During this past school year, I started practicing math facts in a new way with my 3rd and 4th Graders. The name I came up with for the routine was “Forwards and Backwards Practice.”
Like all my classroom ideas, it was lazy and simple. I handed a piece of blank paper to each kid. I told everyone that we’d be doing an activity in two rounds, that they should write “Round 1” at the top of their papers. Then I wrote the “forwards” and “backwards” problems on the board.
The “forwards” problems were pretty familiar to my kids. Solve the equation; put a number in the blank to make the equation true:
4 x ___ = 28, 8 x 4 = ___, ____ x 7 = 42
The “backwards” questions were more open-ended. On the board, I simply wrote three numbers:
21; 42; 81
I explained that for these I wanted the kids to write as many multiplications as they could remember that equaled each number. Accurate “backwards” answers for 21 would be 3 x 7, 1 x 21, etc.
As kids were wrapping these questions up, I called attention back to the board. If there was a common mistake, this is when I mentioned it. I shared accurate answers to each question, emphasizing what I wanted to emphasize.
Then, I erased the board. I told kids that there would be a second round of questions in a minute that were very closely related. Take a minute, I said, and study the multiplication we just reviewed. Try to remember as much of these as you can. When a minute is up, you’ll flip your page over to the blank side for Round 2.
Here’s what I did, basically: I swapped the forwards and the backwards questions. The backwards questions were now forwards equations, and the forwards were now put in backwards form.
That means that the corrections and practice that the kids got in Round 1 are relevant for Round 2. If a kid is just starting to practice 7 x 3, then they get a chance to study it and try to remember it for a problem that is coming right up, moments after they study.
That’s why I like this routine. It packs a pretty virtuous cycle into a fairly quick package:
- Think about what you already know
- Get some explicit instruction in response to your work
- Try to remember it
One thing I like about this routine is that it solves a problem I was having with other whole-group practice, which was some kids were finishing my practice much before others. I didn’t want to end the activity, but the quick finishers needed something to do. Backwards practice is something that sort of “naturally” differentiates. It’s end-goal is vague; kids interpret it according to their understanding of multiplication, so they each student tends to find appropriate math to work on, and my speed-demons don’t force me to call a quick end to the activity.
Depending on the group and their confidence, knowledge, etc., I might vary how closely the questions in Round 2 resemble the questions in Round 1. If kids are really at the beginning of their learning of multiplication, the Round 2 questions might very closely resemble the ones in Round 1. Or we might keep them all “forwards” practice, just knock out different numbers in the equation. Or change the direction (i.e. from 3 x 7 to 7 x 3).
I also use flashcards with my 3rd and 4th Graders.
When I first introduced them, I was very, very nervous. I tend to worry about the most anxious kids in my class, and I had two nervous wrecks in my 3rd Grade group. (One was receiving medical care for his stress.) How would they react to all this? They were already shutting down when I gave out worksheets. Flash cards would only be worse.
So, I introduce flashcards. I get each kid a little plastic decks and I get a ton of colorful index cards. (It turns out that you need both of these things to make this work, because otherwise kids lose their cards or mix them up with other decks. I tried to pull this off with envelopes and white flash cards last year and it was a total disaster.)
We slowly start filling out cards with multiplication (and addition) problems. I ask kids to practice, and I explain what practicing means, and I tell them what good practice looks like. (“None of this stuff where you’re shouting out answers while someone else is thinking. We don’t want to take away someone else’s chance to think.”) And then I give them a good chunk of time to start practicing.
Things looked good in class, but you never know for sure, so I asked kids at the end of class to write a bit about how they liked practicing math with their decks. I’m very interested in what one of my high-anxiety kids thinks, so I grab him at the end. What did you write, O? What were you thinking?
What he tells me is really interesting. He says that he really prefer the cards because they only show one problem at a time. When he sees a page with a ton of problems on it he gets overwhelmed, distracted, stressed out. But cards are significantly less stress for him.
The year goes on. There are a few groups that are getting a bit competitive when they practice, which I come to think is fine as long as I keep an eye on it. I do maintenance on their practice: be nice; you can write another problem as a “starter problem”; make sure everyone you’re practicing with has a chance to answer; you can do this by yourself; throw out a few cards that are too easy. I ask questions: are the cards too easy? are they too hard?
Are you enjoying this practice? I ask that often, because I’m sort of surprised by how much they’re enjoying themselves. But they are, really.
Flashcards are just great for practice. The answer is right there — if you get it wrong you get correction and a nudge in the right direction. (Math facts is the sort of thing that it really does help to get quick corrections on.)
There are other benefits too. Like O said, you only see one problem at a time. You can go fast, you can go slow. You can turn the cards over and do “backwards practice.” You can take the deck home and practice by yourself. You can quickly take it out if you finish an activity quickly — it can go on the menu.
One challenge I’ve had with flashcards is that some kids persist in using really inefficient strategies when practicing with their decks. This is because they are basically choosing how long to spend on each card in their decks. This is attenuated somewhat by kids practicing together but it’s something that I had to keep an eye on while they were practicing.
When I wanted a bit more control over which fact families my students practiced, I used dice games:
It’s another dumb, easy thing. The only problem here is that there are no corrections when kids are practicing. I had my students write down their results for this sort of practice, but I often couldn’t catch mistakes quickly enough to be useful for their practice.
I want to help my students commit as many multiplication facts as they can to memory. I don’t want to feteshize math fact automaticity — some kids do OK without this knowledge — but it’s really useful knowledge for learning more math. Why wouldn’t I try to help my kids commit their math facts to memory?
What’s the best way to do this? Well, you need a theory as to how kids come to commit facts to memory. As I’ve written about before, my perspective is you learn what you practice. If you want to remember facts, you have to practice remembering them. And if you don’t practice remembering them — if you only ever practicing skip-counting to derive them — you’ll probably never come to memorize them.
This helps me navigate the world of multiplication practice, where controversy abounds.
Take, for example, speed practice. Daniel Willingham and Daniel Ansari recently wrote a post defending speed practice. I left a comment arguing that we needed to know why speed can help kids in their practice before we defend it:
In one study I read (about fluency software) I learned that students with learning disability did not improve their addition fluency through untimed practice. Why? Because during untimed practice, the students simply DERIVED the facts rather than trying to RECALL them. In other words, you’d see a lot of kids in front of screen counting out 3 + 9 with their fingers instead of trying to recall them from memory. The kids were already pretty good at using this strategy, and the untimed practice allowed them to keep doing what they were good at.
I see this in my own students too. It’s not so much that timed practice is helpful for learning directly, as much as it creates a context in which kids practice the things you’d like them to practice.
A solution is timed practice with immediate fact instruction. (You got 3 + 9 wrong? OK, 3 + 9 = 12. Try again.)
The worst case scenario is that teachers give kids a full worksheet of problems, and kids can’t directly recall ANY of them. Instead, kids work on using strategies to derive the facts. The teacher says to solve as many as you can, but the students can only correctly answer that many questions using direct recall — with strategies, there’s not enough time. Time pressure (along with the long list of problems) generates anxiety, which makes it harder still to answer problems correctly. None of this produces fact fluency.
Based on talking to colleagues and other math educators, this worst case scenario is in fact prevalent in US classrooms. These “Mad Minute” activities could be used appropriately, but they are instead often given to novices who are not prepared to draw on their mostly memorized facts for the activity. And, I think, this probably does generate feelings of helplessness and anxiety.
As a result of all this, when I think about fact practice I end up asking myself this question all the time: Will the kids be practicing derivation or recall?
And here’s a fundamental follow-up: Kids can’t practice recall unless they are being prompted with the correct answers during the practice.
I really don’t like Mad Minute activities because they don’t prompt you with corrections or instruction in the fact during the activity. So you can’t really learn anything from the activity unless you’re “almost there.” Maybe it helps you practice pulling out the fact from memory, but it can’t help you learn that fact with automaticity without some sort of prompting during practice.
That’s why I like splitting up practice into two rounds, as I do during “forwards/backwards” practice. I get to give prompting/instruction in between the rounds, and then kids get a chance to practice with it during Round 2.
It’s also why I like practice with flashcards, especially if kids are reminded to try to figure out the answer as quickly as they can. (They basically do this anyway.) While I worried that this would be stressful for my kids, I’ve actually found the opposite. Flashcards, the way I use them in class, tend to be less stressful than other conventional practice activities (like long problem sets).
The absence of prompting/corrections is a downside of my dice practice, though it’s attenuated somewhat by the way the problems will reappear as kids cycle through the different boxes and repeat factors. Still, it’s a form of practice that probably would be better at helping kids have a chance to practice strategies rather than remembering.
I think it’s important to be thoughtful here. Math facts aren’t the be-all of school math, but they do make a difference for kids’ future learning.
The fundamental disagreement I have with a lot of people in math education is that I don’t think that practice using a strategy helps kids commit facts to memory. (Though I do believe that having efficient strategies does help kids commit facts to memory. Both knowing efficient strategies and recall practice are important for developing automaticity. I have citations for this. See also the Willingham/Ansari piece.)
And my fundamental displeasure about the debate is how rarely it gets into the classroom details. So, you’ve got a position on how multiplication should be taught? Does it fit on a slide? Do people take pictures of it with their phones during conferences? Tweet it, retweet it, like it?
That’s great, seriously, but let’s talk the nitty gritty. What are your activities? What does your class look like? What is it that you do?
12 thoughts on “What I’ve Learned About Practicing Multiplication Facts”
Michael, Thank you so much for writing this. I suspected that you and I had more in common then we did differences when it comes to teaching multiplication facts, and I was right. I love the backwards and forwards idea. I will model and use it this year. I am thinking about some third graders I work with who naturally discover the relationship between double and halving, tripling and thirding, (etc.) factors. They often stumble on one factor being a fraction and they keep going- they love it. The backwards part of this activity would be perfect them for them. I also appreciate the explanation of how you attended to your students with anxiety. You were so intentional about how you moved those students forward without adding to the anxiety. My son is entering third grade this year and I am toying with the idea of starting a separate blog that chronicles his journey through multiplication fact acquisition. I am most interested in the subtle and nuanced part of learning facts. How does it start? How do you tell when a student is truly starting to think multiplicatively instead of additively – “seeing the groups within groups intuitively”. I am also hoping careful observation of my son will help me learn more about cognitive load. I am just scratching the surface here and really want to learn more about how it effects learning multiplication – from beginning to end. Would love to hear any and all thoughts about this idea. Thanks again. I truly enjoy your blog. You push my thinking and I appreciate your honest reflection.
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Excellent post, very thought-provoking. The distinction between recall and derived facts (and other math concepts) is important for my high school students too. I’m always looking for ways to “Make It Stick”. Thank you.
Love your forward/backward idea.
With flash cards, have you considered ‘triangle flash cards’? You can find some on Teacher Pay Teacher website. I give a set of these to parents to use with their children — when their children can go through the deck with fewer that 5 errors (& in a reasonable amount of time – maybe 5 min) the parents let me know. At that time I ‘test’ for fluency . I found many parents want to help their children but don’t feel confident to help with math. But listening to their children working with the triangle flash cards seems to be a winner.
Thanks Michael. Legendary post.
Questions : Do students make their own cards? What material do you use for the cards? How do you/students decide when to add new cards? How do you /students decide which new cards to add? Are the cards just multiplication facts or other facts/longer problems too? How do/do students time the flashcard activity to avoid the derivation problem? How do you manage this?
Your distinction between recall and derivation opened my eyes to the source of a challenge that I was having last year! Cheers.
Thanks for the comment and questions, Ollie.
Do students make their own cards? What material do you use for the cards?
I give kids about 100 colored index cards (Oxford Ruled Index Cards, 3″ x 5″ Size, Assorted Glow Colors) and a cheap plastic deck that holds the cards. The cards are blank. Whenever we add cards (more on that soon) I write problems and solutions on the board and kids copy the problems onto their index cards.
How do you/students decide when to add new cards? How do you /students decide which new cards to add?
I decide when to add new cards, mostly. (Often kids ask me if they can add their own problems to their decks. My answer is always yes.)
In general, I try to teach efficient mental strategies in advance of memorization. My read of the research literature, along with experience, tells me that efficient strategies support memorization. I’m a bit fuzzy on why, but I think it’s a mix of things. First, I think efficient derivation can create new recall opportunities. (More likely to have a fact fresh in your head if you can quickly derive it.) Second, I think that efficient strategies can sort of scaffold your recall by delimiting possibilities, and preventing you from wasting time recalling wrong results. (e.g. if you know a bit about 9s you’re unlikely to say that 6 x 9 is in the 60s). Anyway, check out this paper by Woodward and my piece about a student of mine for more on strategies/recall.
For multiplication, last summer I collaboratively chunked up the multiplication fact by efficient derivation strategy. (See our doc here.) I took this as a guide for introducing new cards. Both experience and reading tell me that we want a high ratio of success-to-failure with this sort of practice, so I try to add just 5 cards or so at a time, every few weeks. Usually my process is to explicitly teach an efficient mental strategy, assess that my kids have some sort of fluency with it, and then add cards to their decks that give them a chance to practice recalling the fact (ideally, though they often fall into derivation strategies).
Are the cards just multiplication facts or other facts/longer problems too?
I try to include lots of different things from across the curriculum, though I’ve tended to only include arithmetic/algebra and I’ve emphasized multiplication. I always ask kids to put in some fraction addition/subtraction, some division, some addition/subtraction, etc. I’m trying to figure out if it makes sense to expand this to my algebra and geometry courses this coming year.
How do/do students time the flashcard activity to avoid the derivation problem? How do you manage this?
I’m not particularly thoughtful about managing this yet. I’ve been relying on kids’ natural desire to move faster and commit these things to memory, but that doesn’t always work. I could explicitly prompt kids to try to complete as many cards as they can in a given amount of time…honestly, that makes me nervous that I’d make kids nervous. It might be better for me if I just ask kids to try to answer as quickly as they can…or maybe I could try some whole-group practice where I act as a sort of “metronome” to pace their practice?
The flashcard practice prompts most kids to aim for recall, but I think I need to experiment some more this year to sharpen the practice without freaking out my anxious kids.
Just wondering if you’re still doing this? If so, how have you improved it? If not, why did you stop and what are you doing now instead?
I really enjoyed revisiting this post 🙂 O.
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Yes, I pretty much still do this! (This year it’s complicated by COVID. We purchased flashcards for all 3rd Graders to take home.)
There are various changes that I blogged about at times, but it’s all within the same framework more or less: https://problemproblems.wordpress.com/category/multiplication/
Thank you, Michael. This last school year I’ve been thinking a lot about supporting my students in learning basic facts and I am not happy as to how I handled it in my classroom.
I noticed (just like you did) that some students get stuck on inefficient deriving strategies (7+8, start counting from 1) and don’t move from there no matter what. No, I don’t want my students to memorize something they don’t understand but yes, I want them to be able to recall or very efficiently derive.
I love Forwards/Backwards idea, and I will try to make it a routine this year and report how it goes. From comments to Daniel Willingham’s spinner article, I might try some things that John Golden suggested. Solve the facts you know, circle the ones you struggle with. Maybe try flashcards for practice? I hope I can pull it off without anxiety bursts. I’ll be honest though, I had more meltdowns about “this problem is too hard” than “I don’t remember my facts” from my math anxious students last year.
I wonder if a lot of pushback against fact practice comes from experiences when facts practice was not supported by understanding, strategizing, mental math and number sense practice.
Last year I managed to cut 15-20 mins block daily for number sense/mental math routines and it made the difference. I appreciate your posts and reflecting on them helps me to aim for the balance. Fact fluency and procedural fluency don’t have to be separated from conceptual understanding, but teaching them has to be intentional and responsive.