Oliver Steele

What is a “hexangle”?

What follow are some notes from talking about the elections and the presidential primaries with my children, and some metaphors that I found helpful in thinking about the topics. They’re not otherwise related to each other, except that they all came up over the last couple of days.

1. Votes are Agents

Division-free GCD

A few years ago I described an algorithm for computing the greatest common denominator without division. (Euclid’s algorithm requires division, in order to compute the remainder.) Although less efficient than the standard algorithm, I found it easier to teach to my children when they were learning to add fractions.

Here’s a picture I drew to explain addition and subtraction of fractions to the sixth-grader:

We also ended up using a variant on Euclid’s algorithm for finding the GCD. It uses subtraction instead of division and remainder; it’s in general less efficient, but it’s easier to explain and can be easier to do in your head, when the numbers are small.

Construct a series whose first two terms are the inputs, and then continue as follows: each successive term is the absolute value of the difference between the preceding two terms — that is, simply subtract the smaller from the larger. If you reach one, the GCD is one; if you reach zero, the GCD is the previous term. (Or, you could also let the series peter out to zero, but the way I’ve stated it is simpler in practice.)

I posed a second-grader the question of what nine squared was. She reasoned that ten squared is 100, and nine times ten is ten less then that, and nine times nine is nine less than that, so the answer is 81. Then I asked her what eight squared was, and she was flummoxed. She saw that it was a similar problem to the one she’d just solved, but wasn’t sure how to apply the analogy.

Here are the pictures that showed her how to figure out the answer. We drew the location of the squares on a multiplication grid:

and I introduced the idea of a “solution structure”. A solution structure is a graphical representation of the steps of a solution. This is the section that represents the relation between 9² and 10².

Last weekend, I shared some interesting properties of numbers with my kids.

The great thing about explaining something to a non-expert is that you have to actually understand the topic. (This is why making teaching universities and research universities the same actually makes sense.) If you hide behind a formalism, the explanation won’t work. Usually, this means that you didn’t understand why the formalism worked either.

This is why I thought “why are far away things smaller?” was such a great question. “Similar triangles” answers are brittle, and if a tiny error makes far away things come out bigger instead, you won’t catch yourself until you got to the end of the proof.

Mickey Kaus writes:

It’s worth noting that, in the event, not only did successor Arnold Schwarzenegger get more votes (3,744,132) than Davis (3,562,487), he also got more votes than Davis got in November, 2002 (3,469,025) when Davis won reelection.

Dot numbers are a new notation for numbers, that make integer addition look like rational multiplication. They may be useful in primary school math education. The idea is that once you understand integers and addition, you can learn another way to look at it that sets you up to understand fractions and multiplication.

I made up dot numbers a few years ago to try to explain negative numbers to my then-four-year-old son.

Basics

A dot number is a way of writing a number. A dot number is represented as a number of dots above a line. This is the number 3, as a dot number:

Seymour Papert used to tell a story contrasting the practices of medicine and education, in order to illustrate how little the latter has improved. Place a physician from the previous century in a modern operating room and he¹ won’t have a clue about what to do. Transport a teacher forward in time and they’ll fit right in. The moral is that the practice of medicine has made great strides during the last century; the practice of education hasn’t progressed at all.

These questions came up on a family drive last weekend:

• How many posts does a hundred-yard fence with one-yard beams have?

• What if the fence is circular?

• What if it’s a cross?

• What if it’s a figure eight?

The first question illustrates fencepost error. The second relates to the discovery of the benzene ring.