Box 6–11 A Common U.S. Algorithm for Multidigit Multiplication
algorithms has also been reported.75 The data are still insufficient, however, to permit firm conclusions about students’ learning progressions in multidigit multiplication.
Nevertheless, it is useful to examine algorithms students are expected to learn and to consider alternatives that might facilitate understanding. Standard multiplication and division algorithms used in the United States are complex procedures in which multiplying alternates with adding or subtracting (see Box 6–11). In these algorithms the meaning and scaffolding provided by substeps have been sacrificed for efficiency. The algorithms use alignment of place value to keep the steps organized without requiring the student to understand what is actually happening with the ones, tens, hundreds, and so on. Algorithms that might be more accessible to students, and still generalizable and fairly efficient, are presented and discussed below.
Arrays are powerful representations of multiplication. An array or area model is shown on the left in Box 6–12. Such a model provides initial support for the crucial understanding of the effects of multiplying by 1, 10, and 100 (shown by arrows and products around the array). It also shows clearly how all of the tens and ones digits in 46 and 68 are multiplied by each other and then added. The sizes of the resulting rectangles indicate the sizes of these various products (sometimes referred to as partial products). The abbreviated array model (shown on the right in Box 6–12) can be drawn later when the students clearly understand the effects of multiplying by tens and by ones. This abbreviated model summarizes the steps in multidigit multiplication, and the separation into tens and ones facilitates finding the partial products.