Wednesday, December 31, 2014

What is high school algebra for?

Basic algebra is a core part of virtually all high school level educational programs. There are areas of controversy over teaching of math, but the value of studying basic algebra itself is not one of them. The skills and understanding of high school algebra provide a base for much practical and important mathematics, and advanced areas of math build on basic algebra, including probability, statistics, calculus, real and complex analysis to mention only a few.

Still, algebra is only one face of mathematics. Ever since the ancient Greeks, math has had other important faces as well. Euclidean geometry is recognized to this day as a shining example of systematic mathematical reasoning and taught in standard school programs. With the arrival of the computer and the Internet, new kinds of mathematics have come to have great practical importance, including discrete mathematics and mathematics of computation. And mathematics itself has exploded in many directions, especially over the last century or so.

With the computer have also come questions about the math students learn in school. For example now that we have calculators, how important is it to learn to calculate by hand? Powerful computer algebra software now can solve many kinds of math problems, often beyond the abilities of all but the most capable humans. Textbook problems in basic algebra are among the simplest problems of these kinds.

In spite of all this there is still a consensus in favor of basic algebra in schools. It does not seem to be just conservatism or inertia. Where does this continued belief in high school algebra come from? High school algebra can contribute in a number of areas, including:

  • Problem-solving skills. Math has always been a tool for solving problems, and algebra provides excellent arenas in which to state problems, consider the means available, and systematically solve those problems.
  • Fundamental concepts of mathematics. Among these are variables, constants, terms, functions, domain, range, relations and equations.
  • Interpreting mathematical expressions and statements. This means understanding for example the structure of a term, how it transforms its inputs to its outputs, the graphs of equations, and the conditions when mathematical statements are true or false.
  • Understanding and using properties of numbers and operations. These include ordering of numbers, operations such as commutativity, associativity, and distributivity; symmetry, transitivity, and so on.
  • Describing the world mathematically. Typically this means transforming questions about the world into mathematical statements that can allow the questions to be solved or analyzed.
  • Reasoning mathematically. Proper mathematical reasoning is deductive, and the fundamental rules of deductive reasoning apply to all branches of mathematics, both simple and advanced.

Some of the posts on this blog discuss how basic algebra can illustrate principles of mathematical reasoning often overlooked in textbooks and courses in the subject. These principles can also be readily implemented in interactive computer software such as Mathtoys that can support its users in correct reasoning.

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