The State of the Art of Math Education: Let's Move On
9.9.10 – Barry Garelick – I recently took a math teaching methods class in education school–a remarkable class for its embrace of every educational fad I detest. One book we had to read in the class 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. It also hit every hot button I had as evidenced by my copy of the book: it is missing the front cover, which tore off when I hurled it across my bedroom.
Even worse than the book itself were the discussions in class that came out of it. One event in particular stands out. In a chapter that discussed the difference between “knowing” and “understanding”, a chart presents examples of “Inauthentic versus Authentic Work”. In this chart “Practice decontextualized skills is listed as inauthentic and “Interpret literature” as authentic. The black and white nature of the distinctions on the chart bothered me, so when the teacher asked if we had any comments, I said that calling certain practices “inauthentic” is not only pejorative but misleading. I asked the teacher “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. Keeping it on the math level, I then 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 assumed that a word problem that might be contrived didn’t involve “authentic” mathematical concepts. “Let’s move on,” she said.
Both this incident and this book remain in my mind because they are emblematic of the educational doctrine that pervades schools of education. This doctrine holds that mastery of facts and attaining procedural fluency in subjects like mathematics amounts to mind-numbing “drill and kill” exercises which ultimately stifle creativity and critical thinking. It also embodies the belief that critical thinking skills can be taught.
In their discussion of what constitutes “understanding” the authors state that a student being able to apply what he or she has learned 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”. In terms of math and other subjects that involve attaining procedural fluency, employing worked examples as scaffolding for tackling more complex problems is not something that these authors see as leading to any kind of understanding. They blur the distinction between learning a discipline (pedagogy) and practicing it (epistemology). That a mastery of fundamentals provides the foundation for the creativity they seek is lost in their quest to get students doing authentic work from the start.
The authors’ approach to how one teaches for understanding is through a process that they call “backward design”, in which educators plan their courses, units and lessons by starting from what they want the end result to be. That is, what should students know, understand and be able to do? The planning process then entails working backwards from there, identifying the content that goes into this, the big ideas, the questions to be explored and so on.
As the authors state, backward planning is not a new idea. In fact, I was a bit confused as to why it is even needed, given that such work has essentially been done in the writing of the textbooks that cover the course material. Teachers who have had fairly good success using the structure and sequence of a well organized textbook may question why they need to reinvent the wheel. But this brings us to another axiom which I have heard repeated in education school, which states that textbooks are a resource and not a curriculum. The authors pick up on this as well and regard using the textbook for planning as a “sin”, stating that “The textbook may very well provide an important resource but it should not constitute the syllabus.” By using the method of backward planning, the authors believe that teachers are less likely to rely on “coverage-oriented” teaching. They believe this is so because the backward planning process allows the teacher to address the big ideas, enduring understandings, and skills to be acquired in any order that works, thus freeing them from the burden of highly structured, rigid, and largely inauthentic textbooks.
In a paean to constructivism and the abandonment of textbooks, Tomlinson and McTighe, dispose of the notion that sequence of topics and mastery of skills is important, calling such beliefs the “climbing the ladder” model of cognition. “Subscribers to this belief assume that students must learn the important facts before they can address the more abstract concepts of a subject,” the authors state, and then quote Lori Shepherd, a University of Colorado education professor to make their point:
“The notion that learning comes about by the accretion of little bits is outmoded learning theory. Current models of learning based on cognitive psychology contend that learners gain understanding when they construct their own knowledge and develop their own cognitive maps of the interconnections among facts and concepts.”
In fact, this is the crux of how they approach differentiated instruction. Sequence doesn’t matter. Each student constructs his or her own meaning at their own pace, by being immersed in what the authors term “contextualized grappling with ideas and processes”. What does this mean? There are many examples, but the prevalent pattern of instruction to emerge from the book seems to be one of giving students an assignment or problem which forces them to learn what they need to know in order to complete the task. Say it is quadratic equations. Rather than teach them the various methods of factoring first, with the attendant drills, they might start with a problem such as x2 + 5x + 6 = 0. The teacher may then provide some activities that illustrate what factoring is, and then provide some exercises. The goal would be to factor the above equation into (x+3)(x+2) = 0 and, from there, lead the students to see that there are two values that satisfy the equation. This is what they mean by “contextualized grappling” as opposed to “decontextualized drill and practice”. It is a “just in time” approach to learning, (my choice of phrase, not theirs) in which the tools that students need to master are dictated by the problem itself by not burdening the student’s mental inventory with “mind numbing” drills for mastery of a concept or skill until it is actually needed. In the example above, the teacher may differentiate instruction by assigning extra factoring problems for students having difficulty, and provide instruction to the more capable students on how to solve quadratic equations by “completing the square” for expressions that cannot be factored.
While the authors are advocates for constructivism, they lean toward another ideal talked about often in education school classes: the balanced approach. They admit that there are times when direct instruction or ‘teaching by telling’ might work extremely well. “There is a need for balance between student construction of meaning and teacher guidance”, they proclaim. That direct instruction would work even better if topics were presented in a logical sequence is not the message of this particular book, however. Nor are the authors concerned over how many students will learn to hate math and other subjects because worked examples are “inauthentic”. “Just in time” approaches that work as a model for business inventory work just as well in education, they believe. The result is an approach that is like teaching someone to swim by throwing them in the deep end of a pool and telling them to swim to the other side. For the students who may already know a bit about swimming, they may choose to take that opportunity to learn the butterfly. The teacher might advise the weaker students to learn the breast stroke and provide the much needed direct instruction which they may now choose to learn. Or not.
Let’s move on.
This article was published in slightly different form in Educational Horizons, Vol. 88 Number 4; Summer 2010; p. 199
Barry Garelick is an analyst for the Environmental Protection Agency. He is a cofounder of the U.S. Coalition for World Class Math (http://usworldclassmath.webs.com/ .
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