*“Which polygons fit together snugly? Which don’t?”*

[02 Snug Angles download][Note: There is a typo on #6… It refers to problem 2d, but it should refer to problem 2c]

I made this the day before the class I was going to teach it. But I wanted to have a hands-on “playful” component to this. I asked teachers in my school if they had regular polygon tiles with the same side length… I got a set which included triangles, squares, and hexagons. No pentagons, no heptagons, no nonagons, no decagons, nada.

Of course these tiles were probably produced for lower school kids precisely because they fit together “snugly” at a vertex. But no “play” could really happen if sometimes things *didn’t* fit nicely together. So — for future reference — I asked on twitter to my math peeps if anyone knew where I could buy regular polygon tiles of all sorts. No links were forthcoming. Sigh.

In class, I expected #1 to be challenging. I wanted students to come up with a *reason* they had found all the possible regular polygons (of one kind) that fit snugly together. It was nice to see students reason through it, and when we came together as a class, we had a few different cogent explanations. Some involved calculating all possible factors of 360. Some involved recognizing that the more sides you have in a polygon, the fewer of them can fit together “snugly” at a vertex (and the minimum number of polygons that can fit at a vertex is 3).

Although I was expecting #2c to be challenging, I didn’t realize how challenging it would be. I thought I had built a scaffold with the previous problem so it wouldn’t be too hard. What turned out to be the problem? The fact that a regular 7-gon had a non-integer interior angle value. Kids didn’t know that could happen, and that really threw them. Also: setting up the equation was challenging, because kids were confusing “the sum of the interior angles in a regular n-gon” with “the measure of *one* interior angle in a regular n-gon” (a calculation they had never been formally taught, and were supposed to figure out themselves during this exercise).

I’d say only about half the groups could deal with 2c without any help.

However, all groups ended up being successful. And I just graded their assessments on polygonal angles, and almost *every single student* got the problem that was similiar to 2c!

The very last question asks students to discover as many possible combinations of regular polygons that could fit together snugly at a vertex. I assigned this as a nightly work problem — and the next day, students came in with lots of great combinations. Unfortunately, I didn’t do anything with this. I should have — but I felt pressed for time.

We could have talked about why 6 polygons were the maximum number that could fit together, or 3 polygons were the minimum number that could fit together. That could reduce our searching! Then I could have asked how people approached the task. Guess and check? Geogebra? Is there a systematic way they could have approached this problem — if they had infinite time and patience — that they could guarantee they had found all possible combinations? Do all combinations need at least one 3, one 4, one 5, or one 6?

Or we could have spent some more time looking at all possible combinations. Some kids noticed — after looking at the comprehensive list I threw on the board after they finished sharing their values with me — that many of the values had common factors: so 3, 7, 42 is one crazy combination that works. And both 3 and 7 are factors of 42. What else could we find?

What I’m trying to say is: the last question was kind of a dumb question to put on the sheet without having a good way to debrief it, and a meaningful conclusion we could have gotten from it. Sigh.

Okay, on to the exciting part. I said I asked on twitter if anyone had a site to buy these tiles. No responses. *BUT* Christopher Danielson then asked what I was looking for. Kate Nowak jumped on the bandwagon and brainstormed what a teacher might want, ideally. Yesterday, I came home from school and had a box waiting for me. In it:

They are *beautiful*. And gosh do they smell awesome. Real wood, that smells awesome. I was in heaven when I saw them. So beautiful.

And even more satisfying: you’ll notice that the 3, 7, and 42 fit snugly together!

Now the million dollar question: assuming I had however many of each tile I wanted, what would I do with them? How would I restructure the unit to use them in a way that is compelling? I wanted the tiles initially because I thought some “play” with the tiles would be fun, before delving into the algebra to see the justification of why some work and some don’t work. But I want something more! Something that will have them figure out the 3, 7, 42 connection and gasp! And the 4, 5, 20. And the 3, 8, 24. And the 3, 10, 15. And the 4, 5, 20. And GASP with surprise and horror and delight!

I don’t quite know… But maybe envelopes with index cards in them. And some of the index cards have some configurations they have to “check” to see if they work or not. And some of the index cards have two of the tiles, and students have to see if there is a third tile that works. And for each configuration that works, students get to come to the front of the room, grab those tiles, and check to see if their algebra worked by checking to see if the tiles truly do fit snugly. If they do: they record their discovery on the board for all to see. And by the end of the class, students will have had practice, and in the last 5 minutes, we could all gather at the front, and view some of the weird snug angle configurations together. And see how configurations that are “close but no cigar” don’t work (like 3, 10, 16… which is close to 3, 10, 15). When doing this, we could also talk about why 4, 10, 15 is “worse” than 3, 10, 16 in fitting snugly.

That’s all my musings for today! I’m going to be chaperoning a trip to Spain in a few days, and that will last two weeks, so goodbye for a while!

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We are starting to gear up for TMC16, which will be at Augsburg College in Minneapolis, MN (map is here) from July 16-19, 2016. We are looking forward to a great event! Part of what makes TMC special is the wonderful presentations we have from math teachers who are facing the same challenges that we all are.

To get an idea of what the community is interested in hearing about and/or learning about we set up a Google Doc (http://bit.ly/TMC16-1). It’s a GDoc for people to list their interests and someone who might be good to present that topic. The form is still open for editing, so if you have an idea of what you’d like to see someone else present as you’re writing your own proposal, feel free to add it!

This conference is by teachers, for teachers. That means we need you to present. Yes, you! In the past everyone who submitted on time was accepted, however, this year we cannot guarantee that everyone who submits a proposal will be accepted. We do know that we need 10-12 morning sessions (these sessions are held 3 consecutive mornings for 2 hours each morning) and 12 sessions at each afternoon slot (12 half hour sessions that will be on Saturday, July 16 and 48 one hour sessions that will be either Saturday, July 16, Sunday, July 17, or Monday, July 18). That means we are looking for somewhere around 70 sessions for TMC16.

What can you share that you do in your classroom that others can learn from? Presentations can be anything from a strategy you use to how you organize your entire curriculum. Anything someone has ever asked you about is something worth sharing. And that thing that no one has asked about but you wish they would? That’s worth sharing too. Once you’ve decided on a topic, come up with a title and description and submit the form. The description you submit now is the one that will go into the program, so make sure it is clear and enticing. Please make sure that people can tell the difference between your session and one that may be similar. For example, is your session an Intro to Desmos session or one for power users? This helps us build a better schedule and helps you pick the sessions that will be most helpful to you!

If you have an idea for something short (between 5 and 15 minutes) to share, plan on doing a My Favorite. Those will be submitted at a later date.

The deadline for submitting your TMC Speaker Proposal is **January 18, 2016 at 11:59 pm Eastern time**. This is a firm deadline since we will reserve spots for all presenters before we begin to open registration on February 1st.

Thank you for your interest!

Team TMC – Lisa Henry, Lead Organizer, Mary Bourassa, Tina Cardone, James Cleveland, Cortni Muir, Jami Packer, Megan Schmidt, Sam Shah, Christopher Smith, and Glenn Waddell

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*At 3:00, the hour and minute hands on a clock form a right angle. What is the next time that happens?*

The presenting teacher had a pretty darn elegant solution. But I enjoyed working it out using brute force. (That’s pretty much my go-to.) I’m going to type my solution down below the jump.

I saw the hour hand and the minute hand act like vectors coming out of the origin. Their position changed in time.

and , where is measure in minutes.

Now when are two vectors perpendicular? When their dot product is 0.

And lo and behold, naturally occurring in this problem, we see a difference of angles formula for cosine!

And then we just have to find all values that make this true. Well the cosine function has an output of 0 when the input of the cosine function is (where is an integer).

That just leaves us with:

From this, we solve for and we get:

Let’s see what happens for various values of !

And so here we have our answer… around 212 minutes (3:32 and 44 seconds).

Pretty neat, huh? Okay, okay, once you look at the final solution for the general form for all the times, you might suspect there is a much easier thing going on here. And that’s what the teacher presenting the problem shared with us after we finished. But I’m not going to share it with you because I LIKE MY SOLUTION and HECK IF I DIDN’T SEE AND USE A SUM OF ANGLES TRIG FORMULA IN THE WILD! Huzzah!

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I got blocks from our lower school math coach. I told kids (either working individually or in pairs) to play around with them until they found a pattern that looked interesting to them. I didn’t want them thinking about the sequence yet… I wanted them to create patterns that looked neat. The only restriction I put on them is that the pattern had to be three dimensional. If it could be represented in two dimensions, I didn’t want to see it.

They made some really nice sequences! Here are a random set of 4 to look at:

I then had students work on filling out this form. It asks them to articulate their “rule” (for building up the sequence) and has them attempt to come up with both explicit and recursive forms to get the *n*th term. I make it clear to them that if they can’t get the formulae, I’ll give them full marks as long as they show a serious attempt. (Some of the sequences they built involve some mathematical hoops they might not be able to traverse… for example, one group needed to find which is lovely, but not something they are going to easily figure out.

[.docx version here]

If I had time, I’d love to do two more things with this.

(1) I think it would be neat to take the photographs of one person’s sequence and give them to another person, to see what they figured out for the explicit and recursive definitions for these sequences. Why? Not only is it sharing more publicly the sequence the kids created, but many of them got a bit stuck on an explicit formula that they do have the capabilities to find, but couldn’t. I think a fresh pair of eyes, and a conversation, could be beneficial for both the original sequence creator and the new person approaching the sequence. (Additionally, there are often *many* ways to look at these sequences, so even if both got the same formula, there is a good chance they came up with it in different ways.)

(2) Students created a table with the first 5 terms of the sequence in it. I’d love for students to extend the table to 7 or 8 terms in the sequence, and then have students work on finding the first differences, the second differences, the third differences, etc. If students understand that having the same first difference means they have a linear relationship, having the same second difference means they have a quadratic relationship, having the same third difference means they have a cubic relationship, etc., then students who got stuck will have a new tool in their arsenal to find the explicit formula for the sequence. If, for example, they had 5, 9, 15, 23, …, and saw a common second difference, they could do the following:

Since they suspect the relationship is quadratic, they could say: . And then they’d be hunting for the to make this the correct quadratic for our sequence. And then use the following three equations, they could come up with the .

.

In fact, this is an awesome thing to revisit when we get to matrices to solve systems of three variables!!!

UPDATE: One more thought before I lose it! What if I gave students the numerical sequence (e.g. 5, 9, 15, 23) expressed either written out as a list, written out as an explicit formula, or written out as a recursive formula, and had them generate a visual sequence to match it. I’d love to see how many different and interesting sequences might be created that go along with a single sequence!

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Right now, we’re launching it again … but with an awesome twist!

There are going to be two things going on simultaneously.

Those who are comfortable blogging and tweeting, we’d love for you to sign up to be a mentor for someone just dipping their feet into the online math teacher world! You will be a person that newbies can ask questions to, connect ’em with people and blogs they might find interesting, and be a cheerleader as they get involved.

Those who are new are going to have someone help you out. You will be able to have a trusted person to ask questions to, help you find things that will be interesting to you, and encourage you. And through this, you’ll get to see if the online math teacher community has anything to offer that you want. You’ll get to dip your feet in, with no pressure, and a lot of support!

In December, we’ll pair up mentors and mentees. And during that time, we’ll all work on introducing those new to the online math teacher world to what we have to offer.

In January, we’ll have a 4 week “blogging challenge,” with prompts for both new and experienced bloggers.

If you’re interested in finding out more, or you’re ready to sign up to be a mentor or to get your feet wet checking out the online math teacher world (known as #MTBoS which is the unwieldy acronym for mathtwitterblogosphere), check out the exploremtbos website.

Huzzah!

PS. Please spread the word, if you can!

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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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