TL;DR: We fistbumped in precalculus. It was awesome. Super complex math got done.

One of the things I’m working on is improving my questioning this year. One of my strengths is scaffolding, but sometimes — in my desire to be super overzealously prepared — I scaffold too much. Today we had our first “long block” (90 minutes) in Advanced Precalculus. This is how class unfolded, after our warm-up.

I asked students to fistbump everyone at their table.

How many fistbumps did you just do?

How many fistbumps just happened in class?

Then I showed them there were many fantastical ways to fistbump besides the standard “clink knuckles” method. Blow it up. Snail. Squid. Turkey. [1] That was a random impromtu aside. But now, next class, I *must* show my kids the following video:

If you do this in your class, you should definitely have this video queued up. [2]

Then: everyone had 20 seconds of *individual think time for this question:*

If you wanted to devise an efficient way for everyone in the class to fistbump everyone else in the class, what would that way be?

Kids asked what “efficient” meant. I said “it should be as quick as possible, with the least chance of someone not actually fistbumping someone else.” Now you, friend, take a guess. I have 14 kids in my class. How long do you think it would take my kids to fistbump everyone else with an efficient strategy!

Seriously… reader… take a guess! Good. I’ll reveal the answer in a bit.

After the 20 seconds of individual time, each group shared with each other, and had to converge upon their proposal to the class. We went around. The four groups had three ideas:

- Line everyone up. The first person fistbumps with everyone else, then leaves. Then the second person fistbumps with everyone else, then leaves. And so forth.
- We have four groups in our class. The first group goes around and fistbumps with the members of other three groups in order. Then the second group does that with the remaining groups. And so forth.
- Do the exact same thing as proposal #1, except as you don’t wait until the first person is done fistbumping everyone else.
*As soon as the first person is done fistbumping with the second and third person (and continues on down the line), the second person starts fistbumping down the line*. And so forth.

(I had also anticipated students talking about getting in a “circle” and having one person fistbump with everyone, then another person, then another, etc. It’s organized, but not very efficient. One thing kids asked: can we all fistbump each other at the same time, in one giant mass of fists? I nixed that. I also had kids ask if you could fistbump with both hands simultaneously — to two different people. I said yes! But I didn’t give enough time for students to devise something super efficient with that so that never got turned into a proposal.)

As a class, we decided proposal #3 was going to be the most efficient. So I had them all file into the hallway and try out their fistbump method. I got my stopwatch out. And they went at it, after organizing themselves.

You may wonder what all of this has to do with math. That’s coming. This was just the setup. I honestly think by this point in the class, some kids were wondering what the heck we were doing this for…

So how long did it take them?

Yup. Under 12 seconds! I! Was! In! Awe!

Then each group got out a giant whiteboard and markers and answered the following questions:

How many fistbumps did you just do? What was the average time per fistbump?

Once they answered that question, they called me over to discuss their findings with me. Then I had two extensions:

We have 998 students at our school. How many fistbumps would that be? How long would it take, if we used our efficient method and assumed the same average time per fistbump? [3]

Can you find a method to answer that question?

And clearly, this is where the math comes in. This — in case you hadn’t seen it — is the classic handshake problem.

And from this point on, you have to facilitate class based on what your kids are doing. Some advice?

Advice 1: If kids are struggling, have ’em start noticing patterns about the number of handshakes for smaller numbers of people. Two people? Three people? Four people? Continue working up. Make a table. Look for patterns.

Advice 2: If kids have seen the “rainbow method” or some variation (see below), have them think about the difference between an even number of things being added and an odd number of things being added.

Advice 3: Have kids work on coming up with a single formula that works for even and odd numbers of things being added. Then have them explain why that formula works.

Advice 4: Lead kids to the idea of “double counting”: if we have 4 people, then have each person fistbump with everyone else. Since each person fistbumped with 3 people, there were a total of 4*3=12 fistbumps. However we’re *double counting* in this, so there are really only 6 fistbumps. (If kids don’t see the doublecounting, have a group of four act it out.)

Advice 5: If a group needs an assist, have individual members circulate to other groups and gather ideas, and then return and share what they found.

I loved doing this activity. Kids got into it. They felt ownership and camaraderie. Kids were up and moving. Because we had a long block, kids had time to play and productively struggle with the ideas. And most importantly: I didn’t overscaffold. I built up motivation and then sprung a good open-ended question for kids to work on.

[1] If you don’t know what I’m talking about, clearly you’ve never hung out with middle school students.

[2] queue is such a strange word, right? 80% of the letters are unnecessary. “q” is the same pronunciation as “queue.”

[3] ~~The answer is around 18 hours. What I loved is that when a group got that — after we got 12 seconds for our class — I was like “come ON guys? does that make sense? it would take almost a whole day with no breaks? REALLY?” I wanted them to see the answer was kinda absurd. But it is right, because although it might seem absurd on the surface, each time we add more people, we’re making the number of handshakes grow pretty darn fast! (Follow up? How fast? Let’s make a graph! Ohhhh, quadratic? PRETTY! And grows super quickly for higher numbers, unlike linear graphs.)~~ Turns out the answer is much shorter than 18 hours. I had a misconception that someone helped me see on the betterQs blogpost! I liked admitting to my class i was wrong!

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I’ve found the questions on the survey are simple and nonthreatening enough that I get interesting responses. However I find that I do get way more extensive and thoughtful answers from the upper level grades than from freshman.

Here is a link to the survey if you want to check it out.

The thing is… I get tons of interesting information about my kids. They let me know about some horrifying thing that happened in fifth grade math that they still remember, or an amazing feeling they got once some abstract concept snapped into place, or about some lifelong passion of theirs that I wouldn’t know about. Perhaps the most important question — in terms of the information I get from it — is this one:

It’s kinda amazing. The phrasing of the question implies that there is something they are nervous about and are invited to write. (It’s so different than “Are you nervous about math this year? If so, why? And if not, why not?”… It’s like asking “What questions do you have?” instead of “Do you have any questions?”)

I’m not going to copy and paste responses, but I will share some types of responses:

- keeping up with the material / keeping up with classmates / falling behind
- test anxiety
- fractions
- coming across as annoying to classmates
- memorizing formulas
- explaining my reasoning in words

They really open up given the opportunity, especially considering I had only met them for 30 minutes before I asked them to fill this out. And if a kid came to you and had told you they were nervous about any of these things, you would know as a teacher precisely what to say!

So what I do, once I get these surveys, is I write back individually to each kid. The emails aren’t long, but they do talk about things that students specifically referenced in their survey. Here’s one from a couple years ago:

Howdy [Stu],

I’m reading through the surveys that you guys filled out for precalculus, and I wanted to respond to you, just to say hello! I’m thrilled that you’re going to be in our large band of precalculetes for the year. I’m excited about everything! We’re going to be doing a lot of exploring and making a ton of connections. I love love LOVE math and have since I was in high school, and I want to extend myself to you. If you ever feel overwhelmed or unclear about things, and they just are staying foggy, never hesitate to email me to set up a meeting. (Of course, I think you should first try to ask a colleague, because they often are better resources than I am.)

You noted that you’re nervous about keeping up with the workload. It is going to be a solid amount each night, but I very much try to keep it reasonable and I also try to make sure it is all relevant/important. I don’t assign 10 of the same types of problems, but rather I assign a couple of them and expect students to try extra problems if they need extra practice. But please let me know if the workload is getting to be too much for you. Last year I asked for feedback periodically on the workload and for the most part kids said it was fine, except in the third quarter when I think I accidentally asked for too much — and when kids told me, I was able to be more conscientious!

You also said that you don’t talk a lot at first, but you will. I saw you talking in your group! I think maybe because this is going to be a group-based class, you’ll find you’ll come out of your shell pretty quickly! But if you’re painfully shy, definitely talk with me. I’ve worked with kids who are shy before and we’ve come up with ways to help get over that so they can delve into the math!

Glad to have met you, and I’m looking forward to an enjoyable year.

Always my best,

Mr. Shah

It takes up a long time to write to every student. I have smaller class sizes that most of my friends, because I’m in an independent school. But still… I only get 5-6 emails written in an hour.

Why do I do this survey? Mainly because I love reading their responses. Especially to these two questions:

I also take the time to reply individually because I hope — though I never really know — that it helps make me more approachable. I pray that it implicitly tells my kids *hey, I care.* And early in the year, when I stumble through not remembering their names and want to crawl in a hole, this is such an important sentiment to get across.

So in this survey are some of my better questions, and how I deal with them.

[Cross posted on the betterQs blog]

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When we met last week, we decided to rectify this. We brainstormed some general ideas, and I turned those into this activity.

The setup: Kids are so used to looking at “normal” quadrilaterals in geometry. So we thought we’d exploit that. We don’t mention circles. We don’t mention chords.

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!

Here’s the start: we introduce a new type of quadrilateral called a “blermion.” (.docx here)

We had some debate over whether we were giving too much away with this start [1], but we decided we weren’t. (We’re going backwards. The students aren’t deriving the formula. They’re using the formula (which we are calling a “property” of quadrilaterals) to come up with the circle part of the theorem.)

So yeah, we gave kids the ac=bd formula, but in relation to the diagonals of quadrilaterals. And we asked: “which quadrilaterals will this property hold for? We’ll call ’em blermions”

So I ask them to look at the standard quadrilaterals they know — investigating this property using a geogebra sheet — and having them making a conjecture about blermions.

The ggb sheet is here.

So students play on geogebra and come up with some understandings (inductively) about which quadrilaterals are blermions. Then they make a conjecture about all blermions.

This conjecture will fail. Because it is based on students only looking at “nice” quadrilaterals. I want the conjecture to fail. I want to emphasize the point that looking at “nice” examples can often lead to blind spots in your logic.

Students will see it fail when they are asked to drag the four points to specific places (see #5 below). The quadrilateral that results is weird looking. There is nothing that seems special about it. But it does have ac=bd. It is a blermion. Their conjecture about blermions was wrong!

Now students are sent on a chase to find more blermions — and they are encouraged to not just look at “nice” quadrilaterals. They record their results. (If they are stuck, a teacher can have the students fix three points and only drag the fourth point; It turns out you will always be able to drag that point to have ac=bd… and that in fact you can find an infinite number of additional points by doing this dragging of that fourth point.)

At this point, once they have found lots of blermions, students are going to try to make another conjecture about all blermions. I wonder if any student is going to get it. It’s okay if they don’t. At this point, I’m going to have every student plot a different blermion (some “nice” quadrilaterals, but mostly not nice ones). Then I’m going to have them pick any three points and change the color of them. Finally, I’m going to have students go to the “draw a circle with three points” tool, and be surprised by the fact that the circle always goes through that fourth uncolored point.

Why is this good? I hope they *don’t* get it. Because seeing that *every blermion* works like this (a circle goes through all four vertices of the blermion) is the key wow factor for kids. It’s strange, because even though I will be giving away this key fact, I think all this play will make this key fact interesting and weird. [2] Once they all see that, they are going to be curious as to how circles even got involved with these quadrilaterals in the first place. And… that is perfect… because then the kids are going to want to know *why* this happened.

And then we can transition to figuring out how to prove this. Because suddenly the crossed chord theorem is weird and strange and unexpected, and suddenly we kinda want to know why it works!

[1] We had to decide whether students should *discover *the property ac=bd for crossed chords. Motivating that from a circle and crossed chords was hard. We needed kids to somehow *see* similar triangles (which felt like we would be giving away too much) or come up with the multiplication idea of the pieces of chords on their own. We had ways to motivate that multiplication, but they weren’t elegant. So we scrapped that.

[2] Here’s the thing. Most things in geometry are presented to students in such a way that their wonderment about the geometric thing is killed. In a proof, the statement to be proved is given up front — and suddenly it isn’t interesting. It might be something really cool, but the exercise around doing the proof doesn’t highlight that. Or — as I’ve blogged before — theorems like the ones involving all the triangle centers… we tell kids to plot the perpendicular bisectors of all three sides of a triangle and they meet at a single point. It isn’t strange and wonderful. They don’t see *why* that’s weird. They just know we told them to plot the perpendicular bisectors, and they know something will happen because why else would we have them do it? We kill the wonderment of geometry in so many ways.

I want the weirdness and unexpected and unintuitiveness to come back to geometry… that’s where the beauty and curiosity are… and only *then *have my students work on figuring out why the unexpected happens… and get to the point where the weird and unexpected and unintuitive become obvious and natural. Making the unnatural natural. Yup, that’s the goal. But to do that, you have to first get to the unnatural.

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This year I want to spend some time thinking about how to question well. More specifically, thinking intentionally about what questioning looks like (and how it can be improved) in my classroom — both on my end and on my students’ end. I thought I would blog about it throughout the year, and figured it would be fun to blog with others. @rdkpickle had the same idea! So we figured it was a good idea, and set up a collaborative blog. All this is to say:

Check out the blog! Add it to your feedly or googlereaderreplacement. You’ll get posts by many people delivered right to your door.

But more importantly…You are warmly and heartily welcome to join us, and become an author. The blog just started and we’d love to get as many voices and experiences going on the ground floor.

Read a few posts. Browse a bit. It’s only a few days old, so there isn’t too much to gander at! And consider joining us. (If you want, there’s a tab at the top of the blog that tells you how to join, or just click here. We’ll add you as an official author!)

**“But Sam,” you say, “I don’t have time to write every day…”**

Silly goose, I respond! You can write however frequently works for you. Once a week? Once a month? Three times a year? The point is to take some time — however much of it — to think about questioning in your classroom.

**“But Sam,” you say, “I don’t have a lot to write about…”**

Silly turkey, I shoot back! I think it would be cool if you even wrote down a single question that you really loved asking because it provoked discussion. No need to deeply analyze it if you don’t want! Maybe a teacher reading the blog will read that question and think: “YAS! THIS IS EXACTLY THE QUESTION I NEEDED!” And if there were a lot of people just throwing down their good thought-provoking questions, we would soon have an amazing repository.

**“But Sam,” you say, “I have a blog of my own! Why don’t I just post it there?”**

Silly quail, I reply! You can post anything to do with questioning both on your *own* blog, and on *this *blog. No rule against that! In fact, I did that for my first post on the betterQs blog. And that way, someone reading the betterQs blog might get to know you and your own blog!

**“But Sam,” you say, “I’m still scared… I don’t want to sign up and then not do it.”**

Silly emu, I say. Why not take a baby step and just commit yourself to writing one or two things? Just keep a lookout in your school about how you question, or try to script a good question and see how it goes in your classroom, or rewrite a test question and explain how you rewrote it and why… Baby emu steps. And just see how it goes! You just might think: hey, questioning is something I want to pay just a bit more attention to!

Or, silly emu, don’t worry about signing up! As I wrote a couple years ago: “You should never feel guilty engaging with the community in ways that make sense to you. We’re all coming at teaching from such different places in our careers, such different backgrounds, and such different environments. We all need and want different things.” In other words, you do you.

[1] I also love the fact that because I’ve been using the blog semi-regularly, I can see an archive of so many good things of my own (in addition to seeing everyone else’s good things). On down days, it really helps me remember I’m not as bad as my brain tries to convince me I am.

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A former student who was a senior graduating from Northwestern nominated me by writing a reflection about her experience in our calculus class. I then was asked to submit some letters of recommendation and a teaching philosophy / personal statement, which I did. I actually wasn’t going to — another thing on my plate! — but I started thinking how lucky I was to have a former student take the time to write something up about me, and I figured I could put in a couple hours of work to honor that. A few months later, I received an email saying I was a semi-finalist, and had to do a 1 hour video conference interview with six people (administrators, professors, and students at Northwestern) — and that video was then shared with the whole selection committee. Scary!!! I did it, but was a rambling nervous fool. And then: I got a call telling me I was selected, and that I was going to be attending Northwestern’s graduation and be feted.

I was super excited that I got to invite my family (we turned it into a mini-family vacation to Chicago) and a teacher colleague/friend/mentor to join me. This was my favorite photo from the weekend: it was me and my family, my teacher friend, and my former student who nominated me and her family.

The experience… it was once-in-a-lifetime. Memorable moments?

- They put me and my parents up in two “executive suites” at the four seasons. The amount of fanciness was unbelievable, and the view of Lake Michigan from my room was stunning.
*They put little slippers by your bed each night!*I doubt I’ll ever be at a place in my life where I’ll get to experience that kind of luxury again. - There was a luncheon on the first day where the award winners and students (and their families) all got to meet, and the students read aloud their nomination letter (which I had not been shown). I got teary when mine was read. And then I had to give a mini-3 minute speech which I was terrified to do but I think it went well.
- I got to see my former student win an award!
- There was a fancy fancy dinner for long-term retiring faculty and the award winners (where we were again feted), and I got to hear those receiving honorary doctorates give mini-speeches. My favorite was Dan Shechtman who is a Nobel prize winning chemist who talked about teaching young kids about science and not underestimating their abilities.
- Graduation! They had the award winners sit
*at the front of the stage*and we were called out during the ceremony. I was sitting next to the president of the university. This was my view:

When they called out our names during the program, and there was a wave of applause and cheers, I got chills. In a good way. - We (my parents and me) went to my former student’s apartment for lunch with her family. We had falafel delivered and talked about … well, everything. Those two hours were my favorite, actually, of the entire weekend.
*Except for putting on those four season slippers!* - I was invited to go to the “mini-graduation” for the School of Social Policy to see my former student get called on stage and graduate. (But then I was told I was sitting on stage, front and center.) It was so exciting when her name was called that I snuck out my phone to try to get a good photo of it happening!

I secretly was relieved when my name wasn’t called and I wasn’t called out… too much fete-ing can be exhausting!… but right at the end of the ceremony they had awards they were handing out to professors and they then called me out and gave me a plaque with my name inscribed on it. So I suppose I couldn’t avoid betting fete-ed after all. :)

It was an unbelievable experience. For me, the most wonderful part of this adventure was knowing it was all kicked off by a former student — and that I got to share in an important turning point in her life.

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*If your group has a question, everyone in the group must raise their hand to call me over…*This is how I started the last couple years of precalculus (all my kids work in groups). The idea was that if a kid had a question, they needed to first talk with their group so that the math teacher (me!) was not the sole mathematical authority in the classroom. I quickly added on*… and I will call on one of you randomly to ask me the question*. That way everyone in the group had to be comfortable asking the question, and that it was a real**group question**and not just an individual question.Last year, for some reason, I didn’t keep up with this practice, and started answering individual questions. I need to remember to keep up with this practice, because it’s**awesome**and**it works to get kids really talking and explaining without you**.- I taught calculus for seven years, and when I started standards based grading, I used to put after each question testing each skill a little box:

It was useful when I met with students to discuss their tests. If they felt shaky and did poorly, that meant one thing to me. If they felt confident and did poorly, that meant another. If they felt shaky and did awesome, that meant something totally different. It led to some good conversations, and got kids to be more meta-cognitive. It also led to some interesting written feedback on the tests (even if I didn’t meet with the student).But I only ever did that in calculus, and I don’t teach calculus anymore. So I want to incorporate this on my assessments in my other classes — at least geometry and precalculus. When I’m asking a “mathy” question, this is a sort of different additional question that helps me put their response in some context. - Questions can have different purposes for me, even though I don’t (in the moment) think of them this way. Mostly they are to either (a) to get a student to go from a place of not understand to understanding (through asking questions to get them to think and make connections), or they are (b) to help me understand what a kid (or my class as a whole) is understanding.If I’m asking a question to the whole class, and my purpose is to figure out what my kids understand and what they don’t, I’m not going to have my kids raise their hands anymore. I got to the point where sometimes I would call on kids with their hands raised, and sometimes not. I mean: if the kids all raising their hands to answer a question feel they
*know*the answer, then why am I calling on them? Instead, I am thinking of stealing an idea from a friend who taught middle school: THE POPSICLE STICKS OF DESTINY. I am going to have my kids’ names written down on popsicle sticks and pull them out of a mason jar (because I’m such a hipster!) to randomly call on someone. Yeah, index cards work too, but INDEX CARDS OF DESTINY is way less fun to say dramatically.If I do this, however, I need to make sure that the kid who doesn’t know something or is confused feels like the classroom is a safe space. This year I’ll be teaching the advanced sections, so there is a lot of insecurity that these kids have about “being smart” (*cringe* I hate that word) and “appearing dumb” to their classmates. I have to brainstorm how I’m going to publicly reward kids for having good questions or being confused but doing something about that confusion or for being wrong but for owning it and saying “I NEED TO GET THINGS WRONG IN ORDER TO FIGURE OUT HOW TO BE RIGHT. AND I’M AWESOME FOR KNOWING THAT.” Heck, maybe I’ll have a poster made which says that, and have kids read it aloud occasionally when they’re wrong. And I should point to it and say it when I am wrong. Or maybe that’s dumb. I don’t know.

That’s about it for now. Hopefully more to come as I figure things out!

[cross posted on the *betterQs* blog!]

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But while we were all together, something started occurring to me. I do a lot of thinking about this online community, the #MTBoS… who we are, why we came together, why we continue to come together, how are we inviting those who want to join in the fun, what we can do as a collective whole, and what we are doing as a collective whole. For me, this community started 8 years ago, and I’ve seen it grow from a nascent group of bloggers who shared their classroom activities and musings on education and their kids to a much more complex *thingie.* (Yup, I’m awesome at wording, right?!) Two years ago, after TMC13, I wrote:

the main takeaway of the conference was new. It was that **we are a powerful force**. We are not a loosely connected network of professionals, but we are **a growing, tightly-connected network of professionals engaged in something unbelievably awesome. Through this community, we are all – in our own ways – becoming teacher leaders.**

Around that time, I saw a lot of cool collaborative *thingies* just starting to bloom and blogged about how frakking awesome that was:

One thing that is now crystal clear to me is that we’re shifting into a new phase. (“We’re” meaning our little math teacher online community.)[…] Now in the past year or year and a half, there has been an explosion of activity. and this explosion seems to center around (a) collaboration and generating things which are (b) not really centered about us and our individual classrooms. We’re thinking bigger than ourselves.

I’m talking the letters to the first year teachers, I’m talking the Global Math Department, I’m talking thevisualpatterns website, I’m talking the month long new blogger initiation, I’m talking the freaking inspirational One Good Thing group blog, I’m talking Math Munch, I’m talking the collaborative blog Math Mistakes, I’m talking MathRecap to share good math PD/talks with each other. And of course, now we have the Productive Struggle blog, Daily Desmos, and the Infinite Tangents podcast. [1]

We’re still keeping our blogs, and archiving our teaching and sharing ideas, and talking on twitter. But now we’re also moving into creating these *other things* which are crowdsourced and for people *other* than just those in our little community…

It’s been a freakin’ pleasure to see all this stuff emerge out of the fertile soil that we already had. We’re starting to create something new and different… and… and… I can’t wait to see what happens.

At that time, it was just the beginning… So much has happened since.

There are many more people who are jumping in. More initiatives and collaborative projects are happening. People are meeting up more and more in real life tweetups. There has been an NPR story on one of us. Multiple grad students are doing their dissertation and research about our community. The MTBoS has no official organization or centralized structure and doesn’t speak with a single voice (something I value greatly), but it has gotten the attention of the National Council for Teachers of Mathematics (NCTM). The president elect and the executive director of NCTM came to TMC15. They have given us booth space at their last national conference. There are a series of sessions at MTBoS (strands) that have happened (one, two). It’s worth thinking about what this means.

When I was at TMC15, I noticed that there wasn’t as many conversations or mentions of “celebrities” or “rockstars” as in previous years. I think I heard those words at most twice. It’s not like people weren’t excited to meet their math teacher crushes, but something felt different. I think we’re shifting away from “celebrities” and “rockstars” and are moving towards *brands. *Okay, that’s not the perfect term, because there is something pejorative about that, and I mean anything but that, but people have their *thingies.*

Some quick examples:

@cheesemonkeysf is known for *talking points* and how the social-emotional life of a student has everything to do with their ability to learn

@PiSpeak is known for* math debate* in the classroom

@sophgermain is known for *diversity and inclusion* issues

@fawnpnguyen is known for *visual patterns* and her Sage Experienced Teacher Wisdom (aka her funny and emotionally charged stories from her classroom)

@mathequalslove is known for her work on *interactive notebooks* and her *craftiness*

@AlexOverwijk is known for *activity based teaching*

@mpershan is known for exploiting *math mistakes* and *encouraging critical discourse*

and the list can go on and on and on…

I think the idea of “celebrity” is being replaced with “brand” (or niche, or whatever). As the community grows, there are more and more voices. But there are certain ones that get a lot of traction. Of course the more involved they are (via blogging or tweeting), the more noticed they are. But that’s not enough. It’s their messages.

Two things keep ringing around in my head about this.

One came from Christopher Danielson’s amazing keynote at the conference. His message: “Find what you love. Do more of that.” Of course, that’s a little pat, and you need to see the whole presentation to truly understand. It isn’t “I love mathematics” or “I love kids.” He asked us to dig deeper, go a bit farther. What about mathematics speaks to us? What about working with kids makes us tick? His example: he *loves* ambiguity. The space between the certainties. And so a lot of his work as a teacher is exploiting those ambiguities with his students to get them to learn mathematics — but also hopefully appreciate (and dare I say, love) ambiguity too?

The other is from a reddit AMA conversation with Kenji Lopez-Alt. He writes the best food blog posts evar! And in this Ask Me Anything, he was asked for advice on starting a food blog:

I’m not posting this because I want to share his advice on starting and maintaining a blog. But I realized why I love his posts is because he *does* have a specific point of view, and that point of view speaks to me in spades. His passion about the science of foods and sharing his discoveries with others is so apparent. But I suppose what I mean is: he has found something he loves, and is doing more of that. He has a *brand. *

I suppose I’m saying that what I’m seeing is that there are a lot of others out there in the math community who have found that thing they love, that specific thing that makes them the teacher they are, the thing they are passionate about, and their blog and twitter conversations tend to revolve around that. They are doing what Kenji suggested — but I’m guessing without even consciously realizing it.

I don’t know, I’m just musing here. But I think ages ago there were “rockstars” and “celebrities” who were well-known — but some of their rockstarness was from being around for a long time and thus having a large network of people they could communicate with in a tightly knit community that was growing. Now I think that may be shifting. I think as we have more people, the MTBoS has a lot of mini-communities that exist within it — it’s a patchwork quilt. And that is a natural and good thing.

And I’m seeing specific people — old *and *new — speaking with clear voices and messages.* This is what I’m passionate about. This is how I enact that passion. This is what I stand for. This is my brand. Hear me roar. *[1]

And they are going outside of their schools and our smaller community to bring the thing they love to a larger audience. Creating websites, writing books, leading professional development, etc. They are expanding their brand. (And again, I don’t mean *brand *in a negative way!)

These are the people that speak to me. They have a voice. And I’m interested in hearing what that voice is saying. I would venture to say that they speak to others for that same reason.

What is so awesome sauce about this is that they are becoming *teacher leaders*. We don’t have models for what a teacher leader is in the United States. Once you become a teacher, unless you leave the classroom, you will always be a teacher. There are no ranks (except maybe the very expensive National Board certification), and there aren’t well-defined pathways to get more involved in the profession — again, without leaving the classroom. There aren’t a lot of models of those who are effecting change outside their own classroom. Think about it: excluding the MTBoS, can you think of five teacher leaders who are still in the classroom? One? [2]

But I see right now in this community the creation of new models for what a teacher leader can look like. Whether you have five years in the classroom or twenty five, there are pathways that people in the MTBoS are carving out in order to share what they love. Help other teachers. Impact student lives. And more than anything, this is what I predict will be happening more and more as the community continues to grow and mature. [3]

On a more personal reflective note, I realized I don’t think I have that brand. I think if 10 people were asked in the MTBoS, “what is Sam Shah about?” I doubt there would be a general consensus. Why? Because I don’t think I have figured that out for myself… yet. I know many things about myself as a teacher — I can be reflective as heck at times — but I still don’t think I speak with that voice or brand that so many others I admire do. And that’s not a bad thing *at all*. It’s just me still figuring stuff out.

[1] Again, I don’t think many would even say they’re aware of it…

[2] This is not a knock on those who have left the classroom to help our profession. I am just saying it’s hard to be a teacher leader and stay in the classroom. And I want to stay in the classroom.

[3] A lot of this part of my thinking came from @pegcagle and @_levi_’s TMC talk.

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I’ve found this year to be an important transition point:

For the first time, I taught ninth graders, and for the first time, I taught geometry. And in order to do that, I worked an insane number of hours with my partner-in-crime and co-teacher BK in order to write an entire curriculum from scratch, from head to toe. Yup, you read that right. We — in essence — wrote a textbook. We sequenced the course, we wrote materials and designed activities for the course, and we had kids do all the heavy lifting. There are particular moments as a teacher which standout as “big moments.” Moments where we know we’ve developed immensely as a teacher. Transitioning from individual and partner work into total groupwork was one of those moments. Converting my non-AP calculus course into a standards based grading course was one of those moments. And writing a curriculum from scratch, in a single year, with an insanely thoughtful collaborator was the most recent of those moments [2].

The previous two years (before this school year) were two of the hardest years I’ve had as a teacher. We teachers were called on to do a lot in the wake of our school’s five year strategic plan — and it became overwhelming. I had no work-life balance. And I became a bit curmudgeonly because of those tough years. But this year, things have been better. I still have no work-life balance, but the overwhelming onslaught of initiatives have subsided. One of the things I did to actively try to stay positive this year was to write down *every single day* one good thing that happened to me — big or small. From the first day of classes to the last. And those things are archived here. This was especially important because at the start of the school year, my mom was diagnosed with cancer (she is doing very well, fyi, no worries).

That being said, I am going to make a goal: that next year, I am going to just let the things that I can’t control go… There’s no point in getting worked up over something that you can’t do anything about. Instead, I’m going to stay loose, and bring back my frivolity and humor, and go off the beaten path in class more. While organizing today, I was looking through a number of old emails and cards from students, and saw so many inside jokes and fun times that they references… and then I thought about this year… and I came up blank. I couldn’t think of a time that I doubled over laughing in class. I couldn’t think of an ongoing joke that I had with a student. I could think of great lessons and a ha moments, but nothing frivolous and fun. *So my vow is to make sure that next year involves more joy and laughter. For me, and for my students*. *Every day.*

Wow, yes, this braindump led me to something big. With that, I’m out.

[1] That doesn’t mean I’m done with school. I have lucky 13 college recommendations to write. And two summer projects that each will take 25 hours each to complete (revise my multivariable calculus curriculum; plan for our new schedule next year with longer blocks).

[2] I’ve written entire course curricula before. Calculus, for example. But that took a few years to write and get added to. And Adv. Precalculus, which I did in a single year, but lacked the collaboration and innovation that I was able to do this year with BK.

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This year I had six students and these are their projects.

**“Exploring the Normal Distribution Through the Box-Muller Transform and Visualizing It Using Computer Science” (GT)**

This student had never taken a statistics course but was interested in that. We also talked about how to find the area under the normal distribution using multivariable calculus (and showed it was 1). Armed with those two things, this student who likes computer science found a way to pick independently two numbers (one each from two *uniform *distributions), and have them undergo a few transformations involving square roots and sine/cosines, and then those two numbers would generate two new numbers. Doing this a bunch of times will create a whole pile of new numbers, and it turns out that those square roots and sine/cosines somehow create a bunch of numbers that exactly follow a normal distribution. So weird. So cool.

**“XRayField: Detecting Minecraft Cheating using Physics and Calculus” (W.M.)**

This student loves Minecraft and hosts a Minecraft server where tons of kids at our school play. Earlier in the year, there was a big scandal because there were people *cheating* when playing on this server — using modifications to give themselves additional advantages. (This was even chronicled by the school newspaper.) One of the modifications allows players to see where the diamonds are hidden, so they can dig right to them. So this student who runs the server wanted to find a way to detect cheaters. So he created a force field around each diamond (using the inverse square law in 3D), and then essentially calculated the work done by the force field on the motion of a player. A player moving directly with the force field (like on the left in the image above) will get a higher “work score” than someone on the right (which is moving sometimes with the forcefield, sometimes not). In other words, he’s calculating a line integral in a field. His data was impressive. He had some students cheat to see what would happen, and others not. And in this process, he even caught a cheater who had been cheating undetected. Honestly, this might be one of my favorite projects of all time because of how unique it was, and how perfectly it fit in with the course.

**“Space Filling Curves” (L.S.)**

This student with a more artistic bent was interested by “Space Filling Curves” (we saw some of them when I started talking about parametric curves in three dimensions, and we fiddled around with Lissajous curves to end up with some space filling curves). This student created three art pieces. The first was a 2D Hilbert curve which is space filling. The second was a 3D Hilbert curve which is space filling (pictured above). The third was writing a computer program to actually generate (live) a space filling curve which involves a parametrically defined curve, where each of the x(t) and y(t) equations involved an infinite sum (where each term in this infinite sum was reliant on this other weird piecewise and periodic function). I wish I had a video showing this program execute in real time, and how it graphed for us — live — a curve which was drawing itself and how that curve being drawn truly filled space. It blew my mind.

**“The Math Is Right: The Math Behind Game Shows” (J.S.)**

This student, since a young age, loved watching the Game Show Network with his mother. So for his final project, he wanted to analyze game shows — specifically Deal or No Deal, and the big wheel in the Price is Right. I had never thought deeply about the mathematics of both, but he addressed the question: “When should you take the deal? Is there an optimal time to do so?” (with Deal or No Deal) and “If you’re the second player spinning the big wheel (out of three players), how do you decide whether to spin a second time or not?” (for the Price is Right). As I saw him work through this project — especially the Price is Right problem — I saw so much rich mathematics unfold, involving generating functions, combining distributions, and simulating. It’s a deceptively simple question, with a beautifully rich analysis that hides behind it. And that can be extended in so many ways.

**“The Art of Balance” (M.S.)**

This photograph may make it look like the books are touching the wine holder. That is not the case. This wine holder is standing up — quite robustly as we tested — through it’s own volition. And — importantly — because the student who built it understood the principle behind the center of mass. This student’s project started out with him analyzing the “book stacking problem” (which involves how much “overhang” you can create while stacking books at the edge of the table. For example, with one book, you can put it halfway over the table and it will not fall. It turns out that you can actually get *infinite* overhang… you just need a lot of books. This analysis centered around the center of mass of these books, and actually had this student construct a giant tower of books. The second part of this project involved the creation of this wine holder, which was initially conceived of mathematically using center of mass, then that got complicated so the student started playing around with torque which got more complicated, so the student eventually used intuition and guess and check (based on his general understanding of center of mass). Finally he got it to work. The one thing this student wanted to do for his project was “build/create something” and he did!

**“Visualizing Calculus” (T.J.)**

This student wanted to make visualizations of some of the things we’ve learned about this year. So he took it upon himself to learn some of the code needed to make Wolfram Demonstrations, and then went forth to do it. He first was fascinated with the idea of fractional derivatives, so he made a visualization of that. Then he wanted to illustrate the idea of the gradient and how the gradient of a 2D surface in 3D space sort of defined a plane tangent to the surface if you zoomed in enough. Finally, he created an applet where the user enters a 2D vector field, and then it calculates the divergence and curl at every point of the vector field. His description for what the divergence was was interesting, and new to me. About the point chosen on the applet, he drew a circle (and the vector field was illustrated in the background). He said “imagine you have a light sprinkling of sand on this whole x-y plane… and then wind started pushing it around — where the wind is represented by the vector field, so the direction and strength of the wind is determined by the vector field. If more sand is coming into the circle and leaving it, then the divergence is negative, if more sand is leaving the circle than coming into it, then the divergence is positive, and if equal amounts of sand are coming in and leaving the circle, then the divergence is zero.”

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Click on the image to go to the journal and see the cool math and science things kids at my school are working on!

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I’m teaching freshpeople (9th graders) for the first time. And I’ve come to learn how important structure is for them. I’ve realized how useful it is to make topic lists for them (next year, I’m going to ween them off of them and show them how to create their own!). I’ve learned how important it is to be explicit with them about everything. And I’ve learned that many don’t quite know how to study.

In exactly a month, my kids are going to have their geometry final. So I whipped up a guide to explain how they might go about facing this daunting task. It’s not perfect. I hate the fact that it is so long and text heavy. But I want to get it out to my kids soon — so editing will have to wait for next year.

The truth is I don’t know if any of them are going to use it. But I’m going to at least provide them with some ideas — and maybe one or two things will resonate with them. Here it is below (and in .docx form). *If you have any additional advice you give to your young ones that would go well in this, please throw them in the comments. Although I might not be able to add them for my kids this year, I can revise it for next year.*

I am teaching a lot of juniors this year, which means I will be asked to write a lot of college recommendations. I never learned how to formally ask for a recommendation until I was in college — but when I was taught by a professor (who was helping guide me in the grad school application process) it was enlightening. I crafted a cover letter, got my best work together, and set up a time to meet with my professors who I was asking for aletting of recommendation from. At that meeting, I outlined why their classes were important to me, what I took away from them, and things I was proud of — and why I would really appreciate if they would be willing to take the time to do this huge thing for me. In other words, I was “pitching” this. It was thought-out, respectful, and professional.

When I first started teaching, kids would ask me for recommendations as a “by the way” in the hallway, or in a short one line email. I don’t allow for that anymore. I make sure they sit down with me and we talk through it. I ask them to fill out an extensive set of questions which often helps me frame the kids in my recommendation (if I don’t yet have a framing device in mind), and lets me learn about kids in a different way.

This year I sent an email out to my juniors, being as explicit as possible. It isn’t to make their lives harder. It is to teach them skills that are usually never explicitly taught. And all of this helps me craft a better recommendation.

Hi all,

I know it’s about the time that y’all are going to be thinking about soliciting college recommendations. If you are thinking of asking me to craft one, you should read this email. If you are certain you are not, you don’t need to read past this!

I know early in the third quarter I talked briefly about this in class, but I figured you should have it in writing too. First off, you should talk with your college counselor before approaching teachers about recommendations. They will be able to help you figure out if you’re asking the right people, who can write about the right qualities, for the colleges you are considering.

If you are going to approach me about being a recommender, there are some things you need to know. I am not a teacher who is grade-focused. I’m a teacher who values reflection, growth, hard-work, and demonstrated passion. If you’re a student who struggled but has shown a transformation in how you see and appreciate mathematics, or in your approach to effectively learning mathematics, or in how you communicate mathematics, or in your ability to work effectively and kindly in a group, or something else—all that is important to me. On the other hand, if you have done well on assessments, that is all well-and-good… but it is important that you are more than that… it is important to me that you have shown a passion to go above and beyond (inside and outside of the classroom and curriculum), or an enthusiasm for the material, or a willingness/eagerness to help others. In other words, it is important that you have thought about yourself, and can talk to me about how you are more than just grades.

That all being said, just a few reminders of what I said in class about recommendations:

· I do not write recommendations in the fall, so if you’re going to ask for one, you must ask me this year. Fall is a very busy time and is too far away; I like to have students fresh in my mind when I write. You also cannot approach me after our last day of classes (May 22).

· I never learned how to properly ask for recommendations until I was in college. So I want to help you learn that skill. (I’ve had to ask for recommendations in high school, college, grad school, and as a teacher.) If you’re going to ask me, send me an email to set up a meeting to talk formally about it. You need to plan this meeting, because you’re going to be in charge of leading it. Think about what you’re going to ask and how you’re going to pitch it.

· I said in class that you should start keeping a list in the back of your notebook of specific moments that you’re particularly proud of (large and small!), and things that you’ve done that might set you apart or make you unique or interesting! You should be sure to bring that to our meeting. If you have specific things you’ve done throughout the year that you are proud of (large or small!), you should bring those too.

As you might suspect, I write recommendations with great integrity—meaning I am honest and specific in what I write.

In the past I’ve been asked for a lot of recommendations from juniors. This year I may have to put a cap on how many I’m writing for, unfortunately, as each recommendation takes a number of hours from start to finish. After we meet, if I agree to write for you, you will be asked to fill out an extensive reflective questionnaire. I recognize that I ask a lot of students who request a recommendation, but I also know how important these recommendations are – and to do justice in the recommendation, these are important to me.

Always,

Mr. Shah

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For my kids, at this level, I want each term to be a ratio generates a class of similar triangles — which all look the same, but have different sizes. And I want kids to conjure that up, when they think of . But I fear that 0.6428 will stop losing meaning as a ratio of sides… that 0.6428 won’t mean anything *geometric* or *visual* to them. Why? Because the words “sine” “cosine” and “tangent” start acting as masks, and kids start thinking procedurally when using them in geometry.

So here’s the setup for what we’re going to do.

Kids are going to be placed in pairs. They are going to be given the following scorecard:

They will also be given the following sheet, with a clever title (the Platonic part refers to something we’ve talked about before… don’t worry ’bout it) (.docx form). This sheet has a bunch of right triangles, with 10, 20, 30, … , 80 degree angles.

Then with their first partner, on the front board, I project:

The kids will have 3 minutes to discuss how they’re going to figure out which two triangles/angles best “fit” these trig equations. (I’m hoping they are going to say, eventually, something like “well the hypotenuse should be about twice the length of the opposite leg, so that looks a lot like triangle C in our placemat” for the first equation.)

They write down their answers. If they finish early, I have additional review questions from the beginning of the year that will be worth some number of points — to work on individually.

When time is up, they move to a new chair (in a particular way) so that everyone has a new partner. I throw some other equations up. And have them discuss and respond. Then they move again, and have new equations up.

I’ve scaffolded the equations I’m putting up in a particular way — so I’m hoping they lead to some good discussions. And I’m hoping as soon as a few people catch onto the whole “let’s compare side lengths” approach, the switching will allow for more discussion — so soon everyone will have caught on.

At the end of the game, we’ll have some discussion, and through those discussions we’ll reveal the answers. And of course, the student with the most correct answers will win some sort of fabulous prize.

The questions I’m going to ask are here:

The discussion questions are here:

Fin.

I’m super excited to try this out on my kids next week sometime.

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For 1 bag, there is only 1 way.

For 2 bags, there is still only 1 way.

For 3 bags, there are 2 ways.

Here is a picture for clarification:

Can you figure out how many ways for 6 bags? 13 bags?

You are now officially nerdsniped.

A number of people had trouble calculating 4 bags correctly, so I’ll post the number of ways 4 bags could be stored after the jump at the bottom, so you can at least see if you’re starting off correctly…

**Additional Information:** Matt and I figured the solution to this problem together on twitter. It was an interesting thing. We didn’t really “collaborate,” but we both refined some of our initial data (for 5 bags, he undercounted, and I overcounted). It seemed we were both thinking of similar things — one idea in particular which I’m not going to mention, which was the key for our solution. What blew my mind was that at the exact time Matt was tweeting me his approach that he thought led to the solution, I looked at my paper and I had the exact same thing (written down in a slightly different way). I sent him a picture of my paper and he sent me a picture of his paper, and I literally laughed out loud. We both calculated how many arrangements for 6 bags, and got the same answer. Huzzah! I will say I am fairly confident in our solution, based on some additional internet research I did after.

Obviously I’m being purposefully vague so I don’t give anything away. But have fun being nerdsniped!

**Update late in the evening: **It might just be Matt and my solution is wrong. In fact, I’m now more and more convinced it is. Our method works for 1, 2, 3, 4, 5, and 6 bags, but may break down at 7. It’s like this problem — deceptive! I’m fairly convinced our solution is *not* right, based on more things I’ve seen on the internet. But it is kinda exciting and depressing at the same time. Is there an error? Can we fix the error, if there is? WHAT WILL HAPPEN?!

The number of ways 4 bags can be stored is… (after the jump)

4. I promise you this is right, so if you only have 3, you’re missing one!

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However we also gave out the following:

And students made the conjecture that you will always get a right angle, no matter where you put the point. But when they tried explaining it with what they knew (remember this was on the first or second day of class), they quickly found out they had some trouble. So we had to leave our conjecture as just that… a conjecture.

However I realized that by now, students can deductively prove that conjecture in two different ways: algebraically and geometrically.

**Background:**

My kids have proved* that if you have two lines with opposite reciprocal slopes, the lines must be perpendicular (conjecture, proof assignment).

My kids have derived the equation for a circle from first principles.

My kids have proved the theorem that the inscribed angle in a circle has half the measure of the central angle (if both subtend the same arc) [see Problem #10]

**Two Proofs of the Conjecture****: **

Now to be completely honest, this isn’t exactly how I’d normally go about this. If I had my way, I’d give kids a giant whiteboard and tell ’em to prove the conjecture we made at the start of the year. The two problems with this are: (1) I doubt my kids would go to the algebraic proof (they avoid algebraic proofs!), and part of what I really want my kids to see is that we can get at this proof in multiple ways, and (2) I only have about 20-25 minutes to spare. We have so much we need to do!

With that in mind, I crafted the worksheet above. It’s going to be done in three parts.

Warm Up on Day 1: Students will spend 5 minutes refreshing their memory of the equation of a circle and how to derive it (page 1).

Warm Up on Day 2: Students will work in their groups for 8-10 minutes doing the geometric proof (page 2).

Warm Up on Day 3: Students will spend 5-8 minutes working on the algebraic proof (page 3). Once they get the slopes, we together will go through the algebra of showing the slopes as opposite reciprocals of each other as a class. It will be very guided instruction.

Possible follow-up assignment: Could we generalize the algebraic proof to a circle centered at the origin with *any* radius? What about radius 3? What about radius *R*? Work out the algebra confirming the our proof still holds.

**Special Note:**

Once we prove the Pythagorean theorem (right now we’re letting kids use it because they’ve learned it before… but we wanted to hold off on proving it) and the converse, we can use the converse to have a *third* proof that we have a right angle. We can show (algebraically) that the square of one side length (the diameter of the semi-circle) has the same value as the sum of the squares of the other two sides lengths of the triangle. Thus, we must have a right angle opposite the diameter!

I’m sure there are a zillion other ways to prove it. I’m just excited to have my kids see that something that was so simply observed but was impossible to explain at the start of the year can yield its mysteries based on what they know now.

The two semi-circle conjecture documents in .docx form: 2014-09-15 A Conjecture about Semicircles 2015-03-30 A Conjecture about Semicircles, Part II

*Well, okay, maybe not proved, since they worked it out for only one specific case… But this was at the start of the year, and their argument was generalizable.

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**The point of this post is to get you to add yourself to the directory**. If you’re already convinced, do it now. If not, read on to why you ought to…

It not only is beautiful, simple, and sleek, but it has the following features which blew me away:

(1) For each person, it creates a little index-card-like profile, which not only has our twitter picture on it but also has links with our interests. I confuse people easily (and really, why are 30% of math teachers named *Chris*?), and having a little picture icon, and all of their information easy for me to look at is going to be so so so helpful.

(2) It has a map which each person in the directory can easily add themselves to, and this map is searchable. I can, for example, zoom into NYC to see who the NYC educators are… or type my friend’s name into the search bar to remind myself which part of the country (world!) they are in.

(3) The directory itself is crazy searchable. Say you wanted to find teachers who have been teaching since 2000 who are in the Northeast US who teach Geometry and are interested in Groupwork. Done.

(4) If you want to quickly update your information, you can… no muss no fuss it is super easy!

Which is all to say: take 5 minutes and** add yourself to the directory**.

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