study of College Calculus</i>. MAA Notes. Washington, DC: Mathematical Association of America (to be available August, 2015).<br /><br />Donovan, M.S. & Bransford, J.D. (eds.). (2005). <i>How Students Learn: Mathematics in the Classroom</i>. Washington, DC: National Academies Press. <a href="http://www.nap.edu/catalog/11101/how-students-learn-mathematics-in-the-classroom">www.nap.edu/catalog/11101/how-students-learn-mathematics-in-the-classroom</a><br /><br /> Freeman, S. et al. (2014). Active learning increases student performance in science, engineering, and mathematics. <i>Proc. National Academy of Sciences</i>. 111 (23), 8410–8415. <a href="http://www.pnas.org/content/111/23/8410.abstract">www.pnas.org/content/111/23/8410.abstract</a><br /><br />Fry, C. (ed.). (2014). <i>Achieving Systemic Change: A sourcebook for advancing and funding undergraduate STEM education</i>. Washington, DC: AAC&U. <a href="http://www.aacu.org/pkal/sourcebook">www.aacu.org/pkal/sourcebook</a><br /><br />Hsu, E., Murphy, T.J., Treisman, U. (2008). Supporting high achievement in introductory mathematics courses: What we have learned from 30 years of the Emerging Scholars Program. Pages 205–220 in Carlson and Rasmussen (eds.). <i>Making the Connection: Research and Teaching in Undergraduate Mathematics Education</i>. MAA Notes #73. Washington, DC: Mathematical Association of America. <a href="http://www.maa.org/publications/books/making-the-connection-research-and-teaching-in-undergraduate-mathematics-education" target="_blank">www.maa.org/publications/books/making-the-connection-research-and-teaching-in-undergraduate-mathematics-education</a><br /><br />Kober, N. (2015). <i>Reaching Students: What research says about effective instruction in undergraduate science and engineering</i>. Washington, DC: National Academies Press. <a href="http://www.nap.edu/catalog/18687/reaching-students-what-research-says-about-effective-instruction-in-undergraduate" target="_blank">www.nap.edu/catalog/18687/reaching-students-what-research-says-about-effective-instruction-in-undergraduate</a><br /><br />Kogan, M. & Laursen, S.L. (2014). Assessing long-term effects of Inquiry-Based Learning: A case study from college mathematics. <i>Innovative Higher Education</i> 39(3) 183–199. <a href="http://link.springer.com/article/10.1007%2Fs10755-013-9269-9">link.springer.com/article/10.1007%2Fs10755-013-9269-9</a><br /><br />Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com3tag:blogger.com,1999:blog-7251686825560941361.post-37832589267218295992015-07-01T00:01:00.000-04:002015-07-01T10:49:48.425-04:00Calculus at Crisis III: The Client DisciplinesIn my last years at Penn State, I worked with faculty in the College of Engineering on issues of undergraduate education. They had two complaints about the mathematics department. First, we were failing too many of their students. Second, the ones we passed seemed incapable of using the mathematics we presumably had taught them when they got to their engineering classes. Chief among their specific gripes was an inability among their students to read a differential equation, to understand its assumptions of the relationships among the quantities being measured.<br /><br />A decade later, in the <a href="http://www.maa.org/sites/default/files/pdf/CUPM/crafty/curriculum-foundations.pdf" target="_blank">Curriculum Foundations Project</a> workshops with engineering faculty brought from across the country, we heard the same concerns about what their students should be learning from the mathematics department:<br /><br /><div style="margin-left: 1em;">Students “should understand the reasons for selecting a particular technique develop an understanding of the range of applicability of the technique, acquire familiarity with the mechanics of the solution technique, and understand the limitations of the technique.” (from civil engineers, p. 59)<br /><br />“There is often a disconnect between the knowledge that students gain in mathematics courses and their ability to apply such knowledge in engineering situations … We would like examples of mathematical techniques explained in terms of the reality they represent.” (from electrical engineers, p. 66)<br /><br />“In an engineering discipline problem solving essentially mean <b>mathematical</b> <b>modeling</b>; the ability to take a physical problem, express it in mathematical terms, solve the equations, and then interpret the results.” (from mechanical engineers, p. 81)</div><br />From the current <a href="http://www.abet.org/accreditation/accreditation-criteria/criteria-for-accrediting-engineering-programs-2015-2016/" target="_blank">ABET (Accreditation Board for Engineering and Technology) Criteria for Accreditation</a>, all of the references to mathematics under Curriculum talk about “creative applications,” building “a bridge between mathematics and the basic sciences on the one hand and engineering practice on the other,” and the use of mathematics in the “decision-making process.” As ABET moves into the criteria for specific programs, again the emphasis is entirely on the ability to apply knowledge of mathematics, not on any list of techniques or procedures.<br /><br />In the biological sciences, the other big driver for calculus enrollments, the American Association of Medical College and the Howard Hughes Medical Institute have dropped the traditional lists of specific courses that students should take in preparation for medical and instead list the competencies that students will need. First among these is mathematics. Of the seven specific objectives within this competency, six speak of quantitative reasoning and the use of data, statistics, modeling, and logical reasoning. The seventh comes closest to calculus, but what they actually ask for is the ability to “quantify and interpret changes in dynamical systems,” a far cry from the usual calculus course. (For more on this report, see my column on <a href="http://www.maa.org/external_archive/columns/launchings/launchings_11_09.html" target="_blank">The New Pre-Med Requirements</a>.)<br /><br />In the influential <a href="http://visionandchange.org/" target="_blank"><i>Vision and Change</i></a> document crafted by the biological sciences with assistance from AAAS, six core competencies for undergraduate biology education are identified. Two of them are mathematical: quantitative reasoning and the ability to use modeling and simulation. The report goes on to specify that “all students should understand how mathematical and computational tools describe living systems.”<br /><br />These examples can be multiplied in other client disciplines. What we see is a universal need for students to be able to use mathematical knowledge in the context of their own disciplines. In the case of calculus, the challenge is to understand it as a tool for modeling dynamical systems. This is why calculus is required by so many disciplines. But this is an understanding of calculus that is achieved by very few of our students because their focus has been narrowed down to learning how to solve the particular problems that will be on the next exam.<br /><br />None of this disconnect between what we teach in calculus and the needs of the client disciplines is new. It now rises to the level of a force that is bringing us to crisis because these client disciplines are themselves under the same increased pressure to have their students succeed. There may have been a time when there was a sufficiently rich pool of potential engineers that we could afford the luxury of allowing the mathematics department to filter out all but the most talented, the ones who would succeed in spite of how we taught them. If it ever existed, that time has passed. Our client disciplines now have higher expectations for what and how we teach their students.<br /><br />Nothing has driven this point home more clearly than <i><a href="https://www.whitehouse.gov/sites/default/files/microsites/ostp/pcast-engage-to-excel-final_feb.pdf" target="_blank">Engage to Excel</a></i>, the Report to the President from his Council of Advisors on Science and Technology (PCAST). (See my columns <a href="http://launchings.blogspot.com/2012/03/on-engaging-to-excel.html" target="_blank">On Engaging to Excel</a>, <a href="http://launchings.blogspot.com/2012/06/response-to-pcast.html" target="_blank">Response to PCAST</a>, and <a href="http://launchings.blogspot.com/2013/09/jpbm-presentation-to-pcast.html" target="_blank">JPBM Presentation to PCAST</a>.) The frustration of the scientists in PCAST with calculus instruction that does not meet the needs of their disciplines is evident in their call for “a national experiment [that] should fund … college mathematics teaching and curricula developed and taught by faculty from mathematics-intensive disciplines other than mathematics, including physics, engineering, and computer science.” (Recommendation 3-1, p. vii)<br /><br />While there was one particular physicist who was the driver behind this report, it did reflect the concerns of all of PCAST’s members. These are scientists and leaders in technology who deplored the fact that “many college students … often are left with the impression that the field [of mathematics] is dull and unimaginative.” (p. 28)<br /><br />I have yet to find physicists, engineers, or computer scientists who want to take over our calculus instruction. They have better things to do. But some have been forced to do so, and others are contemplating undertaking it as a necessary correction to mathematical instruction that is not meeting their needs.<br /><br />This completes my triad of forces that constitute the reason we are at crisis. It is the nature of a crisis that the solution is not readily apparent. Nevertheless, there are actions that can be taken to improve the situation. Next month, I will explore the first of these: drawing on knowledge of best practices for effective teaching and learning.<br /><br />Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com2tag:blogger.com,1999:blog-7251686825560941361.post-70236762841705795502015-06-01T00:01:00.000-04:002015-06-01T08:30:39.096-04:00Calculus at Crisis II: The Rush to CalculusI began this series last month by explaining how recent economic conditions are sending more students into the primary STEM fields (engineering and the physical, biological, mathematical, and computer and information sciences) while constricting the resources available to meet the needs of educating them. This is just one of a triad of phenomena that are pushing college calculus toward crisis. This month, I will discuss the second of these forces: the rush to calculus.<br /><br />Nothing illustrates the relentless growth of high school calculus better than the graph of the number of AP Calculus exams taken each year (Figure 1), surpassing 400,000 in 2014. According to NCES data [1], 53% of the students who study calculus in high school take an AP Calculus exam, implying that roughly 750,000 U.S. high school students studied calculus this past year. By comparison, this past year only 250,000 students took their first mainstream calculus class at a 2- or 4-year college or university [2].<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://4.bp.blogspot.com/-Gk4e5L3VCvI/VWc-F3pTIlI/AAAAAAAAKWA/aJAD8anD3aI/s1600/Launchings_June-Figure1.tiff" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="275" src="http://4.bp.blogspot.com/-Gk4e5L3VCvI/VWc-F3pTIlI/AAAAAAAAKWA/aJAD8anD3aI/s400/Launchings_June-Figure1.tiff" width="400" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Figure 1: Total AP Calculus exams and fall enrollments in mainstream Calculus I. Sources: The College Board and CBMS Statistical Abstracts.</td></tr></tbody></table><br /><br />What happens to the students who study calculus in high school? We know from the MAA study Characteristics of Successful Programs in College Calculus that about one- third of them retake Calculus I when they get to college. Based on AP scores and common policies for granting credit, roughly 200,000 students accept credit and/or advanced placement for their high school work. From a limited study [3], a clear majority of these students, probably three-quarters or more, do continue on to further courses that build on calculus.<br /><br />All of these patterns intensify at research universities and elite colleges, where at least 70% of Calculus I students are retaking a course they have already seen in high school, and large numbers of students heading for math-intensive majors skip over Calculus I.<br /><br />The result has been a dramatic change in the make-up of Calculus I. At most colleges and universities, it makes little sense to teach this course as if students are encountering calculus for the first time; few of them are. It also makes little sense to teach this course as if the students are heading into the mathematical or physical sciences. Nationwide, only 6% of Calculus I students intend such a major [4]. Finally, it makes little sense to teach this course as if this is where we see our best-prepared students.<br /><br />This last point is clear if we consider how many of the best-prepared students skip Calculus I, but it also is a consequence of what the rush to calculus has done to the middle and high school curricula in mathematics.<br /><br />In fall 2014 there were just over 1.6 million full-time first-year students enrolled in 4- year undergraduate programs in the U.S. [5]. Assuming that most of the 750,000 who take calculus in high school are traditional college-bound students who will enroll as full- time students in 4-year programs, these high school calculus students will constitute 40–45% of traditional first-year college students. The result is a common belief among parents, guidance counselors, and administrators that every college-bound student should, if at all possible, study calculus before high school graduation. I hear this from college students whose reason for taking calculus in high school was that it was expected of their peer group, and I hear it especially from high school teachers who complain of the tremendous pressure they are under to expand calculus classes and admit students they know are not ready for it.<br /><br />Because high school calculus by itself has become such common coin, those students who aspire to an elite college or university try to take calculus, preferably BC Calculus, before 12th grade. Figure 2 shows the exceptional growth in the number of students who take an AB or BC Calculus exam before grade 12.<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://4.bp.blogspot.com/-jalORd6oddw/VWdB8NWOSSI/AAAAAAAAKWM/XEEQ1lWm_q4/s1600/Launchings_June_Figure2.tiff" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="262" src="http://4.bp.blogspot.com/-jalORd6oddw/VWdB8NWOSSI/AAAAAAAAKWM/XEEQ1lWm_q4/s400/Launchings_June_Figure2.tiff" width="400" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Figure 2: Number of AP Calculus exams taken by students in grade 11 or earlier. Source: The College Board.</td></tr></tbody></table><br /><br />We do not know the full effect of this movement of calculus into ever earlier grades, but there is strong anecdotal evidence from teachers at both the high school and university level that many of these students are short-changing their preparation in middle and high school mathematics to join the fast track to calculus. Again anecdotally, this appears to be a significant problem when students attempt a math-intensive major where weaknesses in precalculus material can be disastrous.<br /><br />We can deplore the rush to calculus in high school, but the forces that are sustaining it are formidable. We have neither the authority nor the certain knowledge that would enable us to halt or reverse it. For the foreseeable future, we will have to live with it.<br /><br />Just in the past ten years, the preparation and aspirations of our college calculus students have shifted significantly. We cannot afford to assume that curricula and methods of instruction that were sufficient for the past will be adequate for the future.<br /><br /><br />[1] National Center for Education Statistics (NCES). (2012). <i>An overview of classes taken and credits earned by beginning postsecondary students</i>. NCES 2013-151rev. Washington, DC: US Department of Education. <a href="http://nces.ed.gov/pubs2013/2013151rev.pdf" target="_blank">nces.ed.gov/pubs2013/2013151rev.pdf</a><br /><br />[2] By “mainstream” we mean a calculus course that can be used as part of the pre- requisite stream for more advanced mathematics courses. It usually does not include business calculus, but may or may not include calculus for biologists. The figure of 250,000 is an estimate based on data from the CBMS Statistical Abstracts and the MAA study Characteristics of Successful Programs in College Calculus. Approximately 500,000 students began mainstream Calculus I at the post-secondary level at some point in the past year, and roughly half of them had studied calculus in high school.<br /><br />[3] Morgan, K. (2002). The use of AP Examination Grades by Students in College. Paper presented at the 2002 AP National Conference, Chicago, IL.<br /><br />[4] Source: MAA National Study of College Calculus, <a href="http://www.maa.org/cspcc" target="_blank">www.maa.org/cspcc</a>.<br /><br />[5] Source: HERI, <i>The American Freshman. </i><a href="http://www.heri.ucla.edu/tfsPublications.php" target="_blank">www.heri.ucla.edu/tfsPublications.php</a><br /><br />Mathematical Association of Americahttp://www.blogger.com/profile/10559021045290192742noreply@blogger.com0