Ben Orlin’s recent post on using arrays with multiplication made me reminisce on days past, in which I talked to other educators about foundations of mathematics. I’ve also started watching a great youtube lecture series on the history of math, which also made me rethink how I should be teaching math.

In tutoring, I’ve spent a lot of my time showing students how their current work is just their old work with a twist. Algebra is just arithmetic with a sprinkle of magic. (I’ve seen first graders do 1 + __ = 4). That’s a one-step equation in algebra, and students are forced to subtract one from both sides. WHY?!? Calculus is just algebra with very BIG and very small concepts added in. I spent a while showing a student that the equation of a tangent line to a curve is just point-slope form of a line. Oy. Then she got it.

So let’s look at how arrays can give us the basic operations, without needing any variables.

**Addition**: Hopefully obvious. 3 + 2 = 5. Count the dots.

**Subtraction**: A bit harder, because it involves “take away”. But otherwise obvious (once you know it, anyway).

That’s it for 1-dimensional. Now we move into 2D.

**Multiplication** can be defined as repeated addition (3 + 3 + 3 + 3 + 3 = 3×5) or as equal groupings (5 groups, each one has 3 in it). But instead of drawing numbers on a line, we go into arrays.

I’ll leave the details to Ben’s post.

**Division** is where I haven’t seen too much. We pester kids with the fact that it’s the inverse of multiplication using families (“If 2 x 5 = 10, then 10/5 = 2”). But where’s the JUSTIFICATION for it? Where can I SEE this happen?

The struggle later turns into remainders, and then remainders as decimals and fractions.

There are many great division strategies: standard long-division, partial quotients (one of my favorites), and many others.

But how do we show it? Well, just do multiplication backwards. Multiplication = repeated addition or equal groups. So division = repeated subtraction, or equal groupings.

Then let’s try 12 / 3.

Not bad. What about 13/3?

So, we’re left with remainder 1. That seems reasonable, right? But what about when we want students to represent the remainder as a *gasp* fraction? We can tell them that since it’s in 3 equal groups, we should put 3 as the denominator and then the remainder 1 becomes the numerator. But…WHY?!?

Well, we’re counting rows. How many rows did we get? 4. How many more ROWS do we have? Well…1 piece left over. OK, so if we JUST LOOK AT THE ONE ROW left there, all of a sudden we see it: there is 1/3 of a row! So 13/3 = 4 1/3.

Questions, comments?

Cheers.

Jeff