One of the challenges of teaching mathematics is understanding and appreciating students’ struggles with material that to the instructor, after years of thinking about it, may seem straight forward. Once we understand an idea, it may seem almost impossible not to understand if it is presented clearly enough. Yet experienced math teachers know that presenting mathematical ideas clearly, as important as that is, is generally not enough for students to learn the ideas well, even for dedicated and determined students. At the same time, students who struggle can have insightful and productive ways of solving problems and reasoning about mathematical ideas. Research into how people think about and learn mathematics reveals why this surprising mix of struggle and competence can coexist: learners can use what they do understand to make sense of new things, yet ideas that are tightly interconnected and readily available for an expert may be fragmented or inchoate for a learner.

Consider the ideas surrounding slope and rate of change, which are well known to be difficult for students. To the expert, a slope is a number that expresses a measure of steepness. It connects changes in an independent variable to changes in a dependent variable. This connection is multiplicative and explains why non-vertical lines have equations of the form *y* = m*x* + b. But even students who appear to be proficient—because they can calculate a slope and use it to find an equation for a line—may be missing some crucial connections. They might not see slope as a number, but instead think of it as a pair of numbers separated by a slash, basically “rise slash run.” If the “rise” is 3 and the “run” is 2, then even if they know that 3/2 is a number, they may not connect it to the geometry and algebra of the situation. They might not see this number as a measure of steepness, and if asked to describe steepness, might prefer to subtract the “run” from the “rise.” Students might not see the “rise” as 3/2 *of* the “run” and they might not connect this multiplicative relationship between the “rise” and “run” to the point-slope form of an equation for a line. Mathematics education research is examining the fine-grained details of how students think about ideas surrounding slope. It is investigating how certain ways of representing and drawing attention to ideas can help students extend and connect their ideas. Research-based instruction can then take into account known challenges and opportunities for learning.

We thought readers of this blog might be interested to learn a little about approaches to slope and linear equations that we are currently investigating. Proportional relationships—pairs of values in a fixed ratio—provide an entry point into the study of linear functions and are a focus in the Common Core State Standards for Mathematics at grades 6 and 7 (see [1] and [2]). So consider the proportional relationship consisting of all pairs of quantities of peach and grape juice that are mixed in a fixed 3 to 2 ratio to make a punch. When graphed, these points lie on a line. One way to think about the slope, 3/2, of this line is that for every new cup of grape juice, the amount of peach juice increases by 3/2 cups. This way of thinking is part of what we call a *multiple batches* view, a view that has received significant attention in mathematics education research. From this perspective, we may think of 1 cup grape juice and 3/2 cups peach juice (or 2 cups grape juice and 3 cups peach juice) as forming a fixed batch of punch, and we vary the *number* of batches to produce different amounts in the same ratio. This fits with the image in Figure 1a, which evokes repeatedly moving to the right 1 unit and up 3/2 units. But as indicated in Figure 1b, the general multiplicative relationship, *y *= (3/2)* x,* is less evident, especially for *x* values that are not whole numbers.

*Figure 1:* Slope from a multiple-batches perspective.

Another way to think about the punch mixtures in a fixed 3 to 2 ratio uses what we call a *variable parts* perspective. This perspective has been overlooked by mathematics education research, but we are currently studying how future teachers reason with it. In a variable-parts approach, for any point on the “punch line” (see Figure 2), there are 3 parts for the *y*-coordinate and 2 parts for the *x*-coordinate, and all the parts are the same size. From this perspective, we vary the *size* of the parts to produce different amounts in the same ratio. The parts expand or contract depending on the direction the point moves along the line. In a variable-parts approach, the slope 3/2 is a direct multiplicative comparison between the numbers of parts of grape and peach juice: The number of parts peach juice is 3/2 the number of parts grape juice. Put another way, the value 3/2 is the factor that multiplies the number of parts of grape juice to produce the number of parts of peach juice, regardless of amounts of juice in each part. Therefore the *y*-coordinate is 3/2 of the *x*-coordinate, so *y *= (3/2)* x*.

*Figure 2:* A proportional relationship viewed from a variable-parts perspective.

*Figure 3:* Slope and equations from a variable-parts perspective.

We don’t think there is any way to make the concept of slope easy for students. But we suspect that working with both the multiple-batches and the variable-parts perspectives should help students develop a more robust understanding of slope. In particular, the variable-parts perspective might help students connect the slope of a line and an equation for the line. References [3] and [4] discuss the multiple-batches and variable-parts perspectives in greater detail.

We are currently conducting detailed studies of how students in our courses for future teachers reason from both the multiple-batches and the variable-parts perspectives on proportional relationships*. Discoveries about how future teachers reason about the interconnected ideas of multiplication, division, fractions, ratio, and proportional relationships, and what is easier and what is harder to learn, will help us identify productive targets for instruction in courses for future teachers. But we also hope that others will try the variable-parts perspective with other groups of students. For example, we could imagine a group of college algebra instructors collaboratively designing lessons that use a variable-parts perspective to help students better understand slope and its connection to equations for lines.

We also think that the variable-parts perspective is potentially productive for trigonometric ratios. From a variable-parts perspective, we can think of the radius of a circle as 1 part of variable size, *r*. For a fixed angle, its radian measure, sine, cosine, and tangent can all be viewed as a fixed number of parts (although this number is often irrational). With this perspective, equations such as *x* = cos(θ)*r* and *y* = sin(θ)*r* arise from the very same reasoning that connects slope to the equation of a line.

We think that there are many useful findings of mathematics education research that could help improve mathematics teaching and learning, but that environments and cultures are often not conducive to using the knowledge that we have. We need professional environments and cultures that foster serious discussions about what to teach and how to teach it, where knowledge about teaching and learning mathematics is intertwined with the practice of mathematics teaching, and where knowledge and practice grow together. We applaud the editors and the AMS for starting this blog as a way to nurture and develop such a culture.

[*] We are grateful to the University of Georgia, the Spencer Foundation, and the National Science Foundation, award number 1420307, for supporting our research.

**References**

[1] National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). *Common core state standards for mathematics*. Washington, DC: Author. Retrieved from http://www.corestandards.org/assets/CCSSI_Math%20Standards.pdf

[2] Common Core Standards Writing Team. (2011). *Progressions for the common core state standards for mathematics (draft), 6–7, ratios and proportional relationships.* Retrieved from http://commoncoretools.files.wordpress.com/2012/02/ccss_progression_rp_67_2011_11_12_corrected.pdf

[3] Beckmann, S., & Izsák, A. (in press). Two perspectives on proportional relationships: Extending complementary origins of multiplication in terms of quantities. *Journal for Research in Mathematics Education.*

[4] Beckmann, S., & Izsák, A. (2014). Variable parts: A new perspective on proportional relationships and linear functions. In Liljedahl, P., Nicol, C., Oesterle, S., & Allan, D. (Eds.). (2014). Proceedings of the Joint Meeting of PME 38 and PME-NA 36 (Vol. 2). Vancouver, Canada: PME. http://www.igpme.org

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We thought our readers might be interested to know that nominations are now open for several American Mathematical Society awards related to teaching and learning. The deadline for nominations for the following awards is September 15, 2014.

- Award for Impact on the Teaching and Learning of Mathematics.

- Award for an Exemplary Program or Achievement in a Mathematics Department.

- Mathematics Programs that Make a Difference.

More information about these awards and the nomination process can be found here: http://www.ams.org/profession/prizes-awards/prizes

Our understanding of the importance of processes and practices in student achievement has grown dramatically in recent years, both in mathematics education and education more broadly. As a result, at the K-12 level explicit practice standards are given in the Common Core Mathematics Standards [1] and the Next Generation Science Standards [2] alongside content standards. At the postsecondary level, studies regarding student learning and achievement have revealed the importance of many key practices, and accessible sources exist on this topic [3, 4, 5]. Further, we understand now that not all advanced postsecondary mathematics students are well-served by the same curriculum; for example, pre-service high school mathematics teachers need to develop unique ways of practicing mathematics compared to math majors with other emphases [6, 7]. As discussed by Elise Lockwood and Eric Weber in the previous post on this blog, mathematicians generally appreciate these issues; for readers unfamiliar with mathematical practice standards, their article is a nice introduction to this topic.

All of this leads us to the following question:

*Given the breadth of both content and practices required for students to deeply learn and understand mathematics, what are effective techniques we can use at the postsecondary level to gauge student learning?*

This is an important question for us to reflect on, and one that will never be completely resolved. The stereotypical assessment structure in math courses, especially at the service level, are homework problems (often collected and graded using multiple-choice online homework systems), midterm exams, and a final exam. While these can be useful components of an overall assessment structure for a course, these assessments alone often do not serve students and instructors as well as they could. The main reason for this is that these methods typically assess only procedural mastery and “basic” conceptual understanding, without assessing any aspects of the mathematical processes and practices of students.

I’d like to share some methods of assessment that have been either useful in my own classes or thought-provoking and inspiring. The common theme of these methods is that they are informed by both content and practices, even when they are primarily focused on assessing content. In my personal reflections on the question above, I’ve found the MAA Notes volumes regarding assessment practices in undergraduate mathematics [8, 9] to be helpful sources of ideas and inspiration; readers familiar with those documents will likely notice their influence in the list below (PDF versions of both of these are available for free at the MAA website). I also found reading Alan Tucker’s recent article in the American Mathematical Monthly [10] regarding the history of undergraduate programs in the United States to be thought-provoking in this context, as it provides a sense of how our current curriculum, which is closely related to our assessment methods, developed.

**Allow submission of revised work, grading both the mathematics and the depth of the revision.** I like to reward “honest, productive failure” on the part of my students. While we often assign students exercises on their homework, that is, problems that should be reasonably straightforward given a basic understanding of the course content, it is also good to give students hard problems that they might not succeed with at first. When I give students problems such as this, problems that I don’t expect any of them to solve, I often allow students to revise and resubmit their work after an initial round of grading. Then, when I re-grade their work, I can give credit to students for both their mathematical content and for the depth of their revision, for the degree to which they tried out new ideas and sought to determine the reasons for their initial failure. This allows me one way to reward persistence and self-monitoring, which are important practices. It is a good idea to not accept “first” submissions after the initial due date, so that students take their initial work seriously in order to have their revised work re-graded.

**Assign frequent quizzes rather than infrequent long exams***.* I’ve recently been happily experimenting with giving 10-15 minute quizzes on a weekly basis, rather than 50-minute exams every 4-6 weeks. The class time spent with students taking tests ends up being roughly equivalent in either situation. My experience has been that by assessing student content knowledge more frequently, students feel that they are receiving better feedback regarding their progress and I am able to more quickly identify and respond to student misunderstandings.

**Provide brief peer discussion time a few minutes into quizzes and exams.** One of my regular complaints about quizzes and exams is that students might simply miss an obvious idea that would unlock the door allowing them to succeed. Outside of a classroom setting, people are rarely given a task and then required to work completely in isolation; typically it is important to work as part of a team, and to be able to make individual contributions effectively in this context. Allowing students to think about a quiz or exam problem for 3-4 minutes, then giving students a minute or two for discussion with their peers, better mirrors the reality of the mathematical world while still requiring individual students to do the bulk of the work on a problem.

**Assign short essays.** Essays are one of the best tools that mathematics faculty have to motivate students to reflect on their own processes and practices. Whether requiring students to write personal reflections about their performance in a class, having students critically analyze passages in their textbook, or having students compare and contrast different problems on homework assignments, writing forces students to step back and think about what they are doing in ways that they might not otherwise recognize that they should. The most important practical aspect of using essays in math classes is to identify and clearly communicate a grading rubric to students well before any essays are submitted for grading. Many grading rubrics for general student writing exist (university writing centers are good local resources in this regard), while others have been developed specifically for mathematical writing at various levels [11, 12].

**Assign long-term projects, both expository and open-ended.** Long term projects, such as writing an extensive (10+ page) paper about a major theorem, collaboratively writing a textbook wiki-style with classmates, creating a series of instructional videos, and creating course portfolios, can be incredibly empowering for students. Much like essay writing, long-term projects force students to think at a meta-cognitive level about their own work in a larger context than only considering one homework problem at a time. I’ve assigned 15-page papers regularly in my history of mathematics course, and collected course portfolios in both linear algebra and problem solving for teachers courses. I feel that these activities have had very positive effects on my students.

**Use in-class student presentations of proofs and examples, problem solutions, etc.** While I haven’t used these methods in my classes (yet), I have a positive memory regarding a graded presentation I gave in my college geometry class as an undergraduate. These activities are common in classes taught via Inquiry-Based Learning (IBL) methods [13], and various workshops exist to train faculty in IBL techniques [14]; I’ve heard uniformly positive reports from students regarding their experiences in IBL-style courses. My feeling is that one of the best aspects of using student presentations in class is that it makes errors and mistakes public, which often leads to the counterintuitive result that students feel more comfortable with their mathematical abilities (since they realize that they aren’t alone in needing to persist through mistakes and error-correction during mathematical work).

These are only a few ideas regarding assessment techniques. As the references below illustrate, a wealth of interesting and inventive methods exist for assessment, inspired by both student content knowledge and student processes and practices.

**References**

[1] http://www.corestandards.org/

[2] http://www.nextgenscience.org/

[3] Bain, Ken. *What the Best College Students Do*. Belknap Press, 2012.

[4] Ambrose, Susan A., Bridges, Michael W., DiPietro, Michele, Lovett, Marsha C., and Norman, Marie K. *How Learning Works: Seven Research-Based Principles for Smart Learning.* Jossey-Bass, 2010.

[5] Dweck, Carol S. *Mindset: The New Psychology of Success.* Ballantine Books, 2007.

[6] 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. http://cbmsweb.org/MET2/

[7] Shulman, Lee S. “Those Who Understand: Knowledge Growth in Teaching.” *Educational Researcher, *Vol. 15, No. 2, (Feb. 1986), pp 4-14

[8] *Assessment Practices in Undergraduate Mathematics*, Bonnie Gold et al., editors. Mathematical Association of America, 1999. Available in PDF at: http://www.maa.org/publications/ebooks/assessment-practices-in-undergraduate-mathematics

[9] *Supporting Assessment in Undergraduate Mathematics*, Mathematical Association of America, 2006. Available in PDF at: http://www.maa.org/publications/ebooks/assessment-practices-in-undergraduate-mathematics

[10] Tucker, Alan. “The History of the Undergraduate Program in Mathematics in the United States.” *The American Mathematical Monthly*, Vol. 120, No. 8 (October), pp. 689-705.

[11] Grading Rubric, MA 310, University of Kentucky, Spring 2014. http://ms.uky.edu/~braun/LinkedMaterial_AMSBlog/Sample_Rubric.pdf

[12] Crannell, Annalisa. “Writing in Mathematics.” https://edisk.fandm.edu/annalisa.crannell/writing_in_math/

]]>As students’ mathematical thinking develops, and they encounter more advanced mathematical topics, they are often expected to “behave like mathematicians” and engage in a number of mathematical practices, ranging from modeling and conjecturing to justifying and generalizing. These mathematical practices are distinct from specific content students might learn because they are characteristics of broader behavior, rather than mastery of a single concept or idea. However, these practices represent indispensable components of what it takes to become a successful mathematician.

We think that addressing how students develop the ability to engage in mathematical practices receives far less attention than how students develop content knowledge. In several places in the mathematics education literature, there is a distinction that suggests two different aspects of mathematical knowledge. On the one hand, such knowledge involves knowing mathematical content (such as interpreting the result of dividing a by b), and on the other hand, it entails knowledge of broader mathematical practices (such as generalization or problem solving across domains). The distinction between these two types of mathematical knowledge is reflected in the foundations of the Common Core State Standards for Mathematics (CCSSM), which distinguish between content and practice standards in order to explain how mathematical practices span specific content goals (National Governors Association Center for Best Practices & Council of Chief State School Officers, 2010).

Here, the *standards for mathematical content *(SMCs) are characterized as “a balanced combination of procedure and understanding” (CCSSM, 2010) about certain content, such as the rational numbers, systems of equations, or geometric theorems. On the other hand, the CCSSM highlights eight broader *standards for mathematical practice* (SMPs), which “describe ways in which developing student practitioners of the discipline of mathematics increasingly ought to engage with the subject matter as they grow in mathematical maturity and expertise throughout the elementary, middle and high school years” (CCSSM, 2010). These practices include activities like *Make sense of problems and persevere in solving them, Model with mathematics,* and *Look for and make use of structure.*

We recently had a conversation with pre-service and in-service teachers in a professional development seminar focused on aligning teaching practices with the CCSSM. One teacher asked, “Am I supposed to teach students how to generalize or model, or am I supposed to teach them about exponential functions, or both? Do I approach each type of standard in the same way? When can I say I have taught them mathematics?” These questions highlight the issues the teachers identified surrounding the nature of mathematical knowledge, as well as their struggles with interpreting and coordinating two kinds of standards in their classroom.

Even more, this tension between teaching content and practices extends beyond the K-12 Common Core. In community colleges and universities (and, indeed, in our own teaching as university mathematics education professors), mathematics instructors must attend to incorporating mathematical practice in their teaching, and it is not very clear how to do so effectively.

In our initial exploration of these issues, we talked with seven university-level mathematicians about a) their interpretation of the practices of modeling, problem solving and justification, b) their own experiences with developing these practices for themselves, and c) how they incorporated these practices into their instruction. Here we share excerpts from interviews with two mathematicians, briefly discussing two preliminary themes.

First, the mathematicians we interviewed suggested that mathematical practices are important for students’ mathematical development and can, in theory, be taught and learned. For example, when asked about the teaching of practices (as opposed to specific content), Mathematician 1 said, “So bearing in mind that I don’t know how to teach any of this [the practices], in some ways being able to do these sorts of things is way more important than knowing how to find the intercept of a graph… you can probably get by in life without knowing how to multiply two numbers; you can go get a calculator. But if that stops you from learning problem solving, that’s a huge problem.”

Mathematician 2 also suggested that, in general, the practices are particularly important for students to learn but remain closely linked to content, saying “I think what you really want are the practice [standards], and the way you typically try to do that is by teaching content, but something in the way you teach the content hopefully will teach the standards for practice. So I feel like the goal is the standards for practice, but I have a hard time imagining having anyone doing that in any meaningful way without doing it through some content.”

These comments suggest that the mathematicians value the mathematical practices and feel they are important for students to learn. However, while they view the practices as important, the mathematicians also noted that the practices also pose unique challenges for how they might be developed and measured.

We asked the mathematicians to reflect on the distinction between SMCs and SMPs. In response, Mathematician 1 noted, “Well, it’s a heck of a lot easier to assess content, and these standards are designed with assessment in mind if I’m not mistaken.” We also had the following exchange with Mathematician 2, in which he highlights the inherent difficulty in assessing mathematical practices.

*M2: I don’t mind that division in the standards. I think it’s probably trickier to measure the standards for practice, but I don’t mind putting them out there. Just because they’re hard to measure, to me doesn’t mean that it’s not important to include them as a stated goal.
Int: What makes them harder to measure?
M2: Well I just think it’s a little easier to ask a question and determine from a student’s response whether or not they know a certain piece of content. Where, depending on the situation, it might be easier or more difficult to draw out of them a certain ability they have to approach a problem. Like I can ask a student about whether or not they know the Pythagorean Theorem, and I think it’s not going to be a very long, difficult discussion for me to figure out if they’ve heard of it before and if they can apply it in a problem. Where if I give them a problem about the Pythagorean Theorem and I’m trying to look for how they problem solve or how they model things, that feels a little more context dependent…there might be issues with when that student displays that sort of behavior. They might be very good at that, but not in a geometric context or something. I wouldn’t necessarily say they lacked that skill just if they didn’t bring it to bear on that problem, so I just feel it’s a little harder to say conclusively.*

To summarize, these mathematicians suggested that teaching and learning of mathematical practices are very important, but they are difficult to develop and measure. Teachers at all levels are thinking harder about how to teach and assess their students’ learning of mathematical practice. At the K-12 level this is seen explicitly in the inclusion of standards of mathematical practice in the CCSSM, but the same issues pertain to postsecondary classrooms as well. We encourage readers to think about how to incorporate both content and practices as they teach mathematics, perhaps explicitly having conversations with colleagues and with students about this important distinction.

In the next post on this blog, Ben Braun will discuss specific ideas for assessment that can help with some of these issues.

**References**

National Governors Association Center for Best Practices, Council of Chief State School

Officers. (2010). Common Core State Mathematics Standards. National Governors Association Center for Best Practices, Council of Chief State School Officers, Washington D.C.

The best recruiting tool I have to convince students that they should continue in the study of mathematics is the mathematics that I am teaching, no matter the level. It is all fascinating. In almost every lower division course that I have taught I have convinced at least one student to add the mathematics major. The last time I taught second semester calculus, three students added the math major and one the math minor (and the student selecting the math minor simply could not fit in the last three mathematics courses for the major). One of those students is now a graduate student in biostatistics at Harvard.

I have been fortunate that our department has insisted on teaching mathematics in classes no larger than 35 students. Given the small class size, I require each student to come to my office to talk about his or her career plans. I ask about their goals and I suggest how taking more mathematics can help the student meet those goals. During class time I often mention more advanced courses that will touch on the material being presented. When my students begin registering for courses I offer my assistance in selecting courses for the following semester and beyond. I tell them that I would be pleased to talk to them if they want to stop by in the future to discuss their career plans.

If we viewed our function as communicators of mathematics differently, we would greatly increase the mathematical preparedness of our students. We teach the most fascinating subject in the world. We need to communicate this to our students. Mathematics holds a unique place in university studies. Most programs of study include mathematics as a requirement and mathematics is often part of the recommended first-year course of study. Why?

There are quantitative aspects of programs of study for which mathematics is essential. The growth in the amount of data that is now available is enormous and is a relatively recent event. The mapping of the human genome and the data that the internet generates every day are mined to obtain information about individuals and societies. This amount of data is a new phenomenon and it has impacted many different fields of study, increasing their quantitative needs, and thus increasing their mathematical requirements. This increase in the quantitative needs of other disciplines represents an opportunity for mathematics departments. We can revamp/create mathematics courses to entice more students to include more mathematics in their undergraduate curriculum. All of this is good for mathematics departments, as we are given the opportunity to interact with students at many stages of their careers.

There is another answer to the “Why?” question. Some popular programs, which have limited resources, place mathematics courses as barriers, hoping to keep the size of their program to manageable levels. Mathematics courses, and mathematics departments, now take on a very different function. We are the ones who weed out students from these popular careers. The result is that those students who could not succeed in university mathematics courses come away with a negative view of mathematics. Many of us have encountered individuals who state that they cannot do mathematics, and they seem to say this with pride. However, I have never heard a person tell me that he/she cannot read! It seems to be acceptable in this country to be mathematically illiterate but unacceptable to be illiterate.

The inability of students to pass our mathematics courses has a dramatic impact on their lives. We often hear that a student chose this or that career. This is far from what really happens. The lack of mathematical training precludes entrance into so many careers. It is a fact that the more mathematics a student takes, the more careers are available to that student. This fact places a responsibility on us, as communicators of mathematical ideas. We should not lay the blame on pre-college instruction. It is up to us to address effective instruction in university mathematics courses.

The role of gatekeeper is not a role that we should relish and I want to suggest that mathematics departments embark on a new strategy. Surprise!

All of our courses, no matter how elementary, should be taught with enthusiasm and with the view that we are preparing students for the next mathematics course. We should surprise these popular programs of study by increasing the passing rate of our courses, and at the same time, increasing the number of students pursuing further mathematical studies [1,2]. Our role is not to keep students out, but rather to help students to reach their goals.

The unique role that mathematics holds in our educational system places on our mathematics departments a responsibility. It is up to our teaching staff to provide a broad mathematical education for our students, one that motivates students to the further study of mathematics. In fact, the goal of a mathematics course is not to teach this or that. It is to show the students that the material is so interesting, so germane to their lives, that they have to take the next mathematics course.

**References**

[1] Increasing the Number of Mathematics Majors, Focus, March 2006, pp 24-26. http://www.maa.org/sites/default/files/pdf/pubs/march2006web.pdf

[2] Not business as usual, Opinion Piece, Notices of the American Mathematical Society, May 2003, pg. 533. http://www.ams.org/notices/200305/commentary.pdf

]]>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.

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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.