There’s a lot happening in elementary mathematics and it’s really tough for teachers to keep up with it all…let alone parents and administrators.

I keep reminding myself that the turtle won the race.

It’s my hope that in creating and sharing this video, we can all slow down a little bit more and focus on building fluency through conceptual understanding.

Hope it helps!

This is such a beautiful representation of the progression of multiplication! It makes me so sad I, or rather, my teachers, didn’t know about this when I was learning multiplication in elementary school. But don’t worry, memorizing facts and steps to algorithms that made no sense still didn’t stop me from becoming a huge math dork! I’m super excited to share this with teachers in my district so we can support students’ understanding and discovery of multiplication strategies and algorithms. I have very high hopes for the next generation of math dorks!!

Yep…the struggle is real. But hopefully as the ratio of math dorks to students increases we’ll begin to see some change.

Thank you! I’m fighting the battle of appropriate times to teach the US standard algorithm for both multiplication and division. They are pushing for it to be done in 4th grade so any and every explanation and resource I can use of why this is detrimental is like gold to me right now.

Glad to be of service my Elizabeth and thanks for fighting the good fight. I have to keep reminding myself and the teachers I work with that “the turtle won the race.”

I love how you connect the concrete tools to the visual representation and finally to the numbers so that teachers will be able to mindful of how to implement as their students are learning about these concepts. Thank you for sharing

Grant, this is fantastic! I can see using this w/ teachers and families in my work. Thanks!

err… Graham… not Grant… Was evidently thinking of my nephew…

Thank you for the video Graham. I liked it a lot.

After Descartes began his work that led to the Cartesian Plane and Isaac Newton, (who said if an affirmative number goes to the right a negative number goes to the left) we might have evolved a model of multiplication that spanned both the discrete (integers) and continuous (reals).

I’ve put up a video about a simple multiplication model the world may have missed. And yes, it also reveals why -2 x -3 = +6.

Jonathan Crabtree

I definitely appreciate this Jonathon and the more we can flesh out conceptual understanding of content before we engage with students, the better off we’ll all be.

This is cool…and needed. Have you considered making one for middle and high school, eg the evolution from ratios to promotions to linear functions to quadratics and beyond?

If not, want to? I think it’d be fun to noodle over.

Thanks for the invite Karim…I’ve already began a R&P through linear functions. Love to get your perspective on it.

I know Robert Kaplinsky has also talked about the importance of developing these up through high school. Definitely not my lane of expertise but love to learn from you all. We’ll be in touch.

There are a lot of amazing things on the internet such as pandas playing in snow and babies giggling non-stop and directions on how to fix the improperly wired 3-way switch in my hallway that was driving me crazy, but this might even top all of those. Thank you!

Hi Graham! First time caller. I’ve shared your blogs on my pinterest page too many times to count. I worked with a team of 3rd grade teachers just yesterday who developed their own “progression” for multiplication by deconstructing the standard and doing a little digging. During lunch this video made its way into my inbox, echoing the work these teachers had done (with some added detail, of course)! It was fantastic and so well-timed! As with everyone else, I also have a request. How about a progression in multiplication as it pertains to a number line? (This could include the idea of MIRA using money, just to really get Simon Gregg worked up.) And I agree with others that I wish the last model continued to show how the “area model”/”box model”/”partial products” can be adapted to demonstrate multiplication with decimals. Suddenly the hundred flat represents a whole, and BOOM, decimals into the hundredths. Thank you for constantly pushing my thinking!!

I really appreciate the comment Nova and thanks for posting. I like the idea of using a number line as a progression tool and connecting how it aligns vertically. Adding this to my list now.

Still chewing on how to incorporate decimals. I’m thinking about including them with division but still trying to figure out how to make things align. Division is definitely messier than multiplication.

I’ve finally gotten around to publishing the wonderful lesson I was a part of onto my blog. Enjoy! http://mathiemomma.com/blog/multiplying-decimals-with-area-model

Graham, this is amazing! I hope to share it with all of our elementary teachers. If we could break through and help students conceptually understand multiplication and fractions, all our mathematics woes would end!! Can you just imagine a culture of students really getting this? Awesome! Thank you…

I feel hesitant to inject an issue here that hasn’t been touched on at all: what is multiplication? In particular, is it accurate to claim “multiplication IS repeated addition” (MIRA for short)? The models looked at for 2nd/3rd grade seem to be focused on that notion. There is some question as to what the models for 4th/5th grade are doing (are we no longer teaching multiplication of decimal numbers in 5th grade, getting there via a review of integer multiplication? That seemed to be the case a decade ago, but what you said here makes me wonder if something actually got pushed to LATER in K-12 in the Common Core world, or if you just don’t touch on decimals here for some other reason). Are we still doing repeated addition when we start with arrays, manipulatives, drawings, or what-have-you for multi-digit multiplication? It’s not crystal-clear to me, though it still feels like we’re defining all the partial products as repeated addition and then doing a grand sum at the end to get a product that feels like a sum to me. Isn’t that odd?

Natural analogues to my question are going to be: “Is division just repeated subtraction?” and “Is exponentiation just repeated multiplication? That last one promises to get really hairy when negative exponents, rational exponents, and, eventually, irrational exponents rear their heads, but that will be after K-5 so no need to worry about that for elementary teachers, right? :^)

Getting back to multiplication and MIRA: if multiplication IS repeated addition, why is it that when we multiply numbers with units of measure, we need to have the same units to add/subtract but not to multiply or divide? How come I can’t add workers and hours, but I can multiply them and get worker-hours? That feels a bit puzzling. Or even before units of measure show up, what about fraction arithmetic? Why can’t I add or subtract fractions with different denominators but I can multiply (and divide) them with impunity? (I’m not even going to ask why we have associative and commutative laws for addition and multiplication but not for subtraction and division. That’s just crazy).

I hate to be a troublemaker, but I’m also wondering why if MIRA is true, then why does addition of positive numbers always result in larger and larger sums, but multiplying positive numbers doesn’t always seem to work like that (ignoring for the moment the question of multiplying by 1).

Is anyone getting the sinking feeling that maybe multiplication ISN’T repeated addition?

And if it isn’t, what is it? Is there something that multiplication is that would account for all or at least SOME of the above? Or does it turn out that math is broken? (I tried doing some trig problems involving the law of sines (using Desmos on my iPhone) on Dec. 1 and the answers I got made no sense. I was convinced that math was broken (again) until it struck me that I hadn’t used Desmos for trig problems before and was assuming that the default setting for angle measurements was the same as on TI calculators – degrees. But it turned out that the default in Desmos is radians. Problem solved! Math wasn’t broken after all. Whew!)

But now I’m feeling like someone may have broken math again. Any thoughts?

I don’t think you’re being a troublemaker Michael. This seems like a right place to discuss this kind of thing.

To me as an elementary (or what we in Europe call primary) teacher, it makes good sense that one of our approaches to multiplication is as repeated addition. It seems like a great way to first understand it. And arrays and areas seem to give a really good intuitive grasp of what multiplication means. (You must have seen Ben Orlin’s post on arrays: http://mathwithbaddrawings.com/2014/12/03/the-sixth-sense-for-multiplication/ ) Also, once we have an area, it is different from two lengths in quality or dimmensionality if that’s the right word. Maybe that reflects the situation you talk about with units?

To me, it’s a good idea to approach concepts via multiple routes. And although he’s outlined a main route into multiplication, that will give students a really good grasp of it, I’ve got a feeling that Graham would approach it in other ways that just the ones he shows here. I certainly do. With my students, for instance, we look at factors and prime numbers a lot:

http://pinkmathematics.blogspot.fr/2013/02/important-factors.html

Which you could say is division, but gives insight into multiplication.

Last year we did a series of Grade 3 lessons on scaling:

http://followinglearning.blogspot.fr/2015/05/scaling.html

I’d really like to know how you would supplement (or replace??) these kinds of lessons for elementary children. What kind of pictures come into mind for you? I myself like the idea of stretching a number line – although I’m not sure how this could be a manipulative that the students could experiment with.

Hi, Simon,

What you say in your last sentence about stretching the number line begins to get at a perspective I believe can give kids a sense that multiplication isn’t JUST repeated addition and perhaps isn’t really repeated addition at all. Of course, since we have little choice in the matter as long as we start early grade mathematics with the notion of counting than to focus on the counting numbers and work our way towards integers and rational numbers towards then end of K-5, it’s very difficult for teachers and students to do anything BUT counting ==> adding ==> repeated adding (multiplication) ==> repeated multiplying (exponentiation). And the questions I posed previously begin to rise when you go to irrational numbers and try to make the counting metaphor/model work. I’ve seen folks twist themselves into knots trying to defend MIRA even when discussing the real numbers, but I think it’s a losing approach that leaves students fairly hamstrung when they get to calculus.

But there is another way (probably more than one, in fact). I recommend looking at some of what the Soviet Union was doing with K-3 mathematics back in the ’70s, specifically via the work of V. V. Davydov and his colleagues. They approached mathematics teaching to young children from the perspective of measurement, not counting. While it is very difficult to find much of their work in print in English (I have a set of textbooks by some of Davydov’s colleagues ostensibly still in use in some parts of Russia; someone kindly sent these to me a few years ago, but they are in Russian, and while I have a smattering of knowledge of that language, it is not even close to being adequate for understanding those books well enough to judge what the authors were up to). I do have one book by Davydov in pdf format I can send via email and an interesting 2013 article by Yuri Karpov – “A Way to Implement the Neo-Vygotskian Theoretical Learning Approach in the Schools” that I can send.

There was research done in upstate New York by the late Jean Schmittau on the Davydov approach. She has articles in print about Davydov that you can likely download free, but she was extremely reticent about providing information about the actual materials she used in her work with local schoolchildren. Basically, she would not respond to email or phone requests from me about the materials when I pursued information for several years. A former graduate assistant of hers who wrote a dissertation on the work now teaches at a university not too far from me, but as of a couple of years ago, she said that only Dr. Schmittau could help me see the actual textbooks. So I gave up.

There are a couple of Americans who have created their own measurement-based materials. One is Susan Addington and you can see about her work here: http://www.quadrivium.info/MtWindex.html. I have materials from her but I am not at liberty to share those without her permission.

Another is Peter Moxhay, who used materials based on Davydov in a Maine school district. He sent me copies and I can share those via email.

I hope this gets people thinking about measurement vs. counting (not mutually exclusive approaches, of course) and the implications of the former for how we think about, model, and teach arithmetic operations. Do we really HAVE to start with repeated addition as THE model for multiplication? Would we do better to start with measurement, leading to SCALING as an effective way to think about multiplication (note that scaling has both stretching and shrinking built into it and allows us to avoid some of the paradoxes I mentioned in my previous comment)? Is it simply the case that repeated addition yields the right result for multiplication within some sets of numbers, but that is a deceptive coincidence that, if extended, gets kids into difficulties when they hit real numbers and issues with limits, continuitiy, etc., in calculus? At least one mathematician in addition to Susan Addington thinks so: Keith Devin of Stanford, the fellow whose articles on MIRA got me thinking about all this around 2008 or so. And his writing led me to the work of English researchers Terezinha Nunes and Peter Bryant, whose book CHILDREN DOING MATHEMATICS is worth looking at on these issues.

If you send me your email address (mikegold@umich.edu) I will send you what I can should you be interested.

Wow! Wow again! I don’t know which of my colleagues and admins to send this to first!! This is so powerful for so many reasons. The Progressions are critical for teachers to read and discuss however soooooo dry to read. This really brings the Progression for Multiplication to LIFE! I’m curious if you purposely focused on multiplication, leaving out division except for one mention? Division seems the hardest operation for students, teachers, and parents to teach and learn through a progression. I hope you are planning another video for that since most 4th grade teachers jump right to the long division algorithm with NO concrete-representational approach. Thanks for sharing this!

Thanks for the kind words Lynn! There’s definitely more on the way and division is up next.

Wow. Great visual model to model the progression. I think it deepened my understanding of the models and how they are connected to the algorithm. Thank you!!!

This is great Graham. This really reminded me of reading Chris Danielson’s book Common Core math for dummies, because you and him break it down so well and show the progressions. The guys over at University of Arizona with the progression documents would be proud!

Plus that poster you made is perfect to go up in a faculty lounge. I want all elementary teachers in our district to see this.

Thanks Martin! The poster in itself is a powerful visual! I’ll be sure take a panoramic pic and share that as well.

And YES….Christopher’s book is awesome.

Where can I find the panoramic picture? I would love to have it as a constant visual reference for my teachers! Love, love, love this- swoon….!

I’m curious as to why they’re doing the “representational” rows and columns in second grade — and then use manipulatives in third??

I fervently agree with comments about the importance of words and connecting language to what’s happening (especially for students who might be stronger visual thinkers & be able to imitate what looks right without making that bridge to understanding).

Hooray for the turtles! (This makes plenty room for the more efficient thinkers, too.)

In 2nd grade it is actually listed as a geometry standard:

2.G.A.2 Partition a rectangle into rows and columns of same-size squares and count to find the total number of them.

Then when they begin with the manipulative in 3rd grade it transitions over to an Operations and Algebraic Thinking standard.

Thanks for the comment!

This is wonderful! So clear and concise in such a short amount of time. I’m totally going to share it with my colleagues and even with families! Can’t wait for an addition/subtraction one… : )

This is fantastic Graham. I love everything from the content to how you laid it out. I would love to see you extend this into middle and high school. For example connecting this method to the distributive property and then multiplying polynomials.

This was so well produced! You might also consider taking screen shots of each key slide so that after watching the video you have something to reference quickly as a PDF.

Thanks Roberts! The MS and HS ones are on the way for sure and I really like the idea of attaching PDFs for friends to read and digest as a quick reference. Thanks as always.

You’ve laid it all on the table, Definitely one for all our primary / elementary colleagues to see!

And right in the middle:

“And the big piece here is that we don’t push an understanding on students; we let students explain what their model represents. The big piece is that the context will explain the equation, not the teacher.”

Perhaps too, you’ve inspired the #MTBoS to do similar things…?

Thanks Simon and from the recent comments I’ve seen on Twitter there’s definitely some more of these on the horizon…from myself and others.

A man, some markers, and some chart paper. Brilliant work here Graham. I see this video becoming a PD staple.

Thank you so much for this Graham! This link is definitely being shared with everyone.

Well done, Graham! You nailed this in under 6 minutes. It’s just like the original, but better! I think I want to take this progression through middle school. Everything in your video is applicable for multiplication in 6-8 and beyond! Again, well done.

Thanks Mike. It’s on my radar and I’ll probably be recruiting your help!

Yes, I see the sequel. And all the spin offs. Everything is connected which is the hard part. Y could go off in so many directions. Well done in getting to the core.

Yes, this is great! I had been wanting to either find something or create something like this. This would be great to extend up to 6-8 so they can see what to build on from elementary. I also would love similar videos for the other operations. Any plans for this? THANK YOU!!! I already tweeted this!

Great! I selfishly hope it’s a start of a #progression series…

My Christmas present to my third and fourth grade teachers especially. Beautifully done. Thank you!

AWESOME!!!! Thanks for sharing!

Somewhere along the way they need to figure out, with or without help, that twentythree can be seen as twenty and three, and get a grasp of the place value notation. Words are important, and often overlooked in the rush to use the symbolic notation. And whose idea (CCSS) and others) that we have to stuff their heads with extra jargon – commutative, distributive, etcetera.

@howardat58: at some point (and it’s hardly an absolute one), the issue of communicating with others arises. We can have every student invent his/her own terminology for “it doesn’t matter in what order we add” and a different term for “it doesn’t matter in what order we multiply” and so on. Now, let’s have students in a class write to one another to explain what they’re talking about, using solely or primarily their own terminology (and notation if they so choose). What will be the result of this experiment, do you think? How about writing to a student across the hall? In another school? In another. . . ?

Similarly, at what point would we hope that K-12 teachers stop referring to things like “the top number” and the “bottom number”? Is there a point at which we’d like students to know the difference among “dividend, divisor, and quotient”? Is that point due to pedantry, or is there some other reason or reasons?

@howardat58 & Michael,

I think the important piece is that the formal language of mathematics usually proceeds the understanding, or at least in my world. Once students have began to develop a conceptual understanding of an idea, formal words needs to be introduce for the purpose of communication (like you mentioned Michael).

In my mind I’m picturing a scenario like this, “Alright, today we are going to use the distributive property, let me show you how”. This is usually followed by a series of steps and practiced with rote thinking taking place. The conceptual understanding of place value and what’s happening when we decompose a number needs to take place before the introduction of the term “distributive property”. The introduction of formal vocabulary needs to be in context.

With that being said, the practice standards and mathematics in general, ask that we be precise when writing and communicating mathematically. If the formal language supersedes the conceptual understanding, I can quickly see how students would take the understanding that follows as a series of steps and procedures. I think we need to be strategic as to when we introduce formal language. It all comes back to slowing down the process.

When I work with K-5 teachers, I urge them to refrain from early introduction of official terminology. Why is it important for a primary grade kid to know the term “commutative property” before knowing that order for addition or multiplication doesn’t change the outcome, but it does for subtraction and (later) division? The answer is that it isn’t. However, kids should be thinking about whether or not order matters for a given operation, for other operations, for all operations, (later, of course, the set AND the operation must be considered: matrix multiplication is not, in general, commutative). And they should also be asked to think about why order doesn’t matter for addition and multiplication but does for subtraction and division (and this raises the issue of why it may be vital to teach operations as inverses rather than “opposites” since the former carries an implication of undoing. There are also the fundamental algebraic notions of inverse elements and identity elements, how those two relate, and why it might be problematic if subtraction and division were commutative. None of these things require formal language/terminology to explore.

And no, I don’t think this is way over kids’ heads. It’s been looked at successfully in K-5 by people such as Robert A. Davis in his Madison Project. Materials from that project can be downloaded free. It’s a tragedy of 20th-century math education that the “New Math” became associated in the public mind with the Dolciani textbooks rather than the materials Davis developed for the Madison Project. His teaching and materials were effective where they were employed, but they didn’t receive wide distribution.

It’s the thinking, not the words, that matters. But eventually, terminology and notation become important. Kids can and should have the chance to play and work with the ideas long before the terminology and/or notation is shoved down their throats. But that chance can only be allowed to last up to a point. Then it is important to ask them how they will communicate with other kids or other classrooms. And once the notion of standard/common terminology and notation is on the table, it should be easier in future to get students to want to know the “right” way of talking about or writing down some mathematics. No need to introduce distributivity by stating the term. Rather, I’d recommend asking, “Hey, can we figure out 3(2 +5)? And what if we did it this way? [whichever they didn’t use the first time].”

This is amazing! Thank you so much for sharing this! I’ll be passing this on to the teachers in my district! We have been working on understanding the CRA sequence and how it fits with developing understanding for the traditional algorithm. Very well done! Thank you! Thank you!

Thanks Megan and Christine! I’d love to hear how it goes and I’ll take any feedback you colleagues can offer.

Genius! Seriously!! I can’t wait to share this one.