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.