## Remote proctoring: a failed experiment in control

By Ben Blum-Smith, Contributing Editor

Due to the global health crisis, a huge amount of instruction that was happening in person a year ago is now happening online. One theme highlighted by this change is the question of control. When students are in buildings with us, we[1] have a high degree of control over the environment in which instruction takes place and the materials the students have access to. We even have a significant level of power over students’ movements and choices, at least while they’re in front of us. This is most obvious in primary and secondary school, where there is usually a whole “disciplinary” administrative apparatus designed to support instructors’ ability to dictate the movements and choices of students. But even at college and university, where for example there is often no explicit rule against a student getting up and leaving the classroom or building at any time, physical and social aspects of the classroom setting serve as a mechanism of influence. Continuing the example, to leave a classroom in the middle of class you have to physically stand up and collect your stuff, which means everybody knows you’re not coming back, and then face everyone as you walk past them on the way out. The instructor will certainly notice, will probably be hurt, and won’t necessarily respond kindly. It’s very rare for students to do this—in fairness, this is probably (hopefully) mostly because they don’t want to—but it’s very rare even when they do.

A fundamental aspect of the switch to distance learning is its disruption of all the usual structures and processes by which this control is exercised. In our running example, you can leave a Zoom class just by clicking “Leave”, with no need to awkwardly face anyone and a reasonable likelihood, depending on the size of the class, that the instructor won’t even notice. To cover your bases, you can instead leave without leaving—just mute yourself, turn off video, and go about your business while remaining formally in the meeting.

For a different and much-discussed example, while we are used to being able to design students’ environments rather meticulously during exam proctoring to head off both distraction and temptation, there is no analogous form of control over the exam environment built into distance learning.

How are we collectively responding to the challenges this change presents?

## Happy New Year(?)

Mark Saul, Editor

Mathematics and mathematicians rarely make press.  So it was a bit sweet, but mostly bitter, to read in the New Yorker of the deaths of John Conway, Ronald Graham, and Freeman Dyson, three great losses to our profession.  (Yes, Virginia, Dyson published in ‘pure’ mathematics as well as in physics.)

And of course as soon as this article appeared, friends and colleagues wrote about others we have lost who were not mentioned in the press.  It is likely that each of us has suffered some loss, some grief.  I write here of my own, and what we can learn from it about our work.

## The Calculus of Context

by Yvonne Lai (University of Nebraska-Lincoln)

It is 2020. You are taking a high school mathematics teacher licensure exam. Suppose you see these questions. What do you do? What do you think? (Warning: Your head may spin. These are not licensure exam problems from 2020. Further commentary to come.)

I own a horse and a farm. One fourth the value of the farm is four times the value of the horse. Both taken together are worth $1,700. Find the value of each. Write out a complete analysis. A merchant gets 500 barrels of flour insured for 75% of its cost, at 2 1/2%, paying$80.85 premium. For how much per barrel must he sell the flour to make 20% upon cost price?

Perhaps you are thinking about proportional reasoning and percentages. You might also be thinking: How quaint. These numbers are unnecessarily contrived; and owning horses, farms, and flour barrels is unrealistic to most students and teachers these days.

It is 1895. You are taking a high school mathematics teacher licensure exam and you see those same questions. What do you do? What do you think?

You might still think the numbers are contrived, but the context may seem more realistic.

## 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