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Thinking more deeply about the math that we teach
Posted:
Oct 27, 2001 12:01 AM
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Although I have taught for many years, it is only within the last 5-10
years that I have begun to realize that part of what I need to do is
to think more deeply about what I teach, so that I can "chunk" the
algorithms and concepts into such small pieces that I can begin to
understand how complex the things I teach are, to my students, and why
these things (which seem so simple, to me) are so difficult for them.
For example, we use the Glencoe series (Level 2, for our 7th grade
classes). Today, we were looking at a problem in the book. It told
us that there were 7.2 million people in the state of GA in 1990, and
that the projection for the year 2000 census was a 1/10 increase in
poulation. Several students, having struggled this year, to develop a
clear concept of the meaning of a fractional part (although we have
been working mostly with decimals) volunteered, and I called on one
young man, who proudly replied, "The population will be 7.3 million."
I was, at first, puzzled by his response, but when I asked him to
clarify his reasoning, he quite clearly referred to the "one-block
model," that we use extensively, explaining that "1/10 is one out of
ten (like one row of squares, in our one-block, which contains ten
rows of squares), so we need to add one in the tenths place, the place
just past the decimal point. That means that 7.2 million plus 1/10
becomes 7.3 million, because we've added one in the tenths place."
Because of past experiences involving similar situations (with the
distributive property, and with an equation like 1/2f = 1/5), I
immediately recognized the misinterpretation. I was, then, able to
clear up the misunderstanding by differentiating between the 1/10 of
the population and the 0.1 million that my student was adding. In
fact, I then asked leading questions, in order to guide my class to
see that we could use our "one-block model" to represent ONE million.
Thus, we needed 7 whole one-blocks and another two rows of the eighth
one-block. Then, we needed to take 1/10 of each block, as well as
1/10 of the two rows. When we combined all of those "one-tenths"
together in a single "one-block", my students could easily understand
that we needed to add 0.72 million to the original 7.2 million, to
come up with an answer of 7.92 million, rather than simply adding one
to the tenths place.
I firmly believe that, if we wish to encourage more mathematical
thinking on the parts of our students, we need to learn to think more
deeply about what we teach.
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