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

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.

### Sample topics

**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.

This post made me rethink/rewrite the syllabus for one of my courses this fall. I have always been troubled by the fact that A’s should only be given for extraordinary work— and I didn’t see how just more of the ordinary (i.e., points) should count as extra-ordinary. So, I think you’re idea of having a separate category of work be required for an A is a great one.

How often have you had students complete 2 assignments and then end up with a 79% in the class?

bwfrank:I’m pretty sure it hasn’t happened. But I’ve only been at it for two semesters. Part of the reason it hasn’t happened is that I’ve got the system rigged so that proficiency yields a B-usually in the mid-80s. So a 79 means there’s an important gap somewhere, and that’s unlikely to be associated with a student who completes two A assignments. It’ll happen someday, though.

What has really struck me in the process is the quality of conversation I get to have with those working on an A assignment, and the massive increase in the quality of their work over time. These students are

doing mathematics, with false starts and revisions. And that’s due to a combination of (1) the expectations of the system of A assignments and (2) the tasks I choose for them.Yes, I choose the tasks. But I don’t choose the solution method. My comments on students’ work are intended to help them make their thinking better, not to force them to think about these problems my way.

Perhaps in the next round of the ever-increasing complexity of the structure of my courses, I’ll invent (or steal) a method for getting students to pose their own tasks for A assignments. But then I’ll have to worry again about the students I address in this passage:

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And you do this with college students? Has it been tried with high school or (shudder) middle school students?

Yes. I had a similar system with my middle schoolers back in the day. It was all based on a bulletin board with the title “How do I get an A in math class?” Interesting and unresolved questions accumulated back there and students could take them on at their leisure.

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Christopher,I love your blog. Being a math teacher and having two young children of my own – I have very similar conversations with my kids as you with yours. Quick question – do you have an example of an A Assignment for middle schoolers you’d be willing to share? I’m curious about your formatting and how open these were which in turn means how accessible they might be for all kids. Thanks, Greg

In Britain, students really don’t care if their teacher gives them an A for work done during class. Teachers grade work, of course , but teacher grades really don’t mean much to the students or their parents.

All that really matters is the grade in the external examinations.

So I teach to the test. There is no point giving open-ended questions where the method to reach the answer is not to a large extent already set down.

Because those sorts of questions and that kind of thinking will either not be on the external exam at all, or it will constitute a very , very minor part of it.

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