Wednesday, 9 October 2013

MAT: What are the Chances?

Pic by Errol of Debs and Errol
Intro: Back in September, I signed up for "Explore MTBoS", an eight week event connecting math educators online. This is the first post connected to that event. If this is your first time on my blog, welcome! For the record, I post about writing in addition to math teaching. I also have a second blog, "Taylor's Polynomials", a story about personified math. Find it here:

This week's mission comes from Sam Shah (@samjshah). It asked us to write on one of two prompts: One of our favourite open-ended problems OR one thing that happens in our classrooms which make them distinctly ours.


I could mention the fact that I sing parody songs I've written, but that's the easy way out. Besides, while that may make me unique locally, I know for a fact it doesn't make me unique globally. So let's talk probability instead.

On second thought, let's give it some context first. I don't want people to think "open ended problems" are created out of nothing, or that I'm particularly good at them or anything. That said, you're welcome to skip down to the next subheader.

There's a course in the Ontario Curriculum called "Data Management" (Code MDM 4U). It's essentially advanced probability and statistics, and I've been teaching it now for... uh... five years? A while. One of its curriculum expectations is that:
 "Students will design and carry out a culminating investigation* that requires the integration and application of the knowledge and skills related to the expectations of the course."
*addressing a single problem on probability and statistics or two smaller problems, one on probability and the other on statistics

Education: If you can't measure it, it's probably not important.

I'm in the latter category, separating the probability and statistics. My stats project is in the form of coming up with a question, researching it, designing a survey to run in the school, doing so, and analyzing school results as compared to what their research turned up.

I feel like I'm not getting enough mileage out of that these days. If anyone has a better suggestion, toss it at me.

But that's not what I'm going to talk about.

The probability project is the warm-up to the statistics one. The probability project that I inherited when I started this course involved writing a story (or rewriting a fairy tale) to include probability concepts. This is actually a really good assignment, but after three years, I got tired of reading the same sorts of stories. So I switched it to creating (or investigating) a game.

That was a bad plan. I don't like games.

I feel bad when I lose, and I feel bad when I win because it means the other person lost. I can't see the patterns in Set, I don't have the coordination for video games, and I outthink myself to the point where I'm more stressed playing a game than I am teaching students. Seriously. So go figure, reading about games wasn't relaxing either.

Thus, here's the "open ended problem" aka "rich task". (Yes, finally!)


My task involves asking the students to investigate something that interests them, which is related to probability. They then create a list of assumptions necessary to actually do some math, and finish by writing a brief report.

There you have it. Post done!


Okay, I anticipate questions. Let's field some of them in advance.

"For my next trick, I will read your minds."
1) Isn't that a bit... broad?
Yes. That's why I start by having the whole class write down one or two questions each. I select from those, possibly making them a bit more or less challenging, and offer up a final list of ideas to choose from. Ultimately though, they can choose something else, as long as they clear it with me first.

2) What questions do you mean? Give me a "for instance".
What are the chances of winning in poker after you're dealt two aces. What are the chances of being struck by lightning twice in your lifetime. What is the probability of damaging the screen of your iPhone.

3) Uh... okay... isn't that STILL a bit broad?
Yes. That's where the assumptions come in, which is effectively driven by the data. If they're finding data on US lightning strikes, they can narrow the project down to that region. If they're the only player against the dealer with a fresh deck, the probabilities are easier to calculate. Part of the point is to make them realize that such assumptions are necessary. There's a lot that goes into "your chance of winning the lottery".

4) How much class time do you devote to this?
About three periods, all in the computer lab. Not consecutive, over the span of up to three weeks. After the first week, they need to show me that they've started thinking about what their assumptions are. The expectation where I work is that they can continue out of class time - particularly given that "Gambling" websites are blocked in our board.

5) How exactly do you GRADE this?
Uh. Awkwardly. Next question?

6) You didn't really answer the previous one.

7) You know I can decide to never read your blog again, yes?
Point to you. Okay, so I'm one of those teachers who grades assignments on the "level" system, where "level 1-" is a pass (50%), "level 3" is provincial standard (75%), and the last 25% is showing real proficiency or going beyond. I don't have a formal rubric here yet, I'm still in trial runs, so I basically tell the students that as long as you are making an obvious effort, you can pass. If you are telling me things I already know or could easily find out, level 3. If you take the research and the math beyond that, level 4.

8) That's a terrible system.
Yes. Wait, was that you or me saying that? Either way, I do think it's kind of terrible. If I were grading my own task here, I wouldn't give myself above "level 2". That's partly why I'm blogging about it though. I'm hoping you can say something to make this better.

9) I don't even know where to start fixing this.
Hm, wait, that's definitely me talking to myself. Back to you then - I'm open to suggestion. Another teacher said I might have something here. What do you think?

You probably think I should stop talking to myself.


  1. I think your rich task sounds great. Coming up with question that are both interesting and answerable in a reasonable amount of time might be challenging. I have two students working on an extended essay about probabilities in poker and Set and they found out it is actually a lot harder to come up with the right approach than they initially thought. A lot of learning is taking place there....

    1. Thanks for the thought! Yes, and I think it's really the approach here that I want, not even necessarily an answer. Also, I should possibly add "Set" as a gaming option, not sure why it didn't occur before (probably my aversion to games).