I have always been drawn to probability because of its mysterious qualities. Maybe it is the result of the online poker fad that swept through my high school during the NHL lockout, but the calculation of odds still grasps my attention to this day. What fascinates me the most is how simple rules such as “AND” and “OR” can quickly create a mess of a situation. What begins in high school (or earlier) as a simple fraction that predicts the toss of a coin, soon balloons into factorials, combinations, Pascal’s Triangle, and Probability Density Functions. Despite the complexity of such calculations, they are still only theoretical; anything could still happen. This is a point that I stress to my students whenever we embark on a study of a game of chance.
This morning on twitter, there was a problem that I just had to solve before going out the door. It is safe to say that these types of problems are my vice. Number Theory has always held a special interest to me despite, according to G.H, Hardy, having “absolutely no practical use.” (A Mathematician’s Apology, 2001). This has all changed with the inception of encryption.
I wish just to present the problem and then muse on its educational significance both for my personal learning of mathematics, and for that of my students.
There has been plenty of recent twitter talk about the process of moving the focus of mathematics education away from the “how” and toward the “why”. Traditionally, students have been trained to approach a question–usually given to them by an outside source like a teacher, textbook, or test–with the express intent to show the grader “how” it is answered. Such responses often include the use of algorithms, formulae, or memorized facts we know to be true. (These facts are in no way axiomatic, but constant repetition reduces them to that state. Students have answered them so often, the process loses meaning. Take 2×2 for example.)
There is widespread turmoil among teachers and students when it comes to the practicality of mathematics. School mathematics, at the middle and high school levels, has moved out of the elementary niche of rudimentary skills, but has yet to make it into the realm of complexity necessary to apply it back into the world. Our happy compromise, as teachers, is to go with a two-pronged attack:
1. Tell the students that the practicality comes later
2. Create word problems about trains leaving stations or people tossing balls off cliffs
Teachers in Saskatchewan, Canada have had a lot to deal with lately in the classroom. The ongoing political battle has effected hours of direct instruction in a very real way. I quickly noticed my classes becoming disjointed with large amounts of time between each encounter with the mathematics. Needless to say, I entered today’s lesson in Math 9 with a little apprehension. A Friday morning after 2 days of job action and a long weekend didn’t sound like the most nurturing of environments. I decided that the time was ripe to attempt a lesson that has been in my mind for a couple of months; the following account is the story of the task, presentation, student reaction, and important learnings.
This week marked my baptism by fire into the twitter world. It was not long until I was neck deep in tweets, favourites, re-tweets, and followers. The eternal nerd awoke inside me when I was confronted with my first NCTM “Problem of the Day”. A simple, yet dangerously deep, question was posed. Wanting to cement my reputation as a responsible twit, I sat down and began to tinker with the theory.
The question was as follows:
How many different fractions can you write using only the digits 1,2,3 & 4?
Be sure to include fractions greater than 1.
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