Mathematics departments have long provided the bulk of the mathematics content training for both practicing teachers and those studying to be teachers. This is a tremendous responsibility, and one that presents a variety of challenges and opportunities. In this post, we start early in the mathematical spectrum – with elementary teachers and how mathematics departments impact their mathematical preparation.

Until fairly recently, at many higher education institutions students preparing to be elementary teachers would take one or more general education courses such as college algebra, math for liberal arts, or a version of calculus. It was expected that this would both meet some type of “quantitative reasoning” or “general education” requirement at their institution as well as prepare them with sufficient mathematics to teach elementary school. While there were exceptions, a prevailing thought was that elementary school mathematics was, well, taught in elementary school, so someone enrolled in college should have sufficient mathematics background already.

We have learned that this is far from a truism. While a great many researchers and practitioners have contributed to the development of knowledge in this area, we single out work by Deborah Ball and her many colleagues. Their work on mathematical knowledge for teaching (MKT), that is, the mathematics that teachers actually need to engage in the practice of teaching mathematics, has profoundly impacted courses and programs across the country.

In the practice of teaching mathematics, teachers engage in mathematical tasks such as responding to students’ mathematical statements, addressing students misconceptions, and providing multiple representations of concepts. The following document contains 35 problems covering a broad spectrum of elementary math topics that serve to illustrate the diversity of mathematical knowledge needed by elementary teachers: http://sitemaker.umich.edu/lmt/files/LMT_sample_items.pdf

These skills have both mathematical and pedagogical components and cannot neatly be separated into “content” and “methods” courses. In particular, addressing student misconceptions often crosses into both areas. To address student misconceptions, teachers must recognize the misconception and understand deeply the mathematics behind the topic. However, they must also have sufficient knowledge of student development and student thinking to respond productively to the student to help them grow in their mathematical understandings.

With the changing landscape of mathematics education, it is now well-accepted in the mathematics education community, and increasingly in mathematics departments, that elementary teachers need specialized content courses in mathematics. The latest Conference Board of the Mathematical Sciences (CBMS) recommendations in their Mathematical Education of Teachers II document suggest four such content courses. As a mathematical community, we remain far from this suggested standard in our typical course offerings.

A perhaps surprising challenge is that such courses usually contain content that is not typically familiar to mathematicians. For example, many of us are not familiar with a non-algebraic explanation of why the traditional “invert and multiply” rule for dividing fractions holds, one based only on an elementary understanding of the meaning of fractions, the meaning of multiplication, and the meaning of whole number division. However, building from definitions is solidly in our area of expertise, and we are well qualified to help elementary teachers learn to base their mathematical reasoning on age-appropriate definitions. After all, if the teachers do not have this skill set, then they will not be able to develop it in their students.

Even further removed from our knowledge base may be things like the whole associated with a fraction, unit rates, base 10 blocks or unifix cubes, fraction bars, double number lines, and the partial product or scaffolding algorithms for multiplication and division. Again, mathematicians are certainly capable of jumping in and learning these, but it is specialized mathematical knowledge that we do not just have by virtue of our advanced mathematics degrees.

*So, what can departments and individuals do to contribute further to the mathematical education of elementary teachers? *

First and foremost, we can increase our collective awareness of the importance of our role in preparing future elementary teachers to teach mathematics. At an individual level, we can stay abreast of key documents like the aforementioned CBMS recommendations, we can read articles in the AMS Notices, and we can attend a session or panel related to elementary mathematics education at the Joint Math Meetings or at Mathfest.

As part of our *collective* awareness, we can ensure that the importance of our role is emphasized by both formal and informal leaders within our departments, discussed or at least given genuine recognition at appropriate times during department meetings, and that a culture of respect for this part of our mission is established among faculty.

Going beyond the awareness level, departments can increase their participation and reward faculty participation. Likely there are a few mathematicians in each department who would enjoy and excel at becoming more actively involved in courses for elementary teachers. Encourage, support, and reward them for their efforts. Most importantly, respect their efforts and genuinely accept that it is important work and a much needed contribution to the mathematical spectrum.

Some mathematics departments, for example at the University of Nebraska and the University of Northern Colorado, go beyond the aforementioned faculty involvement, providing opportunities and training for their graduate students to teach courses for elementary teachers. These graduate students then enter the profession as faculty members who already have a basic knowledge base and skill set in this area, able to share their knowledge and contribute their skills to other departments.

Finally, reach out to our partners in education. Find out what the preservice teacher curriculum is at your local institution, volunteer to teach a content course for elementary teachers and put in the effort to learn the specialized knowledge, and consider volunteering at a local school or in some setting where you have direct mathematical interaction with elementary age students. A basic familiarity with where students are in their mathematical thinking can be invaluable as baseline knowledge to being involved in this type of work.

Thus, our level of involvement both individually and collectively can come at many levels, from simply increasing awareness to jumping in and becoming so involved that it becomes a major part of one’s career. Those looking to read further might check out some of the references at the end of this post.

We bear primary responsibility for the content preparation of elementary teachers, and I propose that we should take our responsibility in this area seriously and endeavor to excel at this crucial aspect of our mission. Elementary teachers are providing the initial mathematical training to our future scientists, engineers, and mathematicians. Perhaps more importantly, though, they provide the initial mathematical training to the future adults in our society, including our own children.

** References**

Conference Board of the Mathematical Sciences (CBMS). (2001). *The Mathematical Education of Teachers. *Providence, RI: American Mathematical Society.

Conference Board of the Mathematical Sciences (CBMS). (2012). *The Mathematical Education of Teachers II. *Providence, RI: American Mathematical Society.

Ma, L. (1999). Knowing and Teaching Elementary Mathematics: Teachers Understanding of Fundamental Mathematics in China and the United States, Mahwah, NJ: Lawrence Erlbaum.

Saul, M. (Ed.) (2011). Special Issue on Education, Notices of the American Mathematical Society, 58(3).

]]>In the past nine months, I’ve heard colleagues at three different meetings—an AMS sectional meeting in Louisville, the Joint Mathematics Meetings in Baltimore, and the Contemporary Issues in Mathematics Education workshop at the Mathematical Sciences Research Institute—identify a need for journals focused on publishing useful refereed articles for mathematicians about mathematics education. This raises several questions that get at fundamental issues in the complicated and sometimes uneasy relationships among research mathematicians, mathematics education specialists, and those with interests in both areas (I put myself in the last category).

To begin, why might we need such journals? The most obvious answer is that good work in mathematics education should be identified (hence a refereeing process) and shared in a way that is accessible to mathematicians. Trained as a mathematician, I found some of my first readings in mathematics education research to be inaccessible — full of unfamiliar vocabulary and references to social science research methods that I felt unqualified to evaluate. Naturally this triggered my instinctual skepticism. I have certainly found some writing about mathematics education research to be clear, convincing, and useful to me as a mathematician who teaches and works with teachers. Elise Lockwood’s earlier post on this blog is exemplary, as are [1] and [3]. I’ve also gotten more familiar with the methods and conventions of mathematics education researchers; Alan Schoenfeld’s article [4] from 2000 is a fine place to start. Still, I would appreciate collections of articles about mathematics education that are written with speakers of my native language in mind.

Another answer to the “why” question is that many mathematicians are doing important work in mathematics education, and that work might not get the attention and validation that it deserves from other mathematicians unless it’s being certified by mathematicians. We who teach future teachers, investigate how students learn to write proofs, provide professional development to K-12 teachers, and so on, should have ways to present what we’ve learned in venues that are recognized by the larger mathematical community. As Sol Friedberg, chair of the mathematics department at Boston College, put it at MSRI*, “The coin of the realm in the evaluation of faculty is what? Publications.” The first round of evaluation for a mathematician is in the mathematics department, where skepticism is a professional requirement. Recognition by mathematicians beyond our own campuses might help.

The influence on performance reviews of a mathematician’s activities in math education may be limited, though, even if publications result. Another speaker at MSRI, Steven Rosenberg of Boston University, was blunt on this topic: “We look for math publications; we look for funding in math. So if you have a great love of math education, please wait until you’re tenured. For those of you in math education, when you go to approach colleagues in the math department, please keep that in mind. Is work in math education respected within a research math department? The short answer is, yes, if it’s funded, and even in that case, maybe not as much as research in pure and applied math or statistics.”

At the same MSRI session, however, Brigitte Lahme, a mathematician at Sonoma State University, reported that her department values work in teacher education and professional development, and has revised its tenure criteria accordingly. A contribution to mathematics education, she said, “can’t be just an add-on.” Clearly there is significant cultural variation among mathematics departments. Further, as Lahme added later, “we can challenge the status quo… In my department, it used to be that [publishing] papers was the coin of the realm, and we changed it, and the world has not ended.”

In response to concerns about untenured mathematicians putting themselves at risk by engaging in math education activities, the last word of the session was provided by Bill McCallum: “I think it’s time for us to embrace the contradiction between (a) calls for culture change and (b) our desire to protect everybody. Culture change isn’t going to happen without a certain amount of pain, and I think younger faculty who want to get involved in math education should be encouraged to do so. They’re grownups. Let’s all stop worrying too much about the pain that’s going to be caused; we can all share it, and let’s stand up for the culture change.” Wherever we stand on the question of culture change, mathematicians would do well to examine the cultures in our own departments and institutions.

A second question is this: what journals are already out there for this work? The *International Journal of Research in Undergraduate Mathematics Education* (Springer) will begin publication in 2015. There’s also* PRIMUS: Problems, Resources, and Issues in Mathematics Undergraduate Studies*. For those willing to explore the discipline of mathematics education, the *Journal for Research in Mathematics Education *and *Educational Studies in Mathematics* are respected by practioners. Then there’s the related but distinct area of SoTL: the Scholarship of Teaching and Learning (see [2] for an introduction to SoTL and its publication venues).

Beyond journals, there are conference proceedings as well as edited volumes such as those in the Mathematical Association of America Notes Series. These lack the validation of a fully refereed journal, however, which calls for caution on the part of both writer and reader.

Perhaps we should consider the demand side. Thus my third question: what sorts of papers about mathematics education would mathematicians like to see? I suspect that many mathematicians who teach would like to learn about various approaches to improving student learning, provided those approaches are backed up by evidence that is plausible to them, if short of rigorous proof. Having developed a Math for Teachers course at a small liberal arts college, I’d like to read about math courses for future teachers that integrate pedagogy and math content. I’d like to read about professional development programs for practicing teachers that work by some reasonable measure. I’d like to read about what happens when mathematicians spend time in K-12 math classrooms. I’d like to read more articles that address mathematicians’ skepticism about social science methods used by math education researchers.

It seems to me that there is indeed room for more places to publish peer-reviewed papers on math education for an audience of mathematicians. There’s a lack in particular of outlets for articles on mathematicians’ involvement in K-12 education. This brings me to the last questions: what other venues exist for papers of this type? Are there any new journals in the works? If more journals of this type are available, what are the best ways to bring them to a broad audience? Please respond in the comments section!

*A video recording of this session at MSRI is available here.

Many thanks to Ben Braun, Elise Lockwood, and the Department of Mathematics at Middlebury College for helping me develop the ideas here.

**References**

[1] Ball, D., M. H. Thames, and G. Phelps, Content knowledge for teaching: what makes it special? *Journal of Teacher Education* 59, no. 5 (2008), 389-407.

[2] Bennett, C. D. and J. M. Dewar, An overview of the scholarship of teaching and learning in mathematics, *Primus : Problems, Resources, and Issues in Mathematics Undergraduate Studies* 22, no. 6 (2012), 458-473.

[3] Hill, H., The nature and predictors of elementary teachers’ mathematical knowledge for teaching, *Journal for Research in Mathematics Education *41, no. 5 (2010), 513-545.

[4] Schoenfeld, Alan H., Purposes and methods of research in mathematics education, *Notices of the American Mathematical Society* 47, no. 6 (2000), 641-649.

Almost fifteen years later, Lucy Michal still remembers the exact words Phil Daro told the leaders of the El Paso Collaborative for Academic Excellence as they were preparing to launch the K-16 Mathematics Alignment Initiative, which Lucy would direct: “Find a friendly mathematician.” The goal was to align mathematics in grades K-16, through regular meetings of a working group of a few dozen local teachers of all grade levels. Phil had many contacts, including national authorities in K-12 mathematics, but, for a project like this, he stressed the need for local mathematics experts. A “friendly mathematician” would be respected for mathematics, but would also understand the importance of working with both pre-service and in-service teachers. I became one of those friendly mathematicians. What did I do to live up to this billing?

I had only started working with pre-service teachers two years before this, teaching a “math content” course for prospective K-8 teachers. I was still a little stuck in my academic silo, bracing myself before the first meeting to make sure mathematical content wasn’t going to be given the short shrift in this project by pedagogical concerns. I was also worried what the teachers would think of me, and if they would dismiss me for not having stood in their shoes in a K-12 classroom.

A funny thing happened, though, putting teachers from kindergarten through college in the same room. I was used to the “if only” chorus you often hear when you get enough calculus instructors together: “if only the high school teachers taught algebra better…”. I found out high school teachers have their own “if only” chorus: “if only the middle school teachers taught fractions better,” and so on down the line. But all those “if only”s went away pretty quickly, because we were each sitting next to living, breathing examples of people doing their best to solve the very problem the “if only”s accuse them of causing.

We also found we had a lot more in common than we thought, even regarding the mathematics itself. A striking moment early on was when we discussed how algebraic thinking is (ideally) developed through all the grade levels, starting in kindergarten, which I did not fully appreciate until that day’s discussion. I was sitting next to a kindergarten teacher who described how they teach “clap patterns” (for instance, two claps, then three claps, then two, then three, and so on). To make a long story short, this idea of repeating patterns eventually is extended to the idea of growing patterns, which is an entry point to linear functions and Algebra I. The kindergarten teacher was surprised to see that this activity led all the way to algebra, and I was surprised to learn that ideas about algebra can be started this early. This was where I think I first started to understand the real depth of the mathematics that underlies the K-12 curriculum.

In another early session, Phil Daro came back to talk to the whole working group, and the familiar patterns of “odd + odd = even”, etc., came up, as an example of something or another. I forget if Phil mentioned something like “mathematicians can extend” this idea, but somehow I ended up showing the group, step by step, how modular arithmetic generalizes this whole set up. Yes, this is standard stuff for math majors, but the elementary teachers had not majored in math, nor had some of the high school teachers. Even those teachers who *had* seen it before may have forgotten it, since they don’t teach modular arithmetic in high school, or more importantly, they may not have necessarily made the connection to what they do teach. So this is one of the things I did to earn my “friendly” stripes: Find and share the deeper mathematical structures that extend the topics that show up in the K-12 curriculum.

What else did it take to be a “friendly mathematician”? I wish I could remember more specific examples like the “odd + odd = even” story, but I mostly only remember general behaviors. I think there were a few other opportunities to connect K-12 math to deeper topics, but many more smaller instances of why some topic in K-12 is important for some later topic in calculus or differential equations. I ensured statements the group made were mathematically correct, for instance establishing precise mathematical definitions. Also, once a teacher asked about a new calculus textbook he liked, but wasn’t certain he should use in his AP calculus class because it might not align with what we were doing at the university; it turned out to be exactly the book we were then using at the university.

Of course, there were also basic social norms. I had to do my share of the writing and other work. I had to listen to the teachers, not just talk at them, and value their expertise. Indeed, one of the joys of the experience was how *everyone* valued the different contributions and experiences others brought to the conversation. Let me mention here that I was joined by my UTEP colleague Mourat Tchoshanov, a mathematics education professor in the Teacher Education department, and he also provided valuable insights, different from mine.

When I asked Lucy Michal recently (in preparation for this post), she said that Mourat and I treated the work seriously, as if it was worth our time and attention. Indeed, I quickly saw the value of alignment, and it was clear to me that representing a post-secondary mathematical perspective was an important role I could fill in this project. What surprised me was how much I learned about the K-12 curriculum, and about vertical and horizontal alignment. I also learned a lot about mathematics and education from the outside experts who were brought in occasionally.

I have since jumped at every chance to work with K-12 teachers (well, within the confines of my busy schedule). If you are a mathematician, becoming a friendly mathematician may bring you unexpected profound moments, both socially and mathematically. More broadly, whatever your situation, if you have the opportunity to collaborate with people who have different mathematical expertise, you too can be a friendly mathematician. How do you get this chance? I was lucky; they called. But you don’t have to wait. Find a local school district or professional development effort in your area, and call them — I have no doubt they would be thrilled to work with a friendly mathematician.

]]>

This post is inspired by an article by Karen Marrongelle and Chris Rasmussen [1], in which they discuss the false dichotomy between all lecture and all student discovery as the two exclusive teaching strategies available for mathematics teachers. I’ve noticed that many discussions among postsecondary mathematics teachers lead to a debate of the merits of these two classroom teaching strategies, with the result that interesting teaching practices are left undiscussed. Below I describe three key teaching practices that I’ve learned about and used over the past several years that fit between and beyond these extremes. I’ve observed that when I use these practices, students are generally more engaged in the course, e.g. attending office hours, asking questions in class, forming study groups, etc. Though they appear simple, using these practices successfully has required perseverance and effort on my part, and a willingness to regularly revise their implementation.

**Use student questions as discussion prompts**

When a student asks a question during class, I’ve found that it often reflects an underlying misunderstanding held by a sizeable subset of the class. While it is common for student questions to be answered by the instructor, it can be helpful to provide students with a few minutes of class time to come up with an answer on their own in small groups. While this isn’t an appropriate response for all student questions, such as situations where a negative sign in a computation is overlooked, I’ve found that using student questions as discussion prompts is typically more effective than my answering questions directly. One of the best aspects of this technique is that it either completely resolves the question or else prepares students to seriously think about the explanation that I provide after the discussion time. I’ve found that students are much more attentive listening to my answers if I’ve given them a couple minutes to focus on the problem themselves before I start talking.

For example, when I was teaching a calculus class recently, a student asked a question about computing a limit that required multiplying the function being considered by cos(x)/cos(x). I took an informal poll to see how many students were confused by this problem, and over half of the class was stuck. Instead of telling them to multiply by the appropriate quotient of cosines, I had the students talk with each other for two minutes to share their ideas. Each of the groups had at least one student who knew what to do, and because of this my role in the classroom was changed from being an instructor to being a guide, leading the students to successful peer learning.

A frequent question about this technique (and the next) is how to balance allocating class time for students to collaborate with covering content. In small courses where I am the only instructor, I can rearrange the course schedule as needed, so this isn’t an issue. When I teach large-lecture calculus courses, the online homework, lecture schedule, and examination dates are determined by the course coordinator, so I don’t have much flexibility to rearrange content. In this setting, I generally choose to cover fewer examples in more depth, whereas when I first started teaching I chose to cover many examples with less discussion.

**Collect multiple student ideas for approaching a problem**

A related technique I’ve used is to gather suggestions from students on how to start examples. My goal in this is not to have the students simply take the lead when solving problems, but instead to explicitly discuss as many entry points into each example as possible. My typical approach to this is to write the problem on the board/screen and record a list next to it with ideas that students suggest for how to begin. Regardless of whether or not a valid starting strategy is provided, I continue drawing ideas out of the students until I have five or six items on the list. Once this list is on the board, then I’m in a position to discuss each of these ideas in turn, identifying the ideas that I know will fail, the ideas that I know will work, and any new ideas that I hadn’t thought about before. I’d rather address these in public than in multiple one-on-one discussions during office hours, in the hope that this will guide students to develop better overall strategies for investigating problems.

There are three key aspects of this technique that I’ve learned about the hard way while teaching. First, it is important that I truly value every suggestion given by students. Even if there is an idea for starting a problem that seems rather ridiculous, I consider it carefully and make sure that I provide respectful critical commentary. If students feel that their mistakes are being mocked in the slightest, then they won’t continue sharing them and the entire technique stops working. Second, it is good to frequently remind students of my motivation for going through the process of collecting and critiquing ideas, including mistakes and errors. I want students to have in mind that the purpose of this is both to think critically about errors and to determine the correct answer to a problem. Third, I respond to each student suggestion by thanking them for sharing their idea. I appreciate any student who is willing to risk sharing a misguided idea in public, and I want to make sure students know this.

**Assign critical essays regarding readings from course texts**

When my students complete reading assignments regarding the mathematics they are studying, one outcome is that they learn more and are more engaged in class. However, like many other teachers, I’ve found it difficult to motivate students to regularly read their course texts. The most effective tool I have to address this is to assign short essays regarding reading assignments. Typically these are two to three pages long, and require students to critically analyze/review a section of their text. I instruct students to think of their essays as being similar to a book or movie review, where they have to highlight both the strengths and weaknesses of the reading, and justify their critical commentary.

I give an assignment of this type several times per semester, and try to have the essays focus on the most important topics in each course. The general result of this assignment is that engaged students who are already reading are more focused and retain more from the texts, while disengaged students who ordinarily would rarely open the book are forced to at least complete a reasonable skim through the material while writing their essay. In general, I’ve found that the result of assigning critical essays is that all students get more from the readings than they otherwise would have. To keep the grading of these essays consistent, I use a grading rubric for mathematical writing that I have developed [2]. An interesting side effect of assigning these essays is that it provides me a window into how the students are thinking about the ideas under consideration, which allows me to be more responsive regarding specific issues that students are struggling with.

**References**

[1] Marrongelle, Karen and Rasmussen, Chris. Meeting New Teaching Challenges: Teaching Strategies that Mediate between All Lecture and All Student Discovery. *Making the Connection: Research and Teaching in Undergraduate Mathematics Education*. Carlson, Marilyn P. and Rasmussen, Chris, eds. MAA Notes #73, 2008. pp 167-177.

[2] Braun, Benjamin. Personal, Expository, Critical, and Creative: Using Writing in Mathematics Courses. To appear in *PRIMUS: Problems, Resources, and Issues in Mathematics Undergraduate Studies. Special Issue on Writing and Editing in the Mathematics Curriculum: Part I*, 2014. http://dx.doi.org/10.1080/10511970.2013.843626

As an undergraduate, it was easy for me to assume that as my professors conducted mathematical research, beautiful, complete proofs came to them in moments of epiphany. Their work was mysterious to me, and I believed that somehow their superior intelligence and vast mathematical knowledge gave them immediate access to all things abstract. Had I been asked then, I likely would have said that mathematicians didn’t need to think about examples in their own research – surely they had outgrown the need for concrete examples.

This perspective may be attributable to the fact that throughout my math classes to that point, it had been ingrained in me early and often that showing a statement is true for a few examples is not a valid proof of a universally quantified true statement. The belief that several examples do suffice as a proof has been called the empirical proof scheme (Harel & Sowder, 1998), and a good amount of literature on students’ reasoning on proof has focused on this perspective as a limitation (e.g., Healy & Hoyles, 2000; Knuth, Choppin, & Bieda, 2009; Porteous, 1990; Stylianides & Stylianides, 2009; Zaslavsky, Nickerson, Stylianides, Kidron, & Winicki, 2012). In light of this, teachers expend a considerable amount of effort in making sure students do not incorrectly cultivate this notion, and rightly so, given that we do not want students to wrongly believe that examples are valid substitutes for proofs.

However, my colleagues and I wonder whether the repeated caution against empirical-based arguments has led students to undervalue the role that examples can play in proof. Mathematicians certainly use examples in their development of conjectures and proofs. As Epstein and Levy (1995) contend, “Most mathematicians spend a lot of time thinking about and analyzing particular examples…. It is probably the case that most significant advances in mathematics have arisen from experimentation with examples (p. 6).” Therefore, while we acknowledge the danger of developing incorrect notions of examples as proof, we worry that the emphasis on such limitations, perhaps coupled with assumptions that mathematicians do not use examples, may preclude postsecondary students from engaging meaningfully with concrete examples as they prove.

In graduate school, I had an excellent professor who demonstrated an infectious curiosity. He was always willing to explore problems in front of students, not as someone who had prepared notes or who was simply recalling information, but as a true problem solver who was deeply engaged in the task. As he solved problems or proved theorems with us, he would get his hands dirty with some concrete examples, and all of a sudden the problem would become more real. We realized that he was doing the kinds of things we could be doing ourselves – carefully writing down a handful of concrete examples, searching for patterns, using examples to determine whether a conjecture might be true, and looking at generic examples to see how a proof might be developed. This experience was incredibly illuminating for me, and it helped me to formulate a more accurate view of mathematicians’ activity.

I want to encourage students to be more aware of and open to the valuable roles that examples can play in proof-related activity. The goal is not to encourage an overreliance on examples in the context of proof, or to deride the warnings against an empirical proof scheme. Rather, the point I want to raise here is that there is a key aspect of many mathematicians’ proof-related activity that I am not sure students consider: Examples can play an integral role in developing conjectures and formulating proofs.

There has been a considerable amount of research conducted recently on examples, and more particularly on examples in proof (e.g., Bills and Watson, 2008; Sandefur, et al., 2013; Weber, 2008). In our studies (reported elsewhere in Ellis, et al., 2012; Lockwood, et al., 2012; Lockwood et al., 2013), my colleagues and I found that mathematicians in a variety of fields regularly draw on examples as a part of their proving activity. Indeed, in all of the 250 survey responses and 19 interviews we gathered, no mathematician indicated that they do not use examples. Below are just three of the responses from this data, which reveal instructive insights into the nature of mathematical exploration and proof.

*M1: **“I explore examples to find out what statements mean. For instance, yesterday I was trying to understand the meaning of “If E is an elliptic curve/Q, then there is associated a representation \(Gal(\overline Q/Q) \to GL_2(Z/3)\). So I chose an elliptic curve, specifically the one of equation \(y^2=x^3+x+1\), and tried to find the points of order 3. It took a while, but after I was through I knew what the statement meant. Generally, the difficulty in dealing with a new mathematical concept is to form a mental image of it. Examples help develop such mental images.”*

* M2:**“I start with the simplest conceivable example, then I try to come up with slightly more complicated examples. In parallel to this procedure, I also try to guess counterexamples. This guessing typically fails, and if it does, I try to find specific properties of my guess examples that prevent them from doing what I want them to do. Sometimes this allows a slow “building up” of properties that can eventually say something useful about the conjecture. Other times, it is clear what the counterexamples should be, but it is still unclear how to prove the conjecture.”*

*M3: **“First test the easiest cases. (E.g., for integers, test 2, 3, 4, 5, 10) Then test something that is qualitatively different from the easiest cases. If it still works, make a first attempt at a proof. If you can’t prove it, try to cook up counter-examples that exploit the holes in your “proof”. If you can’t make counter-examples, use what you learned from the failed counter-examples to fix the holes in the proof. Go back and forth between proof and disproof, using the failures of each side of the argument to build up your attempt on the other side.”*

For those readers who are currently undergraduate or graduate students, when you go to prove a theorem, what do you do first? Do you launch into the proof, trying to recall certain techniques? Do you read back through the book looking for similar proofs that you can mimic? Or, do you first play around with some concrete examples, using them to make sense of the statement of the problem? Our research suggests that this kind of experimentation with examples can be a useful first step in understanding a conjecture and ultimately coming up with a solid proof. In fact, as we spoke with mathematicians, we found that they often use concrete examples to make sense of conjectures (M1’s response), or to try to convince themselves whether a conjecture might be true (M2’s response), and even to provide concrete insights into how they might go about proving a conjecture (M3’s response). I would also encourage students to reflect on the metacognitive aspect of these mathematicians’ responses. They are clearly being intentional about how they are selecting and using examples. This kind of flexibility with examples is something that students should develop in their mathematical activity. The takeaway for students is this: There is no substitute for getting your hands dirty with specific examples in mathematics – whether you are solving problems, developing conjectures, or proving or disproving conjectures.

For those of us who teach mathematics, I suggest that we should give explicit attention to the role that examples can play in conjecturing and proving. Students may benefit from being encouraged to work with examples and from seeing how specific examples can actually play a crucial role in proof. This can be modeled for them and also reinforced through tasks and homework problems that develop this activity.

Because mathematicians use examples so regularly and in a variety of ways, students should similarly incorporate example-related activity as a fundamental aspect of their work. Students may greatly benefit from grounding their proof writing and conjecturing in concrete examples that can serve a variety of purposes.

**Acknowledgements**

I would like to thank my colleagues Eric Knuth and Amy B. Ellis, whose collaboration led to many of the ideas in the post. The research described here is supported in part by the National Science Foundation under grants DRL-0814710 (Eric Knuth, Amy Ellis, & Charles Kalish, principal investigators) and DRL-1220623 (Eric Knuth, Amy Ellis, & Orit Zaslavsky, principal investigators). The opinions expressed herein are those of the author and do not necessarily reflect the views of the National Science Foundation.

**References**

Ellis, A. E., Lockwood, E., Knuth, E., Dogan, M. F., & Williams, C. C. W. (2013). Choosing and using examples: How example activity can support proof insight. In A. Lindmeier & A. Heinze (Eds.), *Proceedings of the 37**th** Annual Meeting of the International Group of the Psychology of Mathematics Education.* Kiel, Germany.

Ellis, A. E., Lockwood, E., Williams, C. C. W., Dogan, M. F., & Knuth, E. (2012). Middle school students’ example use in conjecture exploration and justification. In L.R. Van Zoest, J.J. Lo, & J.L. Kratky (Eds.), *Proceedings of the 34**th** Annual Meeting of the North American Chapter of the Psychology of Mathematics Education* (Kalamazoo, MI).

Epstein, D., & Levy, S. (1995), Experimentation and proof in mathematics. *Notice of the **AMS*, *42*(6), 670–674.

Harel, G., & Sowder, L. (1998). Students’ proof schemes: Results from exploratory studies. *Issues in Mathematics Education*, *7*, 234-283.

Healy, L. & Hoyles, C. (2000). A study of proof conceptions in algebra. *Journal for Research in **Mathematics Education*, *31*(4), 396–428.

Knuth, E., Choppin, J., & Bieda, K. (2009). Middle school students’ production of mathematical justifications. In D. Stylianou, M. Blanton, & E. Knuth (Eds.), *Teaching and learning proof across the grades: A K–16 perspective* (pp. 153–170). New York, NY: Routledge.

Lockwood, E., Ellis, A.B., Dogan, M.F., Williams, C., & Knuth, E. (2012). A framework for mathematicians’ example-related activity when exploring and proving mathematical conjectures.

Lockwood, E., Ellis, A., & Knuth, E. (2013). Mathematicians’ example-related activity when proving conjectures. In S. Brown, G. Karakok, K. H. Roh, & M. Oehrtman (Eds.),* Electronic Proceedings for the Sixteenth Special Interest Group of the MAA on Research on Undergraduate Mathematics Education.* Denver, CO: Northern Colorado University.

Porteous, K. (1990). What do children really believe? *Educational Studies in Mathematics, 21*, 589–598.

Sandefur, J., Mason, J., Stylianides, G. J., & Watson, A. (2013). Generating and using examples in the proving process. *Educational Studies in Mathematics*. Doi: 10.1007/s10649-012-9459-x.

Stylianides, G. & Stylianides, J. (2009). Facilitating the transition from empirical arguments to proof. *Journal for Research in Mathematics Education, 40*(3), 314-352.

Zaslavsky, O., Nickerson, S. D., Stylianides, A. J., Kidron, I., & Winicki, G. (2012). The need for proof and proving: mathematical and pedagogical perspectives. In G. Hanna & M. de Villiers (Eds.), *Proof and proving in mathematics education: The 19th ICMI Study* (New ICMI Study Series, Vol. 15). New York: Springer.

Weber, K. (2008). How mathematicians determine if an argument is a valid proof. *Journal for Research in Mathematics Education, 39*(4), 431-459.

This essay describes the changes that have taken place in my teaching philosophy and practice over the past 30 years or so. I have always loved teaching and the satisfaction of explaining concepts to others. However, my understanding and love of the profession has greatly increased over the years, with some pivotal moments emphasizing when those changes occurred. I present these reflections as a possible encouragement to others who may wonder about their teaching and how to make it more satisfying and enjoyable.

**Lecturing**

I had a standard undergraduate and graduate education in the mathematical sciences. This was in Europe but was probably similar to that of many in the United States. All classroom instruction was in the form of lectures with little or no feedback or student interaction. All learning was tested in a final end-of-year exam worth 100% of the grade. This exam was as much a test of memorization as of any deep understanding.

I obtained reasonably good grades through this system but I was never confident that I understood enough. I didn’t really think this was a good system and I didn’t think I was learning very well. I didn’t really enjoy the process and it was more of an endurance test than a learning experience. Deep down I loved mathematics but I wished there was a way that I could understand more and maintain that love rather than simply survive an endurance test.

My first teaching assignment as a graduate instructor was a course in numerical methods to a class of about 100 engineering students. I taught this class in exactly the same manner as I had been taught. I gave a final exam that required memorization. I graded it and gave the best grades to those who could reproduce more of their transcribed notes. I didn’t think I was doing a bad job. I continued this pattern of traditional lecturing for several years. Most courses were applied or engineering based. I didn’t reflect much on what I was doing. Students got the grades they expected and I got the teaching evaluations I expected.

I moved to the United States where continuous assessment and mid-term exams had reduced the impact of the all consuming final exam. I began to see myself in the role of a coach and facilitator of learning. But my overall approach and lecturing style hadn’t changed. However, somewhere in the back of my mind there was a nagging suspicion that there could be a better way and this was occasionally confirmed by encounters with previous students who frankly admitted they hadn’t understood any of the material I presented.

**Beyond Lecturing**

For about 15 years my academic focus was applied mathematics and computing with collaboration with engineers, geologists, and the automobile industry. While this was rewarding interdisciplinary work part of me missed doing pure mathematics. I also noticed more articles about the declining state of math education in the US. So I made a decision to focus more on educational issues. This coincided with some evolving career opportunities and the development of the role of an outreach mathematician (Conway, 2001; Dwyer 2001).

I began to more seriously reflect on teaching and began to read about teaching. A pivotal moment for me was an article describing the processes of learning math in an elementary school classroom (Ball, 1993). This deep analysis of how students processed information and communicated their understanding impressed me greatly. Suddenly teaching was no longer about the transmission of information. It was about the far more exciting venture of interacting with students as they grasped to understand concepts. It became an intellectual challenge to find multiple ways of developing that student understanding.

The first change I implemented was to introduce some group work into the classroom. The advantages and disadvantages of group work are well documented (eg. Morris and Hayes, 1997). This was well received by students. They enjoyed working with one another and learning from one another. They had more time to assess a problem and grapple with it than in a traditional lecture setting. I also found it much more enjoyable to walk around and discuss problems with students rather than just write on the board. This leads to an important change in class preparation. The emphasis is no longer on learning the material and preparing an impeccable set of notes to deliver. The emphasis now is on developing diverse ways of delivering and presenting material and motivating students to work with the material. It was my hope that the classroom become an interactive learning laboratory and not just a location for transcription.

Around that time I became aware of service learning, whereby students perform some service in the community which complements and enhances their classroom learning. In my pre-service teachers’ classes this was best implemented by having students work as tutors in local K-12 schools where they gained increased understanding of elementary school mathematics through explaining the concepts to younger students. The college students liked this option and all of the participants reported increased learning as a result of the service learning activities. This also showed to me that student learning can take place in venues beyond the traditional classroom setting (Dwyer, 2005).

I have firmly believed that quality of learning is far more important than quantity. That means there should be no rush to “cover” all of the material. It is better to fully understand two chapters than to have read (but not understood) five chapters. It seems to me that very few students understand as much as we think they do. From that realization I decided that I would make an effort to keep the syllabus as short as possible and only proceed to the next topic when I feel that all students have some grasp of the material. I think that students enjoy going slowly and really enjoy the fact that they understand as they go along. The amount of material covered is less but the students should leave the classroom with a greater sense of satisfaction than if they had covered several topics with limited understanding. A related point is the observation that a student question is not time consuming; rather it is an opportunity for increased learning.

My teaching still included a combination of lecture and group work. But another change had taken place in my approach to lecturing. Rather than preparing detailed notes I often leave a problem untouched until I am in the classroom. Then I work through the problem with student assistance where possible. This may be risky if I find I am unable to complete the problem in time. But the big advantage lies in the fact that students see me “doing” math on the board. They don’t see an artificial canned presentation but an actual mathematician doing math. Surely this is a nice opportunity to display our real life work and passion, and too many instructors miss this opportunity.

**Assessment of Learning**

I had changed some of the instruction methodology but I still retained traditional exams. My thinking on this issue was also evolving and some reading led to a critical question: how do we know that a student’s writing indicates anything about that student’s understanding of the concept? The area of assessment is very broad but for now my only reaction was a clear understanding that student memorization and reproduction in an exam didn’t tell me anything except that the student had memorized well. So I decided to allow students to bring their books and notes to an exam. Of course, this was welcomed by the students, but they did realize that they could not take advantage of the book if they hadn’t prepared their understanding ahead of time. There are arguments for and against open book exams and when each is appropriate. In my case I could only see the increased emphasis on understanding that resulted from this change.

I was still unsure whether these exams reflected an accurate assessment of learning. I also noticed that students were anxious about exams and despite the relaxed classroom atmosphere they were still hampered by test anxiety. Over a number of years I sought to reduce that anxiety by decreasing the weighting of final exams and increasing numerous alternative methods of assessment. There have been instances where I have given no exams. That increases the challenge to find alternatives. Sometimes I have included the students in the development of their own exams. This has again been well received by students who can relax more and not feel exam pressure. But it doesn’t work for all students as some lack any motivation apart from exams. Ironically this is most prevalent among pre-service teaching students (Ha, 2006) and education majors. As a result my current assessments are a mix of homework, student projects, student essays, in-class work, take home exams, and some judicious open book in-class quizzes.

**Conclusion**

In summary, I believe any instructor can make their teaching more rewarding and productive by considering the following: (a) think deeply about what you want your students to learn; (b) incorporate multiple instruction/facilitation strategies in the classroom and beyond; (c) focus on quality rather than quantity; (d) *Do* mathematics in the classroom rather than *present *mathematics; (e) think deeply about how to assess student learning. * *

My goal is to have my classroom be an open laboratory of learning. I have learned through service learning, group work, and alternative assessments that learning can be achieved in multiple ways. I believe I have made progress in that direction through the strategies described above. I am now more definite about the role of coach and motivator. But I don’t think we should ever stop reflecting on our teaching and improvements can always be made. I do know that I enjoy my teaching much more now than ever before. I approach each class session with a sense of excitement and a desire to facilitate student learning and to pass on my own enthusiasm for mathematics. It is my hope that this essay encourages the reader to reflect on their own teaching and consider ways in which it can be more effective and enjoyable.

**References**

- Ball, D. L. (1993). With an eye on the mathematical horizon: Dilemmas of teaching elementary school mathematics.
*Elementary School Journal, 93*(4), 373-397. - Conway, J. B. (2001). Reflections of a Department Head on Outreach Mathematics.
*Notices of the AMS*48(10), 1169-1172. - Dwyer, J. F. (2001). Reflections of an Outreach Mathematician,
*Notices of the AMS*, 48 (10), 1173-1175. - Dwyer, J. F. (2005). K-12 Math Tutoring as a Service-Learning Experience for Elementary Education Students, in
*Mathematics in Service to the Community*, MAA Notes. - Ha, A. (2006). Alternative assessment in pre-service teachers’ geometry course.
*MA Report*, Texas Tech University. - Morris, R. and Hayes, C. (1997). Small Group Work: Are group assignments a legitimate form of assessment? In Pospisil, R. and Willcoxson, L. (Eds),
*Learning Through Teaching*, 229-233. Proceedings of the 6th Annual Teaching Learning Forum, Murdoch University, February 1997. Perth: Murdoch University. http://lsn.curtin.edu.au/tlf/tlf1997/morris.html

There are major changes afoot in both K-12 and postsecondary mathematics education. For example, the widespread adoption of the Common Core State Mathematics Standards [5] has been a focal point for everyone involved in K-12 education in the United States. The 2012 report Engage to Excel [4] from the President’s Council of Advisors on Science and Technology (PCAST) included substantial recommendations for experimentation and change in mathematics education, including at the undergraduate level for preservice teacher training. A large and growing body of research [6] in STEM disciplines (Science, Technology, Engineering, Mathematics) is demonstrating that undergraduate learning and achievement can be increased by implementing evidence-based teaching practices. Funding agencies such as the National Science Foundation are responding to this seriously, for example through the Widening Implementation and Demonstration of Evidence Based Reforms (WIDER) program [7].

In parallel with these developments, calls for greater engagement by mathematicians in mathematics education at all levels are on the rise. For example, Thomas Banchoff and Anita Salem wrote in 2002 [1] that “the challenge we face in the mathematics community is bridging the divide between… ‘the world of research’ in mathematics education and ‘the world of practice’.” In a 2011 AMS Notices article [2], Sybilla Beckmann called for the creation of a more unified mathematics teaching community, stating that “mathematicians, mathematics educators, and teachers… bear collective as well as individual responsibility for improvement of the mathematics education system as a whole.” The 2012 CBMS report The Mathematical Education of Teachers II [3] makes a similar call, stating that “more mathematics faculty need to become deeply involved in PreK-12 mathematics education by participating in preparation and professional development for teachers and becoming involved with local schools or districts.”

Another important example is The Mathematical Sciences in 2025 [8], a 2013 National Research Council report that was commissioned by the Division of Mathematical Sciences at the National Science Foundation and funded through a $700,000 NSF award [9]. The report states that “it is critical that the mathematical sciences community actively engage with STEM discussions going on outside the mathematical sciences community and not be marginalized in efforts to improve STEM education.” Also discussed in this report are impending challenges facing university mathematics faculty, particularly in the wake of the aforementioned PCAST report; e.g., “the PCAST report should be viewed as a wake-up call for the mathematical sciences community… Change is unquestionably coming to lower-division undergraduate mathematics, and it is incumbent on the mathematical sciences community to ensure that it is at the center of these changes and not at the periphery.” In general, the report states that “A community-wide effort to rethink the mathematical sciences curriculum at universities is needed.”

Given all this, what is a mathematician to do? How do we equip ourselves to handle these new challenges, to become familiar with the existing research literature on teaching and learning, and to view teaching as the scholars we are? Every mathematician wants their students to succeed at doing mathematics, to experience the joy of mathematical discovery, and to learn beautiful mathematics at a deep level. However, the professional education of mathematicians is generally devoted to developing highly-refined scholarly tools for mathematics research. With a few notable exceptions such as Project NeXT [10], the development of a scholarly approach to the teaching and learning of mathematics is often absent, and sometimes even discouraged. The goal for this blog is to be part of the response to these challenges, to stimulate reflection and dialogue by providing mathematicians with high-quality commentary and resources regarding teaching and learning.

Because there is no simple solution to the problems facing mathematics education, this blog will serve as a big tent, giving voice to multiple contributors representing a wide range of ideas. Contributions will range from practical “teaching tips,” to commentary on current mathematics education research, to discussions of social/curricular educational policy, and more. Our focus will include both postsecondary and PreK-12 education, because mathematics education does not abruptly stop and start anew as students make institutional transitions. Issues that affect both high- and low-achieving students will be addressed, as well as issues that affect students who are minoritized in their mathematical communities. We welcome ideas for posts and pointers to interesting materials or events; please feel free to contact one of the editors or add your contributions in the comments.

Stay tuned!

Acknowledgement: The editors would like to thank the American Mathematical Society for supporting and hosting this blog.

**References**

[1] Banchoff, T, and A Salem. “Bridging the divide: Research versus practice in current mathematics teaching and learning.” Disciplinary styles in the scholarship of teaching and learning: Exploring common ground (2002): 181-196.

[2] Beckmann, Sybilla. “The community of math teachers, from elementary school to graduate school.” Notices of the AMS 58.3 (2011).

[3] Conference Board of the Mathematical Sciences (2012). The Mathematical Education of Teachers II. Providence RI and Washington DC: American Mathematical Society and Mathematical Association of America.

[4] Engage to Excel. Report to the President from the President’s Council of Advisors on Science and Technology (PCAST), January 2012. http://www.whitehouse.gov/administration/eop/ostp/pcast/docsreports

[5] Common Core State Standards, www.corestandards.org

[6] Singer, Susan R, Natalie R Nielsen, and Heidi A Schweingruber. Discipline-based education research: Understanding and improving learning in undergraduate science and engineering. National Academies Press, 2012.

[7] http://www.nsf.gov/funding/pgm_summ.jsp?pims_id=504889

[8] National Research Council. The Mathematical Sciences in 2025. Washington, DC: The National Academies Press, 2013.

[9] http://www.nsf.gov/awardsearch/showAward?AWD_ID=0911899&HistoricalAwards=false

]]>