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.
I normally limited myself to one outburst per class and was now at my limit, but I kept on pushing her and put the argument on a math level. I referred to the chart’s characterization of “Solve contrived problems” as inauthentic and “Solve ‘real world’ problems” as authentic and asked why the authors automatically assume that a word problem that might be contrived didn’t involve “authentic” mathematical concepts.
I knew she wanted me to shut up. The class wanted me to shut up. Even I wanted me to shut up. She wrapped the discussion up by saying, “Let’s move on.”
The distinction the book (and the professor) makes between “authentic” and “inauthentic” learning has been around for a while. This concern about “authentic” versus “inauthentic” work comes from progressive education reformers who believe that it’s best for students’ school work to be as realistic as possible—that is, focused on learning about and trying to solve “real world” problems.
Educators who promote “authentic learning” mistakenly believe that novices learn the same way that experts do. They believe that students construct their own knowledge by being forced to make connections with skills and concepts that they may not have mastered. The theory is that they learn what is needed in a “just in time” manner, thus providing the motivation for learning, which they assume would otherwise be a tedious and soul killing exercise.
This approach became evident when we watched a video in class—one of many distributed and produced by the Annenberg Foundation. In the video, a teacher had his students do a variety of tasks, ostensibly to teach them about factoring trinomials, such as x2 + 5x + 6. But rather than teaching factoring techniques, as is done in traditionally taught classes, the session was a mélange of algebra tiles (manipulatives that are plastic squares and rectangles that one can use to represent algebraic expressions and to teach factoring) and a graph of the equation being factored (a parabola).
Then the teacher “facilitates” the class into making a connection between the factored equation and where the graph crosses the x axis. The class had not done factoring before, nor solved quadratic equations before, nor done a host of other things that would be important to understand the lesson.
The professor asked us our reactions to the video. I noted that rather than teach them factoring first and having them practice it, he had them doing things that generally came after such mastery. She seemed delighted at my observation and, smiling, said “Yes, so they were able to make connections between factoring, parabolas and solutions of quadratic equations.”
But when I replied, “There’s so much going on, that I'm not sure what they're learning or if they're learning anything at all,” her face went into a frown and she called on another student. Evidently she was so enthralled with the theory that she was unwilling to consider the possibility that, instead of leading to better comprehension of math, it left students confused.
The ed school approach to teaching math seeks to minimize “inauthentic” learning by replacing it with so called “authentic” exercises. But presenting students with a steady diet of challenging problems that neither connect immediately with their prior knowledge, lessons and instruction, nor develop any transferrable skills results in poor learning.
It is like children playing “dress up” in their parents’ clothing. The education establishment may believe they are producing “little mathematicians,” but the increased enrollments in remedial math courses in universities tell a different and disturbing story.