Strategies for Advancing Children's Mathematical Thinking
Fraivillig, Judith, Teaching Children Mathematics
Ms. Smith, a first-grade teacher, asked her students how they could add thirty cents and twenty-seven cents. The students suggested several different solution methods. One student said that they could use the number grid to count. Ms. Smith asked where they should start, and the student said, "At thirty." Together with the student, Ms. Smith counted twenty-seven more squares on the grid. After finishing, Ms. Smith asked whether the students knew a shortcut. Another student said to start on 27 and count by tens to end at 57.
In this classroom example, the teacher encouraged the students to talk about how they might solve an addition problem, helped one student execute his method, and challenged students to consider alternative methods. This kind of interaction with students stands in sharp contrast with conventional mathematics teaching in which a teacher might ask a closed question, such as "If you have thirty cents and twenty-seven cents, how much do you have altogether?" Here, Ms. Smith engaged the children in mathematical thinking and generated mathematical discussion in the classroom. Her teaching exemplifies instructional ideas presented in the NCTM's Standards documents (1989, 1991, 2000).
What can we learn from this kind of teaching? How can teachers foster problem-solving skills in children? How can teachers advance children's thinking while students are engaged in mathematical inquiry? These questions have no easy answers. Fortunately, humans are naturally adept at learning from examples. By studying examples of effective instruction, we can begin to define instructional methods that have proved successful for other classroom teachers (Schifter 1996). To successfully apply lessons from these models, we must answer two principal questions: (1) What are the characteristics of effective teaching? (2) What general principles of instruction help children make sense of mathematics?
In this article, the practice of one teacher who masterfully engages students in mathematics learning serves as an example from which we can draw generalizations. The general principles of this kind of instruction are organized into a pedagogical framework that can guide other teachers as they move toward providing instruction that focuses on children's thinking. This "Advancing Children's Thinking" (ACT) framework can help teachers design and implement instruction to make mathematics personally meaningful for children. The ACT framework also establishes a structure for the often-complex interactions that occur when teachers and students grapple with real mathematics problems.
Background of the ACT Framework
The ACT framework was synthesized from an in-depth analysis of observed and reported data from one first-grade teacher who uses the Everyday Mathematics (EM) curriculum. This activity-oriented curriculum for the elementary grades draws on the child's "rich store of mathematics understanding and information" (Bell and Bell 1995, p. iii). The data used in this article were a subset of a larger study investigating the implementation of the EM curriculum among eighteen first-grade teachers and a longitudinal study charting the mathematics achievement of a group of students. Priscilla Smith, who allowed her real name to be used for this article, emerged as being distinct from the sample of teachers in the studies because of her ability to engage children in mathematical problem solving and to foster productive classroom discourse about complex mathematical issues.
What made Ms. Smith's instruction so effective? Three aspects of Ms. Smith's teaching contributed to her effectiveness: (1) her ability to elicit children's solution methods, (2) her capacity to support children's conceptual understanding, and (3) her skill at extending children's mathematical thinking. These related teaching components and the supportive classroom climate in which they were used form the basis of the ACT framework (see fig. …