Creative Arithmetic

7 times 7


It’s wonderful to see young kids as they first learn new math concepts.

Very often, they see things in a different way than what we’re accustomed to, and different from how math is often taught.

I believe that such creativity should be encouraged. Isn’t math more exciting when you think of your own way to do something and it works out, compared to being told what to do and how to do it?

Sometimes, though, one method does have an advantage over another method. In that case, it may help to challenge the student with a situation where his or her method doesn’t work out. When the child sees this firsthand, he or she will be more likely to embrace a different method.

Other times, one method is much more efficient than another. Again, it helps when the child can see a specific example where his or her method is inefficient, and to understand why it matters.

Where possible, encouraging a little creativity may be a good thing, even in mathematics. That creativity can help to engage the student in math. Looking way ahead, if the child ever gets far in math, creative problem-solving can become a valuable skill. So it would be a shame not to foster some creativity along the way.


I’ve seen many young kids come up with neat math ideas. My daughter (six) recently thought of a neat way of looking at multiplication. I will use this as an example.

To start out, she was presented only with the definition of multiplication—rather than starting out with a table of multiplication facts. I wanted her to first understand what multiplication means on her own terms and then proceed from there.

She was given a few examples of what multiplication means. For example, 3 times 2 means to add three two’s together (or to add two three’s together). With a few examples like this, she quickly understood what a multiplication problem meant.

The next step was to give her multiplication problems and ask her to figure them out. So, for example, she figured out that 3 times 2 was six by adding three two’s together (and then she could see that adding two three’s together produced the same result).

Now here is where things became really interesting.

My daughter was asked, “What is 6 times 3?”

Surprisingly, after a while, she answered, “9 plus 9 is 18.”

I was obviously curious about this, but decided to be content that she answered the problem correctly (even though her solution seemed questionable) and see how this would play out before investigating this.

A couple of days later, she was asked another question. “What is 5 times 4?”

This time, she said, “10 plus 10 is 20.”

Evidently, it wasn’t a fluke. There must be some method to this madness. So now I asked her a few questions about this to figure out what she was doing.

It turns out that she was visualizing pyramids in her head. She was solving the multiplication problems by grouping numbers together in pyramids.

Below is a picture showing how she worked out 6 times 3.

6 times 3

She started with the bottom row: She knew that 6 times 3 meant to add six three’s together, so she started with six three’s. Then she grouped the three’s into two sets of nine’s, and added the nine’s to make 18.

The following figure shows how she figured out 5 times 4.

5 times 4

Again, she began with four five’s, regrouped them into two pairs to create two ten’s, and added the ten’s to make 20.

Eventually, she will learn and practice her multiplication facts to become fluent in multiplication (that’s the goal, anyway). She will also learn that her solution isn’t efficient if, for example, she wants to multiply 9 times 8—or, worse, 35 times 24. But for now, she has made multiplication her own, and her method works fine for simple multiplication facts.

My daughter doesn’t realize it, of course, but she has the basis here for factoring. For example, she’s writing 6 x 3 as 9 x 2, which both boil down to 3 x 3 x 2.

Copyright © 2014 Chris McMullen, author of the Improve Your Math Fluency Series (which, by the way, is focused on practice and drills—which build fluency—and not on the creative learning aspect that I described in this article)


3 comments on “Creative Arithmetic

  1. Great…Does your daughter tutor?
    I’m very impressed and do believe teaching math in different ways does help those of us who think in different mathmatical ways. I have to admit that I was one of those people who couldn’t understand Algebra until an amazing Statistics professor in college taught me her way. I wish I could remember her secret because it was a charm.

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