“I am sad this class is going to be over,” said one student. “What am I going to do with myself?” asked another during the last week of an Intermediate Algebra class that I taught last summer at the Lincoln Correctional Center (LCC) with Meggan Hass, then a University of Nebraska graduate student.

Meggan and I were sad, too. It’s not often that we hear these types of comments from students, but as I have learned, the unexpected can happen when one teaches in prison.

Here is my story.

It started with a visit by Bryan Stevenson, author of *Just Mercy*, to Nebraska Wesleyan University, where I work. Stevenson is a lawyer who defends incarcerated individuals, many of whom are on death row. In his talk, he urged us to:

“Get proximate. Get uncomfortable. Change the narrative. Have hope.”

I was sold. In the coming months, I put in a request to adjust my Spring 2018 sabbatical.

When Stevenson called for the audience to “get proximate”, he was encouraging us to be closer to those who are suffering or excluded. In response, I arranged to teach classes at the Nebraska State Penitentiary and at LCC during my sabbatical.

The classes were non-credit bearing, so I had liberties in the content. I wanted to pick a topic that wouldn’t require a strong math background but that would be interesting. The 8-week course I developed was “Combinatorics and Probability.”

Not knowing anything about the prison system, I asked basic questions of corrections staff: Will paper and pencils be provided? (yes) Can I move around freely in the class? (yes) Are calculators available? (yes) Can they work in groups? (yes) What can I bring into the facility? (notes, book, pencils, calculator) Can I bring dice and playing cards? (no) How can we determine who is accepted into the class? (basic math proficiency, no recent misconduct)

I was scheduled to meet my students for two hours each week, but the prison staff would regularly let the classes run long. For my students, the longer, the better. The chance to think about something different and be away from the drudgery of the units was welcomed. My students never missed a chance to tell me how much they loved to learn and how education was what they needed more than anything.

Although I had carefully planned the course before it began, I made big adjustments once it started. My students’ backgrounds were quite disparate, creating some unease for me. But I adjusted the material by giving them more time to “count” things using brute force, with the idea that it would facilitate the process of making conjectures. I also asked students with stronger backgrounds to help those with weaker ones. That worked beautifully.

Towards the end of the 8 weeks, one student asked me what surprised me most about teaching them. He wondered if I was scared. No, I wasn’t scared. However, I had a preconceived notion that asking students to participate would make them feel vulnerable; that made me uncomfortable as I walked into class on the first day. The big surprise for me was how willing they were to ask and answer questions. In fact, my main time management issue was trying to address all of their questions. Most questions were about the specific content, but some extended into vocabulary and notation: “Did the question mark first show up in mathematics or in other writing?” (I always researched those questions I could not answer and reported back to them. Incidentally, the history of the question mark is fascinating, and I encourage you to research it.) To cover the main concepts adequately, I occasionally omitted a few examples.

The centerpiece of Bryan Stevenson’s directive to “change the narrative” is to talk honestly about the historical roots of racism and poverty. For me, changing the narrative was more personal. It was about refocusing where I put my energies; I wanted to continue teaching in prisons.

The following academic year, I taught a course in Algebra 1 at LCC. At the prompting of the assistant warden there, I subsequently co-taught, with Meggan Hass, an Intermediate Algebra course in Summer 2019, through a program at the University of Nebraska-Lincoln (UNL), supported by private funding.

Most of the students who had taken Algebra 1 with me applied to take the Intermediate Algebra course. All applicants needed to have a GED or high school diploma. Beyond that, UNL gave me full discretion in choosing participants, so I created a simple application form and held interviews.

Among other things, I asked about the hardest thing that they had ever done. The responses gave a sense of how varied their backgrounds and perspectives were: “Working at Long John Silver’s,” “Your Algebra 1 class,” “Fitting pipes for an oil rig.” But each talked about these hard things with the kind of conviction that conveyed their seriousness about this educational opportunity.

As Meggan and I began preparing the course, a question we had to answer was how much “discovering math” we wanted to put into our course. For example, we needed to decide if we wanted our students to “discover” the rules of exponents working with others, or if it would be better to offer more guidance. The challenge I had had when letting the students in Algebra 1 spend time discovering math was that some students got distracted or overwhelmed. So, we made worksheets for each section and worked through these as a class, asking students many questions along the way. As an example, to show that 3^{4}3^{5}=3^{(4+5)}, we asked our students to write all of the factors of 3 on the left side of the equation to see why it would be the same number of factors of 3 on the right.

I have a fascination with words and so for each class period, we had a “word of the day.” We delighted in vocabulary such as *vociferously*, *chagrined*, *vexillology*, *verisimilitude*, *polyglot*, *polymath*, *apoplectic*, *belie*, *ebullient*, *capacious*, and *pacific*. Riveting discussions ensued with some of the students reaching for a dictionary to uncover the word origins.

On the last day of Combinatorics and Probability, we held a certificate ceremony. I prepared certificates with one of my favorite quotes:

“We choose to go to the moon in this decade and do the other things, not because they are easy, but because they are hard, because that goal will serve to organize and measure the best of our energies and skills”

– President John Fitzgerald Kennedy

I chose this quote because it connected to their eagerness to learn and their willingness to challenge themselves.

The students asked if they could speak. They talked about how important education was to them. One shared that few people are willing to go into the prisons to volunteer and that it was significant when people did. They asked me to tell people about what I was doing so that it would encourage more to do the same. It was emotional.

These thoughts were echoed at the end of the Intermediate Algebra class, too. In this class, there were 26 homework assignments, 3 quizzes, 3 exams, and one final exam. Of these, only one homework assignment was not turned in by one student. In fact, this student had completed it, but accidentally left it in his unit. None of the homework turned in was incomplete and it was done with exceptional care. Attendance was almost perfect; we had one excused absence and no unexcused absences during the entire course.

On the last day of class, the students presented Meggan and me with homemade pop-up cards. They each wrote a personal note of gratitude. We were equally grateful to have been their teachers. In their course evaluations, they talked about the class teaching them much more than math – one said he finally could believe in himself. They shared that the class made them feel like human beings.

It gave them hope.

There are many reminders that you are in a prison when you teach there. But, just like in any class I teach on my campus, each student comes to class with a different history, each student learns differently and each challenges me as a teacher. In my prison classes, I was stretched as a teacher in consequential ways; I had to adjust quickly to different backgrounds, find different ways to explain material and rally around my students to help them build their confidence. I came away seeing the immense value there is in offering incarcerated individuals the opportunity to learn.

* * *

*Note from the author*: If you think you might be interested in such an opportunity, and you would like an experience you might fit into a summer, I am happy to share my course materials from Combinatorics and Probability (class worksheets and homework sets). Email me at kpfabe@nebrwesleyan.edu. They are imperfect but can serve as a starting place.

In Marilynne Robinson’s *Gilead*, Reverend Ames testifies that each person in his flock has “a kind of incandescence in them… quick, and avid, and resourceful. To see this aspect of life is a privilege of the ministry which is seldom mentioned.” I think this is a privilege of the educator too. But our flock’s life cycle, the fourteen weeks from the cradle of the first day syllabus to the grave of the final grade, has a different rhythm to that of a liturgical calendar.

In a semester where I have over 100 students, witnessing each student’s luminosity is challenging, and by the time I get to know even half of them the semester is likely to be halfway through. So I ask my students to help me to see them. I ask what their values are, to tell me about the work they’ve done and are about to do, and to share their hopes and aspirations as people. I ask them about the concrete challenges that they might face as the semester progresses and what they wish previous mathematics teachers understood about them.

I do this because understanding my students holistically aids in my support of each student individually and can cultivate active learning communities. I do this because if the semester gets rough on my end — it invariably does — I have a wealth of prior empathy and insight to fall back on so that my students succeed in their goals. I do this because knowing only their placement scores and/or GPAs is not enough! There’s also a body of research supporting that when students’ values are affirmed in STEM [i], they are more likely to succeed in that course and continue as a STEM major.

I ask for their help on the first day because it is most expedient and it is the most important day of the semester [ii]: Consulting my students on that first day establishes a framework for the class, that we are starting a dance between the content of our discipline and the context from which each person comes; and in this dance both partners matter equally and each will have their turns to lead. The first day is also the liminal space between their previous experiences in mathematics and what this new course promises to be, so it provides an opportune time to reflect and to project, to celebrate and to prepare.

The questions I will discuss below are intended for junior/senior math majors but can be adapted for other populations of students [iii].

*What was your path to becoming a math major? Do you identify as a ‘math major’? How is this choice perceived by others and how do you respond to these perceptions? *

These questions are general yet typically elicit responses about formative experiences, far beyond what I originally anticipated when I first put this questionnaire together. Students tell me if their confidence is low, if they have a passion for physics, if their vocation is in teaching. They let me know if college was not the expected path for their family members, or if it was absolutely expected, each of which exerts its own distinct type of pressure. Students also tell me about their perceptions of local and global mathematics culture and how they situate themselves within it; in doing so they invariably tell me about their values.

Students also share personal issues outside of the major. Some will say they are holding down a full-time job and then I know that they might need some flexibility on deadlines [iv]. Others might be returning from some time away from college necessitated by mental health, financial, or family concerns: I tend to pair these students with partners who are more collaborative than competitive. For example, my typical Quantitative Finance or Engineering students — not inherently, but culturally, competitive — tend to be a poor match for someone who needed a break from school for a period of time.

The responses allow me to tailor my interactions with my students but also to customize some of the content of the course. For example, when I am assigning projects in the second week of Abstract Algebra, I can assign a reading project on matrix groups to the student who is very interested in physics. Similarly, if the class has future teachers then I will assign a project on transformation groups relating to isometries and how that relates to the state’s high-school mathematics curriculum.

The next questions ask the student to reflect on their time at university. It asks for a self-assessment of what courses went well, why they are here for this course, and the extent to which they have learned from positive and negative past experiences in collegiate mathematics. These questions are implicitly asking the students to transition from values to responsibilities, to a sense of ownership of their education. The questions also give me a very practical insight into their mathematical motivation and the gaps in their existing knowledge we might need to address.

*What’s your favorite area of mathematics? Your favorite ideas/results? What course do you wish you could have avoided? What course do you wish you could take again? What would you tell yourself if you could travel back in time and talk to your freshman self? *

My final questions concern mentorship. Knowing my student’s values and motivations, how can I support my student’s future plans? The responses can also be used to calibrate the course’s emphases to where they can have the most impact.

*What are your future plans in terms of vocation? Do you hope to enter into a job in industry? Are you preparing to be a high school teacher? To go to graduate school? To run for the hills, join a commune and live off the land? What non-math skill do you wish you were better at?” *

I distribute these questionnaires on the first day of class and ask the students to return on the second day of class with their responses. At no point do I make these responses public. The questions demand vulnerability on the student’s part so I am compelled to reciprocate by trusting my new students with my own answers to these questions. I do so by responding verbally to these same questions on the first day of class. I answer as I think my twenty-year-old self would have and also how my middle-aged self understands these responses today. I worry that I’ll look like a fool but it never works out that way.

For an “active learning”[v] context, where vulnerability and play are partners of discovery and rigor — a joining of many dialectics in the hopes of achieving a synthesis — the affirmation of students’ values may be yoked to their success in the class.

Miyake et al. provided evidence that when women in an introductory physics class “wrote about their most important values” twice in the semester, the discrepancy between male-female outcomes reduced substantially. Further, that “the benefits were strongest for women who tended to endorse the stereotype that men do better than women in physics.” In a different STEM context, Jordt et al. examined the discrepancy between underrepresented minority students and white students in introductory biology classes and concluded that the relative “underperformance of URM students could be mitigated” by a similar values intervention. The biology education lab of Sarah Brownell has more investigations along these lines.

I would suggest — as an axiom rather than an empirical finding – that the more a person feels affirmed in class the more likely they are to wholeheartedly participate in groups with others: the more likely they are to trust, the more likely they are to take risks. Indeed, group work without intentional individual affirmation is more likely to promote a tepid conformity — the “just get it done” treatment of the worksheet that defers authority to the student perceived as the group’s strongest or, failing that, the instructor — rather than the form of radical consensus that we as instructors wish for. The type of consensus I mean is one where each participant seeks to hear the other and where students’ collective achievements surpass their individual struggles with the material. It’s my suspicion that if we sincerely engage in an active learning practice without paying attention to individual affirmation, we risk cementing real and perceived hierarchies among students and, in doing so, exacerbate the very types of privilege and status quo that we might be trying to supplant — not just in our class but, more broadly, in our discipline’s culture. This is why affirmation matters.

Returning to *Gilead*, the epistolary work referenced in the introduction, Ames writes with the urgency of a dying father anticipating who his young child will be and what he will need from the world. He professes a nuanced and practical faith with decency, introspection, and a transparent and messy imperfection. For the teacher, the stakes are lower than those in Ames’s letter but the exercise, imperfectly anticipating the world as a young person might experience it, requires Ames’s same sense of optimism leavened with a dusting of realism and the acceptance of frequent futility.

To help our students navigate the outer world and the universe of our discipline, we can ask them to share what they think of as their true north, their stars within and beyond our discipline, the terrains they’ve passed through. We can ask which mountains they want to climb and let the contours of our courses, our programs, and our discipline’s culture be influenced accordingly. But if we want to know the answers to these questions, we have to ask our students. And honor the trust they place in us when they answer. In doing so we give our class a greater chance to become a community in which everyone is affirmed as a person, where collective risks are more likely to be taken, and where opportunities are increased for every student to join our discipline fully.

[i] Setting aside my deep reservations about the use of the term “STEM”; I’d prefer if we called it “MASH,” dumping engineering and technology for the arts and humanities, cf. https://sententiaeantiquae.com/2018/11/28/science-and-humanity/

[ii]The importance of the first day is in Chapter 2 of *Geeky Pedagogy: A Guide for Intellectuals, Introverts, and Nerds Who Want to Be Effective Teachers, by *Jessamyn Neuhaus. West Virginia University Press, Morgantown, West Virginia, 2019.

[iii] The questions I ask for introductory classes like Precalculus can be found at my website: http://educ.jmu.edu/~osheaem/teaching/who-are-our-students.pdf

[iv] Art Duval advocates for flexibility too in “Kindness in the Mathematics Classroom,” (this blog, Feb 2018). Duval’s main question is in response to Francis Su’s essay on “grace” in the classroom, how does one practically implement Su’s vision for mathematics praxis?

[v] As expected, “active learning” is quite the broad church. My own style emphasizes the centrality of reading outside of class, some elements of flipped classrooms and IBL, and mastery-based assessment. See “What Does Active Learning Mean For Mathematicians?” by Benjamin Braun, Priscilla Bremser, Art M. Duval, Elise Lockwood, and Diana White, Notices of AMS, February 2017.

[vi] Thanks to Cynthia Bauerle for pointing me to some of the science education literature on affirmation.

]]>Executive Director, Math Circles of Chicago

The New York *Times* recently published an article entitled “The Right Answer? 8,186,699,633,530,061 (An Abacus Makes It Look Almost Easy)”. Its lead photograph features over 100 children seated at desks, facing forward, working individually. This is yet another in a long series of public relations disasters for mathematics. This depiction of mathematics is nothing new, and I most suspect readers experienced no cognitive dissonance in seeing mathematics represented this way.

*Traditional Forms of Math Enrichment — and the Problem with Contests*

Mathematicians collaborate to explore exciting open-ended questions. Unfortunately, this may be the world’s best kept secret. The problem? We have two major gateways to participation in the community of mathematics: the classroom and the contest. The classroom, of course, is universally familiar, and innumerable efforts have been made to improve the student experience in the classroom.

Here I focus on the second gateway, the contest. When I say ‘contest’ I refer to the kind that looks like that Abacus competition pictured in the newspaper. These are quite common. For the majority of children competing for the first time, contests have a number of common features:

- A large group of students gather in one location. They do not talk to each other.
- Students complete closed-ended problems with unique correct answers, which is necessary so that responses can be judged impartially.
- Contest problems are generally predictable. This allows students to memorize formulas and tricks to save time during the contest. Frequently students prepare for contests through practice problems that turn contest
*problems*into*exercises*—known results that can be completed efficiently to save time during the contest, preserving time for checking answers, since partial credit for thinking is not given. Speed is prized. - At the end, four classes of prizes are awarded: 1st place, 2nd place, 3rd place, and no place — with this last group the mode by a wide margin. One can argue that this is a feature of any competition, but the ramifications of competition in a subject as fundamental as mathematics are quite serious. Giving up on math is not same thing as giving up on chess or basketball.

Again, there are many exceptions to this general description of contests, but those exceptions are usually experienced by those who have previously been successful in close-ended contests. One can argue students can get exposed to beautiful problems that they themselves might extend. A colleague once told me that the best part of the contest was after papers were submitted and kids started to “sing the contest” — eagerly talking to their friends about the most interesting problems.

I’ve seen students benefit from contests. But I do think it’s time to reflect on some of the downsides of this kind of enrichment. As we seek to expand access to the world of mathematics, do we really want our main form of math enrichment to narrow the gateway?

*More Subtle Problems*

I’ve known many high school students who have done well in contests and were subsequently motivated to major in mathematics. In turn, I suspect there is a selection bias that supports math contests as the main form of math enrichment. If you didn’t do well in contests, you didn’t major in math, and therefore you haven’t stuck around later on to question the practice.

For others, contests have a negative impact on identity. As I mentioned, “no place” is the most common ranking for the large majority of contest participants. **Contests send and reinforce a fixed mindset message.** Children often compete as part of a team from their school, and schools with more resources tend to perform better. But when less-prepared children from underresourced schools compete, they may see other children achieving at a higher level, and may be led to believe that they just aren’t as good as others out in the wider world. This is particularly pernicious when it comes to underserved communities and communities of color.

Teachers are often inspired by contests, benefit from writing them, and gain personal connections with students that they might not otherwise have formed. But, as I mentioned, preparing for contests prioritizes efficiency over depth. Few contests reward alternative solutions or depth of thinking. Speed matters; hence, game theory dictates that we ought to teach procedures.

Finally, and perhaps most subtly, the world of math enrichment largely embodies a “Field of Dreams” approach: “If you build it, they will come.” My question: *who* comes? If you look at the participants in existing contests, you are more likely to get boys, children from more affluent schools, and people who have the social capital to know where to find the contests. And there is a ‘Matthew Effect’–minor advantages accrue early in life, so that by the time children are participating in contests, what seems like a fair assessment of talent is really just a piling on of advantage.

The math enrichment gateway needs to change, or the usual suspects will be the only ones making it through the gate. We will continue to lose underdeveloped talent—children with latent ability who will never reach their mathematical potential. Career choice, economic mobility, and civic engagement will continue to be unnecessarily limited for many.

*What Might We Do Instead? The Case for Math Circles*

In the last five years, the single greatest impact on my thinking about teaching has come from Alan Schoenfeld’s Teaching for Robust Understanding (TRU) Framework. TRU asserts that to create powerful math learning environments, we need to attend to five crucial dimensions: mathematical connections, cognitive demand, access, agency/identity, and formative assessment. The ongoing program of research and practice of Schoenfeld and his team is working to show that these five conditions are both necessary and sufficient for robust learning to occur, and to explore effective, efficient ways to make them happen.

TRU is being applied in classrooms and it applies equally well to math enrichment settings. Math circles, festivals, and summer camps can be designed to be equitably accessible in a way that contests simply cannot be. Students exploring open ended problems for a math symposium have an opportunity to experience agency at a level that neither the classroom and the contest rarely provide.

Let’s consider math circles. Math circles usually occur outside of regular school hours, where interested children investigate novel mathematics in sessions led by an adult with a strong affinity for math.

Like math contests, math circles can be cognitively demanding. But the phrase ‘cognitive demand’ can be deceptive. Sometimes it’s read simply as ‘hard’. But — ironically — making mathematics hard is not difficult. TRU describes cognitive demand as: “The extent to which classroom interactions create and maintain an environment of productive intellectual challenge conducive to students’ mathematical development. There is a happy medium between spoon-feeding mathematics in bite-sized pieces and having the challenges so large that students are lost at sea.”

Contests rarely provide appropriate levels of cognitive demand to a broad range of students. A math circle, with a more classroom-like environment can be designed to provide that “happy medium” for individual students. Students can work at their own pace and have a personally rewarding experience.

The advantages of a math circle become fully clear in other dimensions of the TRU framework. Because they do not center around competition, math circle sessions are more welcoming spaces for a broader range of students. We can make our sessions accessible. We can improve the likelihood that more diverse students identify with the subject, consider it to be fun and worthy of long-term pursuit. Problems can be fine-tuned to become easier or harder based on the pace at which a given student digests the new material.

And, let’s not forget the mathematics itself. According to the TRU Framework, in order for a classroom to be mathematically powerful, “The mathematics discussed is focused and coherent, …[and] connections between procedures, concepts, and contexts… are addressed and explained.” Typical contests throw disparate problems at students, where mathematical connections between problems can be non-existent. Compare that to a math circle—based, perhaps, in one main rich activity, or a chain of problems that feature a strong connections that can reveal surprising and beautiful results.

*What’s next for math circles?*

While I see the potential of math circles, I also recognize that if we do not implement them thoughtfully, we can end up reproducing many of the shortcomings of contests. The field of dreams approach can still limit who attends, and we may end up serving the same children we served before with contests. Perhaps those children will have a better experience than in competition, but the audience might still be made of the usual suspects. We need to consider where (and when) math circle meetings are held; whether we can hold them free of charge; and how we can forge partnerships with community leaders to thoughtfully recruit children in underserved communities.

Moreover, as we consider serving a more diverse group of children, we need to consider how to serve those children effectively. A math circle leader needs to know how to teach well. They should also know how to build connections with and between students, while also having a deep understanding of the mathematical connections at hand. (I admit this is a significant topic unto itself, and that teaching well is an enormous challenge—but it’s absolutely essential that it be addressed for this enrichment based in student collaboration to be successful.)

Four years ago, I became the Executive Director of the Math Circles of Chicago (MC2). We aim to provide more equitable access to high-quality math enrichment. Since I began, we’ve quadrupled in size and now serve over 800 students.

We strive to make math enrichment accessible by reducing economic hurdles, fighting barriers introduced by geographic and school segregation, attending to student identity during sessions, and promoting interpersonal connections.

In 2015 MC2 had three sites, two on Chicago’s north side (the wealthier and whiter part of the city) and one on the near south side. Four years later we have circles at eight sites that are geographically dispersed around the city. This fall we opened a new site meeting on Saturday mornings in Back of the Yards, one of Chicago’s poorest communities, with an enrollment of over 60 students.

All of these programs are free to families.

Our population of teachers has diversified as we have grown. Initially many of our leading teachers were in graduate programs at universities like UIC, UChicago, and DePaul. While we continue to add more such teachers, we now have many more classroom math teachers, both from middle schools and high schools around the city.

The inclusion of teachers, particularly those teaching middle school, means that as a community that we have much more institutional knowledge of working with younger children (we serve kids in 5th to 12th grades, and in practice more than 80% of those served are middle schoolers).

We also provide workshops for our teachers—from 2015-2018 Dolciani Math Enrichment grants funded the development and implementation of workshops like, “What is a Math Circle?”, “Math Circle Teaching Basics”, and “Intermediate Problem Solving”, along with workshops built around group observations of math circle sessions. These workshops build connections between our teachers and build an esprit de coeur.

MC2 is still very much a work in progress. We need to improve in the evaluation of the work we are doing, particularly to measure whether it’s effective in meeting the needs of the varied children we are trying to serve. It takes time to build relationships within a community, with teachers who can both lead sessions and who can help advertise this opportunity effectively.

Messaging matters, and over time we’ve found that expressing clear core values helps us find the right teachers, community members, and family members.

- Math should be fun and empowering.
- Every child can do and can enjoy rich mathematics.
- Every child deserves equitable access to rich mathematics.
- Students should be agents of their own learning.
- Math can and should be collaborative.

This stance has made us discourage the use of the word ‘gifted’. We certainly draw many well-prepared students (among others), but we are careful to signal that the math experiences we offer are for everyone.

I started by talking about the popular image of mathematics, but changing that image is not enough. The substance of the experiences we provide—Competitive or collaborative? Cooperative or individual? Mathematically connected or disconnected?—will determine whether we can attract a broader audience to the intellectual joy that we know mathematics can offer.

The mathematical community is powerful. High stakes tests make mathematics a pathway to many opportunities, in college, in careers, and in our ability to influence our communities. Many of us spend a significant amount of time writing, coaching, and judging contests. I think it’s time to question how much time is spent this way, and how, proportionally, we might spend more time involved in collaborative, cooperative, connected mathematical experiences that provide access to the many, and a deeper experience for all.

]]>

Texas State University

“[Functions] are completely different, which is what makes this course so challenging.” – Abstract Algebra Student

Functions are hard for students, even students in abstract algebra courses. Even if students have seen the definition and worked with examples of real functions throughout high school and college, their understanding might be stretched to a breaking point when it comes to ideas like homomorphisms on groups or rings. Fundamentally, we might know students don’t understand functions, but the extent to which they don’t understand functions goes deeper than we might think. In this blogpost, we will share some insights from our projects on key places where students’ idea of function can be detrimental to learning concepts of abstract algebra, and what we as instructors might be able to do about this.

The theme of our work is fortifying students’ understanding of functions while also forging through an abstract algebra curriculum. The issue of filling in gaps in prerequisite knowledge while still staying faithful to the current course is a challenge. We take on this challenge and hope that this post spurs others to take on this challenge with us, whether you teach abstract algebra or calculus, or any other course that relies on functions.

We have organized the post by projects led by the authors. We give an overview of each project and then describe what we have learned from each.

Project 1: The Group Theory Concept Assessment (GTCA)

Melhuish began developing this assessment as part of her dissertation, when she realized that there was no instrument that looked at student understanding of group theory (Melhuish, 2019). This assessment’s results, including over 1,000 responses from abstract algebra students over the past 5 years, as well as interviews with students about questions on the assessment, got us thinking: how do abstract algebra students think about function? Here’s one example. (And if this gets you curious as well, there are more examples in Melhuish and Fagan (2018).)

Function Understanding Thought 1: Abstract Algebra Students May Need a “Process” Understanding of Functions

Question (condensed): What is the kernel of ϕ(n)=i^n, ϕ:(Z,+)→(C,*)?

In total, fewer than 50% of students answered this question correctly. Many students, to our surprise, answered this item by providing a singleton element such as “{4}.” Yet, when we interviewed students about their responses, they typically provided an accurate definition that even contained language like “elements that map to the identity.”

We hypothesize that many abstract algebra students may still concieve of functions as “actions” (Breidenbach, Dubinsky, Hawks, & Nichols, 1992), meaning that they are not looking at a function holistically but rather element-by-element. Although this can helpful for direct computations such as “What is ϕ(4)?”, it is less helpful for finding preimages, particularly when the preimage has multiple elements (as kernels often do). For this, students may need to understand functions as a “process,” meaning that the function can be appreciated beyond acting on single elements. Students need to develop a process understanding to master early results in abstract algebra, such as the quotient struture of the First Isomorphism Theorem.

How might we address this? We suggest getting students to get their hands dirty with homomorphisms that are not one-to-one. Although students can recite the fact that a pre-image can have multiple elements, they might not have readily avaliable examples. After exploring examples, instructors can ask questions like, “How do you find all the elements that map to a particular location?” Students then reflect out loud and explicitly on how to generalize what they have explored. We also find it helpful to use relation diagrams during these discussions, such as those shown in Figure 1. Although these kinds of questions and diagrams can be used at many levels, including high school, we suggest that they are still worth taking up in advanced courses.

Project 2: Function Coherence in Abstract Algebra

As a next step, we interviewed a group of students to better understand how they connected functions in abstract algebra with their previous knowledge of functions in calculus and algebra. Our goal was to get a better understanding of their struggles to understand functions in an abstract setting (Melhuish, Lew, Hicks, & Kandasamy, 2019).

Function Understanding Thought 2: Abstract Algebra Students May Not Have a Mathematically Complete Definition of Function

We asked students: What is the definition of a function?

Across the six students we interviewed, only one could provide a complete definition for function (Student B). Here are the definitions the students gave:

Student A: A function is a relationship that maps members of the domain to a member of the range.

Student B: A function is a relation from one set to another where all the elements in the domain should be mapped to at most one element in the co-domain.

Student C: A function is a relation that takes an input and assigns it to exactly one output.

Student D: [An] equation that will do the same kind of operation to an input to get an output.

Student E: A function or a mapping takes a domain to a codomain following set rules.

Student F: A relation between two sets; can be one-to-one, onto, both, and/or neither.

When asked about their definitions, these students often appealed to the vertical line test (or in some cases, incorrectly, a horizontal line test.) However, in a setting like abstract algebra, this graph-based test can cause problems. When we graph a function, we study it analytically rather than algebraically.

How might we address this? We suggest not only stating the definition of function, but also getting students to talk about how this definition relates to maps in abstract algebra. Instructors can draw diagrams of homomorphisms, as well as show tables of homomorphisms, and help students see that one output is assigned to each input. By going back and forth between these diagrams and the formal definition of function, students have a better chance of seeing that the definition applies to abstract algebra.

Function Understanding Thought 3: Abstract Algebra Students May Not See Functions in Abstract Algebra as Functions

We asked students: Are the “functions” in abstract algebra the same as the functions in your prior mathematical experience?

While three students did think functions in abstract algebra were like prior functions, but with domains no longer limited to numbers, the other three students did not. One said, “What I thought of about function is always something to do with graph”; and another said that functions in abstract algebra don’t have a particular “rule.” These second three students—who focused on how different functions seemed in abstract algebra—struggled with the homomorphism tasks. Some believed that non-functions could still be homomorphisms. The first three students had more success with homomorphism tasks.

How might we address this? Here is a task we have found helpful: Which of the diagrams in Figure 1 could be a homomorphism? Which could be an isomorphism? Many students state that a diagram like the top-right could be a homomorphism (even though it is not a function.) This exploration could lead to discussion that both emphasizes the definition of function and explains that homomorphisms must themselves meet this requirement.

Figure 1. Relation Diagrams. Each arrow points left to right, so these are diagrams of relations from G to H.

Project 3: Orchestrating Discussions Around Proof (Easier Said Than Done)

We are currently developing curriculum materials for abstract algebra instructors who want to facilitate deeper classroom discussions while still meeting coverage goals. You may have noticed similarities in our suggestions so far: talk about definitions and properties more explicitly, and use function diagrams. These suggestions show up throughout in our materials. In this section, we share some obstacles that emerged, and also say why these types of moves were ultimately productive.

To begin, we turn to students’ responses to a sample proof that isomorphisms preserve the property of being Abelian.

Figure 2. One sample proof.

Function Understanding Thought 4: Explicitly Analyzing Proofs for Function Properties Can Be Powerful, but Students May Need to Disentangle the Function Definition (well-defined; everywhere defined) from Function Properties (1-1; onto)

We asked students, “Do we need a function to be one-to-one and onto in order for the Abelian property to be preserved? Where do we see these properties in the proof?” (Consider the sample proof in Figure 2.)

Initially, we wrote this question thinking that that students would recognize that, in this proof, the fact that an isomorphism is “onto” is critical, while being “one-to-one” is not. However, we found that students identified the need for onto, but also (incorrectly) argued that onto was also needed to show the existence of ϕ(g) of any g∈G. Later in the proof, the students identified claims like ab = ba implying ϕ(ab)=ϕ(ba) as a consequence of the function being one-to-one.

In conversations with the two groups of students who have tested this unit thus far, we have found that abstract algebra students have not disentangled everywhere defined and onto, and 1-1 and well-defined respectively. The properties of 1-1 and onto play substantial roles in a number of abstract algebra proofs. However, we rarely contrast them directly with function definition properties which may leave properties conflated.

How did we address this? We used the students’ claims about where 1-1 and onto were needed as a springboard to return to the definition of function, 1-1, and onto. This allowed students to discuss what well-defined (everywhere-defined) meant, and how well-defined differs from 1-1 (onto). Through these conversations, the students disentangled the properties and eventually arrived at the conclusion that the property of “onto” is needed, but 1-1 is not.

Function Understanding Thought 5: Function Diagrams Can Be Powerful, But Students May Need to Support to Understand and Leverage Them

We asked students: How can you represent the function in the proof (that isomorphism preserves the Abelian property) using a diagram?

In an attempt to help students sort through these functional attributes and properties, we asked students to use function diagrams to make the ideas more concrete. To our surprise, students struggled with a number of aspects of representing function as diagrams.

Several students seemed to lack attention to the domain (G) serving as the first puddle and the co-domain (H) as the second. Although they labeled the puddles G and H, in numerous instances, the students also labeled image elements in the domain, obscuring where the elements came from, as in Figure 3. The students’ diagrams also illustrated a disconnect between notation such as ϕ(b) representing the image of b. In Figure 4, the student mapped element b to ϕ(a) and not ϕ(b).

Figure 3. Student function diagram displaying difficulty with technical aspect

As the proof contains numerous references to the image of particular elements, the lack of alignment between students’ diagrams and function notation may suggest that students’ understanding of functions (and the image of elements) may obscure important aspects of proofs. At the same time, this lack of alignment gives an opportunity for the instructor to help students develop stronger and more coherent understandings of function.

Figure 4. Student function diagram displaying difficulty with conceptual aspect

How might we address this? Although we have identified some unexpected obstacles that instructors may run into, we found overall that engaging students with function diagrams did serve an important role. Our recommendation is to prompt students to connect proofs to such representations. Through this process, it may become clear that students have some technical and conceptual issues related to functions that may otherwise stay hidden. By having a concrete representation, instructors and students can work to modify the diagrams to have normative features- and ultimatley leverage them as tools to analyze proofs.

Final Thoughts

This post is not meant to provide a thorough analysis of students’ function understanding. Instead we aimed to highlight some of the many roles understanding (or lack thereof) of functions may play in abstract algebra. As instructors, we may overlook something as simple as students having an accurate definition for function or seeing the functions in abstract algebra as examples of functions. However, these oversights might hide the fundamental role that functions play in an abstract course. In a setting like abstract algebra, students are stripped of many of their tools for understanding functions, such as relying on using graphical representations. Moreover, students in abstract courses often have to move beyond an action conception of function to a process conception. Additionally the role of function definition properties are essential when constructing or understanding proofs. However, we often leave the connections to function properties implicit. This is a disservice to students who find themselves in a setting where tools like the vertical line test cannot substitute for a function definition.

We have found much potential for explicit discussion around the definition of function, properties of functions, and function diagrams support students as they engage with various functions in abstract algebra. The biggest recommendation we can make is to not let functions continue to hide behind the curtain. Functions are everywhere in abstract algebra, and students’ understanding of functions can position them for varying levels of success (or lack of success) when engaging with abstract algebra tasks.

References

Breidenbach, D., Dubinsky, E., Hawks, J., & Nichols, D. (1992). Development of the process conception of function. Educational studies in mathematics, 23(3), 247-285.

Melhuish, K. (2019). The Group Theory Concept Assessment: a Tool for Measuring Conceptual Understanding in Introductory Group Theory. International Journal of Research in Undergraduate Mathematics Education, 1-35.

Melhuish, K., Lew, K., Hicks, M., & Kandasamy, S. (2019). Abstract algebra students’ function-related understanding and activity. In Proceedings of the 22nd Annual Conference on Research in Undergraduate Mathematics Education (pp. 419-427).

Melhuish, K., & Fagan, J. (2018). Connecting the Group Theory Concept Assessment to Core Concepts at the Secondary Level. In Connecting Abstract Algebra to Secondary Mathematics, for Secondary Mathematics Teachers (pp. 19-45). Springer, Cham.

]]>**I.**

What do primary/secondary math educators think of the teaching that happens in colleges? And — the other way around — what do mathematics professors think of primary and secondary math teaching?

I’m nearing my tenth year as a primary and secondary classroom math teacher, and every once in a while I end up in a conversation with a graduate student or professor who suggests (politely, almost always!) that math education before college is fundamentally broken. A few weeks ago, a mathematician told me that PhDs are needed to help redeem secondary teaching from its “sins.” Once, at the summer camp where I teach, a young graduate student told me that there is simply no real math happening in American schools.

Well — I disagree! But how widespread is that view? And why does it exist?

The flipside phenomenon is also interesting. When a mathematician criticizes primary/secondary math education, primary/secondary educators sometimes lash back. Often we point our collective finger at pure lecture. Primary/secondary educators tend to think of pure lecture as uniquely ineffective. It gives the teacher no knowledge of whether students understand the material, and students no chance to practice new ideas in class. It is rarely used in primary/secondary math classes. Still, pure lecture was the main teaching mode in my own college classes, across subjects. We therefore bristle at mathematicians critiquing our work; “Let those without pedagogical sin throw the first stone!” I’ve even said this before, or something not far from it.

I have heard this “anti-lecture” critique expressed by some primary/secondary educators, but I wonder how widely held this view is. Is it held even by some math professors? And, in general, do primary and secondary educators tend to see flaws in the way math is taught in colleges?

In short, I wanted to better understand how mathematics professors and educators of younger students relate to each other’s teaching.

Needing a way to approach the question, I created a survey and shared it on social media. I asked people to share their job, the highest degree they have earned, and their views on primary, secondary, and post-secondary education. So far, thirty-four people working in math or math education shared their reactions to these three statements:

- “Primary (K-8) math teaching is generally effective.”
- “Secondary (9-12) math teaching is generally effective.”
- “College (or University) math teaching is generally effective.”

They rated their agreement/disagreement on a scale of 1 through 5 and explained their response at whatever length they chose.

Disclaimer: since the survey made no effort to be representative, themes and patterns that emerged from the responses are at most suggestive. Nonetheless, suggestive themes and patterns did emerge, some of which surprised me.

There were three questions I wanted to more clearly understand:

- What exactly
*are*the issues people have with math teaching at the elementary, secondary and post-secondary levels? - Do people tend to have a rosy view of their own setting while leveling harsh critiques against what goes on in other settings?
- Where do extreme views of different educational settings come from, and why are they sometimes so deeply held?

The background question, though, the one driving this whole project, is whether there is any chance of coming closer together. What are the best ways for us to learn from each other?

**II.**

One mathematics PhD who responded to the survey rated the effectiveness of primary math teaching as just 2 out of 5. This same person rated secondary and college math both at 4. Here is how they explained the ratings:

My experience with a couple of districts is that the primary teachers up through Grade 4 tend to feel uncomfortable with math, so they do their best to opt out of teaching it as much as they can. I have some horror stories, including concerned parents of students who brought their math workbook home at the end of the year untouched.

Here is a college mathematics professor voicing similar views:

I think many elementary teachers are math phobic or have math anxiety, and this impacts how much time they spend on math. Also weak skills and knowledge of teachers could lead to misconceptions for students.

This view — that primary teachers tend to be uncomfortable with (or ignorant of) math, and therefore avoid it — showed up again and again in the survey responses.

I was expecting this, because it is maybe the most prominent critique of math education at any level. It’s the sort of thing that, every so often, pops up in the New York Times. A prominent version of this complaint, for instance, comes from Hung-Hsi Wu:

My own observation is that among teachers, especially elementary teachers, their prolonged immersion in textbook school mathematics has often rendered them incapable of routinely asking why, much less looking for the answer.

What surprised me, though, was how frequently this concern (or ones like it) was also voiced by those working in primary and secondary education. In fact, the math PhD quoted at the top of this section is actually a high school teacher. Meanwhile, a 5th Grade math teacher rated primary teaching’s effectiveness at 2 out of 5, and explained that what primary teachers need most of all is the help of math pedagogy specialists. They can’t handle the math on their own, it seems. Likewise, a high school department chair blamed “lack of content knowledge” for a “strong focus on algorithms” at the elementary level.

The second half of this department chair’s worry — the “strong focus on algorithms” — appeared often on the survey as well, though expressed in slightly different ways. Here is a sample, along with the respondent’s professional role and their rating of primary education’s effectiveness:

Too often students are taught processes instead of concepts (8th Grade math teacher, 3 out of 5)

In the memorization and skills-based way mathematics is taught in most K-8 classrooms, I do not think teaching is effective. (High school math teacher/community college adjunct, 2 out of 5)

Are there shades of distinction between “algorithms,” “processes” and “memorization”? Maybe. But I think there is more in common than not in these complaints, and they are probably trying to say something like what Wu said above — there is no *thinking* going on in too many primary classrooms. The reason? Because teachers fear or misunderstand mathematics. I would say that this picture is the major critical narrative facing primary education.

To put my cards on the table, I think this narrative probably overstates the problem, though it gets at something real. I’ve heard something like this story told by many primary teacher educators and coaches — people who really would know. I do think this critical story misses two important things: (1) what looks like a mindless call for a procedure to an adult is often a thought-provoking and interesting problem for children, when presented appropriately and (2) a lot of good schools do a great job of helping math-phobic teachers teach math at a high level. They provide training, coaching and strong curricular materials that can help teachers overcome their fears and become more mathematically confident. My wife, for one, worked through a lot of her math phobia when she used the TERC Investigations curriculum to teach multiplication to her 4th Grade class.

So much for primary teaching. What about math teaching at colleges and universities?

My own college teaching relied almost entirely on pure lecture as a classroom teaching technique; often the teacher would not even pause for questions. I was fully expecting to hear this come up in the survey. I thought I would hear it from my primary/secondary colleagues, but I was curious to know whether any of those working in higher education would raise the “pure lecture” critique themselves.

The answer was no. The “pure lecture” critique certainly did come up, as expected. It was almost entirely raised by those working in primary/secondary teaching. The high school department chair (quoted above) mentioned “a preponderance of lecture as the instructional strategy” in college classes as an issue. An instructional coach at a middle school bemoaned “lecture-based math classes.” Another response, from someone who works with middle school students, critiqued “lectures that are too difficult to follow, or very hard to be engaged in.”

This “pure lecture” critique was not raised *at all* by professors or graduate students. It came only from primary/secondary educators in my survey.

However, college mathematics teachers did raise other issues. One college teacher (“assistant professor of mathematics at a small private liberal arts college”) described college math teaching as “highly variable, and depends strongly on how much the institution and the individual instructor value teaching as part of the academic job.” Another (“visiting Professor of Math at a 4 year college”) wrote that “the biggest issue for post-secondary mathematics is the mindset of *I know mathematics therefore I can teach it and I will have minimal pushback because I have a terminal degree in mathematics*.”

This narrative is different than the “lecture” complaint. It alleges that some professors either do not value teaching highly or that they are too confident in their ability to teach well. *Strong* content knowledge can perhaps present its own challenges for a teacher of mathematics; deep knowledge can make it difficult to understand the point of view of the struggling student. Curiously, this was only brought up by college teachers; primary/secondary teachers didn’t mention it.

To sum up the situation, in this informal survey, the knock against primary teaching was that its teachers avoid or misunderstand math. This results in students being presented with a distorted picture of mathematics as a subject devoid of thinking but full of procedures to follow. You could even hear this critique from primary/secondary educators themselves. On the other hand, primary/secondary educators were somewhat apt to critique college teachers as relying too heavily on pure lecture. College teachers themselves did not bring up lecture, but did mention the relationship of professors to the work of teaching as an area of concern.

One last thing: I haven’t mentioned what people said about secondary teachers! This is for two reasons: (a) there’s plenty to talk about with primary/university, but also (b) secondary teachers seemed sort of stuck in the middle on the survey. The critiques they (we!) received seem to me best understood as watered-down versions of the concerns leveled at primary teachers. Secondary teachers are taken to have stronger content knowledge, but in the survey we were said to be too procedural, too dry, too focused on memorization and not enough attuned to the needs of the discipline. There were more strong opinions about primary and college than secondary teachers, who seemed to get a slightly different, but weaker, version of the critiques aimed at primary teachers.

**III.**

I had assumed that the responses to my survey would be super-skewed, with everyone defending their own turf but taking issue with the work in other educational settings. But for the most part this was *not* true. Without getting statistically precise about this, people’s ratings stuck pretty close to whatever their own baseline was. People who thought math education was basically working *across the board *didn’t distinguish a lot between primary, secondary and post-secondary schooling. Likewise, there were many others who thought that math education was *across the board *not getting the job done, and they didn’t distinguish very much between settings.

(Getting statistically precise: people’s ratings did not deviate much from the *mean* of the three ratings they provided. The absolute deviation from the mean of their three ratings was 0.493, on average.)

I was surprised by this. I had expected mathematicians would have *much* harsher things to say about primary/secondary education, as compared to their own work. But, at least on my little survey, asymmetrical harshness turned out to be the exception rather than the rule. For instance, one respondent with a Master’s in mathematics said, “The early grades K-3 tend to explore more and most kids are not left behind.” That doesn’t sound damning at all!

Echoing this was a PhD and college mathematics instructor: “Especially with better training nowadays, I generally think that elementary teachers do a pretty good job.” That’s also pretty positive; this person gave primary and secondary teaching a 4 and college teaching a 3. People who were overall cheery about math education tended to be so across the board, even across educational settings.

The flipside also tended to be true. People who were critical of math education *overall *often made significant criticisms of their own educational setting. The source of many strident critiques against primary education, for example, came from those who oversee primary math education. One of my favorite lines on the survey was from a mathematician who explained their middling evaluation of primary education: “I read about it in NCTM literature.”

(NCTM is the National Council of Teachers of Mathematics, the largest professional organization for primary and secondary math educators in the United States. They have been trying to reform primary/secondary math education for decades and are often highly critical of the way mathematics is typically taught to younger children.)

This is interesting! It suggests that we are not exactly a profession divided against itself, as much as a profession that can’t seem to agree as to whether things are fundamentally broken or not. (I tend to side with the “not”s, for the record.)

In general, things were both less critical and less adversarial than I was expecting. People tended to judge the *entire *math education system, from primary through university, as a whole. One high school teacher said, “I’m certain that experiences differ widely (as with primary and secondary), but I had good university teachers and I learned a lot.” One mathematics teacher educator repeated the same comment for primary, secondary and college education: “Too much focuses on procedural knowledge and less on higher-order thinking.”

Both of those views makes a lot of sense to me; whether good or bad, we’re all in this together.

**IV.**

To summarize:

- There wasn’t a great deal of
*quantitative*polarization; people tended to be overall happy or overall unhappy with how things are going in math education. - There was, however, a great
*qualitative*difference in the issues critics recognized in primary, university and (to a lesser extent) secondary math teaching.

It’s worth dwelling on this, if only for a moment. Critics identify *entirely different flaws *in primary and college education. The “bad version” of primary teaching looks almost nothing like the “bad version” of college teaching. The stereotypes, to whatever extent they are believed, are completely unlike each other.

This is fertile ground for extreme views. Under these conditions, people can identify problems with other areas of math education and think to themselves, *nothing *like that is happening *where I teach* — and they would be largely correct. If you are a middle school teacher, none of your colleagues could ever be accused of talking for the whole period straight. And whatever problems exist in college teaching, nobody would ever accuse a professor of turning mathematics into nothing but mindless routines. Quite the opposite! Students are, absolutely, asked to think.

These differences, I feel confident in saying, result from deep differences between our teaching contexts. It is sometimes tempting to see the similarities between our educational settings — there are students, desks, whiteboards and textbooks — instead of uncovering the deeper structural differences. But the differences are vast! Here are just a few of the variables that differ in significant ways between primary/secondary and college classes: the number of students in a class; the frequency of the class’ meeting; whether we are accountable to a test or not; the ability of students to study the material independently; our control over the curriculum; etc. We could easily name more.

These contextual differences should make us slower to come up with educational solutions for other people’s problems. If our educational settings are different enough that *bad *teaching looks different, it seems to me that *good *teaching in primary, secondary and college settings ought to look very different as well. This means that we can’t simply assume that the techniques that are useful for teaching math in one setting will also be useful in another. Which means that we should be cautious before offering advice to those who teach in other contexts based on our own teaching experiences.

It reminds me of something Neil Gaiman says about writing:

When people tell you something’s wrong or doesn’t work for them, they are almost always right. When they tell you exactly what they think is wrong and how to fix it, they are almost always wrong.

When those outside our professional setting tell us that our teaching is not as successful as it should be, we should listen. But as the criticisms and prescriptions get more specific, they almost always grow less useful, at least to me. The culprit is context; we rarely truly understand other people’s constraints.

I don’t intend to sound pessimistic. I think we really can learn from each other’s perspectives. But specific solutions and criticisms can only be supported by a deep understanding of the teaching context. That’s why people who move between these educational settings are capable of doing such important work. These are the PhD graduates who become high school department chairs, the primary teachers who attend mathematics lectures, the secondary teachers who pursue graduate work in mathematics. These are the people who can not only hear the criticisms, but can turn them into something that really works.

In various little ways I have benefited from others who have done this work. When I began teaching math to 3rd Graders at my school, I was committed to sharing “real” mathematics with these students. What surprised me, though, was just how real the mathematics of the curriculum is for these students. Just in the past few weeks I have heard some pretty amazing mathematical conversations about fairly straightforward mathematical questions; things such as 120 + __ = 210 and 3 x 8 = __. But, as mathematical critics of primary teaching have said, there is more to mathematics than arithmetic, and I wanted to expose my students to more. How?

Joel David Hamkins is a professor of logic. At some point I came across his blog, and found materials he posted. For several years, Hamkins had gone in to his daughter’s elementary classroom as a guest math teacher. Each year he put together a pamphlet of problems for the children, and was generous enough to share them online.

For the last several years, my students have loved doing his “Graph Theory for Kids” for a few days each year. They learn about circuits, planar and non-planar graphs, and chromatic numbers. They color maps and hear, for the first time, of the four-color theorem. I feel so grateful that Hamkins was able to really be there, in person, to teach his daughter and her elementary classmates. Do other professional mathematicians do classroom visits? If not, couldn’t they? What if our professional organizations were to organize such things, at some sort of scale?

Primary and secondary teachers have a unique sort of expertise in pedagogy, but I see no way for us to share it unless we tangle in-depth with the context of college math teaching. I have never heard of primary/secondary educators visiting college courses as guest speakers, but why not? Surely there are some who work in primary/secondary settings who could be invited to give lessons or talks in university math courses. We could try to adapt our methods to the university setting, and work together with professors to design different styles of lessons. Could this be a way to learn what primary/secondary pedagogy looks like in a different setting?

The truth is that I am optimistic that something like this cross-pollination is already occurring, though very slowly. For the past few years I have taught at a summer math camp in New York City. The students are all entering the 7th Grade; the faculty are about evenly split between middle/high school teachers, graduate students and college teachers. Every summer, I’m surprised by the sorts of pedagogical discussions we end up having. Assumptions are frequently challenged as we all look for a common language to describe our teaching. For six weeks, we talk daily about how we are structuring our lessons and helping our students. We’ve seen each other teach, we’ve seen the problems each has shared with their students, and we’ve shared our successes and struggles.

When camp is over we say goodbye to each other, taking whatever ideas we have learned over the past few weeks, and help students learn mathematics in all sorts of different classrooms, wherever they happen to be.

]]>After my day-to-day interactions with students, one of my favorite things about teaching is talking with other teachers. There is no shortage of amazing teachers who are working hard to make their classes better and improve student learning. Likewise, there are plenty of opportunities to find inspiration in our colleagues’ work, ranging from attending talks at conferences to simply getting coffee with coworkers to talk about how our classes are going.

A few years ago, I realized that the proportion of inspiring ideas that turned into measurable change in my classroom was essentially zero. As I thought more about this, I realized that *I* was the biggest hurdle to this change. There was a little voice in the back of my head with a constant and emphatic message: *No. I can’t do that, and here are fifteen reasons why. *

I know I’m not the only one who hears this voice. Of course, the reason we have these thoughts is that they are often true. No two people experience teaching in the same way. We have different personalities, different styles, and allow for organized chaos in different ways. As a community, it is easy for us to despair in the challenges we face in our teaching.

Joan Baez said, “Action is the antidote to despair.” At the end of the day we are all mathematicians and we have been trained in solving problems. To be apathetic in the face of the challenges put before us is antithetical to our training as problem solvers. And teaching, particularly teaching well, should be viewed as a problem that desperately needs to be solved. Like many real-world problems, the problem (“What does it mean to teach well?”) is not clearly defined. The data is messy. There is not one single correct answer.

In the rest of this post, I would like to discuss some methods for moving beyond the little voice that says “no” and changing your teaching without reinventing the proverbial wheel. And, as with many real-world problems, I will not answer the question at hand (“What does it mean to teach well?”) and instead I will address a different question – How do I teach better?

In 2012 I saw David Pengelley give a wonderful talk in which he outlined a rather intricate system of assigning and grading calculus problems of varying difficulties over several sections of the textbook, all of which were meant to be done by the students before class and accompanied by an email to the professor reporting on the section of the textbook they had just read. This happened for every class meeting. The mental yoga of keeping track of so much material at once made my head spin, and I did not (and still do not) have the confidence to implement something on such a grand scale. (A summary of David’s method can be found here: https://www.ams.org/publications/journals/notices/201708/rnoti-p903.pdf)

But the ideas in the talk were inspiring. I left the talk telling myself, “I should create opportunities for my students work on math more regularly.” I started giving mini-assignments in all of my classes on a daily basis. The mini-assignments consist of about five relatively simple problems related to the material covered in a given class, which are due at the beginning of the next class.

I can mark the papers and record grades in about 15 minutes per assignment (I do this by hand – if you use WebWork it takes no time at all). It helps make sure the students are engaging with the class material throughout the week instead of just the night before a bigger homework set is due. It also lets me focus my grading efforts on problems with more substance that better measure what the students are learning. This works well for my teaching philosophy and helps make the students stay on top of the course material.

Our teaching experiences are shaped by a large number of variables, including student preparation, class size, resources, research expectations, and other demands on our time. Because of this, the things that my colleagues who work at ivy league schools are able to do in the classroom often are not feasible for me, just as the things that have worked well for me in a class of 20 students might not scale well to a class of 300 students. However, rather than finding all the ways that a technique cannot be applied to your classes, look for ways to transform that solution into something that will work for you.

When I dissect the “no, because…” statements that persistently come to my mind, I find two main themes. The first main issue is one of practicality – what works for person X may not work for person Y – and we have already discussed this. A second main theme, whether we like to admit it or not, can be boiled down to an issue of discomfort or fear.

Change is hard. For most of us, our educational lives consisted largely of lectures to be observed and internalized. This is comfortable because it is what we know best. It is a known quantity in which we hold significant control. Letting go of some of that control is scary. What if it doesn’t work? What if the students don’t like it? What if I don’t do a good job? These are all legitimate concerns, and we can work through those concerns once we are honest with ourselves about their root causes.

Change does not happen overnight. Maybe you are interested in having a more active classroom or having a flipped classroom where students do more work outside of class, but you don’t want to give up complete control to the potential chaos that might come from this. Maybe you don’t have the time or resources to completely re-invent your differential equations course because you’re on sixteen committees and teach three classes per semester. I get it.

You don’t need to do it all at once, and you don’t need to do it all by yourself. Ask colleagues to lend you worksheets or materials. Find activities that help you move towards this goal and try them out a couple times over the course of a term. After a few years of doing this, you will have developed more resources and gained more confidence in this approach, homing in on techniques that work for your personality and style while also better serving your students.

New teaching methods aren’t going to be perfect the first time. You’ll probably mis-judge the difficulty of some tasks. Students might get frustrated. You might get frustrated. But it also won’t be a disaster. Students will still learn. So will you. It will be better next time.

Be prepared for the eventuality that a lesson won’t go as planned. It is hard to know how long an activity will take, but you can bet that it will usually take longer than you think. Don’t be afraid to change an activity in the middle of class if it is taking longer than you expected or the students aren’t getting it. You can always change your course schedule to adapt to this change. Besides, if you try five new lessons in a semester and two of them don’t go as well as you had hoped, that still accounts for a relatively small proportion of the overall class.

At the end of a course where you’re trying new things, reflect honestly about the successes and struggles you faced. Were there common themes in the struggles you faced? How can you fix them next time? Make a list of three things you’d do differently next time. Don’t be too hard on yourself.

There’s a rule in improv comedy called the “yes, and…” principle. If you’re in the middle of an improv comedy skit, you can only react to the material you are given from your collaborators. You may have been ready to tell a very good joke about Care Bears, but now someone has decided that there’s a grizzly bear running around the stage and you need to act on that instead. You can’t stop the skit and ask for a retake, so instead you have to accept the reality of the grizzly bear and add your own brand of humor to it.

The same principle applies to teaching. We can all become better teachers by finding inspiration in others. This takes work, and it can be scary to take a risk and try something new in the classroom. In many cases, we fail to apply the lessons our peers have learned because we feel their experiences do not directly translate to our own. Next time you go to a talk about teaching, I challenge you to move beyond the naysaying gremlins in your head. View the reasons to say no as equations that bound the parameter space of your problem. Say yes to new ideas and apply them in your classroom in a way that works for you. Over time, these small changes can add up to more effective teaching.

]]>Elena Galaktionova sent us this article shortly before she passed away earlier this year.

Elena Galaktionova received her first introduction to mathematics from her favorite middle school teacher in Minsk, Belarus, her hometown. After she had finished her education at the Belarusian State University she went on to receive a Ph.D. from the University of Massachusetts in Amherst. Her area of research was representation theory. She taught mathematics for many years at the University of South Alabama, after some earlier stints at the Alabama School of Math and Science and the School of Computing at USA. In Mobile, Alabama, she was one of the organizers and teachers of the Mobile Mathematics Circle. The Circle has been going strong for 20 years. Later she recruited local teachers and a middle school principal to participate as a team at an AIM workshop on Math Teacher Circles. Upon return to Mobile she founded the Mobile Math Teachers’ Circle. Twice she gave presentations at the Circle on the Road conferences. Her work with local middle schools and her interests in home schooling were motivated by her love for mathematics. She cared deeply about math education. Sadly, Elena passed away earlier this year after a long battle with cancer.

In all my classes I try to teach reasoning, writing and problem-solving skills. I noticed that if a class is heavy on computations and dense in content, such as Calculus, the result of this effort is barely noticeable if at all. I recall a memorable moment in a multi-variable calculus class. The topic was optimization. My students knew just fine how to use the Lagrange multiplier method given a function and a constraint, thank you much. But it turned out they were helpless in the face of even the simplest application problems. Some of these students were studying Calculus with me for almost three semesters and their grades were good and I tried so hard to teach them what matters in mathematics the most. I remember a chilling realization at the moment, that we — the students and I — wasted three semesters.

A very different experience comes from another course. At our university it is called “Foundations of mathematics”. Unlike other math classes it does not have a lot of content. A bit of logic, set theory, relations, maybe some number theory. It is the first class where students are learning to write proofs. This is a writing-intense class. There are essentially no calculations. I collect the homework every class period and grade the same way one would grade an essay. My first requirement is writing in grammatically correct meaningful English sentences. This is not an easy task for most. A lot of students by the time they start this class learned to perceive math as number and symbol manipulations. At the beginning of the course I often see in students’ work words that are strung together in rather random fashion. We go together over some of the responses asking questions like : “Is this an English sentence? What is the meaning? Are all the terms defined? How could it be misinterpreted?”

By the end of the semester I observe a turn-around: there is a palpable effort from even the weakest students to put their ideas into words. The change is most noticeable in weak students. The struggle for finding the right words and writing in grammatically correct sentences may be still there. While they did not suddenly became great at math, their mental activity and learning efforts are much more productive, since they are consciously directed towards comprehension and expressing their ideas verbally with a degree of precision.

I wondered if my students noticed this change themselves; that was until I was approached at the end of the semester by two of my “Foundations” students who emphatically told me how this course entirely changed the way they view and approach math. This is reflected in their grades in other math classes as well. For example, one of my Calculus II students was taking “Foundations” concurrently. Her grade in Calculus II changed from a D at the start of the semester to a B towards the middle. Most notably, she enthusiastically confirmed and told other students how much taking “Foundations” helps with Calculus II, despite having no content in common.

What is most interesting to me is the quantum character of this change and that it was especially noticeable in weaker students.

Young children come to school as a blank slate. Yet they have the innate ability for reasoning, they have curiosity, they are eager to play and explore. Over the years their teachers influence their perception of what math is about. Two of the possibilities are:

- math is a manipulation of numbers and symbols according to a predetermined set of rules;
- math is communicated through meaningful statements.

The students who do not do well in mathematics typically view math as a manipulation of symbols. The “making sense” switch changes this so the students begin to read and communicate mathematics as meaningful, logically connected statements.

To summarize, here is what I observed:

- Both exclusively formal processing of math tasks and making sense of math tasks are learned, eventually habitual, behaviors. Either one becomes a mental process which is practiced and reinforced in every math class.
- Effective learning of mathematics does not happen until mathematical communication is perceived as meaningful statements.
- Students who view math as a formal manipulation of numbers or symbols will habitually direct their effort and mental energy toward this in a math class, unless they are given problems which naturally invite reasoning and stay away from using formulae and rules. In a class with a computational component, such as pre-calculus or calculus, even if a teacher tries to teach reasoning and making sense, it has relatively little consequence: under stress, such as homework due the next day or a test, such students revert to their habits. Some of them spend a fair amount of time studying and reinforcing these habits, often getting frustrated because of the little return for their efforts.
- A dedicated computation-free and writing-intensive class which stays away from problems that may suggest formal manipulation can turn on the “making sense” switch. Students start to perceive mathematics as meaningful statements. They look for logical connections between the statements. Their verbal skills are productively challenged.

The important qualities for such class, assuming the main purpose is to turn on the “making sense” switch:

- The class should be writing intensive.
- The tasks are such that students can rely on their existing reasoning skills, common sense, intuition. They should not be too abstract. For example, it is easier to find appropriate problems in logic, set theory, elementary geometry or combinatorics than in abstract algebra. This allows students to scrutinize math statements using their own “sensometer” and keep working with them until they can make sense of them.
- The course should be light on content and big on thought, allowing sufficient time to think and write about problems.
- The class should not include tasks which could tempt students into formal manipulation.
- Feedback on writing is continuously provided by the teacher: students’ attention is brought to details of their writing, the meaning of what is written, and how the writing could be improved.
- The time required for the “switch” to turn on in such class is less than a semester. This is my experience with undergraduates.
- This works even if this writing intensive class is taken in parallel with other, computationally intensive math classes.
- Once the switch is turned on it stays on in other classes, even in those with computational components, as long as teachers pay attention to making sense and reasoning.
- A practical aside on grading: a class where non-routine and sometimes difficult problems are part of homework presents certain challenges for grading. I told my students that if they could not solve the problem, they should write down their attempts, for example, how they used problem-solving strategies discussed in class, such as looking at related simpler problems or generating examples and trying to find a pattern and showing why it did not work. Adequate effort and quality writing would earn almost full credit. Of course, it is important to also include easier problems which are within reach for nearly everyone. If I did not have sufficient time to grade the full homework, I selectively graded 2 or 3 problems.

Unfortunately, the “Foundations” class is a sophomore level for math and math education majors. In fact, no prerequisites are needed for it. So we started to encourage students to take this class as early as possible, when noticing that it helps them in other math classes. There is no reason why a class with similar characteristics and goals is not taught to seventh graders. It would improve their learning of mathematics for years to come. As an example, in Russia, the class that perfectly fits the bill is Geometry class. Systematic study of Geometry starts in 7th grade and continues through 11th grade. The Geometry class meets 2 or 3 times per week. All statements and theorems are proven based on what is already known. Thus, Geometry is presented as a unified theory and not a random collection of facts. The students are expected to state definitions and prove theorems and they solve problems involving proofs and geometric constructions. Of course, there are Russian students who struggle with writing proofs and deriving formulae. But they are used to the concept of intrinsic reasoning and they know that they are expected to articulate it. In the U.S., ask a class of either seventh graders or freshman Calculus students why a particular fact or formula is true and the answer invariably will be “Because our teacher told us so ” or “Because it says so in the book”.

]]>

Content is essential; so are strategies and craft for teaching; but there’s more. It’s often said that “many elementary teachers don’t really know the content; the content they ‘know’ they don’t really understand; often they don’t realize that there *is* anything to understand.”

However true that characterization of elementary teachers is, we think it’s a distraction. There is no kindergarten teacher *anywhere* who doesn’t know how to count and add and subtract, which is most of what her children will encounter during the year. And if the teacher isn’t sure of the name of some particular geometric shape, that’s way down in the noise of what will matter for teaching. Lack of mathematical *information*—even a lack of *understanding* of why particular algorithms work—is not the biggest roadblock in the earliest grades. The remedy might involve more courses in mathematics, especially mathematics they will teach, but we think that the key issue is not *more* but *different*, even for secondary teachers.

As we see it, what hurts elementary mathematics teaching most—and hurts secondary teaching as well—is some of the ways in which teachers know “too much” math without a tempering sense of what the mathematical enterprise *is* and what *not* to teach. We will give two examples, one from fourth grade and one from first year algebra, to illustrate what we mean.

**Fourth grade. **Over a span of years and in several schools, we’ve watched many fourth-grade teachers as they present , which their curriculum uses as a two-minute warm-up puzzle in advance of a unit that introduces the distributive property that students will apply to multi-digit multiplication. Every time we’ve watched, many children blurted out that had to be 8 as soon as they saw the puzzle, pleased to show that they knew the multiplication fact 7 8. The purpose of the puzzle, at this point, was merely to have kids recognize that the “8” could be the sum of two numbers and to have them come up with several possibilities for and . At this introductory moment, even checking that if *is* 8, then + also gave 56, would be overkill because that’s exactly the work students would next do on their own.

And yet, *all* of the teachers felt a need not only to point that out right away, before the unit started, but also to dive into vocabulary and other formalities. Referring to the and , several teachers asked questions like “What do you think the word ‘variable’ means?” even though the term isn’t used *anywhere* in the lesson. Children generally had no idea. Some teachers then defined variable as “a letter, like *x*, that stands for a number,” even though and aren’t letters! Why?! The teachers recognized something *they* knew and felt compelled to teach the children the “right way.” One teacher wrote out as a justification for the 8 that the kids had already shouted out. One teacher put up a table to show how values for and “should” be recorded. And after kids had offered a few possibilities for and , that same teacher took extra time to say that 4 + 4 — which no kid had suggested — “would not be right because then the equation would have to be written with two squares or two circles.”

The last statement is wrong, of course, but the big problem, in our view, is not the teacher’s error, but the teacher’s apparent feeling, common to all of the observations we made on this puzzle (and a vast number of other observations of other teaching) that everything the teacher knows about the situation is relevant *now*.

Teachers need to know what (and when) *not* to teach.

Of course, good teaching practice does involve looking for learning opportunities, sometimes milking a problem for more than what appears on the surface, but part of the mathematical preparation of teachers *at all levels* must include ways to decide when *not* to do that—what is *not* relevant at a particular time, or is essentially a diversion that will, at best, dilute the focus on an important idea and, at worst, mean nothing at all to the students. “A letter, like *x*, that stands for a number” was certainly one of those; it wasn’t needed and didn’t clarify anything. The *purpose* of the puzzle was to set the stage for a key mathematical idea—a property of multiplication that the nine-year-olds would first explore, then apply, and only after that formalize to extend their ability to multiply. The distraction subverts (perverts!) the mathematics. Worse, because it became nonsense, it can convince students that understanding is neither necessary nor, perhaps, even possible for them. By contrast, when students are given time to solidify an idea first, naming the idea becomes useful, helping them talk about it and even indicating that the idea is important enough to warrant a name.

Starting mathematics lessons with vocabulary and notation seems nearly universal, even among teachers who know from their language arts instructional methods that vocabulary is best learned in context. And elementary school (and often secondary) mathematics teachers seem not to distinguish conventions and vocabulary from what can be reasoned out or understood. We’ve seen that showing and asking “What can you say?,” encourages lively thought and participation even though it’s a piece of notation, because its form allows kids to make sensible guesses about it. On the other hand, asking “What do you think *variable* means?” shuts down logical thought. There’s no context. Kids who know *vary* might come up with a reasonable thought, relevant or not, but to these fourth graders *variable* could mean anything. Vocabulary and conventions are *needed* for clarity and precision of communication, but the mathematics is something else: logic and the inclination to puzzle through a problem and figure it out rather than the disposition to treat each problem as something for which one must first be taught a rule or method. When learning vocabulary and formulas becomes the *focus* of mathematics education children move away from the skills they need to be mathematicians and they don’t develop confidence in their own mathematical abilities. That is because people *can* puzzle through mathematics, but what things are called or how they are notated is convention and *can’t* be “figured out.” Children who proclaim themselves to be “bad at mathematics” are likely not to have seen mathematics as an exercise in logic and reasoning, and have likely not had enough opportunity to see how good they can be at that. Readers of an AMS blog know that memorizing vocabulary and formulas, while it can be useful, has little to do with mathematical aptitude, but many teachers have been prepared to think otherwise and thus emphasize those at the beginning of every lesson, giving students the false and often destructive idea that those *are* the math.

In our view, the fix for *this* particular problem with elementary teaching is not for teachers to learn more mathematical content, but to change teachers’ perception of what mathematics *is*—their sense of how the discipline works—staying mostly within content they already know or once knew.

For example, how many fourth-grade teachers have students do age-appropriate research to find patterns in multiplication facts? Here is a particularly striking pattern that most teachers have never even seen. Presenting to students can be entirely silent—*no “explaining*.*”* On a number line, choose a single number like 4, draw two arrows up from it and write 16, then two arrows from its neighbors 3 and 5, and the product 15. Then *start* the process from one other number (e.g., 3), writing the square 9, draw the neighbor arrows and let students call out their product 8. If students need another example to “get” what you’re doing, give the 8 and start a new pair (e.g., at 6) leaving the numbers to the students. Keep going until kids are bouncing up and down dying to describe the pattern they see.

Then suggest some new research projects for the children to try on their own. For example, what if the outside pair of arrows are drawn from neighbors that are *two* spaces away from the original (squared) number? Or, what if the inside pair of arrows does not come from a single number (squaring it) but comes from adjacent numbers (e.g., 3 and 4) and the outer pair comes from their nearest outer neighbors (e.g., 2 and 5)? Do the patterns hold with negative numbers? What patterns do we see if the line is numbered with consecutive odds? Consecutive squares? Consecutive Fibonacci numbers? Students get plenty of “fact drill” doing research projects like this, and have opportunities to describe what they see.

For teachers in pre-service preparation, this is one example of what it means to *do* mathematics within a territory they already know. There are many others. For teachers, this *does* offer opportunities to develop new mathematical ideas, terms and notation, but if the preservice *goal* is treated as “more math to know,” rather than how to *do* mathematics (research, problem posing, puzzling through to find results), it stamps in the very problems we see so often in classrooms. Teacher preparation cannot ignore content, but it cannot be *about* content; it must be about mathematical ways of thinking, using content as the opportunity to *do* that thinking. Students come to view mathematics the ways their teachers view it. That, in turn, is influenced by the mathematical experience teachers have in their preparation. Though classroom curricula also influence students’ image of mathematics, teachers are key.

**High school. **We observed a class on graphing linear equations using the “slope-intercept” method. For readers outside the culture of middle and high school, this means that you transform whatever equation you have into the form *y *= *mx*+*b*, and, from this, you produce the graph. There is, of course, a sensible and simple method for graphing an equation like 2*x*+3*y*=9 but on this day, the teacher’s goal was the slope-intercept method.

So, students transformed the equation into *y *= (–2/3)*x *+ 3. Then a 3-step procedure is used: (a) go up 3 units on the -axis and put a point; (b) from here, go to the right 3 units and down 2, and put a point; (c) connect the two points.

Most kids followed the procedure and produced the correct graph. Almost as an afterthought, the observer asked one student if the point (1, 2.5) was on the graph. The kid looked baffled, plotted it, and said that it looked as if (1, 2.5) was on the graph. When asked if (300, –595) was on the graph, the kid had no idea how to tell—it was off the paper.

We’ve seen this phenomenon in most classes. For many students, *y *= (–2/3)*x *+ 3 is a kind of code; from it, one obtains three numbers (–2, 3, and 3) and uses them to produce a picture. Completely missing was the idea of determining if a point is on a graph by testing to see if its coordinates satisfy the graph’s equation. Assessments didn’t detect this deficit because, given an equation, students could transform it to slope-intercept form and produce a correct graph. The goal was about procedure, so the gaping hole in students’ understanding remained hidden.

This example might seem just plain weird to many readers, but this kind of thing happens often in secondary classrooms. There’s the “box method” for setting up equations to model word problems, a different box method for factoring quadratics, the “switch *x* and *y* and solve for *y*” method for inverting functions, and a host of other special purpose methods and terminology that have no existence or purpose outside of school.

Note the parallels to the fourth-grade example. In both, the teaching emphasis was on *form*, one particular *way* of writing and doing the problem, not on what the problem meant, which the fourth-graders naturally and instantly gravitated to and which the high-schoolers could have, too.

**What can we learn from this?** Part of teachers’ mathematical preparation *must* include an understanding of fundamental results and methods—content specified in state standards. Missing, though, is the aspect of mathematics that involves research, play, experimentation, sense making, and reasoning. Mathematics is not about how much you know but about how much you can figure out with what you know.

The problem these stories illustrate is not just what’s missing, but what’s *there*––a view of mathematics that most mathematics professionals would not recognize. Wu[1] has written about “textbook school mathematics” as a dialect of the discipline that lives in precollege curricula. Wu’s main criticism is lack of precision, sloppy (or missing) definitions, absence of logical sequencing, and missing distinctions between assumptions (again, ill-formulated) and results that follow from those assumptions. But how best to mend those flaws? In many classrooms that attempt to remediate these deficiencies, the current practice is to put instruction in vocabulary and memorizing forms and formulas first, to teach without first (or perhaps ever) allowing students to build the mathematical sense of the underlying logic. This practice has failed. Wu calls for reducing “teachers’ content knowledge deficit,” remaining “consistent with the fundamental principles of mathematics (FPM).” We would concur, but his FPM seems easy to misread, allowing undergraduate instructors to conclude that it supports what they’ve always done. Wu’s FPM *starts* with “every concept has a definition,” which is not a claim that *teaching* must start that way. But it is easy to interpret as such.

For us, the classroom stories above illustrate something deeper and more fundamental than the “flatness” that raises convention to the same level of importance as matters of mathematical substance. And they are only partly about deficits in content knowledge. What they illustrate is a lacks of the perspective that learning mathematics means developing a collection of practices that help you *figure out* what to do when you don’t *know* what to do—developing the habits of mind that underlie flexible proficiency in the discipline. These classroom examples treat mathematics as a collection of special-purpose methods that allow one to perform specific tasks that are the calisthenics––the finger exercises––of mathematics. Practice is valuable for mastery in any field, but exercise as an end in itself produces muscle-bound results that can impede performance. Knowing how to transform an equation to some canonical form is an important skill, developed best through orchestrated exercise. But knowing *when* to use a particular form is much closer to what mathematics is about. More generally, it’s the *doing* of mathematics that gives people an understanding of the discipline. Learning mathematical facts and methods is absolutely essential but, by itself, builds a view of the subject that emphasizes getting to a particular form, like , rather than understanding the connection between an equation and its graph. By itself, it elevates what you know over what you can figure out.

Is it a stretch to trace the roots of such stories back to teacher preparation? We don’t think so. Yes, other forces are at play––curricula, pressures from high-stakes exams, oppressive working conditions, school lore. But a mathematical preparation that focuses on the doing as well as the learning of mathematics would give prospective teachers some tools to overcome the schoolish nonsense common in commercial curricula, to prepare students for state tests while immersing them in real mathematics, and to downplay the clutter in district syllabi so that there’s time to concentrate on what’s really important. That doesn’t mean *ignoring* the district syllabi—often a teacher can’t. Instead, one might seek a mathematical context, topic or activity of genuine intellectual worth as a venue for presenting the lightweight clutter. Teacher educators could look seriously at school curricula, think hard about how to prepare teachers to find the mathematics within or behind the school-only terms or methods such as “the box method” for whatever, or idiosyncratic or curriculum-specific terms like “number buddies,” or terms like “friendly numbers” that *have* a mathematical definition but appear in school with completely unrelated meanings. Children and teachers *will* hear these in school, and they may sometimes even be useful in school, but they are school-only, and will never be used outside of school. Educators could help teachers learn how to craft age-appropriate research activities that respect time constraints and content requirements but help kids experience the *doing* of mathematics. One way is by giving prospective *teachers* such experiences of doing mathematics.

Teachers know that what they value is communicated to their students. When teachers come to understand and value the heart of mathematics, they communicate this focus to students even when a particular day’s lesson must be about “what you have to know for the test.”

[1] E.g., Wu, Hung-Hsi. 2015. Textbook School Mathematics and the preparation of mathematics teachers. https://math.berkeley.edu/~wu/Stony_Brook_2014.pdf Retrieved September 15, 2019.

]]>The calculus has a very special place in the 20th century’s traditional course of mathematical study. It is a sort of fulcrum: both the summit toward which the whole secondary curriculum strives, and the fundamental prerequisite for a wide swath of collegiate and graduate work, both in mathematics itself and in its applications to the sciences, economics, engineering, etc.^{[1]} At its heart is the notion of the *limit*, described in 1935 by T. A. A. Broadbent as the critical turning point:

The first encounter with a limit marks the dividing line between the elementary and the advanced parts of a school course. Here we have not a new manipulation of old operations, but a new operation; not a new trick, but a new idea.

^{[2]}

Humanity’s own collective understanding of this “new idea” was hard-earned. The great length of the historical journey toward the modern definition in terms of and mirrors the well-known difficulty students have with it. Although it is the foundation of calculus, it is common to push the difficulty of this definition off from a first calculus course onto real analysis. Indeed, mathematicians have been discussing the appropriate place for the full rigors of this definition in the calculus curriculum for over 100 years.^{[3]}

There is also a rich vein in the mathematics education research literature studying students’ engagement with the – definition. Researchers have examined student difficulties coming from its multiple nested quantifiers^{[4]} as well as its great distance from the less formal notions of limit with which students typically enter its study,^{[5]} and have also made an effort to chart the paths they take toward a full understanding.^{[6]}

This blog post is a contribution to this conversation, analyzing in detail three learners’ difficulties with and .^{[7]} If there is a take-home message, it is to respect the profound subtlety of this definition and the complexity of the landscape through which students need to move as they learn to work with it.

Many readers will be familiar with the long struggle to find a rigorous underpinning for the calculus of Newton and Leibniz, leading to the modern definition of the limit in terms of and . In this section, I excerpt a few episodes, which will become important in the later discussion of student thought. Readers already familiar with the history of the subject are welcome to skim or skip this section.

While Newton and Leibniz published their foundational work on (what we now call) derivatives and integrals in the late 17th century, they based these ideas not on the modern limit, but on notions that look hand-wavy in retrospect.^{[8]} To Leibniz, the derivative, for example, was a ratio of “infinitesimal” quantities — smaller than finite quantities, but not zero. To Newton, it was an “ultimate ratio”, the ratio approached by a pair of quantities as they both disappear. Both authors would calculate the derivative of via a now-familiar manipulation: augment by a small amount ; correspondingly, augments to . The ratio of the change in to the change in is , or . At this point, they would differ in their explanation of why you can ignore all the terms involving in this last expression: for Leibniz, it is because they are infinitesimal, and for Newton, it is because they all vanish when the augmentation of is allowed to vanish.

A famous critique of both of these lines of reasoning was leveled in 1734 by the British philosopher and theologian Bishop George Berkeley, arguing that since to form the ratio of to in the first place, it was necessary to assume is nonzero, it is strictly inconsistent to then decide to ignore it.

Hitherto I have supposed that flows, that hath a real Increment, that is something. And I have proceeded all along on that Supposition, without which I should not have been able to have made so much as one single Step. From that Supposition it is that I get at the Increment of , that I am able to compare it with the Increment of , and that I find the Proportion between the two Increments. I now beg leave to make a new Supposition contrary to the first, i.e., I will suppose that there is no Increment of , or that is nothing; which second Supposition destroys my first, and is inconsistent with it, and therefore with every thing that supposeth it. I do nevertheless beg leave to retain , which is an Expression obtained in virtue of my first Supposition, which necessarily presupposeth such Supposition, and which could not be obtained without it: All which seems a most inconsistent way of arguing…

It was a long journey from the state of the art in the early 18th century, to which Berkeley was responding, to the modern reformulation of calculus on the basis of the – limit. The process took well over a century. I will summarize this story by quoting somewhat telegraphically from William Dunham’s book *The Calculus Gallery*,^{[9]} from which I first learned it.

Berkeley penned the now famous question:

… They are neither finite quantities nor quantities infinitely small, nor yet nothing. May we not call them the ghosts of departed quantities?

… Over the next decades a number of mathematicians tried to shore up the shaky underpinnings… pp. 71-72

… Cauchy’s “limit-avoidance” definition made no mention whatever of attaining the limit, just of getting and staying close to it. For him, there

wereno departed quantities, and Berkeley’s ghosts disappeared… p. 78… If his statement seems peculiar, his proof began with a now-familiar ring, for Cauchy introduced two “very small numbers” and … p. 83

… We recall that Cauchy built his calculus upon limits, which he defined in these words:

When the values successively attributed to a variable approach indefinitely to a fixed value, in a manner so as to end by differing from it by as little as one wishes, this last is called the limit of all the others.

To us, aspects of this statement, for instance, the motion implied by the term “approach,” seem less than satisfactory. Is something actually moving? If so, must we consider concepts of time and space before talking of limits? And what does it mean for the process to “end”? The whole business needed one last revision.

Contrast Cauchy’s words with the polished definition from the Weierstrassians:

if and only if, for every , there exists a so that, if , then .

Here nothing is in motion, and time is irrelevant. This is a static rather than dynamic definition and an arithmetic rather than a geometric one. At its core, it is nothing but a statement about inequalities. pp. 129-130

The Weierstrassian definition (i.e., the modern one!) allows the manipulation to which Berkeley objected to be carried to completion without ever asking to be zero.

The core of this blog post is a discussion of three learners’ encounters with the – limit, seeking to illuminate some of the subtle challenges that can arise. I begin with my own story.

I “did well” in my college real analysis class, by which I mean that my instructor (a well-known analyst at a major research university) concluded on the basis of my written output that I had mastered the material. However, I walked away from the course with a subtle but important misunderstanding of the – definition that was not visible in my written work and so went entirely undetected by my instructor, and, for many years, by myself as well.

From my previous experience with calculus, I had concluded that you can often identify the value toward which a function is headed, even if the function is undefined “when you get there.” To the extent I had a definition of *limit*, this was it: *the value toward which the function is headed*.

When I studied real analysis as an undergraduate, I found the class easy, including the – work. I mean, if is where is headed as , then sure, for any -neighborhood around , there is going to be a -neighborhood around that puts inside the -neighborhood. But I related to the notion that is headed toward as *conceptually prior to* this – game. The latter seemed like fancy window-dressing to me, possibly useful post-hoc if you need an error estimate. I did not understand that it was a definition — that it was supposed to be *constitutive* of the meaning of limit. So, I completely missed the point! But I want to stress that you would not have known this from my written work, and of course, I didn’t know it either.

I went on to become a high school math teacher. In the years that followed, I did detect certain limitations in my understanding of limits. For example, I noticed that I didn’t have adequate tools for thinking about if and when the order of two different limit processes could be safely interchanged. But it did not cross my mind that the place to start in thinking clearly about this was a tool I had already been given.

After a few years, I began teaching an AP Calculus course. About 3/4 of the way through my first time teaching it, my student Harry^{[10]} said to me after class, “You know this whole class is based on a paradox, right?” He proceeded to give me what I now recognize as essentially Bishop Berkeley’s critique. At the time, it did not occur to me to reach for epsilon and delta. Instead, I responded like an 18th century mathematician, trying to convince him that the terminus of an unending process is something it’s meaningful to talk about. I hadn’t really understood what the problem was. Of course, Harry left unsatisfied.

The pieces finally came together for me the next year, when I read Dunham’s *Calculus Gallery*, quoted above. I remember the shift in my understanding: *ooooohhhhhh.* The ‘s and ‘s are not an addendum to, or a gussying-up of, the idea of identifying where an unending process is headed. They are *replacing* this idea! It was a revelation to reread the definition from this new point of view. *Calculus does not need the infinitesimal!* I immediately wished I had a do-over with Harry, whose dissatisfaction I hadn’t comprehended enough to be able to speak to.

I concluded from this that a complete understanding of the – definition includes an understanding of what it’s *for*.

Having come to this conclusion, in my own teaching of real analysis I’ve made a great effort to make clear the problem that and are responding to. In one course, I began with a somewhat in-depth study of Berkeley’s critique of the 18th century understanding of the calculus, in order to then be able to offer and as an answer to that critique.

In doing this, I ran into a new challenge. To illustrate, I’ll focus on the experience of a student named Ty. Ty arrived in my course having already developed a fairly strong form of a more intuitive, 17th-18th century understanding of the limit; essentially the Newtonian one, much like the understanding that had carried me myself through all my calculus coursework. He quickly made sense of Berkeley’s objection, so he was able to see that this understanding was not mathematically satisfactory. I was selling the – definition as a more satisfactory substitute. However, Ty objected that important aspects of his understanding of the limit (what Tall and Vinner called his *concept image*^{[11]}) were not captured by this new definition. In particular, what had happened to the notion that the limit was something toward which the function was, or even *should have been*, headed? The – definition of studiously avoids the point “at which the limit is taken,” even speculatively. To Ty, it was the – definition that was, pun intended, missing the point.

Of course, this studious avoidance is precisely how the Weierstrassian definition gets around Berkeley’s objection. The Newtonian “ultimate ratio” and the Leibnizian “infinitesimal” both ask us to imagine something counterfactual, or at least pretty wonky. This is exactly what made them hard for Berkeley to swallow, and as I learned from Dunham’s book, the great virtue of and $\delta$ is that they give us a way to uniquely identify the limit that does not ask us to engage in such a trippy flight of fancy that may or may not look sane in the light of day.

But, at the same time, *something is lost*.^{[12]} What I learned from Ty is that this loss is pedagogically important to acknowledge.^{[13]}

Another subtle difficulty in working with the – definition is revealed when you use it to try to prove something. I think what I am about to describe is a general difficulty students encounter in learning the methods and conventions of proof-writing, but I speculate that it may be particularly acute with respect to the present topic. Consider this (utterly standard) proof that if are functions of $x$ such that and , and , then :

Let be given. Because , there exists such that implies . Similarly, because , there exists such that implies .

Take .

Then for values of satisfying , it follows from the triangle inequality and the definition of that

.

Since was arbitrary, we can conclude that .

Here is a surprisingly rich question: is the in the proof one number, or many numbers?

On one way of looking at it, of course it is only one number: is fixed at the outset of the proof. Indeed, if were allowed to be more than one thing, equations like $\epsilon / 2 + \epsilon / 2 = \epsilon$ would be meaningless. More subtly, we usually speak about as a single fixed quantity when we justify the existence of in terms of the definition of the limit: we know exists because by the definition of the limit, for any there is a , *so in particular*, there is a for , etc. Note the “in particular”: we produce from the definition by *specializing it*.

But on another way of looking at it, of course $\epsilon$ is many numbers. Indeed, it must represent every positive number, otherwise how can it be used to verify the definition *for all *? The singular, fixed with which we work in the proof is a sort of chimera: it actually represents all positive numbers at once. That we think of it as a single number is just a psychological device to allow us to focus in a productive way on what we are doing with all these numbers.

This dual nature of in the above was driven home for me by working with Ricky. Fast and accurate with calculations and algebraic manipulations, Ricky was thrown for a loop by real analysis, which was her first proof-based class, and in particular by the – proofs. After a lot of work, she had mastered the definition itself. But in trying to write the proofs, she found the lilting refrain *for all …* to be a kind of siren song, leading her astray. She was constantly re-initializing with this phrase, so that reading her work, there were 3 different meanings for by the end. “Look at how the proof works,” I would say, referring to the proof of above. “You don’t need to be less than any old . You need it to be less than *the particular* that you are using for .” “What do you mean *the particular* I am using?” she would ask. “I am trying to prove it works *for all !*”

Ricky’s difficulty has led me to a much greater appreciation of the subtle and profound abstraction involved any time an object is introduced into a proof with the words “fix an arbitrary…” In a sense, this is nothing more — nor less! — than the abstraction at the heart of a student’s first encounter with algebra: if we imagine an unspecified number , and proceed to have a conversation about it, our conversation applies simultaneously at once to all the values could have taken, even if we were imagining it the whole time as “only one thing.”^{[14]} But I don’t think I appreciated the great demand that “fix an arbitrary…” proofs in general, and – proofs in particular, place on this abstraction. The mastery of it that is needed here goes far beyond what is needed to get you through years of pushing around.

I offer the above anecdotes primarily as grist for reflection about learning, and especially about the nature of the particular landscape students tread as they encounter and .^{[15]} But I would like to articulate some lessons and reminders that I myself draw from them:

(1) A complete understanding of a concept might require to go beyond mastery of its internal structure and its downstream implications, to include an understanding of its purpose, i.e. the situation it was designed to respond to.

(2) Work that successfully responds to the standard set of prompts may still conceal important gaps in understanding, as mine did in my undergraduate real analysis class. More generally, do not assume because a student is “strong” that they have command of any particular thing.

(3) Conversely, take student thought seriously, even when it looks/sounds wrong. Ricky and Ty were producing unsuccessful work for very mathematically rich reasons; I learned something worthwhile by taking the time to understand what each of them was getting at. Harry’s issue, which I didn’t take seriously at the time, could have pushed my own understanding of calculus forward — in fact, it did, albeit belatedly.

Finally, I hope the combination of these anecdotes with the history above serves as a reminder both of the magnitude of the historical accomplishment crystallized in the Weierstrassian – definition of the limit, and of the corresponding profundity of the journey students take toward its mastery.

[1] There is an important contemporary argument that calculus’ pride of place in the curriculum should be ceded to statistics. (For example, see the TED talk by Arthur Benjamin.) That debate is beyond the scope of this blog post.

[2] The First Encounter with a Limit. *The Mathematical Gazette*, Vol. 19, No. 233 (1935), pp. 109-123. (link [jstor])

[3] In addition to the 1935 *Mathematical Gazette* article quoted above, see, e.g., E. J. Moulton, The Content of a Second Course in Calculus, *AMM* Vol. 25, No. 10 (1918), pp. 429-434 (link [jstor]); E. G. Phillips, On the Teaching of Analysis, *The Mathematical Gazette* Vol. 14, No. 204 (1929), pp. 571-573 (link [jstor]); N. R. C. Dockeray, The Teaching of Mathematical Analysis in Schools, *The Mathematical Gazette* Vol. 19, No. 236 (1935), pp. 321-340 (link [jstor]); H. Scheffe, At What Level of Rigor Should Advanced Calculus for Undergraduates be Taught?, *AMM* Vol. 47, No. 9 (1940), pp. 635-640 (link [jstor]). I thank Dave L. Renfro for all of these references.

[4] E.g., E. Dubinsky and O. Yiparaki, On student understanding of AE and EA quantification, *Research in Collegiate Mathematics Education* IV, 8 (2000), pp. 239-289 (link). In this and the next two notes, the literature cited only scratches the surface.

[5] E.g., D. Tall and S. Vinner, Concept image and concept definition in mathematics with particular reference to limits and continuity, *Educational Studies in Mathematics* Vol. 12 (1981), pp. 151-169 (link), S. R. Williams, Models of Limit Held by College Calculus Students, *Journal for Research in Mathematics Education* Vol. 22, No. 3 (1991), pp. 219-236 (link [jstor]), and C. Swinyard and E. Lockwood, Research on students’ reasoning about the formal definition of limit: An evolving conceptual analysis, *Proceedings of the 10th annual conference on research in undergraduate mathematics education, San Diego State University, San Diego, CA* (2007) (link).

Findings about students’ informal understandings of limits that generate friction with their study of and include that they are often dynamic/motion-based (like Newton), or infinitesimals-based (like Leibniz), and meanwhile, they are also often characterized by a “forward” orientation from to — “If you bring close to , it puts close to .” This is in contrast with the – definition’s “backward” orientation from to — “To make -close to , you have to find a to constrain .”

[6] E.g., J. Cottrill, E. Dubinsky, D. Nichols, K. Schwingendorf, K. Thomas, D. Vidakovic, Understanding the Limit Concept: Beginning with a Coordinated Process Scheme, *Journal of Mathematical Behavior* Vol. 15, pp. 167-192 (1996), Swinyard and Lockwood *op. cit.* (which responds to Cottrill *et. al.*), and C. Nagle, Transitioning from introductory calculus to formal limit conceptions, *For the Learning of Mathematics* Vol. 33, No. 2 (2013), pp. 2-10 (link).

[7] To avoid ambiguity, the learners referred to here are *myself* and the students I below call Ty and Ricky. The student I call Harry illustrates a difficulty one might have *without* the – limit.

[8] The brief account I am about to give represents an orthodox view of the history of calculus, see for example J. V. Grabiner, Who Gave You the Epsilon? Cauchy and the Origins of Rigorous Calculus, *The American Mathematical Monthly* Vol. 90, No. 3 (1984), pp. 185-194 (link). This orthodoxy is not without its detractors, e.g., B. Pourciau, Newton and the Notion of Limit, *Historia Mathematica* No. 28 (2001), pp. 18-30 (link) or H. Edwards, Euler’s Definition of the Derivative, *Bulletin of the AMS* Vol. 44, No. 4 (2007), pp. 575-580 (link).

Readers interested in more comprehensive accounts of the history of the – limit can consult Judith Grabiner’s monograph *The Origins of Cauchy’s Rigorous Calculus*, MIT Press, Cambridge, MA (1981) and William Dunham’s *The Calculus Gallery: Masterpieces from Newton to Lebesgue*, Princeton Univ. Press, Princeton, NJ (2005). A very interesting-looking new book on the topic is David Bressoud’s *Calculus Reordered: A History of the Big Ideas*, Princeton Univ. Press, Princeton, (2019) (link), which takes a broader view, looking at the development of integration, differentiation, series, and limits across multiple millennia and continents, and viewing the limit as a sort of culmination driven by the needs of research mathematicians in the 19th century. Bressoud’s book also considers questions of pedagogy in relation to this history.

[9] Dunham, *op. cit.* (see previous footnote).

[10] All names of students are pseudonyms.

[11] D. Tall and S. Vinner. Concept image and concept definition in mathematics with particular reference to limits and continuity. *Educational Studies in Mathematics* Vol. 12, No. 2 (1981), pp. 151-169. (link)

[12] Relatedly, recovering that which was lost from calculus when and superseded the Leibnizian infinitesimals is often given as the rationale behind Abraham Robinson’s development of nonstandard analysis.

[13] This observation is related to the body of research indicated in note [5]. I think it is subtly different though. As I understand that research, the theme is the difficulties students have with the – definition due to “interference” from their more informal understandings of limits and derivatives. In contrast, my focus here is on a difficulty Ty had not because of “interference,” but rather because he recognized (perhaps more clearly than I did) that this new definition is not actually doing the same thing, so if it was being sold it as a substitute, he was not buying.

[14] To help Ricky contextualize what she needed to do for the proof in terms of things she already understood, I asked her to consider this proof that every square number exceeds by one the product of the two integers consecutive with its square root:

Let be any integer. Then

,so any square number is one more than the product of and .

“I think of the in this proof as every number,” she said. “But you have to relate to it as a single number during the calculation itself,” I replied. “Otherwise, how do you know that ?”

[15] I first encountered the metaphor of a “landscape of learning” attendant to particular mathematical topics in the writings of Catherine Twomey Fosnot and Maarten Dolk.

]]>If you give calculus students graphs, they are going to draw tangent lines. As instructors we often encourage students to rely on tangent lines so heavily that discussions about rates of change become lessons about sliding lines along graphs, rather than about understanding the relationships that these graphs represent in the context of a given problem.

So how do we help students develop a deeper understanding of these relationships? Let’s consider an exercise that you might give your own students during a lesson on tangent lines and rates of change.

Exercise 1: The Growing Cone

The images in Figure 1 (left) depict a growing cone. The graph in Figure 1 (right) represents the relationship between the outer surface area and height of the growing cone. Describe the rate of change of the surface area with respect to the height of the cone as it grows.

A typical solution to the Growing Cone exercise involves drawing a collection of tangent lines along the graph in Figure 1 and exploring the steepness of these lines as the height increases (see Figure 2). One could observe that the tangent lines become steeper, thus the rate of change *increases*.

So, what’s so bad about this strategy? This issue is that students often use this technique successfully without interacting with the quantities of surface area and height. Furthermore, to determine an increase in “steepness” they do not need to measure or determine slope at all, they only need to observe visual properties of the line without actually doing mathematics. Students begin to rely on and practice such techniques exclusively. As a result they are not able to reason in situations where these strategies break down, which we will see in Exercises 2 and 3 below.

What does a response for Exercise 1 with deeper understanding look like? At a beginning level, a student could describe that as the height of the cone increases, the surface area of the cone also increases. At a more sophisticated level, a student could describe that as the height increases in equal sized amounts, the surface area begins to increase by *larger* amounts. This means that the surface area increases at an* increasing rate *with respect to the height of the cone. As a result, the graph of the surface area and height would appear steeper for sections of the graph corresponding to points where the cone is taller.

Supporting students’ understanding of the quantitative relationships we have discussed above provides them with a deeper understanding of tangent lines and other tools in calculus. A student who is able to understand the quantitative relationship in Exercise 1, should be able to illustrate the changes in the surface area across equal changes in the height of the cone both in the picture of the cone (see Figure 3 (left)) and on the graph (see Figure 3 (right)). From here, students can more deeply connect slopes of secant lines on graphs with the changes in quantities of the growing cone as illustrated in Figure 3 (right) and revisit slopes of tangent lines as ratios of infinitesimal changes. A sophisticated understanding of tangent lines then should provide a student with the necessary tools to interpret a tangent line by comparing relative changes of the quantities the line represents.

The “sliding tangent line” strategy is exclusive to graphs in the Cartesian coordinate system with the independent variable on the horizontal axis and the dependent variable on the vertical axis. To help students develop understanding of the co-varying quantities represented in a graph, we can provide them exercises in which tricks like the “sliding tangent line” strategy don’t apply. For example, does this strategy work in Exercise 2?

Exercise 2: The Growing Cone Exercises Part II

Describe the rate of change of the height with respect to the surface area of the cone in the Growing Cone exercise.

A student who is able to quantitatively reason about the situation in Exercise 1 could respond to this question without needing to redraw the graph in Figure 1. The student could instead consider changes in the height of the cone that correspond to equal changes in the surface area of the cone. The cones and graph in Figure 4 illustrate the changes of height in the cone corresponding to an equal partition of the surface area. The changes in height *decrease* as the surface area increases in equal amounts. Thus, the student could conclude that the height of the cone increases at a *decreasing *rate with respect to surface area.

It could be argued that students could redraw the graph of height and surface area in the previous example so that the “sliding tangent” trick still applies. But what about in polar coordinates? Consider Exercise 3 which explores the graph of the cosine function in polar coordinates.

Exercise 3: Polar Coordinate Graph

Describe the rate of change of the radius with respect to the angle in the graph in Figure 5.

Standard tangent lines do not make sense in this problem. We could draw a line that is tangent to the graph as shown in Figure 6. The slope of this line represents the vertical change in relation to the horizontal change, not the change in radius with respect to change in angle.

We could, however, reason with the graph in Figure 5 in a similar way as we’ve previously seen by creating changes in angle and radius. In Figure 7, a collection of equally spaced angles has been marked with brown arcs and the corresponding changes in radius are drawn with orange segments. We can first observe that the radius decreases as the angle increases. We can then compare the changes in radius as the angle increases. The changes in radius get *larger* as the angle moves from 0^{o} to 30^{o }to 60^{o} to 90^{o}. A student could then conclude that as the angle increases from 0 to 90 degrees, the radius decreases at an *increasing* rate.

The focus in calculus should be the measurable attributes, or quantities, involved in rates of change, not on tangent lines drawn on a graph. The tricks and associations we teach students provide them quick ways to draw inferences but leave them unable to understand the implications of these inferences on the quantities. As a result, students come away from calculus knowing how to slide lines along graphs but not knowing how to make comparisons in changing quantities in their world. We can help students develop deeper understandings in calculus by asking questions that focus on the fundamental relationships between changing quantities and challenge students to think beyond memorized strategies.

Several of the ideas and examples here are inspired by the work of Advancing Reasoning, an NSF-funded research project whose mission is to support students’ and teachers’ mathematical thinking and learning by developing products that create transformative learning experiences. For more ideas on how to support students’ quantitative reasoning, see our project page at: https://sites.google.com/site/advancingreasoning/. All figures in this post were created in the GeoGebra application.

]]>