This year, I didn’t teach students to factor the difference of squares. Don’t get me wrong, I did teach how to factor the difference of squares, but I held off on giving this particular case of a quadratic a label.
For the past several years I have been reflecting a great deal on the pedagogy of teaching mathematics and how precise language factors into best teaching practices. So I want to talk about this decision around difference of squares.
Let me start by saying I am a huge believer in using precise and robust mathematical vocabulary. Being able to name something allows us to organize our thoughts better and to communicate effectively with others.
Case in point: this year, I used the language of input-output extensively in reference to functions of all sorts and in all representations. When we got to the midterm exam and students were identifying a function by looking at graphs, I expected an explanation of “it passed the vertical line test.” But to my surprise and delight, out of 30+ students, fewer than 5 left their reason as simply “it passed the vertical line test.” The rest went beyond that to give some variant of “it passed the vertical line test, which showed that for each input there was exactly one output.” Bravo, students! Bravo! I never emphasized memorizing the exact words of the definition of a function, but through using language carefully and consistently, students were able to recall this definition with ease.
Let me mention a second example. Continue reading