I just finished reading A Mathematician's Lament by Paul Lockhart.  Lockhart has an interesting perspective on mathematics and math education because he started out as a research mathematician and professor and then moved on to teach K-12 math.

For much of the book, Lockhart critiques the way math is taught in schools, likening it to teaching music only as music theory (never allowing students to actually listen to or play music until they are in college or even grad school) or teaching art by having children spend most of their early years simply learning paint application techniques and paint-by-numbers and not allowing them to look at real works of art or to free paint because they're not ready for such advanced material.

In many ways, he is exactly right about math education.  There is so much more to math than memorizing math facts or learning algorithms so we can "plug and chug" our way through algebra class without actually understanding what we are doing or why.  The idea that we have to teach all the boring stuff up front because kids won't understand the amazing and beautiful math is absurd.  You don't have to understand Mozart's requiem mass to appreciate it's beauty.  Lockhart says the same is true for math, and he goes so far as to say that Math is actually form of art because we do math not for any practical purpose but rather to create an object of beauty.  We don't need to have PHDs before we can appreciate that beauty or even create it ourselves!

Lockhart may go a bit too far with his point, but I think I understand why.  I agree with him that the way math is usually taught basically squashes all the life out of it. But I think that memorizing math facts is important too.  The truth is, math is unlike music or visual art because it also happens to have several immediately practical applications.  Math has two very different aspects: the art aspect and the practical aspect.  I think we can teach them side by side.  

The way math is usually taught drastically overemphasizes the practical aspect, and it even fails to truly teach how it is practical.  (My favorite word problem began: "You're coasting backwards down a hill..."  Now why on Earth would I be doing that?!  And if I were, do you think I'd need to be able to find my rate of acceleration?)  Lockhart would have the majority of the emphasis placed on the art aspect of math, trusting that students' natural curiosity and desire to learn and explore would lead them to learn the things they needed to know.

Now, I'm proof that only teaching the why and the beauty of math doesn't lead kids to memorize their math facts.  I attended a Montessori school, and I completely understood that multiplication is like making a rectangle and that division is like taking objects and divvying them up between people.  I understood squares and cubes like it was nobody's business.  I did not, however, happen to memorize my math facts in all that playing with numbers.  So when it came to timed math fact tests (we were supposed to do a hundred math problems in a minute), I failed big time.  And since math facts were the only kind of math that was tested at my school, I thought it was the only kind of math that mattered.  Since I couldn't do math facts as fast as the other kids, I thought I was stupid.

Knowing how to do arithmetic in our heads is important--we need it to make change or .  But I think there can be a balance between the practical math and the beautiful math in schools.  Some kids are naturally good at arithmetic and enjoy it.  Some kids understand the why of math but have trouble with memorization and quick recall.  We all should know how to add, subtract, multiply, and divide.  But we all should have seen fractals and tessellations.  We all should have seen some of the most elegant and beautiful proofs in mathematics, even if we don't understand them yet!  We can appreciate their beauty, no matter what.

I'll end with some of Lockhart's closing words: 
"So let me leave you with the only practical advice I have to offer: just play!  You don't need a license to do math.  You don't need to take a class or read a book.  Mathematical Reality is yours to enjoy for the rest of your life.  It exists in your imagination and you can do whatever you want with it."

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