Dr. Jenny Bay-Williams, Productive Ways to Build Fluency with Basic Facts
ROUNDING UP: SEASON 1 | EPISODE 15
保证学生掌握基本的事实依然存在a shared goal among parents and educators. That said, many educators wonder what should replace the memorization drills that cause so much harm to their students’ math identities. Today on the podcast, Dr.Jenny Bay-Williamstalks about how to meet that goal and shares a set ofpracticesthat also support student reasoning and sensemaking.
Eight Unproductive Practices in Developing Fact Fluency
Mike Wallus:保证学生掌握基本事实尸体ns a shared goal among parents and educators. That said, many educators wonder what should replace the memorization drills that cause so much harm to their students' math identities. Today on the podcast, Jenny Bay-Williams talks about how to meet that goal and shares a set of productive practices that also support student reasoning and sensemaking.
Mike: Welcome to the podcast, Jenny. We are excited to have you.
Jennifer Bay-Williams: Well, thank you for inviting me. I'm thrilled to be here and excited to be talking about basic facts.
Mike: Awesome. Let's jump in. So, your recommendations start with an emphasis on reasoning. I wonder if we could start by just having you talk about the “why” behind your recommendation and a little bit about what an emphasis on reasoning looks like in an elementary classroom when you're thinking about basic facts.
Jenny: All right, well, I'm going to start with a little bit of a snarky response: that the non-reasoning approach doesn't work.
Mike and Jenny: (laugh)
Jenny: OK. So, one reason to move to reasoning is that memorization doesn't work. Drill doesn't work for most people. But the reason to focus on reasoning with basic facts, beyond that fact, is that the reasoning strategies grow to strategies that can be used beyond basic facts. So, if you take something like the making 10 idea—that nine plus six, you can move one over and you have 10 plus five—is a beautiful strategy for a 99 plus 35. So, you teach the reasoning up front from the beginning, and it sets students up for success later on.
Mike: That absolutely makes sense. So, you talk about the difference between telling a strategy and explicit instruction. And Iraisedthis because I suspect that some people might struggle to think about how those are different. Could you describe what explicit instruction looks like and maybe share an example with listeners?
Jenny: Absolutely. First of all, I like to use the whole phrase: “explicit strategy instruction.” So, what you're trying to do is have that strategy be explicit, noticeable, visible. So, for example, if you're going to do the making 10 strategy we just talked about, you might have two 10-frames. One of them is filled with nine counters, and one of them is filled with six counters. And students can see that moving one counter over is the same quantity. So, they're seeing this flexibility that you can move numbers around, and you end up with the same sum. So, you're just making that idea explicit and then helping them generalize. You change the problems up and then they come back, and they're like, “Oh, hey, we can always move some over to make a 10 or a 20 or a 30” or whatever you're working on. And so, I feel like, in using the counters—or they could be stacking unifix cubes or things like that—that's the explicit instruction.
Jenny:这是concrete。然后,如果你需要更明确,you ask students in the end to summarize the pattern that they noticed across the three or four problems that they solved. “Oh, that you take the bigger number, and then you go ahead and complete a 10 to make it easier to add.” And then, that's how you're really bringing those ideas out into the community to talk about. For multiplication, I'm just going to contrast. Let's say we're doing add a group strategy with multiplication. If you were going to do direct instruction, and you're doing six times eight, you might say, “All right, so when you see a six,” then a direct instruction would be like, “Take that first number and just assume it's a five.” So then, “Five eights is how much? Write that down.” That's direct instruction. You're like, “Here, do this step here, do this step here, do this step.”
Jenny: The explicit strategy instruction would have, for example—I like eight boxes of crowns because they oftentimes come in eight. So, but, they'd have five boxes of crowns and then one more box of crowns. So, they could see you've got five boxes of crowns. They know that fact is 40, they—if they're working on their sixes, they should know their fives. And so, then what would one more group be about? So, just helping them see that with multiplication through visuals, you're adding on one group, not one more, but one group. So, they see that through the visuals that they're doing or through arrays or things like that. So, it's about them seeing the number of relationships and not being told what the steps are.
Mike: And it strikes me, too, Jenny, that the role of the teacher in those two scenarios is pretty different.
Jenny: Very different. Because the teacher is working非常艰难的教诲与显式策略(笑)tion to have the visuals that really highlight the strategy. Maybe it's the colors of the dots or the exact 10-frames they've picked, and have they filled them or whether they choose to use the unifix cubes, and how they're going to color them, and things like that. So, they're doing a lot of thinking to make that pattern noticeable, visible. As opposed to just saying, “Do this first, do that second, do that third.”
Mike: I love the way that you said that you're doing a lot of thinking and work as a teacher to make a pattern noticeable. That's powerful, and it really is a stark contrast to, “Let me just tell you what to do.” I'd love to shift a little bit and ask you about another piece of your work. So, you advocate for teaching facts in an order that stresses relationships rather than simply teaching them in order. I'm wondering if you can tell me a little bit more about how relationships-based instruction has an impact on student thinking.
Jenny: So, we want every student to enact the reasoning strategies. So, I'm going to go back to addition, for example. And I'm going to switch over to the strategy that I call pretend-to-10, also called use 10 or compensation. But if you're going to set them up for using that strategy, [there are] a lot of steps to think through. So, if you're doing nine plus five, then in the pretend-to-10 strategy, you just pretend that nine is a 10. So now you've got 10 plus five and then you've got to compensate in the end. You’ve got to fix your answer because it's one too much. And so, you've got to come back one. That's some thinking. Those are some steps. So, what you want is to have the students automatic with certain things so that they're set up for that task. So, for that strategy, they need to be able to add a number onto 10 without much thought.
Jenny: Otherwise, the strategy is not useful. The strategy is useful when they already know 10 plus five. So, you teach them this, you teach them that relationship, you know 10 and some more, and then they know that nine’s one less than 10. That relationship ishugely重要的是,知道9 th少一点an 10—um, and so then they know their answer has to be one less. Nine’s one less than 10. So, nine plus a number is one less than 10 plus the number. Huge idea. And there's been a lot of research done in kindergarten on students understanding things like seven’s one more than six, seven’s one less than eight. And they're predictive studies looking at student achievement in first grade, second grade, third grade. And students—it turns out that one of the biggest predictors of success is students understanding those number relationships. That one more, one less, um, two more, two less—hugely important in doing the number sense. So that's what the relationship piece is, is sequencing facts so that what is going to be needed for the next thing they're going to do, the thinking that's going to be needed, is there for them. And then build on those relationships to learn the next strategy.
Mike: I mean, it strikes me that there's a little bit of a twofer in that one. The first is this idea that what you're doing is purposely setting up a future idea, right? It's kind of like saying, “I'm going to build this prior knowledge about 10-ness, and then I'm going to have kids think about the relationship between 10 and nine.” So, like, the care in this work is actually really understanding those relationships and how you're going to leverage them. The other thing that really jumps out from what you said—this has long term implications for students’ thinking—it's not just fact acquisition, it's what you said, research shows that this has implications for how kids are thinking further down the road. Am I understanding that right?
Jenny: That's absolutely correct. So just that strategy alone. Let's say they're adding 29 plus 39. And they're like, “Oh hey, both of those numbers are right next to the next benchmark. So instead of 29 plus 39, I'm going to add 30 plus 40, 70. And I got, I went up two, so I'm going to come back down two. And I know that two less than a benchmark's going to land on an eight to that.” Again, it's coming back to this relationship of how far apart numbers are—what's right there within a set of 10—helps then to generalize within 10s or within 100s. And by the way, how about fractions?
Mike: Hmm.Talk about that。
Jenny: (laughs) It generalizes to fractions. So, let's take that same idea of adding. Let's just say it's like, two and seven-eighths plus two and seven-eighths. So, if we just pretended those were both threes because they're both super close to three, then you'd have six, and then you added on two-eighths too much. So, you come back two-eighths, or a fourth, and you have your answer. You don't have to do the regrouping with fractions and all the mess that really gets bogged down. And it's a much more efficient method that, again, you set students up for when they understand these number relationships. When you get into fractions, you're thinking about, like, how close are you to the next whole number maybe, instead of to the next 10s number.
Mike: It strikes me that if you have a group of teachers who have a common understanding of this approach to facts, and everyone's kind of playing the long game and thinking about how what they're doing is going to support what's next, it just creates a system that's much more intentional in helping kids not only acquire the facts, but build a set of ways of thinking.
Jenny: Mike, that's exactly it. I mean, here we are, we're trying to make up for lost time. We never have enough time in the classroom. We want an efficient way to make sure our kids get the most learning in. And so, to me that is about investing early in the fact strategies. Because then actually when you get up to those other things that you're adding or subtracting or multiplying or whatever you're doing, you benefit from the fact that you took time early to learn those strategies. Because those strategies are now非常usefulfor all this other math that you're doing. And then students are more successful in making good choices about how they're going to solve those problems that are, oftentimes—especially when, I like to mention fractions and decimals at least once in a basic facts talk because we get back, by the time we get into fractions and decimals, we're back to just sometimes only showing one way, the sort of standard algorithm way. When, in fact, those basic facts strategies absolutely apply to almost-always more-efficient strategies for working with fractions and decimals.
Mike: I want to shift a little bit. One of the things that was really helpful for me in growing my understanding is, the way that you talk about a set of facts that you would describe as “foundational” facts and another set of facts that you would describe as “derived” facts. And I'm wondering if you can unpack what those two subsets are and how they're related to one another.
Jenny: Yeah. So, the foundational facts are ones where automaticity is needed in order to enact a strategy. So, to me, the foundational fact strategies are, they're names. Like the doubling strategy or double and double again, some people call it. Or add a group for multiplication, and the addition ones of making 10s and pretend-to-10 strategies. And in those strategies, you can solve lots of different facts. But there's too much going on (laughs) in your brain if you don't have automaticity with the facts you need. So, for example, if you have your six facts, and you're trying to get your six facts down. And you already know your fives, like, automaticity with your fives, then that becomes a useful way to get your sixes. So, if you have six times eight, and you know five times eight is 40, then you're like, “I got one more 8, 48.”
Jenny: That's an added group strategy. But if you're not automatic with your fives, this is how this sounds when you're interviewing a child. They're going to use add a group strategy, but they don't know their fives. So, then they're like, “Let's see, five times eight is 5, 10, 15, 20, 25, 30, 40. Now, what was I doing?” Like, they can't finish it because they were skip-counting with their fives. They lose track of what they're doing, is my point. So, the key is that they just know those facts that they need in order to use a strategy. And that, going back to, like, the pretend-to-10, they got to know 10-and-some-more facts to be successful. They have to know nine’s one less than 10 to be successful. So, that's the idea is, if they reach automaticity with the foundational fact sets, then their brain is freed up to go through those reasoning strategies.
Mike: That totally makes sense. I want to shift a little bit now. One of the things that I really appreciated about the article was that you made what I think is a very strong, unambiguous case for ending many of the past practices used for fact acquisition—worksheets and timed tests, in particular. This can be a tough sell because this is often what is associated with elementary mathematics, and families kind of expect this kind of practice. How would you help an educator explain the shift away from these practices to folks who are out in the larger community? What is it that we might help say to folks to help them understand this shift?
Jenny: That's a great question, and the real answer is, it depends, again, on audience. So, who is your audience? Even if the audience is parents, what do those parents prioritize and want for their children? So, I feel like [there are] lots of reasons to do it, but to really speak to what matters to them. So, I'm going to give a very generic answer here. But for everyone, they want their child to be successful. So, I feel that that opportunity to show, to give a problem like 29 plus 29, and ask how parents might add that problem. And if they think 30 plus 30 and subtract two to get to the answer, whatever, then that gives this case to say, “Well this is how we're going to work on basic facts. We're building up so that your child is ready to use these strategies. We're going to start right with the basic facts, learning these strategies. These really matter.”
Jenny: And the example I gave could be whatever fits with the level of their kid. So, it could be like 302 minus 299. It's a classic one where you don't want your child to implement an algorithm there, you want them to notice those numbers are three apart. And so, there's this work that begins early. So, I think that's part of it. I think another part of it is helping people just reflect on their own learning experiences. What were your learning experiences with basic facts? And even if they liked the speed drills, they oftentimes recognize that it was not well-liked by most people. And also, then, they really didn't learn strategies. So, I feel like we have to be showing that we're not taking something away, we're adding something in. They are going to become automatic with their facts. They're not going to forget them because we're not doing this memorizing that leads to a lot of forgetting. Andbonus, they're going to have these strategies that are super useful going forward. So, to me, those are some of the really strong speaking points. I like to play a game and then just stop and pause for a minute and just say, “Did you see how hard it was for me to get you quiet? Do you see how much fun you were having?” And then I just hold up a worksheet (laughs). I'm like, “And how about this?” You know, again, that emotional connection to the experience and the outcomes.
Mike: That is wonderful. Since you brought it up, let's talk about replacements for worksheets and timed tests.
Mike:所以,你提倡游戏说,an activity-based approach. I think that what I want to try to do is get really specific so that if I'm a classroom teacher, and I can't see a picture of that yet, can you help paint a picture? Like what might that look like?
Jenny: I love that question because [there are] lots of good games and lots of places. But again, like I said earlier, this thinking really deeply about what game I'm choosing and for what. What do my students need to practice? And then being very intentional about game choice is really important. So, for example, if students are working on their 10-and-some-more facts, then you want to play a game where all the facts are 10-and-some-more facts. That's what they're working on. And then maybe you mix in some that aren't. Or you play a game with that and then they sort cards and find all the solve the 10 and more, or [there are] lots of things they can do. They can play concentration, where the fact is hidden and the answer is hidden and things like that. So, you can be very focused. And then when you get to the strategies, you want to have a game that allows for students to say, allow their strategies.
Jenny: So, I'm a big fan of, like, sentence frames, for example. So, [there are] games that we have in ourMath Fact Fluencybook that are in other places that specifically work on a strategy. So, for example, if I'm working on the pretend-to-10 strategy, I like to play the game fixed-addend war, which is the classic game of war, except, there's an addend in the middle, and it's a nine, to start. And then each of the two players turns up a card. So, Mike, if you turn up a seven, then you're going to explain how you're going to use the pretend-to-10 strategy to add it. And I turned up a six, so I'm going to, I'm going to do this, then, I'll—you can do it. So, I turned up a six. So, I'm going to say, “Well, 10 and six is 16, so nine and six is one less, 15.” I've just explained the pretend-to-10 strategy. And then you get your turn.
Mike: And I'd say, “Well seven and 10—I know seven and 10 is 17, so seven and nine has to be one less, and that's 16.”
Jenny: Yeah. So, your total's higher than mine, you win those two cards, you put them in your deck, and we move on. So, that's a way to just practice thinking through that strategy. Notice there's notimefactor in that. You have a different card than I have. You have as much time, and we're doing think-aloud. These are all high-leverage practices.Thenwe get to the games where it's like, you might turn up a six and a five where you're not going to use the pretend-to-10 strategy for that. You've got to think, “Oh, that doesn't really fit that strategy because neither one of those numbers is really close to 10. Oh, hey, it's near a double, I'm going to use my double.” So, you sequence these games to—if you start with one of those open-ended games, it might be too big of a jump because students aren't ready to choose between their strategies. They have to first be adept at using their strategies. And once they're adept at using them, then they're ready to play games where they get to choose among the strategies.
Mike: So, you're making me think a couple things, Jenny. One is, it's not just that we're shifting to using games as a venue to practice to get to automaticity. You're actually saying that when we think about the games, we really need to think about, “What are the strategies that we're after for kids?” And then make sure that the way that the game is structured, like, when you're talking about the pretend-to-10, with the fixed addend. That's designed to elicit that strategy and have kids work on developing their language and their thinking around that particularly. So, there's a level of intent around the game choice and the connection to the strategies that kids are thinking about. Am I understanding that right?
Jenny: That's it. That's exactly right. That's exactly right. And a huge, a lot of intentionality so that they have that opportunity and a no-pressure, a low-stress, think through the strategy. If they make a mistake, their peer or themselves usually correct it in the moment, and they get so much practice in. I mean, imagine going through half a deck of cards playing that game.
Jenny: That's 26 facts. And then picture those 26 facts on a page of paper. And then, and again, in the game that you've got the added benefit of think-aloud, and then you're hearing what your peer has said.
Mike: You know, one of the things that strikes me is, if I'm a teacher, I might be thinking like, “This is awesome, I'm super excited about it. Holy mackerel, do I have to figure these games out myself?” And I think the good news is, there's a lot of work that's been done on this. I know you've done some. Do you have any recommendations for folks? There's of course curriculum. But do you have recommendations for resources that you think help a teacher think about this or help a teacher see some of the games that we're talking about?
Jenny: Well, I'm going to start with myMath Fact Fluencybook because that is where we go through each of these strategies, each of the foundational facts sets and the strategies, and for each one supply a game. And then from those games, they're easily adaptable to other settings. And some of the games are classic games. So, there's a game, for example, called “Square Deal.” And the idea is that you're covering a game board, and you're trying to make a square. So, you get a two-by-two grid taken, and you score a point or five points or whatever you want to score. Well, we have that game housed under the 10-and some-more facts. So, all the answers are like 19, 16, 15, and the students turn over a 10 card and another card, and if it's a 10 and a five, they get to claim a 15 spot on the game board.
Jenny: Well, that game board can be easily adapted to any multiplication fact sets, any other addition. I like to do a Square Deal with 10 and some more, and then I like to do Square Deal with nine and some more. There's my effort, again, to come back to either pretend-to-10 or making 10. Where they're like, “Oh, I just played 10 and some more. Now we're doing the same game, but it's nine and some more.” So, I feel like there's a lot of games there. And there is a free companion website that has about half of the games ready to download in English and in Spanish.
Mike: Any chance you'd be willing to share it?
Jenny: Yeah, absolutely. So, you can just Google it. TheKentucky Center for Mathematicscreated it during COVID, actually, as a gift to the math community. And so, if you type in “Kentucky Center for Math” or “KCM math fact fluency companion website,” it will pop up.
Mike: That's awesome. I want to ask you about one more thing before we close because we've really talked about the replacement for worksheets, the replacements for timed tests. But there is a piece of this where people think about, “How do I know?” right? “How can I tell that kids have started to build this automaticity?” And you make a pretty strong case for interviewing students to understand their thinking. I'm wondering if you could just talk again about the “why” behind it and a little bit about what it might look like.
Jenny: So, first of all, timed tests are definitely a mistake for many reasons. And one of the reasons—beyond the anxiety they cause—they're just very poor assessment tools. So, you can't see if the student is skip-counting or not, for example, for multiplication facts. You can't see if they're counting by ones for the addition facts. You can't see that when they're doing the test, and you can't assume that they're working at a constant rate; that they're just solving one every, you know, couple of seconds, which is the way those tests are designed. Because I can spend a lot of time on one and less time on the other. So, they're just not, they're just not effective as an assessment tool. So, if you flip that—let's say they're playing the game we were talking about earlier, and you just want to know, can they use the pretend-to 10 strategy?
Jenny: That's your assessment question of the day. Well, you just wander around with a little checklist (chuckles), you know? Yes, they can. No, they can't. And so, a checklist can get at the strategies, and a checklist can also get at the facts like how well are they doing with their facts? So, once they do some of those games that are more open-ended, you can just observe and listen to them and get a feel for that. If they're playing Square Deal with whatever fact, you know. So, what happens is you're, like, “I wonder how they're doing with their fours. We've really been working with their fours a lot.” Well, you can play Square Deal or a number of other games where that day you're working on fours. The fixed-addend war can become fixed-factor war, and you put a four in the middle. So adaptable games and then you're just listening and watching.
Jenny:如果你不接受这种方式, then they can be playing those games, and you can have students channeling through where you do a little mini-interview. It only takes a few questions to get a feel for whether a student knows their facts. And you can really see who's automatic and who's still thinking. So, for example, a student who's working on their fours, if you give them four times seven, they might say, “Twenty-eight.” I call that automatic. Or they might, they might do four times seven, and they pause, and they're like, “Twenty-eight.” Then I'm like, “How did you think about that?” And they're like, “Well, I doubled and doubled again.” “Great.” So, I can mark off that they are using a strategy, but they're not automatic yet. So that to me is a check, not a star. And if I ask, “How did you do it?” And they say, “Well, I skip-counted.” Well then, I'm marking down the skip-counted. Because that means they need a strategy to help them move toward automaticity.
Mike: I think what strikes me about that, too, is, when you understand where they're at on their journey to automaticity, you can actually do something about it as opposed to just looking at the quantity that you might see on a timed test. What's actionable about that? I'm not sure, but I think what you're suggesting really makes the case that I can do something with data that I observe or data that I hear in an interview or see in an interview.
Jenny: Absolutely. I mean this whole different positioning of the teacher as coaching the student toward their growth, helping them grow in their math proficiency, their math fluency. You see where they're at and then you're monitoring that in order to move them forward instead of just marking them right or wrong on a timed test. I think that's a great way to synthesize that.
Mike: Well, I have to say, it has been a pleasure talking with you. Thank you so much for joining us today.
Jenny: Thank you so much. I am again thrilled to be invited and always happy to talk about this topic.
Mike: This podcast is brought to you by The Math Learning Center and the Maier Math Foundation, dedicated to inspiring and enabling individuals to discover and develop their mathematical confidence and ability.