I have a problem-solving approach to teaching.
By this I mean that I try very hard to identify the problems of teaching and I look for solutions to these problems. Each solution creates new problems, for which I seek new solutions. Etc. It’s part of what keeps the job interesting for me.
One of the problems I have always had-in my middle school and my college teaching-is how to use the drive to achieve that some of my students have to get them involved in doing interesting mathematics.
This is different from the problem of trying to get the whole class doing interesting mathematics-that’s a curricular problem. I’m talking about getting my high achieving students to do some original and independent thinking.
These are students who are used to getting A’s for mastering techniques. If the teacher tells them what to do, they’ll do it. They are accustomed to meeting external expectations and to being rewarded for it.
I have tried many things to engage these students in thinking beyond the curriculum and I have failed many times.
But now I have something that seems to be working. I call them A Assignments.
Leave aside the rest of my quirky grading system. We’ll begin with the system of A Assignments.
In my courses, there are 100 points in a semester. If you get 60 of those points, you earn a grade of D. 70 is a C. 80 or more is a B. And if you get 80 or more points and complete two A Assignments, you earn a grade of A.
There are four A Assignments to choose from, and they are made available (ideally) at the beginning of the semester. Each A Assignment requires students to dig deeper into some aspect of course content than is required in the standards for the course.
Students read over the assignment, discuss it with me if they like and get to work. They submit a first draft, get comments and feedback from me, then revise and submit a second draft. If this draft is satisfactory (see below for definition of this), then they are done. Otherwise, they do a third draft, etc.-as many as necessary. When in doubt, we return to the definition of satisfactory completion.
Completion of two A Assignments and B work at the level of 80% or better in the rest of the course earns an A in the course.
A Assignments are optional in the sense that they are not required for completion of the course, nor for credit. Students with 95% of the points who have not completed two A Assignments receive a grade of B (but this is extremely rare).
A Assignments have solved my problem. There is enough structure for students who need external direction, but there is enough openness to the task that I feel I am getting these students to push beyond their intellectual comfort zone.
And now the next problem I am working on is how to engage the students who would benefit from working on the A Assignments, but who don’t care about the A. I was that type of student.
Satisfactory completion denotes mathematically correct responses to all prompts in the assignment and adherence to academic norms for written work-including grammar, presentation and academic integrity.
College Algebra: Investigating non-linear asymptotes, Writing an equation modeling the pH of water mixed with orange juice concentrate, Determining a rule for when a^b>b^a for a, b positive integers, etc.
Math for Elementary Teachers: Determining whether all multiples of 12 are abundant, Developing number language for Mayan numeration, Determining under what conditions the standard algorithm for 2-digit by 2-digit multiplication will give identical partial products, etc.