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How People Learn: Brain, Mind, Experience, and School
model as a way for representing the key properties of numbers, particularly those of magnitude and direction. Reading Deborah Ball’s description of her deliberations, one is struck by the complexity of selecting appropriate models for particular mathematical ideas and processes. She hoped that the positional aspects of the building model would help children recognize that negative numbers were not equivalent to zero, a common misconception. She was aware that the building model would be difficult to use for modeling subtraction of negative numbers.
Deborah Ball begins her work with the students, using the building model by labeling its floors. Students readily labeled the underground floors and accepted them as “below zero.” They then explored what happened as little paper people entered an elevator at some floor and rode to another floor. This was used to introduce the conventions of writing addition and subtraction problems involving integers 4−6=−2 and −2+5=3. Students were presented with increasingly difficult problems. For example, “How many ways are there for a person to get to the second floor?” Working with the building model allowed students to generate a number of observations. For example, one student noticed that “any number below zero plus that same number above zero equals zero” (Ball, 1993:381). However, the model failed to allow for explorations for such problems 5+(−6) and Ball was concerned that students were not developing a sense that −5 was less than −2—it was lower, but not necessarily less. Ball then used a model of money as a second representational context for exploring negative numbers, noting that it, too, has limitations.
Clearly, Deborah Ball’s knowledge of the possible representations of integers (pedagogical content knowledge) and her understanding of the important mathematical properties of integers were foundational to her planning and her instruction. Again, her goals related to developing students’ mathematical authority, and a sense of community also came into play. Like Lampert, Ball wanted her students to accept the responsibility of deciding when a solution is reasonable and likely to be correct, rather than depending on text or teacher for confirmation of correctness.
The work of Lampert and Ball highlights the role of a teacher’s knowledge of content and pedagogical content knowledge in planning and teaching mathematics lessons. It also suggests the importance of the teacher’s understanding of children as learners. The concept of cognitively guided instruction helps illustrate another important characteristic of effective mathematics instruction: that teachers not only need knowledge of a particular topic within mathematics and knowledge of how learners think about the particular topic, but also need to develop knowledge about how the indi-