What Algebra?

Each summer, as schools get ready for a new school year, the question returns, “Should we be teaching algebra to our children?” it seems to have been started by Andrew Hacker who has tried to argue and continues to argue that algebra is both difficult and an unnecessary burden for most of our students, and thus should be dispensed with. He continues to ask and the echo continues to resound, but they focus on the wrong question. The question should not be “Whether algebra?” but instead must be “What algebra?” For there are many algebras, but in particular we should focus on two, the algebra of solving equations and the algebra of functions.

Most of the algebra we teach our students is the algebra of solving equations for unknown quantities. Its origin dates back almost 1300 years to the Muslim flowering. It is the algebra we are all familiar with using x as the sign for the unknown quantity and the collection of techniques for finding out what x is equal to. Its fundamental methods, balancing and completion, are found in the title of the great work from Al Khwarizmi from which its name derives. Today it has become little more than the practice of manipulating symbols for the problems that it was designed to solve are no longer the primary problems of the business community which stimulated its practice.

The other algebra that we trace directly back only 500 years to the late 1630’s is the algebra of functions. This algebra does not solve equations, it builds tables and graphs; it is not about unknown quantities, it is about variables, quantities that can and do take on multiple values; it relies not on a series of paper algorithms, but on dynamic engines that graph and change. It is, in sum, not a static algebra but a dynamic algebra and thus the algebra of science. And of most importance, it is the algebra of spreadsheets making it today, the algebra of business too.

The algebra of functions need not and does not rely on the algebra of equations as a foundation, particularly in this time of computers where spreadsheets automatically compute. It does not need the mechanical skills of solving specialized forms of equations, nor the conceptual confusion of this x thing called by the algebra of equations “a variable” but in reality a placeholder for an unknown number. Introduced in schools today as a modest extension of the old algebra for most students, the algebra of functions only makes a serious appearance for those planning to take calculus. Despite its importance in business today, the algebra of functions and their spreadsheet tools are not a significant part of our standard math curriculum even in college algebra courses where the paper and pencil mechanics of functions are studied but not, not their applications. It is no wonder that our students complain so bitterly that the math they are forced to learn is abstract, meaningless, and utterly useless.

We should not be asking whether we should be teaching algebra, we should be asking “What algebra should our students be learning?”

To learn more you may want to read my paper, The Idea that Changed the World.


  1. steveb says:

    It would be great to see a table comparing, on a perhaps a week by week basis, today’s traditional algebra curriculum as taught, with the proposed WhatIfMath algebra of functions curriculum. That would enable a detailed comparison – a powerful tool in evaluating any evidence-based argument.

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