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

• 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

Experiments:

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.

Content:

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.)

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.

Doodling in Math

A few years ago, I blogged about this fun little doodle that students often make — and how another teacher and I found out the equation that “bounds” the figure. I honestly can’t remember if I ever posted how I got the answer. If I did and this is a repeat, apologies.

Tonight I wanted to see if I could re-derive it like I did before — and lo and behold I did. I’m curious if any of you have done it the way I did it, or if there are other ways you’ve learned to approach this problem. (There is a student who I had last year who created this amazing 3-d version of this using the edges of a cube and some string. I love the idea of asking — for this 3-d figure — what surface is generated by the intersections of these strings.)

We start out by having these lines which form a family of curves. But of course we’re not graphing all the lines. If we were, we’d get something more dense like this.

The main idea of what I’m going to do to find that curve… I’m going to pick two of those lines which are infinitely close to each other and find their point of intersection. That point of intersection will lie on the curve. (That’s the big insight in this solution.) But I’m not going to pick two specific lines — but instead keep things as general as possible. Thus when I find that point of intersection for those two lines, it will give me all the points of intersection for all the lines.

Watch.

First we pick two arbitrary lines.

We’ll have one line move down on the y-axis $k$ units (and thus over on the x-axis $k$ units). And the second line will be moved down on the y-axis just a tiny bit more (down an additional $e$ units). Yes, we are going to have that tiny bit, that $e$, eventually go to zero.

The two lines we have are:

$y=\frac{k-1}{k}(x-k)=\frac{k-1}{k}x-(k-1)$

$y=\frac{k+e-1}{k+e}(x-(k+e))=\frac{k+e-1}{k+e}x-(k+e-1)$

A little bit of algebra is needed to find the point of intersection. Setting the y-values equal:

$\frac{k-1}{k}x-(k-1)=\frac{k+e-1}{k+e}x-(k+e-1)$

And then doing some basic algebra:

$k^2+ke=x$

Now solving for $y$ we get:

$y=\frac{k-1}{k}(k)(k+e)-(k-1)$

$y=k^2+ke-2k-e+1$

So the point of intersection is:

$(k^2+ke, k^2+ke-2k-e+1)$

Here’s the kicker… Remember we wanted the two lines to be infinitely close together, right? So that means that we want $e$ to go to zero. Thus, our point of intersection of these infinitely close lines will be:

$(k^2, k^2-2k+1)$ or $(k^2,(k-1)^2)$.

Beautiful! And recall that we picked the lines arbitrarily. By varying $0\leq k \leq 1$ and plotting $(k^2,(k-1)^2)$, we can get any two lines on our doodle.

But I want an equation.

Simple. We know that $x=k^2$. Thus $x=\sqrt{k}$.*

Since $y=(k-1)^2$, we have $y=(\sqrt{x}-1)^2$

Let’s graph it to check.

Huzzah!!! And we’re done!

I wonder if I can do something similar with this cardioid:

I think I must (for funsies) do some investigation of “envelopes” this summer. I mean, Tina at Drawing on Math even introduces conics with these envelopes!

An extension for you. Do something with this 3d string-art.

*Of course you might be wondering why I don’t say $x=\pm \sqrt{k}$. Since $k$ is between 0 and 1, we know that $x$ must be positive.

CUPCAKES! ALGEBRA II! BEST ACTIVITY EVAR!!!

Now that I have gotten your attention, I’m sorry. I don’t have the best activity ever for an Algebra II class that involves cupcakes. But fine, you want cupcakes. Here.

Now for the reason why I lieeeed to you. You know it’s gotta be big, and important. It’s this. I need you to read this, and take a moment, and actually consider it.

We have a math department chair opening at my school, and you or someone you know might be the person who would be perfect for it.

So I have a lot to say. I should probably note at the top that everything I’m saying is my own opinion, and this post doesn’t come from my school or my department. Just me. Now to the other stuff… I am not someone who wants to go into administration. And my colleagues also love being in the classroom full time. We tend to love our little classroom universes, and even though we engage in the bigger picture of the curriculum-at-large, our primary interest is being intellectually stimulated by classroom teaching. So we want to find someone from the outside who can see the bigger picture and wants to shepherd a bunch of thoughtful and awesome-face teachers as we push forward into our next step.

If this even remotely sounds like something you’ve been toying around with, keep on reading.

For some background. I teach at Packer, a fantastic independent school in Brooklyn Heights, New York City. The school is a Pre-K through 12 school. There are so many wonderful things about my school, I don’t know which to list. It is not religiously affiliated, but we are housed in an old church — and there is a chapel where we have meetings, and this chapel has beautiful stained glassed windows. The architecture is Hogwartian. There are about 80 to 90 kids per grade, and class sizes tend to be around 12 to 16 (though sometimes things go under or over). The school underwent a comprehensive renovation of the “Science Wing” and this summer it is going to renovate many of the Upper School (high school) classroom. The kids all have laptops, and all the rooms currently have SmartBoards, but next year they will be upgraded to Sharp LCD boards (and some will have ENO boards). When it comes to teachers being able to get “things” they need to teach, we do. Similarly, I have never been turned down for any professional development opportunity I wanted to pursue, and have always been fully funded. There is a commitment to teachers on that front.

The school is in the middle of an ambitious 5 year strategic plan, which includes a special component involving math and science excellence.  For me, the most exciting thing about the strategic plan is that teachers are thinking more and more about the importance of the process of acquiring knowledge. For me, that’s exciting because I have been wanting to move towards a more “how do we do math?” approach rather than “here, let me show you how to do math, now do some problems.”

Now to speak specifically about the math department, and why I think it’s worth considering. The math department head is in charge of math in grades 5 through 12 (middle school is 5-8, upper school is 9-12). That would mean being the head of 13 or so teachers.

We’re a really well-functioning department, where everyone gets along and are friends with each other. When we’re feeling wonky, I might be in the office with TeacherX , and we’ll close the door, put on the Sound of Music, and we’ll spin around in our chairs. (Because we both love the Sound of Music.) And every single time anyone is going to the photocopier, they ask if anyone else in the office needs something copied. And we all buy diet coke and chocolate share it with each other. We do site visits to other schools to see what they are doing. And teachers of the same class meet regularly. We share materials all the time. We pose puzzles to each other. And we bounce ideas off of each other.

What I’m trying to say is: that would be a concern of mine… coming to a new school and not knowing how the department is. I can say that we is aweeeesome.

I personally see us at a crossroads, and one where someone could come in and do some great work to take us to our next step.

We’ve come a long way in coming up with a solid and coherent curriculum. We have been trying to push our curriculum to get students to articulate their reasoning more… We have made “writing in the math classroom” a goal of ours for the past two years. And although we’re all very busy, we have made a goal to visit each others’s classes a number of times (I think 8?) before the school year ends. (That reminds me… I need to try to a few observations soon!) And we’re now in the process of thinking: how do we get problem-based learning in our classrooms?

And this is the crossroads we’re at. How do we bring our teaching, and our curriculum, to the next level? (I think this is a question the whole school is asking, because of the strategic initiative.) For me, that means learning to focus on letting go more, and developing curricular materials which continue to push students to focus on the fundamental ideas and less on procedures. It means getting kids to do the heavy lifting. It means trying to deconstruct a curriculum so I can figure out what the essential mathematical idea is, and then find ways to really bring that to the forefront. That’s all for me. Different teachers are at different places in their career and have other ideas on what they need to do to get to the next level. But the takeaway for you is that we’re interested in the craft of teaching, and looking to forge forward as a department.

That isn’t to say that everything is all roses all the time. What place is? And better yet, what place filled with teenagers is?

But it’s a place which I’ve been happy and proud to call home since I’ve started teaching. (It is suppose it’s actually a second home to me, since I spend so much time here!) The school took a chance on me — a young kid with only student teaching experience — and gave me a place to grow professionally. I was allowed to experiment with standards based grading (this is my third year doing it in calculus). I felt like I needed to switch one of my courses last year because I was feeling stale with it, and just plain tired, and that happened. I asked for funding to go to multi-day out-of-state conferences and I have always been approved.

The school is going through changes, as we work towards the strategic plan. And I think our department can, with someone with passion and vision and a strong work ethic, help us take our work to the next level!

Our department head is leaving because of reasons unrelated to her job here. And this timing of this is — at least for independent schools — late in the game. That is why I want to reach out to you guys. A perfect audience of math teachers! If you can see yourself or someone you know in a place like this, working with meeeee!, get into gear and apply!

We want someone awesome, and I’m 200% sure that the teachers in the department will do everything we can to support whoever we hire in their new role. You won’t be walking in alone, but rather with the support of everyone in the department who wants you to succeed, and will do everything we can to make that happen. We are a department and we look out for our own.

Because of the lateness in the hiring season, please please please don’t wait a few days before getting around to it. It is (in my opinion) a one-in-a-career opportunity, but the window is not going to be open for long. We are going to be working on this hire ASAP.

Grinning

Today I was grinning irrepressibly.

Last week I received an email from the current faculty adviser to the disciplinary committee — what we call the Student-Faculty Judiciary Committee. It read:

Dear Mr Shah,

You have been referred to the Student Faculty Judiciary Committee for Violation of Dress Code. Your hearing will take place on Tuesday, 4 December, during F band, in the Faculty Lounge. (This is the space above the cafeteria.) If possible, your superhero will join you in the hearing as support. Please arrive on time, and feel free to contact me if you have any questions about your appearance before the committee. Should you be ill on the day of your hearing or need to be absent for any reason, you must contact me via email. Otherwise, the committee may deliberate and reach a disciplinary response in your absence.

For those of you who need some background, I served on the committee for four years… two years as a faculty representative (going to the hearings, giving my thoughts, voting) followed by two years as the faculty adviser (leading the committee). It was hard work. Four years of early morning meetings, dealing with challenging student issues (and sometimes challenging students). We disagreed. We argued. And in the process, in these early mornings, I saw some of the best things I could have possibly hoped to have seen as a teacher. I saw students who came before the committee reflect. I saw students serving on the committee grow in their thinking about responsibility and consequences. I saw committee members show empathy while simultaneously keeping the big picture in mind. I saw students disagree with students, and teachers disagree with students, and students disagree with teachers, and teachers disagree with teachers… and come out the better for it. And I saw, year after year, a committee of students and faculty who were dealing with confidential and difficult and rarely good things band together to form a tight group with a real sense of purpose. To me, the committee truly has been a concrete instantiation of the best kind of work a school can do, and we did it well. [1]

That’s what the committee is all about.

I walk to the faculty lounge. I look up the stairs, and I see a student waving. As I walk up the stairs, I see a ton of people all there and they all start clapping. It was a thank you pizza party in my honor. I got this grin, and the whole time I walked up those stairs, I thought: this is a highlight of my teaching career thus far. There were the current students on the committee, and the current faculty members on the committee, all the old students who had previously been on the committee (who were still at the school), all the old faculty members who had been on the committee, some of the deans, the Head of the Upper School, and even the Head of School. There were many 20 people there, clapping. It was overwhelming. I shook some hands, and I gave a little speech, and we all broke bread together.

I will admit to experiencing symphony of emotions.

One was sadness. Of course I’m happy that I get to sleep in more often and it’s important that there is new leadership and new voices, but seeing everyone made me miss the camaraderie that we had. I also felt guilty. Why hadn’t I created a thank you celebration for the former faculty adviser when I took over the SFJC? She did more in her years to bring the committee to it’s modern form than anyone — she has been the biggest inspiration and mentor I’ve had as a teacher. I also felt undeserving, because so many people do so many great things at our school that go unrecognized. But mostly those feelings were all undertones, and the main feeling in my symphony was elation. And I kept thinking stay in the present, enjoy this, soak it up, because it won’t happen again soon.

So I stayed in the present. I enjoyed every moment. I continued to grin. And I was just so happy.

I’ll end with one thing that someone who had been on the committee for years said to me, a precocious student who loves history. He said that he was thinking that my leadership was analogous to Earl Warren’s leadership in the Supreme Court. Those who know my obsession with the Supreme Court would know why I loved that analogy, and those who know the Warren Court would know why that is such a compliment.

[1] I should also say one of my most trusted colleagues brought me onto the committee, and basically made the committee what it was. My primary goal while serving on the committee was to not let her good work disappear.

A little bit crazy! And some goals!

So I’m feeling totally and utterly overwhelmed with the impending onset of school. I have tomorrow to keep working, and then we have three days of activities with our advisory (Wednesday, Thursday, Friday) and then starting next Monday we have the first day of classes.

My anxiety level is at about a billion. On a scale of 1 to 10.

With the exception of my first year teaching, this is possibly the worst I have ever remembered it being. I think I wanted to post this to let any other teachers who are feeling this way (especially while seeing all the excitement and incredible first-day-activities abounding on the internets) know: it’s okay.

It’s okay.

At least… I think it is.  (Even Lisa has been in a funk.)

For me personally, my anxiety is coming from a few places:

1. the idea of teaching a bunch of new students, and having to develop a positive rapport with them from scratch again
2. teaching a class which is new to me (advanced precalculus) with very little supporting material
3. co-teaching/collaborating with two teachers, when I have never truly collaborated before
4. having a giant class of 19 (in my school, this is monstrous) and not knowing how I’m going to manage
5. being on supervision & evaluation cycle this year
6. anticipating the late nights every day after school, which will come out of having to write/create calculus reassessments, plan precalculus lesson plans and smartboards from scratch, and having to re-work lots of multivariable calculus homework problems since I haven’t taught the course in a year
7. mentoring a math teacher new to the school
8. starting up (with the help of another teacher) a math-science journal
9. not having any concrete goals set for the year, yet

I think the solution for my anxiety is to work a crazy amount (obviously, that will help). But also to set the bar low. Usually by now, I have two or three very concrete but “large” things I want to do this year. It’s stressing me out that I haven’t decided what they are. Maybe, though, this needs to be a year of stasis. While I’m working on a new course, maybe I need to be okay not doing anything dramatic.

Although not set in stone, perhaps my goals this year should be something as simple as:

(1) Be sure to provide formative feedback to kids in all my classes, at least once a week
(2) Really endeavor to use groupwork (and part and parcel of this, whiteboarding) in precalculus, and be sure to give feedback to groups at least once every two weeks so they have a record of their strengths and places they can improve.

These are weaknesses of mine, so they’ll bring me forward as a teacher. But aren’t so overwhelming in their scope as to feel impossible.

There’s a 50% chance that as I try to work out the beginning of classes this year, I will be posting a lot. And there’s a 50% chance is that I go a little crazy and have to hide for a few weeks while I get settled.

With that, night all!

PS. Since it feels weird to post without any equations, videos, documents, I am going to include this picture of me in front of a stained glass window at my school.

Comment Time Is Over!

This is a post of celebration.

This past weekend and this week, I’ve been consumed with writing narrative comments on all my students. In the past two years of teaching, I have been trying to be more thoughtful about what I’m writing. To put all the cards on the table, I don’t think that comments themselves really effect change in students. However, I do think there is a powerful thing that comments can do: it is a way to tell students I see you and I care about you and I am thinking about you and your learning. Not literally, but a comment can send that message implicitly.

So even though I have serious doubts about the efficacy about what I write in helping students to change their practices, I hold firm to the belief that the implicit message is worth it. So I write, and hope that for a few kids, it matters.

It’s almost 9pm. I’m at a coffeeshop now, and I just finished my last (my 49th) comment of the year. 58 pages later, I am breathing a sigh of relief that I’m done.

I’m totally drained.

I’m so tired of writing that I don’t have it in me to talk about how my comments have evolved in the past two years, or how standards based grading has made writing comments so much easier. Or list the places I know I could still improve on. And maybe I will at some later point.

For now, I just wanted to write a post now sharing the good news with everyone:

I am done!

(If  you want to see the type of comments I wrote in my first three years of teaching, I’ve archived that here.)