Algorithms in Everyday Mathematics
Algorithmic Thinking
Mathematics advances in part through the
development of efficient procedures that reduce
difficult tasks to routine exercises that can be carried out without effort of
thought. Alfred North
Whitehead expressed this idea memorably in his book, An Introduction to
Mathematics (1911):
“It is a profoundly erroneous truism, repeated by all copy books and by eminent
people when
they are making speeches, that we should cultivate the habit of thinking of what
we are doing.
The precise opposite is the case. Civilization advances by extending the number
of important
operations which we can perform without thinking about them” (p. 61).
An effective algorithm can be used to efficiently
solve an entire class of problems, without
having to think through each problem from first principles. Knowing algorithms
increases
students’ mathematical power , which is a principal goal of school mathematics
(NCTM, 1989).
The approach described in this paper — invented procedures followed by
alternative algorithms,
with focus algorithms as a backup and a basis for further work — will produce
students who
understand their methods and can carry them out proficiently so that they can
think about more
important things, such as why they are doing what they are doing and what their
results mean.
The approach improves students’ mental arithmetic skills , helps them understand
the operations,
and develops sound number sense , including a good understanding of place value .
The emphasis
on multiple solutions , including both inventing new procedures and making sense
of others’
inventions, encourages the belief that mathematics is creative and sensible. In
Everyday
Mathematics, accordingly, an increase in mathematical power through algorithmic
proficiency is
achieved at the same time that other important objectives are being met.
The authors of Everyday Mathematics have also found that the study of
paper-and-pencil
computational algorithms can be valuable for developing algorithmic thinking in
general. For
this reason, explicit discussions of algorithms occur in lessons devoted to
computation.
Algorithmic and procedural thinking includes:
• understanding specific algorithms or procedures
provided by other people,
• applying known algorithms to everyday problems,
• adapting known algorithms to fit new situations,
• developing new algorithms and procedures when necessary, and
• recognizing the limitations of algorithms and procedures so they are not
used
inappropriately.
By studying computational algorithms, students can
learn things that will carry over to other
areas of their lives. More and more, people need to apply algorithmic and
procedural thinking in
order to operate technologically advanced devices. Algorithms beyond arithmetic
are
increasingly important in theoretical mathematics, in applications of
mathematics, in computer
science, and in many areas outside of mathematics.
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