Every once in a while, a parent will ask, “Why does my child have to learn about number lines (arrays, partial sums) when she already knows how to carry and borrow (or multiply two-digit numbers) using the traditional algorithm?”
Fair question. My response has always been that while procedural knowledge and fact fluency play significant roles in mathematics, conceptual knowledge and the ability to manipulate numbers in a variety of ways lead to greater competency. Our goal is to lead our students to be flexible, efficient, and proficient problem-solvers.
Recently, I spent the day with many Lower School colleagues digging into dynamic math assessment in a workshop led by Cathy Fosnot, whose work on understanding children’s mathematical development has changed the landscape for teachers. Her work examining developmental landmarks, strategies, and big ideas parallels in many ways some of what we know about reading development. We teach our students a variety of word-solving strategies because there are instances when one strategy works better than another. We’ve all witnessed the early reader who tries to sound out the word “laugh” using phonics rules. The highest level of understanding in the addition and subtraction landscape is being able to look at the numbers in the problem first, before choosing the best strategy. Our students cannot be literate if they use only one strategy to decode; likewise, students are not “numerate” if they use the same strategy for every problem.
Two of my own three children chose to major in mathematics in college, and they are constantly talking to each other about mathematical thinking and novel problems. It’s fun to listen as they throw around terms I’ve never heard of and debate the most efficient way to reach a solution. Not once have I seen them whip out a pencil to solve a problem. So it was interesting to learn that professional mathematicians will use a traditional algorithm less than 4 percent of the time to reach a correct answer. The other 96 percent of the time, they choose a different strategy based on the problem.
Sometimes, our kindergarten and first grade math work appears simple to the untrained eye. And yet, when you observe a group of third graders as they manipulate multi-digit numbers in their head, it’s clear that this constructivist approach has paid off.