Our presentation is here:

The crux of the presentation is that we (not just Tina and me, but many in the #MTBoS) have done a lot to make the #MTBoS community more welcoming and accessible to newcomers (the ExploreMTBoS initiative and mentoring program, the mathtwitterblogosphere website).There are conferences (#TMC) and tweetups (all over the place). There is a #MTBoS booth that travels to various (often NCTM) conferences and is manned by #MTBoS participants, to spread the word!

Other #MTBoS created things are available that are useful for teachers who don’t participate in the #MTBoS. There are books that have been written by #MTBoS-ers (e.g. Nix The Tricks, The Classroom Chef). There are website that are created by #MTBoS-ers and used by teachers everywhere (e.g. Visual Patterns, Which One Doesn’t Belong, Fraction Talks, Estimation 180, Would You Rather, Open Middle). There are podcasts (e.g. Tales from the Chalkline, Infinite Tangents). There are webinars and the Global Math Department Newsletter which rounds up and distills stuff from the community.

There are a number of smaller #MTBoS intiative that have happened pretty organically: A Day in the Life initiative, and the Letters to a First Year Teacher initiative, and Virtual Conferences.

And there were fun community building things, like Harlem Shake (Tweep Version) and Twittereen (and the now defunct, for those who remember, “Favorite Tweets”).

** All of this is to say: for those who are interested, there are many ways to help the community. You just have to find something you love about the #MTBoS, and then come up with a way to create/share/expand it with others. (That often involves breaking the idea into smaller chunks, getting other people on board to help, and actually holding each other accountable.) **

**The #MTBoS doesn’t have a set of leaders. It only works because of the members. You don’t need to ask for permission. You don’t need to have been tweeting or blogging for months/years. You don’t need a “huge” project. You simply need to decide you want to do something, and do it. **

That is what our session was about. We shared some ideas that we had for places the community could grow, and ways people could actually do it, and then had people share their own thoughts and ideas.

Personally, the projects I’d love to see someone take on:

(1) **Department presentations:** I’m all about “packaging” something to make it easier for others to use. So I’d love for a group of people to create 3-4 “Introduction to the #MTBoS” presentations/workshops that math teachers can give to their departments. They can be different styles/lengths, and can have different activities involved. (For example, I made my whole department sign up for the GMD newsletter. At another presentation, I made a #MTBoS scavenger hunt, where different finds/activities were worth different points.) Then, anyone who wants can choose one and adapt it to make it work best when they want to evangelize the #MTBoS to their in-real-life colleagues! [Note: A number of #MTBoS presentations have been archived in the comments here.]

(2) **A #MTBoS video:** I saw PCMI (a math teacher conference I’ve been to) created a video to “sell” the program. I would love it if there were a #MTBoS video which captured the essence of what the community is. Maybe 30-60 seconds. Something professional that evokes feelings and excitement, the emotional essence of #MTBoS, rather than outlining what it all has to offer… Capturing lighting in a bottle, that is what I suppose I’m asking for. But if this can be done well, well… I think it could serve our community well.

(3) **So you want to have a tweet up…: **A number of people have held tweet-ups by now. I think it would be good if there could be “instructions on how to organize a tweet up” — from how to find people and contact them about attending to how to find a space to hold it to what to do at a tweet up. Again, perhaps two or three different “packages” for what tweetups could look like! This might make it easier for someone who might want to organize their own tweet up!

(4) **NCTM article: **I’d love for someone to write an article about the #MTBoS community for *Mathematics Teacher *(or another NCTM journal) – to share what the community is about, how it has affected someone’s teaching practice, and to show ways for others who might be curious how to get involved. There is also a call for articles for the 2018 Focus Issue which is on *Tool Kits for Early Career Teachers *which I think a really wonderful article about #MTBoS could be beneficial.

I wonder if two newbie #MTBoS-ers and two experienced #MTBoS-ers could collaborate on writing it! I am personally interested in having this happen because I think it is a way to spread the word through more traditional channels, and might just pique the interest of a lot of teachers!

(5) **Getting Goofy: **In addition to things to expand the reach of the #MTBoS, I think there is room for so much more goofy things that can happen (today I saw a tweet that said #keepmtbosweird, copyright @rdkpickle). I don’t know what this might be, but some sort of goofy community building event like twittereen or the great hedgehog sweater run or needaredstamp. A massive picture-based scavenger hunt? A virtual trivia night? A stupid funny poster contest?

(6) **Appending #MTBoS to Existing Conferences: **A number of people who are going to conferences (e.g. CMC south, Asilomar, NCTM) are planning 2-hour meet-ups with #MTBoS-ers. I think it could even be #MTBoS-ers arrive a day early or stay a day late and have a mini-get together (or even a super mini conference in the hotel!). I’d love a “package” that outlines how to organize one of these meet ups.

(7) **Get more contributors to the One Good Thing blog:** I love the

I had one more idea that I have decided I am going to take on… For those who remember them… I am going to bring back Virtual Conferences. I loved the idea of them, and the person who hosted them is no longer doing them… so I’m going to bring them back from the dead!

The ideas above are things I’ve been mulling over. The ideas that came up in our meeting, or on twitter afterwards (using hashtag #ExpandMTBoS) are below (in the pictures or in the storify):

These ideas include involving Reddit, making a landing page website/app, creating a MTBoS logo, having teachers tell more of their stories, etc.

**Choose something small, like presenting the community to your department or manning the #MTBoS booth at NCTM. Choose something huge, like creating your own conference, or website on (topic x), or writing a book. Or choose anything in between. But if you have the time and inclination, think of a way you can help #ExpandMTBoS! **

If you have an idea of something you want to do, tweet it out with the #ExpandMTBoS hashtag. Get people to help you! And make your idea a reality!

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Sara VanDerWerf (her blog) gave a keynote that was reminiscent of a keynote last year. She said “What are you an evangelist for?” (For her, one of those things is Desmos, because of the equity and access it allows her kids.) Once you know that thing — the thing you are willing to go to bat for, the thing you want to spread — you should think consciously about how to best evangelize it. That might include having an elevator speech ready for you to give, and being conscious of the different audiences you may be talking with about it (students? parents? teachers? admin?). Being an *evangelist* isn’t just being passionate… it includes enacting that passion by finding ways to share “the best… with others who can benefit.”

Sara’s fabulous calculator museum (mausoleum!)… all your calculators are dead… all hail Desmos!

I know I am an evangelist for the #MTBoS. However in terms of math content or math teaching, I don’t quite know what I’m an evangelist for… yet. All this reminds me the end of this blogpost I wrote last year after TMC, where I was trying to figure out what my “brand” was (and came up emptyhanded). But I have faith that with enough time, I’ll figure it out.

Speaking of evangilism… Jonathan Claydon (his blog) shared a “my favorite” about *Varsity Math*, a community he’s built up at his school. I’ve had a teacher crush on this guy for years. There’s something about his energy and style and humor, and the fact that he is good at something I am not (yet) good at (being a “relational” teacher)… he’s a must follow. In any case, Jonathan is an *evangelist* for changing the way kids look at math at his school. Although ostensibly his goal is to increase the numbers of kids taking AP math classes and increase the AP scores of these students, he’s doing it by building a supportive math community — one that feels like a club. He is doing this by creating “shared experiences.” He knows he has succeeded if he can get kids to say “I love (varsity math). (Varsity math) feels like family. You couldn’t understand because you’re not in (varsity math).” The only way the last statement could make sense is if an entire culture is built around (varsity math). Of course what goes in the parentheses is open. Read about his project here. See a photo of @rawrdimus here:

This “my favorite” spoke to me. I’ve been consciously working at my school about raising the math department. Not in terms of teaching and learning (I don’t have much say in that), but in terms of getting kids engaging with math outside of the math classroom. I brought the New York Math League contest to school, I’ve worked (with another teacher) to concertedly increase the number of students taking the American Math Competition each year (from around a dozen to seventy+). I found a non-stressful virtual math team competition that students can compete in so that they can fit it in their busy schedules. I have co-advised math club for years. I started *Intersections *with a science teacher, a math-science journal for students to submit their works to (it’s now four years old!). Lots of things… I want spaces and times for students to engage in math outside of the classroom. But with all of this, I don’t see a culture of kids who geek out about math. There isn’t a community or culture around doing mathematics at my school. And Jonathan’s talk helped me realize that I have to think intentionally about building a community. It is more than “if you create it, they will come.” It isn’t the event or space that I design, but the “shared experience.” What does this mean? What does this look like? I don’t know yet. But perhaps having a student-created chant before each virtual math team competition, bonding field trips (math movies? museum of math? math scavenger hunt?), swag as proud identifiers, a wall of fame…

At the Desmos preconference, I learned about three things

(1) “Listening to graphs.” This feature was included for vision impaired students, but I think many of us teachers started dreaming up other uses for it. To get a sense of it, check out this piece (done by Rachel Kernodle and James Cleveland) playing “Mary Had A Little Lamb” (click on image):

To play (at least on a mac), press COMMAND F5 (which enables voiceover), go to the fifth line and press OPTION T (to tell the computer to “read” the graph with sound), and then press H (to play the graph). When it’s done, you can turn off voiceover by pressing COMMAND F5 again.

Some thoughts… Have the audio for some periodic and non periodic functions, and have kids do an audio function sort? Play audio of graphs (without telling kids that) and have kids do a notice/wonder (before sharing what they are listening to). Have kids identify if a graph has a horizontal asymptote for end behavior from an audio file? Have kids identify which graphs might have a vertical asymptote from an audiofile? Play sine and cosine (or secant and cosecant) and have kids not be able to tell which is which (because they are just horizontal shifts of each other). Have kids devise their own piecewise functions and play them, while other kids have to graph them. Create a piecewise function and have a student who enjoys singing to sing it? I am not convinced that anything I’ve thought up could help a *deeper* understanding of any topic, but I also don’t think it could hurt. Some kids might really get into it and enjoy *playing* with math…

(2) Card Sort: You can create card sorts in desmos now! Check a bunch of them out (that were created at the Desmos pre-conference)! Or if you just want to go to one of them, click on the image of Mattie Baker’s card sort on visual sequences:

To gain this functionality on your desmos account, go to teacher.desmos.com and click on your name in the upper right hand corner, click on LABS, and then turn on Card Sort.

(3) Marbleslides: You can create your own marbleslides in desmos also! Turn it on in labs (see above). Then you have the capability of building your own! If you don’t know about marbleslides, check out this marbleslides activity made by the desmos folk on periodics. At least to me, the use of marbleslides is to help students understand function transformations… so I can see it useful for helping kids gain fluency in transformations. (Anyone see another use for marbleslides, that I’m missing?)

Hedge talked about how she uses SnagIt to display student work. She takes a photo of student work on her phone, and using an app called FUSE, transfers it to SnagIt (on the laptop) — as long as both are on the same wifi network. Here’s her blogpost showing it in action! It costs money ($29.95) but I trust Hedge!

I attended PCMI years ago, and I recall Bowen and Darryl using this technique (kids working on problems, taking pictures of different approaches) to facilitate discussion to bring different ideas together. Nearing the end of a session, they would project pictures of student work, people would explain their thinking. Bowen and Darryl would sequence the pictures in a thoughtful way. They wouldn’t focus on those who “got the answer” but on various approaches (visual/algebraic) — whether they worked to get the answer or not. I liked that so much, and I suspect SnagIt could allow that to work for me in that way.

Fouss gave a wonderful hour long session on making trigonometry hands-on for students. Instead of *telling* us what she did, we got to do some of the activities, and that was powerful. There were activities I’ve read about that I thought “eh, okay, but it would be more efficient to do X, Y, and Z” and then I did them and I saw how the act of doing them could be helpful. Here are three that we got to do: understanding radians with smarties, creating a unit circle with patty paper, and creating a trig wheel to help kids practice converting between radians and degrees *and* visualize what the size of the angles look like.

All her materials are linked to from her presentation, and are easily found on this folder on her google drive. I have to scour them to find my favorites. I did love the radian activity. If you make the radius of the unit circle 7 smarties long, then you can have a good discussion on whether 3 radians is 180 degrees or not… (21 smarties won’t quite make it to 180 degrees… but 22 smarties will fit snugly… nicely giving the 22/7 approximation for . Nice!)

Some of the ideas linked to from her presentations that I want to steal:

(a) Trig Stations

(b) Two Truths and a Lie (useful for more than just trig!)

(c) #TrigIs (useful for more than just trig!)

(d) If I choose to do ferris wheel problems, this ferris wheel comparison [but modified to be more challenging]

(e) Desmos’s Polygraph for Sinusoids and Marbleslides for Periodic Functions

(f) If I teach trig identities, use this matching game (and have kids check their answers once they are done by graphing on desmos!)

(g) Headbandz, trig edition! (for graphing trig functions)

Joel Bezaire presented a great game that can be used in warmups to help students see relationships and patterns. His video on it is here, showing the game and how it is played:

Kathryn Belmont (@iisanumber) gave a great way to have kids really put forth effort on open-ended assignments without using grades as a stick. She will ask kids to do this assignments, and then put their work on their desks. Each student gets posts its, and as they wander around the room, they put post-its on the works they see… They write two accolades for good things, and two ways to push back or improve the assignment. The way I envision this in my classroom, not everyone will see everyone else’s work, but everyone will see 5-6 other students’s work. After the walk about, the teacher says: “Do you have any nominations”? Jake might reply “I would like to nominate Kiara.” If Kiara feels okay about being nominated and “accepts the nomination,” the teacher takes Kiara’s work and puts it under the document camera. Then Jake might say, “Kiara did … and what I thought was so awesome about it was …”

(Her slides for her mini-talk are here. A video of her talk is here.)

The teacher is no longer the sole audience member for the work, and kids are defining what good work looks like. In Kathryn’s classroom, she saw a huge increase in kids putting in effort in these open-ended assignments. (I can see this being useful in my own class, especially when I do my explore math mini-explorations.)

I went to a session by Jessica Breur (@BreurBreur) which was fantastic. Although it was only one hour, I wish it were a morning session. She wants to have teachers establish a culture where students:

- use the group to move the group forward
- talk, trust, and depend on classmates and the teacher
- persist — even in the face of a challenge
- view math as “figure-out-able” and accessible to all

She highly recommended Cohen’s *Designing Groupwork* (a book which I have but haven’t read).

To start, over the first week or two, students will be doing lots of groupwork activities. And at the end of them, they will (in their smaller groups) focus on what the group “looks like” “sounds like” and “feels like.” They don’t necessarily need to focus on all three at once — students could focus on “sounds like” during one activity and “feels like” on another. After the week is done, the class comes up with a set of norms in these three categories — where they talk about what successful/good/fun groups look/sound/feel like.

We did a lot of hands-on work trying out some of these groupwork activities — and she has included all of those activities in her slides. Here is one of my favorites:

This is the red solo cup challenge. A group of 3 or 4 is given 6 red solo cups, stacked inside each other, placed face up on the table (so like a regular drinking up face up). The students are given a rubber band with four strings tied to it (even if 3 students are doing this, keep the four strings). Student must put the solo cups in a pyramid formation. If they finish that, there are other configurations that Jessica includes in her presentations (or students can design their own challenge for others!). Afterwards, the group reflects.

Similar tasks can be done, like 100 Numbers, Saving Sam, Four 4s [but making an emphasis that we want *as many ways* to generate the numbers 1-20, not just one for each], Master Designer, or Draw My Picture.

For more “math-y” things, you can do a Chalk Talk/Graffiti Board– where students answer questions *before a unit* to activate some old ideas. For example, “What do you know about the number zero?” [In fact, any sort of talking point/debate-y statement can be used here.] Kids write anything and everything they know on a poster in their group of four. Then hand the posters up and students walk around and read other students’ responses (if time, writing their own comments down). Finally, for closure, you can ask students aloud or using exit slips “What are two things you didn’t think about that you saw on the graffiti boards?” Another more math-y thing is a donut percent task. An example is here but I’m confident it could be modified for trigonometry (values of trig functions, identities, etc.) or rational functions (equations and graphs) or any number of things! The idea behinds this is that each person in the group is given four slips of paper, and as a group, four complete donuts have to be created.

Sounds simple? But here’s the rub… group members must follow the rules below to each get their own donut completed.

You should keep a poster of the 8 Standards of Mathematical Practice, and every so often during activities or groupwork, ask students which ones they are using.

Once norms are established at the start of the year, you consciously need to be doing activities that practice the norms. Be intentional about it. (If you find that kids aren’t listening to each other, find an activity that promotes listening.)

I loved this session. However what I need now are a set of activity structures that I can fit actual mathematical work into. So things which develop understanding, or practice solving something, etc. And it would be nice not only to have the activity structures, but the activities themselves all in one place (so, for example, activities for Precalculus!).

My morning session was called “Talk Less, Smile More” and was led by Mattie Baker and Chris Luzniak. In the session, they provided various structures to promote math talk in the classroom. I am going to outline some of the ideas that I can see myself using in my classroom.

**DEFENSE MECHANISMS & CLASSROOM CULTURE:** Most importantly, to get talk in the math classroom involves getting over student defense mechanisms. Students fear being seen as stupid, and they fear being wrong. In order to do this, you have to *lower the stakes* so kids can temporarily bracket their defense mechanisms to create emotional safety. These could be by doing things like chalk talks (silently writing responses to questions, and responding to other student responses) or doing notice/wonder activities where all responses are honored. Many of the ideas that Chris and Mattie shared in the session do this, by providing a structure for talking, and a bit of a safety net (often where no response is right, or students are required to give a particular answer and justify it).

When implementing it, you have to be consistent and do these structures fairly often. Start simple, and then get more complicated with the statements/questions. Give a lot of energy and excitement — especially if a student gives a wrong answer or a right answer (“Oh wow, what an interesting thought… let’s explore that…”). If students turn to the teacher and say “Mr. Shah, what about…” sit down and redirect it to the class. (Remember the teacher is not the center… this is about getting kids to be the center!) As teachers, we have to watch our own facial expressions (a.k.a. don’t make a face when you hear a totally wrong answer). You can avoid this (if it’s a problem for you) by looking down at a clipboard when someone is responding.

At the end of a class or a portion of a class with a lot of mathematical talk, do “shout outs” (shout out something they learned, or something someone else said that helped them). And ask kids (to fill out on a card) what they took away from class today (and what questions they still might have). Or “I used to think ____, but now I think _____.”

To give students some crutches when talking, have posters with these simple statement starters to help them (on *all four walls*):

**TALKING POINTS:** In this session I first got to experience Talking Points. I’ve read about them on Elizabeth Statmore’s blog (see links on the right… a bunch of talking points are hosted in one of her google drive folders). But the truth is: I wasn’t sure how much I could get out of them. Now that I’ve participated in one, I feel differently. This is how they work:

(1) students in a group of 4 get *n* statements. The first round involves one person reading the first statement, and then say “agree/disagree/unsure” and then explain why they chose that response. They must give the reason. The next person does the same, then the next, then the last. The important part about this is that no one can comment on another person’s reasons. They can just state their own reasons. They can match someone else’s reasons, but they have to be stated as their own.

(2) The second round involves the first person saying “agree/disagree/unsure” (after hearing everyone else’s thoughts) and *then* they can give reasons involving other people’s thoughts. Others do the same.

(3) The third round is quick and short. Each person says “agree/disagree/unsure” and gives no reasons. Then someone records the tally of the responses.

Here’s an example of what talking points can look like (when they aren’t about math content):

Talking points can also be math content related. Instead of “agree/disagree/unsure,” you can use “always/sometimes/never” or some variation that works for your questions. In our mini-precalculus group, we brainstormed some talking points around trigonometry:

After participating in talking points, we as a group came to the following realizations:

- Talking points were not as repetitive as we thought they would be.
- The more controversial a statement, the more discussion happens.
- You were really forced to listen to each other
- When the talking point includes “I” statements, you learn about other group members
- They are good for pre-assessments (and can be used before a unit starts, as a prelude)
- Give
*n*statements, and then leave 3 blank statements. If a group finishes early, they can write their own talking point statements! - Afterwards, you should have a “shout out” round. Kids should shout out something interesting/great they learned, and/or the teacher should shout out something good they heard/witnessed!

To debrief:

- Don’t go over all of the questions. That debrief will feel boring and repetative. Go over some key things you want to talk about immediately, and then revisit the others during the unit. (You want to make sure that kids don’t leave the unit with misconceptions.)
- Use the tally of A/D/U or A/S/N to see where the controversy lies! (You can collect their slips and talk about them later after seeing their responses…)

**CLAIM AND WARRANT DEBATE: **In a math class, you want students to justify themselves. To build that justification as central to the class, you can introduce the notions of an *argument* which is essentially a statement (a claim) made with sound reasoning (a warrant). (This language comes out of the speech and debate world.)

When responding to a question, a student must stand up (even the teacher should sit down) and say “My claim is _______, and my warrant is ________.” If the student messes up, that’s okay, just have them do it again. You have to build this structure as essential to answering questions. (To reduce the fear, you can give students some think time to write something down, or talk in a pair, before doing the claim/warrant step.) When doing this, I am *not* going to have kids volunteer… I am going to cold call using the Popsicle Sticks of Destiny (names of kids on popsicle sticks… I draw one randomly…).

When introducing claim/warrant, make sure you not only teach the structure, but also have kids who aren’t speaking face the speaker and put their eyes on them. Be explicit about the expectation. You can also have kids summarize another student’s point to make sure they’re paying attention. (If you catch a kid not following the audience instructions, you can walk over near them… if not, you can tap them on the shoulder… or kindly talk with them after class about how “it’s really polite to…”)

To build this up and create this as a routine and class structure, you should do claim and warrant debates every day or every other day at the start of the school year. Use the language “claim” and “warrant” on assessments too!

Types of questions you can ask to get kids started with this:

The best movie is ______.

The most important math topic is ______.

________ is the best method for solving the system y=2x and y=x+1.

[show a Which One Does Belong and say] ______ doesn’t belong.

Notice that each of these don’t have a “right” answer. It lowers the barrier of entry for kids.

One powerful type of question one can create are “mistake” questions. For example:

To extend claim/warrant, you can also create “circle debates” which truly forces listening. One person states a claim/warrant, and then another person *summarizes* that claim/warrant and then makes their own claim/warrant. This continues. It will sound like: “What I heard is that this statement is sometimes true because …. My claim is ____ and my warrant is ____.” I think only *very* open ended questions would be good for this structure.

Another powerful way to extend claim/warrant is to engage is “point-counterpoint.” Let’s say the statement is: “Would you rather have crayons for teeth or spaghetti for hair?” The first person makes a claim/warrant, and the second person (no matter their true feelings) must disagree and make the opposing claim and give a warrant. Then the third person opposes the second person. Etc. It forces students to think of other points of view. In a question like “_____ is the best way to solve this system of equations” it forces students who might only approach a system in one way to consider other methods and justifications for those other methods.

**CREATING DEBATE-Y QUESTIONS/STATEMENTS**: Use the following words:

In the session, we took all types of questions (e.g. Graph ) and came up with debate-y questions based on it (in this example, we said “what number would you change to change the graph the most?” or “what’s the best way to graph a sine function?”). I’m not yet good at this, but I found that even with a little practice and people to bounce ideas around, I’m getting better. We had fun in my group trying to come up with debate-y questions based on this random “do now” that Chris and Mattie found online:

I thought it would be impossible, but the group came up with tons of different ways to convert this to a debate-y statement: (a) without solving, which is easiest to solve? (b) which would you give to your worst enemy? (c) which are similar? (d) rank from easiest to hardest? (e) a 5th problem that would fit this set of equations would be ____ (f) a 5th problem that would not fit this set of equations would be ______ (g) which one doesn’t belong? (h) give -4(x+3)=-6 and ask what the most efficient way tot solve it? and then follow up with “how could you change the problem so that method is not the most efficient?”

After a month or two, the use of claim/warrant may die down. If kids get the idea and are justifying their statements, that’s okay! It’s not about the structure as much as the idea behind the structure!

**QUICKWRITE: **I love this idea because I make writing integral to my classroom. You give kids a prompt and you tell kids to *write nonstop* *for 2 minutes* *without editing. They have to continually write*. Examples:

It can help with vocabulary, but most importantly, I see this as a way to get kids to stop overthinking and looking for “the right” answer, and just write down anything and everything without self-editing of their thinking. It’s like a condensed noticing/wondering done individually. I can be used before a debate — to give kids time to think. Or perhaps depending on the question, kids can “shout out” one part of their quickwrite? But doing it at the start of the year — to help kids get comfortable writing in math class in an non-threatening, non-evaluative manner — is such a great idea!

**RUMORS: ** This idea was stolen from Rona Bondi at all-ed.org. On a notecard/paper, everyone write a response to a question or a couple questions (the one we used is “what is our idea setup of our classroom?” but I think it could be used at the end of class with questions like: “One thing I find easy to understand in this unit is… One question I still have about this stuff we’ve been working on…?” or “The most important mathematical idea from today is …?” or “The best way to approach graphing trig functions is…”).

After everyone is done writing, everyone finds a parter and reads their card, the other person reads their card, and then they discuss. There is a time limit (maybe 60 seconds). Then they swap papers. Everyone finds a second partner, and they read the card in their hand to the other person, and they discuss what is written on those cards (not their own cards) and then swap. This goes on three or four times. This forces listening, it allows ideas to slowly spread, and the papers can be kept anonymous.

**ONE INTENTIONAL MISTAKE: **Each group of students gets a giant whiteboard and a problem (it could be the same problem as other groups or a different problem). They are asked to solve the problem making one “good” mistake (so nothing like spelling names wrong, transposing a number, or labeling the axes wrong). They then present their solution to another group — playing dumb about their mistake. The other group should ask good questions to help students get at the error. Questions like “don’t you need to add 3 to both sides” is too direct… You need to ask questions which lead the group to see and understand the mistake. So perhaps “what is the mathematical step you used to get from line 2 to line 3, and why is it justified?” might be better.

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So without further ado, this is my letter from this year.

***

May 27, 2016 – June 2, 2016

Dear STU,

It’s Friday evening, 9:53pm, and I’m at home listening to Kurt Cobain and his guitar. I know what you’re thinking, and sorry, nope: no showtunes today. Nearby is the book I just put down. It takes place in the nineties, the U.S. is entering a proto-grunge phase, and Nirvana is a recurring theme. The nineties is also when I was in high school and so every so often — usually when one chapter ends and I take a mental pause to regroup — I’ll get flashes of forgotten high school memories. You see, I have a terrible memory. It’s almost comical how much I don’t retain. Almost. So those moments where some feeling-rich memory is drudged up — the heart-pounding anticipation of a wildly-liked senior picking up friendless new-to-town sophomore-me in his car to go to a mock trial practice, or the awe of being perched on the roof of a house with a friend where every word carried into the night sky crackled with deeper meaning — I let them wash over me. Recalling them with any vividness get rarer and rarer as the years pass. (That’s something no one tells you about growing up. Your experience of the world dulls — from vibrant neons to faded pastel watercolor. Your memories become mottled with gaps, like a desiccated leaf chewed up by hungry pests.)

Why am I telling you this? As I now reminisce about me in the nineties, I know you are reminiscing about your lives too. Packer will become a temporary line on your resume, and then — soon into your working lives — not even that. (No one includes high school on their CVs.) You’re moving on, growing up, and you’re losing something and gaining something. You are adults and you are not adults. You are who you are and you are not (yet) who you are.

As you know, in physics there is a wave function. It’s a probability function describing all the possible states of some system. For example, is a particle here or there or waaaay over there? And — here’s the kicker — that wave function is the best that we can do to describe things. The system isn’t *knowable* in any better way. The function within it has all these possibilities, some more probable than others but still, oh so many possibilities. “How many?” I imagine asking you one day in S202, and in unison I hear you all replying “Infinite!” And left alone, the infinite possibilities undulate in time, directed by Schrodinger’s equation. Until one instant it isn’t. It collapses. All possibilities reduce to one actuality. Why? How? The why is easy: someone tries to find out *more* about the system… a measurement is taken. (A box is opened to peek at the cat.) And in that measurement the wave — and all the possibilities — is destroyed. (The cat is either alive or dead.) The how is harder: how does a collection of probabilistic states turn into a single state? That it happens is known, when it happens is known, but how it happens is unknown.

You — right now — are infinite possibilities spread out before you. Right now, you can’t even know what they all are, but they exist. The way you move through the world, the choices you make, the person you strive to be, those all shape the landscape of those possibilities over time.

Like you perhaps, I had grand designs when graduating high school. There were so many things I wanted to accomplish, so many things I wanted to learn. But one thing I did know — the thing that had the largest chance of becoming true — was that I wanted to become a high school math teacher. I truly never know if that nugget surprises students when I share it with them. I always think it does, because in my time at Packer, I’ve only had one student tell me they wanted to be a teacher (and now they are!). But here’s the thing: even then, I knew I *loved* math. Not in a small way, but in a way where I could work on problems for weeks and be in pure bliss. In a way that when I figured something out, I would force my poor mother listen to me outline how I cracked the mathematical nut — even though she had no idea what my excited explanations were all about. I wanted desperately to share with the world that feeling, of the frustrating and seemingly intractable journey ending in deep insight and a joyous satisfaction. I couldn’t *not* share that love with others! I wanted others to have that joyous satisfaction too.

I told my teachers this. And one — the one who looms larger and larger as I get further and further away from high school — got this about me. It was Mr. Parent, my junior and senior year English teacher. He occupies a special place in my limited memory because he was the first person I met who truly and fully embodied the life of the mind. The engine that drove this man was intellectual curiosity, and to bear witness to that sort of person – and his unbridled passion – had a lasting impact on me. At the end of my senior year I bought him a book and wrote him a letter explaining how much he meant to me. In that letter, I offered up a quotation by Richard Feynman, physicist and boyhood hero:

*I was born not knowing and have had only a little time to change that here and there.*

If someone asked me what I wish for my students, I would answer with a pat: “to be good, and to be happy.” I can’t speak to being good part. That’s for you to figure out. But I suspect for you seven, because in you I see parts of me, one path to lasting happiness is to continuously follow your intellectual curiosity. That is our common bond, and one that I have been grateful to have had the opportunity to bear witness to from the first day of class until the very last day. Because we share that, I hope that you remember in the most bleak of days: there is something *magical* about the world around you. Keep an eye out for the magic. It appears as questions… and there are so many questions! How can we – billions of years later – know about the earliest moments of the universe? Where does matter come from? How can the world be probabilistic (quantum) in nature when everything feels so causal? How do we *know* about the smallest worlds we cannot even see? Why are there rainbows on the surface of an oil spill? How do rubber bands work – how do they come back to their original shape? How can we – on this planet – know how far things are, and that there are other galaxies out there? How is it that the natural world somehow can be encoded through simple and elegant mathematical formulas? Does that imply that math is somehow encoded in the universe, and it is being discovered rather than invented? Does the fact that we keep on digging in mathematics and are still drawing connections among disparate sub-fields imply that there is some grand unifying structure undergirding everything mathematical and physical?

Mr. Parent walked up to me on my graduation day and handed me a letter in return — a letter I treasure to this day, keeping it ensconced between the pages of my yearbook. In response to Feynman, he returned one of his own devising: “Stephen Hawking speaks of the thermodynamic, psychological, and cosmological arrows of time that define existence as entropic movement from past to future in an expanding universe. And that seems to define the hero’s journey: the personally expanding possibilities revealed in a courageous life bounded by and aware of entropic time.” I personally read this as an intellectual quest: you – dear students – are in a world that is growing in knowledge and is constantly reshaping itself around you. And you – dear students – have only a lifetime to enjoy it. And I mean “only a lifetime” because the world is vast and time runs short.

As you quest, don’t be afraid of failure. Let failure be a marker of pride, because you tried. You know me, I don’t know much about sportsing, but I do know that you miss 100% of the shots you don’t take. Set the bar slightly higher than you think you are capable of achieving and work extraordinarily hard. Harder than everyone else around you.

Wave functions collapse. But the possibilities of our lives only collapse when we are no more. You are an infinity of possibilities, remember that. You have so much time, and so little time. Make it meaningful.

Always my best, with sincerest best wishes,

Sameer Shah

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Another teacher and I started this journal four years ago (this is the fourth issue). And each year we took less and less of a role, as we trained leaders, taught them to organize themselves, and got them to look for new members. Next year, though, we’ll have to be more hands on because we have only a few people who were on the staff for one year only. The new leaders have some great ideas for next year!

With that, I’m out. Hey, I did say it was a super short blogpost.

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Earlier in the year, I showed my student a video of an augmented reality sandbox that I stumbled across online. She showed interested in making it. It takes in a mapping of a surface (in this case sand in a sandbox) and projects onto the surface colors representing the height of the sand over time (so red is “high” and blue is “low”). The cool part about this is that the projection changes *live* — so if you change the sand height, the projection updates with new colors. Level curves are also “drawn” on the sand.

Here are some videos of it in action (apologies for the music… I had to put music on it so the conversations happening during the playing with the sand were drowned out):

The student was going to design lesson plans around this to highlight concepts in multivariable calculus (directional derivative, gradients, gradient field, reading contour maps) but ran out of time. However upon my suggestion, during her presentation, she did give students contour maps of surfaces, turned off the projector, had students try to form the sand so it matched the contour map, and then turned the projector on to have students see if they were right or not.

During the presentation, one student who I taught last year (but not this year) said: “This is the coolest thing I’ve seen all year!” and then when playing with the sand: “I AM A GOD!” Entrancing!

In my first year of teaching this course, a student was entranced by lissajous curves when we encountered them. These are simple parametric equations which create beautiful graphs. I then suggested for his final project that he create a harmonograph, which he did. Seven years later, I had another student see the original video of my student’s harmonograph, and he wanted to build his own! But he wanted his to have a rotary component, in addition to two pendulums which swung laterally. So he found instructions online and built it!

Here are some of the images it produced:

And here is a video of the harmonograph in motion:

(You can watch another video here.)

During the presentation, the student talked about the damping effect, how the pendulum amplitudes and periods had an effect on the outcome, and how lissajous curves were simply shadows of lissajous knots that exist in 3-space. Because of the presentation, I had some insights into these curves that I hadn’t had before! (I still don’t know how mathematically to account for how the rotary pendulum in the student’s harmonograph affects the equations… I do know that it has the harmonograph — in essence — graph the lissajous curves on a somewhat rotating sphere (instead of a flat plane). And that’s interesting!

A student was interested in creating tools for teachers to illustrate “big” multivariable calculus ideas… Contour lines, directional derivatives, double integrals, etc. So she made a set of five of *super awesome* teaching manipulatives. Here are three of them.

The first is a strange shaped cutout of poster-cardboard-ish material, with four animals hanging from it. Then there is string connected to a magnet on top, and another magnet on the bottom. If you hold up the string and you aren’t at the center of mass, the mobile won’t balance. But if you move the magnet around (and the student used felt around the magnet so it moves seamlessly!), you can change the position of the string, until it balances. This is a manipulative to talk about center of mass/torque.

Another is a set of figures that form “level curves.” At first I was skeptical. The student said the manipulative elow was to help students understand countour plots. I wanted to know *how*… Then the moment of genius…

You can change the height of the level curves to make the “hill” steeper and steeper, and then *look straight down* at the manipulative. If you have a shallow “hill,” you have contour lines which will look far apart. If you have a tall “hill,” you have contour lines which look close together.

Finally, a third manipulative showcases the tangent plane (and it can move around the surface because of magnets also). I can see this also being useful for normal vectors and even surface integrals!

Two students decided to work together on a project dealing with cartographic mapping. They were intrigued by the idea that the surface of the earth can’t perfectly be represented on a flat plane. (They had to learn about why — a theorem by Euler in 1777.) They chose two projections: the Gall Peters projection and the Stereographic projection.

They did a fantastic job of showing and explaining the equations for these projections — and in their paper, they went into even more depth (talking about the Jacobian!). It was marvelous. But they had two more surprises. They used the 3D printer (something I know nothing about, but I told them that they might want to consider using to to create a model to illustrate their projections to their audience) and in two different live demos, showed how these projections work. I didn’t get good pictures, but I did take a video after the fact showing the stereographic projection in action. Notice at the end, all the squares have equal area, but the quadrilaterals on the surface most definitely do not have equal area.

An added bonus, which actually turned out to be a huge part of their project, was writing an extensive paper on the history of cartography, and a critical analysis of the uses of cartography. They concluded by stating:

We have attempted, in this paper, to provide our readers with a brief historical overview of cartography and its biases. This paper is also an attempt to impress upon the reader the subjective nature of a deeply mathematical endeavor. While most maps are based around mathematical projections, this does not exclude them from carrying biases. In fact, we believe there is no separation between mathematical applications and subjectivity; one cannot divorce math from perspective nor maps from their biases. We believe it is important to incorporate reflections such as this one into any mathematical study. It is dangerous to believe in the objectivity of scientific and numerical thought and in the separation between the user and her objective tools, because it vests us, mathematicians and scientists, with arbitrary power to claim Truth where there is only perspective.

Beautiful. And well-evidenced.

One student was interested in fluid dynamics. So I introduced him to the Navier Stokes equations, and set him loose. This turned out to be a challenging project for the student because most of the texts out there require a high level of understanding. Even when I looked at my fluid dynamics book from college when I was giving it to him as a reference, I realized following most of it would be almost impossible. As he worked through the terms and equations, he found a perfect entree. He learned about an equation that predicts the change in pressure from one end of a tube of small radius to another (if the fluid flow in the tube is laminar). And so using all he had learned in his investigation of the field, he could actually understand and explain algebraically and conceptually how the derivation worked. Some of his slides…

It was beautiful because he got to learn about partial differential equations, and ton of ideas in fluid dynamics (viscosity, pressure, rotational velocity, sheer, laminar flow, turbulence, etc.), but even needed to calculate a double integral in cylindrical coordinates in his derivation!

This student works in a lab for his science research class — and the lab does something with lasers and quantum tunneling. But the student didn’t know the math behind quantum mechanics. So he spent a lot of time working to understand the wave equation, and then some time trying to understand the parts of Schrodinger’s equation.

In his paper, he derived the wave equation. And then he applied his understanding of the wave equation to a particular problem:

He then tackled Schrodinger’s Wave Equation and saw how energy is quantized! Most importantly, how the math suggests that! I remember wondering how in the world we could ever go from continuousness to discreteness, and this was the type of problem where I was like “WHOA!” I’m glad he could see that too! Part of this derivation is below.

Overall, I was blown away by the creativity and deep thinking that went into these final projects. Most significantly, I need to emphasize that I can’t take credit for them. I was incredibly hands off. My standard practice involves: having students submit three ideas, I sit down with students and help them — with my understanding of their topics and what’s doable versus not doable — narrow it down to a single topic. Students submit a prospectus and timeline. Then I let them go running. I don’t even do regular formal check-ins (there are too many of them for me to do that). So I have them see me if they need help, are stuck, need guidance or motivation, whatever. I met with most of them once or twice, but that’s about it. This is all them. I wish I could claim credit, but I can’t. I just got out of their way and let them figure things out.

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My multivariable calculus courses was coming to an end, and I got some questions about what college courses in math are about. It reminded me of a comic strip I read years ago, which I frustratingly can’t find again. It has an undergraduate going to meet with his math professor adviser, saying something like “I want to major in triple integrals.” Which is crazy-sounding — but maybe not to a high school student who has only ever seen math as a path that culminates in calculus. What more is out there? What is higher level math about? (These questions are related to this post I wrote.)

So here’s what I told my students to do. They were asked to go onto their future college math department websites (or course catalog), scour the course offerings, and find 3-4 courses that looked interesting and throw these courses down on a google doc.

It was awesome, and made me jealous that they had the opportunities to take all these awesome classes. Some examples?

After looking through all the courses, I highlighted one per student that seemed like it involved topics that other students had also chosen — but so that all the courses were different branches/types of math. I told each student to spend 10-15 minutes researching their highlighted course — looking up what the words meant, what the big ideas were, finding interesting videos that might illustrate the ideas — so they can “pitch the course to the class” (read: explain what cool math is involved to make others want to take the course).

I’m fairly certain my kids spent more than 10-15 minutes researching the courses (I’m glad!). Each day, I reserved time for 2-3 students to “pitch” their courses. And since some of the ideas were beyond them, after the pitches, I would spend 5 or so minutes giving examples or elaborating on some of the ideas they covered.

If you want to see the research they did for their pitches, the google doc they chucked their information into is here.

Some fun things we did during the pitches?

(1) We watched a short clip of a video about how to solve the heat equation (that was for a course in partial differential equations)

(2) I showed students how to turn a communication network into a matrix, and explained the meaning of squaring or cubing the matrix (this was for a course on network theory)

(3) A student had us play games on a torus (a maze, tic tac toe) (this was for a course on topology)

(4) I had students store on their calculators. Then I had each student store a different “r” value (carefully chosen by me) and then type in their calculators. They then pressed enter a lot of times. (In other words, they were iterating with the same initial conditions but slightly different systems. Some students, depending on their r value, saw after a while their x values settle down. Some had x values that bounced between two values. Some had x values that bounced between four values. And one had x values that never seemed to settle down. In other words, I introduced them to a simple system with wacky wacky outcomes! (If you don’t know about it, try it!) (This was for a course on chaos theory)

(5) A student introduced us to Godel’s incompleteness theorem and the halting problem (through a youtube video)

It was good fun. It was an “on the spot” idea that turned out to work. I think it was because students were genuinely interested in the courses they chose! If I taught a course like AP Calculus, I could see myself doing something similar. I’m not sure how I would adapt this for other classes… I’m thinking of my 9th grade Advanced Geometry class… I could see doing something similar with them. In fact, it would be a great idea because then they could start getting a sense of some of the big ideas in non-high school mathematics. Kay, my brain is whirring. Must stop now.

If anyone knows of a great *and fun* introduction to the branches of college level math (or big questions of research/investigation), I’d love to know about it. Something like this is fine, but it doesn’t get me *excited *about the math. I want something that makes me ooh and ahh and say “These are great avenues of inquiry! I want to do all of them!” I think those things that elicit oohs and ahhs might be the paradoxes, the unintuitive results, the beautiful images, the powerful applications, the open questions… If none exists, maybe we can crowdsource a google doc which can do this…

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Now we are at the end of the year, and one of my Advanced Geometry sections had three classes that the other section didn’t have. So I had to come up with something supplemental. Thank goodness for twitter. You see, @jacehan was using my blermion activity, and some of his kids asked him “what if the circle was *inside* the quadrilateral?”

Of course, genius that @jacehan is [his blog is here], he named these creatures merblions.

With this one question, I had the makings of an amazing three days ready for me. You see, in Geoemtry, we had just finished studying angle bisectors (and how they related to pouring salt on polygons). We had also just finished studying triangle congruency. (I know that is usually taught earlier in the year, but when I rearranged the course, it fit best near the end of the year.) So those were two *powerful* tools to analyze merblions.

So I told students to pair up. And they were given the above picture and told that: “A **merblion** has an *inscribed circle* which is tangent to all four sides of the quadrilateral.”

That is all.

Then I told students that in some ways, this is a culmination of everything they’ve done all year. They have everything they’ve learned at their disposal. Geogebra. Paper. Rulers. Compasses. Protractors. But mostly, they need to *make conjectures* and see if they are true — either getting a lot of inductive evidence or by using deductive logic. Anything they wanted to figure out about merblions were fair game.

**I also highlighted that the other geometry teacher and I started investigating these, realized they were very rich and there was a lot to discover, but we purposefully stopped investigating them. We wanted our students to make the discoveries, without us accidentally guiding them****. **

We also told them that they needed to persevere, and be okay trying lots of things. But if they ever felt their wheels were turning and still nothing was happening, they could call us over for a nudge. (I created a list of things I could say to kids to help nudge them along if they got stuck… I didn’t have to use it more than once! Kids were into it.) They knew at the third day, they would be presenting (informally) their findings to the class. So they had to keep track of things, take screenshots, etc.

While they worked (with music!), I saw kids make conjectures, find they weren’t true, and then move on. I then realized kids weren’t recording their “failed” conjectures. But that data is important! So I told kids to keep track of *all* of their ideas, and even if their idea didn’t turn out to be true, it is totally worthy of putting into their presentation! It helps us see their avenues of inquiry. Similarly, I told students to record their conjecture, even if they couldn’t prove them deductively.

The kids were doing so many interesting things — including things I hadn’t thought of. (Two pairs tried finding the smallest merblion, by area, that could fit around a circle of a given size! Three pairs tried to do an “always/sometimes/never” with “A _____ is A/S/N a merblion” where the blank were all the quadrilaterals we’ve studied [kites, rhombuses, trapezoids, etc.]. One pair noted that to use Geogebra to draw a merblion, you only need a circle and two points, but the two points couldn’t be *any* two points — so they wondered where those two points could be located.) It was great.

They continued on the next day, and spend the last 20 minutes of the second class throwing some slides up in our google presentation [posted here, with identifying information of students removed].

What they ended up discovering was *awesome*.

1. The center of the circle inscribed in the merblion is the intersection of the four angle bisectors. And if we cut a merblion out of cardstock and did the “salt pouring activity,” we would see the salt form a pyramid with a merblion base and a single peak (where the peak would exist at the center of the inscribed circle).

2. Kites, squares, and rhombuses are all merblions. However rectangles are only merblions if they are squares, and parallelograms are merblions only if they are rhombuses. Some trapezoids are merblions and some aren’t.

3. No concave quadrilateral can be a merblion.

4. A merblion has two pairs of opposite angles which are acute, and two pairs of opposite angles which are obtuse (unless you have a square).

5. A merblion is secretly composed of four kites. And the four kites have two opposite right angles. (Which means that the non-right angles are supplementary in these kites.)

6. In a merblion, the sum of the lengths of opposite sides are equal.

7. The area of a merblion can be computed by finding the perimeter, halving it, and multiplying it by the radius of the inscribed circle.

8. For all merblions that can be drawn around a given circle, the merblion with the least area is a square.

9. In the other class (not in my class) students found this result… The two angles here are always supplementary.

The kids were totally engaged. They didn’t feel pressure to produce “the right answer” because there was no right answer. (And no grade associated with this work.) I emphasized that all conjectures (even if they don’t work out) were valid, so kids felt okay writing anything and anything down. I didn’t have a specific outcome they had to come up with, so I wasn’t leading. Kids could do anything! They got to work together.

And when some results were presented that explained things that people were wondering about — there were noticeable ooohing and aaahing (for example, result #6!).

And after the presentation happened, it became clear to everyone that by crowdsourcing this problem, we were able to see *lots* of results and then start examining how the different results related to each other (so for example, result #6 explains #2).

This was very fun. Very very fun.

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I didn’t have a clear idea of what to do in multivariable calculus for the block. I still had to cover content, but I wanted it to be “different” also. After many hours of brainstorming, I came up with a solution that has worked out pretty well this year.

**We had a book club.**

The 90 minute block was divided into 50 minutes of traditional class, and 40 minutes of book club. (Or 60 minutes of class, and 30 minutes of book club.)

Now, to be clear, this is a class of seven seniors who are highly motivated and interested in mathematics. I can see ways to adapt it in a more limited way to other courses, with more students, but this post is about my class this year.

We started out reading Edwin Abbott’s *Flatland*.

Why? Because after they read this, they understand why I can’t help them visualize the fourth (spatial) dimension! But it convinces them that they can still understand what it is (by analogy) and makes them agree: if we can believe in the first, second, and third spatial dimensions, why wouldn’t we believe in higher spatial dimensions too? It’s more ludicrous *not *to believe they exist than to believe they don’t exist! A perfect entree into *multivariable* calculus, wouldn’t you say?

After reading this, we read the article “The Paradox of Proof” by Caroline Chen on the proposed solution to the ABC conjecture.

This led us to the notion of “modern mathematics” (mathematics is not just done by dead white guys) and raised interesting questions of fairness, and what it means to be part of a profession. Does being a mathematician come with responsibilities? What does clear writing have to do with mathematics? (Which helps me justify all the writing I ask for on their problem sets!) It also started to raise deep philosophical questions about mathematical Truth and whether it exists external to the human mind. (If someone claims a proof but no one verifies it, is it True? If someone claims a proof and fifty people verify it, is it True? When do we get Truth? Is it ever attainable? Are we certain that 2+2=4?)

At this point, I wanted us to read a book that continued on with the themes of the course – implicitly, if not explicitly. So we read Steve Strogatz’s *The Calculus of Friendship:*

What was extra cool is that Steve agreed to sign and inscribe the book to my kids! The book involves a decades long correspondence between Steve and one of his high school math teachers. There are wonderful calculus tricks and beautiful problems with explanations intertwined with a very human story about a young man who was finding his way. Struggling with choosing a major in college. Feelings of pride and inadequacy. The kids found a lot to latch onto both emotionally and mathematically. Two things: we learned and practiced “differentiating under the integral sign” (a Feynman trick) and talked about the complex relationship that exists between teachers and students.

After students finished this book, I had each student write a letter to the author. I gave very little guidelines, but I figured the book is all about letters, so it would be fitting to have my kids write letters to Steve! (And I mailed the letters to Steve, of course, who graciously wrote the class a letter back in return.)

Our penultimate reading was G.H. Hardy’s *A Mathematician’s Apology*:

I went back and forth about this reading, but I figured it is such a classic, why not? It turned out to be a perfect foil to Strogatz’s book — especially in terms of the authorial voice. (Hardy often sounds like a pompous jerk.) It even brought up some of the ideas in the “Paradox of Proof” article. What is a mathematician’s purpose? What are the responsibilities of a mathematician? Why does one do mathematics? And for kids, it really raised questions about how math can be “beautiful.” How can we talk about something that is seen as Objective and Distant to be “beautiful”? What does beauty even mean? Every section in this essay raises points of discussion, whether it be clarification or points that students are ready to debate.

What is perfect about this reading is at the same time we were doing it, the movie about G.H. Hardy and S. Ramanujan was released: *The Man Who Knew Infinity* (based on the book of the same name).

Finally, we read half of Edward Frenkel’s *Love and Math*:

Why? Because I wanted my students to see what a modern mathematician does. That the landscape of modern mathematics isn’t what they have seen in high school, but so much bigger, with grand questions. And through Frenkel’s engaging telling of his life starting in the oppressive Russia and ending up in the United States, and his desire to describe the Langland’s program understandably to the reader, I figured we’d get doses of both what modern mathematics looks like, and simultaneously, how the pursuit of mathematics is a fully human endeavor, constrained by social circumstances, with ups and downs. Theorems do not come out of nowhere.Mathematicians aren’t the blurbs we read in the textbooks. They are so much more. (Sadly, we didn’t read the whole thing because the year came to a close too quickly.)

I broke the books into smaller chunks and assigned only them. For Flatland, it might have been 20-30 pages. For Love and Math or A Mathematician’s Apology, it might have been 30-50 pages. We have our long block every 7 school days, so that’s how much time they had to read the text.

At the start, with *Flatland*, students were simply asked to do the reading. Two students were assigned to be “leaders” who were to come in with a set of discussions ready, maybe an activity based on something they read. And they led, while I intervened as necessary.

For every book club, students who weren’t leading were asked to bring food and drink for the class, and we had a nice and relaxing time. On that note, never did I mention anything about grades. Or that they were being graded during book club. (And they weren’t.) It was done purely for fun.

Later in the year, I had students each come to class with 3-4 discussion questions prepared, and one person was asked to lead after everyone read their questions aloud.

The discussions were usually moderated by students, but I — depending on how the moderation was going — would jump in. There were numerous times I had to hold back sharing my thoughts even though I desperately wanted to concur or disagree with a statement a student had made. And to be fair, there were numerous times when I *should have* held back before throwing my two cents in. But my main intervention was getting kids to go back to the texts. If they made a claim that was textually based, I would have them find where and we’d all turn there.

Sometimes the conversations veered away from the texts. Often. But it was because students were wondering about something, or had a larger philosophical point to make (“Is math created or discovered?”) which was prompted by something they read. And most of the times, to keep the relaxed atmosphere and let student interest to guide the conversation, I allowed it. But every so often I would jump in because we had strayed so far that I felt we weren’t doing the text we had read justice (and we needed to honor that) or we were just getting to vague/general/abstract to say anything useful.

I mentioned students generated discussion questions on their own. Here are some, randomly chosen, to share:

- Strogatz talks about how math is a very social activity. We see this exemplified in the letters between Steve and Mr. Joffray, but where else do we see this exemplified in math? (papers, etc.) How do you think Strogatz might have felt about Shinichi Mochizuki’s unwillingness to explain his paper and proof to the math community?
- What do you think about Strogatz and Joff using computer programs to give answers to their problems? Are computers props, and their answers unsatisfying? Or are they just another method, like Feynman’s differentiating under the integral?
- Do you like A Square? In what ways is he a product of his society? Does he earn any redeeming qualities by the end of the book?
- Can you draw any connections between things in Flatland and religion? Do you think Abbott is religious? Why/why not?
- When we first read about Mochizuki’s ABC Conjecture, we debated whether or not math is a “social” subject. Perhaps many mathematicians do much of the “grind” work on their own, however, throughout everything we’ve read this year, there has been one common link when it comes to the social aspects of math: mentorship. It appears to me that all of the great mathematicians we know about have been mentored by, or were mentors others. In what ways have Frenkel’s mentors – he’s had a few – had an influence on the path of his mathematical career? Do you think he would/could be where he is today without all of those people along the way? Can you think of any mentors that have had a profound influence on your life? (The last one can just be a thought, not a share.)
- Frenkel talks about the way in which math, particularly interpretations of space and higher dimensions, began to influence other sectors of society, specifically the cubist movement in modern art. This movement was certainly not the first time math and science influenced art and culture – think about the advent of perspective in the Renaissance and the use of technology on modern art now – however math and art are often thought as opposites and highly incompatible. Why do you think that people rarely associate the two subjects? Would you agree that the two are incompatible? Can you think of other examples of math/science influence art/culture/society?

In many ways, I felt like this was a perfect way to use 30 minutes of the long block. After doing it for the year, there are a few things that stood out to me, that I want to record before summer hits and I forget:

(a) I think students really enjoyed. It isn’t only a vague impression, but when I gave a written survey to the class to take the temperature of things, quite a few kids noted how much they are enjoying the book clubs.

(b) For the post-*Flatland* book club meetings, I need to come up with multiple “structures” to vary what the meetings look like. Right now they are: everyone reads their discussion questions, the leader looks for where to start the discussion, the discussion happens. But I wonder if there aren’t other ways to go about things.

One example I was thinking was students write (beforehand) their discussion questions beforehand on posterpaper and bring it to class. We hang them up, and students silently walk around the room writing responses and thoughts on the whiteboard. Then we start having a discussion.

Or we break into smaller groups and have specific discussions (that I or students have preplanned) and then present the main points of the discussion to the entire class.

Clearly, I need to get some ideas from English teachers. :)

(c) I love close readings of texts. I think it shows focus, and calls on tough critical thinking skills. At the same time, I need to remember that this is *not* what the book clubs are fundamentally about. They are — at the heart, for me — inspiration for kids. So although for *Flatland* I need to keep the critical thinking skills and close readings happening, I need to remember (like I did this year) to keep things informal.

(d) Fairly frequently, I will know something that is relevant to the conversation. For example, I might talk about of the math ideas that were going over their heads, or about *fin de siecle *Vienna, or branches of math that might show how the line between “theoretical” and “applied” math is blurry at best. I have to remember to be judicious about what I talk about, when, and why. We only have limited time in book club, so a five minute tangent is significant. And one thing I could try out is jot down notes each time I want to talk about something, and then at the end of the book club (or the beginning of the next class), I could say them all at once.

(e) I usually reserve 30 minutes for book club. But truthfully, for most, 40 minutes turned out to be necessary. So I have to keep that in mind next year when planning class.

(f) Should we come up with collaborative book club norms? Should I have formal training on how to be a book club leader? Should we give feedback to the leaders after each book club? Can we get the space to feel “safe” where feedback could actually work?

And… that’s all!

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You see, I have kids in five groups, and the groups sit at different tables each class. I have folders for each group prepared, and I throw the folders down randomly on the tables. Sometimes I let kids put the folders down — so they are dictating where everyone sits. *THE POWER!*

So yesterday, I had five folders and I wanted kids to put them down. How could I have done it?

Give all five folders to one student to place on the table.

Give one folder each to five students to place on the tables.

Give two folders to one student and three folders to another. Etc.

In other words:

5, 4+1, 3+2, 3+1+1, 2+2+1, 2+1+1+1, 1+1+1+1+1

There were seven ways to divide up the folders to give kids.

(I then said this was like factoring numbers, but with addition. So for example: 20 could be written as: 10*2, 5*2*2, 4*5, etc. But they had never been asked to think about this with addition! That’s what this folder problem was in disguise.)

I asked kids to calculate how we could do this with four folders… and then how we could do this with six folders. When they were done, we created this chart on the board:

1 folder: 1 way

2 folders: 2 ways

3 folders: 3 ways

4 folders: 5 ways

5 folders: 7 ways

6 folders: 11 ways

And then I told them that before class, I calculated 7 folders (lie… I looked it up!) and got:

7 folders: 15 ways

At the end, I asked each group to try to estimate (with reason, but without counting) what the answer for 8 folders would be.

While they were estimating, I pulled up a trailer to *The Man Who Knew Infinity* (a movie about Ramanujan and Hardy which was just released last Friday and was still in theaters…) When I asked each group to share their answer, but without giving any rationale, *every group predicted 23. * Do you see why?

Let me show you what the board looked like after they explained their reasoning… (one image is from one class, one is from the other)…

The second image shows the *differences* between these numbers. The differences were 1, 1, 2, 2, 4, 4… so they naturally added 8 to the last value… so 15+8=23.

Then I had kids watch the trailer to *The Man Who Knew Infinity*:

When it finished, I told them it was still in theaters and it made me cry! And then I said the mathematicians in this movie were very famous — any math major or mathematician will have heard about them… And what were they working on in this film? Our folder problem! (At this point, I had given them the name of the function we were working with: the partition function.) The question was: given any positive integer, could they figure out exactly how many partitions it has without listing them all out?

Then I pulled up the Wikipedia page on partitions and we looked at some more values:

ARGH! It’s at this point the kids see their prediction for 8 folders (23) wasn’t correct (the value is 22!). And we note that the numbers start growing slowly, but then they seem to be growing faster and faster!

I then pull up desmos and type in:

and say that part of the film involves the mathematicians discovering this formula is a pretty good approximation for the number of ways to distribute n folders… if n is large. And we start looking at the graph to see how it compares to the values we find.

It’s actually pretty bad for small n values, but as we try getting to higher n values, we see how high we have to change the window to see the output! Also: weird! Pi is involved?!? And a square root of 3? Whaaaa? (And when I did this in precalculus, kids knew what e was so it was even weirder… but in geometry, kids didn’t so I merely mentioned it was a number like pi but with a value of about 2.71.) [1]

All of this came out of me giving myself permission to go on a tangent… We spent probably 15-20 minutes doing this in each class (I had to do it in my others after I did it on the fly in the first class). But in this 15-20 minutes, I felt like I was alive, and that I was bringing to my kids something I am not able to give them in a traditional curriculum. (It felt like it dovetailed a lot into what I wrote in this post about inspiration and mathematics a few weeks ago.)

Kids were asking great questions, and were talking about other random neat math things. I let that happen briefly… but the ticking clock was getting at me. So then we went back to our regularly scheduled program.

[If I had planned this as an actual lesson, I would have done a lot differently (probably more predicting)… but I’m guessing if I had planned this as an actual lesson, I would have killed it. Part of what I think made this successful was the spontaneity and lack of overthinking/formalism]

[1] In precalculus, we had just finished working with hyperbolas. So after I told kids that this function was good for huuuuge values of n but bad for small values of n, and asked kids what that reminded them of. Two students immediately saw that the equations we came up with for the end behavior of they hyperbola was exactly this… It is terrible for small values of x but great for large value of x!

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For those who aren’t in the know, for me the big idea is that we can conceptualize a parabola as the result of graphing the algebraic equation . But there is a second way to concieve of the same mathematical object: with a geometric argument.

**If you have a piece of paper with single point drawn, and a single line (that doesn’t contain the point) drawn, those two objects uniquely define a parabola.**

That’s a pretty awesome thing, once I started thinking about it. An alternative way to view something that I only ever think about in the standard “graph a quadratic” way!

So given a point and a line, how can we draw this parabola? Here is how…

The point is the blue X. The line is the black line. We want to drag the red point along this vertical line so that the distance from the blue point to the red point is equal to the distance between the red point and the black line. So we use a ruler, some trial and error, and find that red point belongs somewhere here… [1]

And then we leave that red dot there, and start again with another vertical line. And find another point on that vertical line which has the same property!

And again and again and again. Until you have created a whole bunch of red points. Those form a parabola.

I’m still not 100% sure how I’m going to introduce this notion to my kids. I’m pretty sure I’m going to give each kid a printed paper that looks like

And ask them where to place the red dot… And then see if they can find a more efficient way than using a ruler and guessing a checking. (Paper fold! See it? If not, read the footnote.) I will probably do this as a warmup one day — and then have kids go “whaaaaat is this for?” and I’ll shrug and say “Wish I knew, kids…” and then move on not referencing this.

And then the next day for the warmup, I’ll find a way to have the whole class collect points for the same blue point and black line… We’ll generate the locus of all these points which are equidistant from the blue point and perpendicular distance to the black line… and lo and behold… the parabola. And then we’ll do the patty paper folding thing down in the footnote video.

So… Yeah. Now we have an obvious place to go…

Here it is: Given a parabola, can you find the defining point and line? (The fancy mathematical words for these defining objects are “the focus” and “the directrix.”)

And so I created a sheet to have my kids figure out how to find these objects given a parabola. [Note: I haven’t used the sheet. I haven’t even worked out the sheet and made a key. I just whipped it up now! So apologies for any errors, if any.]

2016-04-25 Parabolas [docx form]

Now to be perfectly perfectly honest, there are two things about this sheet I hate.

(1) I give footnote 1.

(2) I give 3c. In fact, partly I think giving 3a is a bit much as is.

Both give away too much. So why didn’t I change it? Do I not have confidence in my kids?

No. It’s because I wasn’t even planning on introducing parabolas. And now I got sucked into them — learning all about them — and I am excited to share some of this stuff with my kids. But I don’t have the time for this. The fact that I’m going to give about a day for parabolas is more than I was planning… so I have to keep things a bit on the crisper side.

What else would I change if I had more time? I would have kids think about if this works for an “upsidedown” parabola. And also have them use what they know about inverse functions to apply this to “sideways” parabolas.

I honestly don’t know if I’m going to use this in class. I probably will because I took the time to make it, and I kinda got excited when I was figuring out for myself all this focus/directrix stuff. I pretty much took this definition of a parabola and figured all this out myself — and I hope kids get the same joy. But have I convinced myself that kids need to learn about a parabola *other* than there is this other way to “create” them that isn’t algebraic? Is there a “big idea” hidden in this worksheet? I don’t think so. This may be a one-time use worksheet.

[1] Now in actually, there is an *easy* geometric way to find that red point. It involves a simple paper fold. Fold the blue point to the point on the directrix below the red point. What that crease intersects the vertical line is where the red dot should be. Perpendicular bisectors FTW! And you can do a quick patty paper demonstration of this to create a parabola! (We did this in my class last year, for parabolas, hyperbolas, and ellipses, thanks to Tina C.)

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Notice that some of the unit cubes have 3 painted faces, some have 2 painted faces, some have 1 painted face, and some have 0 painted faces.

The standard question is: For an *n* x *n* x *n* cube, how many of the unit cubes have 3 painted faces, 2 painted faces, 1 painted face, and 0 painted faces.

[In case you aren’t sure what I mean, for a 3 x 3 x 3 cube, there are 8 unit cubes with 3 painted faces, 12 unit cubes with 2 painted faces, 6 unit cubes with 1 painted face, and 1 unit cube with 0 painted faces.]

Earlier this year, I worked with a middle school student on this question. It was great fun, and so many insights were had. This problem comes highly recommended!

Today we had some in house professional development, and a colleague/teacher shared the problem with us, but he presented an insight I had never seen before that was lovely and mindblowing.

Spoiler alert: I’m about to give some of the fun away. So only jump below / keep reading if you’re okay with some some spoilers.

First off, here are some manipulatives we used in our PD:

We used colors to represent the number of painted faces. So pink = three painted faces, blue = two painted faces, yellow = one painted face, green = zero painted faces.

And we, though standard arguments, saw that for an *n* x *n* x *n* cube

green cubes (zero painted faces) =

yellow cubes (one painted face) =

blue cubes (two painted faces) =

pink cubes (three painted faces) =

There are so many ways to come up with those individual formulas.

And clearly these add up to the total number of cubes. So we have:

Lovely. (Feel free to expand each term out to see that the left side and right side truly are equal.)

Here’s the insight my colleague shared with us.

If you look carefully at the right hand side, you can actually see as the same thing as .

By applying the binomial expansion to , each term in the expansion gives us the number of cubes which have 3 sides painted, 2 sides painted, 1 side painted, or 0 sides painted. Seriously, use the binomial expansion on and see for yourself! WHAAAAA?

So here’s the question he posed to us, and I pose to you…

Why? Why the heck does each term in the binomial expansion for give you the number of cubes with 3 sides, 2 sides, 1 side, and 0 sides painted?

I mean, we *know* it works algebraically. We can expand it out to see. **But why, conceptually?** Where do the and fit into everything?

When I finally figured it out, my mind was blown. So simple and elegant, yet so unintuitive for me. I’m not going to type out my insight here. Yet, anyway. Because I want to leave it for others to think through. But if there is interest, after a short while, I can definitely do an update with the conceptual insight!

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In the introduction, Frenkel criticizes the teaching of math:

What if at school you had to take an ‘art class’ in which you were only taught how to paint a fence? What if you were never shown the paintings of Leonardo da Vinci and Picasso? Would that make you appreciate art? Would you want to learn more about it? I doubt it. [1] … There is a common fallacy that one has to study mathematics for years to appreciate it… I disagree: most of us have heard of and have at least some rudimentary understanding of such concepts as the solar system, atoms and elementary particles, the double helix of DNA, and much more without taking courses in physics and biology. And nobody is surprised that these sophisticated ideas are part of our culture, our collective consciousness.

So many whirling thoughts came up while I was reading these passages. One thought led to another to another to another. Writing this post is an attempt to start recording them and to get them a little more codified in my mind! It is still going to be a hot discombobulated stream-of-consciousness mess. #sorrynotsorry

I wonder if I asked my kids “what is mathematics?” right now, what they would say. I am doubtful that their answers will include the adjectives and verbs that I personally would say.

I wonder if I asked my kids “what is is going on in the field of mathematics?” right now, what they would say. I’m guessing a lot of blank stares.

I wonder what my kids would say if I asked ’em “what courses exist in college for mathematics?”

I wonder what my kids would say if I asked them to name a mathematician who is alive?

I wonder if the word “mathematics” was changed to “astronomy” or “physics” or “biology” if their answers would be different.

There are ideas that my kids learn about modern physics (in popular culture, in classes) which spark their imagination, blow their minds, make them curious and full of wonderment at the weirdness and strangeness of the world. Special relativity. Quantum mechanics. Quarks and the structure of atoms. They are exposed to these ideas, even if they don’t have the mathematical capabilities or abstraction to attack them rigorously. And these ideas have a powerful effect on some kids. (I know I wanted to be a physicist when I first learned about these ideas!)

But what do my kids learn about modern mathematics — from school or popular culture? Are there any weirdnesses or strangenesses that **can** capture their imagination? Yes! Godel’s incompleteness theorem. Space filling curves. Chaos theory. The fact that quintic and higher degree polynomials don’t have a general “simple” formula always works like the quadratic formula. Fractals. Higher dimensions. Non-euclidean space. Fermat’s Last Theorem. Levels of infinity. Heck, infinity itself! Mobius strips. The four color theorem. The Banach-Tarski paradox. Collatz conjecture (or any simply stated but unproven thing). Anything to do with number theory! Anything to do with the distribution of primes! But **do** they capture students’ imaginations? No… because they aren’t exposed to these things.

Where in our curriculum do kids get *inspired*? Where does *awe* and *beauty* fit into things? When do we ever explicitly talk about *beauty* in mathematics? When a kid has a rush of insight and makes a visible gasp, what do we do in that moment? What has to already be in place for a kid to make that gasp?

We need to expand how we frame mathematics in high school so it isn’t seen as “Algebra I, Geometry, Algebra II, Precalculus, and Calculus.” These course names aren’t mathematics.

We need to consciously and regularly introduce a bigger and more modern world of mathematics to our kids. How? Having kids read when the New York Times publishes an article about a mathematician or mathematical result! Using resources like Numberphile and Math Munch and Vi Hart videos. And… I don’t know.

We need to provide space and time for kids to explore an expanded vision of what math is, and have choice in having fun and playing with this expanded vision of math. (My explore math project is an attempt to do that — website here, and posts one, two, and three here.)

We need to have mathematical *lore*, stories we can tell students. Galois duel! Ramanujan’s inexplicable genius! What are mathematical stories that can be passed down from generation to generation? (Does a good resource exist for this? Tell me!) [Update: The internet went down when I was going to edit this post by mentioning we need stories and people who aren’t just white men!]

Do we have Feynman or degrasse Tyson-esque figures we can point to? Dynamic popularizers of the subject that have entered the public consciousness?

***

Maybe what I’m trying to say, if I had to distill everything down to the core, is:

*(1) Can we find a way — in our existing schools with our set curricula and limited time — to expand kids notions of what mathematics is by exposing them to notions external to the Alg-Geometry-Alg II-Calc sequence. And if we can do this well, will it help inspire more kids to be interested in mathematics? *

*(2) Are there ways for us to keep an focus on beauty, the unexpected, awe, and wonderment in our classes? And find ways to record, highlight, and amplify those moments for kids when they happen? Why I love mathematics is because of all of these moments! Maybe focusing on them would help kids love mathematics?*

[1] This notion has so many resonances with Paul Lockhart’s *A Mathematician’s Lament. *Which I highly recommend.

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And finally we did the magic transition where kids saw how ratios *could* be used to find the labeled angle… This is the key turn, from sides of triangles to ratios of sides.

Initially all students used the pythagorean theorem to find the hypotenuse and then they “scaled the triangle down” to find the similar triangle in the book with hypotenuse one. From there, they could find the angle.

But then I asked: *How* could you find the angle *without* using the Pythagorean theorem? They could still use the triangle book, and the basic functions on their calculators (kids obviously at this point don’t know about sine/cosine/tangent). Some were stuck, some saw it right away. But eventually everyone recognized that they were looking in the triangle book for a triangle which was similar to the triangle given. And we know that proportions of corresponding sides in similar triangles stay constant… so they used guess and check to find triangles in the book with a vertical/horizontal ratio that matched 2.2204/0.8082.

Okay, great. **They hated that**. **They had to divide the side lengths on a bunch of different triangles in the triangle book. So annoying. It was much worse than just using Pythagorean theorem and scaling down. **

So here’s where we paused. I said: “okay, fine, agreed. This *is* annoying and horrible, and the Pythag approach is much nicer. Let me ask you this… What if I put the ratio of the vertical/horizontal leg on every page in the book. So you *had* the ratio. Which way would be more efficient to use then?”

Everyone said the ratio. Why? Given a triangle, you simply take the ratio of two sides, and then flip in the book until you see the same ratio. Then you can immediately read off the missing angle. One division, that’s all. (With Pythag, you have to first calculate the hypotenuse without any error, and then scale that triangle so the hypotenuse is one, and only *then* flip in the book! And that might lead to more error.)

So I showed them trig tables. Of course they don’t have sine/cosine/tangent yet on them. And I let them use it on a few problems.

And then… FUN! I asked kids to just look at the table and just “notice” patterns.

They came up with some great things, which I then started playing with on the fly. I told them to call the three columns “Ratio 1,” “Ratio 2,” and “Ratio 3”:

- As the angle increases, Ratio 1 starts close to 0 and goes close to 1.
- As the angle increases, Ratio 2 starts close to 1 and goes close to 0.
- Whoa, wait, the numbers in Ratio 1 and Ratio 2 are “reversed”! Reading Ratio 1 from the top-down, and Ratio 2 from the bottom-up is exactly the same.
- As the angle increases, Ratio 3 starts close to 0 and gets higher… dramatically higher at higher angles!
- The numbers in the Ratios didn’t seem to be going up “proportionally”

While they were looking for patterns, I noticed no one had taken out their calculators, so I told ’em to see if their calculators could help them figure out any additional patterns.

- Ratio 1 divided by Ratio 2 is the same as Ratio 3.

They will be exploring some of these ideas later, and class was coming near to a close, so we didn’t explore everything we could have. But we did talk about a few things.

(1) We briefly discussed why Ratio 1 will never equal 1. (The hypotenuse of a triangle can’t ever equal a leg of a triangle! You wouldn’t have a triangle, but a segment.)

(2) We saw in a triangle why Ratio 1 divided by Ratio 2 yields Ratio 3.

And finally, I most wanted to capitalize on the observation that I hadn’t anticipated… but discussing it would combat a great question kids don’t really grok well in higher grades… What is the shape of the sine curve? Usually they think it is linear from 0 degrees to 90 degrees. That there is a linear relationship between angles and the ratios. So here’s what I did:

I told students that I would be plotting on the x-axis angle number, and the y-axis Ratio 1. If this was a line, then if you pick any two points on this line and calculate the slope, the slope should be constant. [1]

Each kid chose two different angles, and looked at the associated Ratio 1 numbers, and calculated the slope. While they did that, I was doing a little magic in Geogebra to show the data graphed.

Kids were getting different slopes. So they knew it wasn’t a line. But many slopes *very* close to each other! Curious.

A kid saw the graph and said “Hey, it looks linear at the beginning” and that explained why so many slopes were similar but not the same. Kids were mainly ch0sing angles from the first page of the ratio table! Ha! Love it! Last year teaching geometry, I didn’t ever show them a sine curve. But this came up so naturally that I had to!

This was a bit on the fly and haphazard, but this discussion of whether the ratios were linear or not was one of my favorite things I’ve done recently! I should find a way to formalize it and build it into the curriculum in more solid way.

**UPDATE: OMG I am an idiot. I forgot to mention something crucial. I want kids to recognize that if they have a trig table with only Ratio 1 (aka Sine), they can generate the entire trig table. We have an abundance of information! And this discussion of their noticings seems perfect for that. This is the follow up I used last year, and I will again use it this year.**

**The key point I’m getting to: the truth is we don’t need sine, cosine, and tangent. We only need one of them. For example, if I know sine, then cosine can be defined as and tangent can be defined as . So why do we have all three? Life is easier. Look at triangle (g) above in this post. Try using a table with only Ratio 1 to find the missing angle. It is more work than if we had a table for Ratio 3. **

[1] Okay, yeah, so afterwards, I realized I could have just asked if the ratios had a constant difference. But my more complicated approach led to something interesting! Also: if I had more time, I would have asked kids to develop a way to decide if the ratios were growing linearly or not. I bet some would have said common difference, some would have said find the slope, and some would have said graph!

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So I created a totally new approach. Instead of having students discover the theorem, I would work backwards. Here is the TL;DR from my last post — after I had created the activity.

The TL;DR version: students investigate all quadrilaterals where the diagonals satisfy the property that ac=bd. Students are guided to make a conjecture which we as teachers know will be wrong. Then we show a counter-example to blow their conjecture up. And them bam: they have to try again. Using geogebra and some more encouragement, students discover that all cyclic quadrilaterals satisfy ac=bd. And so the circle emerges out of this investigation of quadrilaterals and diagonals. This is, then, the crossed chord theorem. Which students got at by investigating quadrilaterals. Weird. Now they are in a prime place for wondering why the circle shows up. Proof time!

When I shared the activity, I got a couple suggestions from @k8nowak and @bowenkerins and so I modified it with a single tweak which made it oh so much more powerful. **In this post, I will talk about my experience implementing the activity, as well as share the modification I made. However I entreat you to read the original post as I’m not going to outline everything again! So go read!** Okay? Okay.

(1) First off, the change. I made a change to the very last question on the sheet… Instead of having students look for blermions “in the wild,” I had them fix three points and find a bunch of different locations for the fourth point (so the quadrilateral would be a blermion).

[docx: 01 Crossed Diagonals [new]]

At the end of class, I had students fill out a google form with their possible fourth points location.

(2) In class, when students were filling out the conjecture in #4, I saw a number of interesting conversations happening. Their conjectures were essentially: (a) all blermions need at least one pair of parallel sides, (b) all blermions have supplementary adjacent angles, and (c) a blermion has opposite angles supplementary. Most students didn’t find any kites that were blermions, which is why they came up with conjectures (a) and (b). But when the few students who found blermion kites said this to the class, we realized that (a) and (b) couldn’t hold anymore. But conjecture (c) was a possibility still.

Now to be clear, I was expecting conjectures (a) and (b). I was *floored* when not one but **two** groups out of five wanted to persevere and find a good conjecture, and used geogebra to measure angles. It was awesome. And it led to a great discussion later on. More on that later.

(3) I asked kids *why* I had put question #5 on the sheet… what might have been my motivation? I liked asking that question and having groups discuss, because they all recognized that by only looking at “nice” shapes (which, granted, I asked them to do), they could only make limited conjectures. And as soon as they see the blermion in #5, most conjectures would go out the window. The point? To show students that all quadrilaterals aren’t “nice.”

(4) The moment when students saw all their group’s data together in #6… Well, two of my groups got to this point in class. It was … incredible. Kids had their minds blown. Something totally unexpected happened.

For the other groups, I shared the class data (from the data they entered in the spreadsheet):

Holy cow! It is so beautiful! All possible fourth points of a blermion seem to lie on a circle!

(5) Students then wrote a conjecture, and we said if the conjecture were true, we’d suspect (from the Always Sometimes Never questions in #3) that all squares, rectangles, and isosceles trapezoids could be inscribed in a circle. And we discussed how we were going to prove the opposite: *If you have a cyclic quadrilateral (yes, I introduced that term!), that product theorem thingie (a)(c)=(b)(d) holds with the diagonals*. Okay, we were a bit more formal, but that was the crux of things.

(6) Before proving that, I wanted to exploit the conjecture (c). I had students prove that all cyclic quadrilaterals had opposite angles that were supplementary. They struggled a bit with this, but once they had their insight, BOOM. (They used the inscribed-central angle theorem thingie — a central angle is half its corresponding inscribed angle).

(7) Then I left students to prove the crossed chord theorem. I gave them this sheet:

[docx: 01 Crossed Diagonals (proof)]

Almost all kids got to the point where they recognized two pairs of similar triangles. And they recognized that if they could prove one pair of similar triangles were congruent, they could set up a proportion and be done! But the problem was proving the triangles similar. Almost all groups got stuck here — and even though I said: you’re almost there! Think about the inscribed-central angle theorem! — they couldn’t progress. I didn’t do a great job of knowing what to say next.

What I did was show them this (which they had created earlier). For some reason, *this* did not work for them as a hint.

In the future, what I should do is just highlight an arc for them… and say “this arc can guide you!”

Maybe that will work better?

However, eventually all groups got the proof.

(8) At this point, I had students start solving problems. Two with quadrilaterals and two with chords.

Again, I asked them why I included had the second two types of questions, and had the discuss in groups. They recognized that the theorem didn’t need to be stated with cyclic quadrilaterals… Instead it held if we are talking about two line segments in a circle (at that point, I introduced and defined the terminology “chord”). Then I had students write the theorem we had proven *without* reference to the quadrilateral, and we went around and shared and critiqued the wording.

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I don’t always *love* the stuff I come up with. Sometimes it flops. Sometimes it’s pretty good. But rarely do I think it’s so awesome that I would give it the stamp of “highly recommended.” This gets that from me. It is interactive, there is a moment when kids’s minds are blown, and it ties together so many interesting ideas.

For context, I did this *after* we did our unit on similarity. We then proved that the base angles of isosceles triangles were congruent. We then used that to prove the inscribed-central angle conjecture (download here: A Conjecture about Inscribed Angles). And then this. It flows so nicely.

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Yesterday, as we were wrapping this all up, I said to my kids:

“This thing we just proved about circles and chords… this is at the top of a mountain… this theorem is based on lots and lots of other things. If I gave you a bunch of circles with two intersecting chords in it at the beginning of the year, and said, give me some conjectures about this, I doubt you would ever have stumbled upon this… or if you did, it would have taken you a long time and it would have been an accidental discovery. You still wouldn’t have known why it was true. But now you have built so much mathematics throughout the year that this wasn’t an insurmountable feat. What ideas did this theorem lie on?”

And we even had more. We had to know a lot to get there. But wow, it might have seemed impossible at the start of the year, but it was totally doable with all the tools we’ve put in our toolbelts. And how wonderful and inspirational is that?!

**Update: Here is a post about an extension I did on this — involving merblions. **

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Prof. Demaine and his father Martin Demaine both are mathematical artists — playfully using mathematics and art in search of higher truths. The most mindblowing thing that he discovered was that by folding paper however you want, and making only one single cut, you can cut out any polygon. Evenmoreso, the theorem goes further: “Thus it is possible to make single polygons (possibly nonconvex), multiple disjoint polygons, nested polygons, adjoining polygons, and even floating line segments and points.” [1]

Whoa, right? So say you want to cut out each letter of the alphabet? Done.

Or you want to cut out a swan or jack-o-latern?

You can do it. It boggles my mind.

When we went to Prof. Demaine’s talk, on each chair was a packet of paper and a pair of scissors. We were challenged to “fold and cut” each of the shapes out. The shapes were scaffolded well, and so I got pretty far along and was figuring things out. At that time, Brendan and I realized that both angle bisectors and perpendicular lines were *key* for much of what we were doing. We also realized that the puzzle nature of the challenge got us obsessed. We both were stuck on a single page [I’ll write about that in the P.S.] and as I was waiting for the subway home, as I rode the subway home, and all throughout the next morning, I grappled with it. I still have no clue how to solve it.

In any case, we both wanted to expose our geometry students to this puzzle. We figure next year we could turn it into a lesson — having them play *and then have them analyze what they figured out*. But for this year, we wanted to just see what happened if we gave our kids the puzzles.

I faintly recalled my friend Bowman doing this in his class and blogging about it, so I found that post and used his recommendations about what to have the kids cut out in which order, with the scaffolding that Prof. Demaine used in his packet, with some ideas that Brendan had, to create our own packet of fold and cut puzzles.

Fold and Cut Figures [PDF download]

What happened? Well, we gave kids 25-30 minutes. We had extra copies of pages for if kids messed up and wanted to try again. And we said “go at it.” Of all the kids in my class, only one seemed not to get into it… at the beginning. That student was trying too hard to have a “method” and their intuition wasn’t as strong as the others… but they showed me proudly at the end when their star! All the other students were addicted. Paper flew about. Kids called me over to proudly show me their successes, and wailed in frustration when their cut didn’t work (and then hurriedly asked me for another copy of the page they messed up on ). It was exciting to see kids focused but also having fun *playing* with math. I would say that 25-30 minutes was the right amount of time, because at that point, I saw kids start to fade. (It could also be that we met at the very end of the day, and this was the last 30 minutes of a 90 minute block…) No kid in the time given was able to get the scalene triangle (many got close) or the last quadrilateral. But almost every kid was able to get all the figures before ’em.

Next steps from here? I want to turn this into something more formal. I like the play. I *love* the play. But then we need to come up with some general conclusions and talk about why they work. Why are we doing lots of folding along angle bisectors? [Hint: the answer has to do with reflections!] Why are we doing lots of folds perpendicular to the lines of the polygons we’re trying to cut out? [Hint: if we imagine a “vertex” at the place where we have a perpendicular fold, we can consider our fold an angle bisector — bisecting the 180 degree angle of the vertex!] If kids understand those two principles (and the scalene triangle is the most perfect shape to make them both come alive!), I will have a way for kids to tie their puzzling to our geometry curriculum.

What most impressed me was how much intuition kids already had with regards to these. It was amazing to see them take to it as adroitly as they did.

And who knows? Even though I say we should tie this to the curriculum formally next year, maybe I’ll get to it this year after we complete our mountains of salt investigation. Because heck if they aren’t perfectly related to each other!

P.S. So… Here’s where we got stuck. We were given the following paper… no polygon, just a line segment that we had to cut.

You might say: duh, fold it vertically in half and make a half cut. But here’s the thing: you have to make a COMPLETE cut. So once you start cutting, you have to keep cutting until you have completely hit the end of your paper. And BOOM! Suddenly I am perplexed.

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