Saturday, October 22, 2011

Graphing Literacy

My school division has been pushing literacy for a few years now. The division priority has filtered its way down into many programs at the school level. As a basic premise, if students are exposed to literate people and perform literate activities, their skills will grow. 

As the term is dissected, it seems that every stakeholder can find a way to skew the term to mean that their discipline is a crucial part of being literate. Reading and writing skills are an obvious avenue, but the ideas of technological and social literacy have emerged as important parts of every student's school experience. Riding shotgun to these ideas is the idea of Mathematical literacy--Numeracy. 

Most students carry the misconception that mathematics is a unique commodity that is unlike everything else they encounter in schools. Somehow, math educators have managed to make a system created to understand the world seem completely disconnected from it. I believe that the burden of innumeracy is born from this disconnection. Mathematical frameworks can be applied to situations that were otherwise thought of completely innumerate. 

Students learn the Cartesian system of rectangular coordinates fairly early in their High School career. By grade 10, it is used exclusively for the study of functions--algebraic functions. A system of relating variables has been completely transformed into one that is only associated with "x" and "y" and some kind of "slope". Before, and throughout, the unit on relations and functions, my students are encouraged to graph their thoughts--give me a graphical representation of their actions, emotions, justifications, etc.

Students see that changing the dependent and independent variables affects the story that the coordinates can tell. A graph can guide student decisions. Students are encouraged to graph my tendencies as a teacher. If we put "Time into Class" on the x-axis and "Chances Mr. Banting will let you use the Bathroom" on the y-axis, there is a relationship that exists graphically. Students bring a unique experience into the situation. It provides a rich discussion on the shape of the data, because every student is an expert in the field. Examples from textbooks often talk about things so distant from their lives; homework littered with distant topics only further distances mathematics from them. 

Open up a discussion or critique on the topic. As the teacher, I lobby to fix the graph to make me look "nicer" or "more fair". Maybe we graph a "fair teacher" on the same grid. How far is my line from theirs? What does that represent? What if we switched the variables? What if the variable became "Chances of getting a Drink"? Would the line be lower? Higher? What would each represent? 

Students intuitively graph; I open a unit up with this problem, and most jump right on board with strong opinions. They can begin to apply the mathematical relationships because they are familiar with the data. Discussions of continuous and discrete data are easily pulled from an activity like this. We could have two graphs:

1) Chances of going to the Bathroom after a Maple Leafs Loss vs. Time into class
2) Chances of going to the Bathroom after a Maple Leafs Win vs. Time into class

The vertical translation of these side-by-side graphs can be very telling. Soon the class has a "Mr. Banting Mood Graph". What is the Domain? What is the Range? These will begin as qualitative entities. Maybe we apply a scale of 1-10. Do we include fractions? Maybe the data needs to be discrete? Maybe groups break off and graph the situation themselves and them return with a narrative argument for their solution. When the class is ready to move into function notation, a line of best fit could be developed. Could we extrapolate the data if a class was 2 hours instead of 1 hour? How accurate would the data be? How would the graph shift? 

All these questions fit into the curriculum, but emerge much spontaneously. Students feel the independence of changing variables and graphs. Even as I am writing, my script has taken on a tone of freedom and exploration. A simple concept, linked to some powerful emotions, creates a rich mathematical discussion.

I give my students a personal graph assignment every year. After a discussion much like the above, they are given a handout with a Cartesian Plane on it (only quadrant 1). They are asked to pick any two variables and graph the relationship. There is a space for a description of their reasoning below. This is, by far, my favourite assignment of the year. Every inside joke and class dynamic comes out in one form or another.

This year, the staff won intramural volleyball. Two students we beat in the finals are in my class. Needless to say, our volleyball skills come up in many conversations. One student graphed the "Skill level of a volleyball player" vs. "How valuable they are to their team". I made an unflattering appearance on the graph--their ordered pair was very generous. I make sure to re-draw the graph in my interpretation. The assignment became a form of graphical satire.

This activity is very free-flowing, and builds great numeracy skills. It re-inserts a mathematical framework into student consciousness. I get students graphing freelance relationships for me for months after the topic has faded or been taken over by slope y-intercept form. Many mathematical implications emerge by simply playing with graphs. 

NatBanting

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