Beliefs About “Understanding” in Math, Dept.

Here are some of many beliefs about “understanding” in math.  It was hard to choose from so many candidates, but feel free to add some of your own.  

We shouldn’t be teaching kids algorithms before they have the conceptual understanding.

The belief is that standard algorithms for mathematical operations (like adding/subtracting multidigit numbers, multiplying and dividing multidigit numbers, multiplying/dividing fractions, etc) eclipse the conceptual underpinning.  That is, why the algorithm works.

The standard way used to be taught first, and alternate ways later, after mastery of the standard algorithm. Now it’s other way around in the belief that std algorithms eclipse “understanding”. Side dishes now become the main course and students grow confused—sometimes profoundly so. 

Problems are to be solved in more than one way, in the belief that doing so imparts and gives evidence of “understanding”.  You have students being required to solve simple problems in multiple ways supposedly to enhance discovery and impart understanding.   You have students drawing pictures for much longer than necessary, serving as both a means to simultaneously understand and explain an otherwise simple procedure.

Many of these are the same methods for operations that are taught in the traditional manner. But these alternative methods were taught after students had achieved mastery with the standard algorithms—and not for weeks on end. And many students discovered these methods themselves.  Adding 56 +68 and being made to say 60 +60 instead of 6 +6 when carrying and adding the tens column for example defeats the purpose of the algorithm which is to free up working memory.

Robert Craigen, a math professor at University of Manitoba describes this approach as “the arithmetic equivalent of forcing a reader to keep his finger on the page sounding out every word, with no progression of reading skill. It amounts to little more than a “rote understanding” for procedures that unfortunately students probably cannot perform for problems they cannot solve.”  

A blogger recently summarized it this way:

Next year’s teachers that are used to students using an algorithm for multiplication are aghast when students use unsophisticated strategies like counting by ones by drawing pictures or partial product by drawing boxes, or when the students seem to not have any idea what to do. “What do you mean, just multiply!” But to “just multiply” by mimicking an algorithm isn’t part of what students had been doing. These teachers shrug in frustration and teach “the only right way”. Students are left feeling either shafted by the previous teacher or, most likely, that they must just not be “good at math”. “

“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.”

How is “understanding” defined?  And what do they mean by failing to use it or build upon it?  Yes, if a student only knows that 3 x 2 = 6 but does not know what the multiplication represents or what types of problems can be solved, then that student will not be able to use multiplication. 

But does lack of conceptual understanding always have this result?  While progressivists all want to teach for “understanding”, they do so without fully defining what understanding is. A definition of “conceptual understanding” does exist; it appeared in the National Research Council’s 2001 report “Adding it Up”:

Conceptual understanding refers to an integrated and functional grasp of mathematical ideas. Students with conceptual understanding know more than isolated facts and methods. They understand why a mathematical idea is important and the kinds of contexts in which is it useful. They have organized their knowledge into a coherent whole, which enables them to learn new ideas by connecting those ideas to what they already know. Conceptual understanding also supports retention. Because facts and methods learned with understanding are connected, they are easier to remember and use, and they can be reconstructed when forgotten. (pp. 118-119.)

But this has been interpreted to mean that “procedural understanding” is rote memorization and does not entail connections to mathematical ideas.  An example of a student having procedural fluency but lacking conceptual understanding was given in a popular blog about math:

[The student} can accurately compute the area of a triangle, but doesn’t recognize how its formula was derived or how it can be extended to other shapes. (eg. trapezoids, parallelograms, etc.)

and elsewhere, someone referred to calculation as being only procedural:

“Calculation is the price we used to have to pay to do math. It’s no longer the case. What we need to learn is the mathematical understanding.”

Does this mean that a student cannot solve problems that involve areas of various shapes because he or she does not know how the formulas are derived? Is a student who does not know the derivation of the invert and multiply rule for fractional division unable to solve problems involving such operations?  

There is a difference between a novice and an expert. A student who knows a procedure but not necessarily the conceptual underpinning may later gain more understanding as they work with such procedures in solving problems.  And some students may never understand it.  What level of understanding are we talking about here?  Do we expect students to acquire expertise in a top-down fashion, with understanding first and procedure and application later?  Is it wrong to let them solve problems using the standard procedures. Or must we always sacrifice proficiency on the altar of the often undefined but cherished “conceptual understanding” ?

And last but not least, this old chestnut which has appeared in just about every math textbook written from every era you can imagine:

“In the past, math classes were about teaching facts, skills and procedures with no understanding,and mechanized drills.”

Despite this claim, it’s interesting that many adults who were educated in the eras caricatured as “failing thousands of students”, are much more capable at solving the arithmetic problems that today’s students struggle with, even those entering high school.  They are even capable of solving the open-ended, “rich”, depth of knowledge questions that are generally ill-posed one off problems that do not generalize and are assumed to lead students to “deep understanding”.

One example of such a problem is “What are the dimensions of a rectangle with a perimeter of 24 units?” I’ve seen adults who claim they are not good at math get further with such a problem–as ill-posed as it is–than the students judged to lack understanding because they cannot solve them. Apparently, knowing how to calculate the perimeter of a rectangle given its length and width is viewed as “mere memorization”. Subsequent scaffolding of such problems, like “What is the length of a rectangle with a perimeter of 24 and a width of 8?” are viewed as just more memorization. It evidently helped the adults able to come up with answers to the “rich” problem.

Again, we are dealing with levels of understanding along the spectrum of novice vs expert–a spectrum that is conveniently ignored as students are forced to endure a top-down approach to understanding with–it goes without saying but I’ll say it–disastrous results.


Access to Algebra 1 in 8th Grade; the Never-ending Story

“When students take Algebra I matters, but many students do not have early access.”
The Department is encouraging both access to and enrollment in STEM courses. Both aspects are important because, as we will see through the story, even where access to Algebra I classes are available students do not necessarily enroll in them.
The story is mostly grounded in civil rights issues and equity for all, but ignores a key factor in all this: Proponents of the Common Core math standards take a “Common Core wanted it this way” attitude, citing that the standards call for algebra in high school, but not in eighth grade. They take this stance despite Common Core allowing for such option as addresed in the Appendix to the math standards:
A “compacted” version of the Traditional pathway where no content is omitted, in which students would complete the content of 7th grade, 8th grade, and the High School Algebra I course in grades 7 (Compacted 7th Grade) and 8 (8th Grade Algebra I), which will enable them to reach Calculus or other college level courses by their senior year. While the K-7 CCSS effectively prepare students for algebra in 8th grade, some standards from 8th grade have been placed in the Accelerated 7th Grade course to make the 8th Grade Algebra I course more manageable;
But such words do not matter. Algebra continues to be the forbidden fruit of education, reserved for those whose parents can afford to have their kids learn it outside of school–or have enough clout to get their kids in to 8th grade algebra programs when they are offered.  As I wrote about here, the San Francisco school district  did away with algebra in 8th grade. Jo Boaler and Alan Schoenfeld wrote an article in the San Francisco Chronicle, lauding this decision, and stating:
They  (i.e., San Francisco USD) found a unique balance that is now seen as a national model. They decided to challenge students earlier with depth and rigor in middle school. All students in the district take Common Core Math 6, 7 and 8, a robust foundation that allows them to be more successful in advanced math courses in high school.  
Again, an example of an inflexible interpretation of the Common Core Math Standards. And as I discussed in the referenced post, the San Luis Coastal Unified School District limits access to algebra in 8th grade by making it available to the “truly gifted”–a term that went undefined and which I heard  uttered by an official of that school district.  They determine the “truly gifted” by requiring students to receive high scores on two tests given in the 7th grade.  One test has been around for a while–a multiple choice test developed by two universities that did a good job in determining the students who were ready for algebra. 
With the advent of Common Core, the District decided to institute a second test, developed by an outfit called the Silicon Valley Math Initiative (SVMI). The test consisted of questions that in my opinion, were appropriate for formative assessments but not for summative. It did the job, however, and many students were suddenly deemed unqualified (i.e., not “truly gifted”) to take algebra 1 in eighth grade. (Assuming that one has to be “gifted” in order to take algebra in eighth grade; I do not believe giftedness is a necessity for it.)  In the 2015-16 school year only 17% of students took algebra in 8th grade: 88 out of 517, down from about 300 students in 2013.
The report from the Dept of Education is timely.  It is correct that civil rights issues are important, I think the problem goes beyond civil rights. Namely, one no longer needs to be in a minority to be stuck with inferior programs and goals.

The Flawed Approach of Traditional Math, Dept.


A May 29, 2018 article about Common Core from the Yale Tribune,  references another article that appeared in the Washington Post and Chicago Tribune by Jessica Lahey. It summarizes her views about the bad rap she and others feel is being given to Common Core’s math standards:

She believes that the gap between parents and students does not necessarily lie on the Common Core itself, but on the flawed approach of conventional math education where students were taught to memorize and dutifully accept axioms and mathematical rules without completely understanding its application and the principles at work.

This view is shared by many and has become the hobby horse of the math reform movement that gained significant traction with the National Council of Teachers of Mathematics’ (NCTM’s) math standards, first published in 1989 and subsequently revised in 2000.

I have written about the mischaracterization of conventional –or traditional–math many times.   My main message is that the underlying concepts were in fact taught, and students were then given practice applying the various algorithms and problem solving procedures.  I have provided evidence of such explanations in excerpts from the math textbooks in use in the 20’s through the 60’s. Yes, the books required practice of the procedures, but they also showed the alternatives to the standard algorithms.  These were presented after mastery of the standard algorithms as a side dish to the main course.  These alternative methods are by and large the same methods that are taught today under the rubric of “Common Core Math”.  The difference is that the alternatives are generally taught before the standard algorithm in the belief that teaching the standard algorithm first eclipses the understanding of the “why and how” of the procedures.  Delaying the teaching of the standard algorithm by requiring students to use inefficient and often confusing techniques (in the name of “understanding”) can result in a confusion of what is the side dish and what is the main dish. The beauty and simplicity of the standard algorithm is lost among a smorgasbord of techniques that leave students more confused than enlightened.

In short, the ideas expressed in the two articles referenced above represent the groupthink that pervades education schools and other forms of the education establishment. The prevailing mode of thought views drills, practice and the learning of procedures as “rote learning” and prevents true “understanding”. If students “understand”, then everything else follows–the corollary of which is that understanding must come before procedure.

What is left out of such pronouncements is the difference between novice and experts. There are levels of understanding as one goes through school, and depending where one is on the spectrum between novice and expert, the level of understanding may be deep, shallow, or in between.  Procedural understanding is a level of understanding, but students who are at such level are sometimes referred to as “math zombies”.  This term is is relatively current but is what Lahey and others think is the end result of “conventional math.”  And unfortunately, their view seems to rule the roost.

For more on the notion of “understanding” and traditional math see here, and here.  Tell your friends.  Then hire a tutor.