Paul Lockhart says “I am not complaining about the presence of facts and formulae in our math classes, I am complaining about the lack of mathematics in mathematics classes.” An example he uses to illustrate this point is finding the area of a triangle – his argument mainly follows from the point that instead of being told to think of an engaging imaginary triangle enclosed within a rectangle (which would almost certainly in a natural manner lead one to think of a general relationship for the area of a triangle) students are currently made to memorize that the area of a triangle is half the product of the base and the height.
The notion of mathematics as being a stepping stone, or as being the arsenal of some other subject more directly “relevant to the real world” is one Paul Lockhart heartily seems to detest. He argues that mathematics is a world in itself, and its use by people in other fields to further their work does not make mathematics a mere series of special notation and facts solely designed so that the others can use it. If someone agrees with the statement that math is a cold subject in which facts must be remembered and one which leaves no room for intuition or feeling, then they are truly well-trained in following formulae while being truly far from knowing of the true meaning of mathematics. Lockhart cites a number of math graduate students who suddenly end up finding they have no aptitude for actual mathematical investigation, and are only able to follow fixed inflexible directions hammered into them by years of going through results and theorems developed by others without getting any true feel for the subject.
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