I was intrigued last night looking at Dan Meyer's blog post about the power of video to clearly define a problem in a way that a static image does not. I loved the simple idea that his video provoked in me - when does one switch from betting on blue vs. purple? This gets at the idea of expected value in a really nice and elegant way. When the discussion turned to interactivity, Geogebra was the clear choice.

I created this simple sketch (downloadable here)as a demonstration that this could easily be turned into an interactive task with some cool opportunities for collecting data from classes. I found myself explaining the task in a slightly different way to the first couple students I showed this to, so I decided to just show Dan's video to everyone and take my own variable out of the experiment. After doing this with the Algebra 2 (10th grade) group, I did it again later with Geometry (9th) and a Calculus student that happened to be around before lunch.

Each colored point represents a single student's choice for when they would no longer choose blue. Why they chose these was initially beyond me. The general ability level of these groups is pretty strong. After a while of thinking and chatting with students, I realized the following:

- Since the math level of the groups were fairly strong, there had to be something about the way the question was posed that was throwing them off. I got it, but something was off for them.

- The questions the students were asking were all about winning or losing. For example, if they chose purple, but the spinner landed on blue, what would happen? The assumption they had in their heads was that they would either get $200 or nothing. Of course they would choose to wait until there was a better than 50:50 chance before switching to purple. The part about maximizing the winnings wasn't what they understood from the task.

- When I modified the language in the sketch to say when do you 'choose' purple instead of 'bet' on the $200 between the Algebra 2 group and the Geometry group, there wasn't a significant change in the results. They still tended to choose percentages that were close to the 50:50 range.

Dan made this suggestion:

I made an updated sketch that allowed students to do just that, available here in my Geogebra repository. It lets the user choose the moment for switching, simulates 500 spins, and shows the amount earned if the person stuck to either color. I tried it out on an unsuspecting student that stayed after school for some help, one of the ones that had done the task earlier.

Over the course of working with the sketch, the thing he started looking for was not when the best point to switch was, but when the switch point resulted in no difference in the amount of money earned in the long run by spinning 500 times. This, after all, was why when both winning amounts were $100, there was no difference in choosing blue or purple. This is the idea of expected value - when are the two expected values equal? When posed this way, the student was quickly able to make a fairly good guess, even when I changed the amount of the winnings for each color using the sketch.

I'm thinking of doing this again as a quick quiz with colleagues tomorrow to see what the difference is between adults and the students given the same choice. The thing is, probably because I am a math teacher, I knew exactly what Dan was getting at when I watched the video myself - this is why I was so jazzed by the problem. I saw this as an expected value problem though.

The students had no such biases - in fact, they had more realistic ones that reflect their life experiences. This is the challenge we all face designing learning activities for the classroom. We can try our best to come up with engaging, interesting activities (and engagement was not the issue - they were into the idea) but we never know exactly how they will respond. That's part of the excitement of the job, no?

[...] 2011 Nov 29. Evan Weinberg hacked together something that does what Avery described. The results surprised me. [...]

Sweet! Thanks for doing this. A few technical questions and then one comment...

Did you have every student download this file and do this at their own desk (assuming you are 1:1)? How did you migrate the "points" from the students to the overhead?

and a comment...

Instead of describing the angle you stopped at, I might describe the ratio of the circumference you stopped at since (I think) this more easily leads to the idea of weighted averages.

Hey Avery,

When I first quizzed the first group of students, it was with the original sketch on my own computer. I just asked them to tell me when they wanted me to stop the purple region from growing. I then manually entered the results into the spreadsheet view and displayed the list of points. With the geometry students they all ran the sketch on their computer and then shouted out their numbers. It would be just as easy (I imagine) to have a group of people in a todaysmeet.com and enter their answers. A student volunteer could then enter the values into a spreadsheet that automatically plots the points.

After getting a comment similar to yours about the angle from another person during school, I changed the number to showing percentage instead of angle - it absolutely made a big difference. At the time I chose to display the angle, it was for ease of gathering data. The sketch wasn't really an interactive activity that would be used for an individual student to really explore the concept. In the later version I posted at http://bit.ly/rDARUH it is meant for students to explore a bit more, and that includes the percentage of purple rather than the angle.

Thanks for the comments - I love arbitrarily assigning myself Geogebra challenges. Dan tends to be a great source of those....

The excitement of the job, indeed!

Interesting tae on the student versus math teacher view of the problem. It get's into prospect theory versus the expected value models for predicting human choices. The students were evaluating a win of $100 as rougly equal to a win of $200, versus a more painful "loss" - there is some basis for both of these in Kahnemann and Traversky's work, mainly the loss aversion.