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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.]

This article presents a theoretical-didactic perspective on the formation and development of concepts concerning mathematical objects as a result of research on didactic difficulties in dealing with concepts in school. The theoretical foundation that supports the research is based on the contributions of the materialist approach to the theory of knowledge. Specifically, an interpretation of its application in the teaching of mathematics is under investigation, specifically in the process of formation and development of concepts in the field of Euclidean geometry. However, the elements that are derived are essential for the treatment of concepts in any sub-branch of mathematics. The analysis of this process and the factors that influence the development show some fundamental elements to consider in the research activity in the field of mathematics education.

The fact that the mathematics course is abstract, that it is not possible to associate it with daily life, and that it is impossible to concretize abstract expressions causes a prejudice against the this course and leads to a decrease in the academic achievements of students. It is seen that throughout history, various studies have been carried out taking this situation into consideration. A group of these studies is on realistic mathematics education. Realistic mathematics education is an approach that aims to concretize the abstract concepts of mathematics by relating them to real-life situations. The purpose is to make mathematics more understandable by concretizing it. In the study, the subject of calculating the surface area of prisms with the use of realistic mathematics education was taught, and the purpose of the study was to examine the effects of this application. The study group consisted of 20 5th grade students. While determining this group, maximum variety sampling strategy was preferred. In the study, action research, one of the qualitative research methods, was used, and the data were interpreted with the help of descriptive analysis and content analysis. The data collection tools in the study included semi-structured interview forms, video recordings, photos and activity sheets prepared by the students under the guidance of the researcher in accordance with the realistic mathematics education approach. When the findings were examined, it was seen that the students showed a positive development in the meaning and visualization processes; that they could reach mathematical inferences on their own; that they enjoyed the mathematics lesson; and that the realistic mathematics education approach gave positive results on learning. Based on these results, it was suggested that not only the application of the realistic mathematics education approach for different acquisitions but also the inclusion of these activities in the textbooks in accordance with the curriculum will result in improvements in favor of learning.

This study aimed to examine the application of a realist hands-on learning approach and its contributions to optimizing the solid geometry conceptual understanding and problem-solving skills of students. It employed a non-equivalent pre-/post-test quasi-experimental research design. The results showed that there was a statistically significant mean difference between the intervention group and the comparison group, with an effect size of d=0.825 for the students' solid geometry conceptual understanding and d=0.930 for the students' problem-solving skills. Furthermore, a strong correlation was observed between students' conceptual understanding and problem-solving skills in solid geometry, with an r[superscript 2] of 0.6413. These indicate that the use of a realist hands-on approach positively contributes to optimizing learners' conceptual understanding and problem-solving skills. It also provides contextually relevant learning opportunities; therefore, it is recommended that teachers and concerned bodies employ a realist hands-on approach to make solid geometry learning concrete and practical.

Purpose: Ethnomathematics are used in motifs by the Dayak Ngaju Tribe in Central Kalimantan. The motifs contain some Ethnomathematical objects and concepts for Geometry and values. Therefore, this study aimed to describe the motifs based on mathematical objects and concepts of Geometry and value viewed from ethnomathematics aspects. Methodology: This study used a qualitative approach with an ethnography method. The researchers collected information from documents and books. They conducted in-depth observations and interviews with the subjects regarding the culture of the Dayak Ngaju people who reside downstream of the Kahayan River in Central Kalimantan. The subjects were a Dayak Ngaju craftsman and a priest of the Hindu Kaharingan community. The motifs consist of "tingang," "dandang tingang," "jata," "batang garing," "tanduk muang," "buntut kakupu gajah," "taya" tree and "rambunan tambun." Findings: The motifs contain some mathematical objects and concepts. Mathematical objects are created as circles, "batang garing" and "jata" motifs, and a six-sided shape in the "tanduk muang" motif. The concept is the geometric transformations, namely the axis reflection in the mdandang tingang and tanduk muang motifs the axis reflection in the motifs of "tingang," "batang garing," "tanduk muang" and "taya" tree. The rotation on point (0,0) in the motifs of "buntut kakupu gajah" and "taya" tree, and the translation in the motif of "jata." Significance: The study provides some mathematical objects and concepts in the motifs of the Dayak Ngaju tribe in Central Kalimantan. Students learning those objects and concepts should be related to their culture, namely the motifs, and they should acquire the values in such learning. Ethnomathematics plays an important role among Dayak Ngaju people in understanding mathematics concepts such as Geometry and values.

Comprehending and formulating strategies for geometry problems that require higher-order thinking skills (HOTS) is crucial in enhancing mathematics education. This study implements a qualitative case study approach to comprehend how prospective mathematics teachers with varying Adversity Quotients (AQ) solve geometry Higher-Order Thinking Skill (HOTS) problems. We selected 3 participants from 167 Indonesian prospective mathematics teachers to solve the three- and two-dimensional HOTS problems and were invited to an interview session. The three participants represent three types of participants: a climber student (high AQ), a camper student (medium AQ), and a quitter student (low AQ). Our findings show that each student had different responses to deal with the obstacles they faced while solving the problem. The climber student is more adept at solving problems than the camper and quitter students. In addition to identifying specific implications, this study offers a comprehensive understanding of AQ's significant role in solving mathematical problems. This knowledge serves as a concrete foundation for guiding the future advancement of curricula, assessment methods, and instructional approaches in mathematics education, particularly in the field of geometry. This research contributes to enhancing educational practices and policies on a broader scale.

Due to the growing interest in the integration of ethnomathematics in mathematics education, we undertook a systematic literature review and bibliometric analysis of research on the incorporation of ethnomathematics in geometry teaching from 2011 to 2021. The search for 37 relevant research articles was done on Google Scholar, Scopus, and by hand search of some key journals. Data retrieved were analyzed descriptively by using Excel and NVivo. The findings showed that literature on ethnomathematics in teaching geometry is mainly produced and consumed in Indonesia, Zimbabwe, Nigeria, and Israel. Also, the findings revealed that the research topics/themes addressed by literature on ethnomathematics integration in teaching geometry can be grouped into five themes: views, practices, effects, challenges, and development of competence. Most of the studies conducted in this area employed ethnography. From the present study, it was observed that the number of research article publications relevant to the present study has increased exponentially, especially from 2018 to 2021. Therefore, it is our belief that this review will aid in the creation of systematic review procedures and reviews that are specifically concerned with the incorporation of ethnomathematics into geometry instruction.

Linear functional relationships are intended to be symmetric and therefore cannot generally be accurately estimated using ordinary least squares regression equations. Orthogonal regression (OR) models allow for errors in both "Y" and "X" and therefore can provide symmetric estimates of these relationships. The most well-established OR model, the errors-in-variables (EIV) model, assumes that the observed scatter around the line is due entirely to errors of measurement in "Y" and "X" and that the ratio of the error variances is known. If most of the variance around the line is known to be due to the errors of measurement in "Y" and "X," the EIV model can provide an unbiased maximum likelihood estimate for a functional relationship. However, if a substantial part of the variability around the line is due to natural variability, which is not attributable to errors of measurement in "Y" or "X," the ratio of the measurement error variances is not well defined and the EIV model is not directly applicable. The main contribution of this report is the development of a hybrid model that provides plausible estimates for linear functional relationships in cases with substantial natural variability and substantial errors of measurement. An analysis of female and male differential test functioning between an essay test and an objective test used as parts of a licensure examination provides an illustration of the use of the hybrid model.

Issues about poor achievement in mathematics (angles and polygons topics) among Palestinian students has led to the need for developing new effective strategies that can be used to improve mathematics achievement among Palestinian 7th grade students. The aim of study is to reveal the effect of employing the mantle of the expert on the achievement of angles and polygons topics among seventh grade students in Palestine. A quasi-experimental non-equivalent control group design using experimental and control groups with a pre and post-test was conducted. Data were collected using achievement test. The sample of study consisted of (76) students in the seventh grade that were divided into two groups, the experimental group (37) students and the control group (39) students. Data collected from the pre-test and post-test was analyzed using descriptive and inferential statistics. One-way MANOVA test was used to verify the differences of achievement between experimental and control groups. The research found that there is a significant impact of employing the mantle of the expert in increasing the achievement in geometry among seventh grade students. Students who are studied using mantle of the expert had better achievement than those who did not. This study recommended employing the mantle of the expert in teaching other topics in mathematics and in different study stages. This study too showed the necessity of paying attention to the use of classroom and extra-curricular activities in different academic subjects, particularly in mathematics, as well as indicated the necessity of holding courses and workshops to train teachers to use the mantle of the expert in the educational process.

In this investigation, the extent to which differences were present in the mathematics achievement by the ethnicity/race of Grade 3 students in Texas were analyzed. Data obtained from the Texas Education Agency Public Education Information Management System for all Texas Grade 3 students who took the State of Texas Assessment of Academic Readiness Mathematics exam were examined for the 2016-2017, 2017-2018, and 2018-2019 school years. In all three years analyzed, statistically significant differences were revealed in overall mathematics achievement and in all four Mathematics Reporting categories. Inferential statistical analyses revealed a clear stair-step effect. Asian students were the highest performing ethnic/racial group in all four Mathematics Reporting categories and all three performance level standards, followed by White, Hispanic, and Black students. Suggestions for future research and implications for policy and practice were provided.

The purpose of this study was to carry out an analysis of the effectiveness of the Flipped Classroom methodology for teaching mathematics--specifically geometry, in an online environment. Specifically, we measured: (1) the academic achievement of students who were taught based on this method; and (2) the perceptions of satisfaction with this methodology and with the virtual sessions attended through digital resources. In order to gather the data a pre-experimental and ex-post-facto design was used with a sample of 113 secondary education students. Regarding academic performance, positive and significant differences were found between the pre-test and the post-test. In addition, different significant variables were found that affected the academic performance of the students, such as having previously failed the subject of mathematics, the perception of the students about their effort in learning mathematics or the educational level of the parents. In respect of the students' perceptions, the results showed that they were generally happy with the use of the Flipped Classroom methodology, bearing in mind that the environmental context in which the educational process takes place affects the students' perceptions. However, slightly negative perceptions were also found in respect of virtual sessions that made use of digital resources. These findings were significantly affected by the frequency with which resources and electronic devices were deployed. These results would support the integration of this methodology into the teaching of mathematics, where the use of digital resources to study geometry in particular helps to improve the different competences and skills of the students.

The purpose of the study was to determine the association between van Hiele geometric thinking levels and circular achievement exam results for middle school students in 7th grade. The design of the study was a survey method. 157 middle school students were participated into study from three different schools in Kayseri in the spring semester of the 2017-2018 academic year. Van Hiele geometry thinking test which was developed by Usiskin and adapted into Turkish by Duatepe (2000) and a circle achievement test that was developed by the researcher were used to collect the data. Pearson The circle achievement test and van Hiele geometry thinking levels were correlated to find their link. Assumptions were met, hence independent groups t-test and ANOVA were performed. The results revealed that Grade 7. Students van Hiele geometric thinking levels were lower than expected levels. A moderate correlation was found between the circle achievement test and the van Hiele geometric thinking test. The results obtained from the van Hiele geometry thinking test showed statistically significant mean difference across schools 1 and 2. Moreover, there is no statistically significant mean difference across schools at the circle achievement test. Boy and girl middle school students' geometry thinking levels and circle achievement test scores were not statistically significantly different from each other.

Geometry has an important place in mathematics education because it forms the basis of mathematical thinking and allows us to see the emergence of logical theory. As in all fields, academic journals, which are among the official communication languages of science, play an important role in the construction, dissemination and use of scientific knowledge in the field of geometry teaching. With reference to this fact, the aim of the study was to examine articles published in the field of geometry teaching and indexed in SSCI (1975-2020) and ESCI (2015-2020) in the Web of Science Core Collection database by bibliometric analysis method. 109 articles related to geometry teaching (72 within the scope of SSCI and 37 within the scope of ESCI) were reached for the given periods. According to the findings of the research, while there has been a recent interest in geometry education within the scope of SSCI and ESCI, the number of articles still published is limited. Therefore, in mathematics teaching, geometry teaching lags behind areas such as arithmetic and algebra. Especially in recent years, it is noted that the keyword "Van Hiele Levels" and the keywords of innovations that technology brings to geometry education such as "Dynamic Geometry", "Virtual Reality", "Geogebra" and "Geometry Thinking" are frequently used in articles in the field of geometry teaching. It is observed that, while quantitative research methods are widely preferred in studies within the scope of SSCI in the field of geometry teaching, qualitative research methods are preferred within the scope of ESCI.

In high school geometry, students are expected to deepen their understanding of geometric shapes and their properties, as well as construct formal mathematical proofs of theorems and geometric relationships. The process of helping students learn to construct a geometric proof can be challenging given the multiple competencies involved (Cirillo & Hummer, 2021). To support geometry students in the proof-writing process, the authors describe an instructional sequence in which students posed a conjecture about a mathematical relationship, wrote a draft argument, provided feedback/critiques to one another, and revised their argument based on peer feedback. They designed the sequence to provide students with opportunities to engage in multiple elements of the reasoning-and-proving process, to allow them access to proof-writing through multiple modes of entry, and to encourage student agency by having them generate and justify mathematical relationships (Bieda & Staples, 2020).

The German national educational standards state explicitly that students should be enabled to successfully interact with dynamic geometry software. In a feasibility study on providing a standardized assessment instrument by digital means, in order to assess students' mathematical competencies, the implementation of a task with such a dynamic geometry software was investigated. The sample consisted of 101 8th graders and was used to evaluate how students actually interact with the embedded dynamic geometry software. We focused especially on investigating relationships between successfully solving such tasks (evaluated automatically or categorized qualitatively by a human rater) and process data collected during the testing. A multiple linear regression analysis was conducted to identify relevant predictors for success within one dynamic geometry task. On this basis, possible adjustments are discussed so as to enhance the standardization and accessibility of tasks within dynamic geometry systems.

Recognizing, building, describing, extending, and analyzing patterns are key components of algebraic reasoning. Oftentimes pattern-based instruction is used to support the understanding of functions and variables. In this study, elementary preservice teachers (n = 23) were asked to explore, extend, explain, create equations for, and analyze linear, geometric, and quadratic integer patterns using color tiles. Then they created, extended, described, and analyzed their own patterns while working in groups (n = 8). The research question was: What algebraic errors do PSTs demonstrate in the process of constructing, explaining, and analyzing their own linear, geometric, and quadratic patterns using color tiles? The analysis, which focused on errors and misunderstandings preservice teachers demonstrated in their final unit, revealed three specific areas of difficulties: figural errors, explanation challenges, and improper use of algebraic language. Participants failed to connect their algebraic images in a coherent manner, there was confusion in articulating their algebraic thought processes, and algebraic terminology was improperly used. Implications for teaching and further research are described.

Technological Pedagogical Content Knowledge (TPACK) is defined as the teacher knowledge needed for effective technology integration. This study aimed to investigate preservice elementary mathematics teachers' TPACK development. TPACK survey was administered to 33 preservice teachers (PSTs), and six of them were selected via maximum variation sampling to represent as different cases as possible based on their mathematical knowledge and technological self-assessment. These six PSTs implemented four technology-based lessons (two microteaching sessions in the mathematics teaching method course and two lessons in the student teaching). Participants' TPACK levels were examined based on the development model proposed by Niess, M. L., Sadri, P., & Lee, K. (2007, April). Dynamic spreadsheets as learning technology tools: Developing teachers' technology pedagogical content knowledge (TPCK). Paper presented at the meeting of the American Educational Research Association Annual Conference, Chicago, IL. Results showed that participants had not used technology effectively and efficiently in their microteaching sessions and the first lessons in their student teaching. After evaluating their first lessons in schools, PSTs improved their teaching considerably. We suggest giving more opportunities for PSTs to teach with technology in classrooms and to assess their teaching practices reflectively.

"Assumptions" play a fundamental role in disciplinary mathematical practice, especially concerning the relativity of truth. However, much is still unclear about ways to help students recognize key aspects of this role. In this paper, we propose a set of principles for task design to introduce students to the role of assumptions in mathematical activity, with particular attention to the following two learning goals: recognize that (1) a conclusion depends on the assumption(s) underlying the argument that led to it; and (2) making the underlying assumption(s) explicit is crucial to reaching consensus on the conclusion. In the context of a 3-year design research study, we first used existing literature to construct an initial version of task design principles which we then empirically tested and refined by designing and implementing two tasks in Japanese school classrooms. One of the tasks was in the area of functions at the secondary level and the other in the area of geometry at the elementary level. We analyze two classroom episodes to discuss the promise and evolution of our proposed task design principles. In addition, our analysis sheds light on the role of the teacher's instructional moves and the students' mathematical knowledge during the implementation of the designed tasks.

The Great Mini-Golf Project requires students to use a ruler to identify measurements to the quarter-inch, with careful precision and accuracy, which supported the Standards for Mathematical Practice 5 and 6: using appropriate tools to solve mathematical problems with precision (NGA Center & CCSSO, 2010). This project also supported Standard for Mathematical Practice 4 by giving the students an opportunity to model with mathematics. The students were given a real-world problem of designing their own golf course hole, and they had to use what they knew about both playing golf and mathematics to create a prototype of a realistic design. This project sparked students' interest and encouraged collaboration to solve an authentic, engaging real-world mathematical problem. This article examines the goal of this project to introduce and scaffold the learning of geometry and related skills earlier in the school year. This included having students create and analyze a line plot and understand the attributes, classifications, and sub-categories of two-dimensional figures (5.MD.B.2, 5.G.B.3, and 5.G.B.4, respectively [National Governors Association Center for Best Practices & Council of Chief State School Officers (NGA Center & CCSSO), 2010]).

Can You Build It? is an engaging Improve It! task that asks children to construct 3D shapes based on 2D depictions. Students' initial 3D creations often need revision, and as they build and re-build, they develop spatial relationships. This task was implemented in a first-grade class, connected to composing 3D geometric shapes to address the Common Core State Standard 1.GA.2 (National Governors Association Center for Best Practices & Council of Chief State School Officers, 2010). Moving constantly between 2D and 3D leads to a stronger spatial sense that helps students to unveil the shapes' structures, which is an important topic in mathematics and recommended as a mathematical practice. This article examines how first graders' abilities to build and improve their geometry constructions while moving between and connecting 2D and 3D representations. Because the task is primarily about students constructing and exploring, the lesson was organized into three stages: (1) presenting the task to students (10 minutes); (2) students constructing and improving constructions, alone or in peer groups (30--45 minutes); and (3) a whole-class conversation with students presenting and discussing strategies (15 minutes).

The purpose of this study was to compare the concept images and hierarchical classification skills of students at different educational levels (middle school, high school, and university) regarding parallelograms. In this study, a cross-sectional research design was utilized. The study group consisted of a total of 357 students from middle school (5th-grade, n = 80; 8th-grade, n = 73), high school (9th-grade, n = 49; 12th-grade, n = 49), and university (freshmen preservice mathematics teachers (PMTs), n = 54; senior PMTs, n = 52). Data were collected through a diagnostic test consisting of two separate forms, which required drawing different parallelogram shapes and classifying the parallelograms among the provided shapes. Data were analyzed using descriptive content analysis. It was indicated from the results that most middle and high school students classified the quadrilaterals based on their prototype parallelogram concept images, and some also had missing definitions as well as misconceptions regarding parallelograms. Additionally, it was revealed that although the majority of pre-service teachers classified the parallelograms hierarchically, some were deficient in their concept images of parallelograms.

External visualization (i.e., physically embodied visualization) is central to the teaching and learning of mathematics. As external visualization is an important part of mathematics at all levels of education, it is diverse, and research on external visualization has become a wide and complex field. The aim of this scoping review is to characterize external visualizations in recent mathematics education research in order to develop a common ground and guide future research. A qualitative content analysis of the full texts of 130 studies published between 2018 and 2022 applied a deductive-inductive coding procedure to assess four dimensions: visualization product or process, type of visualization, media, and purpose. The analysis revealed different types of external visualizations including visualizations with physical resemblance ranging from pictorial to abstract visualizations as well as three types of visualizations with structural resemblance: length, area, and relational visualizations. Future research should include measures of visualization products or processes to help explain the demands and affordances that different types of visualizations present to learners and teachers.

Background Study: The effects of the flipped classroom (FC) on students' academic performance remain ambiguous, and the use of pre-class videos may be one of the main factors hindering students' progress. A software package called GeoGebra has been proposed as a substitute for pre-class videos to aid students' learning. Objective: This study investigates the effect of the FC using GeoGebra (FCG) on mathematical performance and memory among senior secondary students. Methods: One hundred and sixty students of a senior high school in China were selected. A quasi-experimental design was implemented in which the classes were split into the intervention group (FCG), control group A (FCV) and control group B (TL) by random assignment. Pretest, Post-test 1 and Post-test 2 were introduced. Results and Conclusion: The results indicate that both the FCG and the FC with pre-class videos (FCV) were more effective than traditional lectures in enhancing students' academic test scores, and that the effect of the FCG was comparable to its video counterpart, F(2, 136) = 3.752, [rho] = 0.026, [eta][superscript 2] = 0.052. Although their effects were similar, the qualitative results revealed some hidden benefits associated with the replacement of pre-class video with GeoGebra. Furthermore, although they were equally effective in promoting students' performance in the short-term, the long-term effects on academic performance of both forms of FC were non-significant when compared with traditional lectures.

The Van Hiele model of geometric reasoning establishes five levels of development, from level 1 (visual) to level 5 (rigor). Despite the fact that this model has been deeply studied, there are few research works concerning the fifth level. However, there are some works that point out the interest of working with activities at this level to promote the acquisition of previous levels. Our goal is to describe this level through the construction and validation of a list of indicators for each of the processes involved in geometrical reasoning (definition, proof, classification, and identification). Due to the lack of previous research, we have decided to use the Delphi methodology. This approach allowed us to collect information from a panel of experts to reach a consensus through a series of phases about the indicators that describe each of the processes. The final product of the iterative application of this method is a list of validated indicators of the fifth Van Hiele level of reasoning. In particular, proof and definition processes have turned out to be the most relevant processes at this level.

When justifying a mathematical statement, textbooks often interweave several instances of justification, creating, what we term, "paths of justification." This study aims to characterize the paths of justification of mathematical statements that are offered in eight 7th grade Israeli mathematics textbooks. The study attended to context and compared paths of justification in different mathematical domains (algebra and geometry) and in textbooks intended for students with different achievement levels. Analysis focused on the number of instances of justification in a path, the types of justification of those instances, and their order. Comparing paths of justification between algebra and geometry, and between textbooks intended for students with different levels of achievements, revealed several similarities and differences. The findings portray a complex picture of textbook justifications, underlining the need to be sensitive to context when studying textbook justifications. The study results are discussed in light of relevant literature, highlighting potential factors involved in the way a textbook path of justification of a specific mathematical statement is structured.

This study was conducted to reveal potential sources of students' difficulty and misconceptions about geometrical concepts with the help of eye tracking. In this study, the students' geometrical misconceptions were explored by answering the questions on the geometry test prepared based on the literature and test-taking processes and represented with the help of eye tracking data to make interpretations about their visual and cognitive attention. With this aim, the mixed method research design was used. The data were collected through the geometry test, eye tracking measurements, and the think aloud strategy. Geometry test scores were represented by descriptive statistics such as percentage. The quantitative eye tracking data were analysed by Kruskal-Wallis tests and Mann-Whitney "U"-tests among the choices for each question. The qualitative data collected through the think aloud strategy were analysed by the content analysis. In the findings, the students' geometrical misconceptions were examined and reported in detail from a different perspective with the help of data related to eye tracking. Then, eye tracking data and qualitative data showed the geometry misconception items on which the students had difficulty and cognitive conflict in addition to geometrical misconception based on the students' answers to the geometry questions.

This article reports on the design and implementation of a didactic sequence in the frame of a design-based research study. The research aim is to test the hypothesis that affordances of dynamic geometry may support students' awareness of logical relationships between geometrical properties of constructed figures. We elaborate on the task design principles used in the study and present analysis of four Danish grade 8 students (age 13-14) working in pairs on the very first task of the sequence. The Theory of Semiotic Mediation frames the design of the study and the analysis of data, which was collected in the form of screencast, video and written products. The results show that grasping logical relationships is not immediate, on the contrary, the results indicate that students expect dependencies to be non-hierarchical in DGE and that low achieving students describing the behaviour of constructions during dragging refer to their global appearance, rather than to geometrical properties. Such results suggest that specific prompts in the task design can shift students' attention on specific elements of constructions and that explicitly asking the students to explain any unexpected observation seems to be necessary for active reflection.

This paper reports some results of a study on how Colombian primary-school teachers, experienced in the use of technological tools, integrate digital resources into their practice. Specifically, we investigate the processes of selection and integration (including appropriation and orchestration) of these resources in the teaching of geometry. We aim to illustrate the potential paths (along with associated criteria) that teachers follow when selecting and integrating digital resources into their practice, using the Documentational Approach to Didactics. Accordingly, we present case studies of a 1st-grade teacher and a 5th-grade teacher, which describe their paths during their respective geometry lessons. The paths followed by the teachers were influenced by their knowledge and beliefs regarding the utilisation of digital technology resources and the potential student engagement with such resources. All these aspects come into play as teachers select and orchestrate digital resources. Our empirical data analysis enabled us to infer several criteria for resource selection and levels of orchestration and appropriation manifested by the teachers as they integrated digital resources into their classes. This work can provide valuable insights for further exploration of how teachers integrate digital resources, and can contribute to the development of professional development programmes.

We use Action-Process-Object-Schema (APOS) theory to study students' geometric understanding of partial derivatives of functions of two variables. This study contributes to research on the teaching and learning of differential multivariable calculus and its didactics. This is an important area due to its multiple applications in science, mathematics, engineering, and technology (STEM). The study tests a previously proposed model of mental constructions students may use to understand partial derivatives through a set of activities designed to help students make the conjectured constructions. The model is based on the local linearity of differentiable two-variable functions, and the model-based activities explore the relationship between partial derivatives and tangent plane in different representations. We used semi-structured interviews with eleven students whose teacher used the three-part cycle--Activities designed with the genetic decomposition; collaborative work in small groups and Class discussion; and Exercises for home (ACE)--as pedagogical strategy. The model-based activity set based on local linearity and the ACE strategy helped students construct a geometric understanding of partial derivatives. Results led to reconsider and further refine the model. This study also resulted in improving activity sets and obtaining information on students' construction of second-order and mixed partial derivatives.

Despite the surge in efficacy studies, the relative effectiveness of any two math interventions is largely unknown and replications are rare. To ensure that students benefit from high-quality learning experiences, the current conceptual replication of Wang and Woodworth, a randomized control trial, evaluated the relative impacts of two computer programs used in a school district as supplements to students' regular education math instruction, DreamBox Learning and Zearn Math. For 115 students in kindergarten and first grade, data were collected before and after an 11-week implementation period. Measures of effect size, but not null hypothesis statistical tests, suggested that participants in the DreamBox condition had higher test scores on posttest numeracy and geometry (Hedges g = 0.32 and 0.14). There was also a statistically significant group by pretest interaction, evidence of "Matthew Effects" for numeracy. Although some support for adopting DreamBox as a supplemental method for improving math competencies of students in early elementary school was provided, more research is needed as the true effect may be smaller than what the present study reports. The study's implications are discussed.

The Compulsory Education Course Standards for Mathematics 2022 have highlighted the educational objectives of junior secondary mathematics by emphasizing the development of mathematical competence and practical learning. Model consciousness, as one of the fundamental mathematical competencies to be developed at the junior secondary level, can facilitate students' comprehension of the universal application of mathematics. Teachers of mathematics in junior high school should construct effective classroom activities based on the cognitive qualities of their students in order to enhance their mathematical model consciousness and comprehension of the substance of mathematics knowledge. This paper is a lesson study of the education of Congruent Triangles, and its purpose is to investigate strategies for fostering mathematical model consciousness among junior high school pupils.

Being able to move between different representations levels (macroscopic, submicroscopic, and symbolic) is important to understand a chemical phenomenon meaningfully. The current study investigated how pre-service science teachers moved from symbolic representation to sub-microscopic representation. A total of 205 pre-service science teachers, first, balanced the three chemical reactions and then drew the reactions' sub-microscopic views if they were able to observe them. The findings revealed that pre-service science teachers are most successful at the symbolic level. Yet, this is not a valid conclusion for the sub-microscopic representation level. The drawings were analyzed concerning atomic size and molecular geometry and found that pre-service science teachers have common alternative concepts. Furthermore, they mostly have a superficial knowledge of the molecular geometry of chemical compounds. In conclusion, because preservice science teachers will teach in their future careers, it is important to encourage them to comprehend chemical phenomena at each representation level and shift between them to prevent alternative concepts they may have.

The benefits of technological and theoretical scaffolding were observed when pre-service teachers aiming to teach upper elementary grades were given three learning-based geometrical inquiry tasks involving inscribed circles. They were asked to collaboratively examine the accompanying geometrical illustration and data for some new or interesting feature and then propose a hypothesis resulting from their observations and prove them. Due to the difficulty generally involved in proposing and proving geometrical hypotheses, two forms of scaffolding were provided: theoretical scaffolding based on revising previous learning or specific attributes of the given data and technological scaffolding in the form of specifically designed GeoGebra applets that allowed dynamic observation of the attributes of the geometrical shapes and the changes they underwent during modification. We found that the two forms of scaffolding led to relatively pre-service teachers' high levels of success. They exhibited high levels of interest and participation, were engaged in the tasks, and underwent high-quality learning processes. In follow-up interviews, they confirmed that the exercise improved their inquiry skills, and developed their pedagogical and technological knowledge.

The definition of creativity among professional mathematicians and the definition of mathematical creativity in the classroom context are significantly different. The purpose of this study was to investigate the relationship between students' mathematical creativity (i.e., cognitive flexibility) and figure apprehension when solving geometric problems with novel auxiliary features such as straight lines and curved lines. In other words, this study determined if geometry knowledge influenced mathematical creativity (cognitive flexibility) in problem-solving. Grade-12 students participated in the intervention. The high school that is the research topic attempts to equip students with academic abilities and is, except for vocational schools, the most popular form of high school among all other types. Such a school was chosen for the study so that a significant proportion of students in Makassar could be represented. In this study, we discovered a relationship between cognitive flexibility and the geometric ability of pupils while solving problems involving auxiliary lines. This indicates that the usage of auxiliary lines as a reference for developing pupils' creative thinking skills must be advocated. In addition, good geometric abilities (e.g., visual thinking, geometrical reasoning) will encourage pupils to generate various problem-solving concepts. This finding contributes significantly to future research by focusing on auxiliary lines.

Using origami in the classroom, this study examines the experiences of pre-service middle school mathematics teachers. A case study model was employed in this study as a qualitative research approach. The study involved 39 senior pre-service mathematics teachers. Document analysis, semi-structured interviews, and observation techniques were used to collect data using the triangulation method. Content analysis was used to analyze the data. During their micro-teaching with origami, pre-service teachers preferred learning outcomes focused on geometry. Most participants used origami to teach learning outcomes that were appropriate to their grade levels and learning outcomes. Throughout all the lesson plans prepared according to the 5E model, origami was used to construct mathematical concepts during the exploration step. While some groups utilized origami folding techniques to explore geometric structures, others used rulers or goniometers instead, indicating that origami is partially effective. In order to improve teacher training programs, more practice should be given with origami-based visual proofs of geometric structures. In addition to contributing to professional and personal development, the participants' opinions showed that this process was beneficial to them. Students struggled to explain origami's steps, as well as selecting models and exploring concepts with origami. For pre-service teachers to overcome these challenges, more time must be allocated during their practicum process within the teaching practice courses so that they can practice teaching mathematics with origami in real classroom settings.

This study aimed to reveal the effect of teaching the 4th grade circumference measurement subject with the dynamic geometry software "GeoGebra" on achievement and motivation. In the research, the pre-test -- post-test control group design, which is one of the quasi-test designs, was used. In the Mathematics lessons conducted with the experimental group for three weeks, the subject of circumference measurement was taught with dynamic geometry software, whereas the control group followed the current curriculum. As data collection tools, the 21-item "4th Grade Circumference Measurement Achievement Test" (KR20=0.81) developed by Öksüz and Genç (2021) and the "Primary School Mathematics Motivation Scale" (KR20=0.94) developed by Ersoy and Öksüz (2015) were utilized. The pre-test was applied concurrently to the test and control groups one week before the research was initiated, and the post-test was applied immediately after the research ended. In the study, a paired-samples t-test was used to compare the pre-test and post-test scores of the students in the test and control groups within the groups; whereas covariance analysis (ANCOVA) was used to compare the post-test scores of the related groups. As a result of the research, it was concluded that there was a statistically significant difference between the achievement and motivation post-test scores of the students in the experimental group who learned the subject of "Circumference Measurement" with the dynamic geometry software and the students in the control group who did not use this program on behalf of the experimental group.

Planning a mathematics lesson plan is a complex process. Its quality contributes to the effectiveness of mathematics instruction. Given this significance, improving prospective mathematics teachers' lesson planning ability is essential for teacher preparation programs to produce effective teachers. This study examines how a lesson study within a school-university partnership program (SUPER-LS) could strengthen the lesson planning skills of prospective mathematics teachers. This study employed a case study, which included thirteen student teachers. Data were collected using four initial and four final drafts of mathematics lesson plans developed by four groups of prospective mathematics teachers and reflection reports. These were then analyzed qualitatively using the framework of MKT. The study found that sixteen categories of issues were present in the initial lesson plan, and six remained in the final lesson plan. In the final lesson plan, the 16 problems were reduced to 6, with an improvement of around 62.50%. In addition, prospective mathematics teachers improved their lesson planning skills because of their involvement in the SUPER-LS program, facilitated by intensive collaboration and interaction among participants, especially in-service teachers, and teacher educators, as well as peers' feedback.

Studies that address preservice teachers' knowledge of area measurement emphasize their lack of knowledge and their tendency towards the use of formulas, without offering a body of knowledge that helps to address such difficulties. This study offers an approximation of the mathematical knowledge necessary for preservice teachers to solve area tasks. For this, the Mathematics Teacher's Specialized Knowledge model is used with emphasis on the subdomain of Knowledge of Topics and Knowledge of the Structure of Mathematics. Preservice teachers' resolutions and written justifications are analyzed using qualitative and quantitative tools. The results indicate that those resolutions that manage to mobilize mathematical knowledge are associated with the joint mobilization of different procedures, properties, and geometric principles. Results also indicate that the strategic coordination between different registers of representation allows Preservice Teachers to mobilize categories of specialized knowledge and establishing connections with other mathematical contents.

The aim of this study is to examine 8th grade students' Van Hiele geometry thinking levels, proof writing and justification skills. The research was carried out with sixteen eighth grade students of a private school in Sivas. Participants were selected based on an easily accessible sample. The data were collected from the students who participated in the face-to-face teaching process in their schools and volunteered to participate in the research on the data collection day in the fall semester of the 2020-2021 academic year. The data collection tool is the Van Hile geometry test and the geometry proofing test. The document analysis method was adopted in the study. While analyzing geometry proof writing and justification test data, geometry proof writing test evaluation rubric and justification evaluation rubric were used. As a result of the analysis of the data obtained, there was a linear relationship between the students' Van Hiele thinking levels and their proof writing skills, while there was no positive relationship between their justification skills and their Van Hiele thinking levels and their proof writing skills. [Note: The publication year (2022) shown on the PDF is incorrect. The correct publication year for this article is 2023.]

This paper focuses on the difficulties encountered by Italian students in performing geometry tasks. A quantitative analysis, aimed at understanding the extent of the phenomenon, is carried out using the results of district competitions from the year 2018 to 2020, comparing the scores obtained in geometry questions with those in other areas of Olympic mathematics. In addition, the answers given by the students to a questionnaire administered at the end of the 2020 district competition are analyzed in order to better understand possible motivations behind the phenomenon in question. The results obtained need further confirmation through future research on the topic but represent clear trends worthy of further investigation.

Mathematics instruction in primary school has been influenced by many policy changes and empirical findings in the previous two decades. Geometry lessons, in particular, were re-evaluated due to a paradigm change and, consequently, were attributed a new meaning within the mathematics curriculum worldwide. The present paper focuses on this paradigm shift in the sense of the evaluation to what extent both the didactical potential and the practical value of geometry instruction in elementary education are currently recognised and utilised by primary grade teachers. In total, 120 primary grade teachers participated in the study. The results showed that there had been positive recognition of the didactical potential of school geometry by the teachers over the previous two decades; however, it lacked actual implementation in school practice for different reasons. The results are discussed not only with regard to the latter of these but also with regard to their theoretical and practical implications.

Mathematical proficiency is essential to supporting students' success in learning mathematics. It is the ability to use conceptual understanding, procedural fluency, strategic competence, adaptive reasoning, and productive disposition in problem-solving. Cognitive independence shows students' ability to process information. Meanwhile, motivation is related to things that encourage students to learn mathematics. This study aims to determine the effect of cognitive independence and motivation on mathematical proficiency. It involved 131 students and used a mixed method with Sequential Explanatory Design. The quantitative research part of the study was used to determine the effect of cognitive independence and motivation on mathematical proficiency using a multiple linear regression test. Furthermore, qualitative research was used to deeply scrutinize the effect by using in-depth interviews. The results showed that students with high cognitive independence could process the knowledge that has been possessed logically, while students with low cognitive independence did it illogically. Students with low motivation use trial and error strategies to maximize mathematical proficiency, while students with high motivation use analytical strategies.

Critical thinking ability is a very important basic ability in learning mathematics. In the era of technological advances and digitalization, a comprehensive learning approach is needed that accommodates critical thinking skills. The controversial problem-based multimodal approach with ethnomathematics is seen as being able to improve this ability. This can be seen from the characteristics of the learning model with the Ethnomathematically Controversial Problem-Based Multimodal approach that accommodates visual, auditory, and kinesthetic learning styles. However, there is no empirical evidence that the combination of that approach and ethnomathematics (Balinese culture) in digital learning can improve students' critical thinking skills. This study aims to comprehend the impact of digital learning with a controversial problem-based multimodal approach, incorporating Meru and Jejaitan Bali ethnomathematics on students' critical thinking skills. Quasi-experimental research uses a Randomized Complete Block Design (RCBD) applied to measure students' initial abilities, namely students with low, medium, and high initial abilities. The instrument used in this research is a test of critical thinking skills which contains controversial issues with ethnomathematical nuances in the cases of the sacred building of Meru. The population consists of 202 grade IX students, with 48 in the experimental group and 52 in the control group. The ANOVA test results demonstrated that using this approach was effective in improving students' critical thinking skills. (This can be seen from the value of sig. 0.00 < =0.05 which means that there is a difference in students' critical thinking skills between the experimental class and the control class). According to the t-test results, students in the experimental group had significantly higher critical thinking skills than students in the control group. Tukey's test shows that critical thinking ability varies depending on the student's beginning ability. In conclusion, this combination is effective in improving students' critical thinking skills, especially on the topic of geometry.

The purpose of this study is to explore the effectiveness of the parallel classroom model in comparison to a support facilitation model. Using district-level assessments in geometry, this study assessed student performance when students were taught the same content with the same rigor, where one classroom uses differentiation strategies and small-group instruction to instruct students and the other classroom uses support facilitation. By examining the success of the parallel classroom model, we hope to prove this model to be a viable curricular alternative for the needs of exceptional learners. Specifically, we see parallel classrooms as being more flexible to diverse student needs than traditional classrooms.

This study aims to examine the components of collective argumentation of pre-service middle school mathematics teachers during geometric construction activities. To scrutinize this issue, case study research was utilized. The participants were 14 pre-service middle school mathematics teachers who worked collectively by forming four groups. During the data collection process, the groups worked on four geometric construction tasks by using compass-straightedge or GeoGebra. The findings presented that the collective argumentation processes were depicted by means of eleven components. In more detail, the six components of Toulmin's argument model which are data, warrant, claim, backing, rebuttal, and qualifier were insufficient to represent collective argumentation. Instead of claim, the term conclusion was used in this study since the associated data and warrant were provided in the argumentation. The collective argumentation processes of the groups involved not only the mentioned six components but also the five additional components, which were named conclusion/data, target conclusion, guidance, challenger, and objection. The new components might be used while investigating the argumentation process in other disciplines.