KTM User Pages
In BarModelingVsGraphing, a guest mentioned that variables and equations could be introduced using pan balance problems, in simple cases. Catherine and I were talking about pan balances this past spring, in exactly this context. She asked how I would introduce equations to an absolute newbie; I said that with Ben, I had had luck using the analogy of a pan balance. It's a rather neat analogy, in the initial stages of learning about equations. Emphasis on initial. Not a week later, by coincidence, Everyday Math introduced a whole unit on pan balance problems. These were problems of the following sort: Given the diagram below, tell how many squares are equivalent to a circle. Very neat idea, I thought; it addresses, in an intuitive way, the preservation of equality under both the addition-subtraction and multiplication-division operations. I liked it. The pan-balance problems kept coming home. Pretty soon we had moved on to double pan-balance problems: Given the diagrams below, tell how many squares are equivalent to a triangle. That, I thought, was getting to be a bit over the top -- I was starting to have to coach Ben on how to approach the pan-balance problems that were supposed to be helping him to approach the problem of equation-solving. Then we started seeing pan-balance problems that looked like this: We were getting so close to actually doing real equations, I could feel it.. and the kids were developing such great intuition; they were so ready for the next step, the step to real equations -- and then the unit ended. Fifth grade Everyday Math ended without the kids ever having really been introduced to manipulating equations. But they are good with pan-balances, at least virtual ones. I guess this is a sort of a cautionary tale about the dangers of falling in love with your cool teaching tools. Back to main page.
KtmGuest (password: guest) when prompted.
Please consider registering as a regular user.
Look here for syntax help.
I LOVE THE GRAPHICS! EYE CANDY! -- CatherineJohnson - 11 Jun 2005
OK, now I'm in the Ben category. I completely don't get the first pan balance. To me, this looks like 4 circles/6 squares = 6 circles/2 squares. I can't figure out how that makes sense. When I write it as two equations I get: 4c = 6y
6c = 2y
I'm rusty at this, so I'm probably making mistakes, but here goes: 4c = 6y
c = 6y/4
then, substituting 6y/4 into 6c = 2y I get: 6(6y/4) = 2y
36y/4 = 2y
9y = 2y
9y can't equal 2 y ----- can it? help! -- CatherineJohnson - 11 Jun 2005
ah-hah! Obviously, I am stuck on variables. These are not ratios. These are sums. btw . . . this is a good example of the fact that visual illustrations are not remotely natural or intuitive (at least, not in my experience). I wish to heck I could draw a quickie-bar model here (I'll probably have to upload one), but to Christopher it is not obvious that you can 'add' the 'quantities' shown in a line drawing of a rectangle on a piece of paper. This reminds me of all those old stories about people showing photographs to African natives for the first time, and the natives not knowing what they were. (No idea if that's true!) Another thing: for the past several years our various autism programs have been using 'PECS,' for 'picture-exchange communication system.' The pictures used are made by Meyer Johnson; they're little computer graphics that are basically indecipherable to a lot of the kids. Jimmy never had a clue what they were (and probably still doesn't.) Andrew, on the other hand, was a PECS genius. He could decipher anything -- and he basically taught himself to read just looking at letters and words on the TV screen. (The day he spelled out 'Interpol Warning' on the floor in wood blocks was a bit eerie. He also spelled out 'Osamy' and 'Somaly' in plastic letters on the refrigerator.) I heard a lecturer once say that there are different levels of abstraction, and drawings are pretty far up there in the hierarchy. -- CatherineJohnson - 11 Jun 2005
Another ah-hah moment: I'm not just stuck on variables, I am, again, stuck on a certain overlearned spatial relationship. I see circles and squares side-by-side and I translate to a linear ratio set-up. (Addition, for me, is probably mostly vertical.) btw, Saxon carefully works on avoiding these things by teaching everything in all conceivable vertical and horizontal spatial arrangements. I was shocked the first time I saw a vertical addition problem with a letter variable. -- CatherineJohnson - 11 Jun 2005
Did the PECS pictures make sense to you when you saw them? I know what you mean about having to shut down all those circuits when you see something new. Those pan-balance pictures from EMath flummoxed me when I first saw them, and so did the grid pictures to illustrate fraction multiplication (I'll post something on those pretty soon, too). I decided the pan-balance problems were pretty cute, but I didn't like the emphasis on them, and I didn't like that they led nowhere in the end. -- CarolynJohnston - 11 Jun 2005
Wow. Interesting that you had to 'shut down circuits' (which I think is an excellent way to put it) when you first saw the pans. I'd say that, yes, the PECS pictures did make sense to me. I should find a couple on the web and post them here. I hate them, though; I don't like the drawings at all. (I think they were originally by a mom, or a speech therapist, who just sat down at her computer and figured out how to make little . . . computer drawings. I've forgotten the term for that kind of drawing. I think now they may have shifted to analog(?) drawings that are somewhat more naturalistic. The drawings had jagged edges (they called them 'jaggies,' I think) whenever you enlarged them. Awful. Actually, I kind of love the pan balance pictures . . . though I agree with you that it's NUTS to have your focus being SOLVING TWO-PAN-BALANCE-PAN-BALANCE-PICTURES. Talk about No Common Sense-y. -- CatherineJohnson - 11 Jun 2005