This weekend, I read the book A Mathematician’s Lament: How School Cheats Us Out of Our Most Fascinating and Imaginative Artform by Paul Lockhart with a Foreword by Keith Devlin. It starts with this quote from Antoine de Saint Exupéry:

“If you want to build a ship, don’t drum up people to collect wood and don’t assign them tasks and work, but rather teach them to long for the endless immensity of the sea.”

In the foreword, Devlin says that “It is, quite frankly, one of the best critiques of current K-12 mathematics education I have ever seen.” He recommends that every policy-maker, educator and parent of a school-aged child with any responsibilities of teaching mathematics should read this book.

We’ve all seen children humming songs without care to the key that it’s in or how that song might be notated. We’ve also seen children take paints and crayons and experiment with the different media before they are taught about line, color, tone and other aspects of art. Yet, in math, Lockhart says we do not allow children enough time to enjoy and play with math and we ought to do so. According to Lockhart, the current K-12 curriculum in almost any textbook series only teaches kids a series of steps in how to solve a particular type of problem, along with some special notation. As Lockhart puts it, the current system “[destroys] a child’s natural curiosity and love of pattern making.” He claims that math is “simple and beautiful.”

He gives an example of a triangle in a rectangular box.

How much space do you think it takes up? How do you suppose you can find out?

What Lockhart laments about is that without teachers who understand the beauty of math, we don’t allow children to grapple with this problem long enough before we rush to give them the formula 1/2bh. If we allow kids to ‘play’ with this puzzle, they may actually discover it themselves.

Children will delight when they discover that by drawing a vertical line from the tallest part of the triangle, they will see that they have created two rectangles, and that the area of each triangle part is half of two smaller rectangles.

Today in class we were working with geo-boards and rubber bands. I teach second grade and the main objective was to create a variety of shapes with right angles to measure area (in square units) and perimeter (in units). When one student, who clearly got the concept was done early, I asked her to play with this puzzle for a while. I built a 3X2 rectangle and a triangle inside it. I asked her to think about how she might find the area of the triangle. After about five minutes, she lit up and with much excitement explained that she could divide the shape into two smaller rectangles and found the area of each one to be half of the rectangle. I asked her to try with a different rectangle and triangle, and her response was instant. What I didn’t give her was the traditional formula. She had basically discovered it on her own without realizing it.

Remember, I teach second grade, so this was exciting for me too. With the other children, some were excited in discovering the area of a rectangle to be the base multiplied by its height. This too was a discovery for these students and the joy of math was evident. I also did not provide a formula for them even after their discovery.

There are many critics to Lockhart’s point of view that it took centuries to arrive at many mathematical theories. He would argue that math which is rich “has been reduced to a sterile set of facts to be memorized and procedures to be followed. They are given the formula: Area of a triangle = 1/2 b h and are “asked to apply it over and over in exercises…By removing the creative process and leaving only the results of that process, you virtually guarantee that no one will have any real engagement with the subject…By concentrating on *what* and leaving out *why,* mathematics is reduced to an empty shell.”

Finally, here’s one last quote: “Math is not about following directions, it’s about making new directions.”

In the next few weeks, our school is about to go through a selection process for a math curriculum. While I empathize with Lockhart, there also needs to be a balance. I hope that when it comes time to discuss and debate the pros and cons of each math curriculum, we keep an open mind to process, discovery, and relevance, rather than what’s easiest to implement or what is an efficient way in transmitting a set of rules for practice and compliance.

Anyway, the book is an easy read, very compelling, and makes you think. Whether or not you agree completely with the author’s point of view.

very cool. reminds me of my dividing fractions song–

dividing fractions, my oh my!

Just invert and multiply!

good for solving and tests

bad for understanding

I found your blog this morning via the Freshly Pressed David Brooks post. I stuck around on the page because I liked the sound of your post titles. The one that really got my attention was Math is a Fine Art. I’m now anxiously awaiting the arrival of my amazon order which will include Paul Lockhart’s book; I also passed the recommendation on to my own readers. Thanks for your thoughtful writing.

I was perusing your blog and came across this post. I live vicariously through my kids sometimes. They are being taught in just the way you describe here. Makes me long to return to school and relearn what was taught to me. There was so little joy to be found in older methods of teaching. Catholic nuns did not help matters for me!

I can see that you are a fine and introspective teacher. You bring heart to a profession that is in dire need of a personal touch such as yours. I admire your dedication. Our education system is in dire need of more teachers like you!

Congratulations on being fresh pressed. I will follow!

Jenni in Philly

I asked Will what the answer was and he said 1/2 becaue you make right triangles. I asked him how he knew that and he said “how can I not know that?” sort of like him not remembering not talking. He gets math on an intuitive level that not all kids get. He and I were going over his math homework tonight. His work in 7th grade is stressing my math memory/experience. I asked him to be patient with me – explaining that if he couldn’t keep me up with him, I would have a hard time thoughtfully questioning his answers in the future. I made him show me the answers visualy even though he could understand them conceptually. I like that teachers teach math today in so many different ways to give every student the best possible chance of having that “ah-hah” moment.

It would have been intuitive for Will at second grade. Glad to hear he’s keeping you on your toes with 7th grade math. Is he doing multiple orders of infinity yet?

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