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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 ex pressed this idea memorably in his book, An Introduction to Mathematics (1911):
“It is a profoundly err oneous 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
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 increases
students’ mathematical power, which is a principal goal of school mathematics (NCTM, 1989).
The approach described in this paper — invented procedures fol lowed 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 techno logically 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.

References

Baroody, A. J., & Ginsburg, H. P. (1986). The relationship between initial meaning and
mechanical knowledge of arithmetic. In J. Hiebert (Ed.), Conceptual and procedural
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 knowledge. A
paper presented at the annual meeting of the American Educational Research Association,
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, Student
outcomes in Standards-oriented school mathematics curriculum projects. Hillsdale, NJ:
Erlbaum.

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: 1998
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: 1989
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 Research
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.

Kamii, C., & Joseph, L. (1988). Teaching place value and double - column addition . Arithmetic
Teacher, 35 (6), pp. 48-52.

Knuth, D. E. (1998). The art of computer programming, volume 3: Sorting and searching.
Reading, Massachusetts: Addison-Wesley.

Madell, R. (1985). Children's natural processes. Arithmetic Teacher, 32 (7), pp. 20-22.

National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for
school mathematics. Reston, VA: Author

Resnick, L. B., Lesgold, S., & Bill, V. (1990). From protoquantities to number sense. A paper
prepared for the Psychology of Mathematics Education Conference, Mexico City.

Whitehead, A. N. (1911). An introduction to mathematics. New York: Holt.

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