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Pedagogical style [was Re: The "loop and a half"]

On Thu, Oct  5, 2017 8:18 PM, Ben Bacarisse <ben.usernet at bsb.me.uk> wrote:
Steve D'Aprano <steve+python at pearwood.info> writes:
>> There's no link to the original paper, only to secondary sources that discuss
>> it, e.g.:
>> http://phys.org/pdf128266927.pdf
>> [1] Anecdotes are not data, but for what it is worth, just in the last two
>> days I came across two examples of this. Teaching a boy in Year 10 maths
>> about logarithms, he struggled with purely algebraic questions involving
>> solving exponential equations by using logs, but when given a concrete
>> problem involving an investment he was able to solve it immediately.
>> The second example involved a girl in Year 8 maths, who again struggled with
>> abstract questions about adding and multiplying fractions. In particular, she
>> overgeneralised from fraction multiplication to addition, thinking that 1/4 +
>> 1/4 must add to 2/8. But when put into concrete geometric terms, showing
>> physical shapes divided into quarters, she could instantly tell that 1/4 plus
>> 1/4 must be 1/2.
>> As I said, anecdotes are not data, but when research claims to show that
>> apples fall upwards in contradiction to anecdotal evidence that they fall
>> downwards, we would be wise to be cautious before accepting the research as
>> fact.
>I think the paper is this one:
>(You can find more recent papers by searching the Ohio State University
>>From what I've read, your anecdotes are not in contradiction to the
>paper's claims.

Thank you for finding the paper, since I too was skeptical that
the abstraction-first approach would really he profitable.
I remember all those years in the public schools where
everyone dreaded the 'word problems' in the math text.
The abstractions rarely translated intuitively into the concrete
for many of my classmates.

And after reading this paper, I remain skeptical about its results.
I cannot help but think that the experiment it constructs is inherently flawed.

The claim is that this is an exercise in equivalence classes, obeying the same
rules as modulo-3 arithmetic.  There is the symbolic abstract model with the
circles and diamonds, the fabricated children's game, and the 'concrete'
problems with the liquid containers (shown in the paper), tennis balls, and
pizza slices (not explicitly described).

The paper argues that those who were given the concrete models to start
with did not understand the children's game, simply because they could
not transfer their knowledge to the new environment without the abstract

That's not a flaw in the student; that's a flaw in the model.

Looking at the liquid picture, one can very easily see the concrete values
1, 2, and 3, and can easily recognize the modular arithmetic involved.
It sounds to me that the same thing would be true for the tennis balls and
pizza slices.   And then he tells this group that the children's game is 
exactly analogous to what they have already experienced.

So here they are, all ready to do some more modulo-3 addition, and
the rug gets pulled out from under them.    There is no visible quantity
of 1, 2, or 3 in those circles and diamonds; and probably no 1, 2, or 3
in the children's game either.

How is it any surprise that they did not figure out the children's game
as quickly?

Roger Christman
Pennsylvania State University