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 multiplication . . .12 x 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 representation of grouping 10 sets of four butterflies and having 2 jars not in the group; they recognize that 12 x 4 can be thought of as 10 x 4 plus 2 x 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 x 4 plus 6 x 4 produces the same number as 10 x 4 plus 2 x 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 subsequent lessons, students are challenged with problems in which the two-digit number in the multiplication is much bigger and, ultimately, in which both numbers are quite large--28 x 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.