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. 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 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? 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 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; there is some conceptual context in which a procedure or strategy is taught. 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. . 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”. 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.”
Which is 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 saw how they fit together and solved a more complex problems—which is what transference is about.
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.
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 fetish over conceptual understanding that prevails in the early grades has created a poster child in which “understanding” foundational math is often not even “doing” math
Excellent post. I imagine if we asked surgeons why they make an incision in a particular place many would answer “that’s how this procedure is done”, others might give a more specific reason like “to give access to this particular area” or “this is the most efficient way to reach this” but few would be able to give the reasons in terms of physics or fluid dynamics. Nonetheless they likely all perform the procedure very well. Why we put this extra layer on to elementary math makes no sense to me. If they can adequately understand to perform the math and have heard of the reasoning from a teacher do they need to have the ability to explain it? Are they lesser students if they can’t? The test should be can they progress to the next level with adequate skills to learn the next concept. Also much of our “understanding” comes from doing a variety of problems that allow us to explore the skill at its various limits:bigger numbers versus smaller, negatives versus positives etc. As we develop proficiency we see the concept in greater depth. By focusing on understanding and deemphasizing practice it makes it harder for many students to gain proficiency.
Beyond their fetish over "conceptual understanding" is there fetish for equity (equality of outcomes) rather than equality (equality of opportunity). The combination of the two is absolutely impossible since equity alone is impossible without watering down expectations to a point where everybody - achieves "success" even if they don't bother to come to school very often, much less learn anything of substance needed for admission to almost all STEM majors in college. They are DOA while having been promised reality of opportunity. They had been given a lie. The only way to "achieve" it is to drop requirements for the SAT or ACT. Much less an AP calculus score of 3 or better. These are now being dropped for college acceptance - even at prestigious schools - as they are allowed in only to be allowed to flunk out without massive intervention in order to qualify for beginning courses, most notably a solid calculus base.