Fire Hydrants

In Geometry, we have seen the importance of the perpendicular bisector. It’s kinda amazering! And before we delve too far into rotations of figures (which, again, have a lot to do with perpendicular bisectors), my co-teacher and I wanted to do some sort of investigation.

Now, after doing this, I can see what I did as being something that could introduce the awesomeness of the perpendicular bisector. It could be our anchor problem. However, we had already introduced it. So I thought this little aside would be a fun solidification of what we’ve already learned.

(To be clear, students have learned that the perpendicular bisector of a segment is the set of all points that are equidistant from the endpoints of the segments, that the three perpendicular bisectors of the sides of any triangle meet at one intersection point, and that one intersection point is the center of the circle containing all three vertices of the triangle)

Here’s the problem I gave them… however to turn them off the scent of the perpendicular bisector, beforehand I said “okay, class, now we’re doing to go on a little diversion…Here are 4 fire hydrants”

Let’s say there were fires at these four yellow locations. Which is the closest fire hydrant to each?

Okay! Some are easy to just “see” and some might require some calculations.

So next I asked students to try to color in the picture with all locations which are closest to Hydrant A, all locations which are closest to Hydrant B, all locations which are closest to Hydrant C, and all locations which are closest to Hydrant D.

Unsurprisingly, students filled in the edges of the diagram first (so the top left area was all A, the top right area was all B, etc.).

Unsurprisingly, eventually all students asked “what if a location is the same distance to two hydrants?” (I said you could just color that black.)

Surprisingly, since we had *just* been working on perpendicular bisectors, I was surprised that it took 7-10 minutes of working in groups before I heard the first student say perpendicular bisector. I made sure to have that word repeated and spread from group to group. I anticipated that as soon as students would think about A and C (and how there’s an infinite number of points equidistant, with that horizontal line that can be drawn)… or as soon as students would think about A and B (and how there’s an infinite number of points equidistant with that vertical line that can be drawn), it would unlock the whole problem. But my intuition led me astray. I honestly thought they’d finish this in 10 minutes, but it took around 20-25 minutes. But it was worth it. Eventually we got to this…

Now I will say that this region was tricky for kids…

(Note to reader: we talked more about this in the next class, and pointed out that the intersection point at the top of the yellow circled region was equidistant to A, B, and C (so it was like the intersection of the perpendicular bisectors of triangle ABC), and the intersection point at the bottom of the yellow circled region was equidistant to B, C, and D (so it was like the intersection point of the perpendicular bisectors of triangle BCD)).

After all this, I gave my kiddos a follow up “backwards” task that I invented but I didn’t have an elegant/simple way to solve. I just was curious what they would do…

I said “here’s the diagram… can you find out where the fire hydrants are?”

They had good discussions, but we had such little time left in class, and I wanted to get to some resolution. First, I said, “play with trial and error… all the hydrants are on lattice points.”

They had good discussions. But again, we were short on time. So I said: “okay, I’m going to give you ONE hydrant location. From that ONE location, you should be able to find all the others.”

And indeed, kids were figuring the others out. We had learned about perpendicular bisectors as lines of reflections, so some kids started folding their paper along the lines to get more hydrants…

And then you can get the very last one by reflecting the purple hydrant over the vertical perpendicular bisector. To me, I like that students could see that once you had ANY hydrant, they could get all the rest of the locations. For me, that’s a nice takeaway. As I said, I was just curious to see what they might come up with.

At the very end of class, literally the last 5 minutes (I wish I had more), I gave a little lecture on what these things were. They are called Voronoi diagrams. You can literally see the creation of them by this dynamic image (from Wikipedia):

And then I showed them how a giraffe’s spots are a Voronoi diagram!!! You can click the images to get a bit more

At the start of our next class, to see what students retained, I gave each student this diagram and told them to make a Voronoi diagram for it.

I was proud that most students could do this, fairly easily. Then when we had the solution…

… I asked what did all points on the purple ray represent, what did all points on the red ray prepresent, what did all points on the yellow ray represent, and what did the green point represent. Kids seemed great with this. And then I drew a circle (on the Smartboard, using the circle drawing tool) with a center at the green point, and made it bigger and bigger until lo and behold… it hit points A, B, and C. Huzzah!

Update: David Sabol shared this amazing Desmos activity (by Joel Bezaire) which is like what I posted above but is in many ways better. And deals with food deserts (thus social justice) naturally and seamlessly:

Update 2: And of course Bowman Dickson created a cool project out of this… link to his tweets about it here: of course he did this recently but since I haven’t been using Twitter… I missed it.


PCMI 2022 Post 6: Building Party

Yesterday evening at 7pm, a bunch of math teachers and other PCMI folk gathered in the teacher rooms to build math art. Each table had a different thing one could build (slideshow):

It was sweet to see one participant bring their kid, and another bring their partner, and we all had fun creating delicious little things we could take home. I didn’t end up working on any of these set projects because I wanted to continue to learn to knit. One of my PCMI goals was to learn to knit. I remembered Peg was an expert and so I reached out to her before PCMI to see if she would be able to teach me — which she happily did! (And in fact, last weekend, she took me and a few others to Salt Lake City for a yarn/fabric crawl which was so wonderful). I wanted to learn because I’ve been listening to a lot of audiobooks since I’ve had a hard time concentrating on reading since the pandemic. And I figured having something to do while listening would be neat. So during the entire building party, with 80s music blasting, I practiced my knit stitch and perl stich. I made what I thought was a number of mistakes and so I cast and threw out my initial attempts two different times. The third time I was successful-ish. I did three rows of knit stiches, and then I alternated a bunch of knit and perl stitches. I started out with 10 stitches, but somehow ended up with 11 stitches. And a few times, I looked at my needles and didn’t understand what was going on… I had something twisted or I thought I had made a stitch but I didn’t. At those points, I didn’t know what I was looking at so I asked Peg and Rebecca for help — and so I now have a sense of what to look for. However I realize my next step is to learn how to deal with mistakes… What do I do with the 11th stitch that I didn’t want? How do I analyze my knots/stitches to be able to undo them if there is something wrong? In other words, what are ways to deal with errors? In any case, this is what I have so far.

When I wasn’t knitting, I was going to the different tables, looking at all the colorful things people were making. Here’s a collection of some of the objects that were made, displayed the next day… but not everything!

Being surrounded by all this math-art reminded me of the math-art show I helped organize at my school, which we titled Technically Beautiful.

While knitting, I was sitting at the table where people were making the “straw thingy,” which was actually 5 intersecting tetrahedra.

The first time I heard of these tetrahedra was when looking in an math-origami book and saw a connection to a multivariable calculus project. It turns out to get the tetrahedra to interlock perfectly, so they didn’t jiggle around, is a tricky problem. Years ago a student of mine did a project on this:

What I loved is that all I associated with these tetrahedra was this math — finding the coordinates for the vertices of the points, and finding the optimal strut length. However while I was sitting at the table knitting, I was talking with a math professor who shared with me that he sees a “proof without words” with these tetrahedra. He saw something different mathematically than I did. He told me that one could see those interlocking tetrahedra as representing a particular mathematical group. It isn’t quite the permutation group of 5 objects, but rather if you have 5 objects and permute two pairs of two objects (so if you had 12345, you could do a move like 12345–>21345–>21435, or 12345–>21345–>23145). I think he called that group the alternating group of 5 objects. And then he showed me how if you look at the interlocking tetrahedra, and rotated it around a vertex, face, or edge, you get that same group (like the colos of the straws, after a rotation, swap… but in the way of the alternating group). It was fun to have someone way above me in math explain something to me, who would allow me to ask questions, and use hands on manipulatives (we pulled out straws, and did the rotations!) to make things make sense for me. And apparently, this alternating group of 5 objects deals with the insolvability of the quintic equation, something I learned about ages ago in college, but now is faded, distant memory. [Sorry if any of my descriptions are wrong… It was an informal conversation and I haven’t had time to research it yet to flesh it out.]

What’s neat is that now, these interlocking tetrahedra mathematically for me no longer represent only a mathematical question about optimization (the “optimal strut width”) and an interesting problem about how to find the coordinates of the vertices. These interlocking tetrahedra now also represent for me a group, and connects up with the insolvability of quintic equations! Again, I am reminded of the Francis Su quotation from two PCMI posts ago, which talked about how mathematical ideas don’t exist in isolation. Instead, they build up in time and get richer and fuller when they do. I see something different now when I look at these interlocking tetrahedra than I did before the building party.

Lastly, if you want to have your own math building party, or create something that you see a picture of above, here are all the instructions to the creations!


PCMI Post 5: Ketchup Math

Today I was sitting at a lunch table of 8 people. We somehow got on the topic of grilled cheese. You may know that I’m a vegetarian, but I think a more accurate term for me is a queso-carbotarian because I love cheese and carbs together. So I’m a real fan of grilled cheese. And growing up, when my mom made me grilled cheese made with American singles inbetween two pieces of white bread, she always served it with ketchup.

I thought that was normal. That everyone ate their grilled cheese with ketchup. I mean, yes, you can dip it in tomato soup. And you can dip it in ketchup. And maybe, just maybe, you could eat it by itself. And everyone is going to have their own opinions about what’s best.

What I wasn’t prepared for is that no one at my table of 8 had ever even heard of eating grilled cheese by dipping it in ketchup. I’m not talking about not liking it, or not doing it. They hadn’t ever heard of anyone doing it.

While I was totally certain that everyone at least must have heard of it, and I’d have guessed that most people did it except for just a few. So that fact that of the 8 of us, I was the only one who heard of it led me to believe one of two things:

I was at a very weird table that was not representative of the rest of the American/Canadian population, or that I grew up with this very unique cultural tradition of eating grilled cheese with ketchup — and that was something my mom invented or maybe it’s an Indian thing — but not widely known.

Of course I decided to double down, and claim that my lunch table was TOTALLY WEIRD and that they were the outlier — and more people had to have heard of ketchup and grilled cheese. So I get up to ask another group of around 6 teachers at a neighboring table. And what happened? They all said the combination was new to them. (And later in the day, I ran into author and mathematician Jordan Ellenberg who also said he had never heard of the combination, but understood why it would likely be good.)

Now at this point I feel like my world is crashing down. Something that I thought was so commonplace in the world wasn’t. My sense of reality was being called into question. And instead of accepting it, I double down again. Everyone at my table is interested in this — maybe because I feel totally bonkers and can’t let it go. So I put up a poll on twitter, set for 3 hours, which asks the question below and gets these results from 41 people:

Another person at the table did a similar poll on facebook, and got 25-ish results, and they said about 2/3rds of the respondants said “yes.”

What to conclude?

I figure I can think of it like this. I want to know if my two lunch tables of 14 people (where 13 hadn’t heard of eating grilled cheese with ketchup) WEIRD or if I am WEIRD for doing this. I am using the two polls to be representative of the true population… and to make things easier, I’d even argue there’s some error and so maybe I could even say about 50% of the world has heard of eating grilled cheese with ketchup, and 50% of the world hasn’t heard of eating grilled cheese with ketchup.

Then in a group of 14, for only 1 person to have heard of eating grilled cheese with ketchup IS TOTALLY WEIRD. It’s like flipping a coin 14 times, and only one time you get heads (and 13 times you get tails). Hello binomial distribution.


But… I still don’t know… tonight we had an event at 7pm where a bunch of people showed up. Maybe there were 20 people in the room and the ketchup question was asked and I think only 3 or 4 people had heard of eating grilled cheese with ketchup. So I’m not sure what’s up.

Regardless, I loved this random fun math question that popped up as I was having a mental breakdown involving the internal reality of my world. And that others got into it.

P.S. For another fun condiment digression, our lunch table then started talking about the tiktok trend where people put yellow mustard on oreos and claimed it was so so good. Even Lizzo tried it. Randomly, there was a box of oreos on a table in our classroom, and we had mustard packets from lunch . So later in the day, another person and I tried the combination. I have a screenshot of the video someone took where I tried it. It wasn’t awful at all, but it wasn’t like enhancing the flavor either for me. It was just kinda fine.

PCMI 2022 Post 4: Assessments

At PCMI we’ve been talking a lot in our “reflection on practice” sessions about assessments. Starting broadly, we wandered in the realm of epistemology and started talking about what we mean by knowledge, and how we know if we know something. Which then led to conversations about how we have evidence of knowing (and why it might matter that we teachers think about this and reach a consensus on this).

From this more abstract beginning (where we did try to bring it down to the classroom level), we then started to get more concrete. A very generative question was “what does it mean to be mathematically competent/proficient?” Everyone interpreted the question in a different way, and initially I was stuck – do we mean in a particular skill? or overall, when can we call a person’s body of work “mathematically competent/proficient”? procedurally or conceptually? isn’t competent < proficient? So many layers to unpack in the question itself. However I still started brainstorming and the very first thing I came up with was “can see the idea in a larger web of ideas/connections.” I often think that’s a hallmark of a strong mathematical thinker – where things aren’t this hodgepodge of ideas and procedures but they are tied together in a larger web. And what’s lovely is that one of the most powerful parts for me of Francis Su’s Mathematics for Human Flourishing book so far was in the Meaning chapter:

mathematical ideas, too, are metaphors. Think about the number 7. To say anything interesting about 7, you have to place it in conversation with other things. To say that 7 is a prime is to talk about its relationship with its factors: those numbers that divide evenly into 7. To say that 7 is 111 in binary notation is to have it dialogue with the number 2. To say that 7 is the number of days in a week is to make it converse with the calendar. Thus, the number 7 is both an abstract idea and several concrete metaphors: a prime, a binary number, and days in a week. Similarly the Pythagorean theorem is a statement relating the three sides of a right triangle but it is also, metaphorically, every proof you learn that illuminates why it is true and every application you see that shows you why it is useful. So the theorem grows in meaning for you each time you see a new proof or see it used in a new way. Every mathematical idea carries with it metaphors that shape its meaning. No idea can survive in isolation — it will die. (37) [italics mine]

In addition to this being part of being mathematically competent (which, overall, I think I ascribed to a body of work that a student did instead of with a single skill), I also immediately thought of various mathematical habits of mind.

Something that struck me, after my brainstorming, was that much of what I wrote was about a world of ideas, but it was disembodied from the physical world… and so I ended my brainstorming: “One thing I’m thinking about when I think about “mathematical competency/proficiency” is that those terms seem to be pretty clinical… I’m wondering if joy and appreciation and an emotional connection to math would exist in this or not.” Looking back at this reflection of mine, I think of Rochelle Gutierrez’s eight dimensions of rehumanizing math, and my wondering seems to fit squarely in the “Body/Emotions” dimension.

As I’m writing this blog post, I’m enjoying seeing how lots of different ideas in this conference seem to all be strung together loosely and I’m only now seeing them braid together.

One great exercise we had was just writing down all the different ways we collect evidence of student understanding. All the ideas we came up as a group are here, and a word cloud of what we came up with is:

We talked about bias in various ways we collect and interpret evidence, the ease/difficulty of various ways we collect evidence, and how confident we were in our interpretation. What I liked about this is that we went beyond “paper tests” and “exit slips” and allowed the true range of things that we as real teachers do to get a sense of what a student or class knows… I mean, I do actually use facial expressions to help me get a sense of student understanding, but I also recognize it’s not always the best indicator. So this broad list we generated didn’t seem like a trite exercise because it valued all the ways we as teachers do truly get a sense of things on a day-to-day basis. In other words, it broadened our sense of “assessment” to go beyond “the things that we grade, that often are done on paper.” Assessing is just getting a sense of what kids know – whether formative, summative, or something else (e.g. self-assessing their own confidence on something).

We read a couple articles on assessments as well as having kids self-assess or peer-assess. I liked NCTM’s section on assessments from their Principles to Action — reframing assessments to be something that provides feedback to students but also informs instruction moving forward.

Shifting the primary focus and function of assessment from accountability to effective instructional practice is an essential component of ensuring mathematical success for all students (p. 98)

For me, the big reminder was that we teachers traditionally think about assessment as a noun. That paper thing we give. A static snapshot. But we should think about it as a verb, assessing, and that is part of a learning cycle, a journey — both for student and teacher. However, of course, there’s the ideal and there’s the reality.

There are ways I’ve thought more as a verb, like when I’m did standard-based grading when I taught calculus. But usually, when I give summative tests (which is what I normally do, and I don’t really do projects), I don’t think of them as a part of a cycle or journey. They are the static snapshot.

But I also know in reality, I do lots of assessing in small ungraded ways (self-assessing, start of class problems and walk around, listen intently to students talking, looking at nightly work, etc.), and use that to inform my next teaching move or plan my next class. For one example, I have a general idea of what the nightly work will be each day, but based on what I see in class, I often will alter it based on what makes sense… or generate some new problems to address a misconception or gap I’m seeing… or to have students think about a particular insight that came up in class.

I’m now getting tired, so I want to end with three things before I lose steam.

First, we watched a video of Max Ray-Riek on why 2 > 4. It is an ignite talk, and I’d seen it before, and I realized after watching it that during then pandemic, I started listening for instead of listening to because I felt so stressed for time. I think this is a video every math teacher should watch!

Second, we brainstormed ways that we get real-time evidence of student understanding in our normal everyday classes. And then we each chose a few to share out. I like this document of collective knowledge that we generated, and I want to come back and read through it again to be inspired.

Third, we talked about ways we had students self-assess themselves and their own understanding as a way to become more independent learners, and think metacognatively. I really enjoyed brainstorming this individually and listening to everyone else’s ideas. I wish we had more time for this activity — because I love talking about concrete things we do in the classroom so I can get more ideas and rethink things I already do!

And with that, I’m done! Tomorrow begins our third and final week at PCMI!

PCMI 2022 Post 3: Stupid Square

One of the highlights of my day on Thursday was our first book club meeting to talk about Francis Su’s Mathematics for Human Flourishing. Just the first chapter. At my table, there were five teachers and one graduate student who got the book so he could join us (bonus: he was wearing a Harry Potter shirt and was not intimidated to be surrounded by people who all knew each other). We basically had an hour to introduce ourselves, and then informally talk about two things:

  1. Responding to one, two, or three questions that were posted on the screen, which went something like (my own paraphrase, since I have a terrible memory): “If you were at a coffeeshop and had to define what mathematics is to someone in 2 minutes, what would you say?” & “What is the connection between mathematics and being human?” and “What would you say to someone who asks what’s the point of learning math if you aren’t ever going to use it in real life?”
  2. In the book, we read a letter that someone in prison wrote to the author. It was a letter where the person was vulnerable, and in the letter talked a bit about his journey that led him to prison, but also that he had previously had a proclivity for math and so he was studying it on his own and reached out to the author for assistance. We were asked to think about what we would think and do if we had received the letter.

The looseness of the prompts (and for the first one, we had choice, and our group even modified some of the questions as we talked about them), and the lack of needing to produce something tangible at the end of our discussion, was lovely and freeing. (Over the years, I have led a lot of math book clubs with kids, and you can read some of my advice at the bottom of this article here. This structure for today worked well, and I loved it.) And since the first prompt got at the heart of what we love, why we’d spend our lives devoted to it, our passions, we all had something we could bond over and really feel connected to the other people sharing their thoughts. Or at least I felt really connected with the people at my table.

I want to first write a bit about the second prompt. It reminded me of blinders I often have. Some shared that they would not respond to the letter because of fear for their own safety and fear for people in their lives — a strange unsolicited letter coming to their home — and others shared similar thoughts. It never even occurred to me to think about that, but it reminds me of conversations I’ve had with friends who are women and have to move about the world so differently with a totally different lens (like one told me years ago that when they go into their hotel room, they have a routine where they check under the bed and in the bathroom for someone). I didn’t even consider that aspect of things.

I did think a lot about my friend Sara Rezvi who posted a few months ago about doing this exact thing — communicating with an incarcerated person about math through the prison math project. She tweeted about it here:

Now to the first set of questions, which got us going! A few of us gravitated to the third question (“What would you say to someone who asks what’s the point of learning math if you aren’t ever going to use it in real life?”) which as we discussed it, really seemed to dovetail into the second question (“What is the connection between mathematics and being human?”). One person shared the idea “why do we read Shakespeare if it doesn’t come into our daily lives?” which is often my go-to! There is something inherently captivating about the act of reading it, and analyzing it. And we see the beauty in it. As we talked, I kept on having the idea that mathematics is the act of world building, which seemed to encapsulate much that had been said. Under constraints, we invent, we use creativity to push the things we invent, we explore, we get bored and go somewhere else, we feel emotions as we construct: angst, elation, frustration, anticipation, sadness, and when we’re really lucky, love. There are also lots of other things that we said that don’t fit in here (an informed citizenry, ability to analyze, ability to draw connections, etc.), but that metaphor really resonated with me in the moment.

When we talked about the first question (“If you were at a coffeeshop and had to define what mathematics is to someone in 2 minutes, what would you say?”), we changed the question. Because normally, when we meet someone in a coffeeshop and math comes up, they get turned off. So if we said something like “math is about patterns and meaning in those patterns,” I’m pretty sure that would kinda lead to an end to that part of the conversation for most people. So instead, we changed the question to “what would you say to someone in 2 minutes to express why math is something you want to spend time with?” Like give them something to be captivated by which would let them get sparked and have a glimpse of what we glimpse.

I would have loved to brainstorm this with other teachers for hours — because I think the answers we’d come up with would be amazing for us to use in our classrooms (in addition to random hypothetical coffeeshop situations). Maybe we talk about the 30 second responses, the 2 minute responses, the 10 minute responses, and more! And we teachers have so many ways to do this! (One person said they’d bust out Pascal’s Triangle!) We didn’t get too much time to do this specific brainstorming, but one teacher said something that really encapsulated what I think what many of us teachers feel: in school, we teach under all these constraints and traditionally that amounts to students seeing math as fitting in this “stupid square” when we all see math as (waving all around the square) as this much bigger and beautiful and wondrous thing.

That reminded me of something I heard years ago… In physics, in high school, kids learn about quantum mechanics. They learn about the wave-particle duality, they learn about the probabilistic weirdness, they learn the world is so strange. Learning about that has the ability to captivate the minds of kids (it did me, anyway). But kids in high school can’t do the mathematical parts of quantum mechanics, the wave function. But that doesn’t stop the physics teachers from teaching it. We should be doing stuff like that with modern mathematics in schools, to capture the developing wonder and imagination of kids.

Jordan Ellenberg is at PCMI and I was too nervous to go up to him to say “thank you.” You see, he wrote How Not To Be Wrong: The Power of Mathematical Thinking, which I read when it first came out and loved. (I am super critical about popular math books, in general.) And since then, I’ve done two math book clubs with How Not To Be Wrong with kids at my school — holding a few sessions, each, talking about what we read. It’s part of my trying to get out of the “stupid square.” So instead of talking to him, I send him an email thanking him for giving me another avenue to do this! The day after I sent the email, Prof. Ellenberg gave a talk about “Outward-facing Mathematics” (books, blogs, popular articles, tiktoks, etc.) and why and how to get involved with it. There are so many ways teachers are doing this now… Sidewalk Math, #MathGals, Math on a Stick, Play with your Math, Playful Math Education BlogCarnival, …

When I left his talk, I remembered I actually had done some “Outward-facing Mathematics” with the Big Internet Math Off 2019 (where I came in second place of sixteen competitors!). In case you want to see
my attempts at learning or explaining math for a slightly more general audience, to captivate, they are here:

Entry 1: a confounding conundrum:

Entry 2: a card trick:

Entry 3: a magical property of circles:

Entry 4: an unexpected break in a mathematical pattern:

Entry 5: two beautiful squares:

(And here are all entries to all the Big Internet Math Offs throughout the years, run by the Aperiodical.)

With that, I’m getting tired so I shall bid this blogpost adieu.

PCMI 2022 Post 2: 3D printing

I was so grateful to the neat ideas that I got on twitter about 3d printing, which I included in my last post. (If you want to check the tweet and the replies, they are here.) Some of the responses that really stood out to me are here:

(1) Have students design (using their knowledge of some core functions and transformations) bubble wands using Desmos. Ashley Tewes wrote a moving blogpost about it here (and how she tied it in with empathy and a larger audience than just the students). And just look at how fun and beautiful these are!

And in a similar vein, Martin Joyce has kids use Desmos functions to create objects involving their own names! And @dandersod showed how to 3D print a polar graph from Desmos to be an ornament, which I then did on our school’s 3D printers:

… and I was going to have my kids do our polar graph contest and have the winner’s graph get 3D printed (but the designs were too intricate for that, sadly).

(2) Kids can design their own tesselations (learning the ideas behind how various constructions can build tesselations) and then create 3D printed cookie cutters for them — so they can create “cookie tiles” that tesselate! Or penrose tiles! I initially found a neat blogpost which I’ve lost, but here’s a tweet that showcases it!

(2) Mike Lawler has almost a hundred posts where he and his two sons (who do math together for fun) have used 3D printing. And they are all pretty dang fabulous — an amazing resource. He even chose his favorite ten 3D printed projects here if you don’t want to scroll through all of his posts. The last one he listed in his ten was a model that illustrates Archimedes’ method for deriving the volume of a sphere (without calculus)! I remember learning this in high school and was blown away (so unexpected! so beautiful!), but in all my years teaching, I had never seen this particular manipulative. You can see and download the manipulative here, but I’ll throw down a screenshot of it:

(3) When I taught Multivariable Calculus, we had talked about mappings and coordinate systems, and so one year a student 3D printed this stereographic projection (among other things) and then wrote a paper which analyzed how this all worked:

And I remember showing my multivariable calculus students, in another year, a bunch of optical illusions made by Kokichi Sugihara. They blew my mind, and the kids were smitten. One read some papers on the math behind how you can design these and wrote up a cogent explanation of how this worked using a neat analysis of vector-valued functions.

And goodness knows 3D printing is so cool for surfaces in multivariable calculus, and so much in regular calculus.

But I have to say: after doing a lot of sleuthing, getting things sent to me by others, and just trying to wrack my brain, I’m honestly pretty disappointed with what I think I can do with it in the classroom. It might just be me, but all these schools a decade or so ago were like “WE NEED THESE 3D PRINTERS BECAUSE THEY ARE GOING TO REVOLUTIONIZE STEM EDUCATION.” Maybe so. But after doing an initial foray into them, my current thoughts are: pfft. Maybe I’ll change my mind, but right now: pfft.

Right now, for me, I see the value in 3D printing in two main domains:

MANIPULATIVES: So as I noted, in my last post, there are tons of cool manipulatives a teacher can find and 3D print to illustrate an idea. Like the Archimedes’ proof for the volume of a sphere, or the optical illusion, or creating penrose tiles or printing many of the 15 pentagons that tile (so kids can fit them together and play!), kids will learn. They may be captivated. But kids are learning just from the manipulative, not from the process of 3D printing. That’s just the point of the manipulative — and the 3D printing is one way of getting the manipulative. So great. It isn’t the process of 3D printing that drives student understanding, it is just the manipulative that the teacher finds to illustrate the idea, that happens to be a 3D manipulative. And that’s cool. There’s some value. But in the same value that you can open any math teacher catalog and find lots of hands-on things for kids to play with. This is just a 3D printer printing them, instead of ordering them.

OBJECTS TO SPARK JOY, BUT DON’T HEIGHTEN MATHEMATICAL UNDERSTANDING: Then there are things that I think kids would love doing with the 3D printer in a math class… building bubble wands by using Desmos and function transforemations… developing cookie cutters by learning about transformations… creating polar ornaments by designing creative and beautiful polar graphs. Kids will be able to hold their creations, feel an ownership of mathematics, be proud! So I think there’s a lot to be said for these types of activities. I want to do them! But at the same point, I also truly feel like all the conceptual mathematical learning is happening before the 3D print. The 3D print doesn’t do anything to build on that understanding. What does printing the polar graph ornament from the 2D Desmos polar graph actually teach kids in terms of math? Nothing. I’d argue a kid who printed their bubble wand and a kid who didn’t probably learned the same things. Yes, these things are dang cool, so there’s something to be said for that, but I would argue they don’t build student understanding.

I posited in my last post that there might be a third domain where 3D printing is powerful: where the act of kids actually doing the building in tinkercad or whatever software builds conceptual mathematical understanding. This has been my unicorn, the thing I’ve been really trying to think about or find in the past few days. Because if I’m going to have kids spend time learning new software and troubleshooting finicky 3D printers, there better be a big learning payoff. But at least for Geometry, Algebra 2, and Precalculus, I have yet to anything that really fits the bill.

So for now, as a teacher, I say “hey, 3D printing is cool, but overall, pfft.”

(You might feel differently about this and that’s cool. And I might change my mind. But since I’ve been sending a lot of time trying to think about this and look stuff up, I have just felt a lot of disappointment when I was hoping there was a lot of untapped promise.)

PCMI 2022 Post 1

Haha, I was going to write PCMI 2022 Day 1, but I know I absolutely wouldn’t be able to keep up blogging every day. It’s 8:22pm, and I’m just getting back to my room after leaving it at 7:15am, and golly gee, I’m spent. For those not in the know, PCMI stands for Park City Math Institute and it’s a 3 week summer professional development, and the teacher program is called TLP (Teacher Leadership Program). The information for the teacher program is here, and the constantly updating webpage with the materials we’re using this year is here. I did it two times in the past, in 2010 and 2011, and both were transformative for my teaching, and allowed me to meet and make some amazing math teacher friends around the country! So if you haven’t heard of it, and you want a recommendation from someone who is super critical about most professional development, this is me saying that it’s worth it.

One of the things that happened in the pandemic is that I stopped truly engaging online with other math teachers. Partly, was drowning with trying to figure out how to teach in a totally new way (remote and hybrid), partly I recognized there was no real way for me to teach effectively and I was constantly comparing my teaching to what I had done in the past, and partly my soul was crushed. It was hard to go online and see all the positivity and innovative things that people were implementing when I was just barely keeping my head above water. So I just stopped engaging by blogging which was my form of reflecting (what, a blogpost about how I started doing a bit more lecturing and became less engaging while teaching online? how depressing)… and stopped reading other math teacher tweets.

I’m hoping at PCMI to start blogging again here and there, even if each blogpost is just a mishmash of stuff. I even started engaging with some math teachers again online, and it reminded me why the online math teacher community can be so powerful and why I loved it so much. Here’s an example… one of my two working groups is on how to 3D print. My school has 3D printers. But I have never had any great ideas about how to use them in my classes (except for calculus, which I haven’t taught for a while, but I see lots of connections there). So in my working group, I was learning the basics of tinkercad (which is how you can create basic 3D things to print). But I realized as I was learning the tool, I didn’t know what the motivation was for learning it. In Geometry/Algebra 2/Precalculus, what is a concept that students learn where 3D printing would actually enhance student learning of mathematics — like they would understand the concept better because they learned to 3D print it? And so I threw the question out on twitter:

It was amazing to see people reply! And share links, and ideas! I haven’t yet gotten to look through all of them, but it seems to me like there are probably two or three classes of things: (a) things that kids will make and be proud of and appreciate but didn’t actually enhance their understanding of the math (but would bring them math joy), (b) manipulatives or demonstration thingies that a teacher can make to illustrate or play with ideas (but the making of the manipulatives wouldn’t be so powerful), and (c) the thing where the actual building of the 3D model develops and enhances a kid’s sense of an underlying mathematical concept or idea (where the time spent doing the building is actually worth the payoff in understanding, rather than just using pre-created manipulatives).

So that was fun to re-engage with twitter! And when we saw that Eli Luberoff (founder of Desmos) was coming, it was fun to be able to tweet him to say everyone seemed excited!!!

I also really enjoyed starting to get to know the teachers here in person. I was actually pretty nervous about coming because I honestly think I’m going to just be alone and no one is going to want to hang out with me. But of course that’s never true, but it doesn’t make that fear any less real! (I grew up with no friends when I was younger, so I think that has just scarred me in this respect!) And to find people who want to share groan-worthy math jokes they make in their classroom while we’re eating dinner, or to talk about why someone took the leap to go from 20 year veteran classroom teacher to principal over our breakfast oatmeal, or (fill in any number of conversations here), reminded me how much I love hanging out with math teachers and geeking out.

In the morning, we do fun collaborative math problem-solving. My table (SHOUT OUT TABLE 3) spent a ton of time on this opener:

We found patterns, codified them, refined them, checked them, broke them, fixed them, posed our own questions about them, etc. It was cool because we all had different approaches and styles, but that also made it challenging. I have my kids reflect a lot about how they work in their groups — what they bring to it, what they think they can work on, etc. And so I think that in some ways we were very strong: there was a lot of idea sharing and excitement and conjecturing. But this is a wondering: I wonder if everyone felt like we were all working together to create collective knowledge. Something I tell my kids in my class is that when they’re working in groups, the goal isn’t individual understanding (that they can build at home)… it’s to build collective understanding. And so everyone has to be as invested in the thinking of others as they are in their own thinking. And I’d say that on that metric, there was a lot of interest in the thinking of others, but not for us to build some sort of group collective understanding, but rather to fill it bits and pieces of our own individual understanding. My whole thing about individual understanding versus collective understanding in my classroom… I don’t think this is actually part of the PCMI morning group philosophy, but I think it would be interesting to see if a group I was in all agreed to go in that direction and what we could accomplish both socially and mathematically. I think it would generate some really rich question-asking that would refine our own thinking and understand other people’s thinking, but also help us sortwhat we were having a surface level understanding (more pattern recognition) of to deeper conceptual understanding of (an ability to say why something worked). 

Okay now I’m really lagging, but I want to briefly talk about our “Reflecting on Practice” session. Our focus is going to be on assessments, and I think it’s going to tie into many conversations we’ve been having at my school about grade inflation. Because one of the things we’ve been talking about at our school is “what does a grade mean”, and it’s clearly an artificial construct that flattens a multidimensional thing but is super important in the larger scheme of things because grades matter (at least to my students, for a variety of reasons). And so it was interesting to think about what is “knowledge” and “how do we know that we know something” — because our leader said — we can’t really think critically about assessments until we delve into some of the philosophical underpinnings. We got some reading on assessments from NCTM’s Principles to Action which reminded me if we as teachers reframe and expand our definition and purpose of assessments, they can be much more useful in our teaching practice. 

I also went to a lecture on cryptography which was beyond my level of understanding, but the speaker was excellent and though I needed time to understand the details, I could see the larger argument and zoomed out bigger picture view of what she was sharing. 

With that, it’s 9:10pm, and I am flagging. So tired. So night night I go!