In an attempt to improve student’s mathematical reasoning and understanding, recent research has focused on word problems rather than arithmetic algorithms. Al Rudnitsky, Sue J.M. Freeman, and Therese Gilbert, Smith College; and Susan Etheredge, Springfield (Massachusetts) College, report on a recent study comparing two types of math instruction for word problems.
Their goals were to field-test instruction intended to help students construct knowledge about addition and subtraction word problems that could be applied to actually solving problems, and also to determine whether students continued to apply this knowledge after the instruction ended.
Instructional techniques for word problems
In developing instructional techniques, Rudnitsky et al. were guided by three ideas. First, they believe, students have to actively construct new knowledge by building on their existing experience. Second, students may do their most productive thinking when thinking aloud and attempting to collaborate with others to solve problems. Third, writing, which requires the application or synthesis of ideas, helps develop understanding.
A total of 401 third and fourth graders in 21 classrooms in six schools participated in the study. Four classes served as controls, following the regular mathematics curriculum that included few word problems. They received no explicit instruction in word-problem solving.
The rest of the classes were randomly assigned to one of two experimental groups. The first experimental group was instructed in a writing-to-learn approach developed by these researchers. This “writing” approach involved working in pairs, studying the different types of word problems, and writing and discussing their own word problems.
The second experimental group was taught a currently used approach in which students are shown samples of different kinds of word problems and taught a step-by-step procedure to follow. Problem-solving tips included stating the problem in their own words, drawing a picture or making a list of information, deciding what the question is and which information to use, writing a number sentence, solving the problem, checking the work and deciding if the answer makes sense. Students in this “solving” group received the most practice in actually solving word problems.
Increasing concern about teaching mathematical thinking and problem solving has led to the development of a writing-to-learn approach to math. In previous studies, students have been asked to focus on key words and relationships through rewriting problems in their own words. Researchers have also studied having students write their own problems. In some studies, students worked with peers and were encouraged to think aloud and question one another, defending and refining their thinking. Rudnitsky et al. built on this previous research, using writing to improve math reasoning but adding to it a structured presentation of problem types. They sought to determine if learning problem structure through writing enhances children’s problem solving.
All 21 classes took a pretest, a post-test and a retention test 10 weeks after instruction ended. All teachers attended training workshops, and during the experiment, researchers met with teachers outside of school hours every third day. Researchers provided each day’s plan and materials but were not present during instruction.
Lessons in the cooperative, writing-to-learn classes were designed to engage students in creating math stories and problems. Instructions were detailed, and teachers were told to push students to think and question. The goal was to foster math discussions. Teachers paired students on the basis of social compatibility. They sought variation but not great disparity in math skills when pairing students.
Students were told that math problems came from the real world. A math story was defined as any story that included quantities or amount — any story that had numbers in it. The teachers used sample stories to point out differences and to define three types: the “something happens” or change story, the “altogether” or combine story and the “compare” story.
Working with their partners, children then wrote two examples of each kind of math story. After this, students learned that math problems could be written from each math story. Students were challenged to write two problems for each of their stories. Eventually, instruction included stories with extra information and problems that required more than one operation. Students practiced writing complex stories and problems using extraneous information and multi-step problem-solving.
The solving treatment was designed to reflect current instructional practice in elementary word-problem solving. Students were taught a procedure to address word problems. Instruction began with single-operation problems and progressed in difficulty through various types that eventually included extra information and multi-step solutions.
Problem-solving tips were posted in the room. Students worked both independently and collaboratively on these problems. Teachers modeled a variety of solution strategies and systematically directed students’ attention to the problem-solving steps. Teachers stressed that the focus was on problem solving, not on reading comprehension or computation, They provided all the reading and calculation help students needed.
The pretest, post-test and retention test were equivalent field-tested forms of the same test. Each test contained six word problems worth one point each. If the correct operations were performed, the child received one point even if computation errors were made. No partial credit was given.
Both experimental groups scored higher than the control group, but the “writing” group outperformed the “solving” group. Although the writing treatment stressed writing rather than solving problems, it had a strong positive effect on problem solving. In addition, the method that taught problem structure using writing and student collaboration continued to have a powerful effect over time. After 10 weeks, the writing group’s scores rose on the retention test, while the scores of the solving and control groups declined.
Researchers noted a significant gender difference. The superiority of the writing treatment was particularly apparent with boys. Girls scored equally well in both writing and solving treatments. Boys in the writing treatment, however, significantly outperformed boys in the solving treatment.
Teacher training needed
Writing, especially collaborative writing, is a slow and laborious process and therefore, only a relatively small number of problems were written and analyzed in the writing treatment. Despite this, these students showed superior problem-solving skills.
It should be noted that the writing treatment was demanding for teachers. Most had never used writing in math instruction and were uncomfortable as they learned the process along with their students. The researchers acknowledge that training and ongoing support are necessary to develop teachers’ skills and confidence with this new approach.
Rudnitsky et al. caution that their study is limited by the narrow set of third-grade problems studied, which are considerably simpler than the open-ended, complex problems of everyday experience.
“Learning To Solve Addition And Subtraction Word Problems Through A Structure-Plus-Writing approach Journal for Research in Mathematics Education Volume 26, Number 5, November 1995 pp.467-486.
Published in ERN January/February 1996 Volume 9 Number 1