# Testing probability theories with students

One of the things that has excited me after building computational tools for my students is using those tools to facilitate play. I really enjoyed, for example, doing Dan Meyer's money duck lesson with my 10th grade students as the opener for the probability unit. My experiences doing it weren't substantially different that what others have written about it, so I won't comment too much on that here.

The big thing that hampered the hook of the lesson (which motivated the need for knowing how to calculate expected value) was that about a third of the class took AP statistics this year, so they already knew how to do this. This knowledge spread quickly as the students taught the rest how to do it. It was a beautiful thing to watch.

I modified the sequel. I'll explain, but first some back story.

My students have been using a tool I created for them to sign up for reassessments. Since they are all logged in there, I can also use those unique logins to track pretty much anything else I am interested in doing with them.

After learning a bit about crypto currency a couple of months ago, I found myself on this site related to gambling Doge coins. Doge coins is a virtual currency that isn't in the news as much as Bitcoin and seems to have a more wholesome usage pattern since inception. What is interesting to me is not making money this way through speculation - that's the unfortunate downside of any attempt to develop virtual currency. What I've been amazed by is the multitude of sites dedicated to gambling this virtual currency away. People have fun getting this currency and playing with it. You can get Dogecoins for free from different online faucets that will just give them away, and then gamble them to try to get more.

Long story short, I created my own currency called WeinbergCash. I gave all of my students \$100 of WeinbergCash (after making clear written and verbal disclaimers that there is no real world value to this currency). More on this later.

After the Money Duck lesson, I gave my students the following options with which to manage their new fortune in WeinbergCash:

Then I waited.

After more than 3,000 clicks later, I had quite a bit of data to play with. I can see which wagers individual students are making. I can track the rise and fall of a user's balance over time. More importantly, I can notice the fact that just over 50% of the students are choosing the 4x option, 30% chose 2x, and the remaining 20% chose 3x. Is this related to knowledge about expected value? I haven't looked into it yet, but it's there. To foster discussion today, I threw up a sample of WeinbergCash balance graphs like this:

Clearly most people are converging to the same result over time.

My interests in continuing this experiment are buzzing with two separate questions:

• To what extent are students actually using expected value to play this game intelligently? If you make the calculations yourself, you might have an answer to this question. I haven't parsed the data yet to see the relationship between balances and grade level, but I will say that most students are closer to zero than they are their starting balance. How do I best use this to discuss probability, uncertainty, predictions, volatility?
• To what extent do students assign value to this currency? I briefly posted a realtime list of WeinbergCash totals in the classroom when I first showed them this activity. Students saw this and scrambled to click their little hearts away hoping to see their ranking rise (though it usually did the opposite). Does one student see value in this number merely because it reflects their performance relative to others? Is it merely having something (even though it is value-less by definition) and wanting more of it, knowing that such a possibility is potentially a click away?

I had a few students ask this afternoon if I could give them more so they could continue to play. One proposed that I give them an allowance every week or every day. Another said there should be a way to trade reassessment credits for WeinbergCash (which I will never do, by the way). Clearly they have fun doing this. The perplexing parts of this for me is first, why, and second, how do I use this to push students toward mastery of learning objectives?

I keep the real-time list open during the day, so if students are doing it during any of their academic classes, I just deactivate them from the gambling system. For me, it was more of an experiment and a way to gather data. I'd like to use this data as a way to teach students some basic database queries for the purposes of calculating experimental probability and statistics about people's tendencies here. I think the potential for using this to generate conversation starters is pretty high, and definitely underutilized at this point. It might require a summer away from teaching duties to think about using this potential for good.

# After a hiatus: circular functions

It has been a busy time in gealgerobophysiculus land. By land, I of course mean school, and by busy, I mean what results when you have multiple exciting projects going on, school functions to organize, and the normal operations of a classroom to sort through and organize.

I haven't taught the unit circle in three years. Before that, I took the approach of throwing a definition of the radian up on the board and discussing it as this strange thing that mathematicians decided would be a good idea. When I learned this in high school, we did some cool activities involving string and wrapping functions. At that time, it wasn't clear to me how the string wrapping around a circular object really related to measuring an angle around it. I was always relating the idea of the radian angle back to degrees, because the angle part never made sense.

After some thinking and coding, I put together an activity that I thought would make this concept more concrete for the students in my tenth grade class. You can check it out at http://apps.evanweinberg.org/circlemeetsradius/

It starts with the premise of moving around a circle at distances of integer multiples of the radius. Looking at your own work doesn't really establish how this relates to measuring angles at all. When you look at what happens when many people do the same thing to differently sized circles, the result makes clear that this could be a fairly natural way to measure out angles:

I didn't have the networked part of this applet working when I did this with students, so I collected screenshots of students and their different circles together. I asked students what they expected would be different about the locations of these six points for circles of different sizes, and there was pretty solid agreement that they would be in roughly the same point around the circle, but this was still too abstract to establish the idea that these points measure out angles. The students weren't too surprised by the result, either, but I think the activity in this form still left me as the teacher to connect the dots.

I wish I had an extra day to configure the final screen of this activity. I wouldn't have had to work so hard.

The rest of the unit walked the line from this concrete idea of moving around the circle up the ladder of abstraction to what we ask students to typically do with these functions. We went from identifying points around the circle for a given angle measured in radians, to using our knowledge of 30-60-90 triangles to find the coordinates of some of these points, to formal definitions of sine, cosine, and tangent functions using these points. Every time I could, I related this idea back to the first activity of moving around the circle, but by the time we got to graphing these functions, I think I was demanding a high level of abstraction without also demanding the deliberate practice necessary to connect the angles and coordinates to each other. Students struggled to evaluate the trigonometric functions at different angles not because they couldn't piece it together with time, but because they always felt compelled to go all the way back to the circle. I suppose it's the trigonometric equivalent of going back to counting on your fingers.

I also was a bit disappointed to see that only a third of the class answered this question correctly on the unit exam:

To those that recognized the similarity to our opening activity, it was quite easy. The bulk did not see it this way though.

At this stage, however, I'm not too concerned. Many students admitted immediately after the exam that they did not practice the unit circle as much as they should have. They reported that they understood much of the unit up to the graphing part, where I think I pushed them a bit more quickly to piece together the graphs than would have been ideal for them to get an intuitive sense for them. I'm confident that a second and more rigorous look at these functions next year in IB year one will help solidify some of these concepts for them.