According to the NCTM Learning Principle, proficiency in math requires three elements: understanding of math concepts; accurate recall of math facts; and an ease with executing mathematical procedures. It is the first element, conceptual understanding, that allows a person to use math flexibly, applying skills learned in one situation to a different, yet comparable, combination of variables. Knowing your times tables or the orders of operation do you little good if you don't know when or why to use them.
It is important, therefore, for teachers to build new math understanding upon that which students already possess. Even before starting kindergarten, students have developed mathematical ideas from their daily lives. The simple act of sharing a bag of M&M's between two or three siblings is an exercise in division. Building upon the familiar makes new learning less likely to be rote and then promptly forgotten. Moving from dividing a bag of M&M's to dividing a birthday cake could be a good conceptual progression from dividing whole numbers to fractions.
A student who learns with understanding has the potential to be more independent, fearless, enthusiastic, determined, versatile and successful. To achieve this, teachers must promote class discussion that gets students to recognized the efficacy of their own understanding. Imagine the hoopla of ideas that would spill forth if the 'fictitious' birthday cake mentioned above were to arrive on the scene. Where should we cut the cake first? Why? How many pieces do we need? Imagine dividing the class into teams, each with a template upon which to create their plan. Imagine each group having to present and defend their ideas. What criteria might be used to judge an effective solution? Discourse such as this will promote not only better understanding of math concepts, but also an awareness of how math is relevant and integral to everyday life.
Reference:
NCTM (2011). Learning principle.
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