Algorithms in Everyday Mathematics

skilled at communicating mathematics and at understanding and critiquing others' ideas and

methods. Talking about why a method works, whether a method will work in every case, which

method is most efficient, and so on, helps students understand that mathematics is based on

common sense and objective reason, not the teacher's whim. Such discussions lay the

foundations for later formal work with proof.

The invented-procedures approach to algorithm development has many advantages:

• Students who invent their own methods learn that their intuitive methods are valid and that

mathematics makes sense.

• Inventing procedures promotes conceptual understanding of the operations and of base-10

place-value numeration. When students build their own procedures on their prior

mathematical knowledge and common sense, new knowledge is integrated into a meaningful

network so that it is understood better and retained more easily.

• Inventing procedures promotes proficiency with mental arithmetic. Many techniques that

students invent are much more effective for mental arithmetic than standard paper-and-pencil

algorithms. Students develop a broad repertoire of computational methods and the flexibility

to choose whichever procedure is most appropriate in any particular situation.

• Inventing procedures involves solving problems that the students do not already know how to

solve, so they gain valuable experience with non-routine problems. They must learn to

manage their resources: How long will this take? Am I wasting my time with this approach?

Is there a better way? Such resource management is especially important in complex

problem solving. As students devise their own methods, they also develop persistence and

confidence in dealing with difficult problems.

• Students are more motivated when they don't have to learn standard paper-and-pencil

algorithms by rote. People are more interested in what they can understand, and students

generally understand their own methods.

• Students become adept at changing the representations of ideas and problems, translating

readily among manipulatives, words, pictures, and symbols. The ability to represent a

problem in more than one way is important in problem solving. Students also develop the

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ability to transform any given problem into an equivalent, easier problem. For example, 32 -

17 can be transformed to 35 - 20 by adding 3 to both numbers.

Another argument in favor of the invented-procedures approach is that learning a single standard

algorithm for each operation, especially at an early stage, may actually inhibit the development

of students' mathematical understanding. Premature teaching of standard paper-and-pencil

algorithms can foster persistent errors and buggy algorithms and can lead students to use the