How much deeper understanding do students really need, Dept.

In a recent op-ed in the LA Times, Dan Willingham, a professor in the department of psychology at the University of Virginia, addresses a particular aspect of math education in the U.S.  Blaming poor math performance on bad curricula, he argues, overlooks that elementary school teachers may not have the deep understanding of math that is required to teach it. In fact they may actually fear math.

Students without deep understanding, Willingham argues, may be limited to inflexible thinking. That is, their math knowledge is limited to performing specific operations for certain types of problems but they may falter when presented with problems in new settings or with slightly different wording. The result is an increasing number of high school students floundering in math “because the groundwork of understanding was never laid in elementary grades”.

Willingham suggests that the solution might then be to find and hire those teachers who have “deep math knowledge” and who know how to convey it. I have no problem with hiring teachers who have a thorough understanding of math. What troubles me are the premises that students are doing fine with math facts and standard algorithms. Also I question the disturbingly prevalent belief that student outcomes in math would improve if only they had a deeper understanding of math.

Students are not doing fine with basic math

While high school students may indeed be floundering, I disagree that it’s because it was not taught with understanding in earlier grades. In my opinion the contrary is true; there has been too much of an emphasis and an obsession with understanding in math in elementary schools.

Over the past three decades—in large part propelled by NCTM’s standards that came out in 1989—the preoccupation with understanding has manifested itself with a de-emphasis on learning math facts. Also, standard algorithms for the basic operations are delayed while students are presented with alternate strategies that require making drawings or using convoluted methods. Such methods are nothing new; they were taught in the past, but after students had learned and mastered the standard algorithms. Now, however, they are taught first in the name of providing the conceptual understanding behind why standard algorithms work as they do. Simple concepts are made more complex under what passes as “deeper understanding.” Students I have seen entering high schools do not know their math facts, and use alternate inefficient strategies for simple operations such as multiplication.

 The codification of reform math ideology

The Common Core standards have effectively cemented in the math reform ideology that is increasingly incorporated in today’s elementary school textbooks. Adding to that are the bevy of ineffective teaching methods (inquiry- and problem-based learning, group work, so called differentiated instruction) pushed upon teachers in ed school and in professional development seminars.

Furthermore, they are told in ed schools, in professional development seminars and other sources that memorization is bad and that teaching standard algorithms “too early” eclipses understanding. Teachers who elect to teach standard algorithms and teach in traditional manners are sometimes told to teach their lessons with “fidelity” to textbooks they are required to use. Young teachers who fear for their jobs will do so. Older teachers who may have the understanding that Willingham would like to see are sometimes told the same. Unlike the younger teachers, however, the older ones can simply retire.  And unlike the older teachers, the younger ones are likely the products of the ineffective math teaching and are probably just as confused about math as many of the students we are seeing today.

Given all that, I would agree that having more teachers with a better understanding of math might help the situation. The improvement would come from not only from a larger knowledge base. Specifically, in greater numbers such teachers are more likely to be able to reject the nonsensical approaches foisted on them and use the resources and methods shown to effectively teach students math.

What is “understanding” in math?

Willingham admits that hiring teachers with better understanding might be difficult because state tests have been shown to be inaccurate predictors of who teaches well. Adding to this, is confusion about what constitutes “understanding”. What educationists believe is understanding is in most cases visualization—drawing diagrams that demonstrate what two-thirds divided by three-fourths looks like.  That is not at all what a mathematician means by understanding.  Also, being made to use formulaic “explanations” and dragging work out far longer than necessary with multiple procedures and awkward, bulky explanations is not a sign of understanding. Forcing students to continually stop and explain becomes nothing more than “rote understanding” in the end.

There are levels of understanding that vary depending on where a student is on the novice/expert spectrum.  Novices do not learn like experts; it takes time for knowledge to accumulate with procedures and understanding working in an iterative fashion to support each other. Insistence on understanding at every point where students should be learning procedures while working effectively with their beginning levels of understanding lacks educational value.  With the prevalence of misunderstandings about what understanding is, the criteria for hiring teachers who possess understanding might well result in hiring more teachers with the same misguided views.

Another misconception of understanding is the notion that students who know why a procedure works are in a better position to solve problems via enhanced flexible thinking. In an article by Greg Ashman, he describes a study which suggests that it is a mistake to assume that students in possession of conceptual understanding will use it.

While I disagree with Willingham’s point on understanding in his recent op-ed, he provides, in my opinion, an  earlier article on Inflexible Knowledge that he also authored presents a more apt characterization of the interaction between understanding and domain knowledge, in a  that he authored.  In it, he says:

“Understanding the deep structure of a large domain defines expertise, and that is an important goal of education. But if students fall short of this, it certainly doesn’t mean that they have acquired mere rote knowledge and are little better than parrots. There is a broad middle-ground of understanding between rote knowledge and expertise. It is this middle-ground that most students will initially reach and they will reach it in ever larger domains of knowledge.”

Simply put, no one leaps directly from novice to expert. For sure, teach math with understanding, but don’t obsess over it. Teach the math students need to know.