Reworking the Grading System

Chapter 7 of What the Best College Teachers Do talks about evaluating students. While I can't recall any specific passage or thought from the book that got me thinking this way, it did prompt me to think more about grading systems.

What does it mean for a student to have a problem "80% correct"? If I cannot answer this question, then it makes no sense for me have some conclusion that sounds like "therefore, 80% is an B- in my class." This is one of the reasons I am thinking of grading problems out of 5 points (see the last part of this post). I want to have a grading system that is simple enough to be consistent, but diverse enough that there are enough strata to have an accurate gauge on students in the class.

When I look at student scores right now, I see a spreadsheet that only has the final scores on tests. That means in a hypothetical 4 problem exam, I could not tell the difference between an 80-80-80-80 student and a 100-100-100-20 student. I would feel that the 80 student is probably a stronger student than then 100 student because the 80 student has shown that he has an understanding of all the topics, but the 100 student has not developed the same breadth of knowledge. This is also an indication of me not really knowing what it means to be 80% correct. Alternatively, it could also be that the 100 student just made some small error at the beginning of the last problem and that ruined the rest of his otherwise perfect work.

My current thought is data intensive, but sounds like something I might be willing to do with three small classes. It might be beneficial to keep a record of all the grades on the individual problems (homework and exams). This has multiple benefits:

1. I would be able to see overall trends in the class. If there are specific sections that are giving students more difficulty than others, it will be immediately apparent and give me a chance to cover that material more carefully.
2. It can get students away from thinking about their understanding in terms of percentages. I could assign grades based on the relative numbers of points they got on their problems, and not the sum of their scores (this would also require me to use a grading system of 4-5 points for every problem so that it is a consistent measure).
   
3. It may also provide good feedback for students if this was presented to them in a reasonable manner. It would have to be organized in a nice way and not just a list of numbers.
4. It will also provide historical data to recognize (for example) that most students really struggle with this particular topic, this problem is extra tricky, or other observations that could potentially slip through unnoticed.

As always, the devil is in the details because I have no idea at this point how such a system could be presented to a class in a way that would make sense to them. I need to think about this a little bit more.

Stubborn Students

Here is a brief passage from Ken Bain's What the Best College Teachers Do (Chapter 2, opening pages):

In the early 1980s, two physicists at Arizona State University wanted to know whether a typical introductory physics course... changed the way the students thought about motion...

Did the couse change student thinking? Not really... They had memorized formulae and learned to plug the right numbers inoto them, but they did not change their basic conceptions...

[The professors] wanted to probe this disturbing result a little further. They conducted individual interviews with some of the people who continued to reject Newton's perspectives to see if they could dissuade them from their misguided assumptions. During those interviews, they asked the students questions about some elementary motion problems, questions that required them to rely on their theories about motion to predict what would happen in a simple physics experiment. The students made their projections, and then the researchers performed the experiment in front of them so they could see whether they got it right. Obviously, those who relied on inadequate theories about motion had faulty predictions. At that point, the physicists asked the students to explain the discrepancy between their ideas and the experiment.

What they heard astonished them: many of the students still refused to give up their mistaken ideas about motion. Instead, they argued that the experiment they had just witnessed did not exactly apply to the law of motion in question; it was a special case, or it didn't quite fit the mistaken theory or law that they held as true. "As a rule," [the professors] wrote, "students held firm to mistaken beliefs even when confronted with phenomena that contradicted those beliefs." If the researchers pointed out a contradiction or the students recognized one, "they tended at first not to question their own beliefs, but to argue that the observed instance was governed by some other law or principle and the principle they were using applied to a slightly different case." The students performed all kinds of mental gymnastics to avoid confronting and revising the fundamental underlying principles that guided their understanding of the physical universe. Perhaps most disturbing, some of these students had received high grades in the class.

I earned a B.A. in physics as an undergrad at UCSB. (I didn't earn a B.S. because I didn't take the physics labs; they were too time consuming and I wasn't intending to continue in physics.) I was fortunate enough to be in a specialized program (College of Creative Studies) where we had a very good teacher for our lower-division physics classes. He taught us in ways that forced us to think about how we thought about the subject by assigning very difficult problem sets. What was difficult about them? It wasn't only computationally difficult, but he asked us to interpret the meaning of the results. This forced us to reconcile our results with reality and expanded our ability to think physically.

Here are a couple examples of things he would do:

1. In multi-body problems, we would see what would happen to the results as one of the masses becomes infinitely big or small. Did our new results match the expectations? Why or why not?
2. Some problems resulted in two solutions. Did they both have a physical interpretation?

I would like to find ways to do this with math. I don't know how it can be done, but I expect that it can be done. The details will have to wait for a specific class, or even a specific topic or problem, because I think it's impossible to talk about real life in the abstract. It can only be done in the context of things that are actually happening.

The book continues to discuss the views of the successful teachers with respect to the development of knowledge. I'll give the bullet points here and have you buy/read the book for yourself if you want to find out more:

1. Knowledge is constructed, not received
2. Mental models change slowly
3. Questions are crucial
4. Caring is crucial

The biggest hurdle for the students who are not mathematically inclined (most of them) is that they come in with the presumption of what math is and that they aren't very good at it. They don't really focus on the processes, but the end result. Why? This is how they are trained to think about math in school from the beginning. (Read this post for more on this topic.) The biggest hurdle for me as a teacher is trying to turn away the negative light on the subject. I have some ideas, but I won't know if they will work until I get the chance to try.