# 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 4.10 - Where Joel and Misty talk with Dr. Jenny Bay-Williams about math fluency

Ever wondered what makes a Monte Cristo sandwich so special? Join us on the More Math for More People podcast as we celebrate National Monte Cristo Day with a fun-filled exploration of this delectable sandwich and its (supposed) ties to Omaha (spoiler alert - not true). We reminisce about our own encounters with the Monte Cristo, including the sad demise of a local favorite, and even indulge in some playful speculation about its connection to Alexandre Dumas' "The Count of Monte Cristo."

Next, we're thrilled to welcome Dr. Jenny Bay-Williams, a distinguished professor from the University of Louisville, who joins us to unravel mathematical fluency. Jenny shares her wealth of experience in math education, emphasizing that fluency is much more than speed or automaticity—it’s about reasoning, making smart choices, and choosing efficient strategies for problem-solving. Through examples of fraction addition, she demonstrates how true mathematical fluency involves navigating multiple approaches to find the best solution. This is part 1 of our conversation with her.

You can connect with Dr Bay-Williams on X: @JBayWilliams

To wrap up, we introduce CPM Educational Program's "Instructional Coaching Toolkit," a must-have resource for instructional coaches and leaders. You can purchase your own at shop.cpm.org!!

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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You are listening to the More Math for More People podcast. An outreach of CPM educational program Boom. An outreach of CPM Educational Program.

Speaker 2:Boom Okay. 17th of September.

Speaker 1:Yes.

Speaker 2:What's the national day today? It is National Monte Cristo Day. Monte Cristo Day like the sandwich, the sandwich. Okay. So this is going to be a callback to we were just talking about bowling from the last episode, right, and you and I were talking about how well we might not be able to go bowling because we'll be traveling to Omaha. Mm-hmm, mm-hmm. And I learned when I was researching places to eat in Omaha that Omaha is supposedly, like the home, the birthplace of the Monte Cristo sandwich no.

Speaker 2:That's what I saw Are you serious right now. I am serious right now. Oh my gosh. I love a Monte Cristo sandwich so much. We have to also have a Monte Cristo sandwich while we're in Omaha. Oh my gosh.

Speaker 1:That is going on the checklist for sure.

Speaker 2:I think this is irony. I think this doesn't qualify just as coincidence. I think this is actual irony. I'm not very good at those things do you enjoy a monte cristo? I haven't had a monte cristo in a long time, since I don't eat very much bread or gluten, but I the monte cristos as, as I remember, them are pretty delicious. Actually, I'm trying to decide if I've had one since university.

Speaker 3:I feel like they were one of the things that would be on the lunch menu.

Speaker 2:Cristo sandwich and I never had had one before that, because Monte Cristo is the one. Wait, I'm confused. Is Monte Cristo the one where there's cheese and some kind of meat, meat, and then they like put it in like egg and fry it? Is that the one?

Speaker 3:Yeah.

Speaker 2:What's the one where they have the pumpernickel and the like Thousand Island dressing?

Speaker 1:and sauerkraut on it. That's the Reuben. Oh Reuben, that's right. Totally different sandwich.

Speaker 2:Yeah, well, I know, but they're also sandwiches I never had until I was an adult, because they weren't ones we had as I was a kid.

Speaker 1:I love a Monte Cristo and local establishment here in Salt Lake served my favorite Monte Cristo Like they fried it so good and the fresh. Like they made the berry jam that went with it. There's berry jam on it, oh yeah, and it was so delicious.

Speaker 2:I don't think they made it with berry jam. In went with it. There's berry jam on it, oh yeah, and it was so delicious I don't think they made it with berry jam in Northwestern's tiny halls.

Speaker 1:Well, I was so sad because last time I went I went down to get the Monte Cristo and they changed the menu. What they said, we reduced our menu size and that did not make the list. We're only going to make it on special days. And I said, well, what's the special day? I hope like September 17th is a special day because it's Monte Cristo day, but I was so bummed Like it ruined my whole weekend.

Speaker 2:Apparently you weren't going and ordering it enough.

Speaker 1:I know it's my fault, yeah.

Speaker 2:I'm not sure I mean think about. Yeah, I'm not sure I've actually. I mean think about it. I'm not sure I've had one since university, which is a long time now, and clearly those were not gourmet Monte Cristo sandwiches, they were bulk made Cristo sandwiches. So yeah, monte Cristo is on the list for Omaha. I think it should be by the time everyone on our team hears this, it'll be too late for them to have Monte be.

Speaker 1:That's why everyone on our team hears this.

Speaker 2:It'll be too late for them to have a Monte Cristo. Let's hope they had it. Yeah, exactly, or they could just have one now. Yeah, wherever they are.

Speaker 1:Love that. Delicious sandwich days are kind of like I know I'll be celebrating, I know I'm going to have a Monte Cristo sandwich because it's a delicious sandwich and I can celebrate, and I love delicious sandwiches.

Speaker 2:And Monte Cristo just has like regular bread. There's no special.

Speaker 1:I don't know that it's a special bread, but it's because you fry it and things like that, it's very similar to the French toast when it comes out, you want a thicker bread.

Speaker 2:Yeah, it's your wonder bread, the French toast. When it comes out.

Speaker 1:You want a thicker bread. Yeah, it's your wonder bread, that's right, Well cool. Yeah.

Speaker 2:All right. Well, I don't know what else to say about Monte Cristo sandwich day.

Speaker 1:I can't even say anything because I'm thinking about it.

Speaker 2:Why is it called the Monte Cristo? Well, it has.

Speaker 1:French roots, and I think wasn't there like a count of Monte Cristo.

Speaker 2:Well, yeah, but I don't think it's named after him.

Speaker 1:That was like an Alexandre Dumas book Right, but he had all that time in prison, right, like he had a lot of time to think about stuff.

Speaker 2:He did spend a lot of time in prison, and so he probably thought about this delicious sandwich that he couldn't have. I don't think that is the actual. I mean I'm very skeptical about this story. Oh well. So we don't have a good explanation on your source.

Speaker 1:Well, it's got some French roots here, but this day was initiated to commemorate the long history of Monte Cristo sandwiches that continue to bless our taste buds for decades now. So like in 1910 is when the Monte Cristo kind of made its debut.

Speaker 2:Is it called Monte Cristo sandwich? I can't even spell Monte Cristo, what, okay? The AI overview says the Monte Cristo sandwich is named after the Count of Monte Cristo, an adventure novel by Alexander Dumas. Come on, I'm just saying the sandwich flavor profile pays homage to its French origins and some believe the name was Come on, I knew it. Oh, that's very vague.

Speaker 1:That is vague. I can't believe.

Speaker 2:It also says that the Monte Cristo sandwich is a variation of the Croque Monsieur, a French sandwich that was lightly served in Paris cafes in the 20s. Apparently, it's all from the 20s.

Speaker 1:There you go. I'm trying to click on. How was the Monte Cristo sandwich created and it won't open up, so I have no idea.

Speaker 2:But I think I just told the story that the Count was in prison and that's how it was created. Yeah, it doesn't explain in this AI overview why it would be named after the Count, but it does say also in the 1960s the sandwich was added to the menu at Disneyland's.

Speaker 3:Blue Bayou restaurant and became an American favorite.

Speaker 2:Yes, which just cracks me up. I mean, the AI overview is helpful, but it doesn't really tell me the why. Anyway, all right.

Speaker 1:Well, there's some interesting tidbits of how the Monte Cristo sandwich might be named or not and I hope all of you listening, we have not data checked, are going to go out and get a Monte Cristo sandwich.

Speaker 2:Might be named or not, and I hope all of you listening.

Speaker 1:We have not data checked, are going to go out and get a Monte Cristo sandwich.

Speaker 2:Yep, Enjoy your sandwich. So as a side note, we do want to clarify that Omaha is the home of the Reuben sandwich and apparently my conflation of Monte Cristo and Reuben sandwiches is pretty severe. So if you go to Omaha, don't ask for Monte Cristo, Go get a Reuben Any day, All right. So today we're going to have part one of a conversation with Dr Jennifer Bay Williams. Jenny Bay Williams has been a professor at the University of Louisville since 2006. She teaches courses related to mathematics teaching to pre-service teachers and practicing teachers and is frequently working in elementary schools to support mathematics teaching. Prior to coming to the University of Louisville, she taught in a variety of other places, such as Kansas, Missouri and Peru.

Speaker 2:Dr Bae-Williams is an internationally respected mathematics educator. She is a prolific author and popular speaker on topics related to effective mathematics teaching. Her work has focused on ways to ensure that every student understands mathematics and develops a positive mathematics identity. Her most recent work has focused on fluency and mathematics, communicating that it is more than learning facts and algorithms, but rather that it's about being able to reason and choose appropriate strategies. Her books on fluency and mathematics coaching are bestsellers.

Speaker 2:We're excited to have Dr Bae Williams here on the CPM podcast today to talk with us about her views on fluency. All right, well, while Joel is finishing figuring out how to get himself connected to us, we're just going to start because we can always we can always use AI to insert Joel later. I'll just we'll replace my voice with his voice asking one of the questions. That's pretty amazing. So we're here today with Jenny Bay Williams and, Jenny, you are a professor at the University of Louisville. I have that right. Excellent.

Speaker 2:We have co-workers who are in Louisville. I think one of them is at the University of Kentucky. Might done a lot of work with fluency. You've written some books and many articles, and fluency is definitely a question that comes up, or a topic of concern, I think, for a lot of teachers, and particularly teachers in CPM, because of the mixed space practice that we have and students are interacting with material over time. Then how do we figure out what fluency is right? And there's always these concerns around whether students know their basic math facts and what to do if they don't know them. So we're going to talk about several of those things today. So thanks for joining us on the podcast today.

Speaker 3:I'm excited to talk about all of those things. So thank you for inviting me.

Speaker 2:No, worries and Joel's just going to keep trying to figure out if he can talk to us or not, and we're just going to keep going. So we appreciate your time. The first thing I wanted to launch with is I think that a lot of times when we talk about fluency and what is it? What comes to mind is being able to I think of oh, how fast can you do your multiplication facts right? You have kids doing speed tests and different things, and that's this idea of fluency and I'm wondering what you think about that. I guess, or am I guessing, you don't agree with that.

Speaker 3:Yeah, I don't agree with that. But there is this need to be automatic with something so that you can recall things, information that you need, that you're going to be using without using a lot of thought. But that's not fluency, that's automaticity being automatic. So we want to be automatic with basic facts. And in middle and high school there's other things where automaticity supports the bigger reasoning. For example, equivalencies with a one half, recognizing that 13 over 26,. Hey, that's one half. How do you know? Without a lot of thought? You just recognize fraction equivalencies of fourths, halves and other things like recognizing, I don't know, pythagorean, triples or whatever. But there's these things where you see them so much, you recognize them and you just know. So there are. That's automaticity.

Speaker 3:But fluency is about being able to solve a problem using an efficient method. So I'm going to start with efficiency. So let's just take a fraction. We work on fraction fluency in middle school, right? So let's take two and three-fourths plus two and three-fourths just the same numbers. So we don't have trouble remembering the problem Two and three-fourths plus two and three-fourths just the same numbers. So we don't have trouble remembering the problem Two and three-fourths, two and three-fourths.

Speaker 3:So what you were saying, misty is that somebody would be really fast so they could go about doing the algorithm really really fast. That's not fluency, that is being, I don't know adept at using an algorithm. A fluent person is going to go oh hey, I have some options here for adding that problem. What are my options? And this way looks really fast for this problem and so, and all that happens like in it happens quickly.

Speaker 3:So for two and three-fourths plus two and three-fourths, what they might recognize is they know three-fourths and three-fourths is one and one-half and then add the whole-fourths is one and one-half and then add the whole numbers, add the one and one-half and they're done. They might just move one-fourth over from one fraction to the other and think, oh, that's three plus two and one-half. Again, they got that automaticity with fractions going. Those are then ones they can add in their head, and so the fluent person goes about a problem by looking at it, sizing it up and making a decision about how they're going to solve it. That's what somebody with fluency does. That's different than somebody just being really really fast. It's about making good choices, what we like to talk to middle and high school students about.

Speaker 1:Making good choices with math, you know so I have some questions.

Speaker 2:So I see that as, like one, you talk about the automaticity with fractions and the numbers and making some choices would rely upon a reasonable number sense, right that our ability to just I don't know know, it's just just to know that a half and a half is one, or just to know like that, and and so, which could be called basic facts, but I think it's also, it's that number sense, like I know my tens, I know how to group in fives. I have these like lots of different abstract and more representational ways of thinking about numbers. However, that is, is that? Do you see that?

Speaker 3:or Definitely so. There's more literature, research and discussion about decomposing. I'm just going to pick up on one of the things you were talking about. In, like kindergarten, can you break apart eight lots of different ways, five plus three and that sort of thing but just take like a fraction, like three-fourths right there. To do that strategy that I talked through, you have to think, oh, I could break that three-fourths apart into two-fourths plus one-fourths or one-half plus one-fourths. It's that flexibility and that number sense that allows you to be flexible in breaking numbers apart, putting them back together, knowing that you could apply the distributive property but you don't have to apply the distributive property as your first step in solving algebraic equations, solving for X. So there is this level of looking at number relations and seeing and using those number relations to efficiently solve a problem.

Speaker 3:So, which is harder to understand in the general and easier to understand an example. So if you want some clarification, just ask for an example.

Speaker 2:Well, and the other piece of what I was thinking about, as you were saying, that is that we talk about building procedural fluency from conceptual understanding. So when, in the example you were talking about, how would you and maybe you need a different example, that's okay Like, how would you apply the idea of procedural fluency to that? Because I think of, when I think of procedural fluency, I think, oh, you can do this calculation or this algorithm very quickly or well, but and I think that that teachers often get into that right, it means you can do the long division algorithm, it means you can do all these algorithms. And is that how you see it, or do you see it a little differently than that?

Speaker 3:I do see it that way. I think that the there's this important relationship where the conceptual understanding supports the fluency and the fluency then strengthens the conceptual understanding. So for example, with that two and three-fourths plus two and three-fourths, the more that students are applying this strategy of moving one-fourth over, the better they're getting at the skill of breaking fractions apart. But the more they're getting this idea of basically the associative property that they can take something from one add-in and move it to the other add-in, they still have an equivalent expression after a thing. Another great example is with subtraction. So if we had, let's just do five and three-fourths minus two and three minus I don't know, let's do no, I'm going to change the problem. Let's talk about subtraction and that relationship between conceptual understanding and procedural frequency. So if you take something like five and one-fourths minus I don't know, four and three-fourths, somebody with a conceptual understanding of subtraction understands that it is take away and it's also find the difference or compare. So they look at that problem and they say, oh, those fractions are close together. It's going to be easier to find the difference between four and three-fourths and five and one-fourth it's one-half. Maybe they picture a ruler or a number line. And so, again, that conceptual understanding is coming in and helping them visualize and solve the problem.

Speaker 3:A student with no conceptual understanding of fractions will see those fractions and follow an algorithm. They'll regroup from the five to get four and five fourths and subtract it and get their answer. And that's not fluency, because it took them a much longer time to do all of that work. They have no idea of knowing if their answer makes sense, and so that's the two supporting each other. The conceptual understanding allows for an efficient solution strategy and also to understand if the answer makes sense. And then just the very question of are you going to find the difference or use takeaway is supporting both their fluency and their conceptual understanding.

Speaker 2:Right, right, joel, can I hear you? Can you hear me? Takeaway is supporting both their fluency and their conceptual understanding. Right, right, joel, can I hear you? Can you hear me? Yes, can you hear us? I can hear you. Welcome to the program, joel.

Speaker 1:Thank you.

Speaker 2:Yeah, technical difficulties resolved.

Speaker 1:Okay.

Speaker 3:Cool, so I'm just fascinated by this, this whole idea, because there's so many it the way that I was supposed mathematicians are. As in research mathematicians or engineers or people who are using mathematics, they're trying to be creative about how they're putting things together in order to figure something out, so they're not really following like algorithmic stuff as the way their brain is engaged right. The algorithmic stuff is done usually with technology, so that decision making and creativity is oftentimes the insights into a better way to engineer something or create something or whatever.

Speaker 3:And so to squelch that in school is the exact opposite of what we want to be doing with students. We want to inspire them to take this problem in and come up with an efficient way to solve it, based on what you notice, what the features of the problem are.

Speaker 2:I was going to move to thinking about. So with that in mind, right, we CPM teachers are working with 6th through 12th graders. Mostly, and often I mean I encounter teachers, and they particularly in middle school and they have all this, or even high school. They're quite distressed that their students don't know their basic math facts or whatever that might mean. So I'm wondering, if I'm coming to a teacher and they're saying this to me, what are some things that you might suggest or how could I approach that? How can I help those teachers?

Speaker 3:So what you're sharing is common. I had an email as recently as yesterday where the message was that the middle school teachers are I think the phrase was using the common refrain of the students don't know their basic facts. How can they do fill in the blank, and so basic facts become like a barrier to doing higher math or other math. So they shouldn't be. And so what I want to say about basic facts is it's never too late for a student to learn the facts that they're struggling with. So that means if you're teaching middle school or high school, then don't give up on the fact that the student can learn that fact, because for their life, regardless of what profession they're picking, it's going to be a constant problem for them to not know eight times seven or six times nine or whatever. So that's the first thing is just commit to helping the student to learn those facts, because the facts are going to help solving systems of equations.

Speaker 3:Think of all the. If you're setting up a system of equations, you're looking for like a common factor or whatever, like they're just there all the time everywhere. So it's an investment of time that, even though it's listed as a third grade topic, it's worth the investment of time to come up with a way to teach it. The thing not to do is what we've always done, which is have them retry to memorize. Obviously, that didn't work for them in third grade, fourth grade, fifth grade it's not. It's a weak learning strategy in general, so that's not the way to go about it. So, and when teachers say this is a long answer.

Speaker 2:No, this is a great long answer.

Speaker 3:So the thing is is we have to become diagnosticians. So when we say a seventh grader doesn't know their facts, they do know facts. They know a lot of the facts. There's 100 of them. They likely know 80 of them, 85 of them, they might know 95 of them. There's just some that are getting in the way, but all it takes is a few facts that they don't know to all of a sudden really trip them up with the other work that they're doing. So we have to really figure out what facts they know and go from there.

Speaker 3:So that's my wisdom. And so, like using a test, a basic facts test, isn't insulting to an older child, it's really distasteful to anyone. But you can play a game of any sort. So that involves basic facts and you can observe the students and start to notice. With that age student they can do like flashcard sorts where they're going through and like self-identifying which facts they know and which facts they don't know, so that then you can engage them in their own strategic planning. For oh, six times seven trips me up, but I always know three times seven, aha.

Speaker 2:Mm-hmm.

Speaker 3:Aha, what's the relationship between three times seven and six times seven?

Speaker 2:Right.

Speaker 3:And if that becomes, is that what you want to use as your go-to, Joel? Is that is, that you're going to be your go-to. Right, great. So when you see six times seven and you're stuck, don't try. Skip counting, that's going to you're going to hurt your brain. Skip counting by sevens or six, Think oh, I know.

Speaker 3:my three times seven, I can double it. So I think we have to be really intentional with helping the student realize that they really can learn the facts that they don't figure out, what facts they don't really really know.

Speaker 3:And they come up with a good strategy to get them down, and then they'll be glad. And there's so many games I've written about. I have many of them in my basic fact book, which and many of those are available on the companion website free download in English and in Spanish. Yeah yeah, those games give an opportunity for ongoing practice. So, again, with this idea of the program of the practice over time, these games come in to do that. So if you're working with them to try to help them with their facts that they're struggling with and they come up with these ideas for their method they're going to use when they get stuck on one of their facts that is not automatic for them, then playing those games are enjoyable for the students.

Speaker 3:Some of them have a lot of strategy to them where they're blocking their partner or they're going to bump them off of the game board or whatever. But in the meantime they're getting a lot of. Every time they roll a six and a seven they're like what's six times seven? I forgot? Oh, yeah, but I know three times seven. So then that's where you get to that automaticity and they could for life always think three times seven when they see six times seven. But it becomes automatic. And that's basically. I mean that's me for a couple of facts like six times eight, I think, five, eights and one more, and I can do that in a split second. So I think that's what we have to do because we'll see the benefits in their fraction, fluency, they're solving equations, all that. It's worth that additional time to help them. Help the students sort out what facts are, you know, getting in their way and helping them sort it out.

Speaker 1:So do you think in some 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 2:That's all we'll have time for today, on part one of our conversation with Dr Jennifer Bay-Williams, and you can tune in in two weeks to hear the rest of our conversation, and we'll see you then. Today I have a very exciting announcement cpm. As you know, cpm educational program. We are a mathematics publishing company as well as a professional learning company, and this month we have our inaugural professional learning publication, our first publication designed solely for professional learning that you can purchase. It is the Instructional Coaching Toolkit. So I'll tell you a little bit about it here.

Speaker 2:The Instructional Coaching Toolkit is the launchpad for any instructional coach who wants to strengthen teacher's practice and students' learning while focusing on their own growth. This book is written with you, the instructional coach, as the protagonist. You will learn how to use coaching tools and coaching skills as you journey through a coaching cycle. Using the framework and tools in this book, you can help foster the cognitive dissonance necessary for teachers to challenge their practices, their beliefs and their ways of being. With your guidance and support, teachers can prioritize equity while setting rigorous goals aligned with NCTM's eight effective teaching practices and CPM's three pillars. This, in turn, will support students in embracing learning through the standards for mathematical practice. This essential resource represents the culmination of almost a decade's worth of thought and planning, started by a team of collaborative, visionary educators who recognize the positive impact that job-embedded coaching has on how mathematics is taught and learned. You can find the Instructional Coaching Toolkit in the CPM web store, which is at shopcpmorg, and you can purchase one for yourself. Cheers.

Speaker 1:Hello, it's Joel here. If you listened all the way to the end of the podcast hoping to hear this week's math joke, well, you will be disappointed. We've run out of submitted math jokes, so if you love this segment, then please send us a recording of your favorite math joke to cpmpodcast at cpmorg. Just give us your name, where you live and your math joke Easy peasy. We can't wait to hear them.

Speaker 2: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 and 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 1st, national Homemade Cookie Day, and I'm excited for this one too. I can remember as a kid my mom would leave me to my own devices a lot. So I remember I got a hold of my Mickey Mouse cookbook and I wanted to make some sugar cookies for when she got home. So I remember I made this sugar cookie. I wanted it to be the size of a baking sheet so I put it out on the baking sheet. I added green food colorings because I thought that would look good, and it did. It looked beautiful and when she came home to have some of the cookies and my mom's the best she tried it. She said it was delicious, but I bet she lost three teeth at least because it was so rock hard that cookie. I thought in my early days of making homemade cookies. I can't wait to talk more, thank you.