A Mathematician\u27s Overview of the Virginia Elementary Mathematics Specialist Program

This article discusses the mathematics component of the Mathematics Specialist master\u27s degree program in the Virginia Mathematics Specialist Project (VMSP). It includes my personal views on the significant mathematical knowledge and skills that Mathematics Specialists need, the mathematics that is taught in the Mathematics Specialist courses, and my thoughts on what appear to be the substantial mathematical abilities and aptitudes that are required by successful Mathematics Specialists in their work. The interpretations I present are highly personal and are undoubtedly dependent on my personal history, a short description of which is given (see Appendix A)

Developmental changes in the association between approximate number representations and addition skills in elementary school children

The approximate number system (ANS) is assumingly related to mathematical learning but evidence supporting this assumption is mixed. The inconsistent findings might be attributed to the fact that different measures have been used to assess the ANS and mathematical skills. Moreover, associations between the performance on a measure of the ANS and mathematical skills may be discontinuous, i.e., stronger for children with lower math scores than for children with higher math scores, and may change with age. The aim of the present study was to examine the development of the ANS and arithmetic skills in elementary school children and to investigate how the relationship between the ANS and arithmetic skills develops. Individual markers of children's ANS (internal Weber fractions and mean reaction times in a non-symbolic numerical comparison task) and addition skills were assessed in their first year of school and 1 year later. Children showed improvements in addition performance and in the internal Weber fractions, whereas mean reaction times in the non-symbolic numerical comparison task did not change significantly. While children's addition performance was associated with the internal Weber fractions in the first year, it was associated with mean reaction times in the non-symbolic numerical comparison task in the second year. These associations were not found to be discontinuous and could not be explained by individual differences in reasoning, processing speed, or inhibitory control. The present study extends previous findings by demonstrating that addition performance is associated with different markers of the ANS in the course of development

An emergentist perspective on the origin of number sense

open2noopenZorzi, Marco; Testolin, AlbertoZorzi, Marco; Testolin, Albert

Differential Effects of Virtual and Concrete Manipulatives in a Fraction Intervention on Fourth and Fifth Grade Students’ Fraction Skills

Fraction knowledge has been found to predict later mathematical performance, but many students have difficulty with fractions. Virtual manipulatives (VM) and concrete manipulatives (CM) are effective approaches to teaching fractions, but previous research has not been able to reach a consensus on which manipulatives are the most effective. This quasi-experimental study employed a pre- and posttest design to investigate the differential effects of VM and CM in a fraction intervention on students’ fraction skills. In addition to fraction skills, students’ arithmetic fluency was measured. Fidelity of intervention, social validity, and time-efficiency of the manipulatives were also investigated. Fourth- and fifth-grade participants (N= 115) from Southern Finland were assigned to VM and CM intervention groups. The intervention was implemented during six 45-minute lessons over 2 weeks. Results revealed that the CM group outperformed the VM group in fraction skills, which suggests that CM should be favored in fraction interventions.Fraction knowledge has been found to predict later mathematical performance, but many students have difficulty with fractions. Virtual manipulatives (VM) and concrete manipulatives (CM) are effective approaches to teaching fractions, but previous research has not been able to reach a consensus on which manipulatives are the most effective. This quasi-experimental study employed a pre- and posttest design to investigate the differential effects of VM and CM in a fraction intervention on students’ fraction skills. In addition to fraction skills, students’ arithmetic fluency was measured. Fidelity of intervention, social validity, and time-efficiency of the manipulatives were also investigated. Fourth- and fifth-grade participants (N= 115) from Southern Finland were assigned to VM and CM intervention groups. The intervention was implemented during six 45-minute lessons over 2 weeks. Results revealed that the CM group outperformed the VM group in fraction skills, which suggests that CM should be favored in fraction interventions.Peer reviewe

Weightless: Lossy Weight Encoding For Deep Neural Network Compression

The large memory requirements of deep neural networks limit their deployment and adoption on many devices. Model compression methods effectively reduce the memory requirements of these models, usually through applying transformations such as weight pruning or quantization. In this paper, we present a novel scheme for lossy weight encoding which complements conventional compression techniques. The encoding is based on the Bloomier filter, a probabilistic data structure that can save space at the cost of introducing random errors. Leveraging the ability of neural networks to tolerate these imperfections and by re-training around the errors, the proposed technique, Weightless, can compress DNN weights by up to 496x with the same model accuracy. This results in up to a 1.51x improvement over the state-of-the-art

Learning Mathematics with Technology: The Influence of Virtual Manipulatives on Different Achievement Groups

This study examined the influence of virtual manipulatives on different achievement groups during a teaching experiment in four fifth-grade classrooms. During a two-week unit focusing on two rational number concepts (fraction equivalence and fraction addition with unlike denominators) one low achieving, two average achieving, and one high achieving group participated in two instructional treatments (three groups used virtual manipulatives and one group used physical manipulatives). Data sources included pre- and post-tests of students’ mathematical content knowledge and videotapes of classroom sessions. Results of paired samples t-tests examining the three groups using virtual manipulatives indicated a statistically significant overall gain following the treatment. Follow-up paired samples individual t-tests on the low, average, and high achieving groups indicated a statistically significant gain for students in the low achieving group, but only numerical gains for students in the average and high achieving groups. There were no significant differences between the average achieving student groups in the virtual manipulatives and physical manipulatives treatments. Qualitative data gathered during the study indicated that the different achievement groups experienced the virtual manipulatives in different ways, with the high achieving group recognizing patterns quickly and transitioning to the use of symbols, while the average and low achieving groups relied heavily on pictorial representations as they methodically worked stepby- step through processes and procedures with mathematical symbols