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It is well documented in literature that rational number is an important area of understanding in mathematics. Therefore, it follows that teachers and students need to have an understanding of rational number and related concepts such as fraction multiplication and division. This practitioner reference paper examines models that are important to elementary and middle school teachers and students in the learning and understanding of fraction multiplication and division.

Tablets can be used to facilitate systematic testing of academic skills. Yet, when using validated paper tests on tablet, comparability between the mediums must be established. Comparability between a tablet and a paper version of a basic math skills test (HRT: Heidelberger Rechen Test 1-4) was investigated. Five samples with second and third grade students participated. The associations between the tablet and paper version of HRT showed that these modes of administration were comparable for three arithmetic scales, but unacceptable for a pictorial counting scale. Scores were lower on tablet. Test-retest reliability for arithmetic scales on tablet was satisfactory, but was inferior for a low-performing sample. The overall convergent validity was satisfactory. No effect of test administrator was found. Arithmetic scales can potentially be transferred to tablet with good comparability and maintained test-retest reliability. Precautions are necessary when transferring pictorial scales into tablet. Separate norms for tablet are needed when interpreting scores.

Part of the research community that has followed the Early Algebra paradigm is currently delimiting the differences between arithmetic thinking and algebraic thinking. This trend could prevent new research approaches to the problem of learning algebra, hiding the importance of considering an arithmetico-algebraic thinking, a new approach which underpins the construction of a cognitive structure that links both types of thinking. This paper proposes a theoretical and practical framework for a learning approach that supports the construction of a cognitive structure which fosters arithmetico-algebraic thinking at the beginning of secondary school by means of cultural and technological activities relating to polygonal numbers.

This study aims to understand the differences in student learning outcomes and classroom behaviour when using the interpersonal computer, personal computer and pen-and-paper to solve arithmetic exercises. In this multi-session experiment, third grade students working on arithmetic exercises from various curricular units were divided into three groups. The first group used personal computers (netbooks), the second group used an interpersonal computer (ie, one projector with a screen, one computer and one mouse per child) and the third group used pen-and-paper. The results of the experiment indicate that all three groups achieved an increase in learning, as shown by the pretest and posttest scores. No significant difference was found between the interpersonal computer and personal computer groups. This suggests that the key characteristic shared by the two groups is the provision of feedback. The format that such feedback takes, either private (through a personal screen) or public (through a shared screen), is shown to make no difference. However, the results significantly favour groups that are provided with instant feedback (interpersonal computer and personal computer) as opposed to delayed feedback (pen-and-paper).

This paper refers to Langlands' big set of notes devoted to the question if the (normalized) local Hecke–Tate root number Δ=Δ(E,χ)$\Delta =\Delta (E,\chi)$, where E is a finite separable extension of a fixed nonarchimedean local field F, and χ a quasicharacter of E×$E^\times$, can be appropriately extended to a local ε‐factor εΔ=εΔ(E,ρ)$\varepsilon _\Delta =\varepsilon _\Delta (E,\rho)$ for all virtual representations ρ of the corresponding Weil group WE$W_E$. Whereas Deligne has given a relatively short proof by using the global Artin–Weil L‐functions, the proof of Langlands is purely local and splits into two parts: the algebraic part to find a minimal set of relations for the functions Δ, such that the existence (and unicity) of εΔ$\varepsilon _\Delta$ will follow from these relations; and the more extensive arithmetic part to give a direct proof that all these relations are actually fulfilled. Our aim is to cover the algebraic part of Langlands' notes, which can be done completely in the framework of representations of solvable profinite groups, where two modifications of Brauer's theorem play a prominent role. [ABSTRACT FROM AUTHOR]

Research has shown the importance of helping students, especially those with mild-to-moderate learning disabilities, to offload information during problem-solving. When students can get their thoughts onto paper, number line strategies can help them develop a firm foundation in mathematical problem-solving while understanding the relationships between mathematical operations. These strategies are helpful for the development of addition, subtraction, multiplication, division, and later, fractional mathematics. In this article, we describe the progression of number lines as a supportive strategy for elementary students and those with developmental delays in mathematics to improve mathematical understanding. This strategy is based on students being able to show their work and think about what they have written on paper or how they have used manipulatives.

Analyzes and refutes the arguments made by "back-to-basics" proponents against the use of calculators and for traditional instruction in the algorithms of pencil-and-paper arithmetic. Argues for the value of mental arithmetic in achieving all the aims and more of the traditional curriculum. (Author/ASK)

Against the background of the low mathematical performance of South African learners in international panel studies, there is an urgent need to improve mathematical education. In particular, the curriculum and its structure raise questions. It is logical that the prescribed curricula should align with learners' developmental trajectories. Given the hierarchical structure of mathematics, the curricular requirements should pay attention to learners' current knowledge of mathematical concepts. The aim of this study was to compare the curricular requirements as defined by the CAPS with the conceptual current knowledge of South African learners. South African Grade 1 learners (N = 602) were assessed on a test of numeracy concepts, based on a theoretically informed and empirically validated model of developing mathematical proficiency. The content of the CAPS for Grade 1 was aligned to the model levels by two experts (Cohen's [kappa] = 0.753, p < 0.001). Results show that the curricular requirements go far beyond the current knowledge required to engage with these new concepts of the vast majority of South African Grade 1 learners. The mismatch may to some extent be responsible for the unsatisfactory results in international comparison studies. These results show that the intended curriculum is beyond the grasp of most South African Grade 1 learners. These children are unlikely to develop new arithmetic concepts based on their lack of required foundation knowledge. We therefore argue that the intended curriculum for Grade 1 should focus more on counting skills, ordinal relations between numbers and--most importantly--set-based number representations and part-part-whole relations.

The rapid growth of multimedia pedagogy in the education sector has brought about a game-based technological approach that is shaping the learning of children nowadays. The focus of the approach is to encourage active participation from children; the effect can be seen in their interaction, involvement, and engagement throughout the learning content. Active participation in a learning environment would have a substantial impact on children's self-development and academic achievement, particularly for those with learning disabilities (LD). Therefore, the purpose of this paper is to systematically analyze research conducted on Game-Based Learning (GBL) courseware to support the education of children with LD. A systematic literature review was undertaken, following the PRISMA framework for paper selection. A total of 109 articles were retrieved from the Scopus and Science Direct databases by using a specific keywords search. 14 articles were finalized at the end of the screening based on the inclusion and exclusion criteria. Results reveal the trend of publications, approaches used, and research themes of the selected papers, which include the courseware requirement, student adaptation, and impact of the implementation. The findings demonstrate that GBL is one of the effective methods to be applied in preschool education. It has a significant impact on the development of cognitive skills and assists children who have difficulty in reading, writing, and arithmetic. The findings in this paper can be used as a guide in developing GBL courseware that is developmentally appropriate and effective for children with LD.

Background: The Mental Starters Assessment Project (MSAP) seeks to address poor performance in Grade 3 mathematics. The programme focusses on eliminating inefficient counting methods and promoting strategic mathematical skills, including numerical sense, mental calculation and rapid recall skills. Additionally, MSAP supports teachers' professional growth by providing them with a toolkit of effective calculation strategies to bridge the performance gap and enhance mathematical education. Aim: This paper explores the insight gained from reflections of final year Bachelor of Foundation Phase (FP) initial teacher educators (ITE) students in South Africa. Setting: Grade 3 classrooms. Methods: The ITE students were given training and materials to implement the MSAP, and this occurred over a 4-week teaching practicum, after which they completed a reflective task on the implementation. A total of 20 students were selected from a cohort of 138 based on their academic performance. Results: The analysis of the reflections showed that ITE students benefitted from reflecting on their practice and highlighted important elements of their professional learning. The reflections raised issues related to challenges in their pedagogical content knowledge (PCK) as well as their confidence and competence to teach mathematics and manage the classroom context. Conclusion: With a multi-dimensional model of reflection, ITE students can achieve a deeper understanding of mathematics teaching and learning when building learners' mental strategies, fostering professional growth and elevating the overall quality of mathematics education. Contribution: Overall, the findings provide insight into the benefits of reflective practices for ITE students' professional development and the improvement of mathematics education.

This study identified differences in elementary school students' performance of arithmetic tasks using paper and pencil and computer-assisted instruction. Many were found to perform more poorly using the computer, while others showed the opposite tendency. These findings challenge the validity of decisions made by the computer-based management system in relation to students' advancement. (24 references) (LRW)

This paper is part of broader research being conducted in the area of algebraic thinking in primary education. Our general research objective was to identify and describe generalization of a 2nd grade student (aged 7-8). Specifically, we focused on the transition from arithmetic to algebraic generalization. The notion of structure and its continuity in the generalization process are important for this transition. We are presenting a case study with a semi-structured interview where we proposed a task of contextualized generalization involving the function y = x + 3. Special attention was given to the structures evidenced and the type of generalization expressed by the student in the process. We noted that the student identified the correct structure for the task during the interview and that he evidenced a factual type of algebraic generalization. Due to the student's identification of the appropriate structure and the application of it to other different particular cases, we have observed a transition from arithmetic thinking to algebraic thinking.