While working with third-, fourth-, and fifth-grade teachers in a National Science Foundation--sponsored project designed to enhance the mathematics and science teaching of in-service elementary teachers, we recognized that teaching mathematics problem solving was one of their greatest challenges. Discussions with the teachers revealed that most were using an algorithmic approach to problem solving with an emphasis on facts, rules, and procedures. Their students were being taught to solve word problems in a systematic, single-mode manner. We found that the teachers were most comfortable with the algorithmic approach because that is how they were taught mathematics when they were in school. As one teacher commented, "I was stunned to find out that not everyone worked math problems the same way."
Teachers admitted that the algorithmic approach posed obstacles for their students. Although many students could learn the procedure to solve a particular type of problem, frequently they could not solve related problems. The teachers sensed a lack of conceptual understanding and transfer on the part of the students. One teacher understood the uncertainty that her students faced in working word problems: "My problem in math is that I learn it only one way, and then I can't apply what I've learned to another situation. My [students] do this, too."
On the basis of our own learning and teaching experiences and published evidence that problem solving is facilitated when students are encouraged to visualize (Hembree 1992; Lester 1982; Wheatley 1991; Wheatley and Brown 1994; van Essen and Hamaker 1990; Yackel and Wheatley 1990; Yancey, Thompson, and Yancey 1989), we recommended that the teachers use visualization to help students solve mathematics. Noting differences in visualization techniques reminded teachers that more than one way to solve a mathematics problem is possible.
The Teachers and the Visualization Process
To introduce teachers to the power of visualization, we presented them with a problem and asked them to solve it.
A snail is at the bottom of a 10-foot well. Every day he crawls up 3 feet, but at night slips down 2 feet. How many days will it take for him to get out of the well?
Without exception, all the teachers drew a picture to solve the problem, and although all the pictures were different, the teachers arrived at the same answer. Several teachers commented on the differences in the individual drawings, which prompted another teacher to say, "Just because we're all teachers doesn't mean we think alike. Besides, we all got the same answer." To encourage the teachers to draw pictures as a part of the problem-solving process, we presented many problems for which an algorithm was not obvious. After each problem set, the teachers shared and analyzed their drawings and solution strategies.
Gradually, the discussion shifted from the teachers' problem-solving ability to that of their students. The teachers agreed that visualization could be a powerful tool to enhance their students' problem-solving ability but questioned their ability to teach students to use this tool. Two teachers noted that they had tried to teach their students this method, but neither of them could decide on an effective teaching strategy. Others doubted their ability to teach students how to represent their mental images through the construction of a picture, and some were skeptical that elementary students could visualize mathematics problems at all. One teacher faced an additional obstacle: she had stressed neatness so much that those students who used drawings would erase their work before turning in their mathematics papers because they were fearful of being graded down for messiness. As one teacher said rather succinctly, "It's not enough just to tell my students to draw a picture." The teachers wanted specific how-to s.
Rather than offer suggestions that we had …