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While HS geometry is about thinking logically, the teaching of formal
proofs really does little to help students think logically. Indeed
there is very little evidence to show that the typical math course
actually improves student thinking. There is some evidence that
science courses improve thinking.
In particular Anton Lawson found that while the ability to understand
the Piagetian tasks, and the style of thinking inherent in them does
rise during HS, while formal logical thinking remains very low below
the 10% level.
The big problem with HS geometry is that it generally does nothing to
fix student perceptual problems. As a result students can not look at
some simple geometry and tell you how lines and angles relate to each
other. This can be done by asking what if questions, and by picturing
what happens when you change the geometry. In addition geometry is a
natural place to develop student's ability to do proportional
reasoning, and typical of math courses it does nothing there.
Geometry should be taught in such a manner that students are
frequently drawing and measuring angles and lines, in addition to
learning the relationships. Most geometry courses do little of this.
Informal proofs generated by the students should come first.
Essentially students need to be able to trace the logical connections
without the formal proof machinery. This can be done by having them
write out and explain answers to the important question "how do you
know that?".
A good example of a hard perception problem is looking through a lens
at an image of an object. If the student sees the "real" image which
is in front of a convex lens, virtually all of them say it is behind
the lens. Then if they look through a concave lens 50% say the image
is farther away, and some say there is no difference in the image.
Then there are the students who will tell you that alternate angles
formed by 2 lines crossing are different in size. This is because
they drew the angle arc closer to the vertex of one of the angles.
The list goes on. Perception problems are rampant, and are never
fixed in HS or college.
John M. Clement
Houston, TX
I agree with the below comment about geometry and would add that
when I
teach optics I assume a passing knowledge of high school geometry,
not
to mention just about any problem that is drawn and involves some
trigonometry often deals with the geometry of triangles as well.
Joel R.
|
| Also to address a point that may or may not be in bounds: in
| a lot of places, the HS _math_ program has been going to the
| dogs. Fixing this would be something to be proud of. I am
| specifically referring to the trendy tendency to completely
| gut high-school geometry. Hint: geometry isn't about
| finding the area of a parallelogram. When was the last time
| bought a parallelogram-shaped piece of fabric/lumber/whatever
| and paid on a per-area basis? HS geometry is about learning
| to think logically, and in particular learning to do proofs.
|