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Re: Automaticity and Understanding
Posted:
May 13, 2000 3:25 PM
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> This is exactly why I consider the Saxon Math philosophy so
> innovative. I can open my daughter's fifth grade Math 76 book at
> random and never get the impression that "drill" is even underway.
> There is a rumor floating around that their submissions for the 2001
> California adoption cycle will include some traditional drill because
> of complaints from too many teachers that there needs to be more.
> These teachers may be right, they are the experts here, but starting
> each lesson with a collection of different, although straightforward,
> word problems keeps the focus in the right place, repeated *use*.
Saxon *is* good at revisiting topics to reinforce them and at forcing
students to continue to use things they saw some time ago. But Saxon
is terrible at conveying the notion that mathematical thinking is
really about what to do when you *don't* know how to solve a problem.
Saxon teaches kids that they can't solve problems unless someone has
shown them how. That isn't problem-solving and it isn't mathematics.
> Most of us want to see good presentation, including a sketch if
> appropriate, correct and standard use of notation, etc. That's
> understanding.
I disagree with the "most". In my experience, *few* want these things.
As one piece of evidence, consider the behavior we see when instructors
get together to construct departmental examinations in, say, a
multi-sectioned calculus course. Many a good, meaningful question is
summarily rejected because "it'll be too hard to grade". I recall one
such examination given while I was teaching as a graduate student at the
University of Kansas. One of the problems asked the student to find
something (I don't remember what) that involved the calculation of a dot
product of two vectors. The answer was 5. Those of us who graded that
problem (there were several of us) were horrified to find out that there
were at least six different *and completely incorrect* ways of combining
the numbers given in the problem to get 5, and that students were doing
all of them. (One student, in fact, interchanged addition and
multiplication in the dot product and still got 5.) The reason for our
horror? Why we'd actually have to read everything every student wrote
for that problem instead of just look for the 5's.
I suggest that the attitude I've described above is the norm, and not
the exception, among mathematics teachers. (Can anyone imagine the
faculty in a math department committing themselves, as a department, to
putting the kind of time into their reading of student work that is
required of an instructor in a freshman English course?) Perhaps we do
not test for automaticity (though I think we do); we certainly do the
grading automatically ourselves. In doing so, we encourage unthinking
responses from our students--who are very good at presenting us with the
minimum that we demand. [Who says they don't understand optimization
problems? ;-)]
--Lou Talman
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