By Art Duval, Contributing Editor, University of Texas at El Paso
A popular saying in business (or so I’ve read) is to “eat your own cooking”: Use the products your own company makes. I suppose there are several motivations to do this: to demonstrate faith in your own work; to be your own quality control team; to make your product visible; etc. What does that have to do with teaching and learning mathematics?
The best part about being on the editorial board for this blog continues to be the privilege of working with a talented group of editors and with all sorts of creative authors, who collectively have an incredible variety of important things to say. (F. Scott Fitzgerald: “You don’t write because you want to say something, you write because you have something to say.”) As a result, I sometimes feel like I am drowning in interesting ideas, with not nearly enough time to try them all. Today I would like to tell you about the articles we’ve published here that contain ideas I’ve tried myself and/or shared with students and colleagues. In other words, to answer the question “What have I eaten of our own cooking?” Bon appétit!
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Let’s start with ideas I explicitly share with students. Probably my biggest pet peeve with students is when they find the inverse of some function y=f(x) by first “swapping the x and y variables”. This is both mathematically and pedagogically unsound, as explained so completely by Frank Wilson, Scott Adamson, Trey Cox, and Alan O’Bryan in their article Inverse Functions: We’re Teaching It All Wrong. When my students make this mistake and, worse, see nothing wrong with it, I share this article with the whole class, and briefly summarize the ideas in class.
I also frequently share with students an article I wrote, One Reason Fractions (and Many Other Topics) Are Hard: Equivalence Relations Up and Down the Mathematics Curriculum. The more I look, the more I see equivalence relations throughout mathematics, causing hidden difficulties for students not just with fractions, but also with vectors, similar matrices, limits, and the difference between permutations and combinations. Beyond sharing this article with students, I keep in mind the difficulties caused when we need to work with equivalence classes as objects, so I can head off students’ confusion before it sets in.
Another article I use frequently for myself is Don’t Count Them Out – Helping Students Successfully Solve Combinatorial Tasks, by Elise Lockwood. I regularly teach Discrete Mathematics, and I now follow her advice to have students “focus on sets of outcomes” and not just the number of outcomes. I start each counting technique lesson by having students make a systematic list of all the outcomes. From the discussion that follows afterwards, some (not all) students understand and even sometimes figure out themselves the formulas that they will need to count such sets when they become too large to make an explicit list.
The article, Mathematics Professors and Mathematics Majors’ Expectations of Lectures in Advanced Mathematics, by Keith Weber made a big impression on me when it came out, and I have shared this one as well with students in proof-based courses. I probably need to review this article again, because I see myself slipping back to old habits, such as not writing down enough details of proofs, that I worked hard to reverse when I first read it.
More recently, I decided to give peer assessment a try, in an introduction to proofs course where I can’t give nearly as much individual feedback as the students need. I started with Elise Lockwood’s article Let Your Students Do Some Grading? Using Peer Assessment to Help Students Understand Key Concepts, and with the references it contains, to build out a system. In the end, it seemed that students learned more from when they assessed their peers than from the feedback they got when their peers assessed them.
Finally, a big part of my attitude these days towards students and mathematics comes from the idea of the growth mindset, that being good at mathematics (or other disciplines) is more a result of hard work than of any genetic predisposition. This idea is stated so beautifully in Ben Braun’s article The Secret Question (Are We Actually Good at Math?).
I invite you to revisit these articles, or browse the rest of our collection, to find a tasty morsel of your own from our kitchen of mathematics teaching and learning.