The challenge that “e” poses

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 (1+\frac{100\%}{n})^n as n 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:

pic1

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.
    pic2.png
    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 25=e+e+e+e+e+e+e+e+e+e+0.196986e which has a product of e^{9.196986}. (Or to be super precise: e^{25/e}.)
    And @benjamindickman shared some articles about this:pic10.png
  2. @mikeandallie shared with me this approach by throwing darts. Which includes this gem:
    pic3.png
    and this instantiation:
    pic4.png
  3. @jensilvermath suggested just looking at (1+1/n)^n for larger and larger n, 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:
    pic5.png
    Whaaaa?! But okay! WHOA?!
  5. @roughlynormal suggests:
    pic6.png
    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…pic7.png
  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”:
    pic8.png
    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!
    pic9.pngThe red line is always greater than the blue line (I think… I want to prove it algebraically!). [Note: related, this set of tweets on x^y=y^x 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.sum_{n=1}^{\infty} \frac{1}{n} = 1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+...Apparently, if we look at which n values bring this series to an additional integer, we get:pic11.png

    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.
    pic12.png
    The answer involves 1/e. (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:
pic13.png
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 \pi and when that should be introduced and why.

My conclusions

First and foremost, I knew there was a lot to e — 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 e 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 e^{i\theta}=\cos(\theta)+i\sin(\theta). ‘Nuff said.

So if I want my kids to see what I see when I encounter e, 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 e… but they don’t know it. Like Pascal’s triangle… they can just do the calculations and see e pop out. Or the chess board experiment. Or taking random numbers away from 100. Or the harmonic series. Or the product challenge. Or the x^y vs y^x 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 e number that keeps popping up in all these unrelated problems.

For each of these, they are going to get numbers close to e. 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 \pi. 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 e which we’ll just take as that… an important number that we’ll get to play around with like we do \pi… 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.

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Digits

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 2^{60}. 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 2^{10}=1024. Which is close to 10^3. So 2^{60}=(2^{10})^6 \approx (10^3)^6=(10^{18}). So that puts us at around 19 digits.
  2. Kids noticing this pattern:
    chart
    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!

Going off the beaten path…

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…

shirt.png

… 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 f(x)=e^{-x^2}. 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.

lines.png

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.”

lines2.png

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.

lines3.png

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.

lines4.png

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:

flow.png

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

 

On stepping aside.

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:

tweet

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 friends. 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.

 

 

A simple question

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:

shoutout

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!

End of Year Donations

Throughout the year, I tend to donate to causes that my good friends are spearheading. If they’re going to take the time to spearhead a cause, the least I can do is show my support by throwing a few bucks here and there.

However at the end of each year, around November and December, I think about what I truly value and I find an organization or two that fit a good number of things that I care about. This year, I have two organizations that I am donating to. I’m writing this short post in case any of y’all out there are thinking about places to make end-of-year donations to, and these organizations speak to you in some way.

(1) BEAM: Bridge to Enter Advanced Mathematics.

beam.png

This is an organization that I started volunteering for last year. It was created by a friend of mine. The idea is that there are a lot of underserved kids in New York City who like and can do mathematical thinking, but aren’t getting what they need to excel. BEAM is a summer mathcamp for underserved middle school kids. The organization goes out and finds these kids, and then creates a summer program that’s free for them.

The reason the organization focuses on middle school kids is that’s where the most good can happen from an intervention. And these kids who go through the program get support for many years afterwards. It isn’t a “one and done” experience. Often times, kid return to the camp a second year. But more importantly, the organization helps kids navigate the high school admissions process (which is super complicated in NYC), helps them succeed with open office hours where kids can drop by to ask for help with any subject (this is how I help out… I staff the office hours!), and helps to with the college application and financial aid process. It’s a long-term program that has a huge impact on kids.  (If you want to learn more about it, this New York Times profile is amazing.)

So yeah. A thoughtful, wonderful, amazing program. I’ve worked with these kids and they rock. Right now donating $1 will bring in $3.14 dollars through a matching campaign. If this organization sounds like it fits your values, and you have some bucks to spare at the end of the year, consider donating!

 

(2) Twitter Math Camp (TMC/TMathC)

TMC.png

This organization hits close to home for me. This is a math conference that started up in 2012. It was literally the most grassroots thing ever. A bunch of us math teachers who had been communicating online decided we wanted to all meet and do math together… and that morphed into an embryonic conference. A school donated space, people offered to present on things they were doing in their classrooms, and a small conference was born. It has been running every year since, and has been entirely run through volunteer efforts. Schools or colleges donate space. We have people volunteer to present. Organizers (disclosure: I’m an organizer) work out the logistics, while they’re also working/teaching/coaching/etc-ing full time. It’s pretty awesome. And because of the goodwill of all these people, synergy happens. It is an incredible space for math teachers to grow their own practice and forge life-long relationships with other math teachers.

So as an organization, financially, it’s pretty efficient. We only charge a $20 registration fee because we want to keep costs low. But there are costs associated with the conference (e.g. buses from the hotel to the school/university where the conference is held, custodial charges from the school/university, renting a conference room in the hotel for registration and game night, providing supplies for speakers, etc.). Amazingly, with the few thousand we get from the registration fee, we’ve been able to pay for everything.

But for the first time ever, TMC is hosting its first fundraiser (read more about it here) and is asking for donations. And like with BEAM, TMC has a matching donors program, so every $1 donated will actually be worth $2! Now here’s the thing. When deciding to donate my own money to TMC, I had to take a beat. I thought: “Wait, I’m giving my money to a conference. Which I go to and pay hotel fees and transportation fees for. And I’m asking my friends to give money to a conference. A conference they likely are not even going attend. Does this make sense?”

But here’s what I realized… The conference encapsulates things I value. It gives classroom teachers a voice. It creates a space for teachers to share their thoughts, and maybe realize for the first time that their thoughts are useful to others. By having a powerful conference experience (which often lasts far beyond the conference itself through the relationships forged and communication that happens on social media), TMC is having a positive impact on math education for so many students. My donation is going to go to help increase access to educators of colors and seed a scholarship fund to help provide access to the conference for teachers whose schools might not be able to afford it.

I want to help the conference become financially stable and not have to rely on only the the small $20 registration fees it collects. I want to help increase access to the conference for teachers of color and teachers who can’t pay out of pocket and whose schools won’t help them out financially. It isn’t about me and the conference. I’m giving because I want to make sure that this conference can exist for others, and provide for them the same sort of experiences that helped shaped me as a teacher. [1]

So there you go. I wanted to share these in case you were wanting to donate to some causes, but hadn’t quite hit on the ones that made your heart pitter-patter yet. (Both are non-profits so donations are tax-deductable.) These are the ones that do it for me. And I’d encourage you to give a few bucks if you happen to have them at the end of the year. (You’re probably a teacher if you’re reading this, so you probably don’t have a few bucks to spare, so in that case forget everything you just read!) And if you have any organizations you think I’d be interested in, please throw them in the comments!

 

[1] That’s precisely why I give to the mathcamp I attended when I was in high school, and which made such a large impact on me. Or occasionally why I’ll give to my college (even though I don’t usually give a lot because they are pretty darn flush and already have a lot of wealthy donors… I like to give to smaller organizations where I know a small amount of money will help a lot.).

My First Day, 2018 Edition

I had my first day with kids this past Thursday. We had only 30 minutes with each of our classes, so I went back and forth about what I wanted to do. Some years, I like to get them in their groups and we start right away. I have a compelling question or *something* that starts the first unit, and we charge ahead. When I do this, I’m thinking “I want kids to see what we do every day in class. We do math. We work together. We don’t waste time.” [1] Kids seem to enjoy that. They are usually revved up and excited to start, even though we’re all a little sad that summer is over. (Okay, very sad.) But there’s energy in the air.

This year I decided to do something different. A colleague of mine did this for a class we both co-taught years ago, and I really thought it would be a great way to start this year.

Part I: The Initial Card Sort that Sorted My Kids Into Their First Groups

I said hello for literally only one or two minutes, and then I shared the activity we were going to do for 15-20 minutes. We were going to do a puzzle-y card sort to figure out who was grouped with whom. But in order for the class to be successful, they all needed to work together. I projected a sample card. I said anyone is allowed to use a calculator. But some of the cards might require some laptop assistance. So they had a little laptop symbol on it.

card1.pngSo in this case, for example, I knew almost none of my kids would know what binary numbers are, but using google they could find a converter online that would say this was actually “170.”

Each card had a kid’s name written on back. So each kid got “their” card. And their goal was to find others who were in their group because their cards formed a logical group. Here’s a sample group to show you what the cards looked like and how they link:

card2.png

See if you can tell what the link is among these four cards…

I’ll give you a second.

I will reveal the answer in the next line, so don’t keep reading until you are sure you want to know.

Okay, the link is the number “ten.” So 10! is the number of seconds in six weeks. When the kids type those equations into desmos, they will see the number 10 show up. Neon is the 10th atomic element. And “X” is 10 in roman numerals.

You can see why kids are going to need each other and the class is going to have to work together. Because until someone recognizes that “ten” is a category, these all seem very unconnected. But as soon as you know someone’s card represents “ten,” then things like the neon symbol or the “x” make sense.

I’m kinda proud of these, so I’ll show you another:

card3.png

The theme? “Pi.” The first one is circumference over diameter, the second is a recipe for pie crust, the third is an approximation for pi, and the fourth is a world record holder for reciting the digits of pi.

(If you want to download my cards, here you go: Group Card Sort! And the explanations are Group Card Sort Explanations.)

I only had allotted 15-20 minutes for this. I had no idea if this would go quickly or take forever. In all four classes I did this in, I was able to get them to finish in 20 minutes but only through some careful prodding/help. If I were a bit more hands off, I could see this easily taking 40 minutes and it being time well spent. But alas, I didn’t. Here’s how I intervened:

  1. After 7 minutes, I stopped everyone. I asked who knew what they had. A few people did.
  2. Throughout the time, I gave a “few” hints where I could, but mainly I was acting as facilitator to help others help each other. So for the pie crust recipe one, I had the person go around asking if anyone was a baker (or I would shout out to the room if anyone liked to bake, and had them come to us).
  3. When someone wasn’t doing anything, I had them go help others. They might have been confused about their card, but they could help others (and get help from others).
  4. Sometimes when a kid “got it” but still had some uncertainty, I would put them out of their frustration and tell them they got it. If I didn’t have time pressure, I wouldn’t have done this, but it didn’t ruin the activity or anything.
  5. After 15 minutes, with my proddings and connecting, kids were doing pretty well. So I stopped everyone and had people who knew what their card represented be quiet. There were always three or four people who were stuck. So I had them share their card or write their puzzle on the board and see if anyone could figure it out in the remaining few minutes. (We wrote the different “solved” categories on the board, so sometimes they could figure out their card by seeing what it might be.) They gathered, talked, and some classes barely finished in time and others didn’t. I didn’t focus on that. For the ones that didn’t get them all in 20 minutes, I quickly went through the explanation of the remaining few cards.

It was really fun for me to watch, and I saw kids really getting into the puzzle-aspect of things. The first time a kid figures out their card and finds someone else with the same thing, it’s just a wonderful feeling. It honestly feels impossible to kids at the beginning. They literally start looking for anyone with the exact same card as them, or if they have a picture they’re looking for other people with pictures. But as soon as they realize it’s more challenging and more interesting, I get to see how they react and what they do. Do they sort of back down? Do they go help others? Do they hope someone comes to them? My big goal was having kids realize they can’t do this alone and most cards won’t tell you what they are so you need to hear about others and help others.

Oh! One big thing. I realized in the first class that kids were just kinda sitting with their cards. So I made a rule that until the card sort was over and everyone in the class figured out their cards, no one was allowed to sit down — not even when using their laptops. This actually got kids up and moving. It was a small thing, but I know it was super helpful to making this a success.

I wish we had time for kids to say hi to their first group and do a little group norm setting, but alas with only 10 minutes left, I had to transition.

Part II: New Years

So I totally saw Howie Hua’s first day post and was in love. It was positively inspired. Often times, people post awesome things they do in their classrooms that are awesome but just not me. When I read this, I felt: “OMG THIS IS ME!” He celebrated new years with his classes. Here’s one of his students’ videos/tweets:

And it really got me thinking. The first day IS my new years. My life doesn’t go in January-December cycles. It goes in September-August cycles! And it was the perfect time for kids to make a new years resolution. They had 90 seconds of thinking to come up with something.

new years.png

Then after 90 seconds, I threw up this screen, obliquely referencing the Maurader’s Map from Harry Potter (but opposite-ish) and I had them recite this pledge:

no bad.png

Then I gave each kid a baggie that I prepared. In it was a super fancy piece of origami paper, a mardi-gras necklace that someone had a zillion of and was throwing them away, and a noisemaker I bought from amazon. It mabye took me 45 minutes to put these all together. But totally worth it. For some reason, I believe that being given your own personal goody bag is way more exciting than having someone pass out necklaces, noise makers, and origami paper individually.

I then handed out party hats too (but those had to be returned to me). I actually always keep a stack of party hats in my office, and when it’s a kids birthday, I give them a hat, candy, and we sing a short birthday song. As I said, this idea of Howie’s fit me!!! Anyway, kids had to write their name and their resolution on the origami paper which I collected. (Later that day, I put them together in a ziploc bag and hung them visibly in the room so this doesn’t become a thing we did but wouldn’t return to. I was thinking I’d give them back to kids after the end of the first semester so they can see how they’re doing on their resolution. But I might have another brilliant idea. Who knows!) As soon as the bags were out, the noise makers were making noise. And that was a lovely cacophony of BWWWAAAPP and BAAAAAAAA noises. (That was also why I had kids pledge to do no evil with what they were given… *grin*)

In any case, I was standing at the front of the room when they worked on writing their resolutions. When they were done, they had to bring up the resolution to me and wait at the front of the room with me (with the necklace, hat, and noisemaker). After 2-3 minutes everyone was up. And then… we took a class picture, all decked out, blowing on the noisemakers and just being amazing. And oh yeah, we also took a class boomerang (which is an app that lets you take a 2 second video and plays it over and over).

group.png

The boomerang was my favorite part because kids were jumping up and ducking down and doing fun things. And I kind of am obsessed with boomerangs. So there’s that.

I think I’m going to get these photos printed and framed, and hang them up in the classroom. I don’t know what to do about kids who were missing  (there were a few) or who transfer in after some schedule change, but maybe I’ll list them missing on a caption instead of some awkward photoshop job?

Our first day together. (I did post the boomerang video and our class photograph on the google classroom site in case any kid wanted it.) [2]

And then it was the end of our first 30 minutes together. I was really happy with how it went. I like the feeling that I left each class starting the year with good vibes. Thanks go to my chemistry teacher colleague and friend for the card sort idea which I made into something my style (with my kind of clues!), and to Howie Hua for helping me make a memorable moment to start the year.

***

[1] We do a lot of the logistics things in the following week. They read the course expectations at home and fill out a “get to know you” google form which also asks them questions that require the expectations to finish. And then each day or day, I talk about one or two things I want to be explicit about (like how to write me an email, or that’s it’s okay to go to the bathroom and they don’t need to ask, but they do have to discretely let me know they’re leaving if I don’t see them, or that they need to bring a waterbottle to class because they can’t leave to get a drink).

[2] I just realized this photo could be fun to have up on the screen on parent night, when parents/caretakers come in two weeks to hear me talk about our class.