## Bridging Cultures: An Iranian Woman from an Historically Black College Teaching in a Prison in the US

by Zeinab Bandpey (zeinab.bandpey@morgan.edu)

Morgan State University, Baltimore, MD 21251

Prisoners are provided with a college education so that when they are released, they will adjust easily to society and won’t return to prison. I was fascinated by the idea so much that I wanted to be a part of it. As a result, I have been teaching in prison for two consecutive semesters. In this essay, I will explain how the fact that I am an immigrant from Iran having a single-entry visa helped me to get along with students in a U.S. prison and also motivated them to rely on themselves, focus on their successes and do better in math. I will talk about the challenges my students and I have gone through and, at the end, I will come up with some suggestions that I believe might help any prisoner attending math class in prison.

Posted in testing | 2 Comments

## The Choice to go Asynchronous: Discussion Board Based IBL

by Tien Y. Chih

Montana State University, Billings

Since the COVID-19 pandemic hit during the Spring of 2020, I’ve been nothing short of impressed and amazed at my colleagues’ resourcefulness and creativity in shifting their courses to an online modality.  So when I was asked to teach an online Modern Geometry course this past Summer, I was eager to roll out an inquiry-based version of this course.  But when planning this course  I realized that I would face unique challenges that would make this difficult.

MSU-Billings is a comprehensive regional state university which serves central and eastern Montana and northern Wyoming, parts of the nation that are often very rural and distant from the physical campus.  For this reason, the school has had a strong focus on its online course offerings even prior to the pandemic.  In particular, Modern Geometry serves as the last course in a Math Teaching Minor, which certifies current teachers in the state to teach Mathematics in addition to their other certifications.  Because this minor is intended for current working professionals from across the state, it is necessary for the courses in it to be online.

## MATH ON THE BORDER: Working with unaccompanied migrant children in Federal custody

The events recounted here happened in January 2020. The program described has been suspended during the COVID crisis.  Perhaps there will be no need for it when the crisis is over.

Nadia looked at me with big brown eyes and asked a question.  My Spanish is minimal, so I called over a coworker, one of the caregivers at her shelter.  She was working with tangrams (a geometric puzzle), and was asking whether she could turn a particular piece sideways to form a certain shape.  This was not how the question was translated, and probably not how it was posed.  But I understood it, despite the dual barriers of language and formality.

Nadia is a migrant child who has been separated from her parents and is under Federal custody with the Office of Refugee Resettlement (ORR).  She may have come without authorization with a “coyote”, or been left with a relative and picked up in a raid, or just walked over the border herself.  I do not know how she got here.  But her bright eyes and her engagement with geometry tell me all I need to know.  Her mind is alive, and I want to keep it that way.  Like most of these children, she is resilient and resourceful.  And like most of these children, highly motivated.  These are immigrants, and immigration is a filter.  Only the most energetic and future-minded are likely to pass through.

## 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.

## 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.

## 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

Posted in Classroom Practices, Student Experiences | 7 Comments

## 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

## 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.

Posted in testing | 5 Comments

## 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.

Posted in Classroom Practices, Online Education | 1 Comment

## 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