What do we mean by “understanding” in math? I gave a talk about this at the researchED conference in Vancouver. I have included an excerpt from my talk, and added some commentary at the very end which is designed 1) to further elucidate the issues and 2) to infuriate those who disagree with my conclusions.
One doesn’t need to ‘deeply understand’ a procedure to do it and do it well. Just as football players and athletes do numerous drills that look nothing like playing a game of football or running a marathon, the building blocks of final academic or creative performance are small, painstaking and deliberate.
Many of us math teachers do in fact teach the conceptual understanding that goes along with an algorithm or problem solving procedure. But there is a difference in how novices learn compared to how experts do. Requiring novices to retrieve understanding can cause cognitive overload. Anyone who has worked with children knows that they are anxious to be able to solve the problem, and despite all the explanations one provides, they grab on to the procedure. The common retort is that such behavior comes about because math is taught as “answer getting”. But as students acquire expertise and progress from novice to expert levels, they have more stored knowledge upon which to draw. Experts bundle knowledge around important concepts called “neural links” which one develops in part through “deliberate practice”.
Furthermore, understanding and procedure work in tandem. And along the pathway from novice to expert, there are times when the conceptual understanding is helpful. But there are also times when it is not.
It’s helpful when it is part and parcel to the procedure. For example, in algebra, understanding the derivation of the rule of adding exponents when multiplying powers can help students know when to add exponents and when to multiply.
When the concept or derivation is not as closely attached such as with fractional multiplication and division, understanding the derivation does not provide an obvious benefit.
When the Concept is Not Part and Parcel to the Procedure
One common misunderstanding is that not understanding the derivation of a procedure renders it a “trick”, with no connection of what is actually going on mathematically. This misunderstanding has led to making students “drill understanding”. Let’s see how this works with fraction multiplication.
Multiplying the fractions is done by multiplying across and obtaining But some textbooks require students to draw diagrams before they are allowed to use the algorithm.
For example, a problem like is demonstrated by dividing a rectangle into three columns and shading two of them, thus representing of the area of the rectangle.
The shaded part of the rectangle is divided into five rows with four shaded. This is 4/5 of (or times) the 2/3 shaded area. The fraction multiplication represents the shaded intersection, giving us 4 x 2 or eight little boxes shaded out of a total of 5 x 3 or 15 little boxes: 8/15 of the whole rectangle.
Now this method is not new by any means. Such diagrams have been used in many textbooks—including mine from the 60’s—to demonstrate why we multiply numerators and denominators when multiplying fractions. But in the book that I used when I was in school, the area model was used for, at most, two fraction multiplication problems. Then students solved problems using the algorithm.
Some textbooks now require students to draw these diagrams for a variety of problems, not just the fractional operations, before they are allowed to use the more efficient algorithms.
While the goal is to reinforce concepts, the exercises in understanding generally lead to what I call “rote understanding”. The exercises become new procedures to be memorized, forcing students to dwell for long periods of time on each problem and can hold up students’ development when they are ready to move forward.
On the other hand, there are levels of conceptual understanding that are essential—foundational levels. In the case of fraction multiplication and division, students should know what each of these operations represent and what kind of problems can be solved with it.
For example: Mrs. Green used 3/4 of 3/5 pounds of sugar to make a cake. How much sugar did she use? Given two students, one who knows the derivation of the fraction multiplication rule, and one who doesn’t, if both see that the solution to the problem is 3/4 x 3/5, and do the operation correctly, I cannot tell which student knows the derivation, and which one does not.
Given these various levels of understanding, how is understanding measured, if at all? One method is by proxies involving procedural fluency and factual mastery but which involve some degree of mathematical reasoning.
Here’s an example. On a multiple choice placement test for entering freshmen at California State University, a problem was to simplify the following expression.
In case you’re curious, here’s the answer: (y+x)/(y-x)
This item correlated extremely well with passing the exam and subsequent success in non-remedial college math. Without explaining one’s answer, simply recalling the arithmetic properties of fractions along with being fluent in factoring was enough for a reasonable promise of mathematics success at any CSU campus.
In short, the proxies of procedural fluency demonstrate the main mark of understanding: being able to solve all sorts of variations of problems. Not everyone needs to know the derivation to understand something at a useful—and problem solving—level.
Nevertheless, those who push for conceptual understanding, lest students become “math zombies”, take “understanding” to mean something that they feel is “deeper”. In a discussion I had recently with an “understanding uber alles” type, I brought up the above example of fraction multiplication and the student who knows what the fraction multiplication represents. He said “But can he relate it back to what multiplication is?” Well, that’s what the area model does—is it necessary to make students draw the area model each and every time to ensure that students are “relating it back” to what multiplication is?
They would probably say it’s necessary to get a “deeper” understanding. My understanding tells me that what is considered “deeper” is for the most part 1) not relevant, and 2) shallower.
IN CASE YOU’RE INTERESTED: The entire talk can be obtained here: It is the PowerPoint slides, which if viewed in notes format contain the script associated with each slide.