关键词: 儿童, 数学学习困难(MLD), 数感, 计算流畅性, 动态干预

Abstract:

As a common learning disability among grade school children, Mathematical Learning Disabilities (MLD) typically involve lower arithmetical abilities relative to children’s mental age, However, children with MLD do not necessarily show reading and spelling disabilities. Based on the definition of disproportionate ability and achievement, the common method of MLD assessment is to evaluate the discrepancy between intelligence and mathematical achievement while controlling for children’s reading ability. However, early diagnosis or assessment of MLD seems to be impractical for first to third grade children, due to relatively high math test scores that often yield a “ceiling effect.” Thus, it seems that the achievement discrepancy model is more appropriate for senior grades of primary school (4~6 grade) than for younger children. Unfortunately, it has been suggested that remedial instructions for students with diagnosis of MLD are far from perfect. Clearly early diagnosis and intervention is crucial for young children with potential risk of developing MLD. Number sense refers to an intuitive understanding of numbers that enables a person to solve mathematical problems with flexibility and creativity. It forms a foundation for number concepts and computing methods acquisition. Preschool and primary grades are critical periods for developing number sense. Previous studies supported insufficient number sense as a strong predictor of MLD. Asian essential characteristic of MLD, calculation fluency refers to the capacity of rapid and accurate computation and has been used as a key marker of potential MLD in young children. Given that insufficient number sense can lead to calculation deficiencies which in turn predicts MLD, early assessment and diagnosis of children’s number sense, followed by specialized interventions, are particularly important for prevention of MLD. In order to examine the positive impact of interventions on number sense and calculation fluency among first grade primary school pupils, the current study first constructed the Number Sense Developmental Test, screening for children with number sense insufficiency and then provided dynamic interventions of number transformation and estimation. Children with insufficient number sense was assigned to either intervention group (n = 29) or control group (n = 27). A third group of children with sufficient number sense (n = 40) was also included as the reference group. Using a “pretest-intervention-posttest” design, the current study examined not only the effect of interventions on number sense and calculation fluency, but also the relation between number sense and calculation fluency. Results showed that the 46-item “Number Sense Developmental Test (for 1st grade)” yielded five dimensions: counting, number knowledge, number transformation, estimation, and number patterns. Evidence of psychometric properties of the test includes internal consistency reliability, test-retest reliability, parallel-forms reliability, content validity, and criterion validity, all of which were satisfactory. There was no difference on number sense and calculation fluency between the intervention group and the control group before the intervention, though both groups scored lower than did the reference group. After a six-week intervention that involves twelve blocks of number transformation and estimation training, the intervention group scored higher on both the total and four dimensions of the post-test scores than did the control group (ps < 0.01) with the exception of their scores on the number knowledge dimension. In addition, the posttest scores of the intervention group were comparable to those of the reference group (ps > 0.05). A similar pattern of findings was also found on calculation fluency among the three groups. While children in the control group also made some progress on number knowledge, number transformation and estimation (ps < 0.05), the improvement from pretest to posttest among these children were significantly lower than that of the children in the intervention group. In addition, unlike the intervention group, the control group failed to make significant progress in estimation and number patterns. Further comparison of number sense growth scores among the three groups revealed that the intervention group scored higher than control group on several dimensions, t counting = 3.52, p < 0.001; t number transformation = 4.46, p < 0.001; t estimation = 1.99, p < 0.05; t number patterns = 2.84, p < 0.01; t total = 7.00, p < 0.01. These results confirmed the positive impact the intervention on several key components of number sense (i.e., number transformation and estimation) among children with insufficient number sense. Moreover, the results showed that the pretest scores of number sense was significant related to posttest scores of calculation fluency (β = 0.275, p < 0.01), whereas the reverse relation between pretest scores of calculation fluency and the posttest scores of number sense was not (β = 0.155, p > 0.05), providing some initial evidence on the direction of the relation between number sense and calculation fluency. The findings of the current study provide support for the positive impact of interventions on certain components of number sense, pointing out the critical role an early intervention may play in developing number sense among young children. Such interventions not only are beneficial for helping children develop mathematics abilities, but also are necessary for early screening and diagnosis of MLD among young children.

Key words: children, mathematical learning disabilities (MLD), number sense, calculation fluency, dynamic intervention