Three foundational theorems of elementary school math

By Ben Blum-Smith, Contributing Editor

This post discusses three very familiar facts from grade-school mathematics. In spite of their familiarity, I believe they tend to go under-appreciated, at every level of math education. In the elementary grades, my experience is that if they do get explicit attention, we generally treat them as tools students should learn to use, rather than as the subject of their own inquiry. Meanwhile, in the later grades—middle school, high school, college, and beyond—we are already used to them, so they tend to be seen as trivialities, not worthy of further reflection.

I think this might be a missed opportunity.[1] We lose the chance to delectate with our students in these facts’ surprisingness, their non-obviousness. And we pass up the occasion to ask why they hold.

Therefore, I submit to you three foundational theorems of elementary school math.

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From Teaching Math to Teaching Students Math

by Yvonne Lai (University of Nebraska-Lincoln)

I did not want to present. Someone had selected my solution to a geometry problem to present at a Mathfest 1996 session. I wasn’t sure who this person was, but I knew already that I did not appreciate them. I was 16 years old, my father was ecstatic at the honor, and I wished my father had never found out, because then I could have played hooky and no one would be the wiser.

The problem that I did not want to present

The visual of a solution that I did not want to present

That summer, I was a student at the Canada/USA Mathcamp, a residential program whose name describes its purpose. Mathfest and Mathcamp both took place on the campus of the University of Washington-Seattle that year, and someone had the brilliant idea that a few high school students would be made to present their solutions to the problem in the above figure – which featured in the entrance application to Mathcamp – at Mathfest 1996.

When I met my two co-presenters, I found out that none of us wanted to present, nor had we any idea how to present even a measly 10 minutes of mathematics, nor had anyone told us anything except that we were supposed to show up at a particular room on the appointed day. (Mathcamp was a young program at the time, formed on a whim and a dream, with all but nonexistent infrastructure at the time.) The moment was 10 days away, and we dreaded each closing minute.

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Access To Epidemic Modeling

Kurt Kreith and  Alvin Mendle, University of California, Davis

Covid-19 has left teachers seeking topics that are both engaging and lend themselves to online instruction.   As a guiding force for the measures that have reshaped our lives, epidemic modeling stands out as natural.  For teachers at the secondary level and those involved in teacher education, this leads to the question: How can an understanding of epidemic modeling be made accessible to students at large?

From the vantage point of evolutionary biology, viewing epidemics as a form of natural selection is a good place to start.  An ability to reproduce and mutate rapidly would seem to give the virus a distinct advantage.  The tools Humankind can bring to bear include (1) human intellect and (2) a capacity for social organization. Both have figured prominently into efforts to manage Covid-19 and make an appearance in the model to be developed below.

Modern epidemic modeling began with the S.I.R. model created by Kermack and McKendrick in 1927, introducing the use of S, I, and R to designate susceptible, infected, and recovered demographic variables.   Here we consider a population P of fixed size (it undergoes neither births nor deaths, and those recovered enjoy total immunity). In set theoretic terms, this can be thought of as

P = S ∪ I ∪ R

where people pass from S to I and from I to R at prescribed rates.  The model yields values for all three variables as a function of time.

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Posted in Active Learning in Mathematics Series 2015, Classroom Practices, Curriculum, K-12 Education, Mathematics Education Research | Tagged , , , , , , | Leave a comment

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 , , , , , , , | 3 Comments

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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Posted in Classroom Practices, Online Education | 2 Comments