When a teacher encourages a classroom environment in which students share various solution strategies with each other in written, oral, and visual forms, excellent opportunities arise for students to learn from their peers. Equally important, the teacher is able to develop an understanding of the students' thinking and to use that knowledge to drive the curriculum. This classroom environment is as important at the preschool and kindergarten levels as at other grade levels. However, finding appropriate problem-solving activities at the preschool and primary grades may be more challenging because of the limited knowledge, experiential background, and writing skills of younger students. However, the following scenario illustrates that kindergarten classrooms can become rich environments to develop each student's mathematical power (NCTM 1991).
A problem-solving environment can grow out of contextual situations. If a student does not have a seat, the situation becomes a problem-solving activity with the question, "How can you solve that?" If a student cannot reach the light switch, the same question is asked. In these situations, many solutions are usually suggested, and students soon realize that they must think and that their thinking is valued. Problem-solving activities that focus on mathematical ideas can be word problems centering on classroom routines, such as the distribution of drawing paper or division of crackers at snack time; games requiring the player to find a solution to move a game piece forward (Kamii 1985); and situations involving nontraditional word problems that cannot be solved algorithmically. The animal problem that follows is an example of the latter problem type.
There are some animals behind the fence. You can see four heads sticking out from the top of the fence and twelve legs sticking out from under the fence. What animals could be behind the fence?
This animal word problem presents an excellent opportunity to implement several of the Professional Teaching Standards (NCTM 1991) for teaching mathematics in kindergarten while linking mathematics instruction to a unit on different categories of animals.
Using Physical Space and Materials to Facilitate Learning
The project began with the children helping to decorate a classroom bulletin board that contained the problem. A fence was painted by a group of students and stapled to the bulletin board. Four ambiguous heads appeared behind the top of the fence and twelve ambiguous legs were shown under the fence. Children cut green paper grass and placed it along the bottom of the board. The title read, "Who's behind the fence?"
Task, Discourse, and Analysis of Student Learning
The problem was used with a small group of students each. day during learning-center time. The students initially behaved concretely (Piaget 1963); they wanted to take the fence down to see what animals were behind it. They learned that they were to produce the animals hiding behind the fence by using the number of heads and legs they could see. The students were invited to use manipulatives or paper and markers to help solve the problem. They were asked to record their ideas in writing so that they could be shared with others and eventually be placed on the bulletin board.
While the students solved the problem, the teacher observed their work, made inquiries, and listened to what they were saying. This problem presented a window into the variations that can exist in children's thinking, as illustrated by Kemoy's, Tiffany's, and Brian's solutions.
Kemoy drew pictures of animals with little detail except for their heads and legs. He began by drawing a cat. The next animal he drew was a dog. He was asked how many heads and legs he had so far and how many he still needed. He knew he needed two more heads and four more legs. Being very familiar with the numbers involved, Kemoy immediately divided the remaining number of legs in half so that each remaining head would have two legs. …