Student Guinea Pigs
“A monk weighing 170 lbs begins a fast to protest a war. His weight after t days is given by W = 170e^(-0.008t). When the war ends 20 days later, how much does the monk weigh? At what rate is the monk losing weight 20 after days (before any food is consumed)?” <– That’s an actual problem from our Calculus book, which I find very amusing. I really can’t imagine that this is a viable situation for which you would need to model an exponential equation and use it to make predictions. There are so many word problems that force “real-life” situations into the convenient framework of whatever math topic is being presented in that section. I guess these are supposed to demonstrate to students how useful and relevant math is, but I know that students just find them to be tricky and unyielding disguises to math that they generally know how to do.
There was one word problem that fit an exponential decay model to someone forgetting information, so I decided that instead of just doing the word problem, we would test the model by recreating the experiment. The day after we had a midterm exam, instead of handing back their corrected test, I put them in groups and gave them the following list of 50 three letter syllables that I generated with a random number generator:
SOQ XAC DOB NEB BAR JYS ZYW GEK TUD ZEM GAK KUR BEN XOQ DUX BYR NIT WAP ZIJ HOG HIQ DUW CUD SAM BIM LIH JEV VEZ QEM GUL ZIQ SEQ JYV GUT XYM XAX BIQ DOJ ROM ZIV QEW JEH CYS ZEM FOM KEG DUC GYK WYQ POD
I gave them 15 minutes to memorize as many as they could and then tested them by having them write down all that they remembered. Then, I handed out the midterms and we started going over them. About 5 minutes later, I had them write down as many of the syllables as they could again. Then, we went over a few problems on the midterm… then another memory test…. then more midterm… then another memory test. They had absolutely no idea why we were doing this, so each time they groaned and complained. And they groaned even more when I opened class the next day with another trial. And then again two days after that… And then a last time a week and a half later. All without studying the list after the original 15 minutes.
Finally, I revealed the purpose of the whole experiment. We collected data and used a graphing program to fit various models to their data. There were four different mathematical models to choose from that I found from various psychological studies. Each student picked the one that they thought fit their data best (a function to calculate how many words they would remember over time), used it to calculate their “forgetting function” (a function that tells them how fast they are forgetting words at any given time), and then used both to calculate how many words they will remember in a few weeks and how fast they will be forgetting them at the point.
We graphed all of their functions on the same axes (y-axis = number of words remembered, x-axis = time in hours) to analyze which model was best and analyze how their memories compared to their classmates. The results:
Overall, an extremely successful class experiment. Not only did we do some cool, real, authentic math that was individually tailored to each student, we ended up having a really great conversation on how best to memorize things, which then led to a great discussion about how to learn and study best (especially how you should go about studying math). And it was actually pretty fun. I was amazed at just how many of those syllables the students remembered – the winner impressively remembered 29 a week and a half later. But whenever we were doing the tests and the students were begrudgingly writing down everything they remembered, all I could think was “Oh the power I have. It’s amazing what my little friends will do for me when I have them by the stranglehold of their grades.”