## Sunday, April 1, 2018

### Gaps in Student Understanding of the Fundamental Theorem of Integral Calculus

By David Bressoud

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), 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

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

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

## Monday, January 1, 2018

### Indicators for STEM Education

The National Academies have just released the Board on Science Education report on Indicators for Monitoring Undergraduate STEM Education (available at http://sites.nationalacademies.org/DBASSE/BOSE).

This report is a response to the concern raised by the President’s Council of Advisors in Science and Technology that despite the many initiatives that are seeking to improve the teaching and learning of STEM subjects, we do not have effective national-scale measures of their success. The core of the charge to the committee that produced this report was to identify objectives for the improvement of STEM education, describe indicators that would inform whether or not we are making progress, and catalog what currently exists or could be developed by way of research and data collection to track progress. This extensive report provides this information.

The committee identified eleven objectives, organized into three general goals:

Goal 1: Increase students’ mastery of STEM concepts and skills by engaging them in evidence-based STEM practices and programs.
1.1 Use of evidence-based stem educational practices both in and outside of classrooms
1.2 Existence and use of supports that help instructors use evidence-based STEM educational practices
1.3 An institutional culture that values undergraduate STEM education
1.4 Continuous improvement in STEM teaching and learning
Goal 2: Strive for equity, diversity, and inclusion of STEM students and instructors by providing equitable opportunities for access and success.
2.2 Representational diversity among STEM credential earners
2.3 Representational diversity among STEM instructors
2.4 Inclusive environments in institutions and STEM departments
Goal 3: Ensure adequate numbers of STEM professionals.
3.1 Foundational preparation for STEM for all students
3.2 Successful navigation into and through STEM programs of study
3.3 STEM credential attainment
Each of these objectives is explained in detail, together with indicators of success and suggestions for how these might be measured. To give an indication of the breadth of this report, I’ll summarize some of what it says about the first and third objective, “Use of evidence-based stem educational practices both in and outside of classroom” and “An institutional culture that values undergraduate STEM education.”

The report first explains what evidence-based stem educational practices entail. For in-class practices, the report includes active learning and formative assessments. Acknowledging the lack of a common definition of active learning, this report uses it “to refer to that class of pedagogical practices that cognitively engage students in building understanding at the highest levels of Bloom’s taxonomy,” and then elaborates with examples that include “collaborative classroom activities, fast feedback using classroom response systems (e.g., clickers), problem-based learning, and peer instruction.”

This resonates with the CBMS definition, “classroom practices that engage students in activities, such as reading, writing, discussion, or problem solving, that promote higher-order thinking” (https://www.cbmsweb.org/2016/07/active-learning-in-post-secondary-mathematics-education/). The point being to engage students in wrestling with the critical concepts while in class. Thus the emphasis is not on activity as such, but on the promotion of cognitive engagement in higher order thinking.

I appreciate the emphasis on formative assessment: frequent, low-stakes, and varied assessments that clarify for students what they actually do and do not know. I also have found these helpful in informing me where student difficulties lie. The Indicators report references a 1998 review of formative assessment literature by Black and Wiliam, “Inside the Black Box: Raising Standards through Classroom Assessment,” that presents this as the single most effective means of raising student performance and describes how it needs to be done if it is to have these positive benefits. (Black and Wiliam article available at https://www.rdc.udel.edu/wp-content/uploads/2015/04/InsideBlackBox.pdf.)

Another important insight from this report, also identified in the MAA’s calculus studies, is the importance of course coordination. If a department is to improve instruction, it is essential that its members share a common understanding of the goals of the course. These shape pedagogical and curricular decisions as well as how student accomplishment is to be measured. The degree of coordination is one of the aspects of objective 1.3: An institutional culture that values undergraduate STEM education. As the report states on page 3-12,
A growing body of research indicates that many dimensions of current departmental and institutional cultures in higher education pose barriers to educators’ adoption of evidence- based educational practices (e.g., Dolan et al., 2016; Elrod and Kezar, 2015, 2016a, 2016b). For example, allowing each individual instructor full control over his or her course, including learning outcomes, a well-established norm in some STEM departments, can cause instructors to resist working with colleagues to establish shared learning goals for core courses, a process that is essential for improving teaching and learning.
As I reported last February in "MAA Calculus Study: PtC Survey Results," there is very little departmental coordination around homework, exams, grades, or instructional approaches.

 Table. Of the 207 mainstream Calculus I courses with multiple sections taught in 121 PhD- granting departments and 103 such courses in 76 Masters-granting departments, the percentage of courses that have each feature in common across all sections. Source: PtC Census Survey Technical Report, available at   https://www.maa.org/sites/default/files/PtC Technical Report_Final.pdf.
Of course, the big issue for an institutional culture that values undergraduate STEM education is how teaching is evaluated and role it plays in decisions of promotion and tenure. What is deeply discouraging is how poorly most departments do with just questions of coordination.

## Friday, December 1, 2017

### Essential Questions

For over five years, the Association of American Universities (AAU), representing the 62 leading research universities in the United States and Canada, has been engaged in

an initiative to improve the quality of undergraduate teaching and learning in science, technology, engineering, and mathematics (STEM) fields at its member institutions. The overall objective is to influence the culture of STEM departments at AAU universities so that faculty members are encouraged to use teaching practices proven to be effective in engaging students in STEM education and in helping students learn. (See https://www.aau.edu/education-service/undergraduate-education/undergraduate- stem-education- initiative.)

Products from this initiative that should be of help to every mathematics department seeking to improve instructional practice are now available online. These include a framework for improving undergraduate STEM education with examples of programs at AAU universities that address each of the elements of the framework.
 Figure: Cover of the AAU Essential Questions & Data Sources Report.
This past summer, they released their report on Essential Questions & Data Sources for Continuous Improvement of Undergraduate Teaching and Learning. Data sources include institutional data and tools for its visualization, observation protocols, rubrics, frameworks, student learning assessments, and surveys. The essential questions are separated into questions for institutional leadership as well as at the college, departmental, and instructor levels.

Because I believe that departmental leadership is the critical juncture for effective improvement, I will focus the remainder of this column on the questions addressed to departmental leaders and comment on what we have learned from the MAA’s studies of calculus instruction. By departmental leadership, I mean not just chairs and associate chairs, but all of those who shape the department’s direction. Change does not happen without a chair who is committed to improving the teaching and learning within the department, but it cannot be maintained without the support of a core of senior faculty.

Do all of the courses in the department have articulated learning goals, and are these made clear to students? What process exists to ensure that individual course learning goals connect to learning goals for the program, major, and department?

One of the clearest findings from the MAA calculus studies is that coordination of multiple section classes is essential. A prerequisite for effective coordination is a shared sense of what each course is seeking to accomplish.

What are the demographics of students in the department? What are the progression/retention/completion rates for students in the department or major broken out by relevant demographic categories? How do these compare with other departments and what steps are being taken to improve these rates?

Most departments I have visited have a sense that they are not doing as much as they could or should for students from traditionally underrepresented groups. This is not just a question of race, ethnicity, or gender, but also for students who are first generation, of lower socio-economic status, or from under-resourced schools whether they be inner city or rural. A department cannot know what is working for which populations if it is not tracking success rates by student demographics.

What actions has the department chair taken to encourage instructors to take advantage of both on-campus and off-campus (e.g., through relevant disciplinary societies) resources and professional development related to pedagogy? How many instructors have taken advantage of these resources and what notable improvements have occurred as the result?

The CBMS 2015 survey and other sources have documented that improvements in instructional pedagogy, support services, and course options almost always result from efforts initiated by individual faculty members. This question probes what the department is doing to nurture these faculty.

What resources are available to instructors in the department for encouraging all students to succeed, and what steps have been taken to ensure all instructors take advantage of these resources?

We know that faculty expectations of student ability play a huge role in how well students do, and faculty attitudes toward support services shape how students think about using these resources. The department as a whole must work to ensure the effectiveness of these services and then actively support their use, not as remediation but as a source of support and enrichment.

To what extent do departmental instructors have access to learning spaces that support evidence-based pedagogy? What training in the use of those facilities is available to instructors in the department?

The physical layout of classrooms and access to appropriate technology is critical for implementing effective pedagogies. This means tables where students can work together; sufficient space for instructors to walk around, answer questions, and observe how students are progressing; and sufficient board space for student groups to share their work. It does not have to be high tech classroom, but computer projection that is easily visible by all students is essential.

What is the department chair’s and distinguished faculty members’ support of evidence- based pedagogy? How well-known is this support to instructors and students?

This returns to the issue of nurturing those faculty who are positioned to initiate effective improvements. They need to know that if they are going to sink time and energy into improving teaching and learning within the department, then they will have the support not just of the chair whose term is limited but also of a core group of senior faculty who can ensure that support continues.

What are the biggest barriers to evidence-based pedagogy for instructors in the department and how is the chair working to address them? How often does the chair discuss these issues with the dean or other institutional leaders?

This addresses the chair’s critical role as the bridge between enthusiastic faculty, eager with ideas, and the college or university administrators with concerns to improve instruction and with access to resources that can support change. It is a position that requires insight and discernment on the part of the chair: to understand the priorities of the dean or provost and to comprehend the nature and potential of the initiative that faculty members are proposing. What will it take to implement a particular change? How can it be sold to the dean? What worries of the dean can be matched to ideas from the faculty?

How are all faculty who participate in annual/merit, promotion, and tenure evaluations educated about the meaningful inclusion of measures of teaching excellence in those processes? How closely does the chair review the outcomes of those processes to ensure teaching is indeed meaningfully included?

Finally, there is this elephant standing in the background of every effort to improve teaching and learning: How will it effect promotion and tenure? In my early years at Penn State, I was told that the dean of science was concerned about any faculty member that received high praise for teaching, because that might be a sign that they were neglecting their research. Even in my later years there, I found it necessary to discourage untenured faculty from sinking too much time into educational efforts. Unfortunately, the bifurcation of the faculty that I wrote about in October, separating tenure line faculty from contract faculty, only exacerbates this problem. With the option to “drop down” to a non-tenure line, the pressure to publish and receive research grants is all the greater.

## Thursday, November 2, 2017

### Women in the Profession

In last month’s column, I described the loss of tenure positions and their replacement with other full-time faculty appointments. This month, I will focus on how this has affected women earning PhDs in the mathematical sciences, also drawing on the Annual Survey of new PhDs, made available through AMS.

The first observation is that, while the number of tenured and tenure-eligible female faculty has increased by a third since 1995, most of the employment gains have been in other-full-time positions, which have more than tripled (Figure 1).

The first observation is that, while the number of tenured and tenure-eligible female faculty has increased by a third since 1995, most of the employment gains have been in other-full-time positions, which have more than tripled (Figure 1).

 Figure 1. The number of women employed in U.S. departments of mathematics, applied mathematics, or statistics. T & TE = tenure or tenure-eligible. Other full-time includes post-docs. Source: CBMS Surveys for 1995, 200, 2005, 2010, 2015.

This is particularly noticeable in PhD-granting mathematics departments, where a woman employed full-time is far less likely than a man to be in a tenure or tenure-eligible position (Figures 2 & 3). In 2015, 80% of the men employed full-time in a PhD-granting department were in tenure or tenure-eligible positions, this fraction having dropped from 91% in 1995. For women, the percentage fell from 65% in 2015 to 44% in 2015.

 Figure 2. The number of women employed in PhD-granting U.S. departments of mathematics, applied mathematics, or statistics. T & TE = tenure or tenure-eligible. Other full-time includes post-docs. Source: CBMS Surveys for 1995, 200, 2005, 2010, 2015.

 Figure 3. The number of men employed in PhD-granting U.S. departments of mathematics, applied mathematics, or statistics. T & TE = tenure or tenure-eligible. Other full-time includes post-docs. Source: CBMS Surveys for 1995, 200, 2005, 2010, 2015.
Despite the appearance that women are making substantial gains in tenure and tenure-eligible positions in PhD-granting departments, the fact is that they have only grown from 9% of those faculty in 1995 to 16% in 2015. In comparison, in Masters-granting departments the percentage of women in tenure and tenure-eligible positions rose from 18% in 1995 to 29% in 2015. At undergraduate colleges, it rose from 26% in 1995 to 32% in 2015. Over the same two decades, women rose from 22% of the PhDs awarded by mathematics departments to 26%.

If we look at all PhDs awarded to women in the mathematical sciences, now including departments of statistics or applied mathematics, the situation looks better, rising to 31% in 2015 (Figure 4), with women earning 33% of the PhDs in applied mathematics and 46% of those degrees in statistics.

 Figure 4. Women as a percentage of new PhDs in the mathematical sciences in the U.S. by type of department.  Source: The Joint Data Committee’s Annual Survey available at AMS.org, 1995 through 2015.
CBMS does not collect the data that would enable us to make comparable statements about the type of employment gained by mathematicians from other underrepresented groups and the numbers are so small it is not clear how meaningful they would be, but it does appear that efforts to broaden the diversity of mathematics departments is being stymied by the trend to replace tenure-line positions with contract positions. At least for women, their expanding representation in mathematics faculty is happening primarily in those contract positions.

## Monday, October 2, 2017

### The Loss of Tenure Positions: Threats to the Profession

The preliminary tables from the CBMS 2015 surveys of U.S. departments of mathematics or statistics are now available from the CBMS homepage at CBMSweb.org or by clicking HERE. I am using this month’s column to highlight one of the most dramatic developments: the loss of tenured and tenure-eligible faculty (Figure 1). At the end of this article, I reflect on the implications for our profession.

 Figure 1. Number of faculty in mathematics departments. T  & TE = tenured or tenure-eligible, other full-time includes post-docs.

The year 2015 saw the fewest tenured or tenure-eligible faculty, 15,270, since 1995, a drop of two thousand positions since 2005. Where they have gone is no mystery. The number of other full-time faculty, including post-docs, has tripled over the past two decades, from 2140 in 1995 to 6427 in 2015.

The break-down by type of institution—according to the highest degree offered by the mathematics department: PhD, Master’s, or Bachelor’s—is interesting. PhD-granting universities have seen remarkably constant numbers of tenure positions, Master’s universities have seen the greatest loss, and undergraduate colleges saw a spike around 2005 and have now returned to the number of positions in 1995. The growth in other full-time positions has been most dramatic at the PhD-granting universities, from 758 in 1995 to 2336 in 2015 (Figures 2–4).

 Figure 2. Distribution of faculty in PhD-granting mathematics departments.
 Figure 3. Distribution of faculty in Master’s-granting mathematics departments.

 Figure 4. Distribution of faculty in Bachelor’s-granting mathematics departments.

It is not that we now have fewer students to teach. Since 2005, the number of students studying mathematics in four-year under undergraduate programs has grown from 1.6 to over 2.2 million, an increase of 38% (Figure 5). If we add in the statistics courses taught within mathematics departments, the number of students enrolled each fall has jumped from 1.79 to 2.53 million, almost three-quarters of a million additional students. This dramatic growth holds even when we restrict to students at the level of calculus instruction and above, where the past decade has seen an increase of 262,000 students (Figure 6). To meet this increased demand while dropping two thousand tenure positions, we have added over three thousand other full-time faculty and one thousand part-time faculty.

 Table 5. Undergraduate enrollment in mathematics in four-year programs. Calculus level includes sophomore-level differential equations, linear algebra, and discrete mathematics. Advanced is any math course beyond calculus level. These do not include statistics.
 Table 6. Undergraduate enrollment at calculus level and above.

Not surprisingly, this means that undergraduate courses are now much less likely to be taught by a tenured or tenure-eligible faculty member. Figures 7 and 8 show what has happened at the PhD- granting universities. The 2015 survey was the first time that mainstream Calculus I and Calculus II were less likely to be taught by tenure line faculty than by other full-time faculty.

 Table 7. T &TE = tenured or tenure-eligible, other full-time includes post-docs. For 1995 and 2000, % is percentage of total students taking Calculus I. After 2000, it is the percentage of sections.

 Table 8. T & TE = tenured or tenure-eligible, other full-time includes post-docs. For 1995 and 2000, % is percentage of total students taking Calculus II. After 2000, it is the percentage of sections.

The trends are similar at Master’s universities and Bachelor’s colleges, though not as dramatic (Figures 9–12, following the Reflection).

Reflection. The CBMS data confirm what I have seen in departments across the country, especially in PhD- and Masters-granting departments. More and more of the undergraduate instruction is now the responsibility of contract faculty. In our research universities, it is becoming unusual for a tenured faculty member to teach any undergraduate courses. The unfortunate consequence is that the teacher-scholar, the ideal when I entered the profession, is fast disappearing. Those who are most active in mathematical research receive few teaching responsibilities. The remainder are saddled with heavy teaching loads that leave little time for research.

The reality of this bifurcation of the profession hit home in a recent network analysis of faculty interaction around issues of teaching, undertaken by the MAA’s Progress through Calculus project at a large public university. We found that tenure line faculty only interact with other tenure line faculty, contract faculty only with other contract faculty, with just a few individuals to provide a bridge. In effect, it has become two departments, one for undergraduate teaching and the other for research and the preparation of graduate students.

The teaching faculty are now manifestly second-class members of the profession: earning less money and receiving fewer benefits, carrying heavier prescribed duties, often lacking input in departmental decision-making, and living with the reality that, even with a renewable contract, long-term prospects are uncertain. It is no wonder so many of them have chosen to unionize.

There also are disturbing implications for the research faculty. Unlike Engineering or many of the other sciences, tenured mathematics faculty members seldom receive research grants that cover the full cost of their employment. Our public research universities have justified the size of their departments of mathematics by the large load of service teaching these departments must provide. Administrators are already questioning the wisdom of supporting a large corps of mathematics researchers who contribute ever less to the activities that pay the university’s bills.

We cannot turn back the clock, but there are mechanisms that can mitigate the dangers: involving contract faculty in departmental committees and decision making, involving tenure line faculty in observing and supporting those who carry the brunt of the teaching responsibilities, and ensuring that everyone is respected. There was one simple action that I observed at the Colorado School of Mines, a PhD-granting department. On the bulletin board that posts pictures of the faculty, contract faculty were not segregated from tenure line faculty. All members of the faculty were together in alphabetical order. What a radical idea.

 Table 9. T & TE = tenured or tenure-eligible, other full-time includes post-docs. For 1995 and 2000, % is percentage of total students taking Calculus I. After 2000, it is the percentage of sections.

 Table 10. T & TE = tenured or tenure-eligible, other full-time includes post-docs. For 1995 and 2000, % is percentage of total students taking Calculus II. After 2000, it is the percentage of sections.

 Table 11. T & TE = tenured or tenure-eligible, other full-time includes post-docs. For 1995 and 2000, % is percentage of total students taking Calculus I. After 2000, it is the percentage of sections.

 Table 12. T & TE = tenured or tenure-eligible, other full-time includes post-docs. For 1995 and 2000, % is percentage of total students taking Calculus II. After 2000, it is the percentage of sections.

**Editorial note: Figures 1-4 were updated on October 27, 2017.