There is probably no topic that spurs discussion, incites action, and stirs controversy among educational stakeholders more than computation in mathematics. This attention is understandable because computation is a core element of mathematics, school curricula (and our collective memories of learning mathematics), and the knowledge that is important for day-to-day living in the real world.
In its decades of work to establish comprehensive and coherent standards for mathematics education, the National Council of Teachers of Mathematics (NCTM) has worked to articulate, and rearticulate, its vision of teaching and learning about computation. The most recent version of NCTM's Standards (NCTM 2000) uses the phrase "computational fluency" to embody the council's ideas. As has occurred with NCTM's previous ideas about computation, numerous questions and concerns have been raised about the meaning and importance of computational fluency. Examining the mathematical, instructional, and learning elements of computational fluency is the goal of this focus issue. We hope that this focus issue will facilitate movement toward understanding computational fluency. Ultimately, giving meaning to this phrase will require the work of countless people, including students, teachers, curriculum developers, mathematicians, teacher educators, and researchers.
The Standards describe computational fluency as a "connection between conceptual understanding and computational proficiency" (NCTM 2000, p. 35). Conceptual understanding of computation is grounded in mathematical foundations such as place value, operational properties, and number relationships (p. 32). Computational proficiency is characterized by accurate, efficient, and flexible use of computation for multiple purposes (p. 152). These characteristics make fluency-a word that most people associate more strongly with language development-useful for conveying the aim of computational performance. Just as the fits and starts of word-by-word reading detract from comprehension of a written passage, so will inaccurate, cumbersome, and poorly learned computational strategies ultimately detract from the making of mathematical meaning (National Research Council 2001, chapter 1). The Standards call for regular experience with meaningful procedures so that students develop and draw on mathematical understanding even as they cultivate computational proficiency (NCTM 2000, p. 87). Balance and connection of understanding and proficiency are essential, particularly for computation to be useful in "comprehending" problem-solving situations. The articles selected for this focus issue present a variety of perspectives on computational fluency that converge around three main themes.
Dimensions of Computational Fluency
Two articles act as "bookends" for our thinking about computational fluency. "Meaning and Skill-Maintaining the Balance," an article first published in a 1956 issue of The Arithmetic Teacher, gives us a view of what might have been called computational fluency in that era, with an emphasis on the connections between understanding and proficiency. In "Computational Fluency, Algorithms, and Mathematical Proficiency: One Mathematician's Perspective," a modern-day mathematician states his view of the links between computational understanding and performance. The common theme in these articles sets the stage for investigating both the content and the pedagogy involved in improving students' computational flexibility, efficiency, and accuracy.
Mathematical Foundations for Developing Computational Fluency
Several articles describe the importance of number relationships, place value, and properties of operations as mathematical foundations for computational fluency. "Toward Computational Fluency in Multidigit Multiplication and Division" highlights the importance of students' understanding of the mathematical components of efficient and accurate algorithms. "Subtraction Strategies from Children's Thinking: Moving toward Fluency with Greater Numbers" describes using mathematical foundations to help students bridge existing strategies with more efficient and accurate ones.
Teaching for Computational Fluency
Several articles address important aspects of effective instruction for computational fluency. "Discussion as a Vehicle for Demonstrating Computational Fluency in Multiplication" explores how teachers can assess students' computational fluency from open discussions of carefully selected sequences of computational problems. "Helping English-Language Learners Develop Computational Fluency" extends the instructional strategy of classroom discussion with suggestions for supporting students who are learning English as a second language. Specific instructional strategies for promoting automaticity and accuracy are described in "When Flash Cards Are Not Enough." "Promoting Meaningful Mastery of Addition and Subtraction" also presents instructional practices that promote the development of computational fluency in the primary grades. Continuing the infusion of effective teaching strategies in the classroom, "Developing Teachers' Computational Fluency: Examples in Subtraction" addresses the preservice preparation of elementary teachers.
In addition to the articles, a pullout poster visually represents the dimensions and mathematical foundations of computational fluency. The back of the poster suggests what to listen and look for to determine students' progress in developing computational fluency.
Acknowledgments
We want to thank everyone who contributed to the creation and completion of this focus issue: people interested in mathematics education who submitted articles, NCTM members who reviewed the articles, TCM Editorial Panel members who helped select the articles, and the staff in Reston who provided the assistance required to produce another focus issue for Teaching Children Mathematics. We hope that reading and discussing the articles in this focus issue will be as professionally rewarding for you as editing this issue was for us.
| [Reference] » View reference page with links |
| References |
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| National Council of Teachers of Mathematics (NCTM). Principles and Standards for School Mathematics. Reston, Va.: NCTM, 2000. |
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| National Research Council. "Looking at Mathematics and Learning." In Adding It Up: Helping Children Learn Mathematics, edited by J. Kilpatrick, J. Swafford, and B. Findell, pp. 1-16. Washington, D.C.: National Academy Press, 2001. A |
| [Author Affiliation] |
| Timothy A. Boerst and Jane F. Schielack For the Editorial Panel |
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