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Article Excerpt
Abstract. The present study investigated 90 elementary teachers' ability to identify two systematic error patterns in subtraction and then prescribe an instructional focus. Presented with two sets of 20 completed subtraction problems comprised of basic facts, computation, and word problems representative of two students' math performance, participants were asked to examine each incorrect subtraction problem and describe the errors. Participants were subsequently asked which type of error they would address first during math instruction to correct students' misconceptions. An analysis of the data indicated teachers were able to describe specific error patterns. However, they did not base their instructional focus on the error patterns identified, and more than half of the teachers chose to address basic subtraction facts first during instruction regardless of error type. Limitations of the study and implications for practice are discussed.
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According to the Goals 2000: Educate America Act (PL 103-227), a high level of mathematics achievement for all students is a national priority. According to the National Research Council (2002), all students can and should achieve proficiency in mathematics. Additionally, mathematical skills are fundamental for individuals seeking occupational and educational advancement. Without proficiency in mathematics, students will likely experience difficulty completing other more advanced branches of mathematics (e.g., algebra) and be unprepared for many occupations. Mathematics education should enable students to understand and apply mathematical concepts. With this emphasis on conceptual understanding and higher-order problem-solving skills, teachers must not ignore computation.
Knowledge of basic computation skills cannot be separated from the overall conceptual understanding and forms the foundation for mathematical thinking (Wu, 1999). The National Council of Teachers of Mathematics (NCTM; 2000) emphasizes computation over overall performance in mathematics. According to the NCTM (2000), it is critical for students to know the basic number facts for addition, subtraction, multiplication, and division. Students' fluency and accuracy in methods of computation are equally important. The National Research Council (2002) further articulates the importance of computation by listing computation as the second of five main strands in mathematics. Yet, many students do not learn the basic mathematics skills required for success.
Even more troubling is the mathematics performance of students with learning disabilities (LD). Students with LD experience difficulties learning math, with problems surfacing early and continuing throughout their education (Bottge, 1999; Mercer & Miller, 1992). Deficiencies in mathematics performance are not limited to basic skills. Higher-order thinking skills such as problem solving are also a major challenge for these students (Jitendra, DiPipi, & Perron-Jones, 2002). The average mathematical performance of 16- and 17-year-old students with LD is approximately at the fifth-grade level (Cawley & Miller, 1989). Furthermore, students with LD have documented deficits in the areas of (a) basic facts, (b) subtraction, (c) solving word problems, (d) acquiring concepts, and (e) problem solving (e.g., Garnett, 1992; Miller, Stawser, & Mercer, 1996; Montague & Brooks, 1993).
Many students who are not proficient in the basic mathematics skills demonstrate numerous mathematics misconceptions (Marchand-Martella, Slocum, & Martella, 2004). For example, subtracting the smaller number from the larger number regardless of position is a common misconception not just among low-performing students or students with disabilities (Resnick & Omanson, 1987). When students make errors and formulate mathematical misconceptions, teachers should recognize the errors, prescribe an appropriate instructional focus, and implement an effective and efficient reteaching plan. The first step in this process, recognizing the errors, is completed through a systematic examination of students' mathematics work (Ashlock, 2002).
Error Analysis
Educators typically analyze students' mathematical errors with the intent to improve instruction and correct misconceptions (Mastropieri & Scruggs, 2002). Evaluating students' work to determine an appropriate instructional focus to correct errors is one of the main tenets of remedial or corrective education for all students, but especially for students with LD and low-performing students (Fuchs, Fuchs, & Hamlett, 1994; Salvia & Hughes, 1990; Salvia & Ysseldyke, 2004). Identification and analysis of students' arithmetic errors has the potential to improve instructional planning and, ultimately, student performance.
Although educators and researchers debate the numerous types of errors and their causes, as well as instructional approaches and procedures to correct errors, extensive research, including computer analysis of students' work (Woodward & Howard, 1994), indicates that large majorities of students' errors are consistent and systematic (e.g., Brueckner, 1935; Clements, 1982; Cox, 1975a, 1975b; Newman, 1977; Roberts, 1968).
Subtraction is particularly problematic for many students. Several researchers report that students experience difficulty with problems requiring borrowing (e.g., Cox, 1975a; Drucker, McBride, & Wilbur, 1987; Resnick, 1982). Specifically, many students exhibit an error type called smaller-from-larger (SFL) (National Research Council, 2002; Resnick, 1982). When making this type of error, students subtract the smaller number from the larger number regardless of position (e.g., 326-117 = 211, with the SFL error 6-7 = 1). Another error documented in students' work involves borrowing across a zero digit (BAZ). The BAZ error occurs when a student attempts to borrow from a zero and does not continue to borrow from the column to the left of the zero (e.g., 602-437 = 265, with the student not continuing to borrow from the hundreds column). This type of error occurs less frequently than SFL errors (Resnick, 1982). Both the SFL and BAZ error patterns are classified as incorrect or defective algorithms (e.g., Ashlock, 2002; Resnick, 1982; Roberts, 1986).
In general, an examination of a student's completed subtraction work is important because once a student's errors are pinpointed, a teacher can gear remedial or corrective instruction directly for the specific error patterns. Although identification of errors in mathematics is an important first step for remedial or corrective instruction, there is little evidence to suggest that teachers are able to perform systematic error analysis of students' work.
The purpose of this study was to (a) determine whether teachers were able to identify specific error patterns exhibited in subtraction; (b) establish whether teachers were better able to describe a more commonly occurring subtraction error (i.e., smaller-from-larger or SFL) than a less commonly occurring subtraction error (i.e., borrow-across-zero or BAZ); (c) determine whether teachers were able to prescribe an appropriate instructional focus; and (d) examine the instructional focus that teachers selected to address first.
METHODS
Design
Data were analyzed using...
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