Exponential!

Every so often, you just let your 6-year-old do whatever she wants to during math time, and she ends up calculating 2^30.

I love homeschooling.

At breakfast time on Tuesday, P started with "Two ones is 2, two 2s is 4, two 4s is 8, and two 8s is 16." E then asked me, "What are two 100s?" I doubled for him until we got to 409 600, whereupon folding the laundry and feeding the baby took up too much of my attention.

The interest both P and E had showed in doubling suggested to me that I abandon my original plans for math (which weren't anything special, anyway) and let P practice addition by doubling until she lost interest. She didn't. We started by writing the doubles on the chalkboard, but soon ran out of space. I copied the answers we had so far onto a piece of paper. P struggled to write neatly enough to line up the problems exactly, so I set up each subsequent problem for her on the chalkboard, she solved it, and I wrote it on the paper. By the time we got to 262 144, P was not only not losing interest, she was excited. "Mommy, math is so much fun!" I began to think of possible strategies for ever stopping her, because the boys were getting bored (B was trying to eat the chalk each time P put it down).

Half a year ago or so, we read a picture book in which a girl tricks a greedy king by asking for a grain of rice as a reward, doubled daily for a month. So I suggested that P calculate how much rice the king gave the girl on the 31st day, and then stop. She added excitedly, finally concluding that the total was 1 073 741 824 grains of rice. While P worked, E kept commenting, "That king must be getting worried!" I only pointed out 2 minor errors during the course of this monumental calculation, which P corrected herself.

While P was busy, I had been reflecting on the fact that Sonlight's Core K, which we're using this year, contains a longer book with the same basic storyline. So, once P was done, we read A Grain Of Rice and thoroughly enjoyed it. The story was well told and both P and E followed it with enthusiasm. We compared the amount of rice the emperor was having to give the peasant each day with P's calculations, and it was fun to see them line up perfectly.

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I'm convinced that a good teacher is always learning. Of course, I find math enough fun that I'm reading this book as my Sunday rest book (and at other times during the week too). I just read about a method of multiplying 2 numbers that requires only addition, doubling, and halving. (You can read about it here). We've covered addition and doubling already - when it comes time to teach P about multiplication (which she can already do, she just doesn't know it) this might be a neat thing to show her. I love that even with elementary math, there's something new for me to think about.

Finding Patterns

I noticed that P was having trouble with subtraction. Mainly, she was counting on her fingers inefficiently, and didn't seem to have realized that subtraction is simply the inverse of addition. So I decided to introduce that concept this week. I used Cuisenaire Rods to show her that if 4+2=6, then 2+4=6, 6-4=2, and 6-2=4. We worked on the concept for several consecutive days, and she really seemed to grasp it.

To reinforce this concept and others, on Friday I gave her a blank addition table: a 10x10 chart with the numbers from 1 to 10 on the top and left hand sides. She needed to fill in the sum at each intersection point. At first she just enjoyed the mechanics of pointing to a random space, figuring out which two numbers to add, and writing the sum in the space. Then, all of a sudden, she realized that diagonals had the same answer: 3+5 = 4+4 = 5+3 = 6+2 = 8. She looked up at me, bright-eyed, and said, "Mommy, I love this! There are so many patterns in this!" When the table was filled in, I showed her that if you want to subtract, you find the number you are subtracting (the smaller number) at the top, go down to the number you are taking it away from (the bigger number), and go left to find the answer. I promised that for all future math problems, she may use the table. Since she generated it all by herself, it isn't cheating, and using it will give her practice with the math facts.

While P was busy with her addition table, I got bored. I had been starting to teach her about perimeter, and she did great with measuring the sides of squares or rectangles and adding them together. But to mix it up, I wanted to make her some right triangles (so that I can get all 3 sides the right length - I can't draw a 60 degree angle by eye). So I wondered what the set of all possible integer side lengths for right triangles was (integer because she isn't measuring half inches yet, let alone whatever the square root of 2 is closest to on a ruler). This turned into a fun math problem, which I solved. E wanted me to use his pencil, because he would like to wear it down more quickly (he finds short pencils more attractive, I believe), so every time I stopped to think, he encouraged me with, "Do more math, Mommy!"

If you're wondering what my solution was, leave a comment and I'll give a summary of my reasoning and results. Or, have some fun with it yourself first! (Mwa-ha-ha... trying to infect the world with mathematical recreation...)

Spelling

P wrote me a note:
"I Like sgoL
MoMMy.
SbeshaLy
riting.
I Am Happy."

Once I realized that "sgoL" refers to school, the note made sense.
Although the spelling is not perfect, it shows progress in topics I haven't made an issue of. For example, she correctly used the silent e in "like" and the "ing" in "riting". Since she isn't reading fluently yet, I'm not expecting her to spell well, but it's fun to see that she's picking up some rules without being formally taught them.

I'm also glad that she likes school, and is happy. Given that she usually complains about copywork, I'm also strengthened in my resolve to make her do things that are good for her even if she doesn't appear to like them at the time.

Binary

I wasn't planning on teaching the kids about binary numbers yet, but Ari and I ended up doing it yesterday. It was Barry Simon's fault.

Here's how it happened. We were discussing CBS (Community Bible Study, which the kids and I attend on Thursday mornings) at lunch time. E is never able to remember the story he has heard at CBS - I'm not sure if he just isn't aware that they're telling a story, or if he's decided that forgetting the story is easier than telling us about it. In any case, this led to a discussion of good and bad teachers. And Ari and I started talking about Barry Simon.

When I was a frosh (insert quavery voice) at Caltech, Barry Simon taught Math 1a. He was brilliant, I'm told. So brilliant that there's no way he could stoop to teaching the elementary concept of epsilon-delta proofs in such a way that the average Caltech freshman could understand them. Instead, he found special cases and exceptions, and talked exclusively about those. At least, I think that's what he did. I never really understood anything in Barry Simon's class. I spent one evening determined to understand the topic of the next day's lecture before it happened, so I studied the subject over and over until I was sure I grasped it. I entered the class knowing how it worked; I left class hopelessly confused. The joke was that one day Barry Simon would teach us to breathe, and we'd all suffocate.

As Ari and I discussed his teaching style, I wondered aloud how he would teach a 6-year-old how to add with regrouping. I figured the first thing he would do would be to convert 28+37 into binary, without mentioning that he was doing anything of the sort, or explaining to the 6-year-old that such a thing as binary even existed. I was unsure of whether P was ready to learn binary, but when Ari was done laughing, he decided to prove me wrong.

This actually turned out to be a lot of fun. Ari labeled a piece of paper with columns for 8s, 4s, 2s, and 1s, and we explained that in binary you're only allowed to use 1s and 0s. Ari started by having P identify the values of 0000 (B's age), 0100 (E's age), 0110 (P's age), and 0111. E then wrote a few 1s and 0s at random, and it ended up being 00101. P's ability to easily figure out that that was 5 made me wonder if I really could show her how to add in binary without and with regrouping. We started out with 1+2=3 (01+10=11), and then I explained how 2+3 (10+11) had 2 in the 2s place, meaning 4, so we had to write a 1 in the 4s place instead: 10+11=101. She thought this was really neat - and I suspect that it'll help her grasp regrouping more easily when adding in base 10.

All of a sudden, I started wondering: What does the Fibonacci sequence look like in binary? What about skip counting - what patterns are there? How do you do long division in binary? I can imagine myself forgetting about what I'm trying to teach P and starting to explore this sort of question myself during our math lesson, leaving her hopelessly confused. Please say it ain't so: Could I ever turn into Barry Simon?

Ancient mysteries solved!

Last Friday, I decided to introduce P to addition with regrouping (what you do if the sum of the ones digits is greater than 10, as in 27+64). I carefully laid out the problem, and explained each step thoroughly. As soon as I carried the 1, P cried out, "Oh! I've been wondering about that 1 for years!" Upon being questioned, she informed me that a big kid in our old church in PA (where we lived over a year ago) had demonstrated addition with regrouping, and P had not understood it then. Interestingly, I had tried to explain the same concept about 6 months ago, but P didn't catch on and I left it for later. Half a year's growth and a deeper understanding of place value has made a huge difference. Apparently, Math on the Level's maturation-based approach works!

Willpower!

We spent last weekend in St. Louis for Ari's uncle's wedding. Our plans were quite flexible, and since I didn't know how much time we'd be spending there, I packed most of our school books into the heavy-duty Sonlight tote and brought them along with us. All day Friday, my mother-in-law was preparing to host the rehearsal dinner, and school was a reasonable way to keep the kids occupied. We headed down to the finished basement and put their pencil cases on the ping-pong table. There were fewer toys around to distract everyone (though baby B found a box of rocks and corals to gnaw), so things went more smoothly and quickly than they usually do at home.

We've been using Home School Family Fitness as our PE program. The first step the author suggests is setting up a routine of strength and endurance exercises: sit-ups, push-ups, pull-ups, etc. We've been working on these for a few weeks now. Typically, E has been able to do about 7 sit-ups and 15 push-ups, and he can hang on the pull up bar for about 6 seconds. P has been doing about 15 sit-ups and 7 girl push-ups, and hanging on the pull up bar for more like 25 seconds (neither child can do a real pull-up, but then, nor can I). Before we left for St. Louis, P did 25 sit-ups and 15 girl push-ups. So I was skeptical when she announced on Friday, "I'm going to do a hundred sit-ups!"

I now have extra evidence that this child is related to me (though giving birth to her is pretty strong evidence already). I said, "There's no way you'll be able to do a hundred sit-ups!" She did 105. She then announced, "I'm going to do 70 push-ups." This time, I was even more sure it was impossible. After every 10 push-ups, I asked her if she wanted to quit yet. She did 72. I informed her that she would be in worlds of pain the next day. She wasn't (or, at least, if she was, she didn't say a word about it). She said, "I can feel myself getting stronger. I like being strong."

Motivation is an important factor in what one is able to do. Today, I didn't feel like sitting on P's feet for 10 minutes while she groaned her way through another 100-odd sit-ups, so I told her I'd count how many she could do in 3 minutes. She barely made it to 25 after 2 minutes, and quit. I'm sure if I'd okayed her to do another hundred, she'd have made it. I just have other things to do with our time.

Finding a New Groove With Math

At the end of last week, P complained, "Math isn't fun any more this year." I'd been giving her 5 review problems daily, same as last year, while working on a new concept. It may be that I wasn't approaching the concept of place value in a way that worked with her learning style, or that she had forgotten several of the review items, but I felt that the main problem was that I was requiring too much writing of her. We've been doing math at the end of the school day, right before lunch, and by that time, she has already done a page in her handwriting workbook, written a list of 10 spelling words, often done copywork for language arts, and frequently completed part of a science worksheet.

At the same time, I was busy reading this article. If you don't have time to read 25 pages of sometimes over-emotional diatribe about what ails math education, here's my summary: Math is actually an art form - finding the beauty of patterns in conceptual objects (numbers, triangles, etc). Math education has removed all the art and beauty from math, and turned it into a purely mechanical exercise requiring memorization without creativity. It would be better to teach no math at all than to ruin the subject the way it is ruined by teachers who don't know better because they've never seen math, either.

I don't fully agree with the author, but the article did make me think about how I'm going about teaching math to P. I decided to try to include more unguided discovery, as well as more guided discovery, into our lessons and our review. I started by revamping the 5-a-day review process. Instead of having her write anything with pencil and paper, I'm looking at what concepts we need to review and trying to find games to play that will require understanding of those concepts. Sometimes, failing to come up with anything creative, I simply have her do a problem on the chalkboard, which she at least prefers to pencil and paper.

I've been doing lesson prep for math on Tuesday nights, but Ari and I watched the first half of "Gone With the Wind" last Tuesday night instead, so I had no plan for Wednesday. Fortunately, the math video which once was lost now is found, so I simply let them watch that. "Professor Justin" reviews a number of concepts that we covered a while ago, and even E was really getting into shouting out the answers before Justin said them.

On Thursday, I used an idea gleaned from the Sonlight forums for our science experiment - demonstrating the water cycle. We put water in a pot (the "ocean") and heated it on the stove (the "sun") until it began to evaporate. I then held a bowl about 20cm above the pot and let the water vapour condense inside it ("clouds") until the droplets got big enough to "rain" back into the "ocean". Once the experiment was over, the kids begged to bake something with the boiled water. I had been planning on making bread (and, for vocabulary enrichment and additional science, discussing the differences between whole wheat flour and enriched unbleached flour). Our recipe calls for 3 cups of warm water and 1/2 cup of honey, so I added the 1/2 cup of honey to the boiled water to dissolve it easily. This turned into a lesson in adding fractions - "We have 1 1/2 cups of liquid in our measuring cup, and we need 3 1/2 cups of liquid. We've added all the honey we need, so how much water do we need to add?" P needed a bit of hand-holding, but she grasped it pretty well once I explained it in a couple of different ways. She easily remembered, while helping me make pizza dough this afternoon, that 2 1/2 cups of flour was the same as 5 half-cups of flour. Kitchen math is an excellent way to work with fractions - I plan to incorporate it into our days more often, since both big kids love baking with me. (B does too, if you count him sticking his hand into the dough when I'm not paying attention and then smearing it all over my recipe books).

Today's math lesson, I decided to introduce P to some patterns that I find fascinating. In the RightStart games package which we bought in May, there are games involving the "long chain" and "short chain". I had never heard of these, but they are patterns similar to Fibonacci in that they only require simple addition, but simpler because they only take into account the ones digit. Since we're working on place value and I'd like to help P get more comfortable with her addition facts, I thought they'd be valuable for her as well as enjoyable - she has a thing for patterns. For example, the one I started her out on is "4 2 6 8 4 2 6 8..." - the nth number is the ones digit of the sum of the (n-1)th and (n-2)th numbers. P loved this, and I had her figure out "0 5 5 0..." for herself. I also showed her the trivial case, "0 0 0...". She said, "That isn't a pattern. It's just zero." I neglected to introduce the vocabulary word "trivial", but she clearly grasps its concept. I then demonstrated, with some participation from her, the "long chain", which starts like Fibonacci (0 1 1 2 3 5 8) but, because it only contains the ones digit, repeats after 60 digits. She doesn't have the patience to do that much addition! But she liked the idea of a repeating pattern of numbers, so I imagine we'll play with that again.

Now, I'm off to figure out some hands-on, real-life activities for her 5-a-day reviews this coming week, and see if they lend themselves to any interesting patterns. I like this challenge - it's real mathematics.