Sunday, February 17, 2013

What does it mean to do mathematics? IIb

Previously, we were introduced to the Doing Math Anchor Chart task (here). In this post, I share a recent exemplar that used the metaphor of riding a bike to communicate the preservice teacher's vision of what it means to do mathematics. The next post will offer a more traditional concept map representation.

Artist's Statement
Doing Mathematics is like riding a bicycle. It requires all pieces to work together to move forward. When riding a bike one must stay balanced. This is the same for mathematics. We must employ all the processes to complete a problem and understand if fully.

Pieces
Algebra = Bike Rider
As a student, doing mathematics means doing algebra in some cases. The student must employ all of the tools (the bicycle) in order to perform - just as the rider must pedal as the wheels move and control using the handlebars to ride the bike.

Pedal 1 = Reasoning and Proof
When doing mathematics, reasoning and proof is central to the process. When riding a bicycle, moving the pedal is essential to moving the bike along. When we are reasoning we are investigating a problem and developing arguments. We can also select how we want to reason, like we can change the pace at which we are pedaling.

Pedal 2 = Problem Solving
Problem Solving and Reasoning and Proof go hand-in-hand - just as the two pedals work together to move the bike forward. When problem solving you are building on mathematical knowledge and working to use appropriate strategies (like the appropriate pace of pedaling).

Bell = Communication
The bell on the bicycle is used to communicate with others around. in mathematics we use communication to talk with others in a clear fashion about our work. There are precise signals one can use to tell others you are oncoming when using the bell just as mathematicians must use precise language. Also, the rider must evaluate when the best times are to use the bell and evaluate if others will run into them before using the bell - like we evaluate others' thinking in mathematics.

Gears = Connections
The gears of the bicycle work together with the pedals and the wheels to move the bike forward; they are also the pieces that keep the whole process of riding a bike continuous. The fluidity of a bike is similar to the fluidity of mathematics in which we can find connections and then apply them to doing mathematics.

Wheels = Representations
We use representations in mathematics to communicate or record our ideas. Essentially representations are what help us solve problems through their application. Without the wheels on the bike we would go nowhere, thus we need representations to model mathematics like a bike needs wheels to move.

Handle Bars = Metacognition
When riding a bike we balance on the handle bars. While you can take a hand off now and then, we find we are most balanced with both hands resting on the handle bars. In mathematics, we use metacognition to think about and communicate our thinking. It controls the steps we take when working as we analyze what we have done or what we need to do. The handle bars control in which direction we go. The brakes are also located on the handle bars. At times we may get stuck; this is when we stop our work and think about our thinking once again.