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In the past few years, my school started a “senior speech” program. Five seniors get up and talk to all students and faculty in grades 9-12 about their lives. Coming out as gay. Mental health challenges they face. Racism they face. Code-switching. Being a conservative in a liberal environment. Bullying. Friendship. They usually make me tear up. (Stupid onions.) After each one, I think to myself: “Wow, if I were a ninth grader hearing these seniors be raw and honest and vulnerable, it would help me on a path to become brave and speak my truth.” I also think: “Why don’t we collect these all and publish them somewhere?” I don’t want them lost.

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.

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In August of 2018, I got a message from my friend Mary Bourassa who asked me if I was interested in being a featured speaker at OAME (the annual math teachers conference in Ottawa). I was absolutely going to say no, because I’m terrified of public speaking and I wasn’t sure I had anything of value to say to other teachers…

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).

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I’m teaching Algebra 2 this year and the other teacher and I decided that we should introduce e to our kids. The reason it’s challenging is that it’s hard to motivate in any real way. You can do compound interest, but that doesn’t do much for you in terms of highlighting how important the number is. [1] I asked on Twitter for some help, and I got a ton of amazing responses (read them all here). My mind was blown. This year, though, I didn’t have time to execute my plan that I outlined at the bottom of that post. So here’s what I did:

  1. 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.
  2. 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!

  3. 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.)

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

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I have seen a lot of great stuff on twitter lately, and I’ve missed a lot too, I’m sure. I wanted to just archive some of the things that I’ve saved so they don’t disappear! I also think it might be a benefit for someone who reads this who isn’t on twitter or missed some of these tweets. But that’s just a side benefit. I’m writing this for me!!!

***

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:

A Thing of Beauty: Soap Bubbles - YouTube

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

  1. The right to be confused and to share their confusions with each other..
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I remember when I first heard about Clothesline Math, I was excited by all the possibilities. And in a few conference sessions with Chris Shore, I saw there was so much more than I had even imagined that one could do with it!

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!)

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The Launch

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”

  1. @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 es … 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:
  2. @mikeandallie shared with me this approach by throwing darts. Which includes this gem:

    and this instantiation:
  3. @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.

  4. 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?!
  5. @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…
  6. @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.

  7. @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.
  8. @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.
  9. @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.

  10. @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..

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I’m about to start a unit on logarithms. Kids don’t technically know that yet. To prime them, today I gave both my Algebra 2 classes a warm up. I was super nervous about this, because I haven’t seen a crazy amount of endurance from many of my kids when they get stuck on something. And I was going to give them something totally open-ended! And without a calculator allowed! Gasp!

I asked them to do the following. Think about . It’s going to be a long number when it is all written out. I wanted them to come up with a guesstimate about how many digits there are in the expansion. To scaffold, I asked them for three things:

a) What’s a guess (for the number of digits) that is too low? How do you know? (Can you come up with a larger low estimate?)

b) What’s a guess (for the number of digits) that is too high? How do you know? (Can you come up with a smaller higher estimate?)

c) Based on your work and your intuition, if you had to make a guess, how many digits are in the expansion of $2^{60}$?

Honestly, it was one of the best things I’ve done recently. Kids were showing grit and so much flexibility in their thinking! I had to correct a few misconceptions and nudge a little here and there, but it was all on them how they wanted to go about this. It was beautiful. (At one point, a kid said they wanted to give up, but I came back around a few minutes later and they were rapidly making progress and hadn’t given up.)

At first, kids didn’t know where to start. I told them they were going to get time to work on this, so they could take on strategies that might take a while. (Normally, we start class with something short and quick. I wanted to indicate this wasn’t that.) Initially, I gave 7 minutes, but since so many kids were on a roll, I expanded it to 14 or 20 minutes. I honestly don’t remember how long.

What I adored is that this problem was definitely in their wheelhouse. Most groups were gung ho, and just started writing stuff down — and eventually (sometimes with a little encouragement/prompting from me), they came up with SUPER awesome solutions. Seriously, things I had never thought of.

The main two approaches I saw were:

  1. Kids noticing that . Which is close to . So . So that puts us at around 19 digits.
  2. Kids noticing this pattern:

    So after going up about every 3 exponents, we add an additional digit to the number. (I say about 3 because all groups who did this method saw that a few times, you’d get 4 exponents in a row which keep the same number of digits instead of 3. But it was usually 3.)

    Assuming the number of digits increases after going up every 3 exponents, that means that exponent 12 has 4 digits, exponent 15 has 5 digits, exponent 18 has 6 digits, exponent 21 has 7 digits… etc. So exponent 60 has 20 digits.

    So that puts us around 20 digits (or maybe a little lower because of those occasional 4 exponents in a row).

That’s about all I wanted to share. I was a little out of my comfort zone because I didn’t know if they would all just throw their hands up and give up. But they didn’t, and instead did some phenomenal thinking.

I just realized… you might want to see how this relates to logarithms. It turns out that the number of digits is equal to doing the following: take the log of the number, and then take the floor function of that result, and then you add one. I won’t spoil it by explaining why, though. See if you can figure it out!

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I find myself always pressed for time in class. Doing inquiry-based work, and going at the pace of the kids’ understanding, means that inevitably there are stretches of time when I’m feeling like “ARGH! We need to go faster! We have to be further along to cover the content!” Every teacher I know feels this, and in my office we talk about this, so that makes me feel better. I’m not alone. I also know in the back of my mind that somehow, each year, we manage to get things done. That helps.

That being said, this past week I had one of the very few times that one class was about half a day in front of the other and I didn’t feel the need to forge forward as acutely as I usually do. And in that short exciting window, I saw a student’s shirt which I thought was mathematically beautiful…

… which of course I told to the student. “Math?” He didn’t see it, nor did others in the class. It was an 90 minute class, and when having kids work on some problems together, my mind started thinking… I want to show kids how I see this shirt. The glasses I use to see the world are different from theirs. We take a break in the middle of class for kids to get water or quickly grab a snack, and so 5 minutes before the break I stop everyone. I have them look up. And I say something like: “I know many of you don’t see math in [Stu]’s shirt, but I think if you start looking at the world with math in mind, something like that shirt will pop out as beautiful mathematics. So grant me 5 minutes where I’m on stage, you’re the audience, and I live give you my thought process for creating a version of that shirt on desmos. It may not work, but I think it will be neat to try.”

So they are sitting watching me sit at a laptop. I start by graphing . They ooh. Maybe they don’t. But in my mind’s recreation of the event, I hear them ooh. I explain that this isn’t a random thing I created. I ask who is taking statistics. I mention the “normal distribution.” A few nod knowingly, and for those who don’t, I say “this isn’t genius, this is me seeing [Stu]’s shirt and recognizing one of the most famous equation shapes in the world.” Then I graph the reflection over the x-axis. They understood that. But then I said I need more lines.

So I say: “I am now going to try to make the same curve with a higher peak. I think I should do a slight vertical stretch.”

But then I note that it isn’t just that each curve gets slightly stretched, but also the width of the bump gets slightly widened too. I go to the board and explain how I’m going to do a horizontal stretch too, and write up how I’m going to alter the x-variable in the equation to do that.

I flipped that over the x-axis and then manually entered a bunch more equations that did the same thing — slightly higher peaks, slightly wider bumps. Kids asked me to add in the two circles in the middle, so I did. It looked meh because I only had 6 or 8 curves. I sent them on break and promised them I’d get it to look a bit better when they returned. And that I would do this with just one equation.

During break, I whipped this up using lists.

When they returned, I explained how the list worked in Desmos — so one equation actually plotted a bunch of equations.

I didn’t know what to make of doing this. I wanted them to see how I saw things, how I thought about things. That math is in lots of places if you just look for it. That playing with math can be fun and what they already know mathematically are quite powerful tools. If for just a second one of the kids was like “Oh, yeah, wait, math is pretty neat,” I’d be happy. It might have happened because the next day I was talking with a science teacher who was telling me that my kids who she also taught were talking about it in her class.

Also, you know, I always find that when I deviate from my plans for something I’m excited about, I always feel so good about doing what I’m doing with my life. I have to keep this in mind and try to go off the beaten path more…

PS. Of course when I saw the shirt, I didn’t initially “see” the normal distribution. I saw fluid flow around a cylinder:

But I forgot everything mathematical I know about that. :) So normal distribution it is!

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Note: I debated whether to #pushend or not. Hopefully this helps anyone curious about why I’m stepping aside from TMC.

I posted this tweet yesterday:

I’ve been on the organizing committee for Twitter Math Camp for the last three TMCs and also for this upcoming one in Berkeley, CA. It was initially a surprise — and also upon reflection not as much of a surprise — to me that board member Marian D. stepped down from the board of TMC a few days ago. When I called Tina C., also a board member, to talk about possible next steps, she told me she was ready to step down also.

I will let Marian and Tina talk about their reasons themselves if they wish. But without much information out there, I figure people in the TMC community are confused and filling in the empty space with their own conclusions. I mean three people leaving organizing TMC in the span of a few days? That is a message encoded in an action, right?

Here’s what I want to share about that, from my perspective. I love TMC. I love the people who put on TMC. If I had to point to one thing in my teaching career that has grown me the most as an educator, it would be MTBoS generally and TMC specifically. I don’t have adequate words for my love and appreciation for it.

And mostly, I worry that because of these departures, people are going to vilify TMC. I don’t want TMC to be vilified. And in my opinion, it doesn’t deserve to be vilified.

It was born out of desire and necessity, to have math teachers who knew each other on twitter to meet in person. It provided a space for teacher voice to have value, and a place of emotional support for many who didn’t have any locally. It created one pathway for classroom teachers to become teacher leaders. I can’t say it enough. I love TMC, I love TMC, I love TMC, I love TMC, I love TMC, I love TMC. And for many, many, many people who have attended and have had amazing experiences, they too have a special place in their hearts for TMC.

So know that although I am not going to be a committee member, I am not anti-TMC.

So why, then, would I leave? Before that, I need to say it was a terrifying decision to make. I had committed to helping out, and shirking on something I committed to is a huge sin in my book. The good news is that the work that I commit to doing every year is quite simple logistically and not a huge time suck (organize a new-attendee/supporter mentorship, plan a new-attendee dinner, design some buttons, organize a homespun photobooth). And we’re five months before the conference and I told the committee that I’d work with someone on helping them learn to do these things with all my documentation. I wanted to leave as responsibly as I could.

But yes, why did I leave? As much as I love TMC, I didn’t enjoy the process of planning TMC. I’ve been in so many awesome collaborations where we have the opportunity to plan something awesome (whether it be a lesson or a workshop or whatever). And TMC is one of those. I mean, imagine you got a bunch of dedicated, passionate, awesome, thoughtful math teacher friends together (the committee/board is full of people I call friends!) and you get to put on this awesome event. I mean can you even imagine how fun and exciting that would be? Brainstorming, debating, revising, throwing things out, having a genius moment, outlining, revising? Simultaneously intellectually draining and intellectually invigorating. All the good! But the reality is that the process of planning TMC was broken, and over the years, it has remained so. It shouldn’t be so, and I tried to help change things, but without going into specifics, it never became unbroken. TMC, for me, was amazing. But planning TMC is a different story.

I was on the phone talking with Tina about Marian’s sudden departure, and Tina told me she was thinking of stepping down too. For the past few years, but more so in the last year, I’ve been working alongside Tina on a number of projects. We worked on creating a TMC mission statement with core values (engaging with the community in multiple ways to do this), we worked on improving communication and defining roles in the TMC leadership (beforehand, everything was haphazard and no one knew what everyone else was doing), we worked on figuring out where TMC needed to improve (diversity) and putting together a plan to address that, we worked on spearheading the first TMC fundraiser, and we were working on implementing ideas from the diversity proposal that the TMC committee and board had approved. My earlier work — things like calling a few restaurants and sending out emails pairing up new attendees and supporters — started to feel less important to me. I was excited about working alongside Tina with professionalizing TMC and addressing our deficits. That was the work that felt difficult and juicy but so much more important. Working with Tina on all this was that exhilarating collaboration that I had talked about above. So when Tina told me she was leaving, I sat down and wrote and wrote and wrote nonstop 40 minutes just trying to figure where my head was at with all of this.

What I realized I was losing my mentor-leader who brought meaning to my work. And she was my rock when trying to navigate our dysfunctional planning process. I tried to imagine doing our work without her and thinking about the frustration level of planning with the committee. I couldn’t imagine it. And that’s when I realized I had to leave with her. I thought: better all at once, a little less than half a year before the conference, so the committee/board can regroup.

I was excited about the directions we were taking this year in terms of diversity and equity. Marian reached out to educators of color to present and attend. In the program, a strand of sessions throughout the conference was focused on equity. Affinity spaces for educators of color and LGBTQ+ folk were built into the schedule. Marian kept the idea of “safety” for educators of color on our minds and we were explicitly thinking about people with mobility problems, deafness, etc. In other words, we as a conference knew we were generally inviting, but we were working on being specifically inviting.

The people organizing TMC are good. They might not create a synergistic working group, but they are doing selfless work which has helped so many. There are still people organizing TMC who are hoping to try to make these things listed above a reality. Any one of these things — well-crafted affinity spaces, a strand of strong equity sessions, thinking deeply on small and big ways to put on a conference where marginalized groups are valued and welcome and feel safe — would have pushed TMC forward leaps and bounds. It is a fair statement to say the committee is not strong at thinking about equity and diversity as a whole yet (I’m honestly very far below where I know I need to be at this point, but I’m learning from every conversation I have and pushback I am given). There is a steep learning curve and it’s not always easy to see through these lenses. Mistakes are par for the course. (We’re math teachers, we know this!) But there are those who are invested in this work and they are going to continue pushing TMC forward. They deserve all our gratitude and thanks. And support. If they ask for help, I hope many of you who want to see TMC do better on these fronts charge in and give them an assist. Help make TMC the place you want it to be. This is what TMC has always been to me: a space for passionate people to do good things together.

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I know I haven’t posted a lot this year. I actually have tons to post on because I’m writing a lot of Algebra 2 material, but because I’m doing that work, I haven’t been able to carve out the time to post about it. Blerg.

But today I wanted to write a short but sweet post. Every so often, I ask for feedback from my classes. I’ll create a google form and ask how things are going, if kids’ pronouns have changed, how long their nightly work takes, and other thing I’m curious about. Sometimes I have kids reflect about their own work or their groupwork.

But last year, I started occasionally including this question:

I love that it gives kids a chance to think about who has helped them out. I don’t make it a required question. Only about 1/2 or 1/3 of my kids filled that question out this time. But I really loved the short bits I did get to see. I learned who might have been studying together for tests, or who worked super patiently with another person who might have been struggling, or whatever. And kids got to have a moment where they got to be grateful for someone else.

What was nice is that I actually asked for this survey a week before parent-teacher conferences, so I was able to share with parents who came some shoutouts about their kids (if they got any). Parents really appreciated hearing that their kid received praise from another kid (and why).

And today, I sent short emails to any kid who got a shoutout…

Hi Stu,

In our last check-in survey, I asked students to give a shoutout to someone who was an important part of their learning experience. I wanted to privately share these with students…

Stu2 wrote:

Stu! We work great together because we have different strengths and weaknesses, so when we do a problem together I’m able to understand the whole problem, not just the aspects I’m especially strong at.
Stu3 wrote:
Stu! He is always great at explaining things to me!
Hope this brightens your day!
Always,
Mr. Shah
And… that’s it! A little sweet thing that I came up with that I really like. Short, simple, but for the right kid at the right time, it can be meaningful. (A few kids emailed me back saying that reading the email did make them smile or brought some light in a dark day…)

PS. Once, I had a bulletin board in my room that I had reserved for “shoutouts” or “notices of gratitude.” Where kids could post index cards with shoutouts for other students. I wanted it to be public and for it to “grow.” I would occasionally build in time for students to reflect and add to the bulletin board. That was years ago. It didn’t really take off, which means I didn’t roll it out and implement it well. But what I’m doing with this google form is really nice because it isn’t intensive or involve much planning!

PPS. If you’re at my school and want advice on how to do something like this, feel free to ask. I’m happy to brainstorm with you. I just don’t want everyone doing this which then will take away the “specialness” of when I do it!

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