Geometry Where is it?


Written for Canadian Community News by Mike Sterling

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In 1957, the Soviet Union shocked the world by launching a tiny satellite named Sputnik.  That event sparked a new assessment of education.

Many students were asked to get into engineering.  I can say with some confidence that more than half were not suited for it at all.  They were miserable and did not like it.

I knew a Yale Engineering graduate who told me he was pushed into engineering by his parents. He worked very hard. He quickly saw he was not good at it compared to some of his classmates so he got an MBA and became rich with addition only.  He was an engineer, but in name only.

Again today we hear a lot about the educational system and achievement in fields that offer long term wage security like high technology and medicine.

During the decades that have passed, there has been no dramatic breakthrough in student achievement in North America.  Independent test scores have not shown any miracle at work.

A very small number of people do all the creative work.  Why should we expect more?  How many kids start with piano  lessons and how many turn out to be good at it?  Not, many.  We don't blame the teachers in music.  We don't blame hockey coaches (well, that's a stretch).

Who can say why things just move along slowly.  We can always point to the super star students and those that don't seem able to master the building blocks necessary to go on into fields they like.

I know a educated man who wanted to be an astronomer, but could not do the requisite mathematics.  What was the cause?  Was it teachers?  Were his parents responsible?  Is it genetic?  Was it timing? 

There is never any easy answer in such a complicated area.  How do students learn?  How do they improve?

How can parents, grandparents and teachers get mathematics across to students. Without it, some avenues have a big closed sign on them. How often have we heard kids say: 

 "Why do I have to know that, I'll never use it!".

Sometimes, I think that we concentrate too much on areas that can easily be handled by rote, like multiplication tables and ways to work with fractions.  Old rote learning every day can impact many, but what about geometry?

Depending upon the school, it is taught formally in 8th or 9th grades, maybe 10th grade in some areas.  Really good schools begin much earlier. In the modern curriculum geometry seems to come in and out of focus.

The basics of geometry hark back to Euclid and much, much before.  This subject is bothersome for some students, because for the first time, the child has to think and not do things by a cookbook using memory skills.

Looking over 9th and 10th grade Ontario guides, I don't see the clean almost surgical study of Euclid's Elements (born 325 BC).

It's a surprise to me.  The curriculum kind of integrates analytic geometry (not much content, however) and beginning algebra.  So, classic geometry seems to be lost as they toss in a little trigonometry too.

A good student can learn trigonometry very quickly.  Doing Euclid's Elements should take up much more time because it involves proofs.

The entire curriculum is a surprise.  Graphing straight lines and finding algebraic solutions for intersecting lines comes to the fore.  This is easy stuff and can be handled quickly, but where is classic Euclid?

Neither Euclid nor Archimedes (c. 287 BC to c 212 BC) had our number system to work with at all.  They did not concentrate on algebra. That came later as did Descartes (1596-1650)  So they worked with visuals and mental images of geometry and lines on a slate or sand.  They used a compass and straight edge.  They worked with proofs.

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The current curriculum lacks big time geometry and loses this axiomatic method for seeking proof.  It has proof, but not the elegant Euclid style and depth.

Algebra gets tossed in early and mixed with geometry using simple shapes in graphs like Descartes might do to illustrate his new idea.  Of course he had a wonderful background in classic Euclid and was using it.

There are two curriculums involved in Ontario.  One is called academic and the other applied.  It's an odd choice of words now that I think of it. If you need it for applying it to a problem, then you get an easier standard.  If you are in the academic side, you need it too.  So is it just a sorting of students into two areas?

What it amounts to is that those students not going much further in mathematics and science get just enough to do something, but what?

If they are in skilled trades, they will need more for sure

Here is the outline of what Ontario thinks should be taught and learned. Click for Ninth and Tenth Grade

So what would I do?  I have no magic wand, but I would buy some materials:

  • A big roll of paper 36" wide by 500 feet long.

  • Some scissors.

  • Some clear tape to make big areas of paper maybe 6 feet x 6 feet.

  • Some 2" x 4" x 8' pieces of straight lumber.

  • Some materials to make big beam compasses.

  • Some magic markers to attach to one end of the beam compass.

I'd get permission to use the gym.

Then I'd set the kids to work in pairs to do geometric problems.  I'd have them creating hexagons and equilateral triangles.  This would show them how Euclid works.  They could construct parallel lines and all manner of shapes.

A simple beam compass.  Measurement can be on the top  You can use wood or metal.  Note the sharp points.  One of them can be a marker pen.

It could be fun and it would teach all manner of things like the Pythagorean Theorem and more....



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Monday, April 14, 2014