Back-to-Back Pictionary!

Last week in my highest level class (Calculus A, a class for juniors that combines all of Precalculus and the first half of AP Calculus AB) I was astonished to see how abysmal my kids are at describing graphs using the kind of terminology that their tests and exit tickets would *indicate* they understand - words like increasing, decreasing, concave up or down over the interval ... , symmetric about ... , etc. This occurred to me as I was having a student review a Do Now with the class and when he called on a student to describe the graph of f(x)=x^(2/3), the student described it as "the bird." It was clear then that we had a problem. Students could define the terms above and use them in the context of multiple choice test questions, but they are far from fluent in graphical lingo.

Enter Back-to-Back Pictionary.  I played this today in one of my regular Precalculus classes and it turned out to be a blast and also extremely productive. Partners sat back-to-back, each of them with a different sheet of nine graphs each. Some of the graphs were recognizable based on the types of functions we've studied, and others were strange piecewise concoctions. They also had their mini whiteboards. Students took turns choosing a graph to describe to their partner, the caveat being that they could only use sentence starters from the word box: "The graph is increasing/decreasing over ...", "The graph is concave up/down over ...", "The graph is symmetric about ..." etc. I told students that they weren't even allowed to reference the name of a parent function in their description (let alone make bird noises, as one student asked). The sketcher listened to his partner and drew the graph on his board; when he thought he was finished, he'd show to graph to his partner. The team got a point if the graph on the whiteboard matched the graph on the sheet.

I modeled one with a student, and then they played for around 15 minutes. The room was loud and slightly chaotic, but it was exciting to hear students screaming at each other "I TOLD you it was MONOTONICALLY INCREASING!!!" Not something you hear every day (but hopefully something I'll hear more often, going forward).

The 33-50-95-100% Model

Every so often I have an epiphany about life or about teaching, and even less frequently I have one that lies in the intersection of the two. I realized recently that my OCD, type-A, control freak nature (while often productive and even, some say, endearing) has just as much an effect on my classroom as it does on my life outside of school. Although it would be healthy for me to take a step back and learn to relinquish a little control in both realms, I'll spare you the details of my personal life and stick to my teaching. This "relinquishing of control" is something I'll call the "33-50-95-100% Model".

I'll start off by explaining the model I had been using, which I'll call the "100% Model." In the 100% model, I'd introduce a new concept, design a class period or maybe two of very thoughtful, methodical guided practice, assess the objective on an Exit Ticket, and foolishly hope that 100% of students had mastered the topic. Undoubtedly they wouldn't, and so I'd hurriedly try to patch the holes using Do Now's or spiral homework assignments over the next couple of weeks, so that all 100% would at least master the objective by the time of the unit test. Which, of course, they wouldn't.

Right now my Precalculus class is finishing up their last unit, on limits & continuity. The first topic in that unit was reading the graph of a piecewise function to determine one- and two-sided limits, function values, and types of discontinuities at certain points. And, as with anything graph-related, there was a group of students (about one-third of the class) who immediately caught on (aced that day's exit ticket) while the rest seemed to have forgotten everything they ever knew about reading a graph. Ordinarily I would have done a significant amount of spiral review, but due to timing constraints I had to give a quiz prematurely. Of course, the 33% who understood the material aced the quiz, and the 67% who didn't, didn't.

I had a little more wiggle room during the week after the quiz, so when I handed back the quizzes I did so in groups of 3, basically putting one of the stronger students in charge of tutoring 1-2 peers who were struggling. They had 30 minutes to figure out their mistakes on the quiz and then complete a practice sheet with similar problems. At the end of the period there was an independent exit ticket, again with similar problems. Results of the exit ticket suggest that around 50% have mastered the content, while all but a few are on their way and much closer to mastering it than they were at the time of the quiz.

I suspect that spiral review (through Do Nows, quick drills, etc.) will be much more effective now that a critical mass of students have mastered the objective - hopefully so much so that by the time of the unit test, only a few students will still be struggling. And those are precisely the students that I will be able to focus my attention on during after school tutorials.

I think the only real difference between the two models is that in the 100% model I was operating on the (obviously false) premise that if I just taught a particular topic "well enough" the first time, all students would be able to master it at the level I required and on a timetable that was convenient for me. The latter model is simply acknowledging the unreasonableness of this assumption by allowing kids to move on with a slightly new topic so that they don't feel like they're beating a dead horse every time they walk into class, and then coming back to the original topic in a few days or weeks with a fresh set of eyes and a built-in set of peer tutors. The buzz in the classroom was focused and productive during the peer tutoring session; the strong students were challenged to explain problems and forced to ask questions about the subtler points that even they struggled with, while the weaker students got the one-on-one targeted instruction that they needed but would have never gotten in a large group review.

Of course, this was not easy for me. I had to accept that 100%-or-Bust only has one possible outcome, and I had to trust both my student tutors and tutees to do what they needed to do while I merely facilitated or clarified a point here or there. But at the end of the day, I think I have myself a new paradigm for thinking about how my students actually learn (as opposed to how I wanted them to learn).

An antidote to senioritis?

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!

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

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?