Saturday, June 30, 2018

Departmental Turnaround: The Case of San Diego State University

By David Bressoud

You can now follow me on Twitter @dbressoud

Paul Zorn and I have just published a special issue of PRIMUS on Improving the Teaching and Learning of Calculus (Bressoud & Zorn, 2018) . It contains eight articles that should be of interest to anyone who is discontented with the current state of calculus instruction at their institution. Four of these articles present case studies of universities that have made significant changes within the past few years: San Diego State University (SDSU), the University of Illinois-Chicago, Colorado State University, and the University of Hartford. The most extensive revamping occurred at San Diego State University, which is where I am focusing below.

MAA’s national study of calculus instruction, Characteristics of Successful Programs in College Calculus (CSPCC) , identified seven practices (Bressoud & Rasmussen, 2015; see the Appendix for descriptions) that we observed in the most effective programs. A few years ago, San Diego State University, facing unacceptably high failure rates and low persistence rates in its Precalculus through Calculus II sequence, decided to work on all seven areas. The result has been a dramatic improvement in these courses. Naneh Apkarian, who was a doctoral student in mathematics education within the mathematics department during this process, is the lead author on this account (Apkarian et al., 2018) .

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Figure 1: The landmark Hepner Hall at San Diego State University.

With roughly 30,000 undergraduates, San Diego State University is a large public university, part of the California State University System, and chronically underfunded. It is a Hispanic-Serving Institution where 84% of students are on some form of financial aid. Science, technology, engineering, and mathematics (STEM) majors account for 10% of bachelor’s degrees. The mainstream precalculus and single variable calculus courses enroll about 1,500 students each fall. The Department of Mathematics and Statistics consists of 17 faculty in pure and applied mathematics, seven in statistics, and six in mathematics education.

Michael O’Sullivan was appointed chair of the department in 2014. He made it his mission to revamp lower-division mathematics instruction. The effort began that fall with the creation of a Calculus Task Force charged with proposing a system for coordinating the courses in the Precalculus to Calculus II sequence (P2C2). As Rasmussen and Ellis (2015) have documented, one of the most important characteristics of successful P2C2 programs is coordination of the essential elements of each course including policies, learning objectives, and exams and their scoring rubrics. Coordination also involves regular communication among those teaching different sections. At San Diego State University, total autonomy—to the point where different instructors were using different textbooks, homework systems, and even course content—had been the rule.

As the department expanded its data collection beyond simple pass rates, they discovered that only 17% of those who began with Precalculus successfully completed Calculus II, only 10% within the standard three semesters. This made mathematics faculty aware that something was seriously wrong and needed to change.

Because the discontinuation of large lectures was not financially feasible, the implementation of active learning to address this completion rate was concentrated in breakout sections led by Graduate Teaching Assistants (GTAs). The chair successfully lobbied to increase breakout sections from one to two hours per week and managed to reduce the size of most of these sections.

The chair also tied into a university initiative, Building on Excellence, to fund a new Mathematics Learning Center within the library building, directed by the office of the Dean of Science—ensuring its continued funding—but led by the department. The static 40-question placement exam was replaced by ALEKS Placement, Preparation, and Learning, with the license paid by the California State University System and student payments of $20 per proctored exam.

While these contributions were serendipitous, I have found that—particularly in situations of tightly constrained budgets—deans and provosts are keen to direct resources toward strategic initiatives with the potential for high impact. I have frequently encountered deans who asserted that if only the department would come forward with a well-thought-out and cost-effective plan for improving student outcomes, the money could be found to fund it.

As the authors reported, the effort at revision was successful because of the attention paid to opening and maintaining communication channels with stakeholders in this process (see Figure 2).

Figure 2: Significant communication channels between the mathematics department and various administrative programs as they relate to the seven targeted program features. Source: Apkarian et al. 2018, p. 540.

The result is a calculus program of which the department is justly proud, as reflected in this video. Students find the new Math Learning Center particularly helpful because its work is tightly connected to what is happening in all sections of each course.

The Department of Mathematics and Statistics at San Diego State University is a good example of how a program can be transformed. Its story illustrates the role of leadership from the department chair, buy-in and effort from a core of committed faculty, and strong two-way communication with all of the stakeholders.

References
  • Apkarian, N., Bowers, J., O’Sullivan, M., and Rasmussen, C. (2018). A Case study of change in the teaching and learning of Precalculus to Calculus 2: what we are doing with what we have. PRIMUS. 28:6, 528-549, DOI: 10.1080/10511970.2017.1388319
  • Bressoud, D., and Rasmussen, C. (2015). Seven characteristics of successful calculus programs. AMS Notices. 62:2, 144–146.
  • Bressoud, D. and Zorn, P. (2018). Improving the Teaching and Learning of Calculus. PRIMUS vol. 28.
  • Rasmussen, C., and Ellis, J. (2015). Calculus coordination at PhD-granting universities: more than just using the same syllabus, textbook, and final exam. In Bressoud, Mesa, and Rasmussen (Eds.), Insight and Recommendations from the MAA National Study of College Calculus. MAA Notes #84. Washington, DC: MAA Press. 

Appendix: Seven Characteristics of Successful Programs in College Calculus
  1. Local Data. Regular collection and use of local data to guide program modifications as part of continual improvement efforts.
  2. Placement. Effective procedures for placing students appropriately into their first Precalculus to Calculus II (P2C2) course (both initial placement and re-placing students after the term begins).
  3. Coordination System. A coordination system for instruction that (i) makes use of a uniform textbook and assessments and (ii) goes beyond uniform curricular elements to include regular P2C2 instructor meetings in development of de facto communities of practice.
  4. Course Content. Course content that challenges and engages students with mathematics.
  5. Active Pedagogy. The use and support of student-centered pedagogies, including active learning strategies.
  6. GTA Preparation & Development. Robust teaching development programs for teaching assistants.
  7. Student Support Service. Proactive student support services (e.g., tutoring centers, services for first-generation students) that foster students’ academic and social integration

Friday, June 1, 2018

Explosive Growth of Advanced Undergraduate Statistics

By David Bressoud

You can now follow me on Twitter @dbressoud

The 2015 CBMS Survey is now available. Last month I reported on Trends in Mathematics Majors. This month I am looking at what has happened to enrollments in particular mathematics courses. The column has three section: Enrollments by Category, where we see that the fastest growing category is Advanced Undergraduate Statistics; Calculus Enrollments, noting that the growth here is almost exclusively within the research universities where it is tied to the strong growth in engineering enrollments; and Dual Enrollment, where the story is about the dramatic increase in four-year institutions now offering dual enrollment courses.

Enrollments by Category
The first graph (Figure 1) shows strong growth in course enrollments in 4-year undergraduate programs, exceeding 2.5 million for the first time. This is certainly tied to the rampant growth in the number of prospective STEM majors (Figure 2). The number of prospective engineering majors grew from 108,000 in 2005 to 156,000 in 2010, peaking at 194,000 in 2015. Over the same period, prospective physical science majors grew from 30,000 to 40,000. Students entering with the intention of majoring in the mathematical sciences grew from 10,000 to 16,000.
Figure 1: Undergraduate enrollments by course category in mathematics and statistics departments at 4-year institutions.
Intro Level includes College Algebra and Precalculus; Calculus Level includes sophomore courses in linear algebra and differential equations.

Figure 2: Number of entering full-time first-year students at 4-year institutions intending to major in five core STEM disciplines.
Data from The American Freshman, published by the Higher Education Research Institute.

The most remarkable growth among categories of courses was for Advanced Statistics, any course beyond a first college-level statistics course, almost doubling from 60,000 in 2010 to 110,000 in 2015. This is in line with the growth in the number of Bachelor’s degrees awarded in Statistics, from 858 in 2010 to 1509 in 2015. Figure 3 shows that this growth has occurred primarily within departments of statistics, although there has also been strong growth at Bachelor’s level colleges and a remarkable turnaround in Master’s granting universities.

Figure 3: Enrollments in Advanced Undergraduate Statistics by type of department.
Departments of mathematics are characterized by the highest degree offered by the department.

Calculus Enrollments 

Calculus enrollments have also seen strong growth, driven by increases in prospective STEM majors (Figure 4). The MAA Progress through Calculus study found that for mainstream Calculus I, fall enrollments account for about 60% of all mainstream Calculus I enrollments throughout the year, while fall Calculus II enrollments account for about 40% of all Calculus II enrollments. Thus, about 550,000 students study Calculus I each year at a post-secondary institution. This compares with roughly 800,000 students who study calculus in high school each year (NCES data).

Figure 4: Fall term mainstream calculus enrollments (meaning that they lead to the usual upper division mathematical sciences courses), combined from all 2- and 4-year institutions.

Supporting the claim that most of the growth in calculus enrollments can be attributed to the growth in prospective engineering majors, Figures 5–7 show that the increase in calculus enrollments has occurred at the universities that also offer a PhD in mathematics, predominantly the large research universities.

Figure 5: Fall enrollments in mainstream Calculus I, by type of institution.
Figure 6: Fall enrollments in mainstream Calculus II, by type of institution.
Figure 7: Fall enrollments in mainstream Calculus III&IV, by type of institution.

The connection to engineering is reinforced by an interesting though not surprising observation. In 2005, I plotted the number of prospective engineering majors against the total number of students enrolled in all mainstream calculus classes (single and multi-variable) in PhD-granting departments (Figure 8). The correlation, at slightly over two students enrolled in the fall for each engineering major is remarkably tight, with a Pearson r=0.99.

Figure 8: Number of entering freshman intending to major in Engineering against total fall enrollment in all mainstream calculus (single and multi-variable).
Pearson’s r = 0.99.

The 2010 and 2015 data do not come close to fitting this line. It overestimates calculus enrollments by about 35%. Fitting a line to the data from 1995 to 2015 yields the graph in Figure 9. The multiplier effect of each prospective engineer has dropped to a little over 1, evidence that whereas an engineering major would, in the past, study single or multi-variable calculus in two fall terms, they now usually take calculus in only one fall term.

Figure 9: Number of entering freshman intending to major in Engineering against total fall enrollment in all mainstream calculus (single and multi-variable).
Pearson’s r = 0.97.

Dual Enrollment

CBMS began tracking dual enrollment in 2005, courses offered by a 2- or 4-year college, taught in a high school by a high school teacher, but carrying both high school and college credit. In 2005, 50% of 2-year departments, but only 14% of 4-year departments offered dual enrollment courses in mathematics. By 2015, these percentages had climbed to 63% at 2-year institutions and 26% at 4-year institutions. We conclude this column with Figures 10 and 11, showing the number of fall enrollments in the four most common dual enrollment courses: College Algebra, Precalculus, Calculus I, and Statistics.

Figure 10: Fall term dual enrollment at 2-year colleges.
Figure 11: Fall term dual enrollment at 4-year institutions.

Tuesday, May 1, 2018

Trends in Mathematics Majors

By David Bressoud 

You can now follow me on Twitter @dbressoud

By the time this column appears, the full CBMS 2015 survey of math departments should be available at www.ams.org/profession/data/cbms-survey/cbms2015. I reported some of the data on faculty demographics in my October and November Launchings columns. This month I want to report on what is happening to undergraduate mathematics majors.

From 2010 to 2015, the number of bachelor’s degrees in the mathematical sciences grew by just over 3,000, from 19,242 to 22,265, almost a 16% increase (Figure 1). However, most of the growth was in Actuarial Science (from 849 to 2354), Statistics (from 858 to 1509), joint majors (e.g. biomath, the total rising from 1222 to 1821), and “other” (including Operations Research, from 231 to 907). Degrees in Mathematics Education fell from 3,614 to 2,880. Traditional mathematics and applied mathematics degrees only rose by 326, from 12,468 to 12,794.

Figure 1. Bachelor’s degrees awarded by departments of Mathematics or Statistics.
Source: CBMS Surveys.

For comparison, the total number of Bachelor’s degrees over the years 2010 to 2015 increased by 15%, and the number of degrees in STEM fields (specifically bioscience, computer science, engineering, mathematical sciences, or physical sciences) rose by 34%, from 238,000 to 319,000.

The period 2010 to 2015 saw a decrease in the percentage of Bachelor’s degrees in Mathematics or Statistics earned by women, dropping from 42.4% to 40.8% (Figure 2). This does not include degrees in Mathematics Education awarded by Math departments. If we include them, then women earned 43.3% of the Bachelor’s degrees in 2015.


 Figure 2. Women as % of Mathematics or Statistics Bachelor’s degrees, organized by highest degree offered by the mathematics department. Source: CBMS Surveys.

Figure 3 shows the representation of African-Americans, Hispanic-American, Asian-Americans (including Pacific Islanders), and nonresident aliens. Here we are drawing on data from the National Center for Education Statistics (NCES), which is collected annually. Two trends are particularly interesting: the number of African-Americans has remained pretty much unchanged since the mid-1990s, and the number of nonresident aliens has exploded since 2007. It should be noted that NCES began allowing the designation “two or more races” in 2011. In 2011, 216 Mathematics or Statistics majors chose this designation, growing to 684 in 2016. These numbers are not reflected in Figure 3.

 Figure 3. Number of Mathematics or Statistics majors by race, ethnicity, or resident status.
Source: NCES.

The following graphs, Figures 4–7, look closer at each of these four groups, comparing their percentage of all Bachelor’s degrees, of Bachelor’s degrees in Mathematics or Statistics, and of Bachelor’s degrees in Engineering. Again, these do not include students who designated as two or more races after 2010. We see that until 2000, African Americans were well represented among Mathematics majors in the sense that their representation was comparable to their representation among all undergraduates, but since then their percentage has noticeably dropped off. Hispanic Americans are underrepresented, but the trend is promising. Not surprisingly, Asian Americans are well represented among Mathematics and Engineering majors. Non-resident aliens are growing as a percentage of all Bachelor’s degrees and all Engineering degrees, but their growth among Mathematics majors is remarkable. This attests to the importance of student visas in maintaining our mathematical workforce, but it also suggests that more could be done to attract U.S. citizens to the pursuit of Mathematics, especially African Americans.

Figure 4. African Americans as percentage of all bachelor’s degrees and of bachelor’s degrees in Mathematics or Statistics and in Engineering. Source: NCES.

Figure 5. Hispanic Americans as percentage of all bachelor’s degrees and of bachelor’s degrees in Mathematics or Statistics and in Engineering. Source: NCES.

Figure 6. Asian Americans and Pacific Islanders as percentage of all bachelor’s degrees and of bachelor’s degrees in Mathematics or Statistics and in Engineering. Source: NCES.

 Figure 7. Non-resident aliens as percentage of all bachelor’s degrees and of bachelor’s degrees in Mathematics or Statistics and in Engineering. Source: NCES.

Sunday, April 1, 2018

Gaps in Student Understanding of the Fundamental Theorem of Integral Calculus

By David Bressoud

You can now follow me on Twitter @dbressoud

I have long held the belief (Bressoud, 2011) that we should revert to the original name, the Fundamental Theorem of Integral Calculus (FTIC), for what in the 1960s came to be known as the Fundamental Theorem of Calculus (FTC). The reason is that the real importance of this theorem is not that integration and differentiation are inverse processes—for most students that is the working definition of integration—but that we have two very distinct ways of viewing integration, as limits of Riemann sums and in terms of anti-differentiation, and that for all practical purposes they are equivalent.


Figure 1. Students working on integral as accumulator, reproduced from the homepage of  Coherent Labs to Enhance Accessible and Rigorous Calculus Instruction (CLEAR Calculus

A recent paper by Joseph Wagner (2017) is an insightful study of the confusion experienced by most students about the nature of integration. As he points out, this is not about student deficits, but about common misconceptions that can be traced to the way we teach integration.

Previous work by Sealey (2006, 2014) and Jones (2013, 2015a, 2015b) has shown that
there are three ways in which students describe the meaning of the definite integral,

  •     as an area,
  •     in terms of an antiderivative, or
  •     in terms of a summation.
Overwhelmingly, students employ the first, the second is common, the third is rare.
Nevertheless, when confronted with a problem in physics that requires integration, the interpretation in terms of a summation is more common. Jones (2015b), after reminding second term calculus students that force is pressure times area, asked whycalculates the total force. Of 150 students, 61 (41%) produced an argument that involved summation, although only 25 of them (17%) indicated that any product was involved.

Following up on this insight, Wagner explored the understanding of definite integrals by physics students. He interviewed eight students in an introductory calculus-based physics course focused on classical mechanics and seven third-year physics majors. Of the students in the introductory course, five had completed both single and multi-variable calculus, two were currently enrolled in multi-variable calculus, and one was still in single variable calculus. All were in majors that required this physics course.

When students in the introductory course were asked what Riemann sums have to do with definite integrals, they split evenly between two types of answers: either as something that accomplishes the same task as an integral (usually finding areas) or as a means of approximating definite integrals. As we shall see, the connection between integration as a limit of Riemann sums and in terms of antiderivatives was hazy at best and not recognized as significant. As Wagner reports, several were mystified why they had to learn about Riemann sums, “Because like when they were teaching this, they were kind of like oh, like you’ll do this for the first test, and then you get rid of it and never have to do it again.”

On the other hand, the third-year physics students were much more inclined to explain the meaning of the definite integral in terms of a summation. They were conversant with how to convert an accumulation problem into a definite integral. As Wagner suggested privately, this appears to be the result of repeated exposure to problems from physics in which definite integrals arise from “slice and add” procedures.

But Wagner uncovered an intriguing gap in their understanding. All fifteen students were asked to make up a simple area problem and then solve it. All of them did so correctly, using a polynomial function and antidifferentiation. As an example the area under the graph of y=x^3 from 0 to 2 was calculated as follows,


He then pushed each of these students to explain why this sequence of calculations produced the area. Only one of the fifteen, a third-year physics student, indicated that this was a consequence of FTC. Several of the others struggled to make sense of how the symbols in the definite integral led to the functional transformation implied by the first equality. Wagner argues that many students are looking for algebraic sense-making in that first equality. With two of the third-year students, he documented their growing sense of frustration as they realized that they could not explain why it works. Quoting the first student:

"Yeah, I do it. I don’t–. I’m not proud of it, but I hope there is some way to justify it. […] When I think about integration as a sum of differentials, quantities–. When I think about that, I go, OK, that makes intuitive sense, and it works. Great. But then I wonder, you know, what is, in terms of more modernized math that I’m doing. Because I usually feel like what I’m doing is kind of a trick. And it works. I don’t feel great about doing this, like, intuitively I feel fine."

From the second student:

"So math gives us these sort of weird tools, and they behave differently than any, like, the physical tools we know of, and it doesn’t really make sense to ask why they work or how they work, because they work mathematically, not physically. So this mathematical tool called the integral allows us to change functions, to apply this operation that changes functions into other functions."

Wagner concludes this article with a thoughtful discussion of the distinction between the algebraic equivalence of two expressions, a notion of equivalence with which students are familiar, and the transformational equivalence that is enabled by FTC. As he laments, “Nothing, however, in the standard calculus curriculum prepares students for the sudden transition from making sense of the symbolic processes of algebra to making sense of the symbolic processes of calculus.” He points out that a great deal of attention has been devoted to a Riemann-sum based understanding of the definite integral, but virtually none to helping students understand the transformational aspects of calculus that are so central.

I believe that a shift from FTC to FTIC can help. As Thompson with others (2008, 2013, 2016) has shown, and I have discussed in earlier columns (Re-imagining the Calculus Curriculum, I, and Re-imagining the Calculus Curriculum, II), it makes sense to first develop the definite integral as an accumulator, making it very clear that Riemann sums are neither an introduction to a subject that eventually will be about antiderivatives nor just a tool for finding approximations, but the very essence of what a definite integral is and how it is used. Then, we bring in FTIC to show that there is another—entirely distinct because it is transformational—expression for this same integral and that this equivalent expression facilitates calculation.  Wagner’s third-year physics students were struggling because they failed to realize that integration has these two very different manifestations. It is a very big deal that it does.

References

Bressoud, D. (2011). Historical reflections on teaching the Fundamental Theorem of Integral Calculus. American Mathematical Monthly. 118:99–115. http://www.jstor.org/stable/10.4169/amer.math.monthly.118.02.099?seq=1 - page_scan_tab_contents

Jones, S. R. (2013). Understanding the integral: Students’ symbolic forms. The Journal of Mathematical Behavior, 32(2), 122–141. https://www.sciencedirect.com/science/article/pii/S0732312312000612

Jones, S. R. (2015a). Areas, anti-derivatives, and adding up pieces: Integrals in pure mathematics and applied contexts. The Journal of Mathematical Behavior, 38(1), 9–28. https://www.sciencedirect.com/science/article/pii/S0732312315000024

Jones, S. R. (2015b). The prevalence of area-under-a-curve and anti-derivative conceptions over Riemann sum-based conceptions in students’ explanations of definite integrals. International Journal of Mathematics Education in Science and Technology, 46(5), 721–736. http://www.tandfonline.com/doi/abs/10.1080/0020739X.2014.1001454

Sealey, V. (2006). Definite integrals, Riemann sums, and area under a curve: What is necessary and sufficient. In S. Alatorre, J. L. Cortina, M. Sáiz & A. Méndez (Eds.) Proceedings of the 28th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 46-53). Mérida: Universidad Pedagógica Nacional. http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.500.3209&rep=rep1&type=pdf

Sealey, V. (2014). A framework for characterizing student understanding of Riemann sums and definite integrals. The Journal of Mathematical Behavior, 33, 230–245. https://www.sciencedirect.com/science/article/pii/S0732312313001065

Thompson, P.W., and Silverman, J. (2008). The concept of accumulation in calculus. In M.P. Carlson & C. Rasmussen (Eds.), Making the connection: Research and teaching in undergraduate mathematics (MAA Notes Vol. 73, pp. 43–52). Washington, DC: Mathematical Association of America. https://doi.org/10.5948/UPO9780883859759.005

Thompson, P.W., Byerley, C. and Hatfield, N. (2013). A conceptual approach to calculus made possible by technology. Computers in the Schools. 30:124–147. http://pat-thompson.net/PDFversions/2013CalcTech.pdf

Thompson, P.W., and Dreyfus, T. (2016). A coherent approach to the Fundamental Theorem of Calculus using differentials. In R. Göller. R. Biehler & R. Hochsmuth (Eds.), Proceedings of the Conference on Didactics of Mathematics in Higher Education as a Scientific Discipline (pp. 355–359 ) Hannover, Germany: KHDM.  https://www.researchgate.net/publication/306108323_A_Coherent_Approach_to_the_Fundamental_Theorem_of_Calculus_Using_Differentials

Wagner, J.F. (2017). Students’ obstacles to using Riemann sum interpretations of the definite integral. International Journal of Research in Undergraduate Mathematics Education. https://doi.org/10.1007/s40753-017-0060-7

Thursday, March 1, 2018

A False Dichotomy: Lecture vs. Active Learning

By David Bressoud

You can now follow me on Twitter @dbressoud


On January 31, I published a piece in The Conversation, “Why Colleges Must Change How They Teach Calculus.” The following is one of the statements that I made in this article:

Active learning does not mean ban all lectures. A lecture is still the most effective means for conveying a great deal of information in a short amount of time. But the most useful lectures come in short bursts when students are primed with a need and desire to know the information.



There is no simple binary choice between an active learning classroom and straight lecture. Furthermore, making a class an effective locus for student learning requires more than just active learning. 

An article by Campbell et al. (2017), “From Comprehensive to Singular: A latent class analysis of college teaching practices,” reports on an interesting study of what happens in college classes (not just STEM classes), adding a few layers of complexity that are useful for anyone thinking about how to be a more effective teacher. The authors observed 587 courses in nine colleges and universities, ranging from Research 1 (public and private) to comprehensive state schools to liberal arts colleges at a range of levels of selectivity. They looked for seven types of activities in the classroom.

One of these is lecture, defined as “A presentation or recitation of course content by the faculty member to all students in the class.”

They split active learning into three sub-categories:

  • Class discussion. Back and forth conversation between instructor and students or among students about the course content.
  • Class activities. A structured activity where students engaged with the course content (e.g., case studies, clickers, group work).
  • Student questions. Students asking individual questions of the instructor about the course content.

They also picked up the three practices laid out in Neumann’s (2014) description of cognitively response teaching. Active teaching should be cognitively responsive. Unfortunately, as their observations showed, it often is not. These three practices are:


  • Core subject matter ideas. The instructor introduced in depth one or more concepts that are central to the subject matter of the course, the instructor created multiple representations of “core ideas,” or the instructor introduced students to how ideas play out in the field.
  • Connections to prior knowledge. The instructor surfaced students’ prior knowledge about the subject “core ideas,” or the instructor worked to understand students’ prior knowledge about the subject matter “core ideas.”
  • Support of changing views. The instructor provided a space for students to encounter dissonance between prior knowledge and new course material, or the instructor helped students to realize the difference similarities and sometimes conflict between prior knowledge and new subject matter ideas.

Developing over the past few decades and now accelerating thanks to the work of the community engaged in research in undergraduate mathematics education, there have been remarkable strides in understanding the misconceptions that are barriers to student learning. To cite just two examples that I have discussed elsewhere, students often have difficulty making the transition from trigonometric functions in terms of triangles to the circle definition, and they tend to interpret functions as static objects, impeding an understanding of them as descriptions of the linkage between variables that vary. I discussed this issue of the disconnection between what we say and what students hear in two columns in 2016: What we say/what they hear and What we say/what they hear II. The instructor who does not try to understand the prior conceptions and knowledge that students bring into the classroom is setting a large proportion of the students up for failure.

For the last practice, support of changing views, the physics education community knows how important this is. With their Force Concept Inventory (FCI), Halloun, Hestenes, and Wells (see Hestenes et al., 1992) demonstrated that prior concepts are powerful. Students are reluctant to release them, even in the face of what instructors consider to be clear exposition of the actual state of affairs. Getting students to recognize cognitive dissonance requires skill.

Campbell et al. observed that traditional lecture—what the Progress through Calculus study (Apkarian and Kirin, 2017) has revealed to be standard practice in 72% of all Calculus I classes in university mathematics departments with PhD programs—did a pretty good job on core subject matter ideas, but almost nothing with connections to prior knowledge or support of changing views. And, of course, traditional lecture involved none of the first two active learning sub-categories. Less obvious but not surprising, student questions were seldom observed in traditional lecture.

Active lecture is the second most common form of calculus instruction, found in about 14% of the PhD-granting mathematics departments we surveyed in progress through Calculus (3% of departments relied mainly on active learning practices in the classroom and the remaining departments reported too much variation by instructor to classify their course as one type). These introduced class activities and did not decrease core subject matter ideas. Campbell et al. found that they noticeably increase student questions, but do nothing in and of themselves to improve connections to prior knowledge or support of changing views.

These last two practices were almost never observed in either traditional or active lecture classes. The only classes that were observed to improve these aspects of cognitively responsive teaching were those that made a point of employing all seven behaviors, including lecture. In other words, connections to prior knowledge and support of changing views do not come for free once one is using active learning. They have to be intentionally incorporated, and they rely heavily on carefully guided class discussion.

The lesson is that lecture has its place, and active learning is only one piece of what is needed for a truly effective class. David Hestenes (1998) summed it up nicely in “Who needs physics education research!?”:

Managing the quality of classroom discourse is the single most important factor in teaching with interactive engagement methods. This factor accounts for wide differences in class FCI score among teachers using the same curriculum materials and purportedly the same teaching methods. Effective discourse management requires careful planning and preparation as well as skill and experience … Effective teaching requires complex skills which take years to develop. Technical knowledge about teaching and learning is as essential as subject content knowledge.
References

 Apkarian, N. and Kirin, D. 2017. Progress through Calculus: Census Survey Report. https://www.maa.org/sites/default/files/PtC Technical Report_Final.pdf

Bressoud, D. 2016. What we say/What they hear. Launchings. http://launchings.blogspot.com/2016/02/

Bressoud, D. 2016. What we say/What they hear. II. Launchings. http://launchings.blogspot.com/2016/03/

Bressoud, D. 2018. Why colleges must change how they teach calculus. TheConversation. January 31, 2018. https://theconversation.com/why-colleges-must-change-how-they-teach-calculus-90679

Campbell, C.M., Cabrera, A.F., Michel, J.O., and Patel, S. 2017. From Comprehensive to Singular: A Latent Class Analysis of College Teaching Practices. Research in Higher Education. 58: 581–604. https://link.springer.com/article/10.1007/s11162-016-9440-0

Hestenes D., Wells M., Swackhamer G. 1992. Force concept inventory. The Physics Teacher 30: 141-166. http://aapt.scitation.org/doi/10.1119/1.2343497

Hestenes D. 1998. Who needs physics education research!?. Am. J. Phys. 66:46.5. http://aapt.scitation.org/doi/abs/10.1119/1.18898

Mathematical Association of America. 2017. Instructional Practices Guide. https://www.maa.org/programs-and-communities/curriculum resources/instructional-practices-guide

Neumann, A. 2014. Staking a claim on learning: What we should know about learning in higher education and why. The Review of higher Education. 37:249–267. http://www.ashe.ws/files/Past Presidents/37.2.neumann.pdf


Thursday, February 1, 2018

Getting to Know the IP Guide

You can follow me on Twitter @dbressoud.

In 2015, the MAA’s Committee on the Undergraduate Program in Mathematics (CUPM) produced its latest Curriculum Guide. Extensive as this was, including specific recommendations for most courses and programs offered in departments of mathematics, the steering committee that it left out a big part of what is needed for effective teaching. Spurred by the Common Vision report that outlined what we know about effective teaching and called for their implementation, CUPM set out to describe in detail examples of instructional practices that can greatly improve teaching and learning. The result is the Instructional Practices Guide (IP Guide), now available for free download from the MAA.
From the description of paired board work, page 20 of the IP Guide.
The core of the IP Guide message is that
Effective teaching and deep learning require student engagement with content both inside and outside the classroom.
This puts the emphasis within the phrase “active learning” where its advocates have always intended it to be, on learning, employing those practices that foster higher order thinking skills.

The report is usefully divided into three sections: Classroom Practices, activities that can be used in the classroom to promote engagement with the material; Assessment Practices, how assessment can be used formatively and to probe student understanding; and Design Practices, which get to the broader questions of how to design courses that incorporate the classroom and assessment practices in ways that are most effective. It concludes with two short sections, one on the use of technology and one on equity issues.

Classroom Practices constitutes the longest section, describing how to build a classroom community, use wait time, respond to students, and promote persistence. This section includes explanations and examples of some of the standard techniques of active learning: one-minute papers, think-pair-share, just-in-time teaching, and peer instruction.

Almost as long as the section on Classroom Practices, Assessment Practices goes into detail on what effective, meaningful, and helpful assessment looks like and how it can be accomplished without overwhelming the instructor, even in large classes.

We now have overwhelming evidence of the importance of active cognitive engagement with the mathematics we want our students to learn. Those of us who have succeeded in mathematics have known how to do this outside of the classroom. Most students do not. Most students still approach mathematics as a sequence of templates to be learned for solving specific sets of problems. If we want them to learn anything that will stay with them beyond the term, any knowledge that is transferable, then we must structure our classes so that students are forced to wrestle with the material. The IP Guide should prove to be a useful resource as we reconfigure our courses to meet these goals.

References
Karen Saxe and Linda Braddy. 2015. A Common Vision for Undergraduate Mathematical Sciences Programs in 2025. Washington DC: MAA Press. https://www.maa.org/sites/default/files/pdf/CommonVisionFinal.pdf

Carol S. Schumacher and Martha J. Siegel, co-Chairs, Paul Zorn, editor. 2015. 2015 CUPM Curriculum Guide to Majors in the Mathematical Sciences. https://www.maa.org/sites/default/files/CUPM Guide.pdf

MAA. 2017. Instructional Practices Guide. https://www.dropbox.com/s/xpvkni52tkf0wgt/MAA_IP_Guide_V1-1.pdf?dl=0