Problem solving and word problems are universal concerns. Proficiency testing and math anxiety are issues of the day. When I discuss thin paper with students and adults, they express interest, they want to know how to solve word problems, and how to teach the solution of word problems. They tell about their struggles and their successes, many learned after returning to the task; others have yet to return. Their experiences and observations, however, give insight into our teaching technique.
Of course, some of the interest is a desire for a simple solution, a universal technique. This paper is not such an answer. I can only give reassurance. The process for solving word problems is the same as the processes one uses for reading and writing. Those who teach reading advise reading with questions in mind, reviewing what was read, and thinking out the consequences of what was read. Writing instructors advise listing the main points (an outline, perhaps), writing, and then rewriting. The situation with word problems is analogous.
We read the problems with some questions of our own: What is unknown? What is given? What are we looking for? What relationships exist between the unknowns and the knowns? Then we devise a plan, like the writer’s outline. Then we execute it, getting our solution. Next we review what we have done so as to have greater understanding, as in reading, and to make the solution clearer, as in the rewriting.
This paper is aimed at the solution of the typical Algebra problem. It is a coach reminding us of the steps we should follow regularly. As one gains skill with a technique many steps are squeezed into a few, one forgets the struggle of learning and rushes off to the solution and then the next question.
Word problems are not a topic to be taught at one time and then forgotten. Word problems are a theme of the algebra course. They should be taught throughout the year. They motivate manipulative skills. They give a use for knowing 3x5+8x+6 = 11x + 1; see pickanumber.
In summary the paper exists because word problems are seen as important but difficult. The paper is directed to algebra. As far as problem solving is a universal concern it has broad applicability.
The primary motivation for this paper is George Polya’s book
How to Solve It
. What he talks about and demonstrates in his writing is what teachers want to see happen in their classrooms. Since he draws problems from all of mathematics’ especially geometry, the book is not easily read by Algebra I students. However, I do consider it mandatory reading for adults, particularly those who teach the solution of word problems. Teachers should continually refer to it as they cover word problems, treat it as a manual, Just as when teaching oneself a skill. If we try to teach ourselves we read instructions and practice, then reread and practice some more and so on until mastery is achieved. Each time the reading is compared to the experience, giving more insight Problems that arise from practice send us back to the text for answers. Likewise when teaching word problems, try Polya’s technique, then reread. His book reminds us of what we want to do. It is so easy to tell students solutions rather than leading them to solutions by questions.
Polya and other teachers try to put themselves in the students’ position. This is a way to keep asking the questions the students should ask. However, how o we know the positions of students who do not voice their difficulties?
Two recent books give insight into the minds of students who have difficulty with math:
Mind over Math
by Stanley Kogelman and Joseph Warren, and
Overcoming Math Anxiety
by Sheila Tobias. Both books tell of student reactions to word problems. Some find any statement of student math difficulties as an indictment of math teachers. I see these books as case studies showing what goes on in the minds of some of our students, the ones who do not ask questions, who do not share their ideas. When reading these books, especially the quotes from anxious students in
Mind over Math
, see how Polya’s plan should be applied to avoid the students’ difficulties.
I believe that Polya’s fourth step, “Looking Back” is the place to work on student anxieties. Explore the students’ ideas. If an idea can be extended to lead into a solution follow that lead. Be hesitant to reject any student proposal. Rejection may lead to anxiety. Question the student further, get the student to reject the strategy. If a plan is “wrong” it indicates some misunderstanding; go after the misunderstanding.
I found these two books to be painful. How could students miss the point by so much? In their remarks some seemed to say that they partially listened, that they skipped the math part of any reading. All of them needed a system such as Polya’s. Ironically these are adults who have had successin other fields where they follow a similar process, reading and writing.