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Children with intellectual disabilities experience well-documented challenges in the development of problem-solving, particularly related to mathematical word-problem solving. Such challenges are related to both linguistic and conceptual skills. While interventions to assist children in developing conceptual skills are well reported, linguistic interventions within the mathematical field are limited. This study aimed to determine the effect of an augmented-input intervention, on subtraction word-problem solving for children with intellectual disabilities. A multiple baseline across behaviours design was used and replicated across the seven participants. Intervention was provided in three tiers (one week each) for change, combine, and compare subtraction word-problems. Intervention was withdrawn for a four week period and followed by a maintenance phase. Results suggested that for older participants with higher receptive language and numerical operation scores, the augmented-input intervention provided significant gains across all three types of word-problems. Overall, four of the seven participants showed improvement in subtraction word-problem solving. Augmented input may provide language supports for subtraction word-problem solving for some children with mild intellectual disability.

The purpose of this research was to determine the opinions of hearing-impaired students about the digital teaching material prepared for subtraction, which requires decimal breaking. To achieve this goal, a digital teaching material that includes decimal breaking operations was developed. The case study method, a qualitative research method, was used to determine the opinions about the developed material. Four secondary school students, two primary school students, one secondary school mathematics teacher, and one primary school mathematics teacher participated in the study during the 2020-2021 academic years. Documents, interview forms, system records, observations, and video recordings were used as data collection tools to contribute to the research. The data obtained from the data collection tools were analyzed using the techniques commonly used in qualitative research, including content analysis and descriptive statistics. The results of the research showed that the developed digital teaching material has a positive impact on students' learning, the absence of distracting elements on the material reduces students' distraction, and students showed interest and success in using the material. The teachers' opinions supported the students' opinions and once again demonstrated the effectiveness of the digital teaching material in the education of hearing-impaired students.

It is critical that elementary students develop understanding of fractional numbers because understanding concepts such as fraction magnitude predict later mathematics success. Representing fractions in multiple ways assists students in developing a sense of fraction magnitude. A systematic approach to presenting multiple representations is the concrete-representational-abstract (CRA) sequence. A potential problem with implementation of CRA in a general education classroom is time. The purpose of the current study was to investigate the effects of an integrated CRA fraction intervention. Twenty-eight fifth-grade students received instruction from their mathematics teacher as a tier two MTSS intervention. The researchers measured students' understanding of fraction magnitude using a pretest and posttest. The researchers also compared the students' mathematics achievement to that of their matched peers on the State Assessment.

Expecting teachers to support their students in making mathematical connections is an unreasonable request when teachers themselves have not had opportunities to connect their own understandings. This study investigates how prospective elementary teachers can deepen their mathematical understanding using recommendations developed by mathematics education researchers for students to work with and on prior knowledge, especially when it presents problems for learning. This case study investigated 15 participants as they worked on elementary school mathematics tasks during a 10-day professional development course. The video data and prospective teachers' journals were analyzed using Pirie-Kieren's (1994) theory for the Dynamical Growth of Mathematical Understanding. Tracing two participants' learning pathways revealed a connection-making process that deepened their understanding of elementary mathematics.

Pre-service mathematics teachers' content knowledge is an important issue. Therefore, detailed studies are needed to be conducted on mathematical topics. The study examines preservice elementary mathematics teachers' (PEMTs) special content knowledge (SCK) of integer addition and subtraction in the context of using multiple representations, explaining mathematical reasons lying behind the concepts and justifying them. The findings obtained from the written responses of 42 PEMTs reveal that preservice teachers do not have sufficient and balanced special content knowledge. This is especially more so in the case of addition and subtraction of numbers with opposite signs. The preservice teachers were observed to have more difficulty in using the number line model compared to the use of other representations. The findings offer some indicators about how PEMTs understand integer addition and subtraction through multiple representations and why more emphasis on the SCK components

The current study aims to show the transition skills of sixth grade students between the representations (numerical, model, number line and verbal) given in addition-subtraction operations in fractions and to analyse the ability of the students to construct each representation type. In this research, case study method was used in terms of the subject and process followed. The study group of the research is the sixth grade students of three middle schools in Kastamonu province placed in the northern part of Turkey. The research participants consisted of 59 students, 31 of which are male and 28 of which are female. Data collected through "Multiple Representations in Fraction Operations Test." According to research findings, it has been determined that the achievement of students in using different representations in operations with fractions is higher in addition operation compared to subtraction operation. In addition, it turns out that students are more successful in numerical-numerical, model-model, model-numerical, and numerical-model transitions compared to other transitions in both operation types. It was also found out that students fail in other representation types such as number line and verbal representation transition situations.

Learners in lower primary and even some in upper primary grades grapple to perform mathematical operations which involve fractions. Failure to solve these mathematical operations creates a gap in the teaching and learning processes of mathematics. We opine that this is attributed to use of traditional mathematical approaches of teaching and learning (TMATL) of operations of fraction. With the hope of engaging the reformed mathematical approach of teaching and learning (RMATL), this study investigated the following: How can trainee teachers use the angle model in RMATL operations of fractions? What are the perceptions of trainee teachers in the use of the angle model which engages RMATL to teach the operations of fractions? With the goal to fill the mentioned gap in which learners struggle to perform operations involving fractions, we observed and analysed worksheets on operation with fractions students wrote. Observations and interviews with trainee teachers of lower primary revealed poor performance in problems related to operations with fractions. Observed patterns supported by cognitivism revealed that invented methods or strategies on which RMATL is anchored are suitable enough to engage learner-centred teaching and learning which can prevent the abstractness of the concept of operations with fractions.

This study evaluated an intervention developed through an Institute of Education Sciences-funded Goal 2 research project to teach students with moderate intellectual disability (moderate ID) to solve addition and subtraction word problems. The intervention involved modified schema-based instruction that embedded effective practices (e.g., pictorial task analysis, graphic organizers, systematic prompting with feedback) for teaching mathematics skills to students with moderate ID. The dependent variables included steps performed correctly on a problem solving task analysis, number of problems solved, problem type discrimination, and generalization of problem solving skills. Results of a multiple probe across student dyads design indicated a functional relation between three dependent variables and the intervention. Upon completion of the intervention, all eight participants with moderate ID correctly followed the task analysis, discriminated problem types, and solved word problems. Key discussion items include feasibility of teaching problem solving skills, types of modifications needed, and the generality of these skills to novel formats.

This study aimed to produce learning trajectory with rowing context that can help students understand addition and subtraction of fractions. Subject of the research were students IV MIN 2 Palembang. The method used was research design with three stages, those are preparing for the experiment, the design experiments, and the retrospective analysis. Learning trajectory was designed from in-formal stage to the formal stage. At the informal stage, Rowing was used as a starting point to explore the students' knowledge of fractions. Data collection conducted through video recordings and photos to see the learning process in the classroom, written tests, observation and interviews during the learning process with the students which is the subject of research. Research produced learning trajectory consisting of a series of learning addition and subtraction of fractions dealing with the rowing. The results showed that the use of the rowing can be a bridge of students' thinking and help students in understanding the operation of addition and subtraction of fractions.

Many general and special education teachers teach mathematics word problems by defining problems as a single operation and linking key words to specific operations. Unfortunately, teaching students to approach word problems in these ways discourages mathematical reasoning and frequently produces incorrect answers. This article lists eight common key words, identifies the operation typically associated with each, and provides word problems that illustrate how reliance on key words can result in incorrect answers. Neither of these approaches--defining problems in terms of a single operation or linking key words to specific operations--has evidence to support its use. In contrast, other approaches do promote mathematical reasoning and substantially boost word-problem performance among students with learning disabilities. Two practices that have emerged from research studies as particularly effective for word-problem instruction are: (1) Attack strategies, which provide students with a general plan for processing and solving word problems (Montague, 2008; Xin & Zhang, 2009); and (2) Schema instruction, in which students learn to categorize word problems within problem types (i.e., schemas based on the word problem's mathematical structure), apply an efficient solution strategy for each word-problem schema, and understand the meaning of word problem language (Fuchs et al., 2014; Jitendra & Star, 2012). These strategies are discussed in detail.

Background: The concept of rational numbers is one of the learner's first experiences with a Mathematics concept beyond the basic skills operations on whole numbers. The personal knowledge of fractions that teachers bring to the teaching context is important because teachers mediate the conceptions that their learners construct. Aim: This study was set up to apply Action-Process-Object-Schema theory to study primary teachers' understanding of addition and subtraction of fractions. Setting: The participants of this study comprised 60 undergraduate full-time students, studying to become teachers. The participants were enrolled in a foundational course in Mathematics because they had not passed Mathematics at Grade 12 level. This course was intended to help deepen their understanding of basic numeracy, allowing them to continue with further courses if they wanted to specialise in teaching primary mathematics Methods: Data were collected using written responses of the pre-service students to two tasks that focused on operations with fractions. Ten students volunteered to be interviewed of which three are drawn upon in this article. Results: Many of the pre-service teachers coped well with addition and subtraction of common fractions with the same denominator. However, more than 52% struggled to carry out these operations on common fractions with different denominators, showing that their conceptions had not developed into object-level structures. Conclusion: It is evident that the incorrect procedures have become embedded in the students mental schema. It is crucial that programmes for upgrading pre-service teachers should include opportunities for teachers to interrogate their personal understandings of the basic mathematics concepts.

This study reports errors and misconceptions about division operation in fractions made by eighth-grade students. To investigate these errors and misconceptions, the study examined students' understanding of division operation in fractions using a questionnaire and expectation table. Students' misconceptions were determined for each question and compared with the expectation table. Common misconceptions of this research are; misunderstanding of word problems, incorrectly generalization of learned rules for fractions, considering numerator and denominator as integer numbers, not conceptually learning of division operation in fractions, associating division with addition, subtraction and multiplication operations. Students made same misconceptions with the expectation table.

In this study, it was aimed to analyze the structure of prospective middle school mathematics teachers' problems posed with regard to given symbolic representation including addition and subtraction operations with integers. The study conducted with 96 last grade elementary school mathematics teacher candidates studying in Faculty of Education of Erciyes University in Turkey at the beginning of 2018 years. A Problem Posing Task including five items regarding addition and subtraction operations with integers and semi-structured interview forms were used as data gathering tools. Results indicated that prospective teachers had difficulties in choosing problem types with regard to the given operation structure. Prospective teachers presented less success in problem posing with integers having different signs. Besides, some of the prospective teachers used signs of integers and operations interchangeably and some others multiplied signs of integers and operations, and they gave meaning to the result of this multiplication while they were posing problems.

The aim of this qualitative case study is to investigate prospective mathematics teachers' subject matter knowledge of the underlying concepts of standard and nonstandard algorithms used to solve the problems with whole numbers. Twenty three prospective mathematics teachers enrolled in the Elementary Mathematics Education Program of one of the most successful universities in Turkey were the participants of the study. The data was collected through four tasks containing basic algorithms. More specifically, the Ones Task assessed participants' understanding of the underlying place value concepts of standard algorithms. The Andrew Task and the Doubling Task required participants to conceptualize and interpret nonstandard strategies. In the Division Task, participants were expected to provide in-depth explanation for the difference between multiplication and division and between partitive division and measurement division. The content analysis method was used to analyze the data. The results of the study revealed that more than half of the prospective mathematics teachers had knowledge about the place value of 1 in addition and subtraction, and also multiplication. However, most of the prospective teachers could not explain the underlying principle and the meaning of the nonstandard algorithm in subtraction. Similar to their knowledge on subtraction, prospective teachers' knowledge on division was limited.

In this study, it was aimed to determine the errors and misunderstandings of 5th and 6th grade middle school students in fractions and operations with fractions. For this purpose, the case study model, which is a qualitative research design, was used in the research. In the study, maximum diversity sampling, which is a purposeful sampling method, was used. For this reason, this study was conducted in a state and a private secondary school on a voluntary basis with 105 5th graders and 84 6th graders, with a total of 189 secondary school students from different levels of achievement. In order to determine students' errors and misconceptions about fractions, two tests were prepared by researchers, ten open-ended questions for 5th graders and twelve open-ended questions for 6th graders. Before these tests were formed, the related field was scanned and the existing misunderstandings were determined. Content analysis method was used on the analyzing of the data. The answer papers of the students were coded and students' solving were examined in three categories as correct, wrong and blank. In addition to this the solving belongs to wrong category was examined in detail and mistakes of the students were coded and the reason of these mistakes were dwelled on. It is seen from the result of the study that students have misunderstanding and mistakes about fractions. It is seen that students are not using the modellings on the operations with fractions because of that most of the mistake and misunderstandings are about operations with fractions.

This research aimed to examine whether the activities related to mental computation strategies had any effect on the development of the second-grade elementary school students' problem-solving (PS) skills. Participants of the study are comprised of 14 children aged 7-8 years and enrolled in the second grade of the elementary school of a village. For the collection of data, the Problem-Solving Task (PST) which is composed of 12 problems linked with addition and subtraction operations was utilized. The identification of problems to be included in PST was based on problem types suggested by Sarama and Clements (2009). In this action research which lasted five weeks and three hours per week, 13 mental addition-subtraction activities (MASAs) were applied to children. Responses given by children to PST were examined in the context of time, accuracy, and strategy variables. At the end of executed activities, it was discerned that mental computation strategies used by children in solving problems got diversified, children's response speed was accelerated and the number of correct answers given by children went up. Before MASAs, it was found that children had difficulty in solving problems through mental computation, gave random answers, or tried to count with their fingers for computation. After MASAs, it was deduced that strategies that were selected by children differed on the basis of problem types.

During a time when students are accessing instruction virtually because of COVID-19, reflecting on the sixth of the Standards for Mathematical Practice (SMP 6) is a vital step for educators to take. Although the Common Core State Standards for Mathematics (NGA Center and CCSSO 2010) identifies SMP 6 as Attend to precision, associated expertise includes more than "calculating accurately and efficiently" (p. 7). SMP 6 also states that mathematically proficient students "communicate precisely to others," using "clear definitions in discussion with others and in their own reasoning" (NGA Center and CCSSO 2010, p. 7). Applying exact terminology allows students not only to form and support conjectures but also to engage in mathematical discourse. This, in turn, results in enhanced conceptual understanding because mathematical discourse has been linked to the unveiling of concepts and the debunking of misconceptions (Thompson and Gay 2020). This article describes an eye-opening experience, during which the authors realized that their methods of vocabulary instruction in the virtual environment needed to be revised. Data were collected in the third author's fourth-grade virtual classroom from January through February 2021. The first and third authors were co-teachers, whereas the second author assisted with instructional planning. All students were native English speakers; however, featured practices may also support the cognitive academic language proficiency development of English language learners in a virtual setting.

In the present research, we provide empirical evidence for the process of symbolic integration of number associations, focusing on the development of simple addition (e.g., 5 + 3 = 8), subtraction (e.g., 5 - 3 = 2), and multiplication (e.g., 5 × 3 = 15). Canadian children were assessed twice, in Grade 2 and Grade 3 (N = 244; 55% girls). All families were English-speaking, and parent education levels ranged from high school to postgraduate, with a median of community college. In Grade 2, children completed general cognitive tasks (i.e., receptive vocabulary, working memory, nonverbal reasoning, and inhibitory control). In both grades, children completed single-digit addition and complementary subtraction problems. In Grade 3, they completed single-digit multiplication problems and measures of applied mathematics, specifically, word-problem solving, algebra, and measurement. We found that addition and subtraction were reciprocally related (controlling for cognitive skills). Subtraction fluency predicted multiplication in Grade 3, whereas addition fluency did not. In Grade 3, both subtraction and multiplication fluency were predictors of applied mathematics, with multiplication partially mediating the relation between subtraction and applied mathematics performance. These findings support the view that learning arithmetic associations is a hierarchical process. As students practice each new skill, individual differences reflect the integration of the novel component into the developing associative network.

Students thrive when presented with tasks that are approachable, challenging, and interesting. Remembering to challenge all students is essential, not just those who excel in mathematics; teachers need to open the door to meaningful mathematics for every learner. In this exploration into subtraction patterns, teachers will see how curious young minds can guide discovery and make traditional mathematics concepts lively and exciting. Teachers will also see how taking cues from their students' sense of wonder allows teachers to address vital content and practice standards in their teaching. This example of how a sense of intrigue can lead to a richer teaching opportunity came from third-grade classrooms in a district where teachers promote inclusive teaching for both special education students and English language learners. The lessons were a collaboration between the district mathematics specialist and classroom teachers. The goal was to reinforce subtraction strategies while discovering unique patterns in a common operation. The students enjoy playing with numbers and discovering new and interesting ways numbers work together and the many patterns that emerge.

One of the factors associated with the less than positive mathematics performance of American students could be poorly designed textbooks that fail to facilitate the development of critical mathematical ideas. This study examined two reform-based textbooks (Go math! and Investigations) in reference to essential, mathematical big ideas emphasized in the Essential Understanding Series (EU) presented by NCTM against the backdrop of mathematics education reform. In particular, we analyzed the features of the lessons for teaching addition and subtraction word problem solving. In some aspects, the results indicated that in terms of teaching problem comprehension and algebra readiness, the two textbook series showed improvement compared to formerly used textbooks. However, both textbook series may fall short of reflecting some important EU big ideas (e.g., teaching the underlying structure and part-part-whole relation of addition and subtraction word problems; providing students with chances to deal with exhaustive problem types and situations). In particular, our findings highlight the need for teaching word problem solving through building abstract level conceptual understanding. Implications for future textbook developing and limits are discussed.

Students have difficulties in solving problems of fractions in almost all levels, and in problem posing. Problem posing skills influence the process of development of the behaviors observed at the level of comprehension. That is why it is very crucial for teachers to develop activities for student to have conceptual comprehension of fractions and operations involving fractions. The achievement of such conceptual comprehension can be accelerated through the use of mathematical models. For this, the aim of the study is to identify the errors in the problems posed by primary school teachers with respect to subtractions with fractions, and the models they employ to solve these problems. The present study employs both quantitative and qualitative methods together. This study was carried out with 31 primary school teachers. The teachers involved in the study were selected through random sampling. The study employs the "Problem Posing Test" comprising four items of subtractions with fractions. The test drawn up with reference to the operation of subtraction with fractions includes one item for each: subtracting a proper fraction from another proper fraction, and subtracting a mixed fraction from another mixed fraction. First of all, the answers provided by the teachers were categorized as problem, not-a-problem, or blank. Following such a classification is an analysis of the errors observed in the responses provided in the problem category. At the end, the study reveals that the rate of correct responses offered in the problem category falls as one progresses from item, one where both the minuend and the subtrahend are proper fractions, towards item four where a mixed fraction is subtracted from another mixed fraction. The fact that nine distinct types of errors were observed in the study reveals that the teachers have significant shortcomings when posing problems regarding subtraction with fractions.

In this study, we describe the methodology used to identify and validate a set of expert-defined fraction subtraction related attributes. These attributes are expected to be mastered by 6th grade students toward proficiency in fraction subtraction. This research argues and demonstrates that state standards guiding subject instruction plays an important role in the identification of the domain related fundamental attributes. This study also illustrates complete implementation of cognitive diagnosis model framework, which is used to extract diagnostic information about students' specific strengths and weaknesses.

This study aims to develop a learning trajectory to help students understand concept of subtraction of integers using multimedia in the fourth grade. This study is thematic integrative learning in Curriculum 2013 PMRI based. The method used is design research consists of three stages; preparing for the experiment, design experiment, retrospective analysis. The studied was conducted on 20 students of grade four SDN 1 Muara Batun, OKI. The activities of students in this study consisted of six learning trajectories. The first activity asks the students to classify heroism and non-heroism acts, summarize, and classify integers and non-integer. The second activity asks the students to answer the questions in the film given. The third activity asks students to count the remaining gravel in the film. The fourth activity asks students to count remaining spent money in the film. The fifth activity invites students to play rubber seeds in the bag. The last activity asks students to answer the questions in the student worksheet. The media used along the learning activities are a ruler, rubber seed, student worksheet, money, gravel, and film. The results indicate that the learning trajectory using multimedia help students understand the concept of integer subtraction integer.

The purpose of this study is to present a multimodal languaging model for mathematics education. The model consists of mathematical symbolic language, a pictorial language, and a natural language. By applying this model, the objective was to study how 4th grade pupils (N = 21) understand the concept of division. The data was collected over six hours of teaching sessions, during which the pupils expressed their mathematical thinking mainly by writing and drawing. Their productions, as well as questionnaire after the process, were analyzed qualitatively. The results show that, in expressing the mathematical problem in verbal form, most of the students saw it as a division into parts. It was evident from the pupils' texts and drawings that the mathematical expression of subtraction could be interpreted in three different ways. It was found that the pupils enjoyed using writing in the solution of word problems, and it is suggested that the use of different modes in expressing mathematical thinking may both strengthen the learning of mathematical concepts and support the evaluation of learning.

In this article, the authors summarize results from 5 randomized controlled trials assessing the effects of intervention to improve the fraction performance of fourth-grade students at risk for difficulty in learning about fractions. The authors begin by explaining the importance of competence with fractions and why an instructional focus on fractions magnitude understanding may improve learning. They then describe an intervention that relies strongly on this type of understanding about fractions instruction, and they provide an overview of the intervention's overall effects. This is followed by an overview of 5 intervention components for which the authors isolated effects. They conclude by discussing some of the lessons learned from this research program.

Teaching of addition, subtraction, multiplication and division in mathematics starts from the first years of primary school. The learning output for four operations (addition, subtraction, multiplication and division) affects student success at every level of mathematics education from primary to higher education. At this point errors, misunderstandings and possible misconceptions of students need to be identified, eliminated, and the forms of instruction that prevent its formation need to be investigated. This study aims to identify the errors primary school students make in four operations according to the views of class teachers. The study was designed in the qualitative research design and the semi-structured interview form prepared by the researchers was used in the study. The study group consisted of 48 class teachers. It has been found out that students make more errors in adding, subtracting and multiplying decimal digits, and using zero in division. In addition, classroom teachers stated that the mistakes made in four operations result mostly from student, teacher, program and the student's family and environment respectively. Teachers made course content, teacher, student and family oriented suggestions to eliminate the mistakes in four operations.

In recent times, the endangerment of Yorùbá has highly been speculated among Yorùbá intellectuals, indigenes and enthusiasts alike. In an effort to promote the learning and use of Yorùbá numeral system in carrying out day-to-day activities and transactions, the development of a Yorùbá arithmetic learning system will help bridge the gap between lost Yorùbá traditions of the transaction and modern day business activities. The design of this project was conceptualized using Unified modelling language (UML) tools. The resulting design and software specification was implemented using Python programming language due to its cross-platform compatibility and PyQt graphical user interface (GUI) module for the GUI. System evaluation was done using questionnaire to determine how the system is perceived by potential users. It was observed that 90% were unable to evaluate the addition expressions; 55% were unable to evaluate subtraction expressions, 65% were unable to evaluate multiplication expressions and 90% were unable to evaluate division expressions. It can be concluded from both research and the system developed that in order to make the Yorùbá language more relevant in the world today, attempts such as this project must be encouraged. The issue of the almost non-existent use of Yorùbá numeral must be addressed by producing learning systems or aids to help the teaching and learning of the Yorùbá numerals.

Addressed are four key issues regarding concrete instruction: What is concrete? What is a worthwhile concrete experience? How can concrete experiences be used effectively in early childhood mathematics instruction? Is there evidence such experiences work? I argue that concrete experiences are those that build on what is familiar to a child and can involve objects, verbal analogies, or virtual images. The use of manipulatives or computer games, for instance, does not in itself guarantee an educational experience. Such experiences are worthwhile "if" they target and further learning (e.g., help children extend their informal knowledge or use their informal knowledge to understand and learn formal knowledge). A crucial guideline for the effective use of concrete experience is Dewey's principle of interaction--external factors (e.g., instructional activities) need to mesh with internal factors (readiness, interest). Cognitive views of concrete materials, such as the cognitive alignment perspective and dual-representation hypothesis, provide useful guidance about external factors but do not adequately take into account internal factors and their interaction with external factors. Research on the effectiveness of concrete experience is inconclusive because it frequently overlooks internal factors.

The purpose of the study was to determine whether there was any significant difference in speed, accuracy, retention and transferability between the Decomposition (DEC) and the Base-Complement Additions (BCA) algorithms for performing compound subtraction in non-decimal bases. Fifty-nine students with a mean age of approximately 15 years from two Agona Swedru Junior High Schools in the Agona District participated in the study. The study employed the pretest-posttests non-equivalent design. The two schools were randomly assigned to the treatment groups. The data collected on the four achievement tests namely Pretest, Posttest, a Third test and Retention test administered were then studied and analysed by employing the t-test at 0.05 level of significance. From the study it was found out that the mean performance of the BCA group was significantly higher than the DEC group on measures of accuracy. There was no relationship between the BCA and the DEC groups on the measure of speed. The BCA group produced significantly better computational accuracy than the DEC group. The mean score of the BCA group in the retention measures were significantly higher than that of the DEC group. The third test which was the test for understanding and application of the two methods showed that the BCA group performed better than the DEC group though the difference was slightly significant. Based on the findings, it was established that BCA method has substantial gains over the DEC method and as such recommended that the BCA should thus be included in the Junior Hugh mathematics curriculum as alternative method of solving compound subtraction in non-decimal bases.

How much is 41 - 39? How about 100 - 3? Which of those computations was easier for you to do? It so happens that first graders are much more likely to solve 100 - 3 correctly than 41 - 39. Likewise, second graders are much more likely to solve 100 - 3 correctly than 201 - 199. Our data (Schoen et al. 2016) suggest that the latter problems are more difficult for students to solve correctly, because many students' understanding of subtraction is limited by thinking about the operation only as take-away or by using a default procedure, such as the standard subtraction algorithm in the United States. In this article, we argue the importance of students learning to reason flexibly about subtraction. We highlight a useful but often-ignored way of reasoning, and we offer suggestions for teaching about subtraction. [This paper was published in "Teaching Children Mathematics" (EJ1122656).]

In this study 2nd grade primary school students' skills and success of subtraction especially with borrowing three digit numbers is investigated. Two different models of the subtraction are analyzed. One of them is used in Turkey (1) and the proposed method which is used in Austria (2). The present paper describes the success, both of the subtraction methods to be observed when 16 primary students worked on 10 subtraction problems. These 10 problems were administered repeatedly by means of a class test: In March 2015 firstly the standard Turkish algorithms were introduced and as a pretest examined, afterwards two weeks long two hours in a week Austrian system was practiced and finally they had been examined as a posttest with Austrian system

This study explored the relationships between task beliefs about the empty number line (ENL), mathematical ability, gender, and voluntary ENL use in multi-digit subtraction and addition. One hundred twenty-three Dutch third-grade students and nine teachers from six schools participated in this study. The multilevel path analysis showed that task beliefs about the ENL mediated the relationship between students' mathematical ability and their voluntary ENL use. No gender differences were found in the multilevel path analysis. Finally, the results show that task beliefs about the ENL and voluntary ENL use differed across classrooms. The discussion focuses on the implications of the results for using the ENL in diagnostic assessment.

People tend to underestimate subtraction and overestimate addition outcomes and to associate subtraction with the left side and addition with the right side. These two phenomena are collectively labeled 'operational momentum' (OM) and thought to have their origins in the same mechanism of 'moving attention along the mental number line'. OM in arithmetic has never been tested in children at the preschool age, which is critical for numerical development. In this study, 3-5 years old were tested with non-symbolic addition and subtraction tasks. Their level of understanding of counting principles (CP) was assessed using the give-a-number task. When the second operand's cardinality was 5 or 6 (Experiment 1), the child's reaction time was shorter in addition/subtraction tasks after cuing attention appropriately to the right/left. Adding/subtracting one element (Experiment 2) revealed a more complex developmental pattern. Before acquiring CP, the children showed generalized overestimation bias. Underestimation in addition and overestimation in subtraction emerged only after mastering CP. No clear spatial-directional OM pattern was found, however, the response time to rightward/leftward cues in addition/subtraction again depended on stage of mastering CP. Although the results support the hypothesis about engagement of spatial attention in early numerical processing, they point to at least partial independence of the spatial-directional and magnitude OM. This undermines the canonical version of the number line-based hypothesis. Mapping numerical magnitudes to space may be a complex process that undergoes reorganization during the period of acquisition of symbolic representations of numbers. Some hypotheses concerning the role of spatial-numerical associations in numerical development are proposed.

This study examined professional teacher noticing in the context of written responses 12 pre-service teachers (PSTs) provided in a reflective journal after posing addition and subtraction problems for students in an after-school tutoring program. Professional teacher noticing skills, attending, interpreting, and deciding, were situated within the well-defined mathematics content and associated trajectory of student strategies for the meaning of addition and subtraction. A three-point rubric was designed and utilized to analyze PST attending, interpreting, and deciding responses. Results characterize what PSTs "could do" relative to each skill as well as development of noticing skills over time. Results also highlight how PSTs included a rationale and next steps in deciding responses that related to supporting student understanding of the meaning of addition and subtraction, providing additional practice, and teaching solution strategies. Implications for teacher education are discussed.

In this article, we report the findings of a study conducted with 6 Grade 8 students in the United States. The students posed stories for open number sentences involving addition and subtraction of integers. We analysed the stories posed by the students to build models that describe the conceptual structures behind these posed stories -- the conceptual models for integer addition and subtraction. These four conceptual models for thinking about and using integer addition and subtraction include Bookkeeping, Counterbalance, Relativity, and Translation, and are generated from the students' posed stories. We also provide profiles of conceptual model use for two of the 6 students that describe how the students posed stories to accommodate conceptual model use, such as posing unconventional or unrealistic stories or changing the structure of the number sentences. The conceptual models and descriptions of how the students used them provide perspective into student thinking about integers and contexts, highlighting the mathematics of the students, and calling for a re-examination of contexts used in school mathematics.

Research exists on mathematical interventions to support elementary students experiencing mathematical difficulties, yet little examines the provision of such interventions within an online learning environment. This study explored the online delivery of the virtual-representational-abstract (VRA) instructional sequence, taught via explicit instruction, coupled with the system of least prompts (SLP) to three elementary students with mathematics difficulties. From this single case design study, researchers determined the VRA instructional sequence with the SLP was effective for three elementary students with mathematical difficulties in terms of accuracy in solving double-digit subtraction with regrouping problems. Further, the students were relatively independent in solving the problems. The results of this study hold implications for the online delivery of effective mathematical interventions to struggling elementary students.

The analysis of word problem types and schema-based diagrams are assumed useful for helping students and thus, important for preservice teachers to learn in their preparation programs. Drawing on pre-and post-assessments and videotaped demonstrations, this study analyzed elementary preservice teachers' understanding of addition and subtraction word problem types and how well they could demonstrate their understanding after their exposure in a mathematics methods course. Results suggest that while participants' knowledge of word problem types and their ability to solve word problems were substantially improved, their identification, diagram representations, and solutions for different types of word problems were not substantially developed. Policy and research implications for these findings are then discussed and proposed.

This study aims to examine the differences in numerosity estimation on a right-to-left number line between second- to fourth-grade students and undergraduate students, together with whether number-line estimation is related to basic arithmetic tasks (addition and subtraction). Hence, 53 Arabic-speaking children and 63 Arabic-speaking adults performed a paper-based number-line estimation task and also an arithmetic task. Number-line estimation was represented either by the percentage of absolute errors in positing the quantities on the line, by the accuracy, or by the "r" score for linearity of representation. The results show that, although children's addition (but not subtraction) performance resembles that of adults, the correlations vary between number line and arithmetic scores. Addition scores are positively correlated with error percentages in the children's number-line tasks, while subtraction is negatively correlated with adults' error percentages. These differences are assumed to be attributed to various uses of recall and procedural strategies.

Introduction: Although there is ambiguity about the elements of teacher knowledge, all researchers accept that being able to anticipate what errors can be made, the reasons for and the strategies to overcome these errors, in short, the knowledge of students is important for student achievement. In this study, knowledge of students refers to being aware of students' possible errors and underlying reasons for these errors and knowing how to overcome these errors. Based on this consideration, the purpose of this study is to investigate pre-service elementary teachers' knowledge of students on the subtraction topic. Methods: Considering the purpose, the data were collected from 118 pre-service elementary teachers who were enrolled in a four-year Elementary Teacher Education program via a task-based questionnaire related to the topic of subtraction and semi-structured interviews following the questionnaire. The task-based questionnaire included three completed incorrect subtraction tasks and was prepared considering the related literature and the elementary school mathematics curriculum of Turkey. Each task in the questionnaire contained a different type of error. The pre-service elementary teachers' answers to the task-based questionnaire were categorized as correct, partially correct, wrong, or no answer by means of categorical analysis. The pre-service elementary teachers who gave correct and partially correct answers to the tasks were asked to participate in the second part of the study to learn their possible strategies to overcome the errors made in the tasks. Results: As a result of the quantitative and qualitative analysis of the pre-service elementary teachers' responses, it was found that their knowledge of students for subtraction is limited. Specifically, although the pre-service elementary teachers were partially able to identify the errors in the first and second task, they were not able to identify the error in the third task. Furthermore, they were better able to determine a more commonly occurring subtraction error compared to the uncommon ones. The pre-service elementary teachers could not identify the underlying reasons that led the students to the errors. They could not explain what conceptual knowledge related to the topic of subtraction the student lacked that resulted in the errors. As the pre-service teachers did not attain these reasons, they were not able to provide strategies to overcome these errors different than restating the rules or procedures to overcome students' errors. Discussion: Only being able to identify the errors is not enough to make the instruction effective. Teachers also need to know and provide a rationale for why the errors happen and how to overcome them (Even & Tirosh, 1995). Contrary to this statement, the pre-service elementary teachers in this study could not attempt to understand the students' thinking or could not explain the reasons behind students' errors with the notion of subtraction. Son (2013) emphasizes that pre-service teachers tend to explain the reasons for students' errors as procedural. Similar to this emphasis, the pre-service teachers identified the errors as resulting from not applying procedures carefully or not enough knowing algorithms to find correct solutions. However, without addressing the reasons or focusing on procedures does not promote students' understanding of the related topic (An, Kulm, & Wu, 2004). The pre-service teachers' difficulties in identifying the possible conceptual reasons may result from deficits in knowledge of students (Fennema & Franke, 1992; Ball et al., 2008). Limitations: The findings were limited with the responses of the pre-service elementary teachers participating in this study. Moreover, the pre-service elementary teachers' knowledge of students was investigated within the scope of subtraction. Conclusions: Although the pre-service elementary teachers could identify students' errors on subtraction, they had difficulty in identifying the reasons and suggesting strategies to overcome the errors. Therefore, it can be concluded that the pre-service teachers' knowledge of students for the topic of subtraction is limited. An implication of this is that teacher educators need to include error-analysis tasks to help pre-service teachers be aware of the importance of their knowledge of students.

This observational study used data from 270 second-grade students to investigate the association between students' strategy use for multidigit addition and subtraction and their mathematics achievement. Based on strategies they used during a mathematics interview, students were classified into the following strategy groups: (a) standard algorithm, (b) invented, (c) mixed, and (d) unclassified. We used two-level hierarchical linear regression to investigate the association between students' strategy use and their performance on a standardized test in mathematics. Results indicated that students in the mixed strategy groups had significantly higher mathematics achievement than those in the standard algorithm and the unclassified groups.