Chatting with @jennalaib @kangoff @kvanduzer @heidifessenden about multiplication fluency

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.

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17 thoughts on “Chatting with @jennalaib @kangoff @kvanduzer @heidifessenden about multiplication fluency

  1. Ok, this is not where we are supposed to be going with this, but when I first saw the “non-multiplication stuff” part I immediately thought “state capitals!” This progressed to considering random facts/questions scattered in practice decks based on student preference… favorite superhero etc. before I followed the sentence to its actual ending.
    A mathematical comment is forthcoming, but I’m curious… Would this be too distracting or a neat way to incorporate other parts of the brain while reviewing math facts that may not be intrinsically motivating?

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  2. This reminded me of some conversations I’ve had about the definition of fluency. I think it’s important. What do we mean when we say we want kids to be fluent with facts, either multiplication or addition/subtraction? Fluency is emphasized in the CCSS a fair amount but as far as I know, there’s no clear definition of what it means to the authors (please point me somewhere if I’m wrong about that, because I’m dying to know). Some of my math teacher friends and I have decided that for us, being fluent INCLUDES what you describe here as “informal mental strategies for deriving facts” (e.g. using 5×7 to get to 6×7). But I see, too, the argument for defining fluency as kids truly “just knowing” those facts without having to derive. I was a kid who easily memorized many things, including math facts, but as a result I wonder about the experience of kids who didn’t have memorization as one of their top academic skills. Can kids be considered fluent if they use a deriving strategy, but just use it really fast? I tend to lean towards yes on that, maybe because of my sympathetic imagining what it would be like to have a really hard time memorizing stuff.
    All of that said, I really like your idea about talking with kids about how sometimes you have to go out of your way to try to remember things, and giving them some opportunities to do that in math (and other areas). And I think that doing this in some systematic way (though I can’t stand the Mad Minute business) is a good idea.

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    1. Kim — I hope I was clear that I think Mad Minute is a waste of time, as far as learning is concerned!

      I like your thoughts. Because I am attached to my terminology (I am often attached to terminology) I formulate your idea slightly differently. I ask, is “just knowing” facts important, over and above being able to fluently derive them?

      One reason I like my card practice was that it could serve both roles. Some kids practiced fluently deriving stuff, others practiced remembering it. (And my hope was that they might still practice remembering it on their second time through the deck in that particular session, if not their first.) I can’t think of a lot of 8th or 9th graders I’ve taught who could fluently derive facts without having them memorized, so I’m inclined to think that if we can get kids to fluently derive facts, great.

      I suspect, though, that’s it’s more complicated. A lot of the strategies we have for fluently deriving facts depend on having *some* facts memorized. Otherwise it becomes a Matroyshka doll of multiplication: 6 x 7 requires deriving 3 x 7, 3 x 7 requires 2 x 7, which requires 6 x 2, etc. Instead, I think that a lot of strategy fluency depends on having a repertoire of memorized facts to draw on. That’s why, in my classes, I seek both fluency in deriving facts and promoting memorization.

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      1. I agree that there needs to be some foundation of facts to build upon. That said, numeracy is all about flexibility and making connections — so I think it is reasonable to expect that some facts are automatic or near to that.

        For example, a student could solve 6 x 7 as doubling 3 x 7 — (3×7) + (3×7) — or as 5 groups of 7 and 1 more group — (5 x 7) + (1 x 7). Or maybe that student is fascinated by perfect squares, and can build down from the memorized 7 x 7.

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      2. I am one of those people who has lots of trouble with pure memorization. To this day I still fluently derive 6 x 8. In my head, I double 6 x 2 (which I do “just know”) to get 6 x 4 and double that to get 6 x 8. Because I can double very quickly I can complete this process very quickly, but after 20 years of school that included math classes and 20 years of teaching math, I still need to use my doubling strategy for 6 x 8. Am I fluent? I’m not sure anyone could evaluate whether I am or not unless they were in my head — I can give an answer as quickly as I can say “12, 24, 48” which is within the time limit for every timed test of fluency.

        Based on my own experience, then, I would argue that if the only way one can tell if a student has memorized facts (or is fluently deriving them) is by asking about the child’s thinking, then fluently deriving should, for all practical purposes, be considered having achieved computational fluency.

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  3. Thank you so much for writing out your thoughts! My own quick additions, right before bed:

    Like Kim, I have assumed that fluent includes what you define as “informal strategies.” (I like that phrasing, by the way.) It then becomes pointed that 3.OA.7 calls for students to “know from memory all products of two one-digit numbers.” I tend to be satisfied with the use of informal strategies as long as the student is able to use them rapidly, almost matching the speed of the student who has “memorized” the fact. It cannot be mentally taxing for the student.

    I want to believe that students will commit these multiplication and division facts within 100 through the repeated derivation/practice with informal strategies. It works for tons of kids! Of course, it does not work for all students — and I am usually forced to run an intervention group of these students after that “end of third grade” common core deadline has passed. This does not surprise, or really discourage, me at all: great Tier 1 instruction works is not designed to work for 100% of our student population.

    Somewhere at work, I have the Origo flow chart for multiplication facts. Origo puts out a number of resources for teaching these informal strategies (e.g. doubling and halving, building off of 5s, etc.) There are games that work on these strategies explicitly in the “Fundamentals” series, and a huge “Box of Facts” that includes ideas for introducing, practicing, reinforcing, and extending these strategies.

    I usually do:
    Make generalizations (0s, 1s)
    Use doubles (2s, 4s, 8s)
    Use tens (5s)
    Build up/down (10s–>9s, 5s–>6s)
    I emphasize the commutative property for 7s, but we also do choral counts and such.

    I am never sure where to put our work with 5s: does it come early because students can often skip count by 5s? Does it come after I do the doubles, because it involves halving the 10s? Last year, I went with the latter to reinforce some algebraic thinking. I think I will stick with that this year.

    For me, the hardest parts are:
    (1) communicating a clear, somewhat objective definition of fluent, and
    (2) determining a measure for growth

    I feel confident that the many strategies I use to develop fluency with multiplication and division promote both a conceptual understanding of the operations and some procedural fluency, and even some “Adaptive Reasoning”. (See: the seminal “Adding it Up.”) But how do I measure this growth and give feedback to students? If emphasizing speed puts students at risk for math anxiety (Jo Boaler)…?

    I look forward to hearing more from people tomorrow!

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  4. Are you familiar with ORIGO’s Book of Facts? They cluster the facts by strategy which might be in line with what you’re looking for: https://www.origoeducation.com/product/book-of-facts-multiplication/

    Thank you for barfing your thoughts up for the rest of us to analyze. It gives me some things to think about with regards to supporting my teachers in supporting their students in building fluency and instant recall. I think some are relying on Number Talks as their holy grail solution, and they can be pretty great, but I appreciate what you said about giving students opportunities where the practice is focused on holding a fact in your head and trying to remember it.

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  5. Michael, totally got your disdain for Mad Minute – no worries there! I appreciated your thoughts about fluency as it looks in 8th and 9th graders. Reminds me of Heather Sugrue’s session at NYC TMC about what high school teachers and elementary school teachers can learn from each other! I am so interested in this long view. And very interested in Jenna’s and Brian’s suggestion of the ORIGO book of facts, and the questions about measuring fluency. For me, that’s the hardest part. How can elementary school teachers measure fluency in ways that don’t make Jo Boaler’s hair stand on end?

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    1. Yeah, assessment is tricky here for me too. You don’t want to create a race, a public shaming, or get heart’s pumping too quickly. You also don’t want to position speed as the supreme mathematical value in class.

      At the same time, even if you’re just assessing fluency with deriving facts via strategies, you do care about efficiency and speed is a sign of efficiency.

      TERC Investigations has a series of assessments that give me a model that I like doing. I don’t have a great name for it, but it’s an Untimed Timed Quiz or something. You give kids a bunch of problems and then see how many they can do in 3 minutes. At the 3 minute mark I ask kids to pause, mark the ones they were able to figure out in 3 minutes (“Because I’m curious to know which ones you left for after 3 minutes” or “Because we’re going to try, during the year, to increase the number of problems you can do in 3 minutes”). Then they keep on going. I like this, because it gives me a good sense of where they’re at in overall multiplication fluency/just knowing but the timed factor doesn’t dominate.

      The problem I have is that TERC, in its restraint, only offers two such assessments for the school year and they aren’t structured in the way that apparently ORIGO is. They have a “less than 50” assessment for 3rd and a “less than 144” assessment for 4th. I need to make or find assessments that are scaffolded by group, I think.

      Other than that? I know that in elementary some people are like “just have a one-on-one interview with a kid” when it comes to assessment, but I simply cannot figure out how to pull it off in my teaching context.

      For assessment, that’s all I’ve got.

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  6. Yeah, at my school we used to do a 1 minute/3 minute pen/pencil assessment. Or maybe it was like 2 minutes/5 minutes. The kids would start out in pen and then after the first minute, you would tell them to switch to pencil for the rest of the time. And they were encouraged to skip around on the paper and solve the ones that they “just knew” first. That way you could see which facts were automatic for them and which involved some deriving.

    I totally agree about the tension between caring about efficiency/speed and not wanting to position speed as the supreme value in math class. I wish there was some middle ground. I try to tell my kids (4th and now 2nd graders) that depth is more important than speed in math, generally, but that sometimes being able to know math facts quickly can be really helpful for us when solving problems. But first of all, that’s obviously a mixed message and it’s confusing for kids to hold both of those things to be true. Second of all, it’s not just what you say, it’s what you do, and I think giving timed quizzes, any timed quizzes, does get those little hearts pumping in a way that I don’t really like, and might send a message that speed IS a big part of math, because when you have an emotional response like stress, I think that tends to be what sticks out in your head later. So even if all year we do lovely, wonderful math work that kids enjoy, if we do a bunch of timed quizzes, that might be the main thing they remember about math in the future. At least that’s what I fear. Certainly not saying that you’re doing that … just pondering this for myself, out loud, in your blog comments.

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    1. Do you give non-timed quizzes to your 2nd graders? In my middle and high school classes I give lots of quizzes, as I think they’re helpful for learning. I’ve started giving more (ungraded!) quizzes in 3rd and 4th Grade lately, and so the multiplication quizzes don’t feel much different to me. I don’t know if they feel much different to my kids.

      We’re on the same page, I think, about the tensions and risks of timed activities. All I can say is that I think that this isn’t a point of stress for my 3rd/4th graders, or at least I don’t see it as stressful for them. (No freaking out, no WAIT JUST ONE SEC, no crying, no refusing to hand in their papers, no crumpling it up…all responses I’ve sometimes seen from little kids in other contexts.) That makes me think that it’s possible to thread the needle here. (And, since I’m not particularly skilled as a teacher, it makes me think that it’s possible to make this work in a lot of classrooms.)

      I think there’s no way to look at the quasi-timed activities we’re talking about in isolation. If the rest of your math work supports students in thinking that slow, thoughtfulness matters in math, that they have ideas that you value, that speed is not the supreme value….that gives you a bit more room to do a quasi-timed activity.

      (Also, I think from the student experience a LOT of things in school are timed, even when teachers don’t think they are. Even a turn-and-talk moment is timed, and kids can get cut off in the middle of their thoughts. I often find myself running out of think time in professional development settings. But this is an argument that while being sensitive to speed is important, the quasi-timed quizzes we’re talking about aren’t so different from a lot of what happens in class for kids.)

      So, balance. And you can earn the room for quasi-timed stuff. And a lot of stuff in school is timed in ways that aren’t so different from quizzes. (Really, all classwork is timed.)

      The other thing I’m thinking is that while speed is sort of important, much more important to me is remembering. Speed, I think, does not matter much in math. Creative thought, however, does rely a lot on remembering many things, because when you remember something you can connect it to other things you remember. You can notice patterns. Speed is merely an imperfect proxy for remembering things.

      This is also my response to Ann, above. Speed, coming from fluency, is helpful in some areas of math (like when you need to quickly figure out 8 x 6 in your head). But remembering is valuable all on its own, because it helps you make connections and have other great ideas (like which products are over 50? or which products are odd numbers? or what’s the connection between division and multiplication problems?).

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  7. Some observations, no doubt colored by my own struggles and trauma caused by the constant harping on memorizing facts:
    1. I find it interesting that, as Jenna points out, the CC calls for students knowing ALL products of 2-digit numbers from memory by the end of grade 3. Keeping in mind that students first really begin multiplication in grade 3, this is in stark contrast to the way the CC treats addition facts. They progress very nicely over three grades, increasing from fluency with facts through 5 (K) , 10 (grade 1), and then 20 (grade 2). Then all of a sudden all multiplication facts must be memorized by the end of grade 3. Perhaps if we allowed them to develop over more time we’d have fewer issues.
    2. The increase in emphasis on developing fluency strategies (such as the ones Jenna highlights) through activities like number talks is a tremendous advance and change from what I grew up with, the traditional drill and kill techniques. It’s all for the good! And as Michael points out, there is a set of facts that, if they are automatic (foundational) are really helpful, for example 6 x 6 is a good one. That said, once kids can utilize these to derive facts they may not know from memory, (6 x 6 = 36, and 6 more is 42) we can call it a day and move on. Maybe I’ll be kicked out of the math club, but I sense a fetishization of speed and memory. Everything comes with a cost. Some make it to the final goal, but the road to fact memorization is littered with bodies that have dropped along the way. That’s sad, because they’ll never get to see the good stuff.
    3. Related to #2, too often (always?) memorizing facts is just an end in itself. Why is it helpful to have them memorized? So I can multiply 2-digit numbers faster? Not good enough. So I can get sheet of them done in three minutes? What difference does it make if for you 7 x 6 is automatically 42 and for me it takes a few seconds to derive it from 6 x 6? (That’s not a rhetorical question. I really don’t know.)
    4. I find that engaging multiplication games like Factor Captor, the Product Game, Connect 4 on a blank multiplication table, and Ultimate Multiplication Tic-Tac-Toe are highly motivating, and have the added advantage of being fun. They can scale up, down, and sideways, and they provide compelling reasons to become more fluent and ample opportunity for the practice needed for them to become automatic.

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    1. I love these multiplication games, and I love playing them in class. (As you scale, you move away from practicing remembering multiplication facts to other areas of math. Which is great!)

      I agree with your first point. In a saner world, we wouldn’t try to teach kids as much math as quickly.

      I don’t think I’ll convince you that speed in deriving is important. I waver back and forth on whether speed in mathematics is an important goal for students. But I do feel very strongly that remembering is important for learning new math.

      I think there’s a strong case to make that remembering multiplication facts gives you access to many areas of math. Remembering multiplications gives you a much easier ability to notice the connections between multiplication and division. I think you will struggle to notice patterns *in* multiplication facts if you don’t have some of them remembered. When it comes to algebra, I notice kids who remember multiplication or division have an easier time seeing shortcuts and patterns for solving equations.

      Like you said, this needs to be balanced with the risks that we turn a kid off of math. But maybe because I’m not a coach, I’m just a teacher, my only worry is whether I’m turning kids off of math with the activities that I’m doing in class. And while I know that there are faults in my teaching of multiplication, I have not noticed the sort of intense stress that people describe. I think it’s because I don’t do mad minutes, I’m easy-going about timed stuff, I don’t assign grades based on fact knowledge, I do a lot of number talk-style activities before I ask kids to practice remembering things, and because my kids do a lot of other kinds of math in class.

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      1. My take, in short:

        Speed is mostly unimportant in math. Remembering things is very important in math. When are remembering things we can notice patterns, make connections, compare and contrast those things that we are remembering. Speed is what remembering looks like, but it’s not the speed that’s important.

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  8. Ahhhh! I am finally getting to this and there is so much good stuff. I wish I was in a room with all of you so we could talk about it and I could say: what is that game? How do you play it? But what does that mean? What does it look like?

    I agree with all of you about speed and its importance or lack thereof. I think kids need to know facts relatively quickly, but I don’t mind if they have to derive them. My dirty little secret is that I am pretty good at multiplication facts but there are some addition facts I never got that good at. Like 7+6. I derive them using 10 and it is fine. Or I use my fingers. For real. It has served me well.

    I do want my kids to practice facts, though, and I want to track their progress. I think Michael is right that kids need practice getting the facts in and out of memory. They simply cannot have to skip count their nines every time as they get older (I think). But I don’t want them practicing before they are ready, before they have the conceptual understanding. And I think that is what is happening, especially at home, a lot of the time. Well, and at school too.

    And yes, number talks and multiplication games are key. But not enough, I think.

    For assessment, I do one-on-one interviews but just for 25 minutes per week. It is not a timed test, but I give them facts and they solve them for me and I see how quickly they can answer or derive them. I don’t think it is too stressful for many kids. I am not too strict about the time limits — I just use my judgment about what is too long.

    Most of what I do in terms of learning facts comes from Van de Walle. But I think I need to re-read his chapter on it. It is about strategies and choosing which strategy will help you efficiently find the fact you need. Love the suggestion of the Origo book.

    I love what Joe says about the way addition fact fluency is expected to happen more gradually. Why do the CCSS make multiplication happen so quickly? I think it is crazy. My kids do NOT have the conceptual understanding of multiplication down enough by the end of third grade (because they also don’t have an understanding of subtraction or addition or number sense…). We worked on facts a little in grade 3 but it was HARD. I think it’s fine if they can’t derive them efficiently until the end of 4th. (That’s when I had to learn them, or maybe even fifth grade!). But not all of my students will be fluent even by then.

    Kim, is there nothing in the Progressions documents about what fluency is? I haven’t looked at them in too long.

    Thanks so much for this, all of you! But now, my original tweet was to see if anyone had a perfect system for me for this year. Anyone??

    Heidi

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