Helping third-grade students extend their understanding of numbers from the natural numbers to the integers is a challenge undertaken by an- other teacher-researcher. Deborah Ball’s work provides another snapshot of teaching that draws on extensive subject content and pedagogical content knowledge. Her goals in instruction include “developing a practice that respects the integrity both of mathematics as a discipline and of children as mathematical thinkers” (Ball, 1993). That is, she not only takes into account what the important mathematical ideas are, but also how children think about the particular area of mathematics on which she is focusing. She draws on both her understanding of the integers as mathematical entities (subject-matter knowledge) and her extensive pedagogical content knowl- edge specifically about integers. Like Lampert, Ball’s goals go beyond the boundaries of what is typically considered mathematics and include devel- oping a culture in which students conjecture, experiment, build arguments, and frame and solve problems—the work of mathematicians.
Deborah Ball’s description of work highlights the importance and diffi- culty of figuring out powerful and effective ways to represent key math- ematical ideas to children (see Ball, 1993). A wealth of possible models for negative numbers exists and she reviewed a number of them—magic pea- nuts, money, game scoring, a frog on a number line, buildings with floors above and below ground. She decided to use the building model first and money later: she was acutely aware of the strengths and limitations of each
BOX 7.2
How Many Altogether?
The teacher begins with a request for an example of a basic computation.
Teacher: Can anyone give me a story that could go with this multipli- cation . . .12 × 4?
Jessica: There were 12 jars, and each had 4 butterflies in it.
Teacher: And if I did this multiplication and found the answer, what would I know about those jars and butterflies?
Jessica: You’d know you had that many butterflies altogether.
The teacher and students next illustrate Jessica’s story and construct a procedure for counting the butterflies.
Teacher: Okay, here are the jars. The stars in them will stand for butterflies. Now, it will be easier for us to count how many butterflies there are altogether, if we think of the jars in groups. And as usual, the mathematician’s favorite number for thinking about groups is? [Draw a loop around 10 jars.]
Sally: 10.
The lesson progresses as the teacher and students construct a pictorial represen- tation of grouping 10 sets of four butterflies and having 2 jars not in the group; they recognize that 12 × 4 can be thought of as 10 × 4 plus 2 × 4. Lampert then has the children explore other ways of grouping the jars, for example, into two groups of 6 jars.
The students are obviously surprised that 6 × 4 plus 6 × 4 produces the same number as 10 × 4 plus 2 × 4. For Lampert, this is important information about the students’ understanding (formative assessment—see Chapter 6). It is a sign that she needs to do many more activities involving different groupings. In subse- quent lessons, students are challenged with problems in which the two-digit num- ber in the multiplication is much bigger and, ultimately, in which both numbers are quite large—28 × 65. Students continue to develop their understanding of the principles that govern multiplication and to invent computational procedures based on those principles. Students defend the reasonableness of their procedures by using drawings and stories. Eventually, students explore more traditional as well as alternative algorithms for two-digit multiplication, using only written symbols.
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 model- ing 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 plan- ning 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 depend- ing on text or teacher for confirmation of correctness.