Monday, February 18, 2013

What does it mean to do mathematics? IIc

Previously, we were introduced to the Doing Math Anchor Chart task (here). Then I shared 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. In this post, I will offer a teacher's more traditional concept map representation.

Chartists Statement

My Anchor Chart is in the form of a cycle or a process because I see the act of doing math as a cycle with the central goal of deepening our understanding.  When we do math we begin with a problem that we want to solve.  Using prior knowledge and problem solving skills that we have developed (which could include using representations) we work on the problem to get a response.  We then must evaluate our response and evaluate the process we took to get the response.  If our method isn’t working or we don’t feel our response is correct then we go back using this knowledge we gained to implement a different strategy or to see if it’s possible we responded to a different question.  If we don’t need to go back to rethink about our question or our problem solving method, we make conjectures or generalizations about the responses we obtain.  Many times we have questions about the conjectures about whether or not they work for all cases.  This gives us a new question that we may want to explore.  Regardless of exploring new conjectures we somehow share our thinking and responses with other people to gain their insight.  Because we share our responses, conjectures, and thought processes, we allow other people access to these things so they can ask questions and do math themselves.  In addition, the conjectures we develop may allow us to make connections to problems in other contexts or we can use the problem solving skills we developed in other aspects of life, so the cycle of doing math is not closed, and because we go back and retry different problems and form new questions based on the work we are doing the cycle does not go one way. 

I choose to do develop my chart without specifically writing down the Process Standards in any area because in my cycle of doing math, different aspects of each standard are included in different steps of this cycle.  Here I will explain how the different standards fit into the chart and where. (Note: the numbers correspond to the labeled boxes):

1.     Communication:  the problem we have to solve maybe to analyze and evaluated someone else’s work.
Content: The problem we are working on is based on the content we want to learn, for this unit that that would be Algebra.

2.     Problem Solving: The problem that we have to solve requires that we implement a variety of strategies that are appropriate to the context of the problem.
Reasoning and Proof:  If our problem is to prove something then the strategies we implement will be the different methods of proofs and determining which proof method is most appropriate for the problem presented.

3.     Problem Solving: Implement a variety or strategies determining which one will work best as we work towards a response.
Connections: Recognize similarities between the problem that’s presented and previous problems we have solved.
Representations: Use representations to help you think about a problem and translate between the representations to help solve problems.

4.     Problem Solving: Reflect on the strategy we used.  Did the process we used allow us to effectively find a response?
Connections: Think about how the problem we are solving connects to and builds on other mathematical ideas.
Representations: Use representations as a way to organize our thinking.  Look back that the ones we used when solving problems and think about how they helped us.
Metacognition: interpreting our response and strategies require that we reflect on our process and how we thought about the problem. 

5.     Reasoning and Proof:  Make conjectures about the response we obtained, this may occur though connecting our work with previous work we have done.

6.     Problem Solving: It’s possible that the knowledge we gained and would like to share is the skills we used to solve the problem.
Reasoning and Proof: We can share a conjecture that we have developed or we can share our thinking in the form of a proof.
Communication: Clearly expressing the response and the process to others either formally through writing, possibly a formal proof (Reasoning and Proof), or through discussion with others. 
Connections: It is possible that we use examples and make connections to other problems to help our audience understand what we are communicating.
Representations: Maybe we choose to organize our thinking into some form of representation as way to communicate our thinking with others.
Metacognition: We may decide to share our thinking process; inorder to do this we must think about our own thinking. 

7.     Reasoning and Proof: Throughout the entire process of doing math, we are building an argument.
Connections: We build on the knowledge we gained in order to solve problems in the future.

8.     Connections:  We can apply the knowledge we built through this process to contexts outside of math. 

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