More on mastery exams

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:

  1. Arithmetic operations: Fraction arithmetic, absolute value calculations, decimal operations, squaring and cubing, order of operations.
  2. Linear equations: Solving simple linear equations.
  3. Properties of exponents: Simplifying expressions involving addition, subtraction, and multiplication of exponents.
  4. Additional properties of exponents: Zero, negative, and fractional exponents; solving equations that involve negative and fractional exponents.
  5. Polynomials: Adding and multiplying polynomials, factoring, solving second- and third-degree equations.
  6. Linear and absolute value inequalities: Solving and graphing solution sets of inequalities with and without absolute values.
  7. Rational expressions: Adding, subtracting, multiplying, dividing fractions of polynomials.
  8. 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.

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Filed under Calculus, Teaching

5 responses to “More on mastery exams

  1. Jackie

    Thank you for the detailed explaination! Now I have another question: What are the policies of your department regarding calculator use?

  2. Jackie

    Thank you for the detailed explanation! Now I have another question: What are the policies of your department regarding calculator use?

  3. On the mastery exams, no calculators at all. Elsewhere, we’re pretty liberal with technology. We use Derive and Excel extensively in CP and Calculus, and much of what the students do requires the use of computer software and spreadsheets. Timed assessments tend to be pitched more toward contextual problems rather than pure mechanics.

    As far as calculators proper go, we recommend students own and know how to use a scientific or graphing calculator of their choosing, and we usually allow unlimited use of calculators on everything — except the mastery exams.

  4. Matthew Bardoe

    Here is my question. While I think that mastery of these topics is important, I would like to play devil’s advocate with the following question. What evidence do you have that these tests really indicate mastery? Yes, they do show that students at some point in time were able to do these types of problems, but how is that different from a high school diploma. And please, let’s not beat up on high schools here. It is very likely that students have seen these topics before, and been successful with them at some point. So how does it show “mastery” that can do it on your test now. Will they be able to do it again in 6 months?

    Again, I am playing devil’s advocate here. I am not trying to say that you shouldn’t do this.

  5. “Mastery” of a topic would indicate that the student has complete fluency in that topic. That means, I think, the student can perform exercises on that topic without assistance in a short time frame and make few or no mistakes. Mastery means what it sounds like — the student is in control of that topic rather than at its mercy.

    Calling them “Mastery Exams” is a little strong — some places call them “Gateway Exams” and that might be better. The exams give evidence rather than proof of mastery. But they also certainly indicate the lack of mastery should the student fail them!

    Since students are doing the exams in the moment, they also measure how much of the content knowledge from high school math the student still retains (or at least can get up to speed on his/her own). Will they be able to do these again in 6 months? Maybe, maybe not — but the exams aren’t measuring what a student can do in the future but what the student can do right now.