Math as a Window to God's Character

I got asked today about how it is that I came to see math as a window into the character of God. I'm not sure how to show what I've learned, other than to tell how I came to know it.

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I did not enjoy math in school.

The way I was taught, math was arbitrary: a never ending pile of largely unrelated formulas that must be memorized perfectly and then worked flawlessly. Close doesn't count; it's right --or it's wrong. Teachers seldom had an answer for "When are we going to use this?" They assured us that the upper math has value, but never seemed able to articulate what that value was.

I graduated from high school with a huge sigh of relief: the pre-calculus course I'd taken that year had not gone well, and the hit to my grades carried a heavy cost at scholarship time, and I figured that I'd reached the ceiling of what I was capable of in math. Though I briefly flirted with studying astrophysics, in the end the math intimidated me out of the dream, so I went with Japanese, which required no further math at all.

Then we decided to homeschool.

This meant starting over in math, from the beginning. I was intimidated, not considering myself to be very good at the stuff, but I figured that if I had a particularly "mathy" child, we could outsource math classes when I started feeling like I was in over my head.

But elementary math shouldn't be so hard. I headed to the forums to read about various math curricula. In the process, I ended up discovering how it is that people come to love math: math is patterns. And patterns are both beautiful and fascinating. Math is patterns that can be approached in many different ways, taken apart, and played with, and put back together. On occasion, I got so into a problem -a pattern- that I continued to work it even after my son's interest was spent. (This emphasis on patterns is also the core of the "new math" that everybody hates: my experience was far from unique, unfortunately, and the new "constructivist" approach to teaching math is difficult for parents who were taught with the algorithms only method, like I was.)  We started with Miquon math, which in spite of some weaknesses, taught me as much as it did my children, and then when my oldest outgrew it we continued with MEP, first because it's free, but then afterward we stayed with it because it's just excellent at teaching the kids to find the patterns. And we've all learned a lot about how to see the patterns. I find that I'm actually excited to find out what happens as my oldest gets into the "higher" maths: I am looking forward to the chance to try my hand at it again, this time realizing that there is an underlying pattern, a Real Idea, some bit of reality, that is being described by each type of problem.

I should not have been so surprised by the beauty; math is full of Truth about the world around us, and Truth, Beauty, and Goodness fit together, so where you find one, you'll usually find all three. But the idea that math could be beautiful was so different from the grind of algorithms that I'd always experienced. The reality is, algorithms are only a relatively small part of the story, and if you can work the formula, but you can't see the pattern that makes it function, then you don't really get it, and you haven't learned what it has to teach.

A Math Narration

Everybody's a little bit under the weather, especially me, and I wanted to do math, but not worksheets today. Happily, I follow the Let's Play Math page on Facebook, which is always posting something that's both fun and mathy, and they popped up in my feed, so first, I set up my oldest practicing a couple of his troublesome math facts with the cool math-art thing they had. He and I both did some, and I hung them up on our art door.



Well, Dragon didn't want to be left out of the fun math, so I found one for him, too: a thing where you figure out the differences, and I hoped that he'd learn a cool strategy for figuring out differences, which he did. But it ended up being much cooler than that: we ended up looking at even/odd patterns in the difference problems. It started because I wanted to give a problem: "There are 18 candies, Jon has 5 more than Carl. How many does each boy have?" And you end up with Jon having 11.5 and Carl having 6.5. And I wondered which times you'd have to split a candy. So we made a big thing to show it.

It took us a little while to get it right: we skipped a couple of numbers, and we had all of them with the pattern sides up at first, but that was kind of hard to tell one set from the next, so we fiddled with the boxes for a while to get them to go right. But we got there. At the end, I asked Dragon to tell me what he'd learned. So he did.

More Number Squares

The MEP lesson plans suggested that our Number Square activity should take about nine minutes in a classroom setting. Hero and I have gotten a good deal more milage out of it than that so far! Because the plans said that it was a short activity, I had additional math work ready to go when we started it, but that work sat on the fridge for several days while we played and explored these squares.

The first day, I showed him how to do the activity, using a 3x3 square and the word ONE. Then, we worked together to do a 4x4 square with the word MORE. By the end of that, he had the idea, and I was just watching while he figured things out. Next, he did a 5x5 with his name. That one was more difficult, because he hadn't yet realized that working systematically will make his life much easier. I think this might be the first problem he's run into where working systematically mattered - the activity would have been a success just for that discovery, I think. (There's pictures showing our work in the first post.) For each square, we figured out an equation suggested by the book:

1+2+1=4
1+3+3+1=8
1+4+6+4+1=16

That was as far as we got the first day. I was intrigued, so I played with a 6x6 square that evening, which we talked about the next day. My own weakness with numbers had lead me to misidentify this as an activity working with square numbers, so I started out trying to show him square numbers on our Cuisenaire Rods... which didn't match up. I love the way that I get to (re)learn along with the kids. I never understood how math was related to patterns until I started doing math things with Hero, and it's pretty exciting to me to see how these things fit together. He got to watch me be confused and then work it out -- which is not a bad thing. And looking at the Rods was cool.

I don't know that I'd ever really seen an exponential relationship laid out and made touchable like that. We started out building long skinny towers, and later Hero added the blocked versions as we were starting to run out of space on the table.

Because it meant we were able to use our hundreds flats, the blocked versions allowed us to go even larger (we added another after the photo was taken) without running out of pieces. I was pleasantly surprised at how far our Rods went. That was as far as we got the second day, building the answers to the equations we'd built. I'm not sure if Hero feels like messing around with this anymore, but I am planning to play with at least one more set. We've done a 7 letter word set, which makes 32 solutions. I want to play with an 8-letter word, giving 64 solutions, and see if I can predict what the equation will be. And there's some cool patterns in how the letters fall on the graph that I'm intrigued by. I don't know if Hero's interest is strong enough to keep him engaged at a third sitting for our "9 minute" activity (that would bring us to around 1.5 hours), but if he is, there's plenty to play with still. 

Math is pretty amazing. 

How come nobody ever told me about playing with numbers?!

Number Squares

So, I'm thinking about switching to MEP for our math (Hero's close to outgrowing Miquan) seriously enough that I printed out their lesson plans and we're taking them on a bit of a test drive this week. This was the first activity we did (level 4), and I am having so much fun with it that, after Hero(9) was finished, I did some more to see what the pattern was going to do. It starts like this, with the blue set:

So, that was pretty tricky for him (and me, too, when I was looking it over last night). But you're writing the letters, and then in the final grid, we sort of put all the different variations on a single grid, and counted up how many times the final letter hit each square across the diagonal, where the Es are in that 5th blue grid. Next, we did the same thing with a four letter word - MORE. 

This time, he understood what he was supposed to do with the letters, and filled in the chart completely on his own. I helped with the chart where all the words are "stacked" into the same grid, and we counted them up. At first, he was thinking that there was 2 places where those center diagonal Es are used, so we added that up (the equations are at the bottom of the picture), and it was only six - but we'd clearly filled 8 charts, so that couldn't be the number of times they were used. He went back and found the missing numbers and got the correct equation.

We did the same thing for a 5-letter word, but we used Hero's real name, so no pictures of that on the blog. It was more challenging to keep track of - we had a couple of missed solutions and a couple of duplicates, so it took us some time to sort it out and get our equation to work out the way it was supposed to do. Tomorrow, I want to go back and look at that again. He didn't do them in any particular order, so they were easy to mix up. But when I did the next chart, for a 6-letter word, PUZZLE, I did them systematically, and kept things straight no problem. I didn't want to draw the whole grid anymore, so I printed some graph paper for this one. As I was counting things up to see how many times the final E fell in each square, I used some little squares in the middle to help me keep my place.

By now, I'm noticing a whole bunch of interesting patterns. For instance, the letters always fall in the same places. No matter how I follow the rules - turn right or go down - I never need to put two letters in the same block. That's true all the way through. And the equations that this makes are pretty interesting, too. I think that you could make some kind of equation or generalization for how they go:

1+1
1+2+1
1+3+3+1
1+4+6+4+1
1+5+10+10+5+1

I'm not sure how to generalize how that's going into an equation, but there's clearly a pattern, and you could figure out what it is for any iteration, if you could generalize it into an equation. Which would be cool. I'm tempted to do a 7-letter word, which would be 64 squares, just so that I can make a prediction about what the equation would be and then check to see if  I'm right. This is kind of a fun game.

Also, the numbers are squaring. We haven't done the squared numbers unit in Miquan just yet, but after doing this, I think we're going to in the next few days. This is a pretty cool introduction to squares. (Update: Yes, I eventually realized these are exponents, not squares. There's a follow-up post in the works.)

MEP's lesson plan suggested that this activity would take 9 minutes with a whole class of kids. I don't know who thought that was going to be possible; we took more like 20 to do the 3-, 4-, and 5-letter words, and then I spent a few more minutes doing the 6-letter word on my own. So their time estimate was way off. But I really like the way they are showing kids how to look for patterns and encouraging number play. This is a fantastic activity that we had a good time doing.


Part two is here.