College Mathematics Research Articles

Correction to: How can university mathematics overcome Klein's second discontinuity? Specific course design

Correction to: How can university mathematics overcome Klein's second discontinuity? Specific course design

Students’ ways of using video lectures in a linear algebra course

Abstract Video lectures have become a common element in many university mathematics courses, and students often believe that these support their learning in various ways. However, some studies indicate that students who use such video lectures frequently perform lower in exams. Hence, it is important to investigate how students use video lectures. We interviewed nine students of a linear algebra course about the activities they carried out when watching prerecorded video lectures. In these interviews, all our participants reported in detail lots of activities they engaged in while watching, and our study contextualizes these activities with content-specific examples. However, our study also indicates that even if students are very active while watching video lectures, their cognitive engagement with the actual content might nevertheless be rather low. Possible consequences concerning the implementation of video lectures are then discussed.

How can university mathematics overcome Klein’s second discontinuity? Specific course design

How can university mathematics overcome Klein’s second discontinuity? Specific course design

Analogy and Generalization as a Driving Force of Learning Mathematics – the Case of a Matrix Analog of a Zero of a Polynomial

The concepts of polynomials and matrices essentially expand and enhance the elementary arithmetic of numbers. Once introduced, polynomials and matrices open up new interesting mathematical challenges which extend to new fields of mathematical explorations within university mathematics. We present an aspect of a rather elementary exploration of polynomials and matrices, which offers a new perspective and an interesting matrix analog to the concept of a zero of a polynomial. The discussion offers an opportunity for better comprehension of the fundamental concepts of polynomials and matrices. As an application, we are led to the meaningful questions of linear algebra and to an easy proof of the otherwise advanced and abstract Cayley–Hamilton theorem.

Active Learning Taking Root – A Progress Update

In 2017, Cornell University's Mathematics Department embarked on a project to make active learning the standard teaching method in single-variable calculus teaching, as described in a 2020 PRIMUS article by Bennoun and Holm [Establishing consistent active learning in a calculus I course. PRIMUS. 31(3–5): 565–577]. Here we report on the progress of that initiative, the wider teaching reforms it generated, and the challenges we encountered.

Using Interactive Maps as a Means of Developing Fundamental Mathematical Knowledge

Introduction. Mathematics as a science and academic discipline has rich educational potential. However, due to the lack of motivation among students to master it, difficulties often arise in acquiring fundamental mathematical knowledge. The author believes that questions of the history of mathematics can help to understand how this discipline has developed over the centuries, how it influences modern life, and the tools of the digital educational environment will make it possible to do this in an accessible and understandable interactive form. The aim of the study is to present a way to use digital interactive maps as a means of developing deep mathematical knowledge in students, based on historical aspects of mathematical theory. Methodology: analysis of psychological and pedagogical literature on the research topic, generalization of many years of experience in training future teachers of mathematics and computer science, results of organizing design and research activities of students at Leningrad State University named after A. S. Pushkin (Saint Petersburg). The digital interactive map presented by the author is based on substantive aspects of the development of mathematics from the history of the Saint Petersburg mathematical school. Results. Digital interactive maps are characterized as an option for a teacher to use the functionality of a digital educational environment to fundamentalize students’ mathematical training. Interactive maps increase studentsʼ interest in educational material, integrate various types of data, use various interactivity tools, are the basis for providing detailed reference information, provide productive interaction with the map, and others. The author presents an example of creating an interactive map that reflects various locations in Saint Petersburg associated with historical aspects of the development of mathematical theory. Conclusions. A conclusion is made about the methodologically justified creation and use by the teacher of materials on the history of mathematics for the development of students’ system of fundamental mathematical knowledge, the formation of interest in the subject, the patriotic education of students, which together contributes to the conscious study of each individual topic of the school and university mathematics course, a deep immersion in the essence tasks to be solved.

Representation Process of Prospective Teachers in Solving Mathematical Problems Based on Cognitive Style

This study investigates the correlation between cognitive styles and the thinking process utilized when solving mathematical problems, emphasizing the significance of aligning problem-solving strategies with problem representations. Employing a qualitative approach, the research examines two cognitive styles—Field Independent (FI) and Field Dependent (FD)—chosen based on participants' mathematical abilities. Conducted at Cenderawasih University's Mathematics Education Study Program, the study reveals that students' mathematical problem-solving proficiency remains relatively low, warranting further exploration of their thinking processes. While FI subjects demonstrate competence in aligning thinking processes with mathematical representations, particularly in structuring problem-solving steps, FD subjects exhibit limitations in articulating meaning and verbal representations. These findings underscore the importance of instructing educators not only in teaching mathematical concepts but also in guiding students to articulate problem-solving steps using verbal representations. This study highlights the need for a holistic approach to mathematical education, integrating conceptual understanding with effective problem-solving strategies tailored to individual cognitive styles.

Perceptions of Malaysian University Mathematics Instructors of the Challenges they Face in Implementing Effective Distance Learning

Perceptions of Malaysian University Mathematics Instructors of the Challenges they Face in Implementing Effective Distance Learning

Unveiling the complexities of ‘Abstract Algebra’ in University Mathematics Education (UME): fostering ‘Conceptualization and Understanding’ through advanced pedagogical approaches

Unveiling the complexities of ‘Abstract Algebra’ in University Mathematics Education (UME): fostering ‘Conceptualization and Understanding’ through advanced pedagogical approaches

The mathematics pipeline in England: inclusion and the excellence stream

Abstract In England, there is currently heightened political interest in increasing mathematics attainment and maximizing post-16 participation. The latter is not merely an issue for upper secondary but requires a long-view of students’ mathematical progress. This paper reports such a system-level overview of the mathematics education pipeline in England based on analysis of linked data from the National Pupil Database (NPD) and Higher Education Statistics Agency (HESA). A sequence of mixed-effect logistic regression models at each phase of the education pipeline are developed to predict attainment in General Certificate of Secondary Education (GCSE), A level and undergraduate mathematics and participation in post-16 and university mathematics. In particular, we focus on the impact of prior attainment and student demographics: gender, ethnicity and socio-economic status. The analysis identifies four important findings. Firstly, female participation and attainment in A level Mathematics and Further Mathematics is lower than that of male students, which impacts their representation in undergraduate mathematics, despite them marginally outperforming male students at both GCSE and degree level. Secondly, the attainment and participation of students of Asian ethnicity in GCSE and A level Mathematics is striking in its positive divergence from the trends of other ethnic groups, whereas White students have higher attainment than other ethnic groups at university. Thirdly, many students from poorer backgrounds fall behind their peers during years 7 to 11, and their participation and attainment in A level Mathematics is also reduced compared to their more affluent peers. Lastly, students who study A level Further Mathematics are more likely to study undergraduate mathematics, but they are no more likely to achieve a first-class degree, compared to those who haven’t studied Further Mathematics. The findings should be informative to policymakers and other stakeholders who can put in place enhancements and interventions to improve the outcomes for underrepresented groups.

The meaning landscape of the concept of the total derivative in multivariable real analysis textbooks: an analysis based on a new model of meaning

AbstractIn this paper, we propose a novel conceptual framework tailored for modeling the meaning of mathematical concepts in university-level mathematics, addressing their rigorous nature and their relationships with related concepts as well as interpretations in various contexts. Within this framework, we present a model of meaning for the concepts of total differentiability and total derivative that provides a variety of possible interpretations and aspects. We then use the proposed model of meaning as a tool for analyzing three different textbooks for mathematics majors on the topic of multidimensional differentiability. Our paper is an example of a subject matter analysis of a topic in university mathematics carried out in a structured way. The model of meaning for total differentiability presented in this paper could inspire course design and analysis including the design of assignments and assessments for students. Moreover, it could serve as a valuable research tool for further analyses. For example, it could be used as a framework for analyzing courses taught or as a basis for developing a test instrument to assess students’ understanding. With our textbook analysis, we begin to examine the landscape of textbooks regarding differentiability concepts in the multidimensional case, shedding light on the diversity of meaning facets that are covered in the textbooks. These results could be useful for guiding instructors and learners in selecting and using textbooks for teaching and learning based on their respective needs.

基于OBE理念的线性代数混合式教学模式研究

线性代数作为高校公共数学的一门基础课程，学好该课程对于提升学生的学习和逻辑能力起着至关重要的作用。本文依据OBE教学理念，融合线上线下混合式教学，从教学目标、教学过程及考核评价等三个方面对线性代数课程教学模式进行介绍，以期实现教学质量提高、学生综合素养提升、毕业生满足社会人才需求等目标。Linear algebra is a basic course of public mathematics in colleges. Learning this course well plays a vital role in improving students’ learning and logical ability. Based on the OBE teaching concept and integrating online and offline blended teaching, this paper introduces the teaching mode of linear algebra course from three aspects of teaching objectives, teaching process and assessment. Through the aforementioned teaching process, we aim to achieve goals such as improving teaching quality, enhancing students’ comprehensive literacy, and ensuring that graduates meet the demands of the society for talents.

The Empowering Innovative Mathematics Curriculum and Industry Collaboration in Gearing Malaysia for IR4.0 Era

This paper aims to illustrate the global megatrend of IR4.0 and points out their impacts and gaps faced by today's industries and mathematics applications. Meanwhile, in responding to the MoHE Education Framework 4.0 plan and direction, this paper adopts the qualitative approach to review and investigate the needs of transformation in mathematics curriculum, gears up to the change of industry requirements and adapts to the shift of market demands and operations landscape. Therefore, the outcome of study urges to form a closer tie or partnership between industries and mathematics collaboration in industrial research and problem-solving. With this, the role of industrial mathematics has to become part of the necessity for Malaysia's leading mission to achieve the status of a developed nation. In conclusion, based on the missing links between mathematics and industries, the paper recommends the realignment and rebranding of existing university mathematics curriculum and formation of the National Industries and Mathematics Community (NIMC). It expects industrial mathematics will be one of the key boosters besides engineering and technologies fields, to produce the industries talents and expertise, which will contribute to Malaysia's mission in IR4.0, the migration of industries into high-value services and technological intelligence-based productions era.

The Measures of Cultivating Students' Mathematics Application Ability in College Mathematics Teaching

Under the background of the new situation, the comprehensive ability of advocating quality-oriented education and talents cultivation, puts forward higher requirements on college students, the mathematics as a compulsory course in university, its status is self-evident, it can effectively promote students' thinking ability, innovation ability, the progress of the ability to solve the problem, then, still need professional teachers attaches great importance to the application ability training, as education the main line, To improve the traditional teaching malpractice, create a good teaching environment. This paper takes mathematics application ability as the theme, discusses the existing situation and concrete solutions, hoping to provide opinions and suggestions for the education industry.

Development and analysis of a Numbas diagnostic tool for use in a mathematics refresher program

We describe the development and analysis of an online diagnostic tool implemented in the Numbas e-learning system and used in an Australian university mathematics refresher program. Following the rapid transition to online delivery of the refresher program caused by COVID-19, the diagnostic instruments and methods used within the pre-pandemic, in-person, version of the program were lost. In 2022, we undertook to revive them in a way that would honour their original diagnostic purpose but offer a more sophisticated approach utilising the Numbas diagnostic exam type. Improvement of the tool after its initial deployment has involved the use of Rasch-based item analysis and recursive refinement of the knowledge map underlying the items.

Students’ self-regulated learning of university mathematics in different learning environments

Self-regulated learning is important, due to the large share of self-study time for learning mathematics at university, but it is still underresearched.Theoretically and practically, it is of particular interest to understand patterns that lead to different self-regulated learning processes, in order to identify approaches to support students learning. In this study, we analyse students’ appraisals of different elements of three different learning environments in Germany (on-campus, online) and Finland (on-campus) for their congruence with students’ goals. According to Boekaerts’ dual processing model of self-regulated learning, such appraisals define students’ purposes of self-regulation. We analyse interview data of two students per learning environment, in each environment one with a focus on learning goals and one on just ‘surviving the course’. Our results provide a detailed description of the variety of the six students’ self-regulated learning processes as well as a synthesis of students’ appraisals of different elements of the learning environments, indicating the importance of students’ appraisals of the relevance of mathematical contents, implications of exercise assessments, and coping potential, especially in terms of peer learning. These results have practical implications for ‘supporting students’ self-regulated learning processes which are discussed.

“I’m a bit out of place here.”—Preservice teachers’ positioning in the figured world of university mathematics

Preservice mathematics teachers are sometimes trained in programs so that they have both their own courses and joint courses with mathematics majors from the beginning of their studies. While this is thought to provide them with both deep mathematical knowledge and teaching-specific content right from the start, many of them report disaffection with and disengagement from mathematics during their very first semester. Current approaches often frame this from an individual’s perspective, investigating cognitive and affective individual differences among students. In contrast, we aim to understand this issue from a sociocultural perspective, examining underlying social processes. In this study, we thus explored preservice teachers’ experiences of the mathematics component of their training, using Holland and colleagues’ theory of figured worlds as a theoretical lens. Three group interviews with 14 preservice higher secondary teachers in a common mixed setting in Germany (one course specific to preservice teachers, one general mathematics course together with major students) were analyzed. Our findings displayed that the preservice teachers experienced two dichotomous figured worlds of mathematics and mathematics teaching. Our further analysis of their positioning between these worlds provided new insights to explain their disaffection, disengagement, and consequent learning behavior. We discuss practical implications, focusing on different teaching systems and interventions in teacher education.

Professional collaboration in a lesson study with university mathematics professors

The article analyzes professional collaboration in a lesson study involving university mathematics professors, with the objective of contributing to professional development. The research was guided by the question, What principles promoted professional collaboration among university mathematics professors participating in a lesson study? The research focused on a lesson study involving university professors of mathematics and mathematics education over twelve weekly meetings of two hours each. The empirical material included the researchers’ field notes and the transcripts of the recordings of the lesson study sessions. A qualitative analysis, based on content analysis, revealed that collaboration involved: <i>professional sharing</i> and <i>decision-making</i>. The research shows that by fostering collaboration the lesson study mobilized different principles of teaching professionalism, such as knowledge that is basic to teaching, values and ways of conducting university teaching, a social and moral commitment to teaching, and teaching autonomy, contributing to the professional development of higher education professors.

“Mathematics is a battle, but I’ve learned to survive”: becoming a disabled student in university mathematics

AbstractIn university mathematics education, students do not simply learn mathematics but are shaped and shape themselves into someone new—mathematicians. In this study, we focus on the becoming of disabled mathematical subjects. We explore the importance of abilities in the processes of being and becoming in university mathematics. Our interest lies in how teaching and assessment practices provide students with ways to understand themselves as both able and disabled, as disabilities are only understood with respect to the norm. We analyse narratives of nine university students diagnosed with learning disabilities or mental health issues to investigate how their subjectivity is constituted in discourse. Our analysis shows how the students are shaped and shape themselves as disabled mathematicians in relation to speed in mathematical activities, disaffection in mathematics, individualism in performing mathematics, and measurability of performance. These findings cast light on the ableist underpinnings of the teaching and assessment practices in university mathematics education. We contend that mathematical ableism forms a watershed for belonging in mathematics learning practices, constituting rather narrow, “normal” ways of being “mathematically able”. We also discuss how our participants challenge and widen the idea of an “able” mathematics student. We pave the way for more inclusive futures of mathematics education by suggesting that rather than understanding the “dis” in disability negatively, the university mathematics education communities may use dis by disrupting order. Perhaps, we ask, if university mathematics fails to enable accessible learning experiences for students who care about mathematics, these practices should indeed be disrupted.

Exploration on Hybrid Teaching Mode of College Public Mathematics: Taking Linear Algebra Course as an Example

The rapid development of science and network technology has changed the way people study. Constructing a mixed teaching model by integrating online and offline teaching, and organic integration of knowledge, abilities, and comprehensive qualities, cultivate the mathematical literacy and comprehensive abilities of the students, and it can also efficiently improve teaching efficiency and quality. This work mainly expounds the following two aspects. The first aspect is analyzing the advantages and current problems of the hybrid teaching model in mathematics in college; the second aspect is introducing the application of hybrid teaching mode in linear algebra teaching through case studies.