In this post I mentioned that I was going to be overhauling our Mastery Exams which we give in Calculus Preparation. In the comments, Jackie asked for more details on what these exams cover, so I thought I would oblige — and add some thoughts on top of that.
Calculus Preparation (CP) at our place is 3-hour course that does not count towards graduation for anybody. It is in place for students who math placement test scores put them there. Or, as was the case for my CP classes this spring, you often get students who place into our bottom-level remedial algebra course, who then go on to CP if they pass, who then go on to calculus if they pass CP with a C- or better. These students are primarily business or accounting majors (with science majors a distant second), because those majors require calculus.
Students in CP do pretty typical assessments on standard precalculus material throughout the course. But students also must pass a series of eight Mastery Exams with a 95% or higher. Students get one hour each week during which they may take the exams, with no more than one attempt per exam per week. If they do not get a 95% or higher on an exam, they have to retake that exam; they may do so as often as they like as long as it’s no more than once per week, up through the last day of classes. If a student does not pass all eight with a 95% or higher, their grade in the course is capped at a D+. That means they have to take the course over again (or change majors to something that doesn’t require calculus, which is often a better idea).
The topic structure of the eight exams are as follows:
- Arithmetic operations: Fraction arithmetic, absolute value calculations, decimal operations, squaring and cubing, order of operations.
- Linear equations: Solving simple linear equations.
- Properties of exponents: Simplifying expressions involving addition, subtraction, and multiplication of exponents.
- Additional properties of exponents: Zero, negative, and fractional exponents; solving equations that involve negative and fractional exponents.
- Polynomials: Adding and multiplying polynomials, factoring, solving second- and third-degree equations.
- Linear and absolute value inequalities: Solving and graphing solution sets of inequalities with and without absolute values.
- Rational expressions: Adding, subtracting, multiplying, dividing fractions of polynomials.
- Systems of linear equations: Solving two-variable, two-unknown systems.
Exam 1 is what you’d see in 7th-grade arithmetic. Exams 2-6 are essentially Algebra I topics, and exams 7 and 8 are more like Algebra II. So these are very basic concepts, and in calculus, absolute don’t-have-to-think-about-it fluency is expected in these topics, making the stringent requirements of the exams pretty appropriate. Students hate these things, but having to pass them with such a high rate of mastery drives home an important point: If they can’t demonstrate complete mastery over these concepts, there’s very little in the way of a realistic chance of passing calculus, no matter what they may do in Calculus Preparation.
I did suggest to my department, and will be implementing this summer, an overhaul of the existing exams based on some lessons I’ve learned from teaching CP for the first time this past semester.
First of all, we are eliminating exams 6 and 8, as those really aren’t essential to calculus. You could justify exam 6 if you do a lot with epsilon-delta proofs throughout the course, but we don’t. It’s an important topic, but not one that we can’t live without in calculus. And exam 8 doesn’t show up at all in calculus until you do integration using partial fractions in Calculus II — and if a student makes it that far, s/he probably didn’t need remediation on that topic in the first place.
Second, we are lowering the pass rate from 95% to 90%. This is not to lower standards but to standardize grading. We have a policy that we do not give partial credit on the answers on mastery exams, and that’s good because the idea behind the exams from our perspective is to give quick, almost instant feedback to students. But these exams have as few as six and never more than twelve questions on them, which means that to get 95% correct you really have to get 100% correct. This has led to a dangerous differentiation in the rules for mastery exams among different teachers. Some profs have stuck to the rules and required 100% correct; some fudge the rules and allow a student to miss no more than one problem; some revert to giving partial credit. Students pick up on this, and complain loudly — and justifiably — that Professor A is being unfair in requiring 95% (=100%) correctness because last year Professor B didn’t. So I’m overhauling the exams so that each exam has exactly 10 questions on it. That way we can lay down a consistent rule that students can miss no more than one problem and still pass.
Third, we are allowing students to take these exams out of sequence. It used to be that students would have to pass exam N before being allowed to attempt exam N+1. But I found that some students are actually better at simplifying polynomials than they are at doing arithmetic on fractions, and if that’s the case then I don’t see why they shouldn’t be allowed to demonstrate mastery on the stuff they know first, before they hit the woodshed on the stuff they don’t remember.
I’m happy to elaborate/blog about this further if anybody wants me to.