What do we mean by “understanding” in math? I gave a talk about this at the researchED conference in Vancouver. I have included an excerpt from my talk, and added some commentary at the very end which is designed  1) to further elucidate the issues and 2) to infuriate those who disagree with my conclusions.

Understanding Procedures

One doesn’t need to ‘deeply understand’ a procedure to do it and do it well. Just as football players and athletes do numerous drills that look nothing like playing a game of football or running a marathon, the building blocks of final academic or creative performance are small, painstaking and deliberate.

Many of us math teachers do in fact teach the conceptual understanding that goes along with an algorithm or problem solving procedure. But there is a difference in how novices learn compared to how experts do. Requiring novices to retrieve understanding can cause cognitive overload. Anyone who has worked with children knows that they are anxious to be able to solve the problem, and despite all the explanations one provides, they grab on to the procedure. The common retort is that such behavior comes about because math is taught as “answer getting”. But as students acquire expertise and progress from novice to expert levels, they have more stored knowledge upon which to draw. Experts bundle knowledge around important concepts called “neural links” which one develops in part through “deliberate practice”.

Furthermore, understanding and procedure work in tandem. And along the pathway from novice to expert, there are times when the conceptual understanding is helpful. But there are also times when it is not.

It’s helpful when it is part and parcel to the procedure. For example, in algebra, understanding the derivation of the rule of adding exponents when multiplying powers can help students know when to add exponents and when to multiply.

When the concept or derivation is not as closely attached such as with fractional multiplication and division, understanding the derivation does not provide an obvious benefit.

When the Concept is Not Part and Parcel to the Procedure

One common misunderstanding is that not understanding the derivation of a procedure renders it a “trick”, with no connection of what is actually going on mathematically. This misunderstanding has led to making students “drill understanding”. Let’s see how this works with fraction multiplication.

Multiplying the fractions  is done by multiplying across and obtaining But some textbooks require students to draw diagrams before they are allowed to use the algorithm.

For example, a problem like  is demonstrated by dividing a rectangle into three columns and shading two of them, thus representing  of the area of the rectangle.

The shaded part of the rectangle is divided into five rows with four shaded.  This is 4/5 of (or times) the 2/3 shaded area.  The fraction multiplication represents the shaded intersection, giving us 4 x 2 or eight little boxes shaded out of a total of 5 x 3 or 15 little boxes: 8/15 of the whole rectangle.

Now this method is not new by any means. Such diagrams have been used in many textbooks—including mine from the 60’s—to demonstrate why we multiply numerators and denominators when multiplying fractions. But in the book that I used when I was in school, the area model was used for, at most, two fraction multiplication problems. Then students solved problems using the algorithm.

Some textbooks now require students to draw these diagrams for a variety of problems, not just the fractional operations, before they are allowed to use the more efficient algorithms.

While the goal is to reinforce concepts, the exercises in understanding generally lead to what I call “rote understanding”.  The exercises become new procedures to be memorized, forcing students to dwell for long periods of time on each problem and can hold up students’ development when they are ready to move forward.

On the other hand, there are levels of conceptual understanding that are essential—foundational levels. In the case of fraction multiplication and division, students should know what each of these operations represent and what kind of problems can be solved with it.

For example: Mrs. Green used 3/4 of 3/5  pounds of sugar to make a cake. How much sugar did she use?  Given two students, one who knows the derivation of the fraction multiplication rule, and one who doesn’t, if both see that the solution to the problem is  3/4 x 3/5, and do the operation correctly, I cannot tell which student knows the derivation, and which one does not.

Measuring Understanding

Given these various levels of understanding, how is understanding measured, if at all?  One method is by proxies involving procedural fluency and factual mastery but which involve some degree of mathematical reasoning.

Here’s an example. On a multiple choice placement test for entering freshmen at California State University, a problem was to simplify the following expression.

In case you’re curious, here’s the answer:   (y+x)/(y-x)

This item correlated extremely well with passing the exam and subsequent success in non-remedial college math. Without explaining one’s answer, simply recalling the arithmetic properties of fractions along with being fluent in factoring was enough for a reasonable promise of mathematics success at any CSU campus.

In short, the proxies of procedural fluency demonstrate the main mark of understanding: being able to solve all sorts of variations of problems. Not everyone needs to know the derivation to understand something at a useful—and problem solving—level.

Nevertheless, those who push for conceptual understanding, lest students become “math zombies”, take “understanding” to mean something that they feel is “deeper”. In a discussion I had recently with an “understanding uber alles” type, I brought up the above example of fraction multiplication and the student who knows what the fraction multiplication represents. He said “But can he relate it back to what multiplication is?”  Well, that’s what the area model does—is it necessary to make students draw the area model each and every time to ensure that students are “relating it back” to what multiplication is?

They would probably say it’s necessary to get a “deeper” understanding. My understanding tells me that what is considered “deeper” is for the most part 1) not relevant, and 2) shallower.

IN CASE YOU’RE INTERESTED: The entire talk can be obtained here: It is the PowerPoint slides, which if viewed in notes format contain the script associated with each slide.

6 thoughts on “Misunderstandings about Understanding, Dept.”

1. Keep up the good fight!

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2. Chester Draws

I like to ask the proponents of the area model to do ones with improper fractions, variables and negatives. What does the box for x/2 x -5/4 look like?

Since that is where we need to get to, why would we go down a dead end path on the way, only to do it the traditional way in the end anyway?

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3. Ann in L.A.

Isn’t the answer for the Cal State problem:

+/- (x+y)/(y-x)

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• I should have written it as (y+x)/(y-x), which is really the same, but easier to read. If multiplied by -1, then the terms in the denominator can be reversed, so the expression can be written as -(x+y)(x-y).

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4. SteveH

Rote understanding.

You’ve been pushing this for years. Do you think it’s getting through?

There really is no such thing as successful “answer getting.” Back when I taught college algebra, trig, etal, all you had to do is change a problem slightly to cause students to fail. Answer getting at the algebra+ level for a range of problem variations requires a LOT of understanding.

Answer getting also requires a lot of understanding at the lowest levels of math, including the times table. What is 6*7? 42, isn’t it? Well 5*7=35 + 7 = 42, because 7 is 5+2 and I like to add by fives. Nobody does math only with rote memory. Then there are things like telling time and counting change with quarters, dimes, nickels, and pennies.

Most people forget the incremental understanding process they went through to get there. Telling time is very difficult, what with 365 days per year (plus leap year), 30 or so days per month (do we stop teaching kids the rote rhyme? I don’t even know the reason or understanding of the rhyme.), 24 hours per day (AM and PM), 60 minutes per hour, and 60 seconds per minute. Do they have to know about the solar system, years for different planets, spin on a tilted axis to the sun, and why 24 and 60 as bases? Do they have to know about different bases? My son had a thematic unit in first grade about little Eskimo kids in Alaska without ever knowing why it’s so cold and dark up there most of the year. Clearly, they aren’t worried about rote knowledge.

“Answer getting” is something they made up just to push their beliefs. So what’s driving it? All of that silliness disappeared (for the most part) when my son got to high school. It’s a K-8 thing and in our schools, the fundamental driving force in K-8 is full academic inclusion. Our high school, however, is full environmental inclusion. To get full academic inclusion to work, however, requires lowering grade level expectations and creating a world of magic natural differentiated learning where “kids will learn when they are ready.” I don’t know how many times I heard that. Focusing on concepts and understanding allows them to claim a high ground while disparaging and lowering emphasis on “answer getting.” They are fundamentally and systemically wrong and push all of the critical skill understanding into the home and with tutors.

They focus on very low CCSS proficiency levels that only strive for no remediation for College Algebra courses. I got to Calculus with no help from my parents. That is virtually impossible now. All of my son’s STEM friends got skill help at home in the early grades. It now requires outside help for students to make the change in slope from K-8 CCSS to honors and AP tracks in high school. I remember my son’s classmates (and parents) who felt like they were whacked in the head with a brick when their “exceeding expectations” failed to get them on the Algebra in 8th grade track at the end of sixth grade.

Full inclusion and CCSS have lowered the slope and expectations in K-8 and since the lowest level track in our high school is “College Prep”, then many students and parents are completely unprepared for the reality whack of honors and AP classes to STEM and what we called “College Prep” years ago.

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• “Rote understanding. You’ve been pushing this for years. Do you think it’s getting through?”

I’ve seen the term used in the same way that I use it. I have no trademark on it, and I encourage people to use it!

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