Mathematical Situations Research Articles

Recognizing, supporting, and improving student perseverance in mathematical problem-solving: The role of conceptual thinking scaffolds

Perseverance, or initiating and sustaining productive struggle in the face of obstacles, promotes making sense of mathematics. Yet, engaging in struggle can be grueling and is avoided by some students, and little is known about if and how student perseverance can improve over time. The purpose of this study was to investigate the effect of scaffolding mathematics tasks on student perseverance over the course of six weeks. The results showed that prompting secondary students to conceptualize a mathematical situation prior to problem-solving can encourage re-initiating and re-sustaining mathematically productive effort upon reaching an impasse. Further, encouraging such conceptualization work often improved students’ selection of problem-solving strategies and affect regulation over time, suggesting the malleability of perseverance. For learning mathematics with understanding, these findings suggest supportive learning environments that provide consistent opportunities for students to practice (and improve) their perseverance in problem-solving.

Partial Fuzzy Quantifiers and their Computation

Background: In computer science, one often needs to deal with undefined values. For example, they naturally increase when a mistake such as the square root of a negative number or division by zero occurs. A similar problem occurs in the logical analysis of natural language. For example, the expression “Czech president in the 18th century” has no denotation because there was no Czech president before 1918. Such a situation in mathematics is characterized by partial functions, i.e., functions that may be undefined for specific arguments. Method: In this paper, we will extend the theory of intermediate quantifiers (i.e., expressions such as “most, almost all, many, a few,” etc.) to deal with partially defined fuzzy sets. First, we will extend algebraic operations that are used in fuzzy logic by the additional value “undefined.” Then we will introduce intermediate quantifiers using the former. The theory of intermediate quantifiers has been developed as a special theory of higher-order fuzzy logic. Results: In this paper, we introduce the quantifiers semantically and show how they can be computed. The latter is also demonstrated in three illustrative examples. Conclusion: The paper contributes to the development of fuzzy quantifier theory and its extension by undefined values and suggests methods for computation of truth values.

Analisis Kemampuan Komunikasi Matematis Siswa SMP Kelas VIII pada Materi Relasi dan Fungsi

This research aims to analyze and describe the mathematical communication skills of junior high school students in grade VIII on relationship and function materials in one of the schools in Karawang Regency. The type of research used is qualitative descriptive research with a total of 3 subjects. The taking of the subject is done using purposive sampling techniques. The instrument used in this study is a written test with the number of essay problems as many as 6 questions that contain 3 indicators of mathematical communication skills, namely: 1.) Linking images, tables, graphs into mathematical ideas. 2.) Stating everyday events into a language or mathematical symbol 3.) Provide explanations, ideas, concepts, or mathematical situations with their own language in written form. Instrument results are categorized into 3 categories, namely low, medium, and high. The results of the analysis of mathematical communication skills as a whole are relatively low with a percentage of 32.22%. There is one indicator that includes the medium criterion and two indicators that include the low criterion. Students who have high mathematical communication skills can master all three indicators of mathematical communication skills. Students who have medium mathematical communication skills can only master two indicators of mathematical communication skills. While students who have low mathematical communication skills cannot meet all three indicators of mathematical communication skills.

Theory of Didactical Situations in Mathematics as one of the theoretical approaches to the LPS analysis of classroom data: the case of institutionalization

Theory of Didactical Situations in Mathematics as one of the theoretical approaches to the LPS analysis of classroom data: the case of institutionalization

Measuring Pre-service Primary Teachers’ Shame in Mathematics—a Comprehensive Validation Study

Emotions play an essential role in educational processes. Previous research has mainly dealt with achievement emotions which are experienced in specific situations such as exams or learning situations in mathematics (e.g. enjoyment or anxiety). Some achievement emotions are rather experienced in social contexts in mathematics and are closely related to the self. These emotions such as shame are assumed to be also relevant for mathematics achievement. However, a reliable and valid instrument is missing to measure shame in mathematics. Validity evidence for the newly developed Shame in Mathematics Questionnaire (SHAME-Q) was collected in three studies with pre-service primary teachers. Study 1 investigated the content validity by conducting a systematic expert panel study. Study 2 and study 3 examined with two different samples the factorial structure and relations to other constructs in terms of discriminant (enjoyment) and convergent (anxiety) validity as well as to pre-service teachers’ grade in school mathematics, their intention to teach mathematics at school, and gender. The data supported strongly the validity assumptions as well as reliability and parsimony of the instrument. Psychometric limitations of SHAME-Q and applicability of the questionnaire are discussed.

Let’s tie circumferences up, so we count them all. An unconventional combinatorics problem in compulsory secondary school

Secondary school students (age 13-14) solve a combinatorial task in the Mathematical Olympiad, whose answers are analyzed and discussed using theoretical tools from two didactical frameworks: on one hand, the Onto-semiotic Approach (OSA), and, on the one hand, the Theory of Didactic Situations in Mathematics (TDSM). The statistical study is carried out by tools from Statistical Implicative Analysis (SIA). Results express that most participants possess sufficient arithmetic strategies to solve the task, without turning to combinatorial algebra. At the same time, the algebraization level shown by these same participants in their answers to other tasks of the Olympiad, is strongly correlated to their behaviors in the combinatorial task, and so, a student who masters an algebraic technique is also aware of the limitations of its field of application. Therefore, adaptability is a key element in the analysis of the observed strategies and their success rate.

Understanding Prospective Teachers’ Task Design Considerations through the Lens of the Theory of Didactical Situations

Task design, in general, and task design in a technological environment, is attracting the attention of educational researchers. The present research investigates task design of prospective teachers in the Scratch programming environment. A total of twenty-three female prospective teachers participated in a professional development program. They were in their third academic year majoring in teaching mathematics and computer science in the middle school. The prospective teachers attempted to design mathematics-based programming problems. The present research utilizes the theory of didactical situations in mathematics, specifically the situation types, the paradoxes of the didactical contract and the situation components, to study the task design of the prospective teachers. It does that by focusing on one group of prospective teachers. The research results indicated that the prospective teachers were concerned mainly with the situation of information, situation of reference and situation of action. Doing so, they were concerned with the paradox of the said and the unsaid, the paradox of uncertainty, and the paradox of devolution. In addition, they took care of both algorithmic and creative reasoning. They also took care of students’ devolution, where this devolution was conditioned with following an institutionalization. They were also concerned with giving students autonomy and encouraging decision making regarding the solution of the problem. Furthermore, they planned to enable students’ control over their learning.

High School Students’ Generalization Viewed from Logical-Mathematical Intelligence

Generalization is an important element in understanding, recognizing, and examining mathematical situations. Students' generalization processes can be analyzed according to Mason's four stages of the generalization process, including the perception, expression, symbolic expression, and manipulation of generality. One of the things that may cause differences in students' generalization processes is logical-mathematical intelligence. This research aims to describe high school students' generalization processes in terms of logical-mathematical intelligence. This research is using a descriptive qualitative method. The subjects of this study consisted of 3 high school students in Surabaya with high, moderate, and low logical-mathematical intelligence. The data is obtained from the Logical-Mathematical Intelligence Questionnaire and Generalization Task. The material used in the Generalization Task is curved-face solid. The results of this study indicate that the student with low logical-mathematical intelligence has reached the stage of identifying patterns or regularity in the given information, but the student doesn't use those patterns to determine the next term. In other words, the student with low logical-mathematical intelligence has only reached the stage of perception of generality. The student with moderate logical-mathematical intelligence has reached the stage of identifying patterns or regularity in the given information and using those patterns to determine the next term but the student doesn’t reach the stage of formulating the general formula of the pattern that has been found. In other words, the student with moderate logical-mathematical intelligence has only reached the stage of perception and expression of generality. The student with high logical-mathematical intelligence has reached the stage of identifying patterns or regularity and used those patterns to determine the next term. In addition, the student with high logical-mathematical intelligence also reaches the stage of formulating general formulas of patterns that have been found and using the general formula to solve the given problem In other words, the student with high logical-mathematical intelligence has reached all the stages which is perception, expression, symbolic expression, and manipulation of generality.

The search for the sense of agency in vulnerable situations in mathematics teachers’ initial education

The aim of this study is to analyse the actions of prospective mathematics teachers (PMTs), triggered in preparation for their teaching practicum (TP), to foster their autonomy when seeking their sense of agency in the face of a vulnerability, in the movement to construct their professional identity. To this end, we have carried out a qualitative research about the practicum lesson plans and the simulations of those lessons by PMTs who attended the same class in two disciplines of the Supervised Curricular Teaching Practice, in the last two years of the mathematics teaching degree course of the State University of Londrina (UEL). The identified vulnerability experiences enabled the search for the PMTs’ sense of agency regarding the following actions: defining concepts with which the students had already had contact and problematising them to systematise a mathematical idea; resorting to previous experiences worked with potential basic education students; having support from colleagues for possible actions to be developed in the classroom; problematising the answer to a problem; reflecting on and answering questions; evaluating a classroom situation and concluding that it is necessary to give in so that the other also plays a leading role; assuming a coherent position to what they believe about education; and questioning students. These actions express the reactions of PMTs when exercising their autonomy, in the search for the sense of agency, mediated by other PMTs and teacher educators involved in the formative process. The experiences of vulnerability and the search for the sense of agency articulate altogether singular, individual, and collective aspects that expand the movement of the construction of the teacher’s professional identity.

Academician Jonas Kubilius: works dedicated to the history of Lithuanian mathematics

The rise of the Lithuanian mathematical school in the second half of the 20th century is associated with the development of probability theory and its application, and the foundations of that school were insightfully laid by the famous Lithuanian mathematician Jonas Kubilius. However, the academician also had a second vocation – the history of mathematics. At the end of the 20th century, he purposefully researched the mathematical legacy of the poet, bishop A. Baranauskas, recognizing him as the first Lithuanian mathematician researcher of the second half of the 19th century.
 At the beginning of the 21st century, J. Kubilius undertook a detailed implementation of the idea of a work in the history of Lithuanian mathematics. For this purpose, an informal group of specialists was convened, the content of the work was planned, and the research-based book series ``From the History of Lithuanian Mathematics'' was published. The fourth book in this series, Mathematics in Lithuanian Higher Education Institutions in 1921–1944, presents the research of an academic who reveals the situation of mathematics in universities in Kaunas and Vilnius.
 In addition, the memoirs of mathematics history by J. Kubilius, dedicated to mathematicians Z. Žemaitis, G. Žilinskas and V. Statulevičius, should be mentioned.
 The article, at the end of which fragments of the author's memories are presented, is dedicated to the centenary of the birth of Academician J. Kubilius.

Students' mathematical reasoning ability in solving post-Covid-19 PISA model math problems

This study aims to describe the reasoning abilities of high school students in Pandeglang Regency, Indonesia in solving post-Covid-19 PISA model math problems. The subjects of this study were 80 students of two state senior high schools in Pandeglang Regency, Indonesia. The data collection used a written test consisting of four description questions in the form of PISA model math questions (levels 2, 3, 4, and 5), and were through interviews and documentation. The data analysis followed the steps of data reduction, data presentation, and conclusion drawing. The results show that the reasoning abilities of Pandeglang Regency high school students in post-Covid-19 are fair in following the rules of inference, checking the validity of arguments, proving, drawing logical conclusions, using patterns, analyzing mathematical situations, and generalizing. However, the reasoning abilities were very poor in providing explanations of models, facts, characteristics, and relationships; constructing valid arguments; and studying conjectures. Thus, it can be concluded that the mathematical reasoning ability of Pandeglang Regency senior high school students is poor when measured using the PISA Post-Covid-19 model of mathematics questions.

Encountering Spurious Elements in Electrical Impedance Spectroscopy Data Fitting

The aim is to present situations in which collected impedance spectrum analysis data contain information that reflects not only the studied sample, but pertains also to the fundamental instrument properties or to the method in which the data is fitted or evaluated. Fitting of such data can lead to spurious, phantom elements in equivalent circuits and to frustration in data fitting. While illustrative experiments were done on specific instruments, the conclusions are generic and apply to any instrument with current supplying and potential probing inputs. The described spurious response stems from the existing unavoidable impedance of the reference inputs and from additional resistance, such as reference electrodes, connected to the inputs. Additionally is discussed a purely mathematical situation, when response artifacts are generated from correct impedance spectrum data, but incorrectly applied algebraic treatment.

A model design to be used in teaching problem posing to develop problem-posing skills

A model was designed herein to be used as a common approach in learning environments to teach problem-posing and improve problem-posing skills, which was named the Problem Posing Learning Model (PPLM). It was attempted to design a framework for problem-posing in line with this objective. The PPLM consists of 6 steps, comprising understanding the desired situation to pose a problem for, designing the story, forming the problem statement, solving the problem formed, assessment, and finalizing the problem formed. Understanding the desired situation to pose a problem for includes understanding what the desired situation is about and what can be achieved from it. The designing the story step includes determining or designing a realistic daily-life situation related to the desired mathematical situation to pose a problem for. The forming the problem statement step comprises forming the verbal sentences related to the situation to pose a problem for. The solving the problem formed step comprises solving the problem statement that was formed. The assessment step is the assessment of the problem formed. The finalizing the problem formed step includes finishing the action of problem-posing after checking the problem formed in the assessment and making any necessary modifications. These steps are hierarchical and it is recommended that each step should be implemented in the problem-posing process. The applicability and effectiveness of the PPLM designed to teach problem-posing in learning environments can be examined.

Programming as a language for young children to express and explore mathematics in school

AbstractNatural language helps express mathematical thinking and contexts. Conventional mathematical notation (CMN) best suits expressions and equations. Each is essential; each also has limitations, especially for learners. Our research studies how programming can be a advantageous third language that can also help restore mathematical connections that are hidden by topic‐centred curricula. Restoring opportunities for surprise and delight reclaims mathematics' creative nature. Studies of children's use of language in mathematics and their programming behaviours guide our iterative design/redesign of mathematical microworlds in which students, ages 7–11, use programming in their regular school lessons as a language for learning mathematics. Though driven by mathematics, not coding, the microworlds develop the programming over time so that it continues to support children's developing mathematical ideas. This paper briefly describes microworlds EDC has tested with well over 400 7‐to‐8‐year‐olds in school, and others tested (or about to be tested) with over 200 8‐to‐11‐year‐olds. Our challenge was to satisfy schools' topical orientation and fit easily within regular classroom study but use and foreshadow other mathematical learning to remove the siloes. The design/redesign research and evaluation is exploratory, without formal methodology. We are also more formally studying effects on children's learning. That ongoing study is not reported here. Practitioner notesWhat is already known Active learning—doing—supports learning. Collaborative learning—doing together—supports learning. Classroom discourse—focused, relevant discussion, not just listening—supports learning. Clear articulation of one's thinking, even just to oneself, helps develop that thinking. What this paper adds The common languages we use for classroom mathematics—natural language for conveying the meaning and context of mathematical situations and for explaining our reasoning; and the formal (written) language of conventional mathematical notation, the symbols we use in mathematical expressions and equations—are both essential but each presents hurdles that necessitate the other. Yet, even together, they are insufficient especially for young learners. Programming, appropriately designed and used, can be the third language that both reduces barriers and provides the missing expressive and creative capabilities children need. Appropriate design for use in regular mathematics classrooms requires making key mathematical content obvious, strong and the ‘driver’ of the activities, and requires reducing tech ‘overhead’ to near zero. Continued usefulness across the grades requires developing children's sophistication and knowledge with the language; the powerful ways that children rapidly acquire facility with (natural) language provides guidance for ways they can learn a formal language as well. Implications for policy and/or practice Mathematics teaching can take advantage of the ways children learn through experimentation and attention to the results, and of the ways children use their language brain even for mathematics. In particular, programming—in microworlds driven by the mathematical content, designed to minimise distraction and overhead, open to exploration and discovery en route to focused aims, and in which children self‐evaluate—can allow clear articulation of thought, experimentation with immediate feedback. As it aids the mathematics, it also builds computational thinking and satisfies schools' increasing concerns to broaden access to ideas of computer science.

Situation analysis of mathematical physics learning with online learning during the COVID-19 pandemic

This research was conduct to determine the situation of learning mathematics physics online during the COVID-19 pandemic. Participants in this study were physics semester 4th students of Universitas Muhammadiyah Makassar with 21 students. Data collection techniques include online interviews and questionnaires. The collected data analyzed by reduction, display, and conclusion drawing. The results of the study describe the learning process of mathematical physics not conducive during the COVID-19 pandemic outbreak. The problem that arises in the form of difficulty lectures in teaching explicit physics material that has many mathematical formulas and calculations. The difficulty on the part of students is that students experience limited costs to buy internet quota in online learning. The lecturer provides the solution by creating a mathematical physics module that is given in stages. This module is presented with language and examples of questions that are easily understood by students through the google classroom application. Lectures also occasionally use the zoom application, and Google meets to explain the material that is difficult to understand.

Mathematical Literacy Skills Students of the Junior High School in Solving PISA-Like Mathematical Problems

This study aims to describe students of the junior high school mathematical literacy skills in solving PISA-like problems on number content. PISA-like items are problems created using dimensions in PISA. The method used in this study was descriptive qualitative. The research subject were students of the ninth-grade students in one of the integrated Islamic junior high school in Surakarta, which consisted of six students. The instruments used were tests, interviews and researchers. The results showed that the students’ mathematical literacy skills were classified as good for level 1 to 3 and low for level 4 and 5, and very low for level 6. This categorization was based on the number of students who answered the questions correctly at each level. Students in levels 1 to 3 were able to formulate mathematical situations and translate real-world problems into mathematical representations and then are able to choose the right solution strategy. However, when students faced level 4, 5, and 6 level questions, with a higher level of reasoning and complex problem situations, many students still faced difficulties. It has an impact on the students’ ability to determine strategies, make many mistakes, and not be able to answer the questions correctly.

Mathematical communication ability students based on learning motivation in the COVID-19 pandemic

The research aims to describe Tenth-grade students’ mathematics communication competence of vocational high school of Bina Teknologi, in Purwokerto Indonesia. The research used a qualitative method with Miles and Huberman model consisting of data reduction, data display, and taking a conclusion. The research subjects were Tenth-grade of Network and Computer Engineering class students obtained through purposive sampling. The students were grouped into three categories based on their learning motivation, namely students with high learning motivation, students with medium learning motivation, and students with low learning motivation. Two respondents, from each category, were chosen to be interviewed. The methods of data collection were questionnaires, tests, interviews, and documentation. The used validation test was technique triangulation. The research result shows that Subjects with high learning motivation were able to use language and mathematical symbols in daily events, were able to interpret mathematical ideas based on the graphs presented, were able to provide easy-to-understand arguments, and were able to draw graphs based on mathematical situations or problems. Subjects with learning motivation are able to use language and mathematical symbols in everyday events, are able to provide easy-to-understand arguments, and are able to draw graphs based on situations or math problems. Subjects with low learning motivation are able to use language and mathematical symbols in everyday events and are able to provide easy-to-understand arguments.

MATHEMATICS COMMUNICATION MISTAKES IN SOLVING HOTS PROBLEMS

Student mistakes in communicating mathematical ideas are still widely practiced. Therefore, it is essential to analyze students' mathematical communication errors in solving mathematical problems so that learning planning can be better. This study aims to describe students' mathematics communication errors in solving the higher-order thinking skill's problems in linear algebra and matrix subject. The type of research is a qualitative descriptive study. They were 155 students as subject research. The data analysis started by collecting students' answers and then grouped them according to mathematics communication skills criteria. Later identified and analyzed the errors made by students of each mathematics communication criteria. The results showed that mathematical communication errors on the indicators of writing mathematical situations were concept errors and principle errors. The declaring idea's mathematical communication error is a concept error, a principle error, and an operation error. Furthermore, mathematical communication errors on the indicator state the results of solving-problem using the language itself is a concept error and operator error.

Revisiting Students' Reflection in Mathematics Learning: Defining, Facilitating, Analyzing, and Future Directions

Students rarely reflect on mathematics. Reflection requires higher order cognition that has proven difficult to teach. Much of the literature in mathematics learning has highlighted the significance of reflection without, however, offering much enlightenment on the current research status of students' reflection within mathematical situations. In this study, a wide range of academic literature associated with students' reflection within mathematical situations is reviewed. Three important aspects then emerge from the literature review: how to operationally define, individually facilitate, and analyze students' reflection within mathematical situations. Based on the three emerging important aspects, it would be worthwhile for future studies to develop an operational definition of reflection in mathematics learning, to research how students' reflection relates to their cognition and metacognition within mathematical situations, to invent individual reflective strategies in mathematics learning and to investigate the effect of such strategies on their mathematical performance, to devise techniques to analyze students' reflective behaviors within mathematical situations, and to explore the extent to which students' reflection could be promoted in mathematics learning. It is hoped that this article will develop ideas for future practice and research on students' reflection within mathematical situations about students' reflection and the role it plays in mathematics learning.

ЗМІСТ І СТРУКТУРА ПСИХОЛОГІЧНОЇ ГОТОВНОСТІ СТАРШОКЛАСНИКІВ ДО РОЗВ'ЯЗУВАННЯ ТВОРЧИХ МАТЕМАТИЧНИХ ЗАДАЧ

Informational technologies, the latest challenges in the field of technical development of society condition irreversible environmental and human changes. In the context of these changes a special place is occupied by those activities that are directly or indirectly related to technical, cybernetic, mathematical activities. This actualizes the manifestation of mathematical abilities, giftedness for math activity in youth, features of creative potential, etc. The purpose of the study is to clarify the content and structure of psychological readiness of high school students for creative mathematical performance. The main content characteristics of the creative mathematical problem as a model for the study of creative mathematical activities are considered in the article. Based on the theory of learning problems, the psychological structure of mathematical task is determined. The structure contains the following components: a) the subject of the task (in its initial state or the ascending subject of the task); b) the model of desired state, ie the requirements (set and required) of the task. The contradiction of the creative mathematical situation first of all characterized by the object and subject of the creative mathematical task. The psychological readiness of high school students for creative mathematical performance is an integrated personality trait aimed at realizing of the cognitive abilities of young people in the process of solving creative mathematical problems. Since the process of creative mathematical thinking (creative mathematical activity) is determined by objective (subject-functional) and subjective-personal (motivation, emotional-volitional, value attitude) components, it is worth saying about subject-functional, subjective-personal types of psychological readiness of high school students to perform mathematical creative activities.