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Probability Quizzes with No Questions

A few years ago, I came across the following multiple choice question:

Argue about the solution all you’d like (oh, and people argue about the solution), the beautiful part of this, for me, is that the question is not really the question. The point of the exercise is not to complete the exercise, it’s to dwell a while in the complexities it offers. By constructing the argument, you interact with notions of odds, randomness, probability, and the like. This is similar to the idea of #SandwichChat, where the point is not to define what a sandwich actually is, but, rather, to play with emerging definitions and consider their consequences. I love these sorts of activities, because they, almost unexpectedly, turn our own thinking upon ourselves. They have a way of snapping us out from the familiar ebb and flow of the mathematics classroom, whereby prompts are passed to solvers who manufacture resolutions and, in turn, re-sell them back to teachers at increased costs. Teachers cover this inflation by remunerating the students with a most precious commodity–grades. And so the classroom economy ticks forward.1

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An Improbable Run at the Rosenthal Prize

In early December, I found out that my submission had been selected as the winner of the 2019 Rosenthal Prize for Inspiration and Innovation in Math Teaching. At the time, I had zero reference point to understand what that meant, but have since experienced first hand the hospitality of the international math education community. This post is not a summary of the winning submission; that detailed lesson plan has been posted on the MoMath website. [UPDATE April 2021: Rachel Welbourn a gracieusement traduit les documents de la tâche en français.] Here, I want to take the time to reflect aloud on how this whole thing happened–a process, I think, might be of value for math teachers. I’ve attempted to distill my take-aways into four categories, but, in actuality, they still exist (for me) as a tangled heap composed of equal parts disbelief, gratitude, and empowerment to pursue the next challenge.

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**Update November 2020. Jamie Mitchell–a fantastic teacher from Ontario, Canada–sent me this google doc that his student prepared to justify her solution. After you wrestle with the prompt for a while, take a second to read this brilliant response!

Most probability resources contain a familiar type of question: the two-dice probability distribution problem.

Often times, it is accompanied with questions concerning the sums of the faces that appear on each dice.

For example:

Roll two fair, 6-sided dice. What possible sums can be made by adding the faces together?
What is the probability that:
a) the sum is 6
b) the sum is a multiple of 4
c) the sum is greater than 15?

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Egg Roulette

I find probability to be one of the most difficult topics for students to grasp. Beyond the simple experiments of spinners, coins, and dice, students have issues operating on uncertainty. This issue is compounded when multiple events each involve such a calculation as well as the relationship between them. Soon they find themselves neck-deep in notation and lose all rationality–they forget what they are solving to begin with.

This past week we found ourselves mired in another battle with conditional probability. The initial questions were completed at a high level:

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The difference between what should happen and what does happen is a difficult distinction for students. They are so used to finding exact answers in the back of textbooks, that differing experimental results create an sense of uneasiness. At an early age (Grade 9 in my province) we begin to introduce students to the ideas of sampling and experimental probability.

The topic is usually approached with a project or survey of schoolmates. The results are then tallied and then used to create “probabilities” of various things such as favourite sports team, food, or colour. I love the philosophy behind the project approach; student initiative and autonomy is a powerful thing. I, however, don’t like that the experiment involves humans. Here’s why…
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The recent curriculum renewal has placed a (well-deserved) heightened emphasis on counting, set theory, and probability. Just under a half of a Grade 12 “Foundations of Mathematics” course now covers the three topics. This is a huge improvement from the token, disjointed topics strewn around the last courses. It allows teachers to set a different tone–a tone of curiosity that seems inherent in probability.

I came across the idea of Grime Dice (named and pioneered by Dr. James Grime @jamesgrime) late last year after I knew I was to be teaching probability this winter. I knew right away this was a great task to get students tinkering with probability before defining its inter-workings theoretically. A great description of their function can be found on the PlusMath website written by Dr. Grime himself. They are available for purchase from MathGear.co.uk

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Stations in High School Math

One of the coolest experiences in my university training was the opportunity to invite a kindergarten class into our mathematics methods class for a mathematical field trip. Our class was divided into groups of three or four and were given the task of designing a mathematical activity that the students would try. The afternoon was a hit. Each group set up shop around the room and the kids freely moved from station to station as they mastered each activity.

Somewhere along the way, mathematics becomes formalized and stationary. I imagine it is around the time of fractions. I assume this for no better reason than teachers and students alike seem to blame most of their problems on fractions. That is until Grade 10, when polynomial factoring squeezes out fractions as the most hated mathematical procedure.

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A Declaration of Independence

I used to be roommates with a magician. He kept all of his materials locked up in a trunk in our hall closet. Although he had devoted himself to the study of human psychology, I still convinced him to crack open the trunk and show me a trick from time to time. This experience was one of the most frustrating yet intellectually stimulating experiences of my life. I was a mathematics undergrad immersed in a stressful environment of number theory, numerical analysis, and abstract algebra. I was being trained to reason effectively, and his antics refreshed my perspective on reality. Life often muddies mathematics; such is the unfortunate reality.

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The “Nearly” in Mathematics

Mathematics is the purest form of science, or at least that is what they tell us in university. This ideology carries over into the school staff; it wasn’t long until another member of the staff referred to me as a “math guy”. As much as this label is also self-imposed, I still struggle to understand what it means. The labels “english guy”, “phys-ed guy”, and “science guy” all persist within the building as well, but there is something that about the title of “math guy” that gets me.

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Life’s Not Fair

The school year is now over for me. That is a bittersweet statement, because I still have mountains of grading and report card comments to do, but there will be no more direct lessons in the 2010/2011 school year. I found myself nostalgic this morning, and began to recount the good times in the classroom. I recalled the probability mayhem that ensued with my Grade 11s. It was very amusing to see them come up with ways to describe “fair”. I would always tell them that I would only do something if it was “fair”. This, to them, meant a coin flip, draw from a hat, or a roll of the die. But whose hat? Who rolls the die? On what surface? Do these factors actually have an impact on “fair”?

Most questions with elementary probability include a fair clause. For example: