He was on the way to making a neat Pascal’s Triangle argument. Look at that 70. That’s :

He started working backwards, and saw that 70=35+35.

But each of those 35s came from 15 and 20. So 70=(15+20)+(20+15).

And then going backwards more, we see the 15 comes from 5 and 10, and the 20 comes from 10 and 10. So 70=((5+10)+(10+10))+((10+10)+(10+5)).

And then going backwards once more, we see the 5 comes from 1 and 4, and the 10 comes from 4 and 6. So 70=(((1+4)+(4+6))+((4+6)+(4+6)))+(((6+4)+(6+4))+((6+4)+(4+1)))

In other words, 70=1*1+4*4+6*6+4*4+1*1.

By the time we make our way down from the 1-4-6-4-1 row to 70, we see that:

The first one in the 1-4-6-4-1 row just is added once when making the 70.

The first four in the 1-4-6-4-1 row is added four times when making the 70.

The six in the 1-4-6-4-1 row is added six times when making the 70.

The second four in the 1-4-6-4-1 row is added four times when making the 70.

The second one in the 1-4-6-4-1 is added just once when making the 70.

In other words: 70=1^2+4^2+6^2+4^2+1^2.

The other teacher and I realized we could generalize this. But we were left unsatisfied. It was a Pascal’s Triangle argument, but I wanted to *see* the answer with an understanding of *combinations*. I wanted something even more conceptual. So my friend and I started thinking, and he had an awesome insight. And I want to record it here so I don’t lose it! It made me so happy — little mathematical endorphins exploding in my head!

Let’s assume we have a set of *2n* letters, where *n* letters are A and *n* letters are B.

Blergity blerg, let’s just keep things concrete, and have 8 letters, where 4 are As and 4 are Bs. (We can generalize later, but I want to just see this happen!) Given 4As and 4Bs, there are ways to arrange them to make different 8 letter words [1]. Great! That was the easy part.

Now we are going to construct a whole bunch of different sets of 8 letter words, in a particular way (using AAAABBBB), so that when we add up all those sets, we’re going to get all possible 8 letter arrangements of AAAABBBB.

How are we going to do this? **We are going to create special of 4 letter words and concatenate them together to make 8 letter words. **

**Set 1:** We are going to create a 4 letter word with 0As (and thus by default, 4Bs) and a 4 letter word with 4As (and thus by default, 0Bs).

How many ways can we create 4 letter words with 0As? . To be clear, this is just 1. The word is {BBBB}.

How many ways can we create 4 letter words with 4As? . To be clear, this is just 1. The word is {AAAA}.

And when we concatenate them, we are going to have eight letter words. But we know . So this is simply . And this is just 1, because the only eight letter word possible is {BBBBAAAA}.

This is a degenerate case, so it’s hard to really see what’s going on here. So let’s move on.

**Set 2:** We are going to create a 4 letter word with 1A (and thus by default, 3Bs) and a 4 letter word with 3As (and thus by default, 1Bs).

How many ways can we create 4 letter words with 1As? . To be clear, this is just 4. The words are {ABBB, BABB, BBAB, BBBA}.

How many ways can we create 4 letter words with 3As? . To be clear, this is just 4. The words are {AAAB, AABA, ABAA, BAAA}.

And when we concatenate them, we are going to have eight letter words. But we know . So this is simply . And this is 16 eight letter words. (Each of the first four letter words can be paired with each of the second four letter words… so this is merely 4*4. Just to be clear, I’ll list the first few eight letter words out: ABBBAAAB, ABBBAABA, ABBBABAA, ABBBBAAA, BABBAAAB, BABBAABA, …

**Set 3:** We are going to create a 4 letter word with 2As (and thus by default, 2Bs) and a 4 letter word with 2As (and thus by default, 2Bs).

How many ways can we create 4 letter words with 2As? . To be clear, this is just 6. The words are {AABB, ABAB, ABBA, BAAB, BABA, BBAA}.

How many ways can we create 4 letter words with 2As? . To be clear, this is just 6. The words are {AABB, ABAB, ABBA, BAAB, BABA, BBAA}.

And when we concatenate them, we are going to have eight letter words. And this is 36. (Each of the first four letter words can be paired with each of the second four letter words… so 6*6 eight letter words.)

**Set 4: **We are going to create a 4 letter word with 3As (and thus by default, 1B) and a 4 letter word with 1As (and thus by default, 3Bs). By the same logic as above, we are going to end up with eight letter words. This is just 4*4 eight letter words.

**Set 5: **We are going to create a 4 letter word with 4As (and thus by default, 0Bs) and a 4 letter word with 0As (and thus by default, 1B). By the same logic as above, we are going to end up with eight letter words. This is just 1*1 eight letter words.

Now look at all the different eight letter words created by this process, from Set 1, Set 2, Set 3, Set 4, and Set 5. We have captured *every single possible eight letter word* with four As and four Bs. Let’s check a few random words:

AABABBAB… okay this is in Set 4.

BAABAABB… okay this is in Set 3.

BBBABAAA… okay this is in Set 2.

Cool! I only have to look at the first four letters to decide which set it is going to be in!

But look at what we’ve done. We’ve shown that we can get all eight letter words in these five sets… so the number of eight letter words is:

If we simply write the squares out…

But we saw at the very start that the number of eight letter words is simply

So the two are equal.

All the hard work is done, so I leave it as an exercise to the reader to generalize.

P.S. I take no credit for this amazingly wonderful letter rearrangement solution. I just bore witness as my friend figured it out, and I got giddier and giddier. I love it because it’s abstract, but still understandable to me. But it’s close to my threshhold of abstraction!

[1] If you don’t quite see this, imagine 8 blank slots.

___ ___ ___ ___ ___ ___ ___ ___

You choose four of them to put the As into. There are ways to choose four of these slots. Put As into those four. By default the rest of the slots must be filled with Bs — they are forced! So there are ways to create eight letter words with four As and four Bs.

]]>***

*I mentioned in class that I had stumbled across a beautiful different proof for the double angle formulae for sine and cosine, and I would post it to the classroom. But instead of *giving* you the proof, I thought I’d share it as an (optional) challenge. Can you use this diagram to derive the formulae? You are going to have to remember a *tiiiiny* bit of geometry! I already included one bit (the 2*theta) using the “inscribed angle theorem.”*

*If you do solve it, please share it with me! If you attempt it but get stuck, feel free to show me and I can nudge you along!*

***

Below this fold, I’m posting an image of my solutions! But I say to get maximal enjoyment, you don’t look further, take out a piece of paper, and take a stab at this!

]]>

And then… they got stuck.

You see, I showed them two alternative forms for the double angle formula for cosine ( and ). I *showed* them these forms. And I said: figure out where they came from.

All groups in a few minutes were on yellow cups (“our progress is slowing down, but we’re not totally stuck yet”). I didn’t want to give anything away, but I didn’t have any group have a solid insight that I could have them share with others. I let things remain a bit more, no luck, so then I said: “this looks related to something we’ve seen before… a trig identity… maybe that will be helpful. Bring in something you know to open up the problem for you.” Eventually kids realized they needed to bring in some outside information (namely: ).

I was *sure* that was going to be enough. Totally certain. But after another 5 minutes of watching them struggle, I wasn’t so sure. I didn’t want to give anything more away, but I had to because we had to move forward. But what more could I give without giving the whole show away? Since many groups were trying some crazy stuff, I said: “this is a simple one or two step thing…” Why? I just wanted them to take fresh eyes and see what they could do thinking *simply*. They kept on saying I was trying to trick them, but I told them it wasn’t a trick!

And then, in the span of the next five minutes, all my groups got it.

But what was more interesting was that we had *three different ways *to do it. As kids moved on to the next set of questions (and I breathed a sigh of relief that they figured this out), I reflected on how awesome it was that they persevered and then came up with different approaches. So while they worked, I put up the three different approaches.

And with a few minutes to go at the end of class, I had everyone put everything away and I just pointed out the embarrassment of riches they came up with. And it was great to hear the audible reactions when kids who had one way saw the other ways and say things like “ooooh, I never would have thought of that!” or “that’s so clever!”

I had (have?) so many mixed feelings when I saw how difficult this question was for my kids. And I was hyperconscious about how much time we had to spend on this. But the ending made me feel like it was time well-spent.

]]>So I wasn’t actually alone with Van Gogh’s *Starry Night*. But I went to MoMA this morning and got to tour the museum with other math teachers before the museum opened. Our sherpa? George Hart, mathematical artist. A few months ago, I got an email from two different teacher friends letting me know about this opportunity to take a master class on *Geometric Sculpture* put together by the Academy for Teachers. What an opportunity indeed!

I show up at 8:30 am and me and a gaggle of math teachers (a gaggle is eighteen, right?) are raring to go. We have fancy namecards and everything. (Note to self: at the book club I’m hosting in a bit over a month, create fancy namecards.)

Beforehand, we were assigned a tiny bit of homework. We were asked to go onto the Bridges website (it’s an international annual math-art conference, organized by our sherpa), look at submitted papers for their conference proceedings, select three papers, and then read and reflect on them.

**My Paper Choices and Thoughts**

1. Prime Portraits, Zachary Abel

This mathematician was able to construct *portraits* using the digits of prime numbers. The digit 0 was black and the digit 9 was white, and the other digits were various shades of gray. The digits of a number were put in order in a rectangular array (e.g. 222555777 would be put into 3×3 array, where 222 is the top row, 555 is the middle row and 777 is the bottom row) and an image results. For most numbers, the image will look like noise. But this author was able to use *prime numbers* put into a rectangular array to create images of Mersenne, Optimus Prime, Sophie Germain (using Sophie Germain primes), Gauss (using Gaussian primes), and others. I was blown away. This intersection of math and art doesn’t quite fall neatly into any of the categories that George provided us, but it is close to “mathematics used in calculating construction details necessary for constructing an artwork.” In this case, the portraits themselves are the “art” and the author was using numbers to reconstruct that art. What makes it interesting is that the math version of these portraits *feel* unbelievable. Senses of awe and wonder and curiosity filled me when seeing the portraits for the first time because *how could it be*? It was like a magic trick, because nature couldn’t have embedded those portraits into those numbers. And before reading the paper on how these were constructed, I had a nice few moments thinking to myself how this could have been done.

(If you’re curious, the answer is to start backwards. *First* take an image, pixelate it, and then turn those pixels into a number. Take that number and check if it’s prime on a computer. If it isn’t prime (which is likely), slightly alter the image by the colors by +1% or -1% (some imperceptible noise), repixelate it, and turn those pixels into a number. And again, check if that number is prime on a computer. If it isn’t, do this again. It turns out that you’re going to need to do this about 2.3*n* times [where *n* is the number of pixels]. With a computer, this can go quickly.)

Thoughts/Questions:

(a) *Math:* I recall faintly from college classes that the distribution of primes is related to the natural logarithm. Which explains why the 2.3*n *comes from something involving a natural log. But what is this relationship precisely, and how does it yield the 2.3*n*?

(b) *Content: *I think prime numbers are very rarely taught in high school math in a meaningful way. Number theory is ignored for the “race to calculus.” However there is so much beauty and investigation in this ignored branch of math. Where could I fit in conversations of prime numbers in an existing high school curriculum? Could ideas from this paper be used to captivate student interest (by letting them choose their own image), while showcasing what various types of prime numbers are?

(c) *Extension:* Are there other things that we teach that have visualizations that *look* impossible/unbelievable, but actually are possible? Can we exploit that in our teaching? I’m thinking that often numbers in combinatorics are crazy huge and defy imagination… Perhaps a visualization of the answer to some simple combinatorial problem?

(d) In order to *fully* appreciate this work, the viewer needs to have an understanding of prime numbers. Without that understanding, this is just a pixelated image with some numbers superimposed. All wonderment of these pieces is lost!

2. Modular Origami Halftoning: Theme and Variations (Zhifu Xiao, Robert Bosch, Craig Kaplan, Robert Lang)

*
*I chose my articles on different days, and I didn’t even notice that this article is very similar to the first article! I chose it because I love the idea of a gigantic public art project in a school (I tried once and failed to make a giant cellular automata that students filled in). But this article basically shows how to fold orgami paper (white on one side, colored on the other side) in five different ways to make squares where all of the square is colored, ¼ of the square is colored, ½ the square is colored, ¾ of the square is colored, and none of the square is colored. A number of each of these origami pieces are constructed.

Then an image is converted to grayscale and scaled down to the number of origami pieces you want to use. Then the image is scaled-down image is pixelated with “origami piece” size pixels, and each pixel is given a number based on brightness [0, ¼, ½, ¾, 1].

Then this origami image can be created by putting these five different origami pieces in the correct order based on the brightness of the pixelated image!

Just like with the previous paper, this intersection of math and art doesn’t quite fall neatly into any of the categories that George Hart provided us, but it is close to “mathematics used in calculating construction details necessary for constructing an artwork.” In this case, the portraits themselves are the “art” and the author was using numbers to reconstruct a variation of that art.

Thoughts/Questions:

(a) *Math Classroom: *I really love the idea of having kids take an image with a particular area (*w* by *h*) and figure out how to “scale down” the image to use a particular number of origami pieces. It is an interesting question that will also involve square roots! It seems like a great Algebra I or Algebra II question.

(b) *Extensions:* How could this project be extended to the third dimension? 3D “halftone” origami balloons? Unlike a photograph which can be easily pixelated, can we find a way to easily pixelate the “outside”/”visible part” of a 3D object and create a balloon version of this? Similarly

(c) This is not just a low-fidelity copy of an existing piece of art. If we took a random non-professional Instagram photograph, we might call it “pretty but not art.” But if someone made this Instagram photograph out of origami sheets, we would be more likely to call it art. But why? Just one thought, but there is something about the *intentionality* of the artist (and the *craftsmanship* that goes into creating the origami piece) that isn’t in the original photograph. It also is likely to evoke something different in a viewer – a viewer will instantly wonder “how was that done” when seeing the origami piece (so the art piece evokes *process*) while a random photograph might not do the same (they just pressed a button on their phone and got a cool photo).

3. A Pattern Tracing System for Generating Paper Sliceform Artwork, Yongquan Lu and Erik Demaine

I chose this paper because of the beautiful sliceform image on the first and last page. I had only seen them once before, but forgot what they were called! I wanted to learn how to make them. In this paper, the authors share that most existing sliceforms are created in separate pieces (e.g. the image on the first page, a bunch of hexagons created separately) and then pieced together afterwards. The authors wanted to instead *thread* the paper slices together so they could create the same intricate patterns—but with the paper slices interconnected. So instead of individual hexagons placed together, a giant connected sliceform was created (e.g. the image on the last page). The authors came up with a way to do this for designed created in polygonal tiles, like in many Islamic star patterns, and then created a program to “print” the strips of paper needed – with red lines indicating where folds are, and blue notches indicating where cuts need to be made so the paper slices can be fit into each other.

They accomplished this in two steps. First, they came up with a way to notate the internal structure of a paper slice within one polygon. One notation captured lengths (where slices of paper intersected other slices of paper and where slices of paper needed to be bent/folded), and another notation (not provided) recorded angles that needed to be folded. The second step was more tricky. An algorithm was created that looked at the edge of a polygon (where a paper strip initially ended), and looked to see if it could be extended into another polygon. In that way, one strip could start in one polygon and then enter another, and then another, etc. This is the *threading* that the authors wanted to get. The authors created a three-step algorithm for deciding if a paper strip could enter another polygon at all, and if there were multiple possible paths for this strip to take, which one it should choose.

After doing all of this, the authors then created a program that could take in an image, calculate out the different strips of paper needed to create the sliceform, and with the notation they created, print out the appropriate slice (see image on page 370 for an example).

Thoughts/Questions:

(a) There were two big things I didn’t totally understand when reading this paper. First, how were angles recorded/notated? Second, where did the 3-step algorithm for extending paper slices come from? How do we know if we follow it that all segments in the figure will be created by the paper slices, and no segment will be repeated?

(b) Besides just being “cool,” is there an application to this in a high school math class? What higher level research does this connect up to? (Just like origami was simply beautiful but then it also was exploited to create new and interesting questions for mathematicians, what does this bring up for us?)

Note: When I went to research these, it turns out that Lu and Demaine created a website to help amateurs out: https://www.sliceformstudio.com/app.html

(c) I was wondering what a 3D version of this might look like, but it turns out that this exists! https://www.sliceformstudio.com/gallery.html

**Back to the Master Class**

After getting coffee and pastries, and introducing ourselves to each other in small groups, we all were taken on a tour of MoMA, where George led us to certain pieces to spoke to him as he looked at them through mathematical lenses. There was one sculpture in particular that George stopped us at — a sculpture he remembered seeing as a kid visiting MoMA — that I would have walked right by. It was a figure cast in bronze (?), that had a *lightness* and *movement* despite it’s medium. To me, it screamed that it was a figure in tension. Rooms later, I was still thinking about how it was a collection of oppositions, form and formlessness, fluidity and stability. For George, describing what drew him to it was ineffable.

Here are more photos of George taking us around.

The whole walkthrough, George kept on saying “I’m not an art historian, but this is what I see in terms of my perspective as a mathematician…” which was just what I needed to hear. I know so little about art history and contemporary art, but hearing that let me feel a bit more “free” in looking at something and thinking about it with my own lens, instead of me passively waiting to hear what the piece is “supposed” to convey or what philosophical/conceptual trend it is a part of. In general, I feel ill-equipped to make statements/ judgments about art in museums that go beyond “I like this” or “I didn’t really like this.” But listening to George talk about what he sees as a mathematician and mathematical artist was liberating. Because I can see mathematical ideas/principles (intentional and unintentional) in some of the art too! This walk and talk reminded me a lot of what I imagine Ron Lancaster’s math walk around MoMA would be like!

And as the title of this blogpost suggested, there was something *so special and magical* about being able to have the run of the museum before the general public was let in. And a random fun tidbit: I also learned that there is no simple mathematical equation for an egg. I (of course) had to google that when I got home, and came up with this webpage.

**We Become Card Sculptors**

We get back to the room that was our home base, and some people share out interesting things from the articles they read. I was going to share mine, but I noticed that even though the ratio of men to women was low, more men were taking up airtime than women proportionally. So I kept my hand down.

George gives us a set of 13 cards with notches in them. We only needed 12 but you know how we math teachers really like prime numbers… (Okay, that wasn’t the reason for the 13th card, but I want to pretend it was.) We were asked to crease them like so:

And then… we were asked to put them together somehow, into a freestanding sculpture. No glue, scissors, tape, etc. We were given a hint that you can start with three cards. So I figured we needed to create 4 sets of objects that each take three cards. So with my desk partner we made this:

This was the core object we needed to build the final thing together. It was interesting how it took different pairs different amounts of time to get these three things together. Without instructions, it was a logical guessing game, but it felt so good once we hit upon it.

Then came the tough part. Putting these four building blocks together. That took a *long time *and some frustration, but the good kind. It was one of those problems that you *know* is within your grasp, and you know that you can come out on the other side successfully, but you don’t quite know how much time and how much angst the journey will cause you. It’s that sweet spot in problem-solving that I love so much. And lo and behold:

Many people got it! I would post a picture of mine, but all my photos look terrible. You can’t see or appreciate the symmetry and freestanding nature of this beast. But it was a moment of such pride when we got the last card to slide in the last notch! (And of course when my partner and I tried building hers after finishing mine, it went much faster and we had a better sense of things.)

Oh yeah, this card sculpture is isomorphic to a cube. I was blown away by that. It was hard for me to see at first, but realized that to get my kids to see it, I would give ’em purple circular stickers to have them put on the “corners” and blue circular stickers to have them put on the “faces” and green circular stickers to have them put on the “edges.” It would help me not only count the different things (maybe put the numbers 1-8 on the purple stickers, 1-6 on the blue stickers, and 1-12 on the green stickers?), but also “see” how they are in relationship to each other. (And George told the class he liked the suggestion and would think about trying it out!) George asked the class what the “fold angle” is for each card (what angle the card was bent at in the sculpture). I loved the question because it’s so obvious when you look at the sculpture from just the perfect angle! (The answer: 60 degrees.)

**We See Art and We Build More Art**

Lunch was delivered from Dig Inn, and we ate and briefly chatted. And then George took us on a picture tour of his sculptures and their construction. Some choice quotes:

“Kids need to have an emotional connection to math.”

“Math and art are both about creating new things.”

Finally, we ended our day building our own mathematical sculpture. We had 60 pieces of wood that we set up in trios. And we combined those to create a hanging sculpture.

What’s neat is that this hanging sculpture is going to travel to all the schools of the teachers who were at this session for two week periods. It will come to us disassembled and we’re going to get a group of kids (or teachers!) each to build it up and hang it. And then after two weeks, send it on! I love the idea of this same set of 60 pieces being in the hands of young elementary school kids and my eleventh-grade kids.

**Takeaways and Random Thoughts**

I have recently been into math art. Last year, I helped organize a math-art exhibit in our school’s gallery. I get excited when kids make math-art for their math explorations that I assign in my precalculus class. (In fact, years ago I had two kids make some sculptures and now I know they came from directions George provided on his website.) For me, it isn’t about “art” per se, but about seeing math as more expansive than kids might initially think, and seeing math as a creative and emotional endeavor. That’s why this resonated with me.

At the start of the year, I had intentions of starting a math-art club. Because my mother was sick and I was not taking on any new responsibilities, I decided to put that idea on hold. But now I’m feeling more excited about trying this out. To do this, I want to create 5 pieces on my own based on things I have found online. Things that will kids to say “oooooooooh.” Heck, things that will get *me* to say “ooooooooh.” (Like the origami image I saw in the second paper I wrote about above.) And then show them to students and get a core group of 4-5 who want to just build stuff with me on a regular basis. Maybe as a stress reliever.

What can we make? Who knows! Maybe stuff out of office supplies? Maybe some of the zillion awesome project ideas that George and his partner Elizabeth have put together. Maybe something inspired by the awesome tweets with hashtag #mathart that I’ve been following (and sites like John Golden’s). Maybe something on geogebra or desmos? Maybe something else? The idea of a large visible public sculpture appeals to me. One that random people walking by can add to also appeals to me. (I tried last year to get a giant cellular automaton poster going at my school, with two students in the art club, but it didn’t quite work as planned.)

Maybe this happens. Maybe it doesn’t. I hope I can muster the energy to start thinking this summer and making this a reality next year.

Random thought: Based on all the photos that George posted showing him bringing his math art to little kids in public spaces, I wonder if he’s talked to Christopher Danielson who organizes Math-On-A-Stick? Or if he knows Malke Rosenfeld (we had talked about math and dance earlier in the day)? I’m hoping yes to both!

Random note: George said that among his favorite mathematical artists were Helaman Ferguson, Henry Segerman, and John Edmark. Bookmarking those names to check out later.

Random thing: At MoMA in an exhibit about the emergence of computers to help create art was *fabric* that was created by the artist to hold information in it. What was pointed out to me, which made me go HOLY COW, is that the punch card idea for the first computers came out of the Jacquard loom. So loom –> computer –> loom. What a clever idea. I wish I knew what information was encoded in the fabric I saw! Additionally, this reminded me of one of the artists we had exhibited at the math-art show I helped organize: the deeply hypnotic and mathematical lace of Veronika Irvine. And that of course got me thinking about this kickstarter that I’m so sad I didn’t know about until after it was done: cellular automata scarves!

Random last thing: totally unrelated to this workshop, last night someone posted on twitter that Seattle’s Center on Contemporary Art is about to open a math-art exhibition, and my friend Edmund Harriss is one of the artists in that show! Along with the work of father-son duo Eric and Martin Demaine who both do amazing paperwork (and amazing mathematics). So awesome. Wish I were there so I could go see it.

]]>The hard part is: if we have a function , we can approximate the derivative at a particular point by doing the following.

Find two points close to each other, like and .

Find the slope between those two points: .

There we go. An approximation for the derivative! (We can use limits to write the exact expression for the derivative if we want.)

But that doesn’t help us understand that on any level. They seem disconnected!

But I’m on my way there. I’m following things in this way:

Check out this thing I whipped up after school today. The diagram on top does and the diagram on the bottom does . The diagram on the right does both. It shows how two initial inputs (in this case, 3 and 3.001) change as they go through the functions f and g.

At the very bottom, you see the heart of this. It has

And then I thought: okay, this is getting me somewhere, but it’s to abstract. So I went more concrete. So I started thinking of something physical. So I went to how maybe someone is heating something up, and in three seconds, the temperature rises dramatically. The temperature measurements are made in Farenheit, but you are a true scientist at heart and want to see how the temperature changed in Celcius.

I love this. I’m proud of this page.

And then of course when I got home, I wanted to see this process visualized, so I hopped on Geogebra and had fun creating this applet (click here or on the image below to go to the applet). These sorts of input-output diagrams going from numberline to numberline are called dynagraphs. You can change the two functions, and you can drag the two initial points on the left around. (The scale of the middle and right bar change automatically with new functions you type! Fancy!)

And of course after doing all this, I remembered watching a video that Jim Fowler made on the chain rule for his online calculus course, and yes, all my thinking is pretty much recapturing his progression.

This, to be clear, is about the fourth idea I’ve had as I’ve been thinking about how to conceptually get at the chain rule for my kids. The other ideas weren’t bad! I just didn’t have time to blog about them, but I also abandoned them because they still felt too tough for my kids. But I think this approach has some promise. It’s definitely not there yet, and I don’t know if I’ll have time to get there this year (so I might have to work on it for next year). But I know to get there I’ll have to focus on making the abstract *very tangible*, and not have too many logical leaps (so the chain of logic gets lost).

If I’m going to create something I’m proud of, kids are going to have to come out saying “oh, yeah… OBVIOUSLY the chain rule makes sense.” Not “Oh, I guess we did a lot of stuff and it all worked out, so it must be true.”

A blogpost of unformed thoughts, and an applet. Sorry, not sorry. This is my process!

]]>@rawrdimus shared this applet he made on Desmos for helping kids to understand the idea of a derivative as “slope-iness.” What I like about it is (a) you only get a small line segment instead of the whole tangent line (the whole line would be distracting), and (b) that kids can drag the slider for *a* and get a sense for what’s happening and how that relates to what’s being plotted, and (c) that kids can then make a prediction where the next point will be (and then drag the slider to see if their thinking was correct.

Related to this is something many people worked on earlier this year based on a tweet I wrote (I wanted a surfer or skier to be travelling on a curve, and the surfboard/ski to be the tangent line)… an updated version of this was posted by @lustomatical…

My friend @pispeak posted a nice calculus puzzle that I enjoyed thinking through and solving: “Found this cool question below online (for a challenge) but got stuck…thoughts? help? @samjshah @calcdave @stoodle #mtbos “The line y=0 is tangent to both x^2 and x^3. But there exists another line tangent to both curves. What is the equation of that line?””

I don’t know this teacher, but I like the idea of doing this. Maybe next year I can make it a goal to do write one positive note to each student. Something heartfelt and genuine. A student met with me before school to talk about a “math exploration” she was going to do, and I loved how into her idea she was. I can totally write so many notes saying good things like that to my kids. Like this teacher, doing stuff like this will make *me* feel good.

@mikeandallie retweeted a link to a page that explains the *unsolvability of the quintic *without needing all that abstract algebra. I forgot to dig into this page. But OMG it looks like it’s going to be aweeeeeesome.

@bowmanimal wrote a freaking amazing blogpost about something he did in stats class before winter break. I still am reeling with how awesome it was. The question: *“How can we use basic statistics to examine and tell apart writing styles? What do statistics about your own writing say about your style?” *Doesn’t get you excited? Trust me, click on the link and read how he does this. I don’t often come across lessons that I’m *desperate* to teach, but this is one of them. It also clearly comes from a master curriculum designer.

@dandersod wrote a blogpost ages ago about how to turn a graph into a 3D printed object. I desperately loved it, and had our tech integrator teach me how to do this on our school’s 3D printer.

I wanted to have my precalc kids make mathematical ornaments based on beautiful polar or parametric graphs they tinker around with/discover (maybe have a christMATH tree? haha sorry)… but the timing wasn’t right this year for ornaments (we do polar in the spring). But I still want to make this a reality this year. I hope I remember!!!

@fermatslibrary tweeted out this picture:

I love it because I remember doing something two years ago with my geometry class, arguing that we don’t need cosine and tangent, and that having sine is enough. We showed that we could have done all of trig with just sine. But then we talked about why having cosine and tangent in the mix makes our lives easier. I love this chart because it clearly illustrates what life would be like if we didn’t have multiple trig functions. (On a side note, I wonder what kids would *notice and wonder* about this chart if they hadn’t ever seen or heard of trig before. Like a middle school kid or a late elementary school kid.)

@mzbat (don’t know who this is) wrote a riff on my fav Carly Rae Jepsen, which I feel often enough:

hey i just met you

and this is crazy

but could this meeting

be an email maybe

]]>

This might jog your memory if you don’t know what I’m talking about. You take a piece of paper. You cut out four squares (the same size squares) from the corners. You then fold up the four flaps and tape the box shut. There you go!

You can probably tell that the box’s volume is going to be based on the original paper you start with, and the size of the square you decide to cut out. The question is: **what’s the largest volume you can get for this box. **

If you cut out a teeeny tiiiiny square, you’re going to have a very large base for the box, but almost no height. And if you cut out a giant square, you’re going to have a large height but a teeeeeeeny tiiny base. And somewhere between a teeeeeny tiny square and a giant square is going to be the *perfect square* to cut out which will give you the largest volume.

So the question is: given a specific piece of paper, what size square do you need to cut out to get the maximum volume.

This question has been done to death in middle school classes, in Algebra II classes, in Precalculus classes, and in Calculus classes. And I recognize that this post is just another rehashing of the same old problem. But I remember reading about a teacher who did a variation of this by including popcorn. And I wanted to do the same. No surprise, when I looked it up, it was dear Fawn. But I had such a lovely time in class today watching this unfold that I wanted to share the specific sheet I made up for kids to do this.

[2018-01-31 Popcorn Activity .docx version to download]

**Teacher Moves / Outline**

This activity requires students knowing and using the quadratic formula. My kids (standard level calculus) are pretty weak with algebra, so I started the class with a “do now” that had kids use the QF. So I recommend that.

Show kids the popcorn. (I had two different flavors.) Show your excitement about the activity. (I was genuinely excited!) Get this psyched. Hand out the worksheet but nothing else.

Put a three minute timer on the board. Explain the problem. Show kids a piece of cardstock with 4 squares drawn on it. Show kids a second piece of cardstock with those same four squares cut out and the flaps folded up so it looks like a box (but untaped, so you can unfold it too). Tell them the volume they create is the amount of popcorn they are going to get. *And that you aren’t going to overfill their boxes — just to the brim*. Tell them they have 3 minutes to work with their partner to come up with the best size square they want to cut out. And they are *not allowed to do any calculations. Just visual estimation. *

At that point, give cardstock, ruler, scissors, and tape to kids. Do not let kids start until you press “GO” on the timer. Then… GO!

After three minutes, my kids were done. They measured the side length of the square they cut out and recorded it on the worksheet. They then cut and taped. They weren’t allowed to get their popcorn until they did one more thing… some math…

It was *super *important to me that kids didn’t measure anything, except for the side length of the square, to do these problems. Why? Because this is where I want kids to recognize the side length of the square is the height (that was obvious to all my kids), but also that when calculating the length and width, they were going to be doing *216- 2x* and

Only after checking their volume with me, and I said it was correct, could they fill their boxes with popcorn.

As an aside, when writing this activity, I had to decide what level of scaffolding I wanted to give for this. I decided not much. So I didn’t include any diagrams. (Well, I did put two on the very last page of the worksheet in case a kid needed some additional help. Turns out no one did.) I also initially wrote the worksheet to be in inches, but then changed to centimeters, and then after thinking a bit more, I changed to millimeters. Why? So kids don’t have to deal with fractions (inches) or decimals (centimeters), and we could keep our eye on the prize. It also made the volume huge — and so kids would have to do a little work to get the correct window when graphing.

At this point, I sent them back to their seats with popcorn in their box to then solve the general case. Close to the end of class, I posted the different volumes students got by estimation (it was a tiny class today… kids were absent or at sports).

Overall, I spent about 35 minutes on this in class. One pair finished completely. All the others are at the place where they are in the middle of the calculus work (close to being done).

]]>

Below I have an invitation to submit a proposal to talk at TMC this summer. There are three options: a short 30 minute session, a regular 60 minute session, and leading a 6 hour multi-day session. If you want to come to TMC and haven’t considered giving a talk, I want you to take a moment and think “well… if I did put myself out there, this is what I would talk about… this is what I know.” If you’re a first year teacher, it could be a session called “If I could do it over” and talk about what you learned, to help other early career teachers. If you’re a math coach, it could be about how to wrangle your more challenging teachers and getting them on your side. If you’re an experienced teacher, it could be about how you design your quadratics unit or how you bring outside speakers to the classroom or … I’m just asking you to *consider leading a session*.

We in the online math teaching community and at TMC believe that *everyone* has things of value to share, and we can all learn from each other. TMC is a welcoming place, and if you’re scared of presenting, you’ll know that you’ll be doing it at a small conference to a small and friendly audience (anywhere from 5 to 20 people, usually). It’s a place to just put yourself out there! I personally am terrified of public speaking, but it was at TMC that I first put myself out there, and it turned out to be so much fun to design and implement my sessions, and just a lot less scary than I thought. And I did it with someone else, which made it more fun! So yeah, I’d love for you think about it. Think about what you know, think about what you have to say, think about what you’re strong at… and if you think you don’t have anything, I’d argue you’re being too hard on yourself. We all have things of value to share. And we all can learn from each other.

Now without further ado…

***

We are starting to gear up for TMC18, which will be at St. Ignatius High School in Cleveland, OH (map is here) from July 19-22, 2018. 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/TMC18sessug). 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 nearly everyone who submitted on time was accepted, however, we cannot guarantee that will be the case. 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 Thursday, July 19 and 48 one hour sessions that will be either Thursday, July 19, Friday, July 20, or Saturday, July 21). That means we are looking for somewhere around 70 sessions for TMC18. We are requesting that if you are applying to speak for a 30 or 60 minute session that there are no more than 2 speakers and if you are applying for a morning session that there are no more than 3 speakers.

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 15, 2018 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, David Sabol, Sam Shah, and Glenn Waddell

]]>In one of my Advanced Precalculus classes last week, I saw a group of three students successfully figure something out. To celebrate, one group member taught his group how to do a *three person handshake* which was elaborate and awesomesauce.

Yeah, it moved me. Those are the things that get me.

Because what it showed me, in that moment, was how solid a group that was. They came to a *collective* understanding. They were having fun together — being wonderfully silly. And they were celebrating their success. It was a sign that the group had gone beyond being three people working together; they had created some sort of synergy. It was a lovely instantiation of that synergy.

Sadly for them, two days later, I changed groups. (My groups stay with each other for 6-7 weeks, usually.)

But inspired by this group, I changed what I did when I had kids sit with their new group members (in all four of my classes). They all said “Hi!” but then I dramatically and mysteriously had them hush up, and I showed them the first 11 seconds of this video.

They were entranced. So I asked if they wanted to see it again. They did, so I showed them the first 11 seconds again. They all thought they were going to learn that handshake. Fools! FOOLS!

Instead, I told them about how amazing it was to see this precalculus group develop their own handshake. I shared with them what that handshake meant to me, an outside observer… what it said about their group to me.

And then… I gave kids 3 minutes to develop their own group handshake together. The only thing I said was that the handshake had to involve everyone from the group. (Of course, this took 4 minutes, but saying you are giving them 3 minutes gets them working together very quickly.)

Now I’ll be honest. I thought this could go either way. I thought kids might be hesitant to do something corny/dorky like this, and it would be a huge flop. *But in all four classes, every group did it*. [1] And they were doing SUPER COOL THINGS including pounding the table, incorporating fistbumps, incorporating dance moves, and creating beautifully symmetric hand formations. It was super fun to watch. And some kids wanted to share their handshakes publicly, so those who were comfortable (and that was most of them) demonstrated their handshakes for the rest of the class.

What is going to happen with this?

I don’t know. Maybe nothing. At the very least, it was a great quick way to get kids working as a team on something when switching groups. And if the group can use it when celebrating a collective success, it will make visible and public what fun and friendly groupwork can look like. And that might just inspire other groups to do the same. I like an atmosphere where kids are propping each other up, patting each other on the back, and see themselves working as a team. And the more structures that I can develop that promote this [2], the better.

[1] Okay, one of my calculus classes was a little less enthusiastic as the others, but they all did it too! I didn’t get the same JUMP RIGHT IN feeling in all groups I got from the other classes. Some groups had it, but not all.

[2] Like the hotel bells

]]>This post is going to share the talk. If you scroll to the bottom, you’ll get access to the slides and the handout.

*Abstract*: Don’t like the way the textbook approaches a concept but are intimidated by creating your own content? Bowman and Sam both write their own content from scratch. We’ll share the simple lesson-design tricks we use to write investigations that lead to vibrant discussions and a-ha moments. You will leave ready and excited to write your own content!

* Hack #1: Old Problem, New Problem
The Important takeaway:* This is the simplest of all the hacks. You might already do this naturally, and textbooks sometimes have questions that switch what students are traditionally given and what they are asked to find. If you’re hankering to see if students have gotten what they’re doing conceptually, mix things up. Just look at a problem and see if you can’t refurbish it by maybe giving them some information and “the answer” and asking them for some other piece of information that they traditionally are given. When you do this, kids will think harder, talk a

My favorite slides (one content, one funny):

Relevant blogposts:

- Give students right triangles and have them associate the correct trigonometry equation that corresponds with those right triangles: http://bit.ly/NCTMSamTrig
- Come up with the equation for a parabola given a focus and directrix, and the backwards question: http://bit.ly/NCTMSamParabola
- Give students definite integrals and signed areas but missing the function, and see what functions they can draw: http://bit.ly/NCTMSamIntegral
- Play Rational Function Headbandz with students, where students have a rational function (or trig! or logarithm! or whatever!) on their forehead so they can’t see it, but they ask each other yes and no questions to determine the equation of the graph: http://bit.ly/NCTMSamHeadbandz
- Students use protractors to attack forwards and backward questions on inverse trigonometry on the unit circle: http://bit.ly/NCTMSamInverseTrig
- Instead of giving students visual patterns and ask them to come up with the sequence, why not have them come up with their own visual pattern using blocks?: http://bit.ly/NCTMSamBlocks
- Mathematical Iron Chef using group-sized student whiteboards: http://bit.ly/NCTMBowmanIronChef

* Hack #2: Thinking Before Mathing
The Important takeaway: *Too often, mathematical notation and premature abstractness get in the way of student thinking instead of being the tool for efficiency and communication that it is for those of us that already understand the concept. Let students play around with ideas in their heads, with their own framing, and own vocabulary, before you develop abstract structures. Let them do it their own, inefficient way before you show a better, more efficient, “correct” mathematical way – the right way won’t stick unless they’ve created something in their brain to stick it to!

My favorite slides (one content, one funny):

Relevant blogposts:

- Bowman’s blogpost on the fold and cut problem: http://bit.ly/NCTMBowmanFoldCut
- Sam’s blogpost on the fold and cut problem: http://bit.ly/NCTMSamFoldCut
- Fawn Nguyen’s Visual Patters: http://visualpatterns.org
- Kids conceptualize how solids in calculus are formed before learning technical notation and abstract vocabulary: http://bit.ly/NCTMBowmanVolume
- Have kids “notice and wonder” with a single polar equation to generate lots of curiosity about conic sections: http://bit.ly/NCTMSamConicSections
- Kids are asked to find the center of rotation of a figure, and playing with patty paper first makes it more puzzle-like and fun!: http://bit.ly/NCTMSamPattyPaper
- Dan Meyer: If math is the Aspiring, then how do you create the headache? http://bit.lt/NCTMDanAspirin

* Hack #3: Make Math Magical Again
The Important takeaway: *This hack takes some time, but it is worth it. You are trying to build up a moment of surprise and curiosity for kids – something that will make them want to learn more. (It’s like watching a magic trick. You’re in awe, but you desperately know how the trick was performed because magic isn’t real.) You have to think about something you find interesting and really dig deep to figure out for yourself why it is interesting. That takes some thinking! But once you find the answer, I’ve found it often points directly to a way to get kids to appreciate that thing. Often times, I’ve found that having kids explore uninteresting things is powerful because it gives context for the interesting outcome (e.g. appreciating that the complex solutions to polynomials when plotted aren’t that interesting, but solutions to x

My favorite slides (one content, one funny):

Relevant Blogposts:

- A way to motivate trigonometric identities which will make them feel a bit more interesting/magical: http://bit.ly/NCTMSamTrigIdentities
- A way to get kids to be surprised that the perpendicular bisectors of a triangle always meet at a single point: http://bit.ly/NCTMSamPerpBisectors
- A mistake and observation that a student made can be turned into a fascinating lesson: http://bit.ly/NCTMSamMistake
- The moment Sam vowed to change his teaching so that he put a focus on the artistic and creative aspects of mathematics – because he loves the surprise and beauty and wants kids to get that surprise and beauty: http://bit.ly/NCTMSamCreativity
- A simple way to make the joy/magic students feel public and contagious by using a simple $5 hotel bell: http://bit.ly/NCTMSamBell
- Helping students understand the idea of limits by hiding the value of a function at first: http://bit.ly/NCTMBowmanLimit

* Hack #4: Toss ‘Em An Anchor
The Important takeaway: * Math instruction doesn’t always need to go from skill to practice to application. Instead, application to some interesting context, whether that be abstract or “real world” can actually drive student learning, and help them learn the more mundane skills and contexts. Great anchors are both natural to the mathematical context, and sticky – tangible, novel, memorable, easy to refer back to.

My favorite slides (one content, one funny):

Relevant Blogposts:

- Introducing Inflection with Infection: http://bit.ly/NCTMBowmanInflection
- Modeling Memory with Calculus: http://bit.ly/NCTMBowmanMemory
- Using Income Tax to review piecewise functions: http://bit.ly/NCTMBowmanIncomeTax
- Use inflated balloons to introduce the idea of related rates to make the idea of two rates being connected in different ways “sticky”: http://bit.ly/NCTMSamBalloons
- A series of posts about “pseudocontext” in math textbooks from Dan Meyer: http://bit.ly/NCTMDanPseudo
- An anchor problem for Riemann Sums involving estimating the area of a weirdly shaped object: http://bit.ly/NCTMBowmanRiemann
- When introducing arithmetic series, have every kid in the class fistbump with each other in the most efficient way possible… Kids will constantly be referring to “the fistbump problem” for the rest of the unit:
- http://bit.ly/NCTMSamFistbumpsTwo posts about the concept of letting kids come up with the idea of “Squareness,” basically, inventing a measure for how square a quadrilateral is, one from Avery Pickford, one from Kate Nowak: http://bit.ly/NCTMSquareness1, http://bit.ly/NCTMSquareness2

**Photos of Me and Bowman Presenting:**

**A photo of Bowman, me, and my colleague who came to support me!**

**Some Tweets about the Presentation:**

Click to view slideshow.

**Resources: **

Slides (with one taken out…):

]]>Hi all,

Life is getting away from me with some tough personal stuff. So I haven’t been as active with the online math teacher community/twitter/blogging/etc. for a while, and I sadly probably I won’t be for a while.

That being said, I really wish I could participate in this initiative that Raj Shah (no relation!) shared with me a while ago. But because of life stuff I might not be able to. But one of the biggest things I want to do is bring *joy* into the math classroom as a core value, and this does that. And I love the idea of a collective joyful math moment for students and teachers all around the world! I’ve done a bit of exploration with this initiative — exploding dots — and I think it’s fabulous and full of wonderment. What it takes? At minimum, 15 minutes of classtime! I highly recommend you reading the guest post I asked Raj to write (below), and joining in this worldwide effort to celebrate the interestingness of mathematics!

Always,

Sam

***

The Global Math Project is an invitation to students, teachers, and communities everywhere to actively foster their sense of wonder and to enjoy truly uplifting mathematics. Math is a human endeavor: It’s about thinking creatively, exploring patterns, explaining structure, and solving real problems. The **Global Math Project** will share a unifying, joyful experience of mathematics with people all across the world.

Our aim is to thrill 1 million students, teachers, and adults with an engaging piece of mathematics and to initiate a fundamental paradigm shift in how the world perceives and enjoys mathematics during one special week each year. We are calling it Global Math Week.

This year, Global Math Week will be held from October 10–17. The focus of Global Math Week 2017 is the story of Exploding Dots which was developed by Global Math Project founding team member James Tanton, Ph.D.

Exploding Dots is an “astounding mathematical story that starts at the very beginning of mathematics — it assumes nothing — and swiftly takes you on a wondrous journey through grade school arithmetic, polynomial algebra, and infinite sums to unsolved problems baffling mathematicians to this day.”

The Exploding Dots story will work in any classroom, with a variety of learning styles. It’s an easy to understand mathematical model that brings context and understanding to a wide array of mathematical concepts from K-12 including:

- place value
- standard algorithms for addition, subtraction, multiplication, and long division
- integers
- algebra
- polynomial division
- infinite sums
- and more!

During Global Math Week, teachers and other math leaders are asked to commit to spending from 15-minutes to one class period on Exploding Dots and to share their students’ experience with the Global Math Project community through social media.

1) See Exploding Dots for yourself

Here’s a brief overview: https://youtu.be/KWJVAjONqJM

The complete story: http://www.gdaymath.com/courses/explodingdots

2) Register to Participate at globalmathproject.org

3) Conduct an introductory Exploding Dots experience with your students during Global Math Week

All videos, lesson guides, handouts are available for free at globalmathproject.org. Since everything is available online, inspired students (and teachers) can continue to explore on their own.

4) Share your experience on Twitter during Global Math Week using #gmw2017

That’s it!

The power of the global math education community is truly astounding. To date, over 4,000 teachers have registered to participate in Global Math Week (#gmw2017) and they have pledged to share Exploding Dots with over 560,000 kids from over 100 countries! We already over half-way to our goal

The Global Math Project is a collaboration among math professionals from around the world. Spearheaded by popular speaker, author, and mathematician James Tanton, partner organizations include the American Institute of Mathematics, GDayMath.com, Math Plus Academy, and the National Museum of Mathematics.

]]>Last year, my school’s awesome director of communications contacted the math department to let us know that the one issue of the magazine she publishes four times a year was going to focus on math. And she wasn’t kidding! The cover of the magazine had most of my multivariable calculus kids on it (thinking deeply at the math-art show I helped put on last year)!

One of my favorite things is that the feature article with an alliterative title, ** Making Math Meaningful**, was simply the transcript of a roundtable discussion we had. A bunch of math teachers got in a room around a big table, and we were led by our director of communications who had done her research and come with some questions. There was a digital recorder in the center of the table. And through talking with carefully crafted prompts, we got to think deeply and collectively about our own practice. I can’t even tell you how interesting it was to listen to my colleagues during that facilitated conversation, and how proud I was to be in a school with such like-minded folks that I have the opportunity to learn from. (If you’re a department chair or academic dean, consider doing this!)

I wish I could just post a PDF of the article for you to read, but alas, the whole magazine is online but can’t be downloaded. Here are two quotations to whet your appetite:

…

So if you want to read about a department that is doing strong work moving towards inquiry-based learning, and read the words of real teachers having a real conversation playing off of each other, I highly recommend you:

- Go to this site
- Make the magazine full screen
- Read pages 18 to 29

That is all!

]]>**Marbleslides Challenges**

I love Sean Sweeney. He’s everything good in the world, packaged in humanoid form! He’s so welcoming and kind to everyone… he wants everyone to feel part of things. At the Desmos Fellowship, he was the person I felt most safe saying “I have no idea what the hell I’m doing” and he would hunker down and help. I think many others felt the same. Okay, enough of the love fest. I am going to share his *my favorite* which I desperately want to use in my classroom. First, a little note. There is a difference between *reading *something on a blog and *experiencing *it. More and more, I’m recognizing that. I think if I read about this, I’d think “cool story, bro” and be like “okay, I could do this, but is it really worth it?” But experiencing it like we did during his short presentation, it’s like “I MUST DO!”

Sean has made a number of Desmos marbleslide challenges (if you don’t know about this, google it). Here’s a gif from his blog. The idea is that the marbles drop and you have to create stuff on Desmos to make the marbles hit the stars.

He shared one with us, and everyone in the giant room got obsessed with drawing functions that would let us “win.” For our challenge, people used ellipses, used lines, used piecewise functions, use quartics. It was inspired to see all the different approaches, and all the play that resulted.

What was lovely about Sean’s facilitation is that he *paused* us after a while (note: a teacher trick is to say “I’m going to pause your screens in 5… 4… 3… 2… 1…”). You knew from the cacophony of groans that we were in a good place. Then he shared out different approaches. The diversity of “answers” for the challenge was fascinating.

He made this a regular thing in his classes. I love his poster which shows the diversity of responses:

So how can I use this? I’m not sure yet. I need a way to keep it light and fun, but also with all that my kids have on their plates and their lack of time, I don’t know if they would take the time to do it without some incentive. After teaching kids how to restrict the domain of a function/relation, and reminding them of all they have at their disposal that they’ve learned about (trig, circles, lines, parabolas, step functions, etc.), maybe I need to have a 10 to 15-minute in-class challenge (with kids working in pairs, so they are comfortable). And then do it again two weeks later, in class (but not in pairs). And then… announce that we are going to have regular marbleslides challenges. And the winner(s) will get the bonus question on the next assessment without having to do it. Or maybe buy some cheap plastic trophies which get displayed proudly in class? I want kids to work on the marbleslide challenges *outside of class* because part of this for me is that I want kids who might be slower at processing or coming up with ideas to have the time to execute their vision. I don’t want this to be a timed thing. Though maybe each time I introduce a new challenge, I give everyone 5 minutes in class to work on it.

What I have to make sure to do is share publicly the diversity of answers, like Sean did with his posters.

I also had an idea about how to score it. Something like 1 point for each star. But maybe if we’re learning about conics, or tangent, or something else, I’d give a bonus point for using those functions. And maybe an additional possible bonus point or two for any additional creativity (teacher’s choice)?

Sean’s posts are here and here.

**SQUIGLES**

David Butler also presented a my favorite on *squigles*. The poster and his blogpost are here.

I am not one for acronyms, really. They often are forced. But what I like is that these are used to teach student math helpers how to work with other students. From David’s post:

SQWIGLES is an acronym that we use to help our staff (and ourselves) when teaching in the MLC Drop-In Centre. It is a list of eight **actions** we can do to help make sure our interaction has a better outcome and make it more likely students will learn to be more independent.

It was originally Nicholas’ idea to have something like this. He wanted something to **help the staff choose what to do in the moment, and also to help them reflect on their actions and choose ways to improve**. We noticed that our staff (and ourselves) needed something focused on actions rather than philosophies, because then it could be used on the fly to choose what to do. Telling staff they need to be “encouraging” or “socratic” is not all that helpful when they don’t know how to put it into action. Yet this is what many documents giving advice to tutors do. So we decided to focus on the actions instead.

The reason I wanted to blog about this is because I think it might be helpful to share with the student tutors at my school. We have a peer tutoring program called TEACH (probably an acronym, since I always see it written in upper case… but for what, who knows!). And I haven’t inquired if and how students get trained. But I’d love to do a short 10 minute presentation on this, and maybe do a few scenarios where kids can practice tutoring while other kids watch (fishbowl?) and take notes on which of SQUIGLES happened. (Not all need to happen! Just look for them.)

I think I should also have this on my desk, since I work with students one-on-one a lot and having that reminder can’t hurt!

**Porfolios**

I went to Cal Armstrong’s session on documenting student learning. Over the years, I keep on getting inspired to have kids make portfolios that they turn in to show evidence of different traits. And this came up again in that session. James Cleveland has done it. Tina Cardone has done it. I want to do it. But aaaah! The time to make it into a reality! Argh! But I really would love to make explicit some values — maybe not standards of mathematical practice (… or maybe throw of a few of them in there…), but things like *perseverance* or *active listening** *or *seeing a problem in a different way* or *acting with courage* or *helping someone understand something by asking good questions* or *recognizing your own a misunderstanding *or* changing for the positive as a group member in someway**. *And have kids document these moments or interactions. And then at the end of a quarter, turn them in. (But have a check in halfway through the quarter!) It would mean that they are *looking **for* these things, *looking to do* these things. And recognizing that I value these things. Maybe they have a choice of things they can include — not all of them? Maybe they can take videos or photographs or write paragraphs or draw a comic — it can open-ended how they demonstrated this quality or action.

There is something that I think happens in my school. Kids form facebook groups (or maybe on some other kind of social media) for their classes, and I suspect lots of backchannel communication about the class happens on this group. I suspect a lot of it is positive and uplifting and helpful. I would love to encourage kids to submit that sort of stuff in their portfolio also, if it demonstrates whatever qualities were asked for!

I don’t know if I’m going to do this this year. But maaaaaybe?

**Preview, not Review: Student Intervention**

Kat Glass gave a *my favorite* on intervention with students who were failing. Part of it was a powerful and important note about language and using code-words instead of saying what you mean. We don’t have many kids that fail classes in my school. But one thing that did strike home was that sometimes when working with kids who are struggling, we put all our emphasis on remediation and it’s like we’re always playing catch up. But sometimes we need to remember that with a struggling student, one tack that we can’t overlook is *previewing* upcoming material. It can help kids be more engaged and confident in class, and it sets a good tone moving forward.

I do this sometimes, but I need to remember to do this more frequently. Although I do lots of discovery based work, I don’t think that previewing some of it with a kid, and working through some of the discovery with them one-on-one, and then them seeing some of it happen again in class is a bad thing. I’ll just have to remind them that they need to be careful about not letting other kids have the same insights they had — and their role is to *help without telling*.

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**Don’t play with your food, damnit! Play with your math!**

I love the idea of having kids engaging in recreational math. I don’t have much time to encourage that in my curriculum — or at least the only way I’ve found for that to happen is with my explore math project [posts 1, 2, 3; website]. Some kids get some extra math problems to work on at math club (usually problems from math competitions or brilliant.org), and kids do math problems on our math team. But that isn’t the spirit of what I want to bring to my school. I want to get kids just fooling around with math for fun! Tinkering! Thinkering! Building! Collaborating! So that’s why I fell in love with Joey Kelly (@joeykelly89)’s *my favorite* presentation. Where he shared with us Play With Your Math.

He and a friend created it. Right now it has 15 sheets of paper that can be printed out, each with a challenge. The name, inspired. Design wise, fantastic. But the problems are captivating, easy to dive into, and many have this open-endedness that can lead to obsession. When I was at the Desmos Fellowship a couple weeks ago, they had these for us to work on as a way to get to know each other. Each table had a different one and we were encouraged to play, and meet others who were playing, and then move to a different table and meet and play when we felt like it. The one I spent all my time on, trying to come up with a strategy? One that I know will get my kids in competitive mode? Poster 5:

I liked getting to know people and I liked these problems! At TMC we were given poster 14 and I became obsessed. And eventually, I solved it (and a second more complicated one). But it took A LONG TIME and I DIDN’T CARE. I refused to go play boardgames at gamenite until I had climbed this mountain!

I need to brainstorm if and how I am going to use these in my school. Some initial ideas:

1. Leave copies of these in the library for kids to use. Or put many copies of all of them on a bulletin board for kids to take, so when they’re board and standing there, they just grab one and start thinking.

2. Use these when I need to fill a long block (we have double periods one out of every five times we meet our kids) and I don’t have a good idea.

3. Plan an Upper School math night, where we gather at a space in the school, do math, order pizza. Like PCMI’s “pizza and math” (was that what it was called? we can do better!). These can be the amuse bouche or the main event!

**Math Art!**

Speaking of recreational math, at TMC17 there was so much math art. I just wanted to share some of it!

Captivating! I hope at some point to learn how to make crochet coral. It feels like once I get in the rhythm, it could be so soothing. Actually, I wonder if it would be fun to have a MAKER MATH club where we make math stuff together. And create our own math art gallery. Things like the things shown here, but also like these, and origami (demaine and lang), and a menger sponge made of business cards, and design and 3d print these optical illusions, and carefully color in pictures from Patterns of the Universe, and create our own mathart coloring pages. If you are reading this and have ideas of things that we could make, let me know in the comments! You probably can tell this is something I’m actually totally *feeling* (FYI, for me, the definitive math art page is @mathhombre’s page here.)

**How To Adult: Let’s Buy A House**

So @rawrdimus gave a *my favorite* on how to adult. He was teaching calculus and wanted to keep his seniors engaged. So he came up with this project that had kids pick a few houses and figure out what they’d need to buy it. He was the banker (a hilarious banker) and gave them two different mortgage options (a 15 year and a 30 year, with different interest rates) and they had to figure out their monthly payments.

I know come the spring, the kids in my calculus class will have their attention wane. So I think something like this could work (this investigation on wealth inequality worked a few years ago)! But right now it’s a little bit like trying to put a square peg into a round hole. I need it to have some more calculus before I do something like this though. Maybe we’ll spend some time talking about *e* or we’ll do something with summing (in)finite geometric series, and maybe seeing that as a riemann sum? I think it’s totally doable — I just need to think a bit more! But if you want to get a sense of why I’m trying to make this happen, just watch Jonathan’s presentation and you’ll totally get it. (Here’s his blogpost.)

**Hey, You Guys! Words Matter**

A while ago, I realized when I said “you guys” it was super gendered. So I just sort of said to myself I’ll say “y’all.” When I wrote emails to my classes, I pretty much say “Hi all!” And then… *and then…* someone brought up the “you guys” issue at a faculty meeting at our school, and in my head I was like “I don’t do that!” But for some reason instead of that reminder doing good, and reinforcing what I was doing, I found it impossible to *not* say “you guys.” Like when someone points out you say “um” a lot, or say “like” a lot. You just, um, like, end up, like saying it, um, *more*.

Glenn Waddell spoke about “you guys” at TMC, and it resonated with a lot of people.

So I think I have a plan. Thanks to a huge discussion on twitter (sorry, don’t remember who to cite), here are my options:

The + others was cute… someone recalled they would say: “Humans… and others…” which made me laugh! I think I my lean towards *nerds* and *my loyal subjects* because I like whimsy. And as another teacher I love says about her classroom: “It’s a benevolent dictatorship.” @mathillustrated said it’s fun to mix them up. We’ll see what I’ll do!

A thought: I should post this in my classroom so I can refer to it! And tell students what I am trying to do. And have them catch me if I say “you guys” (which of course will make me say it more!). And have an ongoing tally of how many times I say it. And when they reach a certain amount, I’ll bring them some treat. I like the message it sends: *I care about words because I care about you. For some of you, these words don’t matter. But I’m doing this for the others of you for whom these words do matter. Also: help me get better because I need to be, and I’m happy to be called out when I mess up.
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Other ideas that came up:

@EmilySliman has renamed ‘homework’ as ‘home learning’ [I called it ‘home enjoyment’ because of another colleague, but they have since left me! So I am free to rename it as I please!]

@gwaddellnvhs has renamed ‘student’ (passive) to ‘learner’ (active) [“Learners learn, and students study. I don’t care how much you study. I care how much you learn.” paraphrased from here]

@chieffoulis has renamed ‘tests’ as ‘celebrations of knowledge’ (someone else uses ‘celebrations of learning)

Now do I think things like this will make a difference? Probably not. Calling something “home enjoyment” won’t make kids enjoy it. But it’s stupid and goofy and that’s worth something. And I don’t doubt that making an effort to change language might make a difference to some students. And it can prompt discussion where I get to talk about my values and philosophy around teaching. (“Why do you call tests ‘celebrations of knowledge,’ your majesty?”) I try to live and act those values, but sometimes talking about them can help too.

**Kids Say The Darndest Things: Another Classroom Culture Thing**

I was having dinner at Maggianos with a TMC 1st timer, @pythagitup. Over dinner, he was telling me about a quote board he did where he put funny things kids said up on display. The beaming of his eyes as he recounted his classes and their quote boards made me know he had done something special. I begged him to write a blogpost about it, which he kindly did here. Here are his top 12 quotes:

I just got sad as I was writing this part of the post, because I remembered that I don’t have my own classroom. I usually am in two or three different classrooms and share the space with other teachers. So doing things like this are trickier. Sigh. It did remind me of one year in calculus. Years ago. 2012-2013. Back then, I was actually a funny-ish teacher. Like pretty goofy. And that particular calculus class was gads of fun. Good and strong personalities. I don’t know why but in recent years, I have lost that spontaneousness and goofiness that I used to have. I’m much more even keeled. I don’t know what happened. Does that just naturally happen when you grow older? I am up at the board a lot less now-a-days, so maybe that’s it… less class-teacher-class-teacher interaction? Whatever it is, I’ve changed. But back then, we had a goofy class. And all year, a student was secretly taking notes on funny things I said, or funny things kids in the class said. And she gave it to me at the end of the year. It was one of the most meaningful things a kid has done. You want to read some of it? Thought so. Wait, you said no? TOO BAD MY POST DEAL.

The post about it is here.

**Promoting Kindness & Gratitude**

I want to do this explicitly in my classroom. I tried a post-it wall of kindness/gratitude once, but that didn’t *really* take off in the way I wanted it to. I probably should have blogged about that to share a failed venture, and why it failed (namely: I saw it as a tack on unimportant thing, so I didn’t build time in class for kids to do it, and also kids have difficulty sharing kindness/gratitude so helping them see different things as kindness/gratitude would have helped too). [I see “nominations” as a way to do this too, and also related to the material! post 1, post 2]

But I saw something super nice. @calcdave was wearing a clothespin clipped to the collar of his shirt. I couldn’t read it but I asked about it. He then gave me a huge bear hug… which I thoroughly enjoyed because @calcdave is awesome and who doesn’t want a hug from him… and then looked at the pin. On the front, it said something like “hugs!” and on the back it said:

And then the person take the pin off and puts it on the other person (I think that’s important… they pin it on!). This then continues… from person to person to person. I love that @mrschz got it from me, and has now bought clothespins, painted them, and written on them. She’s all in!

I am not comfortable hugging my kids. I’m not that teacher very often (until they come back from college and visit). But I could see this going in different directions.

(a) Making 10 pins, each with one side blank, and the other side saying things like “high 5! #youmatter” or “two good things! #youmatter” or “fistbump! #youmatter” [and the person who inquires gets a high 5, 2 good things said about them, or a fistbump], and then the clothespin travels. I like the blank side because the clothespin then begs the question… and having different responses

(b) Making a bunch of pins and giving them all out to one class and explaining the purpose. I would have to do this with a class that is totally into stuff like this. I can imagine certain classes having a majority of kids who groan and then throw the clothespin away. So I’d have to choose wisely and come up with a good framing/rollout.

This idea originated with Pam Wilson, who is a true gem.

When this idea made its way on twitter, @stoodle pointed out that @_b_p has done something related in his classroom. And I remember reading this, being like *OH MY GOD I NEED TO DO THIS *and then promptly forgetting about it. The TOKEN OF APPRECIATION. I mean the name itself gets me giddy!

But I like this idea for a few reasons. First: it is done only once a week. It doesn’t take away from classtime. I can do it during my long blocks (once every seven class days). Kids have all week to think about who they are going to give it to. Kids also get to alter it, so at the end of the year, it is a recollection of good.

I know people are going to hate me for saying this, but this upcoming year, I have small classes. I’m at an independent school, so my classes tend to be small. But I think I remember my tentative rosters being even smaller than usual. I like to have larger classes because I like the chaos and interaction and cross pollination of ideas (though not the grading nor comment writing). But I wonder with small classes this year, will this work? I need to think more about this.

**Crouching versus Sitting**

This wasn’t at TMC but I saw it on twitter and wanted to affirm its truth for me.

I am fairly good about this. I have kids sit in groups of 3 but the tables can fit 4, so I tend to just hunker down with groups when talking with them. In most classes, I almost always drag a chair with me from one table to another which doesn’t have one. I agree there is a *huge* difference between crouching and sitting. There is value in crouching… it sends the message “I’m here to sort of briefly check on you and see what you’re doing but I’m likely going to move on… things are on you… so persevere.” I tend to sit when (a) I need to ask the group a set of questions to see their understanding, (b) a group seems to be getting stuck beyond productive frustration, (c) when a group is having a heated or interesting conversation and I want to listen in [I tell kids to ignore me and just continue, which I know they can’t *really* do but they do a pretty good job] or (d) when my feet are tired and I just feel like plopping down somewhere. Ha! Just kidding!

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