Tuesday, November 10, 2015

Why Make Arithmetic Real

Why Look for Active Arithmetic?

Mathematics is powerful. It gets at the core of the matter by removing as many of the particulars as possible in order to get at bare essentials. Even at the elementary level, the resulting abstraction is embraced; those whole numbers stand alone awaiting their interaction to produce an inevitable result that has long ago been abstracted from what it described. Alone, without further meaning, an arithmetic result holds for everything it has been abstracted from, and that is its power; we can reliably use the facts of arithmetic because they abstract facts of the world. 

We all stand on the shoulders of giants who themselves stood on the shoulders of giants. We are a long way from the ground, and that is where the problems that confront us are, even those pesky word problems we remember from elementary school. How do we help students climb down from mathematics to to level of real problems?

A young student's goal is not the ability to do mathematics, it is rather to have the ability to use it; not to create abstractions for the advance of mathematics, but to relate an understanding of mathematics to problems in their world.

In first becoming competent in arithmetic, students learn how to evaluate mathematical expressions before they know where they could have come from, more significantly, before they know how they they relate to the real world?  They can do arithmetic problem that stands in for a large class of real world problems before they can solve any one those. Eventually they must "de-abstract" the mathematics, get at particulars, and solve the one problem sitting before them

Make it Real. Real is Active

At least that is how I remember reality in my youth, the fun and useful parts of it anyway. This is not the way I remember my early formal education. I do remember beginning to see reality in undergraduate mathematics and science. I also remember the stark realization in my freshman year that I just did not get the relation of mathematics to real problems. Up to that point I could more or less fake it since the problems appeared next to the presentation of the mathematics used to solve them: extract the numbers, plug them in, and turn the crank.  Although I worked on and around lots of farm machinery, I never made the connection between it and problem solving. I did not see the corruption inherent in my faked learning; I pretended to learn and school pretended to teach something useful. 

Most of my early formal education was self referential: it taught what you needed at its next stage; and you could never see outside of it. Problem solving was in the service of learning academic mathematics. Real problem solving, even for "real" problems in an academic setting, was never addressed. 

Have things improved, or has the current testing regime made them worse? I do not see how it could make early education less inward looking. My fear is that school education has chosen rigor to aid in the prediction of results it itself determines. How much time does it spend confronting problems as problems where you have to figure out how to understand because you can not fall back on what you have done before? 

Whatever it is, if it is does not work, little room has been left for students to learn and create on their own. Creativity has historically been our strength; I do not think we ever knew if it resulted from the time schools took from children, or from the time it left to them.

Lest I give the wrong impression: I think I learned a lot in my childhood. My early formal education was not pretentious; it did not act like it was the only way to learn and therefore should occupy most of my time. I had time to work, play, and read on my own, and my teachers gave me most of what I ultimately needed from them. 

Who knows, perhaps it was just that there was no TV in my home town until I was in the sixth grade. Or maybe it was because my mother had been a teacher, and my father read in much of his free time. I do know my parents thought learning was important. They did not think sports were, or at least they did not do much to encourage my participation. They did let me solve many of my own problems.

I think I could ultimately relate mathematics to real problems because I had the opportunity to live where problems were active, where imagination could play on an active stage based on the real world. I am not sure that world is still commonly available. I want to present some results of thinking about making arithmetic active, to render active something all students now experience. Failure is an option.  


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