Because of school closings due to Covid 19, there has been a flurry of articles about distance learning, and the difficulties that parents face when having to explain “Common Core” math. The articles take the opportunity to show that parents are just not “with it” and that the new way is actually better because it confers “deeper understanding” rather than rote memorization.

This article is typical as is the following quote from it:

“Amberlee Honsaker remembers learning only one way to add or subtract in elementary school. It was the standard algorithm: stack numbers vertically, add the digits in columns, and carry the ones where necessary. For her daughter, Raegan, math instruction extends far beyond that. In first grade, Raegan is using number bonds, making place-value charts, drawing out 10s and ones — illustrating multiple methods for solving simple addition problems.”

Actually, in my elementary school as well as for many others, there were alternate methods taught. But they were taught *after* mastery of the standard algorithm. The alternate methods in addition to being taught were also often discovered by students themselves as an outgrowth of the mastery of the standard algorithm. A problem like 76 + 85 could be solved by adding 70+80 to get 150, and then 6 +5 to get 11. Adding 150 and 11, the final sum of 161 is obtained.

Number bonds were called “fact families” and place value charts were abundant as a glance through textbooks from the 60’s, 50’s, 40’s, and further back easily show. (See this article for examples)

But now, alternate methods are taught first in the belief that it imparts a “deeper understanding” of what is going on with standard algorithms and procedures which are taught later. Teaching the standard algorithm first is thought to obscure the understanding and is viewed as a “rote” procedure. As a result, what is mischaracterized as “rote memorization” has been replaced with “deeper understanding” as math reformers term it. I think a more accurate term is “rote understanding”.

The so-called “deeper understanding” is measured by having students show more than one way to add or multiply numbers, and to explain in writing why it works.

“Over the past 40 years, education research has emphasized that teaching math should start with building students’ understanding of math concepts, instead of starting with formal algorithms, according to Michele Carney, an associate professor of mathematics education at Boise State University.”

The article does not do us the favor of providing us references to the research but I’ve seen some of it. Most of it is based on “action” research done in classrooms with questionable controls, and authored by the same people who have been taking in each others’ laundry for years. (e.g. Fenema, Carpenter, Hiebert, etc)

Common Core codified much if not most of the reform math ideology that has been at work for more than three decades. Reform ideology got its first big boost with NCTM’s math standards in 1989 which was predicated on the notion that traditional math teaching sacrificed conceptual understanding on the altar of procedural fluency. It put an emphasis on “understanding” and viewed procedures as nothing more than “rote memorization”.

The other catch-phrase of the math reformers is “problem-solving”; so much so, that it has become a verb. It used to be that students solved problems. Now they “problem-solve”. Again, this harkens back to NCTM’s 1989 standards which downplayed the importance of procedural skills, and replaced those with students being able to “explain” their answers. “Math talk” has emerged as an indicator for whether students “understand”. If a student cannot explain how they solved a problem, they are held to lack understanding. Also, if a student cannot solve a problem in more than one way, that too is held to show a lack of understanding.

The typical problems of the past (distance/rate, mixture, number, coin) are being replaced with what reformers believe are problems that students are interested in wanting to solve. These are typically one-off problems that don’t generalize and for which little to no prior problem solving procedure has been taught.

The “problem solve” mentality has made its way into ed schools where I heard the philosophy espoused. That is, there is a difference between *problem solving *and *exercises*. “Exercises” are what students do when applying algorithms or problem solving procedures they know. Problem solving, which is preferred, occurs when students are not able to apply a mechanical, memorized response, but rather have to figure out what to do in a new situation. Moreover, ed school catechism states that students’ difficulty in solving problems in new contexts is evidence that the use of “mere exercises” or “procedures” is ineffective and they are overused in classrooms.

It is more likely that students’ difficulty in solving new problems is because they do not have the requisite knowledge and/or mastery of skills—not because they were given explicit instruction and homework exercises.

Those who make such a differentiation and champion “true” problem solving espouse a belief in having students construct their own knowledge by forcing them to make connections with skills and concepts that they may not have mastered. But, with skills and concepts still at a novice level, students are not likely to be able to apply them to new and unknown situations. Nevertheless, the belief prevails that having students work on such problems fosters a discovery process which the purveyors of this theory view as “authentic work” and the key to “real learning.” One ed school professor I knew summed up this philosophy with the following questions: “What happens when students are placed in a totally unfamiliar situation that requires a more complex solution? Do they know how to generate a procedure? How do we teach students to apply mathematical thinking in creative ways to solve complex, novel problems? What happens when we get off the ‘script’?”

In fact, as Rittle-Johnson, et al. (2015) have shown, procedural fluency does not exclude conceptual knowledge—it can ultimately lead to conceptual understanding. Also, “Aha” experiences and discoveries can and do occur when students are given explicit instructions, worked examples, and scaffolded problems.

While some educators argue that procedures and standard algorithms are “rote”, they fail to see that exercising procedures to solve problems requires reasoning with such procedures — which in itself is a form of understanding. This form of understanding is particularly significant for students with LD, and definitely more useful than requiring explanations that students do not understand for procedures they cannot perform.

**Reference:**

Rittle-Johnson, Bethany; Michael Schneider, Jon Star “Not a one-way street: Bidirectional relations between procedural and conceptual knowledge of mathematics.” Educ. Psychol Review; DOI 10.1007/s10648-015-9302-x

Reblogged this on Nonpartisan Education Group.

LikeLike

And yet with a much “deeper” understanding, why are children much worse at arithmetic today than they were in the past? Surely the whole purpose of the new methods is to make them better at Maths.

In recent years in my area students they have been arriving slightly better than ten years ago — because primary schools are returning to learning rote addition and multiplication. They have seen that the new ways are worse than the old.

LikeLike

The standard answer to your questions is “That’s because they’re not implementing Common Core correctly.” Elizabeth Green wrote a long article published in the NY Times Magazine, and that was her central thesis: https://www.nytimes.com/2014/07/27/magazine/why-do-americans-stink-at-math.html

Then the second part of your question (why are students doing better with “rote” learning) is covered by saying “Yes, they are ‘doing’ math but not ‘knowing’ math. They don’t have deep understanding.” The math zombie thing.

My response: Oh yeah, math zombies. They’re the ones who don’t need to take remedial math in college and end up majoring in STEM fields while those with “deeper understanding” become communication majors.

LikeLike

I think you know where I am going to go with this, Barry – grades. At least in the very academically-inclined Catholic high school at which I teach, grades are the be-all and end-all, the holy grail that will get you into college and happily ever after. The math department works hard to teach procedures and why they work and try to make the connections among algebraic equation-graph-table. But despite the talk of the reformers, the kids want it to be step-by-step memorization. Kids (and their helicopter parents) are only interested in the grade. The only way we have been able to get around that is to construct tests in which kids who are the “math zombies” can at best get a B if they show no understanding beyond the superficial “steps.” This approach “seems” to motivate our students to try to actually learn some concepts, since we have seen a nice correlation with math grades and ACT Math scores in the past 8 years.

It is beyond frustrating how the so-called reformers reduce what we do to mere “monkey see, monkey do.”

LikeLike

In a separate comment that you wrote on another post you give an example of 180 hours spent on direct variation. I’m familiar with this “understanding uber alles” approach. Yes, there is a middle ground and yes we do teach the understanding. Some students will grasp the conceptual more than others. What you are saying is those students get A’s. Are you saying that the B students are just “math zombies” with zero understanding? Do they tend to need remedial math in college?

LikeLike

I am not saying the B students are math zombies. Well, they kind of are. They separate themselves out pretty well though.

After the first quarter, many of the B students start to buy into my mantra, “the name of the class is algebra not answers.” These kids engage in their own “productive struggle” that I do not facilitate. They start looking at math in a different way. They start realizing that math is indeed more than a bunch of steps. Yes, there are algorithms that you need to know, but if you stop there then it gets harder to move on. But this group gets into the A-team by the end of the second or third quarter, because they have finally focused on what leads them to success (well, at least at the algebra 1 level) – reading the math, doing only things you are allowed to do given by properties, and following your own work! I can ‘t emphasize that enough with my students. Don’t try to second-guess yourself. Commit to what you did. You wanted to multiply both sides of that equation by something? Then do it and don’t make stuff up because you think it should look a certain way.

For many of the B students, math becomes harder for them over the years. Because they have ONLY tried to make it procedural, i.e. every day is a new “type” of problem I have to memorize, they basically end algebra 2 having only figured out algebra 1, which makes precalc a challenge. And AP Calculus with its emphasis on conceptual understanding as well as algorithm makes a 3+-score out of reach if all you actually get is the procedure. These students don’t generally stay at B’s at our school either, they tend to go down over time, even in Algebra 1. So yes, some of them do become math zombies, but it is not to their advantage.

LikeLike