Author: samjshah

I am a high school math teacher in Brooklyn, New York. I enjoy getting students excited about math by being math’s loudest and most passionate cheerleader.

#ExpandMTBoS

At #TMC16, Tina C. and I led a session called “Breaking Out of Ourselves.” It was a small brainstorming session which started out with us presenting the ways that the online math teacher community (#MTBoS) has started expanding itself — followed by a call to action.

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 RatherOpen 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.
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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 One Good Thing blog. I would love for there to be more regular and semi-regular contributors. The more voices we have when talking about the joys of teaching math, the better. It has helped me out so much during my saddest and most down days, when I open the blog and see old things I’ve written. And I love reading the joyous elations that other teachers have.

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):

#ExpandMTBoS 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!

 

My Takeaways from #TMC16

I have all the feels, coming back from #TMC16, but they also have paralyzed me. There’s a disconnect between all the feels, my making sense of all the feels, and my ability to express all the feels in words. I felt paralyzed because I wanted to express things right. Since that was impossible, I did nothing. But to get past that, and because I need to collate the gems and thoughts from the conference to learn from them, this post is going to be a random collection of thoughts. It’s more for me — to consolidate my thinking and write down all the little things — so apologies if it feels like a confusing brain dump.

What are you passionate about? 

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!
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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:

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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…

Desmos Features:

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):

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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:

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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.

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(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?)

Showing Student Work

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.

Getting Triggy With It! Hands On Trigonometry

 

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 \pi. 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)

Variable Analysis Game

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:

Nominations: Making Work Public

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.)

Intentional Talk 

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:

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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.

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Similar tasks can be done, like 100 NumbersSaving 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.

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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!).

Talk in the Math Classroom

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):

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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):

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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:

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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:

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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:

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In the session, we took all types of questions (e.g. Graph y=8\sin(2x-4)+1) 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:

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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:

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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.

Senior Letter 2016

Each year, except for one, I’ve written a senior letter to deliver to my calculus classes (when I taught them) and my multivariable calculus classes at our last meeting. I pretty much always give the same sentiment — the life of the mind is important. I always crib a bit from previous years (the perils of being in a time crunch!). I wasn’t going to post it, because it is pretty much the same sentiment year after year. But this year, a student came up to me at prom and said that it meant a lot to him, and got him questioning a bit more about what his future might be. (Usually I hand it out and that’s it.)

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

 

Our Math-Science Journal

A super short blogpost, letting you know about the 2015-2016 issue of Intersections, our school’s math-science journal.

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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.

Multivariable Calculus Projects 2015-2016

Each year, I have students in my multivariable calculus class do “fourth quarter projects.” We continue working with the material during classtime, they have regular nightly work, but I cancel all problem sets and tests. Instead, students choose a project topic they are interested in pursuing that has some relationship to the course (even if the relationship is a bit tenuous). I want the project to be one of passion. Their entire fourth quarter grade is based on these projects. This year, my kids came up with some amazing projects — some of the best I’ve seen in my eight years of teaching this course. (Some previous years projects are here, here, here, and here.)

An Augmented Reality Sandbox

 

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!

Harmonograph

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!

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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!

Teaching Devices for Multivariable Calculus

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.

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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…

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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!

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Cartographic Mapping

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.

Deriving the Hagen-Poiseuille Equation from the Navier-Stokes Equations

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…

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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!

The Wave Equation and Schrodinger’s Equation

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:

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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.

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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.

Pitching college math courses

Ooops. This turned out to be a post with no images. So here’s a TL;DR to whet your appetite: I wanted to expose my seniors to what college mathematics is, but instead of lecturing, I had them “pitch” a college course to the rest of the class.

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?

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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 x=0.3 on their calculators. Then I had each student store a different “r” value (carefully chosen by me) and then type r*x*(1-x)->x in their calculators. They then pressed enter a lot of times. (In other words, they were iterating x_{n+1}=rx_n(1-x_n) 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…

Merblions

Earlier this year in Advanced Geometry, my kids were introduced to Blermions (original post from when I created the lesson; new post after I tweaked and taught the lesson). That lesson gets kids to understand a bit about cyclic quadrilaterals and some of their properties.

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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.

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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.

some big results (some proved, some unproved) found

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).

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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.)

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6. In a merblion, the sum of the lengths of opposite sides are equal. result3.png

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.

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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.

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Why I loved this

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.