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Subject: "creating dissonance"     Previous Topic | Next Topic
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Conferences The CTK Exchange High school Topic #289
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jjjessek
guest
Jul-06-04, 10:31 PM (EST)
 
"creating dissonance"
 
   I am taking an education class now, getting ready to teach high school math. I have come to realize that what math teachers neglect, and what really gets me interested in math, is the dissonance that I experienced when I discovered that there were different sizes of infinity, or that some idea popped up in many places in math. But very little seems to have been written about creating dissonance in math classes, or I am not very good at looking for it. So, if anyone knows where I can find advice on this kind of thing, please help me out. My future students will thank you.


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alexbadmin
Charter Member
1295 posts
Jul-06-04, 11:19 PM (EST)
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1. "RE: creating dissonance"
In response to message #0
 
   I am not sure why you call this phenomenon a dissonance. The fact that the same idea often pops up in different places in mathematics shows, in my view, how harmonious the mathematical edifice is. I may recommend a very nice book How The Other Half Thinks by Sherman Stein. Stein explores verious circumstances that could be described by binary strings. Of course, group or matrix theory, or differential equations provide numerous addtional examples.

At this site, there's a recent page where the same system of equations describes two quite unrelated problems:

https://www.cut-the-knot.org/Curriculum/Geometry/TouchingCircles.shtml

You may also find useful D. Wells' You Are a Mathematician.

The matter of the multitude of infinities is somewhat different. However, Cantor's diagonal process has been used by Güdel, Turing, Chaitin and probably by others to prove different impossibility theorems. Chaitin's The Limits of Mathematics provides a lively reading.

You may want to look for examples of real mathematics that requires very little mathematical knowledge. (This is what Stein's book is about.) At this site, you may want to check, for example,

https://www.cut-the-knot.org/Curriculum/Algebra/BreakingChocolateBars.shtml
https://www.cut-the-knot.org/Curriculum/Algebra/IntergerIterationsOnACircle.shtml
https://www.cut-the-knot.org/Curriculum/Algebra/IntergerIterationsOnACircleII.shtml

But there is more.


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