Here’s a summary of the series (thus far) of posts on multiplication, with links. This allows you to read them in order, from oldest to most recent.

The series grew from a concern raised in the post Middle School Math – What is Hard? which looks some typical difficulties that kids have with math in the middle grade and speculates about an unusual diagnosis – that a rich and deep set of models and representations for multiplication is missing. This leads to the question “What Is Multiplication?” to be looked at in a way that is relevant for middle school level math. The question is not meant as an indictment of middle school nor of elementary school math; in fact, I’m raising it shortly after sharing a strong and encouraging set of observations about kids in middle school working with percentages, a very closely related subject.

I – In the first installment, I look at a conception of multiplication of groups and an equal number of items in each group. This is a common and familiar one, and it is asymmetric: the two quantities involved in the multiplication play a very different role. Is there a way to look at multiplication that is symmetric?

II – In this post, I look at a conception of multiplication as something you see when scaling up something to a larger (or scaling it down to a smaller) size. I show a very simple multiplication machine, and look at how the multiplicative structure is modeled and represented in it.

III – Sending a crew over to work in a yard for a certain number of hours, and then billing in man hours – here is a scenario that is at once very practical and also very abstract. A man hour costs me money – but what in the world is a man hour? And what is the structure that turns men and hours into man hours? What does it have to do with multiplication?

IV – In this post, I compare the difficulty of adding men and hours with the ease with which you can apparently multiply men and hours. Similarly, you can apparently multiply inches and inches (getting square inches), and I compare this to the apparent absurdity of multiplying oranges with oranges (and certainly not getting square oranges).

V – A look at the relationship between two kinds of quantities and focus on the special case where one kind is directly proportional to the other, that is, some multiple of the other. What is so special about that relationship? It may have to do something with the observation – so clear to so many kids – that adding one kind of quantity corresponds precisely to adding the other kind of quantity (the observation we insist on obscuring by calling it the *distributive property*.)

VI – Another look at “square oranges”: a model called the *area model*, or the *array model*.

This conception of multiplication, though still asymmetric with respect to the two quantities, is yet “more symmetric” than the earlier models shown.

VII – More on the array model; though not strictly symmetric, at least you can flip the whole thing on its side and get the same number of oranges: if you change the order in which you multiply numbers you yet get the same result. This has nothing whatever to do with symmetry, and everything with the invariance of the number of oranges when you turn the thing.

VIII – Conceptions of multiplication where one piece is fixed: relationships like “double” and “triple”, where we think of one quantity going in and another quantity coming out. If I count bunnies, and if you count bunny ears, your number will be “double” my number. There is a “two” in there, but the two is fixed, and mostly implied. What if “double” isn’t a special case of multiplication, but a relationship that is learned, and that exists, before multiplication and independent of multiplication?

IX – A look at multiplicative relationships with one quantity going in and another quantity coming out, and bringing to it a metaphor of special purpose tools and general purpose tools – like a set of wrenches, compared to having a single crescent wrench. How would that change our conception of what multiplication is?

X – An excursion to the land of checkout counters, and the dipping of one toe in the controversy whether multiplication is or is not the same as repeated addition. What if we *started *from repeated addition instead?

XI – One way to slice the world of multiplicative structures, based on the idea of units.

XII – Another way to slice the world of multiplicative structures, based on the idea of how many *inputs *the black box is thought to have. Multiplication as a set of lenses through which to look at the world.

XIII – A wrapping up. Multiplication as a landscape of ideas. To entice a student to explore this landscape more fully, the teacher needs to be willing to have the student start from wherever the student is, as this is the only starting point that the student has available. It may be useful for the teacher to have an appreciation for the rich shape of the landscape, and appreciation for all the places where the students may be hanging out.

Not an endpoint, more like a set of benches where we sit down and have our lunch. We’ve explored the area with the zebras and the monkeys and the giraffes; after lunch we may take a look at the darkened house with all the insects. We’re interested in exploring the landscape of the zoo. But, for now, we are just enjoying our lunch.

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Nice series, Bert. It’s been a while since I’ve added a blog to my bookmarks, but I’ll certainly have to visit your site again.

Thanks.

–Josh

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This is a fun read! THanks for not forgetting that middle schoolers and most humans really aren’t prepared to discuss the abstract nature of multiplication (and when we try to take them there, they quickly go to “okay, what does the math teacher want me to say?”) – or, as you’ve noticed, “These kids have given up expecting math to make sense, and they have settled for trying to remember the rules long enough to pass the test.”

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I’m so impressed with your thorough understanding of multiplication. If only memorizing correct answers, children are likely to have trouble with all future math.

I’m retired now but taught for many years. Please see my four models of multiplication that I taught to my advanced first and second graders. The models promote a comprehension of various ways of understanding of multiplication. They loved the exercises and did very well. These models could be used for any age involved in learning multiplication.

http://peggybroadbent.com/blog/index.php?s=Models+of+Multiplication

I also have models of addition and subtraction, if you’d like them.