So my friend @rdkpickle had to start distance teaching already. She’s kinda amazing in all the ways, and so on twitter she shared out how she was doing her lessons — and noted that they’ve been going well. They are low-tech in that they use Zoom and Google Docs, and use a Google Doc as an anchor for the lesson. I love that the doc allows kids who have to miss the lesson for whatever reason (emotional/anxiety issues, having to take care of a sibling, etc.) have a way to keep up.

Before sharing it, I want to say: seeing what she did was the very first thing that I saw that made me feel like: “okay, I can do this. It’s doable.” BLESS. When talking briefly with her online, she was saying right now she couldn’t be all investigatory in the same way she was in class, almost like she was ashamed. BUT *very little of we’re doing is going to be like what we do in class*. The ballgame has changed (from basketball to some other sportsball!). Right now, for me, the question is can I give space and structure and community to kids where they feel they can learn a few things. And @rdkpickle’s low-tech approach allows for that!

Here’s a PDF her googledoc, which she said I could share. (And here’s the google doc.)

Mike Flynn (helped by Sarah Bent) gave two wonderful webinars on distance learning that he has put online — March 11th and March 17th. (If you only have time for one, I’d watch the second one, but both are great.) They were some of the first things that made me realize distance learning was possible, by showing how to do it through his webinar. (Unlike, say, dry powerpoint lectures on teaching active learning strategies. Ahem. We’ve all been there. I just think over and over, “Physician! Heal thyself!”) My takeaways were both about distance learning and about zoom, so I’ll list them here. Fundamentally, though, the best way to learn zoom is to actually just get a few friends and all try it out together (each of y’all practicing being the leaders/hosts of the meeting).

- If you can, start the zoom meeting 10-15 minutes early and let kids know you’ll be there. You can just have informal chats like you do before a normal class, and you can use that as an easy way to start building community.
- You can record your sessions, but if you do that, don’t start recording during that informal chat time. (Right now, since Zoom is overloaded, it’s taking them a long time to get the recorded sessions on their website, FYI. But you can have zoom do a “local recording” on your laptop… so I was thinking if the file were small enough, I could just upload that to a google drive folder my kids could access.) Note that the chat box doesn’t show up on the recording.
- Talk with kids explicitly about the weirdness of talking on Zoom. There are going to be awkward pauses because we can’t use facial cues and body movement to figure out if we’re going to talk or not (we’re all sort of trained to sort of check before we talk so we don’t start at the same time as someone else). So name that, and say that awkwardness is
*normal*in zoom. You should also mention (and give) lots of wait time — just like we should be doing in our regular teaching. - It’s okay if you’re having kids use chat to stop every so often and take a few minutes (in silence) to read over the chat so you can respond to what you’re seeing.
- The chat can be the “lightest lift” for interactivity, but it’s effective! One tip I got on twitter is that you can ask everyone to write a response to a question, but not press enter until you give the command. Then you’ll get a quick flood of responses that you can go through, and students can also read.
- You can also set the zoom meeting to have the chat be private – so students are talking to you but not each other… then as you see the responses, you can say “Nice thinking, Jake!” or “If you’re thinking about a parabola instead of an exponential function, you’re going in the wrong direction!” This came directly from Michael Pershan’s experience teaching online this past week:

- If you have pre-determined questions you want to ask at a particular time during the lesson, have them written in a google doc/notepad, so when you want to ask it, you can just copy/paste them in.
- Have everyone use their own regular names in zoom (and not emails or userhandles) to make life easier for you.
- There is a way to include “polls” in your zoom meetings, but I couldn’t figure that feature out when trying it out!
- You can divide your class into groups (either randomly or pre-determined) and send them to breakout rooms. You can visit any of those rooms and join in the conversations. Each breakout room is given a number when students join. You can have one person in each group (e.g. the person whose last name comes first alphabetically) to create a Google Doc in a Google Drive Folder you share with them in the chat window… And title it “Group 5, March 25, 2020.” Then all participants can write answers in their google doc and you have access to all of them in an organized way.
- When students are first put into a breakout room, if they’re new to working with each other, start with a non-mathy but quick ice-breaker to get everyone talking (e.g. what’s your favorite pizza topping?) and build a tiiiiny bit of community before diving in.
- SUPER COOL DISCOVERY: When I did this in Mike Flynn’s webinar, one person in my breakout room showed me a ridiculously cool feature. In any google doc, you can go to INSERT > IMAGE > CAMERA

And then you just take a picture of your work using the webcam, and it automatically inserts the picture in the google doc!

Bam!

- Don’t go crazy with the new technology. There are so many apps and websites. Limit yourself to just a few, like two, for your own sanity and your students’ sanity. Keep it simple and easy — don’t go down the rabbit hole of looking for “the perfect way to do x, y, or z.” Be okay with the tradeoff of having “good enough.”
- When designing online learning, start with the question “
*how*do we want our students to learn?” Then choose your technology based on that. - Screensharing is awesome (so you can set up a google slideshow, and in zoom you can screenshare that slideshow to the kiddos… And you can show kids how to
*annotate*so individuals or the whole class and write/type/draw on a screen you’re sharing (and you can save that).

Julie Reulbach led a webinar on using desmos for assessments, but basically she outlined all the ways we could create activity builders to actually *teach* content also, and bring students along with us as they navigate the pages, and we talk through what they’re doing. Her resource page is clipped below so you can see what’s there…

But importantly, her page includes links to various activity builders where you can simply copy and paste! Here’s how you copy and paste screens from better activities that your own into your own! They can even have computational layer in them!

Some key tips for creating Activity Builders (but not necessarily for assessments in particular):

- Steal steal steal screens from other activity builder assessments if you’re doing anything fancy (e.g. self checking, anything with computation layer), because there’s no need to reinvent the wheel right now. Julie has curated a whole list of activities that she takes screens from! And desmos has curated a bunch of starter screens(e.g. “graph how you’re feeling today?”) that you can take!
- DESMOS NOW ALLOWS FEEDBACK – so you can write a note to individual students.

https://www.youtube.com/watch?v=P-ebHOily6k&feature=emb_logo - Importantly, let’s say kids are doing a desmos activity or assessment, and you want them to show their work that they’ve done on paper. All you need to do is create a screen which has a blank graph, and instruct kids to insert an image (see sidebar instructions in the image below) and they can actually INSERT a picture of their work! The workflow is a little clunky because they have to take a picture on their phones and then email/airdrop it to their laptops, and
*then*select that file. But it took me only like 20 seconds after I had done it once.

Some key tips for using Desmos for assessments:

- Have kids log in with their name, but “last, first.” That way when you grade their online assessment, you can sort the responses by their username, and that will match your gradebook.
- Have a fun introductory “hi there!” screen
- Have students fill out an honor code/statement screen first if doing a formative assessment
- After you finish the activity, have two screens at the end. First, a feedback screen so you can find out how they felt it went. Second, a screen asking them if they have any questions or anything they want you to know.
- If a formal assessment, you should PAUSE the activity at the end — so kids can’t go back and change their answers or share the class code with other kids

My friend Michael Pershan has been in the thick of online teaching. He wrote a detailed blogpost about what he’s discovered thus far. I highly recommend reading it! Big takeaways:

- His school is using Google Classroom (like ours does), so he’s using that to create a system of organization for the kids, with instructions given day-by-day (within a unit):

He noted: “The most important thing, though, is that each learning activity becomes its own “assignment.” During week 1 I was creating large documents that students were working on over multiple days. This was good in one sense, because I had to post only one thing. But it became very difficult to monitor the progress of kids through the assignment at all. And then it became tricky to modify the plan in the middle of the week by adding on other bits of classwork.” - He’s using google classroom to teach kids how to upload their written work. (Note: my kids always submit PDFs of their work on google classroom, so they’re very familiar with this!)
- To give feedback on google classroom: “Google lets you comment on the work itself via highlighting and commenting, but I’ve found it more useful to give a quick written comment that appears under the assignment itself.”

Lots of great things being shared on twitter. It was so overwhelming that I stopped looking at twitter for a while, but I did save a few things:

What I have below doesn’t mean these aren’t good for others. It just means that for me, I like to jump in and these things didn’t quite pan out fully.

Alice Keeler had a webinar (“Oh Crap, I’m Teaching Math Online Now“) that wasn’t crazy useful for me because it was a brief overview of many things I already knew about. It was just super tech happy (look at Pear Deck! Look at Geogebra! Look at Desmos! Look at …) and didn’t give me the focus or vision I’m searching for.

Global Online Academy (GOA)’s 1 week course on *Designing for Online Learning*. Since this was designed to be “big picture” (so it can accommodate people from many schools and teachers of all stripes and many disciplines), I had trouble getting specifics that I wanted to latch onto. Here’s what I did get:

- They recommended Loom for laptop screen recording, if you were going to be making videos from your laptop. It seemed pretty seamless and easy to use, based on this short video tutorial:

https://www.youtube.com/watch?v=OvRVJ46ffoQ - The basics of good teaching are still important — clarity and being organized is paramount. Specifically for online learning, they highly recommend:
- building routines early (just like with regular teaching!)
- share the “learning goals” at the start of each lesson explicitly
- don’t get over-excited and share too much…
*curate*what you share and make it super easy to follow- using a lot of whitespace and images
- don’t include anything that isn’t super important — focus on key ideas
- not using too many fonts
- everything you share with your students should be “crisp” and “clean” (not “busy”)

- Be present for students. Create or adopt an online persona. Don’t leave them hanging, but show them continual engagement so they know you’re with them on this journey.

Okay, so now kids understand machines have inputs and outputs, and they understand that the “rule” can take different forms: words, equations, tables, and graphs. Wonderful.

**Machines to think about Functions vs. Non-Functions**

So recall that our definition of machines was:

So I had kids try to see what might go wrong with these machines…

From our conversations over these problems, students were able to see which machines were “problematic.” At this point, I told them machines that worked were called “functions” and machines that didn’t work were called “non-functions.” Conversations we had:

- We talked through what made something a function (every allowable input had a single output) and which made something a non-function (there was one allowable input that has multiple outputs).
- I had kids look at the graphs and come up with a quick way to “see” if a graph was a function or not… so from this, they came up with the vertical line test on their own.

We did a lot of practice with this idea. Kids were asked to look at a bunch of representations and decide if they were functions or not. And if they *weren’t* functions, they had to provide a concrete example showing where they failed (e.g. for (g) below, an input of 2 gives two ouptut of 2 and 4.)

By the end of this, I was very confident kids understood the idea of functions and non-functions.

**Combining Basic Machines**

Okay. Here’s where we start to get more abstract. I start telling kids that for now, we’re going to focus on four *basic* machines (machines with add, subtract, multiply, and divide)… but because I’m lazy and can’t make cartoon machines all the time, I’m going to come up with a simplified notation for them…

You, dear reader, might wonder why I’m using “blah” in these machines. That’s because it is helpful when we start *combining machines*:

Yes! It’s like a conveyor belt. Each machine takes in an input and spits out an output… that then becomes the input of the new machine… So they could start figuring out questions like these, which made me happy!

Now #17 was really tough for kids. But I let them struggle before guiding them. From this one problem, *students could start seeing how equations and machines were related*. By the end of our conversation, kids knew the line was . So if they have an input of , they multiply that input by 2, and then they add four to the result. Which means the machine would be:

____[blah*2]____[blah+4]_____

And so students started plugging in various values for the initial input (x value) and saw they got the final output (y value). *Then we substituted x into the machine… and got 2x for the middle blank, and then 2x+4 for the final blank! *Seeing that really helped kids drive home the connection.

Some kids got the second way to write these machines by trial and error. But I was hoping they’d rewrite . And then think if they have an input of , they first add two to it, and then they multiply that result by 2. Which means the machine would be:

____[blah+2]____[blah*2]_____

**Creating Machines from Equations and Vice Versa**

We then became comfortable going from a machine representation to an equation representation, and vice versa.

If I gave students: , they would say: we cube the input, multiply it by two, subtract 4. So the machine would be _____[blah^3]____[blah*2]____[blah-4]_____.

Or if I gave them: , they would say: we take the input and multiply it by -1, we add three, we take the square root of that, and then we multiply by -1 again. So the machine would be _____[blah*-1]____[blah+3]____[sqrt(blah)]____[blah*-1]_____.

And also in reverse, students could be given a machine, and easily come up with the equation by substituting in for the input, and come up with the output. So:

___[blah-3]___[blah^2]___[blah-4]___

would look like: , , , . So the equation represented by this machine is .

Basically, students are seeing how the basic equations are built up and broken down. What’s nice about this is order of operations starts to really get emphasized and naturalized.

**Creeping Up To Inverses**

At this point, kids are comfortable with combined machines. And so I throw them a backwards question, something they’re used to (since they’re my favorite type of question to give kids). First I start off concrete…

… where they were doing a lot of thinking about inverse operations. But then I had an activity where students were trying to create machines that would “undo” another machine. By the end of this activity, students were starting to create their own inverses. They could do problems like this:

I give you this machine which I will call machine : ___[blah-5]___[blah*7]____,

You need to tell me what machines I could append to the end of machine would make *any *input be the same as the final output.

So kids eventually saw the appended machine would be: ____[blah*1/7]___[blah+5]_____

So the big machine would be: ___[blah-5]___[blah*7]____[blah*1/7]___[blah+5]_____

And any input would also be the output (e.g. if we put in 1, we’d get 1 –> -4 –> -28 –> -4 –> 1.)

We called the machine created the *inverse machine*… and we named it machine .

So the inverse machine of ___[blah-5]___[blah*7]____ was ____[blah*1/7]___[blah+5]_____. Kids saw we “read” the original machine backwards, and did the inverse operations.

From this, we started going into inverses of lines:

Eventually, we got to the point where kids would be given . To find the inverse, they would create the machine for this: ___[blah^3]___[blah+4]___. And then they’d create the inverse machine: ____[blah-4]____[cuberoot(blah)]___. And then they’d find the equation that this machine represents by substituting in : .

So questions like these didn’t really faze them:

**Inverses with more representations…**

From here, I started having kids come up with inverses of tables. I reminded them that if we combine a machine with its inverse, whatever we input into the big machine should be the same output… So let’s see what happens…

And this was lovely… We’re combining two tables… and our goal is to create a large machine that when an input goes through both of machines, the output would be the same!

So look at the two tables above as rules. And we’re going to combine both tables to make a big machine. So kids saw from this that if we put an input of -3 into , we’d get 5 as an output… and when we feed 5 into , and we must have -3 as an output (since the output after going through both machines have to be the same as the initial input). So the inverse table is going to look like the original table, but with inputs and outputs reversed.

Why is this so beautiful? Because from this, kids saw that for inverses, the domain and range swap. And they also saw that to create an inverse, you simply have to switch the x-values and the y-values with each other. You get all of this for free!

And then I gave them graphs, and told them (with no instructions) to come up with the inverses… But since they had done tables, and see how the tables just swapped the inputs and outputs to get the inverse, they had no trouble drawing the inverse graphs.

And it’s lovely. Because they figure this out all naturally. I didn’t have to tell them anything but kids were accurately drawing inverse graphs. Putting the graphs right after them doing inverse tables was genius! And some kids came up with the fact that inverse graphs were reflections over the line themselves!

Of course, sometimes inverses exist but aren’t functions… So I threw everyone some curveballs…

And they saw how they could create the inverse… they could fill in the table or graph… But they saw why the inverse was “problematic” (a.k.a. not a function).

So now kids were thinking: okay, what’s the inverse? Is the inverse a function or not?

I drove this home with lots of questioning… We had previously looked at these questions and decided if these each were functions or not. But now kids were able to decide if their inverses were functions or not.

They immediately were looking to see if any outputs had multiple inputs associated with it. And they came up with the horizontal line test on their own. It was glorious.

**Going The Very Last Step with Inverses**

From all of this, kids learned so much. They saw how to graph inverses. They saw the inverse graph is a reflection over the line . And then we drove home the idea that the inverse graph is the same as the original graph, but with every x-coordinate swapped with every y-coordinate. To polish everything off, we saw the equation for the inverse graph can easily be found by swapping the x variable with the y variable.

So the inverse of was .

So finally, my kids could answer questions like:

Sorry this was so long and scattered. But stay tuned. My favorite thing is coming up… whenever I get a chance to write the next post!

]]>But in addition to these representations, I was inspired to include a **fifth representation**. It has a few drawbacks, but I can’t even express to you how many positive aspects it has going for it. It is the “**machine**.” I remember seeing images of these machines in middle school textbooks, and they really emphasize the idea of an input, output, and rule. Here’s one I randomly found online:

In this blogpost, I’m going to share how I introduced this representation, and how I subsumed the others in it. In future blogposts, I’ll share all the ways I’ve exploited this representation. It’s pretty magical, I have to say. So stay with me…

At the very start of the course, I introduced this machine representation also. Just not as fancy and cartooney.

I thought a lot about whether I wanted the machine to allow multiple inputs or allow multiple outputs. In my first iteration of drafting these materials, I did that, but then I backtracked. Things started to get pretty complex with an expanded definition for a machine, and I wanted to start the course simply. And, of course, I really wanted to emphasize the idea of a function and a non-function. So I started with the definition above. And started with things like this…

Notice these are “non-mathy” examples of machines. They eased kids into the idea, without throwing them into the deep end.

What was nice is that we got to understand and interrogate the idea of domain and range from this… where I described the domain as the “the bucket of all possible items that can be put into the machine *and* give you an output” and the range as “the bucket of all possible items that comes out of the machine.”

So for the sandwich one, we know the range is {yes, no}. And the domain might be {all foods} or {every physical thing in the universe}. We talked about the ambiguity and how for these non-math ones, there might be multiple sets of domains that make sense. But then for the math-y ones, we saw there was only one possible domain and range.

In fact, to really drive home the idea of inputs, outputs, domain, and range, I created an activity. I paired up the kids and one kid was the *machine*, and one kid was the *guesser*.

The *machine* got a card like this, with the rule:

The guesser got a card with the domain and range:

And the guesser would give words to the machine, and get a result. And their goal: figure out the rule. Then I would switch the machine and guesser, and give a new set of cards. It was crazy fun! I did it a long time ago, but I distinctly remember kids wanting to play longer than the time I had allotted. (If you want the cards I made, here’s a PDF I created Domain and Range Game.)

Next I showed how the “rule” in the machine could take a number of different forms — **tables, words, equations, and graphs** — and *this* is how I introduced the various representations. Kids were given these and asked to fill in the missing information…

… and then they were asked to find the domain and range for these same rules…

To drive home the various representations, I gave kids questions like these, where kids were given one representation and were asked to come up with the equivalent other representations.

So this was the gentle introduction my kids had to machines. I’ll explain where we went from there in future posts… and I promise you it’s going to be good… I’m really proud of it!

]]>**Kids have to come in knowing: **

(a) what a function is, what interval notation is

(b) what domain and range are conceptually, and how to write them in interval notation

(b) how to read/understand function notation

**Here’s what I did.**

Kids in each group got a giant whiteboard. In one color, they were asked to draw x- and y-axes and put tick marks so each axis went from -5 to 5.

Then they were asked to draw a function. The requirements: it had to be *complicated* and *interesting*. I made it into a small competition, with my subjective interpretation deciding which group won. They also had to be able to determine pretty clearly what the domain and range for their function were. They were told that their graphs would be given to other groups to stump them. So make ’em good!

Kids rose to the challenge. Here are three examples:

Click to view slideshow.Cool, right? I had each group write what the domain and range was for their functions on a post-it note on the back of the whiteboard.

Then I assigned each group a *different group’s* graph. Everyone in class took a crisp photograph of the graph they were assigned. And then class was over.

That night, kids did problems #1-#4 in this sheet I created. I’m pretty proud of this sheet! (Here it is in .docx form to download/edit.)

The next day, kids in groups compared their answers to #1-#4 with each other. They made revisions. They checked to see if their domain/range matched the post it on the back (the post-it the original group made when they created the graph). Then they worked collaboratively on #5 and #6.

When they were all done, I went around and checked their answers. (I had filled out an answer sheet for all the graphs so this part could be smooth.) I had discussions with groups about misconceptions they had. These conversations helped me see precisely where kids were getting tied up.

That’s where we are right now. A great finishing activity to function notation, domain, and range. I was so so so happy with the strong work kids were doing with such tricky functions! It was incredible! I even found a few mistakes in my own answer key!

At the start of our next class, I’m going to project a few questions like these to draw together our understandings and talk through some larger things that I realized I needed to highlight from my smaller conversations with groups:

Overall, this was relatively simple to execute. It broke up the monotony of class. And I love what I got out of it in terms of student thinking/analyzing.

**Some notes from doing this:
**

- I loved kids working on the whiteboards to create their functions, with the easy ability to erase and recreate parts of their graphs. And I’m glad the whiteboards are large. I only wish that the whiteboards had gridlines on them to make the graphs extra neat and easier to read.
- I wondered if, after a group themselves finishes drawing their graphs, they should be given the worksheet to fill out on their own work. (In addition to a new group.) Then the worksheets could be compared and discussions/debates could happen.
- That being said, I liked that the worksheets/questions were hidden from kids, so they felt like extensions beyond the domain/range.
- I thought a lot about how Desmos activity builder could probably be harnessed to make this happen… where kids create their own graphs to challenge classmates with… But even if kids don’t come up with their own graphs, a Desmos activity with well-created graphs could also be neat to have at my fingertips.
- It took kids about 20 minutes to draw the axes and come up with their graphs. And some took a little longer if they had a tough time identifying the domain/range for their post-it.

But I also had my classes engage in such a simple way. I just added this to their nightly work:

Those links, so you can click on them, are:

a) https://blogs.scientificamerican.com/roots-of-unity/q-a-with-autumn-kent/

and

b) https://anthonybonato.com/2017/06/19/on-being-a-gay-mathematician/

The way I facilitated this in my classes was very similar, and really informal. In one, I had people “popcorn” their thoughts about the article they read, share something that struck them, ask questions they had, make note of something they never considered until reading the article. In another, I had groups talk about what they read, and then we talked as a whole class, people sharing out what their groups said. (I think if I did this in the future, I might have everyone pull up the article on their laptops so they could refer to them for quotations, or to de-stress kids who don’t want to mischaracterize something they read.) I did almost no talking. I just kept silent, and let kids share in the whole class. Pro tip: sometimes I find it’s effective to a little silence go for 20 or 30 seconds, and then someone else will say something. And if I think things are coming to an end with a tremendous silence, I say “okay, we have time for one more thing.”

Then after they shared, I brought us together as a class. To talk a bit about allyship specifically, I read this paragraph from Anthony Bonato’s piece on being a gay mathematician aloud (this was something most students mentioned when talking about the pieces… it stood out to them):

I didn’t experience explicit discrimination until I was working on my doctorate. One of the professors in my Department told me to be careful about being open about my sexuality, as it would make professors and students uncomfortable. He thought he was doing me a favor, I think. I nodded politely and buried the incident away in my memory. Being gay often involves so many of these small defeats, these small let downs, that it becomes part of our everyday experience.

In two classes, I asked students “If you wanted to be a strong ally and they were in the professor’s shoes, what are actions you could take to support Anthony?”

In one class, we talked more about Autumn Kent (who is a mathematician who is trans). And I asked where in the piece she mentions how to be an ally, and we reviewed that together

Sometimes we need a shoulder, or an ear. Or just some normalcy.

The thing I think most people don’t see is the constant underlying dread, anxiety, stress, and anguish that a lot of us are carrying around. A lot of the time I am walking to and from my daily tasks, my inner voice hoarse from screaming. After the election I would be out and hear people making small talk about the sunshine and I’d want to tear out my hair. When I am doing bureaucratic tasks at work, I am carrying all of my anguish. When I am teaching and getting a laugh from my class I am carrying my anguish. When I am writing that email. When I am in the elevator or at the water fountain. When you ask how it’s going I am frozen. I am saturated with grief.

Listen to us.

In all my classes, I had to come out during this discussion. And I did it by explaining why I had them do this assignment… I talked about when I was in high school, there weren’t really any out kids, and definitely no out teachers (that I knew about), and no real representation of the queer community. And how I’m so happy times have changed, and hearing stories is such a big part of that. And so I wanted to share these stories, to show that being a mathematician *does* have something to do with sexuality and gender identity… because being a mathematician means being in a community with other people. I ended by sharing the quotation that stuck out the most to me, because I felt it a lot in my first years of teaching. It was from Anthony Bonato’s piece:

We edit ourselves by asking internally a series of questions. While lecturing does the audience think less of me if they know I am gay? When colleagues talk about their family over dinner is it OK for me to join in and talk about [my] husband too? Am I acting too queer in front of my students?

And I mentioned that lots of people in all sorts of marginalized communities do this kind of editing, because they *don’t* know what’s in the hearts and minds of people who are around them. Which is why being an ally can be so important — to *show* what’s in *your* heart and mind.

I remember when I first started teaching, saying I was gay was something I thought I didn’t need to do to students or other faculty members. I’m a math teacher. I was still getting my bearing and earning my stripes. Would it come in the way of me getting my stripes? Would students use it against me? Why would it ever matter in our math class? But I know it does matter. *Because when I was younger, there was no one.* No representation. It’s not that I didn’t think I could be a mathematician. It was worse. I thought that I was alone in the world, and I was wrong in the world. And so in this activity, it would have been disingenuous to not come out, if only to honor the transformation I’ve gone through from my youth to today.

Regardless, if you are a math teacher and are wondering if there is a way to push the needle forward in your classroom on LGBTQ+ issues, maybe try something simple like this. You don’t have to be queer or trans, it only takes 10 or 15 minutes of class time, and it sends a signal out there that you are an ally and care. About these stories, and implicitly to your students, about your students’ stories.

**Update: **It’s Friday evening now. It has been an impossibly long week, with late nights. And parent night on Thursday night keeping me at school until 9pm. I’ve been exhausted. But I want to archive one more moment from Friday.

It was after a lunch meeting. Our school has something called “CCEs” (continuing the conversation events) where student leaders lead discussions on important topics. Today’s CCE was around Ally Week and pronouns. I go to the room. It’s a large classroom for our school, but designed for maybe 20 students. By the time everyone got there, I’d estimate there were over 60 students/faculty in attendance. We watched two videos and then have an interesting discussion. I don’t want to share what was said, since confidentiality was one of our norms. I do want to share what happened to me. I was in the room, and I was overwhelmed by the attendance and the seriousness by which everyone was taking things. And while watching the videos about trans vocabulary and getting everyone on the same page, the video had a section on transitioning. And for some reason, with that word, I was flooded with emotion. My eyes were literally tearing, and I had to keep wiping them for the remaining 20 minutes. I was afraid if I spoke aloud to share my thoughts, I would start speaking and my voice would start warbling and then descend into sobs. So I didn’t speak. I don’t know what evoked this big emotion. In my mind were the multiple memoir books I’ve read about trans women. In my mind was the TV show *Pose* and documentary *Paris is Burning*. In my mind I was surrounded by kids who cared, and maybe a kid or two who identified or were in the process of identifying as trans. In my mind was all the hardships trans people face that we couldn’t even start to understand. In my mind was the many murders of trans people. In my mind was simultaneously hope and despair. And so I kept looking up at the ceiling, and wiping my eyes, hoping no one would see me. Because we were watching videos and everyone was sharing super interesting things.

I was happy when it was over and I could go wipe my eyes properly in the math office. I tissued my eyes dry, no one was around. Then a colleague/friend walked in and we started speaking about the CCE, and just a few words out of my mouth and what I feared would happen in the CCE happened. My voice warbled and the tears just started flowing. I don’t know why. I was exhausted. I was emotional. I couldn’t speak. I tried to explain in the 30 seconds we had, then I wiped my tears away, stuffed a few extra tissues into my pocket, and went to teach my 90 minute class. I walked into the door, took a few breaths, and said “Happy Friday Everyone,” before getting us started on combinatorics.

]]>I asked my students for three things of what *I owed to them*. I loved what they came up with. These kids are already so awesome. And these are all things I know I can work hard on giving them.

I see a lot of commonalities in what they wrote.

Understanding that math doesn’t always come easily to my students. Patience and kindness. Making mistakes and being lost and confused something that is okay and not something to be shamed. Recognizing that math is just one part of students lives. Promising to help students. Clarity. Fairness. Engagement. Encouragement.

I am getting these made into posters which I am going to hang up in the back of my classroom, where I can see them each day… reminders of what my responsibilities are, and seeing where I might be falling short.

]]>So of course, color me insanely jealous. (I think I first heard of this idea from Sara Van Der Werf on this blogpost.) The thing is… I *really* want kids to see math as something that exists outside of the math classroom. And anytime I see an opportunity to do that, I go for it. So things like math club and math team, yes, I’ve led those in the past. Independent studies/work with kids, yes. But I like the idea of opening up the umbrella of what *counts* as math. So a few years ago I helped organize a math-art gallery (with *real* mathematical artists!) at my school — with an exhibition called *Technically Beautiful. *Or organizing math-related book clubs with kids (from *Flatland*, to *Hidden Figures*, to *How Not To Be Wrong*, to whatever.) Or assigning my “explore math” project to some of my classes.

The appeal of the math play space was so strong that last year I decided I would make one for this year. The tricky part is that in my school, we don’t have our own classrooms. Last year, I taught in four different classrooms. But luckily outside of the math office, we used to have a long bench where only a few kids sat on when waiting for class or a meeting. So my plan: remove the bench and make a math play table/space right there.

My colleague and friend Danielle was interested in the idea, so we basically just did it. We asked maintenance to remove the bench. We set up three card tables. And we had the space ready for the first day of school. Ready to see what it looks like?!?

I’ll go through what exists in our space now.

When discussing the space, we agreed that it had to look cozy and inviting. So with our limited artistic skills, we put together this beautiful sign. We tried hard to come up with a better name, but we kept on converging on this simple one… so we went with it. We literally crumpled paper of different colors and tacked them up to write the word space. I’m actually in love with the way it looks. It was what we had around, and we got creative!

Now on the left side we have this:

This little cart was being thrown away by a third grade teacher, so we stole it! We put showerboard on it so it can act as a whiteboard, and if you look closely, we have some whiteboard markers below for students to us. On the board itself is a number game lifted totally wholesale from David Butler (his post about it is here). The idea is that with four small numbers (e.g. 1, 10, 10, 7) and two large numbers (e.g. 60, 120), students should attempt to make the target number 121.

After showing this to one of my precalculus classes, a student was obsessed with trying to get the target number using all six numbers, and came up quickly with a way to do it. He was super proud, and rightfully so!

Next we have two card tables covered with some fun cloth I found at home.

These are books that I brought in for kids to thumb through (though they just have to ask and they can take it home to read!). I have a zillion books that could go here… My criteria was nothing that could turn off a student easily. So a book of math poems, a childrens book about Sophie Germain, a math book based in funny comic strips, women in mathematics book, and a couple “math novels.” I even had a math department colleague/friend write a “recommendation note” that we stuck in *The Housekeeper and the Professor, *like this was a book store! (I asked our school librarians if they had the little book stands, and they were happy to give me some!)

Ikea had some $1 picture frames, so we used them to post some puzzles and jokes!

Click to view slideshow.We also put out some puzzles from Play With Your Math which we thought had a low barrier of entry but that kids might enjoy!

We also have a little estimation station (currently of jars with rice in them):

And of course we saw that Sarah Carter had provided us with a lot of math jokes that we could steal and use in our math space… So we have that up also! Because how could we not?!?!

Lastly, we have a “tinker table” where we have some tiling turtles, other tiles, and a weird set of puzzle pieces which need to get put back into a square shape.

And that… is about it!

At the start of putting things together, we realized we needed a bit of a formal vision for us to stick to… so we drafted this super quickly, but it was something we both felt was approximately right:

*Vision: To create an unstructured public space where kids can relax and fiddle/tinker around with fun math things that might not be related to things in the formal curriculum. The hope is that this allows for the experiencing of math as something casual and playful. We want this space to encourage students to want to talk mathematically with each other.
Through this space, which will be curated and changed periodically, we want to widen the umbrella of what gets counted as “math” and “doing math,” and who gets to be counted as a mathematician. *

We encouraged teachers at the start of the year to share information about the math space with their classes, even writing them a blurb they could read in their classroom but also encouraging them to leave their class five minutes early to bring kids over to just look around. What we wanted teachers to emphasize? “Most importantly, we don’t want you to be scared to sit down there. *We spent time making this space for you.* We want to say that again — *this space is for you**!* Pick up books and see what they’re about. Make designs with the tiles. Flip the joke page over to see what the groan-worthy punchline is. Try the number game puzzle out, or pick up the paper folding puzzle that we have there for you. Make an estimate for the estimation challenge. We want you to feel comfortable here — not treat it like a museum.”

Lastly, you might have noticed that in the vision we mentioned that the math space is designed to changed periodically. That’s the goal. Of course the jokes will change each week as will the numbers for the number challenge. But everything else — books, estimation, picture frames, tiles — will be swapped out. We have a giant list we’ve brainstormed of things that we could put in this space, and we’ll make decisions as we see if and how kids are using it. Some ideas include:

- Instructions for the game of SET, and space for kids to play the game!
- Wooden “put these together to form this neat shape” puzzles
- Legos
- A variety of math poems that students can take and put in their pockets during Poetry Week at our school
- A spirograph or two
- Math and Climate Change coloring books with lots of colored pencils (where we hang up the pages on the bulletin board after things get filled in)
- Towers of Hannoi
- Origami paper and instructions
- 3D printed mathematical objects, including cool math based optical illusions (like these!)
- A museum of WEB Du Bois stunning and eye-opening infographics involving race in America
- Geoboards
- Information on women mathematicians and mathematicians of color and mathematicians that are LGBTQ+ and…
- Fun little math problems (the size of a business card) that kids can pick up and bring with them

I actually have so many more ideas on my list, but it’s all written so informally no one would ever fully make sense of things. But these are just some. **But if you have ANY other ideas that you think would make sense here, I’d love to get a nice long public list for math play spaces — so throw any ideas down in the comments.**

With that, I’m out!

]]>

If you’d like to read my post and my competitor’s post and vote, I’d appreciate it:

The Big Internet Math-Off: The final – Sameer Shah vs Sophie Carr

It will only take a short time (no need to login or anything to vote, the only time it will really take is the reading).

My mathematical tidbit today attempts to have you look at these two squares, a 17×17 colorful square and a 127×127 greyscale square.

Both are… slightly uninteresting.

My goal, through the post, is to show you that both of these squares are *insanely *interesting. I call them the most beautiful 17×17 and 127×127 squares ever. And my conclusion: once you learn about the mathematics embedded in these squares, you’ll never look at them the same way again. You can’t.

It’s like having a huge a-ha moment when learning something. It completely transforms the way you look at something, so you can’t see it in its original form again.

I hope you enjoy!

If you want to see my five entries into the Big Internet Math Off 2019:

Entry 1: a counfounding 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 (the one outlined in this post): two beautiful squares: https://aperiodical.com/2019/07/the-big-internet-math-off-the-final-sameer-shah-vs-sophie-carr/

]]>I taught Advanced Geometry at my school for two years (2014-2016), and I wrote the curriculum with a good friend and dear colleague. We both hadn’t taught geometry before and decided we’d do a super deep dive and come up with a sequencing that made sense to us, and that prioritized conjecturing and noticing. In fact, we were so excited by this process that we shared our thinking both about how we built up the curriculum but also how we collaborated at a conference. Below are our slides, but you can also click here and go to the slideshow and read some of our presenter notes for each slide for more detail.

We were super intentional about everything. We carefully thought through how we wanted to motivate everything, and we didn’t want to give anything away throughout the course. In other words, we wanted kids to do all the heavy lifting and to be the mathematicians that we knew they could be.

Below is a word document with all our skills/topics (you can download the .docx file here: All Topic Lists Combined). The order might seem a little strange (we end, for example, the year with triangle congruence), but it worked for us! Everything was done on purpose (in this case, congruence is just a special case of similarity… so that came beforehand, along with trig which is all about exploiting similarity!). We eschewed two-column proofs for different forms (paragraph proofs, flowchart proofs, and anything else that showed logical reasoning).

Oh wait! For some reason our work on Area and Volume didn’t have a topic list. And I just looked and my core packet for Area and Volume derivations (where kids just figure things out on their own) has handdrawn images in it, but I didn’t scan a PDF of those. Well, at some point in the future if I remember, I’ll try to write a post to share that. (We did it after kids learned trigonometry, so they had a lot of flexibility. For example, I think kids came up with like 6 different methods to find the area of a trapezoid when they were asked to create a formula and justify it!)

I hope this is helpful for anyone trying to think through geometry. As I said before, the best thing might be to just read the blogposts, but this is a bit of an overview.

]]>One of my students this year told me I could put her senior speech on my blog. So here it is, for posterity.

**How I Have Changed the Way I Have Thought about Myself and the World**

When I was a lot younger, I was more hyperactive than I am now and I was always doing something to take up my time. I mainly always liked to keep myself out of my house because then I wouldn’t have to be in the middle of what was going on inside.

There was a lot of fighting that happened between my parents. They’d get physical with each other or throw things at each other. They never seemed to think of my brother and I being there.

I realize now that I was fearful of what was going on, that it put a lot of stress on me. When I was younger my thoughts were not so developed. When I would be with the rest of my family, my brother would be there to protect me. He could not prevent me from seeing or hearing certain things, but he would take my mind off of it the best he could. I don’t know what I figured the behavior between my parents meant or was- but if they were loving to me, I forgot about it and moved on. This was the way life went for me for many years. It was a contorted way of thinking.

At this time I also felt that the person I was defined by was my home life and my immediate family. I felt this way yet I didn’t think moments when my parents were violent towards each other or embarrassing my brother and I were moments that meant anything. I was always taught to let it pass and start new. That way of thinking ran me into trouble. I took what was happening in my home outside of it and I would have unkind thoughts/ engage in unkind behavior towards people I loved and cared for. Afterwards, whether I identified if it was mean or not, I would move on. I didn’t think it said anything about who I was.

Then in seventh grade I was in Ms. Oster’s health class and she showed us this speech by an author named George Saunders. It’s titled Advice to Graduates, and it is centered around the questions why aren’t we kinder? and how do we become more kind? It touched on the idea that becoming kinder comes with age but there are ways you can speed up the process. I didn’t feel like It wasn’t as if I hadn’t heard these words before, or as though this was a new concept to me but they processed differently this time and I began to question who I was.

I realized that I had always thought I was a really nice person but if I wasn’t genuinely acting kind, meaning my actions weren’t speaking for themselves, how could I be a nice person? This caused me to think back to things I had done and ask myself a series of questions like: Did I do the right thing? What did I feel that made me act the way I did? What caused that feeling to arise? Why could that feeling have been brought up?

Seeing that my actions didn’t reflect who I knew I was, I continued in life being very analytical of instinctual tendencies, thoughts, and behaviors. This way of questioning took over and continues to this day. Questioning myself allowed for me to realize the roots of my actions so I can allow myself to change. If I can identify the root, then maybe I can change the way I do things.

And so, through this process of questioning, I came across the concept of time and how I spend my time. I thought to myself what I spend my time doing makes me who I am. I thought, what had I been spending my time doing? School came to mind. I go to school most days out of the year and I am in school most hours out of each of those days. Again, it wasn’t that I didn’t know these facts before this moment but I had never paid attention to the concept. When I began to pay attention I realized that school was a place that nurtured me in all positive ways and I was taking it for granted. I could look to the building as a safe place, the teachers as people who could listen and give me advice and, my peers as those who stand equal to me in our journey to understand and live with ourselves. I am very grateful that I am able to attend school, have teachers who care for their students, students who engage deeply in class conversation, and a place that I can actually call my home.

At this point in my life I feel I have come a long way. Since 7th grade I have realized that a lot of what I was seeing when I was younger was my mom struggling with her mental health. In 8th grade child services took me from living at my mom’s house. They had been in and out of my life until this past summer when I permanently started living with my dad. It has been difficult accepting that my mom was not able to guide me through big developmental parts of my life but I love my mother and I am so thankful she is the woman I call my mom. Now here’s a George Saunders quote that is important to me.

“If we’re going to become kinder, that process has to include taking ourselves seriously — as doers, as accomplishers, as dreamers. We *have* to do that, to be our best selves.”

Now also, Over these years of high school I have come to understand that change is inevitable and a big part of our job is to adapt, learn, and accept it. Through that process I am continuously discovering my best self.

]]>I know the irony of being scared of public speaking and being a teacher, but I know a lot of other teachers also feel this. But my biggest fear was just not being good enough. I wrote to Mary:

*The truth is I can’t help feeling like this might be too big a leap for me. I don’t know if I could do what Fawn or Julie to me when I hear them talk, or do what Chris Shore or John Stevens do when they present. And I don’t want to commit unless I knew I wouldn’t be wasting anyone’s time. *

She replied:

*Firstly, and most importantly, you would not be wasting anyone’s time.*

*I am a big proponent of elevating classroom teachers and giving them a voice and I hope that you will find yours for this event. *

As someone who feels like an evangelist of the online math teacher community, I’m always saying to people hesitant to dip their toes into the water that their voices and perspectives are important and valuable. And when I say this, I mean it with every fiber of my being. So why was I doubting the value of my own voice? I agreed to do the talk as long as I could do it with a collaborator and friend. It was a 75 minute talk (and an associated 75 minute workshop) and planning that individually seemed so not fun. But I thought working collaboratively would be so much more intellectually fun! So I dragged my friend Mattie Baker (@stoodle) into the presentation. We brainstormed for ages, but in the end, finally decided that the idea of “The Teacher Voice” was exactly what we needed to talk about.

It’s now over, so I’ll start with the ending. The lecture hall of 200 people for our talk was almost full. And the 75 minute talk went fabulously. I had to save some tweets for posterity.

And OMG, we got a standing ovation. That was unexpected. And people were crying. That too was unexpected. I am not someone who feels proud about things easily. I usually focus on all that went wrong or ways I could have been better. But when we took a bow at the end and people stood up, *my heart was bursting*. All the work Mattie and I put into the talk for the previous 10 months, the weekends we sacrificed to write and practice and edit felt purposeful because at least for some teachers in the audience, our message was at least temporarily valuable. Weirdly my fear of public speaking disappeared the day of the talk after we had a solid rehearsal the previous day, and my fear of wasting peoples’ time disappeared after the talk ended and people came up to say such nice things to us.

The talk was broken into two parts. First, Mattie and I shared something we each did in our classrooms that was inspired by other teachers, and then adopted by other teachers. We wanted the audience to have something concrete to walk away with in case the rest of our talk didn’t resonate with them. We were breaking down the silos of our classrooms. Second, we each talked about the emotional life of a teacher. We wanted to break down the silos of our emotional worlds. There were so many messages we included in this part of the talk. Mattie shared his first year in teaching, which he previously shared on the Story Collider podcast. But here is one takeaway from my section of the talk:

Teaching is hard. We are going to *feel* bad. We’re going to *be* bad. And that’s okay. It’s okay to not love what you’re doing all the time. I’ve never met a teacher who is putting themselves out there in the important but hard ways who does. But we can be brought closer as we become vulnerable and share these things and realize we aren’t all alone in this.

And a second related takeaway:

Often times, we’re so critical about ourselves, we think of all that isn’t going right, all that we aren’t doing… that we lose sight of* all that we are*…. It’s so easy to be critical of yourself, to set the bar high, to see all the ways you’re not succeeding.

You see yourself in one way. But the reality of the situation is: ** We aren’t really all that good at seeing ourselves.** That’s my big realization, and it only took twelve years. When we’re down and think we suck, yeah, we probably definitely maybe can be doing better. But hell if we aren’t already doing good, and we need to acknowledge that and spread it. We need to believe our friends when they tell us that our ideas our good, that something we did was good… we need to believe our kids when they say something spontaneous and positive about something happening in the classroom… and… we need to be sharing the good and the positive that we see in others. We need to help others see how important they are to you. We need to give cupcakes, send the random email, prop each other up, and help others see how they make your life better.

The talk focused on the hard times in teaching, and what we do when we hit them. At one point I asked the audience to share their coping strategies at the low points. I promised I would share them online, so here are what the audience typed. It’s amazing how similar the responses are…

(All the references to “coffee” in quotations comes from part of Mattie’s talk. You can interpret that to be getting a drink at happy hour.)

I shared my coping strategies afterward, and so many of them were covered by what people in the audience typed! Except for those people who talked about exercise and running and the gym and other evil things like that. Some of my favorites!

In the talk, I also shared the* Explore Math* project that I do with students. The website that I created for the project is here: https://explore-math.weebly.com/

I posted about it early on when I first started it, but haven’t done any additional posts on how I’ve changed it or how it’s evolved or what I’ve noticed when doing it with different grades (sorry, I should). The posts are here, here, and here. The most important thing I can suggest is that you need to adapt it to work for your kids and your school. For example, this year I tried this in 10th grade and it wasn’t as successful overall because I think the kids needed more structure and hand-holding. So I’m going to take that into account for next year.

Two teachers shared their experiences with the project, which I couldn’t fit into the talk. So I’m posting them here in case it entices you to do the project or some variation in your classroom.

At the conference, Mattie and I also gave a 75 minute workshop on the online math teacher community designed for people who were interested in joining in but didn’t know how (our slides for that are here).

]]>

- The core part of what I did to get the number to pop up was to use @lukeselfwalker’s Desmos activity. I like it for so many reasons, but I’ll list a few here. It starts by “building up” a more and more complicated polynomial of the form , but in a super concrete way so kids can see the polynomial for different
*n*-values. It shows why the x-intercept travels more and more left as you increase*n*, so when you finally (in the class discussion) talk about what happens when*n*goes to infinity, you can have kids understand this is how to “build” a horizontal asymptote. It gets kid saying trying to articulate sentences like “this number is increasing, but slower and slower” (when talking about the value of the polynomial when . And they see how this polynomial gets to look more and more like an exponential function as you increase the value of*n*. If you want to introduce*e*, this is one fantastic way to do it. - A few days later, I had everyone put their stuff down and take only a calculator with them. They paired up. (If someone didn’t have a pair, it would be fine… they just sit out the first round.) On the count of three, both people say a number between 0 and 5. (I reinforce the number doesn’t have to be an integer, so it can be 4.5 or something.)Then using their calculators, they calculate their score: they take their number and raise it to their competitor’s number. The winner has the higher number. (If it’s a tie, they go again until there is a winner.)
Then the loser is done. They “tag” along with the winner and cheer them on as they find another winner to play. This goes on. By the end, you have the class divided into two groups each cheering on one person. (I learned this game this year as an ice breaker for a large group… it’s awesome. This is the best youtube video I could find showing it.)

Finally there is a class winner.

So I then went up against them.

And when we both said our numbers, I said:

*e*.The class groans, realizing it was all a trick and I was going to win. We did the calculations. I obviously won.

We sit down and I show them on my laptop how this works:

The red graph is my score, for any student number chosen ().

The blue graph is the student score, for any student number chosen ().Clearly I will always win, except for if my opponent picks

*e*.I tell kids they can win money off of their parents by playing this game for quarters, losing a few times, and then doing a triple or nothing contest where they then play 2.718. WINNER WINNER CHICKEN DINNER!

- After this, I show kids these additionally cool things (from the blogpost), saying I just learned them and don’t know why they work (yet), but that’s what makes them so intriguing to me! And more importantly, they all seem to have nothing to do with one another, but
*e*pops up in all of them!I re-emphasize

*e*is a number like and I showed them this to explain that it pops up in all these places in math that seem to have nothing to do with that polynomial we saw. And that even though we don’t have time to explore*e*in depth, that I wanted them to get a glimpse of why it was important enough to have a mathematical constant for it, and why their calculators have built in*e*and*ln.*

That is all. I honestly really just wrote this just because I was excited by the “game” I made out of one of the properties of *e *and wanted to archive it so I would remember it. (And in case someone out there in the blogoversesphere might want to try it.)

**UPDATE: **Coconspirator in math teaching at my school, Tom James (blogs here) created the checkerboard experiment using some code. You can access the code/alter the code here. The darker the square, the more times the number for the square has been called by the random number generator. And with some updates, you can make more squares! In the future, we can give this to kids and have them figure out an approximation for *e*.

[1] And introducing it with compound interest means you have to assume 100% interest compounded continuously. Where are you going to get 100% interest?!?!

]]>***

Desmos writes interesting job descriptions when they have openings. When someone pointed that out to them, they mentioned that this article on reducing unconscious bias helped informed how they write their job descriptions. It’s pretty great and I highly recommend it if you’re hiring. I have thought a lot about “fit” in the past few years when doing hiring, but it’s tricky to think about it well. I have come to recognize that someone entering our department needs to be open and willing to collaborate and compromise, but also have sympathetic pedagogical beliefs with what our department values (and can’t compromise on those). One way I have tried to avoid it is thinking about these things:

But also I have found it harder to balance these thoughts, which I admittedly have a lot:

Not quite those things, but similar thoughts that get at my own personal views on the what persona/personality traits make an effective teacher. Which I tend to think mirror my own traits. But that’s only because I have these traits because I think they make an effective teacher. But I have worked with enough amazing teachers to know that amazing teachers come in all personas! Just like amazing students don’t all have to have the same personas. But this type of bias is something I am trying to be super cognizant about when on hiring committees.

***

I saved this just because I like the question and wanted to work on it. And I can see all kinds of extensions. A formula for *n* circles? What about spheres? I’m guessing (without working on this problem yet) that this is a classic “low entry point, high ceiling” type problem.

***

I just really liked this quotation, and I need to think about the ways that students can see themselves in the mathematics they do. It is part of a larger thing I want to do which is “humanize math” — but I’m not very good at making it a core part of what I do in the classroom. Small bits here and there humanize and expand what kids think about math, but I’m not there yet. I want to one year leave the classroom and know that kids have looked in the mirror and saw something. (It kind of reminds me in a super literal way of how Elissa Miller put a mirror in her classroom, and I think on the bottom she wrote “mathematician.”)

***

Okay, I love this so much. If you’ve never seen it before, it a great trick. You have someone pick any number between 1 and 63 secretly. They just point to the cards that number is on. In about three seconds, I can tell you your number.

I actually made a set of these cards where the numbers are more jumbled up, so kids don’t see a pattern to it. I do put the powers of 2 in one of the four corners though to make things easier for me. Oh wait, have I said too much?

If you don’t know this trick, or how or why it works, I’m sure you can google it. But I’m going to recommend the awesome book “Math Girls Talk About Integers” (there are a lot of great “Math Girls” books out there, so make sure you get the Integer one.

Not only is the book awesome (and great for kids to read), but it breaks down this trick so well. *Shivers with joy*

***

I was excited with Karen Uhlenbeck won this year’s Abel Prize, the first woman to win it ever! I had my kids read this article in the NYTimes about it, and write down three notes about the article. We started the next class with a “popcorn sharing” of what people wrote down. (I also said that although I liked the article, it was a bit dense and thought it could have been written more lucidly.) One thing that came up in both classes I did this in was what a “minimal surface” was — so I told kids it is a surface with minimal area.

I then showed my kids this short youtube video:

And explained that bubbles, though not “central” to all higher level mathematics, do come up. And then I gave them a question. I’m too lazy to type it out, but watch the first 1 minute and 45 seconds of this video (https://www.youtube.com/watch?v=dAyDi1aa40E) and you’ll see it. Then we talked about some basic solutions. And THEN I revealed the best answer was the answer shown in the video we all watched together.

Of course @toddf9 (Todd Feitelson) used this as inspiration to create his own bubble thingies:

but he also explained how he made them…

and then he EVEN created an awesome desmos activity on this very problem, which I want to archive here for use later: https://teacher.desmos.com/activitybuilder/custom/5cb50bed4dcd045435210d29

(Oh! And Mike Lawler (@mikeandallie) made a mobius strip bubble!)

***

Dylan Kane wrote a nice blogpost about calling on students (and the “popsicle sticks of destiny” — though he doesn’t call them that). My favorite line is this simple question that isn’t about right or wrong:

- After students attempt a problem in groups, or reflect on an idea and share with partners, I call on students asking, “How did your group approach the problem?” or “What is something useful that you or your partner shared?”

It’s so obvious, but even after so many years of teaching, I forget to ask things like this. Or my curriculum isn’t group problem solving based enough for things like this to make sense asking. Or whatever.

***

There’s nothing special about this one… I’ve read it a few places before and it always makes me laugh.

Questions are good. I might have a kid read this at the start of the year and then have a short conversation about why we’re reading it.

It will get at the problematic idea of “obvious,” and when and how learning happens and more importantly when and how learning doesn’t happen.

***

In case you didn’t know, Desmos has a list of all their mathematicians they use when they anonymize in Activity Builder.

https://docs.google.com/document/d/1OY-8dk6vYW1Cags8E6_v3I8YZ-RYROzgsCauW5CZt9w/edit

***

I can imagine putting this picture on a geometry test as a bonus question and asking them why it makes math teachers all angsty… Plus it made me chuckle!

***

I’m so not here yet. Anyone who knows me as a teacher will probably know I’ll probably never get here. I’m such a stickler for making the use of every second of classtime.

***

Crystal Lancour (@lancour28) tweeted out a slide from a session led by Robert Berry (NCTM president) which had this very powerful slide:

Four rights of the learner in the mathematics classroom

- The right to be confused and to share their confusions with each other and the teacher
- The right to claim a mistake
- The right to speak, listen, and be heard
- The right to write, do, and represent only what makes sense to you

***

Love the idea of using marbles/paint to draw parabolas (click here to go to the original tweet and watch the video — it’s not a static picture).

***

Bree Pickford-Murray (@btwnthenumbers) gave a talk at NCTM about a team-taught math and humanities course called “Math and Democracy.” Not only did she share her slides (like *right after* the talk) but also she links to her entire curriculum in a google folder. SUPERSTAR!!!

I’ve gone to a few talks about math and gerrymandering (both at MoMATH and NYU) and listened to a number of supreme court oral arguments on these cases. It’s fascinating!

***

I just finished teaching “shape of a graph” in calculus. But I wish I had developed some activities like this, to make it interactive:

***

I’ve literally been preparing to give a talk next month for… months now. And this one stupid tweet summarized the talk. Thanks.

***

I have so many more things I can post, but I’m now tired. So this will be the end.

]]>It’s basically a number line, that’s all. But it’s a nice public giant number line which can get kids talking. Today I came back from spring break and before break, students learned about logarithms. However I wanted to have them recall what precisely logarithms were… so I created a quick Clothesline Math activity.

I hung a string in the classroom. I highlighted it in yellow because you can’t really see it in the photo…

I then showed them this slide – explaining the string is a number line…

I then showed them this slide, which explains what they have to do if they get two of the same number. (I brought cute little clothespins, but mini binder clips or paperclips would have worked just as well):

And then I gave them the rules of play:

I handed out the cards and let kids go. It was nice to see they didn’t get tripped up as a class on too many of them, but I got to listen to debates over a few trickier ones, which we collectively resolved at the end.

Here are the cards I handed out: .DOC FORM: 2019-04-01 Clothesline Math – Logarithms

Here is a picture of some of the cards. The two on the left are average level of difficulty. The two in the middle caused my kids to pause… it took them time to think things through (they haven’t learned any log properties yet). The one on the right doesn’t belong on the number one (it is undefined) and the kid who got that card immediately knew that. Huzzah!

Here’s a picture of the numberline at the end.

And… that’s it!

I was excited to try it out as a quick review activity. And it worked perfectly for that!

(Other things of note: Mary Bourassa made a clothesline math for log properties and shares that here. The author of Give Me A Sine blog does something similar here, but has *kids* create the cards. I couldn’t find anything with basic log expressions — so I made ’em and am sharing them in this post. Chris Hunter has a nice tarsia puzzle that sticks with basic log expressions here, but I wanted to try out clothesline math so I didn’t use that!) But if anyone has others out there involving logs, I’d love to see them in the comments!)

Last week I met up with my co-teacher in Algebra 2. We’re working on our unit of exponential functions and logarithms, and we were talking about spending a short amount of time introducing “e” to our kids. Personally, this question has haunted me because when I taught Algebra 2 at the start of my career, I couldn’t ever find a motivation for it — except for interest being compounded continuously. That never quite sat well with me because you have to assume that you have an interest rate of 100%.

Like as gets larger and larger.

But where are you getting an interest rate of 100%?!? It isn’t a terrible way to introduce “e” (getting kids to understand the structure of that equation above, there’s a lot of deep thinking that goes on in there). There is also the idea that there is a limiting value for that expression above — instead of the value just going up infinitely — that can be exploited and discussed.

But I never thought “e” and “ln” really belonged in Algebra 2, precisely because I couldn’t motivate them in a way that was intellectually satisfactory. So I tweeted out:

Little did I know I was going to get so many responses! I wanted to archive them here, which is why I’m writing this post, but then share where I’m landing on this whole “e” thing right now.

**Ways To Introduce “e”**

- @retaneri linked to this question from “Play With Your Math” I
*love*that site, but hadn’t worked on this problem before.

Interestingly, @bowmanimal and @averypickford came up with the same problem to share with me! Apparently the answer is to break up the number into a bunch of*e*s … I haven’t figured out why this works yet. But for 25, if I understand this correctly, I think it means which has a product of . (Or to be super precise: .)

And @benjamindickman shared some articles about this: - @mikeandallie shared with me this approach by throwing darts. Which includes this gem:

and this instantiation:

- @jensilvermath suggested just looking at for larger and larger , without reference to a limit. Have kids make predictions about what is going to happen and why, and then let them explore it. At first, I was like “hmmm, would this work?” but I love the idea of kids stumbling upon and wrestling with (1+almost zero number)^(super huge number) might be tricky. Does it have to be a huge result? What data could they collect? What would they do with it?
And @LukeSelfwalker shared this simple but stunning Desmos activity which gets kids to see how polynomials can start approximating exponential functions — a beautiful visual connection to all of this.

- Of course @dandersod showed me a connection between Pascal’s Triangle and e, which I didn’t know about (or if I did, I totally forgot). He sent me this link:

Whaaaa?! But okay! WHOA?! - @roughlynormal suggests:

I have to think about this… Basically the differences relate to the derivative… But I did a quick 5 minute look with a google spreadsheet and I couldn’t make this work. I think for it to work, you need to divide the first differences by the change in x, and also divide the second differences by the change in x, and look for them to be equal. In other words, the derivative… - @jdyer gave this gem which I’ve never heard of before:

“You have a full glass with 1 liter of water. You take gulps from the glass; each gulp is a random real number of liters from 0 to 1. On average, how many gulps do you expect it to take before the glass is empty?”And his discrete version is you start with 100 and kids take away a random number (generated from 0 to 100) per step.

- @bowmanimal also talked about this “game”:

Whoa! TOTALLY new to me. Since we can’t put up e fingers, we could play the game where we each pick a number (doesn’t have to be an integer) from 0 to 5. I had to see this for myself!

The red line is always greater than the blue line (I think… I want to prove it algebraically!). [Note: related, this set of tweets on which @BenjaminASmith alerted me to]. And @mathillustrated shared this amazing presentation (read it!) on scaffolding and formalizing this game with kids. - @CmonMattTHINK shared this fact which I LUUURVE but forgot about:

The probability of a random permutation of n objects being a derangement (no object remaining in its original position) approaches 1/e as n->oo.And @DavidKButlerUoA shared a wonderful presentation he made on where the derangement formula comes from. - @bobloch also shared with me a fact I don’t remember ever learning!
*e*shows up in the harmonic series.Apparently, if we look at which*n*values bring this series to an additional integer, we get:And if you go further and further, and take ratios of these

*n*values, you get a better and better approximation for*e.*I calculated 227/83 and got about 2.735. - @mathgeek76 reminded me of the chapter in Steve Strogatz’s excellent book
*The Joy of X*which talks about how you find the right partner, mathematically.

The answer involves . (Spoiler here.)

And then a wonderful conversation about where “e” belongs in a high school curriculum popped up on my feed, launched off by this tweet:

I should say this was Steve’s launching tweet, but he was open to thoughts on both sides of the discussion that ensued. My favorite part of the conversation was when someone (I can’t remember who) brought up and when that should be introduced and why.

**My conclusions**

First and foremost, I knew there was a lot to — it bridges continuous and discontinuous phenomena. It is important in calculus and (to some degree) combinatorics. There’s a lot I don’t fully understand yet about what people shared with me in their tweets — and I have to work through them to see their connections. I do see lots of connections but I haven’t worked through the math of any of these things to draw them in fully.

Second, in my musings on this, I think I’ve come to recognize that why I have always found so fascinating is that is keeps on popping up in unexpected places. In my mathematical career, my jaw has hit the floor a number of times when I see it suddenly emerge when I never thought there would be a connection. I mean, think of the first time you saw . ‘Nuff said.

So if I want my kids to see what I see when I encounter , that’s what I want them to encounter. A surprise. That it can pop up in totally unexpected places, and you might not initially know why, but it eventually can become clear.

I don’t think I have time to pull this together for this year, but here’s what I’ve decided I want to do in some future year… I want to have a period of time where kids are told “this is a problem solving day” (or set of days). And I give the class different problems that will result in them approximating … but they don’t know it. Like Pascal’s triangle… they can just do the calculations and see pop out. Or the chess board experiment. Or taking random numbers away from 100. Or the harmonic series. Or the product challenge. Or the vs game. Or the compound interest problem. And have them work on them. They all seem unrelated. Yes, this is contrived. Yes, I’m telling them what to do in many of them.

But BOOM. Soon this number that keeps popping up in all these unrelated problems.

For each of these, they are going to get numbers close to . And for me, I can say

*This, *my friends, this is what is so beautiful about this number for me. It is a universal constant. It is like . It pops up in so many unexpected places. There is an underlying structure to why this is all happening, why it pops up everywhere. That is going to start to be revealed in calculus, but that’s only the bare beginnings… It goes much deeper. But I wanted you to get the experience of wonderment and have something that’s you know is true that *begs *the question *why… WHY? WHAAAA? WHYYYYYYY! *Because this desire to know, to figure out why something is true when you know it must be and it feels too unbelievable to be true, that’s a feeling mathematicians get that drive them forward in their work. And making those connections, and we know they must exist, it’s awesome when it happens. So yeah… I wanted to introduce you to this important number which we’ll just take as that… an important number that we’ll get to play around with like we do … but know it’s more than just a number. For you, now, it’s a question that’s begging an answer.

**A realistic ending to an idealist exhortation from me to my students**

Okay, I don’t know if I could really pull this off. But I’d love to.

]]>