What's 3∙4? 4 + 4 + 4. . . err, yeah, so . . .

What's 3∙2/7? 2/7 + 2/7 + 2/7

What's 1/2∙4? Count four half a time, 2.

What's 1/2∙2/7? Count 2/7 half of a time, 1/7.

What's π∙3? 3 + 3 + 3 + (π - 3) ∙ 3, which is the irrational bit of pi times 3, so count 3 the irrational bit of pi times.

So it's a bit rubbish (just a bit), but that doesn't mean it doesn't have value, and that doesn't mean the thinking of the above isn't true; it is true. How do you count irrationally? You just do it. However, that kind of thinking does require a pretty fluent understanding of numbers and multiplication (which goes back to me saying over and over again that number sense is one of the most overlooked math skills (before common core of course)), so from a pedagogical perspective, I'm OK with devaluing the emphasis of the idea-ish.

Aside: I always get more than a little bit frustrated when mathematicians refuse to have any sort of flexibility to their understanding. There's nothing wrong with negative area. There's nothing wrong with describing the derivative as the slope. I'm not saying you should stop there, but as long as you're fine with a certain amount of bend, pretty much everything can be extended to continue to make sense. Definitions are fantastic, but if you can't see how things are analogous, I don't know how you have any understanding of math at all.

Now, that being said, a much better representation of multiplication is area. I mean, the magnitude of the cross product is already the area of the parallelogram of the vectors, so it makes sense to do it this way, and thank god, it works, like, always (so long as you're OK with negative area . . .)

3×4 = the area of a rectangle of length 3 and 4, 12.

3×2/7 = the area of a rectangle of lengths 3 and 2/7; 6/7.

1/2 × 2/7 = the area of a rectangle of lengths 1/2 and 2/7; 1/7.

π × 3 = the area of a rectangle of lengths π and 3; which you know, when you go to actually count those tiles up, you'll have 3 by 3 tiles, and then you'll have 3 fractional tiles that are each 1 by the irrational bit of pi length. . .

err . . . yeah, so, the important thing is that it works right? No, that's never the important thing. LOOK HOW AWESOME THIS IS. This is amazing. Two separate ideas leading to the exact same understanding, and the second idea maybe helps you understand a little more what I meant by counting 3 by the irrational bit of pi times. It's the area of those slivers of rectangles.

Picture:

And it even goes farther than this; think of the Cartesian product between two sets, one set (let's call it

**3**)

**being the set of real numbers between 0 and 3, the other set (let's call it**

**π**) being the set of real numbers between 0 and π. Then the picture above is a picture of the Cartesian product of the two sets!

The Cartesian product is the set of all ordered pairs of both sets, wanna' guess how it's denoted?

**3**×

**π**. This is the set of things like the point (0,3), (3,2), (1.9, π), any ordered pair which can be made with numbers from 0 to 3 for the first coordinate and 0 to pi for the second coordinate.

Of course, it's not commutative or associative, and it would be tempting to describe the set as being of "size" 3π, but in reality, the set is infinitely large.

The differences are interesting, not arbitrary, and mathematics is beautiful.

But yes, by all means, get hung up on describing multiplication being repeated addition.