## Saturday, January 05, 2013

### some common standards for college-level math

As I mentioned in my last post, this spring I will be using standards-based grading (SBG) in my multivariable calculus class. I won’t belabor what that means, since this is my first time using this method of assessment, and others have written far more expertly on the topic. In short, to me SBG means two things:
• refocusing the nature of grades from periodic goalposts to instructive feedback; and
• honing in on specific expectations so that students know where they are doing well and where they need improvement, from our perspective.
It’s often easy, for example, to become frustrated when a student does all the correct computations, but uses parentheses incorrectly. Or when he or she can differentiate polynomials but not factor them. When grading with points and percentages, we can try to leave helpful notes in the margins or at the end of an assignment, but the presence of THE NUMBER or THE LETTER at the top mostly overcomes our efforts to give informative guidance. Number and letter grades obscure the distinction between “generally understands the course material” and “understands this class well but struggles with prerequisites” and “follows set procedures neatly but has only surface understanding”. A set of standards separates out those cases and handles them differently.

After a stimulating conversation last month with T. J. Hitchman, Dana Ernst, and Jon Hasenbank (with some additional feedback from Bret Benesh), I began to rethink what should (or does, or could—there’s never been just one way to implement SBG) constitute a standard in a college/university math class. Up until that point, I had been planning to split up the course material into every possible type of computation or problem that I would expect a student to be able to complete at the end of the class. It became clear to me that this was probably too fine a gradation; it treats the material reductively rather than holistically, and moreover there’s almost no chance I could equitably test all of the pieces in a 14-week semester, with three class meetings per week.

At the time, I had already planned to have two standards appear on every assignment: algebra and presentation. These constantly trip up students at the introductory calculus level, and without SBG I always struggled to convey to certain students how much their performance would improve just by focusing on one or the other of these. During the aforementioned conversation, it was also proposed that problem solving could be its own standard. In some sense, it is an skill that is independent of the particular course material, but requires some kind of content to be implemented.

I began to conceive of two types of standards: those which would apply directly to the new material of the class, and those which would be expected of students in any early-to-intermediate level college math class. My model was inspired by Euclid’s division of axioms into “common notions”—applicable to any mathematical realm—and “postulates”—assumptions tailored to the study of geometry. Or, to use a liturgical metaphor, one might think of the “ordinaries” and “propers” of the mass; they are equally important, but the propers change at each service, while the ordinaries remain the same. The ordinaries provide a constant framework for interpreting and experiencing the propers, which focus on the day or season.

Below is a list of seven common standards I have devised, along with a brief description of what each entails. Next week I will post the content-specific standards for my multivariable calculus class.
• Algebra – accurately simplify, expand, and otherwise manipulate symbolic expressions involving variables and common functions
• Mathematical literacy – correctly use mathematical vocabulary, set notation, equality, and logical implication
• Technological literacy – use computers appropriately for computation, visualization, and research
• Modeling – translate fluidly between verbal, symbolic, and graphical descriptions of both abstract and “real-world” objects
• Problem solving – determine what question or questions are relevant in a given situation and choose an appropriate strategy for answering the question(s)
• Estimation – anticipate the nature and/or size of solutions and evaluate whether a solution makes sense in the given situation
• Presentation – submit neat, organized, clearly written, and independently produced work, with appropriate context and a clear progression of ideas
These are skills I try to develop in all of my classes, but not always so explicitly. Adding estimation to the list, for instance, was inspired by the books Street-fighting Mathematics and Misteaks … and how to find them before the teacher does…, which emphasize what mathematicians often think of as “common sense” guesses as to what solution a problem will yield, as well as checks that a final answer makes sense. It promotes numeracy. It gives me something I can point to when I wish to explain to a student, “It’s less important that you made a sign error early in your calculation than that you failed to notice at the end that your car is going faster than the speed of light.” Modeling is, for some students, one of the hardest skills to learn. I’m partly using Dan Meyer’s notion of the “ladder of abstraction” to shape my thoughts in this area. Both modeling and estimation could be classified under problem solving, but are sufficiently large subcategories that I chose to separate them out and focus the problem solving standard more narrowly.

To some extent, these are standards for myself. Am I providing guidance on when and how to use computers? Am I demonstrating proper mathematical grammar? Am I distinguishing the (neat, organized) presentation of a solution from the (often messy) process of discovering it? I’m looking forward to this way of assessing and highlighting material.