Kevin Knudson: Welcome to My Favorite Theorem, the math podcast with no quiz at the end. I'm Kevin Knudson, professor of mathematics at the University of Florida and I am joined as always by my fabulous other host, co-host? I don't know,

Evelyn Lamb: Co-host. It’s a host but going in the opposite direction.

KK: That’s right. We reverse the arrows. Haha, math joke.

EL: Yes, I'm Evelyn Lamb, a freelance math and science writer in Salt Lake City, Utah. Actually just got back to Salt Lake from a wonderful trip this past week where I got to meet two new additions to my family, ages three months and three weeks. So that was, that was pretty fun, to hold one of the tiniest babies I've ever held. So yeah, very nice little fall trip to take. And now I'm back here and talking about math.

KK: Yeah, well, last Friday night, I drove two hours over to Ponte Vedra, which is sort of near Jacksonville, by myself to a concert. So this is where I am in life. So I went to see Bob Mould, who many people may or may not know, but he was — Yeah, Christopher's shaking his head yeah. He was at Hüsker Dü and then Sugar. He's been doing solo albums forever. And I've been a fan for going on 40 years, which is also weird to say. Had a great time, though. By myself, that's great. This is what one does in his 50s I suppose. Anyway, not as exciting as holding a newborn but, but still pretty good. So anyway, hey, let's talk math. So today, we are pleased to welcome Christopher Danielson to the show. Why don’t you tell us about yourself.

Christopher Danielson: Yeah, I am coming to you from St. Paul, Minnesota.

KK: Nice.

CD: Bob Mould, also a fellow Minnesotan. [Ed. note: Bob Mould is actually from upstate New York.]

KK: He went to McAllister, right. Yeah.

CD: Nice. Right up the street from where I'm standing right now. I work a day job at Desmos Classroom, which is now part of Amplify, designing — working with a number of colleagues to design math curricula. We are currently working on an Algebra I curriculum, about to wrap that up, and moving on to Geometry. And then on the side, I have many projects, some of which will come up in our work today. But I think I understand that you two are familiar with the Talking Math With Your Kids blog that grew into then a large-scale playful annual family math event at the Minnesota State Fair called Math on a Stick.

KK: Cool.

CD: And I also am Executive Director at a small nonprofit that seeks to create playful, informal math experiences for children and families in the same spirit as the work we do at Math on a Stick, but designed for a variety of other sorts of spaces. That nonprofit is called Public Math.

KK: Very cool.

EL: So I'm probably doing that thing where I generalize from a small number of examples. One of my best friends in grad school was from Minnesota, and just loved the State Fair. So I think that Minnesotans just have it this special relationship with the State Fair. And so I did — I am really interested in hearing more about how you do Math on a Stick at the Minnesota State Fair.

CD: Yeah. Should I pick that up right now? Or is there more on the agenda?

EL: Yeah, that would be great!

KK: No, go ahead.

CD: Yeah, so the Minnesota State Fair, it's the second largest state fair in the country behind only of course, Texas.

EL: Where I am from.

CD: Oh, nice. Texas lasts for a month. Ours is 12 days. 12 days of fun ending Labor Day is one of the mottos. The other is the great Minnesota get-together. The location of the fairgrounds is especially convenient for large attendance. The fairgrounds are right, sort of on the border between Minneapolis and St. Paul. And they have been, for probably the past 20 years, have been working on developing some educational and family friendly spaces, out of a perception that it is expensive to go to the fair, which is true, but then once you're in that there isn't much to do besides look at animals and buy a bunch of food.

EL: On a stick.

CD: Yeah, on a stick. So they’ve been working on that. And that led to a lovely literacy space called the alphabet forest that is about 12 years old now. And the first time I sat down there, it was their fifth year and was like the sky, the clouds parted and the angels sang, and I was like, I’ve got to figure out how to build a math version of this. And so together with some organizational support from the Minnesota Council of Teachers of Mathematics and a bunch of expertise from folks that I know through the blog work and through my work in math education, put together a pitch, and after many very boring meetings, it became a thing. So we've got about 15 to 20, different mathematical, playful, creative math activities, everything from a big table full of tiling turtles, to a set of numbered stepping stones that you just see kids jumping up and down happily counting, counting by twos, creating all sorts of fun things to do with. We have a different visiting mathematician or mathematical artist every day, each of the 12 days and they bring whatever sort of hands-on thing they're into. Sometimes that's sort of the standard stuff with, like, Mobius strips and hexaflexagons, and sometimes it is new and new and delightful, creative things that the world has never seen before. So yeah, Math on a Stick, come on out and play with us. 12 days of fun ending Labor Day, always starts on a Thursday, runs through a full week, two weekends and then ends on Monday.

EL: Yeah, that does sound like a neat thing. Sometimes I go to the farmers market here or something like that, and I just think, like, where are there opportunities to kind of create, like you said, these playful, you know, a non-classroom math experience for people?

CD: Yeah, my, one of my Public Math colleagues has a project called Math Anywhere, Molly Daley. She's in Vancouver, Washington, and also does some stuff across the river in Portland, Oregon. And farmers markets are one of the more successful spaces for her. So she'll pay for, for a booth, she has grant money, she'll pay for a booth and just set up a much smaller version of Math on a Stick stuff, as well as some other stuff that she's designed or harvested from other places, but three or four activities, and yeah, delightful times ensue. However, I had a recent experience at the Mall of America, largest shopping complex, also here in the Twin Cities. And it was really interesting, because the way that kids’ families move through the Mall of America is wildly different from how they move through the state fair. So just an invitation to a big STEM/STEAM carnival. And we brought some — one of our favorite things is called a pattern machine or punchy buttons, a nine by nine grid of punchy buttons that you can drop pictures on. And each button is clicky and on a ballpoint pen. So we bought a bunch of those. And then we also had the mega pattern machine, which is just thousands of buttons from all these machines smashed together to make a nice big floor space. But the way that kids come into Math on a Stick is that there’s, like, this long elastic band between parents and children at the fair, not in the super crowded spaces in the fair, but in the less crowded spaces. And so often kids will see those stepping stones that, by the way, start at zero, and then continue on to 23. Yeah, so they'll start on the zero, and they'll lead the way into the space, like we deliberately set up those stepping stones to that the edge of this outdoor space. And by the time kids get to 23, now they're surrounded by eggs that they can put into — little plastic eggs they can put into large egg crates, and tiling turtles and pattern machines and all sorts of fun things to do. And families will sort of follow along behind. At the mall, there’s none of that. There's none of that. Families move in really tight units. There's no, like ,a child leading the family into a space, which is just a really interesting dynamic. And having been out in Portland a couple of weeks ago with Molly when she was at one of the farmers markets, it felt very much more like the fair. A mom or a dad might be much more likely to say, okay, sweetie, you keep playing with these turtles, I'm going to hop over there and buy some apples and I’ll be back in two minutes. They kind of keep their eye on them and everything, but that that elastic band is much longer. Nobody ever says, okay, sweetie, you know, their four year old, I'm going to I'm going to just hop across to, you know, the department store over here, you keep playing with something in the hallway. So Public Math is our project where we're trying to think about how do you design for those kinds of spaces? What would have been a better design than the one we had for something like them our time at the Mall of America?

EL: Yeah. Interesting different kinds of math problems to solve. Different optimization.

KK: That’s right. Yeah. All right. So this podcast does have a name, though. So presumably, you have a favorite theorem. So you want to tell us what it is?

CD: I do! And it is — Yeah, my favorite theorem is, I'll state it simply. And then I guess we get to talk about like, why it’s my favorite and things?

KK: Yeah, sure.

EL: Yeah.

CD: It doesn't have a name. I feel like maybe, maybe it should have — maybe it has a name. Maybe you'll know a name for well,

EL: We’ll brainstorm about it.

CD: But yeah, let me state it simply, which is that the vertices in a polygon are in one to one correspondence with the sides of the polygon. So for example, the three-sided polygon has three vertices. Is there a name for this?

EL: So, yeah, well…

CD: The polygon theorem or something?

EL: I don't know. Yeah, that’s

KK: I mean, a polygon is just a cyclic graph. There must be some graph theory name or something.

EL: I kind of you know, this has a little bit of an interesting linguistic thing, right? Because we call polygons a little bit differently at different sizes, like we call it we talk about triangles not trigons, or trilaterals. When we talk about quadrilaterals, like I think I have heard quadrangle, that must be the tipping point. Then we get to pentagon, so I guess that's not lateral or angle.

KK: That’s just gon. Then it’s gon after five.

CD: But gons are angles. So you are counting —

EL: Okay, is that the Greek word for for angle, and angle is Latin?

CD: So goniometer is the is the thing that you can use to measure your range of motion. I'm gesturing, so that's great on a podcast.

KK: We do it all the time.

CD: Like in your arm or knee? Yeah. So yeah, gon is angle.

EL: Okay. Learn something new.

CD: So it's only the quadrilaterals whose sides you count. Everything else, you count the angles. And, by the way, we also have elided the fact that the vertices and the angles themselves are in one to one correspondence, right? That’s also, maybe a corollary perhaps.

EL: Yeah. Okay. So maybe I'm playing devil's advocate a little bit here. But why is it a theorem that the angles and sides are in one to one correspondence? Why is it not obvious, other than the fact that, like, I've experienced these shapes my entire life and have never experienced one that did not have this property?

CD: Yeah! So I learned that this was a theorem, and its necessity, by working with five-year-olds. So I wrote a book called Which One Doesn't Belong, which was an adaptation, both of the Sesame Street routine, but also playing on some of the routines that I had seen other people playing around with. But for me, the thing that was novel about which one doesn't belong, was that when my children were small, all the shapes books that they had an opportunity to encounter were wildly simplistic. There would be, you know, a triangle page, and then there'd be a square page, and then a rectangle page, and never a square, never a square on the rectangle page. That's confusing for kids. And all of the triangles would be equilateral and oriented on one of their sides, all the hexagons were regular, and again, sitting by their sides, or maybe if they're feeling a little wild, straight up and down balanced on a vertex. But orientation isn't a thing, like, there's all this work that we know is important to come to understand a mathematical idea that just doesn't get doesn't happen in books that get published for young children, even though if you've ever been around four or five or six year old children, they can think about complex relationships, they can think about complex ideas. But somehow we don't understand or value that when we're creating books for kids. So Which One Doesn't Belong was my way of producing, taking ideas that other people had had and condensing them down into what I thought of as a shapes book that was more worthy of children's minds.

EL: I just want to insert that it is a really fun book. I don't remember when or how I obtained a copy. But I have enjoyed going through it myself, and I probably should have asked permission, but I actually used it as an inspiration for one of the pages in this page-a-day calendar I put together a couple of years ago, where I made just one where, you know, it's a bunch of shapes that all have slightly different properties, and you know, you decide which one doesn't belong.

CD: By the way, I’ll give you a little tip before explaining again, why this theorem is important. If you ever try to design a “which one doesn't belong” set, what you want to do is think about whatever your domain is, so say it's shapes, you want to think about four properties of shapes, and then cover up the first one, and design one that has these three, but doesn't have the first one. And then cover up the next one, design one that has those three, but doesn't have this one. And by the time you're done, you'll either realize that your set of four properties is more intertwined than you had originally thought, and now you’ve got to go back and revise, or you'll have a set where you know for sure that there's at least one reason for each not to belong. But then extra, an important key to this is that you have to be open to the possibility that some kid will see a reason for a shape to not belong that wasn't the reason you'd intended. Right?

EL: Yeah.

CD: this isn't a game of “guess which of the four is right.” But it's also not a game of “guess what was in my head when I designed the set.” Instead, we want to offer up something that we know is rich, and then be open to learning from the kids. So I made this book. I was trying to shop it around to get it published, but also needed to, you know, test drive it with children. So I went on what I called my Twin Cities shapes tour. And visited, I think it was three different elementary schools per week for four or five weeks. So I got into just a ton of different situations, worked with kids, kindergarteners, through, like, fourth graders, all in classrooms, like 20 minute bits, and we just had a ball. And frequently, I would hear from kids, like, one kid would say, you know, that shape doesn't belong because it has three sides and the others have four. The opening page of the book is a triangle, and then there are three rhombuses of various types and orientations. So a kid would say that one doesn't belong, because it has the wrong number of sides, right? It has three sides, the others have four. And then somebody else would talk about some other shape. And then another kid would say that one doesn't belong, because it has three corners, and the others have four corners. And in my mind, the first, like, 12 times I heard this from children, I thought to myself, yeah, you're not listening. Some other kid just said that. Didn't say it out loud, kept it to myself. But it was after about the 12th time that I heard it that I said, “Wait a minute. Wait a minute, you heard you heard when this kid over here said said different number of sides?” And they'd be like, “Yeah, and I said different number of angles.” And so it was at that point that I realized that — they’re kindergarteners, right? They haven’t — I know that they haven't seen any good shapes books, right? So they haven't had the opportunity to consider the relationship between the number of sides and the number of angles. And in my adult mind, I had this idea that it was obvious, which is so true of mathematics, like always, right? That if there's something that we ourselves have internalized and experienced for a large number of years, even if it was hard for us to learn at the beginning, we've probably forgotten about that.

KK: Right.

EL: Yeah.

CD: So yes, that's our that's our theorem. And that's why it's important. It's the thing that you actually do have to learn, it isn't obvious when you're first exploring these mathematical objects. I imagine that's true for those who are studying combinatorics. So we were talking about graph theory earlier. Lots of results that feel obvious in retrospect, because you use them all the time, so much that they're sort of internalized, and you don't even think about them anymore. But there is some some point where that thing had to be learned.

KK: So I'm sitting here trying to think of a proof of this theorem. And of course, the dumbest one that just popped in my head is to use the Euler characteristic.

EL: Is that what the five- and six-year-olds do?

KK: I love using sledgehammers to drive nails! Okay, so all right, this is a theorem; it must have a proof. So let's, let's construct one that doesn’t require Euler characteristic.

CD: Yeah, well, I feel like I would start with a line segment that a line segment has two vertices, right? And then every time — so then now I'm going to add another line segment to get what I remember formally being a polygonal curve, right, made up of straight line segments. And when I add another line segment, now I add a segment and a vertex. So I’m always going to have an extra vertex. Until such time that I come back around.

EL: Yeah, and you add a segment and no vertices.

KK: This is exactly the Euler characteristic proof, just in reverse.

EL: Yeah, it's funny, because my mind actually, I think, basically was the dual of what you said, where I swapped out, so instead of that, I was thinking, when you start with an angle, you've got two line segments, and the vertex, and then I was actually kind of thinking, like, the number of angles you have, they each have two segments, but to connect them, you overlap the two. So you divide by two.

KK: Right, so the number of angles is the number of lines.

EL: Yeah, Little, it may be maybe slightly different, but similar sort of idea.

Yeah. Okay. So it's interesting that children see this as two different facts. Children are more literal, right? I mean, in my experience, one of my favorite stories about my son was we were at open house for eighth grade. And he walks in and his soon-to-be math teacher says, “Do you know what eight times seven is?” And he said, “Yes.” Right?

EL: Yeah.

KK: I mean, she was expecting him to say 56. But children will just give you the most literal answer that you can ever imagine. Yeah. So, okay, well, we usually ask if this is a love at first sight sort of theorem. But I don't know. Maybe that's not the right question here. Although maybe it was for you. I don't know.

CD: Well love at first noticing, right?

EL: Yeah.

CD: For me, the noticing that this thing that I had interpreted as being — these two statements that I interpreted as just being equivalent and repetitious of each other, noticing that that was a thing that required learning, and that these kids were absolutely listening to each other. And it gives me an opportunity as a teacher, right? I'm only in there for 20 minutes or so, but it gives me an opportunity to say, “Wait a minute, is that gonna always be true?” The generality is that this one had three sides and three corners? And these all have four and four. Is that always true? Can we imagine a polygon that has some different number of sides and corners?

EL: And what do kids conclude about that? Or do they have, like, ways that they reason about why they have to be the same? Or do they develop pathological shapes that don't have this property?

CD: Yeah, I haven't had time to dig into that in in depth with a group of students. I've had a lot of sort of related experiences. But yeah, I don't know. That would be super fun to to step in. Posed as an offhand question, kids absolutely will both think that it is probably, be willing to believe that it is true, and there will also be kids who will imagine that maybe there is some shape that they just haven't had a chance to meet yet that isn’t. Of course what that investigation with kindergarteners, that's going to get you into a lot of a lot of really interesting kinds of conversations, because they don't have polygon yet as a defined category of mathematical objects. So we're going to have to start to think about whether a circle is a polygon or whether curvy sides count as sides.

EL: Or if you’ve got, like, a square with a handle on it that's just a line segment, what’s that?

KK: Very cool.

CD: But yeah, that kind of, you know, monster creation, from Lakatos’s Proofs and Refutations, that kind of potential counterexample, and then dealing with whether the counterexample is really a counterexample, that kind of stuff goes on at all levels of mathematics, for sure.

EL: All right. I like this. It is not a theorem I have thought about as a theorem ever in my entire life.

KK: Right. Well, I think I see why you love it. Because it actually it's more of a meta-result than the actual theorem. The theorem itself is less important than kind of the questions that it can trigger. And to get kids thinking about things in an interesting way.

CD: But it’s definitely not a Postulate. Like if we're in Euclid, it’s not a postulate, nor an axiom.

KK: No, it isn't. It’s a theorem.

CD: And there are certainly lots of results about triangles in which we know there are three sides, and so there are also three angles, because it was a triangle. Yeah. So if you don't have it, if you get rid of it — like, we can say it's not important, but if you get rid of it, there's a lot of geometry you're not going to be able to do.

KK: Oh, okay. So right. So now instead of non-Euclidean, we might have sort of non-polygonal geometry. So we don't insist that our polygons have the equal numbers of sides and corners.

CD: Yeah, I was just imagining a world in which the theorem is an undecided result, or that we can’t count on. So anything, any place that we assume it, we've got to work around it or prove it again.

EL: Or we can only use theorems about angles.

KK: All right. So the other part of this podcast is we ask our guests to pair their favorite theorem with something. So what pairs well with this?

CD: I have two pairings.

KK: Okay, good. Good.

CD: I don’t know if that counts.

EL: Yes.

CD: Or we need a new word for a pairing.

EL: Yeah. No, that's great.

CD: Yeah. So I'm going to pair it first with a claim and then with an admonition. The admonition is related to what we've already been discussing. But the claim is, it's going to be controversial here, I imagine claim is that a diamond is a shape.

EL: Okay.

KK: A 2-d diamond or a 3-d diamond?

CD: Oh, yeah. So I'm still in plane geometry. Surely there is some corollary for 3-d geometry. But yeah, I got my start in math education teaching seventh and eighth grade. And I used to, when I was a seventh and eighth grade teacher, mid 90s, I was in a camp that is still still very active in which if a child says diamond, I say again, “No, no sweetie, rhombus, you mean rhombus.” Like we call it, we're sophisticated mathematicians, we don't use the word diamond. But again, through working with the kindergarten kids, I came to understand that they don’t — like, diamond and rhombus are absolutely not the same thing to them. So if we treat mathematics as a human construction, right, then the mathematical ideas that a five-year-old has are worth testing and exploring. And one of those ideas that they have is that orientation of the shapes matters, right?

EL: Yeah, I was wondering.

CD: A square standing on its corner is a diamond, a rhombus standing on a vertex is a diamond. But also, if you cut the top off that rhombus, you now have a pentagon. Still a diamond. It's got a vertical line of symmetry, still a diamond.

EL: Right, right.

CD: So not only is there not a correspondence, because rhombus is a thing that doesn't depend on orientation while diamond does, but also that not every diamond has to have four sides in the way that a rhombus does. They don't have to be equal sides. You can you can stretch it. So you've got short sides and long sides.

EL: Yeah, I was wondering if a kite is a diamond.

CD: Yeah, absolutely. Kites are diamonds. And so the thing that I would be very excited about would be a world in which instead of we as math teachers saying, “No, no, sweetie, that's not diamond, you mean rhombus. Diamond isn't the word we use, it doesn't really count.” That it instead be a place where we press on that in all the ways that we press on mathematical ideas and try to get at definitions. Right? So now we're going to make a whole bunch of different examples. Draw me a diamond that looks different from anybody else's diamond. And we create this category. And so I think the best understanding I have of a definition of diamond that would satisfy most kindergarteners, it’s something that has to have a vertical line of symmetry. And it has to be convex. So darts are not diamonds. And somewhere between four and probably, like, eight sides. Triangles are never diamonds. Never, never, never. But four or five.

EL: And it has to have a vertex on the bottom.

CD: Yes, a vertical line symmetry that goes through the vertex at the bottom.

EL: Oh, yeah.

KK: Yep.

CD: Okay, excellent. So that's my claim: a diamond has a shape and therefore worthy of investigation rather than of dismissal.

EL: I’ll buy that.

KK: The admonition?

CD: Sure. The admonition is stop showing children only the special case.

KK: I seem to remember a Twitter like, like you were…

CD: I started yelling at a publisher

KK: You were you were hot about this on Twitter.

CD: Yes. Okay. It’s a really interesting — I think the thing you're remembering was actually almost the reverse, which is something I alluded to earlier, the thing that there's never a square on the rectangle page. So I went to a public library, doing some research on children's books for some work that I'm doing and happened — of course, was in the shape section and happened to see this book about rectangles. Like literally its title is Rectangles. This is a book all about rectangles, it has no other purpose. And I pick it up and just, like, want there to be a witness to this — but of course, there wasn’t — of my predicting, there's not going to be a single square in this in this rectangle book. And I flip through the pages and of course there isn't. So it's just one of these small sort of regional publishers that publishes educational titles for libraries and school libraries and whatnot.

KK: Right.

CD: But I DM them on Twitter to say, hey, maybe we could liven this up a little bit. And they said, Well, no, according to state standards, you know, we're responding to state standards, blah, blah, blah.

EL: Oh no.

CD: I was like, Oh, that's really interesting. I'd love to see the standard that says that you can't say a square is a rectangle. What they came back with was a Texas standard at kindergarten that says at kindergarten, you are supposed to be studying special examples of shapes such as squares being special rectangles. And this publisher was publishing a book for four-year-olds. And so because it was a pre-K title, they couldn't put the kindergarten standard in. It wouldn’t be well-aligned.

KK: Don’t let them get ahead.

EL: Yeah, it would be too advanced to know that a square is a rectangle.

CD: And we have this idea that we can't provide, again, we can't provide complex ideas. We can't give kids interesting things to think about, or conundrums or puzzles. So yeah, admonition isn't quite that, right? My admonition is stop showing them only the special case, but also please, let's show them the special case and help them integrate the special case with the general one. But yeah, all the shapes books with the triangles that are on their bases. And yeah, you know, it's like if we were teaching kids about even numbers and the only even number we showed them was 2, end of story. It seems like maybe we need a little more.

EL: I’m kind of wondering, you know, if, like, guerrilla math person with like square stickers, like going into all the shapes books, putting them in the rectangle pages…

CD: That would be a fabulous public math project.

KK: It really would. That's good. All right. So we like to give our guests a chance to plug themselves and things they're doing. Where can we find you on the line? Where can we purchase your wares? You have excellent wares for sale.

CD: Yeah, thank you. So Talking Math With Your Kids is the blog and also the online store where tiling turtles and pentagons, hexagon puzzles for small children that have widely varying examples of hexagons, are all available there. The Twitter feed is trianglemancsd. Unfortunately, triangleman was already taken by the time I got to Twitter like 12 years ago, and so I had to tack my initials CSD Christopher Scott Danielson.

KK: But not by They Might Be Giants. So who took triangleman?

CD: Yeah, I don't know, some guy who never uses it. I think he lives in Florida. Never tweets.

KK: Sure.

CD: And yeah, by all rights, it should have been turned over to me long ago. But yes, the Twitter handle is in honor of both They Might Be Giants and my love of shapes and geometry. Okay. So that's the Twitter feed. Yeah, and public-math.org for some of the projects, we're up to over there, but you can get to it all through the through the Twitter.

KK: Okay.

EL: Yeah, thanks.

KK: Excellent. Thanks for joining us and for making us think about the fact that it's a theorem. That's, yeah, that's useful.

CD: Truly a pleasure. Thanks for having me on.

[outro]

On this episode, we had the pleasure of talking with Christopher Danielson, who works for Desmos and is involved with several programs to help kids have rich, creative mathematical experiences. Here are a few links you might find useful after you listen.
Danielson's Twitter account
Talking Math With Your Kids
Math on a Stick
Public Math
Math Anywhere
Evelyn's Page-a-Day math calendar, which takes inspiration for August 8's page from Danielson's book Which One Doesn't Belong?