Monday, August 18, 2014

standards for analysis

Writing standards for a proof-based class is a different beast than for introductory calculus, or even probability. In my last post, I described a bit of the structure of the analysis class I’m teaching this fall: inquiry-based, primarily structured around group work, running on a weekly cycle of tackling a problem, agreeing on an approach, and presenting a solution to the class for discussion. My usual way of compiling standards—looking through the course content and breaking it into 20–30 skill sets of roughly equal importance—sort of falls apart here. Do I want students to be able to prove that every Cauchy sequence in the set of real numbers is convergent, and to explain what this implies about the completeness of the reals? Yes, but what I really want is for them to be able to assimilate new concepts and make sense of them by creating examples and fitting the definitions into proofs. Do I want them to be able to compute integrals with respect to both Lebesgue measure and singular Dirac measures? Yes, but what I really want is for them to see how these represent the interplay of mathematics and other sciences—how the exigencies of other fields of science led to the development of both the Lebesgue integral and the Dirac delta—and to feel part of a scientific community, both in and out of the classroom.

While considering these questions, I determined that there are six standards I want students to actively develop during the semester, and on which I want to be giving targeted feedback. These skills will be grounded in the content of the course, but they will also provide the benchmarks of success in mastering the content. Here they are:

  1. Correct use of vocabulary and notation: Using mathematical terminology and symbols, especially those particular to analysis, correctly and appropriately.
  2. Correct and convincing argumentation: Creating and recognizing complete proofs, with their various pieces presented in a logical order.
  3. Clear written exposition: Organizing a paper for the benefit of the reader, making it easy to read and using proper English grammar.
  4. Broad vision of the subject: Providing context in papers, including statements of solved problems, a guide to the structure of proofs, and connections with other ideas in the class (previous work or larger themes).
  5. Effective verbal presentation: Using good speaking habits (e.g., speaking confidently, talking to the class and not to the board, being sensitive to the audience, handling questions well) to present mathematical content.
  6. Collaboration and participation in discussion: Attending class regularly, engaging in discussion through questions and critical feedback, seeking ways to serve the overall community.
(As usual, I’m grateful to Bret Benesh and Theron Hitchman for helping me think through these at an early stage.) As I will acknowledge to my students, some of these standards depend to a certain extent on others. For example, it’s hard to make an effective presentation without mastering the vocabulary of the topic. But I believe these are distinguishable skills, all of which are important for students’ development as mathematicians. And I believe the students should be reflecting on their mastery of these skills as much as their mastery of analysis, and have the chance to show when they’ve improved.

My grading scheme for this class is somewhat of a compromise. I am keeping as many of the features of standards-based grading as I can—including scoring individual assignments by standards and providing opportunities for reassessment—but in order to take into account how well the content has been mastered, at the end of the semester I will weight and total points to determine a final grade. This last step is a kludge made necessary by the continued use of letter grades. If I had my druthers, I would leave the final assessment in terms of the students’ demonstrated mastery of the standards on the individual assignments, so that their focus would always be on improving in those areas rather than reaching a particular grade. I have tried to set this up in a way that, to quote T.J., “if you tried to ‘game the system’ to improve your grade, you would be doing exactly the kinds of things I wanted you to do, and improving your abilities as a mathematician.” (This suggests that we’re having to work against the current grading system to encourage students to grow in the ways we want. I suppose it’s a bit idealistic to believe that we can create a grading and reporting method that will provide both useful feedback to students and a helpful summary to those outside, but I digress.)

Of the standards I’ve listed, 1–4 are basically about writing and 5–6 are basically about active involvement. They will be handled separately in the grading scheme. Each student will write, as part of a group, eleven papers that state and solve a particular problem. These papers will be graded on the basis of standards 1–4, with each standard receiving either a 0 or a 1. After a paper has been graded, the groups will have the benefit of feedback from me and from their classmates, and they will revise, if necessary, until the paper merits at least 3 of the possible 4 points. This final version will be included in a document for the whole class to share. There will be a midterm and a final exam, as required by the college. Both will be take-home, and the individual problems on the exams will be graded according to the same standards as the papers. Following the midterm, students will have the chance to revise their solutions, as they do with the group papers.

Standards 5 and 6 will be graded over the whole semester. Each student will have approximately four chances to present in front of the class; although they will be presenting as part of a group, I will give individual presentation grades, again out of 4 points. The baseline will be 2 points. Grades of 3 or 4 will be achieved based on the quality of the presentation and adherence to the principles stated in the description of the standard. I’ll only consider the highest presentation grade at the end of the semester. For the participation grade, the baseline will again be 2 points, for regular attendance. (This is my first time giving an attendance grade. I generally believe college students should be free to decide for themselves whether coming to class is useful or not. In this case, however, the presence and participation of individual members is essential for the class to work, so I think this grade is justified.) Grades of 3 or 4 will be achieved based on involvement in class discussion, either during meetings or online in the class forum (where each week’s papers will be posted), and in general contributing to a supportive, scientific atmosphere. Since this grade is not given on any particular assignment, I will meet with students individually a couple of times during the semester to gauge their progress and experiences, and to discuss their level of participation.

Now, at the end of the semester, I want students’ work on the group papers and the exams to count about equally towards their final grade, and I want each of those to count about four times as much as their presentation and participation grades. So I will convert everything to a 40-point scale (16 possible points for papers, 16 for exams, 4 for presentation, and 4 for participation Edit: I’ve clarified these numbers in the comments). A letter grade of A will require at least 38 points, with no grades lower than 3 on any assignment (paper or exam problem) or standard (presentation and participation). A B will require at least 28 points, with no grades lower than 3. A C will require at least 18 points. This is as close as I can get to my usual way of assigning final grades: a 4 on 80% of standards (or 90%, depending on the class), with no grade below 3, and so on. It also follows relatively closely the French grading system based on 20 points, with 10 required for passing.

It’s not perfect, but that’s my current grading plan for this inquiry-based Introduction to Analysis course. Thoughts?


Theron Hitchman said...

I like it. I think this has a good chance of encouraging the behaviors you want.

I take it that every student does every assignment?

How will you deal with the fact that students might write twenty-thirty proofs in a term, and thus have 20 different scores in the range of 0-16 for written assignments? Similarly, with n exams, where n > 1?

And is there a way to handle the chance to "redo" a presentation that doesn't go well?

timfc said...

Thinking along the lines of Theron's comment... at the beginning of the semester, students are likely to be not-very-good at the things that you want.

Is it reasonable to either weight later material/presentations more heavily so that they'll have had a chance to develop their communication skills, or, offer, rather than a chance for a do-over, some notion of 'competency based grading' where by X week you need have demonstrated competency in the communication standards that you've outlined. Maybe have some gradations; proficient/advanced/jedi so that kids can see what's an C/B/A?

I guess I'm okay with re-doing a presentation, but, I don't really like revisiting old content unless there's a pedagogical need to do so.

Joshua Bowman said...

Theron: Each week has 3 loosely-related problems assigned to different groups. Every student contributes to their group every week, but generally only to the problem they're working on. The end-of-week presentations give the groups a chance to share their problem and the solution they found with the other groups. Everyone is expected to understand the work that has been done; later problems often depend on conclusions from earlier ones, which the students are expected to recognize.

I might have been unclear on the scaling of points for the final grade. An idealized picture would be one with four written papers and four exam problems. Then each of these would get a grade out of 4 points, yielding a total of 32 possible points, which would be added to the grades for standards 5–6 for a total out of 40. Since there are eleven papers instead of four, I will add up their total points and multiply by 4/11 to get a score out of 16. I don't know how many problems the exams will have, yet, but I'll use a similar scaling process to get another score out of 16. This will mitigate somewhat the scores required for various letter grades; a student could have a 3 on five of the papers and still get 38 points, if all the other grades are 4s.

Each student will have four presentations. I expect them to improve over time, but I'll only count their highest presentation score in the final grade. If at the end of the semester there's a student who's unhappy with her presentation grade, then I'll give her a chance to "redo" one, but I think that's unlikely to happen.

timfc: I think that last paragraph addresses some of your concerns, too. I've included in the syllabus the sentence, "I expect you to improve in each of these areas, and my criteria for success will become more exacting as the semester goes on." I could do a weighted system for the presentations, but the whole thing is complicated enough that I'd like to keep this part simple.

The biggest challenge overall, I think, is going to be in gauging the early grades so that they're neither too strict nor too lax. I want the students to develop habits early on (like revising their papers and giving other groups constructive feedback) that will lead to growth. Much of this is about building a culture rather than being too focused on the grading system.

Good questions. Thanks to you both!

Bret Benesh said...

This looks really great. Funny, I am kind of going in the opposite direction this semester, with a whole lot of content-focused learning goals. For the record, I think that your way is the way I would prefer to do things (I do not think that my course---calculus---is the right place for me to start with this. Also, it is way too close to the semester for me to change things up).

Joshua Bowman said...

Bret: wait for tomorrow. I'm going to post my calculus standards, which will be more numerous and content-based.

Tim said...

Josh, as always I really appreciate your ideas and how much thought you've put into them. The comments have also been very thought-provoking. It's funny, because I am just now putting together some standards for my abstract algebra class. I was worried about how to break up the specific content into different standards, so I really appreciate your situation and your solution.

Bret Benesh said...

How much explicit instruction do you give them in class for "Broad vision of the subject?"

I am looking forward to diving into your next post.

Joshua Bowman said...

The notes that include the weekly assignments also have "Big Picture" paragraphs interspersed (a feature I have kept from earlier versions of the class). So there are examples that model this kind of writing.

I've found that when I suggest students provide a sketch of their proofs in their papers, they manage to draw out the key ideas. We'll see. :-) At first I'll be looking for self-awareness about what concepts and notation were new to them when they started solving the problem.

If I were to use a (poorly constructed) biological metaphor, I would say "vocabulary and notation" form the skeleton of mathematical writing, "argumentation" is the muscle, and "exposition" is the skin. Then a "broad vision" would be the ligaments that bind bone to muscle.