Pre-service mathematics teachers’ concept images of radian

This study investigates pre-service mathematics teachers’ concept images of radian and possible sources of such images. A multiple-case study was conducted for this study. Forty-two pre-service mathematics teachers completed a questionnaire, which aims to assess their understanding of radian. Six of them were selected for individual interviews on the basis of theoretical sampling. The data indicated that participants’ concept images of radian were dominated by their concept images of degree. As the data in this study suggested, pre-service mathematics teachers were reluctant to accept trigonometric functions with the inputs of real numbers but rather they use value in degrees. More interestingly, they have two distinct images of π : π as an angle in radian and π as an irrational number.     

Middle-school students’ concept images of geometric translations

This study explored sixth grade students’ concept images of geometric translations and the possible sources of their conceptions in a non-technological environment. The data were gathered through a written instrument, student and teacher interviews and document analyses. Analyses of student responses revealed two major concept images of geometric translations: (a) translation as translational motion, and (b) translation as both translational and rotational motion. Students who held these conceptions showed various levels of understanding, such as conceiving translations as undefined motion, partially-defined motion, and defined-motion of a single geometric figure on the plane. The findings of the study suggested, in general, consistencies between students’ concept images and their concept definitions. However, most of the students’ concept definitions were inconsistent with the formal concept definition of geometric translations.Data analyses also revealed five interpretations of a translation vector: (a) vector as a reference line, (b) vector as a symmetry line, (c) vector as a direction indicator, (d) vector as a parameter, and (e) vector as an abstract tool. Furthermore, classroom instruction, mathematics and science textbooks, real-life examples and everyday language were the major sources of students’ concept images of geometric translations.     

Mathematical under-preparedness: the influence of the pre-tertiary mathematics experience on students’ ability to make a successful transition to tertiary level mathematics courses in Ireland

Internationally, the consequences of the ‘Mathematics problem’ are a source of concern for the education sector and governments alike. Growing consensus exists that the inability of students to successfully make the transition to tertiary level mathematics education lies in the substantial mismatch between the nature of entrants’ pre-tertiary mathematical experiences and subsequent tertiary level mathematics-intensive courses. This paper reports on an Irish study that focuses on the pre-tertiary mathematics experience of entering students and examined its influence on students’ ability to make a successful transition to tertiary level mathematics. Brousseau's ‘didactical contract’ is used as an intellectual tool to uncover and describe the contract that exists in two case mathematics classrooms in Irish upper secondary schools (Senior Cycle). Although the authors are professional mathematics educators and well informed about classroom practice in Ireland, they were genuinely surprised by the very restrictive nature of this contract and the damaging consequences for students’ future mathematical education.     

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.     

Dimensions of students’ views of themselves as learners of mathematics

Students’ views of themselves as learners of mathematics are a decisive parameter for their engagement and success in school. We are interested in students’ experiences with mathematics encompassing cognitive, emotional and motivational aspects. In particular, we focus on capturing the structural properties of affect related to mathematics. Participants in our study were 1,436 randomized chosen students of secondary schools from overall Finland. In the Finnish upper secondary school, there are two different syllabi for mathematics: the general and the advanced one. Schools were invited to organize the survey by one of their year 2 general syllabus courses and one of their year 2 advanced syllabus courses in grade 11. By means of factor analysis, we obtained seven dimensions in which students’ hold beliefs and emotions about mathematics partly intertwined with their motivational orientations. These dimensions are described by reliable scales, which allow outlining an average image of Finnish students’ views of themselves as learners of mathematics. Moreover, we analyzed relations between the seven dimensions and what kind of structure they generate. Thereby, a core of three high correlating dimensions could be identified, yielding different accentuations with regard to course choice.     

Israeli Jewish and Arab students’ gendering of mathematics

In English-speaking, Western countries, mathematics has traditionally been viewed as a “male domain”, a discipline more suited to males than to females. Recent data from Australian and American students who had been administered two instruments [Leder & Forgasz, in Two new instruments to probe attitudes about gender and mathematics. ERIC, Resources in Education (RIE), ERIC document number: ED463312, 2002] tapping their beliefs about the gendering of mathematics appeared to challenge this traditional, gender-stereotyped view of the discipline. The two instruments were translated into Hebrew and Arabic and administered to large samples of grade 9 students attending Jewish and Arab schools in northern Israel. The aims of this study were to determine if the views of these two culturally different groups of students differed and whether within group gender differences were apparent. The quantitative data alone could not provide explanations for any differences found. However, in conjunction with other sociological data on the differences between the two groups in Israeli society more generally, possible explanations for any differences found were explored. The findings for the Jewish Israeli students were generally consistent with prevailing Western gendered views on mathematics; the Arab Israeli students held different views that appeared to parallel cultural beliefs and the realities of life for this cultural group.     

Puzzle-based learning in engineering mathematics: students’ attitudes

The article reports on the results of two case studies on the impact of the regular use of puzzles as a pedagogical strategy in the teaching and learning of engineering mathematics. The intention of using puzzles is to engage students’ emotions, creativity and curiosity and also to enhance their generic thinking skills and lateral thinking ‘outside the box’. Students’ attitudes towards this pedagogical strategy are evaluated via short questionnaires with two groups of university students taking a second-year engineering mathematics course. Students’ responses to the questionnaire are presented and analyzed in the paper.     

Characteristics of good mathematics teaching in Singapore grade 8 classrooms: a juxtaposition of teachers’ practice and students’ perception

This paper examines the instructional approaches of three competent grade 8 mathematics teachers. It also examines their students’ perception of the lessons they taught as well as characteristics of good lessons. The findings of teachers’ practice and students’ perception are juxtaposed to elicit characteristics of good teaching in Singapore grade 8 classrooms. With limitation, the findings of the paper suggests that good mathematics teaching in Singapore schools centres around building understanding and is teacher-centred but student focused. Some characteristic features of good lessons are that their instructional cycles have specific instructional objectives such that subsequent cycles incrementally build on the knowledge. The examples used in such lessons are carefully selected and vary in complexity from low to high. Teachers actively monitor their student’s understanding during seatwork, by moving from desk to desk guiding those with difficulties and selecting appropriate student work for subsequent whole-class review and discussion. Finally, during such lessons teachers reinforce their students’ understanding of knowledge expounded during whole-class demonstration by detailed review of student work done in class or as homework.     

How concept images affect students’ interpretations of Newton's method

Knowing when students have the prerequisite knowledge to be able to read and understand a mathematical text is a perennial concern for instructors. Using text describing Newton's method and Vinner's notion of concept image, we exemplify how prerequisite knowledge influences understanding. Through clinical interviews with first-semester calculus students, we determined how evoked concept images of tangent lines and roots contributed to students’ interpretation and application of Newton's method. Results show that some students’ concept images of root and tangent line developed throughout the interview process, and most students were able to adequately interpret the text on Newton's method. However, students with insufficient concept images of tangent line and students who were unwilling or unable to modify their concept images of tangent line after reading the text were not successful in interpreting Newton's method.     

Investigating students’ perceptions of graduate learning outcomes in mathematics

ABSTRACT

The purpose of this study is to explore the perceptions mathematics students have of the knowledge and skills they develop throughout their programme of study. It addresses current concerns about the employability of mathematics graduates by contributing much needed insight into how degree programmes are developing broader learning outcomes for students majoring in mathematics. Specifically, the study asked students who were close to completing a mathematics major (n = 144) to indicate the extent to which opportunities to develop mathematical knowledge along with more transferable skills (communication to experts and non-experts, writing, working in teams and thinking ethically) were included and assessed in their major. Their perceptions were compared to the importance they assign to each of these outcomes, their own assessment of improvement during the programme and their confidence in applying these outcomes. Overall, the findings reveal a pattern of high levels of students’ agreement that these outcomes are important, but evidence a startling gap when compared to students’ perceptions of the extent to which many of these – communication, writing, teamwork and ethical thinking – are actually included and assessed in the curriculum, and their confidence in using such learning.     

On students’ conceptions of arithmetic average: the case of inference from a fixed total

There is more to understanding the concept of mean than simply knowing and applying the add-them-up and divide algorithm. In the following, we discuss a component of understanding the mean – inference from a fixed total – that has been largely ignored by researchers studying students understanding of mean. We add this component to the list of types of reasoning needed to understand mean and discuss student responses to tasks designed to elicit this component of reasoning. These responses reveal that inference from a fixed total reasoning is rare even in advance high school students.     

Expanding conceptions of utility: middle school students’ perspectives on the usefulness of mathematics

This research explores how adolescents conceptualize the usefulness of mathematics. Integrating sociocultural theory with the study of utility value, this study uses open-ended survey items and interview tasks to examine conceptions of usefulness among a group of predominantly Latinx middle school students. Findings reveal that students primarily conceptualized the usefulness of mathematics in two ways. First, students considered the applicability of mathematics content, focusing on applications of mathematics in everyday life and future jobs/careers. Second, students considered the usefulness of features of the learning experience, such as the form of interaction and structure of the activity. Both conceptions are compared to existing conceptions of usefulness in the literature, and implications for classroom practice and future research are discussed.     

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.     

The textbook-in-use: students’ utilization schemes of mathematics textbooks related to self-regulated practicing

This paper presents a qualitative study on how students make use of their mathematics textbooks for practicing. The study was carried out in two German secondary schools with 74 students (44 in 6th and 30 in 12th grade). Students’ utilization of textbooks for practicing is analyzed using the theoretical framework of instrumental genesis. The results indicate that students’ choices of contents from the book for practicing can be categorized into three utilization schemes: position-dependent practicing, block-dependent practicing, and salience-dependent practicing. In terms of position-dependent practicing the relative position of the textbook’s contents to teacher-mediated sections guides the students’ choice. Block-dependent practicing relates to the use of contents from the book that belong to particular blocks. Finally, salience-dependent practicing is a utilization scheme of the book where students’ choice is guided by perceptual salience of the book contents. These findings both show how textbook users are influenced by the way mathematics is presented in textbooks and provide insights into students’ conceptions of practicing mathematics.     

Future mathematics teachers’ professional knowledge of elementary mathematics from an advanced standpoint

This paper reports a joint research project by researchers from three countries on an international comparative study that examines the professional knowledge of prospective mathematics teachers in elementary mathematics from an advanced standpoint. For this study, mathematical problems on various topics of elementary mathematical content were developed. Using this instrument, the mathematical knowledge of future teachers from Germany, Hong Kong, China (Hangzhou) and South Korea was measured empirically. The paper presents the design of the study, and also results are discussed. The results show that there are systematic differences among the participating countries; for example, the Korean future teachers outperform their counterparts in other countries. A more detailed analysis of the results suggests that the future teachers often do not seem to be able to link school and university knowledge systematically and cannot achieve the crucial “advanced standpoint” from the teacher training programme.     

Predictive factors of students’ motivation to succeed in introductory mathematics courses: evidence from higher education in the UAE

The purpose of the study is to examine whether there is a significant relationship between students’ motivation to succeed in introductory mathematics courses offered by universities in the United Arab Emirates (UAE) as the dependent variable of the research and another five independent variables including cognitive mathematics self-concept, affective mathematics self-concept, extrinsic motivation as expectations of future career and income, students’ age and the number of mathematics courses taken by students. The rationale of the study is based on the significance of mathematics achievements for students and academic institutions in particular, as well as for the society in general. The study is designed based on a quantitative research methodology and a sample of 685 students participated in completing a survey questionnaire. The sample is drawn from students who were registered in different introductory mathematics courses at four academic institutions of higher education in the UAE. The quantitative correlation analysis among students’ motivation, cognitive mathematics self-concept, affective mathematics self-concept, extrinsic motivation, students’ age and the number of mathematics courses taken by students reveals theoretically consistent interrelationships. The quantitative multiple regression analysis indicates that the five independent variables explain 71.3% of the variation in students’ motivation to succeed in introductory mathematics courses.     

Assessing science students’ attitudes to mathematics: a case study on a modelling project with mathematical software

This article reports the attitudes of students towards mathematics after they had participated in an applied mathematical modelling project that was part of an Applied Mathematics course. The students were majoring in Earth Science at the National Taiwan Normal University. Twenty-six students took part in the project. It was the first time a mathematical modelling project had been incorporated into the Applied Mathematics course for such students at this University. This was also the first time the students experienced applied mathematical modelling and used the mathematical software. The main aim of this modelling project was to assess whether the students’ attitudes toward mathematics changed after participating in the project. We used two questionnaires and interviews to assess the students. The results were encouraging especially the attitude of enjoyment. Hence the approach of the modelling project seems to be an effective method for Earth Science students.     

Live lectures or online videos: students’ resource choices in a first-year university mathematics module

In Maths for Business, a mathematics module for non-mathematics specialists, students are given the choice of completing the module content via short online videos, live lectures or a combination of both. In this study, we identify students’ specific usage patterns with both of these resources and discuss their reasons for the preferences they exhibit. In 2015–2016, we collected quantitative data on each student's resource usage (attendance at live lectures and access of online videos) for the entire class of 522 students and employed model-based clustering which identified four distinct resource usage patterns with lectures and/or videos. We also collected qualitative data on students’ perceptions of resource usage through a survey administered at the end of the semester, to which 161 students responded. The 161 survey responses were linked to each cluster and analysed using thematic analysis. Perceived benefits of videos include flexibility of scheduling and pace, and avoidance of large, long lectures. In contrast, the main perceived advantages of lectures are the ability to engage in group tasks, to ask questions, and to learn ‘gradually’. Students in the two clusters with high lecture attendance achieved, on average, higher marks in the module.