Search

Word problem solving is an approach that can increase knowledge, improve performance and develop learners' critical thinking skills that can assist them in solving mathematical and real-life problems. This study examined the effectiveness of using a word problem solving approach to improve Grade 3 learners' performance when learning addition and subtraction. The study employed this hypothesis to test the difference between the experimental and control groups and between the pre-test and post-test with learners in diverse classrooms solving addition and subtraction word problems. A pre-test-intervention-post-test research design was used to collect the data. Baseline lesson observations and unstructured interviews were conducted to understand the effectiveness of the problem solving approach in teaching number concepts, problems, addition and subtraction. The results of the study revealed a significant difference between the experimental and comparison groups when solving addition and subtraction word problems. Furthermore, there was a significant difference between the pre-test and the post-test in the experimental group. It was concluded that word problems could be taught by following steps for problem solving and equipping learners with essential reading skills to help them comprehend what they are reading. The interviews showed that teachers do not have enough time for learners who experience learning challenges and reading in particular in pursuit of covering the curriculum. Learners in the foundation phase must be taught money problems and problem-solving skills to acquire knowledge to be used in higher grades and real-life situations.

Developing structure sense is an important part of learning algebra. We investigated learners' structure sense of algebraic expressions involving brackets. This led us to propose the constructs "surface structure" sense and "systemic structure" sense. Using a random sample of 58 Grade 10 learners scoring above 40% in a test, we coded incorrect responses for surface and systemic structure errors. The initial analysis revealed that the presence of more brackets supports surface structure sense. However, learners overgeneralised the presence of brackets to represent multiplication in situations involving subtraction. The arrangement of brackets also led to errors in the order of operations. Further analysis suggested that regular application of procedures on familiar algebraic structures may give the illusion of systemic structure sense. We recommend that the teaching of algebraic expressions must emphasise what the arrangement of an expression means before focusing on how to operate on the expression

Despite problem posing's prominence in mathematics education research, its implementation in classrooms is limited. Therefore, teacher educators should incorporate problem posing tasks into preparation programs to help prospective teachers gain confidence in their abilities. One approach to teaching problem posing includes providing examples. The Common Core State Standards Initiative provides taxonomies with examples of one-step word problems for all four basic operations. Through this mixed-methods study, preservice teachers (N = 44) regularly identified and posed word problems within the addition and subtraction taxonomy. Before and after the intervention, the preservice teachers answered survey questions and posed problems to model given equations. The researchers analyzed the preservice teachers' abilities to pose logical word problems and to match given equations before and after instruction. They also quantitatively analyzed Likert-style survey responses on perceptions of posing the various types of word problems and integrated the results with a qualitative analysis of open-ended responses expanding on the same questions. Results showed participants' abilities to pose problems increased after the intervention. Additionally, participants felt problems with the unknown information at the beginning or middle were harder to pose than those with the unknown information at the end.

Mathematical writing is one way for primary students to communicate their mathematical thinking. Research in the field of writing has shown that to become an effective teacher of writing, preservice teachers must have experience engaging in the kinds of writing given to their students. The study reported in this paper explored how 27 preservice teachers enrolled in an elementary mathematics methods course explained their reasons for selecting strategies to solve equations that required making either addition or subtraction calculations. Element scoring was used to analyse the preservice teachers' reasons for selecting the strategy and the composition of their mathematical writing. Results indicated that the majority of preservice teachers included a mathematical reason for selecting their strategy, however; the reasons varied in their connection to the mathematics discipline. This study contributes to the existing literature on mathematical writing by providing insight into how preservice teachers explain their mathematical reasons for selecting a computation strategy; thereby, demonstrating the need for increased opportunities for preservice teachers to write.

This qualitative case study investigates how, and to what extent, primary school teachers notice students' mathematical thinking in the context of whole number subtraction. A task involving a student's invented strategy was used to collect data. Three noticing questions connected to the task were asked to 45 teachers. Their written answers were analyzed to reveal teachers' noticing skills based on attending, interpreting, and deciding how to respond. The study results revealed that most participants provided limited evidence of attending and interpreting skills, and some responses showed a lack of evidence. More specifically, they could not identify the relationship between digits in ones and tens places. Further, their interpretations did not directly focus on students' solutions and consisted of general statements and misconceptions.

The concept of number sense first appeared theoretically in the report published by NCTM in 1989. Although it is an important concept that includes the meaning of numbers and the relationship between numbers, it is very difficult to make a clear definition of it. This could also be true for the concept of integer, which we cannot place in our world of meaning in daily life or which we have difficulty in finding an equivalent for. For integers that have an abstract world of meaning, a misconception may occur between the integer sign and the sign used for operation. In this respect, this study aimed to examine the effect of subtraction, which can create pre-learning, on the concept of integers. Moreover, it was requested to demonstrate that the operational knowledge used by students in subtraction can be utilized in teaching integers. A total of 17 students studying in the 5th grade of a secondary school in Turkey participated in the study. As the research design, the action research method, one of qualitative research methods, was used, and the data were collected with the trilogy technique. The data obtained were analyzed using descriptive analysis and content analysis. The findings revealed that the teaching technique of subtraction opened the door for students to form and make sense of the concept of minus. In this context, it is recommended to provide mathematical skills in a cumulative manner.

Background/purpose: Investigation into the misconceptions of preschool students in mathematics and their differences between the ages of 4-5 and 5-6 years old helps form appropriate developmental mathematics teaching programs. However, although several studies have been conducted examining preschoolers' previous knowledge and misconceptions about mathematics, no corresponding research has been found in Greece. This study aims to investigate preschoolers' misconceptions about numbers and operations and to reveal differences between preschoolers aged 4-5 and 5-6 years old. Materials/methods: Data were collected through semi-structured interviews and analyzed according to content analysis methodology. Results: The study's results showed that 5-6-year-old preschoolers perform better than those aged 4-5 years old. Most misconceptions of the latter group appeared to be related to reverse counting, identifying arithmetic symbols and their matching quantities, adding and removing numbers without using auxiliary objects and multiplication. On the contrary, some preschoolers aged 5-6 years old needed help adding or subtracting two-digit numbers without the use of auxiliary objects. There were also a few cases where cardinality, division, and multiplication were observed. Conclusion: The misconceptions identified in the two groups of students regarding numbers and operations and their distinct needs that emerged through the research will allow teachers to offer differentiated instruction and personalize teaching support.

The notion that mental arithmetic is associated with shifts of spatial attention along a spatially organised mental number representation has received empirical support from three lines of research. First, participants tend to overestimate results of addition and underestimate those of subtraction problems in both exact and approximate formats. This has been termed the operational momentum (OM) effect. Second, participants are faster in detecting right-sided targets presented in the course of addition problems and left-sided targets in subtraction problems (attentional bias). Third, participants are biased toward choosing right-sided response alternatives to indicate the results of addition problems and left-sided response alternatives for subtraction problems (Spatial Association Of Responses [SOAR] effect). These effects potentially have their origin in operation-specific shifts of attention along a spatially organised mental number representation: rightward for addition and leftward for subtraction. Using a lateralised target detection task during the calculation phase of non-symbolic additions and subtractions, the current study measured the attentional focus, the OM and SOAR effects. In two experiments, we replicated the OM and SOAR effects but did not observe operation-specific biases in the lateralised target-detection task. We describe two new characteristics of the OM effect: First, a time-resolved, block-wise analysis of both experiments revealed sequential dependency effects in that the OM effect builds up over the course of the experiment, driven by the increasing underestimation of subtraction over time. Second, the OM effect was enhanced after arithmetic operation repetition compared to trials where arithmetic operation switched from one trial to the next. These results call into question the operation-specific attentional biases as the sole generator of the observed effects and point to the involvement of additional, potentially decisional processes that operate across trials.

Years and Centuries are the measurement units used to quantify a longer time duration, while subtraction is the operation required to determine the duration based on two given time points. However, subtraction of time is a difficult skill to be mastered by many elementary students. To identify the root cause of the student's failure in performing subtraction involving the unit of time, we developed and validated the three cognitive models related to this skill by conducting a descriptive study which involved 119 Grade Five students from three Malaysian elementary schools. The cognitive diagnostic assessment developed based on the three cognitive models was used to elicit the participants' responses. Then, Attribute Hierarchy Method and Classical Test Theory were employed to analyse the data. The findings indicated that the hierarchical structures of all cognitive models are supported by the student's responses. The three student-based cognitive models were also highly consistent with the corresponding expert-based cognitive models. The cognitive models developed could guide diagnostic assessment development and diagnostic inference making.

The aim of this research is to determine the mental processing strategies used by 3rd grade primary school students in addition and subtraction using the empty number lines. The model of the research is a case study, one of the qualitative research methods. The participants consist of 143 students; 75 girls and 68 boys, attending the third grade (8-years-old). Content analysis method was used in data analysis. Considering the results of the research, the mental strategies used by primary school 3rd-grade students were gathered in four categories: counting, balancing, first process step and partitioning. It was concluded that the students mostly used the mental processing strategy of using "multiples of 10", which is in the first process category, and this strategy was followed by "counting all" and "partitioning the second number", the least used mental processing strategy by the students was "using equivalent numbers".

The objective of this study was to evaluate and characterize the argumentation patterns used by seventh-grade students in the context of algebraic addition and subtraction problems. A qualitative case study was conducted using the Claim-Evidence-Reasoning (CER) model (McNeill & Krajcik, 2008) and argument quality framework. To describe the arguments put forth by the participants, a sample of ten students with high mathematics proficiency and ten students with low mathematics proficiency were selected from the target population of junior high schools in a large region of West Java, Indonesia. Data collection was carried out using written argumentation frame for algebraic operations (AFAO). The results revealed that, although none of the students employed the CH+EH+RH pattern (which represents the highest quality of arguments), students with high mathematics proficiency levels exhibited a greater prevalence of these argumentation patterns compared to those with low mathematics proficiency levels. The findings have implications as a valuable resource for teachers in monitoring the advancement of their students and preventing or alleviating diverse difficulties or inaccuracies that they may face. Based on the findings, a specific design for a classroom teaching activity is proposed.

One of the most fundamental subjects of primary school mathematics, the concept of fraction plays a key role in students' educational background. Students experience difficulties and misconceptions regarding this concept. Therefore, it is of great importance to understand this subject well and to use different teaching methods and techniques. Various methods can be used in learning mathematics, such as invention method, computer-assisted teaching, problem solving, creative drama and discussion. Argumentation method is a scientific discussion of a claim on a subject by providing data, warrant and backing, and by revealing the rebuttal and the invalidity of the claim. Students gain more skills by interacting with teachers or collaborating with their peers, and that they possess skills that have not been obtained before or that are too difficult to achieve on their own. Therefore, the argumentation method was chosen for its effect on the development of primary school students' fraction calculation strategies. Having a case study design, this study was conducted with 28 primary school fourth-grade students and 10 open-ended questions prepared by the researchers were used during interviews. The results suggested that primary school students were able to develop fraction calculation strategies through the argumentation method. In general, inventing strategies, using representations, number sense, addition and subtraction, using unit fractions and proportional reasoning were used as fraction calculation strategies. It recommended to use the argumentation method in other critical subjects of mathematics as well.

In mathematics, the expectation to set up and solve word problems emerges as early as kindergarten; however, many students who experience mathematics difficulty (MD) and dual-language learners often present with specific challenges in this area. To investigate why these populations experience word-problem difficulty, we examined the word, problem solving and oral explanations of third-grade dual-language learners (DLLs; n = 40) and non-DLLs (n = 40), all of whom were identified as experiencing MD. Students solved five additive word problems and provided oral explanations of their work, which were transcribed and coded for the number of words in each explanation, type of mathematics vocabulary terms used, inclusion of correct numbers in explanations, and descriptions of addition or subtraction. We identified no significant differences in word-problem scores between DLLs and non-DLLs with MD. For both DLLs and non-DLLs, students who answered problems correctly used more words in each explanation and used more mathematics vocabulary terms within their explanations. For incorrectly answered problems, the most common mistake for both DLLs and non-DLLs involved using the incorrect operation to solve the problem.

The objectives of this research were to construct and verify a mathematics diagnostic assessment on addition and subtraction basis for grade 4 students in Buriram Primary Educational Service Area Office 3. The research samples consisted of 440 Prathomsuksa 4 students selected by adapting the multi-stage random sampling method. The research instruments employed in this research comprised of two types of examinations constructed according to strand 1: number and algebra in Mathematics 1.1. The tests were to examine test and diagnostic test. The examining test was a subjective test, consisted of 80 items which divided into addition part for 40 items and subtraction part for 40 items. The diagnostic test was a multiple-choice with 4 choices tests, consisted of 48 items which divided into addition part for 24 items and subtraction part for 24 items. The diagnostic test. The results showed that the content validity, determined by the expert, had the IOC score at 0.60-1.00. The addition part in the 1st test had a difficulty score of 0.21-0.79, the discrimination score at 0.23-0.89, and the reliability score at 0.89. The subtraction part in the 2nd test had the difficulty score at 0.31-0.72, the discrimination score at 0.32-0.72 and the reliability scores at 0.32-0.81.

As national and international assessments continue to show students struggle with rational concepts, which are seen as a major stumbling block for future career in STEM fields, we used the certainty of response index (CRI) as a method of analysis to investigate a group of 40 United Arab Emirates (UAE) seventh grade students' understanding of basic fraction concepts. The types of errors and misconceptions displayed by students show that over 60% of all participants assumed that adding fractions procedurally is the same as adding whole numbers with many not being able to distinguish a numerator from a denominator. The results lead us to believe that the manner in which students were taught did not ensure the necessary transitional shift from learning whole numbers to learning rational numbers. Applying the theories of Dienes (1960) and Bruner (1966), we propose a method of teaching addition and subtraction of fractions that will allow students to build their own understanding of the rules using the graphing calculator as the medium for concept formation.

Solving arithmetic word problems has been a challenge for primary students due to difficulties in understanding the problem structure and relating the quantities in the problem to each other. This paper reports on an action research study to enhance students' subtraction word problem-solving skills. The authors observed that their students had difficulties in representing the situations in word problems and solving the problems correctly. They designed, implemented, and analyzed an intervention to scaffold their students' subtraction word problem-solving skills. As part of the intervention, a digital subtraction game was developed and used with second and third-grade students. The game involves three different representations: a discrete visual model, a bar model, and a number sentence. The students played the game and solved additional problems to strengthen their skills for representing and solving subtraction word problems. Twenty-four 2nd grade students in China and two 3rd grade students in Turkey participated in the study. Data sources included a pre-test, a post-test, student worksheets, and teachers' filed notes. Data analysis showed an increase in students' subtraction word problem-solving performance. They also effectively used a variety of representations to represent problem situations. The design and implementation processes of the intervention are discussed in the paper. We share suggestions for future implementation.

This paper reports on the design, implementation, and evaluation processes of an activity that focused on the comparison meaning of the subtraction of integers and included the number line model as the primary representation. Finding the answer and determining the sign of the answer in the subtraction questions were targeted as the learning objectives. Four different situations that can be encountered in the subtraction questions were determined considering the magnitude and sign of minuend and subtrahend. Through the designed activity, it is expected that the students work on example questions first and then make generalizations by reflecting on the questions they solve. The activity was implemented in an elementary school with the participation of 27 seventh grade students. The analysis of the implementation process revealed that the activity was effective in supporting the students' performance of finding the difference and the sign of the difference when subtracting integers.

Several working memory processes have been hypothesized to influence different arithmetic operations. Working memory has been compartmentalized into a number of different sub-processes, such as phonological memory and visuospatial memory that are believed to have unique contributions to the performance of two distinct arithmetic operations: multiplication and subtraction. A previous dual task experiment produced these effects, but subsequent experiments have yielded inconsistent results. Because the reasons for these inconsistencies are not immediately apparent, the current study systematically reviewed these subsequent attempts and attempted to replicate this effect in a within-subjects dual task experiment using tasks developed from prior work across a number of different subsamples. In contrast to the original finding, we observed no differential impact of specific working memory secondary tasks by arithmetic operation in any of our analyses. However, our analyses do not entirely rule out the possibility of differential effects of working memory tasks. Our findings suggest that the working memory facet by arithmetic operation interactions observed in previous work may be idiosyncratic in nature and difficult to predict a priori in subsequent experiments.

Central elements of adaptive expertise in arithmetic problem solving are flexibility, using multiple strategies, and adaptivity, selecting the optimal strategy. Research shows that the strategies children actually use do not fully reflect the strategies they know: there is hidden potential. In the current study a sample of 147 third graders from the Netherlands completed a comprehensive assessment of adaptive expertise in the domain of multidigit subtraction, designed to measure, first, the strategies students know and use to solve subtraction problems (potential and practical flexibility). Second, it measured to what extent students know which strategy is optimal and to what extent they use the optimal strategy (potential and practical adaptivity). Findings for flexibility showed that most students consistently used the same strategy across all problems: practical flexibility was low. When prompted, students knew more strategies than they used spontaneously, suggesting hidden potential in flexibility. Findings for adaptivity showed that students hardly ever spontaneously used the task-specific strategy that is efficient for specific problems since it has the fewest and easiest steps. However, almost half of the students could select this strategy from a set of given strategies at least once. Furthermore, an innovative, personalized version of the choice/no-choice method showed that the task-specific strategy was usually not the optimal strategy (fastest strategy leading to a correct answer) for individual students. Finally, students used the strategy with which they performed best more often than the other strategies, but there is hidden potential for the adaptive use of task-specific strategies.

This study examined adults' frequent, efficient and adaptive use of direct subtraction (DS) and subtraction by addition (SBA) in mental multi-digit subtraction with the choice/no-choice method. Participants were offered subtractions in one choice condition (choice between DS and SBA) and two no-choice conditions (mandatory use of either DS or SBA). SBA was used as frequently as DS in the choice condition. DS was most accurate on subtractions with a large difference (e.g., 502 - 18), while SBA was fastest on subtractions with a small difference (e.g., 903 - 886). In general, participants were adaptive for task characteristics and their personal speed characteristics. We further analyzed task-based adaptivity on an individual level via a Latent Class Analysis. Results showed that two-thirds of the participants were adaptive to task characteristics, and that these adaptive participants were the most proficient in accuracy and speed in the choice condition. We further examined whether executive functions (updating, inhibition, shifting) were related to individual differences in strategy efficiency and task-based adaptivity. In line with our hypothesis, updating was related to strategy efficiency, such that participants with higher updating skills were more accurate. In contrast to our expectations, inhibition and shifting were not related to task-based strategy adaptivity. This study highlights adults' efficient and adaptive use of arithmetic strategies, and its association with their proficiency and executive functions.

In this paper it is investigated if the mental computation strategies in the research literature are enough to satisfactorily categorize the mental computation strategy use by preservice elementary teachers (PETs) in subtraction on two-digit natural numbers. The PETs' use of mental computation strategies is measured operationally with a written questionnaire. The paper indicates that the strategies used by PETs are generally seen contained in the research literature on elementary school pupils' mental computation strategy use, but that there are additional strategies used by PETs, which are presented and discussed in the paper.

In mathematics, the expectation to set up and solve word problems emerges as early as kindergarten; however, many students who experience mathematics difficulty (MD) and dual-language learners often present with specific challenges in this area. To investigate why these populations experience word-problem difficulty, we examined the word, problem solving and oral explanations of third-grade dual-language learners (DLLs; n = 40) and non-DLLs (n = 40), all of whom were identified as experiencing MD. Students solved five additive word problems and provided oral explanations of their work, which were transcribed and coded for the number of words in each explanation, type of mathematics vocabulary terms used, inclusion of correct numbers in explanations, and descriptions of addition or subtraction. We identified no significant differences in word-problem scores between DLLs and non-DLLs with MD. For both DLLs and non-DLLs, students who answered problems correctly used more words in each explanation and used more mathematics vocabulary terms within their explanations. For incorrectly answered problems, the most common mistake for both DLLs and non-DLLs involved using the incorrect operation to solve the problem. [This article was published in "Learning Disability Quarterly" (EJ1323542).]

This study replicates and extends prior research on multisensory mathematics instruction (Grünke, Urton, & Karnes, 2018) by integrating a touch points strategy, performance feedback, reward system, and a reinforcing card game into an instructional package. A multiple baseline design across participants was used to evaluate the effects of the touch points package on the subtraction skills of four German female first year students at-risk for learning disabilities. During intervention, the students were administered eight to eleven treatment sessions to learn how to subtract a one-digit subtrahend from a two-digit minuend up to 18 crossing over the tens barrier. Results indicated that all students made substantial increases in their subtraction performance during intervention. Student performance improved from 0 to 2 out of 10 math problems solved correctly during baseline to between 8 and 10 problems correct by the end of the intervention. Effect sizes observed across the four participants indicated the effectiveness of the intervention ranged from high to very high.

The purpose of this study is to electrophysiologically assess the effect of an individualized education program supported with musical activities on the success of primary school students with computational difficulties. A mixed-methods design consisting of qualitative and quantitative methods was employed as the research model. The research group consists of four students with mild special learning difficulties and learning difficulties in mathematics, among the students attending the third and fourth grades at a primary school in Kütahya city center. By examining the primary school mathematics curriculum, 12 mental processing gains were identified for addition and subtraction, one of the third-grade mental processing gains. An assessment form, training module, and math songs were prepared in line with these acquisitions. An assessment form was applied to the participants as a pre-test and post-test to determine the effect of the training module and math songs on the students' success. Additionally, electroencephalogram (EEG) of the participants were recorded before and after the training module, and math songs were applied for 12 weeks. During the EEG recordings, 10 questions were asked to the participants that would enable them to make mental operations. The power densities of the EEG data were calculated using the Welch method in the MATLAB. To analyze the qualitative data of the research, descriptive analysis technique was used. As a result of the study, when the effects of the evaluation form of the participants' mental processing skills were examined, it was seen that the training module and math songs positively affected the mental processing skills of the students. In addition, it has been shown that the prepared training module and math songs increase the success of the participants, supported by electrophysiological evaluations.

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.

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.

Developing textbooks of optimal quality is crucial for enriching the students' learning and understanding. This study examined fraction addition and subtraction problems in the U.S. and South Korean mathematics textbooks according to the types of denominators. In particular, we investigated Everyday Mathematics (EM) and South Korean mathematics (KM) textbooks revised in 2015 and developed an analytic framework encompassing horizontal and vertical dimensions to examine the learning opportunities presented to students by the textbooks. We assessed their topic sequence and frequency of fraction addition and subtraction contents with regard to the former and examined their contextual features, cognitive demands, and mathematical activities with regard to the latter. We observed that EM provided inadequate learning opportunities for fraction subtraction problems, representation problems, and high-order cognitive abilities. However, KM provided more even learning opportunities for fraction addition and subtraction problems, various contextual features, and high and low thinking skills. Moreover, we found that EM emphasized understanding and resolving activities, whereas KM underscored exploring and explaining activities. The findings of this study suggested updating fraction addition and subtraction contents in the U.S., South Korea, and other countries by considering horizontal and vertical dimensions.

The integration of virtual manipulatives in classroom practices facilitates student learning processes. For this, the teacher must understand how to support students in establishing mathematical connections between the manipulation and their interpretations of the representations. We present a learning scenario that integrates the use of virtual manipulatives in mathematical modelling tasks. It was designed and implemented during initial teacher training, with the aim of helping a group of first-year elementary school students to overcome difficulties identified in their comprehension of the meaning of subtraction. The research, following the principles of design-based research, included three distinct moments: an individual written pre-test, an intervention phase with six micro-cycles, and an individual written post-test. The analysis of the collected data--audio, screen recordings, and written productions--allowed us to identify the most influential structural characteristics of the learning scenario: mathematical communication, collaborative learning, self-regulation, and co-regulation of learning. Our results suggest that differentiated instruction, minimal guidance, and scaffolding played an important role in the support provided by the pre-service teacher to students in the mathematical modelling activities that integrated virtual manipulatives.

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 article focuses on how Swedish Year One students (age 7-8) make meaning when working with images in mathematics textbooks. Images include textbook images, but also students' self-drawn images used as support for calculation. The focus was (1) what the images in the exercises were designed to offer (the designed affordance), and (2) what the students discovered when working with them. The data material consisted of video transcripts of 18 students working with subtraction exercises from mathematics textbooks. The results showed that the students sometimes discovered the designed affordance and sometimes did not. The students who discovered the designed affordance sometimes used the image when performing the calculations, while others did not. Some students expressed that images in mathematics textbooks are for those who find mathematics difficult, and that completing exercises without using the images was desired. The students' approaches to images were discussed in two specific cases: First, the students' desire to use mathematical symbols rather than images may lead to students not discovering the mathematics content that the exercise is designed to offer. Second, the use of mathematical symbols rather than images may lead to students not discovering themselves as mathematical individuals.

Focusing on students with mild disabilities, this study aimed to examine the effect of STAR problem solving strategy on their a) solving change problems involving one-step addition and subtraction, b) maintaining their acquisition of solving change problems involving one-step addition and subtraction after 1, 3, and 5 weeks, c) generalizing their performance in solving problems to the classroom environment. Three students with mild mental disabilities participated in the study. A multiple probe across participants design was used in the study. The number of problems that students solved correctly was determined by scoring the data. The data are shown graphically and analysed visually. Findings emphasized the effectiveness of STAR strategy for students with mild mental disabilities when solving change problems that involve a one-step addition and subtraction, indicating that those who acquired this strategy could demonstrate the same problem solving performance 1, 3, and 5 weeks after the intervention. Also, students were observed to generalize their strategy performance to the classroom environment. The findings of the research were discussed within the framework of the relevant literature and theoretical views, and suggestions were made for teachers in terms of interventions and for researchers considering further studies.

In this study, it was aimed to teach the addition and subtraction to a student with dyscalculia who is in third grade at elementary school. Before preparing individualized program, the student was given a pretest to measure about his level of the subject. The instruction was carried out in a room in the school, with a time of 40 minutes per day in the form of individual instruction with the student. In addition, the study was supported by a researcher-developed scenario booklet, concrete materials, computer games, reinforcement award, and assignments. This study was conducted as a case study, one of the qualitative research methods. The data were analyzed by descriptive analysis technique. Results of the study indicate that the student achieved more than 90 % objectives on teaching basic addition and subtraction skills at the third grade. Gains have been actualized by the student with mathematic learning difficulties with over 90 % success in 63 individualized lessons. On the other hand, social validity data were collected through observation and interviews to identify strengths or weaknesses and changes occurring in the student work. As a result of the studies, it has been determined that the student gained self-confidence by enjoying his sense of accomplishment.

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.

Malaysia has undergone several changes in its education policy from KBSR to KSSR over the decades and the changes in school textbooks follow suit. The current KSSR Math textbooks used were developed in 2014/15 and millions of Ringgits has been spent by the government to provide quality education for all. However, the findings from International studies such as TIMSS and PISA have shown a low level of attainment, especially in word problems, among Malaysian pupils in the learning of mathematics. Is there a possibility that this low attainment in mathematics education in schools, among other factors, have something to do with the content analysis of the textbook's pupils use in the classroom? A three-phase descriptive-correlational content analysis design was utilized for the study. In the first phase of the research, a total of eleven mathematics texts (4 textbooks and 7 activity books) from Primary One to Primary Four were analyzed according to eleven types of categories as modelled by the Van de Walle (1998) framework. The findings show that the problem categories were not represented in a systematic manner throughout Year One to Year Four mathematics texts. Some categories were overrepresented while others were underrepresented. The next phase finding depicts a significant relationship between pupils' achievement in the Word Problem Test categories and the distribution of word problem categories across all grade levels. In other words, results showed that pupils face more difficulty on the problems that were underrepresented and least difficulty that were overrepresented. As mathematics syllabuses in KSSR keep expanding and focus more on high order thinking skills among pupils, it is important to make sure that the sources are well prepared for the cause. Therefore, the curriculum and textbook developer should consider taking input from research to provide up to date information related to curriculum matters and in this case to the variety of word problem categories in ration that benefit pupils across levels.

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.

The number line and the neutralization model have been used very extensively in teaching integer additions and subtractions for decades. Despite their advantages, issues concerning subtractions on these models are still debatable. Therefore, the neutralization on an empty number line (NNL) model is proposed in this research to help students better understand the meaning of integer subtractions as well as additions. This report is a part of a design research study conducted in a classroom of 28 elementary school students at the fifth grade. Data were gathered by collecting students' written work, conducting interviews and observations during the teaching experiment. This paper focuses on students' perceptions of the NNL model and what factors that might contribute to students' achievements in understanding integer additions and subtractions. The analysis revealed that most students overemphasized on the use of the NNL model as a procedural method instead of as a model for thinking. Moreover, students' mathematical beliefs and conceptions on the use of the column strategy and the absence of a discussion on the need of using the model are found to be some factors that could cause students' misunderstandings. However, with a thorough planning, the NNL model has a potential to help students developing a meaning of integer additions and subtractions.

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.

Progress monitoring of academic achievement is an essential element to prevent learning disorders. A prominent approach is curriculum-based measurement (CBM). Various studies have documented positive effects of CBM on students' achievement. Nevertheless, the use of CBM is associated with additional work for teachers. The use of tablets may be of help here. Yet, although many advantages of computer- or tablet-based assessments are being discussed in the literature (e. g. innovative item formats, adaptive testing, automated scoring and feedback), there are still concerns regarding the comparability of different assessment modes (paper-pencil vs. tablet). In the study presented, we analyze the CBM data of 98 fourth graders. They processed the exact same computation items once with paper and pen and once in a tablet application. The analyses point to comparable results in the test modes, although some significant deviations can be found at item level. In addition, the children report perceived benefits when working with the tablet.

The paper reflects on a teaching intervention which utilized digital storytelling as an approach for teaching the subtraction algorithm in 2nd Grade. Literature indicates that this is a difficult topic for children of this age. The aim was to examine if through digital storytelling, students find it easier to understand and apply the two subtraction algorithms which are described in the official textbook, namely the "Borrow First and Then Subtract" and the "Austrian Subtraction" algorithms. The results indicate the effectiveness of this approach is, further enhancing the existing literature on the topic.

The inquiry-oriented and cyclic process of action research can lead to the innovative use of technology and instructional strategies to improve teaching practice, particularly for beginning teachers. This action research project examined the impact of the number line and Educreations on second-grade students' verbal and written explanations of three-digit addition and subtraction operations using whole numbers. Pre- and post-surveys were implemented to examine how the second-graders worked on the given three-digit addition and subtraction problems. The students' perceived value of using the number line indicated that the majority of the students found the number line helpful. Classroom observations and the students' use of the number line and Educreations indicated that student skills for solving and explaining the process of addition and subtraction methods increased as demonstrated in the post- assessment and the student videos.

This study aims to produce a learning trajectory using the mathematical modeling in helping students to understand the concept of algebraic operations. Therefore, the design research was chosen to meet the research aims and to give in formulating and developing local instructional theory in learning algebraic operations. Learning trajectory designed in the early phases and tested on 34 seven-grade students in SMP N 10 Palembang. Data collection was conducted through observation by recording the learning process that occurred in the classroom and students' group work was evidenced by video and photos. Data was analyzed qualitatively by describing actual learning which happened in pilot experiment and teaching experiment. There are 3 learning activities in the design of this study. These 3 activities are designed based on the steps of the Mathematical Modeling, activity 1 meaning of algebraic expressions, activity 2 addition of algebraic and activity 3 subtraction of algebraic. Based on the result, it can be concluded that activity which has been designed can help the students in learning algebraic operations using mathematical modeling. Used mathematical modeling can help student solve the problems and understand concept are structured using the assumptions and model start they design so gradually developed into formal mathematics.

The aim of this study is to determine the effect of mathematics teaching with cartoons on the problem solving skills of primary school 2nd grade students based on addition and subtraction. In the research, pretest-posttest control group design of the experimental model was used. In the classroom where the experimental group students were present, cartoon supported education was applied and the current program based teaching method was used in the control group class. The target population of the study consisted of 2nd grade students of all primary schools of the Ministry of National Education of Mersin. The population of the study consisted of 2nd grade students of all primary schools of the Ministry of National Education of Mersin. The study was conducted for 4 weeks in the fall semester of the 2015-2016 academic year. The experimental group consisted of 17 students and the control group consisted of 13 students. In order to determine the validity and reliability of the achievement test used in the study, item analysis was performed with the TAB program. The t-test was used to determine whether there was a significant difference between the pre-test and post-test scores of the groups. As a result of the research, it was observed that the success of the students in the problem solving in addition and subtraction education has increased.

This paper dealt with the design of learning tools, that was the development of learning implementation plan or Rencana Pelaksanaan Pembelajaran (RPP) based on Fun and Easy Math (FEM) approach in the form of a number line model. FEM was an approach in order to teach mathematics effectively to students in a fun and easy way according to the patterns and rules, making it easier for students to understand mathematical concepts and driving them to actively participate in learning activity. Furthermore, the number line model based on the rules aimed to provide the real idea of mathematical concept and avoid the mathematical manipulation. The design of RPP development was intended to create a new RPP which was divided into three activities, namely introduction, core, and closing. The method of lesson plan development used a Plomp's development model which was consisted of three phases, namely preliminary research, prototype, and evaluation. The results of this development were expected to help students in understanding the mathematical concepts, namely integer calculation operations especially addition and subtraction, increasing student activeness in learning process, and avoiding the existence of mathematical manipulation.

A deep understanding of fraction concepts and operations is necessary if pre-service teachers (PSTs) are to present the concepts in multiple forms to learners. Such an understanding needs to be grounded in rich conceptual knowledge. In the present study, we explore the development of this understanding by supporting a cohort of 103 PSTs, who had previously demonstrated poor conceptual understanding of fraction concepts and operations, with a Representational Reasoning in Teaching and Learning (RRTL) approach aimed at strengthening their conceptual knowledge. A comparison of pre- and post-test results indicated that participants showed a significant improvement in shifting the balance of their fraction knowledge to the conceptual end of the procedural-conceptual spectrum. Insights into how this approach assisted in developing PSTs' conceptual understanding were explored through interviews with four participants and an analysis of their pre- and post-test responses. We suggest that the use of teaching strategies such as RRTL are necessary in order to assist PSTs develop strong conceptual knowledge of fractions. [Note: The volume number (1) shown on the PDF is incorrect. The correct citation is v21 n1.]