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This paper presents an example of how paper folding can be used in a geometry class to support conceptual understanding. Specifically, it explains an activity that constructs a rhombus and explores its attributes by using paper folding. The steps of constructing a rhombus are described and some discussion questions are given to consolidate understanding. [ABSTRACT FROM AUTHOR]

The combined use of origami and dynamic geometry software has recently appeared in mathematics education to enrich students' geometric thinking. The objective of this research is to study the roles played by the interaction of two artifacts, paper folding and GeoGebra, in a construction-proving problem as well as its generalization in the Euclidean geometry context. For this, we designed and implemented two mathematical tasks with 52 secondary education students (15–16 years old, 10th grade) during the COVID-19 emergency lockdown period in Italy. The tasks involved four phases: constructing, exploring, conjecturing, and proving. This article presents an epistemic analysis of the tasks and a cognitive analysis of the answers given by one of the students. The theoretical tools of the onto-semiotic approach supported these analyses. Cognitive analysis allows us to confront the intended meanings of the task and the meanings actually employed by a student, thus drawing specific conclusions about the roles of such artifacts in written arguments and give an interpretation of their combined use in mathematics education. [ABSTRACT FROM AUTHOR]

The student experience with mathematical modelling has the potential to differ in nature to the experience of a typical mathematics student. The research reported in this paper forms part of a broader study looking into tertiary learning experiences for mathematical modelling. This paper reports on: What are the student experiences when learning to mathematically model? How might these experiences inform mathematics education? Data were collected across two different tertiary mathematical modelling courses in the form of student interviews. Student interviews were analysed, using reflective thematic analysis, to identify themes relating to the collective student learning experience. The results show that modelling provides an opportunity for students to be actively involved with their learning. Being open-minded was a key behaviour for a productive student experience. [ABSTRACT FROM AUTHOR]

In this paper, I discuss the potential transformations to teach K-12 geometry topics with the technology of 3D Pens, which enabled 3D models to be created instantly via one's moving hands. I draw on an example of using constructionist learning activities with 3D Pens to teach young children, 'how tall is the triangle?' [ABSTRACT FROM AUTHOR]

It is generally agreed that making real-world connections in mathematics teaching increases students' motivation and interest and contributes to meaningful and permanent learning. In this paper we propose a simple and fast activity to find a rectangular hyperbola in real life and we show how to operate the data to retrieve a straight line. Since it deals with simple formulas, we believe that it can be useful in a high-school context, either in math or physics class. The activity consists of assembling a wedge with transparent plates (like glass photo frames), dipping it on a liquid and observing the rise of a hyperbola. Instructions are given on how to obtain the coordinates of the liquid/air interface and how to replot them as a straight line, using just a smartphone. For the physics class, an extension is proposed to measure the surface tension of the chosen liquid. [ABSTRACT FROM AUTHOR]

This paper discusses findings from an ongoing study investigating mental mechanisms involved in the conceptualisation of linear transformations from the perspective of Action (A), Process (P), Object (O), and Schema (S) (APOS) theory. Data reported in this paper came from 44 first-year linear algebra students' responses on a task regarding the range of a linear transformation. Our analysis revealed parallels between Levels/Stages of the range concept and the use of representations of matrix multiplications. More importantly, these representations appeared to have been the enablers of transitions from lower to higher APOS Stages for the range. Conversely, mental mechanisms employing other means showed little to no progressions, some, furthermore, revealed faulty knowledge structures. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to follow the reasoning of high school students when asked to explain the standard trigonometric limit lim θ → 0 ⁡ sin ⁡ θ θ = 1. An observational study was conducted in four different phases in order to investigate if visualization, by means of an interactive technology environment (Geogebra), can contribute in lifting high school students' understanding from a mere procedural understanding to a combination of conceptual and procedural understanding. The obtained results confirm that the students were able to show a conceptual understanding only after using the digital interactive tool. Through comparing, exploring and self-explaining combined with the use of the interactive tool, the students managed to link different concepts together. The students were able to see and interpret the reason making the angle θ and sin ⁡ θ relate under certain conditions, thus leading to the standard trigonometric limit lim θ → 0 ⁡ sin ⁡ θ θ = 1. [ABSTRACT FROM AUTHOR]

The aim of this paper is to present a teaching proposal for the theoretical part relating to the first- and second-order linear difference equations with constant coefficients suitable for the first-year students at various types of universities. In contradistinction to the methods often applied (memorization of algorithms without a proper conceptual understanding), we provide an alternative methodological way-out for lecturers that is partially based on posing suitable questions and constructing the knowledge. Our approach is therefore characterized by the minimal input knowledge imposed on students. In fact, we assume that students are acquainted only with the summation of terms of geometric sequences and with solving linear and quadratic equations. [ABSTRACT FROM AUTHOR]

A carefully constructed method of assessment using short, formative, invigilated in-tutorial quizzes is investigated to see how successful this method is compared to previous assessment tasks. We look at whether student engagement increased, given that students who enter university often face issues with the transition from school, and engagement is a factor affecting their success. First-year mathematics courses see a wide variety of students from various mathematical backgrounds who may not know or have little experience with, how to approach their tertiary study. This paper presents a way of using assessment to engage and motivate students who are studying first-year mathematics at university. Assessment is an essential feature of student learning and it is important to use it effectively to benefit the education and learning practices of students. Our study focuses on a particular method of assessment which we found to be successful. We will introduce this method and analyse its effectiveness using both quantitative and qualitative data obtained from a cohort of students enrolled in a large first-year mathematics unit in an Australian university. Furthermore, we give the academic a list of guidelines should they wish to adapt this assessment into their own teaching practices. [ABSTRACT FROM AUTHOR]

This paper extends work in the areas of quantitative reasoning and covariational reasoning at the undergraduate level. Task-based interviews were used to examine third-semester calculus students' reasoning about partial derivatives in five tasks, two of which are situated in a mathematics context. The other three tasks are situated in real-world contexts (i.e. weight, revenue, and mortgage). There are three main findings from this study. First, interpreting quantities representing partial derivatives in real-world contexts was problematic for most of the students. Second, students' interpretations of partial derivatives were not consistent across tasks. Third, all the students engaged in several levels of covariational reasoning when interpreting quantities representing partial derivatives in the context of weight, revenue, and mortgage. Implications for calculus instruction and directions for future research are included. [ABSTRACT FROM AUTHOR]

This study aims to uncover some key elements of successful home-learning of mathematics, based on students' perceptions, during the COVID-19 pandemic and offer recommendations for mathematics learning beyond the crisis. Throughout our work, we aimed to examine students' reflections on mathematics learning and learning environments that assisted their ease of transition from in-school face-to-face learning to home-schooling during the COVID-19 isolation. In this paper, we will present how mathematics learning has changed for upper secondary school students in Austria and outline students' perspectives in relation to their new learning experiences during the period of school closures. Through the use of case study principles enhanced by some design-based research approaches, we were able to illustrate that (a) familiarity with the context, (b) problems or tasks as learning triggers, (c) mathematics learning as a social as well as individual process, and (d) perceived positive cost-benefit analysis of learning mathematics are key to students' learning success. Although these categories were key for the students when learning mathematics in home-schooling during the current crisis, these categories could be equally considered when education moves back to school or in mixed learning environments post COVID-19. [ABSTRACT FROM AUTHOR]

This paper presents a case study of two sixth-grade students' use of an iPad as an instructional tool for mathematics. Based on their written and oral responses, we investigated and analyzed their meaning making process with mathematical content in a classroom where the iPad was a central tool for teaching practices. The analyses were based on Steinbring's [(2005). The construction of new mathematical knowledge in classroom interaction: An epistemological perspective (Vol. 38). Springer] framework, which we applied to understand how students construct meaning for mathematical concepts. The analysis showed that the students constructed mathematical knowledge by forming associations between contexts represented by the activities and by mathematical symbols and signs. The findings suggest the possibility that touchscreen technology reinforces the link between mathematical content, mathematical representations, and tangible experiences. [ABSTRACT FROM AUTHOR]

Diagnostic testing on entry to university provides an opportunity for students to ascertain individual strengths and weaknesses and allows course teams to gain insight into the competencies of the whole cohort. Coventry University has been using the same diagnostic test since 1991 and a previous analysis showed students coming to Coventry University in 2001 were performing considerably worse than their counterparts in 1991. Given the changes to A level Mathematics that have taken place since 2001 and the recently implemented reforms to A level Mathematics in the UK, it is timely to review performance in the diagnostic test and to create a benchmark before students enter higher education with the new (linear) A levels. This paper therefore reports on the changing entry competencies of students at Coventry University from 2001 to 2017 and shows that there has been a general improvement in performance across all entry grades. However, students are still not performing at the same level as they were in 1991 and the gap between students with higher entry grades (A level A*–C) and lower entry grades (A level D–E) is getting wider. [ABSTRACT FROM AUTHOR]

Flipped classrooms are increasingly popular due to improvements in online technology and expected benefits in student engagement and learning. Although research has increased dramatically in the last 10 years, few studies have assessed the effectiveness of flipping large enrollment, barrier mathematics courses. Creating a comprehensive flipped classroom learning experience that includes high quality in-class activities can be particularly challenging in large classrooms, especially when implemented for the first time. However, improving engagement and learning in these courses would be particularly helpful in improving STEM graduation rates. This paper describes a longitudinal analysis of student performance and opinions in a differential equations course over three years with three different instructional strategies: traditional lecture, initial flipped, and revised flipped. We found that flipping the course reduced withdrawal rates and improved pass rates, but the initial iteration reduced exam performance. We then observed a positive change in exam performance and student opinion from the initial to the revised flipped course. We conclude that the flipped classroom can be effective in large mathematics courses required for STEM degrees, although it may not be possible to see all benefits in an initial implementation. [ABSTRACT FROM AUTHOR]

In application-oriented mathematics, particularly in the context of nonlinear system analysis, phase plane analysis through SageMath offers a visual display of the qualitative behaviour of solutions to differential equations without inundating students with cumbersome calculations of the plane-phase. A variety of examples is usually given to illustrate phase-plane behaviour. We approach these problems by considering a problem containing a single real parameter that exemplifies the various situations clearly and simply. We developed two computer programs in SageMath: one program calculates aspects of the Jacobian matrix and displays the phase plane portraits, the other determines the centre manifold. Computations and images generated with computer codes are useful in understanding dynamic models in biology, physics and engineering that involve planar non-linear autonomous differential equations. This paper covers analytical and computational skills that are helpful for students and teachers. [ABSTRACT FROM AUTHOR]

It is generally accepted that teachers' beliefs impact their instructional choices, but characterizations of that relationship are limited in college settings. Furthermore, examinations of instructor beliefs, instruction, and student beliefs together in one setting are rarely described. Based on interviews with two Abstract Algebra instructors, classroom video from three units of instruction, and survey and interview data from students in the classes, this paper examines instructors' stated beliefs, ways these beliefs manifested in their teaching, and students' beliefs across the course. Both instructors made curricular choices clearly aligned with their stated views of the nature of mathematics, learning, and teaching. Day-to-day instructional choices reflected these stated beliefs as well, but the difficulty of material and tensions with other beliefs like the importance of interactivity manifested. Characterizations of the interactivity of classes and placement of the mathematical authority in class are provided through descriptive and quantitative measures. These characterizations of instruction provide nuanced portrayals of classroom norms and changes in those norms throughout the semester. Furthermore, subtle shifts in student beliefs about teaching and learning are noticeable, suggesting students' beliefs about teaching and learning mathematics can be influenced even by modest changes in instructional practice. [ABSTRACT FROM AUTHOR]

The topic of recurrence relations has recently been introduced in many discrete mathematics textbooks. Recurrence relations are efficient modelling and problem-solving techniques used in mathematics. Combinatorial problems are often used to introduce recurrence relations. However, many textbooks consider problems that can be reduced only to the recurrence relations of the first or second order. To overcome this problem, in this paper, we propose a general combinatorial problem of path tiling with tiles of different sizes and colours. For especially selected tile sizes and for certain tile colours, the problem can be reduced to solving a recurrence relation of the third, fourth, or higher orders. Furthermore, through actual examples, we show that various cases can be derived from the roots of a characteristic equation: different real roots, different real and complex roots, and repeated real roots with different multiplicities. We hope that the presented problems will provide teachers with the opportunity to present this topic with better content. [ABSTRACT FROM AUTHOR]

In this paper we report on the outcomes of two surveys carried out in higher education institutions of Ireland; one of students attending first-year undergraduate non-specialist mathematics modules and another of their lecturers. The surveys aimed to identify the topics that these students found difficult, whether they had most difficulty with the concepts or procedures involved in the topics, and the resources they used to overcome these difficulties. In this paper we focus on the mathematical concepts and procedures that students found most difficult. While there was agreement between students and lecturers on certain problematic topics, this was not uniform across all topics, and students rated their conceptual understanding higher than their ability to do questions, in contrast to lecturers’ opinions. [ABSTRACT FROM AUTHOR]

The present paper describes beautiful conservation relations between areas formed by different geometrical shapes and area relations formed by algebraic functions. The conservation properties were investigated by students at an academic college of education using a computerized technological tool and were subsequently accompanied by justified proofs. [ABSTRACT FROM AUTHOR]

This paper offers insight into teachers' perspectives on the role of manipulatives in the mathematics classroom on either side of the primary to post-primary transition (sixth class teachers in Irish primary schools and first year mathematics teachers in post-primary schools) to examine if discontinuities exist around their usage. The transition from primary to post-primary education is considered the most difficult of all educational transitions that students face, with negative effects more pronounced in the subject of mathematics. A questionnaire involving several open-ended questions was distributed to teachers teaching students in the final year of primary school and teachers teaching mathematics to students in the first year of post-primary education. Results of the qualitative data analysis reveal insights into teachers' perceptions of the benefits, supports and obstacles to manipulative use. It reveals that, in general, teachers on either side of the boundary crossing are in agreement regarding the perceived benefits of using manipulatives and also the potential factors that inhibit their usage. On the other hand, their opinions regarding the factors that support the use of manipulatives are more divergent. [ABSTRACT FROM AUTHOR]

This paper presents research into closed orbits parallel to quadrilaterals inscribed in various geometric shapes that can be represented by mathematical functions: straight lines, circles, ellipses, parabolas, and hyperbolas. Mathematical proofs have been given for the existence of an infinite number of parallel orbits for each of these shapes. Quadrilaterals parallel to a quadrilateral inscribed in a circle were found to have two interesting characteristics. The mathematics was conducted with student mathematics teachers and teacher trainees, accompanied by dynamic research using computer software. [ABSTRACT FROM AUTHOR]

The goal of this paper was to investigate the role of formal constraints (e.g. definitions, theorems) in geometric reasoning. Four students participated in a task-based interview including 2D Euclidean geometric locus problems. Data were obtained from observations, interviews, and video recordings and analyzed by Toulmin's argumentation model. The analysis of the data provided a detailed look at students' abductive, inductive, and deductive reasoning. Results revealed that formal constraints reduced the uncertainty of a claim. Students could transform abduction and induction to deduction if they could reverse the array of formal constraint usage. The co-existence of warrant and backing including formal constraints facilitated this transfer and this co-existence in reversing their usage was the key factor to eliminate the challenge of transforming abduction to deduction. [ABSTRACT FROM AUTHOR]

Many students do not have a deep understanding of slope. This paper defines what a deep understanding of slope is in terms of mathematics-education theory. The various factors which help explain why such a deep understanding is difficult to acquire are then discussed. These factors include the following: the different representations for slope; graphical understanding; ratio and rate; and proportional reasoning. In light of this discussion, the design of a mathematical resource to help give students a deep understanding of slope is described. The resource was used at one university, amongst Chinese first-year undergraduate science and engineering students. Evidence would suggest that those students who did not evidence a correct understanding of all the aspects of slope (deep understanding) before completing the resource improved their understanding of these aspects upon completion of the resource. [ABSTRACT FROM AUTHOR]

Motivated by a problem proposed in a coding competition for secondary students, I will show on this paper how coding substantially changed the problem-solving process towards a more experimental approach. [ABSTRACT FROM AUTHOR]

An introduction is presented in which discuss various articles in the issue on topics including relation between inquiry-based learning, equity and student experiences; enrichment in first-year courses for academically strong mathematics students; and issues surround mathematics education.

This research investigates the influence that gender, single-sex and co-educational schooling can have on students’ mathematics education in second-level Irish classrooms. Although gender differences in mathematics education have been the subject of research for many years, recent results from PISA (Programme for International Student Assessment) show that there are still marked differences between the achievement and attitude of male and female students in Irish mathematics classrooms. This paper examines the influence of gender in more detail and also investigates the impact of single-sex or co-educational schooling. This is a follow on study which further analyses data collected by the authors when they designed a pedagogical framework and used this to develop, implement and evaluate a teaching intervention in four second-level Irish schools. The aim of this pedagogical framework was to promote student interest in the topic of algebra through effective teaching of the domain. This paper further analyses the quantitative data collected and investigates whether there were differences in students’ enjoyment and achievement scores based on their gender and whether they attended single-sex or co-educational schools. [ABSTRACT FROM PUBLISHER]

The school system in both the Republic of Ireland [ROI] and Northern Ireland [NI] is configured to accommodate children into discrete educational bands at different stages on their learning journey. Research has consistently shown that the transition from primary to post-primary, which typically occurs between the ages of 11 and 13, is the most challenging transition with the negative effects more pronounced for mathematics than any other subject. This cross-border study investigates this transition in mathematics education from the final year of primary school to first year of post-primary school from the perspective of teachers involved with these year groups. It examines teachers' knowledge of the mathematics curriculum and teaching strategies employed in the other phase. The majority of primary teachers reported being unfamiliar with the curriculum and teaching approaches employed in the first year of post-primary and the differences between the two jurisdictions were not significant. However, a difference was noted in the responses of second level teachers. Teachers in NI appeared more confident in their knowledge of the curriculum and teaching approaches adopted in the final year of primary school than their ROI colleagues. In this paper, the authors will elaborate on these findings; outline possible reasons for the differences across jurisdictions and share what impact teacher knowledge is having on classroom practices. [ABSTRACT FROM AUTHOR]

This paper compares Revised Bloom's Taxonomy (RBT) and some of the major theories and frameworks that influence the teaching, learning, and assessment of mathematics. This comparison has been made to help students and researchers make decisions about which of the available theories and frameworks might best suit their study. The comparison identifies that RBT has greater potential for analysing the teaching, learning, and assessment of mathematics than many other theories and frameworks because of its broad approach. While other theories and frameworks can be aligned to elements of RBT, they tend to overlook other elements. In particular, the strength of RBT comes from its two dimensional structure, its inclusion of metacognitive knowledge, and its rejection of strict hierarchy. [ABSTRACT FROM AUTHOR]

This work refers to a research project named 'Math walks', whose aim is to investigate a few students' common conceptions and difficulties related to symmetries as well as to guide the student in the transition from visualization to conceptualization. As far as this issue is concerned, we designed a learning activity (LA) according to Dina van Hiele-Geldof and Pierre van Hiele's theory of the development of geometric thought and we have experimented LA with sixth-grade students. Data show that LA seems to be effective in revealing students' difficulties about symmetries, on the one hand, and in providing students scaffolding to overcome them, on the other. From the data collected in the experimentation, new research questions arise. [ABSTRACT FROM AUTHOR]

In this paper, we report on the results of an experiment in teaching discrete mathematics to students majoring in business informatics. We supplemented our problem-based approach to teaching the course with a set of Likert-scale surveys or questionnaires that helped improve the students' performance. On the one hand, these surveys gave us feedback and, on the other, encouraged the students to reflect on the subject-matter. The experiment was quite successful, as the grades obtained by the students on the exam were significantly higher than usual. Here, we describe the structure of the surveys and the method of evaluation of the experimental results. [ABSTRACT FROM AUTHOR]

This paper compares sections on functions and linear functions from two Korean textbooks and an American standards-based textbook (University of Chicago School Mathematics Project [UCSMP] Algebra) to understand differences and similarities among these textbooks through horizontal and vertical analyses. We found that these textbooks provide different opportunities to learn (OTL). UCSMP Algebra places strong emphasis on real-life applications of linear functions rather than on pure mathematics andmathematical algorithms. Also, compared to Korean textbooks, UCSMP Algebra offers more OTL for students to solve, explain, and reason about higher level cognitively demanding mathematics problems than Korean secondary text books. Contradictory results, compared to previous studies about East Asianmathematics textbooks indicate the need for further study to compare secondary textbooks from more East Asian countries. Also, with the results of this study, we need to understand the results of international assessments more carefully. [ABSTRACT FROM AUTHOR]

This paper documents the author’s adaptation of team-based learning (TBL), an active learning pedagogy developed by Larry Michaelsen and others, in the linear algebra classroom. The paper discusses the standard components of TBL and the necessary changes to those components for the needs of the course in question. There is also an empirically controlled analysis of the effects of TBL on the student learning experience in the first year of TBL use. [ABSTRACT FROM PUBLISHER]

In Peru, national assessments repeatedly rank Indigenous mathematics students as the lowest performing across the entirety of Latin America and South America. Whereas lack of financial resources often predicts low measures, the history of educational practice teaches us that students' poverty need not predict their educational outcomes – creative instructional approaches may turn the tables. Here we report on an innovative, body-based arithmetic technique, the Botetano Method, that has been enabling poverty rural children from remote mountainous regions of Peru to match and even greatly surpass their urban peers on comparable test items. The article explains the method's guiding humanistic and cognitive principles and then reports on findings from explorative action research that implemented and evaluated the method. Using observational methodologies, we argue that the students developed in their conceptual understanding of the content as well as in their attraction to the discipline, their professional identity, their personal pride in their achievement, and their general epistemic capacity for concentration and self-regulation. Throughout, we emphasize the methodological limitations of this grassroots proof-of-concept action research, which threaten the validity of the assertions. We speculate on early extensions of the method to literacy studies. [ABSTRACT FROM AUTHOR]

In this paper, we report a pilot study on engaging a group of undergraduate students to explore the limits of sin(x)/xand tan(x)/xasxapproaches to 0, with the use of non-graphic scientific calculators. By comparing the results in the pretest and the post-test, we found that the students had improvements in the tested items, which involved the basic concepts of limits, but had room for further improvement in those that required them to explain their answers. An analysis of the students’ performances in the tests by using the APOS (i.e. action-process-object-schema) theory framework is reported. Reflections and suggestions on how to teach the topic more effectively are provided. [ABSTRACT FROM PUBLISHER]

A questionnaire survey was carried out as part of a PhD research study to investigate the image of mathematics held by post-primary students in Ireland. The study focused on students in fifth year of post-primary education studying ordinary level mathematics for the Irish Leaving Certificate examination – the final examination for students in second-level or post-primary education. At the time this study was conducted, ordinary level mathematics students constituted approximately 72% of Leaving Certificate students. Students were aged between 15 and 18 years. A definition for 'image of mathematics' was adapted from Lim and Wilson, with image of mathematics hypothesized as comprising attitudes, beliefs, self-concept, motivation, emotions and past experiences of mathematics. A questionnaire was composed incorporating 84 fixed-response items chosen from eight pre-established scales by Aiken, Fennema and Sherman, Gourgey and Schoenfeld. This paper focuses on the findings from the questionnaire survey. Students' images of mathematics are compared with regard to gender, type of post-primary school attended and prior mathematical achievement. [ABSTRACT FROM AUTHOR]

Students learn a lot of material in each mathematics course they take. However, they are not always able to make meaningful connections between content in successive mathematics courses. This paper reports on a technique to address a common topic in calculus I courses (intervals of increase/decrease and concave up/down) while also making use of students’ pre-existing knowledge about the behaviour of functions around zeroes based on multiplicities. [ABSTRACT FROM PUBLISHER]

There is considerable variety in inquiry-oriented instruction, but what is common is that students assume roles in mathematical activity that in a traditional, lecture-based class are either assumed by the teacher (or text) or are not visible at all in traditional math classrooms. This paper is a case study of the teaching of an inquiry-based undergraduate abstract algebra course. In particular, gives a theoretical account of the defining and proving processes. The study examines the intellectual responsibility for the processes of defining and proving that the professor devolved to the students. While the professor wanted the students to engage in all aspects of defining and proving, he was only successful at devolving responsibility for certain aspects and much more successful at devolving responsibility for proving than conjecturing or defining. This study suggests that even a well-intentioned instructor may not be able to devolve responsibility to students for some aspects of mathematical practice without using a research-based curriculum or further professional development. [ABSTRACT FROM AUTHOR]

Universities invest significant resources in the provision of mathematics tuition to first year students, through both traditional and non-traditional means. Research has shown that a significant minority of students do not engage with these resources appropriately. This paper presents findings from a study of two groups of students at Maynooth University. Both groups had similar mathematical backgrounds on entry to university. The first group consisted of seven students who had failed first year mathematics and had very low levels of engagement with available supports. The second group consisted of nine students who had passed first year mathematics and had engaged with the supports to a significant extent. It emerged that while both groups initially displayed similar tactics and encountered similar difficulties, their levels of reaction to a number of critical events in their mathematical education were key to their engagement levels and their subsequent progression. Further analysis revealed aspects of the students' behaviour which caused them to approach or avoid difficulties. The reasons behind the different student behaviours were investigated, and the main categories of influence on student behaviour which emerged from the interview data were fear, social factors, and motivation. [ABSTRACT FROM PUBLISHER]

The Feigenbaum constant δ is frequently met by students in a first course on chaos, and discussed with reference to the period doubling within the logistic map. The details of the actual calculation of δ are, however, nontrivial, and form the basis for an undergraduate project which may be used to develop skills in discrete maps and numerical methods. This paper considers how the Ricker map can be used to evaluate δ, and also suggests a number of problems which can be solved by students along the way. [ABSTRACT FROM AUTHOR]

To ensure the competitiveness of the USA in the global economy, and its role as a leader in science and engineering, it is important to cultivate the next generation of home grown mathematicians. However, while universities across the USA offer calculus classes to thousands of undergraduate students each year, very few of them go on to major in mathematics. This paper posits that one of the main reasons is that the mathematical community does not expose calculus students to the beauty and complexity of upper-level mathematics, and that by doing so before they fully commit to their programme of study, the number of students with a qualification in mathematics can be increased. The results show a significant increase in the number of students planning to add a minor in mathematics, and an increased likelihood among freshmen and sophomores to change their major. [ABSTRACT FROM AUTHOR]

In this paper, we consider particular tiling problems suitable for able high-school students. Three alternative student-generated proofs of the solution to the initial problem are given in detail, and various points in connection with these proofs are briefly discussed. [ABSTRACT FROM PUBLISHER]

In this paper, we investigate which aspects are overriding in the concept images of monotonicity of Finnish tertiary mathematics students, i.e., on which aspects of monotonicity they base their argument in different types of exercises related to that concept. Further, we examine the relationship between the quality of principal aspects and the success in solving monotonicity exercises and a few other standard problems in calculus. Our findings indicate that a mathematics student's conception about monotone functions is often restricted to continuous or differentiable functions and the algebraic aspect – the nearest one to the formal definition of monotonicity – is rare. [ABSTRACT FROM PUBLISHER]

In this paper, we present some results of an exploratory study performed with students aged 16-17. We investigate the different uses that these students make of terms such as ‘to approach’, ‘to tend’, ‘to reach’, ‘to exceed’ and ‘limit’ that describe the basic notions related to the concept of the finite limit of a function at a point. We use the interpretive framework of conceptual analysis to infer the meanings that students associate with these specific terms in connection with the effective use of terms in their answers. [ABSTRACT FROM PUBLISHER]

In this paper, we describe the growth of mathematical understanding in university students engaged in mathematics classroom tasks regarding the concept of numerical series. Starting from the Image-Making Pirie and Kieren theory layer, students organize a mathematical concept linking different mathematical elements. In this process, there are important agents that are involved during the interactions to advance in this construction through the mechanism of folding back between different layers. [ABSTRACT FROM PUBLISHER]

In this work, we study the mathematical practice of defining by mathematics researchers. Since research is an important part of many professional mathematicians, understanding how they do research is a necessary step before thinking about future researchers' undergraduate and postgraduate education. We focus on the defining process associated with the generalization of existing definitions as a way of constructing new ones. Data of this qualitative study come from a case study whose subject is a mathematics researcher in the area of differential geometry. We have interviewed this researcher and collected her research documents. From our analysis of the data,we have identified four phases in the defining process (Finding an opportunity to generalize an existing concept, Proposing a new definition, Justifying that the new definition is valid and Continuing the chain of definitions), which we will describe in detail in Section 4. [ABSTRACT FROM AUTHOR]

First-year mathematics instructors at universities across North America and the globe have been noticing a decline in the mathematics skills and preparation of their incoming students, who have been failing out of first-year mathematics courses at alarming rates. Though some universities have implemented placement or diagnostic tests to measure students' preparedness, many still use high school grades as the only indicator of readiness for university mathematics. However, researchers have questioned how effective these high school grades are at predicting success in university mathematics classes due to factors such as mis-aligned teaching methods, curriculum, and grade inflation. This study analyzes the relationship between grade 12 Pre-Calculus grades and first year university Calculus grades at a large Canadian university over the period from 2001–2015 in an attempt to quantify previous research in a large-scale, longitudinal manner. Results show that there is a significant disconnect between these grades, that the disconnect has been growing over time, and that it is is quite significant for students who are not performing well at the Calculus level. Recommendations moving forward include the implementation of placement examination in a wide-scale manner, and increased communication and collaboration between K-12 and university mathematics educators and administrators. [ABSTRACT FROM AUTHOR]

This article is intended to contribute to greater knowledge regarding the importance of flow and the time used to perform an activity, with a focus on students' mathematical experiences of 3D bodies. Thirty-one 9th-grade students took part in the study. Flow and variation theory was used in the analysis of lesson observations, submission tasks, audio recordings, logbooks, tests and nationwide tests. The results indicate that the selected mathematics problem is characterized by seven components, which serve as the basis for identifying intended critical aspects; a variation is evident in the balance between skills and challenges that is characterized by the critical aspects that the students discern; a variation is evident in the experience of flow that is dependent upon the students' approach to their work on various activities; the students' mathematical experiences are based, both short- and long-term, on discerned critical aspects and on the time spent on the activity that generates flow. Theoretical contributions as well as implications for teaching are presented at the end of the article. [ABSTRACT FROM AUTHOR]

A case is made for an early introduction of transformation of data in a first course in statistics. The topic is well received by students and helps pedagogically later in the course. Linear, squaring, and logarithmic transformations are emphasized. Impacts on subsequent parts of the course are discussed. [ABSTRACT FROM AUTHOR]

Polygons are an important topic in geometry. We propose and analyse activities for the exploration of their properties, with extensive usage of a dynamical geometry system. They are based on the analysis of monuments, connecting students' cultural backgrounds with their mathematical learning. Moreover, two opposite directions for work are presented: an analytic way beginning from the monument and exploring its geometric features, and a more synthetic way, using the output of the analysis to build a model with the software. We show how these STEAM activities, based on existing monuments of various shapes, enhance the 4 C's of twenty-first century skills. [ABSTRACT FROM AUTHOR]

In the past, my mathematics students have frequently complained at any suggestion that they should communicate ideas through the medium of a written report. This article discusses student responses when they were asked to write a short report for the mayor of a (hypothetical) small town in response to the mayor's plan to eliminate a contagious disease by locking the town down for three weeks. I discuss the approaches that students took in constructing their reports and summarize some of the great ideas that they had. Many students could see the parallel between what they were asked to do in the assignment and concurrent discussions in the communities that they came from with regard to the spread of COVID-19. The idea that mathematicians might have to communicate ideas in the form of a written report was not dismissed out of hand. I also reflect on ways in which the learning experience could have been improved. This hinges on providing a mechanism by which an individual student has the opportunity to read the reports of all the other students. [ABSTRACT FROM AUTHOR]