Monday, November 15, 2010

Making the Percentages Fit

I know a number of people are struggling with how to turn their SBG scores into report-card-type percentages and/or letter grades. Here is the solution I was excited to find this week using some software which is new to me:

I haven't taught geometry, so software programs like Geometer's Sketchpad (which our school uses) and Geogebra (which most of my twitter friends use) are new to me. They're TOTALLY FUN to play around with. In my free time, I'll just pick up a proposition from Euclid's Elements and see if I can try to prove it. The dynamic nature of it brings technology to geometry in a way that couldn't be done before. It's one of the few uses of technology that I think could not in any easy way be done before.

Anyhow, Geogebra is the one I toy with most these days, so that's what I used for this. It is free and online here.

The issue

In my SBG marking I use scores 0 through 5 to represent various levels of understanding. In my head, I equate 5 with A, 4 with B, 3 = C, 2 = D, 1 = F, 0 = 0%. I have mentioned this to students and they seemed to get that, too. The issue is, when I put it in my online gradebook, it automatically converts to a percentage. So, while the students get it when looking at the problems, they are still not out of the habit of looking at the BIG NUMBER average for the semester and freaking out about how their GPA will change.

Our school changes percentages to letter grades as:
A: 91-100
B: 81-90
C: 72-80
D: 70-71
F: Below 70

So, I wanted to adjust their percentages accordingly. I need to turn one set of numbers (0%, 20%, 40%, 60%, 80%, and 100% for the 1 through 5 scoring) into another set of numbers (0%-100% for my school's grading scale). Enter:

A Mathematical Answer

Sounds like a job for functional analysis! [insert crowd cheers]

Most models require assumptions (sadly):
  1. I don't want any student's grade to decrease from the percentage that the 1-5 system gives them.
  2. Other than 5 = 100% and 0 = 0%, I wanted the other numbers to fall in the middle of the grade range.

So, here are the conversions I want:
SBG = Gradebook -> Converted grade
5 = 100% -> 100%
4 = 80% -> 85%
3 = 60% -> 76%
2 = 40% -> 71%
1 = 20% -> 55%
0 = 0% -> 0%

I used geogebra to plot the points: (100,100), (80,85), (60,75), (40,71), (20,55), and (0,0). Since we have 6 points, we can fit a polynomial of up to degree 5, but for my points, a 4th degree polynomial works out to the same. So, I used geogebra to fit a 4th degree polynomial to the data. (g(x) = PolyFit[{A,B,C,D,F,G}, 4])

Sorry, the GeoGebra Applet could not be started. Please make sure that Java 1.4.2 (or later) is installed and active in your browser (Click here to install Java now)

I wanted to check if that would fit my assumptions. It does indeed go through the required points (I can check by doing g(0), g(20), etc.). It also does not lower anybody's average (checked by looking at the function g(x)-x which is positive for all x between 0 and 100 as can be seen above in green).

And now I have my conversion function. It's not "intuitive," so it adds a bit of mystery to the actual number grade a student will have. It's not something they can figure out without being talked through all these steps and them putting all that effort into setting it up themselves. But, I think it seems "fair" to most students.

It's also not automatic, so at various times (like now when mid-quarter grades are going out), I have to adjust them by hand. I take the grade from the gradebook, open the geogebra file and type in g(84) to get the conversion for a student who has an 84 in the gradebook.

Feel free to adapt the method to your needs if you like. I am open to suggestions for improvement.