This is an area of my 3rd/4th Grade teaching that I want to do a much better job with this year.
Why does this stress me out?
When I first started teaching 3rd/4th, I was at a very fluffy moment in my teaching. I was just going to make sure that students conceptually understood multiplication, and fluency would just happen when it happened.
This worked for some kids, but I remember a number of kids that made very little progress in their multiplication fluency over the course of our time together. At the same time, I was seeing the benefits of the conceptual work because they could find a strategy for tackling bigger multiplication (like 31 x 12) but it would TAKE FOREVER. Because of the fluency.
Around this time I also got hired to do some research reading on multiplication fact fluency. I’m not going to do the citation thing, because no one cares, but that experience along with my teaching helped me clarify my views about how learning multiplication facts happens.
What’s the difference between conceptual understanding, fluency and having facts memorized?
There are lots of terms that we try to use. There are people who have “standard” definitions of all these things. Here is my current understanding of how to promote fluency, and how it relates to practice, timed stuff and conceptual understanding:
- Consider a multiplication problem like 6 x 7. There are a lot of strategies that kids could use to figure out this answer. If a kid can find out the answer using any mathematically valid strategy, then I’m happy to say that this student has conceptual understanding of multiplication. By this I mean the multiplication operation, roughly “what multiplication means.”
- To find 6 x 7 a kid might count by 7s, or by 6s. They might find 5 x 7 and then add a 7. They might find 3 x 7 and double that. And so on. All of these are informal strategies for finding 6 x 7. I have a lot of informal strategies that I use for mental multiplication in my own mathematical life. For example, 12 x 7 I can figure out by doubling 6 x 7. It’s not that I know 12 x 7, it’s that I can quickly figure it out using informal strategies. This is what I think a lot of us mean by “fluency” with multiplication — the ability to efficiently use informal strategies to derive an answer.
- On the other hand, I basically never have to use informal strategies for single-digit multiplication because I’m the kind of kid that had no trouble committing these facts to memory in my elementary years. TERC Investigations calls this “just knowing” what 6 x 7 is, and this is what a lot of us call “having the facts memorized.”
This isn’t news to any of you, but I just wanted to clarify my terms before I used them a bunch.
But how do all of these things — conceptual understanding of multiplication, fluency with informal strategies, and having facts memorized — fit together in kids’ learning?
How do multiplication facts get memorized?
Let’s start with having facts memorized. From experience and my reading, I think the way you come to have facts memorized is by holding a fact in your head and trying to remember it. This is sort of like if I told you my middle name (I’m not gonna) and then it’s rattling around in your memory and a few minutes later you had to remember my name and you’re like….uh, oh yeah, it’s ______. Those moments, piled together, eventually commit my middle name to your long-term memory. It’s memorized.
I’m saying a lot of things and this isn’t really “writing” as much as barfing thoughts. But this is an important point to me: to have facts memorized, you need to try to remember them.
These moments where you’re trying to remember a fact that you’re holding in your head can occur either artificially or naturally. Artificially, through practice. But a lot of practice isn’t actually designed to create moments where you’ve got a fact in your head and you’re trying to remember it. Consider a test that just gives a child a multiplication problem to solve. A student could complete this task by just using some strategy to derive the fact, and they haven’t practiced remembering the fact at all. Or consider those ubiquitous Mad Minute things. They can’t possibly help you practice remembering a multiplication that you don’t already have pretty available to you through fluent derivation with informal strategies.
A lot of harm has been done to kids in the service of creating remembering practice for kids. So why not avoid them entirely?
Kids can get these opportunities to remember facts in more natural settings. Here is a way that this can happen:
- Kids have a conceptual understanding of the multiplication operation.
- You then teach kids a bunch of informal mental strategies for deriving multiplications.
- Kids get fluent at these informal strategies.
- As kids do other multiplication work, they frequently find themselves deriving multiplication through these informal strategies. Then, later in the problem, they have to remember what they derived. This creates remembering practice for kids.
- And if they don’t remember it? They just rederive the multiplication using an informal strategy, and then they’re like “oh yeah!” Another remembering opportunity.
- BUT THE KEY THING IS THAT IT’S NOT THE DERIVING ITSELF THAT LEADS TO THEM GETTING REMEMBERING PRACTICE AND THUS COMMITTING THEM TO MEMORY.
The above, I think, is how some writers, PD people and academics envision how kids could naturally come to have their multiplication facts memorized.
…but the natural approach wasn’t working for enough of my students. So I’ve adopted a more balanced approach that mixes the natural and artificial pathways towards getting remembering practice.
What do I think would be best for kids in my context?
In my 3rd and 4th Grade classes, I never want to ask kids to do something that isn’t meaningful to them. At the same time, I worry a lot about how numeracy impacts the math kids feel confident with in my high school classes. For this reason, I’m not satisfied with just fluency with informal strategies for my elementary students — I also want to help them come to have these multiplications memorized.
My ideal approach goes something like this:
- Conceptual Understanding: Make sure kids have some strategy for solving a single-digit multiplication problem, either on paper or in their heads. I want them to have conceptual understanding of the multiplication operations.
- Using Number Talks to Develop Informal Strategies: Figure out some small group of multiplication facts, and make sure that my kids become fluent in using informal strategies to derive these multiplications.
- Create Practice Cards: I give students index cards and ask them to write that small group of multiplication facts on those cards. I ask them to mix those in with a bunch of cards they’ve mastered. (Flash card key #1: keep the success rate high.) Write informal strategy hints (e.g. “double 3 x 7”) at the bottom of the card, if they want. (Flash card key #2: it’s your deck, you do what makes sense.)
- Practice with Cards: Practice your deck, alone or with a friend. Go through the whole deck at least twice. (Flash card key #3: try to keep it low stress).
- Follow-Up With Quizzes to Assess: Figure out if we’re ready for a new informal strategy, a new set of facts, a new type of talk, a new set of cards to add to our decks, or whatever.
I was doing this in 4th Grade last year, but I was unsystematic and sloppy. I think I know how to do this so that kids don’t get stressed out and that it’s enjoyable and fun. Honest to god the kids loved the cards and often asked if we could work on them. The kids liked them so much that I started putting non-multiplication stuff into their decks. My sales pitch for this sort of practice — some things that we figure out are worth trying to remember — applied to lots of things. By the end of the year I was asking kids to put in a few division problems, some fraction addition and subtraction and multiplication into their decks.
What is wrong with this approach?
I am a sloppy, unsystematic teacher. Partly this is an artifact of my teaching context — I teach 4 grade levels in 4 different rooms, don’t have much wall space, don’t have a consistent stock of supplies in my rooms, etc. — but I know I can do better.
Part of how my sloppiness expressed itself was that practice with cards was whenever-I-remembered. I think it would be better if I scheduled this in to the week somehow.
The bigger issue was that I don’t have a good plan for how to break up the multiplication facts. I’m sure there are resources that I could use here, but what I need is something like Set 1: multiplication involving 2s and stuff you can figure out by using doubling strategies on those 2 facts. Set 2: multiplication by 10s(?) Set 3: Using halving and multiplication by 10s to figure out other stuff, etc. I want to make sure that the informal work with strategies is structured over the course of the school year to cover all the facts. I want the strategy work to precede the remembering practice work. Do you know of a resource for this? I’d be interested in it!
I think those are the major issues that I want to focus on this year.
What do you think?
This isn’t Michael’s Take On Multiplication. The above was “Michael’s response to a tweet.” This is a rough shot at describing how I’ve been handling this in my thinking and my classes lately.