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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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.

Context and mathematics have never had an easy relationship. Continue reading

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

by Zeinab Bandpey (

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.

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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.

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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.

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Posted in Active Learning in Mathematics Series 2015, Communication, Faculty Experiences, Influence of race and gender, K-12 Education, Mathematics Education Research, Prison | Tagged , , , , , , | Comments Off on MATH ON THE BORDER: Working with unaccompanied migrant children in Federal custody