Academic journal article Australian Primary Mathematics Classroom

From Here to There: The Path to Computational Fluency with Multi-Digit Multiplication: Janette Bobis Highlights the Critical Links between Number Sense and a Child's Fluency with Mental and Written Computation When Learning How to Perform Multi-Digit Multiplication

Academic journal article Australian Primary Mathematics Classroom

From Here to There: The Path to Computational Fluency with Multi-Digit Multiplication: Janette Bobis Highlights the Critical Links between Number Sense and a Child's Fluency with Mental and Written Computation When Learning How to Perform Multi-Digit Multiplication

Article excerpt

Drawing upon research, theory, classroom and personal experiences, this paper focuses on the development of primary-aged children's computational fluency with multi-digit multiplication. Getting children from "here" (current strategy use) to "there" (a more efficient strategy) is often not a straight-forward path. The critical links between number sense and a child's ability to perform mental and written computation with ease are examined.

Many readers will know the story of the famous mathematician Johann Carl Friedrich Gauss (1777-1855). As a young boy he was prone to daydream in class. One day his teacher decided to punish him for not paying attention. He was asked to add all the numbers from 1 to 100. Much to the annoyance of the teacher, young Carl was able to derive the correct answer in seconds. Fortunately for Carl, he knew a short-cut. He realised that adding pairs of numbers (e.g., 1 + 100, 2 + 99, etc.) all equalled the same number: 101. He figured that there were 50 such pairs, so calculated the total to equal 50 x 101 or 5050.

Recently I related this story to a group of primary school teachers. One teacher immediately asked, "But who taught him that?" This question sparked a discussion about the critical relationship between a person's understanding of mathematics and their computational fluency. The teachers agreed that Carl's in-depth understanding of mathematics enabled him to see patterns and relationships that made the computation more manageable, but that his knowledge of basic facts and the fluency with which he could compute were equally important. The teachers concluded that understanding without fluency can inhibit the problem solving process.

This paper focuses on the development of primary-aged children's computational fluency. It emphasises the critical links between number sense and a child's ability to perform mental and written computation. The case of multi-digit multiplication is used to illustrate these important links.

Computational fluency: Number sense and the standard algorithm

The idea of teaching mathematics for understanding and for meaningful learning to occur has been advocated for over half a century (Brownell, 1935). However, it was not until the 1980s that the term "number sense" was first used to refer to those who had a deep understanding of numbers. The focus on number sense is manifested in the recent and ongoing emphasis in international curriculum and policy documents on mental computation (e.g., Australian Education Council, 1991; National Council of Teachers of Mathematics, 2000). Research has shown that those who are good at mental computation possess a well-developed sense of number (McIntosh & Dole, 2000).

The increased emphasis on mental computation and number sense has seen a corresponding de-emphasis in curricula on standard algorithms. An algorithm is a specified multi-step procedure that produces an answer for any given set of problems and is characterised by long-term practice. While still recognised as important, some Australian state syllabus documents have delayed the introduction of standard algorithms for around two years to allow a focus on mental strategies for as long as possible (e.g., Board of Studies, New South Wales [BOSNSW], 2002). The worry with an early emphasis on standard algorithms is that students will shift their focus to executing convenient procedures rather than on understanding the mathematics.

A concern is that educators will view the development of number sense and fluency in written and mental computation as separate bodies of knowledge requiring separate instruction. In fact, computational fluency, whether employing mental or written methods, and number sense are intertwined and should be developed together. The aim of the following sections is to examine how children develop proficiency in their computational methods while instruction remains focused on learning with understanding. …

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