In Memoriam N. N. Konstantinov

by Mark Saul

This summer marks the thirtieth year since the end of the Soviet Union.  It also marks the passing of one of the great figures of Russian mathematical culture, Nicholas Nikolayevich Konstantinov.  This note concerns both events, but cannot do justice to either.  Rather, I will here give some personal reminiscences that might contribute to the picture, but not find a place in the historical record.    I leave to other sources the task of a more comprehensive account.  Here’s my story.

The year was 1987.  The Cold War was still smoldering, but no longer raging.  I received a phone call from an American teaching colleague: “I got an email message for you from one Professor Konstantinov in Moscow.”

Just the fact that this message had arrived was remarkable.  The World Wide Web had yet to appear.  Email was new and laptops rare.  And it was not yet clear that the internet could form a bridge between the two camps in the global political stalemate.  Was a graph of the Eastern and Western computer networks even connected?  How did a colleague from Moscow contact me?  And why?

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Outcomes-Based Assessment — Structural Change in Calculus

by Rebecca Torrey

Associate Professor of Math

Brandeis University

Traditional Grading Sends the Wrong Message

For many years I taught Calculus with a traditional structure, in which the students’ grades were mostly determined by a few high-stakes exams (a final and a couple of midterms).  In my classes, I would tell my students:

  • How important it was to practice regularly; 
  • To carefully review their exams and the solutions;
  • That it’s ok to get things wrong and learn from their mistakes;
  • That the idea that we can improve through practice applies in math just as it would in anything else they want to learn.

But the structure of my class was giving them a very different message.  The structure told them:

  • You only really need to study three times during the semester: right before the midterms and the final; 
  • Don’t bother reviewing your work since you will rarely, if ever, get tested on those same problems again;
  • You can only do well in the class if you get all the problems (including the very hardest) right on the first try.

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Mathematics as Logic

by Mark Saul

Maybe it is obvious, but it is something I’ve come to appreciate only after years of experience: mathematics is logic driven, and teaching and learning mathematics is centered on teaching and learning logic.  I find this to be true philosophically, but also practically, in my teaching.  And even in my own learning.

Philosophically, this point of view has deep roots.  Plato’s Academy.  Russell and Whitehead.  Frege, Tarski.  And that’s all I want to say about this area, which is outside my expertise.  I leave it to those who think more deeply about the philosophy of logic to forge connections between my experience and their work.  I think it is probably enough here to think about the ‘logic’ as concerning just the simplest propositional calculus: implication, negation, and perhaps quantifiers.

Because what I want to say is that in my teaching, the closer I look at difficulties that students have the more likely it is that the difficulty is with these basic aspects of logic.  And (conversely!) if students leave my classroom having understood these logical connectives more robustly, I consider that I have succeeded.

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Posted in Classroom Practices, Communication, Curriculum, Faculty Experiences, K-12 Education, Mathematics Education Research | Tagged , , , , , , , | 1 Comment

A Tale of Two Hats (Terrance and Lamar): supporting students in authentic mathematical inquiry

Terrance Pendleton, Drake University

Students who have had me for at least one class are familiar with my alter ego, Lamar. If they were to describe him, they may say that he is the poster child for what not to do in mathematics. They may speak to Lamar’s tendency to arrive at the wrong conclusion by making erroneous assumptions and/or using faulty logic. When a mistake in class is made, it is not unusual for someone to say, “That’s totally a Lamar move to make.”

You may wonder how Lamar came to be. Let me take you back to a chilly spring day in March inside a proof-based linear algebra class. When I first saw this material several years ago, it took me a nontrivial amount of time to understand the difference between a scalar 2 and the identity matrix scaled by 2 when performing matrix algebra. In my current Linear Algebra class, this difference is emphasized through the eyes of Lamar as he tries to prove that a square matrix A is invertible if A^2 – 2A + I = 0. Once Lamar’s name leaves my mouth, students become more alert as they watch for the inevitable misstep that Lamar will make. Indeed, for the example mentioned above, Lamar finds that the inverse of A is given by 2-A. The identification of his misstep leads to a fruitful discussion on Matrix Algebra and I mentally pat myself on the back for what I think is a job well done.

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Posted in Active Learning in Mathematics Series 2015, Classroom Practices, Faculty Experiences, Influence of race and gender | Tagged , , , , , , , | 2 Comments

Building Relationships Before the Semester Begins

By Courtney R. Gibbons (Hamilton College)

We often think about our classes starting on the first day of the semester. But some of our students log on to course management systems and look at the course materials before classes start. I contend that we can start to build relationships with our students well before class begins — with the way we write our syllabi, an initial email to students, and a short first assignment that can be completed before the very first day of class.

A little bit about my own path through college: I went to college right after high school, dropped out in my first semester, went back a year and a half later to finish my first year at a college near home, and then transferred to Colorado College where I (finally!) finished my bachelors degree. To say that my path was bumpy is a gross understatement. I struggled with feeling “behind” my classmates (although, looking back, I doubt anyone realized I was a couple years older than my peers). I had a lot of anxiety about classes, and I often felt like I didn’t belong. (You can hear me talk about this in a pep talk that I recorded for my classes this semester:

I’m keenly aware that many of the things I experienced — that I thought I was alone in experiencing — are becoming more common among our students. Student stress and anxiety levels are rising every year. With that in mind, I’ve been working on centering the humanity of my students in my classes. In this blog post, I’ll share a few things I do before the first day of class.
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Best-Laid Co-Plans for a Lesson on Creating a Mathematical Definition

Steven Boyce, Portland State University
Michael Ion, University of Michigan
Yvonne Lai, University of Nebraska-Lincoln
Kevin McLeod, University of Wisconsin-Milwaukee
Laura Pyzdrowski, West Virginia University
Ruthmae Sears, University of South Florida
Julia St. Goar, Merrimack College

All authors contributed equally to the preparation of the document.

How do students typically engage with new definitions in undergraduate mathematics classes? Are students provided with a definition, and then instructors help students make sense of it? Do students have opportunities to create their own definitions? Often when instructors choose to involve students in the process of creating a definition, the role of the instructor may be to encourage students to structure or word their definitions in a particular way, with the goal of leading students toward a definition found in a textbook. This can be a daunting task. After all, enacting this kind of lesson means anticipating what students may do or say, deciding when to let students keep talking and when to interject, and responding to unexpected contributions. Designing a lesson that is mathematically substantive but also provides opportunities for students to do a lot of the talking (including students providing feedback to other students) is really hard! Even with the most well-laid plans, surprises can still happen.

One way to take on this challenge, and have support as the unexpected arises, is to collaborate with other instructors. The authors of the post are all instructors of geometry courses for prospective high school teachers, who participate in a “GeT: a Pencil” community meeting every other week, and sometimes more often. These community meetings gather university geometry instructors from across the country to collaborate on issues related to the teaching of the geometry course primarily taken by preservice teachers. Among us are mathematics and education faculty, whose academic backgrounds range from mathematical physics to difference equations to hyperbolic geometry to student cognition to teacher education. We saw a pandemic-era opportunity to co-plan and co-teach a common lesson. On Zoom, we can be more than 3000 miles away and learn from each other in the same room. While practices involving the design of lessons (such as the Japanese “lesson study”) have been established for decades in some K-12 settings, it is still rather rare in undergraduate settings, though there are some exceptions.

In this post, we share our experience of developing a lesson that could be taught in any of our courses and how this lesson did not go according to plan. We intended the lesson to focus on creating a new definition. Although the class did not reach a consensus on a definition, the process opened many mathematical questions.

We first show the key example of the concept to be defined. Then we describe why we chose to use this example, how we built a lesson around it, and the unexpected outcomes. Finally, we discuss what we learned (and hope to continue to learn) about collaborative planning and teaching.
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Interactive Teaching ​IS​ Possible with Virtual Learning Technologies

By Enes Akbuga, Drake University (Twitter: @enesakbuga; Email: and Zachariah Hurdle, Utah Valley University (Email:

Many academics and teachers have been struggling with facilitating classes virtually. The 2020 global pandemic has brought many challenges and disruptions to teaching, but opportunities to explore and learn as well. This blog post discusses what we have learned so far, with the hope that these reflections are useful to other higher education instructors.

We teach in two very different university settings. Enes teaches at Drake University, which is a small liberal arts college in Iowa. Zach teaches at Utah Valley University, a public school that is the largest in the state (and open-access, as well). Since the spring of 2020, we have been collaborating on opportunities to use and explore some technological tools. Via frequent discussions over the past year on the new teaching and learning space, we shared some of the successes and frustrations throughout the experience. Specifically in this post, we share some of the highlights of facilitating synchronous​ class sessions using video conferencing tools. From what we have learned so far, most students enjoy real-time, synchronous, virtual interactions and perhaps prefer that over non-synchronous interactions. Like most instructors, we found Zoom (or likewise) to be a useful tool in facilitating online mathematics courses that is an experience shared across other institutions. The main motivation for this post is to share some of our ​experiences​ teaching mathematics online as well as talk about our thoughts on the ​possibilities​ of interactive teaching pedagogies.

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Mastery Grading for Future Elementary School Teachers

By Emily McMillon and George Nasr (University of Nebraska-Lincoln)

We—Emily McMillon and George Nasr—are graduate students at the University of Nebraska-Lincoln. We implemented mastery based testing for two sections of a course on geometry for pre-service elementary teachers during the Spring 2020 semester, and found that our students

  • looked over mistakes on assessments to improve their understanding,
  • felt less stress and testing anxiety,
  • experienced increased confidence in mathematics and greater growth mindset,
  • viewed exams as an opportunity to show knowledge, and
  • reflected on the purpose of assessment in student learning.

In this post, we will discuss what led us to try mastery based testing for this student population, how we implemented mastery based testing in our courses, and some student survey responses.

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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?

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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.

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Posted in Communication, Education Policy, Faculty Experiences, K-12 Education, Mathematics Education Research, Outreach, Student Experiences | Tagged , , , , , , , | 1 Comment