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 ex pressed this idea memorably in his book, An Introduction to
“It is a profoundly err oneous truism , repeated by all copy books and by eminent
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 ope rations 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
students’ mathematical power, which is a principal goal of school mathematics
The approach described in this paper — invented procedures fol lowed by
with focus algorithms as a backup and a basis for further work — will produce
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
The approach improves students’ mental arithmetic skills , helps them understand
and develops sound number sense, including a good understanding of place value.
on multiple solutions , including both inventing new procedures and making sense
inventions, encourages the belief that mathematics is creative and sensible. In
Mathematics, accordingly, an increase in mathematical power through algorithmic
achieved at the same time that other important objectives are being met.
The authors of Everyday Mathematics have also found that the study of
computational algorithms can be valuable for developing algorithmic thinking in
this reason, explicit discussions of algorithms occur in lessons devoted to
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
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 techno logically advanced devices. Algorithms beyond arithmetic
increasingly important in theoretical mathematics, in applications of
mathematics, in computer
science, and in many areas outside of mathematics.
Baroody, A. J., & Ginsburg, H. P. (1986). The relationship
between initial meaning and
mechanical knowledge of arithmetic. In J. Hiebert (Ed.), Conceptual and
knowledge: The case of mathematics. Hillsdale, NJ: Erlbaum.
Carpenter, T. P., Fennema, E., & Franke, M. L. (1992).
Cognitively guided instruction: Building
the primary mathematics curriculum on children's informal mathematical
paper presented at the annual meeting of the American Educational Research
April 1992, San Francisco.
Carroll, W. M. (1996). Use of invented algorithms by
second graders in a reform mathematics
curriculum. Journal of Mathematical Behavior, 15: 137-150.
Carroll, W. M. (1997). Mental and written computation:
Abilities of students in a reform-based
curriculum. The Mathematics Educator, 2 (1): 18-32.
Carroll, W., & Isaacs, A. (in press). Achievement of
students using the University of Chicago
School Mathematics Project’s Everyday Mathematics. In S. Senk & D. Thompson,
outcomes in Standards-oriented school mathematics curriculum projects.
Carroll, W., & Porter, D. (1997). Invented algorithms can
develop meaningful mathematical
procedures. Teaching Children Mathematics 3(7): 370-74.
Carroll, W., & Porter, D. (1998). Alternative algorithms
for whole-number operations. In L. J.
Morrow (Ed.), The teaching and learning of algorithms in school mathematics:
yearbook (pp. 106-114). Reston, VA: National Council of Teachers of Mathematics.
Carroll, W., Fuson, K., & Diamond, A. (2000). Use of
student-constructed number stories in a
reform-based curriculum. Journal of Mathematical Behavior, 19: 49-62.
Cobb, P. (1985). Two children 's anticipations, beliefs,
and motivations. Educational Studies in
Mathematics, 16: 111-126.
Cobb, P., & Merkel, G. (1989). Thinking strategies:
Teaching arithmetic through problem
solving. In P. Trafton (Ed.), New directions for elementary school mathematics:
yearbook. Reston, VA: National Council of Teachers of Mathematics.
Fuson, K., Carroll, W. M., & Drueck, J. V. (2000)
Achievement results for second and third
graders using the Standards-based curriculum Everyday Mathematics. Journal for
in Mathematics Education 31 (3): 277-295.
Greenwood, I. (1729). Arithmeticks. Boston: T. Hancock.
Hiebert, J. (1984). Children's mathematical learning: The
struggle to link form and
understanding. Elementary School Journal, 84 (5): 497-513.
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