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.1 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.

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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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Posted in Mathematics Education Research, Student Experiences | 4 Comments

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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Posted in Graduate Education, Mathematics Education Research, Online Education, Student Experiences | Tagged , , | 4 Comments

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: https://tinyurl.com/OnlineTalkshop.

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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