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)
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.