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This study delves into the intricate relationship between pre-service teachers' (PSTs') Mathematical Knowledge for Teaching Fractions (MKTF) and its influence on their teaching practices. Grounded in the premise that MKTF domains exhibit interconnectivity, shaping the constructs of teaching practices, the study employed the mathematical task framework and the framework for mathematical knowledge for teaching. Utilizing the Mathematical Knowledge for Teaching Fractions test and the Teaching Practices test, data were collected from 171 PSTs. Regression analyses uncovered significant effects of MKTF domains on five teaching practice components, underscoring the pivotal role of a teacher's mathematical knowledge in effective teaching. Notably, among the six MKTF domains, the KCFS domain emerged as the most fundamental, strongly predicting various MKTF domains and influencing teaching practice constructs. This study underscores the significance of the KCFS domain in shaping both MKTF domains and instructional practices. The findings bear implications for the education of PSTs in Ghana and other nations facing similar educational landscapes.

This research aims to 1) develop an effective Active Learning plan combined with skill practice according to the 70/70 criterion, 2) study the learning achievement index through Active Learning combined with skill practice, and 3) investigate the satisfaction towards Active Learning combined with skill practice. The sample group consisted of 27 Grade 3 students from Muang Wapi Pathum School, during the first semester of the 2023 academic year, selected through purposive sampling. The research tools included 1) 10 Active Learning plans totaling 10 hours, 2) mathematics skill practice, 3) an achievement test consisting of 15 multiple-choice questions, and 4) a satisfaction questionnaire regarding Active Learning combined with skill practice, using a 5-point Likert scale with 13 items. The statistical analysis included mean, percentage, standard deviation, and effectiveness index. The research found that the Active Learning plan was effective with an efficiency of 81.00/82.20, which meets the predefined criterion of 70/70. The effectiveness index was 0.6828, indicating that students improved their learning by 68.28 percent. Overall satisfaction was at the highest level, with an average score of 4.54.

In November 2022, ChatGPT, an Artificial Intelligence (AI) large language model (LLM) capable of generating human-like responses, was launched. ChatGPT has a variety of promising applications in education, such as using it as thought-partner in generating curricular resources. However, scholars also recognize that the use of ChatGPT raises concerns, such as outputs that are inaccurate, nonsensical, or vague. We, two mathematics teacher educators, engaged in a collaborative self-study using qualitative descriptive approaches to investigate the procedures, concepts, and cognitive demand of ChatGPT-generated mathematical tasks focused on fraction multiplication using the area model approach. We found that the ChatGPT-generated tasks were mostly procedural and not cognitively demanding. Moreover, despite ten variations of input prompts, ChatGPT did not produce any tasks that used the area model approach for fraction multiplication. Rather, it generated tasks focused on procedural approaches. Alarmingly, some tasks were conceptually and/or procedurally inaccurate and vague. We suggest that educators cannot fully rely on ChatGPT to generate cognitively demanding fraction multiplication tasks using the area model. We offer recommendations for educators' strategic use of ChatGPT to generate cognitively demanding mathematical tasks.

This paper presents a replication study investigating the effectiveness of a strategy to assist students with learning difficulties in adding fractions. Drawing on the research conducted by Grünke et al. (2023), this single-case analysis extends the application of the "Look, Ask, Pick" (LAP) mnemonic technique initially introduced by Test and Ellis (2005) to struggling sixth-grade students. Employing a multiple-baseline design, we implemented the LAP strategy with four participants in a time lagged fashion. Significant improvements were observed in their fraction performance, demonstrating the intervention's potential to enhance understanding and proficiency. The students highly appreciated the strategy, deeming it of great importance. In conclusion, we address the limitations of our study, propose directions for future research, and delve into the implications for teachers. This research contributes to the understanding of fraction interventions and offers valuable insights into facilitating the mathematical progress of students facing challenges.

The purpose of this study is to examine how the categorization skills of the 6th grade students in fractions are distributed and to determine the classification errors made by the students, along with the procedural errors of the students in fractions. The descriptive survey model, one of the quantitative research methods, was used in the study. In this study, 292 6th grade students from four middle schools in two districts of a province in Turkey participated. Developed by the researcher, the fraction operation skill test and classification skills identification test were used. The results showed that students had difficulty in distinguishing examples and characteristics of fractions and fraction types, and even though they partially succeeded in operations, they made procedural errors. The researchers recommend combining procedural knowledge with conceptual knowledge and explaining the basic characteristics of the concept by comparing examples with non-examples.

Fractions are core content of elementary school mathematics, and conceptual knowledge of fractions is essential when developing a comprehensive understanding of fractions. Previous research, however, has indicated limitations in student teachers' fraction knowledge. This study investigated 57 Swedish elementary school student teachers' conceptions of fractions. The data were collected using a paper-and-pencil questionnaire and analysed with an analytical framework building on previous research on four core components of fractions. Using the devised analytical framework, we were able to characterise the conceptual content shown in the student teachers' answers and identify gaps in their fraction knowledge. The most severe gaps were identified in relation to interpretations of fractions, where only the part-whole and the quotient interpretations were identified; the measure, operator, rate, ratio, and number interpretations were missing completely. Aspects of fractions related to representations and procedures were better represented in the participants' conceptions of fractions, but we also illustrate substantial differences between the student teachers. In addition to this quantitative description, we provide qualitative examples. The results raise some questions and implications to be addressed in teacher education programs when developing student teachers' fraction knowledge.

Cognitive demand is a crucial dimension of instructional quality. Its heterogenous operationalizations call for refined investigations, with respect to discursive richness (generic conceptualizations) and conceptual richness (subject-related conceptualizations). Considering not only teachers' intended cognitive activation (operationalized, e.g., by tasks), but also the enacted activation and individual students' participation as realized in the interaction, raises the question of how far the interaction quality is associated with students' prerequisites, school context, and class composition. In this paper, we present a video study of leader-led small-group instruction (in 49 groups of 3-6 middle school students each) with the same fraction tasks, so that differences in interaction quality can be scrutinized in generic and subject-related conceptualizations. In spite of equal task quality, large differences occurred in interaction quality across heterogenous class compositions. The regression analyses revealed that the enacted activation and individual participation were significantly associated with the school context (of higher-tracked and lower-tracked schools), but much less with individual learning prerequisites. These findings reveal the need to capture students' collective and individual engagement in cognitive demands in the interaction and in generic and subject-related conceptualizations and to systematically investigate their association with class composition.

The selection of tasks based on the evaluation of task features can be considered a core practice of teaching and a relevant component of teaching quality. This is typically part of teachers' preparation for their classroom teaching, which prompts the following question: What are the characteristics of the tasks that teachers use when selecting tasks for differentiated teaching? To answer this question, we analyzed systematic differences in the focus of 78 in-service high school and lower secondary school teachers during the evaluation of task features. The teachers had to select eight tasks about the practice of fractions with respect to their differentiation potential--operationalizing their adaptive teaching competence from a mathematics educational perspective. To analyze the differences, we performed a cluster analysis of the task features that the teachers drew upon. Three groups of teachers could be identified with variations in their focus on directly or indirectly relevant, domain-specific or domain-general task features. Taking into account such variations may explain differences in teaching quality and student outcomes and may be relevant when designing teacher professional development programs.

This study's purpose was to examine the effects of the concrete-representational-abstract integrated (CRA-I) sequence on the performance of students who struggled with rational number concepts. Three students in southeastern U.S. fifth grade class participated in the study. The CRA-I intervention was grounded in the principles of explicit instruction and showed rational number concepts related to fractions and decimals using fraction blocks, number lines, base 10 blocks, and coins. Students learned about unit fractions, fraction magnitude, fraction equivalence, addition of fractions with unlike denominators, equivalent decimals, and notation of fractions as decimals. The researchers used a multiple probe across behaviors design and demonstrated a functional relation between CRA-I and three behaviors: decreased fraction estimation error, accuracy in adding fractions with unlike denominators, and accuracy writing fractions as decimals.

Identifying the knowledge resources teachers productively and unproductively draw upon can provide a means by which to create support structures to develop a more robust understanding of the content. To provide more informed grade-level support structures in teacher education programs, this study examined the knowledge resources 20 secondary pre-service teachers (PSTs) and 13 elementary PSTs drew upon when solving a comparison proportional reasoning problem. Data from written work and videos of PSTs' explanations were analyzed using the robust understanding of proportional reasoning for teaching framework. Both elementary and secondary PSTs ubiquitously drew upon the same four knowledge resources (comparison of quantities, ratios, proportional situation, and ratio as measure). Elementary PSTs were more apt to counterproductively draw upon the knowledge resource ratios ? fractions, while secondary PSTs more often counterproductively drew upon equivalence. Mathematics educators can leverage the knowledge resources afforded by this task and strategically highlight productive and counterproductive resources to tailor instruction that develops PSTs' robust understanding of proportional reasoning.

Background: Recent research showed that cross-notation magnitude knowledge of fractions and decimals was related to better performance in fraction arithmetic, but it remains unclear whether it made an independent contribution to fraction arithmetic longitudinally when other cognitive variables are considered. Aims: To examine the extent to which children's earlier knowledge of cross-notation magnitude predicted subsequent performance in fraction addition and subtraction as well as fraction multiplication and division longitudinally. Sample: Three hundred and fifty-four Chinese children (Mage = 112.1 months). Methods: During the first wave of assessment, a range of cognitive abilities of children were measured, including within-notation fraction and decimal magnitude comparisons, whole-number arithmetic fluency, non-verbal intelligence, attentive behaviours, counting recall, word-level reading, and phonological awareness. Twelve months later, the same children were assessed again with two tasks of fraction arithmetic: fraction addition and subtraction as well as fraction multiplication and division. Results and Conclusions: Multiple linear regressions showed that within-notation fraction and decimal magnitude knowledge predicted fraction addition and subtraction longitudinally, after the effects of working memory, nonverbal intelligence, language skills, attentive behaviour, and whole-number arithmetic were controlled. Cross-notation magnitude knowledge made independent contributions to fraction addition and subtraction longitudinally beyond the influence of within-notation fraction and decimal magnitude knowledge and other covariates. However, within-notation fraction and decimal magnitude knowledge were not associated with fraction multiplication and division, whereas cross-notation magnitude knowledge remained a unique predictor. These findings suggest that it may be useful to incorporate cross-notation knowledge in the assessments of children's mathematics abilities and teaching.

Contribution: This article presents a new look at teaching the Laplace transform for engineering students by emphasizing the obsolescence of the current method of finding the inverse Laplace transform when solving differential equations, and by recognizing the important role of a computer-assisted environment in helping the students understand the main idea behind the Laplace transform, instead of asking the students to repeat computational processes by hand. Background: The Laplace transform is a widely used integral transform that has important applications in many areas of engineering, and therefore, has a central place in the curricula for engineering education. However, according to several research articles, many students experience great difficulties understanding the Laplace transform. Research Question: Is the use of partial fractions and Laplace transform tables necessary for a proper conceptual understanding of the Laplace transform method? Methodology: Using the anthropological theory of the Didactic as an educational platform, the current teaching of the Laplace transform method, is analyzed. A parallel discussion of the teaching of logarithms at the upper secondary school level is drawn, where, previously, this also took place using the tables of logarithms, but now the reliance on calculators is overwhelming. The authors suggest a method of teaching the Laplace transform in a computer-assisted environment. Findings: In the light of the shift in computer hardware and software, the authors conclude by calling for innovation in and revision of engineering education through bridging the gap between procedures and understanding, by using computer software, where it is suitable.

After years of noticing the challenge our students have with fraction operations, we decided to implement a scaffold approach that focuses on using benchmarks to better develop both students' understanding of a fraction as a quantity and their ability to think about fraction operations meaningfully. While we found this approach supported students in making sense of fraction addition, we also found that students did not always respond as anticipated. This led to rich discussions tied to important mathematical ideas related to fractions and fraction operations, for example, the importance of the whole and reasonableness of a solution. We showcase student thinking in response to the provided tasks and offer teaching insights to help aid students in making sense of fraction magnitude and operations using benchmarks in mathematics classrooms.

Teaching requires attention to individual student needs by providing both adequate challenge and sufficient support to help students successfully gain academic skills (Shurr et al., 2019). The learning stages framework divides typical learning into four distinct stages: acquisition, fluency, maintenance, and generalization (Collins, 2012; Haring & Eaton, 1978). Thinking in terms of the learning progression can help teachers assess student performance and determine how they can best be supported to progress. This article will lead readers through the process of using the four stages of learning as a framework for assessment (i.e., understanding where students are currently performing), goal setting (i.e., setting the instructional aim), and instruction (i.e., planning for and delivering instruction aligned to individual student needs) within the context of mathematics for students with a variety of disabilities and support needs.

Understanding the magnitudes represented by numerals is a core component of early mathematical development and is often assessed by accuracy in situating numerals and fractions on a number line. Performance on these measures is consistently related to performance in other mathematics domains, but the strength of these relations may be overestimated because general cognitive ability has not been fully controlled in prior studies. The first of two meta-analyses (162 studies, 33,101 participants) confirmed a relation between performance on whole number (r = 0.33) and fractions number (r = 0.41) lines and overall mathematics performance. These relations were generally consistent across content domains (e.g., algebra and computation) and other moderators. The second (71 studies, 14,543 participants) used meta-analytic structural equation modeling to confirm these relations while controlling general cognitive ability (defined by IQ and working memory measures) and, in one analysis, general mathematics competence. The relation between number line performance and general mathematics competence remained significant but reduced ([beta] = 0.13). Controlling general cognitive ability, whole number line performance consistently predicted competence with fractions but not performance on numeracy or computations measures. The results suggest an understanding of the magnitudes represented by whole numbers might be particularly important for students' fractions learning.

The aim of this study was to analyze the joint, relative, and unique predictive value of students' prior knowledge of mathematics (knowledge of fractions and ability to divide natural numbers) and general cognitive ability (fluid intelligence and working memory) upon general mathematics achievement in the last year of primary school. Seventy-five students participated (M age = 11.2 years old, SD = 0.40). Hierarchical regression analysis showed that the ability to divide and fractions knowledge accounted for 41% of the variance in mathematics achievement, both acting as significant predictors. By incorporating working memory and fluid intelligence into the model, fraction knowledge showed to be no longer a significant predictor. These general cognitive skills explained an additional 8% of the variance in mathematics knowledge, both being significant predictors and contributing to mathematics achievement in a unique way. The implications of these results for mathematics teaching are discussed.

Developing mathematics proficiency requires an understanding of mathematics vocabulary. Although previous research has developed several measures of mathematics vocabulary at different grade levels, no study focused solely on fraction vocabularies. We developed and tested a measure of fraction vocabulary for students in Grade 4 to determine the internal consistency and difficulty level of such a measure. Analysis indicated the measure demonstrated high internal consistency. Students, on average, answered less than one-third of fraction vocabularies correctly. We also detected performance differences between students with and without mathematics difficulty and dual-language learners and their peers.

This study investigates the effects of a synchronous online intervention that explicitly taught equivalent fractions using virtual manipulatives to fifth-grade students with learning disabilities. Employing a multiple probe across participants single-case design, this study provided 15 fraction lessons via video conferencing programs to three fifth-grade students with learning disabilities. During these online interventions, participants received one-on-one explicit instruction, practiced key concepts using virtual manipulatives, and solved fraction problems using interactive boards. Researcher-developed probes measured the participants' percentages of correct answers across baseline, intervention, and maintenance phases via an online assessment tool. In overall, students' performance improved as the intervention was introduced; however, the extent and maintenance of improvement varied according to the students' participation and perspectives regarding online instruction.

Recent solar eclipses provide relevant real-world contexts for learning about the scientific phenomena of the lunar phases. News coverage of the phenomenon may have raised questions such as, "Why does the Moon look different at different times, and sometimes as if it is not there?," and "What patterns can be found in the lunar phases?" Teachers can use these recent events to launch investigations into the mathematics of the phases. Connecting learning across the mathematics and science disciplines provides opportunities for students to deepen their conceptual understanding and apply what they have learned in new contexts (Vasquez et al., 2013). In this article, the authors present how they used the Moon Pie simulation in a sixth-grade science classroom to bridge the mathematics of angle measurement, fractions, covariation, and co-splitting (multiplicative/proportional covariation) with the scientific phenomenon of lunar phases.

The aim of the study reported on here was to assess the impact of differentiated instruction in terms of mathematics achievement and the attitudes of secondary school learners to reveal their views on differentiated instruction. The study was designed according to a mixed method design in which both quantitative and qualitative methods were used. The study group, which constituted the quantitative dimension of the study, consisted of 2 control groups and 1 experimental group. The Mathematics Achievement Test, Mathematics Attitude Scale and a semi-structured interview form were used as data collection tools. One-way anova and descriptive analysis techniques were applied for the analysis of the data. We concluded that differentiated instruction in mathematics courses increases secondary school learners' mathematics achievement, but has no effect on their attitudes towards mathematics.

In this study, it is aimed to examine the perceptions of the mathematics teacher candidates regarding the concept of fraction. 32 teacher candidates participated in the research carried out in the screening model. The word association test was used as a data collection tool and the obtained data were analyzed by content analysis method. According to the results of the analysis, it is seen that the perceptions of teacher candidates regarding the concept of fraction are mostly concentrated in the themes of meanings of fractions, operations in fractions, and representation of fractions. In addition, the themes of numbers, notation, and other mathematical topics in which it is used are other themes that emerged. It was seen that the teacher candidates expressed the quotient meaning at most, they did not mention the percent meaning. It was seen that they mostly expressed the addition operation in fractions and the type of compound fraction. As a result, it is seen that pre-service teachers' perceptions about the concept of fraction are limited. Pre-service teachers can be given a more comprehensive and relational education on the concept of fraction.

Since mathematics scores have not improved appreciably in the last 20 years, a need exists to improve fraction instruction for preservice teachers (PSTs), so future elementary students are more successful with fractions (NCES, 2019). Unfortunately, it appears that many future teachers have not mastered fractions to the point they can teach fractions without struggling or repeating the superficial approach their teachers used. How can one expect elementary students to master rational numbers and improve test scores when their teachers still need to improve their own understanding and skill? Educator preparation programs (EPPs) and faculty members have a significant opportunity to address and improve PSTs' fraction knowledge and thus their ability to teach rational numbers before they enter the classroom. Since fractions are so vital in school, STEM careers, and the real world, exploring improved fraction performance among PSTs may also benefit practicing teachers and students alike (Bruce et al., 2013; Gabriel, 2016). Thus, this integrative review of previous PST fraction research seeks to guide future studies and inform EPPs about the improvement of fraction mastery among future educators. The review was guided by the following research questions: (1) What has research revealed about PSTs' fraction knowledge and skill deficiencies? (2) What strategies may not be effective in helping one master fractions? and (3) What strategies and interventions improved PSTs fraction knowledge and performance?

Fractions are an integral part of the mathematics curriculum. Most students acquire proficiency with these concepts during the course of their elementary education and are usually able to perform basic fractions operations when reaching middle-school age. However, a considerable number of students require extra help to not fall further and further behind in the curriculum. In this study, we extended the use of a simple strategy (Look, Ask, Pick; Test & Ellis, 2005) that holds the potential to help students with problems understanding and working with fractions catch up with their classmates. We applied a multiple-baseline design across four struggling sixth graders. After receiving the instruction, all participants' performance on fractions improved significantly; moreover, they viewed the strategy as highly useful. Limitations of the study, future directions of research, and implications for teachers regarding the instructional utility of the intervention are discussed.

There is strong evidence from research conducted in the United States that fraction magnitude understanding supports mathematics achievement. Unfortunately, there has been little research that examines if this relation is present across educational contexts with different approaches to teaching fractions. The current study compared fourth and sixth grade students from two countries which differ in their approach to teaching fractions: Australia and the United States. We gathered data on fraction and decimal magnitude understanding, proportional reasoning, and a standardized mathematics achievement test on whole number computation. Across both countries, reasoning about rational magnitude (either fraction or decimal) was predictive of whole number computation, supporting the central role of rational number learning. However, the precise relation varied, indicating that cross-national differences in rational number instruction can influence the nature of the relation between understanding fraction and decimal magnitude and mathematics achievement. The relation between proportional reasoning and whole number computation was fully mediated by rational magnitude understanding, suggesting that a key mechanism for how reasoning about rational magnitude supports mathematics achievement: proportional reasoning supports the development of an accurate spatial representation of magnitude that can be flexibly and proportionally scaled, which in turn supports children's mathematics learning. Together, these findings support using measurement models and spatial scaling strategies when teaching fractions and decimals.

This study aims to examine the effect of problem posing-based active learning activities on students' problem-posing skills and problem-solving achievement. To this aim, an experimental design with pre-test post-test control group was employed. The participants consisted of two groups of sixth graders, one experimental group (N=23) and one control group (N=25). Students in the experimental group were exposed to seven problem-based active learning activities over the course of six weeks. The study used problem-solving and problem-posing tests to collect data. The results revealed that even though the intervention was not statistically significant, the increase in the problem-solving mean score of the experimental groups was higher than that of the control group. Problem posing pre- and post-test scores of the experimental group differed statistically significantly with a high level of effect size ([eta squared] = 0.80). Finally, educational implications are discussed, and recommendations are made for future research.

This study focuses on the knowledge revealed and developed by Elementary Mathematics teachers, in a teacher education course related to the representation of fraction division and the flexibility of the reference unit. The teachers solved a task aimed at mobilizing (and accessing) their knowledge related to their approaches to the sense of division, representation, and reference unit regarding fraction division. The results suggest that teachers face challenges when representing and justifying fraction divisions using pictorial models, especially when the divisor is a non-unit fraction. This is based in a gap regarding the flexibility of the reference unit to which the numbers refer in their representations, as well as a challenge concerning the sense of fraction division and the different forms of representation. With this research we intend to contribute to reducing the scarcity of empirical studies in the area and the importance of this specialized teachers' knowledge to deal with this topic.

The concept of fractions given in learning is the concept of part of whole and part of unit. The development of student's concepts of fractions can be built through fraction schemes. A partitive fraction scheme is a scheme that estimates the size of the fraction in the form of non-units to the whole that is not partitioned. The concept of fractions using a partitive fraction scheme uses a strong understanding of the concept of the part unit and part-whole fractions. This study aims to photograph the process of developing students' mental actions in constructing the concept of fractions using a partitive fraction scheme. This research method uses a qualitative approach with a case study type on students who are able to express partitive fraction schemes. The participants of this study were fifth-grade elementary school students who already understood the concept of part-whole and part of unit. The results showed that students used two types of mental action processes in the partitive fraction scheme: direct and indirect.

Despite the research regarding the importance of peer review and feedback in pre-service special education teachers, there exists a gap in teaching complex mathematical concepts such as fractional operations. This study sought to address this gap by investigating how pre-service teachers can effectively appraise and revise peer-generated teaching transcripts focusing on fraction operations and compare their feedback with those of experienced educators. The research sought to understand how this integrated approach can contribute to improving the instruction of pre-service special education teachers in the field of mathematics education. A modified version of Crespo's (2018) generating, appraising, and revising of representations was utilized to analyze the video content. Comparisons of the reviews showed that pre-service teachers may not have the content knowledge or experience to provide in-depth feedback to support learning as experienced educators. The article concludes with findings and recommendations for teacher educators who utilize anonymous peer review in teacher preparation for special educators.

This study aims to investigate the effect of using Minecraft on Year 5 pupils' higher-order thinking skills (HOTS) in fractional problems-solving. A quasi-experimental pretest and posttest non-equivalent groups design was used. The study sample involved 65 Year 5 pupils from two intact classes which consists of 31 pupils as the treatment group and the other 34 pupils as the control group. Minecraft was used as the intervention in the teaching and learning of fractions in the treatment group. The research data was collected through the administration of pretest and posttest while the data was analyzed using paired sample t-test and independent sample t-test. The research resulted in several findings. Among them is the significant difference in the ability to solve fractional problems which requires the use of HOTS between pupils who were exposed to the teaching and learning of fractions using Minecraft and pupils who were exposed to the teaching and learning of fractions using conventional methods. This study has proven that the use of Minecraft in the teaching and learning of fractions has the potential to facilitate and enhance pupils' level of HOTS.

The purpose of this article is to validate the relevance of a concept inventory on fractions by measuring the presence and evolution of misconceptions among prospective primary and pre-school teachers, including the overcoming of their misconceptions during and at the end of the instructional intervention. Seven text statements were defined and composed a "Likert scale" concept inventory. This was administered to students at three different stages of the learning process to measure the understanding gain: before the start of the course, after the instructional intervention consisting of a lecture and a seminar on fractions, and finally at the end of the semester. The results from the initial testing confirmed the misconceptions about fractions of the students. Based on the experiment, it is thus obvious the need to allocate a longer time for understanding and fixing the concept of fraction for prospective primary and preschool teachers.

This study aims to develop a didactic design using Engklek games to teach the concept of fractions in elementary schools. This study combines a cultural approach, namely Engklek games with mathematics in the concept of fractions. This study used didactical design research, with 2 research subjects, namely 20 students in grade 5 for learning obstacles as the basis for making didactic designs, and 50 students in grade 4 to determine the impact of implementing didactic designs. Data collection techniques used are tests, interviews, and observation. The data analysis technique uses a qualitative method to see learning obstacles in students and a quantitative method (descriptive and inferential statistics) to determine the impact of didactic design implementation. Based on the results of the Mann-Whitney test that the sig. is 0.000, meaning that there is a difference in students' mean scores on the concept of fractions before and after the implementation of the didactic design, this indicates that there is a positive influence on students getting the implementation of the didactic design using the Engklek game on the concept of fractions in elementary schools. This research contributes to education in an effort to create effective and meaningful learning by involving real contexts in real life through elements of culture.

The present study examined preservice elementary teachers' performance on the problems of multiplication and division of fractions and compared their performances and analyzed the misconceptions. An instrument including 11 fraction multiplication and division tasks was given and the task involved three contexts: making own story problem, computations, representing operation using visual model. The findings reported that among the three contexts, making a diagram was the most challenging task for both operations, and their division performance varied depending on the division problem types. The author suggests that specific emphasis with rich story problem with different whole(s) in fraction, carefully designed context with different types of division concept, and building fractional number sense can help both PSTs and students reduce misconceptions and enhance deeper understanding of fraction operations.

This study contributes to efforts to characterize teaching that is responsive to children's mathematical ideas and linguistic repertoire. Building on translanguaging, defined in this article as a pedagogical practice that facilitates students' expression of their understanding using their own language practices, and on the literature surrounding children's mathematical thinking, we present an example of a one-onone interview and of the circulating portion of a mathematics class from a secondgrade classroom. We use these examples to foreground instructional practices, for researchers and practitioners, that highlight a shift from a simplified view of conveying mathematics as instruction in symbology and formal manipulation to a more academically ample discussion of perspectives that investigate critically both mathematical concepts and their modes of transmission, which involve language practices, that are crucial for educating bilingual children.