Comments for sursumcorda.salemsattic.com/post/math-education-in-the-news

Comment #1753

2008-04-30T15:49:30Z

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.

Comment #1752

2008-04-29T17:08:54Z

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.

Comment #1751

2008-04-27T14:23:00Z

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.

Comment #1750

2008-04-27T06:21:21Z

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).

Comment #1749

2008-04-26T20:25:06Z

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?