You've got the power!
What's the Power Law?
I'm glad you asked, although I don't know a whole bunch about it. If you want a deeper look, check out Transforming Classroom Grading, which goes into the research behind the Power Law of Learning and provides a step-by-step for using it in calculations. But, lemme sum up.
If you were to plot out the scores of a student learning a skill which was brand new to them, which graph do you think would represent their scores over time---the one on the left...or the one on the right?
|Inquiring optometrists want to know: "Which looks more clear: #1 or #2?"|
Both lines have the same start and end point. They describe the same number of assignments. But one is a better representation for what happens during the learning process. Did you guess the one on the right? Yup. You were correct. When new skills are introduced, there is a big gain in learning at the beginning. With repeated opportunities, learning still increases, but at a much slower rate. Devotees of the Power Law believe that the most recent score for a student is the what should be reported as a grade.
Okay. So now what?
Oh, that student scores followed the exact same trajectory every time. But that's not what happens. Learning is impacted by all sorts of factors both inside and outside of the classroom. The graph above shows the average increase. We need to be able to make an individual student's scores apply to that learning curve. This is where the Power Law comes in handy. Here it is:
Calculating m and b requires some statistics, because in a way, we're "normalizing" the data set to the curve. If it's been awhile since you had to deal with this sort of thing, remember that the "E" symbol means "sum of" and "N" = "Number" (in this case, referring to the number of assignments for a given standard. Don't get too sweaty looking at these, because our buddy Excel is going to do all of the heavy lifting for you. I just want you to see them before we look at the gradebook.
And now, a musical interlude:
Back to our regularly scheduled workbook...
I'm going to build this workbook using the finalized Excel for Beginners workbook. Keep in mind that these formulas will work the Intermediate and Advanced versions, too. Just plop the formulas any old place and they'll do your bidding. Here is the first chunk of student scores:
Now, we need to transform them using the Power Law. I've set up a column for the variables we're going to use: x = (natural log of) Assignment #, y = (natural log of) Student Score, then the various combos we'll need using these. Please note that the y here is not the same as the one in the Power Law itself---it is just a statistical variable assigned to calculate m and b. (Confused yet?)
I've probably made this look far more complicated than it needs to be, but I also like the idea of breaking down the steps---especially if you're new to Excel or just need the opportunity to follow along with what is happening. The formulas in D26 and D27 use the natural log (LN) to transform the ordinal number of the assessment (x) and the student score on that assessment (y). Cells D28 - D34 are used to process x and y so they are ready to use in our equations for m (D35) and b (D36). The "predicted score" showing in Row 22 shows you the result of the Power Law formula (y = mx^b) for each student score.
Here is the link for the Power Law workbook with formulas. We'll talk about the graphs for this function in the next post, as well as reveal the final version of the workbook. Download this one first so you have something to play with.
Now, come on and sing it with me:
Want more of The Power by Snap! to listen to while you work on your gradebook? Cue this up.