Teacher: What’s the first thing we do when we see parentheses?
Delighted Teacher: Yes! Distribute!
Add this to the long list of well-meaning techniques taught to students that create an inefficient road block later on. Blindly using the distributive property before considering the structure of an expression may be a very inefficient technique.
“Cluttering heads with specialized techniques that mask the important general principle at hand does the students no good, in fact it may harm them.” ~Jim Doherty in Nix the Trix by Tina Cardone and the MTBoS
Consider the equation: 5(4 + x) = 25.
Wouldn’t it be more efficient to rewrite this equation as 4 + x = 5 than 20 + 5x =25?! The goal is to isolate the variable not to clutter it even more with an increased potential for errors. Compound the problem even more with a literal equation such as:
Solve for n: a(n + ab) = c
And what if there were fractional coefficients? An unnecessary distributive mess!
Standard for Mathematical Practice 6 states that mathematically proficient students are efficient in their calculations. The National Research Council’s report Adding It Up includes efficiency as an integral part of procedural fluency. Students should see mathematics as a tool, not a crux. When students are able to use the structure of an expression as part of their decision making, they develop a facility in problem solving that transcends the silly techniques that only serve to limit us.
Let us strive to equip students with the skills and habits that will make them thinkers, rather than robots. Machines become obsolete over time, but thinkers can adapt to new and different situations, creating new innovations along the way.