*Comment from the Editorial Board: We believe that in our discussion of teaching and learning, it is important to include the authentic voices of undergraduate students reflecting on their experiences with mathematics. This article is our first such contribution. We feel it provides a window into many of the subtle challenges students face as they transition to advanced postsecondary mathematics courses, and that it mirrors many of the themes discussed in previous posts. We thank Ms. Mattingly for being the first student to contribute an essay to our blog.*

In previous math classes, I was the quiet worker who kept to herself and didn’t know when or how to ask questions. After improving my skills in a problem solving class, that has changed. The group work we did each day allowed me to be around other people who think significantly differently than I do. Being in this environment was difficult at first because I actually had to work through problems with other people, which was somewhat unfamiliar to me. My classmates and I were not just sitting down and reading information about specific math problems. We had to analyze and make sense of the best methods and strategies to use and present our ideas to each other. Confusion would set in when other students introduced different approaches. The only way I could understand their ways of thinking was to ask them to explain. Asking questions in math initially intimidated me, especially because my questions had to be directed to my peers. I did not want them to think that I could not keep up with the material or that I did not belong in the class. But I also did not want to misunderstand major mathematical concepts as a consequence of not asking questions. So I started asking my group members each week what strategies they used in their solutions. Although it may have seemed repetitive to them or obnoxious to have to explain their approaches, it helped me immensely. Through my question asking, I was able to talk and think about math in a unique way. I could compare my peers’ techniques to my own, which further stimulated my interest in the particular subjects that were covered in the class. This skill has been and will continue to be essential in my future relationship with mathematics.

With this question-asking skill came a respect for other students’ speed of thinking. As I was learning to think like my group members in certain situations, I also was learning that these students were thinking at different speeds than I was. In many instances, I would attempt to understand what the problem was asking for and in the meantime, my peers were already halfway to a solution. I knew that I wasn’t misunderstanding anything. I simply was not making connections as quickly as others in the class. It constantly surprises me how swift some students are in accurately assessing problems and understanding what is needed to get to a solution. After working with all different types of students, I have learned to respect and accept that others may be working at faster cognitive speeds than I am. In previous classes, I thought that being quick with my math skills was most important. I have now seen that understanding the material is essential to becoming a fast thinker in mathematics.

Through the homework problems and quizzes in the problem solving class, I realized that math problems require perseverance. The problems that are actually difficult, that actually require a student to think about how to apply his or her mathematical knowledge to the solution of a problem, are the problems that take time. I found myself working hours on different math problems, trying to get closer and closer to the right answer. Oftentimes I went through the trial and error process. Failing in math is not as scary as it once was, because of my experience with this problem-solving course. Sometimes I would successfully solve a problem, while other times I didn’t come close to the solution. Either way, I was able to see that simply working with math was enhancing my problem solving skills. I will always remember Paul Zeitz’s quote in his book, *The Art and Craft of Problem Solving *[1], that says, “*Time spent thinking about a problem is always time [well] spent. Even if you seem to make no progress at all,*” (pg. 27). As I encounter future math classes and harder math problems that seem unsolvable, I will keep this quote in mind. Any time spent toying with a problem is enriching my mathematical knowledge.

Understanding Zeitz’s important quote about mathematical thinking prompted me to see the open mind that math problems require. When I took a number theory class in my first year, I was under the impression that all solutions to problems were very obvious and the methods to solve them were evident. Going through a geometry class as a sophomore challenged this belief because I began to see that not everyone knows which method to use immediately to solve or prove a problem. With the problem solving class, this belief was completely put to rest. I have seen firsthand that it occasionally takes experimentation to figure out which method or tool to use in problem solving. Through discussion and group work with my classmates, I noticed that it is not always blatantly obvious that we should draw a picture or use induction or reformulate a hypothesis to find the crux move in a solution. After discovering this, I attempted to open my mind when reading assigned problems. Instead of honing in on one specific method or strategy, I have accepted the fact that one specific method or strategy might not be the only way to achieve my goal.

An open mind in problem solving has allowed me to experience mathematical thinking outside of the classroom. My high school math experiences bred the idea that students don’t necessarily need to think about math outside of the classroom. I brought this idea to college and had no problem passing my classes. But in this math problem solving class, where we were challenged to think in different ways and to explore math on our own, thinking about math strictly inside the classroom was insufficient. One incident that had a deep influence on my problem-solving experience occurred on a walk home from class. In small groups, my classmates and I were trying to determine the difference between permutations and combinations. After working for an entire class period, I did not fully understand the difference between these two basic combinatorics concepts. As I walked home, I reviewed the strategies and methods that I used and compared them with the explanations of my peers. I was still stuck and still frustrated. Upon emailing my professor about my misunderstanding, I realized that although I had encountered an obstacle with the content, I was gaining an invaluable way of thinking. I was extending my mathematical thought processes and interests to outside of the standard classroom environment. I am still constantly left wondering why something in math works. By pondering on my own, whether it is with pencil and paper or not, I am able to see the impact of this class. No class before had prompted me to take my own time to figure out why I did not understand the material. This was a huge win for me as a math student. I was and am still experiencing the effects of struggling with math in a way that will always benefit my problem solving skills.

One of the most interesting proofs that I saw was dealing with the number of subsets of a set with \(n\) elements. Earlier that semester, my probability professor had mentioned that the number of subsets of an \(n\)-element set is \(2^n\). I didn’t think I would be seeing this anymore, so instead of trying to understand why this was so, I just accepted the information. Later on, in my problem solving class, we had a question about the number of subsets in an \(n\)-element set. Obviously I knew it was \(2^n\), but I had no idea why. I eventually understood after I showed interest in the problem through question asking, when another interpretation of all of the subsets of an \(n\)-element set was written up on the board. I saw that another way to write a subset of an \(n\)-element set is by exchanging the actual numbers in the set with 0’s and 1’s. A “0” indicates the absence of a specific element in the subset, while a “1” indicates the presence of a specific element in the subset. For example, the set \(\{1, 2, 3, 4, 5, 6\}\) has the subset \(\{2, 4, 6\}\). This subset can be written as 0, 1, 0, 1, 0, 1, where the 1’s correspond to the numbers found in the subset. Since each space has either the option to be in the subset or not, then each space has two options. Since there are *\(n\)* spaces, then there are \(2^n\) subsets. By asking questions, expressing curiosity, and actually attempting to understand why the answer was \(2^n\), I was able to see a portion of my growth as a problem solver.

I am left with questions regarding my future mathematical experiences. Instead of simply thinking about mathematics outside of the classroom, I am now wondering how to discover and develop new insights about mathematical concepts on my own. Instead of learning about how to use certain strategies, I am wondering how to present these strategies to a group of peers in an orderly and effective manner. Instead of asking questions that do not necessarily prove to be productive, I am slowly learning how to ask the *right* questions.

**References**

[1] Zeitz, P. (2007). *The Art and Craft of Problem Solving*. Hoboken, NJ: John Wiley & Sons, Inc.

*By Oscar E. Fernandez, Assistant Professor in the Mathematics Department at Wellesley College. *

Mathematics is a beautiful subject, and that’s something that every math teacher can agree on. But that’s exactly the problem. We math teachers can appreciate the subject’s beauty because we all have an interest in it, have adequate training in the subject, and have had positive experiences with it (at the very least, we understand a good chunk of it). The vast majority of students, on the other hand, often lack *all* of these characteristics (not that this is their fault). This explains why if I’d start talking to a student about how exciting the Poincare-Hopf theorem is, I probably wouldn’t see anywhere near the same excitement as if I were to, say, let them play with the new iPhone. This may seem like a silly hypothetical, but I believe it brings up all sorts of important points. For one, what does it say about our culture (and our future) when young people would rather be playing games on iPhones (or watching Youtube, or being on Facebook, etc.) than studying math or science? What causes our culture to be the way it is? How did companies like Apple and Facebook get students so interested in these activities? What are they doing that we math teachers aren’t?

First, let me admit that there are many, many differences between getting exciting about the new iPhone and getting excited about math*, but I’m interested in one of them in particular: you can see, feel, *interact with*, and *experience* the iPhone. Moreover, Apple thinks *very carefully* about *every aspect* of the user experience *well before* they release their next phone (there are, after all, *billions of dollars* at stake).

Sadly, the way math is taught in many places, students’ experience with mathematics is often confined to a blackboard or piece of paper. They also spend the majority of their time interacting with math in a very different way, e.g., trying hard to get the right answer before the homework is due as opposed to playing around with the content to discover something new, as a first-time iPhone user might do. And what about the Steve Jobs or Jony Ive of the class—the instructor—who is supposed to make it all magical? Oftentimes that person follows the “definition, theorem, proof” style of teaching, which is likely only “magical” to already math-inclined students. My point: *we* (the math teachers) are the most important drivers of our students’ interest in and excitement about mathematics. Collectively, we are the Apples and Samsungs of the math world. And if we teach math like *we discuss it amongst ourselves***, we’re likely to continue losing the vast majority of students to other careers.

So, what should we do? I say we look to Apple, Samsung, and all the other companies that have successfully hooked our students on their ideas and products. Sure, they have hordes of people whose sole job it is to make their products* fun, cool, and relevant*, but why can’t we do that, too? Why can’t we, for example, give out a survey the first day of class that asks students about their hobbies and interests, and then, at the very least, choose examples and applications for the rest of the course that align with those interests? In fact, why don’t we just structure our courses to make mathematics something that our students can *directly experience* and is *directly relevant* to their lives?*** Let me call this the *Everyday Mathematics* (EM) approach.

Here’s an example. Instead of reviewing the graph of a sine function by drawing a sine curve, explaining what the frequency, amplitude, or period are, showing examples where these parameters change, and finally discussing a Ferris wheel, picture this instead. You pull up a chart of human sleep cycles, you explain that the average cycle length is 90 minutes, that there are four stages of sleep—with Stage 4 being “deep sleep.” You ask your students to find the formula that best fits the sleep chart. Then you ask them: at what times should you wake up to avoid feeling groggy (which happens when you awake near the bottom of a sleep cycle)? You would then guide them to the revelation that they can now use their formula to predict these times and other interesting things, too. Presto! Sine and cosine have now become *relevant*; they are now concepts that help explain *every student’s* sleep cycle and can help them avoid morning grogginess. In other words, this EM approach has made this particular topic at least *relevant* to your students’ lives. I wouldn’t be surprised if, when you move on to tangent, some of your students would start wondering “Hey, what can tangent do for us?” (By the way, how often have you heard a student ask that?)

In general, the EM approach begins with a topic or phenomenon directly relevant to your students’ lives. Then, you (the instructor) build a lesson that slowly guides students through the math you would have taught anyway, except that now there is context, that context is personal for each student, and there is a point to all of it that students can buy into (in the example, helping them sleep better and explain morning grogginess).

From an instructor’s perspective, the EM approach may seem like a lot more work than a more traditional approach. However, I myself was able to generate enough of these EM-like examples (pertinent to Calculus I topics) to write an entire book about it, mainly by just spending a few days being very observant about everything going on around me and then putting on my mathematician hat to see the math behind it. Granted, this approach might not be appropriate for all courses—it probably wouldn’t work in a course on cohomology—but that’s okay, because by that point that student is probably more interested in how that subject relates to other areas of mathematics.

The EM approach may not be the answer to our national crisis in math, but I think it is a step in the right direction. At the very least it realigns our presentation of the content with our students’ interests. It also attempts to emulate the successful efforts of corporations to get people excited about their products, since the approach puts our students—and their interests—first, and then scaffolds on our content goals (as opposed to the other way around). In my experience using the EM approach, I have received some of the most enthusiastic responses I’ve ever gotten after teaching certain concepts. I would love to hear about your own ideas to make math fun, relevant, and something students can directly experience.

_______________________________________________________________

* There are, after all, people who spend weeks in line waiting for the new iPhone; I’ve never heard of a student camping out outside a classroom for weeks waiting for a course to start.

** This would be like Apple unveiling its iPhone by talking *mostly* in technical jargon—after all, that’s how the designers, engineers, and programmers think. I doubt their press events would be so well attended were this the case.

*** No more talking about the largest area a farmer can enclose with a given amount of fencing, or about a ladder falling down the side of a building, for example.

*Editor’s Note: Carl Lee is a recipient of the 2014 Deborah and Franklin Tepper Haimo Award from the Mathematical Association of America. This essay is based on his acceptance speech at the 2014 Joint Mathematics Meetings.*

**My place.** I was born into a family littered with academics, teachers, and Ph.D.s, including a grandfather who was an educational psychologist at Brown serving on one of the committees to create the SAT. My early interest in things mathematical was nurtured in a home stocked with books by Gardner, Ball and Coxeter, Steinhaus, and the like. With almost no exception my public school teachers were outstanding. I was raised in a faith community, Bahá’í, that explicitly acknowledges the presence of tremendous human capacity and the high station of the teacher who nurtures it. I played and experimented with, and learned, mathematics in both formal and informal settings. Thus I grew up in a place in which I was able both to feed my mathematical hunger as well as to have a clear idea of what it was like to teach as a profession. I thrived.

I recount this not to present a pedigree to justify personal worthiness, but rather to emphasize that I enjoyed a perfect match between my personal mathematical inclination and my learning environments. Because of this background, it took me a while to understand the sometimes profound gap between others’ mathematical place, and the consequent care required to pay attention to that place, when designing an effective realm for learning. As a K–12 student I often engaged in math classes at a high cognitive level merely as a result of a teacher’s direct instruction (“lecture”). As a teacher I quickly learned that I engaged few of my students by this process. Not all developed their “mathematical habits of mind” or “mathematical practices” through my in-class lectures and out-of-class homework (often worked on individually). I now better appreciate the significant role of personal context and informal education in the development of students’ capacity.

**The student’s place.** There is an entire discipline of “place-based” teaching and learning, focused on recognizing and making explicit connections with the student’s physical location and social community (an “outer place”). Mirroring and linked with this is a student’s personal cognitive place (or “inner place”)—here, I recall Vygotsky’s writings on the “ZPD,” the zone of proximal development, which he describes as “the distance between the actual developmental level as determined by independent problem solving and the level of potential development as determined through problem solving under adult guidance, or in collaboration with more capable peers.” That is to say, learning can be promoted when the material is above the student’s current state, but not so far above to be unattainable even with scaffolding and assistance. Identifying these outer and inner student places, and making wise and deliberate instructive choices, are major challenges of the teacher.

With respect to the student’s outer place we are all well aware of the encouragement to teach mathematics through “real-world” problems. The Common Core State Standards for Mathematical Practice encourage modeling with mathematics: “Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace.” My work with teachers in central Appalachia has convinced me authentic and locally placed problems can provide powerful stimulus and support for mathematical learning.

On the other hand, with respect to the student’s inner place, another mathematical practice advocates that students must “make sense of problems and persevere in solving them.” Research by the psychologist Carol Dweck, for example, confirms that praise focused on developing a growth mindset positively affects subsequent student achievement, while praise that reinforces fixed intelligence beliefs has the opposite effect. Further, fostering a growth mindset rather than a fixed mindset in the classroom *with the explicit knowledge and understanding of the students* appears to lead to increased academic achievement when students are aware of the value of the struggle. Polyá was an early advocate of the deliberate shift toward raising the explicit awareness of and cultivating mathematical practices among students. Seeing his film “Let us Teach Guessing” while in high school left a lifelong impression upon me. As a result, I feel I must promote *and **observe* struggle in my classroom—deliberately create opportunities in the classroom in which students grapple with mathematics and communicate with each other; carefully listen and use what I learn to shape what is to come; and provide an environment in which mistakes are opportunities for learning and not censure.

The student’s outer and inner places are, of course, deeply connected—where a student is mathematically is not isolated from his or her background and environment. And in Appalachia (as in many other places), struggle is a part of life. The preeminent Appalachian poet and writer Wendell Berry beautifully captures this notion in his essay “Poetry and Marriage” from *Standing by Words*.

There are, it seems, two muses: the Muse of Inspiration, who gives us inarticulate visions and desires, and the Muse of Realization, who returns again and again to say “It is yet more difficult than you thought.” This is the muse of form. It may be then that form serves us best when it works as an obstruction, to baffle us and deflect our intended course. It may be that when we no longer know what to do, we have come to our real work and when we no longer know which way to go, we have begun our real journey. The mind that is not baffled is not employed. The impeded stream is the one that sings.

**The place of community. **There is a continuum of participants and stakeholders in STEM education, including: P-12 students, school teachers, counselors, principals, superintendents, parents, community members, college students taking math and science courses, majors in STEM fields, aspiring STEM teachers, higher education faculty in content departments teaching all of these types of students, higher education faculty in education departments teaching courses for future teachers and engaging in teacher training programs, practicing teachers including those who supervise student teachers or are enrolled in graduate programs, higher education faculty engaging in STEM education research or in outreach to schools, and various local, state, regional, and national agencies and organizations, public and private, commercial and non-profit. There is a natural tendency for each of the diverse participants to operate within a somewhat limited sphere of activity. If we wish to build institutional and regional capacity, there is an imperative need for mathematicians to lend their expertise to this continuum, and for institutions to appropriately reward their contributions. The *Mathematical Education of Teachers II* (CBMS), for example, offers a call to action with explicit guidance and suggestions.

Reflecting on my work with others in these many roles in Appalachia, it is very clear to me that an appropriate understanding of place is essential. Many regard rural Appalachia with “deficit vision” and wish to come in and “fix things.” Yet Wendell Berry’s view is completely opposite—read his poem “The Wild Geese.”

Berry articulates his vision in “The Loss of the Future” from *The Long-Legged House*: “A community is the mental and spiritual condition of knowing that the place is shared, and that the people who share the place define and limit the possibilities of each other’s lives. It is the knowledge that people have of each other, their concern for each other, their trust in each other, the freedom with which they come and go among themselves.”

Bob Wells recalls Berry’s 2007 speech at Duke Divinity School.

“Whatever doesn’t fit a place is wrong,” Berry said. “It doesn’t matter if it is true or false. If it doesn’t belong, it is wrong.” Without a standard of “place” as a measure of real prosperity, Berry said, we will never know what to make of development, technology, research, education, modernization, religion and the environment, or ecosphere.

I have learned that to be more effective I must view place from the perspective of a partner rather than as a knowledgeable outsider (however well-intentioned). Consideration of place must be approached with an authentic attitude of partnership, setting aside such common barriers as “outsider-insider,” “knowledgeable-ignorant,” and “wealthy-poor.” The wealth and strength of Appalachia include rich experience and an abiding sense of community, both of which can significantly contribute to sustainable approaches to educational challenges.

Sentiments such as these were central principles in two recent large-scale NSF funded projects in Appalachia that I had the privilege to work on. ACCLAIM, an NSF Center for Learning and Teaching, focused on “the cultivation of indigenous leadership capacity for the improvement of school mathematics in rural places.” A highlight of this project was the creation of an interinstitutional doctoral program in mathematics education built around issues in mathematics, mathematics education, and rural sociology. Students in this program demonstrated a commitment to rural place and earned their degrees without having to quit their jobs. Their desire to remain in their communities helped sustain ACCLAIM’s impact on future teachers. The resulting dissertations were not required to address rural topics, but often did. I encourage perusal of https://sites.google.com/site/acclaimruralmath for uncovered understandings at the intersection of mathematics education and rural education.

AMSP, an NSF Mathematics and Science Partnership, was an ambitious Appalachian enterprise involving nine institutions of higher education and about 60 school districts. Lessons learned during the earlier years led later to community-based Partnership Enhancement Projects generated by groups of stakeholder partners based on local concerns. The place of the work (e.g., the school, district, or county) provided the explicit context in which participants evaluated challenges, assessed resources, planned, executed projects, and evaluated outcomes.

**The place of mathematics and the mathematics of place. **On the one hand many (including mathematicians) value mathematics precisely because it *transcends* place, even though it may be initially motivated by a particular context (mathematical, physical, or otherwise). On the other hand, the value of place (including rural or urban place, and personal place) offers a rich and meaningful setting in which to nurture the understanding of mathematics and make important connections that can promote mathematical learning and more effective teaching. My present understanding is that the latter view is important to support the former. In teaching and professional development I therefore try to work with others in a spirit of partnership — there are things that I know, and there are things that my partners know. If we abandon a sense of superiority as we approach classroom teaching, professional development, or community capacity building, striving to understand our place, we can dramatically increase the efficacy of our work together.

Chapter 1 of *Make It Stick: The Science of Successful Learning* [2] is called “Learning is Misunderstood.” That is an understatement, as demonstrated by the remaining chapters. The book has received several strong reviews ([3], [5], [8]), so rather than providing a critique, my aim here is to explore the ways in which its account of cognitive science research has validated some decisions I have made about my teaching and gotten me to reconsider others.

Since the early 1990’s, I have been using a form of what we now call Inquiry-Based Learning (IBL) in my Abstract Algebra course; more recently I’ve been doing so in Number Theory as well (using [6]). This all started when Professor Bill Barker of Bowdoin College described an Algebra course built around small-group work, and I was hooked. Surrounded here at Middlebury College by excellent immersion language programs, I realized that Bill was describing a mathematics immersion program. I modeled my course on his so that my students would learn mathematics by speaking mathematics with each other, while I roamed the room as consultant. That first post-conversion semester, there were numerous classes that went overtime before any of us noticed, so engaged were the students.

Meanwhile I began making less drastic changes in my Calculus courses, devoting at most one class session (out of four) each week to small group work. The results were less satisfying; those sessions felt like an add-on rather than an integral part of the course. I assumed that I couldn’t abandon lectures completely because of the list of topics I felt compelled to cover. Last spring, however, considering data showing that few of our Calculus I students go on to Calculus II, I decided to ditch the massive textbook in favor of fewer topics and an interactive format. My goal had shifted from getting them through a fixed set of material to having them engage the ideas deeply enough that they thought differently about measuring change, whether in their economics and biology classes or when reading the news ten years from now.

*Make It Stick* confirmed my preference for an active learning model as soon as page 3: “Learning is more durable when it’s *effortful.* Learning that’s easy is like writing in sand, here today and gone tomorrow.” It’s easier for students to copy my problem solution from the blackboard and then imitate it in a bunch of similar homework exercises, but it’s no wonder that they don’t seem to retain much in that setting. “When you’re asked to struggle with solving a problem before being shown how to solve it, the subsequent solution is better learned and more durably remembered.” [2, p. 88] What I’m reconsidering is the way in which I choose problems; I want the particular struggle to be productive in ways that the authors describe.

Naturally some students resist a shift from passivity to activity. The student evaluations for my first IBL-ish course were quite positive, except for one that said “You’re the expert; you should tell us what to do. I learn better in a lecture,” an assertion that I continue to hear from a few students. According to *Make It Stick*, those students may well be misunderstanding their own learning: “*We are poor judges* of when we are learning well and when we’re not. When the going is harder and slower and it doesn’t feel productive, we are drawn to strategies that feel more fruitful, unaware that the gains from these strategies are often temporary.” [2, p. 3]

To confront such resistance, I put some effort into what I thought of as a sales job: “This way I can help you speak mathematics in real time, and it gives you practice collaborating for later in life, and aren’t we lucky to have small classes at Middlebury,” and so on. These days I think of such effort in the context of metacognition, which I first encountered in [7]. In being explicit about why I structure my courses the way that I do, I’m also encouraging my students to think more critically about their own learning, which is in itself an asset to that learning. This semester I’ve put a page on the course website with information about the science of learning.

The work of the social psychologist Carol Dweck ([1], [4]) comes up in *Make it Stick. *Perhaps I’m biased, but surely mathematics learners are particularly prone to the curse of the “fixed mindset” rather than having a “growth mindset.” This semester, my first assignment in Calculus was to read “Bad at Math is a Lie” [9] and then have a class discussion. First my students shared their “bad at math” moments in groups of three or four, and then we heard some in the full group. I know that one event won’t move everyone into a growth mindset, but it’s a start.

For some reason – the relentless “coverage” drumbeat? – a while back I stopped my practice of taking mini-surveys on Fridays in Calculus classes. These had three questions: (1) What were the important themes this week? (2) What concept(s) intrigued you? (3) What concept(s) are still muddy to you? They helped me know what students were thinking, and communicated to the students that I wanted to know what they were thinking. I’m reintroducing the surveys, not just for those purposes, but also because they ask students to reflect on their learning. “Reflection can involve several activities … that lead to stronger learning.” [2, p. 89]

On the other hand, I’ve always resisted quizzes because of the added stress. According to *Make it Stick*, however, frequent low-stakes assessments that require students to retrieve new knowledge can assist in the learning process. So this term I’ve scheduled weekly quizzes in which anything from the semester so far will be fair game.

The authors of *Make It Stick* suggest that instructors “be transparent.” [2, p. 228] One way in which I convey my intentions to my students is by including this quote at the end of my syllabi: “Trying to come up with an answer rather than having it presented to you, or trying to solve a problem before being shown the solution, leads to better learning and longer retention of the correct answer or solution, even when your attempted response is wrong, so long as corrective feedback is provided.” [2, p. 101] I am still trying to come up with the best ways to provide corrective feedback; that effort might be the subject of a future post. In the meantime, I am grateful to Bill Barker and many others who have been transparent about their pedagogy as I refine my own.

**References**

[1] Braun, Benjamin. Persistent Learning, Critical Teaching: Intelligence Beliefs and Active Learning in Mathematics Courses. *Notices of the American Mathematical Society, ***61 **(January 2014), 72-74.

[2] Brown, Peter C., Roediger, Henry L., and McDaniel, Mark A. *Make It Stick: The Science of Successful Learning.* Belknap Press, 2014.

[3] Christie, Hazel, in *The Times Higher Education*, April 3, 2014.

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

[5] Lang, James N. Making It Stick, in *The Chronicle of Higher Education*, April 23, 2014.

[6] Marshall, David C., Odell, Edward, and Starbird, Michael. * Number Theory Through Inquiry. *Mathematical Association of America, 2007.

[7] National Research Council. *How People Learn: Brain, Mind, Experience, and School: Expanded Edition*. Washington, DC: The National Academies Press, 2000.

[8] Stover, Catherine.“For the most part, we are going about learning in the wrong ways.” *A Fine Line* blog, April 10, 2014.

[9] Waite, Matt. Bad at Math is a Lie. *Math Horizons. * September 2014, p. 34.

One of the highlights of my summer was attending a research conference, Stanley@70, celebrating the 70th birthday of my Ph.D. advisor Richard Stanley. Because it was a birthday conference, many of the speakers went out of their way to say a little something about Richard Stanley, with mathematical or personal anecdotes. One talk in particular, by Lou Billera, did an especially good job giving the history and context of the study of face numbers of simplicial polytopes, in which Richard played an essential role. (The slides don’t totally convey the breadth of the talk, but at least give you some idea of the mathematical story he was telling.) I really appreciated Lou’s talk, and I know (from asking them) that other participants did too. This got me thinking that the mathematical community could do more of this sort of thing, not just at conferences, but more importantly in courses for our undergraduate majors and graduate students. In these courses, we rightfully focus on the truth of mathematical results. Let’s also spend some time sharing with our students *why we care* about the mathematical objects and ideas that show up.

We’ve long been blessed in mathematics to have the freedom to not worry about applications of our discipline. My favorite expression of this attitude comes from Bernd Schröder whose slides at a recent talk jokingly answered the question “Who cares?” with “Who cares who cares? It’s cool.” This was his clever way of expressing his observation that enthusiasm can override pragmatism. (It is only fair to note that Bernd advocates for applications, and that he subsequently gave more specific reasons to care about the topic of his talk.) In other words, we are free to investigate whatever looks interesting to us, and I value that freedom just about every day. But what looks interesting to us, and why?

Of course, in many settings it is the application that makes a result or topic interesting, and I don’t mean to diminish this motivation in the least. Many studies [8, 9] recommend including more applications in mathematics classes at all levels, not just for the sake of the application, or its use for students who are in (or who will be going into) science and engineering, but to help students better understand the underlying mathematics itself. Indeed, the five strands of mathematical proficiency in [8], including “strategic competence” (problem formulation and solving), are specifically described as “interwoven and interdependent”. But these recommendations tend to point towards the K-12 classroom, or towards applied or lower-division undergraduate courses, such as calculus. This same principle seems to me no less relevant in upper-division pure mathematics courses for our majors, and even graduate courses: You can get a better handle on an idea if you know where it came from, or where it is going.

Here are some questions for students to ask or for teachers to answer, even in pure mathematics. The answers don’t need to be long or detailed. Why did people start looking at this topic? What were the motivating examples? How did the ideas develop? How is it used in other areas of mathematics or outside mathematics? Why is this topic in this textbook, or why is this course being taught?

Sometimes a topic, theorem, or definition is just inherently interesting for purely mathematical reasons, which will have resonance for mathematics students who already appreciate abstract thinking. Some quick examples from the mathematics I’m most interested in include symmetry of structures, large matrices whose eigenvalues are integers, and large polynomials that factor linearly. But even when something is inherently interesting to experienced mathematicians, it can be worthwhile to take the time and effort to point this out to students who are just beginning their careers as mathematicians or teachers, and who may not have yet developed that same appreciation.

The needs of future mathematics teachers in this regard may be a little different than those planning to go to graduate school in mathematics. For this cohort, by far the most important context is “How will this show up in my high school (or middle school) classroom?” (See [3] for more detail.) Here, some of the textbooks for capstone courses for teachers, for instance [2, 10], have good ideas, which can be incorporated into other courses as well. For instance, the plethora of structures introduced in an algebra class have important examples in high school, which may be helpful for other students as well: The reals and rationals are fields, integers are a ring, and polynomials and matrices each form an algebra, etc. One textbook we’ve used [10] illustrates the need for all the field axioms by showing that these axioms are exactly the rules we need to solve linear equations.

Students don’t have to wait for instructors to do this for them. Students can ask the questions above, or make up new ones. Students may consult good books with historical and contextual material. Teachers can strongly influence this type of self-study by recommending references, for example the mathematics history books listed in the “What to read next” chapter of [1] and mathematics biographies in the “review of the literature” in [4], and also some of the other references listed below. Teachers can find other ideas for how to guide students to mathematics history at Reinhard Laubenbacher and David Pengelley’s excellent website for teaching with original historical sources in mathematics, which has resources for single projects or entire courses [6, 7].

All this is not to say we should *stop* doing what we normally do in pure mathematics classes. Of course, the careful definition-theorem-proof development of topics is the backbone of mathematics. This is what lets us be certain of our results, which is the other blessing we have working in pure mathematics. But it can also lead to students’ misconception that mathematics is *created* in this order: First come the definitions, which cannot be changed once they are written down, and then theorems are stated, and subsequently proved. Those of us in the business know it doesn’t usually work this way! We should let students in on the secret that the process is a lot more circular than the textbooks let on.

A nice example of this messiness that also illustrates some other ideas here is the notion of an ideal in a ring. It is usually introduced in algebra books simply with the definition, and then some basic results about it are stated and proved. But why would you want to work with this definition? To summarize greatly (see [5] for much more detail), ideals started with Kummer’s introduction of “ideal numbers”, generalizing integers by considering the set of multiples of one number, or, more broadly, a set of numbers. But one reason (which I explicitly share with my algebra students) we see them so much is that if you need to make a quotient ring, then the definition of an ideal gives you exactly what you need in order for this quotient to be well-defined. (This is a good exercise if you haven’t thought about it before.) Each of these extra facts about ideals reinforces points you would probably want to make anyway.

Let me finish where I started, with Lou Billera. When I wrote to him about including a reference to his talk in this post, we had a nice email conversation about these ideas. I’ll give Lou the last word, from that conversation:

When I first started teaching here, I became aware of several older professors (in engineering) whose class lectures consisted of what I called “war stories”,

e.g., how they solved this or that problem for this or that company. I thought they were just wasting their time BS’ing and not “covering the material”, but the students loved it. In the end, for them, the “war stories” were probably much more useful in their professional lives as engineers than the “material” ever could be. (Besides, the “material” was in the book, and they all knew how to read.) To the extent we can get “war stories” into our own mathematical teaching, without sacrificing “the material” (too much), our students will be better off for it.

**References**

[1] Berlinghoff, W., & Gouvêa, F. (2004). *Math through the ages: A gentle history for teachers and others. *Farmington, ME: Oxton House Publishers; and Washington DC: Mathematical Association of America.

[2] Bremigan, E., Bremigan, R., & Lorch, J. (2011). *Mathematics for secondary school teachers. *Washington, DC: Mathematical Association of America.

[3] Conference Board of the Mathematical Sciences (2012). *The mathematical education of teachers II. *Providence, RI: American Mathematical Society; in cooperation with Washington, DC: Mathematical Association of America.

http://www.cbmsweb.org/MET2/met2.pdf

[4] Hersh, R., & John-Steiner, V. (2011). *Loving and hating mathematics: Challenging the myths of mathematical life. *Princeton NJ: Princeton University Press.

[5] Kleiner, Israel, (1996, May). The genesis of the abstract ring concept. *The American Mathematical Monthly*, 103(5), pp. 417-424.

http://www.jstor.org/stable/2974935

[6] Knoebel, A., Laubenbacher, R., Lodder, J., & Pengelley, P. (2007). *Mathematical masterpieces: Further chronicles by the explorers.* New York, NY: Springer.

[7] Laubenbacher, R., & Pengelley, P. (1999). *Mathematical expeditions: Chronicles by the explorers.* New York, NY: Springer.

[8] National Research Council (2001). *Adding it up: Helping children learn mathematics.* Kilpatrick, J., Swafford, J., & Findell, B. (Eds.). Mathematics Learning Study Committee, Center for Education, Division of Behavioral and Social Sciences and Education. Washington, DC: National Academy Press.

http://www.nap.edu/catalog.php?record_id=9822

[9] Schoenfeld, Alan (2007). What is mathematical proficiency and how can it be assessed? In *Assessing mathematical proficiency* (pp. 59-73). New York, NY: Cambridge University Press.

http://library.msri.org/books/Book53/files/05schoen.pdf

[10] Usiskin, Z., Peressini, A., Marchisotto, E., & Stanley, D. (2003). *Mathematics for high school teachers: An advanced perspective.* Upper Saddle River, NJ: Pearson Education.

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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

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