Depth in Math Education: The Limitless Nature of Understanding
As a math teacher, one of my (increasingly many, as I get older) pet-peeves is the view of math education as a one-dimensional, linear process. In this view, students progress from one skill to the next, checking them off one by one, moving along a straight-line path toward the end of their math education. I sense this view from students when they claim, “I’ve learned this already.” I sense this view from parents when they obsess over pacing, upset that their child is not “moving fast enough.” The problem with this view of math education is that it ignores the depth of a student’s understanding: does the student understand why the skill works? Can the student apply it as part of a problem-solving process?
The view of math education as a linear process, without regard for depth, often accompanies a familiarity with a rote approach to math education. Rote learning involves memorizing skills through repetition. The National Council of Teachers of Mathematics (NCTM) uses the term “procedural fluency” to refer to a student’s ability to perform a skill. Standardized tests often set out to measure procedural fluency, leading to an emphasis on rote learning in the classroom and the minds of students, parents and even educators themselves.
Rote learning, which lacks depth, contrasts with meaningful learning in which students understand concepts and can apply them in new and varied contexts. Regarding procedural fluency, the NCTM recommends that it be constructed upon a foundation of conceptual understanding. In this view, math education is not merely a list of skills that students check off, one by one; each of these skills contains many depths of understanding. Instead of rushing from one skill to the next, we must explore them, dig deeper to arrive at a conceptual, transferable, applicable understanding.
To see this difference in action, let’s look at simple subtraction. Subtraction, as a concept, can be carried out in two different ways: by taking away or by counting up. We can find the difference of 8 and 3 by taking 3 away from 8 or by counting up from 3 to 8. A student only needs to know one method to perform the subtraction, but exploring both methods will help a student to understand subtraction as the difference of two numbers. This conceptual understanding will enable a student to apply their learning to new contexts: Jack had 3 apples and Jill had 8. How many more apples did Jill have? Without this understanding, many students must be prompted to apply subtraction to questions like these.
Of course, there exists an infinite potential for depth in any individual’s understanding. Great math minds like Isaac Newton, Leonhard Euler, Euclid, Archimedes or Carl Friedrich Gauss certainly understood concepts to a greater depth than others capable of carrying out the same skills: it is this great depth of understanding that enabled their great contributions to mathematics. When a student says, “I’ve learned this already,” or a parent complains about the pace of their child’s math education, they are unwittingly equating their (or their child’s) understanding to that of these historically great minds: there is nothing left for me (or my child) to learn, here, so let’s move on.
While rote learning often leads students to regard mathematics as mysterious and become dependent upon teachers to lead them through every new problem situation, meaningful learning helps students to become more independent, confident and successful. New skills can be connected to previous skills through conceptual understanding, while rote learning often fails to facilitate these connections and results in a mess of disparate, jumbled steps that lack any frame of reference. Rather than simply checking off skills from a sequenced list, then, a quality math education involves acquiring a depth of understanding. There is no limit to the potential for understanding.