Tertiary Mathematics Research Articles

Tertiary mathematics students: Why are they here?

Tertiary mathematics students: Why are they here?

Writing PISA-Like Mathematics Items: The Case of Tertiary Mathematics Instructors from a State University in the Philippines

<p style="text-align:justify">Mathematics test items in International Large-Scale Assessments (ILSAs) such as the Programme of International Student Assessment (PISA) and the Trends in International Mathematics and Science Study (TIMSS) are nested in contexts defined in their assessment framework (e.g., the Personal, Occupational, Societal, and Scientific contexts in PISA). This study followed the item-writing activities of four tertiary mathematics instructors in the Philippines as they constructed context-based mathematics items. They were tasked to write four items each, following a set of specifications for PISA content and context categories. The data consisted of transcripts from the focus-group discussion which was conducted days after the task. The transcripts were then analyzed using thematic analysis. The results of this study showed that the phenomenon of item-writing in the context of writing PISA-like mathematics items had two themes: the phases of item-writing and the dimensions of item-writing. Findings showed that the respondents struggled to find realistic contexts and that they engaged in a problem-solving task likened to solving a puzzle as they attempted to satisfy the content, context, and process categories in the table of specifications (TOS). This study contributes to filling in the research gap on item-writing activities, particularly those of mathematics teachers in the Philippines- a country whose recent mathematical performance in the PISA 2018, TIMSS 2019, and PISA 2022 was nothing short of dismal.</p>

Changes in mathematical skills among freshman engineering students

This study investigates the critical transition from secondary to tertiary mathematics education, focusing on students’ challenges and the impact of curriculum reforms in Sweden. Through an analysis of diagnostic test scores and admission data from Umeå University over the period 2006–2021, we examine the evolving mathematical competencies of incoming engineering students and the broader implications of these shifts for educational policy and student preparedness for higher education. Our findings indicate a significant improvement in students’ mathematical readiness in the latter years of the study, correlating with the implementation of curriculum changes and variations in admission statistics. This research documents the positive outcomes of these educational reforms on student preparedness. It places these findings within a global context, highlighting the universal challenges and responses in mathematics education transitions. The study underscores the complex dynamics of the secondary-tertiary transition and provides valuable insights into the specific mathematical areas where students have shown varying degrees of improvement.

Identifying mathematics teachers’ competency to look at elementary mathematics from an advanced standpoint: a pilot study

This study evaluates the impact of “the ability to look at elementary mathematics from an advanced standpoint (EMFAS),” which was originally proposed by Felix Klein, in mathematics education. By strengthening the connection between elementary and tertiary mathematics, mathematics teachers are expected to have advanced knowledge to offer students higher mathematical insights. Thus, EMFAS serves as a great indicator for measuring teaching effectiveness. By investigating newly qualified and pre-service teachers’ ability to apply EMFAS techniques, we analyze the logic involved in solving some specially constructed problems and how teachers could deliver mathematical knowledge to students at the elementary level with the advanced knowledge they possess. We explain the importance of the EMFAS course, in terms of pedagogy and content knowledge. This course prepares pre-service mathematics teachers by equipping them with advanced knowledge for their future profession. This study evaluates mathematics teachers’ proficiency in applying EMFAS techniques via a problem set and interviews. We summarize the hierarchical nature of how teachers perceive EMFAS, which could play an important role in guiding pedagogy in mathematics education.

Development and implementation of Concept-Test questions in abstract algebra

In tertiary mathematics courses, students often have difficulties acquiring an understanding of the mathematical concepts covered. One approach to address this problem is to implement so-called Concept-Tests. These are multiple-choice questions whose distractors represent common problems and misconceptions related to the concepts. While there exist lots of such questions for calculus, Concept-Test questions focusing on basic concepts of abstract algebra are still rare, although previous research has shown that students have many problems with these. We therefore developed such questions for important concepts of basic group and ring theory in the years 2020–2022. In this paper, we first want to present the questions and the developmental process. Furthermore, we want to present an empirical study investigating to what extent the questions helped students in a proof-oriented abstract algebra course to acquire an understanding of the concepts covered. This study especially indicates that the developed Concept-Test questions provided good starting points for conceptual changes.

Review of Practice-oriented Research in Tertiary Mathematics Education by Biehler et al. (2022)

Review of Practice-oriented Research in Tertiary Mathematics Education by Biehler et al. (2022)

Career motivations and interest in teaching of tertiary students taking mathematics and science subjects

ABSTRACT Many countries face mathematics and science teacher shortages. There is a need for greater knowledge about eligible individuals’ potential interest to teach and career motivations. We explored career motivations and interest in a teaching career, among 470 tertiary mathematics and science students from four Australian universities. Students with high interest in teaching were distinguished from those with low interest, by high ‘altruistic’ motives, positive prior experiences, career progression prospects, family flexibility and working in a team. They also held more positive perceptions of the status, salary and expertise of teaching. Policy initiatives likely to attract them into teaching were identified, of which those most endorsed were jobs combining teaching with other work, increased permanency and salary, and paid study leave.

“But this is not mathematics!”—mathematicians and secondary teachers explore the affordances of tertiary mathematics for teaching secondary probability

“But this is not mathematics!”—mathematicians and secondary teachers explore the affordances of tertiary mathematics for teaching secondary probability

Epistemological beliefs, motivational values, and study habits among students of tertiary mathematics

Epistemological beliefs about the nature of mathematics affect how a student perceives mathematics in general, whereas high motivation for mathematics gives an individual a personal reason to study the subject. Epistemological beliefs and achievement motives are quite stable at individual level, but previous research has shown that there are significant differences across students with respect to them. This study concerns university students from three different study programs and investigates relations between the students' epistemological beliefs, motivational values, and study habits. The data were collected using a questionnaire and consist of responses from altogether 98 students studying mathematics courses at tertiary level. The study revealed that, when students are grouped according to their study programs, their motivational values vary across these groups more than their epistemological beliefs. The stability of epistemological beliefs was also verified in this study as the strengths of the beliefs hardly correlated with the number of passed credits or self-evaluated study success. However, several sum variables representing the motivational values and epistemological beliefs seem to predict certain study habits in linear regression models. These models can help us to design the teaching of mathematics courses favouring these study habits.

Students’ Strategic Usage of Formative Quizzes in an Undergraduate Course in Abstract Algebra

Besides homework assignments, optional quizzes are a commonly used means for formative assessment in tertiary mathematics education. Instructors, for example, implement these to help students detect gaps in their understanding, or to foster a continuous and active engagement with the content. The extent to which these goals are reached, however, strongly depends on how students actually use such quizzes, which is currently an underexplored topic. We investigated this issue in an undergraduate abstract algebra course with a study using a mixed-methods design. Unlike previous research suggesting that students use optional quizzes in tertiary mathematics courses mainly for rehearsal or for a final self-check of their own performance, our study indicates that students also use such quizzes in many other ways, for example for planning their further learning, or for deepening their understanding of the content of the course. Furthermore, our study shows differences regarding students’ quiz usage during the semester and when preparing for the final exam. Finally, the data propose factors that influence the way students use optional quizzes, for example time constraints due to other obligations during the semester, the perceived difficulty of the questions, or the opportunity to discuss these with peers. This leads to several suggestions regarding the implementation of optional quizzes into tertiary mathematics courses.

Transforming Mathematics Instruction Via Primary Historical Sources: A Study of Influential Factors on Implementation of a Curricular Innovation at the Tertiary Level

Abstract The TRansforming Instruction in Undergraduate Mathematics via Primary Historical Sources (TRIUMPHS) project is producing primary source projects (PSPs) for use in teaching core topics from the tertiary mathematics curriculum and investigating the resulting teaching and learning experiences. This article examines factors related to attributes of PSPs, TRIUMPHS’ dissemination and implementation support strategies, and characteristics of individual instructors that have influenced the incorporation of PSPs into their ordinary practice via case studies representative of two types of end users: those who have thus far limited their PSP usage to a single course and those for whom PSP usage has permeated their instructional practice. Findings affirm that pedagogical orientations within individuals’ instructional practices were critical to both initial use and eventual adoption of PSPs, while the observed differences in usage patterns between the cases were influenced by the perceived relative advantages offered by the core features of PSPs and individuals’ intrinsic motivation. The impact sheet to this article can be accessed at 10.6084/m9.figshare.21205292.

Secondary pre-service teachers’ voices of becoming mathematics teachers

ABSTRACT This paper captures and reports on mathematics pre-service teacher (PST) voices. Mathematics PSTs, in a tertiary mathematics secondary curriculum teaching subject assessment, were asked to perform a small-scale literature review on an issue they identified as important in mathematics teaching. The assessment, written as an article, required literature-based recommendations, a reflection on how this would impact the PST’s future teaching and guidance for all mathematics teachers. The article was aimed for publication by the Mathematics Association of Victoria—a state-based mathematics professional association in Australia. PSTs were advised that articles would potentially be published, and their voice would reach practicing mathematics teachers. The published articles, which capture pre-career mathematics teacher voices were analysed for insight into what four PSTs said of their future teaching to a mathematics teacher audience. The professional teacher standards were recognised to influence comments given the strong emphasis of standards within initial teacher education courses. Findings speak to what mathematics PSTs expect of their future teaching role prior to being in-service. The paper contributes a snapshot of early established professional voices in a discipline where there is currently a shortage and highlights the messages that emerged.

Pre-service Teachers’ Perceptions of Problem-based Learning for Developing Their Mathematics Teaching Pedagogy

The study reported in this paper was part of a larger study that explored pre-service teachers’ perceptions of the effective-ness of a problem-based learning (PBL) teaching approach for developing their mathematics pedagogical content knowledge (PCK) and their ability to enact their PCK in a tertiary mathematics education subject. This paper reports the qualita-tive semi-structured interviews used to capture the student voice with respect to the impact of the PBL approach used on the development of their mathematics PCK in comparison to their previous experiences with teacher-directed instruction. Overall, responses from the interviews revealed the pre-service teachers considered the PBL method used in this study was a more effective way to learn compared to being taught using a teacher-directed instructional approach, and further, that they will most likely use PBL when they become teachers.

THE USE OF PERSONAL RESPONSE SYSTEMS TO RENEGOTIATE THE DIDACTICAL CONTRACT IN TERTIARY MATHEMATICS EDUCATION

Challenges experienced by first-year students transitioning from secondary to tertiary mathematics education are examined through the lens of the didactical contract. The didactical contract describes the expectations of both lecturer and students about their mutual obligations towards teaching and learning. First-year students’ beliefs about the nature of mathematics and mathematics teaching/learning need to be challenged to renegotiate the didactical contract at tertiary level. The study focuses on how to elicit and confront transitioning students’ beliefs in order to support their learning and influence a shift in the didactical contract. A Likert scale questionnaire was deployed at the beginning of students’ first year to gauge their beliefs about mathematics and mathematics teaching/learning and redeployed near the end of the first semester (or term) to observe possible changes in their beliefs and hence the didactical contract. The intervention consisted of personal response system (PRS) sessions regularly incorporated into the traditional transmission mode lecture to flip the classroom and create a student-centred learning environment, aimed at influencing students’ beliefs in order to make them aware of their own learning and their responsibility for learning. Questionnaire data were quantified and compared for the before and after surveys. There is evidence of a shift towards students taking ownership of their learning and a renegotiation of the didactical contract. Qualitative data generated by focus group interviews confirm the role of the PRS sessions in influencing student beliefs and the didactical contract.

Clash of cultures? Exploring students’ perceptions of differences between secondary and tertiary mathematics education

Research has identified several aspects that influence students’ transition to mathematics studies at university, but these aspects have often been studied separately. Our study contributes to the field's understanding of the transition between upper secondary and university mathematics by taking a multifaceted perspective not previously explored. We analyse experiences and attainment in mathematics of 154 engineering students with respect to known aspects of this transition, and our results show that it is important to consider several aspects together in order to understand the full complexity of the transition. It is revealed that students with previous experiences of university studies, when compared with new first year undergraduates, perceive a larger difference between studying mathematics at the upper secondary level and university. Our results also show that the engineering students enrolled in distance programmes experience larger differences between secondary and tertiary levels than engineering students enrolled in campus programmes. Furthermore, our analyses show that students’ success in mathematics is related to their perceptions of the rift experienced in the transition. In all, our results highlight the importance of taking a student perspective in the development of explanatory and useful models of students’ transition between upper secondary and university mathematics.

Online Support for Tertiary Mathematics Students in a Blended Learning Environment

Online Support for Tertiary Mathematics Students in a Blended Learning Environment

Predicting how a disrupted semester during the COVID-19 pandemic impacted student learning

<p style='text-indent:20px;'>Tertiary education faced unprecedented disruption resulting from COVID-19 driven lockdowns around the world, leaving educators with little understanding of how the pandemic and consequential shift to online environments would impact students′ learning. Utilising the theoretical framework of a student′s <i>affective field</i>, this study aimed to investigate how student achievement, achievement-related affect, and self-perceived well-being contributed to predicting how their learning was impacted. Questionnaire responses and academic achievement measures from students (<i>N</i> = 208) in a New Zealand second-year, tertiary mathematics course were analysed. Despite a return to in-person teaching after eliminating community-transmission of the virus, students reported larger impacts of the disruption to semester on both their learning and well-being at the end of the term than during the lockdown. Hierarchical multiple regression revealed that gender, prior achievement, performance on low-stakes assessment, as well as exam-related self-efficacy and hope, made significant, independent contributions to explaining students′ perceived learning impact. Even when controlling for achievement and achievement-related affect, students′ perceived impact to their well-being made a significant and substantial contribution to the impact on their learning. The findings provide motivation to further investigate whether attempts to address student achievement-related affect can help mitigate the effects of major life disruptions on studying. We suggest that frequent, low-stakes assessment can identify students who are more likely to report greater negative impacts to their learning. We finally conclude that student well-being is paramount to how students perceive their own learning, even when controlling for actual measures of and about their achievement.</p>

Traditional lectures versus active learning – A false dichotomy?

Traditional lectures are commonly understood to be a teacher-centered mode of instruction where the main aim is a provision of explanations by an educator to the students. Recent literature in higher education overwhelmingly depicts this mode of instruction as inferior compared to the desired student-centered models based on active learning techniques. First, using a four-quadrant model of educational environments, we address common confusion related to a conflation of two prevalent dichotomies by focusing on two key dimensions: (1) the extent to which students are prompted to engage actively and (2) the extent to which expert explanations are provided. Second, using a case study, we describe an evolution of tertiary mathematics education, showing how traditional instruction can still play a valuable role, provided it is suitably embedded in a student-centered course design. We support our argument by analyzing the teaching practice and learning environment in a third-year abstract algebra course through the lens of Stanislas Dehaene's theoretical framework for effective teaching and learning. The framework, comprising 'four pillars of learning', is based on a state-of-the-art conception of how learning can be facilitated according to cognitive science, educational psychology and neuroscience findings. In the case study, we illustrate how, over time, the unit design and the teaching approach have evolved into a learning environment that aligns with the four pillars of learning. We conclude that traditional lectures can and do evolve to optimize learning environments and that the erection of the dichotomy 'traditional instruction versus active learning' is no longer relevant.

Determining the pedagogical value of code-switching: functions of teacher code-switching in tertiary level mathematics classrooms

ABSTRACT Higher education institutions in the Philippines are directed to teach mathematics in only English or Filipino, implying that code-switching for instruction is neither promoted nor encouraged despite teachers’ and students’ multilingualism. This is true for other multilingual countries with English as an official language. However, literature shows that teachers in multilingual classrooms code-switch to deliver instruction. This study determined whether classroom code-switching (CS) had any pedagogical value by identifying the functions of tertiary-level mathematics teachers’ Tagalog-English CS. Data from four mathematics teachers and their students in a Philippine state college were gathered through interviews, surveys, and class observations. Data were categorised according to the CS functions of Halliday and Matthiessen. Results warranted the creation of fourteen function subtypes and a new function, communicative. Findings show that teachers code-switched for ideational, textual, interpersonal, and communicative purposes. The ideational function is the most elaborated, confirming that CS was used primarily for mathematics pedagogy. Some CS instances were not mainly pedagogical but ultimately contributed to communication and instruction. Moreover, data highlighted the polarity of teachers’ CS, revealing that it can advantage or disadvantage instruction and rapport. Hence, teacher training should utilise findings and develop teachers’ CS awareness to achieve principled pedagogical CS.

The pandemic as a catalyst for rethinking active learning practices in technology intensive instructional environments

Even though K-12 mathematics educators have embraced the role of graphing calculators and computers, student use of technology in tertiary mathematics has been slow to implement in the USA context. Enter March 2020, a global health crisis, and the shift to online and remote learning. The immediate and dramatic cessation of in-person modes of instruction was experienced at all levels of education and in all disciplines. With respect to tertiary mathematics, student use of instructional technology shifted from optional use of homework websites, digital discussion boards and flipped classrooms to an absolute necessity to support student participation in mathematics courses. Our research group has been studying the process of departmental change towards active learning in the undergraduate calculus sequence. During the COVID-19 pandemic, we worked with twenty-three university mathematics departments in the United States as part of a Networked Improvement Community (NIC). During the pandemic, we observed the adaptation of instructional practices to sustain student interaction in remote virtual environments. This paper highlights changes observed within our NIC regarding attempts to sustain active learning during the global pandemic. In addition, a NIC can provide the necessary community for grappling with challenging problems at any time.