Math Epiphany: A Review

PK and I have been working our way through the Interactive Mathematics Program v. 1. Veeery slowly, as I have to struggle with both his $@*! math anxiety and the fact that I’ve forgotten almost all the math I learned in high school :(.

Luckily, however, I actually do have a pretty good head for math (I did well in it, mostly–but my own $@*! math anxiety prevented me from pursuing it once I got to college, and meant that I didn’t actually learn it thoroughly enough to remember it), as does PK. And his father does use higher math pretty regularly on the job and remembers it very well, so when we get stuck we can ask him to help explain what it is we’re missing. More often, though, because this program is all about figuring stuff out with peers, PK and I reason our way through something together and only need the husband to explain what it is we’ve “learned” after the fact (Me: “so is there a way to simplify this problem now that we know how to write it? Is this some form of the quadratic equation?”)

PK and I were really proud of ourselves a couple of days ago, because I had him read the page below out loud to me before we set off on some errands, thinking we could maybe try  to figure out the answer while we ran around town. (Click to enlarge)

"Checkerboard Squares"

I hadn’t yet read it myself, though, and PK pretty quickly said that this wasn’t something we could just reason out in our heads–we’d need pencil and paper, and looking it over myself, I agreed. No biggie; we’d do the pencil-and-paper stuff after we got home, maybe, or the next day.

But while we were erranding, PK was thinking about the problem, and he realized that the smart way to do it was not to start out by trying to figure out how many squares on an 8×8 board, but rather to figure out how many squares on a 2×2, 3×3, etc–and then figure out the formula from the lower numbers. So he started figuring out the lower squares in his head, and when we got home we wrote down the answers (and double-checked the 4×4 board, which was already starting to get hard to hold visually without an actual thing to look at).

So far, so good: despite “we can’t,” PK kept thinking about the problem. More and more I’m realizing that he does that, even when the “we can’t” is more like “WE CAN’T AND YOU CAN’T MAKE ME AND I HATE YOU” followed by door-slamming and crying. Point one for homeschooling: I have the freedom his teachers don’t to just let him walk away when his anxiety peaks, knowing (after much observation) that that doesn’t actually mean he’s giving up (and freeing us both from compounding the problem with a pointless power struggle that just makes him more anxious and more angry).

Awesomeness of IMP program: the book didn’t tell us how to solve the problem, which meant that PK was free to take a different approach, and I felt super accomplished when I said “hey, let’s use an in/out table to figure out the rule here.” (The book has walked us through in/out tables, which were something I never encountered in my own math education but that PK tells me he did in 5th grade math. Wowsers.)

And it didn’t take too long before we figured out the formula: x = 1^2 + 2^2 + 3^2 + 4^2 ….

We both got to feel super pleased that we’d worked through a “Problem of the Week” in one evening (the last one had taken us much longer), and without Papa’s help (the husband being gone on a work trip). So far, so good; I have hope that repeated accomplishments like this one will eventually help overcome PK’s “we can’t” anxiety and reawaken his enjoyment of a tough math challenge.

The epiphanies, however, happened this morning, when I took a risk and opened up the book while PK was eating breakfast. He didn’t immediately start shouting, which is a good sign. I looked over the problem just to make sure that we’d covered it–so far I’m not having PK do the “write up”s to these problems, because writing is another whole ball of refusal that I don’t want to add into the mix quite yet–and said, “hm, we’ve solved the problem, but the book is asking us to explain why the solution works, and maybe how it might be a useful thing to understand beyond the “how many squares on a checkerboard” question. Which is an interesting problem, but not terribly useful in real life.”

We looked at it a bit, and I had my epiphany first: “oh, I get it! The reason that it’s 1^2 + 2^2 + 3^2 + 4^2 and so on is that a “square” in geometry is the same as a “square” in algebra! They just look different! A drawing of a square looks like this, but in the language of math, you write that as x^2!”

(“The language of math” is an analogy I’ve used to try to explain to PK the difference between “getting” math in the reasoning/visual sense–which he’s good at–and being able to “do” math in the algorithmic sense–which he has to learn. IOW, the equations and stuff you see in a “math book” are like words written in English (or any other language) on a page, and learning to “read” math is a slow process, kind of like sounding out words at first–but if you stick with it, I tell him, and don’t let frustration get the better of you, you’ll get fluent at it just like you are as a reader.)

PK’s epiphany–which didn’t feel like one to him, but as you’ll see, it was anyway–was a bored, “Mama, that’s not an explanation. That’s just restating the problem.”

Why yes, it is, I realized. In fact, that’s the WHOLE POINT of this exercise in the textbook–helping us to see that the algebraic expression is just a way of “writing” the geometric/visual drawing (and the abstract figure that it represents)–and vice-versa.

In other words, “the language of math” isn’t just an analogy; it is, in fact, what we’re doing with this algebra stuff.

Once I explained this to him, he “got it.”

This should make the whole thing so much easier, for both of us. I don’t have to be teaching him math; I’m just helping him learn how to read and write it by learning it alongside him.

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7 responses to “Math Epiphany: A Review

  • albe

    Another reason this is an interesting problem is that then he can try to generalize to an n x n checkerboard. He’s got the recursive formula (1^2 + 2^2 + 3^2 + … + (n-1)^2 + n^2). Can he figure out an explicit formula (in other words, a direct formula for the sum of the first n squares)?

    It’s so neat to see this example of PK figuring out an interesting problem in his own way! In great classrooms, this is exactly what doing math looks like.

  • tedra

    Hmm, I wonder if we *can* figure out an explicit formula… maybe we’ll work on it!

    (And again, THANK YOU SO MUCH for putting me onto this math curriculum.)

    I’m glad you liked seeing the example–I thought it was neat, too, and I really wanted to explain how “doing math” was working, because “teaching” it is new to me and figuring out how teaching it works is fascinating.

  • Dave W.

    There is an explicit formula for the sum of squares, but it may be a little tricky to figure out. PK’s general approach of figuring out the answer for small numbers should be helpful – looking at the factors of those numbers should be even more helpful.

    Meanwhile, I’m not sure if you guys worked this all the way out, but in justifying the sum of squares formula, there’s at least one way to do it by thinking about all the places a kxk square can fit within an nxn square, and in particular, thinking about where the lower left corner of that kxk square can fit. It sounds like you guys were at least on track to work that out. There might be other ways to do it, too – that’s one of the neat things about exploring around, you might find another path to a solution.

  • tedra

    Haven’t been able to work it out, so we agreed to look it up and see if we could backwards-engineer it (i.e.,figure out how it worked once we knew the formula). Consulting these two pages to try to figure it out…

    http://mathforum.org/library/drmath/view/59206.html
    http://www.billthelizard.com/2009/07/how-many-squares-are-there-on-chess.html

  • Dave W.

    You might also want to look at these two, that give two different methods for how the closed form for sum of squares formula can be derived. The first is more geometrical, while the second presents the method of finite differences, which can be used to derive the formulas for the sums of kth powers:
    http://blog.jgc.org/2008/01/proof-that-sum-of-squares-of-first-n.html
    http://mathforum.org/library/drmath/view/56988.html

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  • tedra

    Dave, thanks enormously.

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