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Topic: A topic to tune

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Subject:   A topic to tune
Author: George Reese
Date: May 19 2003
Thinking about Varnelle's reference to the Tuning Protocol and the post by Craig
about higher-order and lower-order skills, I have a story that I would like
to share with the group in the hope that it might get some tuning. It has to do
with my personal experience and whether or not there is a more general case to
be made from it.

My daughter, when she was in fourth grade, experimented with me and my teacher
tools. We had a middle school graphing calculator and a distance sensor. The
distance sensor sends out sonic pulses and determines how far away the sensor is
from an object. The calculator connected to the sensor then plots the graph of
the distance (y-axis) against the 15 seconds of time along the x-axis. As
you move the sensor closer to the object (say, a wall) and further away, the
calculator makes a nice graph.

After some initial messing around with the keys to figure out how it worked, Liz
and I took turns drawing graphs and seeing if we could reproduce them using the
distance sensor. After a while, I gave her the loop-the-loop and was very
pleased when she immediately said, "Dad, you can't go back in time." A few weeks
later, over christmas break, I wanted to test what she had learned and I gave
her a problem called "the hurdles race" which asks to narrate a race given the
distance vs. time graphs of the three runners. She did it very well. But the
most interesting part of the story is this.

A year later her teacher, to make sure that the kids knew their basic skills,
gave timed tests on multiplication to her students. Liz failed them. She hadn't
memorized those facts yet, but this was a full nine months after she had
demonstrated the ability to analyze graphs that I know many high school students
and would struggle with.

This, to me, is an example of what Craig was saying. Namely, that we should not
use "mastery of low-level skills as a pre-requisite for exposure to higher
order mathematical thinking skills". Of course, Liz did learn the multiplication
facts, and I'm glad she did learn them.

The artifacts for this story are at
(the file is ~2MB).

I have to add to that Liz, being my daughter and all, is clever ;-) but I
don't think this experience is really so exceptional. I believe students are
often much more capable of this type of thinking than we give them credit for.

While one counter-example is enough to say that it isn't always true that
basic skills are a pre-requisite to high-order thinking, it doesn't provide
much generality. How exceptional is this example? Craig's story is about
economics students is another. I would like to know if there are more out there?

I would also be interested in hearing if this reasoning seems flawed. Are some
lower-order skills an absolute pre-requisite for high-order skills? Which

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