What makes a good math game? We might not know it, but some of our most loved games are steeped in mathematical thinking.

What makes a good math game? We might not know it, but some of our most loved games are steeped in mathematical thinking.

Today we are passing things over to Kent Haines. Kent is a Heinemann Fellow Alum and middle school math educator based in Alabama. He is joined by Dan Finkel, founder of Math for Love, a Seattle-based organization devoted to transforming how math is taught and learned.

Kent and Dan talk about all the ways games can help (or hinder) the development of strong mathematical reasoning.

*Below is a transcript of this episode.*

Kent: I am a big fan of games, both inside and outside the classroom. When I'm at home with my kids or spending time with friends and family, there is nothing I like more than pulling out a board game. But I also love the catalytic effect that games have in the classroom. There's just something about a little strategy and competition to get my students thinking deeply about math. So I am thrilled to talk today with the inventor of some of my favorite math games, Dan Finkle. Dan is a math educator, and the creator of Prime Climb, a board game for upper elementary kids that my middle schoolers love, as well as Tiny Polka Dot a set of games that are perfect for preschoolers and early elementary students. He has thought a great deal about what makes a math game great, how we can use them in the classroom and how parents can instill a love of math in their kids by using games. Dan, thank you so much for being here.

Dan: What a pleasure. Thanks for having me, Kent.

Kent: So, I'll put this first question right to you. What is it about games that make them a good way to learn and practice math?

Dan: I think there are three layers to that… to answer that question. The first is that learning your math facts and being really familiar with it, requires a degree of practice, and games give you a place to practice and a way to make practice more interesting and more fun. I personally remember I was a big cribbage player as a kid. I would play obsessively with my brothers. And if you know cribbage, you know that you make a lot of 15s. And one of the things I realized when I was in school is that I was suddenly better at adding than everyone else because I could find the 10s, find the 15s and add up numbers more quickly and just with more facility, just from playing. So if you have the practice and the skills, the things that you would do on a worksheet, and you can put those into games instead, that is just a win basically and it makes it more fun and more interesting to do those things.

The second layer is that games involve a kind of thinking that is deeply important in mathematics. Kind of an if/then thinking: if I do this, what will my opponent do? If I make this choice, is there a better choice or will I be able to get taken advantage of somehow if I do that? This kind of thinking, to imagine the game from your opponent's perspective, to consider whether you've done the best thing, or if there is an even better way to go, that kind of strategic thinking is really important in mathematics also, and you're practicing, I think, some of the deeper skills of a mathematician.

And then at the third level, and maybe we're getting quite deep here, there's a way that games actually put you in touch with some of the heart of what mathematics is. And by that, I mean that there's a lot of mathematics that has to do with setting some set of rules. Maybe it's the axioms in your system. Maybe it's a kind of technique that involves some particular approach. And there's this question of what is possible here? What is the realm of possibility? And then in a game, you're thinking about that in order to see what's possible.

I think... And similarly, in mathematics, some little rule changed like, oh, what if the parallel postulate were different? What's possible now? In a way you see, okay, rule change and now what can we do? This act of playing with rules and within rules and in the space generated by rules, is in a very real way, what deep mathematics looks like. So really at every level, I think it is a huge win to play games in math classrooms and when you're learning math.

Kent: And when I see students playing math games in a lot of classes or even the games that they try to play in my classroom when we have free time, I see a lot of digital games that to me feel like they're not taking full advantage of those elements of math, that show up in games. So I remember when I was a kid, I played a game called Math Blaster, which was you were shooting asteroids, but the asteroids just had an addition problem on them and you just had to type the sum and then it would shoot the asteroid or whatever.

Dan: I remember it well.

Kent: Yeah. Okay. And then another one was, I think it was called Number Muncher or something, where you had to eat different numbers. And there was one category called the Primes, and I actually memorized all the prime numbers under 100 by pure trial and error. Like just, oh, I ate that one and I died, so I guess that number's not prime. And I had like a sheet of paper that had a list of all the primes, but I had no idea what a prime number was. But people would say, well, those are math games. So what makes a math game rich or interesting or deeply mathematical to you?

Dan: Yeah. There are two things that I think really you're looking for. And the first is that there's some element of choice in the game. And this is often where the paper and pencil curricular games fail is they say, Hey, instead of doing this edition problem, you're going to roll these dice and it'll give you two numbers and then you add those numbers together. And really you're just rolling your own worksheet, but there's no choices, or whoever rolls the biggest number wins. It's just there's a lost opportunity when there's no choices. But the other piece is that math should really be the engine of the game. And this is something that I think about a lot, because every time it's not, every time ... Like any of those games you're describing, it could have been a math game, but it could have been a spelling game. It could have been a historical facts game. It's just a thin veneer of you're playing something and then there's some question you have to answer.

And I always feel like those are a missed opportunity and even worse, they send a message that doing the math in that context is like the quarter you have to put into the game. It's the toil and the annoying work you have to do to actually do the fun thing. The fun thing is shooting the asteroids or munching the monsters or whatever you're doing. The annoying thing you have to do to get to that part is to solve the math problem. And I think that those end up being ... They're very easy to make in a way, because you can slap together some very obvious game and just put that veneer of, okay, here's some arithmetic to do. And because people's conception of what mathematics is so narrow, it's always like, oh, here's arithmetic, here's a new thing to memorize. So we'll just slap that on. And that piece of math not being at the heart of the gameplay, is what makes those games so thin and I think fundamentally does really represent a missed opportunity.

Kent: I would love to try to zero in on an example then, of a great game and hopefully a fairly straightforward game that we can talk about, where math really is the engine as you say, where the interesting decisions or choices that you have to make are mathematical choices. Do you have an example or two of a game that you could use that you feel like does meet your criteria of a good math game?

Dan: Yeah. So there's a really simple game for example, called Don't Break The Bank. I learned this game from a book of dice games, and this was something that they were playing in like 18th century Irish pubs. And that's a good sign that it's actually a good, interesting game.

Kent: Shut the Box….

Dan: Exactly! Shut the box is in this category also. People were playing this not to learn math. They were just choosing to do it because it was fun. And this is a game is a great game for the classroom. Super simple. You just roll a dye, you roll it nine times in all. And over the course of those nine roles, after every roll you fill in nine spaces that form three, three digit numbers, and your goal is to make those numbers sum to the highest amount possible that's not a four digit number. So if you go over 999, you've busted. So you roll a six, where you going to put that? Should that go in the 100s column of your first number? Or maybe you'll put that in the 10s column, or maybe you should put it in the ones column. What's the best place to put it? A very simple choice, but choice is in there and it's fundamental to play the game and math is very much at the heart of the game.

That's not a game that could be a spelling game or any other kind of game. It is very specifically a math game, and yet it is addictive. I've yet to play it with a group of students where they haven't begged to play again. And what's even more interesting is that students want to play it so much, is I had a breakthrough with a teacher once where she taught her third graders this game, and she came back to the professional development session a week later. And she said, well, I found out that a number of my students don't actually know how to add three digit numbers. They just been faking it, but that was the time when they wanted to do it. And suddenly they started asking her and asking each other how you actually add the three digit numbers, because they were genuinely motivated for the first time to actually understand how that worked.

So, it's such a great simple game. And what you find is there's way more to it than you think. That’s an example. And I think there are many, many games for the classroom that are cheap or free to play, that involve just dice or paper and are fantastic.

Kent: Okay. So this is great because I'm imagining a teacher listening to this and they're working with elementary students on multi-digit addition and they're like, oh my gosh, this is perfect. I'm going to use this in my classroom. And I do think that just playing this game would actually do a fair amount of work to get her students excited and engaged about playing the game and therefore practicing their mathematics. But as a teacher, I'm still not 100% satisfied unless I feel like I'm getting every little bit of learning out of this activity. We're deviating from our normal routine to play this game, so how do you see a teacher being able to leverage the underlying interest in this game? Kids want to have fun. They want to play, they want to learn the strategy and how do we make this as strong a learning opportunity as possible, where the kids are really learning something deep about numbers and about addition?

Dan: Partly this is going to depend on what your ambitions are as a teacher. And I will say there are, for some teachers, there is the sense of, I don't have time to spare. And in those cases, I think the idea of the, I'm going to spend five minutes on this opening game or this closing game, I've got an extra couple minutes at the end of class. Let's play this game rather than just do nothing. And that actually represents a real sort of huge possibility and huge motivating kind of place to go. However, if you are now saying, "Okay, I want to go deeper into this game or deeper into any game. "One of the places and one of the questions you can always come back to. And this is when your game has choices involved, you can always do this. Is you can start a discussion around, "What is the best way to play this game?" Or, "What is the strategy you use to play this game? Is there a better strategy that you could have used?"

What you start finding is kids do have things that they do. Just naturally instinctively, you do play a strategy and articulating that is interesting. And it makes for a conversation that everyone can take part in and actually get to start making mathematical arguments about. The other really wonderful thing about this is that if somebody says, "I have this way that I think is the best way to play, and here's my strategy." Then you can actually test it by saying, "Okay, let's play. And I want to see you play this strategy." And do they win or not? Now in games of chance, it's subtler, but… I don't know if you know the game Pig to throw out another one.

Kent: Oh, absolutely. I was going to bring that up. Yeah.

Dan: Right. Which is another beautiful, beautiful game. You roll one die. You can roll as many times as you want and just accrue those points. And at any time you can stop and put all those points in the bank. So I roll a six to five, a four, I've got 15 points. I could keep rolling or I could put it in the bank. If I ever roll a one, I lose all my points that aren't in my bank. And my turn is over. And you can play a 50 or a 100 and there's variations. But that's a game that some people say, "The best way to play this game is to roll ones and then immediately bank your points." And somebody else says, "The best way is to roll three times and then bank your points."

And then you can actually run an experiment and say, "You try that. You try your strategy. You try your strategy, let's see. Or let's have everyone in the class, pick one strategy or the other, see what happens." And you can actually be collecting data and testing those things.

So, I think that's one of the places these goes is by beginning to have conversations about strategy, which are inherent in any game that has any choice to make, we can start having mathematical arguments that we can actually back up with proof and logic. And at least talking about things like place value and whatever else comes up naturally in the game.

Kent: Right. And I've done that in my own classroom where essentially what you've done after you've played the game is, "I've created a set of scenarios and listed out like, here's the scenario, what's the best play? Here's the scenario, what's the best play?" And if you were being uncharitable, you would say, "Well, what I've done is I've made a worksheet." Right?

I mean, what's a worksheet, but a sequence of questions? But the kids don't really interpret it that way. They certainly engage with it on a higher level than if I had just explained the game and then handed that them or something. Because they still have that sort of leftover connection to the fun they were having, that sort of thing.

Dan: Totally. And frankly, in a way we shouldn't be de facto anti worksheet either because you can design good worksheets. What matters is not, was it a worksheet or was it a game? It's, were the students engaged and interacting within a playful way? You can absolutely design a series of questions that for this group of kids grabs them and they are really into it. And you can absolutely have a game that nobody is interested in playing. It's not just that this game works and this worksheet doesn't work and all games are good and all worksheets are bad. It's that I think there was something deeper about the spirit of being invited to play around with something, to play around with ideas. And games, I think are a very natural way to do that and a way that tends to work more easily. But there's all sorts of opportunities in the classroom. And really at the end of the day, I would never argue that everything has to be a game in the classroom, but I think the spirit of play can infuse everything in a math classroom.

Kent: So, let's talk about a couple of these games that you have developed and seen working in the classroom and at home to help kids engage with math on this way. The first game that I believe you made, the first one that I came across is a game called Prime Climb. Now, it's obviously very difficult to explain a game over an audio medium, but perhaps you can give it a shot and try to talk about a couple of the big underlying math ideas in the game?

Dan: For me, the genesis of the gameplay actually came very quickly and kind of all at once. And I think of it as being very simple. You just have two ponds that start at zero. You're trying to get them both to 101. It's almost snakes and ladders-like, except you get to make the ladders and the slides because you roll two dice that are 10 sided dice. And the numbers that you roll, you get to apply to where your pawns are through addition, subjection, multiplication, and division. So if I have a pond on eight and one of my numbers is a five and I would apply it, I could add to 13, I could subtract to three, I could multiply to 40. Or I could, well, I can't divide in that case because it has to come out to an even number.

But what's interesting is when you have two dice and two pawns, there ends up being a lot of kind of surprising moves you can make. If I'm on 23 and I roll a three and a five, I can subtract three and go to 20 and then multiply eight times five and go to a 100. And these moves where you subtract and multiply or divide and multiply or multiply and add, you sometimes can come out of nowhere and either bump your opponent's ponds back to start or suddenly be close to going out at 101. Or getting to 101 kind of by surprise, which makes the really fun, I think.

Kent: What I find interesting about this game is that it does seem like a fairly simple sort of, "This is a great way for my kids or my students to practice their addition, subtraction, multiplication, and division." There's a little strategy along with it but fundamentally they're going to bump their friend, or get to 101 like that sort of thing. And yet, the game also has this underlying structure where there is a benefit to ending your turn on a prime number. And all of the composite numbers on the board are color coded in such a way that shows their prime factorization.

So for example, two is an orange coloration. And so every multiple two has at least one section of orange in it. Every multiple of three, I believe has green and that sort of thing. And so there's this beautiful underlying structure where students can really see, I mean truly like the fundamental theorem of arithmetic. The unique prime factorization of every number. But it's not they're at the surface. It's not something that I would necessarily say that most students will directly engage with, without some sort of prompting of some sort or another. And so I'm curious, how have you seen teachers really leverage the structure of the game in a classroom setting?

Dan: Yeah, so I've seen it really work in two ways and one of them is what you described. You just start playing and then after a little while you have students start to reflect about the game or what they notice about it. And they've seen things pop up. The other way is that you actually look at just a chart of those numbers first. On my website, we have these that you can download. And just asking students what they notice about it. Seeing if you can challenge them to continue the pattern that they see in it, there end up being very, very rich discussions that come out of them. Because as you pointed out, that color scheme is connected to this very deep mathematical principle of unique factorization and the fundamental theorem of arithmetic.

So, starting there where you just look at it and actually prompt students to notice and wonder often gives the beginning of them starting to consider, just to notice what is going on and kind of just see how cool it is. Because it is really cool that using the color scheme alone, it actually tells you everything you need to know about multiplying and dividing. And the idea that, "If seven is purple and I'm on 14, and I want to multiply that times seven, I could do that in my head. But I could also look for a number that combines the colors of those, the orange and purple of 14, the purple of seven. And if I find something that's orange purple purple, which is 98, that's where I'm going."

It's a very interesting thing. The fact that numbers have that property is incredibly surprising. And as with many things that end up being true, once you've accepted them mathematically, you take them for granted. But seeing it for the first time is just so magical. That was definitely one of the early impetuses to have that be an actual game with a board that people would be playing on and using, as well as having those classroom materials where you could prompt the discussions and just think about them a little bit more.

Kent: Because as a model, it is somewhat abstract, right? You're just kind of arbitrarily choosing a color for two and a color for three and a color for five and that sort of thing.

Dan: Right. It's a lot of graphic design choices.

Kent: Yes, but in the context of the game, I've imagined this, I guess maybe a fanciful idea, but having that structure just sort of reappear throughout the year and in one's math classroom. So, you're simplifying fractions and you realize that every time it is possible to simplify a fraction oh, they both have a common factor of three, it would be immediately apparent, they both have a green section and that, and in fact that once we simplify away that common factor of three, the remaining colors would tell you the exact numbers. And it's kind of, I mean, there's a beauty to it. I don't know that I would suggest that it's the key to understanding fractions or what have you, but it is a very deep, and to me, memorable way of understanding this idea of like, what do 14 and 35 and 42 have in common with each other? Well, they have a certain seven-ness that is visually apparent in the game. And I really love that.

Dan: Absolutely. And I think you're right, there's no magic bullet to understanding fractions or understanding any particular thing, but having useful and good visualizations can be super helpful and something to really hold onto and help yourself build those things. And actually, I should give a shout out to two websites that I've collaborated with in the past year or so to that use this coloration in some really nice ways. And one is Mathigon which in their Poly Pad made a prime factor circle images based on the game that you can actually really drag around and combine...

Kent: Oh, that's cool.

Dan: ... Which lends itself to some really neat visualizations and again, using it for fractions and use them to find those common, multiple and greatest common factors… it’s really neat. And the other one is knowledge hook made an online manipulative called the Prime Line, which is a number line that you can drop, you can mark numbers on, and it gives its prime climb coloration scheme for that number. And both of those allow you to go up to much larger numbers, which is really neat. So you can say, well, prime climbing goes up to 101, but I wonder what 1,001 looks like you can just drop it down and see, and it's pretty neat.

Kent: Absolutely. And now we've concluded the portion of the interview where we talk about things like the fundamental theorem of arithmetic. And I want to sort of get back to no less complex ideas, but just done at an earlier age and talk about the other game that you have created called "Tiny Polka Dot". And this is a game. I mean, it's not even a game so much as it is a deck of cards or a resource that you can.

Dan: Yeah, That's right. It's a mathematically enriched card deck. That's exactly right.

Kent: So you can play all kind games with these cards, but if I would describe it, instead of the normal four suits, of diamonds and clubs and what have you, there's a bunch of different representations of the numbers from zero to 10 and mostly it's dots. So there are the sort of familiar 10 frame , there's the dice arrangements that everybody's familiar with from rolling dice there's dots that are of different sizes and just sort of place seemingly randomly around the cards. And there's also the numerals, right?

The actual squiggles that mean one and two and three and four and that sort of thing.

And so I've, I've played this with all of my kids at this point. And because I don't have all my kids are not ready for Prime Climb, but they're all ready for Tiny Polka Dot because whether you're matching two different representations of the same number, or you're playing a matching game where you find two cards that add to five, or add to seven, or what have you, the kids seem to really be able to play a lot of games that they're ready for, but not with a deck of cards that you have from, CVS or what have you.

So, yeah. Could you talk a little bit about where the impetus for this came for and what you've seen done with these cards?

Dan: Yeah, so we built, my wife and I, who collaborate on all of these games originally built tiny polka dot. It came out of our work, actually, both working with teachers who would tell us, oh, I've got, my kindergartners, I've got these kids who are just having trouble on like recognizing numbers or subitizing or doing these things. And what's the thing I can practice. And we kept being like, oh, well, what if you tried a game, that's just like a matching game, and you could make it a little more fun?

So, we would ask the teachers, do you have a deck of cards? Or you could always take a deck of playing cards and take out the face cards and think of the ACE as being a one. And, and at some point we were just like, why don't we just design these? Because we had all these things that we kept encouraging people to do, but it didn't feel like there was a natural deck out there that had all the mathematical kind of clarity that we wanted in there. And so we ended up designing those cards. Originally, we just had stickers of different colors. That was our original prototype is we had four by four piece of paper with note card stock with stickers that we put in different arrangements. And we thought about, which are the most fundamental arrangements and which are the easier ones and the most challenging ones and what corresponds with the standards.

And then, we just started putting together games. And some of them are very simple classic games. Some of them are ones that we invented. Some of them are twists on old games or puzzles, but the idea that, I mean, I go back again. I played Cribbage a ton and that was not the only card game I was playing. I was playing all kinds of card games that I think gave me a greater facility with numbers and arithmetic and mathematical thinking. And just the communication to teachers and parents with young kids that fundamentally doing this kind of play is one of the best things you can do with your kids. And we just can hand you this thing, which makes it easier to do that, which feels like you don't have to make a whole thing or search out what the games are. Here's some great games to play.

Here's this thing, just make it simple for people to get started was really right there. So our original prototype decks were used in that summer program, and we just got rare reviews from the kindergarten and first grade teachers who used them. And then it really felt like it was something that needed to be out there. And it's actually something that I'm most hopeful about of almost everything we do, because I feel like if you can help kids essentially make friends with numbers from the time they are three years old and have ways to play as the defacto way to explore numbers from early on. I just feel like that is a very powerful attentional approach. And I hope that's what Tiny Polka Dot makes easy.

Kent: I think that's, I think it's really important to be respectful of the challenge of all of those fundamental concepts, because, we mentioned at the beginning, these games where math is sort of like the impediment to the fun or what have you. And frankly, it, when, if my seventh grade student is doing addition, wrote addition facts, it's probably not very rich, but this is far from RO this is some of the deepest mathematical thinking that, young kids, preschool, kindergarten, first grade have been doing. And, and I know I gave a deck to my first graders teacher and she's just loved having so many different ways to represent a single number because that's just not, it's not something that is just sort of in a , the default curricular resources is, eight different ways to represent the number five. And she is mentioned to me that just having that has been so helpful for the kids who are still struggling with seeing five, subitizing like being able to look at a set of dots and say that's five dots. You know, that sort of thing.

Dan: I saw a great thing happen at a classroom actually, where it was a second grade classroom with kids who had been stripling a little bit, and this was like a classroom that was meant to remediate some of that and help them. And the kids were playing a game called power dot, which is just like war, except you flip, you can flip up more cards if you want. So they could say flip up two cards. And then whoever has the highest sum, the most dots together. And then they want to flip up three cards and then five cards. And then one of them said, well, what if we flip up all the cards, which on the one hand is the classic , oh, don't do that. That's just going to be a huge waste of time. But on the other hand you kind of want to let kids try stuff out and see what happens.

And these two kids ended up being involved for 20 or 30 minutes, trying to add up all the cards in the deck. And first it was too hard, but then I think I suggested maybe they could group them by tens. And so then they were finding pairs that made 10 and then counting the tens, and then they needed to get some paper because they kept losing track of how many tens there were. And so they had to record how many tens there were. And it ended up being an incredibly rich, project that just came out of, I don't know that strange impulse to just kind of go to the next step. So yeah, that it's surprising I think. And in a way, what I'm always hoping for, with games and really with everything this, I think the sense of play more generally is that somehow kids will feel like they own this.

Kent: Can, can you imagine just for a moment going into that same classroom and handing them a tiny polka dot deck of cards and saying, I want to know how many dots are in this whole thing go. That would've been misery. They would've...

Dan: And, and that is so central. If it comes, if it feels like it’s something you have to do, cause someone else is forcing you to do it's miserable. But if it's your idea and you own it and you want to do it, then it's super exciting. And this is somehow the key. If you get people playing, they start having new things they want to do. They start wanting to try things that sort of spirit of experiment thing, and curiosity comes out and it starts driving them and it makes more things possible. And if you're paying attention, I think as a teacher or parent, some of those things are worth following some aren't, but if you kind of are willing to experiment and not have it always be perfect, you can sometimes have just magical learning experiences happen. And that's part of what we're doing is trying to provide a rich environment to play in order to help students be developing their curiosity and their sense of ownership and really owning what they're doing.

Kent: All right. And so, now that I've gotten you to like really wax philosophical about the importance of playing all these sorts of things, I want to pivot and talk about what I think would be a very surprising project to a lot of people who just heard... Have been introduced to you for the first time here, which is one of your most recent projects was actually working on a set of multiplication flashcards. Which is to me... That and like the mad minute timed test, it's sort of like the epitome of the negative math experience that everybody can remember is having to learn their times tables and that sort of thing.

And so I'm curious what was it that made you feel like this is somewhere where I want to try to find something better, and what is it about your sort of flashcard set... it's called Multiplication by Heart, that you feel help students in a way that other resources don't.

Dan: Yeah. Frankly, if you had asked me five years ago if this was ever anything I would do, I would've said no way. And yet when the idea came, it felt like just the right thing. Partly because I think I kept saying, oh, you can play these games to practice these things. And everyone said, "Oh, I love Prime Climb. It's so great. So what flashcards would you recommend?"

And somehow there's no amount of playing games that would prevent people from getting flashcards. And so many of the flashcards out there were at best, just sort of boring and neutral, and at worst, actively negative, where they were like throwing in weird pneumonic devices that just felt like it was so cluttered. And I don't know, that sense of clutter is just the opposite of what I want as a mathematician for students. I don't want a bunch of garbage; I want to clear it out so it's just the clarity of what is actually meaningful and useful.

So the idea was what if we actually connect the foundational visuals and ways of thinking about multiplication that are key with the process of committing them to memory because you do need to commit your multiplication facts to memory. That's important. You don't want to be a ninth grader and be taking algebra and struggling to remember what six times seven is. That slows you down, and it just makes it harder to actually interfere... it makes it harder to interact with the mathematical ideas that you're really trying to explore.

So I think of mathematics... Or multiplication facts as one of those key things that it is important to commit to memory. And yet, if you're just doing it via pneumonic device, it's not connected. It's not meaningful.

So what we did is we basically said, what's the introductory approach to multiplication? And it is either the skip counting or the understanding of multiplication is equal groups. So think about four groups of three, I have four circles with three in each circle. That's actually what it is. So we actually have groups of four groups of three as our connection to four times three.

So the cards do that from one times one to five times five, which is kind of the second-grade appropriate level. And then it goes to the third grade, which is... The visualization there that's really key is arrays. And we go from a 1 by 1 array to a 10 by 10 array. Again, one of those things that is so fundamental... Not just because it is a good visualization, but it actually connects to the area model of multiplication. It connects to understanding why multiplication follows the commutative property. Having those arrays allows you to see how multiplication works, and that ends up being really important. And it's a model that actually grows with people up until algebra and beyond in terms of thinking about things.

And then the final visualization is the one from Prime Climb, which is multiplication is factors combining. And we use the same colors that we do in Prime Climb. And those are three really different understandings of multiplication and how it works, but three pretty fundamental ones that go at the second grade, third grade, fourth grade level.

In addition to having these different visualizations, it also uses space repetition, which is just the idea that you don't practice everything equally. The ones that you know best... The facts that you know best, you spend more and more time between them. You give yourself a chance to be on the verge of forgetting them, then refresh your memory of them. And that's how your brain actually learns best and retains it and is able to commit them fully to memory.

So we're using some of the science of memory there to help people actually learn them a little bit more effectively rather than just here's all of them and do them all every time. It's like, you only need to do five minutes a day and we're going to be practicing them in a way that has the most impact and helps you actually commit them to memory best.

Kent: And so my natural next question then is, are there any games you can play with the flashcards?

Dan: Yes. And that's actually something... I'm actually in the process of still developing more. So there's a few games that come with it and some explorations and even like looking at, okay, what are square numbers, and how do you see the squares and the array? And can you combine numbers to form squares and different things like that. There's some nice exploration built in. But I'll give you an example of one that I'm planning to make a YouTube video about and publish soon, which is basically a form of UNO or Crazy Eights with the array cards where you can play cards on top of the pile if one of the factors matches the previous card underneath.

Kent: Ooh, I like that.

Dan: Yeah. And then if you really want to get the practice part out of it, in order to play the card, you have to say the product. So I'm looking at seven by eight. Someone else has played seven by one. So I'm like, okay, the sevens match, so I want to play this, but I need to be able to say 7 by 8 is 56 in order to play the card. If I get it wrong, someone can challenge me, and then my turn doesn't count.

But that's an example of a game that you can play.

And I'm hoping to develop some more. I don't think they have to be super new in a way. I think that once you have decks that just have all these numbers on them and neat arrangements, there's kind of some classic games that you can play. Though I think there's some new ones out there too that are nice.

Kent: Well, I've loved getting to dive a little bit deeper into some of my favorite math games, and those are the games... Frankly, your games are among the several that I always recommend to people.

So I'm curious, what are some of your favorite games to recommend to parents or teachers of maybe an early elementary student and maybe a late elementary student. So a game that maybe you kind of feel like, oh, I wish I could have come up with that one or something like that?

Dan: You know it's funny; it's like some of the classics like Yahtzee, I still feel is like such a great game. Farkle is another one that, which is very Pig-like in a way. Pig is a one-die classroom, easy version, but that graduates up to Farkle, which it's just fun and simple and yeah, it just feels very natural. So yeah, I'm a big fan of those.

I actually like games generally. And I would like to say that if you're a parent, especially, I think that you... A lot of games have more math in them than you might even realize. And even playing a game like Scrabble and just keeping score, it's shocking how much just comes up naturally, I think, in those games. And I think it's really worth playing them.

A couple others to mention as games to play at home: Dragonwood is one that I've recently discovered, and I've been giving to people that I feel like is pretty solid at which I kind of wish I had come up with.

Kent: I really like that game. My kids love that game. Yeah.

Dan: Yeah. And another one is King Domino. I feel like I was on the verge of that. I was playing around with that same idea, and then it came out, and I was like, it's too late to do that one, but I feel like that's a really nice game.

Kent: I genuinely credit that game with my son learning multiplication because the way that the... And we don't have to get all the way into the game, but just so if parents have young kids. He was probably first grade, but as you add more territory, he just wanted to know what's my score now, what's my score now? And so it very easily built up and he was doing two times three, and then two times four, and he was skip counting. And over time he just got some of those early skip counting multiplication facts, and it created kind of an architecture in his mind so that when it came time to really understand multiplication, he was just right on board with it.

Dan: Isn't that interesting? I mean, in a way, it's finding the game that does get your kid hooked in like that really is... It does just pave the way for the mathematical concept to be introduced more formally later. It's amazing how valuable that is, I think. Yeah.

Kent: Well, Dan, I can't thank you enough for being on the show with me. As you can tell, I am quite a games nerd myself. And so I've really appreciated the chance to nerd out and talk about math games with you.

Dan: It's been super fun.

Kent: All right.

Dan: Yeah. Thanks for having me on, Kent.

Kent: Sure.

Kent Haines is a National Board Certified middle school math teacher in Birmingham, Alabama. He has spent 10 years in the classroom, as well as two years as a visiting instructor at the University of Alabama at Birmingham. Kent is a 2016 Heinemann Fellow and has helped develop curriculum for The College Board's Advanced Placement program, A+ College Ready and Citizen Math. He writes about math games for parents and kids at Games for Young Minds.

Dan Finkel is the Founder of Math for Love, a Seattle-based organization devoted to transforming how math is taught and learned. Dan develops curriculum, leads teacher workshops, and gives talks on mathematics and education nationally and internationally. His TEDx Talk, 5 Principles of Extraordinary Math Teaching, has been viewed over a million times.

Dan’s curriculum has been used by thousands of students, and is known for its combination of rigor and play. The math games he co-created with his wife, Katherine Cook, have won over 20 awards. They include Prime Climb, Tiny Polka Dot, and Multiplication by Heart.