https://www.maa.org/external_archive/devlin/LockhartsLament.pdf
In the opening, Lockhart wonders what it would be like if music were taught the same way as math. What if, throughout all of grade school and undergraduate college, students were to take classes like like AP Music Theory, AP Musical Notation, and Introduction the Musical Transpositions. Of course, all of this happens without once having the students play music or, God forbid, compose pieces of their own--those are reserved for later in the undergraduate curriculum and graduate school.
What a nightmare! After all, the joy, fun, and beauty of music come from actually being able to experience it! Lockhart uses this example is to introduce the main point of his essay: that this is precisely the state of math education today.
In grade school, students memorize multiplication symbols. In middle school, they learn to mechanically manipulate symbols using a fixed set of rules and shortcuts that come from some magical faraway land called "Algebra." In high school, the same rules are extended and students are taught a new set of rules and shortcuts, this time from a place called "Calculus."
One might argue that math is fundamentally rooted in logic and arithmetic, so we should naturally reinforce "fundamental" concepts in order to better prepare students for more advanced topics. But take, for example, the visual arts. Before allowing students to draw, should we could teach them the benefits of using water colors versus oil pastels or the differences between classical and impressionist styles of painting? After all, if students are to become advanced painters, its only natural that they start by learning all about the techniques which so many artists have dedicated their entire lifetimes to develop. Instead, we do the unthinkable by giving them a paintbrush and letting them paint a self-portrait however they want! Yet, students seem to enjoy art much more than math. No one even has to defend the value of art education by bringing up points like how useful it is or how much more money you can make if you list still-life painting as a skill on your resume. They enjoy it because it's, well, enjoyable!
Imagine if math could be taught like that! Imagine if we could present students with a blank canvas, a very small set of rules to follow, and see where they take themselves just by mixing and matching their preexisting ideas in order to form new ones. Art and music give students something that is almost nonexistent in a math class: freedom. It is that freedom that allows students to develop their own appreciation for the things they do rather than accepting someone else's argument about why it's important or necessary.
I can't say that I have a way to reform the entire grade school math curriculum. But I do think that classes which spend more time talking about the ideas behind math can do so much more for students then classes which ask them to regurgitate a handful of definitions and theorems on a test. What happens if I add all the numbers from 1 to 10? 1 to 11? 1 to 12? Could there be some kind of pattern here? Give a student a chance to think about these questions, come up with answers on their own, and you might discover that even an elementary student is capable of inventing one of the most useful summation formulas in the history of math (spoiler: this problem was solved by Karl Friedrich Gauss in the late 1700s while he was in elementary school!).
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