Mathematics Education Courses Research Articles

Investigating prospective teachers’ skills of predicting and identifying significant student reasoning instances and deciding how to respond to those instances

ABSTRACT The importance of constructing mathematical arguments and critiquing the reasoning of others has been emphasized as a key component for deep mathematical understanding. As a result, acknowledging proof as an essential tool to comprehend mathematics at all grade levels has been suggested by many researchers as well as reform initiatives. To do so, both teachers and prospective teachers should be able to predict how students reason mathematically, identify significant student reasoning instances and decide how to respond productively to those instances. The participants consisted of eleven prospective teachers who enrolled in a teacher education program at a public university in Turkey. All participants enrolled in a two-credit mathematics education course about teaching reasoning and proving in the middle grades taught by the author. The results of this study demonstrated that the participating prospective teachers were able to predict student reasoning productively. However, the results regarding to the participants’ levels of identifying significant student reasoning instances were less successful. This study also documented that majority of the participants listed guiding questions. Although the elective course was not structured toward promoting participants’ noticing skills, all these results could suggest that the course might have influenced the participants’ prediction, identifying, and responding skills.

Including the Brügner Tangram in an Undergraduate Mathematics Education Course: Systematic Search for Solutions and Graphs

This article presents a reflection on a teaching experience involving the use of the Brügner tangram, an interesting but little-known manipulative material. It details an activity conducted as part of an undergraduate mathematics education course for prospective primary school teachers. The main objective of this paper is to present the implementation of a sequence of activities designed to convey to future teachers the importance of systematically solving problems that involve searching for or constructing different cases. Specifically, participants are provided with the three pieces of the Brügner’s tangram and assigned the task of identifying all possible convex polygons that can be constructed. Moreover, they are required to elucidate the methodological approach underpinning their exploratory process, with an emphasis on establishing relationships between the polygons. Graphs are introduced as one of the possible approaches for modelling the problem, offering a graphical representation that aids in the systematic search for solutions. This paper describes different activities involving the Brügner’s tangram which has demonstrated its adaptability as an instructional resource. The teaching sequence adheres to the five-phase structure proposed within the framework of the van Hiele model, which is also part of the course.

(Up)Grading: A (Re)Humanizing Assessment Process with a Focus on Feedback

Researchers across two universities and three different mathematics education courses implemented their vision of a novel grading approach called (Up)grading. (Up)grading shifts the focus of assessment from grades to growth. Key features of implementing (Up)grading included (a) providing students with opportunities to reflect upon and grow from their learning experiences, and (b) giving them a voice in determining their course grade. The findings suggest that most students perceived (Up)grading as a positive experience in their learning and as an assessment approach. The features of (Up)grading students believed contributed to the positive experience included giving them opportunities to reflect on their work and learn from their mistakes, as well as targeted feedback, enabling them to independently move their thinking forward. Tensions in the process did arise, including students’ initial anxiety with the norm shift from grades to growth and instructors’ management of the flow of assignments.

The Online Game Design Uses Scratch, Geogebra, and Geogebra AR Through Mathematics Education Multimedia Lectures

This study aims to design online mathematics games using Scratch, Geogebra, and Geogebra AR through multimedia education mathematics courses. This study used the multimedia development life cycle. Participants in the study were 41 students (13 male and 28 female) of fifth-semester in a multimedia course in mathematics education at a university in West Java, Indonesia. The research instrument was a questionnaire through Google Forms, performance appraisal, observation, and interviews. Based on the research results it is known that Scratch, Geogebra, and Geogebra AR can be used in math game design even though previous students have not studied both, but the results are good. The author suggests that students can try out these math games at school during research or teaching practice. In addition, the authors hope that the game can be published on the internet so that it can be used by many students.
 Keywords: game online, scratch, geogebra AR

Mathematics students’ self-reported resources used for solving real-life problems in a challenge-based modelling course

There are new educational approaches that are moving towards student-centred teaching and learning. One of them is challenge-based education, where students acquire and develop disciplinary knowledge and competencies by collaborating on solving real-life problems. Hence, in this paper, we report on a study of applied mathematics students’ learning experiences in the context of challenge-based education in a higher education mathematics course. Using a case study approach, we investigate how students perceive the role of the different resources to help them solve real-life problems during a one-week challenge-based modelling course. We collected data, from four groups of students (5–7 students each) who worked together and interacted with their mathematics supervisors and problem owners. Results show (1) the crucial importance of using and integrating different resources, such as social resources (e.g. mathematics supervisors and problem owner), for such learning environments to link mathematics to an authentic situation and develop the skills of an ‘applied mathematician in the real world’; (2) the relevance of mathematical knowledge in creating models to solve real-life challenges.

From a teacher education course to upper elementary classrooms and back: Revealing innate abilities of children to do teachers' mathematics

One of the key ideas of the modern-day elementary mathematics teacher education deals with mediating learning by visual thinking to enable transition from seeing and acting on concrete objects to describing the visual and the physical through culturally accepted symbolic representations. This paper shares mathematical activities designed originally for teacher candidates and used with students in upper elementary classrooms at a school in Upstate New York with minority student enrollment 97%. Because successful use of conceptual thinking by young students does have positive impact on their future teachers, connection of work in the school to a master’s level elementary mathematics education course taught by the authors is discussed. It is shown how using a spreadsheet and Wolfram Alpha allows for the research-like extension of the activities to the secondary level of mathematics education.

EDITORIAL: Humanizing Learning

This volume contains three articles that address the idea of humanizing online learning. learning technologies as platforms. The first by Olivia Stewart investigates work done to increase engagement and participation in an asynchronous course. The second is an implementation paper by Kam Moi Lee and her colleagues that investigated the role that online personas and journey mapping through the lens of equitable design practice. The third is a review of higher education mathematics homework and course platforms.Finally, April Burton contributed a book review of Caitlin Tucker’s recent book on Blended Learning.

Enhancing educational research pedagogy: Integrating in-house peer review in a flipped/hybrid classroom environment

This paper introduces a novel approach to teaching educational research—a two-semester course employing an in-house peer review approach within a flipped/hybrid learning environment. The study engaged 21 pre-service mathematics teachers enrolled in the Research in Mathematics Education course. Employing a concurrent triangulation mixed methods design, quantitative analysis from peer review ratings was triangulated with quantitative data from focused group discussions. Results showed that the in-house peer review feedback provided significant improvement to both Research 1 proposals and Research 2 full articles, from initial drafts to final versions. The findings highlight the efficacy of the in-house peer review approach in enhancing student research outputs within the course activities. The hybrid classroom format fosters in-depth research discussions during consultations. Complementing these interactions, the flipped classroom approach facilitated additional learning opportunities. The integration of qualitative and quantitative data enabled an understanding of the peer review process as a distinctive pedagogical approach in educational research courses. Policy directions aimed at further advancing these practices were delineated.

Exploring Polygonal Number Sieves through Computational Triangulation

The paper deals with the exploration of subsequences of polygonal numbers of different sides derived through step-by-step elimination of terms of the original sequences. Eliminations are based on special rules similarly to how the classic sieve of Eratosthenes was developed through the elimination of multiples of primes. These elementary number theory activities, appropriate for technology-enhanced secondary mathematics education courses, are supported by a spreadsheet, Wolfram Alpha, Maple, and the Online Encyclopedia of Integer Sequences. General formulas for subsequences of polygonal numbers referred to in the paper as polygonal number sieves of order k, that include base-two exponential functions of k, have been developed. Different problem-solving approaches to the derivation of such and other sieves based on the technology-immune/technology-enabled framework have been used. The accuracy of computations and mathematical reasoning is confirmed through the technique of computational triangulation enabled by using more than one digital tool. A few relevant excerpts from the history of mathematics are briefly featured.

Aligning numeracy task design with SDG goals: Nutrition facts as a context for prospective mathematics teachers' problem posing

Nutrition facts are the details on food packaging that describe its nutritional value, including serving size, calories, macronutrients like carbohydrates, protein, and fat, and micronutrients such as vitamins and minerals. These facts have mathematical concepts that can be utilized as a context for prospective teachers to create mathematical problems. By leveraging this real-world data, educators can contribute to multiple Sustainable Development Goals (SDGs), particularly SDG 3 (Good Health and Well-being), by promoting nutritional awareness, and SDG 4 (Quality Education) by enhancing student engagement and understanding through relatable examples. This study examines the profile of mathematical tasks created by prospective mathematics teachers using nutrition facts as a context that meets numeracy task criteria. Data were collected from 62 mathematical tasks created by 31 prospective teachers attending a realistic mathematics education course on numeracy based on a problem-posing task at a public university in Surabaya, Indonesia. The posed tasks were categorized into solvable or unsolvable tasks and the domains of the level of context use and the level of cognitive processes. Results revealed that the level of context use in the posed tasks varied from zero to second order. Surprisingly, most of the tasks were in the first-order level context. Most posed tasks reached the application level, with only a few identified as reasoning tasks. Interestingly, some tasks coded as second-order context were classified as reasoning tasks. The study provides implications for designing numeracy tasks using nutrition facts and interventions in teacher education related to numeracy task design.

Emerging perspective of a mathematics teacher educator on connections between advanced and school mathematics

Contemporary research on mathematics teacher education reveals attention by researchers on mathematics connections (MC). Nevertheless, researchers have not investigated experiential views of mathematics teacher educators (MTEs) around MC. This article addresses the scenario by relating findings of an intrinsic case study involving a purposively chosen MTE. MTE responded to interview questions regarding experiential viewpoints on connections between advanced and school mathematics. Narrative data analysis suggested that MTE prefers special mathematics education courses (MECs). Envisaged MEC are supposed to align with school mathematics but comprised of subject matter, which is neither advanced mathematics nor school mathematics. Such courses coupled with school-based mentorship are expected to acquaint mathematics student teachers with MC. MTE could not provide subject matter-based connections except for pedagogic ones. Study of advanced mathematics was encouraged to foster understanding of MC. These findings motivate further research concerning how advanced mathematics ought to be connected to school mathematics concepts.

Psychology - an inherent part of mathematics education

On the chronology of individual stations of psychology and their effect on mathematics education designed as working document for use in teacher training.The article is structured as a literature survey which covers the numerous movements of psychology towards mathematics education. The current role of psychology in mathematics education documented by different statements and models of mathematics education should provide a basis for the subsequent investigations. A longitudinal analysis pausing at essential marks takes centre of the continuative considerations. The observed space of time in the chapter covers a wide range. It starts with the separation of psychology from philosophy as a self-contained discipline in the middle of the 19th and ends with the beginning of the 21st century. Each stop states the names of the originators and the branches of psychology they founded. These stops are accompanied by short descriptions of each single research objective on the one hand, and their contributions to mathematics education on the other hand. For this purpose, context-relevant publications in mathematics education are integrated and analysed. The evaluation of the influence of concepts of psychology on teaching technology in mathematics is addressed repeatedly and of great importance. The layout of this paper is designed for the use as a template for a unit in teacher-training courses. The conclusion of the article where the author refers to experiences when teaching elements of psychology in mathematics education courses at several universities in Austria is intended for a proof on behalf of the requested use.
 Subject Classification: 01A70, 01-XX, 97-03, 97D80

Exploring Conceptions of ‘Number Sense’ Evident in Pre-service Programmes for Primary School Teachers: A Review of Texts Used Across 11 University Programmes

This paper examines the conceptions of ‘number sense’ as promoted in pre-service primary mathematics education courses at 11 South African Higher Education Institutions through the texts used by academics or prescribed for students. While all the participating institutions agree that the development of primary school learners’ ‘number sense’ is central to their mathematics methodology courses and that there is an overwhelming amount of research and literature on ‘number sense’ nationally and internationally, their conceptualisations of the nature of ‘number sense’ vary. Teacher educators, who develop pre-service teacher education courses, were asked to provide the texts, used to underpin the 11 universities’ mathematics education modules in the Bachelor of Education (Foundation and Intermediate Phases) and Post Graduate Certificate in Education (Foundation and Intermediate Phases) programmes. These texts were analysed drawing on Whitacre et al.’s emphasis on three ‘number sense constructs’ identified as Innate Number Sense, Early Number Sense and Mature Number Sense. The results show that there is no common language of description for ‘number sense’ across the 11 universities. This research implies that there is a need to develop a consistent understanding of ‘number sense’ and how it is developed across institutions.

The role of peer-review workshops in prospective teacher training

In this paper we present a model for formative assessment based on peer-review workshops on MOODLE, designed by researchers of te University of Salerno and  the University of Torino. The model has been implemented in the Mathematics and Mathematics education courses in Primary Education at the University of Torino. The peer-review workshops are intended to pursue the following objectives: (1) strengthening the argumentative skills of prospective teachers to direct them towards a relational view of mathematics; (2) provide them with models of formative assessment rooted in peer-review feedback managed by the university lecturer. We present an example of peer-review that intertwines effective feed-back with a relational understanding of mathematical thinking.

Assessing the Relevance of Information Sources for Modelling Student Performance in a Higher Mathematics Education Course

In recent years, most educational institutions have integrated digital technologies into their teaching–learning processes. Learning Management Systems (LMS) have gained increasing popularity, particularly in higher education, due to their ability to manage teacher–student interactions. These systems store valuable information which describes students’ behaviour throughout a course. These data can be utilised to construct statistical models that represent learner behaviour within an online LMS platform. In this study, we aim to compare different sources of information and, more ambitiously, to provide insights into which source of information is most valuable for inferring student performance. The considered sets of information come from (i) the Moodle LMS; (ii) socio-economic data about students acquired from a survey; and (iii) subject marks achieved throughout the course. To determine the relevance of the incorporated information, we use artificial intelligence (AI) methods, and we report the importance measures of four state-of-the-art methods. Our findings indicate that the selected methodology is suitable for making inferences about student performance while also shedding light on model decisions through explainability.

Predicting Student Success in Co-remediated General Education Mathematics Courses at a Large Urban Public University

Placement and support of students in first-year mathematics courses at institutions of higher education has long been a consequential issue. In a bid to address it, many systems and institutions of higher learning have elected to implement a co-remediation framework in place of pre-remediation, due in large part to the prohibitive cost of the latter, both in terms of financial resource, as well as student academic progress. Accompanying this evolution has been the expansion of the introductory mathematics curriculum beyond algebra to include statistics and quantitative reasoning. The present study discusses three distinct introductory mathematics courses at a large urban public M.S. granting institution in the U.S., with the goal of identifying the characteristics that correlate with success in each. Conditional probability and non-completion rate analyses were implemented to compare student performance in each course. Predictive models were then trained and validated, building insight concerning the differential relationships of demographic and academic covariates with course completion.

Special issue on mathematics in early childhood education

"Mathematics is an important part of children’s daily life. The value of mathematics for young children is reflected in curricula for early childhood in the Nordic countries and consequently, mathematics education courses are included in teacher education for early childhood. However, mathematics in early childhood education is understood in different ways, influenced by both politics and research, and practitioners’ approach to the field. As a field of research, mathematics education for young children is fairly new in the Nordic countries."

How narratives about the secondary-tertiary transition shape undergraduate tutors’ sense-making of their teaching

Drawing on the commognitive framework, we construe the secondary-tertiary transition (STT) as a distinctive element in the pedagogical discourses of various communities. Our interest rests with university tutors in light of the emergent recognition of their impact on undergraduates’ mathematics learning in many tertiary contexts worldwide. We aim to understand the roles of STT communication in tutors’ reflections on incidents that took place in their tutorials. Our participants were undergraduate students in the advanced stages of their mathematics degrees in a large New Zealand university and who were enrolled in a mathematics education course. Throughout the semester, the participants led tutorial sessions for first-year students and wrote reflections on classroom incidents that drew their attention. Our data corpus consisted of 58 reflections from 38 tutors collected over four semesters. The analysis revealed that STT communication featured in tutors’ descriptions of classroom incidents, assisted them in making sense of unexpected events, positioned their instructional actions as replications of what was familiar to them from their own STT experience, and contributed toward generating new pedagogical narratives. We situate these findings in the literature concerning undergraduate tutoring and teachers’ perspectives on STT.

The Effectiveness of Case Method Learning in View of Students' Critical Thinking Ability

This study aims to describe the effectiveness of the case method in terms of students' critical thinking skills. This type of research is a quasy experiment with a one-group pretest-posttest design. The population of this study was all students of the Elementary School Teacher Education Study Program (PGSD) at the University of Mataram. The sample used in this study were 29 semesters 2 students taking elementary school mathematics education courses. Data collection techniques using test and non-test techniques. The instrument for collecting data uses a critical thinking ability test. The analysis technique uses the normality test for the prerequisite test, the t-test to find out the difference in the pretest and posttest averages, and the N-Gain score test to determine the effectiveness of the case method. The results showed that the results of the pretest and posttest of students' critical thinking skills obtained successive significance scores of 0.088 > 0.05 and 0.094 > 0.05 meaning that the data was normally distributed. Furthermore, the statistical results of the t-test showed a significance value of 0.000 <0.05. This means that there is a significant difference in average between the pretest and posttest. While the result of the N-Gain score is 0.713 and is in the high category, it can be said that the implementation of the case method has a high level of effectiveness on students' critical thinking skills.

Analisis Kesalahan Mahasiswa PGSD pada Materi Aritmatika Mata Kuliah Pendidikan Matematika Kelas Rendah

This study aimed to identify the errors of PGSD students in the arithmetic subject on the Low Grade Mathematics Education course at Efarina University, Pematang Siantar. The subjects of this study were semester III PGSD students at Efarina Univesity, Pematang Siantar in the academic year 2022/2023 as many as 33 people. Errors in mathematics in this study are divided into factual errors, conceptual errors, and procedural errors. This type of research is a quantitative descriptive research. The data collection method used was a test. The test instrument used was a conceptual understanding test to be able to find students errors. A total of 5 items in the test validated by three experts with the Lawshe’s CVR technique. The results showed that (1) The general error rate of PGSD study program students in Arithmetic material for Low Grade Mathematics Education was 57% which was in the medium error category, (2) There was a highest error of 82% (very high error category) and lowest 30% (low error category), (3) factual error average 34.5% (low category), (4) conceptual error 82% (very high category), and (5) procedural error average 67% (high category).