More Math for More People
CPM Educational Program is a non-profit publisher of math textbooks for grades 6-12. As part of its mission, CPM provides a multitude of professional learning opportunities for math educators. The More Math for More People podcast is part of that outreach and mission. Published biweekly, the hosts, Joel Miller and Misty Nikula, discuss the CPM curriculum, trends in math education and share strategies to shift instructional practices to create a more inclusive and student-centered classroom. They also highlight upcoming CPM professional learning opportunities and have conversations with math educators about how they do what they do. We hope that you find the podcast informative, engaging and fun. Intro music credit: JuliusH from pixabay.com.
More Math for More People
Episode 4.11: Where Joel and Misty talk about cookies and continue their conversation with Dr. Jenny Bay-Williams
Join us at the CPM 2025 Teacher Conference in sunny San Diego, featuring an inspiring keynote by Dr. Tyrone Howard. With sessions led by CPM authors, professional learning team members, and experienced teachers, this conference promises to equip you with practical classroom strategies and activities. Plus, don’t miss the pre-conference day with seven diverse options covering leadership, coaching, inclusion, and more.
Then Joel and Misty celebrate National Homemade Cookie Day, indulge in some light-hearted fun with us as we reminisce about our favorite cookie recipes.
And Part 2 of their conversation with Dr. Jenny Bay-Williams about math fluency. If you missed part 1 of the conversation, head back to Episode 4.10 (Sept 17, 2024) to give it a listen.
You can connect with Dr Bay-Williams on X: @JBayWilliams
Send Joel and Misty a message!
The More Math for More People Podcast is produced by CPM Educational Program.
Learn more at CPM.org
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Speaker 2:Announcements, announcements. It's time for some announcements. Doop, doo-doop. All right, that's an announcement song, it's not that great, but I want to let you know that registration is open for CPM's 2025 Teacher Conference. This year, the Teacher Conference will be on February 22nd and 23rd 2025, and it's going to be in beautiful San Diego, california, this year. So remember, the CPM Teacher Conference features all kinds of classroom-ready ideas and strategies and activities that you can use right away. There's sessions led by CPM authors, professional learning team members and teachers, and, additionally, you can also sign up for the pre-conference if you'd like.
Speaker 2:The pre conference is on Friday, february 21st. It is a lovely lead-in to the main conference. We're going to have seven different options this year, including leadership, coaching, inclusion, merging, multilingual learners, equity and a session on the California math framework. Our keynote speaker this year is Dr Tyrone Howard. He is a Pritzker Family Endowed Chair and Professor in the School of Education and Information Studies at UCLA, and he is also the former Associate Dean for Equity, diversion and Inclusion. Dr Howard is one of the most frequently cited scholars in the nation on issues tied to race, equity and educational opportunity. He has been listed by Education Week as one of the 200 most influential scholars in the nation informing educational policy practice and reform to cpmorg backslash conference. Click register and you can get in on the early bird prices. We're excited and we'll hope to see you there. Today is October 1st 2024. True Good. What is the national day of?
Speaker 1:It's National Homemade Cookie Day.
Speaker 2:Oh my goodness, after all of the difficulties we've been having trying to get this recorded, I could use some homemade cookies, right.
Speaker 1:I do like cookies. I actually last night I was cleaning out cupboards and I found two no three uneaten packs of Girl Scout cookies. Not homemade cookies, I understand, but just think of the restraint that I've had to put on myself. Just sitting in your cupboard Just sitting in the cupboard. And now it's like a treasure that I get to open and have it's like how many months were they in their cupboard?
Speaker 2:Are they still good?
Speaker 1:I bought them from our coworker, Jocelyn, her daughter, about three or four years ago. Oh wow, they're Girl Scout cookies. They got to be good.
Speaker 2:The fact that they can live in your cupboard for several years and still be good does not make them sound more delicious.
Speaker 1:I think it's. I think it's gonna be good.
Speaker 2:I'm excited to try all right, well, we'll report back later. Yeah you, the opposite of homemade, because homemade cookies would never survive in your cupboard.
Speaker 1:For three to four years. The last homemade cookie that I made was a peanut butter cookie and it was flourless and it was delicious. It was really good well, cool, yeah, I've.
Speaker 2:So my grandma, my favorite cookies. I had two favorite cookies growing up when I was a kid. One was cowboy cookies, which were basically like fancy oatmeal chocolate chip cookies, and my other favorite cookie was this no-bake chocolate and oatmeal cookie which had peanut butter also.
Speaker 1:That would be good too, yeah.
Speaker 2:I saw they actually sell those in the store too. I always had them homemade. I didn't know, they were like cookies that you could buy, but I have seen them and I have bought them.
Speaker 1:That's awesome, I used to get tricked. That's awesome, I used to get tricked. I had a friend who would make chocolate like a chocolate cookie and instead of chocolate chips would put raisins in it. I was like, who puts a raisin in a chocolate cookie? And it would trick me.
Speaker 2:They would have raisins and chocolate or just like it was a chocolate, it was a chocolate cookie, and then it has-. Like a chocolate flavored cookie, like the whole cookie is chocolate and then raisins were the condiment.
Speaker 1:Do you call?
Speaker 2:it a condiment. Well, that would be addition.
Speaker 1:We'll call it yes, it could be addition, huh interesting.
Speaker 2:I mean, I like chocolate covered raisins, but making it into a cookie seems strange it was odd.
Speaker 1:I didn't like it.
Speaker 2:That does seem it does seem I haven't made any homemade cookies in a while. Have you made any homemade cookies in a while, after a?
Speaker 1:while, but it is. I just made my first soup of the season, so perhaps it's time for cookies yeah, yeah, okay, you consider cookies as seasonals. I do. I don't do a lot of summer cookies. Feels more wintery fall-ish to me, yeah.
Speaker 2:Those comfort foods for winter. What Are you going to make cookies today?
Speaker 1:Absolutely.
Speaker 2:All right, what kind of cookies are you going to make?
Speaker 1:I think I'm going to try. I don't know what I'm going to try. I was going to make something up right there, but I think I'll just go into my recipe book One of my favorite cookies. I used to be a sales person for Pampered Chef and I got a lot of good materials like baking stones and stuff like that, and two. I got a lot of great recipes. So I'll go into my Pampered Chef Rolodex it's literally index card Rolodex. I'll pick one of those. How about you? Okay, Are you going to celebrate?
Speaker 2:Probably not.
Speaker 1:All right.
Speaker 3:Well, I'm not going to have homemade cookies, that's for sure.
Speaker 2:That'd be way, way too many cookies for me. I'm not a big you know cookie fan to begin with, gotcha, but I might have a cookie. Okay, I like it. I can find one that I like.
Speaker 1:Excellent.
Speaker 2:All right. Well, hopefully everyone else enjoys homemade cookie. Please do, okay. Okay. So this week we have part two of our conversation with Dr Jenny Bay Williams from the University of Louisville in Kentucky, and if you missed part one, then please go back and take a listen to our podcast from two weeks ago so that you can hear part one of our conversation with Dr Bae Williams about fluency. Here you go, part two.
Speaker 1:So do you think in like basic fact instruction then? So I'm hearing about the games and stuff like that. Oh, I noticed my students aren't getting facts so I should stop and take a day to play games. Or do you play a game in the middle of the lesson? Or what type of instruction strategies would you say?
Speaker 3:So I like for strategy instruction, because you're really trying to teach strategy to address the facts that they're working on. So, as an example, let's go with the doubling strategy so you could show like images. I had these images from the grocery store of the little cheeses that come in six packs, so six times seven, just by coincidence, because we were already talking about it but they come in a six pack.
Speaker 3:So so, if you have. But I'm going to go to a different strategy. Okay so, because now you have a group of six, so you've. So let's say, they know, they don't know their fours facts, but they know they're double. So you have two bags. So there's like maybe an image on screen that has two of the bags of six If you can picture two bags of six on a screen of any objects, it doesn't matter and then they're like there's 12.
Speaker 3:And then the very next screen has four bags of six. And so you pause and you ask how many little rounds of cheese? Students say 24. Well, how did you think about that? And they'll say, well, I skip, counted 6, 12, 18, 24. And then somebody will say, well, I just doubled the last one. Oh, okay. So then we do this again. But now there's some other object. The first screen has two of the groups could be boxes of crayons that come in eight. So there's two of them. And then the next screen has four of them, and so they're learning the doubling, and then we can do the same thing. Getting back to the threes and the sixes, where there's three on the screen and then there's six on the screen. There's three on the screen and there's six on the screen. So they're saying oh hey, what are you noticing? Well, if I know my three facts, I can use them for my six facts. Are you following this?
Speaker 3:It's hard to do without the visuals, but I think, teach that. And then I like to play a game that actually is very focused on that fact we're working on before we just do an all-out game that uses all the facts. So a game, for example, that I've included in my book is Trios, which is basically connect four, but it takes too long to get four in a row so it's just three in a row. So the name is Trios and let's not stop with one. Let's just keep getting as many trios as you can, as much as the time allows. So a teacher might have eight minutes for them to play the game. That's enough that everybody's playing the whole eight minutes. So trios, they can block their partner or not.
Speaker 3:No-transcript. So when they roll a dice, they're multiplying it by six. But they're thinking, oh, I'm going to use that idea that I know my three's fact and I'm going to double it. You can give students sentence frames to work on that, but they're going to practice that fact unless they just remember it. So then when they see six times seven, they might say 42, because I just saw somebody else rolled that two turns ago. I remember it. Or they might forget six times seven and think oh, three times seven, I'm going to double it because the game board is six facts, right? So they're just going to be practicing their six facts by using the doubling strategy. So you're tearing up the strategy of doubling with the six facts and then, I don't know, in two weeks let's play that game again. Sure, and maybe a month later we'll play that game again. Ooh, let's pull that game out, but this time it's got the seven facts on it.
Speaker 3:Now what strategy are you going to use? And so you're like just interweaving. They know how to play that game, they like that game, they got beat the last time, they want to win the next time and then you're just changing out the facts. So I think you help them see an idea, not memorize them.
Speaker 2:See an idea, not memorize, but see an idea visually, through representations, and then they practice it, talking through it aloud, and just practice, practice, practice through that interwoven gameplay over time and and I could see that connecting to a couple of different things that teachers might be doing right, what you were describing at the beginning is a lot like a form of number talks or dot talks, right, right, where you're looking at different things and making those connections and that you could use as a classroom opener and then interweaving this, games at different times throughout and not just stopping. Great, we're gonna spend two weeks on math facts now and, before we do anything else, making it a barrier to the rest of what we're doing.
Speaker 3:Exactly when you just take that time out. First of all, not everybody needs that time, but also it feels like you're remediating them and what you're really trying to do is help them just bring to automaticity the facts that aren't there yet. And so most of their facts are there. We got to get all of them there because we're going to be just using these facts so often in our grade level work. So we're just going to work on this strategy and we're going to practice it, and then we're going to keep coming back to it and set a goal for how often you want to check in on how they're doing with their automaticity and which facts they're really struggling with.
Speaker 2:Yeah, I wrote down one word here. I wrote down confidence. Right, that it's not necessarily the speed at which I'm doing it at this moment, it's also building that confidence, which then not going oh, three times six and then doubling, that's just the process. But I can do that more quickly and with more confidence as I practice it.
Speaker 3:That's right and when you can do it with competence and you're efficient at doing the doubling strategy so that it doesn't distract you from the other thing you're doing. If you have to go over here and really think about four times six, well then you've lost track of this other thing that you're really working on. So you want to get to where the production of that strategy is automated so that you can do it over time, which happens over time.
Speaker 3:You have to distribute that practice over time to get to automaticity. It's not going to happen quick and likely. If there are students in middle and high school that are struggling with facts, it is because that's been like the September, September, October popular topic and so, honestly, when they don't know what the next year, it isn't that they forgotten it since May. They might not have been working on it since the year before.
Speaker 1:So no wonder, right so for anything that we want.
Speaker 3:Automaticity, where we want students to be adept at something, be able to revisit it over time through gameplay and through just quick problems here and there, just helps us continue to have that, that fluency with it, that we recognize it, we know what our choices are, we can, we're adept at what method we want to use to solve it.
Speaker 1:I like that, and it also makes me think about flexibility and about so. There's not just one way to look at those patterns or the solutions to that game, right, like they could think of it in a variety of different groupings or a variety of different ways.
Speaker 3:No-transcript. And isn't the calculator, because they're just going to encounter it too often and both of those other things are too distracting to what they're trying to focus on.
Speaker 2:Whatever the problem is, yeah, yeah, and I see that also helping build their own, their own ownership of their ideas, and sharing math authority. Right, if I get to choose how I want to do it, it's not just memorize, it's not just know it. Those parts haven't worked for me, for whatever, for this particular one. Oh, here's a way that I like and I'm going to do it that way and and then build my understanding around that.
Speaker 3:Exactly, it's their go-to way. And then simultaneously they're building their confidence and their competence. So they're like all right, now I have a way that I can multiply this. Now I'm not going to get nervous when I see six times seven because I've got my way. I got my way. I might even have a second way. That's like my backup, but I got my way. I might even have a second way. That's like my backup, but I got my way. Now I don't have to sit here with my fingers under the table embarrassingly trying to skip count. That takes us back to the bigger problems, right? So if you have four times two and three-fourths, you could double you could do.
Speaker 3:That's not a great problem for that strategy but you could use a doubling strategy to solve it because the four you could do two groups of two and a fourth and then double again. Again, it depends on how flexible you are with your fraction work, because we haven't really done a lot of flexibility work with fractions, but as soon as you said it, I had to look left and think about the three-fourths part like that was right
Speaker 2:there, yeah, yeah, totally what are some of the other things we said we were going to talk about. Jenny, can we mention to you that we haven't got.
Speaker 3:We've been all over the place how about what flexibility looks like with solving for x?
Speaker 2:oh, I love that, I love that this is one of my least favorite things to ever see on it. Well, I have many unfavorite things to see on teachers walls, but one of them is the rule of algebra that what you do on the left you must do on the right. And I'm like, unless you do the, the equation are the same amount.
Speaker 3:Yeah, and so that helps us. But where we go awry in terms of fluency is we think about solving for X as sort of going through the order of operations. Now, like you've got to eliminate parentheses first which, by the way, that is applying the distributive property, to call it the correct thing and then, after we do that, the next thing we're going to do is add and subtract on both sides, and the last thing we're going to do is we have the sequence that we tell students to follow. But that's not the fluency approach, that's sort of an algorithmic approach. So a fluency approach would be to let them know you have four options of things you can do. Actually, I'm going to say five things, five things that you can do and you get to pick from that menu.
Speaker 3:So the first thing on the menu is use relational understanding like reason, think backwards. So if you have like 2x plus 1 equals 11, do you really need to subtract 1 from both sides and then divide by 2 to solve it? You don't. So could we just reason through that If 2x plus 1 equals 11, 2x has to equal 10, x has to equal 5. And here we see back again the conceptual understanding and the procedural fluency hand in hand, because you have to understand that 2X that term means two groups of X, right, that's the meaning of it. And so two fives is 10.
Speaker 3:Can you use relational understanding? Do it in your head? Great, go for it. Do it that way. That's the most efficient. That doesn't work. You got four actions you can take, still on the menu. So one of them is distribute is apply the distributed property, which in the example I gave doesn't apply, in fact the one that I gave you. Only one of them really makes sense, which would be to subtract from both sides. But let's do a different one. Let's say, parentheses, x plus one half equals 25. Okay, I could solve that during relational understanding. If I get any longer, no listener is going to be able to track the problem. Let's just say you don't want to do it relational understanding. So you have this thing in parentheses, it's multiplied by five and you got 25 on the other side. So one option is to apply the distributive property. Then you had to multiply that five times that one half.
Speaker 2:So yeah, who wants to do that? What middle or high school student?
Speaker 3:wants to end up with five halves after they've used the distributive property.
Speaker 3:So, if they would have noticed that one of the things on the menu is to divide both sides by the same quantity. They could divide both sides by five. Now they have x plus one half equals five, at which point they can go back to the menu and do relational understanding or decide to subtract one half from both sides. So it's that sort of menu approach. That is again confidence and competence. So you can choose. If you love applying the distributed properties, your first step, you go for it. But what happens is and the teacher can have students who started the problem differently share their way and compare. That's where the flexibility comes in. Which way worked out better for this problem? Why did it work out better? What's your takeaway for the next time you solve a problem? That will help you decide whether you want this to be your first step or that to be your first step. So that is the flexibility that is important to being fluent at solving equations.
Speaker 2:And one of the things I loved about the way you said that is having two students who did it different ways, share what they did, talk about what they did. What do you like other students? As opposed to the teacher going up and saying, well, here's a way you could have done it differently and showing them a new way that now they take as oh, that's the way I was supposed to do it. Right. It's sharing that authority again and really using the student voice in the classroom.
Speaker 3:A hundred percent, and part of it is choice. But there's also the point at which choice might not be efficient, where we get to come in and say, all right, which of these is more efficient? Make a case right for which one's efficient, and the answer could be both. Or they could make a case that this other way was two steps shorter, so it's more efficient. Right, so that we're having that as part of our own development of fluency, recognizing that if this problem can be solved in I don't know 15 seconds using three steps, that that is a quote-unquote better method than using seven steps over here in two minutes. That is sort of the norm in the field. Right, so that follows the norms, without putting time pressure on the students, but just helping them weigh in on. Oh yeah, this way is way too clunky. This way is more straightforward, less prone to error. This is a quote-unquote better method and the number one's better than the other, and sometimes they're not.
Speaker 3:So it's just a good dialogue to engage the students in, because you're teaching them to think like a mathematician.
Speaker 1:Exactly. It makes me think of like quadratics and we think about the different forms and then go into solving. Do I want to have it in standard form or factored form or text form or complete the square? However, we want to do it Quadratic formula and having the students think about that.
Speaker 3:Exactly, exactly, and so here we are, back up to the higher math that goes all the way back to basic facts when they're choosing. How, then, you recognize characteristics that a fluent person knows? I see these characteristics, so I'm going to use this strategy.
Speaker 2:What does a problem look like? Looking at different ways that people solved it or did it and thinking about, well, what makes sense to me? What do I like better? Why do I like it better? Yes, sometimes, oh, that was more efficient. I didn't see that I still had the skills to do it this other way or not, but having then be able to expand my toolbox and my repertoire of things that I've seen and understand to be able to do, yeah, and on that note, when you say repertoire of things I've seen and understand, we have to be super intentional that they see and understand the methods that are useful.
Speaker 3:So there can be students who never want to divide by five on that problem that I gave, because they are comfortable with applying the distributive property, because that's what they were told to always do first, if you see parentheses. So there's this discomfort with the freedom of being able to choose from a menu. So they have to have enough experiences with the methods that are possible so that they will feel like choosing from that menu. They have to feel comfortable with what's on the menu.
Speaker 2:A number of times I've seen students solving some. Quadratic was like something in parentheses squared equals 36 and they're multiplying it out.
Speaker 3:I'm just like oh, what are you doing? What are you doing?
Speaker 1:But this is what I'm supposed to do.
Speaker 2:Yeah.
Speaker 3:Notice the features of the problem Like just pause, that's going back to this hurry up piece about fluency.
Speaker 3:Fluency actually typically starts with a pause, like let's pause and take in what we're noticing. What are we noticing about this problem Features? Do we see any shortcuts? Is there anything that gives me a hint into how I should approach this? It's those that pause at the start that tend to be the really have that strategic competence, you know. So that's why we're not trying to say hurry up, get busy, do this. That's sort of like a whole different direction that was maybe appropriate before technology ever existed, if you were training people to be I don't know, accountants or something where they were competing by hand. But press to be fast now doesn't really play out in terms of the discipline.
Speaker 2:Yeah, it's disappointing that there's so much about being good at math as means being a quick calculator. When I hear adults say I'm not very good at math as means being a quick calculator, you know, when I hear adults say I'm not very good at math and I ask them why they're like I could never memorize my math facts that much and it's ah.
Speaker 3:Or they struggle with understanding fractions.
Speaker 2:I'm like that's because your brain was never really designed to understand things that aren't whole. We like whole, number things. So it is a stretch, it is a challenge.
Speaker 1:It is a stretch.
Speaker 3:I have sort of I build a case when I work with teachers for why not? Why should you not use time tests? But one of them really is that it sends a message that being fast is being good at math, and that's just so far from the truth. If you watch people doing mathematics in their careers, they're making good choices. That choice might be using technology. That choice might be solving something mentally. That choice might be pencil and paper, getting away for hours, but it isn't. But it's very intentional. It's this intentionality that comes in the discipline. That is why the time test is such a misdirect into what's really important.
Speaker 3:Automaticity is important. I don't want to A hundred percent.
Speaker 2:Yeah, yeah.
Speaker 3:They need the automaticity Right, but the automaticity is something that can be better assessed in other ways.
Speaker 2:Mm-hmm, mm-hmm. Yeah, oh, my goodness, thank you so much for this conversation.
Speaker 3:I think we're probably going to wrap it up, I think that we got to all the things, at least that I know that I remember we were going to talk about and we've talked about so many more, so we really appreciate you coming and having this conversation with us.
Speaker 3:We're glad Joel was able to get his volume and his microphone working and thank you so much. Thank you for having me. I'm happy to talk about these ideas at the middle and secondary level and how they really whether it's basic facts or it's quadratics that this fluency focus is critical to students' competence.
Speaker 1:do at CPM is with our work. We do some modules to help teachers work through issues with students with exceptionality, so strategies and things like that. And we got to listen to your talk and your research when you spoke at Mead about your facts and fallacies around fluency, so we'd love to talk to you again and continue the conversation. One last thing, one thing that we've been starting this season on our podcast is a math joke.
Speaker 2:I'm going to put you on the spot.
Speaker 1:I'm going to put you on the spot. Do you have a math joke? And if you don't, that's okay. Math's not that funny.
Speaker 3:I don't have a math joke.
Speaker 2:And.
Speaker 3:I'm terrible when I'm being put on the spot.
Speaker 2:Sorry, we can end that part the.
Speaker 3:thing is my colleague that was in my office. I had to shoo him out when you all came in. He has an endless and I'm talking infinite jokes. He was a high school math teacher and he literally, if you just even specifically name a topic, he's got one.
Speaker 1:You're good. I appreciate you playing along.
Speaker 3:I'll be ready for the next podcast.
Speaker 2:Awesome, all right, thank you so much. So that is all we have time for on this episode of the More Math for More People podcast. If you are interested in connecting with us on social media, find our links in the podcast description, and the music for the podcast was created by Julius H. It can be found on pixabaycom. So thank you very much, julius. Join us in two weeks for the next episode of More Math for More People. What day will that be, joel?
Speaker 1:It'll be October 15th, national Cheese Curd Day, and, being from the Midwest myself, I do love me some cheese curds. I know that my favorite is in my memory anyway is probably fried cheese curds at the Minnesota State Fair. I've had them for breakfast, for lunch, for snacks, for dinner. They're so good there. But even here now, living in Utah, there's many cheese places. Here you can buy curds local curds in the store, in the grocery stores, or you can go to the dairy themselves. Curds in the store, in the grocery stores, or you can go to the dairy themselves. There's even a program where you can learn to be a cheesemaker and I've been thinking that I might apply for that as another thing to put on my recipe be a cheesemaker, no-transcript.