On the Write Path: Improving Communication in an Elementary Mathematics Classroom
Anderson, Melissa A., Little, Deena M., Teaching Children Mathematics
"You have three quarters, one nickel, and one dime. You go to the store and buy a cookie for $0.69. How many nickels do you have left?"
Wide-eyed Bobby stared at his paper, wondering where to begin. He thought, "Should I make a graph, work backward, look for a pattern? My teacher says I have to use a strategy. How do I know which one to use? When I pick one, what do I do next?" Frustrated that he did not know where to start, Bobby chose to add the coins, remembering this technique from recent mathematics lessons on money. Even if he could not answer the question, at least he knew the total amount of money. He thought, "That will be fine. My teacher will be proud of me for finding the correct sum." As he showed his paper to his teacher, his frustration returned when she began to question him about the steps he took to solve the problem. She said, "Under your explanation, you wrote, 'I added the coins to get $0.90. Then I subtracted $0.69 and got $0.21. I know I'm right because I checked my math.' Can you tell me how this answers the original question?" "Oh, no," Bobby thought. "Isn't that the correct answer? I know my math is correct. What else does she want?"
Although Bobby is a composite character based on many of our fourth-grade students, his thinking processes are not wholly fictitious. We have found his problems to be the norm in our own classrooms. As this example proves, young learners in this age group struggle with transitioning from an algorithm to reasoning and justifying.
Why Focus on Communication and Reasoning?
As mathematics teachers in a suburban district mainly composed of middle- to upper-middle-class families, we have discovered that reasoning and communication are our biggest areas of concern in mathematics instruction. Our district requires fourth-grade students to do more justifying of their answers and explaining of the processes they followed. We have found that the students often have trouble expressing their ideas in writing. This was very evident when we attempted to work on problem-solving skills in mathematics. Looking at the students' standardized mathematics test scores, we noticed that the problem-solving and reasoning strand was the greatest area of weakness for the vast majority of our students. The reports showed that 88 percent of the students in our classes had their lowest level of mastery in this strand. This alarming statistic caused us to focus on reasoning and communication in our mathematics instruction.
Problem solving is woven throughout mathematics as well as across the curriculum. Students must learn to reason through situations and develop a problem-solving system that is successful for them. Being able to effectively communicate their solutions and defend their reasoning are life skills that students need not only in school but also beyond the classroom walls throughout their lives. NCTM's Principles and Standards for School Mathematics (2000) emphasizes these ideas, and we used them to evaluate and improve our problem-solving instruction.
Students' Struggles with Problem Solving
The vision of Principles and Standards includes that "orally and in writing, students communicate their ideas and results effectively" (NCTM 2000, p. 3). We noticed that our students struggled with effectively communicating their mathematical thinking in problem solving, and we chose to make this our initial area of focus. In previous years, we devoted Friday's mathematics instruction to "problem solvers" activities. This entailed the teacher modeling a strategy and students independently practicing the solving of a similar problem. Students moaned every week as we turned on the overhead. Frankly, most of their teachers had a similar silent reaction. The problem solvers consisted of a one-paragraph problem. The students robotically restated the question, identified the important information, selected a strategy, and showed their work. …