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This paper argues for teaching pre-service teachers about remediation strategies for learners who encounter problems in mathematics in the early grades. The premise is that all teachers should be equipped with theory-based practical knowledge to support learning. A few teaching sessions to develop the concepts that underlie the mathematical operations of addition and subtraction are introduced in this paper. An empirically validated, comprehensive model of cumulative arithmetic competence development from the ages of four to eight years forms the basis for the construction of the suggested teaching unit. The model distinguishes five competence levels of arithmetical conceptual development, and proposes that concepts build on one another hierarchically. A 'part plus part is equivalent to whole' model was constructed based on this hierarchical structure and the understanding that the concept of addition is a dynamic process. The teaching examples include exercises for all children, not only ones who struggle. Possibilities for adapting the exercises to the individual development level of slower or faster learners are also included. All exercises are accompanied by a reflection on the procedure and strategies applied in order to support meaningful and sustainable learning and to give student teachers the opportunity to use knowledge of mathematical cognition theory during their pre-service years.

This paper aims to describe how students develop understanding of one-digit decimals. To achieve the aim, Local Instruction Theory (LIT) about the process of learning decimals and the means designed to support that learning are developed. Along with this idea, the framework of Realistic Mathematics Education (RME) is proposed. Based on the aim, design research methodology is used. This paper discusses learning activities of three meetings from teaching experiment of the focus group students of the fourth grade elementary school in Surabaya: SDIT Al Ghilmani. The data indicated that the learning activities promoted the students' understanding of one-digit decimal numbers.

A key aim of mathematics teaching is for children to develop appropriate and efficient strategies for solving tasks. The analysis presented in this paper moves beyond the exploration of changes in the strategies that children employ to solve tasks and extends to observation and exploration of changes that occur when their overall solving approach remains seemingly stable. We present an analysis of data from two qualitative studies, each of which combined a microgenetic design with task-based interviews, to examine changes that occur in 5-6-year-old children's verbal reports when solving an additive task, and in 9-10-year-old children's verbal reports when solving fraction word problems. Children's verbal reports were analysed through the lens of the theory of shifts of attention. We found that phases of stability are underlain by dynamic changes in how the same strategy is communicated and conceptualised over a number of sessions and these changes appear to be accounted for by changes related to shifts in the object and structure of children's attention, i.e. "what" children attend to and "how," when reporting on their solving approach. The paper extends the theory by revealing and studying microqualities that underlie different learners' structure of attention during phases of stability in arithmetic tasks. The findings provide new, significant insights for understanding qualitative dynamics of change in learning. Sensitivity to differences and changes in learners' shifts of attention is essential for teachers to make sense of what learners experience and identify opportunities conducive to further learning.

Advances in technology have seen a proliferation of touch-screen interfaces available to young children. These screens have changed the way in which young children engage with digital technology; with increased exposure and use, it raises debates about the suitability of integrating digital technology within early childhood settings. There are limited empirical studies that investigate the appropriateness of emergent digital technology within the field of early childhood. Based upon the TPACK model, this timely paper thus discussed the Digital Play Model (Numeracy) and reported on a project investigating the integration of digital technologies into childcare settings. In particular, the focus of this paper was to report on the use of games using a gestural interface device have on 3- to 4-year-old children's ability to learn numeracy concepts. This study found that young children's numeracy learning outcomes were improved. This paper also provided empirical evidence of the use of Apps via iPad technologies on young children's numeracy learning, with an implication to a need for a pedagogical model for successful integration of digital technology within early childhood settings.

Whole-number arithmetic is a core content area of primary mathematics, which lays the foundation for children's later conceptual development. This paper focuses on teaching whole-number multiplication (WNM) to build a stepping stone for children's proportional reasoning. Our intention in writing this paper is to obtain a practice-based perspective on the teaching of WNM, given the growing attention being given to significant connections between children's understanding of multiplication and their proportional reasoning. To this end, we review studies on teaching WNM that were conducted in Japan. Our review highlights several aspects of teaching WNM that are integral to building an initial understanding of proportional relationships. Linear representations such as tape diagrams and number lines are also found to help make explicit the important properties of proportional relationships that can be used progressively across grade levels.

The goal of this paper is to present a succinct consolidation of the why, what and how of the 'Model' method as there is a lack of such documentation for researchers, practitioners and policy makers thus far. The 'Model' method is a tool for representing and visualising relationships when solving whole number arithmetic (WNA) word problems. The paper traces the history of the method in the Singapore school mathematics curriculum, exemplifies the three basic models of the method, explores its efficacy and assesses its role as a problem solving heuristic in the Singapore primary school mathematics curriculum. The findings show that the method, a tool for representing and visualising relationships between quantities when solving whole number arithmetic word problems, was introduced in 1983 as part of the primary school mathematics curriculum for schools in Singapore. The method comprises three basic models, namely the part-whole model, the comparison model and the change model. Past studies carried out both in Singapore and other parts of the world, related to the use of the method and its impact on student learning show that the method does help students, including those with mathematical difficulties, improve in their ability to solve whole number arithmetic word problems. Perceptions of expert primary school mathematics teachers show that they engage students in solving whole number arithmetic word problems through Polya's framework and also Newman's strategies. They encourage students to represent on diagrams the givens and goals of a problem as this is a necessary step in visualising and recognising relationships. To do so they may use the method. Students may also use any other heuristic to solve the whole number arithmetic word problems.

The probability is exceptional in the teaching of mathematics because students often have difficulties to understand the basic terms and the problem solving strategies. Understanding lacks of the probability concept and various types of misconceptions arise from the misleading intuition and misinterpretations of experience with the stochastic phenomena. The probability concept seems too abstract to some students therefore it is advisable to use mathematical problems based on real-life ideas, such as drug efficacy testing, tests for diagnosing of diseases in medicine, sports competitions, and games. By eliminating misconceptions and improving understanding of the problem solving strategies, it is possible to use various types of visualization to solve problems from this field of mathematics. Tables and different types of graphic diagrams can help students to understand the basic rules and problem solving techniques. This paper describes the main objectives and the structure of an interactive worksheet, prepared in spreadsheet environment, in which students are guided to use the visualization to solve probability problems. The implementation of an automatic feedback enables to evaluate the students' answers. In the case of incorrect answer, solutions of the additional tasks using tree diagram or a tangram are recommended to the students. Students can decide which type of the visualization is more understandable for them to solve the probability of random events. Solving different task sequences using the selected types of visualization allows more learning paths for students. The final part of the paper contains an evaluation of the results and experiences of problem solving in the teaching of probability at grammar schools.

This paper details an exploratory study of 14 elementary prospective teachers (PTs) solving purposefully crafted, two-digit addition problems. The numbers in each problem were chosen to elicit diverse solution strategies. We coded 95 responses based on the ways the PTs completed the calculation (for example, by referring to the number as individual digits and/or by using strategies that attended to the value of the number). The findings suggest that although an overwhelming number of strategies explicitly referred to the numbers as distinct digits while adding, there was some evidence that PTs utilized more diverse strategies or sub-strategies (e.g., counting up or down strategies, derived facts of doubles or base ten knowledge). Our study seeks to unpack for mathematics teacher educators the typical narrative of PTs' reliance on algorithmic strategies while highlighting their use of more nuanced strategies as well.

Students require knowledge of fraction, arithmetic and integer to tell the time. The purpose of the current study is to discover the conceptual knowledge related to hours, how the students read the clock and use half and quarter concepts to read the time in Turkey. A qualitative research method with a case study approach was used for this research. The participants of the study consisted of 12 students studying at the third grade of a public school located in the west of Turkey. The data of the study were collected through semi-structured interviews. A form consisted nine questions was used in the interviews. The data were then analysed using content analysis. As the result of analysis, it was determined that the students used fraction knowledge, number of minute and absolute time while telling the time, but mostly for fraction knowledge and some students could not associate the knowledge in the subject of the fractions with the reading the clock. [This study was presented as an oral presentation at International Congress on Science and Education (ICSE-2018), Turkey.]

Children with learning disabilities face problems in acquiring the basic skills needed for learning. Dyscalculia is one among those learning disorders which affects the ability to acquire arithmetic skills that are needed to perform mathematical calculations. However this is a learning difficulty which is often not recognized. The objectives of this paper are to review studies and related literature on different types of dyscalculia; dyscalculics at various levels; relating dyscalculia to other learning disabilities; methods for overcoming dyscalculia; and comparison of dyscalculics and normal children. The dissertation abstracts international (1990-2010), the educational and psychological journals both at national and international levels, websites and related books have been reviewed for fulfilling the objectives of this paper. Among the studies reviewed, ten were taken from dissertation abstracts international, eleven from journals and six were from websites. The related literature was also collected from five books and from an Indian newspaper. Review of studies reveals the fact that less research has been done in the area of mathematical deficiencies. The need to conduct more researches on dyscalculia is essential in order to mitigate the problems of dyscalculics.

This study aims to examine the effect of instructional materials developed on the basis of 4 stage constructivist model on the success of 6th grade students in the subject of decimal fractions. For this purpose, a worksheet on decimal fractions, a conceptual change text and an anology map were developed. These instructional materials were administered to 62 students selected from two different 6th grade classes in a primary school in Trabzon in 2008-2009 school year. Semi-experimental method was used in this study. At the end of the study, it was seen that the instructional materials developed increased students' success. [Paper presented at the World Conference on Educational Sciences (WCES) 2011.]

This study investigates the effect on conceptual change of instructional materials about "decimal fractions" developed by adopting 4 stage constructivist learning model. The study was conducted with 30 students 6th grade at a public elementary school in Trabzon during the spring term of 2008-2009 school year. Case study method was used in the study. The clinical interviews were made with 6 students just before the application and one month later. When the students' responses to the interview questions after the application were investigated, it was identified that the number of students' responses containing misconceptions was decreased at the end of the study. [Paper presented at the World Conference on Educational Sciences (WCES) 2011.]

This paper describes the difficulties faced by a group of middle school students (13- to 15-year-olds) attempting to translate algebraic statements written in verbal language into symbolic language and vice versa. The data used were drawn from their replies to a written quiz and semi-structured interviews. In the former, students were confronted with a series of algebraic statements and asked to choose the sole translation, of four proposed for each, that was semantically congruent with the original. The results show that most of the errors detected were due to arithmetic issues, especially around the distinction between product and exponent or sum and product in connection with the notions of perimeter and area. As a rule, the error distribution by type varied depending on the type of task involved.

Mathematical giftedness (MG) is an intriguing phenomenon, the nature of which has yet to be sufficiently explored. This study goes a step further in understanding how MG is related to expertise in mathematics (EM) and general giftedness (G). Cognitive testing was conducted among 197 high school students with different levels of G and of EM. Based on our previous studies, we perceive MG as a combination of G and EM. Exploratory factor analysis of test results revealed five main cognitive factors: visual-serial processing (VSP); arithmetic abilities (AA); pattern recognition (PR); auditory working memory (AWM); visual-spatial working memory (VSWM); and Structural equation modeling (SEM) based on the factor analysis revealed clear differences in the role of cognitive abilities as predictors of EM, G, and MG. The study demonstrates that visual components are especially important for the development of EM and that G students are less dependent on their visual cognitive processing. Based on the study results, we argue that EM, G, and MG, which are often considered equivalent characteristics, are interrelated but different in nature. The paper presents a research-based justification that not any gifted is an expert in mathematics and not any expert in mathematics is gifted.

Introducing the distributivity property of multiplication over addition is a well-known challenge in mathematics education, especially in primary school. As a contribution, this paper presents the results of a cycle of design research that focuses on the design, implementation and evaluation of a modelling activity in which 2nd-grade students are introduced to the key concept of distributivity of multiplication over addition. The results show that modelling activities designed through the heuristics of didactical phenomenology, guided reinvention and emergent modelling may support the accessibility of distributivity, as these students were able to reinvent this concept solving an experientially significant problem. This result can be attributed to a combination of several factors: the choice of a realistic and rich problem, that stimulated students to elaborate formal mathematical concepts mathematizing their informal solving strategies, rooting new understandings in experientially real phenomena; the use of a suitable artifact, that presented mathematics as a means of interpreting and understanding reality and increasing the opportunities for observing mathematics outside of the school context; the role of the teacher, who guided students in reinventing mathematics in an active way.

The paper is the result of extensive research carried out among Czech students and focuses on a conception of infinity. The questionnaire survey was taken by 861 students ranging from grades 7 to 13. The aim of the research was to describe the development of students' conceptions of infinity. These conceptions are built on the intuitive phenomenon of the horizon. We monitor the proportional representation of these conceptions in four combinations of views (into the distance and in depth) and contexts (arithmetical and geometrical). It can safely be maintained that the development of the proportional representation of the earliest conception comprehension of the concept of infinity, the so-called natural infinity, is not concurrent with the students' age. The development of the proportional representation of the conception of actual infinity is non-decreasing, at least in the view into the distance, in both contexts with respect to age. In general, it can be stated that the proportional representation of individual conceptions of infinity is strongly dependent on both context and view.

This paper explores how student performance across the full set of multiple modular assessments of individual standards, which we refer to as mini-assessments, from a large scale, operational program of interim assessment can be summarized using Bayesian networks. We follow a completely data-driven approach in which no constraints are imposed to best reflect the empirical relationships between these assessments, and a learning trajectory approach in which constraints are imposed to mirror the stages of a mathematic learning trajectory to provide insight into student learning. Under both approaches, we aim to draw a holistic picture of performance across all of the mini-assessments that provides additional information for students, educators, and administrators. In particular, the graphical structure of the network and the conditional probabilities of mastery provide information above and beyond an overall score on a single mini-assessment. Uses and implications of our work are discussed.

We investigated understanding of the linear independence concept based on the type and nature of connections displayed in seven non-mathematics majors' interview responses to a set of open-ended questions. Through a qualitative analysis, we identified six categories of frequently displayed connections. There were also recognizable differences in the way the connections were applied by the participants. Overall, our findings pointed to an understanding in the form of two main clusters of connections. The two clusters were connected only by linear combination ideas. Each cluster, furthermore, was distinguishable via representation types. The first cluster contained arithmetic/algebraic modes and the second cluster included, mostly, geometric ideas. This paper discusses similarities and differences within and between clusters supported by participant responses. In light of the findings, we provide suggestions for the improvement of linear algebra education of non-mathematics student population.

Open math problem solving is critical to help students deepen the understanding and promote transfer of mathematics knowledge. However, the cognitive mechanism for open math problem solving, particularly the role of spatial abilities, has not been paid enough attention. This study recruited 192 junior middle school students (14.30 ± 0.48 years old). Results showed that both spatial visualisation (measured by paper folding) and spatial working memory (measured by spatial 2-back) significantly associated with open math problem solving with controlled variables including age, sex, nonverbal matrix reasoning, and arithmetic principles. Moreover, spatial working memory was more associated with easy open math problem solving, while spatial visualisation more associated with difficult open math problem solving. These findings suggested that as the difficulty of open math problems increased, the construction of problem space mattered more than retaining and updating problem space in open math problem solving.

Recently, there has been a growing emphasis on basic number processing competencies (such as the ability to judge which of two numbers is larger) and their role in predicting individual differences in school-relevant math achievement. Children’s ability to compare both symbolic (e.g. Arabic numerals) and nonsymbolic (e.g. dot arrays) magnitudes has been found to correlate with their math achievement. The available evidence, however, has focused on computerized paradigms, which may not always be suitable for universal, quick application in the classroom. Furthermore, it is currently unclear whether both symbolic and nonsymbolic magnitude comparison are related to children’s performance on tests of arithmetic competence and whether either of these factors relate to arithmetic achievement over and above other factors such as working memory and reading ability. In order to address these outstanding issues, we designed a quick (2 minute) paper-and-pencil tool to assess children’s ability to compare symbolic and nonsymbolic numerical magnitudes and assessed the degree to which performance on this measure explains individual differences in achievement. Children were required to cross out the larger of two, single-digit numerical magnitudes under time constraints. Results from a group of 160 children from grades 1–3 revealed that both symbolic and nonsymbolic number comparison accuracy were related to individual differences in arithmetic achievement. However, only symbolic number comparison performance accounted for unique variance in arithmetic achievement. The theoretical and practical implications of these findings are discussed which include the use of this measure as a possible tool for identifying students at risk for future difficulties in mathematics. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to present a summary of selected facets of geometry and measurement in elementary school programs and to describe curricula content options designed to demonstrate the feasibility of seeking high level outcomes and meanings for students with learning disabilities. While there are a multitude of published papers relative to arithmetic, the literature specific to geometry and measurement is scant indeed. The illustrations of the area of the circle and the volume of cubes, pyramids, cylinders, and cones are presented to encourage researchers and teachers alike to seek higher-order outcomes for geometry and measurement and avoid the rote drill and practice that often dominates arithmetic instruction at the elementary grade levels. (Contains 8 tables.)