Download Secondary Algebra Education: Revisiting Topics and Themes by Paul Drijvers, Aad Goddijn, Martin Kindt (auth.), Paul PDF

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However, he initially provided rules for the transformation of the ‘equations’ that were created in this way; for example, he says the following about simplifying: If a problem leads to an equation in which specific terms are equal to terms of the same type, but with different coefficients, then it is necessary to subtract equivals from equivals until on both sides one single term is equal to one single term. ’ We should not read this as a rule that means a minus times a minus is a plus; negative numbers are not present in Arithmetica.

This is an important phenomenon which we will now explore more deeply; it is also a good time to reflect briefly on the role of various notations in algebra. There are two extreme standpoints regarding the relationship of algebra to its notation: 1. Without notation using letters, there would be no algebra. 2. Algebra concerns relationships and structures; notation is only a memory aid. The first standpoint is a superficial definition of algebra; the second standpoint appears to go far beyond the superficial appearance, but does disregard the fact that we require such a memory aid in complex situations.

ALGEBRA IS SOLVING EQUATIONS What we now refer to as solving equations is actually arithmetic, but in reverse. In the equation situation, ‘someone or something’ has begun with one or more numbers or quantities and has conducted some arithmetic operations with a numerical final result. The challenge is to retrieve the original number, numbers, or the lengths of segments. This is the perspective from school: that someone already knows the answer and asks you the ‘trick question’. This school perspective is also present to some degree in history: there was much emphasis on polynomial equations, much time was spent linear and quadratic equations, and then within a much shorter time period mathematicians raced through the cubic and quartic equations as if it were a contest.