Algorithms in Everyday Mathematics

common basis for further work. All students are expected to learn the focus algorithms at some

point, though, as always in Everyday Mathematics, students are encouraged to use whatever

method they prefer when they solve problems.

The following sections describe this process in more detail. Note, however, that although the

basic approach is similar across all four operations, the emphasis varies from operation to

operation because of differences among the operations and differences in students' background

knowledge. For example, it is easier to invent efficient procedures for addition than for division.

There is, accordingly, less expectation that students will devise efficient procedures for solving

multidigit long division problems than that they will succeed in finding their own good ways to

solve multidigit addition problems.

Invented Procedures

When they are first learning an operation, Everyday Mathematics students are asked to solve

problems involving the operation before they have developed or learned systematic procedures

for solving such problems. In second grade, for example, students are asked to solve multidigit

subtraction problems. They might solve such problems by counting up from the smaller to the

larger number, or by using tools such as number grids or base-10 blocks, or they may use some

other strategy that makes sense to them. This stage of algorithm development may be called the

invented procedures phase.

To succeed in devising effective procedures, students must have a good background in the

following areas:

• Our system for writing numbers. In particular, students need to understand place value.

• Basic facts. To be successful at carrying out multistep computational procedures,

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students need proficiency with the basic arithmetic facts.

• The meanings of the operations and the relationships among operations. To solve 37 -

25, for example, a student might reason, "What number must I add to 25 to get 37?"

Research indicates that students can succeed in inventing their own methods for solving basic

computational problems (Madell, 1985; Kamii & Joseph, 1988; Cobb & Merkel, 1989; Resnick,

Lesgold, & Bill, 1990; Carpenter, Fennema, & Franke, 1992). Inventing procedures flourishes

when:

• the classroom environment is accepting and supportive;

• adequate time for experimentation is allotted;

• computational tasks are embedded in real-life contexts; and

• students discuss their solution strategies with the teacher and with one another.

The discussion of students' methods is especially important. Through classroom discussion,

teachers gain valuable insight into students' thinking and progress, while students become more