Reflecting on mathematics as the art of giving the same name to different things (Part 2): Averages finite and continuous

by Bill Rosenthal, Queens, NY; Whitney Johnson, Morgan State University; Daniel Chazan, University of Maryland

The July 15 blog post by Dan Chazan and two colleagues referred to Poincaré’s enigmatic remark: “Mathematics is the art of giving the same name to different things.” Poincaré called “giving the same name” an “art,” no doubt referring to the beauty and depth of showing mathematical relatedness and also the care with which that must be done. The post explored how the word tangent in the contexts of circles and graphs of functions connotes different things for learners and argues that, for learners, “the same name” can add depth, or confusion, and that teachers must be alert. In this post, we return to Poincaré’s remark and consider how reuse of names may be seen differently by students who come to the mathematics classroom with disparate experiences.

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THE ZOOM ROOM: Vignette and Reflections About Online Teaching

Mark Saul

A child’s insight

“I know how to find out how many divisors a number has. You factor it into primes….” Alejandro was with a virtual group of four enthusiastic ten year olds, in the midst of exploring a problem. He gave the usual result, using his own somewhat makeshift words. But not too distant really from what I would have said: If $N$ factors as $p_1^{a_1}p_2^{a_2}p_3^{a_2} \dots$, then the number of divisors is $(a_1+1)(a_2+1)(a_3+1)…$. His description was less economical, but still accurate.

His virtual friend Xue said: “That’s great. Let’s look it up on Wikipedia.”

Then, “No. Let’s not look it up. Let’s pretend we don’t know it and see if we can prove it.” It is this insight into his own learning, not any mathematical breakthrough, that I remark on in the subtitle to this section.

Dear Reader: I swear to you, on Galois’ grave, that I am not making this up. Nor the rest of the vignette I will be recounting here.

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A K-pop dance routine and the false dilemma of concept vs. procedure

By Ben Blum-Smith, Contributing Editor

For reasons that will not be considered here, I recently learned this dance:

Although I have no background in any style of dance, I can now do the whole thing, start to finish. I am very proud.

My purpose in attaining this objective was unrelated to mathematics or teaching. Nonetheless, the experience put an eloquent fine point on a certain basic dialectic in math education.

Procedural vs. Conceptual

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Writing Good Questions for the Internet Era

Jeff Suzuki

CUNY Brooklyn

The forced conversion to distance learning in Spring 2020 caught most of us off-guard. One of the biggest problems we face is the existence of free or freemium online calculators that show all steps required to produce a textbook perfect solution.  A student can simply type in “Solve ” or “Find the derivative of ” or “Evaluate ” or “Solve ,” and the site will produce a step-by-step solution indistinguishable from one we’d show in class.  With Fall 2020 rapidly approaching, and no sign that distance learning will be abandoned, we must confront a painful reality:   Every question that can be answered by following a sequence of steps is now meaningless as a way to measure student learning.

So how can we evaluate student learning?  Those of us fortunate enough to teach courses with small enrollments have a multitude of options:  oral exams; semester-long projects; student interviews.  But for the rest of us, our best option is to ask “internet resistant” questions.    Here are three strategies for writing such questions:

●       Require inefficiency.

●       Limit the information.

●       Move the lines

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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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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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