How DO We Learn Math?

What makes it possible to learn advanced math fairly quickly is that the human brain is capable of learning to follow a given set of rules without understanding them, and apply them in an intelligent and useful fashion. Given sufficient practice, the brain eventually discovers (or creates) meaning in what began as a meaningless game.

It seems obvious that our children must have a wide range of experience with real world objects before counting, addition, or subtraction mean anything to them. But are other topics, such as calculus, better learned as abstract rules — as a game that we play with symbols? And what about the topics in the middle? For instance, how best can we break our algebra students of common errors such as distributing the square or canceling out addition terms?

To teach effectively, I need to understand how students learn. Do different approaches work best with different concepts? Or at different ages or stages of development? I can think of at least 3 ways that I have learned math — what about you? How do you and your children learn?

Quotations XVI: Back to the Blackboard

Classes are back in session at our homeschool co-op, so I am again collecting short quotes for the blackboard. Here are the ones I used in September:

Any fool can know. The point is to understand.

Life without geometry is pointless.

You don’t understand anything until you learn it more than one way.

Confession: I Am Not Good at Math

I want to tell you a story. Everyone likes a story, right? But at the heart of my story lies a confession that I am afraid will shock many readers. People assume that because I teach math, blog about math, give advice about math on internet forums, and present workshops about teaching math — because I do all this, I must be good at math.

Apply logic to that statement. The conclusion simply isn’t valid. …

Update: This post has moved.

Common Sense and Calculus

One more quote from W. W. Sawyer’s Mathematician’s Delight before I have to return the book to the library:

If you cannot see what the exact speed is, begin to ask questions. Silly ones are the best to begin with. Is the speed a million miles an hour? Or one inch a century? Somewhere between these limits. Good. We now know something about the speed. Begin to bring the limits in, and see how close together they can be brought. Study your own methods of thought. How do you know that the speed is less than a million miles an hour? What method, in fact, are you unconsciously using to estimate speed? Can this method be applied to get closer estimates?

You know what speed is. You would not believe a man who claimed to walk at 5 miles an hour, but took 3 hours to walk 6 miles. You have only to apply the same common sense to stones rolling down hillsides, and the calculus is at your command.

Mathematics and Imagination

Comments by W. W. Sawyer, in his wonderful, little book, Mathematician’s Delight:

Earlier we considered the argument, ‘Twice two must be four, because we cannot imagine it otherwise.’ This argument brings out clearly the connexion between reason and imagination: reason is in fact neither more nor less than an experiment carried out in the imagination.

It’s Elementary (School), My Dear Watson

[Rescued from my old blog.]

From Time magazine, June 18, 1956:

“[M]athematics has the dubious honor of being the least popular subject in the curriculum… Future teachers pass through the elementary schools learning to detest mathematics… They return to the elementary school to teach a new generation to detest it.”

Quoted by George Polya in How to Solve It. I finally got my very own copy of this excellent book, so I can quit pestering the librarian to let me order it from library loan again…

Blogger Rudbeckia Hirta teaches math to pre-service teachers, and it seems that not much has changed since 1956. Hirta says the test answers shown were representative of her class — for instance, 25% of her students missed the juice problem. Too bad these students never read Polya’s book, in which he discusses a four-step method for solving problems. Step four is to look back and ask yourself whether the answer makes sense. Good advice!

The “Aha!” Factor

[Rescued from my old blog.]

For young children, mathematical concepts are part of life’s daily adventure. A toddler’s mind grapples with understanding the threeness of three blocks or three fingers or one raisin plus two more raisins make three.

Most children enter school with a natural feel for mathematical ideas. They can count out forks and knives for the table, matching sets of silverware with the resident set of people. They know how to split up the last bit of birthday cake and make sure they get their fair share, even if they have to cut halves or thirds. They enjoy drawing circles and triangles, and they delight in scooping up volumes in the sandbox or bathtub.