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The aim of this study is to examine how middle school mathematics teachers diagnose and attempt to eliminate students' misconceptions in algebra. The study employed a case study method and embedded single-case design. The research was conducted with three mathematics teachers working in different state schools and having different professional experiences, as well as ten students from the eighth-grade classes of the same schools. The data collection instruments used in the study included the Diagnostic Test developed by researchers to identify students with misconceptions, the Assessment Framework prepared for the evaluation of teacher performances in diagnosing and eliminating student misconceptions, and Semi-Structured Interviews conducted between students and teachers, which were recorded and made available in audio/video format. The study revealed that teachers generally resorted to conventional methods in the process of diagnosing and eliminating students' misconceptions. In most cases, teachers superficially addressed students' errors and did not fully focus on students' thinking. Regarding the processes aimed at eliminating student misconceptions, teachers preferred to directly inform students that their answers were incorrect, rather than facilitating students in recognizing their own mistakes. The findings highlight the need to increase teachers' awareness and student knowledge regarding misconceptions in algebra.

In this article, we explored Grade 11 learners' algebraic and geometric connections when solving Euclidean geometry riders. A qualitative interpretive case study design was followed. Thirty Grade 11 learners from a non-fee-paying secondary school in the Capricorn North district of South Africa were conveniently sampled to participate in this study. Data were collected through learners' responses to classwork, homework exercises, and task-based interviews. Data were analysed thematically. The findings revealed that to solve Euclidean geometry riders successfully, learners need to establish the feature connections embedded in the given figure or diagram. The ability to make feature connections provides a point of departure in the solution process of a geometric problem. Contribution: Once the feature connection is established, other connections will subsequently emerge. In addition, the reversibility connections become a form of feature connection when solving Euclidean geometry riders. Therefore, we recommend that mathematics teachers emphasise and use mathematical connections in their daily teaching of mathematics.

This paper will provide several examples of science and mathematics integration: navigation/map-reading, ecology/ecosystems/population growth, and chemistry/molecular structures. This paper underscores integrating STEM subjects with problem-based learning with technology such as video/computer simulations/programming/coding and the dynamic free online software GeoGebra. The paper suggests some GeoGebra-based connections to the science and math that is taught and a way to connect the disciplines with technology. Science involves and uses mathematics, and both subjects involve problem solving, and math is the tool that science uses. Subsequently, integrating science and mathematics using problem-based learning with technology might be a way to improve science and mathematics instruction while building confidence for STEM. Recommendations for STEM teachers are made along with some GeoGebra illustrations.

We conducted a descriptive study among first-year engineering students at the Anton de Kom University of Suriname. We analyzed students' errors regarding necessary prior knowledge in a calculus A exam. We found that the stage of the solution in which prior knowledge is required impacts the importance of prior knowledge. We also found that many errors concerned basic algebra and trigonometry concepts and skills. We concluded that even though the required prior knowledge concerns basic algebra and trigonometry, the stage of the solution in which prior knowledge is needed is of great importance.

Algebra is critical in shaping future mathematics success and is integral to the K-12 curriculum. Despite its inclusion, a common challenge arises as students' progress to higher grades without a solid foundation, resulting in challenging learning experiences. This action research study focuses on the algebraic learning experience of a Grade 7 student. The research explores various teaching and learning approaches, including consideration of cognitive development milestones, real-world applications, interactive learning, differentiated instruction, personalized learning plans, technology integration, and innovative assessment methods. Findings suggest that aligning instructional methods with cognitive development, incorporating real-world applications, facilitating interactive learning, offering differentiated instruction, personalizing learning plans, integrating technology, and utilizing innovative assessment approaches enhance the engagement and comprehension of algebraic concepts.

This study examines the direct and indirect effects of some affective constructs, such as self-efficacy (SE), mathematics anxiety (MA) and mathematics interest (MI) on algebraic problem solving achievement (PSA). The sample of the study consists of 400 class IX secondary school students in Morigaon district of Assam, India. The instruments employed in this investigation were the SE scale, the MA scale, the MI scale and the algebraic PSA test. The relationship between affective constructs and PSA in algebra is investigated using two-stage structural equation modeling. The results reveal that SE is the only affective construct that had a direct effect on algebraic PSA, while MI has an indirect effect on algebraic PSA through SE. Also, MA has an indirect effect on algebraic PSA through MI and SE. The findings suggested that mathematics educator should adopt innovative strategies to make the subject matter more interesting and increase students' self-efficacy, which may reduce math anxiety. These will assist the students in enhancing their problem-solving abilities and their achievement.

In this study, the Mathematical Knowledge for Teaching-Elementary Patterns Functions and Algebra-Content Knowledge (MKT-PFA) test, originally developed in English as part of the "Learning Mathematics for Teaching Project" at Michigan University, was adapted into Turkish. The test comprises two equivalent forms, A and B, each translated into Turkish and culturally adapted through consultations with two mathematics education academics and five secondary school math teachers pursuing doctoral studies. A total of 328 preservice teachers at a Turkish public university's elementary school mathematics teaching department were administered form A (14 questions, 29 items) and form B (12 questions, 27 items) at a one-week interval. Psychometric analyses revealed high reliability (KR-20: A=0.712, B=0.735; Lord reliability: A=0.733, B=0.756), and strong correlations (r[subscript pbi]) with the original English forms, indicating suitable adaptation. Item difficulties analyzed using a one-parameter Item Response Theory model showed a normal distribution, affirming the tests' validity for assessing preservice teachers' algebra teaching knowledge in Türkiye.

Objectives: The main objective of this study is to understand the factors that encourage the use of mobile ebooks among mathematics postgraduate students. Method: This study employed a qualitative case study approach. Eight mathematics postgraduate students from the algebra research group participated in the study. Data were collected via interview and analyzed by using constant comparative analysis. Results: Utilitarian, cognitive, affective, social norm, and content qualities were the main factors influencing the use of mobile ebooks among the participants. Conclusions: Mobile ebooks are easy to use and provide a meaningful platform to find information. In addition, the mobile ebook provides instant access to new information with a single finger click, thus enabling students to extract the needed information and make constructive comparisons on the subject matter. Theory and/or Practice Implications: Results expand understanding of the factors that motivate the use of mobile ebooks for learning activities among mathematics postgraduate students. In addition, the study also provides support for institutions planning to incorporate mobile ebooks as an important reference material for postgraduate learning and research activities.

Mathematics is a collection of cognitive products that have unique characteristics from other scientific disciplines. As cognitive products, mathematics, and thought processes are two things that cannot be separated. Although in the literature there have been many approaches proposed to support the analysis of students' thinking processes, not many have used Harel's theory of thinking to reveal the characteristics of students' mathematical thinking. Therefore, this research aims to describe the thinking process of class IX students in solving algebra problems based on Harel's thinking characteristics. To achieve this goal, qualitative research was conducted with a case study design. The questions on the topic of algebra were adapted and developed from questions from the National Examination (UN) and the International Program for Student Assessment (PISA). Next, a test was given to 30 class IX students followed by interviews. The results of the data analysis concluded that in general students still use non-referential symbolic thinking and only a few have algebraic invariance thinking or proportional and deductive thinking. This shows that there are still many students who have difficulty understanding algebra. In increasingly complex problems, students' thinking processes become less flexible, and concepts are understood separately (procedural).

This research article is about "Introducing a Supportive Framework to Address Students' Misconceptions and Difficulties in Learning Mathematical proof techniques (MPT): A Case of Debark University". This study aims to develop, introduce, and implement a supportive framework to overcome students' misconceptions and difficulties in MPT. The framework, named IR[superscript 2]CP[superscript 2]CE, was developed, introduced, and implemented at Debark University in Ethiopia using various data-gathering instruments such as questionnaires, interviews, classroom observations, and document analysis from students and instructors. The study collected data over four months, including the implementation of a supportive framework using mixed, quasi-experimental, and pragmatism research approaches, designs, and paradigms respectively. The internal reliability of the data-gathering instruments was interpreted using Cronbach's coefficient, Spearman-Brown, Spearman correlations, Kuder-Richardson 20 and 21, and difficulty and discrimination indices. The results showed that the implementation of the supportive framework led to significant improvements in students' academic performance in MPT, regardless of factors such as gender, academic year category, and preliminary knowledge and proving skills. This study recommends additional imperatives for practice and future research.

The paper describes the results of a study whose main aim was to find the interrelationship between pupils' school grades in Czech language (native), mathematics, and physics and pupils' cognitive predispositions to problem-solving in science and mathematics diagnosed by the Lawson Classroom Test of Scientific Reasoning and the Culture of Problem Solving test. A total of 180 pupils from the Czech Republic aged 14-15 took part in this study. The results show that pupils with better grades in the monitored subjects achieve better results in both tests. It also turns out that there are generally statistically insignificant differences between the results of pupils assessed by grades 1 or 2 and between those assessed by grades 3 or 4. Pupils' performance in the two tests might help to strengthen the objectivization of grading at school. They might also help identify the indicators important for developing problem-solving skills. The research specifically points to the need to develop algebraic thinking, the conception of infinity, spatial imagination, geometric imagination in the plane, proportional reasoning, and the ability to control variables.

Developing algebraic thinking is a key factor in learning mathematics. Despite its importance, many students still struggle with algebraic concepts. This research investigates students' achievements in algebraic thinking using Demetriou's test across 7th (approximately 12-13 years old), 8th (approximately 13-14 years old), and 9th (approximately 14-15 years old) grades. The study analyzes performance in different levels of algebraic tasks (i.e., [1] extrapolation of relationships, [2] coordinating simple structures, [3] operating with undefined symbolic structures, and [4] coordination with undefined structures), revealing a significant developmental leap in algebraic abilities during the 9th grade. While no statistically significant differences were found in the first level, 9th grade students demonstrated superior performance in levels 2, 3, and 4, suggesting cognitive readiness for abstract algebraic concepts around the age of 14. The research unveils a disjointed development in algebraic abilities, indicating a progression from basic arithmetic operations to proportional reasoning before the full integration of algebraic thinking. Notably, tasks involving variables in the third level pose persistent challenges for students. The findings contribute to understanding the optimal age for introducing algebraic concepts and underscore the importance of considering cognitive development in mathematics education. The study proposes implications for educators, such as emphasizing proportional reasoning in earlier grades and employing differentiated instruction based on individual students' abilities.

Foundation programmes provide an alternate access route for prospective students whose prior academic results exclude direct entry to undergraduate studies. Bridging courses within foundation programmes address gaps in prior knowledge while developing content knowledge and requisite skills to equip students for the rigour of undergraduate degree study. This study looks for relationships between motivation and learning strategies at course commencement and the final course results of 796 purposively chosen participants across four iterative cycles (cohorts) enrolled in an algebra course within a foundation programme at a private higher education institution in South Africa. Data were collected with the "motivated strategies for learning" questionnaire, and cohort responses were analysed using correlational statistics. Statistically significant differences mainly were detected in the motivation subscales, and the academic performance was largely related to gender and prior mathematics syllabus. Where cohorts are similar, generic interventions designed to equip one cohort may equip others. Specific intervention strategies that target the needs of students based on the needs identified in this study may equip future students to improve their algebraic knowledge. Contribution: The research contributes by augmenting the exiguous literature of studies of students in algebra courses in foundation programmes who aim to progress to undergraduate degree studies. Investigating relationships between motivation and learning strategies at course commencement and final course results within multiple cohorts promotes the development of flexible, relevant intervention strategies that can be implemented timeously. A study of multiple cohorts further allows for improved validity and reliability in conclusions relating to scalability.

In learning mathematics in the 21st century and after the COVID-19 pandemic, a multimodal role in the process of drawing conclusions involving mathematical symbols or signs is very much needed. This process is called multimodal semiotic reasoning. This research aims to study the literature on multimodal semiotic reasoning research articles. Until April 2023, there were 32 articles on the keyword multimodal semiotic reasoning with the subject of mathematics in the Scopus, Science Direct, and ERIC databases. Furthermore, it is identified based on a) suitability of criteria; b) the similarity of the articles in the three databases. Finally obtained 24 articles that meet the requirements for further research. A literature review on multimodal semiotic reasoning is explored based on a) kinds of mathematical reasoning involving symbols and signs using multimodal, b) multimodal used in research, c) the most researched mathematical concepts and the mathematization process that occurs, d) the role of multimodal semiotic reasoning in mathematics education. The results showed that a) the most dominant type of mathematical reasoning involving objects, symbols, and signs using multimodal was Algorithmic Reasoning (AR), b) the most multimodal source was verbal answers, c) the concept of geometry through a process of vertical-horizontal mathematization was the most researched, and 4) the biggest role of multimodal semiotic reasoning for the benefit of knowledge of mathematics education. In conclusion, multimodal semiotic reasoning has had a major impact on mathematics education and research on this multimodal semiotic reasoning will continue to thrive in the future. [Note: The publication year (2023) shown in the citation on the PDF is incorrect. The correct publication year is 2024.]

Reversibility thinking carried out mentally in mathematical operations has an important role in the process of understanding concepts as it involves developing a thinking process from beginning to end and from end to beginning. This qualitative research aims to describe students' reversible thinking processes in solving algebra problems, specifically in climbertype adversity quotient. A total of 2 students who have the climber type were selected as participants from 36 potential subjects involved in the research. Data collection was carried out by providing Adversity Response Profile questionnaires, Test of Thinking Reversible, and semi-structured interviews. Our participants solved problems according to polya stages and through forward and reverse processes. The results of data analysis show that in the forward process, there are 2 aspects of reversible thinking, namely negation and reciprocity, while in the reverse process, they involve 2 aspects of reversible thinking, namely the capability to return to initial data after obtaining the result and negation. The results suggest that student's ability to perform mental reversals indicates a strong reversible thought process, leading to more precise cognitive thinking.

Mathematical creative thinking skill often becomes the orientation of mathematics learning, aiming to enhance students' creativity in mathematics. Recognizing that creativity encompasses the capacity for thinking creatively and creativity disposition is essential. Building on this conceptual foundation, the primary objective of this study is to develop a comprehensive model illustrating the relationship between students' aptitude for mathematical creative thinking and their creative disposition. The research methodology employed in this study aligned with the framework of cause-and-effect analysis. The study cohort consisted of 36 students, carefully selected by a cluster random sampling technique. The research instruments included a mathematical creative thinking ability assessment and a creative disposition scale. The data was analyzed using the Non-Recursive Structural Equation Modeling. The results showed the reciprocal cause-and-effect dynamic between mathematical creative thinking ability and creative disposition, exhibiting a mutually influential relationship with determination coefficients of 21.83% and 21.05%. This shows that mathematical creative thinking ability is better at explaining mathematical creative disposition than mathematical creative disposition explaining mathematical creative thinking ability, with a relatively small difference (0.78%). This study also concluded that an optimal approach to mathematics pedagogy entails a balanced and simultaneous focus on nurturing mathematical creative thinking ability and disposition.

Teaching "out-of-field" is a present obstacle in mathematics education in many countries, and developing professional learning programs aimed at upskilling non-specialist teachers is urgent. A teacher study group was established wherein two non-specialist teachers of mathematics (Years 7-10; aged 12-16) engaged with algebra to develop a deeper understanding of the subject and its teaching. The study group lasted one school year, during which the teachers actively participated in the learning. Multiple data sources were collected, including reflection forms, open-ended questionnaires, and a storytelling form. Analysis of the teachers' solutions to a sample of algebra problems and self-reports suggests that the study group enabled teachers to acquire new knowledge/skills in mathematics pedagogy, gain a new understanding of how student thinking and understanding develop, adapt new classroom instruction strategies, develop ability/confidence in problem-solving, and develop new knowledge/ability in algebra. The study findings contribute to our understanding of supporting non-specialist mathematics teachers' professional development.

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

There is a dearth of empirical data to support the positive effects of problem solving (PS) combined with digital technology in the classroom, despite claims that these activities improve students' algebraic thinking abilities. Therefore, the purpose of this study was to evaluate how the teaching method known as Polya's problem solving with digital bar model (PSDMB) affected the seventh graders' ability to think algebraically. Ralston's framework, which covers Generalised Arithmetic, Function, and Modelling within the topic of Linear Equation, served as the foundation for the evaluation of algebraic thinking abilities. A quasi-experimental pre-test and post-test control group design was employed. A total of 90 seventh graders, aged twelve- to thirteen-year-olds, from a secondary school in Tambunan, Sabah, Malaysia, made up the sample. Three teaching groups were formed out of these randomly chosen students: PSDMB (n = 30), Bar Model (MB) (n = 30), and Conventional Problem Solving (CPS) (n = 30). Both the pre- and post-algebraic thinking tests were taken by students. The post-test results were analysed using MANCOVA with the students' pre-test results acting as covariates. The results indicated that students in the PSDMB group performed notably better in Generalised Arithmetic, Function, and Modelling than those in the MB group, who, in turn, outperformed those in the CPS group. These results imply that incorporating digital bar model into problem-based learning is a successful strategy for improving seventh graders' algebraic thinking abilities.

Algebraic thinking is the ability to generalize about numbers and calculations, find concepts from patterns and functions and form ideas using symbols. It is important to know the student's algebraic thinking process, by knowing the student's thinking process one can find out the location of student difficulties and the causes of these difficulties. This study aims to analyze students' algebraic thinking processes in constructing new knowledge of high-ability students based on the Cognitive System of Marzano's Taxonomy. The subjects in this study were twenty one mathematics teacher candidates who took linear programming courses. The algebraic thought process of prospective instructors in solving linear equation problems is described using a qualitative descriptive technique in this study. The data collection technique starts with giving algebraic thinking questions and interviews/observations. Data reduction, data display, and deriving conclusions are the data analysis techniques employed. The results of the research show that algebraic thinking processes with types "Generasional Representasi Sequential Concrete" students are able to extract conclusions and organize better. Algebraic thought processes with types "Generational Representation of Concrete Random" students are able to build models and form generalizations but their representation is not good enough that they cannot be communicated properly.

This study examines the impact of early Algebra I coursework on advanced Carnegie credits among 12th graders from diverse racial/ethnic backgrounds, using data from the NCES HSTS (1990-2019). Findings indicate that early Algebra students, particularly Black and Latinx, earn more advanced credits, revealing a widening gap in advanced course attainment. The study underscores the importance of early Algebra in college readiness and STEM preparation, highlighting the need for interventions to address educational disparities. Future research should explore additional factors influencing college readiness.

This article presents a study investigating the use of games in undergraduate mathematics instruction, with a specific focus on students' perspectives regarding post-instructional quiz games. The study explores students' opinions on the implementation of a linear algebra quiz game as a supplementary learning tool. A survey was administered to 78 students to examine their perceptions of the game's usefulness in preparing for their linear algebra examinations. Students reported high levels of attention and interest while engaging with the game and described it as fun, dynamic, exciting, and beneficial for reviewing linear algebra concepts. The findings suggest that quiz games can be a valuable addition to undergraduate mathematics education, fostering student engagement and providing an enjoyable and effective means for reinforcing course material.

This research investigates errors and misconceptions among learners in algebraic education by utilizing Koch's error analysis method alongside the Structure of the Observed Learning Outcome (SOLO) taxonomy. The primary aim of the investigation is to discern the kinds of errors and cognitive stages demonstrated by Grade 9 students when engaged in algebraic problem-solving tasks. The studies' outcomes uncover several prevalent error categories, including conjoining, cancellation, and problem-solving errors, indicating deficiencies in conceptual comprehension and procedural execution. Moreover, applying the SOLO taxonomy elucidates learners' diverse levels of understanding, with a majority position within the uni-structural or multi-structural stages. Theoretical implications underscore the necessity for tailored instructional approaches to mitigate learners' obstacles and foster a deeper grasp of algebraic principles. Consequently, this research contributes significantly to the advancement of algebraic pedagogy and provides valuable insights for curriculum enhancement, thereby facilitating improved mathematics learning outcomes.

Some countries, including Indonesia, have a framework for understanding how students receive and process math concepts as new knowledge through learning styles. Learning style, particularly Kolb's model, is one of the learning styles that contribute to students' success in learning. Experts have explored the characteristics of Kolb's learning style and found many effects on student learning outcomes as a starting point in learning mathematical concepts. However, the research still focuses on exploring integer operation materials in specific math abilities. The researchers hardly found any discourse to study, such as exploring number patterns in computational thinking and semiotic mathematics. Therefore, this study aims to explore mathematical semiotics and computational thinking on number patterns in terms of Kolb's learning style model. This research uses hermeneutic phenomenology to explore through written tests and interviews. An explanation of the components, characteristics, and semiotic characteristics of mathematical and computational thinking seen in students who have the Kolb model of learning in solving problems of number patterns is part of the interpretation. These findings can be used as benchmarks in developing mathematics materials. Thus, this knowledge is a concrete foundation to guide future advances in curriculum, assessment methods, and learning approaches in mathematics education, particularly in algebra.

Research about the optimal methodologies for guiding the training of primary school instructors in the instruction of early algebra remains in the process of determination. This manuscript delineates the methodology involved in developing and validating an assessment tool to evaluate the mathematical acumen of prospective primary educators in the domain of early algebra during their initial pedagogical training, drawing upon the constructs delineated in the Mathematical Knowledge for Teaching (MKT) model. To this end, an instrumental inquiry comprising four distinct phases has been executed: an exhaustive review of extant literature concerning the mathematical proficiency of primary school instructors and the pedagogy of early algebra; formulation of the preliminary version of the assessment instrument; validation of said instrument via expert appraisal and a preliminary application involving ten pre-service primary educators enrolled in a Spanish academic institution; and subsequent refinement and finalization of the assessment tool. This endeavor's outcome culminated in the MKT-Early Algebra Questionnaire (6-12-year-olds), comprising six open-ended inquiries meticulously designed to explore diverse facets of early algebra while aligning with the various sub-domains delineated in the MKT model. It is deduced that the resultant instrument holds promise as an effective diagnostic apparatus, serving dual purposes: elucidating the mathematical proficiency of primary school educators in the context of early algebra and fostering introspection regarding pedagogical strategies conducive to the cultivation of algebraic cognition at this developmental juncture. By furnishing a comprehensive questionnaire that systematically addresses all facets of pedagogical knowledge requisite for the effective instruction of early algebra, this study furnishes invaluable insights into the specific components that should be integrated into teacher education curricula in the domain of algebraic didactics.

Solving systems of linear equations is a core concept in linear algebra and a wide variety of problems found in the sciences and engineering can be formulated as linear equations. This study sought to explore undergraduate students' development of the schema for solving systems of linear equations. The triad framework was used to describe the schema development in general and the system of linear equations was used as an example. A case study of an undergraduate class doing a linear algebra course in 2020 was considered in this study, where fifteen students participated in the study whereby they responded to a task with three question on solving a system of linear equations. The findings revealed that albeit minor manipulation errors, students were able to solve given systems of linear equations using the Cramer's rule and Gaussian elimination. However, students could not adequately go beyond the algorithmic computations to attain appropriate mathematical reasoning and establish underlying relations required to solve systems of linear equations, as no-one attained the trans-stage of conceptualization. It follows from this study that the identifications of the challenges that students encounter when solving systems of linear equations empowers course instructors on how to overcome the challenges.

This study aimed to interpret and describe students' mathematical thinking processes of non-routine mathematical problems that were solved based on didactic situation theory. This study uses a qualitative method, a phenomenological hermeneutics study for grade 8 students at a junior high school in Banda Aceh in the 2021-2022 academic year. Research data obtained through data were collected using instruments, namely written tests based on the didactic mathematical situation theory framework, structured observation, documentation, and clinical interviews carried out after the action. The results of the study show that the students' mathematical thinking processes in the critical reflection category can reach the convincing stage with algebraic arguments in validation situations. Subjects in the explicit reflection category can reach the convincing stage by providing arithmetic arguments in validation situations. Meanwhile, the category of students who cannot solve problems and can only specialize by giving examples of what is being asked. Students in this category have difficulty identifying relevant patterns and formulating the mathematical models needed to solve the problems. To support students in developing the level of mathematical thinking, the teacher can present contextual problems that are in accordance with the level of student thinking, can predict possible responses or ways of thinking of students to the problems given and present problems according to the structure of the concept sequence and the functional order of students' thinking. To support students in algebraic thinking category, teachers can start learning by presenting contextual problems that are easily recognized by students, then expand that context in symbolic form.

Gaming the system is a persistent problem in Computer-Based Learning Platforms. While substantial progress has been made in identifying and understanding such behaviors, effective interventions remain scarce. This study uses a method of causal moderation known as Fully Latent Principal Stratification to explore the impact of two types of interventions--gamification and manipulation of assistance access on the learning outcomes of students who tend to game the system. The results indicate that gamification does not consistently mitigate these negative behaviors. One gamified condition had a consistently positive effect on learning regardless of students' propensity to game the system, whereas the other had a negative effect on gamers. However, delaying access to hints and feedback may have a positive effect on the learning outcomes of those gaming the system. This paper also illustrates the potential for integrating detection and causal methodologies within educational data mining to evaluate effective responses to detected behaviors.

This article aims to analyze the process of the progressive mathematization of the concept of the function by secondary school students through the resolution of a free fall situation. A free-fall mathematical situation was designed and experimented using the Tracker software to obtain movement data. Worksheets and conversations were collected between students and the teacher, and experiments and class discussions were filmed. The analysis consisted of identifying the development from informal strategies associated with the context of the problem to formal procedures. The results show that the process brings meanings to mathematical ideas such as variation, variables, and the relationship between variables, allows coordination and transition between representations and contributes to the learning of functions as a representation of the variation of physical phenomena.

High school math projects center on solving real-world examples in a wide variety of situations that require a creative skill set. Creativity in high school mathematics takes place in a subtle form. In contrast, creativity presents itself conspicuously in art. Therefore, this paper will explore the utilization of polynomial equations in drawing various figures as part of algebra class projects. Additionally, these projects incorporate integrated digital tools in mathematics. The paper will analyze three algebraic projects: the first project involves drawing a bird, the second project focuses on drawing a butterfly, and the third project entails drawing a bat using polynomial equations with the technological tool Desmos. All three projects are products of student work that encompass figures requiring up to forty polynomial equations. The students' impressive work demonstrates the power of mathematics in pushing the boundaries of other subjects. As digital technology becomes increasingly integral, mathematical tools are positioned as universal tools for the future and are poised to rival artists in producing visual art. This raises the intriguing question: can a mathematician create better art with mathematical equations through digital technology than an artist?

The researchers conducted a qualitative case study on a learning community comprising special education and psychology majors. This community was strategically organized to foster interdisciplinary learning among its members and to design interventions for students struggling with mathematics anxiety, specifically in classes like Algebra 1. Working alongside a university faculty member and a graduate student, the undergraduates applied mental health principles from psychology research to real-world situations. Their efforts included teaching Algebra 1 to a student grappling with both a learning disability in mathematics and mathematics anxiety. In their approach, the group utilized mindfulness principles to tailor conversations to the high school student's interests, effectively incorporating mindfulness techniques into the mathematics tutoring sessions. This approach proved successful, particularly when the student was experiencing high levels of anxiety, enabling significant progress in his Algebra 1 work. The collaborative efforts also provided substantial benefits to the college students, who learned about each other's fields of study. They effectively combined this knowledge in practical ways and applied it in real-world settings. The findings of this study offer important insights for the training of pre-service special education teachers and college students in related fields.

Given the relevance of graphs of functions, we consider their inclusion in primary education from the functional approach to early algebra. The purpose of this article is to shed some light on the students' production and reading of graphs when they solved generalization problems from a functional thinking approach. We aim to explore how 3rd and 4th graders construct graphs associated to functions and what elements they use; and how they read function associated graphs and whether they connect pairs of values to see beyond the data. After four working sessions about functions, we designed and implemented individual interviews to 12 students. Through a qualitative analysis, we highlight that the students can read data in a graph on two different cognitive levels and also construct it from different elements of the graph initially provided. Regarding data reading, we evidence two levels: (a) literal reading of a given element in the graph, and (b) reading beyond the data. The construction of the graph is described with base on the axes, values and labels on the axes, scale of the axes, and construction techniques. We present examples of students' work that evidence that graph construction varied depending on whether it was created from a blank sheet or it was necessary to provide help regarding the axes or the scale of the graph. We describe several techniques used by the students in the representation of data that yield non-canonical representations of a graph and that help glimpse how students are interpreting this representation.

This survey paper presents recent relevant research in mathematics education produced in Spain, which allows the identification of different broad lines of research developed by Spanish groups of scholars. First, we present and describe studies whose research objectives are related to student learning of specific curricular contents and process-oriented competencies, namely arithmetic, algebra, geometry, functions and calculus, probability and statistics, and argumentation or proof in geometric contexts. Next, we present characteristics and foci of investigations dealing with different aspects of mathematics teacher education, encompassing a large part of Spanish research in mathematics education. The descriptions of other transversal lines of research complement the previous two big blocks: research on students with special educational needs and the effects of using technology in different curricular contents and educational levels. Finally, we report on the research activities and advances of Spanish research in mathematics education from two main theoretical frameworks created or developed by Spanish researchers. This plurality of research strands also corresponds to a wide range of international collaborations, especially with Latin American colleagues.

Spanish educational curriculum adopts a mathematical process-based approach, which encompasses problem solving, reasoning and proof, communication, connections and representation. A fundamental role in the integration of these processes in mathematics teaching is played by teachers' professional practice of designing tasks. According to this, our aim is to analyze the ways in which pre-service early childhood teachers understand the mathematical processes in the professional practice of designing early algebra tasks and to identify how they intend to promote these processes through the tasks. Content analysis techniques were used to examine the designed tasks. To illustrate the data analysis and results, six tasks are presented. As a result, pre-service early childhood teachers associate problem solving with challenges and questions. They understand problems as unfamiliar situations but ignore the relationships between students and tasks. Moreover, they do not encourage exploration of phases of problem solving and tend to use strategies more suitable for routine tasks. Communication is identified in all the tasks designed, encouraging interaction and discussion. However, only one task explicitly promotes mathematical language. For reasoning and proof, pre-service teachers begin to use questions to elicit explanations and justifications, but do not encourage verification strategies and various modes of reasoning. The process of connections is only present in one task, reflecting the fragmented nature of mathematics teaching. We conclude that the professional practice of designing mathematical tasks is a powerful in teacher education. However, training programs should place greater emphasis on the meaningful use of mathematical processes.

Well-designed online educational games can improve students' math knowledge, skills and engagement; however, more research is needed to understand how to formatively assess components of students' mathematical understanding and learning as students solve problems in online educational games. In this study, we examined how 7th-grade students' (N = 760) in-game performance (i.e., strategy efficiency, mathematical validity of the first action, mathematical errors) and help-seeking behaviours (i.e., hint requests) correlated with their algebraic knowledge in an online mathematics game. The k-means cluster analysis identified four groups of students based on their in-game metrics, and some variabilities in their in-game performance were found. Although hint requests were available, only a few students showed a high percentage of hint requests during their gameplay. The regression analysis revealed that students' in-game performance and help-seeking behavioural patterns explained a significant amount of variance in students' algebraic knowledge above and beyond their prior knowledge. Our results suggest that how students interact with the online learning game is related to their later math performance, and qualitative aspects of student behaviours captured using log data provide meaningful information to students' algebraic understanding.

When analyzing instructional quality in mathematics, observational measures can consider domain-specific instructional approaches (e.g., effective teaching of mathematics) and content-focused practices (e.g., effective teaching of a specific mathematical topic). In this study, we analyze lesson videos from secondary algebra classrooms using two different instruments--one focused on ambitious mathematics teaching practices and one focused on instructional features that support students' learning opportunities in algebra specifically. We consider the information provided by overlaying these instruments to raise implications for instructional improvement efforts, particularly related to how information from the instruments might inform formative feedback to teachers. We consider how alignment between instrument and instructional emphasis can influence the strengths and areas of instructional improvement identified in a given lesson. We further model a collaborative process for looking across observation instruments to deepen understanding of the information provided.

Despite the prominence of analogies in mathematics, little attention has been given to exploring students' processes of analogical reasoning, and even less research exists on revealing how students might be empowered to independently and productively reason by analogy to establish new (to them) mathematics. I argue that the lack of a cohesive framework for interpreting students' approaches to analogical reasoning in mathematics contributes to this issue. To address this, I introduce the Analogical Reasoning in Mathematics (ARM) framework. Constructed from an analysis of interviews with four abstract algebra students, ARM identifies several analogical activities that serve to analyze students' analogical reasoning with a finer grain size than was previously possible with existing frameworks. Using this framework, I present an analysis of the students' constructions of a ring-theoretic analogy to subgroup, thus revealing that even constructing simple analogies can elicit diverse pathways of analogical reasoning across students. Implications for further research related to analogies and analogical reasoning in mathematics education are discussed.

Comparison is an important mechanism for learning in general, and comparing two worked examples has garnered support over the last 15 years as an effective tool for learning algebra in mainstream classrooms. This study was aimed at improving our understanding of how Modified for Language Support-Worked Example Pairs (MLS-WEPs) contribute to effective mathematics learning in an ESOL (English to Speakers of Other Languages) context. It investigated a novel instructional approach to help English Learners (ELs) develop better understanding in mathematical reasoning, problem solving, and literacy skills (listening, reading, writing, and speaking). Findings suggest that MLS-WEPs not only enhanced ELs' ability to solve algebra problems, but it also improved their written explanation skills and enabled them to transfer such skills to different mathematical concepts. Moreover, when controlling for ELs' prior knowledge, the effectiveness of the MLS-WEPs intervention for performing and explaining calculations did not vary by their English proficiency.

The question of precollege physics access and performance has been a persistent concern when considering the goal of diversifying participation in post-secondary STEM study and careers. This observational study examined school-level academic and demographic predictors of high school physics enrollment and performance in the USA. Due to the sequential and hierarchical nature of high school science and mathematics coursework, related predictors were included in multivariable regression and structural equation models to understand whether selectivity constrains the STEM pipeline. Descriptive and inferential analyses of state-level data were conducted (N = 663 high schools enrolling 559,044 students) to define the scope of physics enrollment and performance, and the predictive value of ethnicity, socioeconomic status, and science and mathematics course enrollments and performance. Physics was taken by fewer students when compared to biology, earth science, and chemistry. Correlations, multiple regression, and structural equation modeling indicated that although socioeconomic status was the main predictor of student "enrollment" in physics, this was mediated by schoolwide enrollment in chemistry and algebra II as well as schoolwide performance in chemistry and geometry. School-level "performance" in physics was negatively predicted by the percentage of students traditionally underrepresented in STEM, yet the predictive value was mediated by chemistry and algebra II performance. Results suggest that the prevalent science and mathematics sequence correlates to diminished physics enrollment, particularly with regard to students traditionally marginalized in STEM. US school leaders and policy makers should implement more proactive interventions earlier in the STEM pipeline to promote diverse physics participation and more equitable performance outcomes.

The objective of this paper is to establish comparisons between the Linear Algebra (LA) approach practiced in Engineering courses offered by Chilean and Brazilian Higher Education Institutions and, for that, teachers from both countries were interviewed. From a methodological point of view, the research consists of two case studies. For the analysis of the interviews, we used the precepts of Content Analysis. The data were qualitatively analyzed considering the MTSK theoretical model. In this analysis, we sought to identify what specialized knowledge these teachers have regarding AL and about its teaching and learning. The results have shown that, in relation to the Mathematical Knowledge (MK) domain, which refers to content knowledge, the Knowledge of Topics subdomain (KoT), linked to the way that teachers know the topics they teach, is the most present among interviewees. In the Pedagogical Content Knowledge (PCK) domain, covering specific pedagogical knowledge linked to Mathematics, the Knowledge of Mathematics Teaching (KMT), regarding strategies, teaching theories, resources and materials, and Knowledge of Mathematics Learning Standards (KMLS) subdomains, concerning the curricular specifications and the moments in which the student must or can learn certain content and with what level of depth prevail.

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.