The Obsession Over Conceptual Understanding in Math

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. Among other things, the progressivists argue that understanding of a procedure or algorithm must precede the procedure/algorithm itself.  I am a middle school math teacher and am in the former camp.

Arguments people have made in support of this 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 two-third ounce servings of yogurt are contained in a three-fourth 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.

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.” This all or nothing attitude ignores that both are using some understanding to solve the problem: i.e, they understand what fractional division represents, and how to use it to solve problems. 

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 view 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.

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? 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 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. While there are some concepts that a student may not understand, there are still connections that students make to previously learned material and concepts which serve to inform a recently learned procedure—and ultimately may lead to further understanding. In freshman calculus, for example, 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? 

Going to more basic mathematics, let’s look at multiplication. Learning multiplication—what it is and how to operate with it, provides an example of various levels of understanding. Efrat Furst, a cognitive neuroscientist, designs and teaches research-based classroom-oriented curriculum for educators and students addresses this. She writes:

Memorization usually means the ability to recite a certain fact like “four times three equals twelve” – a student that is able to do that is not considered to demonstrate understanding of multiplication. However, the student does understand “four times three” in a basic level that would allow effective communication at a low level and in a very 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 that demonstrate the meaning of the concept we can establish a higher level of meaning for the concept “Multiplication”.

In other words, one can operate at a very basic level of understanding that grows over time. While some basic levels 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 has the mathematical tools and the skills to use them to solve 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.

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 what needed to be done: "You divide the two fractions on top, by flipping the second one and multiplying, and since one is negative you'll get a negative answer. You get -5/4.  Then you do the bottom, and you have to convert  2 and 1/3  to a fraction by multiplying 2 by 3 and adding 1, so you get 7/3, and you multiply -3/4 x 7/3  and you cross cancel and get -7/4.  Now you divide -5/4 by -7/4  so you flip the -7/4  and multiply. You have two negatives so your answer is positive. And you can cross cancel. You get 5/7.”

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 by using reasoning (which is one form of understanding) and saw how they fit together and solved a more complex problems—which is what transference is about.

Problem Solving

Does understanding ever actually help in solving problems? In my experience, it does when the concept is part and parcel to the procedure. An example: knowing what procedure to use to simplify a² × a³ versus (a²)³ .  Students often have trouble remembering in which case exponents are added and in which one they are multiplied. The concept of multiplying powers is helpful; in the first case, the student remembers it is (a ×a) × (a × a × a), and it is easily seen that the exponents are added.  In the second case, raising a power to a power, the same principle applies: a² ×a² ×a² , which lends itself to understanding that the exponent “2” is multiplied by 3. When the concept or derivation is not as closely attached, understanding the derivation does not provide an obvious benefit, such as certain trigonometric identities. 

Likewise, in solving word problems, worked examples provide students a direct access to solving problems that are similar, and in the same category.  By scaffolding such problems;, that is varying the problems slightly beyond the initial worked example, students then are forced to stretch and to make connections that aren't as obvious. 

For example, students may be shown how to solve this type of problem: Two trains, 360 miles apart, head toward each other, one going at 100 mph and the other at 80 mph.  How long will it take them to meet?  The student can be shown that the sum of the two distances represented by 100t and 80t, where t is the time traveled by each train makes up the initial 360 miles.  A variation of this problem is: After the trains pass each other, how long will it take for them to be 90 miles apart?  In this case, the same concept is at work: the sum of the two distances represented by 100t and 80, again where t is the time traveled by each train, makes up the future distance of 90 miles.

In the words of Dylan Wiliam (Emeritus Professor of Educational Assessment at the University College of London Institute of Education): "For novices, worked examples are more helpful than problem-solving even if your goal is problem-solving"

Ending the Fetish Over Understanding

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.

The reform movement has succeeded in foisting its beliefs upon ever growing populations of new teachers who believe this is the only way. It is so entrenched, that even teachers who adamantly 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 math reform movement has created a poster child in which “understanding” foundational math is often not even “doing” math.