Believing in Mathematics

By Benjamin Braun, Editor-in-Chief, University of Kentucky

In my experience, many students in K-12 and post-secondary mathematics courses believe that:

• all math problems have known answers,
• failure and misunderstanding are absent from successful mathematics,
• their instructor can always find answers to problems, and
• regardless of what instructors say, students will be judged and/or assessed based on whether or not they can obtain correct answers to problems they are given.

As long as students believe in this mythology, it is hard to motivate them to develop quality mathematical practices. In an effort to undercut these misunderstandings and unproductive beliefs about the nature of mathematics, over the past several years I’ve experimented with assignments and activities that purposefully range across the intellectual, behavioral, and emotional psychological domains. In this article, I provide a toolbox of activities for faculty interested in incorporating these or similar interventions in their courses.

Psychological Domains

A useful oversimplification frames the human psyche as a three-stranded model:

The intellectual, or cognitive, domain regards knowledge and understanding of concepts. The behavioral, or enactive, domain regards the practices and actions with which we apply or develop that knowledge. The emotional, or affective, domain regards how we feel about our knowledge and our actions. All three of these domains play key roles in student learning. In post-secondary mathematics courses, our classroom activities and assessments often focus primarily on intellectual knowledge and understanding, with emotional and behavioral aspects of learning addressed either implicitly or not at all. A partial antidote to this is found in the many active learning techniques being used in post-secondary mathematics courses, such as think-pair-share, “clicker” systems, one-minute papers, inquiry-based learning, and service learning, among others. A strength of active learning methods is that they challenge students’ unhelpful beliefs and practices through public dialogue and activities. What active learning techniques might not explicitly do is frame these discussions and activities within a broader context involving the nature of intelligence and the process of successful learning.

A goal for my courses is to incorporate direct interventions that provide students with three things:

1. language that supports articulate reflection and discussion in the context of emotional and behavioral domains,
2. an environment in which such reflection and discussion arise naturally and effectively, and
3. a contemporary “external source” motivating this language and environment so that our discussion is not driven by the will of the instructor.

The ways in which these interventions are realized in my classes will change over time, and I am willing to follow current educational trends if they are effective tools for my students. Many of the interventions I have used are based on research in psychology regarding mindsets, a topic that I’ve written about previously on this blog. While the literature on mindset research contains contradictory empirical findings, this is not a problem for me since my main goal is to use the language and motivation that this research provides as a tool for engaging students across psychological domains. Mindset research is only one among many possible sources of motivation for meeting the goals above; what is critical is to make sure that my mathematics courses include activities that explicitly promote student development across all three of these psychological domains.

A Toolbox of Interventions

What follows are student assignments and activities that I’ve used in classes ranging from 20-student upper-level courses for math majors to 150-student Calculus courses for STEM majors. They have a common purpose of promoting student development in one or both of the emotional or behavioral domains, complementing other work that my students do to develop intellectually in mathematics. An important disclaimer: none of these activities are original with me; rather, these are all adaptations of the work of others, to whom I will always be indebted.

Introductions. On the first day of class each semester, I begin with students introducing themselves to each other. In a small class with less than 30-50 students, there is time for everyone to take turns sharing with the entire class their name and the reason they are taking the course. In a large-lecture course, I tell students to do the same thing with 4-6 people sitting next to each other. I teach at the University of Kentucky, and many of our STEM majors are primarily enrolled in large lecture courses during their first year. By beginning every course with a 5-minute activity that recognizes the students and promotes discussion, a collaborative tone is set for the remainder of the course, and some of the isolation that students feel (especially as one among many in a large lecture) can be countered.

Day 1, small classes: reading and autobiography assignment. During the first week of class, I assign an article regarding mindset research by Carol Dweck along with a one-page autobiographical essay. I have used Dweck’s articles “The Secret to Raising Smart Kids” and “Is Math a Gift? Beliefs that put females at risk” for this with success. I assign a grade to the essay based on completion only, completely ignoring the quality of the writing, editing, or ideas. The goal is to get students to reflect and be honest, not necessarily to train them to write well. If students respond to the prompt in a relevant manner, they get full credit.

Day 1, large classes: video and small group discussions. In large classes with 150 or more students, especially in courses that are coordinated across sections, the autobiography assignment is harder to implement. Another way to introduce students to the language of mindsets (or other tools) is to have students students watch a 10-minute video about mindset research during class on the first day. Following the video, have students spend 2-3 minutes free response writing about the video. Following the writing, have students spend 2-3 minutes discussing their response with a neighbor in the class.

Course policy on supportive language. I have a course policy on supportive language that I use in all of my classes: Students are not allowed to make disparaging comments about themselves or their mathematical ability, at any time, for any reason. I give students a variety of examples of “banned” phrases and suggested replacements that can be found here. The important aspect of this policy is that it must be enforced — if I hear students making negative comments, I say “course policy” and have them create a neutral rephrasing of their negative self-comment. This is tougher to implement in large lectures, but even in this context the policy sets a positive tone for the first month of class. In large lectures with accompanying recitations, it is important that graduate student teaching assistants are aware of this policy and enforce it during their recitation sections. It is also important that students know that the policy applies to faculty and teaching assistants as well. I had a student in a large Calculus II lecture call me out for violating this policy last semester when I was frustrated at making errors during an example, and it was an excellent moment for the class.

Video regarding effectiveness of science videos. During class, I have students watch a video about research regarding the effectiveness of science videos. As with the video on the first day of class, students complete a two-minute free writing followed by a two-minute discussion with their neighbors regarding their response to the video. For many students, a common behavioral practice is that if they are stuck on a math problem, they immediately search the internet for videos that explain how to do this type of problem. This is typically an unproductive behavior, and dedicating some class time to confront it directly sets the stage for further discussions regarding the processes students use for completing homework and solving problems.

Assign an unsolved problem as homework. As I’ve written before, assigning an unsolved math problem as homework can serve as a gateway to discussions about the nature of high-level mathematical problem solving and the processes, practices, and attitudes that students bring to authentic mathematical challenges. When I assign an unsolved problem, e.g. those given in the article linked to above, I provide students with the following prompt.

This is a famous unsolved problem in mathematics. Work on it for a while — the goal isn’t for you to solve this, but rather to get a feel for the problem. Create an essay by recording your thoughts and attempts as you work. Focus on responding to the following questions: What did you try to do? Why did you try this? What did you discover as a result? Why is this problem challenging? (Seriously, write down everything you’re thinking and every idea you try, even if it doesn’t go anywhere.)

It’s good to grade this problem generously regarding mathematical content, keeping in mind that the goal is for students to be rewarded for demonstrating persistence and good mathematical processes.

Reflective essay about homework. In most of my upper-level courses, especially those in which I assign an unsolved problem as homework, I have students write a 2-3-page essay explaining what they found most and least challenging in the homework so far, and what their most and least favorite homework problems have been. The prompt can ask them to directly link to mindset or another external topic, or can be left relatively open-ended to see what connections students make on their own. This can be either graded with a rubric for writing or graded based on completion. The majority of my students have discussed at length their experience working on the unsolved problem, both what they did and how they felt about their work.

Create-your-own homework assignment. A recent assignment that I’ve used is to have students write their own homework assignment toward the end of the semester. The specific prompt I used was this:

Create your own homework assignment containing three problems. The homework assignment should be typed. There should be a mix of easy and hard problems that represent a broad spectrum of ideas from the entire course. For each of these problems, type a paragraph explaining why you chose that problem, whether you think it is easy, medium, or hard in difficulty, and what area of the course the problem represents. Once you have created the homework assignment, you should include complete solutions to each of the problems. Your solutions to the problems may be either typed or handwritten, but they should be complete and correct.

It was fascinating to see what the students came up with for their homework. What I found particularly noteworthy was the large number of students who included as one problem a critical analysis essay or short reflective essay similar to what I had assigned in the course to complement mathematical content work. I had honestly expected their assignments to contain a range of standard problems focused on mathematical content, and was pleasantly surprised to see the students incorporating into their homework tasks that addressed behavioral and emotional aspects of doing mathematics.

End-of-course reflective essay. In my smaller classes, I assign as the final homework assignment the following short essay prompt. The grade is based only on completion, because I want students to write honestly without fear of being penalized for their opinions.

What were six of the most important discoveries or realizations you made in this class? In other words, what are you taking away from this class that you think might stick with you over time and/or influence you in the future? What have you experienced that might have a long-term effect on you intellectually or personally? These can include things you had not realized about mathematics or society, specific homework problems or theorems from the readings, etc. These can be things that made sense to you, or topics where you were confused, points that you agreed/disagreed with in the readings or class discussions, issues that arose while working on your course project, etc. Explain why these six discoveries or realizations are important to you.

I have found that reading through these essays is a fascinating exercise, because of the wide range of messages that the students perceived as being central to the course. Using this assignment consistently over time has helped me improve my ability to create focused courses with clearly defined and communicated learning outcomes.

Final Thought

If you experiment with any of these activities in your own courses, I would love to hear about your experiences!