#mtbos Book Club! Danny Martin edition

Hey folks,

Yes, yes, I’ve been terrible and I haven’t posted or made any decisions, but school is out, and here I go.

The next book is Jacqueline Leonard & Danny Martin’s The Brilliance of Black Children in Mathematics.

It’s not cheap ($41 on amazon, $39 from the publisher), but Danny Martin is THE PERSON YOU NEED RIGHT NOW. I can only assume Jacqueline Leonard is equally astonishingly mind-altering.

We would like to do something slightly different for this book. A means to testing whether we can get deeper conversations in a different format. Maybe it will work, maybe not. Here is the plan:

  1. Post a blog reflecting on the book as you are about half way through BY JUNE 30TH.
  2. Respond to other folks blogs on that “half way” point. By JULY 7TH
  3. Post a blog when you are done with the book. BY JULY 31ST
  4. Respond to other folks blogs when they are done. BY AUGUST 7TH
  5. Twitter chat. FIRST OR SECOND WEEK OF AUGUST?

EVERYBODY’S welcome, and we would love to see even more participation. Please comment or dm me if you have questions.

Val Brown’s Question

Val Brown posted this question the other day:

She wrote a whole moment on it, and you should spend 10-15 minutes reading.

I’ll be honest, when I first saw it, I scrolled past. It seemed like a trick question, so many “what-ifs” came up, and I didn’t really have the mental energy to put into it. That said, as I see more and more people commenting, and some friends have encouraged me to respond or consider, I’m now sitting down to think. And as others have been pressed into answering with a specific number, I’ll do the same: my number is roughly $500,000.

Rationale: Over the course of my expected career as a teacher, I expect to earn  ~$1,750,000. ($50,000/year for another 35ish years). If I all of a sudden become Black, I expect things are going to get a lot harder, and I would love to have a cushion against the times I’m likely to lose a job or not get a job, have health problems, and struggle to get a “fair” (read: white person’s) shake. I don’t expect it will become impossible to get a job, nor that my chosen field of education will make it impossible for me to continue on a similar path. I expect I’ll have to start watching what I say a lot more, and that I’ll have to proactively protect myself with more vigilance than I currently do.

I should point out that I’ve had the benefit of not answering first, have been able to read other people’s responses and consider them, and I’ve given myself most of the day to ponder this. I absolutely get folks hedging and saying they’d want tens of millions or more – that was my gut reaction, too, but I suppose I don’t think that me turning Black means I need enough protection to have the means of Beyoncé.

I do, however, concede that I personally feel absolutely no right to ask for that half-a-million dollars (a HUGE amount of money). Black people are born Black into this world (especially America) as is and they’ve gotten no reparations at all. (I do think this mental exercise makes a strong case for them.) It’s me desperately clinging to my privilege that made me instantly think I would need the money. That it could possibly buffer me from the loss of privilege.

I wish I could say zero without lying. I get why reading these responses have been hurtful (Sherri Spelic wrote a really thoughtful blog on this that you should DEFINITELY read). That my number is so high is a signal to me not just that we have a problem around race in this country (Captain Obvious, reporting for duty), but that I have a lot of work to do on two fronts. 1) Doing my part to rectify the situation, by using the privilege I have to bring attention and change to the issues and 2) Continuing to analyze, acknowledge, and understand my privilege. If I’m very honest with myself, I can confront it when I’m “in the mood”, but today, I really just wanted to be excited that summer was starting! Yay! I can do what I want with my time! And honestly, sitting down to write this is not what I had in mind. I was thinking about drawing and wine and a patio. That’s privilege.

I feel VERY uncomfortable publishing this. I really want to buy some time to think it through more, hedge more bets, and double check that it’s okay. But as a dear, beloved friend of mine has routinely pointed out to me, it is only when we get to a vulnerable place that we can learn and grow.

I apologize deeply if my response hurts folk. I’m certainly not done thinking about it, and I would greatly appreciate it if any of you want to challenge, question or push me to expand on any part.

Danny Martin’s Vision

I had the great fortune to speak at the NCTM Regional conference in Chicago last fall. Easily the most impactful piece of my time there was meeting and learning from Danny Martin. I have rarely heard someone so willing to cut through niceties to get to the heart of a matter. I immediately bought several books of his, and started asking around to see who might be able to record his talk at the NCTM Annual conference in DC. Luckily, NCTM itself recorded, and you can listen to the talk, “Taking a Knee in Mathematics Education” here. (And you really, really should.)

Finally having found some time, I started to live-tweet my viewing of it. The start of that thread is here.

I was excited I was able to pause and rewind – able to digest a bit more slowly what he said. Dr. Martin’s intellect is so far beyond mine it’s laughable to think I can fully comprehend his message, but I am trying to because it is so necessary and so compelling.

Per the normal, while I was watching it I got a text from a colleague about a time-sensitive matter, and I had to pause my listening. Life has gotten away from me and I hadn’t gotten back to it, but with some persistent (because she’s wonderful) prodding from Marian Dingle, I sat down to finish it. I’ve also perused Wendy Menard’s post on the same talk. Here are just some of my reactions to it.

White people on the whole have absolutely no understanding of just how bad things are for Black children in mathematics education. And we keep trying to minimize it. 

I experienced this when I listened to Dr. Martin talk in Chicago. Not only does Dr. Martin detail physical violence enacted on Black children, he also provides startling examples of emotional trauma.

Almost more upsetting than this, however, is his (accurate, to my experience) pointing out the high threshold White people have for accepting that harm is being done to Black children. We are unwilling to believe Black students telling us things are bad without proof right under our nose. (Perhaps that is why Dr. Martin felt it necessary to include such blatant examples in his talk.) “You feel you’re not included? Well, how? When exactly? What did the teacher/school/students do?” We don’t just take it on faith that students feel excluded or ostracized from their school environments, we demand proof. Would we be so demanding of a White child upset about their treatment in schools? Would we make them retell and offer proof of how bad things are or would we simply take the child’s word that their experience was negative and see how we could improve it?

I’ve personally seen this at work. (And if I’m honest, I’ve probably done it, too.) Black students at my school have been particularly vocal at our school this year, and I have heard teachers I know say that the students are exaggerating. I have told non-school people in my life what my students are telling me, and had them say, “well that’s just how it is, they’ll get through it.”

I would kindly ask that you start listening for this. Do we let our Black children experience their world how they experience it? Or do we put pressure on them to experience it how we hope they will?

Dr. Martin’s argument of “Equity as Compromise” absolutely cut me to the bone. 

At the end of his speech, Dr. Martin makes a very compelling case, quoting Carol Anderson, that while the civil rights movement may have earned Black people some rights, it negated the deeper issue that human rights must supercede civil rights. The energy that went into attaining civil rights is a mask that allows America to deny Black people basic human rights, beginning (but not ending) with dignity. We say, “You can vote now! Segregation is illegal!” and pat ourselves on the back while calling the police on Black people (Black children included) for just living their lives, and refusing to see their full humanity. If a White person is getting “unruly” (loud voice, exaggerated body movements) we think that they’re acting so silly. Must be having a great day! If a Black person is acting “unruly”, we call the police. Or prevent them from dancing to celebrate earning their college diploma. Or ask our students to be more quiet and compliant.

A White child struggling is supported and the root cause of their non-compliant behavior is sought, while a Black child struggling is “defiant” and “misbehaving” – we approach them from deficits rather than with compassion. We don’t see their strengths unless we are forced to.

Dr Martin summed it up well in his conclusion:

“Mainstream math education has traditionally invited Black people to participate on its terms. Expecting the system to reform from its foundational purpose and fundamental character to a new state of validating and valuing the humanity of Black people is unrealistic in the face of evidence otherwise. Traditional discourses of equity and inclusion have been self-serving within liberal white imaginaries, white supremacy, and anti-Blackness. But they have been inadequate for Black liberation. The kinds of changes advocated for within mainstream math education discourses are welcomed and accommodated within the self-correcting systems of white supremacy and anti-Blackness because they represent no real threat to these systems.” -Danny Martin

What cuts me to the bone here is that I can see how much of my work toward including social justice in my classroom may have fallen into the category of me patting myself of the back – saying that I’m doing the work to improve things, while still maintaining the system as it is. A system that so often refuses our Black (and often any other non-white students) their full humanity. I have been guilty of praising Black exceptional figures. Praising their exceptionalism rather than seeing them as representative of a whole. I have absolutely done things to support the system as it stands.

Dr. Martin is calling for a true revolution. An absolute abolishment of the system of math education as it stands today and building it anew from the ground up for Black children, supervised, supported, and approved by the Black community. And I gotta say, the math education he advocates for sounds amazing. Capstone projects that address issues in the world. Teachers informed by their community. Teachers to serving in the communities they actually live in. A focus less on technical math skills and more on applicable skills students will use to understand and improve their lives. It sounds so fantastic. And it’s just so so far off.

Wendy suggested in her post that Dr. Martin’s vision may not have a place for white educators like her and me. That’s totally possible (although I don’t think he completely closed that door, he advocated the community having approval power over the teachers of their children). And I guess I agree that it’s not the job of white educators to create the world Dr. Martin dreams of. It will be better if its built by the people who inhabit it. After all, white people built this one and it’s not doing too great.

I do think, however, that White educators have an important role, and that is to start by acknowledging the current system as it truly is and not how we hope it is. I live within a really great online community of educators that regularly challenge their assumptions, beliefs and attitudes in the hopes of attaining a more just world. But outside that lovely community, I find it exhausting trying to drag white educators into conversations about how we impact our students of color.

There are just SO VERY MANY math teachers out there who don’t want to think about this at all. 

I say this from my own experience trying to talk to other math teachers in my district and at conferences. I have the privilege of being a white woman familiar with a fairly wide network of math teachers, and it’s been dawning on me over the past year (and there’s some privilege for you, it’s only really the past year that I’ve noticed how wide spread this apathy is) just how few teachers seem really willing to do “the work”. There absolutely is a dedicated core, but I have yet to see evidence that the majority of teachers are willing to truly examine their place in the education of Black children. We’re happy to shake our heads and say that yeah, golly gee, things are bad, pay lip service to social justice in our classes, but then we need to get back to making a worksheet about linear systems. It’s just so very exhausting to try to drag anything deeper out of them. (And sometimes it’s exhausting to drag it out of myself.) Any reflection on how we might be contributing to the damaging experiences that Black students experience day in and day out is too often absent. But dragging that conversation out is something I can do, and although I am already exhausted by it, my tiredness is absolutely nothing compared to the exhaustion of educators of color trying to get us to do the same.

 

 

Reflecting on DingleTeach Wisdom

Saw this today, and pretty immediately started listening.

I hope that all of you are now familiar with Marian Dingle @dingleteach, either way, you will learn a lot from this conversation. She’s an elementary teacher with a seemingly infinite depth of wisdom and eloquence. Passionate about bringing social justice to her classroom and yours. Relentless about making sure our conversations aren’t superficial. Go listen to this conversation. It’s long, but it’s totally worth it. You will learn a lot.

Taking a Knee in Math Class

Here are a few thoughts I had while listening and a few personal reflections on myself and my teaching.

I have a lot of work to do to broaden the impact of any work that I do. 

I’m currently at a pretty self-centered place when it comes to teaching, learning, and my classroom. I struggle to effectively collaborate with other teachers, and I’m not nearly as vocal as I could be, even when I know my voice will be heard. (Not necessarily heeded, but it is often heard.) In the podcast, Marian asks us to challenge the conscious and unconscious expectations we have of students. I like to think that I’m trying to do that. When I grade, I am always in my head asking whether I’ve scored students fairly based on the work in front of me, or whether I’ve graded them based on what my perceived expectations are for them. I regularly catch myself being unfair, and I try to rectify that. I think about that all the time, but I think this is the first time I’ve said it “aloud”. I don’t regularly talk to other teachers about it, because it’s uncomfortable. I should get over that. I try to also be in my head during class – who am I talking to? Who am I giving wait time to? With whom am I being too helpful and with whom am I not helping enough?

Marian also asks what teachers are doing to center their students of color (SOC). I like to think that I’m trying to do that, too. I have consciously and intentionally invited several SOC to join my “advanced” classes and had conferences with them after they’ve joined to see how it’s going. This I do bring up in staff meetings, and I do try to regularly ask other teachers how they’re helping to make the advanced classes more representative of our student body (read: more SOC).

I pay close attention to whose voices are heard in my classroom, and try to lift up the voices of SOC. Students have talked to me about this being a problem in the past, and I’m trying, in daily discussions, to make sure SOC are heard every time they want to speak. I also try to actively seek their voice when they aren’t volunteering. This of course, is touchy because I don’t want to put them on the spot, but I do things like asking SOC during work time if they’re willing to share their ideas, and or I ask if I can display their work. I am positive that I could do better, that I can do more, but I am trying to do this. I have not, however, tried much, to talk to other educators about it. I should do that. (This is a start.)

I have a RIDICULOUS amount of privilege. Despite my efforts to try to be more conscious of it, I still don’t see it all. 

Early on in the conversation, Marian talked about how she wanted to start talking this year about race and social justice in her classes, so she went to her administrator and held parent meetings. My initial reaction to this was to wonder why she felt she needed to go through these hoops. When I’ve wanted to talk about race, or any other perceived “touchy” subject, I’ve just done so. I never doubted that I would be able to deal with any blow back, nor that it would be doubted I had the right to talk about it in class. Now, it’s true that Marian is just better than I am at foresight and she is more cognizant of the community at large. As I alluded to above, I think that I’m working primarily at an individual level rather than a collective. But it’s striking to me that while I feel I can just start talking about these things, Marian prepared.

See that? Not only has Marian, a black woman, (and she talks about this in the podcast) had to repeatedly justify her expertise to her staff, admin, parents, and students, she anticipated the possible issues with discussing race and social justice in her classroom and took preemptive efforts to address those. I’ve never felt the need. I haven’t had parent meetings, felt like I needed to head’s up my administrator, nor that it would be reasonable of anyone to question that I want to bring these issues to my classroom. That’s privilege, and I didn’t see it until listening to this conversation.

History, especially misguided history, is a palpable thing that dramatically impacts us. When it’s not questioned, when parts are left out, it is so damaging. 

Marian mentioned Hidden Figures in the podcast, and how angry she was that this part of history had been hidden from her. (She also noted frustration that the math community wasn’t as hype about it as she expected. She’s not alone. I was jarred by the lack of enthusiasm in the math teaching world for the movie.) The host of the podcast, Dr. Angela Dye, also mentioned that she has only recently discovered she enjoys math, and that had life been different, she might have taken a different path, citing a lack of access to knowledge about math careers or role models. They also open the podcast talking about what is present in American history texts and what isn’t. I’ve only recently started reflecting on my own historical education, and although I’m really not that old (I’m 32), I have experienced some pretty messed up history in my time. I absolutely recall being told that the civil war wasn’t really about slavery, it was about “economy”. I remember the day. I’m only starting now to be able to dismantle the damage done to my biases from being taught growing up “to not see color”. When I think of the scope of how often these things were taught and how embedded they are into my generation (and each generation has its own), it is horrifying.

Marian mentions the Mathematician’s Project in the podcast, and more importantly, that while it’s a first step, it’s just that. I have so many feelings about it. I firmly believe it to be a good thing, but I will share that my perception of it has changed so much over time. When I first started doing it, even though it was my student’s idea, I’ll admit that I totally patted myself on the back. As I’ve shared it with more and more educators, though, I’ve started to get a sinking feeling. It has always felt kind of weird to me, that this thing, a trove of mathematicians of many backgrounds, hasn’t already had a place, an institution, where it existed before. There are TONS of similar projects out there, but they all seem disconnected and fledgling. Mine, too. What makes me feel so uneasy is that it seems so revolutionary to so many people I talk to. It should not be revolutionary. It’s not something one should pat ourselves on the back for, it’s something we should be ashamed wasn’t cooked into schools from the first. We’re not doing “extra” good work here, we’re making up for the failures of how we present mathematics and who gets the accolades for it. I will likely continue to do the mathematicians project for the rest of my career, and it will only just be a start.

There is so much more in the podcast. Please go listen to it. We are so blessed that Marian is part of our community, and so blessed that she’s dedicated so much time to reaching out to have conversations with colleagues.

 

MTBoS Books: Pushout (March 3) & For White Folks Who Teach in the Hood (April 14)

Hey everyone! We’re now on to MTBoS BookClub #4 & #5.

Book Club #4: Pushout: The Criminalization of Black Girls in Schools by Monique W. Morris on March 3rd.

Pushout Cover

Book Club #5: For White Folks Who Teach in the Hood…and the Rest of Y’all too: Reality Pedagogy & Urban Education by Christopher Emdin on April 14th.

for white folks book cover

EVERYONE is welcome to join in. We’ll meet in person in Minneapolis at Urban Growler, and for a twitter conversation online right after.

The New Jim Crow: Discussions

TL;DR Black and White people use and sell drugs at remarkably similar rates. The New Jim Crow does and EXTRAORDINARY job of laying out the systematic ways that Black and Brown people have been unjustly locked up in the “war on drugs”. It goes (deep) into the history of the war on drugs, the ways police are incentivized to over arrest, how and why Black communities are targeted even though data would point police elsewhere, and how courts have been used to solidify the system. We don’t allow Jim Crow laws anymore, and we call out overt racism, but racism has just morphed to fit this new system. Honestly, don’t read this blog post, go read the book. 

The next #mtbos book club meeting is next Saturday, January 20. In person in Minneapolis, on twitter if you can’t join us here. It’s on The New Jim Crow by Michelle Alexander. Full confession, I am not yet done with the book. I’m about 2/3 through it, but I want to get the ball rolling, and Marian Dingle (@dingleteach) made the very good point that MLK day would be a good one for this book. 

Instead of summarizing the book here, what I’d like to do is share conversations I had with my class over the past week. I’ll be tweeting out lots of things from the book throughout the day once this is posted.

On January 8th, “The New Jim Crow” was trending on twitter. Marian forwarded me a tweet from Sean King.  Continue reading “The New Jim Crow: Discussions”

#MTBoS Book Club! The New Jim Crow January 20th

Join us! January 20th, 3PM in person in Minneapolis, 4PM for a twitter chat, and on this blog (and yours! Or guest post here!) to discuss The New Jim Crow by Michelle Alexander.

New Jim Crow

For those unfamiliar, this is a book club primarily for math teachers (but honestly, EVERYONE is welcome!), because math teachers, like everyone else, have a responsibility to educate themselves on race and equity in America. We’re reading, we’re learning, we’re talking, and you should totally join us.

If you’re around Minneapolis, we’ll meet at 3pm at Urban Growler. (RSVP here so I know how big of a table to grab!)

Let me know how excited you are (then I’ll try to tag and actively engage you) in the comments, and join us!

I would love to hear suggestions for the next book, too!

The #MTBoS is Exhausting and Exhilarating #unsexy

 

I.

am.

so.

tired.

Oh my gosh. It’s 6:30pm, and I’ve been wondering for a least an hour if I can go to bed. This time of the school year is totally exhausting. And just for fun-zies, my school decided to LOSE THEIR GOSH DARN MINDS by concocting a torturous schedule today that included…

  • Late start (read: 2 hours of meetings)
  • FIFTEEN MINUTE CLASSES
  • An assembly that included a 10 minute video with no words
  • An hour long advisory period
  • And, oh, did I mention that I found out about this schedule yesterday?

I tweeted about this last night, and true to form, MTBoS did their darndest to give me excellent suggestions.

It just wouldn’t end. My god. All of these thoughtful, helpful suggestions. I’m not even including them all because I’m tired of copying and pasting the darn links.

Allow me to be perfectly clear. I adore each and every person who gave me suggestions (especially Bowen, I really needed that laugh). It was nothing if not thoughtful of them to see, “Geez, Annie needs help planning for such absurd classes, allow me to remind her of all our MTBoSy awesomeness!” (I imagined each and every one of these people wearing a cheerleading outfit as they typed to me.) I am not here to disparage them. I am here to declare instead of planning an awesome, amazing lesson using their super duper helpful suggestions: I spent 15 minutes answering kid’s questions and had them make a cheat sheet for their test tomorrow. That’s right. I went #UNSEXY, and I feel freaking fabulous about it. 

When I’m up and excited, MTBoS is constantly cheering me on and totally there for me. When I’m grouchy and exhausted…MTBoS is….constantly cheering me on and totally there for me. Which, frankly, is occasionally dreadful. Yesterday, I just wanted to vent that I was going to have an horrifically disrupted day, but MTBoS was all, “Come on, Annie! You can do it! Think about how awesome all these teachers are, and they seem to never ever get tired or ever feel overwhelmed!” Like a twerpy little sick kick that’s always trying to pump you up. All thoughtful and s***. Ugh. (But seriously, I love you guys.)

I completely understand that I am a jerk for whining that I have a supportive community that’s always trying to pour energy and love into me. I know. And I know that I have been that twerpy side kick many, many times. But I just wanted to make sure that OTHER people also know that while I love the MTBoS completely, am SO stinking grateful for all it has given to me and my students, it’s okay to occasionally plug your ears, say, “LALALALALALA!” and give the kids a worksheet. It is emphatically NOT okay to do this on the regular, but if it’s what you need to do in the moment, don’t necessarily feel like you’ve let them down, nor that you’ve utterly failed your students.  Teaching is a long game, and if you’re volunteering for every play, you’re going to run yourself down and burn out. I truly believe that.

And you know what? The day was fine. By not having a phenomenal activity planned for our fifteen minute sprints, I had time to chat with students that I don’t have every day. I got to have some conversations about what I really love about math and what I think is ridiculous. I got to hear about what they’re up to. So, honestly, I feel pretty good about phoning it in on the lesson planning today.

And then when I got home, I saw this kind mention, which brought me back to a blog post I’d written over a year ago, that reminded me of all the good stuff.

Teaching take a lot of energy, and it’s okay to conserve for a while. I’ve learned my lesson that MTBoS is not always the best place to get an empathetic “that sucks”, (although I’m sure that if I asked for it, I would get it) but it is a great place to turn for help when you need to know that someone out there believes in you and wants you to do well for your students.

Guest Post! Marian Dingle on “Weapons of Math Destruction”

I have been extraordinarily blessed to have Marian Dingle join me in reading and discussing books for a #mtbos bookClub, and she has graciously written a reflection on the recent book, Weapons of Math Destruction by Cathy O’Neil. Enjoy, then follow Marian’s blog and chat with her on twitter @dingleteach

Weapons of Math Destruction: Post-Chat Thoughts

By Marian Dingle

First, I’d like to thank Annie for her work and dedication in starting and maintaining this #mtbos book chat series. I am humbled she has allowed me to share my thoughts here. I am afraid that I have far too many more questions than I have answers. But we are all here to learn together, right?

Briefly defined, a WMD (weapon of math destruction) is an algorithm that seeks to quantify certain traits in order to predict outcomes. This alone is not a new concept; we are taught to model in this way throughout our K-12 mathematical experience through algebraic relationships, calculus maximization, and even micro- and macroeconomics. What separates the modeling in WMDs is the curious ways it enters our livelihoods and the scale at which it occurs.

Initial reactions ranged from shock to validation, mixed with an urge to act.

An important point is that the author, Cathy O’Neill, a former quant who participated in creating and applying these WMDs, began as one certainly meaning no harm, but had an epiphany, ultimately leaving this lucrative field. Sherri, below, made a great point that the fact that O’Neill is female, and probably a Wall Street outsider, enabled her to see things with slightly different eyes.

Now for my (tangential?) thoughts. Of the many topics O’Neill discusses, I was struck by college selection. Although I do look at rankings, they were not much of a factor when considering college choices for my two children. As a person of color, I have learned not to solely rely on such rankings, as the information that is crucial to my family is often not captured there. Yes, I want my children to attend a “good” school, but my definition of good includes support of marginalized students, their graduation rates, and the number of faculty members of color. A brand-name university can potentially be more harmful than beneficial. This is a reality that many people of color face.

Informal algorithms like these are often generated through social networks and aid in other decisions such as where to live, work, and enroll children in K-12 settings. Would it be helpful to have more quants of color designing algorithms for big data? Perhaps, but is it even more important to control the question the algorithm seeks to answer? Would this help us with results of standardized testing? Are tests designed to justify the existence of an achievement gap? Can we design one to dismantle oppressive systems?

As we think about our roles going forward, I think it’s worth pondering our roles up to this point. More and more educators are agreeing that education and teaching, even in mathematics, is not neutral. What we choose to discuss, and not to discuss, reflects our politics, and affects our students. Do we discuss the purpose of mathematics with our students or colleagues? Have we created a space for them to discuss how mathematics can be used to support bias? Do we even ask them what they think? Do we know what we think?

What I know for sure is that we can no longer afford to be silent. Courage is required to analyze our own agendas and roles in this work.

Geez, These Links are Complex

Many of the drawings I do are the ones that simply seem to me as though they’ll be interesting. Sometimes they meet my expectations, sometimes they don’t, and occasionally they wildly exceed them. This drawing is of the latter quality.

2017-11-12-19-38-13.jpg

I’m not actually that pleased with what it looks like. It’s not particularly pretty. I also spaced out and mixed up the purple and red in the bottom right image. But the mathematical results are really neat. Here’s what I was hoping to do with it. These are what I would call 4×4 grids, each that has exactly 1 break in it. For reference, here’s what a 4×4 looks like with no breaks.

4 by 4

Up until now, I’ve been thinking of this as 4 distinct links. In my head, I have labeled them 1, 2, 3 and 4 starting at the top left of the image, and moving right along the top. Thus, in this picture, dark green is 1, light green is 2, pink is 3 and purple is 4.

link numbersI now think this might be incorrect, and I should instead think of it in symmetric terms where 1 and 4 are the same link with a plurality of two, and links 2 and 3 are the same, also with a plurality of 2. Allow me to explain why.

In planning the image at the top of this post, I wanted to try putting a single break in all of the unique places a single break might exist in a 4 by 4 grid. Those are shown below. Each yellow line is a unique breaking point, with the white lines representing the duplicate breaking points that could be obtained by reflections or rotations. (Note that all the vertical breaks can become horizontal if you just rotate.)

break-drawings.png

I also disregarded the any breaks on the farther outside spaces because the resulting link would have to have a section traveling vertically rather than at the usual 45 degree angle. Maybe I should have considerd these, but they seem to be an expansion beyond the type of link I wanted to consider for now.

disregarded break

Okay. So, I started drawing. Part of what I like about the length of time these drawings take is that I have built in time to ponder what’s happening. I decided on the fly to color in all of the links affected by the break first. I wasn’t particularly surprised to find that each break had essentially connected two of the four links. 2017-11-12 18.11.38

In fact, I thought, “That makes perfect sense!” Because if the break connected 2 links, and if I had, in fact, found all the unique breaks, I should have 4 choose 2 different images! Brilliant. So I should have a graph that connects links 1 & 2, 1 & 3, 1 & 4, 2 & 3, 2 & 4, and 3 & 4! I dutifully started to identify them aaaand… well, crap. For one thing, link 1 appears FIVE times. What the hell?!

link 1

I mean… that’s ridiculous. Link 2 appears twice, Link 3 appears three times, and Link 4 appears only twice. So are some links… more…. important (?) that others?

For that matter, there are TWO images that connect Links 1 and 4. Yet there’s only one image that connects links 2 and 3. Upon inspection, I can see why this is. There are two totally different breaks in the 1 & 4 images. One of the breaks (bottom row middle) connects to the center opening, and one of the breaks (bottom row right) doesn’t. Clearly, those are different. And the image connecting links 2 and 3 only has one unique type of link because I had disregarded the second type since it would give me a weird looking link. breaks connecting

Had I not abandoned the links on the furthest outside, I would have gotten two more images that would connect links 1 & 2, and links 2 & 3. The images I currently have shown are in grey below. The missing images are shown in green.

link net

I think I can resolve some of this by no longer naming the links 1, 2, 3 & 4, but rather naming them in symmetric pairs, but it’s getting late and I should finish up some school work before tomorrow.

In the meantime, here’s a time lapse for your viewing pleasure.

 

 

If you would like to join in on the fun, please add thoughts or comments below. Please do not join in if you happen to have already solved this problem and just want to show off or give out answers.

 

UPDATE (11/13/17)

I have created the ghastly disregarded links in the name of mathematical inquiry. I feel justified in abandoning them as horrible, but the data may be useful, so here they are.