How Can We Teach Problem Solving?

Posted on by Denise Gaskins

We continue to plan our co-op courses for next fall. Some of the classes I had hoped for will not happen, and my children are going to have to make some tough choices between the remaining topics. Unfortunately, they have not yet mastered the ability to be in two classrooms at once.

I have three math courses to plan, and I think I will focus as much as I can on teaching math through problems, even at the elementary level. These are once-a-week enrichment classes for homeschooled students, so I assume they have a “normal” math program at home. I want to introduce a few topics they might not otherwise see, to deepen their understanding of the topics they have studied, and to give them a taste of that “Aha!” feeling that comes from conquering a challenging problem. Has anybody done something like this, and can you recommend some good resources?

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Quotations IX: A Good Student Is…

Posted on by Denise Gaskins

Aaaargh! My Internet service is on the brink again. But since I had to run into my husband’s office to check email, I’ll take a few minutes to post a quick note. Here are a couple of quotes especially for teachers:

The only teaching that a professor can give, in my opinion, is that of thinking in front of his students.

— Henri Léon Lebesgue

A good student is one who will teach you something.

— Irving Kaplansky

Quotations VIII: The Essence of Mathematics

Posted on by Denise Gaskins

Our homeschool co-op classes are done for the semester, so this will be my last compilation of blackboard quotes for awhile. I love collecting quotations, however, so I will be treating you to some of my favorites “just for teachers” over the summer. Stay tuned!

It is better to solve one problem five different ways, than to solve five problems one way.

—George Polya

Education is the key to unlock the golden door of freedom.

—George Washington Carver

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Happy Pi Day I

Posted on by Denise Gaskins
photo by jorel314 (cc by 2.0)
photo by jorel314 (cc by 2.0)

It can be of no practical use to know that Pi is irrational, but if we can know, it surely would be intolerable not to know.

— Edward Titchmarsh

I don’t remember anyone ever mentioning Pi Day when I was in school, but any excuse to celebrate math sounds like fun. March 14 at 1:59 (a.m. or p.m.) is about as close as the calendar can get to 3.14159…

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In Honor of the Standardized Testing Season…

Posted on by Denise Gaskins

[Feature photo above by Alberto G. (CC-BY-SA-2.0) via flickr.]

The school experience makes a tremendous difference in a child’s learning. Which of the following students would you rather be?

I continued to do arithmetic with my father, passing proudly through fractions to decimals. I eventually arrived at the point where so many cows ate so much grass, and tanks filled with water in so many hours. I found it quite enthralling.

— Agatha Christie
An Autobiography

…or…

“Can you do Addition?” the White Queen asked. “What’s one and one and one and one and one and one and one and one and one and one?”

“I don’t know,” said Alice. “I lost count.”

“She can’t do Addition,” the Red Queen interrupted. “Can you do Subtraction? Take nine from eight.”

“Nine from eight I can’t, you know,” Alice replied very readily: “but—”

“She can’t do Subtraction,” said the White Queen. “Can you do Division? Divide a loaf by a knife — what’s the answer to that?”

“I suppose—” Alice was beginning, but the Red Queen answered for her. “Bread-and-butter, of course.”

“She can’t do sums a bit!” the Queens said together, with great emphasis.

— Lewis Carroll
Through the Looking Glass

…in other words…

If you could lead through testing, the U.S. would lead the world in all education categories. When are people going to understand you don’t fatten your lambs by weighing them?

— Jonathan Kozol
at Westfield State College’s 157th Commencement

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Common Sense and Calculus

Posted on by Denise Gaskins

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

Posted on by Denise Gaskins

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.

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