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Purpose: The language load within mathematics test items can lead to inaccuracy in assessing "Orang Asli" pupils' mathematical ability due to their struggle in comprehending the academic language. This study aims to determine the validity of using test accommodations in the form of oral academic language and oral native language when administering mathematics computation and word problem test items among the "Orang Asli" pupils. Methodology: By employing a quantitative approach, this study utilised random equivalent group design to examine the validity of using oral test accommodations among 334 Grade Four "Orang Asli" pupils. Three testlets were constructed and used in this study: written test in the academic language (WAL), oral test in the academic language (OAL), and oral test in the native language (ONL). The three testlets comprised Paper 1 and Paper 2, each consisting of 20 mathematics computation and 20-word problem test items, respectively. One-way ANOVA was used to analyse and examine if there were differences in the mathematics performance between the three groups of tests administered to the "Orang Asli" pupils. Findings: The results indicated that the test scores of the pupils on the oral tests were higher compared to the written test administered to the pupils for both mathematics computation and word problem items. For computation items, the pupils performed better in the OAL testlet, followed by the ONL testlet. On the other hand, the pupils obtained the highest test scores in the ONL testlet compared to the OAL testlet for word problem items. Furthermore, the results illustrate that there is a difference in the mathematics performance of the "Orang Asli" pupils between the oral tests in both languages and written tests in the academic language. Significant: This study clearly shows that language plays a prominent role in affecting the "Orang Asli" pupils' mathematics performance. By utilising the oral test as a promising test accommodation, this measure could help address the language barrier faced by the "Orang Asli" pupils in mathematics testing.

Language as a cultural element influences Aboriginal pupils' mathematic performance. This study examines the utility of oral test in native language for computation and word problem mathematics items using score comparability. The sample includes 230 Grade 5 Orang Asli pupils from eight Malaysian primary schools in two states. By employing equipercentile linking using RAGE-RGEQUATE, the oral mathematics test was easier and more helpful, especially in native language for word problem items. The oral mathematics test is a valid measure for Orang Asli pupils in capturing their mathematical ability by reducing language as an erroneous secondary construct while assessing the primary dimension, which is mathematics knowledge. In addition, incorporating Aboriginal pupils' cultural elements can mitigate their challenges in learning Western mathematics.

Because word problems present mathematical information through a scenario, they are language-intensive and require mathematical and reading comprehension skills to solve them. In addition, they are linguistically complex, which makes them challenging for all learners, especially multilingual learners. Given the rising number of dual-language bilingual programs in the United States--which have the goal of simultaneously supporting content and language learning--it is important to understand the linguistic complexity of word problems and their translations. Therefore, we adopted a sociolinguistic perspective to recognize the inextricable relationship between language and mathematics and conducted a linguistic content analysis of Grades 1 and 4 addition/subtraction word problems in a widely used U.S. elementary curriculum. We analyzed the ways in which word problems were linguistically complex and how complexity differed between English, French, and Spanish translations. We found that word problems were complex due to vocabulary inconsistencies, assumed cultural knowledge, verb tenses, and errors. The French and Spanish translations also had more errors and inconsistencies than the English source text. Findings have implications for curriculum and assessment developers related to multilingual editing and linguistic and contextual consistency, as well as for teachers to support students' engagement with language and mathematics.

Background: Problem-solving as a vehicle to develop independent thinking skills is mostly underestimated and is often either overlooked or not given adequate attention within the existing South African mathematics curriculum. Consequently, numerous learners often display limited skills or lack skills to adequately crack Mathematics problems by applying methods put forward in class. This generally results in under-achievement. Aim: This study aims to explore and emphasise the problem-solving methods applied by Grade 4 learners involved in solving measurement word problems, and to reveal what transpires when the selected learners apply these methods to arrive at meaningful solutions. Setting: Data were collected from a class of 42 Grade 4 learners at a primary school in Cape Town South Africa. Learners were conveniently selected. Methods: A qualitative case study research design was adopted. Data gathering instruments of the study included observing learners solving, measurement word problem activities and focus group interviews. Results: The study revealed that singular methods were applied by Grade 4 learners, such as, adding, multiplying, creating a sketch or diagram, grouping, dividing, subtracting, logical reasoning, guessing and tabulating values. Conclusion: Grade 4 learners are prone to applying methods such as clustering or organising into groups, tabulating numerical values and logical reasoning were all applying mathematically sound methods. Such learners, however, needed a degree of supervision and instruction to indicate the way in which such methods were applied successfully as these methods were not necessarily dealt with in classroom context or in textbooks. Contribution: The findings emphasise the need for tackling learners' limited problem-solving competencies and accentuate the necessity for greater attention to develop and grow methods for optimal and successful solving of problems in context.

Mathematics proficiency in the Philippines is a persistent concern, seen by many as a sign of an educational crisis. Teachers are responsible for improving learning outcomes in any discipline, including math. Thus, the study intended to conduct an error analysis of verbal problems among pre-service mathematics teachers. The researcher employed descriptive research and Newman's Error Analysis to suffice the research objectives. The findings revealed low levels of error in reading stage, but moderate levels of error in comprehension, transformation, process skills, and encoding stages. In addition, underlying factors contributing to these errors were incomplete solutions and answers, incorrect or incomplete processes, grammatical errors, conversion errors, and failure to indicate an answer. The researchers concluded that pre-service teachers are proficient in the early stage of problem solving, but challenged in properly constructing equations, the utility of operations, and interpreting their results.

It is important to work on verbal mathematical problem-solving skills, which is one of the high-level skills with which students have difficulty. Supporting problems with visuals has been suggested by mathematics educators to develop this skill. This study examined the effect of the visualization of mathematical problems on the problem-solving performance of primary school first grade students. The study group consisted of 41 students attending first grade in a public primary school located in a low socio-economic status region of Istanbul. Semi-structured individual interviews were conducted with 8 students. Problem-solving questions developed by the researchers were used for pretest and post-test, the results of which were analysed using the Wilcoxon Signed-Ranks test. For the experimental process, the Visually Supported Problem-Solving Worksheet was used as a data collection tool. Students' views on the implementation were analysed using a content analysis method. The results indicate that the problem-solving worksheet supported with visuals was effective in improving the mathematical problem-solving performance of the students. The majority of the students stated that the implementation contributed to improving their performance and their satisfaction.

Connecting mathematical concepts to real-life situations is a crucial skill in math class. Using mathematical literacy problems in classes may be an option to make connections with real life. In this context, the aim of this research is to reveal the experiences of preservice mathematics teachers in the process of writing mathematical literacy problems and to determine the difficulties they encountered. The participants of the research were 28 preservice teachers who were taking the mathematical literacy elective course in the 2022-2023 spring semester. In this qualitative research design, preservice teachers were provided with essential theoretical information on mathematical literacy over nine-weeks. They were then presented with mathematical literacy questions and their analyses were conducted. Following five weeks, participants worked in groups of 3-4 individuals, each group creating four mathematical literacy problems relating various content areas. At the end of their studies, in order to determine their experience regarding problem posing process, preservice teachers were given a survey consisting of open-ended questions. As a result of the analysis of the data obtained from the survey, it was seen that the issues in which the preservice mathematics teachers found themselves inadequate were more than the issues in which they found themselves competent. It was observed that participants focused on choosing context and writing the problem in accordance with the real life. While one group found themselves very competent in this regard, the other group found themselves quite inadequate. In addition, preservice teachers thought that they needed to examine more mathematical literacy questions and made writing attempts in order to pose qualified mathematical literacy problems.

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.

This study aimed to evaluate the value of justice, one of the 10 core values in the mathematics course curriculum, in the context of an exemplary activity in classroom practice and student interviews, where it was integrated into mathematics learning goals. The case study model was employed. The sample of the study included eight 7th-grade students in the 2018-2019 academic year. An activity prepared about the value of justice and a semi-structured student interview form were used as data collection tools. The data obtained following the implementation of the activity were analyzed using content analysis. Students stated that it was necessary to carry out value-based activities in mathematics classes to understand the importance of the value of justice and to increase social awareness about it. According to the results of the study, it is recommended to plan, implement, and discuss in-class activities in which students can explain their value perceptions according to mathematical problems and actively participate in the process so that their awareness of the value of justice can be increased.

This study was conducted while 9th grade students learn to solve inequalities and seeks to understand their approach to solving problems with a real-life context. Specifically, the aim is to understand: (1) What are the main characteristics of the students' approaches to the proposed problems? (2) What is the impact of the real context on the students' resolutions? A qualitative and interpretative methodology is adopted, based on case studies, with data collected through documentary collection and audio recording of discussions between a pair of students while solving problems. The main conclusions suggest a trend to approach problems without establishing immediate connections with what was being done in the classroom, with students' decisions being essentially guided by criteria of simplicity. The real context of the problems seems to have the potential to develop in students a more integrated mathematics, focused on understanding and not so much on the repetition of mechanical and meaning-independent procedures. The students' familiarization with the context in question is one of the aspects highlighted by this study.

The ability to relate mathematics teaching to real life is the common goal of many countries' mathematics curriculum National Society for Experiential Education's (NSEE) [2011] "Eight Principles of Good Practice for All Experiential Learning Activities." (intention, originality, planning, clarity, monitoring and evaluation, reflection, evaluation, and acknowledgement) recommend that students' experiences and the meaning and learning they derive from those experiences should be consolidated. Students' ability to develop mathematical skills is directly linked to their ability to relate, which includes skills such as "problem solving, reasoning, communication." Bringing real-world contexts to mathematics lessons can affect students' attitudes towards mathematics and also support their mathematical association skills. In this context, it is important to analyze how the activities in the textbooks, which are the first resources in the classroom, are prepared and to what extent they connect the student with real life. In this regard, the focus is on the activities in the primary school mathematics textbook. In this study, the relationship of the activities in the Greek Primary School 3rd-grade textbook with real life was analyzed based on Gainsburg's (2008) classification. The document analysis method was used in the study, which was conducted on the basis of qualitative research. The data source of the research is the Greek Primary School 3rd grade mathematics textbook. The data obtained was analyzed using the descriptive analysis method. As a result of analyzing a total of 147 activities, it was determined that the activities were collected in the type of classical word problems and partially included activity types associated with real life.

Students' difficulties in differentiating the direct proportion and inverse proportion problems cause interference. Proactive interference is the error that occurs when old information (concept of direct proportion) interferes with new information (concept of inverse proportion). In solving the problem of inverse proportion, students often use the concept of direct proportion. The student's mental structure regarding the concept of proportion as a result of previous learning is referred to as met-before. Therefore, this study aims to describe the met-before of students who experience proactive interference. This research is a case study involving 32 8th-grade students in Malang, Indonesia. These subjects were students who experienced proactive interference with specific fluency of communication and willingness. Data was collected through proportion problems and interviews. Students' work was analyzed based on the description of the met-before. The results showed that students who experienced proactive interference with the non-flexible type had suppressed problematic, while students with the flexible type have focus supportive met-before in solving direct proportion problems. Both students with non-flexible type and flexible type have focus problematic met-before when solving inverse proportion problems. This is because met-before about cross multiplication strategy interferes with students' problem-solving.

Special tutorials both online and off-line were experimented in order to provide extra support for the senior pre-service mathematics teachers at an Australian regional university to improve their learning experience and achieve the best possible learning outcomes in an advanced mathematics course focusing on solving ordinary differential equations and applying mathematical modelling. Two types of special tutorials were offered to the students, the progressive tutorials on solving the same problem with different methods according to the learning progression and student's instant requests, and the targeted tutorials to address the common problems shared by many students in attempting questions in the formal assessments. The experiments on these special tutorials indicated that the targeted tutorials were immensely useful for the students to either expand the scientific knowledge related to a real-world scenario described by words so as to begin problem solving with correct setting-ups or streamline multiple mathematical processes together to solve a complicated real-world problem described in words. This approach motivated most students to achieve their best possible learning outcomes. The progressive tutorials were effective in addressing student's curiosity of solving the same problem by multiple techniques and hence improving student's mathematical thinking and problem-solving skills in general. This exploratory study also found that there were common problems with a lack of general science knowledge and retention of the previously learnt mathematical techniques among most students. There also existed a portion of students who showed no interest in engaging with learning regardless of how much extra learning support provided to them.

This study compared the problem-solving abilities of ChatGPT and 58 pre-service teachers (PSTs) in solving a mathematical word problem using various strategies. PSTs were asked to solve a problem individually. Data was collected from PSTs' submitted assignments, and their problem-solving strategies were analyzed. ChatGPT was also given the same problem to solve with various prompts, and the correctness of its solutions and problem-solving strategies were assessed alongside those of PSTs. The results indicated that PSTs used diverse strategies and achieved accurate solutions, but not always relevant strategies to children's level of understanding. ChatGPT employed similar strategies to PSTs but mostly produced incorrect solutions, and its performance needed to be contextualized in the primary school context. The study highlights the potential of ChatGPT in mathematics teaching and informs teacher education programs about the possibility of using it in teaching problem-solving strategies.

Single-subject research can provide adequate justification when developing successful evidence-based educational practices for children with autism spectrum disorder (ASD), which can help such students thrive academically. This research investigates whether point-of-view video modelling effectively improved the word problem-solving addition performances of primary-school-aged students with ASD. The effectiveness of the intervention on each participant's ability to solve single-digit addition by a single digit, effectively view the video on a tablet and generalise a learned skill was investigated by utilising a design that included several participant baselines. All participants saw an improvement in their ability to solve simple addition problems due to using point-of-view video modelling. Between the baseline and intervention phases, each participant's rate of digits correctly entered per minute and total number of steps completed significantly increased. A generalisation phase was performed at home. After receiving the intervention, people with ASD could independently complete word problems involving addition with a single digit. The findings suggest that this technology could practically support the education of the growing number of children and young people with ASD by mollifying the particular learning obstacles their impairment brings.

Students encounter mathematics word problems as early as kindergarten and continue to see them throughout their schooling experience. Schema instruction with an attack strategy can support students to successfully navigate word-problem solving. Schemas help students categorize word problems by similar characteristics. To better support students making minimal word-problem-solving growth with schema instruction, teachers can consider five strategies to increase support in schema instruction: mathematics language, paraphrasing, gestures, visuals, and organizers. These five strategies pair well with all the additive and multiplicative schema problems, making them flexible tools to work across instruction. This article discusses the evidence base for, how to implement, and cautions for the five strategies when pairing them with schema instruction.

Typically, vocabulary instruction is associated as part of reading instruction. However, vocabulary instruction is also a necessary component of mathematics instruction. In fact, state and national standards (e.g., Common Core State Standards) require elementary students to communicate about mathematics to provide clear explanations, construct coherent arguments, and critique mathematical reasoning. Students need to become proficient in mathematics language and vocabulary to meet these standards. This article provides guidelines for selecting which mathematics terms to teach, describes the vocabulary map routine used in Stevens et al. (2023), and provides a sample vocabulary map with step-by-step modeling.

This study reports a secondary analysis from a randomized control trial study conducted with third graders with mathematics difficulty (MD; N = 75). We used simple linear regression to investigate whether students' performance on a word-problem vocabulary assessment predicted their performance on a measure of word-problem solving. Additionally, we used multiple linear regression to determine whether English learner (EL) status (n = 21; 28%) moderated this relation. Students' word-problem vocabulary knowledge significantly predicted their word-problem performance (R[superscript 2] = 0.27), but this relation was not moderated by EL status ([delta]R[superscript 2] = 0.003). These findings suggest word-problem vocabulary knowledge is important for word-problem solving for students with MD, regardless of EL status. Given the results showed that improved word-problem vocabulary was associated with higher levels of word-problem solving, researchers and teachers might consider including word-problem vocabulary support as part of word-problem interventions.

This study assessed the effectiveness of modified schema-based instruction (MSBI) in teaching students with autism spectrum disorder (ASD) how to solve additive compare word problems. The study was conducted in Spain, and it involved three students with ASD who were 6, 7, and 9 years of age. A single-case, multiple-baseline across-students design was followed. Results indicated a functional relation between MSBI and improved performance in solving additive compare problems of type difference amount unknown and compare amount unknown, maintained for 4 to 6 weeks post-intervention. Additionally, two students successfully applied their skills to reference-amount-unknown compare problems, demonstrating successful generalization. However, their generalization to problems involving two operations was somewhat limited by difficulties with one of the operations. Social validity data collected from the students' families and teachers indicated that both groups had positive views on the use of the MSBI for mathematical word problem-solving skills.

This article uses the concurrent mixed methods design to explore the errors made by 171 Grade Seven learners in algebraic problem-solving within the Assin Central Municipality in Ghana. The participants were categorized into low-achieving and high-achieving groups based on their performance in a pretest, to help provide a detailed examination of the discrepancies in error occurrence between these groups. The Newman error analysis framework was used to unveil distinct patterns of errors among learners when tackling algebraic tasks. Quantitative data from the test were complemented by qualitative insights employing the Think Aloud Protocols (TAP). The analysis revealed that low-achieving learners struggled with reading, comprehension, and transformation errors, while high-achieving learners mainly encountered transformation and process skill errors. The findings contribute valuable insights into learners' challenges in mastering algebraic concepts, offering implications for educational interventions and curriculum development in mathematics education. The study recommends implementing differentiated instruction strategies and providing additional support for comprehension, and problem-solving skills to improve algebraic proficiency among learners.

Mathematical word problem-solving skills are crucial for students across their lives, yet solving such tasks poses challenges for many. Therefore, understanding the characteristics of mathematical word problems that are associated with students' performance is important. The objective of this systematic review and meta-analysis was to evaluate the effects of linguistic and numerical task characteristics associated with mathematical word problem-solving performance among elementary school-aged children (Grades 1 to 6). The systematic review was based on five electronic databases and citation searching. Reporting was conducted following The Preferred Reporting Items for Systematic reviews and Meta-Analyses (PRISMA). The findings (K = 69) showed that five of the six investigated linguistic task characteristics (i.e., the position of the unknown, schematic structure, irrelevant information, realistic considerations, and lexical consistency) and one of the two numerical task characteristics (i.e., number of operations) were related (g = 0.39 to 4.26) with elementary school-aged children's mathematical word problem-solving. However, the findings did not provide support for a general association between a familiar situational narrative or the required operation with mathematical word problem-solving. The findings highlight that elementary school-aged children especially struggle with mathematical word problems requiring realistic considerations or multiple mathematical operations, containing lexical inconsistency, and problems in which the position of the unknown is the first value. This further understanding of elementary schoolers' word problem-solving performance may guide the design of appropriate and progressive instruction and assessment tools and steer research into the interactions within task characteristics and with individual characteristics.

Recently, automatic question generation (AQG) has been researched extensively for educational purposes. Existing approaches generally lack relevant information on the authentic context and problem diversity with various difficulty levels, so we proposed a new AQG system for generating contextualized and personalized mathematic word problems (MWP) in authentic contexts using the Generative Pre-trained Transformers (GPT). Our proposed system comprises (1) authentic contextual information acquisition through image recognition by TensorFlow and augmented reality (AR) measurement by AR Core, (2) a personalized mechanism based on instructional prompts to generate three different difficulty levels for learners' different needs, and (3) MWP generation through GPT with authentic contextual information and personalized needs. We conducted a quasi-experiment with the participation of 52 students to evaluate the effectiveness of the proposed system on geometry learning performance. Further, the learning behaviors were analyzed in the aspects of authentic context, mathematics, and reflective behavior. The findings showed better results in geometry learning performances from students who learned with contextualized and personalized MWPs than those who were taught without contextualization and personalization on MWPs. Moreover, it was found that student's ability to comprehend the practical situation or scenario presented in a problem (problem context understanding) and students' ability to recognize relevant information from the problem context (identifying contextual information) significantly improved their learning performance. Moreover, students' ability to apply math concepts and solve medium-level MWP also contributes to the improvement of learning performance. Meanwhile, learners showed positive perceptions toward the proposed system in facilitating geometry learning. Therefore, it is useful to promote an authentic context setting for mathematical problem-solving.

Word problem-solving is one major area in mathematics that has been identified as being particularly challenging for students, specifically for those with learning disabilities (LDs) and emotional and behavioral disorders (EBD). This study aims at evaluating the effects of a strategic math intervention with concept maps on the ability to solve word problems (addition and subtraction problems, number range of thousand) among students with LDs and EBD from the eighth grade. A multiple-baseline design across participants (N = 9) was applied to evaluate the intervention, which was held three times a week over a 6-week period. Overall, the results demonstrated a functional relation between the amount of correctly solved word problem tasks and the intervention. All nine students improved in word problem-solving, as evidenced by the fact that more tasks were solved, with a higher score in the intervention phase compared with the baseline (between-case standardized mean difference was 1.84; 95% confidence interval [1.24, 2.44]). The social validity data display that all students found the intervention helpful but also partly exhausting. The limitations and implications of this study are discussed.

Word problem solving is the premise for students to generalize their problem solving skills to real world problems. This includes students with autism spectrum disorder (ASD) and intellectual disability (ID). Modified Schema-based Instruction (MSBI) is one strategy that research has shown is effective for teaching students with ASD/ID to acquire and generalize problem solving skills. The key word strategy is an ineffective strategy for teaching students to problem solve, especially since problems in everyday life do not contain key words. This article provided a practitioner friendly 7-step strategy for implementing MSBI with learners who have ASD/ID

Automatic generation of math word problems (MWPs) is a challenging task in Natural Language Processing (NLP), particularly connecting it to real-life problems because it can benefit students in developing a higher level of mathematical thinking. However, most of the MWPs are presented within a scholastic setting in a decontextualized way. This paper describes a prototype system that generates authentic math word problems (i.e., real-life problems) using authentic contextual recognition with controlled difficulty levels. Our innovative approach includes the acquisition of authentic contextual information through recognition technology, an instructional-based prompt generator for three different difficulty levels, and question generation through the Generative Pre-trained Transformer (GPT) model. We evaluated the performance of the system in terms of the quality of generated questions using automatic evaluation and human evaluation. Further, we assessed the usability of the system using heuristic evaluation. The automatic evaluation showed our generated MWPs were relevant for geometry topics and varied in sentence generation. The human evaluation found our system generated more realistic problems with satisfactory language quality and mathematical validity. Our system also produced more questions with higher difficulty levels based on human evaluation. Heuristic evaluation captured the usability of the system and highlighted the potential to be applied in a pedagogical context for mathematics learning.

We conducted a descriptive exploratory study in which we analyzed 313 sixth to eighth grade students' answers to a word problem, accompanied by diagrams, involving generalization in an algebraic functional context. In this research, we jointly addressed two objectives: (a) to determine the strategies deployed by students to generalize and (b) to identify the types of representation used to express their generalizations. We integrated how regularities are produced, evidenced in structures and represented by students. One of the most prominent findings was that functional strategy was used by almost all the students who generalized. They expressed the generalization using verbal, symbolical, or multiple representations. Ways of expressing regularities that are not restricted to algebraic symbolism are also shown. Although the potential to identify functional relationships was observed in sixth graders, seventh and eighth school students were able to represent more varied and structurally complex relationships. However, no relevant differences in generalization strategies were found between students of different ages with and without previous algebraic training.

Coming from the commognitive standpoint, we consider proof-based mathematics as a distinct discourse, the transition to which requires special rules for endorsement and rejection of mathematical statements. In this study, we investigate newcomers' learning of these rules when being taught them explicitly. Our data come from academically motivated high-school students who took a special course in undergraduate mathematics. The course teacher dedicated three academic hours to introducing and explaining selected rules of proof to support students' shift to the new discourse. The homework assignment consisted of typical proof-requiring problems and a scriptwriting task, asking students to compose a dialogue between fictional characters about a proof-related mistake of their choice. We analyzed the differences and similarities between the rules discussed in the classroom and those that students addressed and implemented in their proofs. The analysis showed that while students' solutions to proof-requiring problems required rule implementation, fictitious dialogues opened the space for rule formulation and substantiation. In many cases, the students discussed the rules presented in the classroom, extending, elaborating, and specifying the teacher's formulations. Furthermore, while the students' proofs were mainly consistent with the teacher's expectations, some of their rule formulations were more radical and overgeneralized than expected. These findings suggest that newcomers' communication about the rules of proof may lag behind their capability to implement those rules to prove mathematical statements.

Primary school textbooks can enhance the acquisition of arithmetic word problem solving skills by offering diverse problems based on their semantic-mathematical structure with targeted reasoning aids, including schematics highlighting their mathematical structure. While certain countries, such as the USA and Singapore, have made progress in improving the problems and aids found in their textbooks through the use of specific theoretical-methodological approaches, textbooks from other countries, such as Spain, have included a very limited variety of problems, with hardly any aids to reasoning. Recently, however, two of the most widely used Spanish publishers have released textbooks that adhere to these theoretical-methodological approaches. To assess whether these textbooks progressed past their predecessors in relevant aspects related to the resolution of arithmetic word problems, we conducted an analysis of the quantity of problems and their variety in terms of semantic-mathematical structure and level of difficulty, as well as the inclusion of schematic representations of their mathematical structure. The study demonstrated improvements among textbooks when publishers adopted a theoretical framework, suggesting that a reference framework could enhance textbook design. This is particularly relevant in countries such as Spain, where there are no applicable standards or official curricula for designing textbooks related to solving arithmetic word problems.

Recent studies have shown that mathematical vocabulary is an important predictor for mathematical performance in kindergartners, primary school students, and secondary school students (e.g., King & Purpura, 2021; Peng & Lin, 2019; Ünal et al., 2021). However, a recent meta-analysis showed that mathematical vocabulary might be more important in some mathematical domains than in others (Peng et al., 2020). The current manuscript focusses on proportional reasoning, a mathematical subdomain containing a high amount of proportional vocabulary (a subtype of mathematical vocabulary), and consists of two studies. The first study describes the development of an instrument specially created to measure proportional vocabulary, as there was no good instrument available. The results provide indications that the scores on our new proportional vocabulary task are valid and reliable. This instrument was used in a second study to investigate whether second graders' proportional vocabulary is associated with (1) proportional reasoning abilities, and (2) performance on proportional and additive word problems. The results showed that after controlling for gender, age and class, proportional vocabulary uniquely contributed to proportional reasoning abilities and performance on both proportional and additive word problems.

Previous research has shown the importance of conducting early interventions in mathematics in disadvantaged children. Solving arithmetical word problems is a field in which children particularly fail. In this study, preschoolers from disadvantaged French public schools (n = 101; M[subscript age] = 5-6) were taught strategies for using fingers to solve arithmetic word problems and compared with a control group. The intervention consisted of collective rituals based on learning finger patterns and 7 sessions spread over 4 weeks, for about 20 min, focusing on explaining how to use the fingers to solve problems. The results showed that the intervention has a significant post-test impact on the targeted transformation problem-solving skill and that children with lower performances in problem-solving at the pre-test benefited more from the intervention. The intervention also indirectly benefited the other problem-solving skills. However, there was no intervention effect on the arithmetic addition task. Our research highlights that an intervention focused on the explicit teaching of finger strategies for problem-solving can be successfully implemented into ecological learning contexts, especially in disadvantaged areas.

Replications provide credibility by demonstrating under what conditions experimental findings can be repeated, the premise behind evidence-based practices. Replications in single-case research also investigate generalization of findings across groups. For groups with high variability, such as individuals with autism, assumptions of generalizability should be based on learners who are similar in critical ways. The purpose of this study was to use Coyne et al.'s framework for replication and the next generation guidelines for single-case research to extend understanding of "for whom" and "under what conditions" modified schema-based instruction (an established evidence-based practice for individuals with autism) is effective. In this distal conceptual replication of Root et al., contextual and instructional variables of theoretical and practical importance were intentionally manipulated or maintained and reported to model transparency and support replicability. Four high school students receiving special education under the Individuals With Disabilities Education Act category of autism were taught mathematical and social problem-solving behaviors within the context of percentage-of-change word problems. Researchers used modified schema-based instruction and augmented reality in a one-on-one setting and assessed generalization to purchasing in the food court of a mall biweekly. We frame our discussion around the recommendations for replication research from Coyne et al. and recommendations for single-case research from Ledford et al., concluding with suggestions for future replications that use single-case research designs.

The aim of this study is to explore the possibility of introducing the general notions of function and inverse function through a mathematical activity on linear functions, focusing on the quantitative meaning associated to the connection between a relation and its inverse. We present a genetic decomposition, that is, a viable cognitive path for learning these introductory concepts, in terms of mental structures and mechanisms, from the viewpoint of APOS (Action-Process-Object-Schema) theory. One of the innovative aspects of this study consists in the design of a genetic decomposition that involves the construction of two concepts. Through interviews, we explore the conceptions that two high school students--who had not been introduced to the function concept before--developed in connection with the notions in question. Our results confirm that not only it is possible to study a relation and its inverse together, but also doing that can enhance the understanding of the meanings involved in the mathematical relation. This leads us to question the learning of mathematics being linear, and to open a discussion about the organization of mathematical content for instructional purposes. We offer suggestions for further research both involving other mathematical concepts, and in terms of theoretical constructs.

This cohort-sequential study explored the working memory (WM) structures that underlie growth in mathematical word problem solving (WPS) performance in elementary school emergent bilingual children whose first language (L1) is Spanish. To this end, children (N = 429) in Grades 1, 2, and 3 in southwest U.S. school districts at Wave 1 were administered a battery of cognitive (short-term memory [STM], WM, rapid naming, inhibition), domain-general nonmath (reading, vocabulary, and fluid intelligence) and domain-specific math measures (arithmetic computation, estimation, and magnitude judgment) in both Spanish (L1) and English (L2). These same measures were administered 1 and 2 years later. Four important findings emerged: (a) A three-factor structure (phonological STM, visual-spatial WM, and executive WM) captured the data across three testing waves within and across both language systems; (b) growth in phonological STM and executive WM uniquely predicted WPS growth, but these two WM structures interacted within the English language system; (c) Spanish verbal WM (phonological STM, executive WM) enhanced the magnitude of predictions of English verbal WM on English WPS; and (d) growth in language-specific phonological STM and executive WM predicted cross-sectional and within-child changes in WPS independent of growth in other domain-specific and domain-general academic areas. Taken together, the results suggest that language-specific age-related phonological STM growth and executive WM growth rates underlie Spanish and English math word problem-solving performance. The results extend the contribution of development models as a function of WM structures across two language systems as they apply to complex math performance.

Mathematical skills significantly predict students' educational paths. Mathematical achievement varies depending on the student's socioeconomic status (SES). However, the extent of the SES gap for specific mathematical skills remains unclear. In this cross-sectional study, we examined age variations by SES in three mathematical skills, applied problem-solving, arithmetic fluency, and written calculations, among students aged 7-12 in a socioeconomically segregated educational context. The contributions of the home environment and schools on SES gaps were also explored. The analytical sample comprised 10,665 students (49.2% girls, M[subscript age] = 10.1, SD = 1.3 years, 13.4% from Indigenous ethnic groups) from the Chilean Early Childhood Longitudinal Survey (Encuesta Longitudinal de Primera Infancia). Linear models revealed SES gaps across the three mathematical tasks at the beginning of primary education. The widest gap was observed in written calculations, which was moderated by the home environment. The findings also showed an exponential increase with age in the SES gap for written calculations and arithmetic fluency. However, for applied problem-solving, the initial gap increased and remained constant. Furthermore, schools accounted for 19%-21% of the variance related to the change of the SES gap in all three mathematical skills. The findings shed some light on the role of the home environment and the schools in maintaining, increasing, or decreasing socioeconomic gaps in specific mathematical skills at different ages.

The study examines language dimensions of mathematical word problems and the classification of mathematical word problems according to these dimensions with unsupervised machine learning (ML) techniques. Previous research suggests that the language dimensions are important for mathematical word problems because it has an influence on the linguistic complexity of word problems.Depending on the linguistic complexity students can have language obstacles to solve mathematical word problems. A lot of research in mathematics education research focus on the analysis on the linguistic complexity based on theoretical build language dimensions. To date, however it has been unclear what empirical relationship between the linguistic features exist for mathematical word problems. To address this issue, we used unsupervised ML techniques to reveal latent linguistic structures of 17 linguistic features for 342 mathematical word problems and classify them. The models showed that threeand five-dimensional linguistic structures have the highest explanatory power. Additionally, the authors consider a four-dimensional solution. Mathematical word problem from the three-dimensional solution can be classify in two groups, three- and five-dimensional solutions in three groups. The findings revealed latent linguistic structures and groups that could have an implication of the linguistic complexity of mathematical word problems and differ from language dimensions, which are considered theoretically. Therefore, the results indicate for new design principles for interventions and materials for language education in mathematics learning and teaching.

Transitional engineering students are those who are academically ineligible to enter a bachelor's engineering program but are enrolled in an associate engineering program with a university. Successful completion of such an associate engineering program allows the higher achievers to transfer to a full bachelor's engineering program. The associate engineering program is taken commonly by self-employed tradesmen, technical workers, and young apprentices in regional, rural, and remote (RRR) areas. The foundation engineering mathematics course in the associate engineering program, particularly knowledge and skills in solving word problems involving triangles, plays a key role for the smooth transition of these students to the engineering disciplinary courses. However, there is little we have known about the performances of the transitional engineering students in solving problems involving triangles as the associate engineering programs are not among the mainstream of undergraduate programs. This study analyzed the 27 transitional engineering students' performances in solving word problems involving triangles assigned to the students in the foundation mathematics course at a regional Australian university and found that the RRR transitional engineering students demonstrated a higher level of study ethics and achievement in solving word problems involving triangles, compared with the RRR student mathematics teachers. This seems mainly due to the professional experiences in delivering real-world projects prior to the start of their mathematics learning. Further research should be expanded to more areas of mathematics to gauge the overall performances of the transitional engineering students in mathematics learning and progression.

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

An assessment of the possible effect of mathematics symbol instruction on achievement in word problems was the main thrust of the study. Two null hypotheses were tested in the study. The non-equivalent group pre-testpost-test quasi-experimentalresearch design was adopted. The multistage sampling technique was used to select a sample of 387 participants from 6,740 senior secondary class one students in the 81 government-owned coeducational secondary schools in Calabar education zone of Cross River State. The treatment class was exposed to mathematics symbol instructional strategy for solving word problems while the control class was exposed to regular word problem teaching, otherwise called the conventional method. "Students mathematics achievement test" was the instrument used for data collection after validation by experts. The reliability coefficient of 0.90 was determined using the Kuder-Richarson 20 formula. Primary data were obtained after informed consent from the participants. The null hypotheses were all tested at the 0.05 alpha level using the analysis of covariance. Findings revealed a significant effect of the mathematics symbol instructional strategy on students' word problem achievement in secondary schools. Male students perform significantly better in word problems than their female counterparts when taught using the mathematics symbol instruction. These findings discussed fundamental theoretical, practical, and research implications. It was recommended, among other things, that mathematics teachers concentrate on demystifying the abstraction in mathematics by deliberately adopting mathematics symbol instructional strategy in interpreting mathematical statements and sentences into their symbolic notations before engaging in their computation.

This study aimed to determine the perceptions of preservice elementary mathematics teachers about creating and evaluating real-life connections in story problems. The study was conducted according to the case design. The study group consists of 35 preservice teachers taking Mathematical Connection Teaching course. Study instruction was used as a data collection tool. While analyzing the data, content analysis and the codes of similar studies were used. According to the findings, real-life problem preferences of the teacher candidates are generally about numbers and operations learning domain at the seventh-grade level. Preservice teachers' aim while creating story problems is to show students how to use them in real-life and to create awareness. The preservice teachers mostly paid attention to the suitability of the student and grade level while creating problems. While preservice teachers' real-life associations were predominantly in the form of examining accurate data, only one study met the evaluation criteria. Most preservice teachers thought that the story problems could be associated with real-life situations. Consequently, it was determined that the preservice teachers found the reflection of real-life situations in the story problems useful for students, and the mathematics subjects can be associated with real-life situations.

The aim of this study is to determine the primary school fourth-grade students' understanding of four operation symbols, using them in arithmetic operations and word problems. Phenomenology research design, one of the qualitative research methods, was used in the research. The participants of the research consist of 123 students, attending the fourth grade of primary school (9 years old). The participants of the study were determined by using convenience sampling, The "four operation symbol study form" developed by the researcher was used as a data collection tool. Content analysis method was used in the analysis of the data. According to the results of the research, it was determined that the students formed meanings in different categories for the meanings of symbols, they were more successful in writing arithmetic operations by using symbols than in solving arithmetic operations. It was concluded that the students showed lower success in writing and solving word problems by using symbols compared to writing and solving arithmetic operations. It was determined that the lowest success in the use of symbols in arithmetic operations and word problems was in the use of the "=" symbol. [Note: The volume number (5) shown in the header on the PDF is incorrect. The correct issue number is 15.]

The present study explored 285 11th-grade students' preconceptions, misconceptions, and errors in solving mathematics tasks by graphical method. A descriptive-explorative study design was adopted. Cluster sampling was used to select students from sampled secondary schools in eastern and central Uganda. Students' paper and pen solution sketches together with the task-based interviews were used to identify students' preconceptions, misconceptions, and errors in linear programming (LP). Students' responses were analyzed thematically and interpreted as students' learning gaps. The results indicated that students lacked proficiency in relating basic algebraic concepts and procedures to the mathematical language with which LP is conveyed. Generally, most students could not adequately use their previous knowledge and connect it to the learning and solving of LP tasks. Besides applying wrong mathematical algebraic concepts, students had difficulties interpreting and writing correct models (inequalities) from LP word problems. Misconceptions and errors were common and peculiar to and in individual student's solution sketches especially in applying the concepts of equations and inequalities to graphically solve and optimize LP tasks. Students held extremely weak concept images of graphing equations, and inequalities and their linkage to optimizing feasible regions. This research provides insight into the learning of mathematics word problems(LP) and recommends that mathematics educators should effectively apply students' preconceptions, misconceptions, and errors as opportunities for enhancing students' LP conceptual changes. For mathematical proficiency, suitable learning approaches, methods, and strategies should be adapted to address specific individual student's flawed conceptual and procedural knowledge and understanding. These approaches may guide educators in helping students to construct the connections between the old and the new knowledge.

The aim of this study is to reveal the meanings that fourth grade students in elementary school attach to the four operation symbols (+, -, x, :) that are abstract, in different representations. The pattern of the study is a case study of qualitative research methods. In the study, the evaluation tool developed by the researchers was applied to elementary school fourth grade students. It was tried to present the situations of writing a mathematical sentence and expressing terms, formulating a problem, displaying the problem with a figure/diagram from students related to the symbols (+, -, x, :) for four processing skills. Based on the findings obtained in the study, it is seen that students attach concept and operation meanings to four operation symbols (+, -, x, :), create more routine word story problems related to symbols, and are more successful in writing a mathematical sentence appropriate for the problem they are writing than drawing a figure/diagram. In addition, it can be said that students made mistakes throughout the process from a linguistic point of view in terms of; using punctuations correctly, writing letters, numbers, units and expressions correctly, naming terms, and legible and understandable writing.

This study investigates the extent to which student and task-related characteristics are associated with different types of note-taking and analyzes how task success depends on these elements. For this purpose, a sample of n=866 students (age: mean=13.99) completing two reality-based tasks as part of a paper and pencil test are considered. The results demonstrate that the note-taking type differs significantly between the two parallel constructed tasks. For example, language skills (r=0.26), interest in mathematics (r=0.13), and the socio-economic statuses (r=0.12) are observed to be significantly correlated to greater note-taking frequency. Based on linear regression (dependent variable: successful task solution), 34% of the variance is attributed to note-taking and other student characteristics. The most relevant predictor for a successful task solution (ß=.36) is notes containing an elaboration of the given task information.