# Intersections, 2013-2014

Today we had our launch party for Intersections, our school’s math-science journal. Last year a science teacher and I gathered interested students to produce this journal — and they worked tirelessly and did a spectacular job. This year, we have some new students and some old students who served as editors. Here they are giving their speech at the launch party (which was also a pizza-soda party).

More than anything, I have enjoyed watching the editors become independent leaders, organizing something involving so many people and moving parts, and presenting their creation to administrators, math teachers, science teachers, computer science teachers, and other students. I feel like I’m coming to understand the niche I play in my school: I find ways to make math exist outside of the formal curriculum for kids who want to get more involved. Intersections is one of those spaces — both for editors and for those students who submitted.

If you want to check out this year’s issue, please click on the cover photo (designed by a student) below and it will take you to the website.

More than anything, if you have the time, just click around and see what cool things you discover!

Although it’s a lot of work, if you have any thoughts about starting something like this at your school, I highly recommend it.

# Intersections, a high school math-science journal

At the end of last year, a science teacher and I came up with the idea of creating a math-science journal for students to publish their work in. Our school has a literary and art journal and even a foreign language journal. But there is no forum for students to show off their more mathysciency work…

There was a secondary reason for wanting to start it. We are in the middle of implementing our school’s strategic initiative, and that involves doing more project-and-problem-based-learning, and more interdisciplinary work. We thought that by having the journal, it might help drive some of these curricular changes by encouraging teachers to assign rich problems or projects that could be submitted to the journal.

It took a long time and a lot of work of many people to produce, and I wanted to take this post to outline the process of it’s development. However before I do that, take a moment and head over there and check it out! And if you want to see a few specific articles, here are some that I think you might enjoy to whet your appetite: the very model of a modern natural satellitea challenging chalkboard problemb-text, counting art, running in circles, transcendent fractions.

## The Initial Stages

The other teacher and I crafted an initial document outlining what we thought the journal could look like, after brainstorming. We did this at the end of the last year. The one thing we both agreed on is that we didn’t want the journal to be ours but we wanted it to be the kids. These were just our initial ideas, but that’s all. When kids would get involved, they would take charge of what the journal would look like.

Here’s the document.

This is also the document we both shared with our head of school, our division head, and our department heads (math and science). We realized that if we didn’t have the support of these people, it wasn’t worth going forward with. But (as one would expect) they were all ecstatic about the idea. We also got permission to take some of the math department meeting and some of the science department meeting to introduce the journal to get teachers on board. I made a little slideshow with some ideas for how we could use it as a department, in an attempt to get the buy in from teachers. We encouraged teachers to make assignments next year that students could submit. Or if a student had an ingenious question, we could encourage them to research it and submit it to the journal. Or a student who had an out-of-the-box way of thinking about a problem, they could submit that. We emphasized that we wanted a low barrier for participation. We didn’t want kids to feel like they had to write a huge research paper or do all this independent work or be the best and the brightest to submit.

We knew without teacher buy in this would have been dead in the water…

But we both have awesome colleagues, and everyone was on board.

## Getting Kids Involved

So at the end of the year, after we saw this was something other people were willing to get behind, we sent an email about to a number of kids. We figured for the first year, to get it started, the other teacher and I would reach out to individual kids we thought might enjoy the process. (We figured that after the journal was established, the student editors would decide how the leadership would pass from year to year.) So we made a list of kids in various grades (because we wanted kids in various grades) and emailed them to see if any of them were interested. We got four responses, and so it worked out perfectly…. two juniors and two sophomores. Yay!

This was all at the end of last year. We left for summer.

And then we came back. After about a month into school, we had our first meeting with the editors.

## Initial Stages with Kids

I looked at our meeting notes for the first meeting. They included us brainstorming a list of all the possible things that could go in the journal. Brainstorming names for the journal. Brainstorming all the clubs and classes that do things that we could envision having connections to the journal (e.g. computer science club, trebuchet club, earth club, math club, chem club, the science research seminar, the math applications elective).

## Creating Structures and Actionable Goals

We came up with adjectives to describe the journal, and listened to various ideas. And through our discussion, we were able to come up with a name for the journal. It just kinda popped out. This was important because we needed to start reaching out to others about it, and we needed something to call it.

We brainstormed ways to promote the journal (ideas: a facebook group, individually reaching out to clubs, having students make presentations in department meetings to get teachers on board, getting t-shirts made).

And we left with the following list of things we knew we needed to do.

The next big thing for us is that we needed to create a website where we could send people to, so they could learn about the journal.

## The Website Gave Us Direction

It actually turned out to be that working on the website is the thing that gave concreteness to the abstractness. I met with a student who was going to lead the website effort after school. We did some planning:

And as we did that, we realized that we needed a statement of purpose for the site written by the student editors, as well as an explanation as to why they agreed to work on it. In addition, we agreed that we needed to have a page with some example ideas of what students could submit, and finally, we needed a submission form and a method for students to submit.

So at our next meeting, coming up with these things became our focus. We needed these things for the website, yes, but we also needed these things ready so we could start soliciting submissions.

It took a while, but eventually we created a way to get submissions (a form that students fill out and email to one student editor, who was going to be our submissions pointperson), a statement of purpose and a video trailer expanding upon it, and a page filled with the types of things that students can submit. We also started working on coming up with ideas for the cover and for the logo.

## Soliciting Submissions

We still didn’t have an editorial policy. Would we accept everything that was submitted? How would we edit these articles? What if there were mistakes, or the submission was really unclear? However, we quickly realized we couldn’t make those decisions in this first year without having seen what sorts of things people were going to submit. So we tabled those more theoretical discussions and decided it was time to start soliciting submissions from students. We had the website, we had a process for kids to submit. Now we just needed to get buy in.

To do this, the students decided to reach out in various ways. Each student editor decided to become responsible for soliciting submissions from different groups.

Math teachers. Science teachers. Numerous Clubs. Various classes working on things.

And they did. And now that we had our website up, we could direct people there.

We also had signed up for a “Community Meeting” presentation. Every other week, the entire high school gathers together while a club or group gives a presentation about something or another. To get the word out, the kids created a 10 minute presentation to share with the community what the journal is all about, what sorts of things can be submitted, and how they can actually submit. The big thing we kept on remembering when designing this was that we wanted the journal to appeal to all students. As we had decided earlier, we wanted a low barrier for participation, and the kids emphasized that.

We also had come up with a submissions deadline, and we made that public for the first time.

They kids rocked it, and gave a hilarious and informative presentation.

## Waiting… but not Patiently

Now the word was out. Teachers knew about it. Clubs knew about it. Students knew about it. We had to wait for submissions.

But we weren’t passive. The kids brainstormed projects that students already had done in class, and went to those teachers. The kids heard when someone was going to submit something, and encouraged that student to follow through. I encouraged more than a few kids to take something they were interested in and turn it into something they could submit. We kept an unofficial ongoing list of people working on things that we knew were going to come to us, and for a few months, the student editors were each responsible for getting those people to follow through.

And through this process, the editors were being proactive, and because of that we got a good number of submissions!

## Putting Everything Online

We got a lot of great things, and we didn’t want to turn anything away, so we accepted all the articles that came to us. The student editors came up with a way to divy up who was going to edit each article, and they were going to have a second editor double check the work.

A couple editors took put the initial drafts of the articles online (a very time-consuming task), and then everyone else helped out with fixing things up and formatting them.

And finally, we came up with a way to divy up the articles into four different categories, I taught the kids how to make menus on the website, and then I let them go at it. You can imagine how much work that involved. But without a single complaint, they put everything online.

## Celebrate

To launch Intersections, we planned a pizza party. We invited all people who submitted articles, as well as all the math, science, and computer science teachers, and the head of the upper school and the head of school. A ton of people came and partook in the pizza delight. The kids had planned a presentation to talk about the process of putting the journal together, and to highlight a few of the things in it. At the end of their presentation, the kids had secretly brought flowers for me and the other teacher!

Most importantly, the launch party was the time where the kids who put in so much of their time for this journal were able to be seen, and get recognized for their hard work.

It was a primary goal of this project — for both me and the other teacher — to have this be a very student led enterprise. For it to be meaningful, they had to be in charge of it. They had to make the big decisions. They had to take responsibility for doing much of the heavy lifting. And I was more than impressed. They not only rose to the challenge, but surpassed every expectation I had. When we would talk about one thing, they not only would do it, but then have twelve more ideas that would come out of it, and do those too.

Fin.

## Post Script

There are lots of next steps with this. The biggest thing that I feel needs to happen next year is to have more teacher participation. Although a few teachers clearly mentored students and encouraged them to submit, I feel we could have done a lot better on that front.

# A High School Math-Science Journal

In my first year of teaching, fresh from my haze from history grad school, I remember approaching the history and English department chairs about creating a high school level journal for those subjects. I mean, our school has a literary magazine, and also even a publication for works in foreign languages (seriously!). But nothing for amazing critical analyses and interpretations in English and history. I figured having something like this might encourage students to revise already excellent work for publication, and also make the audience of their paper be an audience of more than one. I even contacted the literary magazine student editors to see if they would feel like the journal would encroach on their domain (they said no). For reasons that are still quite beyond my understanding (because I still think it’s an amazing idea), both department heads rebuffed my idea. (Also, if they said yes, they would have gotten an enthusiastic first year teacher who would have taken on all this work!)

And so, I let this idea pass. One of many that I have, think are awesome, and then languish and die, either due to my own laziness or due to external circumstances beyond my control.

Until last year. When I was thinking: I’m a math teacher. Why not start a math and science journal? It’s so obvious that I don’t know why the idea didn’t hit me over the head years ago. So I found a science teacher compatriot who I knew would be interested, and we came up with an initial plan. And at the end of last year, we presented it to some students who we thought might have been interested (as this was something that is something that has to be for them, by them… if they don’t want it, there’s not point in doing it… it’s not about us…). They were, and we were officially off to the races.

We shared with the students the following document we made, with a brief outline of one vision for the journal. But with the understanding that this was their thing so their ideas reign supreme. This was, in some sense, a mock-up that the science teacher and I made to show them one possibility. The one thing that the science teacher and I were really aiming for in our mock-up was that the journal shouldn’t just be for superstar students. We wanted to come up with an journal that has a low barrier of entry for students submitting to the journal, and that if a student has interest or a passion for math or science, that’s really all they need to get started. To do this, original and deep research wasn’t really the primary focus of the journal. So here’s our brief proposal:

The additional benefit of having this journal is hopefully it will cause curricular changes. Teachers will hopefully feel moved to create assignments that go “outside of the box” — and that could result in things being submitted. Students who express an interest in some math-y or science-y idea (like why is 0/0 undefined… something that came up in calculus this week) could have a teacher say “hey, that’s great… why don’t you look it up and do a 3 minute presentation on what you find tomorrow?” … and if they do a good job, encourage them to write it up for the journal. Or a teacher might assign a group project on nuclear disasters, and encourages the students who do extraordinary work to submit their project to the journal. (Which can be showcased by teachers the following year!) Or a student who notices a neat pattern, or comes up with an innovative explanation for something, or who wants to try to create their own sudoku puzzle, or decides to research fractions that satisfy $\frac{1}{a}+\frac{1}{b}=\frac{2}{a+b}$. Or whatever. Knowing there is a publication you can direct the student to, as a way to say “hey, you’re doing something awesome… seriously… so awesome I think you kind of have to share it with others!” is going to be so cool for teachers. (As a random aside, I was thinking I could enlist the help of the art and photography teachers, because of the overlap between math and art… They might make an assignment based around something mathematical/geometrical, which students can submit…)

I honestly have no idea how this is going to turn out. What’s going to happen. How the word is going to get out. If anything will be submitted. If kids get excited about it. Lots of questions. But I have a deep feeling that the answers will come and good things are going to happen with this.

I’m soliciting in the comments any thoughts you might have about this. If your school does a math journal, a science journal, or a math-science journal, what does it look like? What works and what doesn’t? Do you have a website/sample we could look at? If you don’t have one, and you are inspired and think of awesome things kids could put in there (e.g. kids submitting their own puzzles! kids writing book reviews of popular math/science books, or biographies of mathematicians/scientists! getting kids to create photographs or computer images of science or data visualization or just making geometrical graphing designs! trust me — brainstorming this is super fun!) I’d love to share any and all ideas with the kids involved with this project at my school.

# Students communicating mathematics has opened my eyes to mathematical ugliness (and what that means to me)

This year, as I have been in the past few years, I’ve been attempting to incorporate more writing in my math classes [note: Shelli found a post from 2009 I wrote on this endeavor]. It’s been extraordinarily enlightening, because what this has done is show me two things: (1) kids don’t know how to explain their reasoning in clear ways, and (2) I’m usually extraordinarily wrong when I think my kids understand something, and the extent to which I am wrong makes me cringe.

(wow, been too busy to shave, have we Mr. Shah?)

For the first point, I don’t actually do much. I ask them to write, they write, I comment. And we discuss (more at the start of the year, but I always let this go and I forget to talk about it a lot). In Algebra II, they get one or two writing questions on every assessment. And each quarter they had problem sets where they had to write out their thought processes/solutions comprehensively and clearly. Even though I didn’t actually do anything systematic and formal in terms of teaching them to write (mainly I just had them write), I can say that I’ve seen a huge huge improvement in their explanatory skills from the beginning of the year. What I used to get just didn’t make sense, honestly. A random string of words that made sense in their heads, but not to anyone reading them. But now I get much more comprehensive explanations, which usually include words, diagrams, graphs, examples. They aren’t usually amazing, but they’re not ready to be amazing.

For the second point, I realized that the types of questions that we tend to ask (you know, those more routine questions that all textbooks ask) don’t always let me know if a student understands what they’re doing. It just lets me know they can do a procedure. So, for example, if I asked students to graph $y < 2x+3$, I would bet my Algebra II kids would be able to. But if I showed them the question and the solution, and ask them to explain what the solution to that question means, I would expect that only half or two thirds of the class would get it right. (Hint: The solution is the set of all points (x,y) which make the inequality a true statement.) They can do the procedure, but they don’t know what the solution means? That’s what I’ve found. And you know what? Before asking students to write in the classroom, I had deceived myself into conflating students being able to answer $y < 2x+3$ with a full understanding of 2-D linear inequalities. [1]

Before having students write, I actually believed that if I asked that question (“What does this solution mean?”), almost all the students would be able to answer it. (“Like, duh, of course they can!”) But since asking students to explain themselves, explain mathematics, I’ve uncovered the nasty underbelly to what students truly understand. The horror! The horror! But now that I recognize this seedy underworld of misconceptions or no-conceptions, I’ve finally been able to get beyond the despair that I originally had. Because now I know I have a place to work from.

The counterside to this point is that when kids do understand something, they kill it.

This simple question I made for my calculus students early in the year, and this student response, says it all. I have no concern about this kid understanding relative maxs and mins. No traditional question would have let me see how well this student knew what was up.

For me the obvious corollary is that: we need to start rethinking what our assessments ought to look like. If we want kids to truly understand concepts deeply, why don’t we actually make assessments that require students to demonstrate deep understanding of concepts? I am coming to the realization that the more we keep giving the same-old-same-old-assessments, the more we are reinforcing the message (implicitly) that we don’t reallyreally care to know about their thinking. We are telling our kids (implicitly) that we are content if they show their algebraic steps. But as I’ve noted, my big realization is that students performing those algebraic steps don’t necessarily mean that the student knows what they’re doing, or what the big picture is.

I don’t know have an example of what I think a truly ideal assessment might look like, but I do know it isn’t anything like I gave when I started off teaching five years ago (has it really been five years? why am I not better at this?), and I do know that each year I am slowly inching towards something better. Right now, my assessments are fairly traditional, but with each year, they are getting less so.

Sorry if I’ve posted something like this before. I have a feeling I have. But it’s what’s been going through my head recently, and I wanted to get it out there before I lost it.

[1] Another good illustration might be having students solve $-3x<6$. Sure, they can get $x>-2$. But does doing that really mean they understand that whole “if you divide by a negative in an inequality, you switch the direction of the inequality” rule that has been pounded in them since seventh grade? Nope. The traditional questions don’t tend to check if the kids know why they’re doing what they’re doing.

# Absolute Value

So I taught absolute value equations in Algebra II. And so far I think things have gone fairly well. I read Kate Nowak’s post on how she did absolute values, and I thought I would change my more traditional introduction to them… but I didn’t. I realized that the way Kate was motivating it (with the distance on the numberline model) was great, but I felt I could still get deep conceptual understanding with the traditional way she eschewed in her post.

So I stuck with that.

I used exit cards to see how they could do… and they were okay.

But after learning how to solve $|2x-3|=5$ or $|2x-3|=-10$, I asked kids to solve things like $2-5|5x+6|=5$ or something similar. Many students said on their home enjoyment:

$2-5(5x+6)=5$ or $2-5(5x+6)=-5$.

It is unsurprising to me, and yet, it makes me want to throw up. Because what’s coming more and more into focus, and I’m sure you’re going to hear me complain about this more and more in the coming months, is how reliant students are on “coming up with rules” and “applying rules” — without thinking. They desperately want unthinking rules. And this year, because I can’t handle throwing up all the time, I’m vowing to really not give rules to them.

I really got to the heart of this “I LIKE PROCEDURES” thing with them with a true-false activity that I did, using my poor man clickers. I think this exercise highlighted how dependent my kids are on procedures and coming up with simple rules that help them in the short term… but that can hurt them in the long term… It’s a bunch of True-False questions. And when we talked about each one of them, my class saw concretely how reliant they were on misconceptions and false rules. EVERY SINGLE QUESTION led to a great short discussion.

So here they are, for you to use. Sadly, I don’t have the blank slides to share with you, because my school laptop is not with me at home now.

These were great for asking “so who wants to justify their answer?”

# And So It Begins…

The year in full swing, and it feels like I’ve been teaching for days upon days, even though it has only been two days, so I suppose I should have said “day upon day.” It shocks me (BZZZ!) that a person can go from lazing about, jaunting off for coffee, picking up a book and reading it through in a day, watching an entire season of real housewives of (insertanycityhereandit’sprollytrueforme), going to the restroom whenever you please… to being trapped in a building (no AC!) with a hierarchy, having to answer to a lot of someone elses, having inhaled and not having the opportunity to exhale until hours later. And then you remember: oh yeah, I have to plan for the next day.

So it’s like I’ve never left. And I love it. There are things I cringe at, but heck if seeing my kids and my colleague friends, and getting to think about how I can do what I do but less sucky: it’s thrilling. I suspect this glow will be gone in a week, so don’t worry: my normal self will return soon enough.

Glow Self:

I just wanted to talk about the first two days of Algebra II. I usually start out the year with a honest but (upon reflection when I looked at it a few days ago) boring exhortation about mathematics and why it’s useful, beautiful, interesting. Then I talk a bit about the course expectations. And then we jump straight into talking about sets. I did it this way because I wanted to dive right in and show them what I valued: doing math. This wasn’t going to be a class where we get derailed with non-math things.

Well, I was unsatisfied with that, because it was boring. A boring set of slides with me speaking (albeit with a wildly inflecting voice, which can make anything less boring), followed by possibly the most boring topic: union and intersection of sets. It also was me lecturing about sets.

This year I vowed to take risks in how I teach. Less lecturing. Less partner work. More group work. More deep thinking and problem solving. And since I made a post saying some of the things I wanted to try, I decided to scrap everything and start anew.

I looked through the Park School of Baltimore’s curriculum and found a perfect thing to transition us into sets: mathematical symbols.

So on the first day, I sat kids down in their seats, I explained how they were to move their chairs to get in their groups. I asked them how they were feeling, I told them my goal was to make them feel good about math. Then, suddenly, I asked students to get in their groups. I projected the first page of the Park School packet that I photocopied. We did one part of one problem together (I had kids read the problem aloud and work in their groups to come up with the answer). Then I set them free, after handing out the packet, with only the following instructions.

Then they started (some faster than others) and I went to the following SmartBoard page [update: here if you want to download it]…

… and started the participation quiz (what I’m calling “groupwork feedback”). [To understand what comes next, you have to read the link above.] I didn’t explain anything. I just typed and dragged and typed and typed and walked around. Kids would ask me questions, and I would just shrug. They stopped asking me questions and started relying on each other and their brains. I didn’t stop groups which were off task. One group of four broke up into two groups of two, and then rejoined. I just kept on filling in the grid, not talking about it.

Honestly, the idea that I would have to be filling in this grid scared me. I didn’t know if I was going to be able to do it. I didn’t know if I would have the heart to put “off task” if a group were off task. I didn’t know if I could keep up, or if I could hear the kids talking, or keep track of everything. But it was easier than I thought. Students worked for about 30 minutes. I think that’s the right amount of time, because I wouldn’t have gotten a critical mass of feedback if they had worked any less.

Then I stopped them. What I noticed after doing it in two classes is that engaging in this type of observations of groups is super interesting and helpful for me. I had a good sense of which groups knew how to do groupwork already and which groups didn’t. I heard some great conversations, really great conversations, about some rich problems (“does it mean that the only way to get an odd number is with …”). I saw group dynamics at work (especially the difficulties that present themselves with groups of 4). I also saw that one of my two classes already has a good handle on how to work in groups, and the other is going to need some time and coaching.

We spent 12 minutes talking about the results. We talked about if “I don’t know” is a good or bad thing to have on that chart (it depends…) and finally I asked groups to look at this thing that I whipped up (not great, but I needed something) and to classify themselves, and to think of some ways they could improve and think of some things they did well. And we went around and had each group explain.

Although terrifying, I’m glad I did it on the first day. It was scary to try something new (new problems! groupwork feedback!). I feel confident that I showed my kids what I hope to value in the classroom this year. Communication. True thinking. Independence. Collaboration in the learning process. (I don’t see the last two things as contradictory.)

That was the first day. Today (the second day) I saw only one of my classes. And what I did in it didn’t unfold nearly as well, in my opinion. I wanted kids to present their solutions. The night before I had them do a few more problems on their own, so I gave groups 8 minutes at the start of class to talk through their work, telling ’em that they were going to be asked to explain.

Then I had individual students come up and explain their work for some of the problems (after a short discussion on how it’s great to not get something and to have misconceptions / confusions, because that’s where we learn, and a discussion on how to be a good audience for the explainers).

They put their work up under the document projector. And talked. But what I learned is: I need to work on having students be effective presenters. And how to encourage the audience engage with the presenters more. And how to balance me intervening versus letting the student go on. (It’s hard for me to let go of the “explain” part of class.) So now I know I have to work on this. (Luckily I was meeting with my teacher friend mentor for lunch, who does a lot of modeling work in her classroom, and she had a lot of good things to suggest. )

So there we are. I’m trying to be very intentional (thanks @bowmanimal for the word) in how I start the year. I also printed out “exit slips” for my classes tomorrow because my goal is to get formative feedback at least once a week in each class. And I tried to do “What’s the Question?” (known in my class as “Que es la Pregunta?”) in Calculus to activate prior knowledge on rational functions. However it kinda totally fell flat. It did what it should have, but it wasn’t as enjoyable/fun as I hoped. I think I might need to rethink how I set it up.

And there you are. Some words on the first couple days of school.

# Math Taboo

I participated in a great twitter conversation the other day where we brainstormed a few strategies to help make our courses more accessible to English Language Learners (we used the hashtag #ELLmath, the approximate transcript is here if you are interested). It was a great start to what needs to be a running dialogue for me, as I teach almost 100% students for whom English is not their first language. If anyone has any ideas about #ELLmath, I would love to hear them in the comments. The conversation reminded me of a little idea I had last year, playing the game Math Taboo to help students expand “definitions” to actual understandings of concepts. Now, I’m sure other people do this, and a quick Google search leads me to believe it’s not all that novel, but while discussing #ELLmath, it struck me as a particularly good exercise for ELL students.

The idea of the real game is to get your partner to guess a word by describing without using any of the five taboo words, which are usually the first words that anyone would go to in a description. So the obvious math equivalent is to pick a term that you are throwing around in your class and get students to describe it without using their go-to math descriptors.

We played during beginning-of-the-year-review as a class, with the word to guess already known to everyone, and I gave students a chance to take a stab at verbalizing a definition without using the taboo words, one at a time until we got an acceptable description. However, this could easily be adapted to be a much more interactive activity (though its creation might take just a bit of time).

#### So why play this?

Whenever working one on one with students, I found myself trying to diagnose why they were not understanding a problem. I would ask them things like, “Well, what is a derivative anyway?” and they would often answer with something that I found acceptable, but perhaps could have been just something that they had figured out should be said as the “correct” answer. Even if they weren’t saying book definitions (which would actually be easier to deal with), many times they were using my informal definitions – words that they had internalized about the concept that might not actually display a deep understanding, but that I had been mistakenly accepting as evidence of learning. Definitions are important, but assuming that those are indicators of deep understanding is, of course, very problematic, no matter where those definitions come from.

So, this Taboo game serves a two-fold purpose: learning for the students (by forcing them to think deeply about a mathematical concept; by having them trade in math jargon for conceptual understanding; and by hearing classmates describe something in more accessible vernacular) and learning for me (by seeing how well students actually understand a concept; and by seeing what language students use to talk math in the hopes that my mathematical narrative can better reflect theirs in the future).

#### Alternative game: In how few words can you express this definition?

I have never tried this game I’m about to describe, but the idea is to start out with a long definition from a math textbook and see how few words you can use to express the same idea. Delving into the Twitter world this summer I have realized how wordy I am, and the process of editing my tweets down has made me realize how many words I use that are unnecessary. Twitter forces me to think about what is the core of my idea, which led me to think up this exercise. This could be done competitively (give groups 5 minutes to brainstorm), or you could do it countdown style, trying to lower the number of words by one each time. This could get students to really consider what is important about a mathematical concept and to get them to realize that the thing itself is more important the words you use to express it.