Knots, Links, & Learning

For those of you who follow me on twitter or instagram, you’ll be familiar with the fact that I’ve been drawing lots of these pictures, recently. Rainbow Links.jpg

To me, mathart is all about getting surprising results. I make a lot of mathy art, and while I totally appreciate other types of art, I like math art because it helps me make sense of something. Here’s the first knotty (not technically a mathematical knot, apparently, but rather links) image that I created.


At the time, I don’t think I thought a whole lot about the math behind it, but it hung around in the back of my head for a while, and I eventually decided to do a few more drawings.


On the left, you’ll see a 5×5 grid, with 5 distinct closed links (2 pink, 2 blueish, 1 brown). The middle has 3 links, and is a 3×9 grid. The right is a 6×8 grid with 2 distinct loops. Seems likely that greatest common factor of the grid is going to be the number of loops. But I have no idea how to prove that. I just have anecdotal evidence.

A bit overwhelmed by the actual proof of the thing, I veered off to the side for a while to investigate how holes affect the number of loops, too.


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The above investigation was super fascinating to me as I saw the number of loops forming and the shapes of the links changing as the holes got in their way. Except I apparently have no idea how to count the holes, because I’m getting weird nonsense answers.

One of the reasons I take time-lapse videos of these is that I think it helps other people join in. Looking at the final image, you miss a lot of the chances to think through what goes through my head while making them. It was surprising to me to have the blue loop jump from a rectangular shape to a jazz-hands-out-to-the-side shape in the slideshow above – it would take a lot of looking at the final image to see that. I like that it just pops out when I do the time-lapse.

Time-lapse is also just very satisfying to watch at the end of two hours of drawing.

I occasionally show these videos to my students, and by far the most common question I get is, “Wait, how long did that take?” followed by general incredulity that I would spend that long just drawing something. When I tell non-math-teacher-adults about this, the question is often, “What class is this for?” followed by a narrowing of eyes and shaking of head when I say I’m just doing it for myself. Both groups, however, are still pretty entranced by the time-lapse.

Back to the question of how many links based on the nxm grid.

There’s a lot happening here, and I don’t yet have conclusions, but I do have a LOT of notice and wonders.

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  • This is a study of twelves. The 12×2 has 2 loops, 12×3 has 3 loops, 12×4 has 4 loops and 12×6 has 6 links. See a pattern? These all agree about the GCF.
  • Look at the dark purple color throughout. Notice how in the top image, it bounces from top to bottom and back 3 times? In the 12×3 it does so twice. In the 12×4 it does 1.5 times, and then just one full time in the 12×6. There’s gotta be something there.
  • I’ve noticed that if you consider the perimeter of the grids, that each link has to consume an even number of pointy bits.
  • My boyfriend, Joe, has to put up with more of these discussions that he’d perhaps like, but I’ve cajoled him into thinking through a lot of these. He asked the reasonable question of whether each link is equal in length, which is worth investigating. In the 12×6, each link has 21 distinct parts, so… yes? But each distinct “part” is not equal. If you look at the corners, those are lengthier “parts” compared to the inside rectangular pieces. I’m not totally sure how to calculate the exact length of the pieces without getting deep into calculus. I do wonder if it’s possible to make an argument about equal lengths based on structure rather than calculation, though.
  • I also wonder if it’s possible to mathematically predict the shapes of the links. The bright green link is a different shape than the tealy blue link. And the lilac colored link in the 12×3 is fascinatingly symmetric. Hm…

If any of this is compelling to you, I would love it if you wanted to join me in this investigation. I don’t want someone to show up and say, “here’s the final solution, dummy, it’s already all been solved,” because I’m sure that this is already a chapter in a textbook somewhere. I don’t care about that. I care about investigating for myself and with friends what’s happening. Feel free to share ideas below, and I would love it if you did some of your own drawings and joined in on the party.

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Here’s a rough attempt at explaining how to draw these:



Here’s one I did now that I’m actually crocheting the knots.

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Black Stats Chat summary & Thoughts

This will be quick. We had our first #mtbos book club on Black Stats a couple weeks ago and I just wanted to wrap that up with a few thoughts. I attempted a storify of our chat, and I encourage you to click on and browse that. For the tl;dr crowd though, here is my summary in several tweets, followed by two things I’ve been thinking about since the chat.

Continue reading “Black Stats Chat summary & Thoughts”

Book Club: Black Stats by Monique W. Morris

Welcome to the first of the #mtbos Social-Justice/Racial-Equity/Let’s-All-Read-a-Darn-Book-Club posts!

Be sure to join or pop into the twitter chat THURSDAY, OCTOBER 5TH 7PM CST, using #blackstats


Seriously, we want you there. Even if you didn’t finish the book. Join!!! (Also, as I have never hosted one of these things, if you know how to do that or can give me some pointers… please do. I have no idea what I’m doing….but I’m excited nonetheless.)

Continue reading “Book Club: Black Stats by Monique W. Morris”


I’m mostly  housing this here to record them for myself, but if it’s of any help to any of you then all the better! There’s probably a fair amount of overlap – I’m not vetting everything as I put it here, but if I don’t get the links together, I’ll almost definitely not get around to sifting through them anyhow.  Please add stuff in the comments! 

Continue reading “#CharlottesvilleCurriculum”

Geometry! Social Justice! MCTM Follow Up! You’re Invited!

UPDATE! Here are the resources from our session!

Also, we’re doing a follow up book club on Weapons of Math Destruction 


Continue reading “Geometry! Social Justice! MCTM Follow Up! You’re Invited!”

A Love Letter to TMC (few facts, lotsa feelings)

So. I’m now back from TMC (Twitter Math Camp) and am prepared to vomit my thoughts onto this blog. In absolutely no particular order, here they are:

I am wildly impressed by the wisdom of the MTBoS (Math Twitter Blog-o-Sphere) community. 

It’s getting just ridiculous. It would have been an utterly reasonable thing for me to go about the whole conference slack-jawed at the most recent brilliance shared with me. From Grace Chen‘s passionate truth-bombs about how algorithms can perpetuate systemic racism (lobbed from a serene, thoughtful exterior) to Glenn Waddell‘s “subversive as fuck” (direct quote from him) insight into the power of how words impact our community (try replacing the word “guys” with “gals”, and you’ll get the gist of his talk), all of MTBoS and TMC humble me with their excellence and dedication to mathematical exploration and teaching.

I am so grateful for the generosity of the MTBoS community. 

Edmund Harriss made my year by designing a brand new curvahedra piece and bringing all of the pieces for us to attempt a hyperdodecahedron curvahedra build.

We got to stage 3 and while there was much mockery over my enthusiasm, it was not dampened. And SO MANY people joined in to help make it!!! Joey, Emma, Chris, Andrew, Kent, Bob, Sean, Justin, Christopher, Henry, Marsha, (and many others that just aren’t in any of the pictures I took) all pitched in and it was fantastic. There are few words to describe my joy about this build. I love math, I love art, and I love people who also love those things. This project pulled all of those together and my heart nearly burst from joy.

The generosity doesn’t stop there, though. During my talk about the Mathematicians Project (via Matt VaudreyJohn Stevens volunteered himself and Jedidiah to make an AwesomeTable for the Mathematicians Project.

And THEY JUST SAT DOWN AND STARTED. Seriously. Sessions ended, and the two of the sat down for an hour or so and started building it. Unbelievable. So generous. Basically turning my mathematician list into a searchable database. Here’s a screenshot of what’s coming… Get excited! awesome table

I am awed by the welcoming nature of the MTBoS community.

There’s been a lot of talk about #iteachmath and #mtbos, and I’m not diving into that debate beyond saying that I think everyone’s heart is in the right place and that’s fantastic. And THAT is the core of my experience with the MTBoS. At the very end of the conference, Lisa asked us to think about why we think TMC is so different (and let’s be honest, just plain BETTER) than other conferences, and David Butler, my personal hero, stood up with gusto and said “Everyone is worthy to present here.” Brought tears to my eyes. And it’s true. There is NEVER a question as to whether you or your friend or that person on the other side of the room has something to contribute. It’s assumed. And sought. I had a friend ask me if TMC is clique-y at all, and after a second of thought I said that sure, there are definitely groups of people who tend to hang out more. BUT BUT BUT… There is absolutely NO EXCLUSION. If you walk up to any group of people at TMC, you will be welcomed with open arms, a hug and a compliment. Seriously. There is such a concerted effort on behalf of the organizing committee to welcome people that it infects everyone else and the whole conference is spent with everyone actively engaging everyone else.

I am hopeful about TMC and MTBoS addressing equity.

I think about equity a lot. Last year, my advanced classes were overwhelmingly white and my non-advanced classes were decidedly more colorful. And no one was more aware of that than my students. I’m here for them and equity is what they care about, thus equity is what I care about. I have a long way to go on the equity train, but I am starting to recognize just how white all of education is, and TMC is no exception. I also would speculate that the schools represented by the teachers present tend toward majority white students, but I have zero evidence on that. I also have zero evidence on the public/private divide, but it seemed like a lot of private schools were present. Maybe I was hyper aware of people saying they worked at private schools because that’s not my experience.

All that said… I heard a lot of conversations about equity and the TMC board seems ready to tackle the issue. I will contribute by proposing sessions centered on equity and being present for those discussions.

TMC is such an excellent place to do math(art).

I realize that among the tweeps who know me, I have developed a bit of a reputation for getting hysterically excited over math. And one of the reasons I love TMC so much is that it is at TMC that I can find others to match my enthusiasm. I had so much stinking fun with Justin Aion, Suzanne von Oy, Sean Sweeney, Wendy Menard, James Cleveland & the others in the ExeterMath morning session. I mean, I knew that optimal sphere packing was going to be and excellent problem, but I didn’t realize it was going to be THAT excellent. For those interested, know that concave cubes/octagons are going to be prominent in your solving.

I also love that there was so much art getting passed around. People were showing off crochet and sculpture and patterns they’d discovered and it was just excellent. Plus Justin brought his hexagons and I fell in love.


I’m sure that I’m missing a whole lot of things, but it’s half a week later and I hadn’t posted this so here I go. I may dig deeper into some of the sessions in future posts, but at least I’ll get this out there.

Is the algorithm for Racially Biased?

So, at TMC (Twitter Math Camp) I had the privilege of attending two separate talks that got me wondering. First was Grace Chen’s absolutely phenomenal Keynote, “The Politics(?) of Mathematics Teaching”. In it, she mentioned that the age of black children is often overestimated by 5 years.

Later, Bob Lochel shared the hilarious website  Upload/take a picture and it will estimate the age of people in the photo. The whole room had a great time playing with this. I personally (actually 32 years old) received everything from 26-42 as an age, depending on the picture.

Recalling Grace’s talk, I thought I might investigate. My boyfriend’s niece is Haitian and has very dark skin. She’s absurdly adorable and 5 years old. My nephew is white and has very light skin. He is ALSO absurdly adorable and 5 years old. 2017-07-08 10.34.48

Sure enough, did a good job with my nephew and greatly overestimated the age of my niece.

This matters. Although it’s a silly website, by overestimating the age of black children, and seeing white children as more innocent, we perpetuate racism. I can’t say it better than Robin Bernstein did in his article, so go read that. (Pointed out to me, naturally, by Grace Chen.)

Knowing I needed more data points, I started entering a ton of photos and recording the information. If you would like to help out, head over to and add the data to this quick google form! I’ll update the graph. Below are the current results.

On the x-axis are the people’s actual ages in the photos. On the y-axis is the age shown on The diagonal line shows x=y (or if got it right). It’s color coded by race. What do you notice? What do you wonder? Anything else we should investigate? (Add in comments!)

How old net graph

For what it’s worth, Emma Veach also happened to find this gem on the same day!

Students > Teaching Math

I just finished It Won’t Be Easy: An Exceedingly Honest (and Slightly Unprofessional) Love Letter to Teaching by Tom Rademacher, and I have a lot of feelings. It had me hooked pretty early on.

It had me hooked for several reasons. Tom (I’m pretty sure I’m good with first names here, we follow each other on twitter, so we’re basically best friends) kicked his book off by noting several things that matter a lot to me: humor, Race is necessary to talk about in teaching, you must teach to write about teaching, and no research. He also curses a lot, which I appreciate.

Teaching is a shit show. It’s a wreck. It has been necessary for me to teach middle school boys that drawing on a girl’s face is not effective flirting. I once seriously considered making the sign, “Please do not eat Clorox Wipes” because it would save me not having to repeat myself so often. No matter what you do, Hot Cheetos dust is everywhere. Penis drawings are an accepted fact of life.

Teaching is also a painfully beautiful, human activity. My students and I cry, for good reasons and bad, often. There have been harsh, unnecessary words from both of us (more often from me), and ugly truths are revealed. I have been called some names I’m not willing to retype here, and I have been told I will forever be someone’s hero.

Because it is all of these things, teaching is beloved by Hollywood, and I really cannot emphasize how much I hate that. There are a lot of teaching movies I loved before I became a teacher, and now I detest them all. I hate talking about teaching with people who are not teachers, because unless you are in it, it is nearly impossible to get right. It has happened more than once that a well meaning person has praised me for the work teachers do to “save those kids!” Setting aside (ha!) the ugly implications of what they mean by “those” kids, just how wrong people are about that makes me equal parts nauseous and angry. I hate talking to non-teachers about teaching, because they don’t get it. So I don’t do it. Except when I do. Which is most of the time. It’s like people who are wrong on the internet. You know you shouldn’t engage with them, but they. are. just. so. WRONG.

I say all of this because while reading It Won’t Be Easy I often thought, “I gotta get _____ to read this.” Rademacher is brutally honest about teaching and articulates a lot of things I think about but struggle to say. I want a lot of my non-teaching friends to see that.

In the first, Rademacher is really honest about being a White teacher. Because I am also a White teacher struggling to figure out what that means and how it affects my students, I really appreciate that. Rademacher acknowledges that he really wants to exclude himself from the “White men” category, but that doing so is exactly the problem. He tells us about listening to a student who has had all White teachers for two years in a row and comments, “It’s just too many White people.” Rademacher acknowledges how uncomfortable it is to look this stuff in the face but how incredibly necessary it is to do.

Round about this chapter (White Guy Bullshit), is when I silently revoked my earlier wish that, “I gotta get _____ to read this.” I can see recommending this book to well meaning non-teachers and having them totally shut down at this chapter. But, of course, that’s the problem. As exhausting as it is to try honestly confront Race as a White teacher, it’s more exhausting to talk about Race to White people who aren’t actively trying to examine their Whiteness. But I have been trying to listen to the voices of People of Color and two things I have heard loud and clear is that if I want to be a White ally, I first have to own my Whiteness, and then I have talk to other White people about Race. I have this insane privilege that when it gets to be too much I can just keep being a White person who doesn’t have to think about Race all the time. My Students of Color don’t have that privilege. The discomfort of White people is something we need to accept and not run away from because People of Color are uncomfortable all the damn time. I know that because my Students of Color have told it to me over, and over, and over again. I don’t get to judge or excuse or try to forget that. I have to listen.

Rademacher also called me out on the fact that I do not engage nearly as much with my school as I should. I have avoided as many meetings as possible, do not know the names of most of the adults in the building, and am not advocating for my students outside the classroom. Fair enough. I’ve been called. I’m going to think about how I can change some of that this next year. My students are not best served when I try to pretend that things outside of my classroom do not affect them or me. I cannot do everything, I know that. But I can do more.

Rademacher is also unwavering in the fact that while yes, content is important and stuff, students are more important. When I think about some of the worst moments I’ve had as a teacher they are often when I have insisted that a worksheet is the most important thing a kid does that day, when both the student and I know that’s a lie. Times when I’ve noticed a kid is really emotionally upset, but I have been too wrapped up in myself to give that kid the space and time to deal with it. Or even just acknowledge that its happening.

I have been thinking more and more about what my actual goals are in the classroom. If you spend more than 10 minutes with me, I don’t think you’ll have any doubt of my love for or dedication to math as a subject. I spend a lot of time talking to teachers about teaching. About how to approach this standard vs the next standard. A lot of those conversations are fascinating, and I can totally get into a rabbit hole about how to best introduce students to quadratics or philosophize about how deeply my students need to understand the concept of zero. Unfortunately, as fun as all of that is, teaching math is less important than my students. These students we interact with every day are people. There are humans in my classroom and if I really think about it, my best days of teaching are when that fact is front and center. I’ll sometimes talk to non-teachers about my classes and how I changed the topic that day because kids couldn’t care less about the boring standard but cared that day about something else. The intense fear that my students will probably fail the ACT is often the first reaction of said non-teacher. Sigh.

Students do not learn in classes where they are not acknowledged as people. They might memorize a few things to get you off their back, but they will not open up and really start thinking in a place where they cannot be themselves. After the election, my kids needed space to talk about that. I could have shoved a worksheet down their throats, but good god, why? They weren’t going to learn math that day. Even less emotionally fraught situations like when a kid asks a question that’s interesting but has nothing to do with what I’m supposed to do that day, you bet your butt I’m going to help my students explore that if that’s what they’re genuinely interested in. To heck with the lesson plan.

Yes, yes, I’ll make sure they can solve equations. Fine. But the school year is actually pretty long and there’s plenty of time for that. We, as teachers, get so caught up in ourselves and our lesson plans that we sometimes forget that the students in our rooms are people. We have to get through this standard and figure out how to grade a billion things and we need data on how our students are doing for god’s sake!

I have never had a kid say to me, “Ms. Perkins, I really want to do this math because you told me how important this worksheet is and I really want to improve my score on that standardized test.” I have (actually) had kids say to me, “Ms. Perkins, I do not care about this stuff at all, but I’ll do the math because of you.” The not quite articulated bit there is that they do it because they know I care about them. The content stuff comes along, it really does. But when I’m not doing a good job of letting them know I care about them, guess what? No math. Game over.

This doesn’t mean that there aren’t days when I really do push kids down one path when they want to do something else. There are. Rademacher does a good job of helping to draw the line between teacher and friend in his book, too. Like it or not, those are totally separate things, and there are actions we have to take as teachers that we would not take as friends, and that is as it should be. Our students are humans, but they are also children. They’re going to do stupid, ridiculous things and we have a responsibility to be the grown ups in the room. It’s just that some people (these are the people I mentioned before who are WRONG) think that being a grown up means being in charge and controlling everything. When really, being the grown up means helping to build a safe place where kids can learn and figure out both how to factor a quadratic and how to be a good human.

Rademacher’s book is really good. There are a ton of other, fantastic books out there about teaching, but this is the first one I have personally read that was as honest about all of the things I think we need to be honest about and put front and center the fact that our students are people first and students second. You should probably all read it. I’ll lend you my copy if you like.