Mathematical Experiences Research Articles

Understanding of base-10 concept and its application: a cross-cultural comparison between Japan and Singapore

ABSTRACT It has become increasingly clear that the early use of decomposition for addition is associated with later mathematical achievement. This study examined how younger children execute a base-10 decomposition strategy to solve complex arithmetic (e.g. two-digit addition). 24 addition problems in two modalities (WA: Written Arithmetic; OA: Oral Arithmetic) with sums less than 100 were administered to 22 Japanese and 22 Singaporean 6-year-old kindergarteners. Our findings reveal that they were able to solve complex addition. For instance, Japanese kindergarteners tended to solve complex arithmetic using base-10 decomposition across the modality, whereas Singaporean kindergarteners used standard algorithms and basic counting to solve complex WA and OA problems, respectively. We speculate that Japanese kindergarteners might have a clearer understanding of the base-10 concept and were able to use this knowledge more readily than Singaporean kindergarteners. Mathematical experiences in kindergarten and number-naming systems have been put forward as two of the crucial contributors for such cross-cultural differences. This study also provides new directions for future research on the understanding of the base-10 concept and its application among young children.

Preservice teachers’ experiences with digital formative assessment in mathematics

ABSTRACT This study investigated the experience of preservice mathematics teachers working with example-eliciting tasks (EET) on an online formative assessment platform. The study focused on the effect of working with EETs on the preservice teachers’ (PSTs) professional noticing of learners’ mathematical thinking. Participants included nine PSTs studying for their master's degree in a teacher education college in Israel. The PSTs were presented with their own responses and peer's responses about an angle bisector activity, and semi-structured interviews were conducted following the completion of course meetings. Qualitative methods were used to analyse the data. The findings show that the PSTs’ experiences of engagement with EETs were reflected in their noticing skills: attending, interpreting and responding. The findings also show that PSTs paid special attention to the variance between submitted examples and noticed the differences between answers. Their experiences also affected their knowledge of content and teaching, namely the proper sequence of instruction and the choice of examples for use in the classroom to facilitate deeper understanding of concepts. Finally, the findings indicate that using EETs on an online formative assessment platform enhanced the noticing skills of PSTs.

An exploration of teacher and student perceptions of blended learning in four secondary mathematics classrooms

The COVID-19 pandemic forced many teachers around the world to make a sudden switch from face-to-face to online teaching. This shift in practice has provided an opportunity to reconsider how technology use in mathematics education can be utilised to improve student engagement. In this study, we explore four case studies of Australian secondary mathematics classrooms conducted prior to the COVID-19 pandemic to examine how teachers are using blended learning approaches and how their students perceive these pedagogical practices. Findings across all four sites indicate that technology use expands student opportunities to engage with mathematics learning through the provision of multiple pathways and methods of access. Specifically, we find evidence supporting the use of blended classroom teaching strategies to provide differentiation and personalised learning approaches; visualisation and dynamic manipulation of mathematics concepts; and alternative methods for teacher-student feedback and communication. We argue that the student learning experience in mathematics can be enhanced through a variety of blended learning approaches by allowing for diverse points of access to learning opportunities which are more closely aligned to individual learning needs and free from the temporal constraints of the classroom.

Data informed leadership: the work of primary mathematics lead teachers in New Zealand

ABSTRACT Increasingly, teachers are considered accountable for continuous improvement in student achievement. They are expected to integrate and analyse data from multiple sources in order to make decisions aimed at positively influencing student outcomes. Given the growing recognition of the importance of data-informed decision making in teaching, it is important to understand how teachers can best be supported in this. Currently, there is little research into how teachers can facilitate data literacy development for their in-school colleagues. This study employed an interpretive qualitative methodology to focus on the experiences of mathematics lead teachers (MLTs) in leading change to support the development of their colleagues. MLTs were found to use a range of tailored strategies, providing ‘on the job’ and ‘just in time’ support, and demonstrating evaluative and adaptive capabilities. They were found to provide leadership in the development of data-informed decision making through their collaborative approach.

Facilitating collaboration between mathematics methods and content faculty through cross-departmental lesson study

PurposeThis paper reports on experiences of university-level mathematics teacher educators (MTEs) from participating in lesson study to improving mathematics teacher preparation. The authors investigate the questions: “What did MTEs report as benefits of participating in cross-departmental lesson study?” and “What considerations did MTEs report for stakeholders interested in lesson study?”Design/methodology/approachThis study employed qualitative methodologies on one cycle of lesson study where the authors served as lesson study participants. Debrief and focus group interview data were analyzed for emergent codes on benefits, supports and challenges from a university-level lesson study.FindingsEngaging in university-level lesson study provided MTEs firsthand professional development, created community, enabled attention to detail, improved knowledge of issues related to teaching mathematics, improved curricular materials and extended beyond one lesson. Institutional support, a natural evolution of the process, alignment of goals among members and support within the group supported their university-level lesson study. Challenges of time and funding, imposition, misalignment of goals and building trust are discussed.Originality/valueThe paper contributes insights about benefits for MTEs who use lesson study to examine their own teaching practice for mathematics teacher preparation. The importance of supports and constraints is highlighted.

Using a Superfactorial Problem to Provide Extended Problem-Solving Experiences

ABSTRACT The purpose of this paper is to describe the use of a problem involving superfactorials (a specific product of factorials) that provides an in-depth and comprehensive mathematical experience, encompassing skills that mathematicians view as tantamount to exploration. The problem is easily accessible and fosters creativity and perseverance, thereby allowing participants to practice standard steps in developing a mathematical argument: from the problem, to simplifications, then conjecture, and finally proof. We present a brief solution to this problem and then discuss how the experience in trying to solve this problem may evolve during a professional development session for teachers. By providing teachers with extended time to investigate a single problem, we aim to provide professional development opportunities beyond the standard mathematical content or instructional practices. This problem also makes a welcome addition to any undergraduate mathematics curriculum and could be used in a mathematics for liberal arts course, capstone course, or any course with a focus on developing problem-solving skills and the process of doing mathematics.

Epistemology and rationality of intuition and imagination in the field of mathematics

This paper aims to contribute to the clarification of the role of mathematical intuition and imagination in the constitution of mathematical knowledge, evidencing its epistemological and procedural characteristics. For that, an "epistemology of intuition and imagination" in the field of mathematics is outlined (suggested) emphasizing the need to adopt a dynamic conception of mathematics. In this context, intuition and imagination present themselves as forms of mathematical experience that give access, through paths that are not purely logical, to mathematical knowledge. Its epistemological and rationality characteristics, a rational of a non-logical nature, are highlighted by several examples, resources for moving the ideas involved. The epistemological study of intuition and imagination also allows highlighting its ontology, constituted of more relations than objects. From the pedagogical point of view, we discuss the formative character of philosophical studies involving intuition and imagination, mainly related to creativity in mathematics. Keywords: Mathematical knowledge; Mathematical experience; Epistemology of intuition and imagination; Creativity in mathematics.

From High School to College Calculus: Beliefs about Sense-Making and Mistakes

Despite longstanding efforts in the K-12 STEM education community to create meaningful mathematical experiences across disciplines, mathematics continues to be a siloed subject which is tracked based on ability. With an increasing number of high-achieving students enrolling in Algebra I in Grade 7 or earlier, there is a need for research on the readiness of accelerated STEM-intending students to persist in mathematics at the college level. A mathematical mindset framework was employed to explore the relationships between level of Algebra I acceleration, student (n = 2111) and instructor (n = 141) beliefs about sense-making and mistake-making, and attitudes in first-semester college calculus. Findings from a series of multilevel analyses indicate that interactions between student mathematical mindset and perceived progressive teaching practices influence attitudes toward mathematics. While student-centered instruction had a slightly negative effect on attitude, there was a differential effect in relation to student beliefs about sense-making as a metric of success in mathematics. These findings contribute to empirical understandings of mathematical mindset and the complex transition from high school calculus to college calculus. Implications for interdisciplinary STEM education and persistence in STEM undergraduate study are discussed.

Moments of critical self-reflection of a transformative mathematics teacher

I have been travelling through a transformative research journey since the beginning of my MPhil study. It allows me to understand my past and present strengths and limitations in learning and teaching of mathematics, thereby envisaging alternative practices for futurist education. As my research involved critical self-reflection on my professional praxis, I used a multi-paradigmatic (interpretivism, criticalism, postmodernism) research approach and autoethnography as a research methodology. Thus, the main purpose of the paper is to portray the moments of critical self-reflection on my experiences of doing mathematical activities during my childhood and learning mathematics during my early days of schooling, aiming at improving my practices as a teacher, a practitioner-researcher and an educator.  I used Habermasian knowledge constitutive interests (i.e., technical, practical, and emancipatory) and Schubert’s curriculum images (i.e., content or subject matter, experiences, cultural reproduction, etc.) to interrogate my experiences of doing and learning mathematics. As a mathematics teacher and practitioner-researcher, reflections on my childhood experiences as well as early days of schooling ultimately opened up somewhat closed box of my personal and professional practices. This paper in/directly indicates the enhancement of the students’ engagement in mathematics through context-based activities. Moreover, the selection of the contents should be based on learners’ experiences which might be experienced through mathematically rich activities such as games, daily household works, etc.

The Joyful Experience in Learning Mathematics

This study conducted to determine the effectiveness of joyful scaffolds in teaching grade VIII Mathematics in the Department of Education of the Philippines. Two intact groups were used as the subjects of the study. The experimental and control group were exposed to joyful and traditional scaffolds in teaching respectively. This study includes the pre – post assessments, significant improvements, and significant mean gain differences of students exposed to traditional and joyful scaffolds in teaching Mathematics in terms of performance level. The findings of the study revealed that the joyful scaffold in teaching is more effective in improving the students’ Mathematics performance as compared to the traditional scaffold in teaching and the skills acquired is sufficient enough to prove that these experimental group of students was able to gain experience from joyful approach and a positive change of attitude towards their view on Mathematics was realized after the exposure. The cognitive progression was deepen and attainment of knowledge was enhanced, psychomotor skills were activated and augmented through enjoyable mathematics learning activities, and students’ interests and learning manners were maximized. Furthermore, the joyful scaffold in teaching in this study was immensely efficient in improving the students’ Mathematics performance as compared to the traditional scaffold in teaching. Moreover, the outcome of this research supports the assertion that employing, relating, or incorporating several joyful media to learning boosted up students’ academic performance especially those who are detached during the classroom discussion.

Mentors Views on Mathematics and Science Student Teachers School Experience

Mentors Views on Mathematics and Science Student Teachers School Experience

What’s the math in Minecraft? A Design‑Based Study of Students’ Perspectives and Mathematical Experiences Across game and School Domains

This paper presents empirical findings from a qualitative study on Minecraft as a mathematical tool and learning environment. Even though Minecraft has been used for several years in classrooms around the world, there is a lack of detailed empirical studies of how students learn subject‑related content by working with the game. This study is based on a design experiment with an inquiry‑based teaching unit for fifth graders, which focused on using the coordinate system embedded in Minecraft, as a means to navigate and explore the game in order to solve mathematical problems. Based on student interviews, we explore how the students experienced and switched to new perspectives on mathematical knowledge through their participation in the teaching unit. Using thematic analysis, we explore data from six group interviews. The theoretical framework is based on domain theory, dialogical theory and notions of students’ mathematical agency. The key analytical findings regard the students’ experience of the coordinate system as part of both the academic domain of mathematics and their everyday domain of playing Minecraft, how they actively use the coordinate system to improve play in Minecraft and how they experience new ways of participating in mathematics. The article concludes by offering design principles for the future use of computer games in mathematics education.

Teaching to Listen, Question, and Ponder: An Investigation into Promoting an Enquiring Atmosphere in a Mathematics Classroom

This study reports on a classroom-based intervention that lasted eight lessons with the particular focus of exploring whether and how specific activities, strategies, and questions may promote an enquiring atmosphere in a mathematics classroom, i.e., a classroom environment in which students feel safe to investigate new mathematical ideas in order to produce, share, and test their conjectures in search for a proof. Although many researchers agree on the central role that conjectures and proofs should play in all students’ mathematical experiences, many students find it difficult to engage with conjecturing and proving. In this paper we argue that the classroom atmosphere has a crucial influence in helping students to overcome these difficulties and we focus our investigation on an English secondary mathematics classroom taught by the first author. Students’ responses to questionnaires, prompts, tasks, and interview questions were analysed in order to evaluate the impact of the intervention. The analysis suggests that specific activities, strategies, and questions can be used by teachers to facilitate an enquiring classroom atmosphere. However, for these to be effective, the teacher needs to explicitly teach students how to listen, question, and ponder in order to give students the tools to formulate conjectures and proofs.

Identity evolution of STEM teachers in Egyptian STEM schools in a time of transition: a case study

BackgroundThe study was initiated as a response to the growing Science, Technology, Engineering, and Mathematics (STEM) experience in Egypt in terms of the growing number of schools, number of students admitted, and the effect of this experience on the overall education system in the country. Viewing teachers as the backbone of the STEM experience and teaching as a relational sociocultural practice (Biesta, G. & Stengel, B. (2016) AREA Handbook of research on teaching, 5th edition), this research explores the evolution of teachers’ STEM identity during their professional journey in the Egyptian STEM education reform initiative.AimThe purpose of this study is to explore the dynamic dialogical interaction between the different factors impacting the development/evolution of the STEM teachers’ identity. The study is guided by the following research questions:How did Egyptian STEM teachers’ identity evolve over the course of the STEM experience in model STEM schools in Egypt?How did the political and structural characteristics of the STEM schools and teachers developing STEM identities interact and co-evolve?What do Egyptian STEM teachers identify as being important characteristics of STEM teachersMethodologyA multiple holistic case study design was used to explore the research questions within the bounded context of an established Egyptian STEM school.MethodsData for this study were drawn from (1) semi-structured interviews conducted with seven teachers from an Egyptian STEM school and (2) document analysis of Ministry of Education decrees and the published reports of Education Consortium for the Advancement of STEM in Egypt.ResultsUsing inductive and deductive data analysis, teachers were found to have undergone a deep transformative change process from a traditional teacher into teachers with a progressive mindset and student-centered classroom practices. The teachers developed strong conceptualizations of the STEM education, stressed the importance of collaboration, critical thinking, and motivation for teachers seeking to work in a STEM setting, and considered themselves moving towards established STEM teachers’ identity.ConclusionThis study concluded that STEM teachers’ identity can be viewed as a dialogical, dynamic, and evolving process that results from the interaction of personal and professional traits within new educational experiences exemplified by the new and different experiences in their STEM schools.Despite several challenges, such as limited resources and influence from the bureaucracy and norms of the traditional education system, the teachers still persist in their professional growth and desire to impact the larger context of the Egyptian education system.

Pre-university informal engagement with mathematical activities and the decision to study mathematics at university

A survey was created to investigate the experiences of mathematics undergraduates with informal mathematical activity prior to starting university, and links these with the decision to study mathematics. A questionnaire was completed by a small sample of first-year undergraduates at two UK universities. Generally, incoming undergraduates are shown to have a high level of enjoyment of mathematics and engagement with informal mathematical activity. Popular activities included mathematical puzzles and games, and online videos about maths. Students were often engaged with family or via social media, playing computer, tablet or phone games, watching TV game shows with mathematical aspects and participating in organised competitions. Only around half engaged via talks or workshops organised through school and watching more structured documentaries or videos of lectures. Few participated in organised clubs. It seems there was greater engagement with ‘fun’ aspects of mathematics than with activities which demonstrate mathematics linked to career choice. The link to goals of outreach and similar initiatives is discussed, with further research indicated.

Domain-specific experience and dual-process thinking

Prominent dual process models assert that reasoning processes can transition from effortful (Type 2) to intuitive (Type 1) with increases in domain-specific experience. In two studies we directly examine this automation hypothesis. We examine the nature of the relationship between mathematical experience and performance on the cognitive reflection test (CRT; Frederick, 2005). We test performance and response time at different levels of experience and cognitive constraint. Participants are required to complete a secondary task of varying complexity while solving the CRT. In Study 1, we demonstrate changes in thinking Type across real-world differences in mathematical experience. In Study 2, convergent with Study 1, we demonstrate changes in thinking Type across a mathematical training paradigm. Our findings suggest that for some individuals low experience is associated with Type 1 processing, intermediate experience is associated with Type 2 processing, and high experience is associated with Type 1 processing. Whereas, for other individuals low experience is associated with ineffective Type 2 processing, intermediate experience is associated with effective Type 2 processing, and high experience is associated with Type 1 processing.

Assigning mountain-valley fold lines of flat-foldable origami patterns based on graph theory and mixed-integer linear programming

Traditional origami design is generally based on designers’ artistic intuition and skills, mathematical calculations, and experimentations, which can involve challenges for crease patterns with a large number of vertices. To develop novel origami structures for engineering applications, systematic and easy-to-implement approaches capable of generating diverse origami patterns are desired, without requiring extensive artistic skills and experience in origami mathematics. Here, we present a computational method for automatically assigning mountain-valley fold lines to given geometric configurations of origami structures. This method is based upon a geometric-graph-theoretic representation approach combined with a graph-theoretic cycle detection algorithm, taking the subgraphs of a given structure as inputs. Then, a mixed-integer linear programming (MILP) model is established to find flat-foldable origami patterns under given constraints on the local flat-foldability and degree of vertices, leading to the identification of crease lines associated with local minimum angles. Numerical examples are presented to demonstrate the performance of the proposed approach for a range of origami structures with degree-4 or -6 vertices represented by their corresponding subgraphs.

What’s the math in Minecraft? A Design‑Based Study of Students’ Perspectives and Mathematical Experiences Across game and School Domains

This paper presents empirical findings from a qualitative study on Minecraft as a mathematical tool and learning environment. Even though Minecraft has been used for several years in classrooms around the world, there is a lack of detailed empirical studies of how students learn subject‑related content by working with the game. This study is based on a design experiment with an inquiry‑based teaching unit for fifth graders, which focused on using the coordinate system embedded in Minecraft, as a means to navigate and explore the game in order to solve mathematical problems. Based on student interviews, we explore how the students experienced and switched to new perspectives on mathematical knowledge through their participation in the teaching unit. Using thematic analysis, we explore data from six group interviews. The theoretical framework is based on domain theory, dialogical theory and notions of students’ mathematical agency. The key analytical findings regard the students’ experience of the coordinate system as part of both the academic domain of mathematics and their everyday domain of playing Minecraft, how they actively use the coordinate system to improve play in Minecraft and how they experience new ways of participating in mathematics. The article concludes by offering design principles for the future use of computer games in mathematics education.

Enhancing the mathematics experiences of 3-year-old children in daily routines

Enhancing the mathematics experiences of 3-year-old children in daily routines

Modelling Experiences And Its Factors In General Mathematics: The Case Of Grade 11 Students

Student’s experiences in learning plays an essential role in producing quality academic achievement. This study aimed to develop multiple regression models on the students’ experiences in learning mathematics in regards to its influencing factors. A simple random sampling of 112 grade 11 students in the first semester of school year 2018-2019 from Visayas State University were used as the respondents of this study. Result of the study reveals that grade 11 students’ learning experience in general mathematics is challenging and logical. However, it is found out that it is satisfying and rewarding since it develops their critical thinking and decision making in real life. The number of hours in studying and studying with internet positively influence the students’ learning experience in mathematics. It is also revealed that STEM students are more fond learning mathematics compare to non-STEM students. Furthermore, result also shows that a conducive learning environment and religious activities in the campus helps the students to have a peaceful and positive learning experience.