By Barry Garelick
I am a mathematics teacher. I majored in math and, prior to going into teaching, used it throughout my career.
My facility with math is due to good teaching and good textbooks. I fully expected the same for my daughter, but after seeing what passed for mathematics in her elementary school, I became increasingly distressed over how math is currently taught in many schools.
Optimistically believing that I could make a difference in at least a few students’ lives, I decided that after I retired, I would teach high school math. To obtain the necessary credential, I enrolled in George Mason University Graduate School for Education in the fall of 2005.
The ed school experience did have some redeeming features. Most of my teachers had taught in K-12, and had valuable advice about classroom management problems and some good common-sense approaches to teaching that didn’t rely on nausea-inducing theories.
Those theories are inescapable, unfortunately.
Specifically, many education theorists hold that when students discover material for themselves, they learn it more deeply than when it is taught directly. In this vein, the prevailing belief in the education establishment is that although direct instruction is effective in helping students learn and use algorithms and mathematical procedures, it is ineffective in helping students develop mathematical thinking.
According to the establishment, students should be “led” to their discovery of the answers. Providing explicit instruction is considered to be “handing it to the student” and prevents them from “constructing their own knowledge.”
“Discovery learning” isn’t bad. Most teachers use some discovery learning and group work in their classes. Also, staging problems so that they vary slightly from the worked example—so that the students are essentially applying prior knowledge in a new situation (called scaffolding)—has the “look and feel” of discovery. The problem is that the reigning education theory focuses mostly on discovery, with only a nod to direct instruction. That’s mistaken.
The worst class I took in education school was on “math teaching methods.” It was taught by my advisor at the time. (I say “at the time” because shortly afterward they changed advisors on me, and she no longer taught courses, but worked with Ph.D. candidates. From what little biographical information I have seen about her, she has not ever taught any classes, math or otherwise, in K-12.)
The math teaching methods class was remarkable for its embrace of every educational fad I detest.
One book we had to read was Integrating Differentiated Instruction and Understanding by Design by Carol Ann Tomlinson and Jay McTighe. This book is popular in the education school and professional development circuit. Despite its popularity it only served to infuriate me, as evidenced by the missing front cover of the book, which tore off when I hurled it across my bedroom.
The book is emblematic of the doctrine that pervades schools of education. That doctrine holds that mastery of facts and attaining procedural fluency in subjects like mathematics amounts to mind-numbing “drill and kill” exercises that supposedly stifle creativity and critical thinking.
In their discussion of what constitutes “understanding” the authors state that a student being able to apply what he or she has learned (for example, using the invert and multiply rule to carry out fraction division) does not necessarily represent understanding. “When we call for an application we do not mean a mechanical response or mindless ‘plug-in’ of a memorized formula. Rather, we ask students to transfer—to use what they know in a new situation."
If you accept that, then in math (and other subjects that involve attaining procedural fluency), using worked examples as scaffolding for tackling more complex problems does not require mathematical reasoning nor lead to understanding. That view embodies the educational establishment’s notion that procedural fluency obscures understanding. The fact that a student can recognize when, say, fractional division may be required to solve a problem requires some reasoning, as well as application of the procedure itself (mechanical though it may be). Both fluency and understanding work hand-in-hand. As students increase their expertise more non-routine problems appear to them as routine.
Worse than the book itself were the discussions in class that arose out of it. One event stands out.
In a chapter that discussed the difference between “knowing” and “understanding,” a chart presented examples of “Inauthentic versus Authentic Work.” In that chart “Practice decontextualized skills” (otherwise known as “reading”) was listed as inauthentic while “Interpret literature” was listed as authentic.
The professor asked if we had any comments. I asked, “Do you really think that learning to read is an inauthentic skill?” She replied that she didn’t really know about issues related to reading."
Read the rest here.