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?