This fall, I get to teach Finite Mathematics and Precalculus. While I plan on using Standard Based Grading for both, my focus so far has been on the former. A little background on the course is probably helpful.
At Suffolk, every student has to take mathematics as far as calculus for management and social sciences, calculus for scientists or finite mathematics. On average, I would guess the calculus for scientists class gets two classes of 12 students per semester. The other two, meanwhile, get the remainder of the student body. Consequently, both of those courses expect to run several dozen classes per year (the classes are capped at 25 students!) To avoid anarchy, these courses are coordinated courses. That is, there is a course coordinator who sets general policy, chooses the course content and writes the final exam taken by all students in the course. The course coordinator will come back in a later episode of this saga, so keep him in mind. The content of finite mathematics is all over the place. There are three main topics in the course, separated by midterm exams. The topics are linear equations and linear programming, financial mathematics, and combinatorics and probability.
Before the spring term ended, I got a little excited reading about SBG and read through the entire finite mathematics text, taking notes and writing down standards. I then created this spreadsheet. Although it is about 3.5 months from being finalized, I’d love some feedback on it. My next step was to read through the book again with a copy of the final exam from last year (along with the formula sheet the students get). I noticed that the final exam reduces nearly the entire section on financial mathematics to inputting data into the formulas on the formula sheet.
Do you really want your kids to be the kind of people that won’t attempt to make a pizza from scratch for lack of the 1 tsp of anise seed that has little to do with the overall success of the dish? – Cornally
Problem 1: How do I motivate myself (much less, the students) to honestly cover that section knowing how it is tested?
Next, I noticed that the book is in the wrong order when it comes to the combinatorics and probability: they teach combinatorics first and then probability. The problem is that the interesting questions in the course that use combinatorics are all in the probability section. Wouldn’t it make more sense to teach the probability section first and when you get to a problem that requires to simply state that we have a problem we don’t know how to solve, i.e., create a crisis? Now, if the original problem is worth solving, the students have an interest in solving my otherwise contrived problem of choosing n things from r things (although it did seem less contrived before the university made me stop using my Kama Sutra example).
Problem 2: Convince the students to not worry about textbook order. Meh, most of them never open the textbook anyway.
The next thing I noticed is that the book treats the formulas for counting and computing probabilities as distinct formulas, when they are really the same thing. Reading Gerd Gigerenzer’s Calculated Risks made this quirk even more annoying. If you followed the recent New York Times series of Steven Strogatz, then you may already know about the book. If not, check out the article Chances Are from that series. It is stunning how many ways the students (or doctors or lawyers) can avoid mistakes in computing conditional probability by simply computing in terms of frequencies instead of probabilities.
Problem 3: Convince the students to not worry about the formulas in the textbook.
Finally (for now), I was discussing with one of my students how to plot the feasible region in a linear programming problem.
In the diagram above, we plot the following system of linear inequalities:
The feasible region is where all the shaded regions overlap. Using the picture, it isn’t so hard to spot the region. However, imagine drawing this with a single pencil. All of a sudden, it isn’t as clear. The book suggests that instead of shading first, to use arrows that point in the correct direction and only after finding the region where all the arrows simultaneously point do you then shade the feasible region. After explaining the process to my student, he informed me that in Vietnam (his country of origin) the regions are sketched in reverse.
Essentially, you shade the regions you don’t want and are left with a hole: the feasible region. You only need one color and you can avoid drawing the arrows.
Problem 4: Convince the students that they won’t be marked down on the final exam for drawing their solutions differently than the every section of the course.
That is enough for now. The next post will be my first Tricks of the Trade post and then I’ll come back and talk about my learning a foreign language fetish and how it relates to teaching (I doubt anyone can correctly guess where I’m going with this!) Sorry for the length; there is a lot of thought bottled up. The next post will definitely be shorter.