Infinigons, etc.

An antidote to senioritis?

2012-03-03T03:49:00.000-08:00

The state tests are coming up next week, so I've spent the entire week cramming (er ... "re-accessing prior knowledge") with my juniors. To be honest, it's actually been a nice break from a jam-packed and rather tedious precalculus curriculum - HSPA, New Jersey's state exit exam, actually tests a lot of good math and my students have tackled some legitimately interesting open-ended problems (more on those to follow). But, in any case, the point of this post is not my juniors (who I'm really hoping rock the HSPA next week (not that I really believe in the validity of a single standardized test (but still ...) ) ), but my seniors. I had to find something to do with them during the two weeks (one for review and one for the test) devoted to junior testing. Since I've been griping (mostly to myself) all year about why these particular seniors are taking precalculus (which is essentially algebra for the THIRD time) and not, say, statistics or computer science, I decided on a two-week statistics unit.

Now, it's interesting for me to work at a place where having a two-week window open up in the curriculum is an extreme rarity, and it was made clear to me that this unit was to last two weeks, period. I knew I wanted my students to do some sort of mini-project the second week so I had to really hone in on a few specific topics for the first week, which we just wrapped up. I decided to introduce the bell curve (of course) and focus on teaching students to use the z-tables for the standard and non-standard normal distributions. What I really wanted to get to by the end of the week was calculating margin of error and constructing confidence intervals, because that's what they'll need for next week's project. The idea is similar to a project I did at my old school, but in about 25% of the time. Students will be designing an experiment (like a Pepsi challenge) or a survey, writing an analysis of their results, and making a presentation. In their analysis they need to do things like construct their own confidence intervals and determine whether there is a statistically significant difference between two subgroups of their choice, like males and females.

So, back to the content: margin of error and confidence intervals. While I had used the "guided practice" model to teach students about normal distributions and the z-tables - "guided practice" is just my school's nomenclature for showing students a new skill and then gradually loosening the reins until they are doing it on their own - I decided to go for a college lecture on confidence intervals. Again, "college lecture" means something very specific at my school, but in essence the point is to give students a taste of what a 300-person college class will actually feel like. The teacher takes on the role of "professor" (which, I'm not going to lie, is a lot of fun) and delivers a PowerPoint lecture, preferably at super-speed and without much, if any, audience interaction. Of course we scaffold good teaching strategies in to make sure that our students don't flounder, such as intermittent note checks during the lecture and a comprehension check exit ticket afterwards.

My favorite part about the college lecture format is what happens the next day: students work in study groups on a college-style problem set, interrupted only by a brief chance to ask "the professor" questions during "office hours" (okay, so maybe we take the analogy a little far...). This brings me to the actual point of my post - sorry you had to read all the other stuff - which is: giving my seniors this independence and responsibility turned them from slouchy, grouchy second-semester seniors into a spitting image of actual college students. The transformation was unreal. They were engaged with the material 100% of the time (which is not usually the case in this class), challenging each other's understanding and use of terminology, referring back to lecture notes and the text when they got stuck ... essentially, everything we'd want them to do as college students in just a few short months. I use another teacher's room for that class, and that teacher actually asked me in the middle of the class if she could commend the students at the end of class because she's seen so many classes where their performance has been ... less impressive.

As I write this, I'm realizing that it's not rocket science. There's not much in the way of "guided practice" in college. And not to glorify some rather shoddy teaching methods, but maybe there is actually one good reason (albeit many bad reasons) for that - when people get to be a certain age (like, say, 18?) they crave less hand-holding and more independence. Based on some previous classes with my seniors, it might seem that they get easily frustrated with difficult math and take every opportunity to zone out. Now, I'm not so sure. These statistics topics are probably the most conceptually challenging ones we've done all year, and not even the usual suspects could be found zoning out. So maybe doing more of the lecture/problem set/legitimately interesting project or discussion is what they need? In other words: If I keep treating them like college students, will they continue to act like college students?

Finally, something to talk about!

2011-11-23T02:41:00.000-08:00

Have I committed blog suicide by not posting in several months? I'll take the fact that you're actually reading this as a plus (thank you!) and start with a feeble excuse: I moved across the country this past summer and began teaching at a new school whose culture has taken some ... getting used to. This school's driving mission is to close the achievement gap between low-income, urban students and the rest of the country. This is an important task in a city where only 4% of high school freshmen will ever graduate from college, and they do an incredible job at it. The flip side is that my days of dedicating several class periods to wacky problems or to projecting balls off the roof seem to have come to an end, at least for the foreseeable future. For this reason I've felt at a loss for what to write - I'm teaching a fairly standard and very rigid Precalculus curriculum to juniors and seniors, with very little time for exploration or out-of-the-box discovery. What could I possibly blog about?

Well, finally with a few hours to collect my thoughts over Thanksgiving break I now realize that the same rigid nature of the school that has forced me to bite my philosophical tongue for several months has actually allowed me to experiment with some cool ideas that I'd love to get more input on, even if they're not as (dare I say?) glamorous as projecting balls off the roof.

Idea #1: Study groups. This is by no means novel, but I had never used study groups in my own classroom until my students were studying for their big end-of-quarter assessment a few weeks ago. I put them into groups of four and gave them a choice of six activities; their first item of business was to set their agenda and pick 3 activities that they would prioritize. They had 20 minutes for each activity and had to stick to their agenda, even if they weren't completely finished with an activity after the allotted time. Two options included reviewing previous exams, and I provided them with solution guides that they could use. A few of the activities were practice sets on particularly difficult topics that I knew most students were struggling with (like graphing transformed sinusoids - does anyone have a great way of teaching this?). The remainder of the activities involved making study materials of some sort. One such idea came from one of my seniors, who struggles tremendously in math but has found success in other classes making "process cards" and wanted to give it a try. Process cards are similar to flashcards, but instead of emphasizing one fact or formula each card provides a quick reminder of how to execute a particular process or solve a recurring type of problem (like graphing a transformed sinusoid). These are great for those complicated problems that involve a series of steps of which students invariably forget one (like factoring out the period to find the phase shift), because they can tailor the cards to their needs with individualized reminders using language that makes sense to them.

In any case, students were on task for the entire time and the illusion of choice (I mean, let's be honest - in the end, they were just doing practice problems) seems to have been effective. I also like that once I set them up, the remainder of the period was entirely student-led. To add a measure of accountability, I had students complete an exit ticket in which they graded their peers according to a study group rubric and wrote down one specific thing they learned during each activity.

Idea #2: Hands-down discussions. I stole this idea from an English teacher colleague whose room I share and whose classes by default I spend a lot of time observing. I adapted it to my math class as follows: Students solved a problem as a class while I served only as the "scribe", writing exactly what they said on the board. As an example, one problem was to simplify the expression cot(arcsin(1/5)). Students had to take turns providing steps or asking clarifying questions. As the name of the activity implies, they didn't need to raise their hands but instead took their turn when they felt they had something to contribute. Removing myself from the action, so to speak, had some positive effects:

Students were forced to direct questions at each other and to be critical of each others' work, since I gave little indication as to whether a particular step was right or wrong;

They had to be precise and specific with their language; I was obnoxiously literal in transcribing what they said, which was handy in getting them to struggle with algebraic nuances.

On the other hand, it took an awfully long time to do just one problem, so I'd like to find ways to speed up this process while preserving its organic nature. My English teacher colleague uses the hands-down discussion as a way to review several questions that students have had time to work on individually, which is nice because it gives the weaker students a chance to process their ideas and decide what they want to contribute to the discussion beforehand. He also sets a timer (we set a timer for everything at this school), and the students only have the allotted time to complete the problems. Turning the hands-down discussion into a race against the clock by offering some sort of class points as a reward also increases the sense of urgency.

Seeing as this is the time of year to give thanks and not to complain, I need to remember that there was a reason I came to this school, and that there is so much I can learn within their framework if I stop harping on what I am not able to do.

Happy Thanksgiving!

Putting myself in my kids' shoes

2011-06-18T14:49:00.000-07:00

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)

2011-05-21T16:27:00.000-07:00

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?

2011-04-25T16:47:00.000-07:00

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!

2011-04-09T11:23:00.000-07:00

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:

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?
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)

2011-03-20T19:53:00.000-07:00

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?

2011-03-07T21:18:00.000-08:00

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

2011-03-01T20:30:00.000-08:00

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.

Whaddya know? Math in the "real world"!

2011-01-29T08:17:00.000-08:00

I've been helping out a bit with the FIRST Robotics team at my school. By "helping out," I really mean learning a ton from the kids and the amazing advisor (I really couldn't have told you the first thing about building a robot a month ago), and occasionally being useful on the programming side of things. Last Saturday, we were writing the code (in a program called LabVIEW, a "graphical development environment") to make the robot's two motors spin in various directions in order for it to move forwards, backwards, and turn right or left when the driver moves the joystick accordingly. After banging our heads against the wall with "if" statements and such, we realized that what we were really trying to do was a simple rotation/change of coordinates - enter linear algebra!

The joystick has two axes: let's call the horizontal axis A1 and the vertical axis A2 (see diagram). The program can read one input from each axis, so in the extreme case if the driver is pushing the joystick forward as far at it will go but not moving it side-to-side at all, the "coordinates" of that move are (0,1). Similarly, moving the joystick completely to the right but not forwards or backwards at all gets read as (1,0).

Now, of course all of this information has to get sent to the motors somehow. There is one motor on each axle of the robot, which we'll think of as the left motor and the right motor. In motor-land, the coordinates will be (left motor, right motor). We figured that in order for the robot to go forward, both motors need to spin forwards. In other words, (0,1) in joystick-land should correspond to (1,1) in motor-land (see diagram below). Similarly, (0, -1) in joystick-land means we want the robot to move backwards, so that would be

(-1,-1) in motor-land since both motors need to spin backwards. To make the robot turn to the right -- (1,0) in joystick land -- the right motor needs to spin forwards while the left motor remains still, so that would be (0,1) in motor-land. So you can see that it was really a matter of reading off the coordinates in joystick-land and transforming them over to coordinates in motor-land, as in the chart.

Knowing that a change of coordinates simply corresponds to a matrix transformation, we just had to figure out the matrix. That turns out to be really simple because of all the zeros and ones; for example, setting up one of the matrix equations:

tells us that b=1 and d=1. The others tell us that a=0 and c=1. So, we've figured out that when the program reads input from the joystick as (A1,A2), we want those coordinates to get multiplied by the matrix we found before getting sent to the motors. This means:

which tells us exactly what to do with the input from the joystick before sending it over to the motors!

I loved everything about this - the fact that transformations and matrices just "came up" when we were actually solving an interesting problem, the fact that I was able to explain this to the programmer on the robotics team (a 10th grader who had never seen matrices before), and the fact that the solution turns out to be so simple. What kills me is that I spent a month last year doing matrices with my 11th graders via some marginally interesting linear programming problem that didn't necessarily get at the essence of what matrices are. I know how I'll be teaching matrices in the future!

Maybe this will rile up your geometry class

2011-01-11T20:36:00.000-08:00

I was reading this awesome unit on "other dimensions" courtesy of Mathematics Illuminated. In it, they make the following deceptively simple observation:

If we take two points in 1-D space and connect them, we form a line segment. This line segment has a property that no single point has, length. The length of a line segment in 1-D space can be found from the positions of the two endpoints via subtraction.

But if no single point has length (instead, it is defined only by a location), yet a line segment consists only of single points, how is it that a line segment itself has measurable length?

Of course this isn't really a paradox once we introduce infinity and stop trying to interpret infinity as just "a really big number" with all the same properties as numbers. The segment contains infinitely many points, so apparently infinity*0 is not zero (unlike "numbers" we are used to, all of which satisfy the property that: # times zero is zero). A beautiful revelation just waiting to be plucked from an awfully dry geometry standard.

From experience, my kids love talking about infinity in all its forms. Maybe yours will too!

Is programming the new math?

2011-01-09T21:40:00.000-08:00

This past semester I taught a programming elective for seniors We used Python because it's already installed on all of the school computers and we have a resident Python expert who I could turn to for support.To be honest, I didn't have very much programming experience before the class began. I had taken a C class in grad school when I was too sick of math to care about my research, and that's the only formal programming I had done. This summer I taught myself some Python using this fantastic book by Michael Dawson, which teaches the basics through programming games. By the end of the summer, I knew enough to plan the first couple of months of class. I was unsure of my own programming abilities, let alone my ability to teach programming.

When all is said and done, this may have been my favorite class this semester. It turns out that programming is just so much fun that students can't help but get engaged, which is a far cry from what usually happens in math class. Sure, we can make math fun with activities, and once in a while you hit upon a topic or a problem that kids are naturally drawn to. But much of the time I would loosely equate teaching math with pulling teeth, and programming couldn't be more different.

There was certainly a learning curve to my teaching. I'll spare you the messy details and cut to the chase:

I learned to minimize lectures. Kids (and all people) learn programming by doing. Giving them a few examples to follow and execute on their own is much more effective than parsing code as a class in lecture format (at least with my group, which tended to get squirrelly really quickly).
When I did lecture, I used PowerPoints that the students could upload onto their laptops so they could follow along at their own pace. I also tried to include as many opportunities as possible for them to try out commands along the way. I finally started getting the hang of this when we were doing Visual Python; see my lecture notes here if you are interested.
Tiered programming assignments rock. There are easy, medium, and hard programs in a single assignment. Students choose whichever ones they want and aim for a certain total point value. Harder programs are worth more points. See my programming assignments here if you are interested.

Something that became obvious very quickly (and was integral in quelling my fears that I was under-qualified to teach a UC-approved programming class) is that almost every student in the class was into it. This was bizarre, having a class where 25 out of 27 students were really trying to figure out a problem and would literally groan when I told them they had to shut down their computers at the end of class. This shouldn't have been a surprise; I also find myself so engrossed in programming that I don't even notice that several hours have passed. I guess I forget that students are just like us - they like things that are inherently interesting, and dislike things that aren't.

Programming is great - it's hands-on, there's the "wow" factor (even early on, when kids can add up all the numbers from 1 to 1000 in a second!), and most importantly there's an element of immediate gratification. Oftentimes with math there's this fuzzy feeling students have when solving a problem ... ["Am I doing this right?" / "Do you think you're doing it right?" / "I thought so, but it looks really weird."] or ["Is this the right answer?" / "Looks like it!" / "Oh, so that's it?"] ... of course there are exceptional problems whose answers are so beautiful as to be undoubtedly correct, even from students' perspectives, but for the most part students struggle with knowing whether they are on the right track and whether they have solved a problem. In programming, there is no such ambiguity. They are "right" when their program does what it's supposed to do, and they can check their steps along the way simply by executing their code.

Moreover - and this is risky for me to say, because if this sentiment went viral I'd no longer have a job - it seems pointless to be teaching kids math en masse when we could be teaching them more programming instead. Our "justification" for making kids suffer through 12 years of math when most of them will never "use" math beyond algebra (of course this is a gross oversimplification. Statistics, for example, should and could be used by everyone and it's not clear whether this comes "before" or "after" algebra ... Despite what the state standards would have you believe, there is no total ordering on the set of math concepts) is that in enhances their quantitative thinking skills, teaches them creativity, and gives them experience solving difficult problems using out-of-the-box thinking. I buy that, I really do. I love math and I've seen the amazingness that happens when students collaborate on a challenging math problem. However, this can be a rarity in the math classroom. So much time is spent on skills that some find interesting and some find worthy of eye-gouging, yet whatever you believe you have to admit that fewer than 1% (an admittedly unfounded estimate) of adults will ever use any of the algebraic manipulations or shortcuts they learn in math class. Do they use the creativity and problem solving? Maybe. But programming might be a better way to go about cultivating those less black-and-white skills. It involves all of the logical thinking, all of the precision, all of the creativity -- yet without the tedious rules and algebraic manipulations that many students (rightfully) find mind-numbing.

I don't think we should cancel all math classes tomorrow. But I do think there's something behind the following observation: even those kids in my programming class who are not mathematically-inclined and/or completely disengage in math class, were engaged for a majority of the time in programming. And over the course of the semester, I saw a definite improvement in their ability to think logically and focus for long enough to try several strategies for solving a single problem. This is something I tried to get them to do in math last year, yet never felt terribly successful. In programming, I didn't even have to try very hard for them to accomplish those things.

One more thing: I was absolutely amazed by what my kids were able to do by the end of the semester. Most of them literally had no idea what programming was when the semester began, yet for their final projects they created programs that were pretty complex! Some examples of student work follow. All of these are interactive games made in VPython (Visual Python). (In retrospect, I might have taught VPython from the beginning. Even the few stragglers who weren't really engaged in programming at the beginning couldn't help but get into it when we started the visual stuff.)

A soccer game where the player controls the blue goal and the
ball moves randomly about the field

A classic 2-player game of dots & boxes (one
surprising challenge associated with this game is having
the computer recognize when a player has won)

A car game where the user has to navigate the red
"car" around a series of moving obstacles

An asteroid field where the user controls the
space ship and tries to avoid oncoming asteroids

If you have any say in this whatsoever, teach your kids programming! (Even if that means you teach yourself programming along the way.) Python is free and open-source and available for download here. Also check out a bunch of great resources for teaching programming.

A really fun physics/calculus lesson

2010-12-18T08:59:00.000-08:00

This was the last week before winter break, and my honors calculus classes spent it working on The Great Calculus Challenge. In a nutshell, I built a wooden ramp and I told them we'd be putting the ramp on the roof of the school and letting a metal cube (I used density cubes - brass in one class, copper in another) slide off of it. They had 3 days to figure out where on the ground it would land, and as a class they'd get to place one blue plastic cup on the ground. The goal, of course, was for the block to land in the cup. I made a big show of the fact that the entire class would get only one try to get the block in, so they all had to agree on their answer.

This was a sort of "culmination" of our first semester of calculus. We've spent a lot of time talking about derivatives and antiderivatives in the context of motion - position, velocity, and acceleration. My students had done tons of problems about motorcycles screeching to a halt, potatoes being projected off of cliffs, etc. The new mathematical element here was that students had to calculate the velocity of the block leaving the ramp, which required them to take into account acceleration other than that due to gravity (like friction).

So what happened? In the first class, as my students were feverishly perfecting their calculation, my boyfriend (who is finishing his Ph.D. in math, and who I'm trying to convince to become a high school physics teacher - hence dragging him to school for the day) did his own calculation in about 10 minutes. (He actually wrote a little Python code to help him.) When we went outside, the class put down their cup and my boyfriend put down his (it was just short of theirs ... very "Price is Right" of him!) and, lo and behold, the block landed in his cup! I would say that the excitement this caused was a very close second to what would have happened had the block landed in their cup. In the second class, we did the same and this time everyone agreed on where the block should land. However, it fell about 2 inches short of the cup. We talked about why this might be, and I blame it on the shoddy craftsmanship (and therefore variable initial conditions) of the ramp (for which I am completely responsible). In any case, it was a fun way to spend the week before break:

Some students chose to solve the problem
by experimenting from different heights...

Others took a pencil and paper approach...

A good time was had by all...

Especially by my colleague Kyle, who got to
scale the building and drop the block for us!

The anticipation was INTENSE...

And in the end, the block fell just a smidge short.

For me, this turned out to be an experiment in what happens when you tell a group of 25-ish bright, motivated students that they have three days to come up with one answer to an open-ended problem. The two classes approached the task completely differently - one class relied heavily on a couple of "leaders" and many students were quiet or worked mostly independently, while the other class naturally split into a few truly collaborative groups. It was fun for me to be a bystander, observing the classroom dynamic and occasionally giving a cryptic nod of my head or raise of my eyebrows to indicate whether or not they were on the right track.

I really liked this activity for one main reason: when a couple of students asked how they would be graded, I got to be really dramatic and say something like "Graded??? This is SO much more than a grade! This isn't me versus you, it's you versus the laws of physics!" and that kind of silenced that conversation. I would love to have more of these activities in my back pocket for next semester, where there's a high level of intrinsic motivation, especially because I'll have second-semester seniors ... any thoughts?

I can prove it, but I don't believe it!

2010-12-07T21:55:00.000-08:00

There's a problem I really like whose result always surprises me. First, imagine you have a piece of string that’s long enough to stretch all the way around a basketball (the circumference of a basketball is 30 inches). Then you realize you have an extra 24 inches of string in your pocket, which you want to add to the string. So, you cut the circle of string somewhere, add exactly 24 inches, and then smooth it out until it makes a circle all around the ball (kind of like a ring orbiting a planet). The question is to figure out how high is the string off the basketball? It's a simple geometry calculation, and you wind up getting around 3.8 inches.

Got it? Now, try the same problem, except instead of a basketball imagine that you start by wrapping a string around the equator of the earth. Then, just as before, you find an extra 24 inches of string in your pocket, which you add on to the string, and then smooth out the resulting string until it makes a circle around the earth. How high is the string off the earth?

My intuition always tells me that the gap should be minuscule - after all, what is a mere 24 inches compared with a 25,000 mile equator? But of course, every time, the answer comes out to be ... around 3.8 inches.This definitely falls under the category of "I can prove it, but I don't believe it!"

It occurred to me that there just HAD to be some calculus in this problem (is there a problem for which that couldn't be said?), and lo and behold I found it. For their weekly challenge set, I gave my students the basketball/earth problems, and also this last one: Express the radius of a sphere as a function of its circumference, and then find the derivative of this function. Why does this make sense in light of your previous answers?

The answer is that r(c)=c/(2*pi), so r'(c) is just the constant 1/(2*pi)! So, regardless of the starting circumference, a constant change in circumference will result in a constant change in the radius. Neat, huh?

So I get it. I can prove it with calculus and without calculus. Yet somehow, I still don't really understand how this can be true. And I have to admit, after years doing math calculations that "give" the answer, it's always refreshing to come across one that doesn't...

Order of Operations?

2010-12-03T23:38:00.000-08:00

One of my calculus classes just finished a long introduction to the derivative -- we learned it (or rather, I taught it) once, at which point I realized that the students were still completely confused about the concept of the derivative even though they were making progress in actually computing derivatives. So, we spent another week reviewing both the concept and the skill (as I attempted to explain, possibly in vain, why both of those things are important ... I told them that if they left my class without being able to explain the derivative to someone, I would have failed at my job). After another week of review, I am confident that they are ready to move on to the next topic.

This of course brings up the question of what the "next" topic is. Calculus is not linear, and at a school with absolutely no prescribed curriculum the thought of moving on often propels me into a panic-ridden tizzy. What's the best next topic - the one that will tap into the students' current understanding, keep their interest, draw on their mathematical strengths, and work on their weaknesses? (Am I asking too much of a single topic?) My two thoughts are:

Having them discover the power rule. They're primed to appreciate it now that they've spent weeks calculating derivatives using limits. Also, my other classes have known about the power rule for awhile now and have already tried to ruin the surprise (I have to admit, part of me loves that there even is a math rumor mill!), so I'd feel a little slimy hiding it any longer.
Having them practice graphing functions and their derivatives. This will get back to the concept of the derivative as the "rate of change," which is something I really want to hammer home. I would have them start out simply by making observations using this super-rad calculus grapher, which I just read about on Sam Shah's blog. Gosh I love the internet.

I teach three calculus classes - two honors and one regular. Teaching this non-honors class in particular is really forcing me to hone in on the essence of calculus. Somehow I'd be missing the point if I tried to push the "standard" calculus curriculum on these students, because they're still getting comfortable with so much of the algebra that successful calc students take for granted. I'm sure that we could eventually get to the point where they could apply the quotient rule and the chain rule to complicated functions, but it would be at the cost of a greater conceptual understanding. This reminds me of something my office-mate Kyle said the other day: "A year from now, I'd rather have students say that they understand what derivatives are and they used to be able to calculate them, than that they know the derivative of x^2 is 2x and they used to actually understand why."

This brings up the juicy debate over how and why calculus is taught in high school, but that's for another day...

The Agony of Math Review Games

2010-11-30T22:13:00.000-08:00

My calculus classes took a quiz on the power rule and "reverse power rule" before Thanksgiving break. (The "reverse power rule" is my name for the elementary anti-derivatives they've been taking ... it's always funny to make up terminology on the spot, and then all of a sudden realize that the "reverse power rule" and the "freeloader rule" are now recorded in 50 calculus notebooks). The results of the quiz suggested that my kids needed more time to digest the material, so that's what we've been doing for the past couple of days. They have a re-quiz tomorrow, so I thought that a good way to practice in a relaxed, positive atmosphere would be to play a review game. Turns out, I was wrong.

I stole the game from one of my colleagues Ted. Students work in pairs to solve a problem on the mini white-boards. For the first problem, Partner A is the only one who can write on the board; Partner B can help and offer advice, but cannot touch the marker. The roles switch for each problem. When they are done, they hold up their board. The first team to hold up the correct answer gets to throw a hacky sack at a target I've drawn on the board, and depending on where the sack lands they either get 1, 2, or 3 points. The throwing of the hacky sack clearly has no purpose other than getting the students up and excited, which kind of reminds me of Solve Crumple Toss, except that they don't have to throw away their work when they finish a problem.

I'm not sure what it was -- maybe I made the problems too hard, maybe I should have given them time in between rounds to finish up their solution even if they weren't the winning team, or maybe I should've figured out a way to get rid of the time pressure. In any case, a couple of teams wound up dominating the game and many students expressed frustration after class that they weren't able to get any of the problems right and they didn't like the time pressure. I tend to agree about the time pressure - after all, it's not about how fast you can do a problem, but how well - but if there's one calculus skill that lends itself to this kind of a game, it has to be taking derivatives using the power rule. It's like the multiplication tables, but for calculus.

I'm really curious if anyone out there has developed the perfect math review game -- one that is fun and competitive, but encourages (or better yet, forces) everyone to participate and doesn't automatically favor the quickest kids in the in the class.

"When will we ever use this?"

2010-11-17T06:56:00.000-08:00

At my school, there's a huge emphasis on "adult-world connection." A question that teachers (or at least I) often hear when introducing a new concept is "How is this used in the real world?" (As an aside: How much do teachers at other schools encounter this?) Last year, I taught 11th grade math: an ill-defined combination of pre-calculus, algebra II, statistics, and random topics that don't really fit into any standards. I tried to create projects that touched on several different applications of math (such as linear programming to solve a land use problem, and statistical analysis of a survey the students created). Granted, I'm sure I could have done a better job creating and executing projects, but based on my limited experience I came to the following conclusion: real real-world math is really messy ... mostly too messy to survive the attention span of the average high schooler. So the real-world math we did was really fake real-world math. And guess what? The students totally didn't fall for it. It felt like a completely contrived textbook word problem that was extended into a month-long project.

Lately, I've been feeling that the question "How is this used in the real world?" is a bit misguided. Sometimes there is a really good answer that fits into a 2-minute sound bite without completely disrupting the flow of your class. Quadratics are the classic example of this: How will you ever know where your rocket/football/rubber chicken will land if you don't understand projectile motion??? However, once you step into the realm of calculus it's difficult to encapsulate the usefulness of something like the derivative in 5 sentences or less. (Or not ... please let me know if I'm totally off-base here!) Most often, I find myself wanting to give one of these two responses to that dreaded question (although I never would):

Asking how [insert complicated math concept here] is used in the real world is like asking how multiplication is used in the real world. It's everywhere, but could you sum up how it's useful in 2 sentences to a fourth grader who was just learning his times tables?
The real-world connection is that you're going to have to be able to [insert tedious technical skill here] in order to pass your intro-level calculus/physics/engineering classes in college so that you can move on to bigger and better things.
Does it matter??? Can't we just do it because it's interesting in its own right? A little abstract thinking never hurt anyone! (Shockingly, this argument doesn't actually seem to make concepts more interesting.)

My point is that its connection to the real world is only one (small) reason why I love math. What I really love is the feeling of success when I finally figure out something that I never thought I'd understand. I love discovering beautiful patterns. And yes, I sometimes even love the rules and the stability that they offer in the midst complete chaos. I'm also convinced that (most) high school students don't really want to learn about real-world math, except for maybe the occasional pretty picture of a fractal or 2-second snapshot of some grotesque computation that is used to solve a "real" problem. Instead, they (like all of us) just need a hook. Usually that hook is sinking their teeth into a problem that captures their attention, not doing a "realistic" mathematical analysis.

Don't get me wrong -- I do want my students to understand why mankind has spent hundreds of years studying the derivative. I want them to understand the equations of motion, and optimization, and differential equations -- but it's going to take an entire year of calculus to really answer that question, and I'd rather answer it well in a year than answer it superficially in 5 minutes. In the mean time, I can try my best to make the concepts engaging in their own right so that my students want to learn them, even if they don't instantly see the "real-world connection."

I swore I'd never do this...

2010-11-14T07:32:00.000-08:00

Before I started teaching a year ago, I didn't understand the blogosphere. What made people think that there was an audience out there for their every musing? I swore that I would never be one of "those people" ... and then I started teaching. It took me about five minutes to realize that I had absolutely no idea what I was doing and I sought solace in - where else? - the internet! I quickly found an online community of math teachers whose stories made me laugh, whose struggles I could identify with, and whose ideas I started stealing (er, "borrowing") by the handful. I survived my first year with my head (barely) above water, and now I'm a seasoned (ha) second-year teacher. As my profile indicates, I teach math and computer science at a public charter school in San Diego, CA.

I'm blogging now because I have a lot to say and, to be quite honest, my boyfriend is sick of hearing me blather on about teaching. I am now one of those people who knows there's an audience out there for my every musing. :) Seriously, though, I feel that if I can contribute in some small way to this amazing online collaboration that's going on, then it's worth doing.

I'll keep my first post short. I know that many have said not to let a catchy name stand in your way of creating a blog, but I just couldn't do it until I had the right name. And then, a couple of months ago, one of my students inspired me with a comment he made in class. My calculus class was starting limits by figuring out the area of a circle by taking the limit of the areas of inscribed polygons with more and more sides. At the end of our discussion, one students made a comment that was so absolutely perfect that I couldn't believe that in all of my years of studying math it had never occurred to me: "So a circle is really an infinigon?" I got super-excited as I tend to do when my students astonish me, and I told him that I would include that in my book one day ... for now, a blog will have to suffice.

What makes a "good" challenge problem?

2010-11-13T16:42:00.000-08:00

I've been giving my calculus students challenge sets every couple of weeks or so (8-10 total for the semester). I'm requiring that they complete at least 5 of them, and their challenge set grade will count for 10% of their final grade. Many students work on all of them (for some combination of enjoyment and extra credit, I think) and I've learned that before each set is due there's a group of them who loiter at a local chain establishment and argue for hours about the problems, probably spending a grand total of $1 on a bottle of water. I'm not going to lie -- this brings back fond memories of math study groups as an undergraduate, and it makes my heart smile to know that they've started their own little geek-fest.

It turns out that some of my students are pretty demanding and tell me when I've given them a "good" problem and when I haven't. They didn't like their last set so much - it basically asked them to investigate the limits of a function similar to f(x)=sin(1/x). I have lots of thoughts as to why they didn't enjoy this one as much, and it's primarily because they didn't really understand what I was asking for and started it too late to ask me any questions. However, it makes me incredibly happy that they even have an opinion one way or another about the problems: passion in the classroom = a math teacher's dream.

I fear that I've ventured far, far away from the course content with these challenge sets. On one hand I feel like I should be able to come up with some honest-to-goodness calculus challenge problems; on the other hand, it's actually more important to me to generate some excitement about math, period. It's probably also not a bad idea to give the kids who aren't loving calculus another point of entry into math.

I really, really like this week's challenge problem:

Alice and Bob need your help! They have been captured by pirates and will only be released if they can accomplish the following task:

A pirate will deal Bob five random cards out of a standard deck of 52 playing cards (no jokers). He gets to choose one card to put aside as the “mystery card.” He must use the other four cards to communicate the identity of the mystery card to Alice. They may not talk or look at each other’s faces once the cards are dealt. However, they can communicate and agree on a strategy beforehand.

Alice gets one guess – if she’s right (about the number and the suit of the card), the couple will get released. If she’s wrong, it’s bad news for Alice and Bob.

What should their strategy be?

Note: Your strategy shouldn’t contain anything shady like “If Bob points his index finger a certain way, the card is a heart.” There is at least one purely logical/mathematical strategy.

I'm trying to figure out what makes a "good" challenge problem - one that's rich with mathematical ideas but also approachable and engaging. So far, I've got:

Anything having to do with infinity (like this back-handed approach to the harmonic series);
Something they can "play around with"/brute force until they figure out a more "elegant" solution

Finally, I also tapped into current grad students in my old math graduate program for help. Every grad student has their favorite puzzles and a desire to do anything other than their research, so not surprisingly I got a ton of great problems from them. Here they are, for your perusing pleasure.