The Five Components of Math Proficiency

I recently wrote about the long-running debate about how to best teach math, aka the Math Wars. One side advocates for clear, step-by-step instruction of math procedures while the other side advocates for dedicated time to explore mathematical concepts through inquiry. As I discussed in that post, the problem with this debate is that both sides are presenting a view of math instruction that is incomplete. (If you missed that post, you can read it here.)

Now, let's dive into what it truly means to be proficient in math because an understanding of how the research defines math proficiency helps us define the best way to teach math. In Adding It Up: Helping Children Learn Mathematics, the research report written by top math education researchers in 2001 to try to settle the Math Wars once and for all, the authors define five essential components of being proficient in math, what they call the five strands of math proficiency: “Mathematical proficiency is not a one-dimensional trait, and it cannot be achieved by focusing on just one or two of these strands.”

The first two strands are terms you may have heard before: conceptual understanding and procedural fluency. In order to be proficient in math a child must both understand the mathematical concepts and accurately perform related procedures with ease. For example, to be proficient with multiplication of fractions a child must understand the concept that multiplying 6 by 2/3 means you're finding the portion of 6 that is equal to 2/3 of it or splitting 6 into 3 equal parts and taking 2 of those 3 parts (conceptual understanding). They must also be able to find the answer efficiently through a procedure such as multiplying 6 by 2 and dividing by 3 or dividing 6 by 3 and then multiplying by 2 (procedural fluency). 

A third strand is something I reference regularly on Counting With Kids: productive disposition. This means a child sees math as both purposeful and doable. They also see themselves as capable of doing math well. We help our children build a productive disposition when we approach mistakes in math with phrases like, "Hmm, let's figure out where we went wrong," and when we help our children see how the math they're learning is useful in the world around them.

The final two strands are strategic competence and adaptive reasoning. Adding It Up defines strategic competence as "the ability to formulate, represent, and solve mathematical problems" which is what we often think of as math problem-solving skills. A child with strategic competence can use what they know about math to solve a novel math problem (a math problem that isn't identical to one they've already done before).

Adaptive reasoning is defined as "capacity for logical thought, reflection, explanation, and justification." This means that a child is able to see the relationships between math concepts and use this understanding to explain their thinking or justify their work. For example, if a child is asked to find 25% of 12 and they divide 12 by 4, they've used adaptive reasoning to understand that 25% of 12 is the same as 1/4 of 12, which can be calculated by dividing by 4 and taking one of the 4 parts. A child with strong adaptive reasoning would also be able to explain why that process works.

Adding It Up calls these strands interwoven and interdependent, meaning that they are developed and strengthened alongside each other and true strength in one strand requires strength in the other four. But how do we know if our child is actually building all five of these strands of proficiency in their math class at school? In my next post, I'll discuss the necessary components of a well-structured math class designed to support all five strands of proficiency. I'll also share tips for how parents can support each of these strands at home in the event that a child isn't getting enough work with one or more of them in class.