I have written about understanding in math, and the education establishment’s view of it. With all this talk about how it is important for students to “know math” and not just “do math” the question arises: “How do we measure a student’s understanding as opposed to their ability to go through procedures?” That is, how do we differentiate someone who truly understands from that of a “math zombie”.
In my opinion, the most reliable tests for understanding are proxies involving procedural fluency and factual mastery but which involve some degree of mathematical reasoning. Here’s an example.
On a placement test for entering freshmen at California State University, a single item on the exam correlated extremely well with passing the exam and subsequent success in non-remedial college math. The problem was to simplify the following expression. (Multiple choice test):
Without verbalizing anything or explaining one’s answer, simply recalling the arithmetic properties of fractions along with being fluent in factoring well enough to complete the task correctly was enough for a reasonable promise of mathematics success at any CSU campus. For those who are curious, the answer is (y+x)/(y-x).
Yet, the education establishment often proceeds from the belief that “Getting answers does not support conceptual understanding.” In the teaching of math in K-12, we are seeing more interest in the process by which students obtain the answers to “authentic” problems. If students really understood, the thinking goes, then they could apply prior knowledge to problem types they have never seen before.
But we have to be aware of level of understanding. Novices don’t think like that. Novices learn how to solve problems from worked examples. Subsequent problems are varied slightly beyond the initial worked example, forcing students to make connections to prior knowledge. This process is called “scaffolding”.
For example, if we ask what is the perimeter of a rectangle, with sides that are 5 and 7 inches, the student applies the formula he has (yes) memorized: 2W + 2L = perimeter, where W and L represent width and length and comes up with 24 inches. Subsequent problems are variations on this theme: A rectangle has a side that is 7 inches with a perimeter of 24 inches. What is the length of the other side? … and so forth.
But such scaffolding is sometimes held in disdain, viewed as rote memorization of procedure. To counter this, we have students working on problems that can’t be readily solved by formulas or previous learned procedures. These are called “rich problems”.The best I can do at a formal definition of “rich problems” comes from someone who disliked “algorithmic” problems: “It’s a problem that has multiple entry points and has various levels of cognitive demands. Every student can be successful on at least part of it.”
My definition might be a bit clearer: “One-off, open-ended, ill-posed problems that supposedly lead students to apply/transfer prior knowledge to new or novel problems that don’t generalize.” (See figure)
For example: “What are the dimensions of a rectangle with a perimeter of 24 units?” A student who may know how to find the perimeter of a rectangle but cannot provide answers to this (and there are infinitely many) is taken as evidence of not having “deep understanding”. In their view, the practice, repetition and imitation of known procedures as illustrated in the original example about perimeter of a rectangle, and variations thereof, relies on “imitation of thinking”.
But imitation is key as one goes up the scale from novice to expert. As anyone knows who has learned a skill through initial imitation of specific techniques, such as drawing, bowling, or learning a dance step—watching something and doing it are two different things. What looks easy often is more complicated than it appears. So too with math. The final accomplishment often does not resemble how one gets there. Like playing a game of football or running a marathon, the building blocks of final academic or creative performance are small, painstaking and deliberate.
As the cognitive scientist Dan Willingham has said, only experts see beyond the surface level of a problem to its deeper structure. “But if students fall short of this, it certainly doesn’t mean that they have acquired mere rote knowledge and are little better than parrots.” Rather, they are making the small steps necessary to develop better mathematical thinking. Simply put, no one leaps directly from novice to expert.