Lesson in decimals, lesson in metacognition.

2013-06-19 10.16.43

Alex has really been enjoying her first exposure to decimals in MEP 4b, so I didn’t expect any trouble today, when the lesson introduced decimal addition. That was my first mistake.

As often happens in MEP, the first step was to give a problem orally and encourage the child to come up with ideas for how to approach it. The problem went like this: “Alex was digging a trench in her garden to plant a hedge. The first day she dug 2 meters, 70 centimeters. The second day she dug 3.8m. The third day she dug 4m, and the fourth day she dug 3 6/10m. How long was the trench altogether?”

Alex wrote down 2m 70cm, 3.8m, 4m, and 3.6m and announced that the sum was 12.84m.

“Okay, it looks like you added the whole meters first, and then you added the centimeters, which was a good strategy,” I said. “The whole meters you added and got 12, and then you added 70 and 8 and 6 and got 84. Let’s take another look at that part. This .8 meters right here – how many centimeters would that be? Eight-tenths of a meter is most of a whole meter, so could it be eight centimeters?”

Light dawned. She changed the .8 and .6 to 80 and 60cm, added them together, and gave the correct answer: 12m 210cm, or 14m 10cm. Awesome.

What was supposed to happen next: I was supposed to show another couple of ways of solving the problem – converting all the lengths to straight-up centimeters, and then making a place-value table and slotting the numbers into it so that the numbers are lined up properly for column addition.

sample solution

What did happen next: Alex started sighing and rolling her eyes during my explanation of alternate solutions, and then I snapped at her for being rude, and then she started complaining that she was confused, and then I tried to illustrate by walking her through another problem but going straight for the place-value table, and she escalated to crying and yelling that it was just getting more and more confusing.

I put the math lesson away. I broke out the Base 10 blocks. We agreed together that if the “flats” were 1′s, the rods would be tenths and the little unit cubes would be hundredths. And then we worked through several decimal addition problems, each time with the blocks first and then on paper, until she fully understood that you can only add hundredths to other hundredths, and tenths to tenths, and so on. I took the picture at the top of this post, figuring that I’d write a nice little “method” post about using Base 10 blocks to teach decimals.

…Until we were in the car on the way to pick Colin up at day camp, and Alex started a different sort of conversation.

“Mom, you know when you were showing me the different ways of doing the same problem? I didn’t understand that all that was about adding decimals, so first I got really bored and then it made me think that the way I did it was wrong.”

“Ah,” I said. “And it seems to me that that’s where things started to go badly with our math. Because you got bored and stopped paying attention, and then you were confused and frustrated, and I got mad because you weren’t paying attention.”

“Yeah,” she said eagerly. “I got frustrated and then I got really mad, and that usually means tears.”

“Uh huh. Let’s think about whether that could’ve gone differently. Like, if we could jump in the TARDIS and go back in time, what would’ve helped? It seems like I should’ve been more clear at the beginning, like, this is why I’m showing you these other ways, because these are the steps to learning to add decimals.”

She agreed, but couldn’t contribute anything she might’ve done differently.

“Well, how about, what if when you first started to feel frustrated, you told me, Mom, I’m don’t understand why you’re showing me all these different ways. Would that have made things go differently?”

She was dubious. She explained that she’s just the kind of kid whose feelings explode. I suggested that she may not be able to control how she feels, but she can learn to control what she does.

“I guess so, but when I get upset it’s really hard to think of what else to do. I’ve tried bottling up my angry feelings, but I only have so much bottle and then I explode!”

“Yeah,” I agreed. “I totally know what that’s like. So we know that bottling isn’t going to work. Usually the best thing is trying to notice before the feelings get really strong, and doing something then, at the beginning, while you still have options.”

That made a lot of sense to her. We went on to have a nice, sympathetic conversation about how tricky that is – including that I can’t always do it myself, which is why I sometimes yell at her. A huge part of becoming a grownup, I explained, is learning to understand yourself and notice your feelings so that you can have more control over how you act. But I’m not perfect at it, and I don’t expect her to be either.

I’m amazed that she was able to initiate, and apparently benefit from, this conversation. The real lesson today didn’t involve decimals; it involved metacognition – “thinking about thinking.” That ability to be reflective about your own mental processes is hugely, hugely important – especially to a kid who’s a bundle of nerves, like Alex. I am unbelievably proud of her.

Accelerating without a net.

2013-03-15 18.49.23
Sushi is our traditional reward for finishing a math book.

On Thursday, Alex finished MEP 4a, which is theoretically the first half of fourth grade math. I looked ahead in math to see what our likely sequence might be. On the pre-algebra pretest at the Art of Problem Solving website, the only things she can’t do now are multidigit divisors, operations with decimals, and negative numbers. Allowing for plenty of practice, she could realistically finish the elementary math sequence in another year. Which would put us on pace to start pre-algebra somewhere around her ninth birthday.

That scares me.

I am grateful that homeschooling allows us to proceed at Alex’s own pace. I am glad that we can calibrate her math work based on our own observations, without having to justify our case to an educational bureaucracy. And yet it’s also scary to be accelerating without a net. What if we’re missing something?

What if we’re self-deluded?

After all, one of the most common tropes in modern American parenting is the parent who overestimates her kid’s talent. I’ll admit that I’ve seen things written by other parents that have made me cringe. So it’s uncomfortable for me to talk about giftedness or acceleration; I vividly remember the scornful condescension with which an anonymous commenter once explained to me that Alex, while “cute” and “obviously well-exposed,” was certainly nothing unusual.

In general, I’m a fan of a “deeper, not just faster” approach to math; rather than race Alex quickly through the levels of a standard curriculum, I’ve sought out the most challenging programs I can find. I’ve been planning to run her through the majority of MEP and Beast Academy, so that she’s exposed to different teaching strategies, emphases, and enrichment topics. I’ve looked to add in fun enrichment and have contemplated substituting logic for math one day a week. And even though we’re doubling up on curricula, I have avoided compacting either program very much. After our experience with Beast Academy 3a-c indicated that she does fine with less intensive practice, I did approach MEP 4a with greater willingness to eliminate problems – but it wasn’t until near the very end that I dared to eliminate a few whole lessons.

Part of what’s been in the back of my mind, through all of that, is discomfort with the whole idea that she might hit algebra at ten or eleven years old. I’ve found myself assuming that “slowing her down” is inherently a good idea, without looking at that too closely. I haven’t, after all, wanted to be “one of THOSE parents.” Really, when it comes down to it, I’ve been afraid to accelerate in any significant way. It feels safer to have her be no more than a year or so “ahead.” It’s scary to be her parent and her teacher, making the call about sending her flying out there without the “net” of some official validation.