What’s in a name?

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

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Tiny Change, Big Difference.

One of the jobs I have as a math teacher is to coach students to become more and more confident in their own ability to think and reason.

Recently, I was working with a student.  She is an incredibly hard worker, but lacks confidence when it comes to math.  So naturally, when she comes for extra help, she looks to me constantly to reassure her that she is doing the “right thing.”  She will say things like:

“Would I add x to both sides now?”

“Should I set them equal to 180?”

“Do I use CPCTC now?”

And yet, I don’t want her to be looking to me for whether or not she’s on the right track.  I want her to first look to her own reasoning abilities.  So we have worked to change her language from “Should I,” “Would I,” and “Do I” to:

“I think  _______________ because _______________.”

Starting that very day, she really took my suggestion to heart.  For example: she was working with a complicated diagram trying to find angle measures. She started to say: “Would I add these up to equal 180?” But she caught herself and said instead: “I think I should add these angles up to equal 180 because they make a straight line.” To which she responded (to herself) “oh wait…no they don’t.  They make a line with a third angle, though.  I could add all 3 up to equal 180.”

It seems like a small thing, but a teacher is there to coach students through the process of learning math, not to be the arbiter of all truth.  Students are empowered when they learn to think of themselves as critical thinkers, capable of reasoning through complicated problems.