The Conceptual Understanding Fixation in Math

Melting the golden calf of math reform

Barry Garelick taught 7th and 8th grade math as a second career after retiring from the federal government where he worked in environmental protection. He majored in math at University of Michigan. He is the author of several books on math education, including “Traditional Math,” and "Out on Good Behavior: Teaching math while looking over your shoulder,” both published by John Catt. He and his wife reside in the central coast area of California.

We are excited to reprint this 2023 piece from his Traditional Math Substack, which will remain relevant so long as reformers continue to insist that conceptual understanding must precede procedural fluency.

The Schoolhouse is a series from Education Progress featuring articles for and from teachers, parents, education officials, and others working in the education system.

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Conceptual understanding in math has served as a dividing line between those who teach in a conventional or traditional manner, and those who advocate for progressive techniques. (I am a middle school math teacher and in the former camp.) Among other things, the progressivists frequently argue that understanding a procedure or algorithm must precede using or applying the procedure/algorithm itself.

Arguments made in support of the above statement not only border on the ridiculous, but often cross it. For example, a teacher stressing over how to assess understanding with students learning to add and subtract told me: “When ‘your work’ consists of counting and adding and subtracting, there isn’t a whole lot of work ‘to show.’ At these basic levels there has to be another way to ascertain whether a student understands these basic concepts in a meaningful way.” So if a student consistently gets addition and subtraction problems correct and applies them to solve problems, what basic concepts does she feel the student lacks? Apart from such an extreme, one of the most popular arguments that has emerged as the poster child for the reform math movement’s push for understanding is the “invert and multiply” rule for fractional division. They argue that students know how to use the rule but have no idea why it works and that such conceptual understanding always helps students to solve problems. What is frequently left out of such discussions is that the teaching of fraction operations is not devoid of understanding; fractional division is presented in the context of what such operation represents, and what types of problems can be solved using it.

A discussion I once had with a math reformer about this provides an accurate picture of the two sides of the “understanding” issue. I said to consider two students solving the following math problem: How many 2/3-ounce servings of yogurt are contained in a 3/4-ounce container? One student knows why the invert-and-multiply rule works and the other doesn’t. Both students solve the problem correctly. I maintain that I cannot tell which student knows why the rule works and which one doesn’t. What I do know is that both understand what fractional division represents, and how to use it to solve problems.

The math reformer responded that the student who did not know why the invert-and multiply-rule works “obviously” does not understand fractional division. I failed to understand that reasoning, but I have heard variations of it through the years. Generally it goes like this: “Students who fail to understand a concept are unable to know how to use it or build upon it. They will end up with misconceptions that can go undetected for months or years.”

Informing math teaching with this kind of thinking can result (and often does result) in holding up a student’s development when they are ready to move forward. Students who show mastery of procedure but cannot explain the concepts behind them are viewed as “math zombies,” to use a phrase coined by a math teacher clearly in the “students must understand or they will die” camp. A math teacher I know who is not in that camp responds to such views by stating that “Worrying about math zombies is like worrying that your football players are too good at passing the ball — on the basis that their positional play is no better than the rest of the team, and therefore they obviously don’t understand what they are doing when they pass beautifully.” In this article, I hope (but realistically do not expect) to put the arguments about understanding to rest. Or at least place them in a conceptual context.

Important Caveat and Disclosure

I will state at the outset that I, like many teachers, do in fact teach the underlying concepts for algorithms, procedures, and problem solving strategies. What I don’t do is obsess over whether students have true understanding. And I don’t stop them from using a procedure or algorithm if they don’t “understand” it.

Conceptual understanding and procedural fluency often work in tandem — sometimes with understanding coming first, sometimes later. One feeds the other, and usually after a person has more mathematical tools and procedures that make understanding more accessible. (Case in point: many procedures and rules of arithmetic are easier to understand once one has a facility with algebra and symbolic manipulation.)

And sometimes, people can proceed without ever understanding a particular concept.

What is Understanding?

How one defines mathematical understanding is a large part of the problem. There is no one fixed meaning. Does it mean to know the definition of something? In freshman calculus, students learn an intuitive definition of limits and continuity which then allows them to learn the powerful applications of same; i.e. taking derivatives and finding integrals. It isn’t until they take more advanced courses (e.g., real analysis) that they learn the formal definition of limits and continuity and accompanying theorems. Does this mean that they don’t understand calculus?

Does understanding mean transferability of concepts? Or, as a teacher I had in ed school put it: “What happens when students are placed in a totally unfamiliar situation that requires a more complex solution? What happens when we get off the ‘script’?” Dan Willingham, a cognitive scientist who teaches at University of Virginia calls being able to transfer knowledge to new situations “flexible knowledge.” There is no simple path of understanding first and then procedural skills — and no simple path to flexible knowledge. Willingham explains that it is unlikely that students will make such knowledge transfers readily until they have developed true expertise. Understanding is an important goal of education, he argues, “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.

Levels of Understanding

There are different levels of understanding. One can operate at a very basic level of understanding that grows over time. While some basic level are thought of as “rote memorization,” lower-level procedural skills inform higher-level understanding skills in tandem. Reform math ignores this relationship and assumes that if a student cannot explain in writing a process used to solve a problem, that the student lacks understanding. Testing students for understanding in this manner, particularly in the K-8 grades, will often end up with students parroting explanations that they believe the teacher wants to hear — thus demonstrating a “rote understanding.” How is understanding best measured, then? I maintain that understanding is not tested by words, but by whether the student can do the problems. At the K-12 levels, understanding is best measured by the proxies of procedural fluency and factual mastery. The mastery serves as evidence that higher skills grow out of lower ones. I expect that this last statement will raise hackles on those who work within the educationist domain and try to build into their studies a confirmation that higher-order thinking is at odds with lower procedural skills, and that focusing on procedures prevents understanding.

Math is not taught in a vacuum, in which students are told “Do this, and never mind what it means.” When students learn about multiplication, they are shown that 3 + 3 + 3 + 3 can represent four groups of three things, or 4 x 3. For fractional division, students are shown first what it means to divide an orange, say, into halves, quarters, and so forth. That context is further extended to include fractions divided by fractions; e.g., how many 3/8-ounce servings are in a 15/16-ounce container of yogurt.

In teaching math, we teach a procedure within a context as the examples above illustrate. While there are some concepts that a student may not understand, there are still connections that students make to previously learned material and contexts which serve to inform a recently learned procedure — and ultimately may lead to further understanding. Efrat Furst, a cognitive neuroscientist who designs and teaches research-based, classroom-oriented curriculum for educators and students, addresses this. She writes:

Memorization usually means the ability to recite certain facts like “four times three equals twelve” — a student that is able to do that is not [yet] considered to demonstrate an understanding of multiplication. However, according to the formulation above, the student does understand “four times three” at a basic level that allows effective communication in a specific context (i.e., answering a question in a math quiz). To create a higher level of understanding, additional concrete examples are required (e.g., “Jess has three baskets, four balls in each”) as well as explicit connection to the new concept (“so we can say Jess has four balls multiplied by three”). By adding more familiar (concrete) examples to demonstrate the meaning of the concept, we can establish a higher level of meaning for “Multiplication.”

One proxy that teachers use for understanding and transfer of knowledge is how well students can do all sorts of problem variations. A student in my seventh grade math class recently provided an example of this. As an intro to a lesson on complex fractions, I announced that at the end of the lesson they would be able to do the following problem:

The boy raised his hand and said “Oh, I know how to solve that.” I recognized this as a “teaching moment” and said “OK, go for it.” He narrated step by step what needed to be done: “You flip the -3/5 to become -5/3 and multiply and you get -5/4. Then on the bottom you change 2 1/3 to 7/3 and multiply it by -3/4 to get - 21/12. So then you have -5/4 on top and -21/12 on the bottom, and you divide them. So -5/4 ÷ -21/12 is the same as -5/4 x -12/21. When you get a positive, and the answer is 5/7.” And that happens to be the correct answer.

He had certainly never seen this exact same problem before. And while he did not know why the invert and multiply rule worked, nor could he explain why multiplying two negatives yield a positive product, he was able to orally dictate the method, taking it apart mentally and explaining it verbally. He put together basic skills that he learned and used reasoning to see how they fit together in order to solve a more complex problem.

Ending the Understanding Fixation

The belief that teaching procedures prior to understanding will result in “math zombies” is entrenched in educational culture. The people pushing these ideas view the world through an adult lens which they’ve acquired through the very practices that they feel do not work. They become angry that their teachers (supposedly) didn’t explain all these things to them and are certain that they would have liked math more and done better if only their teachers would have focused on understanding. Their views and philosophies are taken as faith by school administrations, school districts and many teachers — teachers who have been indoctrinated in schools of education that teach these methods.

These ideas are so entrenched that even teachers who oppose such views feel guilty when teaching in the traditional manner so reviled by well-intentioned reformers. Given that today’s employers are complaining over the lack of basic math skills their recent college graduate employees possess, the fixation on conceptual understanding that prevails in the early grades has created a poster child in which “understanding” foundational math is often not even “doing” math at all.

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