# Developing Math Automaticity in Learning Handicapped Children: The Role of Computerized Drill and Practice ARTICLE

Number Sense: Rethinking Arithmetic Instruction for Students with Mathematical Disabilities By: Russell Gersten and David J. Abstract We describe the concept of number sense, an analog as important to mathematics learning as phonemic awareness has been to the reading research developing Math Automaticity in Learning Handicapped Children: The Role of Computerized Drill and Practice ARTICLE. Understanding the concept of number sense and relevant research from cognitive science can help the research community pull together fragmented pieces of earlier knowledge to yield a much richer, more subtle, and more effective means of improving instructional practice.

Although subsequent psychometric studies identified the flaws in their conceptualization, our current understanding of the importance of phonological processing and its contribution to reading development suggests that Kirk and Bateman were at least partly accurate in their analysis. We believe that there may be an analog as important to mathematics learning as phonemic awareness has been to the development of reading. Our goal in this article is to introduce this analog. To accomplish our goal, we briefly review the concept of phonemic awareness and its crucial role in helping students with learning disabilities to learn to read. Our model indicates how the number sense concept provides a sensible middle ground in what is becoming an increasingly heated controversy about how to teach mathematics. In this article, we draw analogies between phonological awareness and number sense. We also draw analogies between earlier research on ways to remediate mathematical disabilities and earlier research on reading disabilities.

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The goal here is to provide a brief overview of phonological awareness concepts and number sense before introducing the concept of number sense, rather than to attempt to provide a comprehensive review of either topic. As a consequence, sustained efforts were critical to transform students with learning disabilities into fluent readers. IQ, readiness test scores, or socioeconomic level. Phonemic awareness is the insight that words are composed of sounds. Phonemic awareness is not always easy for children to obtain. Although most young children have no difficulty segmenting words into syllables, many find it very difficult to segment at the phoneme level.

Current thinking about special education reading instruction, both remedial and preventive, now invariably notes the importance of explicit instruction in phonemic awareness skills. For example, we have learned from research that explicit training in sound blending is useful to students. The concept of phonemic awareness has helped connect the pieces of the puzzle of reading acquisition. Phonemic awareness provides greater precision and helps to inform instruction in a way that earlier concepts of phonics instruction, which rarely included instruction in either blending or segmentation, did not. Just as our understanding of phonemic awareness has revolutionized the teaching of beginning reading, the influence of number sense on early math development and more complex mathematical thinking carries implications for instruction.

Those who plan mathematics instruction for young children fail to take fully into account that, along with increased competence and fluency with basic addition and subtraction facts, children also develop- or fail to develop- a number sense. Number sense is difficult to define but easy to recognize. Students with good number sense can move seamlessly between the real world of quantities and the mathematical world of numbers and numerical expressions. They can invent their own procedures for conducting numerical operations. They can represent the same number in multiple ways depending on the context and purpose of this representation. Most children acquire this conceptual structure informally through interactions with parents and siblings before they enter kindergarten. For example, one child may enter school knowing that 8 is 3 bigger than 5, whereas a peer with less well-developed number sense may know only that 8 is bigger than 5.

This number sense not only leads to automatic use of math information, but also is a key ingredient in the ability to solve basic arithmetic computations. However, more than 100 basic addition facts must be memorized to automaticity before students can experiment with this type of interesting problem. 1970’s has provided evidence of a preverbal component to number sense. By age 3 or 4 years, most children can compare two small numbers for size and determine which is larger and which is smaller. Number sense is facilitated by environmental circumstances. As with phonemic awareness, the environmental conditions that promote number sense are, to some extent, mediated by informal teaching by parents, siblings, and other adults. The notion of number sense has enjoyed intuitive, almost romantic, support previously.

It is even common to hear educators comment that some students are “just good with numbers” or generalize about the mathematics prowess of certain groups of students. We contend that number sense is more than a common parlance notion. The number sense construct that is the focus of our attention here seems analogous to phonemic awareness in several ways. We want to stress that, like all analogies, this one is far from perfect. There are numerous differences between the development of an understanding of and proficiency in mathematics and the development of the ability to read with understanding. Similar to phonemic awareness instruction, we are not sure of the best approach to use to teach number sense. But this “brute force” approach made mathematics unpleasant, perhaps even punitive, for many.

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Researchers explored the devastating effects of the lack of automaticity in several ways. Essentially, they argued that the human mind has a limited capacity to process information, and if too much energy goes into figuring out what 9 plus 8 equals, little is left over to understand the concepts underlying multi-digit subtraction, long division, or complex multiplication. This focus on the devastating effects of weak automaticity on the ability to solve problems and understand mathematical concepts is a direct parallel to the reading research of the early 1980s, which demonstrated that students who are slow or plodding decoders tend to be poor comprehenders. Processing moves from a state in which demands are made on a limited attentional resource pool to a state where fewer demands are made on those resources. A sophisticated instructional program was established in which students were provided with individualized daily practice for approximately 10 minutes per day. The program used an interspersal of target facts with already automatic facts to maximize practice time. Practice continued until the student consistently used retrieval as opposed to counting on his or her fingers.

Controlled response times were also used to force students to not rely on counting. The ultimate goal was to “free up” mental resources for performance. Although this implication is not a revelation to many educators, it is in contrast to the approach typically used to teach arithmetic. Other research offers valuable additional information to help guide instruction.

Adults often underestimate the time it takes for a child to consistently use a newly learned mathematical strategy. These are problems that are very difficult to solve without use of the strategy and fairly easy to solve with the strategy. This finding, too, would seem to have important instructional implications for special education. In their comparative research on arithmetic knowledge of low- and middle-income kindergartners, Case et al. The goal of this instruction is in large part, for students to develop and elaborate an integrated schema that centers on a mental number line, allowing students to solve a variety of addition and subtraction problems.

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These preliminary findings are extremely promising and the cornerstones of this approach appear to have relevance for special education mathematics instruction. Early interventions focusing on prenumeracy skills attempt to expose children to experiences lacking in their home or in preschool. For example, parents can help children develop early number sense by asking them to ascend and count four steps and then count and descend two steps. Similarly, parents might ask their children to set the table and count the correct number of place settings. However, many children need explicit, consistent help in understanding the specifics of the system. In other words, they possess very crude levels of phonemic awareness and need help developing the sophisticated awareness necessary for fluent, nonstressful reading.

It is important to note at this point that strategies such as the “min” strategy are not easy to teach. To recall quickly that 8 is bigger than 3, a child must have some factual automaticity. Also, a child needs a sense of numbers to assist in access or automaticity. Examples of these differences have been identified in research. 100 middle school students with learning disabilities. Careful analysis of these tests revealed that more than half the students showed systematic error patterns, many of which revealed limited conceptual understanding of the algorithms and strategies taught to them. The first is a high frequency of procedural errors.

5, children first typically attempt to retrieve the answer from memory. The procedural deficits may disappear in later elementary grades if quality instruction is provided. By way of a brief definition, cognitive psychologists use the term representation to refer to coding information in any form for storage in the brain. A brief discussion of the link between math and reading- related deficits is appropriate. Too often, special education math instruction focuses on mastery of algorithms, repeated practice with limited opportunity for students to explain verbally their reasoning and receive feedback on their evolving knowledge of concepts and strategies.

In other words, special education mathematics instruction continues to focus on computation rather than mathematical understanding. By considering the analogy between phonemic awareness and number sense, it is hoped that we can focus attention on significant, necessary shifts in how mathematics is taught to young children, especially those with learning disabilities or those entering school with limited familiarity of arithmetic concepts. At some point in time, even basic arithmetic facts are problems to be solved by naive learners, Therefore, mere drill and practice on basic math facts will be insufficient for developing students who are competent in mathematics. Even if students are not automatic with basic facts, they still should be engaged in activities that promote the development of number sense and mathematical reasoning. It has been our goal in this article to review research that explicates the number sense construct, helps link it to recurring issues in special education math instruction, and provides some guidelines for further research in special education math instruction and the development of more effective instructional approaches to reduce the difficulties that students with disabilities often experience in mathematics. We envision a wave of research and development on math instruction for students with disabilities that parallels research and development of instructional strategies and approaches related to the concept of phonemic awareness.

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Important first steps are the development and validation of measures that have reasonable predictive validity. To date, most of the assessments have been more clinical in nature. Equally important is increased research on effective beginning mathematics instruction using number sense as a construct in both curriculum development and assessment of effectiveness. Also germane are the important findings about the relationship between procedural and conceptual knowledge of Siegler and colleagues based on more than a dozen years of research on young children’s development of arithmetic reasoning. Their findings suggest an intricate relationship between conceptual understanding and consistent use of efficient strategies for computation and problem solving.

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Conceptual and procedural knowledge may develop interactively, with gains in one leading to gains in the other, which in turn trigger new gains in the first. Thus procedural knowledge could also influence conceptual understanding. Under some circumstances, children first learn a correct procedure and later develop an understanding of the concepts underlying it. The authors extend their thanks and appreciation to Scott Baker and Sylvia Smith for their thoughtful feedback on earlier versions of this manuscript. The authors also thank Kate Sullivan, Janet Otterstedt, Jon Gall, and Karie Hume for their editorial assistance. References Click the “References” link above to hide these references.

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