Understanding secondary–tertiary transition in mathematics

In Clark and Lovric (Suggestion for a theoretical model for secondary–tertiary transition in mathematics, Math. Educ. Res. J. 20(2) (2008), pp. 25–37) we began developing a model for the secondary–tertiary transition in mathematics, based on the anthropological notion of a rite of passage. We articulated several reasons why we believe that the educational transition from school to university mathematics should be viewed (and is) a rite of passage, and then examined certain aspects of the process of transition. Present article is a continuation of our study, resulting in an enhanced version of the model. In order to properly address all aspects of transition (such as a number of cognitive and pedagogical issues) we enrich our model with the notions of cognitive conflict (conceptual change) and culture shock (although defined and used in contexts that differ from the transition context, nevertheless, we found these notions highly relevant). After providing further justification for the application of our model to transition in mathematics, we discuss its many implications in detail. By critically examining current practices, we enhance our understanding of the many issues involved in the transition. The core section ‘Messages and implications of the model’ is divided into subsections that were determined by the model (role of community, discontinuity of the transition process, shock of the new, role of time in transition, universality of transition, expectations and responsibilities, transition as a real event). Before making final conclusions, we examine certain aspects of remedial efforts.     

Pupils’ attitudes towards mathematics: a comparative study of Norwegian and English secondary students

Comparing English and Norwegian pupils’ attitude towards mathematics, in this article I develop a deeper understanding of the factors that may shape and influence ‘pupil attitude towards mathematics’, and argue for it as a socio-cultural construct embedded in and shaped by students’ environment and context in which they learn mathematics. The theoretical framework leans on work by Zan and Di Martino (The Montana Mathematics Enthusiast, Monograph 3, pp. 157–168, 2007) to elicit Norwegian and English pupils’ attitude of mathematics as they experience it in their respective environments. Whilst there were differences which could be seen to be accounted for by differently ‘figured’ environments, there are also many similarities. It was interesting to see that, albeit based on a small statistical sample, in both countries students had a positive attitude towards mathematics in year 7/8, which dropped in year 9, and increased again in years 10/11. This result could be explained and compared with other larger scale studies (e.g. Hodgen et al. in Proceedings of the British Society for Research into Learning Mathematics. 29(3), 2009). The analysis of pupils’ qualitative comments (and classroom observations) suggested seven factors that appeared to influence pupil attitude most, and these had ‘superficial’ commonalities, but the perceptions that appeared to underpin these mentions were different, and could be linked to the environments of learning mathematics in their respective classrooms. In summary, it is claimed that it is not enough to identify the factors that may shape and influence pupil attitude, but more importantly, to study how these are ‘lived’ by pupils, what meanings are made in classrooms and in different contexts, and how the factors interrelate and can be understood.     

The relevance of advanced mathematics studies to expertise in secondary school mathematics teaching: practitioners’ views

This study investigates the different ways by which secondary school mathematics teachers view how advanced mathematics studies are relevant to expertise in classroom instruction. Data sources for this study included position papers and written notes from a group interview of 15 Israeli teachers who studied in a special master’s program, of which advanced mathematics courses comprise a sizeable share. Data analysis was iterative and comparative, aiming at identifying and characterizing teachers’ different perspectives. Overall, all participating teachers thought that the advanced mathematics studies in the program were relevant to their teaching of secondary school mathematics. Moreover, teachers specifically mentioned the importance of studying contemporary mathematics from research mathematicians. All teachers pointed out at least one specific feature that they viewed as relevant to their work: advanced mathematics courses (1) as a resource for teaching secondary school mathematics, (2) for improving understanding about what mathematics is, and (3) for reminding teachers what learning mathematics feels like.     

Changing students’ beliefs about the relevance of mathematics in an advanced secondary mathematics class

ABSTRACT

This study shows that using authentic contexts for learning differential equations in a differentiation-by-interest setting can enhance students’ beliefs about the relevance of mathematics. The students in this study were studying advanced mathematics (wiskunde D) at upper secondary school in the Netherlands. These students are often not aware of the relevance of the mathematics they have to learn in school. More insights into the application of mathematics in other sciences can be beneficial for these students in terms of preparation for their future study and career. A course differentiating by student interest with new context-rich curriculum materials was developed in order to enhance students’ beliefs about the relevance of mathematics. The intervention aimed at teaching differential equations through guided small-group tasks in scientific, medical or economical contexts. The results show that students’ beliefs about the relevance of mathematics improved, and they appreciated experiencing how the mathematics was applied in real-life situations.     

Post-primary students’ images of mathematics: findings from a survey of Irish ordinary level mathematics students

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.     

Prospective mathematics teachers’ use of linguistic signifiers in the context of angles formed by two lines cut by a transversal

Embracing a multisemiotic approach, this case study addresses the ways in which prospective middle school mathematics teachers use linguistic signifiers idiosyncratic to the Turkish language to construe mathematical meaning of angles formed by two lines cut by a transversal. Also, students’ mathematical referents to explain angle relationships were characterized. Six students (3 female, 3 male) volunteered to participate in an individual task-based interview. The results indicated that students used morphological units of meaning when they explained the mathematical concepts. Also, most students used the parallelism of the two lines cut by a transversal as a qualifier to be able to talk about the angle pairs on a transversal. They most often recited properties, such as the U property, to explain the angle relationships. Implications for future research are provided.     

Comparative studies of mathematics teaching: does the means of analysis determine the outcome?

This paper addresses four questions concerning the influence of culture on mathematics teachers’ professional practice. Firstly, drawing on categorical data yielded by the application of low inference coding schedule to video recordings of sequences of lessons taught by case study teachers on four common topics in England, Flanders, Hungary and Spain, we undertook an exploratory factor analysis to examine the ways in which such coded variables interact. This process yielded five factors, each of which was interpretable against the literature and highlighted the extent to which dichotomisations of mathematics teaching as reform or traditional are not necessarily helpful, not least because all project teachers exhibited characteristics of both. Secondly, factors scores were analysed by nationality to reveal culturally located practices resonant with the available literature. Thirdly, cluster analyses yielded four well-defined cross-cultural clusters of episodes, each indicative of particular didactical perspectives that appeared to challenge the exclusivity of these culturally located practices. Finally, the key methodological finding was that the manner in which data are analysed influences greatly the outcomes of comparative mathematics research.     

Students’ self-regulation of emotions in mathematics: an analysis of meta-emotional knowledge and skills

Over the past decade, the concept of self-regulated learning has broadened to include motivational, volitional, and emotional components next to (meta-)cognitive ones. In this article, we present a meta-emotion perspective as an essential component of a conceptual framework on self-regulation that fully acknowledges the role of emotions. Against this background, a study is presented that attempts to contribute to the clarification of the relevance and the functioning of students’ meta-emotional knowledge and emotional regulation skills in school-related mathematical activities. It investigates the coping strategies that 393 students of the second (age 14) and fourth (age 16) year of secondary school report to use to regulate their emotions in three different mathematical school settings (i.e., a mathematics test, a difficult mathematics homework, and a difficult mathematics lesson). More specifically, it aims (1) to document the nature and frequency of the reported coping strategies, and (2) to explore—for the three different mathematical school settings—relationships between these reported coping strategies and personal characteristics (i.e., students’ familiarity with the particular school settings, their track in secondary education, their achievement level, their age, and gender). The results indicate that students report to know and to make use of several coping strategies in school-related mathematical activities, and reveal that the use of these strategies is related to specific person-related characteristics. In conclusion, we elaborate on how schools and teachers can stimulate students to acquire appropriate strategies and skills to self-regulate their emotions.     

Presenting mathematics in a first‐year engineering course at university

The approach used to teach mathematics to first‐year engineering students at the University of Warwick is described. The instruction is programmed, with weekly assignments and tutorials, so that the learning is self‐paced within each week. Our experiences with this method of teaching are discussed.     

A university basic course on mathematics: MAT‐100

Simón Bolívar University has been experimenting for the past three years with a new concept in the teaching of Traditional Calculus, a method of open studies that puts more emphasis on learning than on being taught. This achieves an independent way of thinking in the students.

MAT‐100 is taken by all the students that enter the University. It is divided into 26 learning units that go from the Cartesian Plane and Functions to Differential Equations in Partial Derivatives. Every Learning Unit is divided into learning modules, each of which is subdivided into activities with their own learning objectives and exercises of auto‐evaluation.

The students are distributed in sections of approximately 50. Each section is attended by a professor and a student aid. In class the students work in groups where they do workshops and consult, studying the Units, solving exercises and resolving auto‐evaluation tests. Lectures are given periodically on the most difficult topics.

When the student feels prepared, he presents an exam which is given weekly. The exams are corrected by a computer DEC‐10 with a system programmed in SIMULA.

At the moment the course is being taken by 3439 students distributed in 56 sections, employing 29 professors and 103 students aids. We have evaluated the course and have found that the students have developed a great ability in Calculus.     

Solving equations with fractions: An analysis of prospective teachers’ solution pathways and errors

We analyzed the solution pathways and errors found in the written responses of 469 prospective teachers solving an equation containing fractions. The majority (332, or 70%) used an algebraic method; 141 of the 332 (42%) were correct, and 22% of the algebraic methods were abandoned before a solution was obtained. We identified the steps in the written solutions, determined which solution pathways led to the correct solution, and identified common errors in the solution pathways of respondents who incorrectly solved the equation. Respondents initially attempted different methods. The most common method was solving by using fractions, but the majority of respondents who solved by using mixed numbers were able to correctly solve the problem. Common errors related to fraction arithmetic and the distributive property. Nearly all of the abandoned pathways contained no errors, but ended with a step that likely would precede an operation with fractions. Our findings suggest that the ability to solve an arithmetic equation with no fractions was necessary, but not sufficient, to solve an arithmetic equation involving fractions, and that the problem of solving equations with fractions was more closely tied to one's difficulties with rational number arithmetic and less with one's understanding of algebra.     

Declining participation in post-compulsory secondary school mathematics: students’ views of and solutions to the problem

We analysed multivariable calculus students' meanings for domain and range and their generalisation of that meaning as they reasoned about the domain and range of multivariable functions. We found that students' thinking about domain and range fell into three broad categories: input/output, independence/dependence, and/or as attached to specific variables. We used Ellis' actor-oriented generalisations framework to characterise how students generalised their meanings for domain and range from single-variable to multivariable functions. This framework focuses on the process of generalisation – what students see as similar between ideas in multiple contexts. We found that students generalised their meanings for domain and range by relating objects, extending their meanings, using general principles and rules, and using/modifying previous ideas. Our findings suggest that the domain and range of multivariable functions is a topic instructors should explicitly address.     

A case study: assessment of preservice secondary mathematics teachers’ metacognitive behaviour in the problem-solving process

The purpose of the present study was to investigate preservice secondary mathematics teachers’ metacognitive behaviour in the mathematical problem-solving process. The case study methodology was employed with six preservice mathematics teachers, enrolled at one university in Ankara, Turkey. We collected data by using the think aloud method, which lasted for two sessions. It was found that there was no relationship between academic achievement and frequencies of metacognitive behaviour. However, the types of problems could affect these frequencies. Furthermore, there was no pattern in metacognitive behaviour with respect to achievement and type of problem.     

E-learning in secondary–tertiary transition in mathematics: for what purpose?

Mathematics is a challenging subject for many science freshmen, and failing mathematics exams is a major factor in university dropout rates. Many factors, including metacognitive, affective and linguistic ones, play a role in students?? difficulties in mathematics. Starting from this perspective, we conducted a theoretical exploration of the potential of online environments in helping students counteract the mathematical difficulties they face in the transition from secondary school to university. Although this secondary?Ctertiary transition and the use of technology are both widely researched issues in mathematics education, the potential of technology in helping students in the ??rite of passage?? to tertiary education has not yet been researched. This paper reports on the developed theoretical framework and on the preliminary findings from the implementation of an e-learning course that we designed with the aim of supporting students in the critical phase of transition from secondary school to university.     

Assigning mathematics instruction time in secondary schools: what are the influential factors?

Similar to countries such as the Netherlands and the United Kingdom, secondary schools in Ireland can decide how to allocate instruction time between curriculum subjects. Although there are national guidelines available from the Department of Education and Skills (DES), the majority of schools make their own decisions about how much time they allocate to different subjects. This results in variations between the amounts of time allocated to teaching mathematics in different schools and between different year and class groups within the same school. Decisions regarding time allocation are generally taken by the school management. This means that the ethos of the school and the individual opinions of school management can determine the amount of mathematics that students experience throughout their second level education. The aim of this study is to evaluate the most influential factors that management considers when assigning instruction time in Irish secondary schools. For the purpose of this research, seven possible factors were identified and 400 deputy principals from a stratified sample of secondary schools around Ireland were asked to select their top three. Timetabling constraints, the availability of mathematics teachers and the perceived importance of the subject were found to be the most influential factors.     

Instructors’ use of technology in post-secondary undergraduate mathematics teaching: a local study

In this study, instructors of undergraduate mathematics from post-secondary institutions in Newfoundland were surveyed (N = 13) and interviewed (N = 8) about their use of, experiences with, and views on, technologically assisted teaching. It was found that the majority of them regularly use technologies for organizational and communication purposes. However, the use of math-specific technology such as computer algebra systems, or dynamic geometry software for instructional, exploratory, and creative activities with students takes place mostly on an individual basis, only occasionally, and is very much topic specific. This was even the case for those instructors who use technology proficiently in their research. The data also suggested that familiarity with and discussions of examples of technology implementation in teaching at regular and field-oriented professional development seminars within mathematics departments could potentially increase the use of math-specific technology by instructors.     

Phase‐plane analysis of simple nonlinear autonomous systems

The aim of this paper is to encourage the teaching of phase‐plane analysis. This topic has practical applications in a wide variety of fields including national economic modelling, biology, technological innovations, sociology and modern medicine.     

Mathematics teachers’ specialized knowledge: a secondary teacher's knowledge of rational numbers

Recognising teachers’ knowledge as one of the main factors influencing their practices and student learning, we aim to contribute to obtaining a better and deeper understanding of the specificities of teachers’ mathematical knowledge. A case study involving one 8th-grade Chilean mathematics teacher is presented in the context of rational numbers. Using video and audio recordings of classroom practices, questionnaires, and an interview, we sought to characterise, and better understand the content of the Knowledge of Topics from the perspective of the Mathematics Teachers’ Specialized Knowledge (MTSK) theoretical framework. The results reveal some critical aspects that teacher education should focus on, while also identifying lost opportunities and examples of “good” practices, thus contributing to the refinement of the MTSK conceptualisation. The conclusions can be considered in a broader perspective, with implications for teacher education in other contexts.