At some point during the process of teaching multiplication to our children, we really need to come to terms with this question:

What IS multiplication?

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“What’s my answer? It’s not one that society’s going to like. Because society expects — demands, even — that mathematics be concrete, real-world, absolute, having definitive answers.

I can’t give a definitive answer.

Multiplication manifests itself in different ways. So maybe the word ‘is’ there is just too absolute. And it’s actually at odds with what mathematicians do.

Mathematicians do attend to real-world, practical scenarios — by stepping away from them, looking at a bigger picture.”

Continuing on my theme of times table facts, here’s the inimitable James Tanton:

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“If our task is to memorize this table, please make it about mathematics — about thinking your way through a challenge, and what can I do to make my life easier.”

You may also enjoy my blog post series about working through the times tables, paying attention to mathematical relationships (and a bit of prealgebra) along the way.

A comment from a friend got me playing around with multiplication. I found a few videos from some of my favorite math people, so I’ll be sharing over the next few days.

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“When students have to drill multiplication facts, it’s frustrating, unproductive and it makes them hate math. A better way to master the multiplication table is work on the skills that allow students to multiply quickly and efficiently.”

Making doubles and halves are a great foundation for all sorts of math.

Do you ever play the doubling game with your children? One player picks a starting number, and then you take turns doubling it until your mental math skills run out. How far can you go?

Or try the halving game: One player chooses a starting number, and you take turns cutting it in half. How tiny can you go?

As Sonya demonstrated, these skills help your child master their multiplication facts. And they are fantastic preparation for exponents and logarithms, too!

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How Is This Math?

The idea that math is only about numbers, calculations, and textbook exercises is one of the greatest lies we learn in school. Of course, nobody ever comes straight out and actually says that. But the whole system teaches us every day what counts for math and what doesn’t.

James Tanton’s math salute is a physical puzzle.

How in the world did he do that?

Physical puzzles don’t fit into our cultural understanding of math. But the process of figuring out the puzzle is the same problem-solving process we use to figure out other puzzles — including the puzzles we call math.

In fact, real mathematics is all about figuring out puzzles without a teacher showing you what to do. Problem-solving is a universally useful skill.

As master teacher W. W. Sawyer said:

“Everyone knows that it is easy to do a puzzle if someone has told you the answer. That is simply a test of memory. You can claim to be a mathematician only if you can solve puzzles that you have never studied before. That is the test of reasoning.”

So tackle the puzzle of the math salute. Show it to your kids. (And don’t be surprised if they figure it out before you do!)

[THE FINE PRINT: I am an Amazon affiliate. If you follow the link and buy something, I’ll earn a small commission (at no cost to you). But this book is a well-known classic, so you should be able to order it through your local library.]

The best way to practice math is to play with it—to use the patterns and connections between math concepts in your pursuit of something fun or beautiful.

Diffy Inception puzzles have their own symmetric beauty, but mostly they are just plain fun. Students can practice subtraction and look for patterns in the difference layers.

I just published four new activity books to our online store:

My publishing company runs this online store, so you can find all my playful math books there — including an exclusive pre-publication ebook edition of my newest title, Prealgebra & Geometry: Math Games for Middle School. Click here to browse the Tabletop Academy Press store.

Homeschooling friends, check out this new homeschool math program that’s fun, rigorous, and engaging — a delightful, hands-on course that helps parents (and their children) understand math.

I had the privilege of previewing this class as Sonya and Lacy put it together. I highly recommend it to anyone who struggles with math, or who wants to take a non-traditional approach.

By focusing on making sense of number relationships, and by teaching algebra before arithmetic, this course provides a stress-free path to rich mathematical mastery.

And for all they provide, including weekly live workshops and a slew of printable math journal pages that prompt deep thinking, the price is a steal!