Senior Letter 2013-2014

Every year I write a letter to my seniors. Each year the message is pretty much the same, though the way I deliver it may change a bit based on the class and what I’m feeling at the time. Each year I hate my letter when I’m done, but I decide I’m going to give it out because it’s a tradition and I don’t want to break it, and I convince myself it is not that awful. I hand it out. I’m grateful after I do, because… I suppose I need closure. I have worked with these kids closely for a year (sometimes more). And I have come to care about them all. And although it happens every year — they leave and I stay — and from this point on they slowly begin fading from my memory, right now they are in my life in saturated colors and I know I’m going to miss them and I want the best for them.

So even though I currently hate it, here is this year’s senior letter.

It came packaged with their “who I am” sheet that they wrote about themselves on the first day of class, and two cards I had printed.


Switching Things Up, Need Help: Geometry is on the Horizon

For the past seven years, since I started teaching, I have been teaching calculus. When I started, I had 8 students in my class. Now I have 36 students. I’ve had to shift how I’ve thought about the course tremendously, and I’ve undergone a dramatic transformation in the content I teach (it is non-AP) and in the style in which I teach it. Seven years with a class is both a blessing and a curse. And honestly right now, I’ve reached the end of my usefulness for the course. I’m spinning my wheels. The only way I would be able to do a better job with it is to leave it for a few years and come back to it with a fresh pair of eyes.

And luckily, I have the opportunity to try something new. Next year, I will be giving up Calculus to teach an Advanced Geometry course for the first time. In fact, it’s the first time I’ll have ever taught geometry at all.


When I first began teaching, I was scared of geometry. Partly because as a student in high school, I found geometry to be uninteresting. It certainly didn’t have the elegance of algebra, at least the way I was taught it. Partly because I realized in that course — more than any other course — you as a teacher really have to focus on hard things. If you want kids to be able to do a proof of any kind (two-column or not), you are really teaching intuition building and connection making. Which is tough, and daunting for any new teacher, and this is why I recoiled at the thought. Now, years later, I see this as such an exciting challenge.

Right now, I am not anywhere about how to teach this course. And in fact, I’m only teaching one section and the other teacher is teaching three sections. But he’s very open to really revitalizing the course. So now we’re in exciting territory. Before I go bananas on scouring everything out there, I thought I’d crowdsource.

For any of you geometry teachers out there, if you have time to answer one or two (or all!) of these questions in the comments, I’d be ever so grateful!

1) What are your favorite geometry teaching resources — both online and offline? I’m talking books, websites, applets, manipulatives, whaever?

2) What are your favorite math teacher blogs that focus on geometry?

3) Is there a lesson you absolutely could not imagine teaching Geometry without?

4) Do you teach the course with a connective thread? Like: We are studying space and the properties inherent in space as we build space? Or: We are studying exactitude –and in particular, how we define mathematical entities so they yield uniquely understandable creatures? Or: We are studying “measurement” (in the vein of Paul Lockhart’s book).

5) I’m concerned that our kids lose a lot of their Algebra I skills when they take geometry. The other teacher and I have talked about putting coordinate geometry front and center from the beginning to help with this. Do y’all do anything else that helps keep their algebraic skills sharp, and maybe even push them forward?

6) Anything else? Problem solving? Sangakus? Geogebra use? Things you throw out because you feel strongly it’s only taught because it’s always been taught? Incorporation of Euclid’s Elements or math history? Graphic-design-y projects? Math art?

UPDATE: WOW, everyone, thank you so much for your resources and advice and for taking the time to type out so much great stuff. Now I’m genuinely THRILLED and CHOMPING AT THE BIT to get started re-learning geometry (and then teaching it). I am going to sort through things this summer!

Experimentum Crucis: A Symposium Course


This January, for seven days, I taught a seven day course with a friend and fellow teacher. Our school eliminated midterms and instead instituted different programs for different grades. Juniors and seniors were given the opportunity to sign up for full-day courses designed and taught by faculty on topics of interest. Faculty were given the opportunity to design courses which got kids to think about topics in a different way.

My co-teacher and I developed a course that was designed to be interdisciplinary (we were working at the intersections of history, science, and philosophy), hands-on (students would be working in the laboratory), and rigorous (meaning kids would be expected to think and work at a high level).

Designing and teaching this class was one of the hardest things I’ve ever done as a teacher. And I don’t know — honestly, I don’t know — if we were successful or not. Even with the feedback we received. Thus even though it was challenging, I’m not sure I felt it was rewarding. In fact, the reason I’m writing this blogpost now, months after this, is because I was so exhausted with the whole thing I couldn’t bring myself to even think about it in a reflective or objective way.

The origins of the class go back to the previous year, when my co-teacher and I started trying to envision precisely what the big picture ideas were, and how we were going to get kids to go from point A to point B in their thinking. This also was coupled with the question: how the heck do you design seven days with the same group of kids, from 8:3o to 3:15. Seriously put yourself into our shoes for a second. Initially, it’s pretty exciting! All this time! Do what you want! But then you realize: you are going to have 12 to 16 kids in your charge, and you need to fill up that time with multiple activities! Quickly this went from exciting to daunting and anxiety-filling. For months, the co-teacher and I would have meetings, read books and articles, come up with ideas, refine our ideas, and throw out our ideas. Coming up with a lesson plan for a single day took weeks of work. The agony, the hours, the frustration… I don’t wish that upon my worst enemy. But we finished.

Our course abstract:

Can you imagine building a battery without the concept of electrons?  What would it be like to describe chemical reactions without discussing atoms?  Would you believe Einstein’s theory of relativity if no text book told you to and there were no way to test it?

In this course, you will have the opportunity to put yourself in the shoes of scientists who (in retrospect) revolutionized the way people viewed and understood the natural world.  By carrying out famous historic experiments, you will explore the process of creating “scientific models” and “scientific facts,” many of which we now take for granted as self evident. This course will be hands-on and interdisciplinary. In addition to lab work, we will read primary and secondary sources that will allow you to place science in historical context and understand scientific knowledge making as a process and a product of its time.

Our course objectives:

Through this course, students will explore:

  • science in historical context
  • how science is influenced by and a product of its time
  • that the process of science involves models changing over time
  • that what we take for granted is often messy, weird and sometimes illogical
  • that science is a human endeavor
  • that the making of science is a process
  • how scientific “facts” get accepted/discarded –  that ideas are nothing without the acceptance of many people

and ask the big questions:

  • What is an experiment?
  • What is a scientific fact?

Anchor Texts:

Thomas Kuhn’s The Structure of Scientific Revolutions
Original papers by Robert Boyle and Alessandro Volta
Secondary texts


Originally, we planned to have a number of experiments: Proust, Boyle, Volta, Oersted, Einstein. However because we had a snowday (there went Einstein and the discussion of thought experiments), and because some of the experimentation took much longer than expected, we had to eliminate more (Proust and Oersted). Thus, we only ended up working extensively on Boyle and Volta.


One day was spent on a field trip to the Chemical Heritage Foundation in Philadelphia, but the rest of the days were spent having deep class discussions and carrying out two in-depth experiments in the labs. We did Boyle’s Law experiment, and they had to bend glass to make their own J-tube, and play carefully with mercury. (We inducted all our kids into the Royal Society, after reading bits of the original charter, and administering the oath that the initial founders took.) Our kids saw that our modern instantiation of Boyle’s Law (PV=k) was nothing like the original formulation (they only were given Boyle’s original paper to guide their research and help them figure out how to reproduce the original experiment), and they started to get at the idea that Boyle was looking at his experiment through a totally different lens (“the springiness of air”). My favorite part was when kids saw how their little sidebar about Boyle in their chemistry textbooks was just a black box for so much! And how it wasn’t just “one crucial experiment” that suddenly worked and changed our understanding. Mwahaha, the title of our course is precisely the thing we aimed to get our kids to debunk. 

Our second experiment was building (well, improving upon) the first voltaic pile. Again they only had Volta’s original paper to work from, they were given many materials that Volta mentioned in his paper to play around with and test (e.g. lye, silver, zinc, tin, coins, leather, cardboard, salt water, etc.), and they were working to win le Prix Volta (a real prize Napoleon and the French Academy of Science offered for research in electricity, after Napoleon saw Volta’s original battery demonstrated). This contest was good to talk about collaboration and competition in science, but my favorite part was having kids read a challenging history of science article about what actually was behind the creation of the battery (a torpedo fish!) and what sorts of things had to have happen for there to be the physical and intellectual space for Volta to even have the conditions for him to come up with his Voltaic Pile. That the battery is historically situated, and tools, ideas, and people had to come together in a specific way for the battery to emerge and look the way it did. I also really liked that students could understand that there could be an explanation of electricity that didn’t center around electrons.

That dovetailed really nicely into how we were talking about Thomas Kuhn. We used Kuhn’s Structure of Scientific Revolutions as our core text that they were reading extensive bits here and there each night, and although I was worried it would be too abstract for them, they grappled with it and came out victors. And I think (hope) it was a real mind-blowing experience when they realized that “old” theories weren’t “bad” because those scientific practitioners who adhered to them were dumb (or at least, weren’t smart enough to see the Truth with a capital T). And listening to them discuss Kuhn, grapple with the idea of Normal Science, and start to see glimpses that (1) science isn’t accumulative in the simplistic way that textbooks tend to say it is, and that (2) we always are looking at data, theories, experiments, observations through specific eyes, and what we see is dictated by the paradigms we accept.

Images: Here are images from the Symposium, without student faces in them. (Hence, we don’t have the majority of my favorite pictures.)

“Explore Mathematics: Part II”

I felt like my first venture into “Explore Mathematics!” was so successful last quarter with my Advanced Precalculus kids that I wanted to build upon that. So this is what I’m doing for “Explore Mathematics!: Part II”

  • Last quarter students scoured the web and did 5 different mini-explorations which exposed them to all the neat math that exists outside of our standard curriculum. This quarter students will be doing up to two more in-depth explorations.
  • Because I don’t want this to be seen as busy work, doing “Explore Mathematics!: Part II” is going to be completely optional. I was glad to read that almost every kid who did the five mini-explorations last quarter didn’t end up finding it busy work, but I suspect doing it a second time would feel tedious.
  • To have some sort of incentive for those who do it, I am going to make each of the two explorations worth 12 points. These explorations will count as a mini-assessment (normal assessments are around 50 points). This is useful for kids because our fourth quarter only has 18 days of instructional time (seriously) — so there are only two major assessments and one minor assessment scheduled. Doing these explorations can act as a way to get another mini-assessment grade in there, that will be low-stress, high-reward. [1]
  • I’m not framing it around the grade boost it will likely provide, but around the fact that it’s an opportunity to do some awesome math explorations, for anyone who wishes to do so.
  • It is still pretty open-ended, but I’m now looking for students to write something to get others to see what they find interesting/intriguing/awesome about something.

Here’s the document I just emailed my kids:

Here it is in .docx form in case you want to modify it.


[1] Yes, I do SBG with my calculus kids. Yes, I know how ridiculous this sounds, me playing the “point game.” I almost wanted to make it so that there was no external reward, but our kids are so busy with so many things that I know even a little incentive will go a long way. I’ve been at my school long enough, and know our kids well enough, to know this is doomed to failure without a little external reward.

My Wunderkammer: A Visual Resume

About 6 years ago, I remember receiving a stack of resumes for a math teaching job. We were looking to hire someone to join our department, and there were so many resumes and cover letters to go through. Over 50, maybe around 100. And my eyes started glazing over. The resumes looked similar, and the cover letters were banal. And then: one applicant stuck out.

It was a cover letter that gave a link to a really simple website, and on that website was an educational philosophy, a few sample tests, and some student work. Although it was pretty basic, what I liked was that on that simple site I got a much better sense of who this candidate was. I loved the idea. And I decided then and there that I would create my own teaching portfolio online that would capture who I was as a teacher.

This past summer, I did it.

To be clear: this isn’t a reflective teacher portfolio.  It’s a descriptive teacher portfolio. It is something that I put together — a mishmash of snippets — that together hopefully gives a solid sense of who I am, what I do, what I believe in. I think calling it a visual teaching resume or a wunderkammer best describes it. (Click on the image to go to the site.)


There are a few missing things that I would like to add to this site at some oint:

  • I would like to add everyday samples of student work. Not projects. Just everyday stuffs.
  • I would like to add a section about the two week history of science course I designed and implemented with another teacher this year. (See Days 80-87 on my 180 blog for more.)
  • I would like to add a section about the “Explore Math” project (more info here and here) I did in Precalculus this year.
  • I would like to finish the student quotation page. I actually have quotations typed for a number of previous years, but I do not have more recent years ready.

It was pretty simple to make (I used the free website creator weebly) and I hope if I ever were to go on the job market, it would catch the eyes of whoever had the giant stack of cover letters and resumes in front of them. I wasn’t really going to make a post about my visual resume, or share it with anyone, because I thought: who would care?

But heck: maybe someone out there is going on the job market and thinks the idea is worth replicating? So I decided to post.

Explore Math (Reprise)

At the beginning of the 3rd quarter, I did an experiment in my Advanced Precalculus classroom: Explore Math. This post is the compilation of the survey results from my kids on this experiment. So if you don’t know what the activity was, read up here, and then see what this survey is all about. I will share examples of some of student work for this experiment later. Part of the assignment for students included submitting one exploration to our school’s math-science journal, Intersections. When this year’s issue of the journal comes out, I hope to link to my kids’s explorations!

The question in the survey:

The “Explore Math” project is something I’ve never done before. I explained my reasoning behind it — which is I wanted to encourage you to see that there is so much more than our curriculum covers, and let you just have fun looking at math stuff outside of our curriculum… and get some easy credit for it (almost everyone is getting full credit for the first batch of things I’ve seen). However, as a teacher, I know something like this could easily be seen as busy work, and that was my big concern — that it would feel like a chore rather than something you actually want to do.

This is me laying my cards on the table. If I came to you in the student center and told you this and asked you for your thoughts, what would you say?

Every Student Response In Entirety:

I really liked the Explore Math project and I definitely would say it was an overall success. I loved how many options we were given for what we could do, and the fact that you gave us the options was great because otherwise it can feel like you are just trying to desperately research and find a topic to write about. My Explore Math topics I thought were extremely interesting, and it was cool to even connect some to the stuff we were learning in class. It was a lot of writing, which is something foreign for math classes, and also made it kind of difficult to grasp exactly how to format what we were writing (five page essays for each topic?). One other thing that was a little stress-inducing was the deadline and I know it was for a problem for most people that it often happens that when there are multiple assignments due on one day, students leave them all and do them in bulk. Because of this, having the deadline of the first three due in February was definitely helpful. Overall, I really loved the assignment.

I really liked this project! I found a lot of things about math that I would have never known about if we weren’t assigned this project. I learned new formulas, new (very addictive games), great youtube channels and informative popular articles. I found an entirely new community online that I did not know existed.

At first I expected it to feel like a bit of a chore but when I actually sat down and did it, it was pretty fun. I think it was great that there were multiple ways you were allowed to “explore math.” I also thought it was amazing I could play around with the project a little bit to find areas of math that are aligned with my personal interests. Being able to think about how math affects our society, in a math class, was an amazing interdisciplinary activity. I think it’s good that not every option was a math puzzle — that would have felt constrictive.

I would say as long as the students are innovative, interested and patient people the project sounds wonderful. The student, if very interest in math, should be encouraged to further their mathematical understanding, and find means in which math is even more interesting to them as it was prior. Emphasizing the point that one (the student) does not need to seek the more difficult problem or most tedious theorem is also very helpful, as the student will be encouraged to explore areas of math in which really interests them.

I would say that I absolutely love the explore math project. I have always been a person who enjoyed math that connected with the world. Being in a classroom memorizing formulas was never my interest and I was psyched when you announced the project. I think that this project can be very helpful in putting math on the global scale for students who only see it as a class in a school. This opens their eyes to new heights math can taken and how much math actually helps outside of the classroom.

I agree it felt like busy work some. I find it weird that something that’s supposed to be us having fun exploring math had a grade and time constraint attached to it. That’s one thing I didn’t like.

All I have to say is that this was not busy work; in fact it was productive and learning work. I found this to be incredibly intensive and interesting, and it broadened my horizons of the understandings of applied mathematics and sciences, and introduced me to things that I had previously trembled [at] before, like string theory, for instance. I thought this was a great project and a simple and easy way to get us thinking in a mathematical mindset, and I am definitely reaping the benefits from it, because I have come away with much more knowledge about certain aspects of math that I had previously not known. I really wouldn’t know what to change because I liked these individual explorations so much and they intrigued me so much. Thank you for giving a projected that I was thoroughly interested in, seriously!

 For someone who is very interested in math in and out of the classroom, I am generally engaged with math concepts that are not a part of our curriculum. Thus, this was a good experience for me in that I was able to get credit for simply enjoying and exploring math; it also perhaps pushed me a little bit to go further than I normally would in exploring mathematical concepts online. However, for students who don’t love math outside of the classroom, I could definitely see how this might have seemed like busy-work. If you don’t genuinely enjoy math, then writing a lot about it and research about it is going to be cumbersome, but if you do, it’s enjoyable.

I really liked doing the explore math assignment. I liked that you were giving us an outlet for us to not just do the math that needs to be done in order to complete the class. This assignment allowed me, personally, to dive deeper into how math can be applied to the world and that math is actually occurring all the time. Also, I remember not really understand[ing] infinite series and then I did an explore math with infinite series that really helped me because it was a visual representation that really clicked with me.

I think that initially I thought the project might just be busy work and I didn’t really understand what we were expected to be doing. Once I read over the assignment and saw the scope of the projects we were allowed to do, I was much more interested and saw the project completely differently. I think that it is important to highlight, when giving the assignment, how broad a range of options you have when doing this, and that there are so many math projects that relate to everyday life that could be interesting if you just think about it, rather than relying on the assignment sheet completely to guide you.

Personally, I have enjoyed what I have done so far. Just recently, I voiced my concerns about the state of math in America and was able to comprehensive research about the bitcoin that I would not have done on my own. That being said, some of this has seemed like busy work and stuff “I just have to do for credit.” Since it seems like you genuinely want us to enjoy the project, it might be made better by making it extra credit. That way, we could be able to explore as much as we want without worrying about our grade.

I had a really awesome time doing my Explore Math assignments, but the one thing you could do to make it less busy work is make it 3 different assignments, rather than 5, and make them a little more in depth, and more interesting in that regard. I think that if the students only had to do 3, they could expand more on what they were interested in.

I really like the idea, but for me personally, it turned into busy work. Not because I find it boring but because I have so much other work that it gets pushed back towards the end of my load. I would like to spend more time on them, so possibly have it on top of the nightly work for math, designate a night specifically to explore math.

This is practically the farthest thing from busywork we can do! Repetitive problems often seem like busywork. Practice is always good, but once you have something down, it can be quite annoying to practice it over and over again. Sometimes i feel that way about homework, but with this project we’re choosing any math-y thing that interests us! We have a lot of freedom, and hopefully it piques an interest in math outside of the curriculum. This project is great, personally, I wish I had taken more time with it. As long as you don’t procrastinate too badly with it, I don’t see how this project could be a chore, unless you claim to hate math.

I LOVED this project, and I wish we got to do more things like this throughout the year. (I know we can do things like this whenever we want, but it’s really nice to get some recognition and the chance to formally share your math ideas with others.) As a side note, this project was also interesting to be doing while looking at colleges for the first time. I know that sounds like a really strange thing to say, but getting to enjoy math in new contexts, such as music theory, has given me new ideas of things I would like to pursue and take classes [on] while I am at college because we don’t always get to learn about things like this on a daily basis in high school.

I do admit that I wasn’t very enthusiastic at the start of the project, but as soon as I started I completely changed my mind. Most of the work that I did was stuff I had never done before and might never do again. I was genuinely interested in what I was doing, and it was great to be able to choose what I focused on instead of being told what to look at.

I understand why you assigned this project, and I think it is very important to see the relevance math has in the world. This breathes life into the abstract “why are we learning this” type that doesn’t appear to have anything to do with life outside the classroom. However the problems with this assignment are that I didn’t know what I was searching for. When I found the Sloane’s Gap video and paper I felt like I struck gold after seemingly endless mining. However the mining part is very un-exciting. Not un-exciting enough to undo the excitement of finding the cool stuff, but it’s not very encouraging either. I wouldn’t want this assignment to turn into a chose 5 of these pre-determined projects because that wouldn’t make anyone feel like anyone feel like they’re venturing outside the classroom. I’m not really sure what I would do to change this assignment, but I think it really is a good idea that with some refinement could become a really dynamic way to get into math. I think keeping it low pressure and “easy credit” is the way to go because stress + ambiguity about an assignment is a terrible combination that would only end in resentment from your students, and students not enjoying their work.

Honestly, I had quite a bit of fun with the “Explore Math” project as I saw many cool analogies of real-world applications of math. For example, one of my five “research topics” was the probability and randomly guessing on every SAT multiple choice question. I learned that the probability is horrifyingly low — I already knew this, but not to such an extent. Furthermore, I saw some very cool analogies in this SAT topic; for instance, if a computer were to take the SAT 1 million times a day, for five billion years, the chance of any of the SATs resulting in a perfect score on just the math section would be about 0.0001%. Crazy, I know!

Mulling things over

Tonight at school we had our inductions into Cum Laude (like national honor society) and Mu Alpha Theta (the math honors society). And our guest speaker gave a rousing speech about his life, and how it’s okay to have fear, but the biggest hurdle to doing something significant with your life is accepting the fear and moving on in spite of it. Accept it, own it, be afraid, and then still forge forward.

At the end, he said something powerful. The first thing one needs to do to when leading a purposeful life is to say what it is that you want to do. Articulate it aloud. And that is scary. Making it public so you can hear yourself say it, but also so someone else can hear you say it. So it becomes real instead of this thing that bounces around in your head but never gets out. And so at the end, he told everyone to be quiet, and he was going to say something he wanted to do, and then afterwards there should be silence… and when anyone else wanted to say something they wanted to do — something they would declare out loud — they should stand up and say it, and then remain standing. This was an open invitation to the students in these honors societies, but also to the parents and teachers there as well.

The speaker said: “I want to change the world.”


A little more silence where everyone looked around and felt uncomfortable.

Then a student — one courageous student — got up and said something. And remained standing.

And then another. And another.

The head of the upper school said something. Then more students. Then a parent. Then me. Then another math teacher. Then more students.

At the end, every student made a declaration, and a few adults too. It is scary. But it also showed me how much courage our kids have. Their declarations ranged from showing others that girls can do math and science to spreading love to making people laugh to promoting peace to inventing something to becoming a biochemist to making a mark on the world. Big things and small things, lofty things and concrete things, but all things that share with the room a sense of self and a sense of purpose.

I loved watching this.

I also loved and hated how hard it was for me to come up with my thing. My purpose in life. I said:

I want to make it so that kids see math as an artistic and creative endeavor.

And I meant it. Because you know what has been bouncing around in my head that I have been having trouble articulating? I am now pretty good at coming up with deep and conceptual approaches to mathematical ideas. And I’m okay at promoting mathematical communication. And I’m transitioning to having kids do groupwork all the time, to learn from each other — so I am not the sole mathematical authority in the room.

But all of that said: I don’t think I teach math in a way to shows how it is an art form, how deeply creativity and mathematics are intertwined. And I know that this is one of my charges as a teacher moving forward. It’s going to be an uphill challenge, and one that will likely take me many years to wrap my head around. The hurdles are significant. Having a set non-problem-solving-based curriculum which doesn’t allow time for much mathematical “play,” nor for the inclusion of rich problems with multiple entry points, is the largest hurdle. But there must be ways — activities or units here and there — that can illuminate the artistry and creativity of doing and discovering mathematics. And I want to be involved in finding ways for this to happen. Yes, this happens at math circles. Yes, this happens at math clubs. Yes, this happens at summer math programs. That’s where the love and excitement and understanding of the beauty of mathematics unfolds for many students. But I want to find a way for this to happen in a normal classroom, with normal students, with the normal constraints. That (one of) my purposes.