Many people don’t have to think very hard to remember a math “trick” they learned in grade school. For instance, third graders learning their multiplication tables will likely be taught the following “trick”: When multiplying a number by 10, add a zero to the end of that number for the answer. At first glance, this does not appear to be a trick at all. For most people, adding a zero is how you multiply by 10. However, this shortcut, or “trick,” fails to explain why this works. Getting an answer is not the same thing as knowing how you arrived at the answer. As explained in the article Nix the Tricks (found here), many third graders who know this shortcut may not understand that the basis for this rule can be explained with a place value chart, in which a 1 becomes a 10, a 10 becomes a 100, and so forth.
In previous posts, we examined the differences between instrumental and relational understanding. At the heart of Nix the Tricks is the notion that simply teaching students math shortcuts without providing foundational explanations deprives them of math concept development. Essentially, the “tricks” represent instrumental understanding (knowing what to do, not why to do it), and replacing these tricks with instruction of foundational concepts promotes relational understanding.
By using relational understanding, educators teach students to think for themselves. This benefits the students in the long run, giving them problem solving and analytical skills unable to be developed through memorization and instrumental learning. In many ways, the benefits of relational understanding can be compared to the following Chinese proverb:
“Give a man a fish and he eats for a day. Teach a man to fish and he eats for a lifetime.”
As giving someone a handout is to instrumental teaching, so preparation for real-world application is to relational understanding.
The difficulty, then, for most teachers (and students) is recognizing such tricks in order to replace them with meaningful teaching. Glancing through Nix the Tricks, several shortcuts stood out which I had been taught, including PEMDAS for order of operations and FOIL for multiplying binomials. Nix the Tricks catalogs a huge array of common math shortcuts, offering helpful alternatives to meaningfully replace these tricks. Investigate to see how many tricks have snuck into your classroom!