Student Mathematical Thinking Research Articles

Connections within the Multiplicative Field: A Case Study of Adán’s Mathematical Thinking

ABSTRACT Mathematical coherence is a goal within the Common Core State Standards for Mathematics. One aspect of this coherence is how student mathematical thinking is developed across concepts. Unfortunately, mathematics is often taught as isolated ideas across grades. The multiplicative field is an area of study that needs to be examined as a space to develop connections across concepts. This paper uses a case study of mathematical thinking from one 5th-grade student over a school year to demonstrate the connections in his thinking across whole numbers, fractions, and graphing work. Analyzing his solution strategies with three frameworks detailing multiplicative, proportional, and functional thinking, his strategies show a developing understanding of proportional reasoning, and functional thinking as he is building his multiplicative thinking and connections from whole numbers to fraction problem-solving strategies to graphing. Examining this student’s mathematical thinking points to the opportunity for connections across the multiplicative field through teaching mathematics in a manner more responsive to student sense-making.

Animations and depictions as a tool to improve pre-service elementary teachers noticing of students mathematical thinking

Animations and depictions as a tool to improve pre-service elementary teachers noticing of students mathematical thinking

A framework to support teacher noticing of students' mathematical thinking in technology‐mediated environments

AbstractThe practice of teacher noticing students' mathematical thinking often includes three interrelated components: attending to students' strategies, interpreting students' understandings, and deciding how to respond on the basis of students' understanding. This practice gains complexity in technology‐mediated environments (i.e., using technology‐enhanced math tasks) because it requires attending to and interpreting students' engagement with technology. Current frameworks implicitly assume the practice includes noticing the ways students use tools (including technology tools) in their work, but do not explicitly highlight the role of the tool. While research has shown that using these frameworks supports preservice secondary mathematics teachers (PSTs) developing noticing practices, it has also shown that PSTs largely overlook students' technology engagement when they are working on technology‐enhanced tasks (Journal for Research in Mathematics Education, 2010; 41(2):169–202). In this article, we describe our adaptation of Jacobs et al.'s framework for teacher noticing student mathematical thinking to include a focus on making students' technology‐tool engagement explicit when noticing in technology‐mediated environments, the Noticing in Technology‐Mediated Environments (NITE) framework. We describe the theoretical foundations of the framework, provide a video case example, and then illustrate how the framework can be used by mathematics teacher educators to support PSTs' noticing when students are working in technology‐mediated environments.

Preservice teachers equitably attending to and interpreting multilingual students' mathematical thinking

AbstractResearch indicates that teachers may hold biases against multilingual learners (MLs). However, there is limited research regarding how teachers notice MLs' mathematical thinking. Given the complexity of teacher noticing and multilinguality, the lead researchers developed the Equitably Attending to and Interpreting a Student's Mathematical Thinking (EAST) framework to support preservice teachers' development of equitable noticing skills. Using a pre‐ and post‐intervention study design and EAST based lessons, we explored differences in preservice teachers' equitable noticing of MLs' mathematical thinking. We collected data on a noticing assessment from two groups of preservice teachers, 21 of whom received the EAST lessons, and 39 of whom received lessons on eliciting and interpreting student thinking without a focus on equity. Preservice teachers in both groups noticed components of the EAST framework, but more often noticed issues of access and achievement, and rarely attended to issues of identity or power. Findings from the study provide insight into what equitable noticing of a MLs' mathematical thinking entails, as well as areas for further development in both teaching and research regarding equitable noticing.

To what extent can teachers and preservice teachers predict students mathematical thinking A qualitative analysis

To what extent can teachers and preservice teachers predict students mathematical thinking A qualitative analysis

Analisis Kemampuan Berpikir Kreatif Matematis Siswa SMPTQ Al-Matin Dalam Menyelesaikan Soal Bangun Ruang Sisi Datar

This research aims to analyze the mathematical creative thinking abilities of 8th-grade junior high school students in solving problems related to two-dimensional objects. This study is a case study research with a qualitative approach, and the research design utilizes qualitative descriptive methods. The subject selection for this research utilized probability sampling, resulting in one student out of 34 students selected as the research participant, referred to as participant R1. The research instruments include a test of students mathematical creative thinking abilities and data collection method such as observation, test, interviews, and documentation. Data analysis includes data reduction, data presentation, and drawing conclusions, data validity is ensured throught the use of triangulation techniques. Students' mathematical creative thinking ability in solving flat-sided space building problems is still low.

Shifting the ways prospective teachers frame and notice student mathematical thinking: from deficits to strengths

Noticing the strengths in students’ mathematical thinking is a critical skill that teachers need to develop, but it can be challenging due to the prevalence of deficit-based thinking in mathematics education. To address this challenge, a teacher education course was designed to encourage prospective teachers to engage in critical reflection on their own and others’ framings of students’ thinking and shift their focus towards noticing students’ strengths. The study analyzed written responses from the prospective teachers, collected at the beginning and end of the course, to investigate their framing and noticing of students’ mathematical thinking. The analysis focused on the aspects of students’ thinking that the prospective teachers paid attention to, the stances they took when interpreting students’ thinking, and the instructional moves they proposed in response to their thinking. Furthermore, the study established a spectrum of deficit-based and strength-based framings on students’ mathematical thinking. This spectrum allowed for the identification of each participant’s written noticing responses within a range of possibilities, contributing to a more nuanced understanding of the changes in teachers’ framing and noticing of students’ thinking over time.

Noticing elementary students’ mathematical reasoning and diverse needs: investigating the impact of different noticing tasks

Providing equitable mathematics learning opportunities in inclusive settings requires pre-service teachers (PSTs) to strengthen their skills to notice students’ diverse needs and mathematical reasoning. However, helping PSTs develop noticing skills is not an easy task for teacher educators. In our study, we examined the impact of different tasks (i.e. a written artefact task, an interview task, and a video analysis task) on elementary PSTs’ skills to notice the mathematical thinking of students with diverse needs. Our findings indicate that the sequence of three different noticing tasks scaffolded elementary PSTs’ ability to interpret and respond to students’ mathematical reasoning and individualized needs. Particularly, the interview task supported PSTs in effectively and purposefully interpreting and responding to mathematical reasoning and the struggles of elementary students with diverse needs. To articulate our findings, we share cases of three PSTs interviewing elementary students: one who struggles in mathematics, one who has an Individualized Education Program (IEP), and one who has been categorized as gifted and talented. We discuss our findings in relation to previous research, share implications for mathematics teacher educators, and provide recommendations for possible future studies.

APPLICATION OF INFORMATION AND COMMUNICATION TECHNOLOGIES IN THE ACTIVITIES OF A MATHEMATICS TEACHER

The article discusses the use of information and communication technologies in the activities of a mathematics teacher. We all know that information and communication technologies are of great importance for the effective organization of the learning process, the development of mathematical thinking of students, that is, the learning process cannot be imagined without information and communication technologies. In our article, it is precisely because of these advantages that information and communication technologies are used to develop students’ mathematical literacy in mathematics lessons.; in the organization of independent work of students; in the implementation of an individual, personality-oriented approach.

Mathematic Creative Thinking Processes Through Mind-Mapping Based Aptitude Treatment Interaction Learning Model: A Mixed Method Study

<p style="text-align: justify;">This study aims 1) to determine the effectiveness of the Mind-Mapping based Aptitude Treatment Interaction model towards creative thinking and 2) to explain the mathematical creative thinking process based on the creative level. The number of participants was 26 students who took the Multivariable Calculus course in the odd semester of 2020/2021. This research used the mixed-concurrent embedded method. The data collection techniques were validation, observation, creative thinking tests, and interviews. The results showed that 1) the Mind-Mapping based Aptitude Treatment Interaction model was effective in developing creative thinking, as indicated by the average creative thinking score of the experimental class, which was higher than the control class and 2) the characteristics of students mathematical creative thinking process varied following the creative thinking levels. The students mathematical creative thinking level consists of not creative (CTL 0), less creative (CTL 1), quite creative (CTL 2), creative (CTL 3), and very creative (CTL 4). Students at the CTL 2, CTL 3, and CTL 4 can meet the aspects of fluency, flexibility, and originality.</p>

A Meta-analytic Review of the Relationship between Mathematics Anxiety and the Mathematical Thinking of Africa Students

The development of mathematical thinking is a crucial component of Mathematics education. Mathematics is needed for students to have the ability to manage, acquire, and make use of information to survive in competitive, uncertain, and ever-changing circumstances. Mathematical thinking is the primary goal of learning Mathematics. Despite the relative importance of Mathematics, it is highly disheartening to see that, due to Mathematics anxiety, students’ mathematical thinking has remained persistently subpar in Africa. Mathematics anxiety, which is the feeling of worry, stress, and trepidation when people engage with mathematics, has long been a topic of concern in education. This study investigated the relationship between Mathematics anxiety and the mathematical thinking of students in Africa. A meta-analysis method was used to statistically analyze the results of studies yielding 1789 samples that examined the relationship between Mathematics anxiety and the mathematical thinking of students. These studies were conducted from 2005 to 2021. The findings showed a strong negative correlation between Mathematics anxiety and the mathematical thinking of students in Africa. This study's statistical findings demonstrate how the problem-solving ability of students and the reasoning of students are affected by Mathematics anxiety. These results can help education stakeholders to create environments that will enhance students' mathematical reasoning abilities. This study recommends that curriculum developers should come up with strategies for incorporating fresh perspectives into the Mathematics curriculum that will make it engaging for African students and boost their thinking to become better not just in their academics but also in their day-to-day living.

Enhancing students mathematical thinking using applets

This article presents two different classroom situations in which two pairs of students, one in grade 2 and the other in grade 6, attempted mathematically challenging problems through GeoGebra applets. Their reasoning, conjecture making, and argumentation is analyzed using Pea’s (1985, 1987) theory of technology as amplifier and reorganizer and the notion of internal and external representations. GeoGebra played the role of amplifier and reorganizer in enabling students’ explorations and led them to make conjectures. Furthermore, it was observed that students’ static internal representations were enhanced to more dynamic external representations and this played a significant role in developing students’ thinking. The three aspects of fidelity have been discussed with regard to the use of GeoGebra based applets and how such applets need to be critically designed and assessed in order to support students’ learning.

Компоненты дидактической сложности математической задачи и их оценка

The problem of estimating the didactic complexity of a mathematical task, which together with the solution is a multidimensional object characterized by a set of applied methods, is studied. The formation degree of the student's mathematical thinking is also determined by the set of mathematical methods he/she has mastered.
 Therefore, it is logical to consider the complexity of using certain methods of solution as the didactic complexity components of a particular task. It has been established that: 1) each mathematical problem can be characterized by a one-dimensional matrix, the components of which are proportional to the complexity of applying the corresponding method to solve it; 2) as the main methods for solving a mathematical problem, we can choose: methods of reading text, arithmetic calculations, algebraic transformations, method of the geometric constructions and operations with vectors use, combinatorial method, the trigonometric method, the method of logarithms and exponential functions, the differentiation and integration method, the operator method; 3) the complexity of applying a particular method in solving given task can be determined using a special computer program by counting the number of corresponding marker-terms and taking into account their complexity. The article evaluates the various components of the didactic complexity of 9 mathematical tasks on various topics and determines their general didactic complexity. It has been established that task 9 for calculating the divergence and rotor of a vector field is about 20 times more difficult than task 1 for solving a first-degree equation with one variable.

Metacognitive Strategies Related with Logical–Mathematical Thinking for Adolescents with ADHD

This article focuses on the contributions of the still-scarce corroborations available on the social nature of the metacognitive regulation of joint attempts in order to offer systematic means to operationalize and analyze shared regulation. The mathematical knowledge aims to achieve the metacognitive needs of students and, in particular, those with learning difficulties. The present research process aims to explain the relationship between the logical and mathematical thinking of students with ADHD in secondary education schools in Heraklion (Crete) and metacognitive awareness and academic motivation, including questions about pupils’ logical–mathematical skills and logical decisions of life and problem solving. Appropriate psychometric tools are used to evaluate their performance as well as their short and medium-term and consequently their long-term goals. The results of the current study imply that, when students realize that teachers and their parents emphasize the essential process of learning, appropriate strategies can be shown to them to allow them to learn how to solve problems on their own. As a result, it is of great significance to point out the relationship between students’ academic achievement and academic motivation.

Making sense of student mathematical thinking: the role of teacher mathematical thinking

Making sense of student mathematical thinking: the role of teacher mathematical thinking

Noticing Student Mathematical Thinking: Self-Contemplation of a Pre-Service Teacher

This study focused on a pre-service teacher’s self-contemplations about the two opportunities provided in a field experience course, Formative Assessment Interviews (FAIs) and Model Building (MB), which were designed to support pre-service teachers developing the skills of noticing students’ mathematical thinking. In the field experience, pre-service teachers conducted FAIs with a pair of children and participated in MB sessions to hypothesize students’ mathematical thinking. The findings suggest that although the pre-service teacher, Gloria, experienced some challenges, FAIs and MBs created a supportive medium for the pre-service teacher’s development of noticing student thinking and appreciating the value of this practice, which is important to gain in teacher education.

Системный подход к методике тифлопедагогики на примере задач математического анализа

The article discusses some of the problems that arise when working with students with visual defects. In particular, the authors discuss ways to improve the culture of mathematical thinking of students with visual defects. The authors rely on the experience of working at the Faculty of Information Technologies of Moscow State Moscow State University of Psychology & Education.

Осязательные модели в аналитической геометрии

The article discusses some of the problems that arise when working with students with visual defects. In particular, the authors discuss ways to improve the culture of mathematical thinking of students with visual defects. The authors rely on the experience of working at the Faculty of Information Technologies of Moscow State University of Psychology & Education.

Providing Virtual Mathematics Feedback: Connecting Research to Practice

Feedback is an essential form of communication between the student and teacher. Research has documented the importance of feedback in advancing student mathematical and critical thinking, with renewed recommendations to provide and use feedback in mathematical instruction during the era of COVID-19. Giving personalized feedback in an online environment can be a challenge – especially in a mathematics class. This article summarizes five core principles of feedback, associated strategies for mathematics teachers to provide students virtual feedback, and notes on how we have implemented these strategies in middle school mathematics classes.

The Effectiveness of Using the Cooperative Learning Model of FSLC Type on Students' Mathematical Reflective Thinking Ability

<em>The purpose of this study was to test the effectiveness of the cooperative learning model of Formulate Share Listen Create (FSLC) type on students' mathematical reflective thinking ability on the Pythagoras theorem. The research method used was quantitative. The population of the study was the VIII grade students of SMP Negeri 17 Singkawang, which amounted to 166 students. The sampling technique used was purposive sampling. The number of students who became the sample was 58 students. Data collection techniques in this study used measurement techniques through posttest questions, direct observation techniques through activity observation sheets, and questionnaire techniques through motivational questionnaires. The data analysis technique used was a one-sample t-test, Man-Whitney U-test, the percentage of activity, and the average analysis. The results showed that: (1) students completeness scores KKM ≥ 75 reached 75% with Z<sub>count</sub> > Z<sub>table</sub> = 0.12 > -1.96, (2) There was the difference of students' mathematical reflective thinking ability between using the cooperative learning model of FSLC type and the direct learning with an average score of 82.35 for the experimental class and 47.42 for the control class, (3) students’ activity was classified as active when the cooperative learning model of FSLC type was applied with a total percentage of 80.18% in the very high category, (4) students' learning motivation was high when the cooperative learning model of FSLC type was applied with an average of 4.15 in the high category. Thus, it could be concluded that the cooperative learning model of the FSLC type was effective for teaching students mathematical reflective thinking ability.</em>