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The current research aimed to unpack teachers' knowledge of integers by investigating how they used the number line and counter models to represent the two meanings of division (i.e., partitive and measurement). The participants were three middle school mathematics teachers working in different cities in Türkiye. Data consisted of teachers' written responses to an open-ended questionnaire consisting of four division operations and interviews conducted thereafter. Findings revealed that although two teachers could accurately model all division operations with the number line model, one teacher could neither provide a problem context nor a model displaying one of the division operations. For the counter model, only one teacher could accurately model all division operations in the questionnaire.

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.

The present research focuses on analyzing how fourth-grade elementary school students (ages 9 to 10) solve and interpret the result of non-routine problems, precisely division measurement and division-partition with remainder. The methodology is qualitative, with a descriptive and interpretative approach. The information was collected using a questionnaire consisting of three problems (two of quotitive division and one of partitive division) and a clinical interview. The results showed the importance of using the division, multiplication, and addition algorithms to give a realistic answer to the problems. In the same way, it was possible to demonstrate the graphic strategy combined with counting to give a realistic answer to the problem. However, students were found to use division correctly but without an interpretation of the remainder or quotient. Likewise, they struggled to choose the correct procedure to solve the problem. These data suggest including realistic problems in mathematics classrooms to make sense of mathematical concepts in real life or the student's context. Likewise, this study provides implications on the mathematical problems that the teacher proposes in the classroom, where not only the division algorithm should be taught mechanically, nor focus on posing routine problems that lead the student to use a single heuristic resolution strategy. Essentially, the teacher is required to include real-world problems, where the student can awaken creativity to represent in different ways the understanding of a problem and, therefore, different strategies to solve it. In addition, that the student has the ability to check the result of the problem, with the conditions, situations or circumstances imposed by reality or everyday life. [This is a reprint of an article originally published in "Mathematics Teaching Research Journal" v14 n5 p159-180 Win 2022.]

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.

Background: Internationally, the teaching of division has noted that the use of sharing situations with sharing actions (one-by-one distribution) is the predominant division model at the beginning of schooling. In South Africa, research suggests a sharing situation with sharing actions is also preferred in the early grades. Aim: This paper aims to look at the predominant approaches to the use of division actions that teachers offer in teaching division tasks. Setting: The study is set in three government schools in Gauteng, South Africa. Methods: In this qualitative study, the teachers were observed through video recording, and then the video recording was transcribed, and semiotics was used to make sense of their teaching. Results: The findings of this article suggest that grouping actions and group-based approaches to teaching division tasks are more prevalent than sharing through one-by-one distribution actions, even when sharing situations are used. Conclusion: This study concludes that grouping actions and group-based approaches are part of how teachers solve sharing situations. Contribution: This study concludes that in a South African context, identifying the grouping actions and group-based approaches linked to sharing situations is a more efficient way of solving sharing situations and will assist teachers in explaining division tasks more coherently.

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 used traditional and holographic video, along with eye-tracking technology, to examine how preservice teachers' physical act of looking is associated with how they attend to and assess students' fraction reasoning. Findings revealed that, although viewing of holograms may have influenced more focus on students' work area, there was little observed difference in the written noticing of preservice teachers in each condition. Additionally, by examining eye-gaze data for how teachers focused rather than simply what they focused on, findings were able to distinguish between how different participants perceived students' fraction work. Results indicated that the more participants described students' conceptual reasoning, the more likely they were to focus on the mathematical work students were doing and not simply on the region they did such work.

We present an external close replication of the 1985 Fischbein et al. study about intuitive models of multiplication and division. We administered two batteries of mathematics items developed in the original study, via a spiralling process, to a quota sample of 903 students attending grade 7. Compared with the analytic strategy based on the count of correct answers employed in the original research, our study goes a step further as we propose a methodological approach that guarantees measurement invariance, thus allowing for the direct comparison of different groups of students and/or items. The advantages of Rasch estimates compared to percentages of correct answers over the total are critically discussed to show why the former should be considered as more robust than the latter.

In prior work, we fit the mixture Rasch model to item responses from a fractions survey administered to a nationwide sample of middle grades mathematics teachers in the United States. The mixture Rasch model located teachers on a continuous, unidimensional scale and fit best with 3 latent classes. We used item response data to generate initial interpretations of the reasoning characteristic of each latent class. Our results suggested increasing facility reasoning about fraction arithmetic from one class to the next. The present study contributes two further arguments for the validity of our initial interpretations. First, we administered the same survey to a new sample of future middle grades mathematics teachers before and after 20 weeks of instruction on multiplication, division, and fractions, and we found that from pretest to posttest future teachers transitioned from one latent class to another in ways consistent with increased proficiency in fraction arithmetic. Second, we interviewed 8 of the future teachers before and after the instruction and found that future teachers' reasoning during interviews was largely consistent with our original interpretation of the 3 latent classes. These results provide further support for our original interpretation of the mixture Rasch analysis, demonstrate the utility of our approach for capturing growth and change in future teachers' reasoning during teacher education coursework, and contribute innovative applications of psychometric models for surveying teachers' reasoning at scale.

This study used units coordination as a theoretical lens to investigate how whole number and fraction reasoning may be related for preservice teachers at the conclusion of a math methods class. The study contributes quantitative evidence that units coordination provides a common foundation for both mathematical knowledge for teaching whole number addition and subtraction and for mathematical knowledge for teaching fraction multiplication and division. An analysis of PSTs' written responses to two different types of tasks (referent unit and appropriateness of proportional reasoning) revealed that some aspects of mathematical knowledge for teaching rational number and the multiplicative conceptual field are associated with units coordination (referent unit tasks), whereas others are not (appropriateness tasks). A logistic regression of PSTs' conceptions of multidigit number on their scores on these tasks provided empirical evidence that the most sophisticated conceptions of multidigit number (corresponding with having interiorized three levels of units) were associated with referent unit tasks but not associated with appropriateness of proportional reasoning tasks. Together, these findings support the claim that units coordination has promise for explaining the relationship between teachers' mathematical knowledge for teaching multidigit addition and subtraction and fraction multiplication and division. Implications for teacher education are discussed.

The purposes of this study are to report current Preservice Teachers' (PTs) decimal abilities regarding the types of error patterns that they commit and to analyze characteristics of the errors in the problems posed for multiplication and division in decimals. We have determined common concept- and procedure-based errors made by the PTs. 37 PTs in the USA were asked to complete two decimal multiplication problems and two decimal division problems. Results indicated that four major types of error patterns could be detected: Misunderstanding of the place value, misunderstanding of mathematical equivalence, arithmetic errors, and reversing the positions of divisor and dividend. PTs were more likely to have concept- than procedure-based errors. This indicates that PTs may have limited knowledge on the definition of decimals and the meaning of mathematical equivalence, as well as a lack of understanding of place value. Consequently, we believe that the learning of the concept of decimals requires conceptual change. The concept-based errors found intersect with those of students in primary school, so this helps Teacher Education programs in ways that promote PTs' mathematical understanding of decimal operations on a conceptual basis to prevent their future students from making errors.

The present research focuses on analyzing how fourth-grade elementary school students (ages 9 to 10) solve and interpret the result of non-routine problems, precisely division measurement and division-partition with remainder. The methodology is qualitative, with a descriptive and interpretative approach. The information was collected using a questionnaire consisting of three problems (two of quotitive division and one of partitive division) and a clinical interview. The results showed the importance of using the division, multiplication, and addition algorithms to give a realistic answer to the problems. In the same way, it was possible to demonstrate the graphic strategy combined with counting to give a realistic answer to the problem. However, students were found to use division correctly but without an interpretation of the remainder or quotient. Likewise, they struggled to choose the correct procedure to solve the problem. These data suggest including realistic problems in mathematics classrooms to make sense of mathematical concepts in real life or the student's context. Likewise, this study provides implications on the mathematical problems that the teacher proposes in the classroom, where not only the division algorithm should be taught mechanically, nor focus on posing routine problems that lead the student to use a single heuristic resolution strategy. Essentially, the teacher is required to include real-world problems, where the student can awaken creativity to represent in different ways the understanding of a problem and, therefore, different strategies to solve it. In addition, that the student has the ability to check the result of the problem, with the conditions, situations or circumstances imposed by reality or everyday life.

Comics engage students in learning mathematics through their interactive nature and the relatively simple language they use. Not only that, it also helps expand students' thinking through the introduction of contextualized mathematics. This is also in line with the latest math education trend to engage students with math problems in a real-world context. As a result, there is a great potential for the use of comics as a teaching tool in primary school mathematics lessons. The current research works pointed out that comics increase the efficiency of teaching and strengthen the relationship between teacher and students. The purpose of this study is to determine the effect of educational comics-supported instruction on academic achievement of primary school students in division concept. The study used quasi-experimental design with pretest-posttest control group. The study group consists of 42 second grade pupils (21 for experimental and 21 for control group) studying in a public primary school located in Nigde, Turkey. As for data collection, academic achievement test was developed based on the learning outcomes of the mathematics curriculum. The pilot study was conducted to determine the statistics and the quality of test items. In order to ensure the validity of this test, evidences were collected and the reliability of the test calculated. The results of this study showed that the students who learn division via educational comics-supported instruction is more successful than the students who learn division via traditional instruction. Two of the reasons could be listed as (1) division is a complex and abstract concept so that traditional instruction have some difficulty in making the understanding easier, (2) educational comics strengthen the communication between student and teacher, attract attention of the students. This study is considered important in terms of emphasizing that educational comics are a preferred tool for some concepts of mathematics instruction.

In the current study, it is aimed to determine the activities that need to be done to eliminate the mistakes made by primary school fourth grade students in multiplication and division operations and to present solution suggestions for eliminating these mistakes. The study employed action research, one of the qualitative research methods. The study group was constructed by the criterion sampling method, one of the purposive sampling methods. The study group is comprised of 10 fourth graders attending a primary school in the spring term of the 2016-2017 school year in the city of Istanbul and making similar mistakes. A student information form, clinical interview form and student worksheets were used as data collection tools in the study. Activities prepared in line with the principles of Realistic Mathematics Education (RME) were applied in order to eliminate the mistakes made by the students in the multiplication and division operations. When the mistakes made by the students in the multiplication and division operations were examined, it was revealed that the source of the mistakes was the operational, conceptual and problem situations. During the implementation of RME activities, it was determined that the mistakes of the students started to be eliminated. After the implementation, it was found that the mistakes of the students committed in the multiplication and division operations decreased. Thus, it can be said that RME is an effective application in reducing the mistakes in multiplication and division operations made by students in primary school.

In this paper, we have tried to clearly explain the meaning of fractions and operations with fractions. Also, we have tried to illustrate with some examples how to apply the rules of operations with fractional numbers, giving different practical techniques through different figures, visualization methods, and concretizing problems. The paper presents with examples of the most common mistakes made by students, giving suggestions for their avoidance, as well as through a questionnaire, which considered the survey of 60 students of grade 6 of school, "Elena Gjika" City Prishtina, Kosovo, with the help of the statistical test we have concluded that practical techniques such as figures, visualizations, and video lessons have a close dependence on the meaning and operation of fractions.

In this paper, the effects of the online learning of the mathematical area of divisibility are studied, by comparing the achievements of students who have learned this mathematical topic in the online and in-class environments. Data for this study were collected in seven schools at the beginning of the seventh grade of elementary education, with 383 participants aged 12 to 13. The test with four questions was designed according to the standards and levels set by APOSO (Agency for Pre-Primary, Primary, and Secondary Education in Bosnia and Herzegovina). Data were analyzed using a two-sample t-test and a Chi-square test. The results highlighted that there was no statistical difference in the total scores between the students who learned divisibility in the in-class environment and those who learned it in the online environment. When comparing students' achievement in each question separately, statistical difference appeared only in the question of the highest level according to APOSO. The mistakes that students made when solving divisibility problems were also part of this research.

The Akita approach is the guideline in learning that responds to rapid changes in the 21st century. This research aimed to determine the effectiveness of learning management based on the Akita approach in accordance with the criteria efficiency of 75/75 and to explore the index of effectiveness of learning management pursuant to the Akita approach. Furthermore, to discover students' learning retention and to study the students' satisfaction with learning management based on the Akita approach. The participants were 34 primary school students in the northeast of Thailand, selected by cluster random sampling. Research instruments comprised of learning management plans, achievement tests, and a questionnaire. Statistics were used, including percentage, mean and standard deviation, and dependent samples t-test. The results found that, firstly, learning management plans based on the Akita approach identified students' efficiency (E1/E2) at 86.46/85.29, which was higher than the specified criteria. Secondly, the index of effectiveness of the learning management plans based on the Akita approach on multiplication and division of students was equal to 0.6835 or 68.35 percent. Thirdly, for students who studied by using Akita-based learning management plans on multiplication and division topics, the post-test scores after studying and the post-test scores after they had finished studying for two weeks appeared to reveal no difference, which showed the persistence of students in learning. Lastly, students were satisfied with studying multiplication and division by Akita-based learning management plans with high overall scores and each item at the highest ([x-bar] = 4.65-4.88) level.

The number one plays a special role in mathematics because it is the identity element in multiplication and division. The present findings, however, indicate that many middle school students do not demonstrate mathematical flexibility representing one as a fraction. Despite possessing explicit knowledge of fraction forms of one (e.g., 95% of students indicated that 36/36 = 1), most students did not recognize and apply knowledge of fraction forms of one to estimate numerical magnitudes, solve arithmetic problems, and evaluate arithmetic operations. Specifically, students were less accurate in locating fraction forms of one on number lines than integer forms of the same number; they also were slower and less accurate on fraction arithmetic problems that included one as a fraction (e.g., 6/6 + 1/3) than one as an integer (e.g., 1 + 1/3); and they were less accurate evaluating statements involving fraction forms of one than the integer one (e.g., lower accuracy on true or false statements such as 5/6 × 2/2 = 5/6 than 4/9 × 1 = 4/9). Analyses of three widely used textbook series revealed almost no text linking fractions in the form n/n to the integer one. Greater emphasis on flexible understanding of fractions equivalent to one in textbooks and instruction might promote greater understanding of rational number mathematics more generally

The aim in the current study is to examine the conceptualizations of zero in arithmetic operations among students with learning disabilities (LD) and no learning disabilities (N-LD). The similarities and differences in the understandings of students with LD and N-LD of zero in arithmetical operations will be discussed. The study is a multiple case study with a qualitative research design. Six students, 3 students with LD and 3 with N-LD aged between 10 and 12 years participated in the study. The data were collected through clinical interviews. The data were analyzed by content analysis. Students' limited understanding of zero and operations affects their interpretation of arithmetic operations with zero. The conceptualizations of students with LD and N-LD regarding specifically division by zero show similarities with the exception of their use of the knowledge about operations. It has been observed that LD students have developed a different algorithm when it comes to addition and multiplication with zero. Through this study examining the differences in the understandings of students with LD and N-LD on a specific concept in terms of underlying conceptions, it is thought to provide an insight in terms of discovering LD and a more detailed recognition of these students' mathematics.

The positive effect of peer assessment and self-assessment strategies on learners' performance has been widely confirmed in experimental or quasi-experimental studies. However, whether peer and self-assessment within everyday mathematics teaching affect student learning and achievement, has rarely been studied. This study aimed to determine with what quality peer and self-assessment occur in everyday mathematics instruction and whether and which students benefit from it in terms of achievement and the learning process. Two lessons on division were video-recorded and rated to determine the quality of peer and self-assessment. Six hundred thirty-four students of fourth-grade primary school classes in German-speaking Switzerland participated in the study and completed a performance test on division. Multilevel analyses showed no general effect of the quality of peer or self-assessment on performance. However, high-quality self-assessment was beneficial for lower-performing students, who used a larger repertoire of calculation strategies, which helped them perform better. In conclusion, peer and self-assessment in real-life settings only have a small effect on the student performance in this Swiss study.

Skills and understanding of operations with negative numbers, which are typically taught in middle school, are crucial aspects of numerical competence necessary for all subsequent mathematics. To more swiftly and coherently develop the field's understanding of how to foster this critical competence, we need shared measures that allow us to compare results across studies with diverse populations and theoretical perspectives. Yet, to date no validated instrument exists to assess all four primary operations (addition, subtraction, multiplication and division). Thus, we conducted a Rasch analysis of the Integer Test of Primary Operations (ITPO) with 187 middle school students to provide a valid and reliable assessment with good person and item fit. The implications of this study are numerous for multiple stakeholders including scholars, test and textbook developers, as well as teachers. First, we validated three forms of the ITPO to foster future longitudinal studies of how integer arithmetic knowledge ismaintained or decays as well as how such knowledge might be related to success in STEM disciplines. Second, our analysis provides trustworthy insights about relative difficulty of integer problem structures because regardless of test form similar problem structures loaded together. For instance, sums of additive inverses were the easiest structure, whereas division by -1 was more difficult than multiplying or dividing by any other integer. We discuss each of these and other findings that have practical implications for learning and teaching integers. Third, for broader mathematics assessments in which minimal items can be included to measure integer knowledge, this study informs which items would serve the intended assessment purpose. Finally, we provide the three forms as an appendix in printable formats to ensure these validated tests are practical to implement for teachers as well as scholars.

Background: In several countries, children's math skills have been declining at an alarming rate in recent years and decades, and one of the explanations for this alarming situation is that children have difficulties in establishing the relations between arithmetical operations. Aim: In order to address this question, our goal was to determine the predictive power of previously taught operations on newly taught ones above general cognitive skills and basic numerical skills. Samples: More than one hundred children in each school level from Grades 2 to 5 from various socio-cultural environments (N = 435, 229 girls) were tested. Methods: Children were assessed on their abilities to solve the four basic arithmetic operations. They were also tested on their general cognitive abilities, including working memory, executive functions (i.e., inhibition and flexibility), visual attention and language. Finally, their basic numerical skills were measured through a matching task between symbolic and nonsymbolic numerosity representations. Additions and subtractions were presented to children from Grade 2, multiplications from Grade 3 and divisions from Grade 4. Results and Conclusions: We show that addition predicts subtraction and multiplication performance in all grades. Moreover, multiplication predicts division performance in both Grades 4 and 5. Finally, addition predicts division in Grade 4 but not in Grade 5 and subtraction and division are not related whatever the school grade. These results are examined considering the existing literature, and their implications in terms of instruction are discussed.

Over the last years, there is a growing interest in studying students' difficulties with rational numbers from a cognitive/developmental perspective, focusing on the role of prior knowledge in students' understanding of rational numbers. The present study tests the effect of the "whole" or "natural number bias" (i.e., the tendency to count on natural number knowledge to interpret information about rational numbers and deal with rational number tasks), on students' expectations about the "size" (i.e., bigger or smaller) and the "type" (i.e., natural number or decimal) of the results of multiplication and division. Items that were congruent and incongruent with students' assumed expectations were administered to 91 seventh and eighth graders, asking them to evaluate equalities presenting multiplication and division between given and missing numbers. The results showed that besides the already well-documented effect of the size of results (i.e., "multiplication makes bigger" and "division makes smaller"), students tended to think that the numbers involved in multiplication and division should be of the same type (i.e., natural or non-natural, e.g., decimals). Both size and type of the numbers involved in the operations were significant factors affecting students' evaluations, with size being stronger than type.

A one-on-one dialogue-based mathematics intelligent tutoring system (ITS) for learning multiplication and division of fractions was developed and evaluated in this study. This system could identify students' error types and misconceptions in real-time by using a block-based matching method. The adaptive dialogue-based instruction was supported by a response-driven tutoring model, which was constructed based on the diagnostic teaching methodology. Instructional strategies including provoking cognitive conflict, problem simplification and representational teaching were used in the tutoring model of the system. Effectiveness of the math ITS in remedial instruction was evaluated through a quasi-experimental study. The participants of the study were 66 sixth graders chosen from central Taiwan. They were divided into an experimental group of 35 and a control group of 31. One week after the pretest, the experimental group received 2-h one-on-one instruction via the math ITS, while the control group took a 2-h conventional teacher instruction with the same teaching content in the classroom. All participants took a post-test within 2 days after the remedial instruction. The results showed that the experimental group using the math ITS significantly outperformed the control group. Further analysis indicated that the math ITS had a significant effect on the lesser-performing group (the lower 75% in the pretest score). In addition, a usability and user experience survey showed that students were willing and likely to learn mathematics using the dialogue-based math ITS.

The aim of the study was to determine the concepts that primary school fourth-grade students reveal with their mind maps related to four operations (addition, subtraction, multiplication, division) and to compare the mind map and achievement test scores. In the study, a mixed method was used in which quantitative and qualitative data were collected and presented together. The study was carried out with a total of 14 students. There were eight girls and six boys, who studying in the fourth-grade of a primary school in Mentese district of Mugla, in the spring semester of the 2019-2020 academic year. In the research, mind maps created for four operations and success test were used as data collection tools. The data collection process was carried out simultaneously. In the analysis of the data, qualitative data were transformed into the quantitative analysis, and quantitative analyzes were made. Concepts in mind maps created for four operations were subjected to qualitative analysis and photos were added as evidence. As a result of the research, it was determined that the majority of the students adopted the concepts of addition, subtraction, multiplication, and division. Besides, it was revealed that there was a high-level positive relationship between the scores of the students obtained from the achievement test and the scores from the mind maps they created. The evaluation made using the mind map for primary school fourth-grade students is more advantageous than the achievement test in terms of determining the conceptual understanding.

Mathematics is still viewed as a culture-free subject. This forms a negative perception for students on mathematics. Most students assume that mathematics and culture are not related. This may occur because mathematics taught in school is not contextual and far from the reality of everyday life. Historically, mathematics has become a part of daily life. As a maritime nation, Indonesia has a diverse culture. But many teachers are not yet aware of the integration of the culture into mathematics learning. Barongko cake is one of the Buginese cultural heritages. Buginese people have unconsciously been practicing mathematics in making these cakes. Therefore, this research aims to explore activities in making Barongko cakes in the Buginese community that involves mathematical concepts. This research is a qualitative descriptive with an ethnographic approach. The data collection methods are carried out through observation, documentation, interview with an expert in making Barongko cake. This research found that Barongko making process involves mathematics in the concept of division, congruence and similarity, as well as a triangular prism, and half sphere. This cake has the potential to be used as a source of contextual mathematics learning in schools.

Explore the impact technology has on mathematical identity and agency when students use mathematical action technology to engage in cycles of proof and support case-based reasoning. This article showcases a mathematics task used in a fourth-grade class that allowed students to develop their conceptual understanding of division. The authors designed this mathematical challenge to address the content standard and create a learning experience during which all students meet the following objectives: (a) make conjectures about relationships among the divisor, dividend, and quotient; and (b) explain and justify their conjectures.

Common divisor is one of the concepts that started to be learned in secondary school and forms the basis of many concepts. But students generally have difficulty in making sense of it. The purpose of this study is to investigate concept formation processes of common divisor through a case study on seven sixth grade students. To do this, we designed a learning environment based on Realistic Mathematics Education and tried to determine how students construct the concept. The data collected through group and individual study papers and semi-structured clinical interviews were analyzed with content analysis within the APOS theoretical framework. The findings showed that one student was unable to internalize the action and her thoughts depended on the contextual problem. Other students internalized all common divisors and coordinated them with prime factorization. At the end, students were able to encapsulate all the processes into object by finding the product of common prime numbers. Moreover, students were able to identify greatest common divisor among all common divisors and clarified its meaning.

The aim of this study is to examine the semantic structures used by mathematics teacher candidates to transform algebraic expressions into verbal problems. The research is a descriptive study in the survey model, which is one of the quantitative research types. The study group of the research consists of 165 teacher candidates studying in the primary school mathematics teaching department of a state university in the south of Turkey in the 2019- 2020 academic years. 73.2% of the teacher candidates in the study group are female and 26.8% are male. Criterion sampling method, one of the purposeful sampling methods, was used in the selection of teacher candidates in the study group. While the Algebraic Expression Questionnaire Form was used as the data collection tool, the evaluation rubric of verbal problems was used in the analysis of the data. As a result of the research, it has been revealed that pre-service teachers are more successful in transforming algebraic expressions into verbal problems, but they have problems in creating problems with algebraic expressions that make up systems of equations. Again in the study, it was concluded that pre-service teachers used addition and subtraction problems more than multiplication and division problems. On the other hand, when the problems in the type of addition and subtraction are examined in the study, in the type of combining and separating; it has been concluded that the category of equal groups is mostly used in the problems of multiplication and division.

This paper presents an analysis of fractions errors displayed by learners due to deficient mastery of prerequisite concepts. Fractions continue to pose a critical challenge for learners. Fractions can be a tricky concept for learners although they often use the concept of sharing in their daily lives. 30 purposefully sampled learners participated in this study. The research instrument consists of fractions test whose questions were selected from various Annual National Assessment examination papers containing in addition, subtraction multiplication and division of fractional operations. The different types of errors displayed by learners were then identified, coded and categorized. The analysis showed that the main sources for errors were lack of understanding of the basic concepts, learners' prior knowledge, misconceptions and misapplication of rules. This study recommends that teachers should help their learners to develop fractions conceptual understanding. Learners need to be explicitly taught that errors are opportunities for learning, and that they are springboard of inquiry.

This study is a qualitative research which was conducted in order to reveal the instructional explanations of class teachers and primary school mathematics teachers working in state schools about division. A semi-structured interview form with three open-ended questions about division, prepared for this purpose, was examined by the experts. The semi-structured interview form inluded three questions asking the teachers to solve the long division operations of 3385: 13 = ?, 1238: 12 =? and 102102: 12 =? using the mathematical table of digits with a descriptive language as if they were telling the primary school students the solutions. While the first two questions were suitable with the 5th grade learning outcomes, the third question was suitable with a high level learning outcome. The main purpose of asking the 3rd question was to evaluate the instructional explanation of the teachers in a problem of different difficulty. The study group consisted of 34 teachers, 16 of whom were primary school mathematics teachers and 18 of whom were class teachers, working at central primary schools in a province located in Eastern Anatolia region of Turkey. The content analysis of the data showed that not all of the teachers could interpret the operation of division regarding the concept of digit accurately, and their division was result and reasoning oriented. However, it was found that few teachers made generalizations in a similar way. It was also seen that teachers who were at problem-solving level according to Kinach's (2002b) comprehension level framework could not make sense of the logic underlying the division. In addition, the reason why zero (0) was moved to the quotient and when the divisor sought in remaining number should be completed by the teachers could not be clarified because they did not know the logic of the division.

This article highlights the main ideas that underlie the differences in potential pragmatic knowledge constructs students experience when solving problems, between the zone of proximal development (ZPD) and the zone of potential construction (ZPC). This qualitative research is based on a phenomenological approach to finding the meaning of things that are fundamental and essential from the ZPD and ZPC phenomena. Researchers observed mathematics learning by a teacher on 24 fourth-grade students who were divided into groups A (high IQ) and B (low IQ). Data collection through tests, observation, and interviews. While the validity of the data is done through triangulation of methods and triangulation of sources. The results showed that students of the Upper (A) group had high IQ but small ZPD and ZPC. In contrast, students in the Lower (B) group have low IQ but large ZPD and ZPC. This result means that intelligence (IQ) is measured not only logically-mathematically but also in the verbal-linguistic and spatial-visual fields. The conclusion is that there are differences in the construction of students' knowledge in the learning zone. This difference occurs because the knowledge constructs that the students have previously had an effect on the accommodation process of the schemes that students have built while in the proximal development zone (ZPD) where scaffolding works. Meanwhile, the potential construction zone (ZPC) is not sufficient to describe the real development of students. However, it only reflects what students have accomplished.

This study compared two goal-setting approaches found in the Precision Teaching literature, namely the minimum celeration line and the beat your personal best during the mathematical practice of three male students diagnosed with autism, aged 8-9. An adapted alternating treatments design with a control condition was embedded in a concurrent multiple baseline across participants design. Each approach was randomly allocated to either the multiplication/division (×÷) table of 18 or 19, while no approach was allocated to the ×÷14 table that acted as a control. Instruction utilized number families and consisted of (a) untimed practice, (b) frequency-building, (c) performance criteria, (d) graphing, and (e) a token economy. Upon practice completion, an assessment of maintenance, endurance, stability, and application (MESA) was conducted. Participants improved with both conditions and maintained their performance well, while improvements with the control condition were weak. The beat your personal best approach was highlighted as slightly more effective in terms of average performance and more efficient in terms of timings needed to achieve criterion. No differences were identified in terms of learning rate (i.e., celeration) or performance on the MESA. More research is warranted to identify which goal-setting procedure is more appropriate for students in special education.

Children's multiplicative thinking as the recognition of equal group structures and the enumeration of the composite units was the subject of this research. In this paper, we provide an overview of the Multiplication and Division Investigations project. The results were obtained from a small sample of Australian children (n = 21) in their first year of school (mean age 5 years 6 months) who participated in a teaching experiment of five lessons taught by their classroom teacher. The tasks introduced children to the "equal groups" aspect of multiplication. A theoretical framework of constructivist learning, together with research literature underpinning early multiplicative thinking, tasks, and children's thinking, was used to design the research. Our findings indicate that young children could imagine equal group structures and, in doing so, recognise and enumerate composite units. As the children came to these tasks without any prior formal instruction, it seemed that they had intuitive understandings of equal group structures based on their life experiences. We argue that the implications for teaching include creating learning provocations that elicit children's early ideas of multiplication, visualisation, and abstraction. The research has also shown the importance of observing children, listening to their explanations of their thinking, and using insights provided by their drawings.

This study aimed at investigating how preservice teachers' understandings of division and reasoning about ratios support and constrain their formation of proportional relationships in terms of quantities. Six preservice teachers from a middle-grade preparation program in the USA were selected purposefully based on their mathematics performance in a previous course. An explanatory case study with multiple cases was used to make comparisons within and across cases. Two semi-structured interviews were conducted with each pair. The results revealed that preservice teachers who did not explicitly identify different meanings for division struggled to differentiate between the two perspectives on ratios. The results also showed that those teachers had difficulty forming proportional relationships while solving the proportion tasks. These results suggest that explicit identification of the meanings for both types of division is critical to keeping the two perspectives on ratios separate, which is a key aspect for a robust understanding of proportional relationships.

This study aimed to reveal the conceptual and operational conceptions of sixth-grade students in the process of division. The focus of the study included the strategies used in the division process, the students' understanding of the division algorithm, and their ability to interpret the remainder in a real-life context. Being qualitative in nature, the current study adopted the case study methodology. The sample of the study consisted of 64 sixth-grade students studying at two middle schools in the province of Kastamonu, Turkey in the 2018-2019 academic year. The data collection tool was a test consisting of five open-ended questions presented to the students. According to the research findings, while most of the students used the division operation in problem-solving, some students used different strategies, such as multiplication, addition, subtraction, and mental calculation. The majority of the students using the division algorithm were successful in applying the steps of the division operation but had difficulty in interpreting the remainder. In this research, it was also seen that the students had difficulties regarding the use of zero as a placeholder in the division operation. The students also encountered more difficulty in the division problems requiring the use of zeros in the last digits of the quotient than using zeros in other digits of the quotient.

The objective of this research was to analyze the causes and consequences of the systematic process of teaching and learning of the division of whole numbers, a study aimed at students and teachers of the 3rd and 4th level of basic education. For its development the qualitative paradigm was used through the modality of field research. As informants, twenty (20) students and two (2) teachers participated. All participants are enrolled in the 3rd and 4th level of the Giordano Bruno School located in the city of Quito, Ecuador. To collect the information, the observation technique and the semi-structured interview were used, and the instruments were subjected to a validity process, where the triangulation was applied. The results were analyzed to give an answer to the proposed objective of the manuscript, where the teaching-learning method applied to the students is presented as a critical-constructivist model, but in reality, a mechanic-academicist education system is used. It was evidenced that teachers shorten the logical processes of the division of whole numbers, by the time-space factor.

Students show deficient understanding on fraction division and supporting that understanding remains a challenge for mathematics educators. This article aims to describe primary students' understanding of partitive fraction division (PFD) and explore ways to support their understanding through the use of sequenced fractions and context-related graphical representations. In a design-research study, forty-four primary students were involved in three cycles of teaching experiments. Students' works, transcript of recorded classroom discussion, and field notes were retrospectively analyzed to examine the hypothetical learning trajectories. There are three main findings drawn from the teaching experiments. Firstly, context of the tasks, the context-related graphical representations, and the sequence of fractions used do support students' understanding of PFD. Secondly, the understanding of non-unit rate problems did not support the students' understanding of unit rate problems. Lastly, the students were incapable of determining symbolic representations from unit rate problems and linking the problems to fraction division problems. The last two results imply to rethink unit rate as part of a partitive division with fractions. Drawing upon the findings, four alternative ways are offered to support students' understanding of PFD, i.e., the lesson could be starting from partitive whole number division to develop the notion of fair-sharing, strengthening the concept of unit in fraction and partitioning, choosing specific contexts with more relation to the graphical representations, and sequencing the fractions used, from a simple to advanced form.

Purpose: Since there are a limited number of studies on how to develop relational thinking in secondary school students in mathematics education literature, this study will contribute to the field both in theoretical terms and concerning the implications for in-class applications. In this respect, this study aims to examine how to develop the relational thinking skills of 5th-grade students. Research Methods: The participants of this study, which was adopted as a research design of the teaching experiment, were six students attending 5th grade in secondary school. The teaching process was eight sessions per week with one session. The main data source of this study was in-class teaching videos. The data were analyzed descriptively. Findings: The questions, the in-class dialogues directing relational thinking and activities in each session of the teaching experiment conducted with the fifth-grade students were presented under related themes. Implications for Research and Practice: The most general result was that at the end of the teaching process based on numbers, relationships between numbers, operations and properties, the students made use of equality axioms to evaluate the true/false and open number sentences without any calculation. It was also seen that the students made connections between addition-subtraction, addition-multiplication and multiplication-division and that they made effective use of commutative, associative and distributive properties.

Teaching fraction operations for conceptual understanding is a challenging task. For the topic of fraction division especially, teachers need support because teachers find this difficult to teach and elementary and middle school students struggle to learn this concept. Well-structured professional development can assist teachers in exploring their own understanding as well as how to incorporate multiple representations in their instruction. This naturalistic inquiry study investigated fourth to sixth grade teachers' invented real-world problems and pictorial representations for fraction division. The teachers completed a written assessment focused on conceptual understanding of fraction operations through multiple representations at the beginning and conclusion of a weeklong professional development experience. The questions on the assessment were taken or modified from prior research studies. The results in this paper focus specifically on a task in which the teachers had to develop a real world story problem for 2 1/2 ÷ 3/4 and solve the problem by drawing a pictorial representation. The teachers showed statistically significant improvement on developing appropriate real-world problems and using pictorial representations to solve their problems. Implications for teachers' usage of multiple representations are discussed along with needed structures for professional development targeted on fractions.

This study reports an analysis of how pre-service teachers (n=34) made sense of fraction division with remainders using pictorial modeling strategies, and how small-group and whole-class discussion helped them develop conceptual understanding. One and a half class sessions were video recorded, and 12 interviews were conducted. Results indicate that pre-service teachers can develop a conceptual understanding of fraction division with remainders using modeling strategies, and their understanding emerges in three levels: a) level one: ignoring the remainder or labeling it incorrectly; b) level two: interpreting the remainder in the original unit but not relating it to the new unit; and c) level three: interpreting the remainder both in the original unit and the new unit flexibly.

This research investigated students' sources and causes of errors and misconception in solving routine problems involving addition, subtraction, multiplication, and division of integers. This qualitative study involved observation of eight Year 7 classes and interviews with the respective classroom teachers. Sixteen Year 7 students who exhibit errors when solving the problems given in the Error Identification Integer Test (EIIT) were also interviewed to probe their thinking. The different types of errors were categorized according to how they were construed. The sources of errors were found to stem from carelessness, poor basic knowledge such as the inability to multiply and divide even the whole numbers, inability to assimilate concepts of integers since they are used to the schema of whole numbers, and rule mix-up which is also the result of surface understanding. Teachers were questioned about common errors and possible reasons for these errors made by their students. The main cause of errors and misconceptions is superficial understanding, which was most probably due to teachers rushing to complete the extensive syllabus, and consequently, students resorted to memorizing rules because of surface understanding. Teaching episodes were found to lack multiple-representation, creativity, as well as cooperative learning and active learning.

This small study sought to determine students' knowledge of multiplication and division and whether they are able to use sets of bundling sticks to demonstrate their knowledge. Manipulatives are widely used in primary and some middle school classrooms, and can assist children to connect multiplicative concepts to physical representations. Qualitative data were generated from semi-structured interviews with 32 primary and middle school children aged nine to eleven years. Participants were asked to work out the answer to multiplication and division examples and explain their thinking using bundling sticks. Results suggest that the majority of participant students may have a limited knowledge of aspects of the multiplication process and even less knowledge of the division process. The study also identified that many of the students appeared uncomfortable and/or unfamiliar with using bundling sticks and a number of them had difficulty in using bundling sticks to explain the multiplication and division processes. We conclude that manipulatives such as bundling sticks do not magically lead children to mathematical learning but are sufficiently powerful to warrant teachers familiarising themselves with how manipulatives can be used to develop conceptual understanding.