The right answer

How did the answers in the answer key get there?

We’ve been thinking a bit more about the student we mentioned last week, who didn’t quite understand the idea behind an “error bound.”  We’ve concluded that part of her misunderstanding lay in the way she (like just about everyone) had been taught mathematics.  For most of our school existence we are given problems to solve as a way of testing whether we’ve mastered the concepts and techniques.  If we get the right answer, it’s a reasonable (though not infallible) indication that we understand the subject.  To give us a check along the way, often the answers to the odd-numbered problems are provided in the back of the book.  This is actually a quite effective and even efficient way for a student to work, and teaches the valuable skill of tracking down one’s own errors.

However, it implants the belief (conscious or not) that there is always a sort of answer in the back of the book, the right answer, toward which we’re all working.  That is, there’s always a Right Answer, which is what we reach if we’re diligent and good.  At any rate, someone knows it.  This idea is reinforced by Physics labs in which students measure, say, the acceleration of gravity, or Chemistry labs working out molecular masses.  At the end there is almost invariably a comparison of lab results with those from the textbook and a calculation of “percent error.”  Students are often invited to speculate on the source of their disagreement with the Right Answer.  Our astronomer, when he was classroom teaching, tried unsuccessfully to keep his students from listing “human error” as a cause.  They firmly believed that, had they made no mistakes, they would have attained the golden fleece.

Of course, as soon as our consultants started doing actual research, they had to jettison the idea (if they hadn’t already).  They were calculating things no one else had calculated before, or did it a different way, and there was no answer in the back of the book.  They developed a whole range of techniques to check their work for actual errors, and absorbed the standard scientific ways of working out how uncertain they actually were: uncertainties, not “errors.”

Indeed, the “back of the book” conviction gets the actual process of science and much of that in mathematics exactly backwards.  The Right Answer is unknown; depending on your particular strain of Philosophy, it may be unknowable.  What we do know is our own result.  That’s generally the easy part.  The hard part (and certainly the most tedious) is to work out how far from the Right Answer we may be.  With state-of-the-art labs and experienced scientists it will not be as far as with High School labs and students, and the former are the results printed in High School textbooks.  But they are arrived at by what is, in principle, the same method.