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Monday, November 21, 2011

I Hate Algebra...

A Feynman Diagram
I have been watching a series of interviews with Richard Feynman on youtube.  Fenyman, who I wrote about previously in "Your Child's Mind...", is a Nobel Prize winning Physicist.  Feynman, along with Sin-Itiro Tomonaga and Julian Schwinger won the prize for in 1965 for "fundamental work in quantum electrodynamics, with deep-ploughing consequences for the physics of elementary particles."

Quantum Electrodynamics (QED) involves a lot of complex ideas.  Quantum mechanics, for example, has the notion that specific particles, like a photon, may be in more than one place at once.  To represent this a mathematical idea of a "wave function" is used to describe the probability of where the particle might be found - think of a line with two bumps or peaks at different points representing a higher probability that the photon might be at one of those two locations.

Fenyman and his colleagues took the notion of quantum mechanics further by introducing the notion of what happens to the particles over time at relativistic (light) speed.  This was a breakthrough in regard to making quantum mechanics and special relativity, developed by Einstein, come to full agreement with regard to matter, light and experimental results.

In watching this video I was struck for how Feynman, clearly an able mathematician, disparages lowly algebra:

He describes how, as a child, he watches his older brother being tutored in algebra.  The tutor is reviewing how to solve for x in 2x + 7 = 15.

For Feynman the answer, 4, leaps immediately to mind.

But as he observes the brother laboring over the algebra process of subtracting seven from both sides and dividing both sides by two he realizes how stupid the notion of teaching children to reason about mathematical notions in this way is.

Now obviously Feynman needs mathematics to develop QED.

So what is he saying in this video about algebra?

I think basically he is objecting to the notion that you are teaching children how to do something through an algorithmic process that they really don't understand in the first place.  While I can teach monkey the process of steps in changing a tire the monkey, not understanding what it means to drive, really cannot see the point.

One thing that I have seen over the last few decades is that high school algebra seems to be a deciding point in the educational process for most kids.  Either you figure out the model of algebra enough to get by and move on to other mathematical and scientific courses or you don't - in which case those types of course are basically forever out of your reach.

But I have also come to believe that many of the problems that kids have with things like mathematics or music have to do with their perceptions of their ability as much as anything else.  Particularly in the context where things are presented without meaning.

As a child I never liked my "music education."  I always liked music but not how it was taught.  I wondered as a child why it seems so complicated: you had to learn to read notes, staffs, sharps, flats, etc. - all to play a simple song.  It was very discouraging.  Being dyslexic (or maybe lysdixec) just made toiling through the notation a misery.

In algebra I found the rules and algorithms more understandable but I was prone to making foolish errors and mistakes which discouraged me.  My friend and I somehow got a hold of a teachers addition of the algebra book and we were able to use it to work out the steps that were required and check our work (the teachers guide, interestingly enough, would spell out each and every step).

Sadly at the time I did not realize that the teachers did not all know these steps (if they did there would be no need of detailed explanation in the teachers addition).

Since the objective was learning how to apply the steps we did not view it as cheating - but instead as a way to check what we had done.  The tests were not in the book yet we learned to ace them.

Without this experience I would have never passed beyond algebra in high school.

Today, for example, the world of software development runs on this model.  Download some sample code - work out what's missing - publish a program.  Companies from all sizes (Apple down to Lexigraph) all publish "sample code."  Sure we could all start from scratch each time but why bother?

In later years I found the same issues with calculus as with algebra and music.

As an adult I came to realize that musical notation, like algebra, and like Feynman says, is for those that do not have grasp of the basic meaning.

When I see 2x + 7 = 15 I innately realize that x is 4.  Now there are certainly more complex problems where the rules of algebra help to simplify things - but only if you know that you need to simplify things and why.

And that's Feynman's point in the video.

With music I could always hear the notes and pick them out.  As a child I was stymied by believing that somehow reading music was required to play music.  I did not realize that the ability to hear and find the notes is musical talent.  Notation is secondary.

Later as I began to take music seriously I was shocked to discover this.  I spent a great deal of time learning to play what I was hearing - or to harmonize with what I was hearing.  Something I could apparently do all along.

The teaching model for music, like algebra, however, shut off my interest as a child.

Now most everyone will probably not need algebra for much in their lives - unless they are involved in some sort of science or technological endeavor.

But its a shame that the standard teaching model is so discouraging of all forms of talent.

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