Readiness-based differentiation in primary school mathematics: expert recommendations and teacher self-assessment

The diversity of students’ achievement levels within classrooms has made it essential for teachers to adapt their lessons to the varying educational needs of their students (‘differentiation’). However, the term differentiation has been interpreted in diverse ways and there is a need to specify what effective differentiation entails. Previous reports of low to moderate application of differentiation underscore the importance of practical guidelines for implementing differentiation. In two studies, we investigated how teachers should differentiate according to experts, as well as the degree to which teachers already apply the recommended strategies. Study 1 employed the Delphi technique and focus group discussions to achieve consensus among eleven mathematics experts regarding a feasible model for differentiation in primary mathematics. The experts agreed on a five-step cycle of differentiation: (1) identification of educational needs, (2) differentiated goals, (3) differentiated instruction, (4) differentiated practice, and (5) evaluation of progress and process. For each step, strategies were specified. In Study 2, the Differentiation Self-Assessment Questionnaire (DSAQ) was developed to investigate how teachers  self-assess their use of the strategies recommended by the experts. While teachers (N = 268) were moderately positive about their application of the strategies overall, we also identified areas of relatively low usage (including differentiation for high-achieving students) which require attention in teacher professional development. Together, these two studies provide a model and strategies for differentiation in primary mathematics based on expert consensus, the DSAQ which can be employed in future studies, and insights into teachers’ self-assessed application of specific aspects of differentiation

Cognitive predictors of children’s development in mathematics achievement: a latent growth modeling approach

Research has identified various domain-general and domain-specific cognitive abilities as predictors of children’s individual differences in mathematics achievement. However, research into the predictors of children’s individual growth rates, i.e., between-person differences in within-person change, in mathematics achievement is scarce. We assessed 334 children’s domain-general and mathematics-specific early cognitive abilities and their general mathematics achievement longitudinally across four time-points within the 1st and 2nd grade of primary school. As expected, a constellation of multiple cognitive abilities contributed to the children’s starting level of mathematical success. Specifically, latent growth modeling revealed that WM abilities, IQ, counting skills, nonsymbolic and symbolic approximate arithmetic and comparison skills explained individual differences in the children’s initial status on a curriculum-based general mathematics achievement test. Surprisingly, however, only one out of all the assessed cognitive abilities was a unique predictor of the children’s individual growth rates in mathematics achievement: their performance in the symbolic approximate addition task. In this task, children were asked to estimate the sum of two large numbers and decide if this estimated sum was smaller or larger compared to a third number. Our findings demonstrate the importance of multiple domain-general and mathematics-specific cognitive skills for identifying children at risk of struggling with mathematics and highlight the significance of early approximate arithmetic skills for the development of one’s mathematical success. We argue the need for more research focus on explaining children’s individual growth rates in mathematics achievement

Inhibition, friend or foe? Cognitive inhibition as a moderator between mathematical ability and mathematical creativity in primary school students

It is still unclear which cognitive factors stand at the base of mathematical creativity. One factor could be inhibition, but results are inconsistent. A possible explanation is that this relation is more complex than the direct relations tested, until now. In the current study, the hypothesis was tested that cognitive inhibition moderated the relationship between mathematical ability and mathematical creativity. The sample included 82 primary school students between 8 and 12 years of age. Mathematical creativity was measured with a multiple solution task and scored on fluency, flexibility, and originality. While there was a direct relation between mathematical ability and mathematical creativity, inhibition did not have a direct effect on mathematical creativity, but it positively moderated this relationship for flexibility and originality. These results indicate that reduced inhibition strengthens the relationship between mathematical ability and mathematical flexibility and between mathematical ability and mathematical originality, but not the relation between mathematical ability and mathematical fluency. These findings are discussed in relation to children with high and low mathematical abilities, measurement of inhibition, and the domain-general/domain-specific discussion of creativity

The effects of digital learning material on students’ mathematics learning in vocational education.

This study investigates the e ects of Digital Learning Material (DLM) including instructional clips, online guidance, structuring of content, and a col- laboration tool on students’ mathematics learning in Dutch vocational education. A pretest–posttest design was used. Apprenticeship students were asked to com- plete assignments and to discuss them with their peers and the online teacher. The results showed that DLM can enhance students’ mathematics learning in vocational education. The learning enhancement was mostly due to the use of instructional clips and structuring of the content of the mathematics tasks. Elaborations of these results, implications, limitations and recommendations for further research are provided

The effects of computer-based virtual learning environments on nursing students’ mathematical learning in medication processes

Computer-based virtual learning environments (CBVLEs) are potentially useful teaching tools for training nursing students in professional duties such as the mathematical tasks associated with medication processes. In this study, a CBVLE was designed with well-structured instructional activities such as interleaved practice and feedback. Mathematical medication scenarios and basic arithmetic exercises were integrated into the CBVLE. Four training conditions were used in the CBVLE to facilitate extra support for mathematical medication learning: (1) learning without worked examples, (2) learning with worked examples involving domain-specific knowledge, (3) learning with worked examples involving regular thinking strategies, and (4) learning with combined worked examples. This study was conducted with 118 nursing students enrolled in post-secondary nursing education and Bachelor’s nursing programmes. Students were pre-tested and post-tested on their mathematical medication learning. Training in the CBVLE improved mathematical medication learning for all students from pre-test to the post-test stages, but no differences were found among the four different conditions. Nursing students’ prior knowledge, non-verbal intelligence, and number of correct tasks predicted mathematical medication learning outcomes. When controlling for non-verbal intelligence, students in the condition 1 benefited more than students in condition 3 in terms of their mathematical medication learning outcomes. The same accounted for the support of the low-achieving students in the CBVLE. The support conditions for the high-achieving group appeared to be unimportant for mathematical medication learning. It seems that technology is taken over some of the capacity of working memory, which accounts for the benefits to the low-achieving learners

Nursing students’ satisfaction with the instructional design of a computer-based virtual learning environment for mathematical medication learning

Computer-based virtual learning environments (CBVLEs) have attracted attention as a learning innovation that can foster students’ self-efficacy and intrinsic motivation. Research on the instructional design regarding these aspects of learning in a virtual learning environment is rather piecemeal. This study investigates the instructional design of a CBVLE for mathematical medication learning by nursing students in vocational education. The instructional design was based on a task-centered approach, and students’ future learning tasks formed the backbone. We examine the extent to which the CBVLE fostered the nursing students’ mathematical learning, self-efficacy, and intrinsic motivation, and the ways in which the design components of the CBVLE met nursing students’ satisfaction. In total, 118 nursing students were assigned to four groups, with or without extra support from worked examples, and were trained via the CBVLE on mathematical medication learning tasks over four consecutive weeks. Students were pre- and post-tested on their mathematical medication learning, self-efficacy, and intrinsic motivation. Students also rated their satisfaction with the instructional design. Our results showed that the CBVLE fostered nursing students’ mathematical medication learning, self-efficacy, and intrinsic motivation, but no significant differences were found between the four conditions. Overall, student satisfaction was above average. The design components were able to predict nursing students’ mathematical medication learning, self-efficacy, and intrinsic motivation

THE ROLES OF ARITHMETIC FLUENCY AND EXECUTIVE FUNCTIONING IN MATHEMATICAL PROBLEM-SOLVING

This study is conducted to further understand the direct and indirect contributions of executive functioning (visuo-spatial updating, verbal updating, inhibition, shifting) and arithmetic fluency to mathematical problem-solving in 458 fourth-grade students. Arithmetic fluency along with visuospatial and verbal updating were significant predictors of mathematical problem-solving at the end of grade 4. When the growth in mathematical problem-solving during grade 4 was analyzed, only arithmetic fluency directly and strongly contributed to students’ problem-solving at the end of grade 4. Inhibition and shifting (in combination with inhibition) were indirectly connected to mathematical problem-solving at the end of grade 4 via their arithmetic fluency. Arithmetic fluency plays a critical role and continues to do this in mathematical problem-solving. Furthermore, a decline in importance for visuospatial and verbal updating and increasing importance of inhibition and shifting (com-bined with inhibition) were found with regard to stu-dents’ ability to solve mathematical problems during grade 4

Impact of Children's math self-concept, math self-efficacy, math anxiety, and teacher competencies on math development

We examined to what extent children's development of arithmetic fluency and mathematical problem-solving was influenced by their math self-concept, math self-efficacy, and math anxiety but also teacher competence, specifically: actual teaching behavior, self-efficacy, and mathematical teaching knowledge. Participants were 610 children and 31 teachers of grade four. Multi-level analyses showed children's math self-concept to be a positive predictor of arithmetic fluency and actual teaching behavior to be a negative predictor. The development of mathematical problem-solving was predicted: positively by mathematical teaching knowledge; negatively by actual teaching behavior and teachers’ self-efficacy; and not at all by the child factors of math self-concept, math self-efficacy, or math anxiety. Promoting the self-confidence of young children is essential for their mathematical development. More research into the relationship between teaching behaviors and children's math development is needed

Inhibition, friend or foe? Cognitive inhibition as a moderator between mathematical ability and mathematical creativity in primary school students

It is still unclear which cognitive factors stand at the base of mathematical creativity. One factor could be inhibition, but results are inconsistent. A possible explanation is that this relation is more complex than the direct relations tested, until now. In the current study, the hypothesis was tested that cognitive inhibition moderated the relationship between mathematical ability and mathematical creativity. The sample included 82 primary school students between 8 and 12 years of age. Mathematical creativity was measured with a multiple solution task and scored on fluency, flexibility, and originality. While there was a direct relation between mathematical ability and mathematical creativity, inhibition did not have a direct effect on mathematical creativity, but it positively moderated this relationship for flexibility and originality. These results indicate that reduced inhibition strengthens the relationship between mathematical ability and mathematical flexibility and between mathematical ability and mathematical originality, but not the relation between mathematical ability and mathematical fluency. These findings are discussed in relation to children with high and low mathematical abilities, measurement of inhibition, and the domain-general/domain-specific discussion of creativity