Quick question....

[conuly]

Let's say you had to do some math in your head. No calculator or pen and paper, though you could also use your fingers and toes (or other body parts as appropriate. As a kid, I'd count to twelve by closing my eyes!)

There's a lot of ways to do this, and they don't all rely on what you do when writing with pen and paper.

Like, take 8 + 5. Working with Ana, I've found that I usually add 5 + 5 and then 3 more. I could also add 8 + 2 and then 3 more, but even though that's the same equation I would not do that. I don't know why. It's just easier to break apart the numbers that aren't fives.

Ana starts with 8 and starts counting upward. Evangeline... well, that's a bit big for her, but with smaller numbers she counts with her fingers, even if she's not using her fingers. So she would go "One, two, three, four, five. One, two, three, four, five, six, seven, eight" before counting straight up to 13.

But I have no idea what other people do. I'm sure there's a huge range of things. (I'm sure plenty of people have, for the sake of convenience, all their single digit combinations memorized, for example.) What do you do?

Comments

[leora.livejournal.com]

Start at 8 and add 5. For some situations I'd use tricks with subdividing a number or rounding up and subtracting that amount. But for 8 + 5 if I weren't sure, I'd start at the larger number, use my fingers to keep track, and add up to 13.

On a side note, both kids sound developmentally normal to the best of my memory. I believe that counting all of the numbers is the typical first step vast numbers of kids use when doing addition and then starting with one number (preferably the bigger one) and then adding the next number is one of the most common early refinements kids will develop. The other techniques tend to be later additions as math skills develop, although some kids obviously don't develop many techniques and some kids develop very sophisticated ones very young.

But what you're describing from them sounds very much like the sort of thing I'd hear in a psych class on normal development.

[sayga.livejournal.com]

Yeah, my kid is getting held back in Kindergarten (for many reasons, but) partially because she hasn't developed her math concepts yet. She can't yet start at 8 and add to by saying "eight...nine, ten, eight and two is ten." She has to start from one and go all the way up.

She also doesn't have Conservation down yet. She says, "What is 6+4?" I show her my hands, 6 fingers up 4 fingers down. "6 up, 4 down," I say. "It uses up all my fingers, you know how many fingers I have." But she still has to count them. She had an assignment that was essentially x + 3 = 10. Everything equalled 10, she just had to find X in each case. (It was an empty box, not an x). I put out 2 bowls and we counted 10 candies into one bowl. "This problem says, "3 plus something equals ten. Move 3 of these candies into the other bowl, and whatever is left over in the first bowl is the answer." She moved them, and I said, "Do we still have 10 altogether?" "No," she told me. "We didn't take any away or add any, we just moved them around. Are there 10, more than ten or less than ten?" "More than 10," she said, and even after counting them, she was surprised there were 7 in one and 3 in the other. And she continued to be surprised as we did every other combination of 10. I thought conservation developed around age 4, but she clearly doesn't have it yet.

[leora.livejournal.com]

I always forget the expected ages part of my studies in child development. But what I do recall about conservation is that it is environmentally influenced. Experiences are important. They found that while most of the kids they studied (probably US maybe Europe) had conservation of number of items first (like with the candies you were doing) and then got conservation in liquids and finally conservation in more amorphous substances like clay, when they studied kids who were in homes of potters that saw working with clay and were exposed to that a lot, they got conservation in clay much younger and out of order. So, conservation is probably more affected by experiences than most of the early skills and may have a lot more potential variability in when it is picked up.

I'm mildly curious if she would appear to have conservation if you ran the experiment with much smaller numbers, specifically ones she was able to subitize (count without counting, an amount small enough the person seeing it can just know how many are present, rather than where you need to number them out, how many a person can subititize is variable and also affected by experience). It might be a lot more obvious when she really can see it as she is looking, rather than needing to calculate it out to see it's the same amount.

Of course, I hear these things and I think... I couldn't add when I was in kindergarten. We didn't even start doing addition problems til first grade. Then when we did, I was utterly horrible at them. For me my first few years of education were a matter of being told I was gifted, but being really bad at reading and basic math, which was almost all of what school was then. So, part of me just wants to grab up other kids going more slowly and say, it's okay!

I'm also reminded that I tried a conservation problem in Middle School (because it was the set-up to a joke) and in my gifted class none of the kids could understand that the amount of liquid was conserved. By the end of the period, I was finally able to convince the ~teacher~ that conservation held. I ended up drawing circles to represent molecules of liquid, just to make the point that the liquid had to be conserved (because you assume no evaporation or spillage or condensation for that matter, which isn't actually the way the real world works, but we pretend for the sake of math problems and jokes).