Putting myself in my kids' shoes

I always preach mathematical fearlessness to my kids, my colleagues, and just about anyone who will listen. I go on and on about the importance of being able to sit with a new problem for more than 5 minutes, or even 5 days, and mulling it over, poking around, trying whatever you can to solve it without giving up or losing interest or Googling the answer. I always want my students to try something in lieu of just staring, and I am usually stingy with hints.

One thing - quite possibly the only thing - I miss about math grad school is the feeling of sitting down and figuring out a really, really difficult problem. At the time it felt a bit like "intellectual masturbation" because that's all I would do and I felt like I was contributing absolutely nothing to the world, but it definitely helped me develop my mathematical fearlessness. Since I started teaching it is rare that I actually sit down with a truly challenging problem and force myself to think it through from start to finish. So much of what I do is in a rush - I don't have time to really think about a problem because I am trying to plan two lessons for the next day, so instead I think about it for 5 minutes and then look at how someone else did it. Shameful, I know. The question presented itself to me: Can I actually practice what I preach? Am I still mathematically fearless?

My office-mate presented me with this problem about a week ago: You have a square dartboard. What is the probability that a randomly-thrown dart will land closer to the center of the dartboard than to an edge?

I sat down to solve it and was absolutely stumped. I had no idea where to start besides drawing a little picture. Said office-mate told me that he had banged his head against the wall and couldn't figure it out, which made me a little disheartened because I consider him to be a much more clever problem solver than I am. [Lesson one about how my kids feel: It's difficult for them to actually believe in their abilities when they look around and see classmates who they consider to be "smarter" who are also struggling with the problem.]

I took the problem home and presented it to my boyfriend, who really is the smartest math guy I know. He struggled with it for a little bit, went inside the bedroom, and came out about 30 minutes later announcing that he had solved it and that he would not tell me how he did it. Fine. Be that way. At that point I was still really struggling; I didn't even feel like I had a solid starting point. I went into the bedroom and I have to admit that I glanced over at his clipboard where he had solved the problem and saw some nasty math that I didn't like. My heart sunk even more. [Lesson 2 about how my kids feel: When they see another student's solution and don't immediately understand it - and how could you really immediately understand someone else's solution to a problem? - they tend to give up because they think that they could never think of that solution. The thought doesn't even cross their mind that maybe they can come up with another way to solve the problem.]

So I gave up for a few days, thinking about the problem a little bit here and there but never hard enough so that I'd feel like a failure if I didn't figure it out. [Lesson 3: Not trying hard is how kids avoid feeling like failures.] This went on until yesterday afternoon. My boyfriend and I spent the afternoon at my favorite neighborhood coffee shop, basking in the San Diego sunshine. I was working on my end-of-year comments when I suddenly remembered the dartboard problem. I asked him to tell me how he had solved it. He looked at me somewhat disappointedly. "Really? But then you'll never keep thinking about it your way." At that point I still didn't have a "way" but a small fire lit inside of me - How could I not have a "way"? Some idea, some line of reasoning? What would I say to a kid who asked me for a help with a problem and didn't have anything of his own to show? So I told him to wait a sec, and I took out a piece of paper and started working. I came up with what seemed like a great solution with a simple answer. Boyfriend checked the work and agreed, but then asked me to look over his solution because he had gotten a completely different answer and had been sure he was correct. As he was explaining it, he immediately found an error in his reasoning which made his method quite complex to carry out. However, we then went back to my method and found an error, so we worked it through again together. I now have a solution that I'm pretty happy with and that is completely different from his solution, and it feels so good that it is mine. He was right - if I had looked at his solution first, I would have never had the guts or desire to come up with my own. [Lesson 4: The lengthy process really is worth it! Any human being - teacher, student, adult - feels amazing after coming up with a clever answer to a problem that once seemed insurmountable.]

I would be happy to post my solution if anyone is interested; I would also like feedback because I wouldn't say I'm 100% confident. More importantly, though, I feel like this was an exercise in my own mathematical fearlessness. I gained a new respect for my students. As I know they do, I felt anxious, inadequate, and angry at various points in this problem-solving process. My new question is: How can I better support them so that they actually want to stick it out until the end?

An amazing week (or, why I love my seniors)

Last Friday, I thought we had "finished" differential calculus (modulo taking the final exam and watching Stand and Deliver - essentially a requirement for high school calculus, no?). We had just spent about a week on optimization and students had a one-question quiz, something trivial about minimizing the surface area of a box with a fixed volume. I was agonizing over what do do with only 12 "real" school days left before their final- not enough time to start integration, as I had wanted to, yet too much time just to review for the final. And then I went home to grade the quizzes and found that a majority of my students could not correctly solve the simple box problem from start to finish.

They clearly needed more practice with optimization (so that's what "formative assessment" is!), but I worried that another problem set would cause immediate-onset senioritis. So, I decided to use a structure I had used before, printing out a bunch of different problems on separate sheets of paper for students to work on. The problems would range from easy (1 point) to devilish (15 points) and students would work with a partner to solve whichever ones they chose (you can check out the problems here). When students brought up a correctly solved problem, they'd get a stamp. (It always blows my mind that 17 and 18 year-olds still get really excited about stamps.) Then, they'd choose a new problem to work on.

In the past, the group with the most points at the end would be the winner. The problem with that is the problem with most math review games - everyone knows before the game even starts who is going to win. I needed a way to keep all groups working hard the entire time ... and then, behold! The Class v. Class Showdown. I have two honors calculus classes, and they would compete to get the most total points. The kids loved this, and I can honestly say they were engaged for the entire three days. Some things I really liked about the Showdown were:

• Everyone's points really did matter since we were totaling them all up. There was an incentive for everyone to work hard.

• Many students started with easier problems to build up their confidence, which is what they needed. However, they would have been reluctant to practice those easy problems without the incentive of racking up points.

• Since the challenging problems were worth a ton of points, students didn't give up on them. Many spent an entire period working on a single difficult problem. 

• There was the perfect amount of peer support. More advanced groups would give a hint to groups who struggled with the more challenging problems, but wouldn't do the entire problem for them because they wanted to accumulate more points of their own. 

After three days, the competition was intense. I happen to know that there was trash-talking going on in advisory and some potentially shady business on Facebook. And while I certainly don't condone these things in general, I couldn't help but bask in the knowledge that Calculus had achieved a pretty rad social status. On Thursday, we counted up the points in first period - 209. Then my second period students came in and worked their tails off for their final hour. Their points totaled 220, and when they realized that they had won they went CRAZY. Jumping out of their seats, cheering, high-fiving, back-slapping crazy ... over Calculus! I'm sure this wouldn't work as well if I did it too often, but it was a nice way to spice up humdrum skills practice.

On a completely unrelated note, enter Friday morning: a colleague and I drove up to school only to realize that the sole entrance to the parking lot had been sealed off by three pick-up trucks plastered with "Class of 2011." Senior prank! As we drove around the block looking for a place to park, we saw that every single chair in the school had been lined up on the roof. I have to admit that I shed a tear of pride over this. My kids had succeeded in pulling off a good, clean prank that wouldn't get any of them expelled but was still clever and required immense teamwork and organization. Talk about project-based learning at its finest!

And finally, this morning (Saturday) was the culmination of two of my students' senior project. (At my school, all senior teachers oversee 25-ish individual senior projects.) These two amazing young ladies had organized a conference entitled "She is..." for young girls. The conference consisted of keynote speakers, a career panel with successful women, a workshop dealing with body image & the media, and so much more. The impact of the conference can be summed up by a comment made by one tenth-grader during the closing activity of the body image workshop: "Today I feel beautiful. I don't always feel beautiful, so I want to always be able to think back to when I did." The absolute best part was that I had nothing to do with this; it was my students who had organized this entire transformational experience. The months of senior project-induced stress and tears now seem unimportant. [Perhaps more so even than the chairs on the roof] this was project-based learning at its finest.

All in all, a week that I thought would drag with senioritis and boredom turned out to be one of the best of  the year. Next year, I'll be transitioning into a new school on the other side of the country with a very different culture. While my school's chaos of late often has me looking forward to next year, this week my seniors managed to remind me of why it is I wanted to teach at this crazy school in the first place.

What is the goal of math education?

There's been quite the buzz of late over this article, A Better Way to Teach Math, published in the NY Times' Opinionator Blog last week. If you haven't read it, the author discusses the idea that maybe math achievement doesn't have to be distributed along a bell curve at all, and that we're actually just not teaching math in a way that allows most students to succeed. The method highlighted in the article is a curriculum called JUMP Math. According to its website, "JUMP Math is a charitable organization working to create a numerate society." I certainly have no beef with their mission. The emphasis of their method seems to be confidence-building and breaking each mathematical procedure down into its most basic component pieces and "assess[ing] each student's understanding at each micro-level before moving on."


There are many claims made in the article that I agree with, and many ways in which I applaud the JUMP program. There is a huge achievement gap in math and I agree that "for children, math looms large; there’s something about doing well in math that makes kids feel they are smart in everything. In that sense, math can be a powerful tool to promote social justice." In the end, I am a proponent of any program that effectively levels the playing field and allows all students to reach their potential, mathematical and otherwise. However, these words - "potential", "achievement", etc. - are riddled with bias and my fear is that programs like this one pander to our current paradigm of math education instead of questioning its rather tenuous premises. What are some of those premises? Standardized testing as a measure of numeracy. The AP obsession. The glorification of calculus as the be-all and end-all of high school math.


The author states, "In every math class I've taken, there have been slow kids, average kids, and whiz kids. It never occurred to me that this hierarchy might be avoidable ... Can we improve the methods we use to teach math in schools - so that everyone develops proficiency? Looking at current math achievement levels in the United States, this goal might seem out of reach." My immediate response to that is: When we measure "achievement" as a single proficiency score between zero and 100, then of course the scores are going to fall along some kind of a bell curve. That is the nature of such simplified quantitative data. In some ways, it seems like our system is set up to produce high-achievers, middle-achievers, and low-achievers.


There is currently a lot of amazing brainpower being devoted to developing strategies for helping kids succeed in the current system - Khan Academy and JUMP Math are two examples. I wonder where we'd be if there were similar amounts of brainpower devoted to shifting the paradigm of math education and creating an actual, tangible resource bank that is in line with the paradigm shift. In my own little math edutopia, math classes would look a lot like the ones presented in Lockhart's A Mathematician's Lament. Students would do mathematics as mathematicians do  - by collaborating, by posing natural questions, and by attempting to answer them. Mathematics is meant to be critiqued and refined just as a piece of creative writing is, and the art of proof is meant to be taught as such (an "art") and not misrepresented as an exact science. This is of course oversimplified summary and I encourage you to read the Lament. It's a beautiful piece of writing that may just change the way you think about education. 


My esteemed colleague at Broken Airplane (who I also have the privilege of working with every day) makes a great point: Sure, Lockhart's Lament sounds great and provides lots of food for thought, but where's the stuff? Where's the curriculum, the activities, the books full of usable tangible things? Until he's got the goods to back them up, his ideas are somehow destined to take a back seat to the current system (for which there are a plethora of really effective resources).


In the end, this tension between skills-based math and inquiry-based "pure" math exists because we haven't yet decided what the goal of math education really is. Why is it that we make our kids study math for at least twelve of their formative years? Is it so that they can be good little calculus students in college and maybe even good engineers? Or is it so that they can develop an intellectual appreciation for inquiry and patterns and proof and abstraction, ultimately applying that creativity and critical reasoning to the endeavor of their choice? If it is the former, then breaking down every mathematical concept into skills-based components is certainly the way to go. If it is the latter, then doing so might just obfuscate the very beauty of math that we are trying to impart.


It is my impression that a lot of us are trying to strike a balance between the two. We want to prepare our students for college-level mathematics and engineering because that is our duty, but we also want them to experience why it is that we fell in love with math. What I find, though, is that I wind up betraying that second goal so that I can adequately cover all of the content that I feel compelled to. Of course there must be some ideal balance between the two, but it seems to me that right now the pendulum has swung much too far in the skills-based direction. In my humble opinion, this is because (a) it's much easier to assess, and (b) it's much easier to teach. [One could argue that (b) is a direct corollary of (a).]


My question is: are these two goals mutually exclusive? Can one both help students develop a great skills-based mathematical toolkit while simultaneously creating a classroom where students really become little mathematicians? Am I missing something? What do you do in your classroom to strike a balance?

Festival del Sol - Cuckoo for Calculus!

I teach at a project-based school, yet in 12th grade math I rarely do an actual "project." Of course, the meaning of that word is completely subjective and I don't mean to say that I don't do anything interesting or creative in my classes, just that I don't try to stuff content into a contrived project just for its own sake. However, this past week was our annual "Festival del Sol" and each class was expected to exhibit something. I had been racking my brain for a a calculus project all year - one in which students would truly learn the content through the project - and couldn't come up with anything. (The closest I've come to this was the "Great Calculus Challenge" where we dropped a block off the roof of the school - see previous post about that one.)

So, I stopped stressing out about it and figured that I'd give my kids a "break" for a couple of weeks with the following project: Pick any concept or problem that you've enjoyed this year, write a short technical paper explaining the concept / problem, and figure out a cool way to present the concept / problem at Exhibition. At some students' suggestion, we called this project "Cuckoo for Calculus!" (To my surprise, no one volunteered to dress up as the crazy Cocoa Puffs bird.) You can see the actual project handout and specifications for the write-up here.

I rationalized spending two and a half weeks on this by telling myself:

1. My kids are learning a ton of math this year, most of which they'll probably forget anyway - so why not spend time delving deeper into a topic they enjoyed with the hopes that they might actually remember it?
2. It's probably worth doing something fun and rejuvenating that might ward off the inevitable post-spring break Senioritis.

I did several projects last year as an 11th grade teacher, yet this was the first one the kids were somewhat excited about - and I'll admit, that felt rad. I think that the student choice element was key, as was the fact that my students are generally motivated and enjoy the class. Many students chose to return to a problem from a past challenge set, which was kind of cool. (My office mate commented that we try to get kids to work on these cool problems, and some of them get 'em and some don't, and then too often the problems just "die" and we never return to them.) You can browse through the challenge sets here.

There were two distinct pieces to this project: the write-up, and the exhibition product. I'll talk about the latter now, because it's easier. Basically, I got some really creative products. Some of my favorites were:

A giant Tower of Hanoi game made of a PVC base and handmade pillow "discs"

The background poster for
the 3-D product rule

A physical representation of the
proof of the 3-D product rule
(from a challenge set)

An artpiece demonstrating the "picture proof" that any triangle constructed
with the diameter of a circle and any point on the circumference is a right triangle.
The piece opens up to a full circle in order to demonstrate the proof, which is
inspired by the famous problem from Paul Lockhart's A Mathematician's Lament. 

A giant pop-up book explaining derivative shortcuts with
the help of "Deric the Differentiating Duck"

This one was exceptional - a painstaking model of the notorious
(among my students) Ferris Wheel / Water Cart problem
presented by Goonies and made of pipe cleaners, complete with a
diving ballerina and a Lady Gaga-esque emcee.

Two pirates present the box-method for the chain rule
with nested boxes, ending in a treasure chest with gold coins for
those who are successfully able to take the derivative of a
complicated composite function.

A comic (a la xkcd) presenting the challenge problem involving pirates and a secret
language for communicating the identity of a card using only four other cards.

Pretty fun, huh? I loved seeing the responses of people who came to check out my kids' projects - in general, they were impressed with their creativity and with their understanding of the material. Even though math isn't necessarily the most exhibit-able subject, it's fun for the kids to get to show off their fancy math once in awhile (whether or not they're at a "project-based" school).

Musings on the Chain Rule (Sorry, Newton)

Very, very rarely do I teach something and think, "Wow. That went really well!" In fact, I think it's happened exactly once. And it happened with the chain rule, which for some probably brings back horrible memories of first-year college calculus. For whatever reason, it couldn't have been more different in my calculus classes and I wanted to share my strategy in case someone else might find it helpful.

I was inspired by Think Thank Thunk's use of gears to teach the chain rule and by Sam J. Shah's box method. I started out with a simple gear example. We worked as a whole class because I didn't have enough gears set up for all the kids to play around, but they had actually just finished discussing gears and gear ratios in their engineering classes (yeah, my school's that awesome) so it worked fine.

The moral of the story is that when you compose gears, speeds multiply. They got that. I then invoked a little poetic license to use "gears" as a metaphor for functions and "speeds" as a metaphor for derivatives. They were confused. I don't blame them. But THEN we put it all together and the confusion turned to glee! (Okay, maybe not quite glee, but you get the point...) After asking them what kinds of functions we don't yet know how to differentiate, we came to the conclusion that even though we know how to take the derivative of 2 to the power of x, we don't know how to take the derivative of 2 to some function of x.

So we connected those functions back with when we did function composition at the beginning of the year, and I reminded them that to make a complicated function like f(x) = 2^(6x+3) we had to take a trip to the function factory, where we immediately went to the assembly line where they made f(x)'s. Yes, I actually drew the following picture on the board, conveyor belt and all (I find that when it comes to cheese, go all the way or go home):

The conversation goes something like this: "What's the first thing that happens to x?" "It gets multiplied by 6 and added to 3." "Right, so it goes into the 6 blah plus 3 machine. What does it come out as?" "6x+3" "Then what happens to it?" "It gets raised by a power of 2" "Right, so it goes into the 2 to the blah machine. What does it come out as?" "2 to the 6x+3." "Are we done?" "Yes."


When we go to take derivatives, we'll say stuff like: "The derivative of '2 to the blah' is 'ln of 2 times 2 to the blah'." Although I'm pretty sure this "blah" nonsense is standard language when it comes to the chain rule, it's still kind of funny to see my kids at the board saying things like "2 to the blah." When they parrot those funny things back at me, I have that strange realization that sometimes they're actually listening to what I say, and if they are then it goes down into their notebooks as Calculus with a capital "C". And what would Newton think of "2 to the blah"? Sometimes I worry about these things.

Anyway, we loved the chain rule. By the end of the first class, they were begging me to put a really crazy one on the board. At the beginning of the period, I had made them repeat after me: "I will not be afraid of the chain rule." At the end of the class, they asked why I had made it seem like it would be so scary - it was the easiest thing they had learned all year! I must admit that part of me wished they had to suffer through the chain rule just a *tiny* bit more by learning about u-substitution or whatever awful way I first learned it, but of course as teachers we must resist the urge to do things a certain way simply because that's how we learned them.

How do I foster inquiry?

One of my classes has spent the last month or so on basic trigonometry: right triangle trig, the unit circle, and graphs of trig functions. We have a school-wide spring exhibition coming up in exactly one month, and since trig seems to be one of the few high school math topics that has actual real-world applications that are both authentic and accessible to teenagers (a balance that's difficult to strike) I've decided to do a  project surrounding trigonometry and sound (specifically music).

I've spent the last couple of days just fiddling around with Audacity, a great free audio editor that you can download here. I've experimented with importing entire songs and just with playing single notes or chords on the piano. You can also record yourself, which gets really fun for obvious reasons. In any case, any sound will generate a neat image which, if you zoom in enough, is just a complicated sound wave: 

If you keep zooming in it looks more like a combination of the sine waves that we all know and love, and if you go to Analyze --> Frequency Analysis, you get a nifty frequency distribution of all the different notes in your clip. Even niftier for those of us who dig this sort of thing is that Audacity is really performing a Fourier transform on the sound wave! Gosh I wish someone had showed me this when I took Linear Algebra. Orthonormal, schmorthonormal.

What's cool is that even if you play a single note, like an "A", you don't just get a single peak at 440 Hz (a perfect A). Instead, you get a bunch of overtones that contribute to what you hear on different instruments.Once I had the basics (and I mean, the very basics. I know literally nothing about music) of Audacity down, it was just so much fun to play around. You can generate tones, superpose waves on each other, and create all sorts of nifty fade and filter effects. I wonder about having my kids try to re-create sound waves in GeoGebra by superposing simple waves until they get close to the original complex waveform. What about looking at what happens to a sound wave when it has to travel through a glass of water? Or, recording their voice and a friend's voice and trying to harmonize them by changing the pitch - a little experiment in intervals.

A colleague and I spent a good hour this afternoon recording ourselves saying simple words or phrases, and then playing them back in reverse and trying to repeat the reversed word and then play that back in reverse to see if it sounded anything like the original word - which was a jolly good time. It turns out that the word "potato" is incredibly difficult to do this with because of the little "p" at the beginning which is just a puff of air that is somehow really difficult to capture in reverse. But I digress.

... or maybe I don't. Maybe this is actually the point. I still can't think of a "project". But what I really, truly want is just for my kids to play around like I did, generate some interesting questions or "what if?"s or ideas, and try to tackle them. From experience, my fear is that when I ask my kids to simply play around and write down some observations, they freak out a little. They want you to ask a specific question so they can come up with a specific answer. It strikes me as a little odd that research - for some, that coveted culmination of one's academic career - is almost exclusively about asking good questions, yet K-12 education (and even college, in many cases) is all about getting the right answer to questions that are given to you. 

So my conundrum: I want my kids to experience the "wow" factor of playing around with some neat software and exploring the connection between the mathematics of sound and why we hear things the way we do. I don't want to give them a laundry list of questions to answer because that immediately decreases the coolness by a factor of at least 100. But at the same time, I want them to have enough direction that they don't immediately flounder. Any words of wisdom out there?

Counting my Blessings

Our school is currently going through its WASC accreditation, which means we've had a committee of several outside teachers/administrators strolling our halls, meeting with teachers and students, and poring over a  long "self study" document that our staff has been putting together all year. During one of our WASC meetings today we were praised on what we do, which was really nice to hear - apparently, our students are happy and successful, our teachers work hard to make sure that they learn, and all in all there's lots of great stuff going on at our school. Unfortunately, our scores on the state tests aren't necessarily our best attribute (the reasons for this are a whole post unto themselves, and probably don't even need to be explicitly stated for most of the people reading this). As a result, our AYP, API, etc. are lower than we'd like -- which means that this became a major topic of discussion at the aforementioned meeting because these scores are what the state uses as evaluative metrics.

While the WASC committee was incredibly helpful and generally positive about our school, they really wanted to impress upon us how much we want to raise those AYP/API scores to avoid becoming an "NCLB Program Improvement (PI) School." I was incredibly curious about what this meant, so I came home and looked up the requirements for PI schools here. Both committee members at this meeting were able to speak to some of the consequences of becoming a PI school - having state-appointed administrators come in to implement programs, being forced to read lesson plans that have been scripted down to the minute, and even - yes, I could not believe it - BANNING NOVELS IN ENGLISH CLASSES. Folks, I was floored by this. True, no one's "banning novels" explicitly. But apparently PI schools are required to cover so many writing and reading samples in English classes that it's virtually impossible for teachers to teach entire novels, and those who do receive threatening memos from their administrators. I realize that struggling schools need change, and that change isn't always easy, but - for goodness sake - does anyone out there really, truly think that preventing our kids from reading novels is going to make them more educated??? Just like a bad teacher can "get in the way" of a beautiful subject, it seems that the government have managed to "get in the way" of education. I'm 100% for accountability at all levels (school, teacher, student) but I also believe that this accountability happens locally. And certainly not by banning novels. There are very few things that shake me out of complacency in my old age (ha) but this really did it. What can I do to change the status quo?

I left the meeting a little shaky (which I believe was the intention) but also feeling incredibly lucky. Lucky to work at a school with people who are so passionate about education and about constantly refining their practice. Lucky to have the freedom to experiment with new ideas and to make changes in my classroom as I see fit. I always knew I was lucky to begin my teaching career at such an amazing school, but today the point was really hammered home.