The importance of learning mathematical concepts over rules and methods is supported by extensive brain research where it has been concluded that the brain can only compress and retain concepts; it cannot compress rules and methods. Thus, concepts are retained whereas rules and methods not connected to a concept are forgotten. Once students learn a mathematical concept they can apply connected math facts, algorithms, and procedures to new situations that are related to the concept. Application of a learned concept is connected to the concept itself and is then compressed by the brain as part of the concept.
The difference between rule and concept is summarized by Jo Bohler in her book Mathematical Mindsets. Bohler notes three comparisons as a way to think about concept vs. rule:
When I attended the December 8, 2016 session of Minnesota Math Leaders Networking at the Minnesota Department of Education (MDE), I came away with some constructive ideas for the use of inquiry to help students attain mathematical concepts. The session focused on changing mundane mathematical tasks into focused inquiry based tasks where students grasp mathematical concept before practicing an accompanying algorithm. I have highlighted some of the big ideas from the workshop below:
THE DEFINITION OF A RICH MATHEMATICAL TASK
A rich mathematical task should include the following features:
GOOD INSTRUCTIONAL ROUTINE FOR CONNECTING STUDENT REPRESENTATIONS
This routine encourages students to look at their peers work and make connections, look for patterns, and find solutions or algorithms that can be used for future math study.
THINGS THAT TEACHERS SHOULD BE AWARE OF WHEN USING STUDENT REPRESENTATIONS
I invite you to watch this Teaching Channel video representation of instruction focused on teaching concept before rule. You will see that the teacher is using a rich mathematical task, follows a routine for connecting student representations, and is aware of student perspective for displayed work.
This post brought to you by Dan Devine, Secondary Implementation Associate
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