*Or, Seeing Arrays (L**ess Cinematic Than Seeing Dead People, B**ut More Useful)*

This year, I’m teaching younger students than I’ve ever taught before. These guys are 11 and 12. They’re newer than iPods. They watched YouTube before they learned to read.

And so, instead of derivatives and arctangents, I find myself pondering more elemental ideas. Stuff I haven’t thought about in ages. Decimals. Perimeters. Rounding.

And most of all: Multiplication.

It’s dawning on me what a rich, complex idea multiplication is. It’s basic, but it isn’t easy. So many of the troubles that rattle and unsettle older students (factorization, square roots, compound fractions, etc.) can be traced back to a shaky foundation in this humble operation.

What’s so subtle about multiplication? Well, rather than just tell you, I’ll try to show you, by using a simple visualization of what it means to multiply.

Multiplication is making an array.

If you think of multiplication as “repeated addition”—that is, 5 x 3 as 5 + 5 + 5, then great! That’s the same thing.

And if you think of multiplication as “groups”—that is, 5 x 3 means 5 groups of 3, or 3 groups of 5, then even more great! That’s *also* the same thing.

Now, what’s the benefit of this visual model? Ah, where to begin! Without it, you’re multiplying from behind a blindfold. Tear that cloth from your eyes, and begin to see!

Take the *distributive property*, a seemingly opaque bit of symbolism that says a(b+c) = ab + ac. Its misuse haunts algebra teachers’ nightmares. But it’s no mystery—just a simple fact about adding two arrays together.

Or what about the *prime numbers*, those invisible atoms of the mathematical world? Well, under this view, they’re as pleasingly tactile as moss on a stone.

First, *composites* (the opposite of primes) are numbers that can form arrays.

And primes? They’re numbers that *can’t* form arrays.

By the way, why isn’t 1 prime? (It’s a fact that confounds students, and “because we want prime factorization to be unique” isn’t a satisfying answer.) Well, visually, it doesn’t quite fit either pattern. It can’t form an array of smaller numbers, so it’s not composite. But it isn’t an awkward misfit like the primes. It’s just an indivisible little unit. So it’s neither prime nor composite.

Ever heard the word “commutative”? It means you can switch the sequencing of an operation without changing the outcome. For example 5 x 3 = 3 x 5. And this makes perfect visual sense for multiplication, because rotating the array shouldn’t change its size.

We can dig deeper. A “factor” or “divisor” of a number is a side length you can use for its array.

And a “common factor” for two numbers is a width that they share. It lets you combine them into one larger array.

Similarly, a “multiple” of a number is a larger array that contains multiple copies of it.

And a “common multiple” of two numbers is an array inside which they can each fit repeatedly, like tiles.

On a simpler level, why is it called *squaring* a number? Because it makes a square!

We can take this further. Replace your dots with squares.

Suddenly, we’re talking about *area*. We’re looking at the formula for the area of a rectangle, from which we can derive all other basic area formulas.

This way, we can even multiply fractions! Multiplying 4/5 x 3/5 splits a unit square into 25 pieces, and picks 12 of them—hence, 12/25!

I find mixed success when inflicting drawings like these on my students. A few Y9’s are so excited they practically leap out of their seats. Others squint skeptically. “So you want us to draw a picture every time we multiply?” they ask. “Isn’t that a waste of time?”

“Yes,” I tell them. “It is.”

“Do *you* ever use these?” they ask.

“No,” I say. “I don’t.” Fluency means taking simple steps mentally, without resorting to fingers or scratch paper. A kid who can’t use the distributive property in his sleep is as doomed as a kid who doesn’t know 9 x 4 = 36.

“So why do these pictures matter?”

They’re an early-stage ingredient. When you’re first developing a concept, they’re something you throw into the batter. Later, when you think about multiplication, you’ll rarely need to draw these pictures. Their flavor will have suffused your understanding, like vanilla extract baked into your cookies.

But trying to add this visual understanding later is often fruitless. You can’t pour vanilla extract over the top of your finished cookies. The concept is done baking.

For now, I’m enjoying my days with the young’n’s, these kids who are younger than the Euro. It’s never been so fun feeling so old.

Reblogged this on So, You Think You Can Teach ESL?.

I love the way you cut to the quick with maths, with the target “What does it mean?”. Without that the whole business is rather a waste of time. One more possibility with dots or squares is to use them to show rearrangements explicitly, for example 4×7 = 2×14,

I have a question. With all this Modern Algebra jargon about “commutative” and “distributive”, are the kids expected these days to know these terms, or is this for the teachers only?

Students are expected to know these terms.

That is so so stupid!!!!!! They belong to Abstract Algebra. There is NO POINT in knowing a term such as associative unless you are considering situations in which the operation is not associative.

There’s enough garbage to fill their minds with already, and it is real bad in the USA.

Howard, ease off a little. The reason one uses terms such as commutative and distributed is that it makes it easier to refer to the use of these concepts. And it is far easier to learn these concepts in algebra (Algebra 1, not Abstract Algebra) if one has some exposure to them in earlier years. As for “associative,” which was not part of the original question, you make a valid (but unnecessarily harsh) point.

Enough students proceed as if subtraction is associative, for example, so there are counterexamples at their level of math. If they still don’t appreciate the usefulness of the property, we can ask if a rotation then reflection of their iPhone isn’t the same as that reflection then rotation.

It’s interesting that you use Abstract Algebra to say we shouldn’t reify these concepts. My state elevates the ‘addition property of inequality’ to the same level as the associative property, and I argue that anything not named in higher math or short of 3 million Google results shouldn’t get an ‘official’ name for 8th graders.

This is a reply to Timteachesmath, (and anyone else !)

I read your reply to my rather abrasive comment on Math with Bad Drawings. I am about to do a full post about the proper meanings of “binary operation”, “associative, commutative etc”, and the I think somewhat inappropriateness of using these terms in elementary school math. My site is

howardat58.wordpress.com

Tim, your email thing doesn’t work.

My 3rd grader was required to know these terms, which I thought was silly because as someone who’s taken maths through differential equations, I do not. I know the principles but not the terms.

isn’t it easier to just use the right words from the beginning? that way they become familiar with them in the start. we call rectangles, rectangles from the start not 4 sided shapes.

As someone who doesn’t do any “proofy” math, I have never seen any utility in knowing these terms. The principles are useful, but there’s no need to name a rule as obvious as “you can regroup addition all you want” if you’re only interested in applications.

This may vary by learning style, though. I learn by understanding how pieces and concepts fit together. Memorizing names doesn’t help me. For some people, naming the concept might be useful.

seem to be only able to reply here?

isn’t it easier though when your referring to something to call it a tree rather than the tall wooden thing with leaves? putting it in maths terms when talking about triangles its easier to say equilateral triangle rather than the 3 sided shape when all the sides are the same length and the angles are all 60 degrees.

That logic doesn’t always follow, or it would be easier to say “Nothotsuga longibracteata” when talking about a tree – and it isn’t. Latin tree names aren’t of interest to the general public, only to people who make a profession, or hobby, of studying them.

It’s more proper to say “equilateral triangle” sure, and it saves you the bother of describing in other words that the triangle has 3 60degree angles. But to extend that to saying “by the Commutative Property, we change B + A to A + B,” just no. There’s no need to explicitly state the property, except possibly for someone doing a proof. Nobody else would need, or even want, and explanation for two additive terms changing places.

“by the Commutative Property, we change B + A to A + B,” just no. There’s no need to explicitly state the property, except possibly for someone doing a proof. Nobody else would need, or even want, and explanation for two additive terms changing places.

well you would need and want the term if you were using it.

How often in will people need or want a lot of things, words such as heptagon, regular (in terms of polygons),isosceles and co-efficient don’t have amazing use in the average persons day to day life.

easier to say “Nothotsuga longibracteata”

I would say it is easier to say tree that the Latin for common every day use, but if I was studying trees and in particular is a genus of coniferous trees in the family Pinaceae (if im wrong about this, im not a tree guy) then it might make more sense to use the Latin.

There are lots of times in my class when I use this law and so it would be much easier to say “I used commutative properties” than “because it doesn’t matter which way round you add numbers, I switched the numbers”

BTW, the reason to specify that multiplication is commutative isn’t for the sake of saying 5 x 4 = 4 x 5.

It’s for the sake of saying xaxb = abx^2 (or similar things).

Patterns that felt obvious with specific numbers can feel mystifying when done in the abstract. It helps to call attention to the patterns early on, during arithmetic, so they’re not so surprising later, during algebra.

re alex:

“How often in will people need or want a lot of things, words such as heptagon, regular (in terms of polygons),isosceles and co-efficient don’t have amazing use in the average persons day to day life.”

Well that’s the point, isn’t it? They aren’t teaching any of those terms in 3rd grade.

By the way I’m not angry about it or anything. Just think it’s a silly use of time for 8 year olds to be required to know those terms when they don’t yet know one-digit numerical division.

I’ve been interested to read this debate!

My own feeling: ‘Commutative’ isn’t urgent to learn, nor is it poisonous.

I can imagine a classroom where the word never comes up, because students just say, ‘You can switch around a x b into b x a without affecting the product.’ It’s slightly slower, but not such a pain, and it’s nice when math can feel colloquial rather than technical and arcane.

I can also imagine a classroom where bringing in the official vocabulary allows for quicker, clearer, and more precise communication. After all, we’re not talking about some dazzling feat like learning the Latin names of a hundred plants; we’re talking about getting comfortable with two or three highly useful words. The burden isn’t so immense.

In general, I encourage students to use words like ‘commutative’ and ‘distributive,’ but I don’t require it.

I don’t have a big problem with “commutative”, apart from the difference in pronunciation between England and the States. It is the extra layer involved in the term “distributive law” that bothers me.

Take 3 x (4 + 5)

Aah, let me distribute the 3 over the 4 and the 5

Now, is it 3 x 4 + 3 x 5, or is it 3 + 4 x 3 + 5, and should I have put some brackets (parentheses, in brackets) in ?

Another damn thing to remember, and I haven’t.

But STOP!

I’ll do a Ben Orlin dot pick.

But I don’t need to, I can see one in my head !

Yes, of course, 3 lots of (four and five) is 3 lots of 9, which by the dots is clearly 3 lots of 4 + 3 lots of 5

And that is 3 x 4 + 3 x 5

“By George he’s got it !” (after My Fair Lady)

And from the horrors department: I have read “Using the distributive law in reverse…”

The language of distributivity is extremely useful; without it, students who learn to multiply two binomials (e.g., in algebra 1) by FOIL are at a loss when trying to multiply trinomials.

I never got the impression that you were angry about it and I like and welcome the chance to think about these things especially when they make me re-question how I think about things which this has.

I live in England so I’m not sure how old 3rd grade is, however, I do have a 5 year old and when I speak to her about everything she learns about I try to use the technical language where its needed, and I guess that’s where me and Ben agree.

I guess my only point is that if they get used to words like this when they are younger, it makes more sense when they are older.

Reblogged this on The Fate of Empires and commented:

The crux of problems faced in middle school maths – now in visual form.

Reblogged this on MathSugarOff and commented:

In this case, the drawings are great, as is the commentary. Lately I’ve been seeing headlines such as “Parents go Back to School with the Common Core” and “Teachers Teaching Math to Students and Their Parents”; I wonder about the vanilla extract analogy at the end of this post. Is it really too late for math-challenged parents? I’d like to say no, but then again I occasionally have college students with a poor grasp of these concepts. Of course it may be easier to hear “come see what math your kids are working with” than “your calculus problem is actually an arithmetic problem; you need to go back to fifth or sixth grade math.”

Yeah. It’s not strictly too late, but it’s a damn sight harder to un-learn something that it is to learn it right the first time.

I see this with graphing in high school. Once students have learned to see graphs as holistic shapes, generated according to arbitrary rules, it’s almost impossible to change their perspective so that they see graphs as xy pairs satisfying the equation.

I love dots… I taught my daughter basic math using an abacus and it’s incredible how quickly she can mentally solve math problems now!

Another interesting way to use arrays is to estimate square roots of non-square numbers. Since square roots represent side lengths of square, you can work towards the square with the arrays.

For example, square root of 20 is approximately 4.5

Arrays of 20 (repeated in reverse order for dramatic effect):

1*20

2*10

4*5

5*4

10*2

20*1

The one that is closest to a square is 4*5…4*4 is a little small, and 5*5 is a little big…so you can estimate the square root to be right in between 4 and 5.

I like that a lot! May use that next time I’m introducing square roots.

I love your posts. I’m the only licensed science teacher on staff at at project-based learning school and since we don’t have a math teacher, I’m the one who most often gets to help our students who are struggling with our online math program. Math is my second love after science but I don’t have any training with actually teaching math, so I kind of fly by the seat of my pants and do the best that I can to help students create a conceptual understanding so they’re not blindly applying methods they don’t understand. I can see a lot of our students lacking some of these fundamentals and do my best to help them “relearn”. I’ll definitely use this with my students who struggle with multiplication.

Have you ever thought about writing a book? Because I would buy it in a heart beat.

Thanks for reading! Glad to hear that you’re pushing them to reach that conceptual understanding.

I have a couple of book ideas in the early, early, early stages – but someday, I hope!

Re: Mayflower

I’m an undergraduate who tutors my peers in calculus and I see students struggling with fundamentals, too. I often wonder what are the best ways to respond. For example, when I was working with a student and we decided to “take the integral,” she surrounded the function with the integral symbol and then +dx. Do I just say, no, that’s not how the notation works, or take her back to Riemann sums? I can’t tell if she just lacks practice with the notation or doesn’t understand the mathematical concept. I find it really hard to strip away what I just take for granted, which is a common difficulty for teachers. Any suggestions on how to reach a happy medium?

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I agree with Priscilla that these concepts can be very useful to parents. An important aspect of all of these pictures is that it helps students to create mental models, and these mental models provide insight in and of themselves. In the past, when I’ve asked adults why a x b = b x a, a common response is that it is obvious. When I prod them to explain, they say that it is a property of multiplication. In reality, they mean that it is something that they know (like knowing that the sky is blue) and they don’t feel a need for further thinking on this topic. But the pictures make it obvious why a x b = b x a, and help people to make other interesting connections. Most importantly for parents, the pictures show the importance of mental models and conceptualization in math. The pictures reveal that arithmetic is much more than learning how to add, subtract, and multiply. Indeed, arithmetic sets the stage for all further topics in math.

Agreed.

It’s sad to see students who excel at arithmetic, but can’t do algebra. I find it’s often because their arithmetic knowledge is intuitive and instinctual; they can’t explain their thinking, or identify the patterns they’re exploiting. So when they hit algebra, and the number intuition must be replaced by explicit rules, they’re lost.

Making math transparent! So glad you get a chance to guide a group younger than normal. Even if you do it for only a year, I hope your many insights will help guide your instruction when you return to an older group AND these kids will have the benefit of making connections later! Vertical teams were always so insightful for me and all teachers need to understand what happens in grades prior to and following their own.

As far as offering insights for adults who are so ingrained with rote procedures….good luck! At least we have a shot with the open-minded ones! Love your stuff! Keep it up!

Thanks for reading!

It’s definitely useful for me to teach these younger grades. And it works both ways – having taught high school, I think, makes it easier to identify the important bits where the younger ones are likely to struggle (i.e., where I’ve seen gaps in my older students).

Reblogged this on Math Mama and commented:

I love the thoughts he presents here. With the disconnect we see as students reach grades 7 and 8 in regards to multiplication, changing how it’s taught in the younger grades might help students make the connections easier–and in ways they will remember.

Reblogged this on Saving school math and commented:

A drawing is worth a thousand symbols !

And do get as far as this line and what follows:

“I find mixed success when inflicting drawings like these on my students.”

Something worth thinking about – in the section with prime numbers you mention their dot grids do not form complete rectangles. Yet you could form a complete rectangle of side length 1. I think it would be fine to ignore this possibility, but later under Factors of 12 you draw rectangles with side length 1 and call them factors.

It’s hard to get satisfying answers where 1 is concerned (you touch on this well in your Aside on 1) and the number of students who will actually be bothered by this detail is sure to be small, but I think it would be simple to just suggest that prime numbers can only be laid out in a grid of side length 1.

And maybe that’s why 1 is so special and is neither prime nor composite? Now I’m getting hand-wavey, so I’ll stop.

I think it’s going to be a bit hand-wavey when introducing it. The inclusion/exclusion of 1 as prime really is kind of an arbitrary decision (I think) until you add in the “unique factorization” part (1 = 1×1) but a prime gets defined later adding in that its unique factors are 1 and itself. Unfortunately part of lower level math is accepting it (or you could start a debate, and see what kids think).

The discussion of 1 as a factor does become important when factoring by grouping. I think it’s worth the effort, whether or not it’s discused as not being prime or composite. Right?

Yeah, I think the conversation is good to have with kids.

The real reasons why 1 isn’t prime seem to be to be the following:

1. We want prime factorization to be unique, and if 1 is prime, then it’s not.

(This is clear and unambiguous, but also a little unsatisfying.)

2. The number 1 is special. It’s the multiplicative identity. It’s not a proper factor; it’s a trivial factor that all numbers share. In some sense, it’s the ABSENCE of factors.

(This is a little mushier and more conceptual, but to me, gets closer to the heart of the matter, from a group-theoretic perspective.)

Love your “Multiplying Fractions” doodle. I think it helps the why-smaller-not-bigger part stick in the brain.

Good point – hadn’t thought about that!

“lets graph it”

Ack! Fourths!

*shakes fist at sky*

“Hip hip, array!”

Oh man. That would have been a much better title for this post.

My decision to follow your blog was based solely on its name. Sorry if I disappoint you, but I think your drawings are wonderful. I have noticed that many mathematicians seem to share a gene that makes their drawings look hauntingly similar.

Perhaps the way you are reintroducing students to the idea of seeing their multiplication equations as pictures may help prime your students for being able to look at the shapes on graphs as equations?

Well, thank you, and I’m glad you’re not thrown by my inability to draw, say, a proper circle!

Graphs are a very different sort of picture than this (each point representing an xy pair, rather than a unit), but I hope the practice in connecting multiple representations of an idea will pay off.

Warm and Witty. Insightful. Thank you.

Whoaaaa, awesome! I am fairly comfortable with multiplication, but this made those ideas so much clearer. I’m going to come back to reread this, probably multiple times.

This is another great visualization of prime factorization. I think it’s great, though I’ve never actually used it with a class. http://www.datapointed.net/visualizations/math/factorization/animated-diagrams/

Ooh, there’s a lot going in there! At first I was annoyed by the way it shows different factors (showing 6 as a peace sign, for example, rather than a 3×2 array), but as it went on I saw how they used it for nested factorisations, and that’s pretty cool. You could definitely have a long conversation about that with a class.

I really like this post. Thanks, Ben.

And now for more math discussion. It seems more reasonable to exclude 1 as a prime number if prime numbers are viewed as building blocks of integers. Every integer greater than 1 can be created from these building blocks through multiplication. A number greater than 1 is prime if it cannot be created from smaller building blocks, and thus is one of the building blocks. In terms of creating integers, 1 doesn’t work as a building block since multiplying by 1 doesn’t change the number. The difficulty with the usual definition of prime (divisible by no positive integer other than 1 and itself) is that this definition does not give a sense for why primes are of great interest.

I think that’s a really good point – if all you hear is the ‘divisible by no positive integer other than 1 and itself’ definition, then the primes sound like a mostly meaningless toy problem, rather than the deep concept that they are.

‘Building block’ is definitely a useful phrase.

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I wish you had been my teacher when I was 11!

Too kind! I’m sure you would’ve been a fun student to teach.

I love this post. I also began working with younger students after teaching MS/HS and enjoyed expanding the idea of multiplication from just repeated addition to the many uses you have addressed. I found an iPad app called visual multiplication that will demonstrate some of these ideas as well. Part of app is just fancy flash cards, but it allows the user to see all of the different arrays for a number and shows patterns in the multiplication table as well as a 100’s chart. It works as an additional tool for making connections after the kids have used manipulatives and illustrations.

Ooh – that sounds really useful. Apps like that seem like a good place to turn when you start reaching numbers where manipulatives and illustrations get unwieldy, but you still want some visual intuition.

One of the hardest things is the rhetoric and justification for using it. Do we use it just because it’s the “right” word? I would disagree. Do we not use it because otherwise we would end up with long-winded description of the concept? I would disagree.

I would never call a tree a “tall wooden thing with branches”, but I don’t think that’s an apt analogy to what’s going on here. When I talk about the tree in my yard, I don’t always use the specific type of tree (birch, holly, etc) because we all agree on that fact. In the same way, I don’t need to use commutative just because it’s specific.

There should be a justification for using it – then I would go for it. If the classroom is designed/focused on developing the appropriate, consistent language of mathematics, then go for it. But every time you use it, you don’t have to justify it. I don’t justify everything all the time – I just do what I know is right, as long as I can explain (and often, more concisely and accurately is better).

Commutative property, at least in needing the language for it, is better defined (in certain case) by what is NOT commutative. Once matrix multiplication is introduced (and the necessity for that is another discussion), then commutativity is an issue. So AB =/= BA. Not commutative. Now we have a framework for discussion. Before it was just an inherent property of numbers, now it’s a topic of discussion.

I do see, which I didn’t consider before, Ben’s example of algebraic manipulation. I think that would also require associativity – you can only commute later terms because multiplication is associative and you can do it in any order you’d like (up to the precedence of operators).

tl;dr = It’s great to use appropriate language, but if it’s not mandated (by having things that also AREN’T commutative), then it’s really class-dependent.

What a great teacher you must be; I would have enjoyed math instead of struggled with it if someone had gone about it that way for me to visualize it better.

As for using proper terms for the youngest learners, why not? It serves no purpose to dumb it down for them.

Keep up the fun!

I love the picture that goes along with multiplying fractions. My son is 12 here in the US and his class is adding and subtracting fractions. And I think they are very soon to be multiplying and dividing fractions. As he has dyslexia that affects math (and reading), using pictures like this is very helpful. Awesome!!

This is good. This is very good. Good good good. You are good.

I like the post overall.

However, there is one glaring inconsistency. You implicitly disallow 1 as a side length of an array when talking about primes and composites. On the other hand, you slip in 1 as a side length when talking about factors without even calling out this difference. That in my mind is … unmathematical.

Have you seen Paul Lockhart’s “A mathematician’s lament?” The book form (an expanded version of the pdf that has circulated on the internet for some years) includes this as an example. It’s a great book (and a great way to think about multiplication).

I have just done a post on pictorial representation of “The Laws of Operations”.

What else is needed? Check it out:

https://howardat58.wordpress.com/2015/02/03/commutative-distributive-illustrative-ly/