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[xmca] Good Examples of Bad Examples

The hors d'oeuvres at our Tuesday seminars are kind of epicurean joke. We started off with "Dunkin' donuts", because it's a seminar on "Immersion". On Tuesday night we had apples, because we were discussing two sets of data on teaching mixed numbers through English, and both of them started by counting and cutting apples. 
I argued that in many ways an apple is a good example of a thing, a concrete object, a piece of matter. But for that very reason, it's not necessarily a good example of a fraction, a decimal, a mixed number, or, for that matter, the concept of number generally.
The reason why apples are not a good example of the concept of a mixed number is clear as soon as you cut the apple into halves. The teacher sees an apple and a half, but the child sees one and a half apples. 
Notice the problem: one expression says it is singular and the other says it is plural. The teacher sees a single quantity, but the child sees two objects. Verily, even when we sleep on the same bed (or in the same classroom) we dream different dreams!
So what IS a good example of a mixed number? We looked at a couple. One of them was AGE, expressed in years, decades, twelfths of a year (months), or even three hundred and sixty fifths of a year. Another was TIME, expressed in hours, half hours, quarter hours, thirds of an hour, sixtieths of an hour. 
These turned out to be pretty singable too ("How old are you?" when it is NOT your birthday, and "What Time Is It?, which is a popular song for fourth graders, and can be sung around the room in a clockwise fashion with each child offering a precise answer in minutes and seconds. 
A couple of years ago I was asked to edit a book on physics for children for a publishing company. I found it really infuriating that the book uses AIR as an example of a gas, and WOOD as an example of a solid. 
This is a very good example of a bad example: it tells the child exactly what the child already knows (wood is solid and air is gassy) and then stands there like a barrier presenting the child from learning any more (in particular, that solid is a potential state of ALL matter, and so is gas).
In contrast, our fourth grade science book uses WATER. This is (I think) a good example of a good example. You can use water as an example of a solid, a liquid, and a gas. And once the child has grasped this good example, it becomes possible to generalize to ALL the different things that have water in them (including the child's own body) and well beyond. It's a very good example of why TEACHERS and not PUBLISHERS should write textbooks!
Obviously, every example has to have TWO ends: one attached to the child's concrete experiences and the other attached to theoretical generalization. It's possible to err on either end, and as Mike points out in his note on "teaching/learning" Soviet education consistently errs on the theoretical teaching end while we tend to err on the empirical, learning one.
This morning I finally got around to starting the Davydov text Andy made available ("Types of Generalization in Instruction"). On page 5 he gives a pretty pedestrian example of conservation of number straight from Piaget (5 + 3 = 8). And then he's got what has to be one of the best examples of a bad, Stalinist, example that I've ever seen:
"In a history lesson, the teacher might ask the children why the caption 'Prince Igor collects the tribute' is not under a picture entitled "The Community". The existence of a generalization, the children's understanding of the typical nature of the fact represented, is expressed in the following response, for example (?): 'Because it is not only Igor' (sic) who collected tribute in this way".
No indeed. Some teachers collect tribute like this too. What is enraging is that they imagine that they are somehow developing "the child's ability to abstract himself (sic) from certain particular and varying attributes of an object." 
David Kellogg
Seoul National University of Education

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