Author: samjshah

Bragging about my school

This is a milestone for me. I have been at my school for ten years, and this is the start of my eleventh. It’s the only school I’ve worked at. That’s a testament to my school, but more specifically, to my colleagues.

Last year, my school’s awesome director of communications contacted the math department to let us know that the one issue of the magazine she publishes four times a year was going to focus on math. And she wasn’t kidding! The cover of the magazine had most of my multivariable calculus kids on it (thinking deeply at the math-art show I helped put on last year)!


One of my favorite things is that the feature article with an alliterative title, Making Math Meaningful, was simply the transcript of a roundtable discussion we had. A bunch of math teachers got in a room around a big table, and we were led by our director of communications who had done her research and come with some questions. There was a digital recorder in the center of the table. And through talking with carefully crafted prompts, we got to think deeply and collectively about our own practice. I can’t even tell you how interesting it was to listen to my colleagues during that facilitated conversation, and how proud I was to be in a school with such like-minded folks that I have the opportunity to learn from. (If you’re a department chair or academic dean, consider doing this!)

I wish I could just post a PDF of the article for you to read, but alas, the whole magazine is online but can’t be downloaded. Here are two quotations to whet your appetite:

quote 0

quote 1.PNG

So if you want to read about a department that is doing strong work moving towards inquiry-based learning, and read the words of real teachers having a real conversation playing off of each other, I highly recommend you:

  1. Go to this site
  2. Make the magazine full screen
  3. Read pages 18 to 29

That is all!


Marbleslides, Squigles, Portfolios, Previewing: My Third TMC Recap Post

Another blogpost about takeaways from TMC17 which I may be able to use in my classroom.

Marbleslides Challenges

I love Sean Sweeney. He’s everything good in the world, packaged in humanoid form! He’s so welcoming and kind to everyone… he wants everyone to feel part of things. At the Desmos Fellowship, he was the person I felt most safe saying “I have no idea what the hell I’m doing” and he would hunker down and help. I think many others felt the same. Okay, enough of the love fest. I am going to share his my favorite which I desperately want to use in my classroom. First, a little note. There is a difference between reading something on a blog and experiencing it. More and more, I’m recognizing that. I think if I read about this, I’d think “cool story, bro” and be like “okay, I could do this, but is it really worth it?” But experiencing it like we did during his short presentation, it’s like “I MUST DO!”

Sean has made a number of Desmos marbleslide challenges (if you don’t know about this, google it). Here’s a gif from his blog. The idea is that the marbles drop and you have to create stuff on Desmos to make the marbles hit the stars.


He shared one with us, and everyone in the giant room got obsessed with drawing functions that would let us “win.” For our challenge, people used ellipses, used lines, used piecewise functions, use quartics. It was inspired to see all the different approaches, and all the play that resulted.

What was lovely about Sean’s facilitation is that he paused us after a while (note: a teacher trick is to say “I’m going to pause your screens in 5… 4… 3… 2… 1…”). You knew from the cacophony of groans that we were in a good place. Then he shared out different approaches. The diversity of “answers” for the challenge was fascinating.

He made this a regular thing in his classes. I love his poster which shows the diversity of responses:

marble.PNGSo how can I use this? I’m not sure yet. I need a way to keep it light and fun, but also with all that my kids have on their plates and their lack of time, I don’t know if they would take the time to do it without some incentive. After teaching kids how to restrict the domain of a function/relation, and reminding them of all they have at their disposal that they’ve learned about (trig, circles, lines, parabolas, step functions, etc.), maybe I need to have a 10 to 15-minute in-class challenge (with kids working in pairs, so they are comfortable). And then do it again two weeks later, in class (but not in pairs). And then… announce that we are going to have regular marbleslides challenges. And the winner(s) will get the bonus question on the next assessment without having to do it. Or maybe buy some cheap plastic trophies which get displayed proudly in class? I want kids to work on the marbleslide challenges outside of class because part of this for me is that I want kids who might be slower at processing or coming up with ideas to have the time to execute their vision. I don’t want this to be a timed thing. Though maybe each time I introduce a new challenge, I give everyone 5 minutes in class to work on it.

What I have to make sure to do is share publicly the diversity of answers, like Sean did with his posters.

I also had an idea about how to score it. Something like 1 point for each star. But maybe if we’re learning about conics, or tangent, or something else, I’d give a bonus point for using those functions. And maybe an additional possible bonus point or two for any additional creativity (teacher’s choice)?

Sean’s posts are here and here.


David Butler also presented a my favorite on squigles. The poster and his blogpost are here.


I am not one for acronyms, really. They often are forced. But what I like is that these are used to teach student math helpers how to work with other students. From David’s post:

SQWIGLES is an acronym that we use to help our staff (and ourselves) when teaching in the MLC Drop-In Centre. It is a list of eight actions we can do to help make sure our interaction has a better outcome and make it more likely students will learn to be more independent.

It was originally Nicholas’ idea to have something like this. He wanted something to help the staff choose what to do in the moment, and also to help them reflect on their actions and choose ways to improve. We noticed that our staff (and ourselves) needed something focused on actions rather than philosophies, because then it could be used on the fly to choose what to do. Telling staff they need to be “encouraging” or “socratic” is not all that helpful when they don’t know how to put it into action. Yet this is what many documents giving advice to tutors do. So we decided to focus on the actions instead.

The reason I wanted to blog about this is because I think it might be helpful to share with the student tutors at my school. We have a peer tutoring program called TEACH (probably an acronym, since I always see it written in upper case… but for what, who knows!). And I haven’t inquired if and how students get trained. But I’d love to do a short 10 minute presentation on this, and maybe do a few scenarios where kids can practice tutoring while other kids watch (fishbowl?) and take notes on which of SQUIGLES happened. (Not all need to happen! Just look for them.)

I think I should also have this on my desk, since I work with students one-on-one a lot and having that reminder can’t hurt!


I went to Cal Armstrong’s session on documenting student learning. Over the years, I keep on getting inspired to have kids make portfolios that they turn in to show evidence of different traits. And this came up again in that session. James Cleveland has done it. Tina Cardone has done it. I want to do it. But aaaah! The time to make it into a reality! Argh! But I really would love to make explicit some values — maybe not standards of mathematical practice (… or maybe throw of a few of them in there…), but things like perseverance or active listening or seeing a problem in a different way or acting with courage or helping someone understand something by asking good questions or recognizing your own a misunderstanding or changing for the positive as a group member in somewayAnd have kids document these moments or interactions. And then at the end of a quarter, turn them in. (But have a check in halfway through the quarter!) It would mean that they are looking for these things, looking to do these things. And recognizing that I value these things. Maybe they have a choice of things they can include — not all of them? Maybe they can take videos or photographs or write paragraphs or draw a comic — it can open-ended how they demonstrated this quality or action.

There is something that I think happens in my school. Kids form facebook groups (or maybe on some other kind of social media) for their classes, and I suspect lots of backchannel communication about the class happens on this group. I suspect a lot of it is positive and uplifting and helpful. I would love to encourage kids to submit that sort of stuff in their portfolio also, if it demonstrates whatever qualities were asked for!

I don’t know if I’m going to do this this year. But maaaaaybe?

Preview, not Review: Student Intervention

Kat Glass gave a my favorite on intervention with students who were failing. Part of it was a powerful and important note about language and using code-words instead of saying what you mean. We don’t have many kids that fail classes in my school. But one thing that did strike home was that sometimes when working with kids who are struggling, we put all our emphasis on remediation and it’s like we’re always playing catch up. But sometimes we need to remember that with a struggling student, one tack that we can’t overlook is previewing upcoming material. It can help kids be more engaged and confident in class, and it sets a good tone moving forward.

I do this sometimes, but I need to remember to do this more frequently. Although I do lots of discovery based work, I don’t think that previewing some of it with a kid, and working through some of the discovery with them one-on-one, and then them seeing some of it happen again in class is a bad thing. I’ll just have to remind them that they need to be careful about not letting other kids have the same insights they had — and their role is to help without telling.


Play! Create! Adult!: My Second TMC17 Recap Post

Here are some more TMC17 notes!

Don’t play with your food, damnit! Play with your math!

I love the idea of having kids engaging in recreational math. I don’t have much time to encourage that in my curriculum — or at least the only way I’ve found for that to happen is with my explore math project [posts 1, 2, 3; website]. Some kids get some extra math problems to work on at math club (usually problems from math competitions or, and kids do math problems on our math team. But that isn’t the spirit of what I want to bring to my school. I want to get kids just fooling around with math for fun! Tinkering! Thinkering! Building! Collaborating! So that’s why I fell in love with Joey Kelly (@joeykelly89)’s my favorite presentation. Where he shared with us Play With Your

He and a friend created it. Right now it has 15 sheets of paper that can be printed out, each with a challenge. The name, inspired. Design wise, fantastic. But the problems are captivating, easy to dive into, and many have this open-endedness that can lead to obsession. When I was at the Desmos Fellowship a couple weeks ago, they had these for us to work on as a way to get to know each other. Each table had a different one and we were encouraged to play, and meet others who were playing, and then move to a different table and meet and play when we felt like it. The one I spent all my time on, trying to come up with a strategy? One that I know will get my kids in competitive mode? Poster 5:


I liked getting to know people and I liked these problems! At TMC we were given poster 14 and I became obsessed. And eventually, I solved it (and a second more complicated one). But it took A LONG TIME and I DIDN’T CARE. I refused to go play boardgames at gamenite until I had climbed this mountain!

I need to brainstorm if and how I am going to use these in my school. Some initial ideas:

1. Leave copies of these in the library for kids to use. Or put many copies of all of them on a bulletin board for kids to take, so when they’re board and standing there, they just grab one and start thinking.

2. Use these when I need to fill a long block (we have double periods one out of every five times we meet our kids) and I don’t have a good idea.

3. Plan an Upper School math night, where we gather at a space in the school, do math, order pizza. Like PCMI’s “pizza and math” (was that what it was called? we can do better!). These can be the amuse bouche or the main event!

Math Art!

Speaking of recreational math, at TMC17 there was so much math art. I just wanted to share some of it!

Captivating! I hope at some point to learn how to make crochet coral. It feels like once I get in the rhythm, it could be so soothing. Actually, I wonder if it would be fun to have a MAKER MATH club where we make math stuff together. And create our own math art gallery. Things like the things shown here, but also like these, and origami (demaine and lang), and a menger sponge made of business cards, and design and 3d print these optical illusions, and carefully color in pictures from Patterns of the Universe, and create our own mathart coloring pages. If you are reading this and have ideas of things that we could make, let me know in the comments! You probably can tell this is something I’m actually totally *feeling* (FYI, for me, the definitive math art page is @mathhombre’s page here.)

How To Adult: Let’s Buy A House

So @rawrdimus gave a my favorite on how to adult. He was teaching calculus and wanted to keep his seniors engaged. So he came up with this project that had kids pick a few houses and figure out what they’d need to buy it. He was the banker (a hilarious banker) and gave them two different mortgage options (a 15 year and a 30 year, with different interest rates) and they had to figure out their monthly payments.

I know come the spring, the kids in my calculus class will have their attention wane. So I think something like this could work (this investigation on wealth inequality worked a few years ago)! But right now it’s a little bit like trying to put a square peg into a round hole. I need it to have some more calculus before I do something like this though. Maybe we’ll spend some time talking about e or we’ll do something with summing (in)finite geometric series, and maybe seeing that as a riemann sum? I think it’s totally doable — I just need to think a bit more! But if you want to get a sense of why I’m trying to make this happen, just watch Jonathan’s presentation and you’ll totally get it. (Here’s his blogpost.)

You Guys, Funny Quotes, #YouMatter, Sitting Down: My first TMC17 recap post

ARGH! I have too much in my head and don’t even know where to start. I want to blog about all the large and small things I want to take away with me from TMC17, but they are so disorganized in my brain. So I’m just going to do a Faulknerian stream-of-consciousness style post and get some of it out now.

Hey, You Guys! Words Matter

A while ago, I realized when I said “you guys” it was super gendered. So I just sort of said to myself I’ll say “y’all.” When I wrote emails to my classes, I pretty much say “Hi all!” And then… and then… someone brought up the “you guys” issue at a faculty meeting at our school, and in my head I was like “I don’t do that!” But for some reason instead of that reminder doing good, and reinforcing what I was doing, I found it impossible to not say “you guys.” Like when someone points out you say “um” a lot, or say “like” a lot. You just, um, like, end up, like saying it, um, more.

Glenn Waddell spoke about “you guys” at TMC, and it resonated with a lot of people.

So I think I have a plan. Thanks to a huge discussion on twitter (sorry, don’t remember who to cite), here are my options:

all, y'all (2).jpg
The + others was cute… someone recalled they would say: “Humans… and others…” which made me laugh! I think I my lean towards nerds and my loyal subjects because I like whimsy. And as another teacher I love says about her classroom: “It’s a benevolent dictatorship.” @mathillustrated said it’s fun to mix them up. We’ll see what I’ll do!

A thought: I should post this in my classroom so I can refer to it! And tell students what I am trying to do. And have them catch me if I say “you guys” (which of course will make me say it more!). And have an ongoing tally of how many times I say it. And when they reach a certain amount, I’ll bring them some treat. I like the message it sends: I care about words because I care about you. For some of you, these words don’t matter. But I’m doing this for the others of you for whom these words do matter. Also: help me get better because I need to be, and I’m happy to be called out when I mess up.

Other ideas that came up:

@EmilySliman has renamed ‘homework’ as ‘home learning’ [I called it ‘home enjoyment’ because of another colleague, but they have since left me! So I am free to rename it as I please!]
@gwaddellnvhs has renamed ‘student’ (passive) to ‘learner’ (active) [“Learners learn, and students study. I don’t care how much you study. I care how much you learn.” paraphrased from here]
@chieffoulis has renamed ‘tests’ as ‘celebrations of knowledge’ (someone else uses ‘celebrations of learning)

Now do I think things like this will make a difference? Probably not. Calling something “home enjoyment” won’t make kids enjoy it. But it’s stupid and goofy and that’s worth something. And I don’t doubt that making an effort to change language might make a difference to some students. And it can prompt discussion where I get to talk about my values and philosophy around teaching. (“Why do you call tests ‘celebrations of knowledge,’ your majesty?”)  I try to live and act those values, but sometimes talking about them can help too.

Kids Say The Darndest Things: Another Classroom Culture Thing

I was having dinner at Maggianos with a TMC 1st timer, @pythagitup. Over dinner, he was telling me about a quote board he did where he put funny things kids said up on display. The beaming of his eyes as he recounted his classes and their quote boards made me know he had done something special. I begged him to write a blogpost about it, which he kindly did here. Here are his top 12 quotes:


I just got sad as I was writing this part of the post, because I remembered that I don’t have my own classroom. I usually am in two or three different classrooms and share the space with other teachers. So doing things like this are trickier. Sigh. It did remind me of one year in calculus. Years ago. 2012-2013. Back then, I was actually a funny-ish teacher. Like pretty goofy. And that particular calculus class was gads of fun. Good and strong personalities. I don’t know why but in recent years, I have lost that spontaneousness and goofiness that I used to have. I’m much more even keeled. I don’t know what happened. Does that just naturally happen when you grow older? I am up at the board a lot less now-a-days, so maybe that’s it… less class-teacher-class-teacher interaction? Whatever it is, I’ve changed. But back then, we had a goofy class. And all year, a student was secretly taking notes on funny things I said, or funny things kids in the class said. And she gave it to me at the end of the year. It was one of the most meaningful things a kid has done. You want to read some of it? Thought so. Wait, you said no? TOO BAD MY POST DEAL.


The post about it is here.

Promoting Kindness & Gratitude

I want to do this explicitly in my classroom. I tried a post-it wall of kindness/gratitude once, but that didn’t *really* take off in the way I wanted it to. I probably should have blogged about that to share a failed venture, and why it failed (namely: I saw it as a tack on unimportant thing, so I didn’t build time in class for kids to do it, and also kids have difficulty sharing kindness/gratitude so helping them see different things as kindness/gratitude would have helped too). [I see “nominations” as a way to do this too, and also related to the material! post 1, post 2]

But I saw something super nice. @calcdave was wearing a clothespin clipped to the collar of his shirt. I couldn’t read it but I asked about it. He then gave me a huge bear hug… which I thoroughly enjoyed because @calcdave is awesome and who doesn’t want a hug from him… and then looked at the pin. On the front, it said something like “hugs!” and on the back it said:


And then the person take the pin off and puts it on the other person (I think that’s important… they pin it on!). This then continues… from person to person to person. I love that @mrschz got it from me, and has now bought clothespins, painted them, and written on them. She’s all in!

I am not comfortable hugging my kids. I’m not that teacher very often (until they come back from college and visit). But I could see this going in different directions.

(a) Making 10 pins, each with one side blank, and the other side saying things like “high 5! #youmatter” or “two good things! #youmatter” or “fistbump! #youmatter” [and the person who inquires gets a high 5, 2 good things said about them, or a fistbump], and then the clothespin travels. I like the blank side because the clothespin then begs the question… and having different responses

(b) Making a bunch of pins and giving them all out to one class and explaining the purpose. I would have to do this with a class that is totally into stuff like this. I can imagine certain classes having a majority of kids who groan and then throw the clothespin away. So I’d have to choose wisely and come up with a good framing/rollout.

This idea originated with Pam Wilson, who is a true gem.

When this idea made its way on twitter, @stoodle pointed out that @_b_p has done something related in his classroom. And I remember reading this, being like OH MY GOD I NEED TO DO THIS and then promptly forgetting about it. The TOKEN OF APPRECIATION. I mean the name itself gets me giddy!


But I like this idea for a few reasons. First: it is done only once a week. It doesn’t take away from classtime. I can do it during my long blocks (once every seven class days). Kids have all week to think about who they are going to give it to. Kids also get to alter it, so at the end of the year, it is a recollection of good.

I know people are going to hate me for saying this, but this upcoming year, I have small classes. I’m at an independent school, so my classes tend to be small. But I think I remember my tentative rosters being even smaller than usual. I like to have larger classes because I like the chaos and interaction and cross pollination of ideas (though not the grading nor comment writing). But I wonder with small classes this year, will this work? I need to think more about this.

Crouching versus Sitting

This wasn’t at TMC but I saw it on twitter and wanted to affirm its truth for me.


I am fairly good about this. I have kids sit in groups of 3 but the tables can fit 4, so I tend to just hunker down with groups when talking with them. In most classes, I almost always drag a chair with me from one table to another which doesn’t have one. I agree there is a huge difference between crouching and sitting. There is value in crouching… it sends the message “I’m here to sort of briefly check on you and see what you’re doing but I’m likely going to move on… things are on you… so persevere.” I tend to sit when (a) I need to ask the group a set of questions to see their understanding, (b) a group seems to be getting stuck beyond productive frustration, (c) when a group is having a heated or interesting conversation and I want to listen in [I tell kids to ignore me and just continue, which I know they can’t really do but they do a pretty good job] or (d) when my feet are tired and I just feel like plopping down somewhere. Ha! Just kidding!


Problem Solving with Trig

So I’m at #TMC17 and Rachel Kernodle nerdsniped me. Or rather, I asked to be nerdsniped. Her session is at a time when there were a lot of other amazing sessions I wanted to go to, so I wanted to know if hers was one where I could hear about it and get the gist of things instead of attending. After some internal debate, she said that since it involved working on a problem, and then using that problem solving to frame the session, the answer was maaaaybe not. But then she thought: maybe I can try the problem on you and see how it goes. As long as you’re willing to put in the time to problem solve. Of course I said yes.

First, you can see her session description, which then framed how I approached the problem:

triangle 2.png

And then this is what she gave me (but it was hand drawn):


From the session description, I knew I had to find the ratio of the side lengths, so I could find exact trig values for angles other than 30, 60, 90, 45.

Rachel also gave me a “hint page” which she told me to look at when I was stuck (and to time how long it took me before I opened it). Let’s just say I’m extremely stubborn, and so as long as I think I have the capability to solve something and I am not completely stuck, I knew I wasn’t going to open it. Turns out my stubbornness paid off, and I ended up solving it.

In this post, I wanted to write a little bit about my experience with the problem. Because now when I look at that triangle, I have an duh, there’s an obvious approach to use here and everything I know points at that obvious approach. And the answer feels really obvious too. It is funny that I’m almost embarrassed to post this because there are going to be people who see it right away, and I worry (irrationally) (math pun) that they are going to judge me for not seeing it as quickly as they did. Even though I know being good at math has nothing to do with speed. And that it was important to go through the steps I did!

It took me over an hour to solve this problem. I had to do a lot of play and make a lot of random leaps before I stumbled across the “obvious approach.”  And I needed to do that in order for me to mine it for lots of things. It was true problem solving. And I know I really deeply understand this because at first the problem looked flummoxing and interesting, and now it looks obvious and somewhat trite. That’s my metric of how I know I deeply understand something. There are still certain things that I teach that I don’t deeply understand: like how the cross product of two 3D vectors yields a third vector perpendicular to the original two. I have done the math, but it’s non-obvious to me why the crazy way we compute cross products give us something perpendicular.(When I only understand something by doing brute algebra, I rarely feel like I get it.)

I’m going to try to outline the messiness that was my thought process in this triangle problem, to show/archive the messiness that is problem solving.

  1. The first thing I noticed was 36 and 36 sum to 72. So I was like: obviously put two of those figures together, and just play around. Something nice will happen. I remember when seeing the problem that approach felt immediate, obvious, and would lead to the solution. I was like yes! I have an inroad! This is going to rock, and I’m going to solve it quickly! And I’ll even impress Rachel!


    That appraoch didn’t work. Nothing popped out. I saw 54s and 18s and 144s pop out. But those weren’t angles that helped me. But I did then realize something nice… 36 is a tenth of 360! So I was going to use a circle somehow in this solution. Obviously!

  2. So I drew this:
    and I was like, I have something here! But after looking around, I was getting less. You can see I was trying to draw in some other lines lightly and play around — I thought maybe creating other triangles within these triangles would work. But nothing seemed to pop out. At one point, I thought I had possibly created an equilateral triangle in this (even though I saw one of the angles was 72! I was clearly desperate!). I started to get dejected at this point. I knew the circle had something to do with it…
  3. But seeing that 54s and 18s and 36s and 72s kept appearing, I thought maybe algebraically I should play around with the numbers (adding in 180 also, since I can draw a straight line wherever) to see if algebraically I could get a 30, 60, or 45. I tried adding and subtracting numbers from the set {18, 36, 54, 72, 180} looking for 30, 60, or 45. I figured if I could somehow do that, then I could find a diagram that would have angles I could get side relationships from. And then like a domino effect, I could get others. I don’t know. But after like 2 seconds, I got bored with this and didn’t see it as very efficient. My intuition was strongly saying I was going in the wrong direction. So I stopped:


  4. At this point, I was pretty dejected. I was slightly losing interest in the problem, thinking it was too hard for me. I tried to “force” a 60 degree angle in a diagram of that original blasted triangle. Hope! And then hope dashed!

  5. Damnit! I know the circle had something to do with it. It is just too nice to abandon the circle! Maybe…

    At first I drew all ten vertices for a 10-gon. I started connecting them in different ways. I thought I could exploit the chord-chord theorem in geometry, but that wasn’t good. I tried in that second diagram to extract part of the circle diagram and investigate it more. And the third was just more of the same. At one point, I was like e^{i\theta}=\cos\theta+i\sin\theta and was thinking I could somehow think of this as a problem on the complex plane, where each vertex was e^{ni\pi/5} and then look at the real parts for the x-coordinate and the imaginary parts for the y-coordinate. Clearly my mind was whirring, and I was going anywhere and everywhere. I actually thought maybe this complex plane thing seems ugly but it will be so elegant. But then I realized I didn’t know where to go if I labeled each of the points on the complex plane. Done and done and doneAt this point I put the problem away. Nothing was working.

  6. But after a minute, I couldn’t let it go! I wanted to solve it!!! So I went back. I thought I was getting too complicated, so I went simple.


    Nope. Didn’t help. But for some reason, this diagram and looking at the 72 reminded me of something I hadn’t thought of before. This is the leap that helped me get to the answer. And I can’t quite explain why this diagram sparked this leap. Which sucks because this is that moment that led to the rest of the problem for me! But I immediately remembered something about 72s and pentagons. And it hit me.

  7. So I drew what this connection was. My brain was whirring, and I was somewhere good…  

    I remembered the 72 degree angle appeared in a star. And this star was related to a pentagon. And that the pentagon had something about the golden ratio tied up in it. So I knew that maybe the golden ratio was involved in the answer. And where does the golden ratio appear? When there are similar triangles and proportions. I had my new approach and my inroad that I thought would work. Two triangles next to each other failed. Circles failed. But star/pentagon might work!

  8. So I looked at the original triangle and tried to figure out where I could find a similar triangle. And so I drew one line and created a similar triangle. I labeled the two legs as having length “1.”

    Initially, I was thinking I could do something with the law of sines. Because if you think about it, this is the ASS case — where you have that 36 degrees (circled), the side I labeled 1 (circled), and the other side I labeled y (circled). But you note that last side could be in two different places, which is why there are two ys circled. I still think there is something fun that I could do with this. But as I was doing this, I realized I was making things more complicated.

    I knew that the golden ratio came out of a proportion. So I abandoned the law of sines for the proportion. I simply set up a proportion with the two similar triangles. I first found “?” by doing 1/y=y/?. So ? was y^2This was exciting. I knew the golden ratio came out of solving a quadratic. Yeeeeee! At this point, my excitement was growing because I was fairly confident I was almost at the solution.

    Then I labeled the part of the leg that wasn’t ? as 1-y^2 (since the whole leg length was 1). Finally I looked at the third triangle in the diagram that wasn’t similar to the original triangle. It was isosceles and has legs of y and 1-y^2 so I set them equal and solved and not-quite-the-golden-ratio came out! (There was a mistake I made where I set y^2=1-y^2 and got y=\sqrt{2}/2. But I then found it and rewrote the equation y=1-y^2. This was the most depressing part of it. Because I couldn’t find my error because I was so tired. I went through my work multiple times and nothing. But taking some time away and then looking with fresh eyes, it was like: doh!)

    And so that was the end. I found if the original triangle had leg lengths of 1, the base was going to have a length of \sqrt{5}/2-1/2.

    I was so proud. I was on cloud nine. I was telling everyone! SO COOL!!! 

It probably took me in total 90 minutes or so from start to finish. So many false starts at the beginning, and one depressing transcription error that I couldn’t find.

The point of this post isn’t to teach someone the solution to the problem. I could have written something much easier. (See we can draw this auxiliary line to create similar triangles. We use proportions since we have similar triangles. Then exploit the new isosceles triangle by setting the leg lengths equal to each other.) But that’s whitewashing all that went into the problem. It’s like a math paper or a science paper. It is a distillation of so freaking much. It was to capture what it’s like to not know something, and how my brain worked in trying to get to figure something out. To show what’s behind a solution.



A curriculum is more than a set of papers

I wrote, with my friend Brendan, an advanced geometry curriculum. I was insanely proud of some of it. For those of you who know me, you know I love writing curriculum. It takes time, so much time, but it flexes the best part of my teacher brain. I’m forced to think backwards (“what am I trying to really do here? what matters?”) and requires creativity (“how can I get kids from point A to point B by having them do the heavy lifting, but in that sweet spot where I’m not necessary but their collaboration is? where that moment of invention and surprise is real?”). It is tough, and a lot of what I do isn’t great. But even my worst is better than any textbook I’ve seen.

Back to geometry. A few weeks ago, I met with one of the teachers at my school who is going to be teaching advanced geometry. I shared all my materials with her electronically, but I met to talk through things in more detail. But this meeting reminded me of something I’ve felt acutely for a few years: a curriculum is more than a set of papers.

As I wrote each piece of the geometry curriculum (or as I worked with my colleague as he took the lead), I had so much whirring around in my mind. I knew the intentionality of the questions and their ordering. I knew where kids would stumble. I knew where I asked questions that had no answers — on purpose — to get kids to think. I knew that I included a particular question in order to prompt a class discussion. I knew there were placed I needed kids to call me over to have a discussion with each group individually.  I knew I had included questions which were designed for me to verbally ask follow up questions. And of course I knew which things were hastily designed and didn’t work out so well when teaching.

But as I was attempting to go through my materials with her, it struck me pretty hard how hidden and implicit all those things were in that collection of papers that she had.

A real curriculum needs so much more, if someone else is going to successfully use it instead of me. When creating materials for other people in my department, who are teaching the same material, I started writing comments/notes in Word when I had a teacher move that I had in mind when crafting the problems:

teacher notesteacher notes2teacher notes3

It’s also a good reminder for me in the future. These notes help me and my colleagues remember what I was thinking of when writing my stuff. When I started doing this, I realized how a curriculum is a set of problems/activities with the intentionality behind the problems and teacher moves spelled out

In the past few years, I’ve had the fleeting and recurring thought: hey, I should organize all my geometry, precalculus, and calculus files neatly, and put them online in a systematic order for anyone to access. Maybe all of it will be useful to someone, maybe bits and pieces. I still sometimes think that. But what keeps me back from doing it is that gnawing feeling in the back of my mind: things need to be spelled out so someone else understands the flow and intention of each thing. And how to use it in the classroom. Where to stop. How to start. If there were any important “do nows” that weren’t captured in the sheets. Or knowing that someone was written as extra practice or to reinforce an idea that a class in a particular year wasn’t getting.

Over the past two years, it’s become harder and harder for me to open my feedly app and read blogposts. (I find most of my blogposts through twitter now.) It’s just been hard to find the time, and I get overloaded. And I haven’t had time to blog much either. And that sucks. But one thing I love about blog posts — that you can’t get on twitter/facebook/ed research — is that they often illuminate hidden ideas and bring to life something inert. Like when I read a blow-by-blow about an activity/problem set/ worksheet. Something that shows me the thinking that went into creating it, or better yet, how things unfolded in a classroom. What teacher moves happened? What were students thinking? [1]

If I wrote materials… and had a blogpost about how each day unfolded with those materials… that would be a curriculum at its best in my eyes. Because life is breathed into it. It becomes three dimensional. It involves people. The teacher. The students. And it makes explicit what is happening and why. [2]

Note: Funnily enough, Sadie posted a great piece on the idea of “curriculum” the day after I started writing this one! It is definitely worth a read.

[1] I like writing these kinds of posts — though they take a long time. Here’s a recent one:

[2] Obviously I won’t ever have the time to do this. But it’s nice to fantasize about. An extensive 180 curricular blog. Writing this post also reminds me that I need to get back to regularly reading blogposts.


The Friendship of Calculus: A Girl’s Journey Into the Unwavering Depths of The Third Dimension

One of my multivariable calculus students did her final project based around a book we read and discussed in class. It is called The Calculus of Friendship by Steven Strogatz. In it, the author writes each chapter about his own life and relationship with his former calculus teacher through the lens of some mathematical puzzle or concept.

My student wanted to do something similar, exploring her her multiple identities with her mathematical experience through the lens of multivariable calculus concepts. With her permission, I am putting up her three chapters here. It was a powerful experience listening to it as she read it aloud during her public presentation. I entreat you to read it. And although it may seem strange, there are many parts of it that are worth standing up and reading aloud. If you do that, you can inhabit my student’s voice for a while and really hear what she’s trying to say.


The Friendship of Calculus: A Girl’s Journey Into the Unwavering Depths of The Third Dimension

by Brittany Boyce


Chapter One: The Fourth Dimension

The fourth dimension as described in the dictionary is “a postulated spatial dimension additional to those determining length, area, and volume.” The key word in that definition is postulated. The fourth dimension is not something we can see, hear or touch, it comes from our imagination. In the times of early human life, the Mystics saw the fourth dimension as a place where spirits resided, since they did not inhabit our 3-dimensional world and were therefore not limited to our earthly confines. Albert Einstein, in his theory of special relativity, called the fourth-dimension time, but also concluded that time and space were inseparable. But what truly is the fourth dimension? In life, we try to make meaning of the world, what it will bring, what it will mean, how it will help us grow or not, and how it will change. Although we have a certain plan on what we want our world to look like, it is not something tangible that we can hold on to or grasp. The 4th dimension is something we can only imagine. We use the 3rd dimension, what we know and live through to help us envision the 4th. We assign colors and densities to certain points in space, and that helps us paint a picture that we can live with, but we are never truly satisfied.

In 1884 Edward A. Abbott, published a book about the problem of seeing dimensions that are not our own. In “Flatland: A Romance of Many Dimensions,” Abbott describes the life of a square living in a 2-dimensional world, which means he lives with triangles, rectangles, circles, and other two dimensional creatures, but all he sees are other lines because everything is flat. When the square finally has the chance to visit the third dimension with the help of a trusty sphere, a new world opens up to the square. Yes he is a shape like his 3rd dimensional counterparts, but he never took the chance to step out of his world and never sought to understand other worlds because he was never encouraged. At first, the square did not have the ability to comprehend the 3rd dimension, because for his whole life he only knew two dimensions. When the sphere takes the square out of the 2nd dimension, the square is finally able to see that there is a lot more to the world than just flat shapes like himself. The square was able to learn that other shapes have depth, color, height, etc. and because he was so amazed he turned to the sphere and asked what was beyond this dimension. The sphere, like the square, was appalled, unable to comprehend a world that wasn’t his own.

In this way, the sphere is like each and everyone of us. We are unable to comprehend other worlds, simply because we haven’t lived in other worlds. Our levels of privilege and different experiences explicitly prohibit us from knowing what each other’s lives are like. But does that mean we shouldn’t try? Does that mean we should just sit down and not try to understand anything simply because it is different from our experience? The answer to that question my friends, is a simple no.

∫  ∫  ∫

It was spring of junior year 2016 and I was sitting with my dean at the time, Mr. Brownstone, in his office going over my course registration. Now, this was it. This was the end, the last course registration I would ever have to do, the icing on the cake that would make me and my resume look appealing to the college of my dreams.

We were going over each class to make sure they looked okay, english looked good, history looked good, languages looked good, and art looked good. The only problem was if I was going to decide to jump into the deep end that was Multivariable Calculus. At first when I heard that Multivariable Calculus was an option I avoided that conversation like the plague with all my previous math teachers.

“You’re taking Multi right?” Mr. Brownstone said.

“Can I take Math Apps instead? They’re both different types of advanced math right?” I replied with a slight chuckle. He looked and me and laughed and replied with a hard “No.” There was no way he wasn’t letting me take Multivariable Calculus, and there was no way he wouldn’t make me step up to the challenge. As a kid who was already succeeding, I did not see the point in taking something extremely hard, but I went along with it anyway.

See that’s the thing about Mr. Brownstone and many other faculty members at Packer. They look out for you by pushing you to your limits and although in the moment you hate them, it’s always worth it in the end. Multivariable Calculus had already had its reputation of being a class, that would really “challenge you,” to put it nicely. Mr. Shah also already had a reputation of being one of the hardest teachers in Packer, so just thinking about this class was making my stress levels rise.

As a junior going into what would be the second half of the hardest year of my life, I didn’t think I was ready for this level of mathematics. I had always prided myself on being good at math and I enjoyed the subject as a whole but all the new variables, operators, and symbols in calculus had opened the door to a whole new side of math that scared me to be honest. Not that an integral sign is physically scary in anyway, but I was scared of the fact that I might not be able to do it. I was scared of needing help because growing up I was taught to be independent. Help was a foreign concept to me because I’ve always been told that based on my skin color no one was willing to help me and so I always had to fight for myself.They had always taught me to be independent because independence was power, and power was success.

Multivariable Calculus had always been a puzzling topic to me. What is it? I still couldn’t tell you. I was already confused by the addition of the alphabet, Greek and English, into the mathematical world, so when I heard that there could be multiple variables added into equations that I would soon be required to solve, I was even more worried. I remember thinking to myself that Mr. Shah would be too hard of a teacher for me and that the material would be too confusing. There was a part of me that thought that I would lose my status of “intelligent” and that I would let down all the people who told me I could be successful regardless of my background. In taking this class, I felt a certain pressure to do well as a poor, young, black, gay woman because not many others like me had this opportunity to study at such a high level in high school. Going to a place like The Packer Collegiate Institute, where I was one of few, always reminded me of my duty to the marginalized communities.

This type of math, meaning calculus, had always felt like a very distant topic to me. I never could picture myself being a “mathematician” because even though I was passionate about math and I had always been good at it, when I looked in the mirror, I never saw a mathematician.

∫  ∫  ∫

So it was September of 2016 and my first day of Multivariable Calculus with Mr. Shah had finally arrived. I had no idea what to expect and I was scared out of my mind. It was my second day of classes as a senior in highschool. The pressure was on. I had a chance to prove that I could be as great as everyone thought I could be. So here was my shot, my ticket to the big time academia.

Overall, looking at my new math teacher, Mr. Shah, he didn’t look so intimidating. However, his reputation still preceded him. See that’s the thing about Packer teachers, there are some that you can’t mess with. Some that are so passionate about what they study that they try to imbue you with that same passion in the form of school work. They expect so much of you, and give you so much work to better you, that you can’t help hate and respect them for it.

Every Packer upper schooler knows who I am talking about. Firstly in the sciences, there is Dr. Lurain, an exceptional chemistry teacher who often appears and often is very serious, but will light up and burst out in laughter in appreciation of a good chemistry joke. Next, in the languages there is Mr. Flannery, an inspiring Latin teacher who pushes his students to the breaking point every week with his famous tests. You will always catch one of his students learning lines, memorizing vocab, or reading some famous classical story. Mr. Flannery is no joke, but he has a devout dedication to each and every one of his students. The list goes on and on, but Mr. Shah was one of those teachers. Students told me how they were required to write essays on their tests or be so thorough in their answers to get full credit. But, he didn’t have the demeanor of a mean and strict teacher, he was very passionate about math and he didn’t look like he planned to intentionally make my life a living hell.

∫  ∫  ∫

In the first few days of Multivariable Calculus with Mr. Shah, I remember thinking “okay, come on hit me! I can handle it.” I was expecting some complex problem that I couldn’t handle or some other problem that required, some “higher math” that required prior knowledge I didn’t get a chance to grow up with. Instead, Mr. Shah nurtured us, all of us.  He taught us not to be scared of Multivariable Calculus. He taught us that we were prepared for 3-dimensional calculus, and the third dimension was just a step up from the second dimension. He made us aware that we already fought hard enough through Calc I & II with Mr. Rumsey, which was a battle of its own, to be sitting here together taking the same class. He never said it was going to be easy, but he made us feel like we were prepared from the bottom up. But, this comfort and reassurance is not something everyone in the world has the privilege to have.

At times, going up a dimension can seem scary. Most often, in our world things can be complicated enough, which causes us to forget that there are things that are higher than ourselves and more important than ourselves. If you’re like me, you use the fact that two-dimensional calculus was already hard enough, so why study 3-dimensions? Why go beyond what you already know? What’s the point?

The point wasn’t to solve the problem right every time or to be able to understand the most complex things first. It was to be willing to take that step into the unknown in the first place. I had an amazing opportunity to try to understand a world that didn’t necessarily welcome me with open arms. I wasn’t lucky because I had the intellectual ability to take Multivariable Calculus. I was lucky because I was one of few students who had an instructor that made me feel like I could understand the higher maths. Not many kids my age have the ability to study the higher maths, or to even believe that they could study the higher maths, especially students of color, women, and LGBTQ+ students. Today’s education system lacks mentors that have the ability to push kids in the right direction and to make them believe in themselves regardless of their social status. What is unique to my experience is that as a woman of color, low socioeconomic status, and who is proud to say that she is a part of the LGBTQ+ community, I had people around to support me. There was never one time I felt that my peers or teacher didn’t think I was worthy enough to be there taking that class because of my gender, race, sexuality, or socio economic status.

However, although my reality was brighter and more positive than other students who share my identity and do not have the same support system I do, I cannot just be grateful and move on with my life. I must think about those who have to fight harder, speak louder, and do better than I do to hold their place in the classroom and the community of the higher maths. I must bring attention to their fight even though I only know my own.

Chapter Two: Line Integrals

A line integral is essentially integration of a function along a curve. But, that means nothing to most of you. On each curve there are an infinite number of points that trace the path of the curve, determining what it will look like, how it will behave, and how it can be analyzed. Not each point is worth more than another in value or in status, but each plays an integral role in defining the curve. Let’s just say, all points are created equal. But what does that curve really mean? What can it do for us and what can we do for it? Sure it can be pretty to look at or cool to trace, but it all means nothing if we can’t make something out of it or give meaning to it.

That’s where our friend the line integral comes in. To many, it looks like a weird “s.” To my readers, three of these majestic creatures in a row means that I am switching directions or switching to a different moment in time. But to a mathematician, the line integral gives meaning to the curve. It takes the path traced by the infinite amount of points and cuts it into infinitesimally small pieces and adds it all together into the culmination of a single amount, quantity, and meaning. The line integral represents the culmination of everything we’ve been through and the addition of all those infinite moments into one big picture called life. But, while you may have all the pieces and the trajectory, solving the line integral and finding the meaning behind the trajectory, will not always be easy.

Often times, in school we as children are set on a given path or a chosen trajectory, let’s call it f(x). We are given a curve C, and we are told to follow it. We get the grades, play the sports, and be the children our parents want us to be. But what does it all mean when we have hit all the points, traced the path, and completed it? What is it supposed to mean? How are we supposed to evaluate our lives when we haven’t even begun to make any choices for ourselves? And how are we supposed to deal the the fact that we may never make meaning of our chosen path even though we might have all the tools?

The creators of calculus dared to confront this problem through math, because of course, it was the only option. To them, doing the work, solving the integral and making meaning of such a path, was more important than perhaps what the integral meant numerically. Frankly, to be the most cliche, it’s about the journey, not the destination. Not all integrals are meant to be solved in the most complex way or with calculus; sometimes it only takes simplest geometric proof or the simplest meaning of life that can propel you in the right direction, or help you move forward in the problem.

Do you ever wonder how long it takes to change your life? What measure of time is enough to be life altering? Is it four years like high school? One year? A 2-semester calculus class? A semester long, history course? Can your life change in a month? A week? A single day? We’re always in a hurry to grow up, to go places, and get ahead. But when you’re young, one hour or even 50 minutes can change everything.

Through integration, a curve becomes a series of tiny straight lines, working together towards one common quantity. Through integration, life becomes a series of tiny moments working together towards the culmination of you and what your life means. However, sometimes it may be hard to make meaning of a certain time in your life. Sometimes that moment may be unsolvable and that can be frustrating. But, the important thing to remember is that each infinitesimally small piece or small moment works to affect the meaning of your life. Each small experience adds something to your journey.

∫  ∫  ∫

I had made it through the first semester of Multivariable Calculus feeling like I could actually pursue mathematics in college. But, I wasn’t completely sure what helped me get here. There was some small moment along my path, where it just clicked. There was something about the elegance of Multivariable Calculus that caused me to light up during every class. Surely there were days that I was tired and defeated, and felt that I could not take anymore of Mr. Shah’s high expectations; But, something about the math itself always brought me back to that stillness I felt. The stillness that was almost calming at the sight of an elegant proof or after spending time doing hard rough algebra, fighting and wrestling with exponents, variables, and symbols to finally get an answer. I didn’t know it then, but that stillness was my ability to feel passionate about math. I had a willingness to understand the concepts behind the algebra I was doing, and had come to appreciate the conceptual approach rather than the hard hitting, laborious algebra I was used to my whole life.

For the more complex conceptual solutions, sometimes I felt cheated, when the very complex parts of the problems were reduced by simple geometric approaches. I saw the immense power of calculus, and I didn’t want it to be reduced or lessened by geometry. There was something about putting my head down and jack hammering through the hard work that always pleased me, but I soon learned that it wasn’t cheating, nor did it lessen the power of calculus in any way.

One simple solution to a complex integral we often faced in class was the integral of cos^2(x) from zero to kπ, k being a multiple of ½. Now for all you mathematicians out there, you know that this integral is no joke. There is no simple u-substitution or power rule you can use to solve this, it must be solved with integration by parts, which is a method that requires some of that “jack hammering” I loved so much.

The proof of ∫ cos^2(x)dx using integration by parts, goes as follows:


Using this integral, the area under the curve on the interval 0 to π/2, makes the integral equal to π/4. While this solution did take intuition and elegance, such as turning the ∫ sin^2(x)dx into ∫ 1 – cos^2(x)dx. Then, adding the ∫ cos^2(x) to both sides to make the proof simple algebra. We learned to tackle complex integrals like this using integration by parts in Calculus I & II,  but 3-dimensional calculus builds on 2-dimensional calculus, so complex integrals always popped up in daily problems. I admired the hard work that calculus required, and the instincts that one could gain from solving such problems, but let’s be real, no one is going to remember the solution to a random integral because one random integral is not that important to all of math. So the question we’re faced with is do we fill our minds with random memorizations of quantities representing areas under curves or do we find another way to remember?

One day, Mr. Shah gave me and my fellow peers a new tool to add to our mathematician’s tool belt. He gaves us geometry. He took us back to our roots and showed us that sometimes simplicity is the ultimate sophistication. So we tackled the same solution.

What is ∫ cos^2(x)dx?


We were essentially tasked with finding the area shaded above. Sometimes when you’re in the middle of solving a problem, and this integral pops up, you can’t result to algebra every time. Sometimes the matter is too urgent and the problem can’t wait for you to do all this algebra. So Mr. Shah showed us one single shape that would change the way we would approach any integral for the rest of our math careers.


Now look closely. The area under the curve is equivalent to exactly ½ the area of the blue rectangle. Now the graph tells us that the length of the rectangle is 1 and the width of the rectangle is π/2. That makes the area of the rectangle  π/2 • 1 =  π/2, making half the area of the rectangle π/4. BOOM. One complex integral simplified with the power of geometry. This proof amazed me. I was astounded by the elegance of such a simple solution. I mean a seventh grader could do this.

∫  ∫  ∫

Calculus was never meant to be unreachable. Renowned mathematician Edward Frenkel once said, “mathematics directs the flow of the universe, lurks behind its shapes and curves, and holds the reins of everything from tiny atoms to the biggest stars.” The beauty that math holds has become a privilege unreachable to those who are marginalized everyday for their skin color, race, and sexuality. Everyday students of color and women are told that they cannot or should not see this beauty, the beauty of math is held from them until they climb and fight to the point where they are so bruised, broken and beaten that they give up. There are increasingly low percentages of black and Latinos in high-paying, high-status jobs in finance, science and technology. Since  perceived intelligence in the higher math communities are increasingly influenced by racial prejudices it is getting harder and harder for students of color to believe that they can be something more than the stereotypes. Fundamentally, this is a question about power in society.

Being a student of color who had to claw tooth and nail and go to highly selective programs to even be in a place like Packer, I have experienced that loss of a love for education. Being a black girl who was able to show her intelligence at such a young age, I was set on the path to success. Do your school work, get a good job, be successful. But at the time, I didn’t really know what it meant to be successful. I still don’t know what it means. Most of the time, success is dependent on whether or not I beat the system. I was never told to do what makes me happy. I was told to do what makes me money. I never had the privilege of growing up studying what interested me, or what I was passionate about, and I never knew that having the chance to delve into European history or a new language was a privilege. I was too busy preparing for survival. I was busy getting a head start on the material I needed for the future, so the pressure and the rigor of a predominantly white and male setting wouldn’t defeat me.

There are kids out there who don’t get to enjoy and love knowledge because they are not taught that knowledge is beautiful, they are taught that knowledge is power, and that power is the key to success. Academics never become leisure activity because survival is more important than leisure. They are set on their own path, and asked to make meaning of that path without loving the path in the first place. At the end of their trajectory, they are left at a crossroads, choose another path that they truly love with the possibility of failure or never love a path at all.


Chapter Three: Path Independence

Path Independence shows that the value of a line integral of a conservative vector field along a piecewise smooth path is independent of the path; that is the value of the integral depends on the endpoints and not the actual path C. Now wait a second, am I hearing that vector calculus thinks that it’s about the destination not the journey? Frankly, I don’t blame the creators of this theorem. Most of our world thinks life is path independent. People think that they can see past their privileges and just go on with their lives and that every accomplishment they achieve is independent of a third party. But is our world truly conservative? No pun intended. Do we live in a world where, as one of my favorite bloggers puts it, “instead of recognizing our unfair privileges, we just build walls around us and project out way of life as normal. Any story you tell about how you got where you are that doesn’t include land theft, profiting off of forced, unpaid labor, illegal occupation, murder, assault, theft, psychological and physical warfare, exploitations, and a culture of complicity is, you know, a lie.”

If it is then what’s the point of me fighting so hard to hold onto my passion for mathematics? What’s the point if my journey, which might be ten times harder than someone else’s is recognized in the same or even a lesser fashion than someone who got to the same endpoint. Isn’t there supposed to be beauty in the struggle? Value in someone’s journey? What’s the point in finding the meaning of your path if it is weighed the same as everyone else’s path who started and finished at the same places you did? How are we supposed to try to learn and value the experience of others if we just value where we’ve ended up? Does this mean that the situation you are born in, something that you can’t control, has some type of influence on the overall meaning of your path? It shouldn’t.

∫  ∫  ∫

The problem of the conservative vector field that is our world always had a place in our Multivariable Calculus classroom although we sometimes didn’t know it. Every Day 4, when we had class for 90 min, we would hold a book club. Mr. Shah would assign us a different piece of literature to read regarding math, whether it was Flatland by Edward A. Abbott, or The Calculus of Friendship by Steven Strogatz, or Love and Math by Edward Frenkel. As a senior, already up to my eyeballs in work, I disliked him for giving me this reading on top of all the math problems he had already assigned me. I never knew it then, but what Mr. Shah was doing was important work. He was showing us what is was like not to be path independent. He made us value the stories of the mathematicians before us, so that we could know how hard it could be for the person sitting right next to us to be successful in the mathematical community. He made sure to make us feel the responsibility we had to the ethics of the math community. We discussed the politics of math, the religion of math, and the inequities of math every week.

He showed us that while learning the material itself was important, the story behind the material is just as important. In life and in math, there are multiple approaches to solving problems. Often times in math class, Mr. Shah highlighted when two students had different approaches to the same answer and would even have them write it on the board for the whole class to experience. Each approach would have something different. Maybe a trick, a new tool, or even a slight adjustment. When I thought about the way a problem was solved, I never really saw the value in the different approaches, all that mattered to me was that the same answer was achieved.

∫  ∫  ∫

In my last quarter of my Packer career and a Multivariable Calculus student, we returned to pure learning. We watched a series of lectures, which was considered our preparation for college math and the whole host of difficulties that would come with it. As the time was winding down, and I began to think about what my final project would look like, I admired Mr. Shah for making us do something that we were interested in and that was meaningful to us. I had never gotten the chance to do so, while still preparing for “survival.” Once again, I hated all the stress and work it brought me. But, I was very grateful that I had gotten a chance to make meaning out of my experience. While the culmination of my mathematical trajectory or “path” at Packer was not completely numerical or quantitative, the experience of finding meaning through math has been life changing.

Math is beautiful, and I only hope that seeing this beauty no longer becomes a privilege in this world, but a necessity. Everyone deserves to believe that they can be passionate about something and not be deemed a failure. No kid should have to carry the weight of their struggle alone. We must not be path independent, we must be aware of the stories that are around us.