Stella in My Classroom
by Rudd Crawford
Stella's Stunners are problems that are tough but tantalizing. At first glance they are daunting, with insufficient information given, or with seemingly contradictory or impossible conditions, or simply of enormous difficulty. Their purpose is to go well beyond the routine problems of a course and to put students into situations where they don't know what to do. The question, "What do you do when you don't know what to do?" is ever present, and leads to substantive work with heuristics. In working with the Stunners, students become ever more crafty in their approaches — their minds wake up and the "aha" moments become more and more satisfying.
Stella and her Stunners were a part of my classroom from 1984 until I retired in 2005. The problems have been woven through the courses I have taught in various ways, depending on the course. Here is the classic scheme of how I used them.
One Way to Go
Imagine a course with students of ostensibly similar levels of ability and motivation. On a Friday I would hand out a set of 5-8 problems. The material ranged each week from beginning level visual puzzles, through arithmetic problems, and on up so that the last two problems were related to the material from the current course, one from earlier in the course and the other from more-or-less current material.
I should note here that these problems were assigned on top of the regular coursework, so progress in the course was not slowed down by the "Stellas," as the students call them.
Students were charged with working on the problems with no contact with anyone else, including other students and especially including parents. Each student kept a notebook of the problems and solutions. By the following Friday the notebook was due. Each problem was cut out from the handout and pasted into the notebook, followed by the written solution or partial solution that the student had come up with. On Friday I collected the notebooks and passed out the next set of problems. I read the notebooks over the (busy) weekend and handed them back early in the week — I tried for Monday. See Grading Tips for how I graded the notebooks.
So the students had their new problems over the weekend. On Monday we read through them together, just to make sure that everybody understood what each problem was asking for. There was probably no further discussion of any of the problems that day. Then, as the week went along, I checked in each day to see how things were coming. I might give a hint about one or more of the problems if everybody was stuck, or I might not.
On Friday, the students turned in whatever they had done.
One variant of this system, as explained so far, was to permit collaboration among students. But parents were specifically forbidden to help, in spite of the fact that the problems make great dinner table conversation.
Using the Heuristics
I never taught the heuristics directly. I would wait for students to get stuck on some problem and then suggest, "Well, here's something we might try." It seemed better to me to pull out heuristics when they were needed and would be appreciated, rather than showing them a particular heuristic and then apply it right away to a problem. Too prescriptive!
Having students keep the full list of heuristics in their notebooks is a good idea.
A colleague who also used the Stella problems made a filing card for each of the 25 heuristics and put them up around the room. Then when a student devised a nice solution using one of them, she would tack the student's paper up by the card. Students could see the heuristics in use that way.
Variations. In a heterogeneous class, such as we have at tiny Oberlin High School, there is a wide range of ability and motivation, with very weak 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 will describe what I did and then do some thinking about what I might have done.
In the first place, I did not assign as many problems, sometimes only two or three at a time. I would try to find particularly interesting or outrageous problems to pose. I would give more time in class for discussing them and teasing out possible approaches, and then on Thursday we would actually solve at least one of them as a class, and for particularly weak students, I would write a full solution on the board that they could copy and use for their write-up. The hope was that the more talented and motivated students would be finding their own solutions. Many of my students could hardly write a coherent sentence on their own, and I had to believe that the process of copying out a logical argument helped them. But the top students did not get the challenge they deserved, no doubt of that.
What I Wish I Had 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, etc. This is sort of like using Stellas for extra credit, except that the grading structure made it effectively a requirement for the top students. What held me back from doing this, I guess, is that I never found class time for discussing the hard problems.
The underlying issue is how to implement differentiated instruction in a heterogeneous class. I was never very good at it, particularly with the enormously diverse levels of ability and motivation that resulted from the ever-increasing mathematics graduation requirements.
In whatever way they manifested themselves, Stella and her Stunners were a strong presence within the OHS student body. Stellas were a known part of the curriculum — outrageous, crazy, and demanding — something everybody suffered through, with ultimately buoyant feelings. There were several years when the problems were part of every mathematics class taught at OHS. I would sometimes recruit the dazzling brainy cheerleaders to come into the freshman class to motivate the star-struck beginners. By the time I retired in 2005, I was the only teacher still using them, at the reduced level of intensity. I was sad about this, because students from the 1980s would come back for reunions and tell me, "Stella got me into law school," "Stella got me high marks on the SATs," and so forth. I hope that by the end there was enough voltage in the way I was using the Stunners that recent graduates could still say the same things.