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: https://teacher.desmos.com/activitybuilder/custom/5d2a410855693a4619850fd5

**Update 2**: And of course Bowman Dickson created a cool project out of this… link to his tweets about it here: https://twitter.com/bowmanimal/status/1589804876686675969 of course he did this recently but since I haven’t been using Twitter… I missed it.

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!

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]]>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.

SO MY TABLES WERE WEIRD.

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.

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!

]]>- 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?”
- 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: https://aperiodical.com/2019/07/the-big-internet-math-off-2019-group-2-jorge-nuno-silva-vs-sameer-shah/

Entry 2: a card trick: https://aperiodical.com/2019/07/the-big-internet-math-off-2019-group-2-vincent-pantaloni-vs-sameer-shah/

Entry 3: a magical property of circles: https://aperiodical.com/2019/07/the-big-internet-math-off-2019-group-1-marianne-and-rachel-vs-sameer-shah/

Entry 4: an unexpected break in a mathematical pattern: https://aperiodical.com/2019/07/the-big-internet-math-off-2019-semi-final-1-lucy-rycroft-smith-vs-sameer-shah/

Entry 5: two beautiful squares: https://aperiodical.com/2019/07/the-big-internet-math-off-the-final-sameer-shah-vs-sophie-carr/

(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.

]]>(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.)

]]>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!

]]>I’m going to try to outline my approach/solution, because I sometimes like deconstructing my thinking to see how I actually think/learn… so from this point on… **SPOILERS**.

Some things that stood out to me… First, it looked like there was initially a single circle in a square, and the circle got cut in half and then it started sliding. So I initially drew the full circle in the square (before sliding), I drew the diagram shown, and then I drew the two semicircles in a rectangle after they fully “slid”… I saw the cut circle “in motion” — but after a short while I didn’t see how that would help me.

Then I drew the image and solved the problem and felt proud about it. But then I realized I drew the picture wrong. I circled the wrong part in my diagram, so you can see. I had the “slice” hit the corner of the rectangles, and then I was able to use similar triangles to come up with a solution.

I was proud but for some reason, probably because Matt’s initial tweet suggested to me that it would be harder than this, something was nagging me about it. So I went back and quickly saw my error. *But I have always found that taking a wrong approach can help eliminate pathways to a solution, but might also help me see possible tools to use in a solution. And in fact, this idea of using that “cut line” and similar triangles was important in my pathway to the end. *

So when I went back to the drawing board, I wanted to really see *how* this diagram worked… Some things were fixed (the 12 by 19 rectangle, the fact that the semicircles sort of “slid,” and importantly, the fact that the semi-circles were tangent to the rectangle at two places). So I decided to build this diagram in geogebra (with only one of the semi-circles), and as I built it, I saw that everything hinged on the movable point “G.”

I made the line where the semi-circles touched movable, based on the location of point G. Play around with moving point G here on this web-based geogebra page, and try to get it so the semi-circle on the bottom is tangent to the right and bottom side of the rectangle!

So to me, everything hinged on location of point G, or in other words, the distance from A to G (which is the same as the distance from H to C). We are looking for the location of point G which makes the semi-circle perfectly tangent to right and bottom sides of the rectangle. So to me, those appeared to me as “keys” to the problem. [1]

Sooooo I drew my diagram, and importantly labeled the distance from point A to point G with a variable, *a*. And then I labeled lots of things in my diagram in terms of that variable and the radius of the semi-circle, *r*.

I had two variables, so I needed two independent equations. And here is something nice… because I initially went down a wrong path earlier with my mis-drawing, I had already gotten similar triangles in my head! So I got one equation from that.

I hunted and hunted, and found another equation I could get… using the Pythagorean Theorem!

So now I had two equations and two variables.

… and since I knew this was going to be a beast to solve, I just used Desmos, and got that the solution is *a=1.5* and *r=7.5*.

I did a little of the algebraic gymnastics to try to work this out by hand, but it was pretty uninteresting to me and I was pretty convinced that if I really wanted to, I could. To me, getting the equations was the interesting part, and the rest felt like pencil-pushing. So I stopped there. It was nice that the geogebra applet I created seemed to confirm my answer for me:

So that was my process to solving this mathematical puzzle. Who knows – I could also be totally wrong! I’m left thinking of the following:

(1) Is there a more elegant way to come up with the answer? Because the answer is so nice (a diameter of 15?!?!) but it comes out of such an ugly set of equations, I bet there is a nicer way. In other words, is there a better “conceptual” approach that gives a stronger insight into the geometric nature of the setup?

(2) How did Matt come up with this puzzle? How did he come up with the 12 and 19, so that the answer worked out so neatly to a diameter of 15 (radius of 7.5)? Based on my playing around with this puzzle, I wouldn’t have expected a nice answer — so that shocked me. I would have anticipated nice side lengths and an ugly diameter, or ugly side lengths and a nice diameter.

Finally: If you like puzzles like this, you might want to google “Sangaku” and look at the twitter feed of Catriona Agg.

[1] At this point, I had a small detour where I briefly tried to work this problem on a coordinate plane, where I was finding the intersection of the two lines to find the location of the center of the circle, point I, based on the coordinates of G… but when I realized that once I had the intersection point, I’d still have to find find the right coordinates for G to make the circle tangent to the edges, I realized that would be annoying. So I abandoned the coordinate plane work, though I could always return to it if I needed.

]]>The first one came from my Precalculus co-teacher James. We had been finishing up our unit on combinatorics and also creating new groups, and he devised a great question. So here’s the two-part problem I posed to my kids:

First Problem:We have a class of 14 students, with two groups of 3 and two groups of 4. If I were to have a computer program randomly create new groups: (a) what is the total number of different configurations/outcomes we could have? (b) what is the probability thatyourentire group was the exact same if you were in a 4-person group?

I thought I solved it successfully and was feeling really confident. Then James told me I was wrong. Then I tried but didn’t understand his logic. So I made a simpler case, and then I thought I understood it. My brain hurt so much. I kept switching back and forth between a couple different answers. It was marvelous! Finally, I felt like I understood things and felt confident. I shared it with my class, and lo and behold, a couple students got what I got, and a couple students didn’t. But the students who didn’t convinced me with their logic. And then I shared their thinking with James, who didn’t have the same answer, and he too was convinced. And I *thoroughly* enjoyed being wrong and telling the kids that this problem messed with my head, and they helped me see the light!

The second problem came from a student who emailed me about wanting to become a better problem solver. And they shared this old entrance exam for this summer camp they were thinking of possibly applying for, and wanted some guidance. The problem that I got nerdsniped by and ended up spending hours working on over Thanksgiving break was as follows:

**Second Problem:**

This is from the 2019 entrance questions for a summer program. I think I was able to successfully solve (a) and (b). And then I think I solve (c) for n=3 and n=4 (and got an answer for n=5, but haven’t proved it is optimal). And I have no way to even start thinking about (d). But what I thought was lovely is how many different places my brain when went trying to think through this problem. And the neat geometric structure that arises out of the setup. (Even though I wasn’t able to *fully* exploit this structure in my thinking.)

I hope you enjoy thinking about these!

]]>She said I could share that part of her essay here on my blog when I asked. I like to archive good things in teaching, and this is something I’d like to archive. So here it is.

***

I’ve been thinking about ways that coaching, questioning, and telling played out in my education at Packer. Packer was the ideal setting for these methods of learning, as we had small, seminar-style classes and teachers with the capacity to work with students one-on-one and develop individual relationships. Upon reflection, my eleventh grade math class, Advanced Pre-Calculus with Mr. Shah, exemplified coaching, student-telling, and questioning, and also included exhibition-style projects.

This was the hardest class I took in high school. The content was very challenging, and Mr. Shah’s approach to math drove me crazy. Mr. Shah sees math as a creative field, one that demands critical thinking and a deep conceptual understanding of topics that many see as surface-level and robotic. Mr. Shah’s packets used broad questions as the benchmarks of understanding, pushing students to explain concepts using their own words. His problems had a playful tone, and we spent most of our class time working through material in small groups while he played music in the background and buzzed around answering questions and challenging students to think more deeply. Why are conics important? What is the meaning behind this geometric sequence? Why did you choose to solve this combinatorics problem this way? At the time, these questions made the class a nightmare for me – the content was already challenging enough (it was in this class that I failed the only test I’ve ever failed), and the way he forced us to think about it made it even harder. But, after reading Sizer and thinking about coaching, telling, and questioning, I see what Mr. Shah was doing.

In addition to trying to make math more fun and meaningful, he was pushing us to develop mathematical skills that built upon each other. Sizer writes that the “subject matter chosen should lead somewhere, in the eyes and mind of the student” (Sizer, 111). The curriculum progressed when we made “mathematical discoveries,” and those discoveries led us to mastery of complicated skills and a deeper understanding of concepts. Mr. Shah never talked at us. Discoveries came through telling, but, it was table-mate to table-mate telling. As the teacher, Mr. Shah’s role was to encourage our discoveries and offer support as we worked through our own questions and explained concepts to one another.

Through his approach, we were also practicing broader skills like critical thinking, creativity, perseverance, and thoughtful reflection. He centered the class around group work and we spent a significant amount of time reflecting on our individual contributions to the group and the strengths and weaknesses of our team. I honed these skills in my other classes at Packer, and I am sure they are part of my academic success at Tufts. Mr. Shah’s class was enormously challenging for me, but in writing this paper I have found an appreciation for his approach, and I know I owe him a thank-you.

Lastly, Mr. Shah also incorporated exhibition-style projects into our curriculum. He called them “Math Explorations,” and we had to do four of them throughout the year. They were not as big as these example exhibitions and they did not center around a presentation [like someone mentioned earlier in the paper], but they provided an opportunity for individual exploration in a subject area of our choosing. Being the English-lover I am, for one of my Math Explorations I wrote a series of math poems – a sonnet, some haikus, an ode, and a free-form poem. I was proud of these poems. It was exciting to take ownership in a class in which I often felt overwhelmed, and pursue something that made the content relevant to my interests. These projects were empowering, and they helped me feel connected to the material and confident in the class. If this is the power of exhibition-style learning, then I’m in full support, because the takeaways made a difference in my learning. If only more schools had teachers like Mr. Shah and the resources and the capacity to make classes like his more widely available.

–R.L.

]]>It has profiles and posters of Latinx and Hispanic Mathematicians! And so many interview podcasts, where kids can hear what mathematicians *do*, but also hear stories that either might resonate with them because they’ve experienced similar things as the interviewee, or inform them of something they might not have experienced or even knew was an experience.

At our school, another teacher and I created a “Math Space” outside of our math office. (Read more about that here.) Since we’re now in COVID-times, we don’t have the table and stools there for students. But we still have the bulletin board. So I printed out two of the posters from the Lathisms site, and created my own poster of a passage from Evelyn Lamb about why we should care about these stories.

It’s a little barren, but this is what it looks like:

I haven’t decided what I’m going to do in my classes yet… But I am thinking of having them leave class 7 minutes early to come to the hallway, and have different students read different paragraphs aloud for the poster I made and also for one of the mathematicians. (And of course share the Lathisms site with them, and encourage them to listen to a podcast or read more.)

Wait, maybe the reading aloud in the hallway thing isn’t a good idea because with the masks it’s just hard to hear things… I might need to re-think! But now, I have to go back to planning for classes and looking at nightly work. Adios!

]]>But now we’re about to start a new school year. And I’m vaccinated. And my students are vaccinated. And we’re wearing masks. And my classes are going to be with all my kids together in a single room [1], which is such an awesome thing compared to last year.

One of the classes I’m teaching this year is Advanced Precalculus. Another teacher, my friend James, is also teaching the same course. And he’s new to my school this year, and so when talking about the course, he shared with me how he formally incorporated *Mathematical Habits of Mind* in his teaching in previous years. And interestingly, last year, I toyed with the idea of formally getting kids to be metacognitive about problem solving strategies — but decided to focus on something else instead. So when James shared this idea with me, I got excited.

Right now I have an inchoate idea of how this is going to unfold. Hopefully I’ll blog about it! But for now, I wanted to share with you posters I made using James’ *Mathematical Habits of Mind*. Most importantly, here is a link to James’ original blogpost with his habits of mind and rubric.

Photo of the posters hung up in one of my rooms:

I know, I know, the lighting is terrible. The key words are:

If you want these posters, the PDF file is here.

And here are all of them shared as a single sheet, and not as a poster.

Of course, if you’re a math teacher, you know there are a lot of lists of mathematical habits of mind. We agreed to use the ones James had already been using. But there are many alternative or additional things we could have included.

- Common Core has their Standards of Mathematical Practice.
- NCTM’s Process Standards
- Cuoco, Goldenberg, and Mark’s Habits of Mind
- Park School of Baltimore’s Mathematical Habits of Mind
- And this outline by Kien Lim of so many different schemes.

At the very least, I know that as we get kids to think about what strategies they’re using to solve problems, we’ll also see where there are lacuna in our curricula in terms of using those strategies. Or maybe we’ll discover it doesn’t have as much problem solving as I imagined in it. All entirely possible, since we — the kids and James and I — will all be looking through what we’re doing through our metacognitive *Mathematical Habits of Mind *lens.

[1] The reason I note this is because at the end of last year, I was teaching students live simultaneously in three places: they were in two different classrooms and there were a few at home on zoom. Yes, seriously. When I mention that to teachers and non-teachers alike, they asked how that was even possible. It was… a lot.

]]>I encouraged some of my students to enter the contest for the Strogatz Prize for Math Communication at the end of last year. Even recently, I went to a wonderful virtual event they held called *Bending the Arc, *which featured a panel of Black mathematicians and scientists and allowed participants to talk with them more intimately in smaller groups in breakout rooms. The name for the session came from Martin Luther King Jr.’s quotation which I sincerely hope is true: “*The arc of the moral universe is long, but it bends toward justice.” *

(The reason I mention that is because previously the museum had planned an event in honor of Martin Luther King Jr. day which somehow connected his “Letter from a Birminham Jail” to a session devoted to the Prisoner’s Dilemma problem. For so many reasons, this was a poor decision that I was surprised no one caught when setting it up.)

Also, over the years, I had heard grumblings about the museum from people who worked there or volunteered there as interns. Recently someone shared with me a letter that was sent to the Museum of Math’s Board of Directors that was concerning.

My understanding was it was written a while ago, but only recently shared with the Board. I’ve been told that it is now officially okay to make this letter public. It’s short, but packs a punch. Here is the top-line conclusion, written in a signed letter by two former “Chiefs of Mathematics” at the Museum along with others who work/ed there.

**“With respect to our educational mission, race and class discrimination are embedded in the Museum’s practices.”**

To me, knowing that there are multiple people — including people who were high up in the organization — who felt the need to write an open letter to describe some of their concerns speaks volumes to me. They didn’t have to. It is easier not to. It puts them at some risk, publicly speaking out.

To me, one of the most problematic charges in the letter is that students from Title 1 schools who visit MoMATH often get lessons that end up being 20-25 minutes instead of the normal 45 minute sessions. The letter states “We cannot remain silent while the Museum chooses to offer sub par services forthe least fortunate students who are vastly more likely to be people of color.” I hope that with this letter, those at the Museum take a close look at their practices to ensure there is equity for all the students visiting the Museum. To me, more than anything else, this is of paramount importance before the museum opens its doors again.

Also damning is this paragraph:

*Unfortunately, the Museum actively discourages any form of negative feedback, and the staff has virtually no autonomy. This repressive culture has been described on social media and in the many letters you have received from former employees about mismanagement, abuse of hours, and general lack of respect for staff. In fact, the Museum’s formal Employee Policies Document warns staff members against contacting the Board. Staff members who have advocated for improvements in the Museum’s operations have seen obstacles set up in their paths and have been pressured out or fired. The rapid turnover of MoMath staff, which has an average tenure of less than a year and a half, is evidence of this. Joe Quinn, former Chief of Mathematics for MoMath, was fired shortly after expressing his opposition to the discriminatory provision of education services to Title 1 schools. We believe this to be unlawful retaliation against him*

It is important for an organization which promotes diversity and inclusion to make sure that concerns can be heard safely, that feedback can be given. To these letter writers, it sounds like that hasn’t been the case and there is a culture at the museum which sounds, frankly, oppressive.

I went to GlassDoor, to see reviews of the museum from people who work there.

Most museums I looked at had star ratings in the 3+. (I recognize that those who leave reviews on GlassDoor are likely to be those who have a lot to say in either direction.) It was disheartening to read through the reviews.

I only hope that this letter prompts some sort of investigation into the working conditions at the museum. How long do people work there? Why do they leave? Are employees being taken advantage of in terms of their hours worked? At the very least, this letter suggests that the Board can and should look into this.

]]>

First, you need to install the Zoom Scheduler extension for chrome.

Then, all you do to set up appointment slots for kids is to go to the day you want to on your google calendar and create a new event. Now to make appointment slots for kids to use, it’s just this simple!

That’s literally it. You’ve done it! Once you’re done, your google calendar will look like this:

So how do students sign up for an appointment? You just send them a link to your *appointment calendar,* and they click on the appointment slot they want. To be clear, your appointment calendar is different from your regular calendar. The appointment calendar *only* lists the things that you have set up appointment slots for, *not* your entire calendar. (So you don’t have to worry that kids will see the happy hour that you included on your google calendar!)

To get the link to your appointment calendar to share with kids, just click the appointments you just created on your google calendar and you’ll see this:

Clicking “go to appointment page for this calendar” will lead you to a page with just appointments. The URL on that page is the link that you give to your kiddos. So they use that one link for the whole year! That’s it!!!

*But if you’re like me, you want a bit more info… so…*

This is what the appointment page looks like for your kiddos:

They click on the time slot they want, and it will reserve it in their name. This pops up when you click on it [note: it shows my name and not a student name because I signed up for an appointment with me! It will really show your student’s name.]

Now on the appointment page, that slot has been taken away so no other student can claim it:

When that happens, I as a teacher get an email that alerts me to the fact that a student has signed up. That email gives me the zoom link to the meeting — but I can get that zoom link when I go to my regular google calendar and click on the time slot that was taken. This is what my calendar looks like after someone signs up for a meeting:

Also, the student who signed up gets an email in their inbox with the information for the meeting (including the zoom link)… and the appointment automatically shows up on their own personal google calendar. AWESOMENESS!

Okay, that’s all!

In case this is helpful! It worked for a lot of us at my school.

PS. A couple pieces of advice…

I did this during my required “office hours” which were usually 3-4pm. I created 15 minute windows for them to meet with me. But I told kids they had to sign up before 2pm on the day of. That way I wasn’t just waiting around for them in case they signed up at 3:44pm.

Even though these were individual meetings, I figured sometimes kids might want to come in a pair or trio (if they were working on something together and got stuck). I told them one of them had to sign up but they could share the zoom link with others, so we could all meet at once! No one actually did that in the spring, but I suspect that that’s because I only mentioned it once or twice… I might make more of an effort to have kids to do that in the fall.

**UPDATE: Recurring Office Hour Meetings**

If you want to create Office Hours that kids can sign up for *every* Monday, you don’t have to do this every week. You can do this when setting up your office hours…

After entering the basic information, click “MORE OPTIONS”

You’ll “Make it a zoom meeting,” and then you’ll click on the “DOES NOT REPEAT” option to bring up the options to create a repeating meeting. I always choose Custom — because even if you want office hours weekly every Monday, if you click “Weekly on Monday,” they’re going to appear on your calendar every Monday until the end of time. :)

So then you just decide how you want the office hours to recur! (Yup, you could even say Monday, Wednesday, and Friday if you want! Or every other week!)

This screenshot above shows office hours being created every Monday but will stop showing up on the calendar on December 14th (so, for example, at the end of a semester or some reasonable date).

But WAIT Sam! What if there is a particular Monday you can’t have office hours?

Just click on the office hours on your google calendar and press the *delete button* (trash can). It’ll ask you if you want to delete ALL of the office hours, or just this one day of office hours. Just pick “This event”:

]]>

What I can say is that I did similar things in both Algebra II and in Advanced Precalculus. I would say based on the regular feedback I was collecting that all students really appreciated my organization, consistency, and clarity. That being said, even though I had pretty much the same structure in both classes, things seemed to go better in Advanced Precalculus. I have some conjectures as to why, but I don’t really know the cause.

**Update: A Disclaimer and Caveat**

I wanted to write this up for me, to archive my process/thinking. It helps me. But I also want to make clear that this *isn’t* a how-to guide for anyone else. This is just how I figured out things to work in the situation I was in. In teaching, *context is key.* There is never a one-size-fits-all approach. I have so many friends who had to teach “but not any new content” or weren’t allowed to expect that kids would be able to join class “live,” or had to do everything asynchronous. Of course much of this wouldn’t work in many situations. And more importantly, I have so many friends who have kids or other obligations that took up much of their time. If I had, for example, a kid, I would need to come up with a totally different plan. To be clear, this was *not* sustainable for much longer. It worked for me for three months, hopefully for my kids, and I say overall it went “fine.” But I couldn’t do this over the course of a year. I didn’t talk about my mental state in the original post because I did a lot of that processing with friends in the moment. But let’s just say this whole sudden online teaching took its toll. There were so many evenings I wanted to break down and cry. I was frustrated, mad, angry, overwhelmed, drowning, and felt like no one could really understand. Like this was a Sisyphean task. If I shared with you some of the texts I wrote to friends, I’m guessing even though we were in different situations, you would totally point and say “yeah… me… that was me.”

**The Planning**

I spent a good part of my Spring Break trying to envision what class online could be. I used my friend Alice as a sounding board and I realized I had to figure out what my core values are that I wasn’t willing to compromise on — and build from there in the space we had available. I hit on these three things:

Initially, we were given very short classes (30 minutes) and then later they were extended to 35 minutes.

**The Setup**

I opened a document to write a revised set of course expectations. And as I thought about each section, I started to be forced to imagine what our class was going to look like, how students were going to be assessed, how I was going to make things manageable for me, how I was going to provide support for my students. I didn’t quite know what to expect before we entered this phase. It could have been only for a few weeks, or (as it turned out) it could last to the end of the year.

Here’s what I came up with:

Online Learning! – Adv. Precalculus – Google Docs

The main highlight of this is that I switched our courses to Standards Based Grading. Our school went to Pass/Fail for the second semester and I wanted a way to assess that would support my kiddos. This also gave me a way to determine Pass vs. Fail. I’m really familiar with making SBG work because for years I taught standard Calculus and I learned how to change the flavor of SBG so it worked for me in my particular school.

Most importantly, although we switched to virtual school, my goal was to keep our classes as consistent as possible in terms of how students would learn. I didn’t want to immediately make students work individually since they were used to collaborating in teams. I didn’t want to give them videos showing them how to solve some sort of mathematical problem since they were used to figuring that out themselves.

Everything wasn’t perfect, but I can say that overall the feedback was pretty positive. Here are a few comments from a reflection/feedback form I gave to kids a few weeks into our online learning:

Honestly, it’s working so well for me. This class feels the most structured and like I’m engaged and getting something out of each class. Thank you for all the effort you put into making the Demos activities!!! I also really like the structure of watching videos outside of class, and then coming back to any questions and building off of what we watched.

I personally find that math class is working really well for me. We’ll see how the upcoming assessment goes, but I feel like I’m understanding the material we are being taught almost as well as I did in live school.

I think the structure of our virtual math classes is pretty successful. I really enjoy working in breakout rooms together with my classmates. It allows me to “spend time” with people and work on math together, which is awesome. I also really like when you spend a few minutes explaining concepts by sharing your screen and using a virtual whiteboard. It feels pretty close to the normal organization of our math classes.

**The Constraints**

Our schedule allowed us 30 minute classes for the first few weeks we did online learning. Then, when we refined the schedule, we were given 35 minute classes. They were short.

Almost all of my kids had working internet and a school-issued laptop. Access wasn’t a huge problem.

I decided I had an obligation to hit all the major ideas I would have covered, but I felt comfortable paring things down to smaller and more essential bits, and eliminating the things that felt more minor.

We have a weird 7 day rotating schedule where we had 5 times we could meet. We had a choice for how many of those classes we wanted to be “live.” (Some teachers, like history or English teachers, would have kids read or work on papers for some of the days and then meet live only a couple times each cycle. Or they would set longer one-on-one meetings up with their kids to talk through ideas.) For the structure I set up, I usually had my kids meet “live” 4 out of every 5 classes — and I had something for them to do that didn’t require collaboration for the fifth day. It’s important to note that these classes were a mere 30 minutes (later extended to 35 minutes), so every minute was precious.

We were using zoom as our communication/video platform.

I didn’t want to give a lot of nightly work, since kids were going to be on their computers a lot. Since we were meeting live a lot, my goal was in the range of 5-30 minutes, depending on the day and their level of understanding.

**Organization**

I used the “Classwork” tab on Google Classroom to be our central hub. At the top of the page was:

I had a revised version of our course expectations, an ongoing skill list for what we were learning, and a link to my google calendar where kids could reserve a time to meet with me individually. (The other links aren’t as important.)

Then below that I created a different “topic” for each week of learning:

We were asked to create an assignment for each live class meeting we had — so that it would appear on student’s google calendars (since we had the option of holding a live online class or not). Then each day I would add the nightly work. Notice I would have the nightly work due by 7am the next day we had a live class. I’ll explain why that was so below.

**The Planning**

Here’s how it worked. I centered the learning using Desmos Activities. I didn’t want kids to have to learn a new platform (they had used Desmos Activities a number of times before). And Desmos had instituted a way to give students feedback.

So the crux of every live class was students working on Desmos Activities that I had adapted or created from scratch. They worked together in breakout rooms, where one kid would share their screen and they would work through the activity together. Some of the slides were “practice” — so not much talk would happen — but some of the slides included exploration and investigation and conjecturing and explaining conceptually what’s happening.

Here are all my Desmos Activities for Advanced Precalculus used during remote learning: https://teacher.desmos.com/collection/5e80e25ec9089c33af3d954f

Here are all my Desmos Activities for Algebra II: https://teacher.desmos.com/collection/5e80e247431047086cf42c54

I kept two evolving separate google documents with my lesson plans for each day. They looked something like this — with easy access to links that I could copy and paste quickly into the zoom chat box when I needed them to go to an activity.

I’m a teacher that likes to go at the pace of my students — so my different sections weren’t always perfectly aligned. I would design the next class based on where kids got.

Here’s what a “normal” class might look like from a student perspective (remembering we only had 30 or 35 minutes):

- Kids join the zoom. Near the end of the year, they started hearing me playing music as they were admitted into the class. It gave me something to bop along to and put me in a good mood! :)
- Kids hear me say “hi!” I send kids (in the chat box) the link to the Desmos Activity they had been working on and ask them to go there and spend a couple minutes silently looking at the feedback I left them. I do this for just a couple of minutes — most of the feedback is short, and I tell them to look more seriously at it after class. We don’t have much time together.
- Kids hear me outline what I took away from the work they did during the previous class and what they for nightly work after the class. If there were issues that more than a couple kids in the class had, I made sure to address it in the whole class. I would do this by screensharing a particular slide of a desmos activity and talk through it, or sharing my iPad and talking through an idea. During this time, I might occasionally preview an idea or remind students of something they had seen previously that might come in handy. This would take 2-7 minutes. (But with 30 minute classes, I wanted to have kids work together during the majority of the time.)
- Before kids go to the breakout room, they hear me say: “Okay, you’re going to log into this Desmos activity. Write this down in your notebooks — today you’re going to call me over so I can talk with your group at Slides X and Slide Y. Remember if I’m busy to keep working and I’ll come by when I’m done with the group I’m with.”
- Kids work together in their breakout rooms. Sometimes they’ll see me pop in when I’m following along on their work on Desmos and see something I want to point out, correct, or compliment. (I didn’t have much time to compliment, honestly, though I tried to do that so me popping in always didn’t seem like it would be a critique.) When they get to particular screens where they were asked to call me over, I’ll join and give them feedback, ask a few questions I’ve prepared to assess they know what they’re talking about, and then have them contiinue on (or ask them to discuss more after I nudge them forward, and then call me over if they didn’t seem they got an idea).
- Three minutes before the end of our time together, I’ll either send kids in breakout rooms a message saying they can leave at the end of the class straight from the breakout room, or I’ll call them back to the main room to say something and then dismiss them.
- The nightly work will be posted on google classroom pretty soon after class. The assignment will look like this:I’ll ask them to review my feedback from the previous night’s work some more. Sometimes I share with them a resource if they struggled with that work (usually a video I created going over some of the problems.) I post what they’re supposed to do. Sometimes I’d include DeltaMath practice for more routine problems, which I love because it gives students feedback on how they’re doing.

On my end, this is what my side of things look like to make this all happen

- School starts at 9am, so I wake up at 7am and shower and get ready by 7:30am. I sit down at my table and look at my calendar. I tell me Amazon Echo to send me reminders 5 minutes before every class/meeting I have.
- Then I look through the Desmos activities for the classes I was seeing that day. I always made the nightly work due by 7am of the day we’re having class, so I could look it over and understand where kids were at, and give them immediate feedback on their work/thinking. I go over every student’s slides (choosing key slides to give feedback on). As I do this, I make a note of which topics are worth bringing up in class — if anything. There were a good number of days when kids seemed to get most of the material!
- As I do this work, I also fill in a nightly work spreadsheet to keep track of whose doing the work. I also had a column where I started keeping information that might be useful about things I noticed in their work, but truth be told, keeping that additional column wasn’t sustainable so I ended up using it for notes about when kids didn’t do their work — if I emailed them, what they said, etc. (As a side note, if a kid didn’t do their work, I let them complete it later.) Here’s a sample of what my spreadsheet looked like.

- If I see a bunch of students are struggling with an idea, I quickly prep a short iPad presentation to talk about a concept or work a problem — a mini-lecture I’m going to deliver. I add that into my lesson plan for the day. From start to finish, looking through the nightly work for the kids and doing any last minute mini-lecture prepping usually takes me a little over an hour.
- I open the classroom 5 minutes before class starts. As I admit kids into the zoom room (two or three at a time), I mark them present on my attendance spreadsheet. Sometimes when kids come too quickly right at the start time of class, I’ll just admit everyone and fill this in when kids are in their breakout rooms. (This is a fake spreadsheet to illustrate.)

When everyone has arrived, I say hi and then tell kids to check the feedback I left for them on their nightly work (on a Desmos activity). I put the link in the chat box. - After they look at the feedback, I gather us together. I go over the things I noticed from their work in the morning, and give any mini-lectures I feel is necessary for that section. I send them off to work on the Desmos activities — telling them to call me over at one or two pre-chosen slides. Usually, I set them up in random groups of 3-4 students, though occasionally I’d do pairs for certain activities.
- At the very start when they’re working in breakout rooms, I’ll take a piece of paper and write down the answers for each slide I think they might get to. That way when they enter their work into desmos, I can quickly check it. This usually only takes me a few minutes and kids are still settling into working together.
- Then I start keeping tabs of what’s going on by using the teacher dashboard. I can see which group is on which slides. I write down on a piece of paper the name of one student per group, and I tend to follow along using that student’s work as a representative for the group. As they’re working, I’m noting down which slides they’ve completed correctly and if there is anything I need to talk with them about when their group calls me over. I’ll occasionally pop into breakout rooms when I see a group is stuck and needs some help.
- After groups get to a particular slide and call me over, I’ll look over their work (if I wasn’t able to keep up as they were working) and ask them questions I had pre-scripted to check their understanding. I ask if they have any questions for me, and then they go on. These pop-ins are short — as short as a minute, but if we start discussing, we can get to three or four minutes.
- I’m following along and checking in for pretty much the entirety of the time kids are working on the Desmos activity.
- At the end of class, if I call kids back to the main room, I remind them of anything that might be upcoming or encouraging them to see me in office hours if they were feeling lost, and then I dismiss them. I always remind them I’ll stay after if anyone needs to talk about anything, and a few times students did hang back and ask some questions to shore something up.
- After class ends, the first opportunity I have, I go to google classroom and think about what I saw, where students got to in the activity, and then decide what the appropriate nightly work should be. I would often have kids work up to a particular slide in the Desmos activity. I might choose a DeltaMath assignment. I might make and include a video of me working through a slide or two that kids had difficulty on the night before (if any) so they would have something to look at if they struggled.
- After that, when I have another small expanse of free time, I’ll look through where we are and whip up a new lesson plan for our next class based on where we got, and add it to my ongoing lesson planning document.

And the cycle starts over again the next day.

I’d use evenings and weekends to think through and create the Desmos Activities, and create videos of me working through specific slides that I thought kids would need help with. (That way after kids attempted them, I could lead them to the video and have them watch it for assistance.) (The videos were more for Algebra 2. I didn’t find I needed many of them for Advanced Precalculus since kids seemed to get the ideas fairly quickly.)

[Note: at the start of the time we had online, I would include “check in” screens to see how kids were doing/feeling. I would also reach out to individual kids if I thought something was wrong, or check in with their advisor or dean. Trying to understand and attend to the emotional well-being of my kids was really tough. But that’s a post for another time.]

**Assessments**

The assessments I gave were fairly traditional. I kept an ongoing skill list, like the one below:

Adv. Precalculus Skill List (Ongoing) – Google Docs

Then on assessment day, I would upload a test for kids to work on. I was pretty standard in terms of what the test would look like — though I was super duper extra explicit about everything in terms of how I wanted students to format their answers. (For example, I wanted the work for each skill to be written on its own page. So for a five skill assessment, they’d submit five pages.) Students were given a fair amount of time to take it on the honor system in one sitting. I didn’t have the energy to think of all the ways kids might cheat — it felt like such a low priority in terms of what I wanted to give my mental energy to. I figured it was better to just trust my kiddos, because they hadn’t given me any reason not to trust them during my time with them in-person.

Kids used the CamScanner app on their phones (they used this throughout the year to submit their nightly work, so the process was familiar to them) to submit their test on Google Classroom.

I would mark it up and give feedback using the iPad and pencil that my school got for me (bless them!), and then email it back to kids after I had marked them all up and recorded their scores.

Since we were doing Standards Based Grading, if kids didn’t show a solid understanding of the material, they had the opportunity to sign up to reassess that skill. I had a system set up that was easy to manage, but it did mean that for every test I created, I had to create two versions (one for the original go-around, one for the reassessment).

**Feedback Loops**

I was very intentional to make sure that I had a way for kids to understand what they knew and what they didn’t. Here are the ways that played out:

When given assignments on DeltaMath, if students got something wrong, they immediately know and they also are given a complete solution to the problem to learn from. The way DeltaMath is set up is that you keep working problems until you show competency — which could be doing a few problems or it could mean doing a bunch.

When given assignments on a Desmos Activity, I would go through each morning it was due and give feedback. I’ll leave no feedback on slides that kids were getting right, but on a slide where kids did a bunch of work, if they got it all right, I’d make a note of that. I’d also point out if there were mistakes. I also would have videos made (more for Algebra 2) with me working through particular key slides, so if I saw a student was struggling with something, my feedback in Desmos would include “Look at the video I created and will post on the nightly work today! I think that will help!” I would also encourage kids to meet with me in office hours to talk through things that I saw they were struggling with.

Based on looking at the whole class’s work, I would address common misconceptions or point out different interesting approaches at the start of every class.

If students messed up on a skill on an assessment and didn’t show a solid understanding, they could look at my feedback, go back to our Desmos activities, set up a time to meet with me, or talk with friends… and then ask to rework it to show a stronger level of understanding.

Close to the end of our online learning, two weeks before we ended, I asked kids explicitly about the feedback I was providing them. Here are some of their responses:

I love this structure! I love having structure in general. It’s so helpful when you go over common errors at the beginning of class, and I’m able to take notes on it. I also like the little desmos feedbacks if it was just a personal issue.

I feel as though this feedback look is extremely helpful. I particularly like the specific comments you leave on our Desmos activities – I find them super targeted and helpful. Additionally, I really like it when you share your iPad/give general feedback pertaining to the whole class in class (and sometimes start w/practice problems if you think that we need them).

It has been working really well! The comments on desmos at the beginning of class have really helped direct my questions that I ask in breakout room, and my meeting with you after school really helped me understand the material on the first test better.

I chose some of the ones that were more detailed, but almost all students said they found the feedback system helpful. It was awesome to read.

**My Own Organization**

I had everything for online learning in a single Google Drive folder that I linked to from my bookmarks bar.

In here, I had a folder for everything related to assessments, folders for reflection forms and feedback forms, my attendance/nightly work/check-in-with-kids spreadsheets, a google doc keeping all my individual meetings with kids and what we talked about, my ongoing lesson planning documents for both classes, and my course expectations for online learning.

I kept all attendence, nightly work, and notes on individual check-ins for students all in the same Google spreadsheet. Each got different tabs. So I would open a spreadsheet and see this at the bottom.I didn’t want to have information spread out over a thousand documents. My goal was to be as consolidated as I possibly could be.

Five minutes before I taught each class, when Amazon’s Alexa reminded me, I opened the following windows to get prepared and ready to go:

I’d have my google calendar up, because I often needed to refer to it to find the time the class ended. I had my ongoing lesson plan document open so I could execute the plan I came up with. I had the attendance spreadsheet ready so I could take attendance, and I had any Desmos Activities tabs open (for what kids did the previous day and/or any new activities we’d be doing).

I’ve never been a person who scheduled my life using google calendar, but during this time, I came to fully rely on it. Here’s a screenshot of what a random week looked like on my calendar:

**Final Thoughts**

Wow, that took longer than I anticipated to type out. I honestly figured this would just be a 30 minute blogpost where I throw up a few screenshots. Sadly I think most of this structure won’t be useable next year if we’re in hybrid mode. And I wouldn’t say it was perfect or even great. It was… fine.

The biggest thing that I felt was after a few weeks, it started to feel monotonous to me — and so I assume it was the same for kids. We’d do the same thing in most classes. I needed to find ways to break things up — different activities or ways to learn or engage with the material. But I was so fried from juggling everything and creating everything and worrying about covering key content that I didn’t have the opportunity to mix things up in the ways my kids deserved.

I should also mention that this was a lot of work that isn’t outlined here. Planning and creating the desmos activities took massive amounts of time. I had to collaborate with my teaching partner. Reach out to kids and adults when I was worried about kids. Create the skill lists and plan out the content we’d get through for the year. Write assessments and mark up assessments. Write reassessments, set them up, and mark up reassessments. Work with kids during office hours. Not to mention plan our daily advisory and attend meetings (including some of my own doing… like a book club I helped kids organize). There were many days where I’d be on my laptop every moment from 7:30am to 7 or 8pm with only a short break for lunch and dinner. Being on my laptop so long gave me headaches sometimes. Weekends were super important for me to organize myself and get as much preparation as I could for the following week. It was a lot. I found ways to make it streamlined and sustainable, but doing this work — even just “fine” work — took a lot out of me.

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