Category: Pedagogy

I’m currently reading “Burn Math Class,” and it’s got me thinking about language. Yesterday, I saw an item about teaching students why cancelling works in this case: \[5 + 3 – 3\] but not in this case: \[5 + 3 = 5 – 3\] The conclusion that the students were led to is that the…

This morning, I was reading the NCTM blog, and the subject was on students struggling with systems of linear inequalities. First, as background: I don’t have any difficulty with systems of linear inequalities, and I don’t remember ever being taught such things (although I may have, I just don’t remember). But then I get two…

As a high school teacher, I struggle routinely with getting students to understand that \(x/0\) is undefined. Students don’t seem to understand that division with a remainder is incomplete. I have long attributed this to the way that division is taught in the elementary school years. For instance, \(17\div 5 = 3R2\) is considered perfectly…

One of the questions my mind keeps returning to is: What is a number? I don’t mean this in a high-level set theory way. I’m not talking about aleph-numbers or other sorts of concepts. I’m restricting my thoughts here to the sort of numbers that are the domain of high school mathematics, nothing trickier than…

One of the challenges that I see with students learning mathematics is their confusion with what qualifies as the content of mathematics and the language of mathematics. In a famous and enduring article, “Relational Understanding and Instrumental Understanding”, Richard Skemp pointed out that teaching concepts instead of procedures will be difficult if students think that…

I’m currently reading Howard Eves’s Great Moments in Mathematics After 1650 (1983, Mathematical Association of America), a chronological collection of lectures. The first lecture in this volume (the second of two) is on the development of probability as a formal field of mathematics as it was driven by Pascal and Fermat, with regards to a specific problem…

This morning, I’ve been watching YouTube videos. I started with Tarleen Kaur’s video on Middle Term Splitting. What I find interesting about Kaur’s Chapter to Chapter videos is that, because she’s a student in India, her methods are often different from those I’m familiar with. That’s the case in this video as well. I haven’t…

They say that when you’re a hammer, everything looks like a nail. Since I’m currently thinking about conceptual vs procedural teaching, I’m noticing examples. Here’s a good definition of absolute value: “the magnitude of a real number without regard to its sign; the actual magnitude of a numerical value or measurement, irrespective of its relation…

I have been thinking about procedural vs conceptual thinking, which Skemp’s seminal article refers to as relational vs instructional. One of the questions on this year’s geometry final asks: Given a triangle ABC with sides AB = 5, BC = 6, and AC = 7, what is the smallest angle? (Edit for clarity: The question is simply…

A persistent topic in mathematics education is whether to focus on conceptual or procedural knowledge. After reading Kris Boulton’s recent post that argues, “It depends,” I found myself thinking about the disconnect between arithmetic and algebra. What is needed to understand algebra? The first leap that students need to be able to make is from the…