Study Suggests Math Teachers Scrap Balls and Slices is a New York Times article about research at Ohio State University that may have profound implications for the teaching of mathematics to young children. Unfortunately, just what those implications may be is completely obscure to me, as it is, I daresay, to the researchers.
I've written before (here, and here) about my belief that many children can learn abstract math concepts much earlier than is commonly supposed, and do not need to be limited to practical, real-world mathematics until they reach Piaget's magical age of abstract thinking. So I suppose I should have been pleased to read about a study that purports to show that children actually learn mathematical concepts better through abstract symbols than they do through practical examples. Not so; I'm also a fan of practical math, and believe both to be important. In any case, it's hard to draw any conclusions one way or the other from this research—or at least from the Times report.
In the experiment, students were taught a mathematical concept either through abstract symbols or through concrete examples such as combining liquids in measuring cups. They were then asked to apply that knowledge to figure out the rules of a game. Those who had learned the concept abstractly did better at that application than those who had learned through the concrete examples, and also better than those who had been taught first the concrete examples and then the abstract ones. Supposedly the real-world examples actually obscured the underlying math.
I'm going to break the post here, because some of you will want to look at the puzzle yourselves before reading about my experiences. I strongly encourage you to read the article, but if you don't do that, at least follow the link and click on the link in the upper left under "Multimedia" that will show you the problems. (It's a javascript pop-up, and I can't figure out how to insert it here without offending someone's copyright.) Then click the "more" link after this paragraph to continue reading this post.I quarrel with the application of this research to any conclusion other than "We need to study this further." For one thing, The laboratory rats in question were college students, whose brains might reasonably be expected to have already become capable of abstract thought. It would be risky to apply anything learned from them directly to six-year-olds. For another, the "concrete" examples appear—thought it's hard to tell for sure from the article—to be abstractions themselves. Thinking about a measuring cup filled 1/3 full of liquid is more similar to thinking about a circle than it is to pouring liquid from an actual, physical cup.
Believing my own brain to be reasonably competent at both mathematics and abstract reasoning, I was shocked at how difficult I found the problem. I couldn't get my mind around what they were trying to show. What finally gave me the necessary clue was actually the measuring cup example, but thought about abstractly, as numbers. Thinking about its physical properties was, indeed, distracting from the underlying mathematical concept, but the abstract symbols were no more helpful. When I finally thought about the problem in terms of numbers alone, then answer was obvious, and to my mind much more easily explained abstractly—though I do still have a problem with the third example in the measuring cup system, so I may be missing something even though I got the correct answers. I'll be very interested to hear what other people think. Especially my reader from Oswego. :)
I wouldn't change the way I taught anyone mathematics based on this study alone, but I would be very interested in hearing the results of a similar study done with other demographics: not only children of different ages, but also adults of various occupations and levels of education.
Abtract thought is not my thing I guess. I had to convert the symbols to the fractions shown in the measuring cups to come up with the correct answers.
Otherwise I could make no sense of what the symbols were supposed to tell me.
What is the problem you had with the third example in the measuring cup system?
I had to use the cups as well to find the addition / modulo rule. The symbols did exactly nothing, because unlike the bugs and vases they had nothing that represented any numerical value. If they were dots on a circle, that would have helped - the one corresponding to the vase with the dot at 4 o'clock, the bug at 8 o'clock, and the ring at 12 o'clock. Once I got the rule, it was easiest with the vase (one protrusion), bug (two protrusions), and ring (three stones).
I would say that numbers are at least as abstract as symbols, albeit a form with which we all are very familiar. For a young child to discern the pattern from the numbers 1, 2, and 3 might be just as hard as with the circle, rhombus, and, um, whatever it is that looks like a flowchart symbol.
Clever thought, considering the protrusions on the items in the game; I didn't realize that, but saw them simply as symbols, as I did the first examples, once I figured out the pattern: vase/circle = 1, bug/rhombus = 2, etc.
The problem I have with the measuring cups is that I think the full cup answer should be an empty cup. I spent a lot of time at the beginning mentally pouring one cup into the other, and actually was quite close to the answer by considering what was left over if you excluded full cups -- but in that case the third answer should have been an empty cup, and it wasn't. And, indeed, 3 mod 3 is 0, not 3, so if the pattern is supposed to teach modular arithmetic, it's not quite right.
This brings up another reason why I'd like to see the study repeated over various demographics. Is it possible that the preference for symbols is an artifact of the move away from literacy? People do a lot less reading than in previous generations; could it be that the computer/video game/TV-taught students understand better through symbols?
I was taught modular arithmetic through words, which I still consider the most efficient mechanism, though if I were teaching the concept I think I would supplement the lesson with a physical demonstration of the measuring cup example. I learned the concept solely in the abstract -- and enjoyed it, because I like that sort of thing -- but really appreciated seeing the concrete example here, which I had never before considered.
I think that gets us into the discussion if for instance the word "vase" is an abstract symbol, too, as well as the depiction of a vase. But I'll grant that numbers are more abstract than pictures.
3 mod 3 is indeed 0, but I suppose that didn't bother me because 12 o'clock noon looks like 12 o'clock midnight, so in a mod 3 situation 3 and 0 are equivalent in my mind, whether right or wrong.
I didn't learn anything from the symbolic examples. I don't know how you would be able to distinguish between circle and circle = diamond, and circle and circle = wavy thing. The cups are "easier" since they have the addition aspect. (and it isn't really a modulo operation, (that would be too abstract...)
1/3+1/3=2/3. 2/3+1/3=1. 2/3+3/3=5/3. but you only have one cup to show the result, which makes it partially modulo.)
Since the final game is an abstract game that has arbitrary rules, with multiple possible answers, until you memorize each rule, it makes sense that someone taught with the arbitrary rules would understand it better, where someone who only saw the straightforward "concrete" example would take longer. They don't mention how much time they took to train on the first puzzle. I'd expect the time "saved" at the final were "wasted" in learning the first example.
Said a different way, I didn't need the third column of the concrete example, since they were intuitively obvious (though if you were allowed two containers, there would be two possible answers for the last one).
Whereas the first example, you can't possibly know the answers prior to seeing all of the examples.
Though, I have no idea how to solve either of the two questions at the bottom, so I am clearly not understanding what is supposed to be taught here. They gave us four rules involving two sources, and one answer. The questions involve three and four sources, and depending on how you group them, you get different answers, or undefined sources.
Ah, now reading your comments, I see that you have to assign numbers to each symbol in order to solve it.
And I think your point about numbers being just as abstract is a good one - I am not convinced that teaching abstract concepts are better than the concrete examples. Seems to me that teaches you to memorize random rules that have no meaning, and the next puzzle could have equally abstract unrelated rules, and learning the first one does nothing to help you with the second one.
