*Editor’s note: The editors thank David, Kelly, Micah, Andres, and Nina for taking the time to share their thoughts and insights with us. Biographical information for each of these students is included at the end of this article.*

“Something that I’ll remember the most is there were a couple epiphany moments where we just all of sudden we seemed like we just stumbled into this, you know magical place of thinking of something we never would have thought or come up with that was really important for our project and the reason I think that I’ll also remember that for a long time is that it gave me a lot of confidence that I could do research because being able to come up with a creative way forward is sometimes I think one of those important parts.”

— REU student

Why should students participate in a summer Research Experience for Undergraduates (REU)? What do undergrads gain from such programs? What has driven their growth and popularity over the past several decades? In this post, we share highlights of a conversation that the editors had with five undergraduates at the 2017 Joint Mathematics Meetings about their experiences at five different REUs (described in the final section). If you are a faculty member, we hope this inspires you to share information about REUs with your students. If you are an undergraduate student, we hope this inspires you to apply for an REU! (Lists of REUs can be found here and here.)

In our conversation, five major themes emerged regarding the students’ REU experiences:

*Collaboration:*the importance of collaboration, friendships, and networking*The Nature of Mathematics:*an appreciation or gained understanding of the nature of mathematics and mathematical research*Self-Beliefs and Agency:*heightened awareness and/or insight about oneself as a learner or person in general*Back to the Classroom:*the positive impact of REUs on subsequent coursework*Graduate School:*an increased or decreased interest in graduate school or insight into graduate school

While there were some additional comments off these themes, which we include below, in this article we hope to tell a story of the impact of REUs on undergraduates through the students’ own words. Note that all student quotes in this article have been lightly edited for clarity.

**Collaboration**

In the interview, a repeating theme was the value of collaboration for students faced with the task of engaging with challenging mathematics at a new level of depth, starting from the first day of the REU.

- The first two days were just lecture and these were all new ideas to me, words I… hadn’t heard before. Much less know how to compute things using them and that kind of thing… The comfort came in the fact that the other students felt the same way.
- There’re really solid friendships that come out of that struggle.
- What I think I’ll remember the most is the friendships.

The students’ friendships with each other were valuable because they found other people from different places and different backgrounds, in some cases finding a mathematical community for the first time.

- I appreciate the REU because it introduced me to a lot of people really serious about math and that’s something I don’t get too much at my college. There are a couple of people interested in grad school but not nearly as many as being at an REU and… now when I come to conferences I know a lot more people and it’s, we’ve kind of like supported each other through the application process of applying to grad school. So I guess, I mean like a lot of people I see myself being friends with pretty much for the rest of my life, being that we plan on being in the same field and I really, I really do appreciate that.
- It’s very easy to make good friends because you drop the pretense in some sense so it’s just a lot of fun. But yeah, definitely walking in ‘cause the different backgrounds you’re kind of able to collaborate in a different way from class … So you’re trusting someone else’s background, their knowledge, their expertise in these kinds of things and it allows you to work better as a group but also simultaneously it makes it easier to pull back.
- I think meeting different people from different places is the other thing you get out of REU and that should stick with you.

Eventually, in some cases, students formed efficient teams, creating “magical” moments.

- A lot of what I learned really happened outside of the hours in which we did research back at someone’s apartment asking, “Hey what does this mean again?” or “Can you explain this to me?” or “This came up today and at the time I didn’t know what it really meant,” and so I think that’s just a rewarding experience that I’m sure I’ll be able to take with me for the rest of my life.
- Working on the actual problem… there was little bit more of that give and take that [another student] spoke about. One was an applied math major I believe and the other a statistics major and I’m just a regular math major. So I guess I kind of contributed helping put stuff in “math speak”, and I guess that they contributed in other ways: ‘cause we were looking for patterns so one of the people had taken a number theory course and so he helped me and the other person in other areas.
- Attending the REU really kind of opened my eyes into the magic that can happen when you have a whole group of people who are interested in the same topic… I saw that change happen this past semester coming back from the REU, just my ability and my willingness to kind of invite that collaboration and to search for it in a greater way than I ever had before
- There are seven other people on this planet that have seen me at my most frustrated moment and at my happiest moment. You know you come to care about these people a lot… it’s just such a rewarding thing to know there are other people out there who have had the same passions as you and who will always kind of be that support team for you and who are going through the same, the same sort of I guess struggles that happen when you’re an undergrad looking toward a career in mathematics

**The Nature of Mathematics**

At their REUs, students learned a lot of new mathematics, delved deeply into the process of doing mathematics, and discovered how research is different than classwork.

- I think because these programs are well-organized [by] people who are really interested in helping and working with students, it [was] a little bit easier… I think for the most part trying to understand this background and the context with… the end goal of genuine research is much different than, I don’t know, someone sitting in a class and proving theorems that have been proved, I don’t know, hundreds of years before or something.
- If you’re just used to taking math classes the problems that you have experience solving are more straightforward and you have an idea of what tools you’re supposed to be using and what you’re drawing on. But then once you get into math research it’s like, you have everything anyone’s ever done to draw on and that can feel very overwhelming but also very exciting… Progress is not linear necessarily so it might feel like you’re making no progress at all for like several weeks and then suddenly you have a breakthrough and everything comes together.
- Our group had a few people who were much more experienced and then a couple of juniors and we just hadn’t seen a lot of things. So there were some things that came up all the time and we just didn’t know what they were and so getting to finally learn that stuff was so exciting.

Many students discovered that the world of mathematics is bigger than they had realized.

- Getting introduced to the math world and what math research is, is like almost bigger than learning the math than you need for your project.
- One thing I’ll definitely remember is a trip that we took to MathFest that summer and that was my first math conference and that was such an eye opener, because the REU was actually also at my college. So I had only, you know, really experienced math through this very small community at my school. And to suddenly be opened up to this world of thousands of mathematicians and all these different areas of math research that was all super exciting and something I’ll definitely remember.
- I felt, after my REU, that a lot about math not just as a subject but also culturally as a community in a lot of ways became demystified… for example, like, reading math research or something like that or reading math papers and this kind of thing just feels like something far off on the horizon or something like this and then you just sit down and do it.
- There’s so much math out there so it’s kind of, you know the saying “you don’t know what you don’t know”… I just learned that there’s so much more out there involving math that I had no idea existed. Which is really exciting for me so I guess what will affect me the most is it introduced me [to] something that I think will motivate me for the rest of my life.

**Self-Beliefs and Agency**

A key feature of these REUs was that students gained confidence and independence, became more comfortable admitting what they don’t know, and learned new ideas.

**You go to an REU and now you’re working with people from all universities. Everyone has a sort of different knowledge base, some people are more interested in things that you’ve never heard of and so what I found was this past semester these terms coming up, “oh hey I remember talking about that at the REU and I didn’t know what it meant then but now I have this motivation to learn what it means” because those conversations kind of made you want to learn these things that these other students knew.**- You find this sense of independence and you start learning these things and it is a very rewarding experience to look back after those two months time. You know, I didn’t even know what this word meant two months ago and now I like to talk to people about it kind of thing.
- I wish I had been a little more comfortable not knowing every last bit of the topic we were working on… trusting that you’re not going to spearhead every aspect of your project. There are other people in the group who are going to understand some things better than you.
- You can’t really have an ego, it’s very obvious what you know and what you don’t know.

In particular, students had to learn when and how to ask for help.

- We had very different levels of experience [with] the area that we were working in and so I think one of the difficulties was trying not [to be] discouraged by that. Because a lot of the time it can feel like you’re not contributing equally but just because you’re not, you know, the one like coming up with something new every single day, that doesn’t mean that you don’t have a role to play. And one other thing that I sort of learned over that summer was how to be assertive and make sure I knew what was going on and be willing to ask my group members to explain something in more detail so that we could all be on the same page. And that’s something they’re always willing to do but you sometimes have to ask for it, ‘cause they’re not always going to realize that they’re moving really fast and not everyone’s following.
- The REU experience, first off, is that it’s a very humbling experience, right. You find out what you know and what you don’t know very quickly… the most progress I think is when you finally admit that to yourself … It can be an uncomfortable thing but I think once you allow yourself to admit what you know and what you don’t know it opens the door, it opens the door to learn a lot more.
- It’s ok to ask people for help and there’s always going to be someone who knows something you don’t know. And you don’t have to look at that as any sort of discouragement towards yourself. It’s just an opportunity to learn something new. So I think that’s a very valuable group dynamic.

**Back to the Classroom**

When students returned to their home institutions following the REU, their summer experience influenced their work in subsequent mathematics courses.

- It was just very motivating and I found a lot of tie-ins to the course work I did the semester after the REU.
- [The REU] improve[d] my mathematical maturity so when I went to take analysis… it was much easier to read the analysis book than it had been to read the math stats book last year.
- I think it was from the first time I ever had to read some very difficult books. The books we had to read from were like really really tough to crack and they were using kind of a language I wasn’t very comfortable in… this made it easier I think to go back into the classroom and maybe read books.
- For me it kind of made it harder to go take classes because I enjoyed doing research so much.

**Graduate School**

The REU experience motivated students to consider whether or not to go to graduate school.

- It got me more motivated to go to graduate school
- Some of us have already decided “I don’t want to do research”… I’m really glad I did that [REU] so I found that out.
- Going into my REU I was sort of thinking of it as something, like, tell me whether or not I would like math research and whether or not I should think about going into [it]… and I think now, a year and a half later, I don’t think [this is] at all how you should approach your REU. I mean if you really really like it that’s definitely a good sign (chuckles) but if there are some things that you don’t like, that doesn’t necessarily mean that research is not for you.
- There are just so many factors going on in REU that I think even if it doesn’t go exactly how you would of wanted it to, you should still think about giving research another chance.

**Overall Observations**

Students agreed that one should not judge an REU by the first few days, and that the total REU experience was worthwhile and rewarding.

- One thing I wish I had known is that just as much as the details of your project, one thing that can determine like how the REU goes is your advisor’s advising style.
**Your experience the first two days is not representative of the whole experience.**- I don’t think you’re going to get the chance to work with professors who are so patient, so knowledgeable and so interested in their fields, I mean the amount of patience that it takes on the part of REU by the advisors is just entirely astounding to me. (laughing) It’s, to get someone from not knowing the definition of the basic object that you’re working on and getting them to produce a result I just I don’t even know what that takes.
- You know what everyone is facing and so it’s a really, it’s just a rewarding experience and I think that persevering past those initial discomforts is key to success.
- The graph of the REU experience is like the graph of the sine function: you’re going to have highs and lows and I promise you those positive parts are going to stick with you at the end.

**Interview Participant Bios **

David Burton completed his undergraduate degree at East Tennessee State University during the fall of 2016 and has just started as a graduate student there. He participated in an REU at the University of Connecticut Health Center for Quantitative Medicine during the summer of 2016, where he worked on an ongoing project entitled “Functional data analysis of copy number alterations in bladder cancer tumor chromosomes.” Next year, he hopes to enter a Ph.D. program in statistics.

Kelly Emmrich is a junior at the University of Wisconsin, La Crosse. She participated in the REU at California State University, Fresno, during the summer of 2016, where she worked on a project in complex analysis called “Sufficient conditions for a linear operator on R[x] to be monotone.” After graduation, she plans to enter a Ph.D. program in mathematics.

Micah Henson is a senior at Spelman College. She participated in the Mathematical Sciences Research Institute Undergraduate Program during the summer of 2016, where she worked on a problem in algebra called “The sandpile group of thick cycle graphs.” After graduation, she plans to pursue a Ph.D. in mathematics.

Andres Mejia is a junior at Bard College. He participated in the WADE Into Research REU at Wake Forest University during the summer of 2016, where he worked on a project in number theory called “Classically integral quadratic forms excepting at most two values.” After graduation, he plans to enter a graduate program in mathematics.

Nina Pande is a senior at Williams College. She participated in the SMALL REU at Williams College during the summer of 2015 and an REU at the University of Michigan during the summer of 2016. At Williams she worked on a problem in commutative algebra called “Controlling the dimensions of formal fibers,” and at Michigan she worked on another problem in commutative algebra called “First Koszul homology over a local Artinian ring.” After graduation she will teach high school mathematics in North Carolina before entering a graduate program in mathematics.

]]>Two of my favorite pieces of mathematical writing are recent essays: Francis Su’s January 2017 MAA Retiring Presidential Address “Mathematics for Human Flourishing”, and Federico Ardila-Mantilla’s November 2016 *AMS Notices* article “Todos Cuentan: Cultivating Diversity in Combinatorics”. ** If you have not yet read these, stop everything you are doing and give them your undivided attention.** In response to the question “Why do mathematics?”, Su argues that mathematics helps people flourish through engagement with five human desires that should influence our teaching: play, beauty, truth, justice, and love. In a similar spirit, Ardila-Mantilla lists the following four axioms upon which his educational work is built:

Axiom 1.Mathematical talent is distributed equally among different groups, irrespective of geographic, demographic, and economic boundaries.

Axiom 2.Everyone can have joyful, meaningful, and empowering mathematical experiences.

Axiom 3.Mathematics is a powerful, malleable tool that can be shaped and used differently by various communities to serve their needs.

Axiom 4.Every student deserves to be treated with dignity and respect.

These essays are two of my favorites because they provide visions of teaching and learning mathematics that are rich with humanity and culture, visions that welcome and invite everyone to join our community.

The ideals and aspirations offered by Su and Ardila-Mantilla are inspiring, emotional, and profound, yet also fragile — for many mathematicians, it can be difficult to balance these with the sometimes harsh reality of our classes. An unfortunate fact is that for many of us, a significant part of teaching mathematics consists of the struggle to support students who are uninterested, frustrated, inattentive, or completely absent. We are regularly faced with the reality that large percentages of our students fail or withdraw from our courses, despite our best efforts, and often despite genuine effort on the part of our students as well. How does a concerned, thoughtful teacher navigate this conflict between the truth of the tremendous potential for our mathematical community and the truth of our honest struggle, our reality?

In my practice of teaching, I have found that the only way to resolve this conflict is to simultaneously accept both truths. This has not been, and still is not, an easy resolution to manage. In this essay, I want to share and discuss some of the mantras that I have found most helpful in my reflections on these truths.

**1. Excellence is possible, perfection is not**

While perfection is impossible, excellence as a teacher is achievable, though elusive. The most difficult part of teaching for me is that I deeply want *all* of my students to succeed. In reality, this rarely happens, for many different reasons. Nevertheless, it is possible to reach excellence in teaching and learning, even if that doesn’t translate into an idealized outcome for every student in every course. By holding ourselves to the standard of excellence rather than perfection, it also becomes easier to hold our students to more reasonable standards of excellence as well. I have found that it is easy for me to slip into a mode where I am disappointed when my students don’t reach what I feel is their full potential, but by doing so I can also miss the opportunity to recognize the successes that they do achieve. To seek excellence rather than perfection, in ourselves and others, allows us to maintain our ideals while accepting the challenge of our reality.

**2. All human systems have flaws**

Colleges and universities are complex institutions, many of which serve diverse communities of students and employ faculty in a broad range of positions. Like many of my friends and colleagues, I have at times become focused on specific institutional flaws that are impossible to effectively address, often at the expense of turning my energy toward reaching more tractable goals. It is common to hear people say “pick your battles,” but at an institutional level I prefer to phrase this as choosing to engage with certain challenges and to yield to other challenges. There are many times when yielding to a challenge can provide significantly more freedom than fighting “the good fight.”

Consider Richard Tapia, a mathematician at Rice University, who received the National Science Board’s 2014 Vannevar Bush Award for “his extraordinary leadership, inspiration, and advocacy to increase opportunities for underrepresented minorities in science; distinguished public service leadership in science and engineering; and exceptional contributions to mathematics in the area of computational optimization.” In a video produced by the NSF, Tapia states:

When I started, I was so naive I thought I could change my colleagues, OK. You don’t change colleagues. You get them to maybe tolerate things you’re doing. You know, “Richard Tapia does ‘this’.” But where I see things changing and things going on is through my students. Without even directly telling them “you have to do ‘this’ and ‘this’,” they see it by example. And so I am really satisfied when I see how many students of mine are doing exactly what I was doing.

What Tapia describes is not a direct confrontation with the cultural norms and reward structures that influence his colleagues, but rather a yielding to these forces and a redirection of his energy in more effective directions. While there are certainly times when we must directly challenge flawed systems, we must also recognize that for many institutional problems, we create a higher impact by yielding to them in the short term and making progress through a different approach.

**3. Maximize student learning within a set of constraints**

In addition to large-scale institutional flaws, there are many additional constraints on our teaching. It is important to remember that our goal is not to have perfect learning from every student, but rather to maximize student learning given these constraints. The way we do this will vary dramatically given our situations, but there is a core principle that we can and should always rely on: *focus on the experiences of our students*. Here is an example of what I mean.

At the University of Kentucky, our first-year courses in Calculus (for students majoring in engineering, mathematics, and the physical sciences) are taught in large lectures of ~150-180 students with ~32-student recitation sections led by graduate teaching assistants. The first three times I taught these courses, the outcome was mediocre at best. My original strategy with large lectures was to import the best methods I had developed for small-scale teaching into the large lectures, but they were not effective. The constraints for teaching large lectures are completely different from those for my small courses, and the solutions I had used to maximize student learning in my small courses were not optimal solutions for the large courses.

The fourth time I taught a large-lecture calculus course, I completely yielded to these constraints. I was not excited about this, and was not expecting the course to be enjoyable for anyone, myself or my students. I could not have been more wrong — this was one of the most memorable courses I have ever taught, and my students were both successful and happy with their experiences. In hindsight, I realized that yielding to hard constraints had led me to a profound change in my perspective about large lectures: my primary focus should be to identify positive aspects of the large-lecture environment *from the perspective of my students* and take advantage of these as much as possible. Previously, I had focused almost exclusively on the negative aspects of large courses *from my personal perspective as a teacher*. This caused me to overlook most of these potential positive aspects, such as the effectiveness of a well-organized teaching team, the vibrancy of student excitement in a large class, and the broader range of peer interactions students can have among a large group.

A concrete example of an in-class change I made is my method of presentation. Like most mathematicians, I prefer to use the chalkboard when I teach; with large lectures, this was never as effective as it is when I teach 20-30 students. I also never wanted to use a microphone, as a personal preference. I finally gave up on all of these teacher-focused preferences. I now use Crayola markers to write on blank white paper projected using a document camera, use desmos.com for all my graphing, and use the lapel microphone. With the microphone, the students hear me clearly and the class is more relaxed since I am not straining my voice. By using markers and desmos, students can see better, I can scan my actual in-class notes and post them after class, and the lectures are literally colorful. I had dramatically underestimated the impact of these simple changes — my student evaluations are now consistently full of positive comments about how my use of colorful markers and dynamic graphics are uplifting in a drab room and help students pay attention.

Do these things make every student learn perfectly? Of course not. However, by thinking more purposefully about working within constraints to maximize student learning, leaving some of my own personal preferences aside, I have developed an approach to teaching large lectures that is more successful, and which my students and I feel reasonably positive about.

**4. Students can have meaningful mathematical experiences without us**

In my early teaching, I had bought into a false idea that student contributions were most meaningful when I could provide feedback about them. This was one reason for my preference for whole-class inquiry-based learning courses, and my distaste for large lectures. As with many other things in life, sometimes less is more in this regard. In my courses for first-year graduate students and in my large-lecture calculus courses, students are engaged and report positive experiences when I give 7-10 minute lectures followed by a 2-3 minute pause where students can discuss any points of confusion with their neighbors. The most effective prompt that I have found is to tell the students to turn to their neighbor and ask “do you have a question, yes or no? If no, why does this make sense?” It actually does not seem to matter whether or not I hear these conversations, what matters is that the students are talking about mathematics, struggling with the ideas, and are regularly engaged in conversation about what we are doing.

Similarly, in my small courses, I am less concerned than I used to be with having every small group report on their work, or check with me. This does increase the risk that students might have a misconception that is not immediately addressed, but it gives students more agency and authority in their own learning. It also recognizes the reality that students can achieve excellence in their learning without being perfect, and have meaningful experiences in mathematics without me being intellectually present at every moment.

At a deeper level, when we recognize that students can have meaningful mathematical experiences without us, we allow ourselves to embrace our most important task, to guide and inspire students, rather than to seek a false sense of control over their learning. Our most fundamental role as teachers is not to transmit truths to our students, but to create and sustain supportive environments in which students deeply learn, to create opportunities for students to engage with mathematics at a fundamental and profound level. We balance the tension between the aspirations and ideals that Su and Ardila-Mantilla offer and the reality of teaching by honoring this fundamental role we play, while simultaneously allowing students the choice of whether or not to take advantage of the opportunities they have. This leads to my final mantra.

**5. Do not be afraid of honest failure**

This has been the most important mantra for me. Like most people, I want to reach my goals. I want my students to succeed in my courses. However, the dichotomy of “success versus failure” is not sufficient when we set challenging goals, and goals in the context of mathematics are almost always challenging! Instead, we should strive to succeed or fail honestly. It is debilitating to have dishonest failure, where we fail because we choose not to put in our best work, where we fail because we do not risk anything. It is also a waste to have dishonest success, through cheating or gaming the system. If we succeed in our teaching, if our students succeed in their learning, these successes are most meaningful when they are honestly earned. If we fail in our teaching, or if our students fail in a given course, that is still a meaningful experience as long as the failure is honest.

I have been fortunate that I have not yet encountered epic failures in my mathematical life. However, this is not true of my life overall; whether in mathematics or something else, each of us has stories to tell of when things went awry despite our best efforts. If our students are doing what we hope they will, are pursuing challenging goals, chasing after dreams, learning beyond what they thought they were capable of, there will be honest failure along the way. We must honor those failures, and value them, and make sure our students know this. If we as teachers are striving to realize our aspirations and ideals, we will have honest failures as well.

**A final thought**

These mantras and my approach to teaching have been influenced by the concept of strengths-based practice in social work, by my interest in mindfulness practice, and by my readings about history and politics. At a fundamental level, all of these are about the challenge of resolving the tension between ideals and reality. I am far from unique in having had a significant influence on my teaching come from non-mathematical sources. For example, in his essay mentioned at the beginning of this article, Francis Su describes how his teaching has been informed by the ancient Greek idea of *eudaimonia*, and Federico Ardila-Mantilla’s essay describes how his work with students has been informed by research in social science and psychology. While it is worthwhile and meaningful for us to look inward and see how the strengths of mathematics and our community can be used in the practical pursuit of our ideals, we should remain open to inspiration from all aspects of our lives.

**Acknowledgements**

Thanks to my colleague Serge Ochanine for his insightful comments about Francis Su’s article, which inspired me to put these thoughts into coherent form. Thanks to my father James Braun for introducing me to strengths-based practice in social work. Thanks to the other editors of this blog for many thought-provoking conversations and their helpful comments on a previous version of this essay.

]]>Due to connections with mathematics education, some of our readers might be interested in the following statement issued by the members of the Board of Trustees of the American Mathematical Society regarding the Executive Order on Immigration issued by the President of the United States. To read the statement, see this link: http://www.ams.org/news?news_id=3305

]]>Some of our readers might be interested to know that the February 2017 *Notices of the American Mathematical Society* contains an article on active learning that is based on the six-part series on active learning published on this blog in Fall 2015. See the Notices article here:

“What Does Active Learning Mean for Mathematicians?” Benjamin Braun, Priscilla Bremser, Art M. Duval, Elise Lockwood, Diana White. *Notices of the American Mathematical Society*, Vol 64, Number 2, February 2017.

What is the first meaningful mathematics problem you remember solving? For me, it was the nine dots, four lines puzzle. When my fourth grade teacher assigned it as an extra credit problem, I spent several days of recess scribbling out attempted solutions in the sandpit, erasing, and trying again until, at last, I found a solution!

I believe this geometric puzzle still sticks out in my memory nearly three decades later because it was one of the first experiences I had with trying to answer a question that didn’t simply involve mimicking previous work. For practitioners, informed trial-and-error is a key step in doing mathematics, so the idea of “thinking out of the box’’ (or in my case, literally thinking in the sandbox…) to build intuition seems natural. However, this is a far stretch from the view of many students who see mathematics as an opportunity to memorize formulas and execute repetitive tasks.

Where do students learn the process of refining mathematical conjectures? Certainly, teaching (via) inquiry in the mathematics classroom has generated much discussion, but often the conversation about inquiry is attached to particular material in the curriculum, with an inquiry-based approach to calculus or statistics, for example. Despite being fundamental to doing mathematics, the majority of the time the inquiry process is a means to an end, rather than a focus of an entire class, and it’s rarely addressed directly. In this environment, some students internalize the inquiry process by indirect exposure. Others finish their education without a true sense of how mathematics is actually developed.

Experimental mathematics courses are one answer to the need to celebrate and study inquiry for the sake of inquiry. In particular, an experimental mathematics course is not a course about a particular set of material; it is a course about a particular approach to doing mathematics.

Courses in experimental mathematics have been offered by at least 7 different colleges and universities [1]. Outside of those who have taught or taken these courses, there is not widespread understanding of what “experimental mathematics” means in the undergraduate curriculum. My goal in this post is to give a better idea of what such a course looks like.

Comparing the syllabi of various experimental mathematics courses quickly shows the material isn’t standardized (nor does it need to be). However, these courses have some common themes.

First, students gain experience with programming and/or computer algebra systems throughout the course. While computer work is common to most experimental courses, it is not a necessary feature. Experimental mathematics could happen even without a computer in the room. (My solution to the nine dots, four lines problem occurred in a sandbox, rather than in a computer lab, and it still used experimental techniques to hone in on a solution.) However, in many cases, especially involving functions, number theory, combinatorics, and more, the use of a computer accomplishes the same thing one might do by hand in considerably less time. The computer can generate data and help sift through the results, quickly locating an example or counterexample. The goal, then, of using computers in an experimental math class is not just for the sake of using computers. Machine computation is a tool to greatly expand students’ reach as they explore.

Further, students build intuition via experimentation and conjecture, and, most importantly, students produce projects where they develop their own solutions to open-ended mathematics questions.

It’s tempting to say “my students experiment when I introduce a new concept with a group activity.” But the key word in that sentence is “introduce”. In the traditional syllabus, the focus is on material. Students can learn the material in a variety of ways, but a calculus syllabus is generally less focused on *how* students learn and more on the fact that they should learn about limits, derivatives, integrals, and their applications. Even if it is introduced in an interactive inquiry-based fashion, the star is the content. When preparing for course assessments, students don’t study strategies for building intuition; they study the theorems and computations that class activities led them toward.

For example, in a calculus class, students learn the limit definition of the derivative. However, once they learn the conceptual idea that derivatives compute slope or rate of change, they look for speedups. We compute the derivatives of \(x^0\), \(x^1\), \(x^2\), and \(x^3\) using the limit definition and look for a pattern. State the pattern and practice computing with other functions like \(x^7\), \(x^{42}\), \(x^{1/2}\), or \(x^{-5}\). This approach is computational. In fact, it can be done in a way that builds intuition and uses active learning pedagogy. But this is still a content-centered lesson. The takeaway: \(d/dx(x^n)=nx^{n-1}\).

In experimental mathematics, the star is the *approach* to new information. The content could be different each time the course is taught, but the method of figuring out new information involves a sequence of experiment, conjecture, and repeat.

A “typical” experimental math class meeting might look something like this:

1. The instructor presents a new mathematics problem, leading class discussion long enough to make the problem statement clear.

For example, one topic that lends itself well to exploration with experimental methodology is continued fractions. An entire class can be built on the idea that any real number \(r\) can be written as \(r=a_0+1/(a_1+1/(a_2+1/\ldots))\), where the number of integers \(a_i\) required to write \(r\) could be finite or infinite. The instructor presents the definition of continued fraction and computes the continued fraction form of a few well-chosen real numbers by hand. Then students are asked to look for (families of) real numbers whose continued fraction expansions have predictable structure.

2. Students brainstorm in small groups to determine what data might help them better solve the problem and then gather the data, often with computer assistance.

In this case, students will find it useful to write code that inputs a real number \(r\) and outputs the first \(n\) terms \(a_0, a_1, a_2, \dots\) in the continued fraction expansion for \(r\). They may also find it helpful to input the terms \(a_0, a_1, a_2, \dots\) in a continued fraction expansion and output the simplified corresponding real number. Students use the data to conjecture a solution or patterns for special cases. At this point, the experimentation begins in earnest.

Clearly any finite continued fraction represents a rational number, but is the converse true? What patterns are there in the continued fractions for irrational numbers? Students play with expansions for\(\pi\), \(e\), and powers of \(\pi\) and \(e\); \(e^n\) has some nice structure when \(n\) is an integer that \(\pi^n\) does not. The continued fraction for \(\sqrt{n}\) (where \(n\) is a positive integer) has particularly attractive eventually-periodic structure. The instructor’s role during this exploration can vary. I circulate around the room and talk with individual students as they work. I also periodically ask students to share interesting observations with the rest of the class. Then, if one person hits on a promising idea, it can quickly be shared across the room, encouraging other students to explore related avenues of inquiry.

3. If possible, students prove their conjectures. If not, students refine or revise their conjectures by iterating steps (2)-(3). Or students try the same process for a related or generalized version of the same problem.

In this case, students may notice the eventually-periodic structure of the continued fraction for \(\sqrt{n}\) and conjecture patterns for entire families of continued fractions for square roots. This could lead students to ask a converse question: if I have a periodic continued fraction, is it necessarily the continued fraction for a square root? If so, which one? If not, what other kinds of numbers have periodic continued fractions? Students could also look at generalized continued fraction expansions, where the numerators in the fractions aren’t all 1s.

The big idea from this class meeting is not “square roots have periodic continued fractions”. The takeaway is: you made a conjecture and confirmed or refined it; what’s the next natural conjecture? The instructions given at the beginning of class have some specific mathematical questions to get started, but they also involve reflection. For example, the final part of students’ written work from class could be to write a paragraph response to: “if you were to continue this problem, what question would you investigate next and why?” The particular material is less important than the process of asking and revising questions.

Classrooms require structure. Often, that course structure is dictated by a list of content and computational skills that are required in subsequent courses. Many days of experimental mathematics courses also require structure so that student learning can be assessed. In both the calculus example and the continued fraction example above, one could argue that structure exists because the instructor has a clear end result in mind, whether students take different routes or a prescribed route to get there. However, it is also possible to provide structure without having a pre-determined final content goal in mind.

The beauty of experimental mathematics is that it gives students the tools to conduct open-ended inquiry, which is often described in terms of a “project”. These could consist of several weekly projects or could take the form of a single long-term project. Projects involve questions where a student can’t just conduct a literature search and find answers to all the questions they generate. On the other hand, it’s ok if their work isn’t all new to the literature, but has some overlap of re-discovered results. In my experimental mathematics class, I have students complete one large, semester-long project. I make sure that no two students have the same project area on any given iteration of the course. There is a project deadline approximately once per month during the four-month semester.

- Month one: Pick a project topic. Each student selects a broad area they want to investigate, but not necessarily a specific research question yet.
- Month two: Each student presents a question or two they’ve independently decided to explore and gets feedback from their classmates.
- Month three: Each student submits a preliminary written report on their progress to me for feedback.
- Month four: Each student gives a 15-minute conference-style presentation on the results of their project to the rest of the class.

At the beginning of each semester, I provide a list of suggested project areas, including some resources for finding tractable open problems. Students may choose from the list, propose a twist on a suggested project, or propose their own problem. The delightful thing about these projects is: in the four iterations of the course I’ve taught at Valparaiso (representing over 40 different student projects), some general topics have been selected more than once, but each time, the students went in completely different directions with their experimentation. For example, several students have chosen to study cellular automata, but one student may look at variations of rules that generate the automata and study entire families of automata, while another student may become very interested in a particular automaton and study iterations of that one automaton over time. Either way, by the end of the course, each student has true ownership of their project and the direction it took over the duration of the semester. While they’ve had me and their classmates as a sounding board, the final result of the project was never prescribed to them. It is the result of doing what we do each day in class: program, experiment, conjecture, and repeat, run over the course of an entire semester to see what happens.

As long as we only discuss inquiry in the context of standard course material, we’re missing half of mathematics! The true joy of doing research is trying a problem that has never been solved before and then experimenting and refining conjectures until you hone in on something that works. Experimental mathematics lets students experience the process of how mathematics is actually discovered. Far beyond generating examples together or following inquiry-based activities to arrive at an expected theorem, experimental mathematics pushes discussion and inquiry past the standard classroom boundaries and owns up to the process of making your own conjectures without an intended final answer. Ultimately, students will be richer when we encourage them to create and answer their own questions instead of only leading them through our own.

[1] Links to experimental math courses:

Dartmouth College (https://math.dartmouth.edu/archive/m56s13/public_html/)

Grinnell College (http://www.math.grin.edu/~chamberl/courses/444/syllabus.html)

Ithaca College (http://www.tandfonline.com/doi/abs/10.1080/10511970.2013.870264)

Lynchburg College (http://lasi.lynchburg.edu/peterson_km/public/Courses/Fall%202016/Math350_f16.htm)

Rutgers University (http://www.math.rutgers.edu/~zeilberg/math611.html)

Tulane University (http://129.81.170.14/~vhm/syllabus.pdf)

Valparaiso University (http://www.tandfonline.com/doi/abs/10.1080/10511970.2016.1143899?journalCode=upri20

Recent calls to bring Computer Science to K-12 schools have reached a fever pitch. Groups like Code.org and Girls Who Code have become household names, having raised tens of millions in funding from Silicon Valley luminaries and small donors alike. In February of 2016, President Obama announced the “CSforAll” initiative, and asked for $4 billion of funding from Congress to pay for it. Even in today’s divided climate, this initiative found bipartisan support, and mayors and governors from coast to coast have made sweeping commitments to bring CS Education to all students.

This effort has serious consequences for math education. Adding a new subject is easier said than done: recruiting, training, hiring and retaining tens of thousands of new CS Teachers will take decades and cost billions, and the finite number of hours in the school day and rooms in the school building make it difficult to find space for these courses. To meet these commitments, many schools and districts have employed three strategies: (1) take time out of existing math classes for CS, (2) take math classes out of a teacher’s schedule, and instead have them teach a CS class, and/or (3) have CS classes count as a math credit [1]. All of this is done because there’s a widespread misconception that “computer science is just like math”, and that skills from one will transfer to the other. Unfortunately, most of the programming languages being taught in these classes have little to do with mathematics, and embrace concepts that are explicitly *math-hostile.* In this article, I will discuss some of the challenges and opportunities faced by K-12 mathematics educators in our efforts to develop an authentic incorporation of CS into the K-12 curriculum.

**Some Challenges**

A core example of a challenge facing math teachers is that numbers themselves behave differently in most programming languages. Math places no limit on how small or large a number can be, yet programming languages frequently truncate values without warning, leading to unpredictable results. Any 5th grader should know that 2 ÷ 4 equals ½… but in Java the teacher will have to explain why the same expression evaluates to zero!

Making matters worse, programming languages like Java, JavaScript, Python, Scratch and Alice all rely on the concept of *assignment*. Assignment means that a value is “stored in a box”, and that the value in that box can be changed. Here’s a simple JavaScript program that demonstrates this:

x = 10

x = x + 2

The first line of code *assigns* the value 10 into a box named “x”. The second line reads the value back out, adds 2, and assigns the new value back into x. When the program finishes, x contains the value 12. Unfortunately, the semantics and syntax are completely incompatible with mathematics! In math, names are given to *values*, not boxes. In fact, there’s no notion of “boxes” in algebra (or “assigning” values into them) at all! Moreover, the written syntax of “x= x + 2” translates to a statement that is mathematically wrong. Adding insult to injury, computer scientists refer to x as a *variable*, despite the fact that it behaves nothing like a variable in math. The problem is made even worse when it comes to *functions*. In most programming languages, functions can (and often do) fail the vertical line test, producing different values for the same input or perhaps *no *value for *any* input. Students typically struggle with the concepts of function and variable when they get to algebra. Now, they are confronted with incompatible definitions of the same terms – in a class taught by a math teacher, for math credit.

It should come as no surprise that there is little evidence supporting the proposition that programming leads to higher performance in critical classes like algebra. Asking math teachers to cut back on math to make room for programming is problematic in and of itself. When numbers, variables and functions behave contradictory ways, all in the context of a “math-credit class”, the problems are far greater.

**Some Opportunities**

While the risks of bad integration are significant, the opportunities for an *authentic* integration are tremendous. I would argue that an “authentic integration” between math and programming has three characteristics:

*Tools*– The language itself must include (and enforce) basic mathematical concepts like Numbers, Variables, and Functions. At the very least, we need to get our tools right (within reasonable limits).*Curriculum*– The curriculum offered alongside the tools must be aligned to national and/or state standards for*mathematics*, with a clear scope and sequence that addresses the needs of a mainstream math teacher. It should include homework assignments, rubrics, assessments, and handouts that address mathematical concepts. Demanding that a math teacher find the time to figure out the alignment and make these resources on their own is a non-starter.*Pedagogy*– There is more to great teaching than having a great curriculum. A CS course that aims to address math content must also address pedagogical techniques that matter in a math class. How is an activity differentiated? How is a concept scaffolded? How should student break down a word problem broken down? The answers to all of these questions (and more) must be explicit, and must also fit within recognized best-practices for math instruction.

I firmly believe there are ways to do it right, and there’s tremendous potential for teachers who are able to do so. Authentic alignment of mathematics and computer science requires significant time to develop materials and integrate them with existing math curricula, and significant intersectional experience between computer science, mathematics, math instruction, curriculum development, software engineering, and teacher professional development. And while there are almost certainly multiple pathways to get here, I can speak from experience about one of them.

I’m a former math teacher, math-ed researcher, and the co-director of an organization that has spent nearly a decade researching this challenge and developing evidence-based solutions. Bootstrap (http://www.BootstrapWorld.org) is a research project at Brown University that offers a series of curricular modules built around *purely mathematical programming*. Our introductory module is carefully aligned to standard algebra, and after a decade of research has been shown to significantly improve students’ performance on standard, pencil-and-paper algebra tasks (http://www.BootstrapWorld.org/impact). The win for students is twofold: they’re learning real algebra, and they’re doing it in a way that is 100% hands-on and applied. Bootstrap gives math teachers a chance to teach algebra in a new way, and to makes their experience teaching math an asset rather than a liability when it comes to teaching programming. By leveraging the experience math teachers already have, Bootstrap makes it possible for math teachers to deliver rigorous programming education without years of re-training. And since every student takes algebra, it allows schools to bring computer science to *every child *without having to find room in the budget for a new teacher or room in the schedule for a new class.

**Conclusion**

Computer Science is coming, most likely in a form that finds its way into math classes across the country. As members of the math-ed community, we have a responsibility to make sure this integration happens authentically, and in a way that supports math instruction instead of undermining it. Doing this takes careful attention to the tools we use, the curricula we teach, and the pedagogical techniques we employ. If we withdraw from this conversation, it will happen without us – and recent history shows that it is likely to happen in a way that risks harming math education. If we are active participants in the conversation, the enthusiasm and energy surrounding CS education bring enormous potential to math classrooms everywhere.

[1] – Kentucky counts CS as a math credit, Georgia counts CS as a math credit, Pennsylvania counts CS as a math credit

]]>During my freshman year of high school, my geometry teacher came into class one day and challenged us to trisect an angle with a compass and a straight edge. Anyone who was successful would receive an A in the class for the rest of the year. We wouldn’t have to do any more homework or take any more tests. Nothing. Of course, this should have seemed too good to be true. But I was in ninth grade and didn’t know any better, so I set off to solve this seemingly innocent problem.

I came up with a dozen or so false proofs, all of which included reasoning like “well, now you just move the compass a bit over here and then you draw this line, and it works!” Of course it didn’t work, but this is the kind of non-proof you would attempt to make if you had only just learned what a proof is.

But rather than simply tell me I was wrong and insist that I was doomed to failure, my teacher let me share the ideas behind every failed proof so that I could see the shortcomings in my arguments. He sat with me and we talked more broadly about what does and does not constitute a proof. He knew I was going to be wrong. He knew this was an impossible assignment. But he still listened.

My teacher’s openness to hearing my ideas inspired me to keep working and to keep trying new approaches. As I learned more math, I kept coming back to this problem. I tried using trigonometry. I tried using calculus. I tried making up a unit distance that I would call “1.” After watching *Good Will Hunting*, I decided that it would probably help if I drew all of my diagrams on mirrors. None of these things helped. Along the way, I learned about quantifiers. I learned about proofs. I learned to identify the errors in my attempted proofs on my own. Ultimately, I think I shed a tear of joy when I finally saw the proof of impossibility in my graduate algebra class.

This story can lead to a lot of different discussions. Ben Braun wrote a beautiful article for this blog about the value of having students work on difficult and unsolved problems, which I highly recommend. Instead, I’d like to explore the value of talking about mathematical ideas informally, especially when they are ill-formed and possibly incorrect; the value of encouraging our students to share such ideas with one another; and the value of participating in these discussions with our students.

The practices of active learning, inquiry based learning, project-oriented group learning, and others have become quite popular as means of addressing the fact that simply lecturing about math is not effective for students (Deslauriers 2011), (Freeman 2014), (Lew 2016). Encouraging mathematical communication is a byproduct of many of these methodologies, and I would like to start by arguing in favor of talking about math because there are added educational and cultural benefits to encouraging open discussion of mathematics both inside and outside the classroom.

*By discussing mathematics together, our students develop their own language and intuition for mathematical ideas.*As I circle around the classroom listening to my students as they work, I sometimes hear conversations that can best be described as the opposite of the game of “telephone.” One student tries to describe a solution in a way that, to me, is completely incoherent. But his groupmates kind of get the idea and someone else chimes in with a more coherent explanation. Then someone else adds more clarity. And by the end they have a solid basis upon which we can introduce more formal language and definitions. If I had simply interjected with a litany of corrections and edits after the first incoherent attempt, I would be robbing my students of the opportunity to learn by developing their own ideas. I think we would be fooling ourselves if we claimed that our private research moments and meetings with collaborators did not follow this similar “telephone-in-reverse” phenomenon.*Mathematical conversations encourage multiple ideas, multiple perspectives, and different solutions.*It seems fair to say that most of us, as educators, want our students to appreciate that a single problem may have many possible solutions. Through traditional lecture, we may only present one such solution, leaving students who had different approaches to wonder if their solutions are correct — or worse yet, to believe that their different ideas are wrong.By giving our students time to work together and talk together, we give them the time to learn from one another by discussing different solutions and approaches to the same problem. Students are more likely to ask the question “I did the problem in such-and-such different way. Does that still work?” in a small group or private conversation than they are in front of an entire lecture section.*By discussing our students’ ideas, we can provide more personal attention to their learning.*When we talk with our students we can very quickly assess the difference between someone who can complete all but the hardest problem on a homework set and someone who is struggling with the first problem. Talking about the first problem with the former student or talking about the hardest problem with the latter student is a waste of everyone’s time. By talking individually with our students, we can concentrate on what they need to learn based on what they already understand.

Most of all, we empower our students’ learning by giving value to the questions and mathematical ideas that are at the front of their minds, rather than by taking a scatter-shot approach and hoping that we address something that is meaningful to everyone at some point during each lecture.

William Thurston wrote an article “On Proof and Progress in Mathematics” for the Bulletin of the AMS (Thurston 1994) where he says:

Mathematicians have developed habits of communication that are dysfunctional…we go through the motions of saying for the record what we think the students “ought” to learn, while the students are trying to grapple with the more fundamental issues of learning our language and guessing at our mental models.

He goes on to explore this idea further in an example. If Alice and Bob are researchers within a given subfield, Alice may be able to communicate the overall ideas behind a recent research development to Bob over coffee. But in contrast, Bob may struggle to glean similar insights from an hour-long colloquium talk or over the course of several hours of reading Alice’s paper. Thurston continues:

Why is there such a big expansion from the informal discussion to the talk to the paper? One-on-one, people use wide channels of communication that go far beyond formal mathematical language. They use gestures, they draw pictures and diagrams, they make sound effects and use body language. Communication is more likely to be two-way, so that people can concentrate on what needs the most attention.

In contrast, research talks and written papers require far more formalism, and they prevent the audience from interacting with the material in such a personal and intuitive way.

As professional mathematicians we have all experienced this. We have all sat through talks without understanding anything after the first five minutes. We have all read the same sentence in a paper 20 times without understanding its meaning. And we have all asked a question over coffee to find illumination in a well-phrased answer from a colleague, collaborator, or friend. So if this is the case when we, the so-called experts, are trying to learn new material, how then can it *not* be the case for our students as they are trying to learn mathematics?

In theory this argument may resonate with a lot of people, but implementing these ideas may seem difficult for any number of reasons. Here are a few concrete tips that can be implemented anywhere:

- Set aside 5 minutes of each class for your students to work on an example problem. This example can be as simple as “What is the derivative of \((3x+1)^2\)?” Have them compare their answer with their neighbors and give each other a high five if they agree. If you have more time, set aside more time and give the students more problems. An example the students work out together is far more valuable than another example you do on the board.
- Encourage students to attend your office hours, your TA’s office hours, and a campus math help center. Remind them about these resources every day. Be open and approachable. Your students are human beings, and many of them are interested in doing cool things. If you engage with them on a personal level, they will feel more comfortable in asking you math questions.
- Share your mathematical struggles with your students. One reason that many of us have been successful as mathematicians is that we are willing to keep working on a problem that seems impossible at first. But in our students’ eyes, we can seem to be omniscient solutions manuals who know how to solve every math problem. We need to strive to bridge this divide.
- Solicit student input in helping you present solutions to problems. Ask them to articulate why they did certain things, and develop diplomatic reactions to incorrect ideas. Rachel Levy posts some great suggestions for accomplishing this here.

And Mr. Pelzer, if you’re out there reading this — thanks for letting me share my ideas with you. Failing to trisect an angle sparked a lifetime of mathematical curiosity.

Deslauriers, L. et. al. “Improved Learning in Large-Enrollment Physics Class.” *Science* 332 (2011): 862-864.

Freeman, S., et. al. “Active Learning Increases Student Performance in Science, Engineering and Mathematics.” *Proceedings of the National Academies of Sciences* 111, no. 23 (2014): 8410-8415.

Lew, K., et. al. “Lectures in advanced mathematics: Why students might not understand what the mathematics professor is trying to convey.” *Journal for Research in Mathematics Education* 47, no. 2 (2016): 162-198.

Thurston, W. “On proof and progress in mathematics.” *American Mathematical Society. Bulletin. New Series* 30, no. 2 (1994): 161-177.

What would you do if you discovered a popular approach to teaching inverse functions negatively affected student understanding of the underlying ideas? Would you continue to teach the problematic procedure or would you search for a better way to help students make sense of the mathematics?

A popular approach to finding the inverse of a function is to switch the \(x\) and \( y\) variables and solve for the \(y\) variable. The strategy of swapping variables is not grounded in mathematical operations and, we will argue, is nonsensical. Nevertheless, the procedure is so ingrained in textbooks and other curricula that many teachers accept it as mathematical truth without questioning is conceptual validity. As a result, students try to memorize the strategy but struggle to “accurately carry out mathematical procedures, understand why those procedures work, and know how they might be used and their results interpreted” (NCTM, 2009; Carlson & Oehrtman, 2005). As we will illustrate, this common process for finding the inverse of a function makes it *harder* for students to understand fundamental inverse function concepts.

**Foundational Ideas about Functions and Their Inverses**

A function \(f\) describes the relationship between two covarying quantities represented by variables \(x\) and \(y\). Without loss of generality, let \(x\) be the independent variable for the function \(f\) and \(y\) be the dependent variable for the function \(f\). The inverse function \(f^{-1}\) also describes the relationship between the quantities represented by variables \(x\) and \(y\) except \(y\) is designated as the independent variable for the function \(f^{-1}\) and \(x\) is designated as the dependent variable for the function \(f^{-1}\).

The following properties hold:

**Concept #1: **The domain of a function \(f\) is the range of its inverse function \(f^{-1}\) and the range of the function \(f\) is the domain of its inverse function \(f^{-1}\) (Wilson, 2007).

**Concept #2:** \(f^{-1}(f(x))=x\). In layman’s terms, the inverse function *undoes* whatever the function does (Bayazit & Gray, 2004).

These two concepts form the foundational ideas of the inverse function concept and hold true for functions represented in equations, graphs, tables or words.

**Problematic Conceptions Arising from the Switch x and y Approach to Finding Inverse Functions**

We define a conception as “problematic” if it describes an understanding that obscures connections to related ideas, introduces mathematical inconsistencies, and/or is likely to hinder students from developing powerful meanings of future topics. There are two problematic conceptions that emerge from the

**Problematic Conception #1:** The inverse of \(y=f(x)\) is \(y=f^{-1}(x)\).

In this statement, the independent variable for both \(f\) and \(f^{-1}\) is \(x\) and the dependent variable for both functions is \(y\). This problematic conception develops out of the procedure of switching \(x\) and \(y\) to find the inverse of a function, as illustrated in the following example.

Given \(f(x)=86x+15\), find \(f^{-1}\). \[\begin{align*} f(x) &= 86x +15\\ y &= 86x +15\quad \text{since}\ y= f(x)\\ x&=86y+15\quad \textbf{switch x and y}\\ x-15 &=86y\\ y &= \frac{x-15}{86}\\ f^{-1}(x) &= \frac{x-15}{86} \end{align*}\]

To some educators, calling this statement a *problematic conception* may seem like heresy. However, it is easy see the conceptual problem when the variables are assigned real world meanings.

In 2016 – 2017, tuition at the Maricopa Community Colleges was $\(86\) per credit hour. All students registering to take classes were also required to pay a $\(15\) registration fee. The function \(y=f(x)\) where \(f(x)=86x+15\) (introduced earlier) relates the number of credit hours, \(x\), to the total tuition cost (including the registration fee), \(y\). For clarity and emphasis, we change the variables in this equation to \(c\), for the number of credit hours assigned, and to \(t\), for the total tuition cost in dollars. The resultant equation is \(t=f(c)\) where \(f(c)=86c+15\). No matter what we do to mathematically manipulate this equation, the meaning of the variables \(t\) and \(c\) will remain unchanged. Suppose we are asked to calculate and interpret the meaning of \(f^{-1} (445)\). Using the *switch \(x\) and \(y\)* approach, we concluded earlier that \(y=f^{-1} (x)\) where \(f^{-1}(x)=\frac{x-15}{86}\). In terms of \(c\) and \(t\) this is \(t=f^{-1} (c)\) where \( f^{-1} (c)=\frac{c-15}{86}\). So \[\begin{align*}f^{-1}(445) &= \frac{445-15}{86}\\ f^{-1}(445) &= \frac{430}{86}\\ f^{-1}(445) &= 5 \end{align*}\]

What is the meaning of the result? Since \(c\) is credits and \(t\) is tuition cost in dollars, the result must mean that \(445\) credits cost $\(5\). This statement is false because credits cost $\(86\) per credit hour! To make sense of \(t=f^{-1}(c)\), we would have to change the meaning of the variables \(c\) and \(t\).

The confusion is easily remedied by applying an alternate strategy to finding the inverse. The strategy of *solve for the dependent variable* is demonstrated in the following example. As stated earlier, \(t\) represents the total tuition cost in dollars and \(c\) represents the number of credit hours assigned. For the inverse function \(f^{-1}\), \(c\) is the dependent variable so we solve the equation for \(c\).

Given \(f(c)=86c+15\), find \(f^{-1}\).

\[\begin{align*} f(c) &= 86c+15\\ t &= 86c+15\quad \text{since}\ t=f(c)\\ t-15 &= 86c\\ c &=\frac{t-15}{86}\\ f^{-1} (t) &= \frac{t-15}{86} \end{align*}\]

Note that \(t\) is the independent variable and \(c\) is the dependent variable for the inverse function. \( f^{-1} (445)=5\) implies that when \(t=445\), \(c=5\). In other words, when the total tuition cost (including registration) is $445, then 5 credits are purchased. This statement is true.

By referring to basic inverse function concepts, we can also detect the fallacy in the statement, “The inverse of \(y=f(x)\) is \(y=f^{-1}(x)\).” Let \(x\) be the independent variable and \(y\) be the dependent variable of a function \(f\). Then \(y=f(x)\) . We know \[\begin{align*}f^{-1} (f(x)) &= x\quad \text{Concept 2}\\ f^{-1} (y) &= x\quad \text{since}\ y=f(x)\end{align*}\]

Notice that the independent variable for the inverse function \(f^{-1}\) is \(y\) and the dependent variable is \(x\). So the inverse of \(y=f(x)\) is \(x=f^{-1}(y)\) not \(y=f^{-1}(x)\).

The tuition example represents a traditional exercise where students focus only on a memorized procedure. Carlson and Oehrtman warn that “this procedural approach to determining ‘an answer’ has little or no real meaning for the student unless he or she also possesses an understanding as to why the procedure works (2005).” The conceptual weakness of the problematic approach to finding the inverse becomes clearly evident with functions representing real world contexts.

Keeping track of the meaning of variables is essential when working with exponential and logarithmic functions. Understanding that \(y=b^x\) is equivalent to \(\log_b y = x\) is key to understanding logarithms conceptually. The *switch \(x \) and \(y\)* approach to finding inverses obscures the inverse relationship between exponential and logarithmic functions. For example, suppose that \(f(x)=3^x\). Find \(f^{-1}\).

\[\begin{align*} \textit{Switch x}\ & \textit{and y}\ \text{approach} & \textit{Solve for the}\ & \textit{dependent variable}\ \text{approach} \\ f(x) &= 3^x & f(x) &= 3^x\\ y &= 3^x & y &= 3^x\\ x& =3^y\quad \text{switch}\ x\ \text{and}\ y & \log_3 y &= x\\ \log_3 x &= y & f^{-1}(y) &= \log_3 y \\ f^{-1}(x) &= \log_3 x & & \\ \end{align*}\]

Using the *switch \(x\) and \(y\)* approach, it is common for students to conclude incorrectly that \(\log_3 x=3^x\) because of the statements \(\log_3 x=y\) and \(y=3^x\) included as part of the problem solving process. No such confusion exists when the *solve for the dependent variable* approach is used.

**Problematic Conception #2:** With the horizontal axis representing the independent variable and the vertical axis representing the dependent variable, the graphs of \(f\) and \(f^{-1}\) may be drawn on the same axes. The resultant graphs are symmetric about the line \(y=x\).

It is true that the graphs of \(y=f(x)\) and \(y=f^{-1} (x)\) are symmetric about the line \(y=x\) but, as established earlier, there are inherent issues with saying that \(y=f^{-1} (x)\) is the inverse function of \(y=f(x)\). The result \(y=f^{-1} (x)\) comes from *switching the \(x\) and \(y\)* variables in the inverse function. In fact, switching the variables in any mathematical relation will create a graph that is symmetric about the line \(y=x\). The practice of graphing \(f(x)\) and \(f^{-1}(x)\) on the same axes should be avoided (VanDyke, 1996) because it muddles the concept of inverse. Instead \(f(x)\) and \(f^{-1}(y)\) should be graphed on separate axes labeled appropriately with \(x\) or \(y\) on the horizontal axis.

The conceptual problems which occur when graphing \(f(x)\) and \(f^{-1}(x)\) on the same axes are evident when modeling even the simplest real-world context. The weekly earnings, \(y\), of an employee earning $\(10\) per hour who works \(x\) hours in a week is given by \(y=10x\). The independent variable for the function \(f\) is \(x\) and the dependent variable is \(y\). For the inverse function \(f^{-1}\), the dependent variable is \(x\) so we solve \(y=10x\) for \(x\) and get \(x=\frac{1}{10} y\). We have \(y=f(x)\) with \(f(x)=10x\) and \(x=f^{-1} (y)\) with \(f^{-1} (y)=\frac{1}{10} y\). If we switch the \(x\) and \(y\) variables in the inverse function equation, we get \(y=f^{-1} (x)\) with \(f^{-1} (x)=\frac{1}{10} x\) . We graph \(f(x)\) and \(f^{-1} (x)\) on the same axes and label the axes with the variables \(x\) and \(y\) as is customary. We include the units associated with the variables \(x\) and \(y\).

From the graph, we see that \(f^{-1} (20)=2\). The \(x\)-axis is labeled *hours worked weekly* and the \(y\) axis is labeled *weekly earnings (dollars)* so this must mean that when the employee works \(20\) hours the employee earns $\(2\). But this doesn’t make sense because we know the employee makes $\(10\) per hour! We could remove the labels from the axes, but this does not help someone understand a function’s graph as a visual representation of a relationship between two quantities and is likely to make it even harder to comprehend the meaning of a point on the graph. Graphing \(y=f(x)\) and \(y=f^{-1} (x)\) on the same axes created confusion and did nothing to help us understand inverse functions.

There are two equally viable strategies for representing functions and their inverses graphically. The first strategy is to graph each function on its own pair of coordinate axes with the horizontal axis representing the independent variable and the vertical axis representing the dependent variable of the function.

From the graph of \(f\), we determine that \(f(2)=20\) means that working 2 hours weekly results in weekly earnings of $\(20\). From the graph of \(f^{-1}\), we determine that \(f^{-1} (20)=2\) means that when weekly earnings were $\(20\) the number of hours worked was \(2\). Both results make sense in the real-world context.

The second strategy for graphing a function and its inverse comes from changing the way we think about graphs. With this approach, we use the same graph to represent a function and its inverse but designate the horizontal axis to represent the independent variable for \(f\) and the vertical axis to represent the independent variable for \(f^{-1}\) (Moore, Liss, Silverman, Paoletti, LaForest, & Musgrave, 2013). Observe that to determine \(f(2)\) we start at \(x=2\) on the horizontal axis and move vertically until we touch the graph of \(f\). We then move horizontally until we touch the vertical axis at \(y=20\). We conclude \(f(2)=20\). To determine \(f^{-1} (20)\) we start at \(y=20\) on the vertical axis and move horizontally until we touch the graph of \(f^{-1}\). We then move vertically until we touch the horizontal axis at \(x=2\). We conclude \(f^{-1} (20)=2\).

This way of thinking can be powerful for students who recognize the equation \(f(x)=30\) is equivalent to \(x=f^{-1} (30)\). The student finds \(30\) on the vertical axis and determines the corresponding value on the horizontal axis is \(3\). The student concludes that the solution to \(f(x)=30\) is \(x=3\) because \(f^{-1} (30)=3\).

Bayazit and Gray (2004) claim that learners with a conceptual understanding of inverse functions were able to deal with the inverse function concept in situations not involving formulas whereas learners limited by a procedural understanding of inverse functions (e.g. *switch \(x\) and \(y\)*) were less likely to be successful in a context without a formula.

A side benefit of discarding the *switch \(x\) and \(y\)* approach is that it frees learners from the \(x\)-addiction – the notion that only \(x\) can be the independent variable. In graphing, the \(x\)-axis becomes the *horizontal axis* and the \(y\)-axis becomes the *vertical axis*. The reality is that disciplines outside of mathematics rarely use \(x\) to represent the horizontal axis and \(y\) to represent the vertical axis. Rather, they use variable names (perhaps even complete words) that make sense in the context of the situation. Since, as we propose, the axes are no longer tied to \(x\) and \(y\), learners think more deeply about the concepts of independent and dependent variables when graphing real world data models such as \(p=f(t)\) where \(f(t)=298,213,000(1.009)^t\) and \(\textit{height}\ = f(\textit{time})\) where \(f(\textit{time})=-8.99 \cos(\frac{\pi}{6}\cdot \textit{time})+12.74.\)

When students understand the concept of inverse function in the context of a real world situation, they engage in reasoning (the process of drawing conclusions on the basis of evidence or stated assumptions (NCTM, 2009)) and sense making (developing understanding of a situation, context, or concept by connecting it with existing knowledge (NCTM, 2009)). This connects directly with the Standards for Mathematical Practices – specifically Math Practice #1 (make sense of problems and persevere in solving them) and Math Practice #2 (reason abstractly and quantitatively) (National Governors Association, 2010). The Mathematical Association of America encourages similar ways of thinking in their Committee on the Undergraduate Program in Mathematics Curriculum Guide (MAA, 2015). Cognitive Recommendation #1 states that *Students should develop effective thinking and communication skills*. All such connections help students understand and retain new information, something that is more challenging if students are not engaged in reasoning and sense making (Hiebert et al., 1997).

**Summary**

A correct understanding of inverse functions empowers learners mathematically. By eliminating the *switch \(x\) and \(y\)* approach and implementing the *solve for the dependent variable* approach, teachers can reduce confusion and enhance student understanding. By recognizing that the inverse of \(y=f(x)\) is \(x=f^{-1}(y)\), learners can make sense of inverse functions in multiple mathematical contexts including real world data analysis and modeling.

*Adapted from an article by the same authors, listed in the references below.*

**References**

Bayazit, I. and Gray, E. (2004, July). Understanding inverse functions: the relationship between teaching practice and student learning. *Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education: Vol. 2*. (pp. 103–110).

Carlson, M. & Oehrtman, M. (2005). Key aspects of knowing and learning the concept of function. Mathematical Association of America Research Sampler, No. 9, March 2005.

Hiebert, J., Carpenter, T., Fennema, E., Fuson, K., Wearne, D., Murray, H., et al. (1997). *Making sense: Teaching and learning mathematics with understanding*. Portsmouth, NH: Heinemann.

Mathematical Association of America (2015). *2015 CUPM Curriculum Guide to Majors in the Mathematical Sciences*. Carol S. Schumacher and Martha J. Siegel, Co-Chairs, Paul Zorn Editor. Washington, DC: MAA

Moore, K. C., Liss II, D. R., Silverman, J., Paoletti, T, Laforest, K. R., and Musgrave, S. (2013). Pre-Service Teachers’ Meanings and Non-Canonical Graphs. In Martinez, M. & Castro Superfine, A. (Eds.), *Proceedings of the 35th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education* (pp. 441-448). Chicago, IL: University of Illinois at Chicago.

National Council of Teachers of Mathematics (2009). *Focus in High School Mathematics: Reasoning and Sense Making*. Reston, VA: NCTM

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

United States Census Bureau. (2006). Table 96. *Expectation of Life at Birth, 1970 to 2003, and Projections, 2005 and 2010*. (NTIS No. PB2006500023)

Van Dyke, F. (February 1996). The inverse of a function. *Mathematics Teacher*. 89, pp. 121 – 126.

Wilson, F. (2007). Finite mathematics and applied calculus. Boston: Houghton Mifflin Company.

Wilson, F.C., Adamson, S., Cox, T., and O’Bryan, A. (March 2011). Inverse functions: What our teachers didn’t tell us. *Mathematics Teacher*. 104, pp. 500-507.

World Health Organization. (2006). *World Health Statistics 2006*. WHO Press. Geneva, Switzerland.

This article is intended to serve as a rough “proof” of the statement, “There exist many resources and opportunities supported by the National Science Foundation (NSF) to improve the teaching and learning of undergraduate mathematics.” We present a curated, annotated list of projects funded by the NSF’s Division of Undergraduate Education (DUE) that readers of this blog might be interested in. Additionally, we demonstrate the remarkable diversity of projects and institutions that are funded by DUE to improve the teaching and learning of mathematics, and share professional opportunities for people who share these goals.

The NSF is an independent federal agency tasked by the United States Congress to “promote the progress of science.” With a budget of 7.5 billion dollars in fiscal year 2016, NSF received approximately 50,000 proposals and made almost 12,000 awards. NSF is organized into seven Directorates that support research in various disciplines in science, technology, engineering and mathematics (STEM) as well as in education. Each of the Directorates is further organized into Divisions. For example, the Division of Mathematical Sciences (DMS) is situated in the Directorate for Mathematical and Physical Sciences (MPS). The Directorate for Education and Human Resources (EHR) houses DUE, which manages the awards that are the primary focus of this article.

DUE’s current signature program is Improving Undergraduate STEM Education (IUSE). IUSE is the latest incarnation of DUE’s programmatic efforts to actualize its mission “to promote excellence in undergraduate STEM education for all students.” Former DUE programs include “Transforming Undergraduate Education in STEM” (TUES) and “Course, Curriculum and Laboratory Improvement” (CCLI). The current IUSE solicitation is 15-585, and the next deadline for full proposals is January 11, 2017.

In the next section, we will describe (primarily using excerpts from publicly available abstracts) several active IUSE awards that illustrate variation in topics, institutions, budget size, grant duration, and project type supported by NSF.

*Progress Through Calculus***(1430540****), $2,250,003, PI David Bressoud, Mathematical Association of America. **

This project is a follow-up to the project *Characteristics of Successful Programs in College Calculus* (DRL-0910240) which “undertook a national survey of Calculus instruction and conducted multi-day case study visits to 20 colleges and universities with interesting and, in most cases, successful calculus programs.” ** Progress Through Calculus** has two focal areas of research guided by the following questions: (1) What are the programs and structures of the pre-calculus to calculus sequence as currently implemented? How common are the various programs and structures? How varied are they in practice? What kinds of changes have recently been undertaken or are currently underway? (2) What are the effects of structural, curricular, and pedagogical decisions on student success in pre-calculus to calculus? Success will be assessed on a variety of measures including longitudinal measures of persistence and retention, performance in subsequent courses, knowledge of both pre-calculus and calculus topics, and student attitudes.

** Collaborative Research: Data-Driven Applications Inspiring Upper-Division Mathematics (**

This is a collaborative project (which consists of linked awards at multiple institutions) involving investigators at , Hendrix College, Kenyon College, and Lewis Clark State College. The goals are to (1) introduce current cutting-edge research and practical data problems from science, industry, and government to students in undergraduate upper-division mathematics courses and (2) lead these students to develop the problem-solving, collaborative, and research skills that are so crucial in today’s work environment. The focus of this project is to create a body of applied data-driven instructional modules to motivate student research as well as to generate a deeper understanding and appreciation of the mathematical theory needed to solve these problems.

*Collaborative Research: Improving Conceptual Understanding of Multivariable Calculus Through Visualization Using CalcPlot3D ***(****1524968****), $456,993.00. PI Paul Seeburger, Monroe Community College. **

Three investigators at a community college (Monroe Community College), a public 4-year college (State University of New York at Buffalo), and a private 4-year college (Robert Morris University) are collaborating with faculty across the United States and Mexico to: (1) design and test a series of new visual concept explorations and applications in CalcPlot3D to improve student understanding of multivariable calculus; (2) expand the features of CalcPlot3D to accommodate the new concept explorations and address applications in differential equations, linear algebra, physics, and engineering; (3) create new visualization apps, including a new version of CalcPlot3D, that work on more platforms, including tablets and phones; (4) conduct and publish research investigating how student understanding of multivariable calculus concepts changes through the use of visualization and dynamic concept explorations; and (5) extend and diversify the user base by disseminating project materials through papers, workshops and conferences, by creating a Spanish language version of project materials, and by promoting the exchange of user feedback and research.

*Collaborative Research: Professional Development and Uptake through Collaborative Teams (PRODUCT): Supporting Inquiry Based Learning in Undergraduate Mathematics***, (****1525058****), $2,842,393. PI Stan Yoshinobu, California Polytechnic State University at San Luis Obispo. **

This collaborative project between California Polytechnic State University at San Luis Obispo and University of Colorado Boulder intends to greatly expand the capacity of faculty to implement the specific active learning strategy of inquiry-based learning (IBL). PRODUCT will conduct 12 four-day intensive IBL workshops, as well as 15 short workshops and five Professional Development (PD) Preparatory Meetings, and will host a PD Summit for mathematics faculty developers. Through these activities, PRODUCT will directly provide professional development for 320 undergraduate mathematics faculty, adapt and improve IBL PD materials, develop multiple new teams of faculty developers who will be prepared to engage additional faculty in the future, and develop a framework for building professional development capacity. A research-with-evaluation study will provide formative feedback, study the process and outcomes for development of the professional development teams, gather data to benchmark workshops led by new teams against a model known to be effective, and investigate the classroom practices of workshop participants to understand how the professional development experience shapes their teaching.

*MATH: CONFERENCE: Active Learning Approaches in Mathematics Instruction: Practice and Assessment Workshop (***1544374****) , $25,000, PI Ron Douglas, Texas A&M University.**

This project is an example of a workshop award (which are typically less than $50,000 and are submitted at any time after communicating directly with a program officer). 1544374 supports the implementation of a workshop entitled Active Learning in Mathematics Instruction that was held in conjunction with the Mathematical Association of America’s 2016 Mathfest conference. The workshop was designed to survey and investigate the characteristics, challenges, and evaluation of active learning approaches to collegiate mathematics instruction and to expand the community of individuals who are knowledgeable about both the methods and important questions involving active learning. Participants in the workshop included experts in education and social science research methods as well as active learning mathematics practitioners and departmental leaders.

*MATH: EAGER: Developing a Learning Map for Introductory Statistics (***1544481***)***, $299,832. PI Angela Broaddus, University of Kansas. **

This award is an example of a special funding mechanism at NSF called EAGER (Early-concept Grants for Exploratory Research) that is intended to support potentially transformative research that is considered “high risk high payoff.” The goals of 1544481 are to create and validate a “learning map” (Stat-LM) for the content of undergraduate introductory statistics. This learning map will be a graphical representation of statistics concepts with connections among the concepts suggesting effective learning sequences. Use of Stat-LM is intended to improve undergraduate learning by providing diagnostic information to instructors about students in their statistics courses, informing professional development for undergraduate statistics instructors, and modeling how critical prerequisites taught in high school connect to postsecondary learning expectations.

*Assessing the Impact of the Emporium Model on Student Persistence and Dispositional Learning by Transforming Faculty Culture (***1610482***), ***$299,999, PI Kathy Cousins-Cooper, North Carolina Agricultural & Technical University.**

This is an example of a project co-funded between two programs found in two divisions of EHR. 1610482 was submitted to the IUSE program in EHR/DUE but also received funds from the Historically Black Colleges and Universities – Undergraduate Program (HBCU-UP) in EHR/HRD (Division of Human Resources Development). The investigators will employ, study, and assess an instructional and student learning model, called the Mathematics Emporium Model (MEM), to improve students’ performance in introductory mathematics courses. These gatekeeper courses are normally taken during an intense and often difficult transition for students, from high school to college. The MEM eliminates lecture and uses commercially available interactive computer software combined with personalized on-demand assistance and mandatory student participation. The underlying principle of the Emporium Model is that students learn by doing. Research reveals that the shift to student-centered instructional practices enhances students’ attitudes and beliefs about learning in mathematics courses and increases student-learning gains. The project will directly reach a combined annual enrollment in traditionally low-pass-rate courses of more than 4,000 students, who will be mostly from underrepresented groups.

*Professional Development Emphasizing Data-Centered Resources and Pedagogies for Instructors of Undergraduate Introductory Statistics*** (StatPREP)**

This project responds to a recommendation found in *A Common Vision for Undergraduate Mathematical Sciences Programs in 2025*, a report funded by an award from EHR/DUE (1446000) and issued jointly by the American Mathematical Association of Two-Year Colleges (AMATYC), American Mathematical Society (AMS), American Statistical Association (ASA), Mathematical Association of America (MAA), and Society for Industrial and Applied Mathematics (SIAM). Specifically, StatPREP will catalyze the widespread use of data-centered methods and pedagogies in undergraduate introductory statistics courses. It will work directly with 240 college-level instructors by (1) offering an extended professional development program for mathematics instructors, particularly at two-year institutions, who teach introductory statistics; (2) establishing regional communities of practice to support instructors who teach introductory statistics; and (3) establishing a national online support network comprising instructors who teach introductory statistics and statistics education experts.

The above awards are just a fraction of the dozens of active awards we manage in EHR/DUE. We are excited about the work of the DUE-funded projects and their impact on improving the future of teaching and learning in undergraduate mathematics. We encourage readers to examine the full list of EHR/DUE active awards in undergraduate mathematics and mathematics education. If readers of this blog have ideas or suggestions for proposed activities that could improve the teaching and learning of mathematics, feel free to contact us.

]]>

I think that mathematics draws in some people and repels others in large part because of the distinctive role of authority in our discipline and teaching, especially when we act as content experts and discussion leaders in the classroom. For instance, consider the following phrases from students, distilled from my interactions with college students over the past 15 years.

I’m not a math person. I learn best when you show me a bunch of examples and then I practice them. It’s true, so why do I have to prove it? That’s just how my last teacher told me to do it. I always liked math because there was one right answer. I just want to teach high school; why do I have to learn this? Wait, what – you want me to ask my own question!? Do I have to simplify my fractions? Well, that’s what the computer said was the answer. The test was unfair because it had problems we didn’t discuss in class. ~silence~

I expect that these comments are also familiar and painful to the reader. I think that each of these comments is in part a symptom of ways students have internalized a relationship with authority from our teaching. In this post, I will illuminate the role of authority in mathematics teaching, argue that taking a more overt stance toward it can better support both the students we repel and the ones we attract, and offer a handful of strategies for taking such a stance.

Depending on whom you ask, the truth of a mathematical conclusion can stand independently of a human authority or based entirely on the word of an authority. Mathematicians will often claim (e.g., [8]) that we depend only on proof to develop reliable knowledge and will dismiss student efforts to use empirical evidence or the word of an expert, but researchers [13] are showing that mathematicians use these strategies as well. I certainly agree that deductive reasoning occupies a special place in our discipline, but the absence of methods, evidence, and theoretical frameworks in the discussion of mathematics quietly places the math itself in the position of perfect (Platonic) authority. I do not take issue with this perspective except when our silence about authority leaves students to grapple with it alone, often painfully. This post will focus on these implications for our students.

**Models of Student Development**

My thinking about authority begins with Williams Perry’s work from the 1950s and 60s on the epistemological development of college students [14]. In broad strokes, Perry’s scheme describes a sequence of positions from which the students he studied viewed truth or knowledge; I will use a condensed version of this scheme with three main positions. From *Dualism*, students view knowledge as binary, and Authorities know the difference between true and false. From this position, students equate learning with memorizing any information that these Authorities pass on to them. From *Multiplicity*, students notice that much of knowledge is context-dependent and come to believe that any perspective is a valid source of knowledge. As a result, while in this position, students become less interested in the perspectives of others, including authority figures, instead focusing on their own authority. From *Relativism*, students begin to demand that other perspectives be justified; as a result, knowledge becomes the result of argument and evidence. In this position, students initially focus on learning HOW authorities want them to think, rather than WHAT. Eventually, these students acknowledge that their arguments must be grounded on accepted assumptions, and they become concerned with establishing appropriate and personal precepts. Common elements of college programs, such as first-year writing seminars and introduction-to-proof courses, encourage our students to adopt a relativistic position in their collegiate work by helping them practice appropriate modes of argumentation. For the purposes of this post, I will group students as though they have a predominant or preferred position.

My dualist students often find enticing or comforting the idea that mathematics is a field in which truth is absolute and completely known because they feel that it allows them to avoid pesky ambiguity, especially in their course assignments. These same students regularly say dismissive things about other disciplines to me, most commonly about literature courses. They seem to believe that the work of engaging literature reduces to forming a personal opinion and that all opinions are equally valid; they will often go further to rail against grading in these courses because they see it as relying on whether their valid opinion happens to match the one held by the teacher. To be clear, these are deleterious conceptions of both disciplines; I think that one of the most important goals of a college (liberal arts) education is helping students change these stereotyped views of disciplines. While these dualist students sometimes come to college liking math, they have been set up for a painful bait-and-switch when their math work shifts suddenly from execution of provided algorithms to generation of original arguments, leading to their comments about the good old days of Calculus. Perry saw evidence in his data of students retreating to earlier positions when facing difficulty, which can explain student resistance in their first proof courses, not to mention the first time they are asked to do something original and creative in math. I’m not surprised that many of these students consider switching to engineering or economics because they believe the math used there is aligned with their earlier perceptions of mathematics.

The other large group of incoming students, those for whom multiplicity is the preferred epistemological position, curiously say essentially the same things about mathematics and literature to me, with exactly opposite emotional value attached to the descriptions. They seem to hate that math has no room for their individual perspective and prefer discussion-oriented courses in the humanities and social sciences because they offer spaces in which they are encouraged to consider their own perspective. The departure of this group from studying mathematics is one of the leaky joints in the mathematics pipeline discussed by CSPCC [2] and even the Obama administration [7]. Perry’s data suggested that some students in this position aligned with authority figures and others against and that this alignment impacted their trajectory. I think that the importance of this split path can be understood by considering research that responds to one of the major critiques of Perry’s work. The population on which he based his scheme, namely students at Harvard at the time, was overwhelmingly male and non-representative of today’s college students in many other ways.

Belenky, Clinchy, and Goldberger, using a population of female interviewees whose ages varied widely, developed a scheme called “Women’s Ways of Knowing” [1]. These two schemes share many common features, but WWK illuminates two important observations. First, some of the women in this study talked about ending educational experiences because they perceived educational environments as placing no value on their voices and offering no pathways to expertise. In contrast, it seems important that the students in Perry’s population were male and enrolled at an elite college in a time when going to college was a rarer and more intentional choice than today; in other words, Perry’s population was likely selected to contain participants who already envisioned themselves as future authorities, an attribute that may help a person persist in (collegiate mathematics) courses when they experience friction between their epistemology and their course work. Combining this with the under-representation of women in our discipline, it’s hardly surprising that female students switch out of math and STEM more than male students [6]; they must endure years of study in an environment that seems not to value any student voices, and when students begin to find their voices, their male peers may have more support for identifying as future authorities themselves. Second, the WWK scheme also adds an important, fifth position (before the others) called Silence, from which interviewees experienced a world in which they had no access to knowledge or truth. I think the multiplicitous and silent students are driven from math similarly, though the “myth of genius”, which correlates to heavily with gender representation in STEM fields [10], might be a silencing factor well before college.

There is a third, much smaller group of students who view mathematics as relativists. I believe I was in this group; I liked how I could avoid memorizing almost anything in math because I knew that everything could be derived from the definitions. Importantly, this perspective seemed to maintain but reframe the dualist and multiplicitous things I liked about math. Yes, our knowledge and truth seemed to be of a different sort than in other disciplines, but this characteristic came from the kinds of arguments that are possible about abstract objects, not by declaration. And yes, there wasn’t much room to disagree with a theorem, but I felt empowered to make my own arguments given our egalitarian access to definitions, to try to extend algorithms or make them more efficient, and to have my own personal way of thinking about problems. I conjecture that most of us who have persisted in mathematics made similar transformations of our love for math that helped us persist and that many who did not persist did not make. Significantly, I believe that math classrooms can be set up to be welcoming to students in all of these positions while also helping them move forward.

**Using these Models to Inform Pedagogy**

Fortunately, Perry’s research also illuminates at least three kinds of experiences that impel students to move to later positions: encountering questions without known answers or about which Authorities disagree, engaging a pluralism of ideas among peers, and rigorously justifying claims and questioning assumptions. To me, this feels like a description of a classroom organized by inquiry. I further claim that mathematics makes it particularly easy to include these experiences in our classrooms: each new definition gives us instant access to an unlimited supply of perfect copies of objects into which we can explore, often very quickly and cheaply, while entertaining open conjectures that require justification. In contrast, work in the natural and social sciences seems to require data collection that can be slow and expensive, work in the arts and humanities can require nuanced analysis that uses a holistic perspective that is not immediately accessible, and work in applied fields often interacts with a great deal of information from its context. I’m not suggesting that these aspects of other disciplines are negative; in fact, I think each discipline has opportunities to highlight different facets of authority and epistemology more easily and that contrasting these themes across disciplines is a key mechanism for supporting student growth. I am saying that I see a way that our disciplinary context allows me to put the epistemological work of education front and center in the math classroom, in contrast to placing it in an invisible role.

Building a classroom that supports these epistemological themes can be challenging because it requires course materials and teaching skills that are consistent with this goal. A complete discussion of such course materials and general teaching skills is beyond the scope of this post, but I would suggest the reader start with the writings coming from these two threads [17,4]. Instead, I would like to offer two suggestions for strategies that engage authority in your classrooms overtly but don’t require an immediate reimagining of your teaching practice. First, I suggest that you read and discuss some mathematics education literature with your students — here are some of my favorites. Harel and Sowder [9] describe categories of students’ perspectives on mathematical justification called “proof schemes”; these categories align extremely well with Perry’s described above, and discussing this paper (or another that uses it) can help students understand professors’ expectations in advanced mathematics courses. Weber and Alcock [18] discuss student and expert behaviors when validating proofs; this paper always leads to discussion of subtle but important points including the role of communal expectations for proofs. If your students don’t yet have experience with proof, then articles or videos about Carol Dweck’s work on “growth mindsets” [15,5] or Paul Lockhart’s provocative (and informal) “A Mathematician’s Lament” [3] are both strong choices; these papers are not overtly about authority and epistemology, but they help students talk about the discipline in a way that allows an instructor to engage their perspectives. And while it’s not about mathematics, I also enjoy discussing summaries of Perry’s scheme with students in many courses, most commonly using Chapter 1 of this reference [11]. You may worry that these activities would take away time from other learning objectives, but the opposite has been my experience; through these reflection opportunities, students are able to see course activities in a new light that deepens understanding of the past work and makes future work more effective and efficient.

Second, I suggest that you humanize the mathematical practices in your classroom. I think that small changes in our language can help students adapt a more productive stance toward authority. Using “we” and “us” communicates that the people in the room are engaged in mathematics and have some local authority. Tagging ideas, questions, and conjectures with student’s names when we reference them highlights the fact that individuals are impacting the development of the mathematics; asking students to use their peers’ names suggests that these people have perspectives that we will consider. This language makes space for the psychological and social aspects of our work that we may crush with our silent authority otherwise. In addition to investing students with some authority, I think it is important to humanize ourselves to resist becoming the abstract authority. As a first step, we can talk about moments inside and outside of the course during which we are or were uncertain mathematically. However, I think that telling students that they can only know mathematical things about us as people communicates a preference to be seen as a distant authority. Personally, I feel an ethical obligation to go further and be a multifaceted person with my students. For example, I end up coming out in some of my courses because doing otherwise feels hypocritical given the level of openness I’ve asked of students about their lives and minds [12]; I also appreciate that they are now aware of at least one queer mathematician as a potential role model.

**Conclusion**

I would like to reiterate the importance of mathematics educators taking a more overt stance toward authority. I think we are compelled to use the distinctive interaction of mathematics with authority to help students mature generally in their lives, and I think the lenses discussed here help me see opportunities for this work. For example, Perry noticed changes in the most common starting position of first-year students in the 1960s, which he connected to the marked changes in the national dialog and the role of Authority therein. Analogously, I believe that these schemes are going to be key tools for understanding the waning public faith in higher education and for responding to the changing needs, perspectives, and skills of students entering college during the next decade, especially regarding the authority of hands-on parents and of information technology. In the context of our classrooms, I have argued above that when we vest all authority with mathematics in the abstract, we seem to create a vacuum that leads students to engage with us as Authorities instead, in ways that attract some students while setting them up for struggle and make other students feel unwelcome. I have made some conjectures about how this might perpetuate the under-representation of women in mathematics, and I would make similar connections and conjectures about under-representation of people of color (e.g, [16]). In summary, I think that being intentional and overt about the role of authority in our teaching can transform our classrooms into more inclusive and equitable spaces, and I hope you feel empowered to use the ideas in this post to do this work.

**References:**

[1] Belenky, M. F. (1986). *Women’s ways of knowing: The development of self, voice, and mind*. Basic books.

[2] Characteristics of Successful Propgrams in College Calculus: Publication & Reports – Retreived from http://www.maa.org/programs/faculty-and-departments/curriculum-development-resources/national-studies-college-calculus/cspcc-publications

[3] Devlin, K. (2008, March) *Lockhart’s Lament*. Retreived from https://www.maa.org/external_archive/devlin/devlin_03_08.html

[4] Discovering the Art of Mathematics – Retreived from https://www.artofmathematics.org/

[5] Dweck, C. (2014, November). *The Power of Believing that You Can Improve (TED talk)*. Retreived from http://www.ted.com/talks/carol_dweck_the_power_of_believing_that_you_can_improve

[6] Ellis, J., Fosdick, B.K., and Rasmussen, C. (2016). *Women 1.5 times more likely to leave STEM pipeline after calculus compared to men: Lack of mathematical confidence a potential culprit*. PLoS ONE 11(7): e0157447. doi10.1371/journal.pone.0157447

[7] Feder, M. (2012, December 18) *One Decade, One Million more STEM Graduates (based on a statement by the President’s Council of Advisors on Science and Technology)*. Retreived from https://www.whitehouse.gov/blog/2012/12/18/one-decade-one-million-more-stem-graduates

[8] Fischbein, E. (1982). Intuition and proof. *For the learning of mathematics*, *3*(2), 9-24.

[9] Harel, G., & Sowder, L. (1998). Students’ proof schemes: Results from exploratory studies. *Research in collegiate mathematics education III*, 234-283.

[10] Leslie, S. J., Cimpian, A., Meyer, M., & Freeland, E. (2015). Expectations of brilliance underlie gender distributions across academic disciplines. *Science*, *347*(6219), 262-265.

[11] Love, P. G., & Guthrie, V. L. (1999). *Understanding and Applying Cognitive Development Theory: New Directions for Student Services, Number 88* (Vol. 27). John Wiley & Sons. (Chapter 1)

[12] Lundquist, J. & Misra, A. (2016, October 18) *Establishing Rapport in the Classroom*. Retreived from https://www.insidehighered.com/advice/2016/10/18/how-engage-students-classroom-essay

[13] Mejia-Ramos, J. P., & Weber, K. (2014). Why and how mathematicians read proofs: Further evidence from a survey study. *Educational Studies in Mathematics, 85*(2), 161-173.

[14] Perry Jr, W. G. (1999). *Forms of Intellectual and Ethical Development in the College Years: A Scheme. Jossey-Bass Higher and Adult Education Series*. Jossey-Bass Publishers, 350 Sansome St., San Francisco, CA 94104.

[15] Popova, M. (2014, January 29) *Fixed vs. Growth: Two Basic Mindset that Shape Our Lives*. Retreived from https://www.brainpickings.org/2014/01/29/carol-dweck-mindset/

[16] Robinson, M. (2011, Spring). *Student Development Theory Overview*. Retrieved from https://studentdevelopmenttheory.wordpress.com/racial-identity-development/

[17] Teaching Inquiry-Oriented Mathematics: Establishing Supports – Retreived from http://times.math.vt.edu/

[18] Weber, K., & Alcock, L. (2005). Using warranted implications to understand and validate proofs. *For the Learning of Mathematics*, *25*(1), 34-51.