How Can We Teach Problem Solving?

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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Bill Gates Proportions II

[Feature photo above by Remy Steinegger via Wikimedia Commons (CC BY 2.0).]

Another look at the Bill Gates proportion… Even though I couldn’t find any data on his real income, I did discover that the median American family’s net worth was $93,100 in 2004 (most of that is home equity) and that the figure has gone up a bit since then. This gives me another chance to play around with proportions.

So I wrote a sample problem for my Advanced Math Monsters workshop at the APACHE homeschool conference:

The median American family has a net worth of about $100 thousand. Bill Gates has a net worth of $56 billion. If Average Jane Homeschooler spends $100 in the vendor hall, what would be the equivalent expense for Gates?

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Putting Bill Gates in Proportion

[Feature photo above by Baluart.net.]

A friend gave me permission to turn our email discussion into an article…

Can you help us figure out how to figure out this problem? I think we have all the information we need, but I’m not sure:

The average household income in the United States is $60,000/year. And a man’s annual income is $56 billion. Is there a way to figure out what this man’s value of $1mil is, compared to the person who earns $60,000/year? In other words, I would like to say — $1,000,000 to us is like 10 cents to Bill Gates.

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The Case of the Mysterious Story Problem

[Feature photo above by Carla216 via flickr (CC BY 2.0). This post was rescued from my old blog.]
I love story problems. Like a detective, I enjoy sifting out clues and solving the mystery. But what do you do when you come across a real stumper? Acting out story problems could make a one-page assignment take all week.

You don’t have to bake a pie to study fractions or jump off a cliff to learn gravity. Use your imagination instead. The following suggestions will help you find the clues you need to solve the case.

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Mastering Miquon: Top Ten Tips

cuisenaire rods
Image via Wikipedia

[Rescued from my old blog.]

I love Miquon math, but the program does feel odd to many homeschoolers, especially at first. It is so different from the math most of us grew up with that it takes time for the teacher to adjust. DJ asked for Miquon advice at a forum I frequent, but I thought enough people might find these tips useful to justify an expanded repost. If you have more advice on teaching Miquon, please chime in!

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Elementary Problem Solving: The Early Years

[Rescued from my old blog. To read the entire series, click here: Elementary Problem Solving Series. Photo by Studio 757 via Flickr (CC BY 2.0).]

You can begin to teach your children algebraic thinking in preschool, if you treat algebra as a problem-solving game. Young children are masters at solving problems, at figuring things out. They constantly explore their world, piecing together the mystery of how things work. For preschool children, mathematical concepts are just part of life’s daily adventure. Their minds grapple with understanding the three-ness of three blocks or three fingers or one raisin plus two more raisins make three.

Wise homeschooling parents put those creative minds to work. They build a foundation for algebra with games that require the same problem-solving skills children need for abstract math: the ability to visualize a situation and to apply common sense.

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Story Problem Challenge Revisited

Well, I didn’t get any takers with the last story problem challenge. But school is in full session now, and we’re doing story problems in Math Club this Friday, so I thought I’d try again.

Here’s the challenge: Can you and your students make up some original math problems?

In Math Club, we always start by reading part of the book Math by Kids for inspiration. I can’t print those stories here, however, because of copyright rules, so I’ll share some of the stories my past students have made, arranged in roughly increasing order of difficulty. After you solve a couple of these problems with your children, encourage them to try making some of their own.

And please, share their gems with us!

Update

The problems below are now available as a printable handout: Story Problem Challenge.

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Percents: The Search for 100%

[Rescued from my old blog.]

Percents are one of the math monsters, the toughest topics of elementary and junior high school arithmetic. The most important step in solving any percent problem is to figure out what quantity is being treated as the basis, the whole thing that is 100%. The whole is whatever quantity to which the other things in the problem are being compared.

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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.

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Percents: Key Concepts and Connections

[Rescued from my old blog.]

Paraphrased from a homeschool math discussion forum:

“I am really struggling with percents right now, and feel I am in way over my head!”

Percents are one of the math monsters, the toughest topics of elementary and junior high school arithmetic. Here are a few tips to help you understand and teach percents.

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