Conceptual understanding in math has served as a dividing line between those who teach in a conventional or traditional manner (like myself), and those who advocate for progressive techniques. The progressives/reformers argue that understanding of a procedure or algorithm must precede the procedure/algorithm itself; failure to do this results in what some call “math zombies”.
I will state 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 nor do I hold up a student’s development when they are ready to move forward.
For many concepts in elementary math, understanding builds from procedures. The student practices the procedure until it is realized conceptually through familiarity and tactile experience that forges pathways and connections in the brain. (Efrat, 2018). Daniel Ansari (2011), maintains that procedures and understanding provide mutual support. Sometimes understanding comes first, sometimes later. And I’m fine with that.
There is No One Fixed Meaning to Understanding
What does understanding mean? 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 taking derivatives and finding integrals. It isn’t until they take more advanced courses that they learn the formal definition and theorems of limits and continuity. 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?” Dan Willingham, a cognitive scientist who teaches at University of Virginia calls being able to transfer knowledge to new situations “flexible knowledge”. Willingham (2002) explains that it is unlikely that students will make such knowledge transfers readily until they have developed true expertise. He argues, “[I]f students fall short of [understanding], 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.
How Do You Test for Understanding?
One proxy that teachers use for understanding and transfer of knowledge, is how well students can solve problems and their 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 can do that now; I know how to solve that.” He then narrated what needed to be done. He had certainly never seen this exact same problem before. And while he did not know why the invert and multiply rule worked, he put together basic skills that he learned and saw how they fit together and solved a more complex problem—an example of knowledge transfer.
Is Understanding Always Necessary to Solve Problems?
When does understanding help in solving problems or doing procedures? In my experience, it does when the concept is part and parcel to the procedure. An example: knowing what procedure to use to simplify 𝑥2 ∙ 𝑥5 versus(𝑥2)5. Students often have trouble remembering when exponents are added and when they are multiplied. The concept of multiplying powers is helpful; in the first case, the student remembers it is(𝑥 ∙ 𝑥 ) ∙ (𝑥 ∙ 𝑥 ∙ 𝑥 ∙ 𝑥 ∙ 𝑥), and it is easily seen that the exponents are added. In the second case, raising a power to a power, the same principle applies: (𝑥2)5 = 𝑥2 ∙𝑥2 ∙𝑥2 ∙𝑥2 ∙𝑥2 which lends itself to understanding that the exponent “2” is multiplied by 5.
When the concept is not as closely attached to the procedure, (e.g., some trigonometric identities) the conceptual underpinning may not be as accessible. In such cases, the understanding may not necessarily help to solve problems.
Ending the Fetish over Understanding
While some basic levels of understanding 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 and is a math zombie.
As a former college football player and high school football coach told me recently:
“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.”
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” basic math.
Shameless Self Promotion, Dept.
For actual examples of hands-on, real-world experiences of understanding vs procedure as it happens in the classroom, then read my book “Out on Good Behavior” before it’s ruined by the Hollywood movie version. Plus it has the intrigue of school politics wrapped in the enigmatic axiom of “You never really know for sure what’s going on.”
References
Ansari, D. (2011). Disorders of the mathematical brain: Developmental dyscalculia and mathematics anxiety. Presented at The Art and Science of Math Education, University of Winnipeg, November 19th 2011
Furst, E. (2018) Understanding ‘Understanding’ in blog Bridging (Neuro)Science and Education https://sites.google.com/view/efratfurst/understanding-understanding?authuser=0
Willingham, D. (2002) Inflexible knowledge: The first step to expertise, in American Educator 26, no. 4 7 (2002): 31–33, 48–49. https://www.aft.org/periodical/american-educator/winter-2002/askcognitive-scientist
traditional math 