Algorithms in Everyday Mathematics

ability to add strings of single-digit numbers, and yet it is more efficient than the traditional long

multiplication algorithm for all but the simplest multidigit problems. Students are urged to

6

experiment with various methods for each operation in order to become proficient at using at

least one alternative.

The alternative-algorithms phase of algorithm development has significant advantages:

• A key belief in Everyday Mathematics is that problems can (and should) be solved in more

than one way. This belief in multiple solutions is supported by the alternative-algorithms

approach to developing computational proficiency.

• Providing several alternative algorithms for each operation affords flexibility. A one-sizefits-

all approach may work for many students, but the goal in Everyday Mathematics is to

reach all students. One algorithm may work well for one student, but another algorithm may

be better for another student.

• Different algorithms are often based on different concepts, so studying several algorithms for

an operation can help students understand the operation.

• Presenting several alternative algorithms gives the message that mathematics is a creative

field. In today's rapidly changing world, people who can break away from traditional ways of

thinking are especially valuable.

Teaching multiple algorithms for important operations is common in mathematics outside the

elementary school. In computer science, for example, alternative algorithms for fundamental

operations are always included in textbooks. An entire volume of Donald Knuth's monumental

work, The Art of Computer Programming (1998), is devoted exclusively to sorting and

searching. Knuth presents many inefficient sorting algorithms because they are instructive.

Focus Algorithms

The authors of Everyday Mathematics believe that the invented-procedures/alternativealgorithms

approach described above is a radical improvement over the traditional approach to

developing computational proficiency. The Everyday Mathematics approach is based on decades

of research and was refined during extensive fieldtesting. Student achievement studies indicate,

moreover, that when the approach is properly implemented, students do achieve high levels of

computational proficiency (Carroll, 1996, 1997; Carroll & Porter, 1997, 1998; Fuson, Carroll, &

Drueck, 2000; Carroll, Fuson, & Diamond, 2000; Carroll & Isaacs, in press).

In the second edition of Everyday Mathematics, the approach described above is extended in one

significant way: For each operation, one of the several alternative algorithms is identified as a

focus algorithm. All students are expected to learn the focus algorithms eventually, although, as

usual in Everyday Mathematics, proficiency is expected only after multiple exposures over