Friday, January 25, 2008

Decomposing Word-Problem Solving

A couple of years ago, I was tutoring a sixth-grade boy in mathematics. I noticed that although my student was fairly adept at arithmetic, he had difficulty in knowing which operations to perform in sequence for multi-step computations, which was especially evident in solving word problems. One manifestation of this divide between knowing how to perform a computation and knowing what the computation signifies or how it functions in a larger context came in his stubborn refusal (conscious or otherwise) to retain terms associated with operations. For example, when multiplying mixed numbers, I attempted to remind him several times first to "convert the mixed number to an improper fraction." Each time he would look at me or the page for a moment and then say something like, "Oh, you mean 2 times 3 plus 1?" (if the mixed number was 3 1/2), frequently pointing to the numbers with his pencil as he spoke. "Blank times blank plus blank"---that is, the concrete sequence of operations contextualized in the specific problem at hand---seemed to be the extent of his understanding of what he was doing, and the rejection of proper terminology seemed to suggest that the concepts of improper fractions and converting between different representations of the same number were irrelevant to him. Now, although I love words and expressing ideas succinctly, precisely, and elegantly, I don't intend to suggest an unnecessary proliferation of terminology. However, discipline-specific terms, in moderation, facilitate communication in the discipline, particularly at a level of abstraction necessary for understanding how the various component concepts of the discipline function with respect to one another, which is vital in constructing multi-step solutions to non-trivial problems.

When teaching complex skills---that is, skills that require more than one action, simultaneously or in sequence---it's frequently necessary to break the complex task down into simpler skills, teach the simple skills independently first, and then have the student blend the simple skills together to complete the complex task. Let me refer to this intentional, metacognitive pedagogical sequence as decomposition of the complex skill, and to instruction that does not involve decomposition as holistic. Decomposition is frequently latent in math instruction sequencing: One teaches whole number addition, then multiplication, then least common multiples, before blending these component skills into addition of fractions. Unfortunately, in other areas, traditional instruction has not always achieved sufficient decomposition; and, while bright students can figure out the component skills for themselves in the process of being taught the complex skill holistically, students who struggle with complex skills could potentially benefit greatly from training first in simple skills and then (metacognitively) in blending the simple skills together to achieve a complex task. (For an example outside of mathematics, consider Why Our Children Can't Read, where Diane McGuinness claims that instruction based on phonemic awareness---a decomposition of reading---is necessary for the 30% of students that whole language and traditional phonics, both more holistic with respect to reading as a complex skill, will inevitably fail.)

Now, word problems have long been used in math classes to connect pure operations with real-life applications, develop reasoning skills, and increase student interest. Unfortunately, when many people recall their school days, they remember word problems as the bane of their math class. This, I imagine, is due in part to the fact that solving word problems is a complex skill and that traditional instruction has not sufficiently decomposed word-problem solving. Although students may master simple skills such as arithmetic operations, they may not figure out on their own how to select and blend them in the context of a word problem. From my limited exposure to recent elementary-school math textbooks, I have the impression that textbook authors are beginning to move toward decomposing word problems with periodic special pages with metacognitive tips, such as identifying unnecessary information and selecting a problem solving strategy; but, I believe more work can be done toward this end, and indeed must be done to raise achievement levels in mathematics.

I envision a sequence of graded exercises that would train students in the skills of (1) selecting the operation required for a computation, (2) identifying the information required for a computation, (3) identifying what is being asked, (4) identifying what information is given, (5) identifying extraneous information, (6) identifying missing information, (7) constructing a "knowledge chain" to fill in missing information, and (8) constructing a sequence of operations to arrive at the desired result. Each exercise would focus only on one of these skills. For example, the exercise for (2) would consist of problems of the form "You would like to purchase some oranges. What do you need to know to find out how much to pay?" (the number of oranges and the unit cost) or "You would like to put a fence around your yard. What do you need to know to find out how long the fence will be?" (the length and the width of the yard) but not "You would like to put a fence around your yard. What do you need to know to find out how much it will cost?" which is a compound problem (the unit cost and the perimeter, which comes from the length and the width) and fits under (7). The exact wording of each problem could, of course, be varied or stay the same according to the learning styles and needs of the students. As the students progress through the sequence of exercises, the teacher would also provide explicit (metacognitive) instruction in how to combine the simple skills. For example, "When you read a word problem, first determine the question, then ask yourself what information you need to know in order to answer the question like we practiced last week." To emphasize the metacognition, students should explain verbally the steps as they perform them (something akin to "I read the problem, and now I have to identify the question. In the problem, John wants to find out how much to pay. Next, I have to figure out what information is necessary for figuring out how much to pay. To figure out how much to pay, I have to know the number of hot dogs and the price of each hot dog. The problem tells me that here. Next, I have to decide what operation to use. I have to multiply the number of hot dogs and the price. Now, I can calculate the answer.")

Of course, the trick comes in finding developmentally appropriate ways of expressing and teaching each component skill, since the ability to understand abstractions comes only over time; but, I suspect that there are exercises that can facilitate even handling abstractions in the decomposition of learning about problem solving (as a skill distinct from problem solving itself). Nevertheless, the guiding principle of decomposition remains: break complex skills into component parts, practice the component parts, train students to think explicitly about how the component parts fit together, practice blending the pieces into complex skills. With this kind of simple, explicit, step-by-step, cumulative method, the most daunting of complex skills can come within students' grasp.

No comments: