Stella in My Classroom

by Rudd Crawford

Stella's Stunners were an official part of my high school mathematics curriculum at Oberlin High School from 1984 until I retired in 2005. The problems were woven through the courses I taught, in various ways, depending on the courses and the nature of the students in them. Here is the scheme, the big one, the most complicated way I used them.

Imagine a tracked course, with students of ostensibly similar levels of achievement and motivation. On Friday I hand out a set of eight problems. In each set the material ranges from beginning-level visual puzzles, through arithmetic problems, and on up so that the last two are loosely related to the material from the current course, one from earlier in the course and the other from more-or-less current material. You see that most of the problems don’t have anything to do with what’s going on in the course at the time. The eighteen suggested problem sets are designed this way on purpose.

Some of the problems are very hard, with a certain outrageous quality to them, either because they seem outrageously difficult or because their stories are ridiculous. I got over my own fear of the problems and got better at solving them myself. I would occasionally give the students a problem I couldn’t do, and over-rode their objections with, “Look: if I’m the best problem solver around here, this school is in trouble.” So they’d work on them, if only just to show me up, which occasionally they did.

I should note here that these problems are assigned entirely on top of the regular course work and are mostly unrelated to it. Interestingly, the regular progress of the course is not slowed down by the "Stellas", as the kids called them.

Students have a week to work on the problems. They are charged with having no contact with anyone else, including other students. Each student keeps a notebook. On the following Friday the notebook is due: each problem has been cut out from the handout and pasted into a page in the notebook, followed by the written-out solution or partial solution that the student has come up with. He also writes a "Dear Stella" note, reflecting on his feelings about the problems and his week's work. I collect the notebooks and pass out the next set of problems. I read the notebooks over the (busy) weekend, write comments, assign some points, and hand them back early in the week – I try for Monday. See below for how I grade the notebooks.

So the students have had their new problems over the weekend. On Monday we read them through together, just to make sure that everybody understands what each problem is asking for. The students are wide-eyed; I laugh at them. As the week goes along, I check in briefly each day to see how things are going. I may give a hint about one or more of them if everybody is stuck, or I may suggest a heuristic to try.

On Friday, the students turn in whatever they've done, get their new problem set, and so it goes. I skip the weeks when we have a test, or when there is otherwise too much going on. (We've provided eighteen suggested problem sets, so you can give one out on the average of every two weeks as the year goes along.)

One variant of this system, as explained so far, is to permit collaboration among students. But parents are specifically forbidden to help until the problems are more than a week old; students are then free to bedevil their parents with them to their hearts’ content.

Grading:

The goal is a fair and efficient process.

Each of the week's problems gets from 0 to 4 points for mathematics, and either 0 or 1 for grammar and English usage (I am relentless about this). I put the scores on a little pink tally sheet, along with my often-extensive comments, taped by each problem in the notebook. Then, to tabulate a math grade, I use Steve Meiring's brilliant point system, as follows.

The problems are hard enough that a student getting half of them has done very well. But that's only 50%, which would be failing in a regular percentage system. Steve's system gives, say, 28 points (out of 100) for the first correct solution out of the eight attempts, no matter which problem it is, 20 for the second, 12 for the third, and so on in decreasing order until the last one gives only 2. Thus getting three problems right out of the eight is 60 points already, getting five is 80, and when you're struggling with that last toughie you're only losing 2 points if you can't do it. I make an overall judgment about the spelling and grammar. This sounds complicated and time-consuming for the grader, and yep, it is. The tiny pink grading ticket makes the process efficient. See the illustrations provided. I’m not sure why they were on pink paper; they just were.

Besides the solutions of the problems themselves, maximum total 92, I give a maximum 8 points for pasting in all of the problems whether a solution has been attempted or not, for logging the amount of time spent on the whole set, for listing their collaborators if any, and for writing a "Dear Stella" note commenting on the problems in general and how they felt – good or bad – about doing them. These comments will range from "I couldn't believe I actually got the third one!" to "Dear Stella, when are you going to die?"

The English score is kept separate from the math score. All in all, there are 100 points available for the math plus eight or so for English.

At the end of the marking period, I usually give the total Stella grade about the same weight as one test. This depends on the type of class that it is, though. If the class is anxious about how they’re doing, I say that I'll include the Stella grades only if it will help a student’s overall grade for the marking period. (Everybody has to turn in every set, though. No skipping.) (Actually, if a student misses even one homework or Stella assignment during the marking period, that student will fail the whole course for that marking period, but that’s another story, never mind...)

I found it enormously satisfying to have this written back-and-forth with each student, keeping in closer touch with both their minds and their feelings than I could have done otherwise. I felt as if I was really being their teacher.

Variations:

The mathematics requirements for graduation have increased over the years, and math classes in our very small high school have to be untracked for scheduling reasons (the tracking had previously been automatic, with many students simply not taking more than a year or two of math). Thus in our classes we faced a wide range of achievement and motivation, with struggling students in the same class with hot-shots. Running a Stella program in those classes was a serious challenge for me, and I am not entirely happy with what I did. I'll describe that first, and then say a bit about what I might have done (OK, what I wish I'd done).

What I did:

In the first place, I didn't assign as many problems, sometimes only two. I tried to find particularly interesting or outrageous problems to pose. I gave more time in class for discussing them and teasing out possible approaches, and then on Thursday, if necessary, we actually solved at least one of them as a class, and I’d write the full solution on the board that students could copy and use for their write-up, due on Friday. Some of the students would come up with solutions of their own – but their Stella experience was thin soup – two problems instead of eight? Sheesh. But read on:

What I wish I'd done:

I would like to have told the class that doing well on all of the regular work of the course, including the minimal Stella program as described above, was enough to earn a B in the course. Any student wishing to get an A would need to do the Stella program in the classic sense, all eight problems, the whole thing. This is sort of like using Stellas for extra credit, except that the grading structure made it a requirement for the students who were going for an A. What held me back from that, I guess, is that I never dealt with how to find class time for the discussion of the hard problems since not all of the students were working on them.

What's coming on stage here is the general problem of differentiated instruction in an untracked class. I was never very good at it, particularly with regard to the enormously diverse achievement/motivation levels within in math classes at OHS that resulted from the ever-increasing graduation requirements.

HEY! If you, yes you, come up with a way of using Stellas in a mixed classroom that you like, let's hear about it! I would like to think of all of us Stella users as a community. You can be a help to all of us.

Back to my experience. In whatever way they manifested themselves, Stella and her Stunners were a strong presence within the student body – the talk of the school, as a history teacher put it, a known part of the curriculum, outrageous, crazy, and demanding, that everybody suffered through, with generally buoyant feelings. There were several years when the problems were part of every math class taught by anybody in the math department; I'd sometimes recruit the dazzling brainy cheerleaders to come into my freshman class to help the star-struck beginners. It was a golden age.

By the time I retired I was the only teacher using them, at the reduced level of intensity. I was sad about this, because students from the 80's and early 90’s would come back for reunions and tell me "Stella got me into law school," "Stella got me high marks on the SAT's," so forth. I'm not at all sure that by the end there was enough voltage in the way I was using the Stunners so that recent graduates could say the same. There’s a collection of feedback comments below.

I am eager – beyond eager, actually – to hear of your experiences as you experiment with ways of introducing your students to Stella's Stunners.