Pedagogical implications of Mathematics as the art of giving the same name to different things

by Daniel Chazan, University of Maryland; William Viviani, University of Maryland; Kayla White, Paint Branch High School and University of Maryland

In 2012, 100 years after Henri Poincare’s death, the magazine for the members of the Dutch Royal Mathematical Society published an “interview” with Poincare for which he “wrote” both the questions and the answers (Verhulst, 2012). When responding to a question about elegance in mathematics, Poincare makes the famous enigmatic remark attributed to him: “Mathematics is the art of giving the same names to different things” (p. 157).

In this blog post, we consider the perspectives of learners of mathematics by looking at how students may see two uses of the word tangent—with circles and in the context of derivative—as “giving the same name to different things,” but, as a negative, as impeding their understanding. We also consider the artfulness that Poincare points to and ask about artfulness in mathematics teaching; perhaps one aspect of artful teaching involves helping learners appreciate why mathematicians make the choices that they do.

Our efforts have been in the context of a technology that asks students to give examples of a mathematical object that has certain characteristics or to use examples they create to support or reject a claim about such objects. The teacher can then collect those multiple examples and use them to achieve their goals.

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Flip Your Class: Social Distancing Edition

Flip Your Class:  Social Distancing Edition

by Jeff Suzuki

Unless you’ve been living under a rock for the past decade, you know that one of the buzzwords in education is active learning:  Be the guide on the side, not the sage on the stage.  One of the more common approaches to active learning is the so-called flipped or inverted classroom.  In a flipped classroom, students watch lectures at home, then come to class to do problems.  This is actually a 21st century implementation of a very traditional approach to pedagogy, namely reading the textbook before coming to class.  Many of us embraced this idea, and shifted our approach to teaching.

Then came the era of social distancing and forced conversion to distance learning.  It might seem that those who switched to the flipped classroom model had an advantage:  Our lectures are already online.  And that’s true.  But the second part of the flipped classroom involves working problems in class.  This is now impossible, and those of us who had embraced the flipped classroom model have spent the past few months in existential agony. The “sage on the stage” can still give lectures through Zoom, but the “guide on the side” can’t guide.

The New Normal?

And yet…it’s now more important than ever to be the guide on the side.

We don’t know how long the current phase of social distancing will last, but even after it ends, we can expect that distance learning will be the new normal:  it’s a trend that began long before the pandemic.  And this forces us to deal with a new problem:  It is impossible to monitor student activity remotely.

We accept this when we assign homework, and expect students will do the work with their books open, their notes in front of them, and a half-dozen math help sites open in different browser tabs.   Before the pandemic, we told ourselves it didn’t matter, since they’d have to do the exam without all these study aids.  But in the post-COVID world, there is now no difference between the resources available to students on homework assignments and on exams.

Don’t believe the hype about lockdown browsers (which work fine for the students who don’t have smartphones).  Live webcam monitoring can be defeated by taping cheat sheets on the wall behind the computer.  And if a student turns in a textbook perfect answer, it’s possible they listened to us when we explained how the answers should be written.

Will students cheat on exams?  We’ve found copies of our exam questions posted to Chegg (with answers).  This shocked me:  Who would pay for a Chegg subscription, when there are so many free sites that show all steps to solving a problem and, unlike Chegg, leave no evidence behind?

The bad news is that every exam question that can be answered by following an algorithm is now obsolete, because such questions can no longer distinguish between the student who understands the material and the student who knows how to use Google.

Here is where the flipped classroom can be our salvation.  A key component of the flipped classroom is letting your students figure things out for themselves, and not giving them a step-by-step algorithm for solving a problem.

For example, let’s consider a basic problem in calculus:  Finding the derivative of a function.  In the internet era, any function that can be described algebraically can have its derivative found, with steps, by a free online problem solver.  So we have to ask questions that can’t be resolved by typing the problem into (not, so far as I know, a real website, but a thirty-second Google search will give you a plethora of possibilities).

Transcending the Machine

The good news is that computers are good at exactly one type of problem:  problems that have algorithmic solutions.  If you can describe the exact sequence of steps needed to solve a problem, then a computer can implement those steps faster, more accurately, and more cheaply than any human being.  The real moral of the story about John Henry is don’t compete with the machine in the machine’s areas of strength.  Instead, find the things the machine is bad at.  In this case, the easiest way to neutralize these problem solving sites is to make every problem a word problem.

Of course, “students can’t do word problems.”  This is a meaningless objection:  at the start of calculus, students can’t integrate, but we still ask them integration questions on the final exam! Our job is to teach these students how to do these things.  Here’s where the flipped classroom becomes a key part of the solution.  Don’t spend class time lecturing:  students can view lectures on their own time.  Instead, class time should be spent working problems, especially those that can’t be solved by following a sequence of steps.

It’s helpful in the discussion that follows to think of problems as falling into one of two categories:

  • Routine problems, where the mathematical question and the relevant information are explicitly given: “Find the derivative of y = tan(3x).”
  • Non-routine problems, where this information is not given explicitly. Roughly speaking, every word problem is non-routine, and such problems form the bulk of the questions in “reform-oriented” textbooks.


Flipping Your Class, Social Distancing Edition

Here’s one possible structure for such a class (where “class” means any time you’re working with students in realtime).  All of the following takes place before class:

  • Students watch assigned lectures on the topic.
  • Students complete routine homework problems, using some online homework management (OHM) system. If you’re using a commercial text, there is an OHM associated with your text.  If  you’re using your own, there are free products (MyOpenMath is my go-to) that can be used.
  • Students are also assigned a set of non-routine problems to consider. These don’t have to be separate from the OHM:  again, almost every word problem should be considered non-routine.

How should you run class itself?  Class time is the most valuable resource available to students; using it efficiently and effectively can be challenging.  Here’s a few things that may help.

At the start of class (online or in person), take down a list of student questions.  One risk is that the more outspoken students tend to dominate the discussion; taking down a list of all questions at the start of class is a way to make sure that every student has a chance of getting their question answered, and to ensure that a sufficient variety of problems are presented.

Establish from the start that the routine problems have lowest priority:  these are problems that should be solvable by students who followed the assigned lecture.  It is vitally important that you keep to this rule:  The biggest challenge to running a flipped classroom is students who don’t watch the lectures beforehand.  Depending on how you’ve set things up and the system you’re using, it might even be possible to determine whether a student has watched the assigned lecture (though trying to do this realtime requires a bit of practice); another option, which I use, is to assign simple 1-point problems that students answer after they’ve watched the lecture.  Remember:  Class time is the single most valuable resource available to the students; it should not be spent on things that can be done out of class time.

One way to efficiently use class time is to focus on the setup.    For example, let’s consider the following problem, which probably appears in every calculus text ever written:

A 25-foot long ladder rests against a wall.  The base of the ladder begins sliding away from the wall at 2 ft/second, while the top of the ladder maintains contact with the wall.  How rapidly is the top of the ladder falling when the base is 10 feet away from the wall?

The “sage on the stage” would identify the relevant parameters and write down the mathematical problem to be solved.  The “guide on the side” would lead students to the mathematical problem.   For this, it’s important to ask leading questions and not give outright answers.  For example:

  • What’s going to answer the question “How rapidly?” (Students should identify that this is an instantaneous rate of change, so it’s a derivative)
  • What other things are changing? (Students should recognize that the distance of the base of the ladder from the wall is also changing, but the length of the ladder is not)
  • Is there a relationship we can write between the quantities?
  • Which derivative do we want? (Students should identify that they want $\frac{dy}{dt}$; it’s also worth making them explain why $\frac{dy}{dx}$ is not relevant).

In the end, we have the mathematical problem, “Find $\frac{dy}{dt}$ when $x^2 + y^2 = 25$ and $\frac{dx}{dt} = 2.$”     At this point, it becomes a routine problem—and if you’ve established that minimal class time will be spent on routine problems, you can leave the problem at this point, perhaps with a directive of “Finish the problem after class.”

It’s worth noting that, at this point, the problem can be handed off to an online calculator, which can then solve the problem.  You might even go so far as to point students to the online calculator, lest they develop a mistaken belief that you’re unaware of the existence of such things.  This epitomizes the idea that humans should do what humans are good at, namely extracting the mathematical problem to be solved; while machines should do what machines are good at, namely applying an algorithm.

As the preceding example suggests, it’s possible to teach a flipped class with very little change in how you’re already teaching.  The main difference is establishing the expectation that students watch lectures before class.

Let’s see how we might take a larger step, using a standard topic:  finding the extreme values of a function.  A traditional approach might be to have students find derivatives, then critical values, then apply some test to decide whether a critical value corresponds to a maximum or minimum.

In a flipped classroom, students wouldn’t be given this algorithm.  Instead, they’d create their own approach, typically through some guided exploration of a question.  Coming up with good questions is challenging; fortunately, thirty years of reform calculus have provided us with an abundance of material, and many of these questions have been incorporated into every standard calculus text, so you needn’t write your own.

For example, I like to give students the following question:

An accelerograph records the acceleration of a train (assumed to be moving in a straight line); some of the data values are shown below.   Assuming the velocity of the train at t = 0 was 0 m/s, estimate when the train was moving the fastest; defend your conclusion.

t (seconds) 0 1 2 3 4
a(t) (m/s2) 3 2 1 -1 -2


A sequence of leading questions can guide students to creating their own approach:

  • What does the acceleration have to do with the velocity? (The student should identify that it’s the rate of change of the velocity)
  • So a(0) = 3 and a(4) = -2. What does that tell you?  (The student should identify that the velocity is increasing at t= 0, and decreasing at t = 4)
  • So when is the velocity increasing, and when is it decreasing? (The student should identify the velocity’s increasing at t = 0, 1, 2, and decreasing at t = 3, 4.  I usually find I have to ask “Is the velocity increasing or decreasing at t = 1?  t = 2?  )
  • So where is the velocity going to be the greatest? (The student will probably say t = 2, at which point remind them that they just told you velocity is increasing at t = 2)

and so on, leading to an answer like:  The train’s velocity appears to be increasing until at least t = 2, and is decreasing from t = 3, so there’s a maximum velocity between t = 2 and t = 3.

What’s worth noticing here is that none of these questions can be answered by appeal to a formula or an algorithm.  Consequently, any attempt to use an online calculator on this type of question will result in, at best, a nonsense answer.   The closest thing to an “algorithm” is recognizing that the change from increasing to decreasing is where the local maximum value will occur, but even then, since that change occurs “offscreen”, students must consider how they know that the change has occurred.


A Return to Normalcy

Suppose, against all predictions and the entire trend of human history, we go back to how things were at the beginning of 2020:  traditional in-person classes, no social distancing, exams where we could control the resources used by students.

None of the preceding needs to change.  In fact, all of the preceding alterations in our pedagogy and our assessment are worth doing regardless of how we will give exams.  The hard truth is that sooner or later, our students will leave the classroom.  If what they’ve learned from our classes can be done by a free internet application, then their education is worth a free internet application.

We owe it to our students to give them something more.

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A Geometric Approach to Functions

by Karen Hollebrands, Allison McCulloch, Daniel Scher, and Scott Steketee

Fostering an understanding and appreciation of the deep, beautiful threads that unite seemingly disparate areas of mathematics is among the most valuable outcomes of teaching. Two such areas that are ripe for bridge building—functions and geometric transformations—are the focus of our NSF project, Forging Connections Through the Geometry of Functions. In this post, we describe the pedagogical benefits of introducing students to functions through the lens of geometric transformations.

Geometric Transformations as Functions

The most common representations of functions are symbolic and numeric in nature. This emphasis on number limits students’ images of the variety of mathematical relationships that can be represented as functions. As such, it contributes to common student misconceptions. Students may conclude that:

  • every function turns an input number into an output number;
  • every function can be expressed as an algebraic formula;
  • a formula is the primary representation of a function, and all other representations derive from it; and
  • the ultimate test of a function requires graphing it in rectangular coordinates and applying the vertical line test.

Although students investigate reflections, translations, rotations, dilations, and glide reflections in a geometry course, they typically do not regard them as functions; the functions they encounter in algebra always have numbers as input and output. We can expand students’ horizons and deepen their concept of function by treating geometric transformations as functions that take a Euclidean point as input and produce another point as output. Coxford and Usiskin pioneered Continue reading

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Active Learning and the Transformation of a Graduate Student Instructor

by Sarah Hagen

Recently as a graduate student I taught a week-long boot camp for incoming mathematics graduate students at Oregon State University. It was my first foray into teaching under the active learning model and it was a completely transformative  experience for me as an instructor. The change in my own attitudes towards teaching and pedagogy were so abrupt, so all-encompassing, that I felt compelled to immediately record my thoughts on the experience. Continue reading

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The things in proofs are weird: a thought on student difficulties

By Ben Blum-Smith, Contributing Editor

“The difficulty… is to manage to think in a completely astonished and disconcerted way about things you thought you had always understood.” ― Pierre Bourdieu, Language and Symbolic Power, p. 207

Proof is the central epistemological method of pure mathematics, and the practice most unique to it among the disciplines. Reading and writing proofs are essential skills (the essential skills?) for many working mathematicians.

That said, students learning these skills, especially for the first time, find them extremely hard.[1]

Why? What’s in the way? And what are the processes by which students effectively gain these skills?

These questions have been discussed extensively by researchers and teachers alike,[2] and they have personally fascinated me for most of my twenty years in mathematics education.

In this blog post I’d like to examine one little corner of this jigsaw puzzle.

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Starting Earlier on Lifelong Learning

By: Matt Stamps, Yale-NUS College

When Yale-NUS College reviewed the curriculum for its Mathematical, Computational, and Statistical (MCS) Sciences major in the autumn of 2018, I spent several weeks reading about mathematics programs at similar institutions.  A common learning objective among many of the programs was a variation of “preparing students to become lifelong learners.” I really like this goal because, among many other reasons, it reminds teachers that students are human beings who have lives beyond their studies, and it reminds students that learning is not confined to the early years of one’s life.   As I reflect on my life of learning thus far, I cannot help but notice how significantly the way I learn has changed since I was a student.  Some of these differences arose naturally with changes in my circumstances over the years, while others could have been addressed while I was still a student.

In this post, I want to share some observations about how my approach to learning has changed since I started working as a professional mathematician, and how I have changed my approach to teaching with the hope of helping my students develop more effective and relevant learning strategies earlier in their mathematical journeys.

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Square peg in an octagonal hole

Interview with Ari Nieh, with commentary from Yvonne Lai

Like many of us, I began teaching online this Spring. Unlike many of us, I began doing so at the start of the semester. I am co-teaching a class at Michigan State, and I live in Nebraska. One of the most useful conversations I had in preparation for this assignment happened in 2013, well before the current coronavirus epidemic. The math department at the University of Nebraska-Lincoln had been considering a synchronous online version of a mathematics course, for rural teachers. I chatted with Ari Nieh, then an instructor for Art of Problem Solving, about what it would take to teach online, especially via chat forum technology. (Ari then became a lecturer in Writing, Rhetoric, and Professional Communication at MIT; and now he is a game designer at Wizards of the Coast.) In the end, that course was run asynchronously (and in many ways consistent with the advice given in a previous post). Nonetheless, much of the advice I received 7 years ago aged well. With Ari’s permission, I share snippets of our conversation in this post, edited for readability, and with commentary from present-day me.
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Online learning in the time of Coronavirus: Tips for students and the instructors who support them

Abbe Herzig, AMS Director of Education

In the midst of the upheaval due to the Coronavirus, students and faculty are transitioning to new virtual classrooms. Many of us haven’t chosen to learn or teach, but here we are, making the best of this new reality.

In this post, I describe some guidelines that may help students manage the transition to online learning as smoothly as possible. Instructors can support students by helping them to learn online, and I encourage instructors reading this to pass it along to your students. I offer these suggestions with a caveat: Some of these ideas may not be feasible for everyone, and that’s ok. We all have unique living, learning, and life situations, and what works for one of us may not work for others. Take what you can, and leave the rest. Keep realistic expectations of yourself, understanding that these circumstances are less than ideal. While the suggestions in this post are directed toward students, I also offer “teaching tips” to help instructors support their students.

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Surprise! Transitioning to online teaching

 By Abbe Herzig, AMS Director of Education

Many of us are experiencing stress as schools, colleges and universities move instruction out of the classroom. Fortunately, even if distance learning is new to you, it isn’t new, and there is a lot of wisdom to draw on.

This document describes some practical strategies that will hopefully get you started, along with some helpful web-based resources. From there, you can do a deeper dive by accessing the open community on MAA Connect called “Online Teaching and Distance Learning.” MAA members can log in with their member credentials, and anyone who creates a free profile can join this group. This is an extensive platform to exchange ideas with other faculty and to access resources and advice for developing your courses. The STEM faculty blundering through remote teaching in a pandemic FaceBook page is another great place for faculty to share ideas and figure all this out together. Continue reading

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Help! I need to teach my course online and I’ve never done this before

By: Yvonne Lai, University of Nebraska-Lincoln
Ray Levy, Mathematical Association of America

This is cross-posted in MathValues and Abbe Herzig has written a companion post. Additional resources and future meetings are also available here:

In times of crisis we need community. With schools, colleges and universities mandating online teaching and learning in response to COVID-19, often with only a week of preparation time, people are scrambling for resources and information. Dr. Ray Levy, aås Deputy Executive Director of the MAA, asked an online group whether they would like a Zoom space to discuss online learning. With only 12 hours of notice, Dr. Jeneva Clark helped co-facilitate, and 36 people gathered. The next day, with only a few hours of notice Dr. Abbe Herzig and Dr. Yvonne Lai joined as co-facilitators, and 86 people gathered. Below is some of what we learned in the second meeting.
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