Balancing Knowing Math Basics with Developing Mathematical Thinking

So, what does it mean to think mathematically?

Hindsight being 20/20, I understand now that I used to equate “mathematical thinking” with whether or not a student could follow a set of prearranged steps (algorithm) I synthesized for them.  For instance, as an educator in my first few years of teaching, I believed that if a student could perform the long division algorithm, then they knew “how to divide”.  Additionally, if a student could then read a word problem and correctly choose to complete long division to solve the problem, then I believed they really got it.

Want to do a quick check of your students’ grasp of division based on their proficiency with the long division algorithm?  Ask them what they are counting at each step of the algorithm.

Consider 257 divided by 4.  To solve, one student might suggest that we first need to count how many times 4 can go into 2.  A different student suggests that we first need to consider separating 200 pieces into 4 equal groups, then count how many pieces there are in a group.

Which student has a more developed sense of division and relationships between numbers?   

As I spent more time reflecting on my math practice, with the help of  mentors and professional resources aimed at enhancing math practice, I began to realize that I was only providing a very narrow opportunity for students to develop their mathematical thinking by focusing on math computation ‘basics’ that I found useful.  I did not provide students with a broader opportunity to make connections between math ideas or by extension an opportunity to develop as a mathematician.

In Ontario, we have a math curriculum document that outlines what students essentially need to ‘know’ and ‘do’.  An important part of this curriculum is a description of seven math process expectations that can help students ‘know the math’, deeply.  Here is an excerpt listing the process expectations, found on page 11 of the Ontario mathematics curriculum document.

math process expectations

Based on this list of mathematical processes, the narrow opportunity I used to provide students with to learn math was essentially through the lens of selecting tools and computational strategies.  This learning experience I provided for students “fit” into my main instructional model which was predominantly lecture style teaching where I ‘showed’ students how to solve problems.

At that early period of my teaching I felt that if a student could read a ‘word problem’ and solve it by applying an algorithm I taught them, then I was teaching through ‘problem solving’.  In time I realized, “teaching though problem solving is not the same as solving word problems”.

What I try my best to do now is provide broader opportunities for students to engage and interact with math.  Opportunities where students problem solve, develop their intuitive math thinking, share and prove their thinking to a partner, listen to the ideas of others within a large group setting, reflect on the validity of strategies, and try to make connections between math strands or ideas.

In my opinion, mathematical thinking is more than applying an operational strategy.  It is at the very least being able to think through a problem and account for additional information needed to make sense of the question.   Mathematical thinking involves students devising a plan to investigate the problem, reflecting upon the “reasonableness”of a solution, then communicating and proving to an audience why they are right.

By providing students an opportunity to interact with math through the process expectations listed above, then perhaps students can do more than learn math content and instead, more importantly, develop as mathematicians.

It might be time to replace ‘teaching math’ with an ‘inspiring math learning’ mindset.

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