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Improving Skills of Addition and Subtraction Involving Negative Numbers Based on Cognitive Task Analysis and Assessment of Mental Representations of Negative Numbers: A Case Study of a Seventh-Grade Student - eScholarship

Improving Skills of Addition and Subtraction Involving Negative Numbers Based on Cognitive Task Analysis and Assessment of Mental Representations of Negative Numbers: A Case Study of a Seventh-Grade Student Atsushi Terao (atsushi@edu.hokudai.ac.jp) Risa Sawaki (sawaki@edu.hokudai.ac.jp) Mayu Hasegawa (hasegawa@edu.hokudai.ac.jp) Harumitsu Murohashi (muroh@edu.hokudai.ac.jp) Graduate School of Education, Hokkaido University, Kita 11 Nishi 7 Kita-ku, Sapporo 060-0811 JAPAN Abstract The aim of this paper is to show how we can use cognitive task analysis and the SNARC effect (e.g., Fisher, 2003) to refine students’ skills of addition and subtraction including negative numbers. We taught these problems to a seventh- grade student. She had frequently made errors in solving the problem. Several possible ways of calculation were identified based on a cognitive task analysis. Some were pictorial solutions involving a mature mental number line; others were purely algebraic solutions not requiring direct use of a mental number line. It also turned out that her errors were due to bug rules and the lack of a critical production when executing a purely algebraic solution. The direction of the SNARC effect suggested that the student had an immature mental number line, containing only positive entries. Considering the bug rules and her immature mental representation of numbers, we focused on refining her solution procedure without emphasizing the use of a mental number line. It worked very well. This case suggests that cognitive task analysis and assessment of the mental representation of numbers can provide a foundation for effective teaching of integer addition and subtraction. Introduction The authors run a social and academic activity under the supervision of the fourth author of this paper, called the Saturday classroom, in which we teach school subjects and play with students who suffer from a mild developmental disorder, such as autism, AD/HD, or a learning disorder. The goal of our research group is to understand these disorders from the viewpoints of psychology and brain science. The Saturday classroom gives us a chance to observe these disorders and to understand the difficulties the students are experiencing. Each student comes to our department twice a month, on Saturday. From the beginning of the 2004 school year (from April 2004 to March 2005), the first three authors taught a 7 th grade-student (initials AY) addition and subtraction incorporating negative numbers. In Japan, this calculation is the first topic of junior-high-school mathematics (7 th to 9 th grade). A doctor has diagnosed AY with a learning disorder, stemming from a car accident when she was an older preschooler. Her brain was severely damaged, and the difficulty she has in learning calculations involving negative integers may be due to her disorder. However, we believe that the errors she made can be observed in normal students, and what was done in AY’s case is also useful for normal students. Moreover, the aim of this paper is to discuss usefulness of the information obtained from cognitive task analysis and the assessment of the mental representation of numbers in teaching calculations on integer numbers. For these reasons, hereafter we will not refer to her disorder. It is surmised that the mental number line should be used as a critical base of calculation on real numbers { x | −∞ < x < +∞ } . The mental number line is a language-independent representation of number magnitude (Dehaene, Spelke, Pinel, Stanescu, & Tsivkin, 1999). All the textbooks used in Japan, as far as we know, first teach this calculation on integer numbers as a movement on the (written) number line. For example, a textbook teaches 2 − 6 as moving 6 to the negative direction (left) from the point of 2 along the number line. AY was initially taught to solve this kind of problem in the same way as the textbook does. When students reach a certain level of expertise, they no longer write a number line on paper. However, it is thought that an internal representation of the number line still plays a role in executing calculation. Several studies have shown that the mental number line is a critical source of mathematical thinking, including approximation (Dehaene, et al., 1999), elementary addition and subtraction (Griffin, Case, & Siegler, 1994), and solving word problems (Lewis, 1989; Paige & Simon, 1966; Terao, Koedinger, Sohn, Qin, Anderson, & Carter, 2004). AY had often made errors in executing the integer calculation. For example, her answer to 2 − 5 was 3. Using a written number line, she was taught that the answer to this problem should be a negative number (actually -3) because from the point of 2 we have to go over zero to reach the point of the answer on the number line. She seemed to understand the calculation and was able to solve similar problems correctly. But in the next Saturday classroom (two weeks later) she made the same errors again. Given her relapse, it became imperative to understand when and why these errors occurred, and to consider whether how she was being taught was in order.