# Education

## The Problem with Problems

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…

## Some Thoughts on Teaching Mathematics

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…

## Is Zero a Factor of Zero?

Generally speaking, if $$a \times b = c$$, then $$a$$ and $$b$$ are factors of $$c$$. This concept appears at the secondary level in two contexts: The factors of positive integers, and the factors of a polynomial. If we limit…

## Pascal, Pacioli, Probability, and Problem-Based Learning

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…

## Geometry for multiplication, division, and roots

Contemporary plane geometry of the sort taught in the standard American high school is most heavily informed by two books and a third mathematician. The first of these is Euclid’s Elements, which is so conceptually tied to planar geometry that…

## Factoring and long division

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…

## Positive numbers and absolute value

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…

## The smallest angle

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,…

## Concepts vs procedures

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…

## Multiplication Table Slide Rule

Using Publisher, I’ve created a slide rule for multiplication tables (up to 10×10). To use it: — Print it out and cut along the dotted line. — Move the 1 on the bottom part to any single digit on the…

## Town Squares problem

A friend of mine, a father, recently posted this item on his Facebook feed. It’s from Pearson, and he was struggling figuring it out. I also had to read it several times to figure it out. This is a large…

## The Geometric Proof of Infinite Primes

I was recently wondering why Euclid, the geometer, published a proof that there is an infinite number of primes. I should have known that his proof is geometric. It is: “Let A, B, and C be distinct lengths that cannot be…