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 
Reblogged this on Nonpartisan Education Group.
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For most people, math is a tool, not an art form. I was one. Letting them skip over the hands-on part — the algorithms and the practice — in order to have them “think conceptually” is like giving a child a piano but rather than teaching them how to use it to make sounds and tunes, instead lecturing them on music theory.or acoustics or music history. All fine things to learn, but most people want to start with using their fingers to make sounds that make patterns. And benefit from this approach.
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May I agree with Barry and EB?
I think the first question is: What is the goal?
For most students, the goal is to be able to solve math problems correctly.
Cognitive experts tell is that to be able to do that, they need to be taught new concepts, and practice applying them to simple problems they can solve by mental math (memorized math facts). Recallable mental math leaves room in the brain’s working memory to process what the concept is about. Processing moves the context cues of the concept into long-term memory (LTM). That’s what learning is: Storing information in LTM so it can be recalled and applied to solving problems
Cognitive experts also tell us that students canNOT use reasoning based on conceptual understanding to solve anything more than simple problems. Why? Working memory (where the brain solves problems) is stringently limited in duration and capacity when trying to reason with information that has not been well-memorized.
This means, the cognitive experts tell us, once the student is taught the concept (which does not take long), to apply the concept to solve a problem of any complexity, the student must memorize by practice and apply an algorithm.
An algorithm is simply a stepwise procedure which it has been proven the brain can manage to solve a problem without overloading working memory. Without an algorithm to manage the data of a problem, working memory overloads quickly, and confusion results.
The goal or learning math or science or a musical instrument or driving or the use of ANY tool is to become a zombie: To be able to use it correctly “with automaticity,” which means quickly, effortlessly, and without conscious thought.
Higher level skills can only be learned after lower level skills are automated. If lower level skills are not automated, they take up room in working memory and leave no room for higher level facts and procedures.
Cognitive experts say that we want college math majors to be able to explain the “why” of how math works. But the NSF tells us that each year, America graduates about 20,000 math majors. America also graduates about 200,000 majors in the sciences and engineering. For those 200,000 – they need automated recall of the facts and algorithms that solve math problems so they can focus their mind on the science.
Math majors need automated recall of facts and procedures so they can focus their attention on explaining the why.
If zombie means getting the right math answer every time without being able to say why, for all of us who work in science and engineering, becoming a math zombie is the goal. We need our attention focused elsewhere.
The folks who claim students can do mathematical reasoning of any complexity without applying memorized algorithms have ignored what science says about the brain’s proven strengths and limitations. “Progressive” math educators need to read at least the basics of cognitive science. Working memory limits determine what students can and cannot do.
For more that agrees with Barry on what science says about why students MUST master math algorithms, and use algorithms not concepts or reasoning to solve problems, see https://bit.ly/2NzwrYR .
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I agree with everyone who has posted at this point – Barry, EB, and eanelson2014. I have used the term “math zombie” in the past, but I don’t think I meant it in the way it is being used here.
i don’t mind the math zombies described above; understanding usually follows once you do something enough times, or when you are sitting in the math class above the one where you “mastered” whatever algorithm had to be done. I tell my precalc students on the first day of class that precalculus is hard to teach, not because the stuff is necessarily hard, it’s that it won’t make any sense until you are sitting in calculus!
That out of the way, I think what I always meant by math zombies was “grade chasers” or “try-hards.” These are the kids that want the grade to come from doing a bunch of stuff that may or may not be related to the task at hand but want an A anyway. As a teacher who uses mastery learning as the basis for grading, I don’t care if the kid understands or not; if they can do the task, then they have mastered it. Of course, I have “A-level” questions on my exams that discriminate between the zombies and those who understand. What I have a problem with is the parents who don’t like my method of mastery learning and how the grades flow from that.
There is a real problem in the education world with what grades are supposed to communicate. Parents want the grade to be about being a try-hard, but then they want to be able to say about that same grade that bubba shouldn’t be failing calculus because he got a B+ in precalc where the grade was not based on mastery.
From the mastery learning standpoint, the grade is about being able to successfully do the task. Therefore, at my school, math grades are 65% or more from tests and quizzes (usually around 45% or more for summative, 20% or more for formative). “Doing the work” is only 35% or less of the grade. But the lesson here is to do the work that doesn’t count so you can excel at the work that does. It takes the grade-chasers and their parents awhile to understand this. The math zombies get it, and that is fine with me.
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