Via Terence Tao, Terrytao.wordpress.com

“High school algebra marks a key transition point in one’s early mathematical education, and is a common point at which students feel that mathematics becomes really difficult. One of the reasons for this is that the problem solving process for a high school algebra question is significantly more free-form than the mechanical algorithms one is taught for elementary arithmetic, and a certain amount of planning and strategy now comes into play. For instance, if one wants to, say, write as a mixed fraction, there is a clear (albeit lengthy) algorithm to do this: one simply sets up the long division problem, extracts the quotient and remainder, and organises these numbers into the desired mixed fraction. After a suitable amount of drill, this is a task that can be accomplished by a large fraction of students at the middle school level. But if, for instance, one has to solve a system of equations such as

for , there is no similarly mechanical procedure that can be easily taught to a high school student in order to solve such a problem “mindlessly”. (I doubt, for instance, that any attempt to teach Buchberger’s algorithm to such students will be all that successful.) Instead, one is taught the basic “moves” (e.g. multiplying both sides of an equation by some factor, subtracting one equation from another, substituting an expression for a variable into another equation, and so forth), and some basic principles (e.g. simplifying an expression whenever possible, for instance by gathering terms, or solving for one variable in terms of others in order to eliminate it from the system). It is then up to the student to find a suitable combination of moves that isolates each of the variables in turn, to reveal their identities…”

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