A Common Vision

Five major mathematical societies—AMATYC, AMS, ASA, MAA, and SIAM—have just released a joint report, A Common Vision for Undergraduate Mathematical Sciences Programs in 2025, 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.

The central message, repeated throughout this document, is that “The status quo is unacceptable.” Specifically, A Common Vision issues a joint appeal to

1. Update curricula, 
2. Articulate curricula across the critical divide between high school and college mathematics,
3. Scale-up the use of evidence-based pedagogical methods, 
4. Remove barriers at critical transition points, and 
5. Establish stronger connections to other disciplines.

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:

Curricula. 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.

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.

Course Structure. 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.

Workforce Preparation. 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.

Faculty Development and Support. 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.

Other Issues. In addition, other issues have arisen in one or more of the society reports. These include the need

1. To attend to issues of student diversity, particularly the retention of students in at- risk groups,  
2. To ease difficulties as students move between institutions, 
3. To recognize the special needs of contingent faculty, 
4. To devote energy toward the preparation of K-12 teachers, 
5. To properly prepare graduate students for their contributions to the teaching mission of the department, 
6. To recognize and address the issues that lead to high failure rates, 
7. To look for ways of improving courses in developmental mathematics so that they retain and adequately prepare students, 
8. 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,  
9. To be aware of technology-enabled models of delivery of course content and to critically consider when and where they might be beneficial, 
10. To gather and use empirical data to refine programs and improve student learning, 
11. 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.

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.

———————————————————————————————————

Karen Saxe & Linda Braddy. 2016. A Common Vision for Mathematical Sciences Programs in 2025. Forward by William “Brit” Kirwan. Washington, DC: Mathematical Association of America. www.maa.org/sites/default/files/pdf/CommonVisionFinal.pdf

What we say/What they hear. II

In 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).

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,

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)

Figure 1. Summary of intersubjective operations involved in the communication of meaning. (Thompson, 2013, p. 64)

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:

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)

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.

To the instructor, this proof was an opportunity to showcase general approaches. The fact is that the theorem that was proven, “If a sequence {x_n} has the property that there exists a constant r with 0<r<1 such that |x_n–x_n–1| < rⁿ for any two consecutive terms in the sequence, then {x_n} 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.

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.

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.

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.

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.

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.

References

Lew, K., Fukawa-Connelly, T., Mejia-Ramos, J.P., and Weber, K. 2016. Lectures in advanced mathematics: Why students might not understand what the mathematics professor is trying to convey. Journal for Research in Mathematics Education. Preprint retrieved from pcrg.gse.rutgers.edu on January 24, 2016.

 Thompson, P. W. (2013). In the absence of meaning… . In Leatham, K. (Ed.), Vital directions for research in mathematics education (pp. 57-93). New York, NY: Springer.

Thompson, P. W. (2015). Researching mathematical meanings for teaching. In English, L., & Kirshner, D. (Eds.), Third Handbook of International Research in Mathematics Education (pp. 435-461). London: Taylor and Francis.

What we say/What they hear

An important paper is about to appear in the Journal for Research in Mathematics Education, exploring why lecture is so ineffective for so many students: “Lectures in advanced mathematics: Why students might not understand what the mathematics professor is trying to convey” 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.

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_n} has the property that there exists a constant r with 0 < r < 1 such that |x_n–x_n–1| < rⁿ for any two consecutive terms in the sequence, then {x_n}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.

1. Cauchy sequences can be thought of as sequences that “bunch up”
2. One can prove a sequence with an unknown limit converges by showing it is Cauchy
3. This shows how one sets up a proof that a sequence is Cauchy
4. The triangle inequality is useful in proving series in absolute value formulae are small
5. The geometric series formula is part of the mathematical toolbox that can be used to keep some desired quantities small

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.

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.

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.

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.

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.

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.

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.

References

Goodstein, G. & Negebauer, G. 1995. Preface to R. Feynman’s Six Easy Pieces. pp. xix–xxii. New York: Basic Books.

Lew, K., Fukawa-Connelly, T., Mejia-Ramos, J.P., and Weber, K. 2016. Lectures in advanced mathematics: Why students might not understand what the mathematics professor is trying to convey. Journal for Research in Mathematics Education. Preprint retrieved from pcrg.gse.rutgers.edu on January 24, 2016.

MAA Calculus Study: Building Networks

I 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.

Announcing an NSF-sponsored MAA Conference on

Precalculus to Calculus: Insights & Innovations 

June 16–19, 2016  

University of Saint Thomas, Saint Paul, Minnesota 

To be followed immediately by a workshop on

 Curriculum, Instruction, and Placement in Algebra and Precalculus 

June 19–20, 2016

Same location

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:

Focus on Curriculum. Content of and alternative approaches to precalculus, articulation issues, preparation for downstream courses

Focus on Students. Placement, early warning systems and support services, formative and summative assessment, supporting students from underrepresented groups

Focus on Pedagogy. Active learning strategies, making the most of large lectures, use of Learning Assistants, assessing effectiveness of innovations

Focus on Instructors. Building communities of practice, training of graduate teaching assistants, working with adjuncts, getting faculty buy-in for innovative practices

The workshop will be an opportunity to learn from the work of Marilyn Carlson, Bernie Madison, and Michael Tallman on Using Research to Shape Instruction and Placement in Algebra and Precalculus (NSF #1122965).

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.

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 www.maa.org/cspcc.

Review of applications will begin March 15. Those accepted will be notified by April 1.

This conference combines the efforts of the two studies on which I have been PI: Characteristics of Successful Programs in College Calculus (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 Insights and Recommendations. 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.

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:

• 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)
• What is “precalculus?” (content, purpose, function)
• Aligning precalculus so that it is truly a preparation for calculus
• Dealing with the multiple purposes for a variety of students (e.g., preparation, gen. ed., STEM, business)
• Encouraging/Supporting/Implementing Active Learning
• Especially when the institution insists on large classes 
• Information about flexible and/or non-standard models for the precalculus/calculus 
• GTA Training Programs 
• Specifically with regards to issues surrounding active learning
• Student skill retention within and across courses 
• Making calculus accessible for students from varying backgrounds
• Can it be done in one classroom, or are “flavors” needed? 
• Placing students into appropriate courses and then supporting them
• Establishing what various high school calculus courses really are
• Early warning systems
• Various pathways through calculus
• Professional development/Increasing faculty buy-in
• With respect to active learning
• Identifying ways of supporting faculty interested in using active learning strategies
• With respect to utilizing technology to support student learning
• Strategies for increasing administrative support/handling administrative pressures 
• Collecting and managing data 

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 www.trelliscience.com, register, then search for the PtC Discussion Group and request to join.

Strategies for Change

One of the most striking findings from the MAA’s survey of university mathematics departments undertaken this past spring (see last month’s column) 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.

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: “Increasing the Use of Evidence-Based Teaching in STEM Higher Education: 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.

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 prescribed, meaning that they try to implement specific solutions, versus those they describe as emergent, 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.

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).

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 Diffusion and Implementation 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. Implementation 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.

At the opposite corner are the emergent strategies that focus on environments and structures. Here Borrego and Henderson consider Learning Organizations and Complexity Leadership Theory. 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.

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.

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.

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.

Bibliography

M. Borrego and C. Henderson. 2014. Increasing the use of evidence-based teaching in STEM higher education: A comparison of eight change strategies. Journal of Engineering Education. 103 (2): 220–252.

C. Henderson, A. Beach, N. Finkelstein. 2011. Facilitating change in undergraduate STEM instructional practices: An analytic review of the literature. Journal of Research in Science Teaching. 48 (8): 952–984.

C. Henderson, N. Finkelstein, A. Beach. 2010. Beyond dissemination in college science teaching: An introduction to four core change strategies. Journal of College Science Teaching. 39 (5): 18–25.

MAA Calculus Study: A New Initiative

With the publication of Insights and Recommendations from the MAA National Study of College Calculus, we are wrapping up the original MAA calculus study, Characteristics of Successful Programs in College Calculus (CSPCC, NSF #0910240). This past January, MAA began a new large-scale program, Progress through Calculus (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.

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?

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.

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.

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.”

1. Student placement into the appropriate initial course 
2. GTA teaching preparation and development 
3. Student support programs (e.g. tutoring center) 
4. Uniform course components (e.g. textbook, schedule, homework) 
5. Courses that challenge students 
6. Active learning strategies 
7. Monitoring of the precalculus/calculus sequence through the collection of local data 
8. Regular instructor meetings about course delivery.

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”).

Figure 1. PhD universities. What they consider to be important versus how successful they consider themselves to be.

Figure 2. Masters universities. What they consider to be important versus how successful they consider themselves to be.

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.

Table 1. Departments that consider themselves to be very successful as percentage of those that consider the practice to be very important.

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.

Evidence for IBL

Special Note: The AMS Blog On Teaching and Learning Mathematics has started a six-part series on active learning.

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 Evidence of Improved Teaching (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.

In Implementation and outcomes of inquiry-based learning in mathematics content courses for pre-service teachers (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.

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.

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.

The second recent article is Transforming learning: Personal empowerment in learning mathematics (Hassi and Laursen, 2015). In Adding It Up (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 cognitive empowerment, and productive disposition, which they separate into self-empowerment and social empowerment, the last of which also incorporates effective communication.

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.

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.

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.

References

Hassi, M.-L., and Laursen, S.L. 2015. Transformative learning: Personal empowerment in learning mathematics. Journal of Transformative Education. Published online before print May 24, 2015, doi: 10.1177/1541344615587111.

M. Kogan and S. Laursen. 2014. Assessing long-term effects of inquiry-based learning: A case study from college mathematics. Innovative Higher Education 39 (3), 183–199. http://link.springer.com/article/10.1007/s10755-013-9269-9

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. International Journal of Mathematical Education in Science and Technology. Published online before print July 25, 2015, doi: 0.1080/0020739X.2015.1068390

National Research Council (NRC). 2001 Adding it up: Helping children learn mathematics. J. Kilpatrick, J. Swafford, and B. Findell (Eds.). Washington, DC: National Academy Press.

Sonnert, G. and Sadler, P. 2015. The impact of instructor and institutional factors on students’ attitudes. Pages 17–29 in Insights and Recommendations from the MAA National Study of College Calculus, D. Bressoud, V. Mesa, and C. Rasmussen (Eds.). Washington, DC: Mathematical Association of America Press.