Addition, multiplication and exponentiation (raising to powers) are related operations.

Multiplication is repeated addition.

Exponentiation is repeated multiplication.

Look at the photo to see what I mean.

Math is hierarchical. Skills build on top of other skills. In this way, math is similar to learning a foreign language. That basic vocabulary and simple rules and verb tenses you learn at the beginning are essential for your future fluency.

Matt and I see many students who are quite good at “the hard stuff,” the algebra and geometry that they are learning now in high school, but they can’t truly excel because they’re still struggling with earlier skills such as multiplication and fractions.

More often than not, kids wind up with weak math foundations because their elementary-school and middle-school learning didn’t progress at the same pace as did the school curriculum.

For example, many kids memorize their times tables without understanding what they are doing. Most kids can tell you that three times five equals fifteen. But do they know what that means?

Here’s a quick assessment to help you see what your student really knows about multiplication:

• Write a multiplication problem, say 5 x 3 = ?, on a piece of paper
• Your student should tell you the answer is 15
• Now ask the student to write a word problem to fit the equation.
• Kids with multiplicative thinking can write an appropriate story. There were five cats and each one caught three mice, how many mice is that?
• Kids who are still thinking additively write addition stories. There were five cats and three more cats arrived, how many cats are there now?

And here’s something students of any age (including adults) can practice, to increase math fluency, exercise your brain, and feel better about math!

• Take a multiplication fact. Let’s use 8 x 7 = 56
• Find as many different ways as you can to get that answer:
• 8 x 5 = 40 and 8 x 2 = 16, 40 + 16 = 56
• 10 x 7 = 70 and 2 x 7 = 14, 70 – 14 = 56
• 4 x 7 = 28, 8 x 7 is twice as much, so 28 x 2 = 56

Can you find some more?

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    Last reviewed: 12 May 2011