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The capacity for mathematical reasoning, pivotal in grasping core math concepts, directly shapes a student's success in their mathematical studies. This study aimed to detail and analyzed how the search, solve, create, and share (SSCS) instructional framework impacts mathematical reasoning skills within the context of linear equations. We adopted a quasi-experimental research approach using a non-matching control group design. To gather data on mathematical reasoning proficiency, a validated and reliable essay-type test was employed. Data collection on mathematical reasoning abilities used an essay test instrument that has been valid and is reliable. This research involved 102 eighth grade students whereby they were selected by using a purposive sampling. To measure and describe the effect size, we utilized a specific method, yielding a result of 0.97. The t-test was employed to assess the influence of the SSCS learning model, and the significance value obtained was less than 0.05. The effect size, as determined, stands at 0.97, suggesting a substantial impact. The t-test results underscore a significant relationship between the SSCS learning model and the enhancement of mathematical reasoning abilities. This research delineates a framework for employing the SSCS model to enhance students' mathematical reasoning capabilities. It is evident that such reasoning abilities play a pivotal.

Eye-tracking (ET) method provides a promising channel for educational researchers to connect learning outcomes to cognitive processes. The main principle of ET is that our gaze and our focus of attention are connected. Due to the advent of digital technologies, eye tracking studies are increasingly growing in different fields and in mathematics education. We assume that the provided task is one of the ways to analyse deeply cognitive processes, and we wonder how ET studies are dealing with the integration of digital technology in task-design in geometry. Starting from Strohmaier et al. (2020) we found three studies, one focused on geometric proofs and two based on construction of geometric objects. We highlight the contributions of integrating digital technology in learning and instruction, and in research in geometry.

Mathematical reasoning has been a critical concern in Turkey especially since the structure of the student selection examination for high schools changed six years ago. The ability to solve the questions in the new exam requires high level reasoning and argumentation skills. Schools, whether they are public or private, prepare 8th graders for this exam with intense educational programs. They frequently use skill-based questions--similar questions to the ones in the new exam, which require high level mathematical reasoning. This study indicates that students gain better mathematical reasoning skills in learning environments promoting collective discussion and argumentation. Hence, these questions need to be solved in these kinds of environments. In this study, two cases are analyzed to understand the argumentation process in classrooms in depth. Audiotapes of two 8 th grade classrooms, one from a public school, another from a private school, are analyzed. The same teaching material--a worksheet including skill-based questions--is used in the classrooms. During analyzing, qualified argumentation pattern--a pattern including students' claims and justifications/evaluations for those claims--is defined. Analysis of classroom audiotapes revealed that that there are significant differences between two classrooms' argumentation structure and type of dialogues emerged in the classroom. In one classroom there is a more qualified argumentation process than in the other classroom. In addition, analysis of the type of the dialogues indicates that two classrooms have different types of dialogues. Overall, the study reveals that despite the differences, both classrooms mostly included teacher-individual interactions and they have little collective discussion.

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.

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.

This work is the result of a master's investigation in Brazil, which discusses the teaching of the parabola in the initial training of mathematics teachers. Our theoretical framework addresses the relationship between intuition and the dialectics of the theory of didactic situations, which supports the analysis of the results of this study Its objective was to identify and register categories of intuitive reasoning manifested by the subjects when solving a didactic situation involving the parabola with GeoGebra software contribution. The methodology adopted was didactic engineering, experimented with eight students in initial training at a Brazilian public university, among 6th and 9th undergraduate semesters. The posterior analysis and validation allowed us to verify the need to discuss the parabola, articulating its geometric, algebraic, and analytical views, as well as to reinforce the importance of its teaching with the use of technology.

GeoGebra is a dynamic software that is frequently used and of increasing importance in mathematics teaching processes in our digital age. Accordingly, in this study a new perspective has been brought to the proofs of the "two square difference identity" expressed for the square, which is a flat polygon, made with different approaches. With side lengths a, b, and a>b, it has been shown that the identity given by the equation (difference of area) a[superscript 2] - b[superscript 2]=(a-b)(a+b) is true for other regular polygons as well. In the study, direct proof method was used within the framework of the principle of conservation of measure, which is one of the basic principles of geometry teaching. GeoGebra program, which is a dynamic geometry software, was preferred for drawing geometric shapes used in proofs. In order to generalize the number n, a different fragmentation technique was preferred to the proofs made using different drawings for equilateral triangle and square, which are the simplest regular polygons. It has also been shown that this identity is true for circles viewed as polygons with an infinite number of sides.

The primary goal of advanced mathematics courses is to equip students with proof skills. That is why the competence of pre-service mathematics teachers in proof is considered an assessment of their performance. Despite the emphasis on mathematical proof in undergraduate education, students studying higher mathematics at university often face difficulties with proofs. Therefore, it is essential to examine students' perceptions and thinking processes regarding proof. Accordingly, this study aims to explore the types of proof perceptions and thinking processes of pre-service elementary mathematics teachers based on their Van Hiele geometric thinking levels. The study utilized the case study design, one of the qualitative research designs. The study group comprised 67 fourth-grade pre-service elementary mathematics teachers studying at a medium-sized education faculty in western Turkey. The study group was determined using the convenience sampling technique, one of the non-probability-based sampling methods. Van Hiele's geometric thinking level test, proof selection scale, and scale for determining proof thinking processes were employed as data collection tools. The data were analyzed descriptively. Consequently, it was found that pre-service teachers were predominantly at the fourth geometric thinking level and exhibited the B-proof perception type. Only a few students were observed to reach the level of proof in their thinking processes. According to the results obtained from the study, the geometric thinking levels and proof perception types of pre-service teachers significantly affect their thinking processes.

Abstraction is essential to learning mathematics because the mathematical concepts obtained through abstraction will be more meaningful than directly receiving these concepts. This study aims to describe the pre-service teachers' abstraction in constructing relationships among quadrilaterals. This research method was explorative qualitative research with a purposive sampling technique. The subject of this research was a pre-service mathematics teacher who had taken a geometry course. The data analysis techniques used in this study were data condensation, data display, drawing and verifying conclusions. The research results showed that the participant used epistemic actions in an abstraction, such as recognising each quadrilateral type, building-with their properties, constructing relationships among them, and consolidating the abstract results made. Thus, the abstraction in constructing relationships among quadrilaterals can be observed from the epistemic actions: recognising, building-with, constructing, and consolidation, known as RBC+C.

This study aimed to examine the effect of using the flipped classroom model on students' proportional reasoning. In this context, the study focused on students' achievement and solution strategies for proportional reasoning problems. The participants of the study were 56 seventh-grade students, who were determined by convenience sampling method. In line with the purpose of the research, the concept of proportion was taught to the experimental group using the flipped classroom model and to the control group through teaching in accordance with the mathematics curriculum. The data of the study were collected through the Proportional Reasoning Test. The results of the study showed that the flipped classroom model was more effective in terms of mathematics achievement than the teaching method in line with the curriculum. In addition, the experimental group students used more correct solution strategies and fewer incorrect solution strategies than the control group students while solving the problems.

Science, technology, engineering, and mathematics (STEM) education has emerged as a powerful tool for boosting students' mathematics achievement. By integrating these disciplines, students are equipped with essential skills for the 21st-century workforce and can develop a deeper understanding of mathematical concepts. Through hands-on experiments, problem-solving activities, and real-life applications, STEM education offers a dynamic approach to learning mathematics. This quasi-experimental study aimed to investigate the impact of STEM education on mathematics achievement among 8th-grade students. Although the study lacked the full randomization typically seen in experimental designs, it meticulously designed and validated its measurement tool for assessing mathematics achievement. The pre-and posttests administered to both the experimental and control groups provided valuable data for evaluating the effectiveness of the STEM education intervention. The statistically significant difference in mathematics achievement between the two groups, with a significance level of = 0.05, strongly suggests that the STEM education program had a positive and meaningful impact on the experimental group's math skills. As a result, the study offers valuable insights into the potential benefits of integrating STEM education into the curriculum for middle school students. Students are encouraged to think critically, collaborate with their peers, and utilize their creativity to solve complex problems. This interdisciplinary approach fosters a deeper engagement with the subject and ignites a passion for learning.

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

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

The purpose of this study is to examine the performances of university students' using dynamic mathematics software GeoGebra in argumentations and proving processes. A task related to the limit involving "sinx/x" was designed and 18 university students worked on the task during the collaborative learning, scientific debate, and self-reflection stages. Students' argumentations and proofs were analyzed based on Toulmin's model and Habermas' construct of rationality. The results revealed that the rationality components can be used to explain the implicit features of warrant in Toulmin's model. It was found that the semiotic mediation opportunities offered by GeoGebra and the teamwork and debate stages in the teaching method enabled the students to create geometrically the ranking relation between the value of an angle and the sine and tangent values of this angle. They also facilitated using the various kinds of representation and enhanced the transformations within and/or between the representations. GeoGebra tools as instruments of semiotic mediation helped the students to construct productive arguments, validate them, be productive by providing backing and rebuttal to the collective arguments put forward during the teamwork stage, and reach consensus thanks to scientific debate.

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

This theoretical article explores the affordances and challenges of Euler diagrams as tools for supporting undergraduate introduction-to-proof students to make sense of, and reason about, logical implications. To theoretically frame students' meaning making with Euler diagrams, we introduce the notion of logico-spatial linked structuring (or LSLS). We argue that students' use of Euler diagrams as representations of logical statements entails a conceptual linking between spatial and non-spatial representations, and the LSLS framework provides a tool for modeling this conceptual linking. Moreover, from our Piagetian epistemological perspective, reasoning with Euler diagrams entails engaging in spatial mental operations and making a logical conclusion from the result. We illustrate the utility of the LSLS framework through examples with two undergraduate students as they reasoned about the truth of the converse and contrapositive of a given logical implication, and we identify specific spatial operations that they used and coordinated in their problem solving.

This study was carried out in order to determine how the 3rd grade students of the Department of Elementary Mathematics Education structured their "if and only if propositions". The data were obtained by examining the students' answers given to the midterm exam questions and discussing the solutions with the students in the classroom. The study is a case study. As a result of the application, it was found out that the students had difficulty in determining the parts of the hypothesis that are included in "if and only if" proposition and therefore dividing the proposition into two "if" proposition. Some students think that the part or parts given as hypothesis should also be proved. When defining propositions, in addition to their "if and only if propositions", it is suggested to define new types of proposition in the form of "hypothesis-containing if and only if propositions".

Number line estimation has been found to be strongly related to mathematical reasoning concurrently and longitudinally. However, the relationship between number line estimation and mathematical reasoning might differ according to children's level of performance. This study investigates whether findings from previous studies that show number line estimation significantly predicting mathematical reasoning replicate, and whether this relationship holds across several points of the mathematical reasoning distribution. Participants include 324 Singaporean children (162 girls, M[subscript age] = 74.1 months, SD[subscript age] = 4.0) in their second year of kindergarten who were assessed on the number line estimation task (0-10 and 0-100) and mathematical reasoning skills. The results replicate previous findings showing that higher accuracy on the number line estimation task is predictive of higher mathematical reasoning, for both the 0-10 and 0-100 number line. Quantile regressions show that performance on the number line task similarly predict mathematical reasoning across the performance distribution. Possible differences between the 0-10 and 0-100 number line's predictive capacity are discussed.

This study investigated novice mathematics teachers participating in an online teacher education course focused on covariational reasoning and understanding the behavior of functions. The analysis centered on documenting the emergence of participants' sociomathematical norms for engaging in online asynchronous discussions. In this paper, we characterized participants' initial mathematical discourse and documented two emergent sociomathematical norms, namely "explaining why" and "emergent shape discourse." When participants "explained why," they used specific quantities or symbolic representations of functions to justify why function graphs have particular visual features. When participants engaged in "emergent shape discourse," they coordinated change between covarying quantities to justify why function graphs behave in certain ways. This study provides evidence that online settings can provide context for mathematics teachers engaging in legitimate collaborative mathematical activity and that activity can be enhanced by participation in discourse featuring specific sociomathematical norms. We discuss conjectures regarding the potential of reflective discussion activities paired with the Notice and Wonder Framework to support the emergence of generative sociomathematical norms. We also discuss potential relationships between characteristics of participants' mathematical discourse and their membership with the core and periphery of a social network.

This study aims to reveal the mathematical reasoning process of 8th-grade students and the reasoning indicators they use in this process. The study was carried out in line with the data obtained from the Mathematical Reasoning Test (MRT) administered to 292 8th-grade students. The mathematical reasoning performances of the participants were generally at low and medium levels. Evidence was found that students used reasoning indicators such as "… for/because of…, /therefore/so/thus…", "if … then…", "because…", "should be/should be..." and "equal..." when reasoning. It is important that when evaluating the solutions to questions that require mathematical reasoning, students must be interviewed and confirmed how they reached those solutions. The limitations of the research are, firstly, that it is limited to 8th graders, and secondly, that students will get bored if the number of questions in the MRT is high. Students must be asked to justify their reasoning while solving problems that require mathematical reasoning.

It has been over thirty years since the nuclear reactor meltdown at the Chernobyl nuclear power plant, but why there is still an officially designated exclusion zone? The Chernobyl Disaster Task combines the learning of exponential functions with properties of radioactive substances to help students understand the ongoing effects of the meltdown. Mathematical modeling is "a process that uses mathematics to represent, analyze, make predictions or otherwise provide insight into real-world phenomena" (Garfunkel and Montgomery, 2019, p. 8). Mathematical modeling tasks provide multiple points of entry for students, many solution pathways, and the opportunity for students to express various ways of thinking as they use mathematical reasoning to justify their models. Integrated mathematics and science modeling tasks, like the Chernobyl Disaster Task, offer a way to make difficult-to-study, real-world phenomena accessible for a rich and authentic learning experience (Hjalmarson et al., 2020). Following a description of the task's implementation with high school students, this article will offer insights to support teachers in using this task with their students.

Instructors manage several tensions as they engage students in defining, conjecturing, and proving, including building on students' contributions while maintaining the integrity of certain mathematical norms. This paper presents a case study of a teacher-researcher who was particularly skilled in balancing these tensions in a laboratory setting. We introduce sociomathematical scaffolding, which refers to the scaffolding of normative aspects for mathematical discourse. We found that the teacher-researcher's sociomathematical scaffolding entailed inquiring into the students' intended meaning of their draft and then supporting students in revising their draft to adhere to mathematical norms. We illustrate this pattern in three episodes in which the teacher-researcher supported a pair of students to revise their drafted (1) definition of unbounded above sequences, (2) conjecture of the Archimedean Property, and (3) proof by contraction of the Archimedean Property.

This paper addresses the recent body of research in algebra and algebraic thinking from 2018 to 2022. We reviewed 74 journal articles and identified four clusters of content areas: (a) literal symbols and symbolizing, (b) equivalence and the equal sign, (c) equations and systems, and (d) functions and graphing. We present the research on each of these content clusters, and we discuss insights on effective teaching practices and the social processes supporting algebraic reasoning. The research base shows that incorporating algebraic thinking into the elementary grades, emphasizing analytic and structural thinking processes, and emphasizing covariational reasoning supports students' meaningful learning of core algebraic ideas. We close with a discussion of the major theoretical contributions emerging from the past five years, offering suggestions for future research.

The calls for reasoning and proof to have a prominent place in secondary mathematics classrooms raise the question of knowledge that teachers need to support student learning of reasoning and proof. Several conceptualizations of Mathematical Knowledge for Teaching Proof (MKT-P) have been proposed over the years, yet some key questions about the nature of MKT-P remain unclear, such as whether MKT-P is a special kind of knowledge specific to teaching or whether it is just common mathematics knowledge. Another question is whether MKT-P can be improved through targeted intervention. Our study attempts to respond to both questions. An MKT-P questionnaire was administered to in-service secondary teachers, undergraduate mathematics and computer science (M&CS) majors, and pre-service secondary teachers before and after taking a capstone course Mathematical Reasoning and Proving for Secondary Teachers. The results suggest that MKT-P is indeed a specialized type of knowledge, which can be improved through targeted intervention.

In recent years, professional organizations in the United States have suggested undergraduate mathematics shift away from pure lecture format. Transitioning to a student-centered class is a complex instructional undertaking especially in the proof-based context. In this paper, we share lessons learned from a design-based research project centering instructional elements as objects of design. We focus on how three high leverage teaching practices (HLTP; established in the K-12 literature) can be adapted to the proof context to promote student engagement in authentic proof activity with attention to issues of access and equity of participation. In general, we found that HLTPs translated well to the proof setting, but required increased attention to navigating between formal and informal mathematics, developing precision around mathematical objects, supporting competencies beyond formal proof construction, and structuring group work. We position this paper as complementary to existing research on instructional innovation by focusing not on task trajectories, but on concrete teaching practices that can support successful adaption of student-centered approaches.

Undergraduate students are expected to produce and comprehend constructive existence proofs; yet, these proofs are notoriously difficult for students. This study investigates students' thinking about these proofs by asking students to validate two arguments for the existence of a mathematical object. The first argument featured a common structural error while the second was a valid argument of the claim. We found that the students often considered the logical structures of the arguments when validating them. They provided reasons for their evaluations, including why they thought the structure of the first argument functioned to prove the claim and why they thought the structure of the second argument did not function to prove the claim. We discuss how these reasons provide insights into why constructive existence proofs might be challenging for students. We end the paper with implications for the teaching and learning of constructive existence proofs and their proof frameworks.

Reasoning with mathematics plays an important role in university students' learning throughout their courses in the scientific disciplines, such as physics. In addition to understanding mathematical concepts and procedures, physics students often must mathematize physical constructs in terms of their associated mathematical structures and interpret mathematical entities in terms of the physical context. In this study, we investigate physics students' reasoning about mathematics in relation to physics content addressed in two quantum mechanics problems. Through qualitative analysis of interview data from twelve students, results show that 1) students use intricate, nonuniform problem-solving methods with reasoning that moves fluidly between structural (mathematizing and interpreting) and technical (conceptual and procedural) skills in quick succession, and 2) student reasoning about orthonormal bases, change of basis, inner products, and probability informed their flexibility in choosing problem-solving approaches. We illustrate the results with examples of student reasoning and discuss the inextricability of mathematics and physics in students' reasoning.

The study aimed to investigate the effect of the flipped classroom strategy based on ADDIE Model on Prince Faisal College students' development of algebraic thinking skills. To achieve the goal of the study, the researchers developed a measure of development of algebraic thinking skills. The study used the quasi-experimental approach with a pre-post design, and the study was applied to (53) students, who were divided into two groups: an experimental group of (27) students, and a control group of (26) students. The results showed there is an effect of the strategy on the dimensional measurement of the test of developing all algebraic thinking skill. Flipped Classroom helped develop algebraic thinking skills. The study recommended using the stages of ADDIE model in applying the flipped classroom strategy, and flipped classroom based on ADDIE model enhances student learning, because the strategy integrates student learning at home and in the classroom.

This study examined opportunities provided for preservice secondary mathematics teachers (PSMTs) to learn reasoning and proof in algebra from the perspective of college instructors. We analyzed interview transcripts of 15 course instructors recruited from three teacher education programs in the United States. We examined the reported opportunities provided for PSMTs to engage in proving-related activities, including making conjectures, investigating conjectures, developing arguments, evaluating arguments, and disproving by using counterexamples. We also analyzed instructional strategies reported by the instructors. We found the inconsistency between instructors' perceptions of the importance of reasoning and proof in algebra and instructor-reported opportunities to learn. Findings also indicated that developing arguments was reported the most frequently. In addition, instructors reported more pedagogy-focused general teaching strategies than proof-specific teaching strategies.

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.

Creativity is not only for gifted students, but also for regular ones. This case study was aimed to analyze the appropriateness of tasks and the elaboration of multiple solutions to occasion fourth-graders' mathematical creative thinking through a documentary multiple-solution counting task in a figurative setting. The data came from the written report of 48 fourth graders in two classes in Taiwan, China. The appropriateness of creativity was reflected in the appropriateness of tasks and solutions, particularly suitable for complex problem solving. Elaboration was detail-dependent, and visualization was beneficial to the analysis of elaboration. The regular students who had just entered the fourth grade could show their creative thinking through different angles (horizontally or vertically) and starting points (holistic or partial), but with slightly more partial and horizontal than holistic and longitudinal, more adaptation than transformation. These fourth-grade students have had the basic mathematical creative thinking capability of adaptation, combination, change, rearrangement, extension or going back by using counting, combining, adding and reducing, overlapping, moving, and diagonal division strategies.

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.

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

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 study aims to examine the effects of a course module aimed at improving pre-service mathematics teachers' (PMTs) proof schemes. This study used designed-based research. Participants of the course were 22 PMTs in a teacher preparation program. We obtained data from two surveys, one of which consists of proof questions. The other requires identifying the proof schemes of ten 9th grade students in a scenario including an excerpt of a hypothetical discussion with their teacher. We implemented both surveys before and after the intervention. In addition, semi-structured interviews were conducted with four PMTs. Findings indicated that the module had a significant and large effect on PMTs' proof schemes in favor of post-intervention. Most of the PMTs transformed their external and empirical proof schemes to the analytical ones. Findings also indicated that PMTs had difficulties identifying students' symbolic, example-based, transformational, and axiomatic proof schemes but overcame these difficulties after the intervention.

The study of loops and spaces in mathematics has been the subject of much interest among researchers. In Part 1 of "The Theory on Loops and Spaces," published in the "Mathematics Teaching Research Journal," introduced the concept and the basic underlying idea of this theory. This article continues the exploration of this topic and aims to advance the understanding of the theory through observation and analysis of patterns. A systematic examination of intersection points and their numerical Sum is conducted, and the effect of the order of numbering on the analysis is analyzed. Furthermore, the physical implications of the theory are discussed, and the validity of the theory in the third dimension is confirmed through analysis. This article provides a solid foundation for the understanding of the elementary principles of Graph Theory and paves the way for the development of more advanced theorems in the field. Additionally, the article demonstrates how patterns in nature can be analyzed and expressed mathematically, offering a unique perspective on the interplay between mathematics and the natural world. The work is inspired by the video posted by mathematician Dr. James Tanton on his YouTube channel on September 27th, 2021. [For "The Theory on Loops and Spaces. Part 1," see EJ1350656.]

With the objective of identifying intrinsic forms of mathematical production in complex analysis (CA), this study presents an analysis of the mathematical activity of five original works that contributed to the development of Cauchy's integral theorem. The analysis of the mathematical activity was carried out through the identification of the types of expressions used and the way they were used by the historical subjects when communicating their results, to subsequently identify transversal elements of knowledge production. The analysis was refined by the notion of confrontation, which depicts the development of mathematical knowledge through the idea of building knowledge against previous knowledge. As a result of the study we established epistemological hypothesis, which are conceived as conjectures that reveal ways in which mathematical knowledge was generated in CA.

This paper presents the results of an exploratory case study that examined a veteran secondary teacher's knowledge for teaching non-routine mathematical problem-solving with digital technologies. Data was collected through the observation of a veteran mathematics teacher, in real time, solving a mathematical problem with the digital tools of his choice. The descriptive model Mathematical Problem-Solving with Technology (MPST) was used to analyse the teacher's utterances and actions while solving a mathematical problem, with GeoGebra and a spreadsheet, and expressing his reasoning also with technology. Our findings reveal the complexity of expert problem-solving with technology, through regulation processes and several micro-cycles that mainly involve the processes integrate and explore. Mathematical problem solving with technology entails relevant mathematical knowledge as well as knowledge about the mathematical affordances of the digital tools available, and the ability to combine them to develop a conceptual model of the solution to the problem. Thus, the teacher's techno-mathematical fluency seems crucial to successfully solve the problem and express the reasoning with technology. Based on the findings, we discuss the teachers' knowledge for teaching mathematical problem-solving with technology as including a particular kind of proficiency, techno--mathematical fluency for solving-and-expressing problems with technology. The limitations of the study, further research topics and implications for professional development and teacher education programmes are discussed.

There is strong evidence from research conducted in the United States that fraction magnitude understanding supports mathematics achievement. Unfortunately, there has been little research that examines if this relation is present across educational contexts with different approaches to teaching fractions. The current study compared fourth and sixth grade students from two countries which differ in their approach to teaching fractions: Australia and the United States. We gathered data on fraction and decimal magnitude understanding, proportional reasoning, and a standardized mathematics achievement test on whole number computation. Across both countries, reasoning about rational magnitude (either fraction or decimal) was predictive of whole number computation, supporting the central role of rational number learning. However, the precise relation varied, indicating that cross-national differences in rational number instruction can influence the nature of the relation between understanding fraction and decimal magnitude and mathematics achievement. The relation between proportional reasoning and whole number computation was fully mediated by rational magnitude understanding, suggesting that a key mechanism for how reasoning about rational magnitude supports mathematics achievement: proportional reasoning supports the development of an accurate spatial representation of magnitude that can be flexibly and proportionally scaled, which in turn supports children's mathematics learning. Together, these findings support using measurement models and spatial scaling strategies when teaching fractions and decimals.

A critical part of supporting the development of students' algorithmic thinking is understanding the challenges that emerge when students engage with algorithmatizing tasks--tasks that require the creation of an algorithm. Knowledge of these challenges can serve as a basis upon which educators can build effective strategies for enhancing students' algorithmic thinking skills. This paper presents three illustrative cases of emergent challenges evident as students grapple with the process of creating an algorithm. The first challenge highlights discrepancies between the method with which students solve a problem and the algorithm they create, and claim would, when implemented, solve the same problem. The second challenge pertains to the persistence of students' normatively incorrect algorithms, despite going through multiple iterations of testing and revising. Finally, the third challenge concerns issues around the use of test problems for supporting students in their creation of generalized algorithms. These three challenges are discussed using student data (illustrative cases) from three different mathematical algorithmatizing tasks. Suggestions are put forth for addressing some of these challenges, with a particular emphasis on practical pedagogical suggestions for cultivating students' mathematical thinking in the context of algorithmatizing tasks.

In this study, it is aimed to reveal what kind of relationship exists between prospective teachers' thinking structures and their level of writing proofs. This study was carried out as a case study. The study was carried out with 82 prospective teachers enrolled in the Department of Secondary School Mathematics Teaching of a state university located in the Eastern Black Sea Region of Turkey. The first data collection is the mathematical processing instrument developed by Presmeg (1986). The second data collection was developed by the researcher and consists of five proofs that are also covered in the curriculum about the geometric shapes circle and circular region. As a result, when the thinking structures of the participants are examined, it can be understood that harmonic thinking is the most common thinking type among the study participants. When the proof-writing levels in this study are looked at, it becomes obvious that most of the prospective teachers are weak in terms of this skill. In this study, the participants' proof-writing levels and thinking structures were found to be correlated.

In this study, we aimed to find problematic and supportive issues in Afghan undergraduate students' proofs of the irrationality of [square root]3 and [square root]5/8 while using the indirect proof method. Collecting and analyzing produced proofs of 30 sophomore and 48 senior undergraduate students on the irrationality of [square root]3 and [square root]5/8, respectively, revealed that the majority were not able to extend indirect proof beyond showing the irrationality of [square root]2 due to not being able to apply the supportive 'multiples of an integer' notion in their reasoning process of the irrationality of [square root]3, and this notion was not supportive in the case of showing irrationality of [square root]5/8, and finally we propose the method of finding nonzero integral solutions to the resulted Diophantine equation and using the rational roots theorem to prove that [square root]p/q is irrational.

This study was intended to examine the progress of the geometric reasoning skills of pre-service mathematics teachers (PSMTs) in the College of Teacher Education, Oromiya, Ethiopia. In the study, there were three different classes: EG1 (n=48), EG2 (n=38), and CG (n=30), where PSMTs in EG1 were treated with technology into guided inquiry-based learning (TGIBL), EG2 with guided inquiry-based learning (GIBL), and CG with a conventional learning approach. A geometric reasoning test was used to collect data. The data were analyzed using a one-way ANOVA and a paired sample t-test. The findings showed that the effect of using the GIBL model assisted by technology was higher when compared to the other two models. On the other hand, a paired t-test showed significant mean gains in reasoning skills in the TGIBL and GIBL approaches. Hence, technology must be integrated into teaching and learning geometry at CTEs to promote PSMTs' geometric reasoning skills.

This study aimed to determine the mathematical thinking levels of gifted and non-gifted students and to compare them according to some variables. In this study, screening model was used from quantitative research methods. The study group of the study consists of 194 gifted students and 168 non-gifted students studying in grades 5-8 in a province in the southern region of Turkey in the 2022 to 2023 academic year. According to the results of the research, it was found that the mathematical thinking levels of gifted students were higher than their non-gifted peers. In addition, the mathematical thinking levels of gifted students showed a significant difference compared to grade level. This does not apply to non-gifted students. It was found that the mathematical thinking levels of female students were higher than the mathematical thinking levels of male students in gifted and non-gifted student groups, but this difference was not statistically significant. In addition, it was seen that the success grade of the mathematics course made a significant difference on the mathematical thinking level of both the gifted and non-gifted student groups.

Technology is recognized for its potential to implement exploration tasks. The ease and speed with which it becomes possible to observe many cases of a situation, allows the development of conjectures and brings conviction about their veracity. Mathematical proof, assumed as the essence of Mathematics, tends to appear to the students as something dispensable. Based on KTMT -- "Knowledge for Teaching Mathematics with Technology" model, this study intends to understand the impact of the teachers' knowledge on mathematical proof in a context of technology integration. The study adopts a qualitative and interpretative methodology, based on case study, analyzing the practice of one teacher. The conclusions emphasize the relevance of the teacher's MTK -- "Mathematics and Technology Knowledge," and TLTK -- "Teaching and Learning and Technology Knowledge." The teacher's MTK guides her decisions, leading her to focus on helping students understand the meaning of conjecture and proof, valuing, at the same time, the relevance of algebraic manipulations. However, the teacher's TLTK guides her practice, where the knowledge about the students is determinant. The study provides evidence about the difficulty of articulating proof and technology, but it also clarifies the relevance of this articulation and of how the teacher's KTMT can impact the teacher's decisions.