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This paper discusses the relevance of using physical situations to introduce the concept of angles at elementary school. We compare the effectiveness of two geometry teaching sequences. In the first one (physical sequence), the pupils learned the angle concept by experimenting on the playground (i.e. mesospace) and then modelling the situation. In the second one (paper-and-pencil sequence), the pupils worked solely in the space of a sheet of paper (i.e. microspace). In both sequences, pupils compared areas of space delineated by an angle between two directions. Pupils in two Grade 3 classes were exposed to one of the two teaching sequences. The unfolding of these sequences was videotaped and analyzed, and the pupils were tested individually, before and after teaching, to measure each sequence's effectiveness. Results showed that both sequences are effective to grasp the angle concept: Most pupils overcame the common erroneous conception of comparing angles' sides' lengths instead of angle openness. The comparison of areas of space delineated by an angle between two infinite directions, which is the two sequences' common core, seems to be the key factor underlying angle conceptualization. This paper ends with a discussion of these results' teaching implications and the merits of each sequence. [ABSTRACT FROM AUTHOR]

The use of digital technologies in teaching and learning processes has opened new and wide frontiers in mathematics education. Research in education moves towards teaching and learning (and assessing) twenty-first-century skills and computer-based testing seems an interesting possibility for assessing such skills. From this perspective, comparing computer-based with paper and pencil testing is an important issue for the educational community. It could allow us to better understand differences, innovations and weaknesses of classical assessing strategies and methods. The purpose of this study is to provide an example of the differences that can be found in migrating a task from one environment to another through a qualitative analysis of task and students’ solution processes. The sample involved in this study includes grade 6 and 8 students. The analysis shows that the environment migration process is not neutral: the tasks present differences. Students’ performances in the tasks are the same, but differences can be observed in solving processes. In particular, students of our sample differ in the mathematical resources applied, and in the assessing processes. In addition, it seems that students activate similar problem-solving processes as they become more familiar with the task and the environment in which it is administered. [ABSTRACT FROM AUTHOR]

The purpose of this note is to discuss how paper folding can be used to find the exact trigonometric ratios of the following four angles: 22.5°, 67.5°, 27°, and 63°. [ABSTRACT FROM AUTHOR]

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 purpose of this paper is to provide examples of ‘non-traditional’ proof-related activities or theorems that can be explored through paper folding by university and high-school students. These theorems were encountered through playful acts of paper folding by the author. The author used these activities successfully with preservice teachers. The paper contains proof outlines for each theorem. [ABSTRACT FROM PUBLISHER]

It is well-known that the Lotka–Volterra predator-prey model has a family of periodic orbits, but does not possess limit cycles and therefore the model is said to be structurally unstable. The Lotka–Volterra model is a special case of a much larger group namely the quadratic population models and it can be shown that none of them can produce limit cycles. The surprising finding is that by combining two quadratic models a quadratic population model with two limit cycles is uncovered. Although the model looks simple at first glance it provides a rich source of dynamics and deserves attention. In this paper, we revisit a model that has its origin in the work of Dubois and Closset. A set of two quadratic population models interact as piecewise defined differential equations. The model has been discussed by Ren Yongtai and Han Li, cryptically written and showing some linguistic and typographical errors, but providing an excellent vehicle for developing skills in mathematical modelling, differential equations and technology for the young researcher. We explore the model in clearer detail and supplement the theory with rich graphical illustration. The paper has the purpose of providing an example of how a young researcher, such as a postgraduate student in biomathematics, can expand on an existing model by making use of current technology. [ABSTRACT FROM AUTHOR]

In the article 'How long is my toilet roll-a simple exercise in mathematical modelling' several models of increasing complexity are introduced and solved to calculate indirectly the length of paper on a toilet-roll. All these results are presented without errors. The authors of this comment believe the error analysis of measurements made in a laboratory is an important part of the experiment. The calculation of the absolute errors in indirect measurements provides us with information about the confidence that we must have in our experimental results. For that reason, the error calculation for all approximations has been made. [ABSTRACT FROM AUTHOR]

In this note we discuss how a modular triangular origami gift box can be made with six rectangular sheets of printing paper. We also derive a formula for the volume of the box based on high school geometry and algebra. [ABSTRACT FROM AUTHOR]

This paper set out to contribute to mathematics education research by elaborating on a methodology developed during a study, trying to understand, view and follow shifts in prospective teachers' discursive patterns. The methodology aims to illustrate and describe how prospective teachers adapt to the context of teaching through a flexible process. This flexible process is then described in the result as a narrative. It is argued that the methodology can be used in relation to different theoretical directions, such as research about beliefs, knowledge, or identity. Another contribution is that the methodology presented gives insights into bridging the gap between different analytical levels, micro and macro. With a theoretical foundation in 'Cultural Worlds' [Holland, D., Skinner, D., Lachicotte, W., & Cain, C. (1998). Identity and agency in cultural worlds. Harvard University Press.] the Social Semiotic approach of Systemic Functional Linguistics, SFL [Halliday, M., & Hasan, R. (1989). Language, context, and text: Aspects of language in a social-semiotic perspective (2nd ed.). Oxford University Press] is used as a methodological tool. SFL offers a toolkit that allows the analysis of meaning at the clause level to uncover how and why a speaker produces a particular wording rather than any other in a specific social practice. The paper aims to illustrate and describe how to go beyond findings in the micro-analysis and then present the result as a narrative case. [ABSTRACT FROM AUTHOR]

Operations Research (OR) is a branch of applied mathematics that deals with optimization problems arising from different real contexts. The solving process of its problems is based on the construction and resolution of mathematical models, showing the possible connections between mathematics and the real world. Nevertheless, OR is not typically included in most curricula of higher secondary schools (i.e. Grades 9–12), but it is usually presented mainly at university level. To show how OR could be significant for these school students' education, the authors of this paper developed an educational project consisting of three teaching units. In this paper, we share the result of the teaching experiment related to the first unit, addressed to Grade 10. Qualitative and quantitative analyses show how it is appropriate to include OR and its typology of problems in regular school mathematics lectures. Second, these data also show how modelling and problem-solving skills, developed working with OR, can be fostered by implementing a collaborative way of working, also by making use of digital technologies. Last but not least, we demonstrate the positive impact such activities have on students' appreciation of OR. [ABSTRACT FROM AUTHOR]

The research question studied in this paper is how teachers' task designing can be enhanced in professional development. We show that this can be done by scaffolding. Scaffolding is used in problem solving and we apply it by considering designing a task as a problem. In the paper, we describe a professional development of four teachers led by a mathematics educator. The professional development consisted of 10 meetings. The teachers were first exposed to a well-designed task in geometry as learners and then were asked to design new geometry tasks for their students. The mathematics educator helped them by supplying scaffolds such as modelling, reflection, articulation and worked examples. The data consisted of video-recordings of the workshops and a semi-structured interview with each of the teachers at the end of the project. The data analysis shows how the use of particular scaffolds was helpful while preserving the teachers' autonomy as task designers. [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]

Epidemiological models have enhanced relevance because of the COVID-19 pandemic. In this note, we emphasize visual tools that can be part of a learning module geared to teaching the SIR epidemiological model, suitable for advanced undergraduates or beginning graduate students in disciplines where the level of prior mathematical knowledge of students may not be very strong. Visual tools – phase portrait, flow field and trajectory and line plots – available in the R software are presented in a step by step manner, moving from the exponential growth model to the logistic growth model and then to the SIR model. Code for numerical simulation of differential equations and estimation of parameters is presented for the SIR model. Suggestions for students to connect the learning from these examples with research papers on COVID-19 are provided. [ABSTRACT FROM AUTHOR]

The article discusses papers within the issue adapted from the Southern Hemisphere Conferences on the Teaching and Learning of Undergraduate Mathematics and Statistics with topics such as engineering, finance and group theory.

We present a method to compute the power series expansions of $ e^x $ ex, $ \ln {\left (1+x\right )} $ ln⁡(1+x), $ \sin {x} $ sin⁡x, and $ \cos {x} $ cos⁡x without relying on mathematical analysis. Using the properties of elementary functions, we determine the coefficients of each series through the method of undetermined coefficients. We have validated our formulae through the use of mathematical induction. The Newton binomial formula is obtained using the expansions of the exponent and the logarithm in their power series. We build upon the approach outlined in a paper by L. P. Mironenko and O. A. Rubtsova (Approximations of Some Functions by Polynomials and the Method of Undefined Coefficients, Scientific Papers of Donetsk National Technical University. Series: Computing and Automation, No. 2 (25), p. 128–135, 2013) and complement their approach. This approach does not depend on the application of Taylor's theorem to the expansion of functions. Consequently, these expansions can be integrated into the educational process before students become acquainted with the concept of derivatives. It simplifies the study of the theory of limits, especially in the computation of standard limits such as $ \lim _{x\to 0}\frac {\sin (x)}{x} $ limx→0sin⁡(x)x, $ \lim _{x\to 0}\frac {1-\cos (x)}{x^{2}} $ limx→01−cos⁡(x)x2, and $ \lim _{x\to 0}(1+x)^{1/x} $ limx→0(1+x)1/x. [ABSTRACT FROM AUTHOR]

The aim of this paper is to contribute to the ongoing discussion about the role of intuition and ambiguity in doing, teaching and learning mathematics. We start by discussing the ways in which limits and continuity are presented in Calculus textbooks, to illustrate some of the ambiguities, and to contrast them with precise and rigorous definitions and statements. Next, we offer a brief overview of the topological terms necessary to introduce the notion of a continuous function in the topological setting. We conclude by suggesting a way of presenting the concept of a continuous function appropriate for Calculus instruction, while still staying true to its topological roots. This paper is also a suggestion to mathematics instructors to consider modifying the way they introduce limits and continuity in their classrooms, by creating a balance (appropriate for their students!) between rigour on the one side and intuition and ambiguity on the other. [ABSTRACT FROM AUTHOR]

This paper reports the results of a didactic engineering proposal for statistical modelling and inquiry in the context of teacher education. The tasks and situations we focus on are related to Brousseau’s guessing game, which was originally proposed for the teaching of probability and randomness to primary school students. In this paper, we describe two different didactic engineering designs based on the guessing game, which were independently managed by two different research teams in the context of pre-service teacher education. One of them is aimed at primary school teachers in Spain, and the other at secondary school teachers in Japan. The theoretical tools and design principles employed come from the anthropological theory of the didactic, especially praxeologies and study and research paths. The description and comparison of the two cases reveal the potential of the guessing game for the teaching of statistics in teacher education, as well as the limitations of the dominant epistemological models in statistics, for researchers and educators when planning to work on statistics 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]

This paper presents a study aimed at investigating the didactic potentiality of the combined use of two different kinds of artefacts for the purpose of constructing and conceptualizing mathematical meanings related to the notion of axial symmetry. In our view, the process of meanings construction can be fostered by the use of adequate artefacts, but it requires a teaching/learning model, which explicitly takes care of the evolution of meanings, from those personal, emerging through the activities, to the mathematical ones, aims of the teaching intervention. The main hypothesis of this study is that a potential synergy may occur between the use of different artefacts, synergy that can foster the integration of different and complementary meanings providing a rich support to the development of the expected mathematical meaning. The Theory of Semiotic Mediation offers the theoretical framework suitable to design the teaching sequence and to analyze the collected data. Specifically, the construct of semiotic potential provides the tool for describing the potentialities of the two artefacts, while that of didactic cycle offers a model for the organization of the different activities. The paper reports on a teaching sequence and its implementation in a teaching experiment, involving pupils at fourth grade level. We describe them, within the chosen theoretical framework, and provide the analysis of key episodes of the teaching sequence. We show evidence supporting our main hypothesis about the combined use of an artefact that can be manipulated (paper and pin), and a digital artefact (Dynamic Geometry Environment) in the development of the notion of axial symmetry and its properties: the combined, intentional and controlled use of the two artefacts may develop a synergy, so that each activity enhances the potential of the other. [ABSTRACT FROM AUTHOR]

In this note we describe the mathematics that emerges from the construction of an origami box. We first construct a simple origami box from two rectangular sheets and then discuss some of the mathematical questions that arise in the context of algebra, geometry and calculus. [ABSTRACT FROM AUTHOR]

An introduction is presented in which the editor discusses articles in the issue on topics including relationship between research and scholarship; mathematics educators; and challenges of mathematics teaching at university level.

This study is framed within the anthropological theory of the didactic. The first part is a comparative study of the mathematical organisation of concave/convex functions on the one hand at university, and on the other hand in secondary education. Data for this analysis are from a university textbook on calculus and a secondary school textbook. The analytic results show how the mathematics (to be) taught at secondary level consists of a largely demathematised logos block. The corresponding institutional fragmentation of knowledge reveals a missing praxeological organisation from which the organisations of knowledge we consider in this paper could be derived by transposition. Responding to this insight, in the second part of the paper, we have outlined what we call an archeorganisation for concave/convex functions, a framework that teacher education institutions could refer to when trying to satisfy the specific constraints they are subjected to (notably because of its publics), without for that matter isolating themselves epistemically from other academic institutions. Formulating an archeorganisation represents a meta-curricular study, with the objective of transforming the institutional conditions and constraints that guide mathematics educators and curriculum designers in their deliberations about the concept of concave/convex functions. [ABSTRACT FROM AUTHOR]

Two groups of mathematics faculty, one from a four-year college and one from a two-year college, redesigned their respective introductory college mathematics courses following presentation of observational data regarding how faculty had been teaching the courses. This presentation emphasised how infrequently faculty teaching introductory college mathematics employed recommended pedagogical practices. This work was part of a multi-year federal grant project designed to increase the numbers of underrepresented students majoring in STEM disciplines. While the two teams developed very different redesigned course activities, in both instances the primary motivation for initiating the work was the information provided to faculty in a professional development workshop regarding how they had previously been observed teaching the mathematics content and how infrequently they utilised the pedagogical practices recommended by STEM education experts. The paper also highlights faculty resistance to curricular reform and enumerates some ways of addressing this resistance. The paper also discusses why faculty from two-year and four-year institutions resisted working together on course redesign and provides recommendations for future efforts addressing course redesign. [ABSTRACT FROM AUTHOR]

This paper reflects on the Inquiry-Based Mathematics Education approach and the professional development of university mathematics lecturers. The main focus is on task re-design as a key aspect of subject matter focussed lecturer training. The professionalisation activity we have established is rooted in a framework developed by the European project Partnership for the Learning And Teaching IN University Mathematics (PLATINUM). The focus of this paper is a didactic instrument developed within this project and implemented in the training activity. We focus on challenges, leeways and scope for actions that arise in the design and implementation of mathematical tasks for an inquiry-based teaching. For the empirical analyses, we use data from a professional activity carried out at the Complutense University of Madrid in Spain, jointly by a team from Madrid and Hannover. [ABSTRACT FROM AUTHOR]

This paper describes a teaching experiment in a Numerical Methods course for Master of Science students. The experiment uses scientific papers to develop modelling studies in the context of wine fermentation microbial interactions. The course involves theoretical and laboratory classes that focus on implementing numerical methods using Matlab for Initial Value Problems and Boundary Value Problems. The students are asked to formalise the mathematical model and build their own experiments using the information provided in the papers. Additionally, a parameter estimation experiment is organised, which involves generating synthetic data and computing noisy data to estimate the natural death rate of sensitive yeast. The results show that data noise significantly affects the parameter estimate and that scaling the data can help reduce the impact of measurement errors. The presented results can be used to investigate other possible assignments, such as how the evaluation of the Jacobian affects the estimation performance and compare different optimisation algorithms. [ABSTRACT FROM AUTHOR]

Mathematical modelling has great potential to motivate students towards studying mathematics. This article discusses several different approaches to integrating research work with a second-year undergraduate, mathematical modelling subject. I found sourcing papers from the areas of epidemiology and ecology to be a fruitful source area, particularly models involving only two or three coupled differential equations. These models were amenable to students as well as interesting and relevant to students because they came from real research papers. I will describe the use of scenarios and case studies in lectures, and group projects for assessment. The scenarios and case studies were published in a textbook that I wrote. Scenarios, case studies and projects provided an opportunity to expose students to some novel applications of differential equations. One example is developed here as a Classroom Note: modelling the dynastic cycles in Chinese history. [ABSTRACT FROM AUTHOR]

Improving the mathematical skills of undergraduate Engineering and Information Technology (EIT) students has been an enduring challenge for educators. Lecturers continually implement new strategies that try to overcome the challenges encountered in teaching university students with varying levels of success being encountered. This paper provides insight into how educators have handled the challenges students face in their mathematics learning journey. In addition, the paper highlights some of the methods used to improve the performance of these students as problem solvers, as well as the development and implementation of creative techniques that may have made the situation more successful and less challenging. The literature was searched through using a structured literature review approach based on the primary databases: Scopus; the Education Research Complete (ERC); and the Education Resources Information Centre (ERIC). This study followed the three main steps of a systematic literature review: retrieval, selection, and analysis of the literature. Analysis of the literature showed that there is still a need to further develop and deliver innovative pedagogical strategies for teaching mathematics to EIT students. It is recommended that an interdisciplinary approach, inclusive of technology, will lead to increased understanding, improve performance and, ultimately, enhance student satisfaction. [ABSTRACT FROM AUTHOR]

An editorial is presented on INDRUM being the acronym for the International Network for Didactic Research in University Mathematics. Topics include contributing to the development of research in didactics of mathematics at all levels of tertiary education; and focusing on building research capacity in the field showing concern for the development of early-carreer researchers and dialogue with the mathematics community.

As educational systems move from onsite to online due to the COVID-19 pandemic, teachers face the difficulty of designing online examination formats which minimize opportunities for dishonesty. In this paper, we expose our design of such a format: a protected Microsoft Excel spreadsheet containing short-answer questions, which was implemented in a calculus module taught by us. This format allows examiners to randomize questions with the aim that each student receives each question with different numerical details, making plagiarism impossible, while keeping the marking effort very low. [ABSTRACT FROM AUTHOR]

This paper discusses the mathematical and didactical problems of teaching indefinite integral in the context of the ubiquitous availability of online integral calculators. The symbol of indefinite integral introduced by Leibniz, unfortunately, does not contain an indication of the interval on which the antiderivatives should be calculated. This yields mismatched results of calculating the same integral on different online calculators. The paper justifies the expediency of including the interval of integration in the designation of indefinite integral or, at least, in the settings of problems of its calculation. Indication of the integration interval ensures the uniqueness of the solution to the problem of calculating the indefinite integral, the uniqueness that does not occur in the traditional setting of the integration problem. This is important because online calculators provide users with formulas that differ from each other and can represent both different and coincidental antiderivatives of the same function. Analysis of the domains of definition of the obtained by calculators antiderivatives allows users to understand how they relate to each other. [ABSTRACT FROM AUTHOR]

The dynamics of a simple pendulum are often presented to undergraduate engineering students in introductory courses in dynamics. It is usually the first dynamic system considered by students that is modelled by a differential equation. This paper presents the standard material given to students. It is fair to say that students are accepting this material, but many do not fully appreciate its importance as a basis for the analysis of other dynamic models. Students also believe that the behaviour of a forced simple pendulum is obvious and is not worthy of a deep analysis. This paper presents many interesting results that are available in the open literature, but is not usually presented to undergraduate students. Some of these are counter-intuitive results such as unstable points becoming stable when forced and deterministic models giving chaotic pendulum trajectories. Many results demonstrate that simple pendulums are very interesting and worthy of analysis. The extension of a forced simple pendulum to an application that all students have experience of a child playing on a swing is presented. Only a limited analysis of the swing models is presented. This leaves some open questions for students and lecturers alike to explore. [ABSTRACT FROM AUTHOR]

The paper describes active learning strategies used in undergraduate college algebra and business calculus courses. There are a variety of active learning strategies described in the literature. We wanted to implement a few that together, satisfy the key characteristics of active learning strategies. The active learning strategies described in this paper are, interactive presentation style, group-work with discussion and feedback, volunteer presentations of solutions by groups, raise students' learning interest towards specific topics, involve students in mathematical explorations, experiments, and projects, and last but not least, continuous motivation and engagement of students. We analyse the relationship between the average results of 27 college algebra and business calculus sections and the effects of the level of use of these active learning strategies. We demonstrate that the application of these strategies has a positive effect on the average results of the sections and the passing rates of the students. [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]

The purpose of this paper is to describe the mathematics that emanates from the construction of an origami box. We first construct a simple origami box from a rectangular sheet and then discuss some of the mathematical questions that arise in the context of geometry and algebra. The activity can be used as a context for illustrating how algebra and geometry, like other branches of mathematics, are interrelated. [ABSTRACT FROM AUTHOR]

This paper reports on data obtained during semi-structured interviews with students who had recently experienced a guided inquiry approach to learning ordinary differential equations (ODEs) in a service-taught module. Previous research identified the strengths and weaknesses of similar cohorts of students. The results from that research informed the design of an intervention (15 guided inquiry tutorials) which sought to improve the outcomes for students taking a subsequent version of the module. This paper reports on the data gathered during interviews with these students that describe their perceptions of the guided inquiry approach used. The students noticed the change in emphasis toward conceptual understanding that the intervention was trying to instil. They cited the change in questioning style most frequently as being where they saw this change, but also noted the prevalence of group work and change in interaction pattern as keys to its success. Thus, by probing students' opinions in this way, we find validation for a guided inquiry approach to teaching ordinary differential equations in third level that emphasizes active learning and lateral interactions among students. The students' personal goal orientation and the goal structure of the learning environment are also discussed. [ABSTRACT FROM AUTHOR]

In 2010 a mathematics curriculum was introduced in Irish second level schools entitled 'Project Maths' (PM). It aimed to refocus second level mathematics teaching and learning away from an over emphasis on procedural mathematics towards problem solving and real understanding [Department of Education and Skills (DES). (2010). Report of the Project Maths implementation support group. ]. This paper aims to examine the performance of 1st year undergraduate students' procedural and problem solving skills after the introduction of PM. A diagnostic test was developed to determine students' skills in each area and findings demonstrated that students perform statistically significantly better in the procedural skills in mathematics when compared to problem solving skills. These findings raise concerns around the lack of anticipated improved problem solving skills of a cohort of students exposed to this style of teaching and learning. The paper raises discussion surrounding the intended and actual teaching and learning taking place in second level classrooms along with consideration for the potential role of learned helplessness and the literacy issues. Recommendations are made for follow up qualitative research with stakeholders in mathematics education to better understand the 'why' of the results presented here. [ABSTRACT FROM AUTHOR]

Many studies have investigated students' problem-posing activities. However, there has not been a strong focus on affective aspects in mathematical problem-posing. Also, students' understanding of the function concept whilst generating functions has not previously been investigated. This paper investigated students' misconceptions of the function concept through problem-posing tasks. It also delineates students' views on the problem-posing experiences in a functional context. The participants of this study were 74 female high-school students, of whom six were selected according to their performance on a problem-posing test to air their views. The findings show that problem-posing was a good approach for pinpointing students' misconceptions about function concept, some of which are presented in this paper. Students' interviews also indicate that, besides being productive and enjoyable for students, problem-posing activities helped them to develop a deeper understanding of the function concept through reflection. Indeed, exploring students' views in depth made it possible to provide a better analysis of the factors that can inform the effective use of problem-posing in mathematics education. [ABSTRACT FROM AUTHOR]

Take a family of independent events. If some of these events, or all of them, are replaced by their complements, then independence still holds. This fact, which is agreed upon by the members of the statistical/probability communities, is tremendously well known, is fairly intuitive and has always been frequently used for easing probability calculations and allowing important developments in probability theory. However, can this result be proven without advanced elements from probability theory, such as independent σ-fields and related results? This paper aims primarily to fill this possible gap. Two elementary proofs that require only introductory probability content are given in detail and compared with the 'technical' proof found in books on advanced probability theory. A discussion regarding how these proofs could be included in regular probability courses is also included in the survey. [ABSTRACT FROM AUTHOR]

The paper introduces the notion of collateral creativity as an unintended outcome of the appropriate use of technology, both physical and digital, in the teaching of mathematics. The notion is motivated by Dewey's conceptual framework of collateral learning which enables one gaining knowledge beyond the intent of the traditional curriculum. Creative thinking by the learners of mathematics may emerge accidentally, in a collateral way, within a classroom culture encouraging reflection on a teacher-assisted technological representation of mathematical concepts. The use of technology is described in terms of Vygotsky's concept of the instrumental act. The main argument of the paper that tokens of collateral creativity can be observed in all students resides at the confluence of various theories which, in the context of mathematics, converge to describe creativity as a slumberous skill with potential to be awakened through the age-appropriate pedagogical mediation supported by the teachers' awareness of often hidden complexity of seemingly mundane problems. The paper discusses different scenarios conducive for the emergence of collateral creativity through the study of mathematics with physical and digital tools. The ideas of the paper stem from the authors' work with schoolchildren and with teacher candidates both in the classroom and in the field. [ABSTRACT FROM AUTHOR]

This paper describes a study for teaching number system in a computer arithmetic course and addresses the existing gap in the current course by focusing on the characteristics of applications. A redundant residue number system (RRNS) is an efficient and innovative number system that inherits the interesting properties of both residue number system (RNS) and redundant number systems. It breaks the operands into residues by considering a moduli-set and then signifies the residues by a redundant representation. So, by limiting the carry propagation chain inside and outside the moduli, RRNS proposes more efficient arithmetic units which could be employed in digital signal processing (DSP) applications. Despite the applicability of RRNS, there is not well-organized teaching on RRNS covering the recent achievements. Besides, there is an important question for students and researchers what is the most appropriate number representation for each application category? In this paper, we present a step-by-step education process for RRNS that includes basic concepts of RNS and the redundant number system. Then we address characteristics of different applications that RRNS is suitable to be employed for them. So, the method gives a well-organized proceeding to computer arithmetic designers and students. [ABSTRACT FROM AUTHOR]

This paper provides some insights into the use of example-generating tasks in the design of a technology-rich learning environment to enhance students’ mathematical thinking. The paper reports on an early stage of a design-based research project concerning the design of tasks and associated feedback utilising the affordances provided by a combined use of a dynamic mathematics software environment and a computer-aided assessment system. In example-generating tasks, students are asked to generate examples that fulfil certain conditions. Based on data in terms of examples generated by 491 first-year engineering students, taking a first course in calculus, the paper examines patterns of student response to three example-generating tasks. As a theoretical lens, the notions of dimensions of possible variation and associated ranges of permissible change are used. In light of the observed patterns, the paper provides some guiding principles for designing example-generating tasks and associated formative feedback to foster students’ mathematical understanding by enriching their example spaces. For example, this paper illustrates occasions where it might be instructive to start by asking for two examples, followed by adapted feedback before requesting a third example. [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 short paper is to describe a new proof of the Pythagorean Theorem that involves paper folding. [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]

In this paper, we introduce a technology-enhanced pedagogical sequence for supporting lower secondary school students' sense-making of the concept of volume in a non-procedural and non-formula-driven way. Specifically, we illustrate a novel approach of using dynamic geometric environment (DGE) to introduce the meaning of volume and then deriving the volume of a hemisphere with minimal algebra but simple trigonometry. This method aims to support students' geometrical reasoning through comparing the volume of different three-dimensional shapes and analysing their cross sections two-dimensionally, thus drawing on the concept of area. We highlight the affordances of the designed DGE sketches for fostering a dynamic conception of volume. [ABSTRACT FROM AUTHOR]

Assessment is among the inevitable components of a curriculum and directs students' learning. E-assessment, as prepared and administered with the use of ICT, provides opportunities to make the process easier in some aspects, but also brings certain challenges. This paper presents an e-assessment framework from a student perspective. Our study lasted for one academic year and incorporated a total of 631 students' responses to questionnaires conducted three times in two different mathematics courses. The courses are compulsory for undergraduate informatics students. The research included a three-step factor analysis: two exploratory factor analyses and a confirmatory factor analysis. It yielded a model of a student perspective on e-assessment consisting of four factors: Transparency and fairness of assessment, Formative and summative assessment and feedback, Meaningful use of technology in assessment and Difficulty of learning outcomes. The students' opinion about the use of e-assessment was generally positive. The students appreciated the student-centred approach based on sound pedagogical alignment of all teaching and learning elements with assessment. They found it important that technology should not distract students during assessment tasks and considered cheating in a controlled e-assessment environment to be no more frequent than in face-to-face assessment. [ABSTRACT FROM AUTHOR]

Mathematics teacher candidates enter teacher education with knowledge that could support or limit their learning. It is therefore important to gain insights of this knowledge to inform teacher education. This paper offers such insights for prospective mathematics teachers' initial knowledge of problem solving (PS) for teaching at the beginning of their teacher education programme. It reports on a study that investigated three categories of problem-solving knowledge for teaching: problem characterization, the PS process, and PS disposition. Participants were 109 prospective primary teachers at a university in Spain. Data sources were questionnaires designed for the study to explore the initial knowledge held by the participants for three categories of problem-solving knowledge. Findings indicated that the participants' knowledge was stronger regarding the PS process and disposition than for problem characterization but had limitations that needed to be addressed in their teacher preparation programme. [ABSTRACT FROM AUTHOR]

In this study, we redesigned paper-and-pencil activities and adapted them to an introductory multivariate calculus course supported by 3D dynamic geometry software. We used semi-structured interviews and students’ written productions during the semester to analyse the use of technology in supporting students’ learning. Considering the work of two students, we find that the software has transformative potential but that it did not promote students’ learning as expected. We discuss possible reasons for this. [ABSTRACT FROM AUTHOR]

This paper explores the position of the vertex centroid for a generalisation of Van Aubel’s theorem: specifically we look at what happens to the vertex centroid when directly similar quadrilaterals are placed on the sides of an arbitrary quadrilateral. After giving a simple proof that the position of the vertex centroid remains unchanged, the result is further generalised to directly similar triangles (or other directly similar shapes) on the sides of polygons using vectors. Not only are the results mathematically interesting, but can also provide an appropriate classroom opportunity for dynamic geometry exploration, and to build 2D models with clay and drinking straws (or thin wire) to illustrate and check the theoretical solutions. [ABSTRACT FROM AUTHOR]