Why do we need to study the history of mathematics?

If you answered instinctively -- Because I need the course credit for my major -- I recommend that you view this captivating lecture (you might also download the audio only) from the famous Gresham College in the U.K.  Professor Robin Wilson in his lecture Keep Taking the Tablets on the history of ancient Egyptian and Mesopotamian mathematics explains eloquently:

First of all, mathematics is part of our heritage – as central as literature and painting – and we need to study a culture’s mathematical artefacts for the same reasons as we study its cave paintings, its buildings or its literature: indeed, it’s been claimed that mathematics is one of the two oldest professions in the world, and so we certainly need to investigate its origins.

 Secondly, mathematics is not just the tidied-up God-given subject that we find in our textbooks. On the contrary, it reveals itself as a fundamentally human endeavour – whether we’re interested in the counting of sheep, the collection of taxes, or the measurement of time. On the other hand, if we’re primarily interested in the mathematics itself, then our understanding may be significantly increased if we can explore the context from which it developed – whether as a practical problem arising in technology or astronomy or the biological sciences, or whether it arose out of a desire to solve a mathematical puzzle or develop a mathematical idea.

And finally, going back to the original sources can provide new motivation – our reasons for studying a subject now may differ markedly from the original purpose in studying a problem. Perhaps bored schoolchildren would find quadratic equations more fun if they’re told that the methods originated in Iraq some four thousand years ago, and if they’re encouraged to decipher the solution themselves by studying the markings on a clay tablet.

You can download the video, audio, lecture notes and print the transcript
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Keep Taking the Tablets. Robin Wilson. 2004