A case study of pedagogy of mathematics support tutors without a background in mathematics education

This study investigates the pedagogical skills and knowledge of three tertiary-level mathematics support tutors in a large group classroom setting. This is achieved through the use of video analysis and a theoretical framework comprising Rowland's Knowledge Quartet and general pedagogical knowledge. The study reports on the findings in relation to these tutors’ provision of mathematics support to first and second year undergraduate engineering students and second year undergraduate science students. It was found that tutors are lacking in various pedagogical skills which are needed for high-quality learning amongst service mathematics students (e.g. engineering/science/technology students), a demographic which have low levels of mathematics upon entering university. Tutors teach their support classes in a very fast didactic way with minimal opportunities for students to ask questions or to attempt problems. It was also found that this teaching method is even more so exaggerated in mandatory departmental mathematics tutorials that students take as part of their mathematics studies at tertiary level. The implications of the findings on mathematics tutor training at tertiary level are also discussed.     

Varied Meanings and Engagement in School Mathematics: Cross‐Case Analysis of Three High‐Achieving Young Adolescent Girls

This paper presents an in‐depth cross‐case analysis of three high‐achieving young adolescent girls who had contrasting mathematics learning experiences during the first year of middle school. In particular, this study examines the foundation for their motivation, as well as the dominant mode of learning and academic engagement in relation to three sociocultural factors, family background, the role of peers, and the level of teachers' understanding of the students and instructional support provided. Our data analysis revealed that the three girls possessed motivation structures and learning dispositions that are more or less prone to conceptual or procedural understanding in mathematics. This resulted in a significant variation in the mode of their academic engagement with the subject, and this provided a different set of challenges in each girl's pursuit of higher level of mathematics learning.     

Are beginning calculus and engineering students adequately prepared for higher education? An assessment of students’ basic mathematical knowledge

Are students transitioning from the secondary level to university studies in mathematics and engineering adequately prepared for education at the tertiary level? In this study, we discuss the prior mathematical knowledge and skills demonstrated by Norwegian engineering (N?=?1537) and calculus (N?=?626) university students by using data from a mathematics assessment administered by the Norwegian Mathematical Council. The assessment examines students’ conceptual understanding, computation skills and problem solving skills on the basis of the mathematics curriculum of lower secondary education. We found that calculus students significantly outperformed engineering students, but both student groups struggled to solve the test, with the calculus and engineering groups scoring an average of 60% and 46%, respectively. Beginning students who fail to master basic skills, such as solving arithmetic and algebra problems, will most likely face difficulties in their further courses. Although few female students enrol in calculus and engineering programmes compared with male ones and are thus underrepresented, male and female students at the same ability level achieved comparable test scores. Furthermore, students reported high levels of intrinsic and extrinsic motivation, and a positive relationship was observed between intrinsic motivation and achievement.     

A support learning programme for first-year mathematics*

ABSTRACT

This article is a follow-up to an earlier paper on the mathematics support learning tutorial programme (SLT), an intervention programme at The University of Queensland that targets students considered to be at risk of failing Calculus and Linear Algebra I, the first tertiary level mathematics subject at The University of Queensland. The first paper (Hillock, P., Jennings, M., Roberts, A., & Scharaschkin, V. (2013). Amathematics support programme for first-year engineering students. International Journal of Mathematical Education in Science and Technology, 44(7), 1030–1044) reported on the inaugural programme implemented in 2012. This article provides an update of the progress of the SLT since 2012. We provide statistics for the subsequent 12 semesters to Semester 2, 2018 and describe the evolution of the SLT since its implementation. Statistical analysis of the additional data and student feedback indicate that the SLT continues to have a positive impact on student learning, with weak students making significant gains from attending the programme.     

The Effects of a Teacher In‐Service on Low‐Achieving Grade 7 and 8 Mathematics Students

Previous research (e.g., Woodward & Baxter, 1997 ) found that Standards‐based mathematics teaching provides marginal or no benefits for low achievers, in contrast with positive effects for middle and high ability students. A randomized quasi‐experiment in 52 Canadian schools found that low achieving grade 7 and 8 students who received support consisting of placement on a learning continuum, instruction focused on their specific learning needs, and concrete materials to represent mathematical constructs, benefited from teaching that emphasized construction over transmission of knowledge. Treatment students showed small but statistically significant improvements over controls in student achievement, and controversially, in mathematical beliefs, and attitudes. The latter finding raised issues of the appropriate balance between Type I and Type II error in educational research.     

A hybrid model of mathematics support for science students emphasizing basic skills and discipline relevance

The problem of students entering university lacking basic mathematical skills is a critical issue in the Australian higher-education sector and relevant globally. The Maths Skills programme at La Trobe University has been developed to address under preparation in the first-year science cohort in the absence of an institutional mathematics support centre. The programme was delivered through first-year science and statistics subjects with large enrolments and focused on basic mathematical skills relevant to each science discipline. The programme offered a new approach to the traditional mathematical support centre or class. It was designed through close collaboration between science subject coordinators and the project leader, a mathematician, and includes resources relevant to science and mathematics questions written in context. Evaluation of the programme showed it improved the confidence of the participating students who found it helpful and relevant. The programme was delivered through three learning modes to allow students to select activities most suitable for them, which was appreciated by students. Mathematics skills appeared to increase following completion of the programme and student participation in the programme correlated positively and highly with academic grades in their relevant science subjects. This programme offers an alternative model for mathematics support tailored to science disciplines.     

Conceptions of mathematics and student identity: implications for engineering education

Lecturers of first-year mathematics often have reason to believe that students enter university studies with naïve conceptions of mathematics and that more mature conceptions need to be developed in the classroom. Students’ conceptions of the nature and role of mathematics in current and future studies as well as future career are pedagogically important as they can impact on student learning and have the potential to influence how and what we teach. As part of ongoing longitudinal research into the experience of a cohort of students registered at the author's institution, students’ conceptions of mathematics were determined using a coding scheme developed elsewhere. In this article, I discuss how the cohort of students choosing to study engineering exhibits a view of mathematics as conceptual skill and as problem-solving, coherent with an accurate understanding of the role of mathematics in engineering. Parallel investigation shows, however, that the students do not embody designated identities as engineers.     

MATLAB in first-year engineering mathematics

The paper details the integration of the mathematical software MATLAB into the teaching of core mathematics to first-year university engineering students. The engineering faculty requested that the mathematics staff allocate one hour per week for students to learn and use MATLAB in a computer pool and that group projects involving the use of MATLAB and other similar tasks be included in the overall assessment of the subject. The central concept in achieving this integration was the production of a guide A Focused Introduction to MATLAB which provides a bridge between the software and the core subject material covered in the textbooks and lectures. Recent extensions involve the construction of a Web site dedicated to this subject where students can obtain all information concerning the subject, including a copy of all MATLAB code and corresponding graphical demonstrations used in the lectures. Feedback has been very positive.     

Mathematics support--support for all?

Mathematics Support Centres are to be found in various formsin the majority of UK higher education institutions. They havebeen established in order to ease widespread and serious difficultiesthat a significant number of students have with mathematics,particularly at the school–university transition. Theyusually offer mathematics and/or statistics support to studentsacross the full range of disciplines studied. Anecdotal evidencesuggests that those students who make good use of such centresare not just those who struggle with mathematics. Many frequentusers are quite competent and simply want to do better. Thestudy reported here describes and analyses data from one cohortof engineering students. A novel aspect is the quantificationof the proportion of support centre visitors who fall into these,and other, categories. We conclude of the cohort in the study,mathematics support has improved the pass rate by 3%. Of thefailures, about half (4% of the sample total) could well havepassed had they attended the mathematics support centre regularly.Furthermore, the majority of those attending were not studentswho were in danger of failing. This has important implicationsnot only for the design of mathematics support provision, butalso for the performance of the high fliers. The methodologyoffers one way tackling the difficult task of evaluating theeffectiveness of mathematics support initiatives.     

Challenges students identified with a learning disability and as high-achieving experience when using diagrams as a visualization tool to solve mathematics word problems

This article addresses a much understudied topic and concern regarding how students of varying ability levels employ visualization as a strategy in mathematics learning. The importance of this topic can be found in its connection to students’ ability to solve mathematical word problems. Many students, particularly students with learning disabilities, often struggle to use visualization as a strategy and this impacts their mathematics performance. The purpose of this article is to present findings from a study that examined the challenges that students—those identified as learning disabled and high-achieving—displayed when using one visualization form, a diagram, to solve mathematics problems. Overall, nine challenges related to the use of diagram proficiency to solve problems were identified. Further, students with learning disabilities were found to be more likely than their high achieving peers to experience these challenges. Implications for practice are provided.     

Integrating history and philosophy in mathematics education at university level through problem-oriented project work

Through the last three decades several hundred problem-oriented student-directed projects concerning meta-aspects of mathematics and science have been performed in the 2-year interdisciplinary introductory science programme at Roskilde University. Three selected reports from this cohort of project reports are used to investigate and present empirical evidence for learning potentials of integrating history and philosophy in mathematics education. The three projects are: (1) a history project about the use of mathematics in biology that exhibits different epistemic cultures in mathematics and biology. (2) An educational project about the difficulties of learning mathematics that connects to the philosophy of mathematics. (3) A history of mathematics project that connects to the sociology of multiple discoveries. It is analyzed and discussed in what sense students gain first hand experiences with and learn about meta-aspects of mathematics and their mathematical foundation through the problem-oriented student-directed project work.     

Applying an alternative mathematics pedagogy for students with weak mathematics: meta-analysis of alternative pedagogies

Student mathematics performance and the need for work-ready graduates to be mathematics-competent is a core issue for many universities. While both student and teacher are responsible for learning outcomes, there is a need to explicitly acknowledge the weak mathematics foundation of many university students. A systematic literature review was undertaken of identified innovations and/or interventions that may lead to improvement in student outcomes for university mathematics-based units of study. The review revealed the importance of understanding the foundations of student performance in higher education mathematics learning, especially in first year. Pre-university mathematics skills were identified as significant in student retention and mathematics success at university, and a specific focus on student pre-university mathematics skill level was found to be more effective in providing help, rather than simply focusing on a particular at-risk group. Diagnostics tools were found to be important in identifying (1) student background and (2) appropriate intervention. The studies highlighted the importance of appropriate and validated interventions in mathematics teaching and learning, and the need to improve the learning model for mathematics-based subjects, communication and technology innovations.     

Can't do maths--understanding students' maths anxiety

The number of students continuing with their mathematics educationpost GCSE level has declined in recent years and hence studentsentering Engineering degrees are reducing. The University ofBirmingham recognized this problem and introduced the Suiteof Technology programme (STP) which no longer requires studentsto have A-level mathematics. Therefore lecturers at universityare now faced with teaching A-level mathematics in order togive the students the mathematical skills for their technologydegree. With little experience of teaching at this level, lecturersfrequently face the challenge of choosing the most appropriatelevel for a lecture that encourages students to engage withand learn a subject that they are novices in. It turned outthat some students have a mathematics anxiety and hence thebiggest challenge for a lecturer is supporting the studentsovercoming this fear of mathematics. Choosing the appropriatestarting level for any lecture and the fear of mathematics wasacknowledged during a peer learning group meeting as part ofthe Post Graduate Certificate in Learning and Teaching in HigherEducation (PGCLTHE) at the University of Birmingham. In orderto explore this further, a session of mathematics was taughtby a Civil Engineering lecturer to fellow peers who do not havean engineering background. This article describes the mathematicsteaching session, reflections from the lecturer and the learners,and the impact that this had on teaching mathematics to undergraduates.Further, the article explores the difficulties and challengesexperienced by lecturers when teaching mathematics as a servicesubject.     

工科复变函数中的迁移教学

迁移是指一种学习对另一种学习的影响.影响学生学习迁移的主要因素有:学生原有知识的准备状态、概括水平、认知结构变量、学习材料的性质、学习的指导.为了能有效促进学生学习正迁移的发生,同时防止学生学习负迁移的发生,在工科复变函数教学中针对其特点采取以下教学策略:注意加强工科复变函数与高等数学的衔接;突出转化和类比的数学思想方法的教学;加强复变函数学习方法的指导.     

Assessment of numeracy in sports and exercise science students at an Australian university

The effect of high school study of mathematics on numeracy performance of sports and exercise science (SES) students is not clear. To investigate this further, we tested the numeracy skills of 401 students enrolled in a Bachelor of Health Sciences degree in SES using a multiple-choice survey consisting of four background questions and 39 numeracy test questions. Background questions (5-point scale) focused on highest level of mathematics studied at high school, self-perception of mathematics proficiency, perceived importance of mathematics to SES and likelihood of seeking help with mathematics. Numeracy questions focused on rational number, ratios and rates, basic algebra and graph interpretation. Numeracy performance was based on answers to these questions (1 mark each) and represented by the total score (maximum = 39). Students from first (n = 212), second (n = 78) and third (n = 111) years of the SES degree completed the test. The distribution of numeracy test scores for the entire cohort was negatively skewed with a median (IQR) score of 27(11). We observed statistically significant associations between test scores and the highest level of mathematics studied (P < 0.05), being lowest in students who studied Year 10 Mathematics (20 (9)), intermediate in students who studied Year 12 General Mathematics (26 (8)) and highest in two groups of students who studied higher-level Year 12 Mathematics (31 (9), 31 (6)). There were statistically significant associations between test scores and level of self-perception of mathematics proficiency and also likelihood of seeking help with mathematics (P < 0.05) but not with perceived importance of mathematics to SES. These findings reveal that the level of mathematics studied in high school is a critical factor determining the level of numeracy performance in SES students.     

Beliefs and Attitudes Toward Mathematics Among Third- and Fifth-Grade Students: A Descriptive Study

A survey on attitudes toward mathematics of third- and fifth-grade students enrolled in a large urban school district was conducted in the spring of 1994 as part of a review of the mathematics program. Student responses to survey items were analyzed with respect to gender and grade level. Although girls and boys were equally likely to indicate that they like mathematics, in both Grades 3 and 5, boys were more likely than girls to report being good at mathematics. This result has often been reported for older students but has rarely been explored in younger children. Some gender and grade differences were observed in students' attitudes towards the specific strands of mathematics. Both girls and boys in Grade 5 were more likely than students in Grade 3 to believe that mathematics was relevant to their lives. Virtually all children thought that both girls and boys needed to study mathematics. No gender or grade differences in students' beliefs regarding the process of learning mathematics were observed. The results support the need for further research to identify variables that influence the development of student perceptions and attitudes toward mathematics during elementary school.     

Content Knowledge,Attitudes, and Self‐Efficacy in the Mathematics New York City Teaching Fellows (NYCTF) Program

The purpose of this study was to understand the mathematical content knowledge new teachers have both before and after taking a mathematics methods course in the NYCTF program. Further, the purpose was to understand the attitudes toward mathematics and concepts of self‐efficacy that Teaching Fellows had over the course of the semester. The sample included 42 new Teaching Fellows who were given a mathematics content test, attitudes toward mathematics questionnaire, and teaching self‐efficacy questionnaire at the beginning and end of the semester. Further, the teachers kept teaching and learning journals. Findings revealed a significant increase in both mathematical content knowledge and positive attitudes toward mathematics. Additionally, Teaching Fellows were found to have positive attitudes and high self‐efficacy at the end of the semester, and relationships were found between attitudes and self‐efficacy. Finally, Teaching Fellows generally found that classroom management was the biggest issue in their teaching, and that problem solving and numeracy were the most important topics addressed in their learning. Future studies should address self‐efficacy differences between preservice and in‐service teachers and the effects of alternative certification teacher knowledge, attitudes toward mathematics, and self‐efficacy on students in the classroom.     

Examining undergraduate student retention in mathematics using network analysis and relative risk

Higher education faces challenges in retaining students who require a command of numeracy in their chosen field of study. This study applies an innovative combination of relative risk and social network analysis to enrolment data of a single cohort of commencing students from an Australian regional university. Relative risk, often used in epidemiology studies, is used to strategically investigate whether first year mathematics subjects at the university demonstrated a higher risk of attrition when compared to other subjects offered in the first year of study. The network analysis is used to illustrate the connections of those mathematics subjects, identifying service subjects through their multiple connections. The analysis revealed that attrition rates for eight of the nine subjects were within acceptable limits, and this included identified service subjects. The exception highlighted the issue of mathematics competencies in this cohort. This combined analytical technique is proposed as appropriate for use when investigating attrition and retention at faculty and institutional levels, including the determination of levels of intervention and support for any subject.     

A three-year perspective on conceptions of and orientations to learning mathematics of prospective teachers and first year university students

The paper reports a compilation of results from three studies conducted over three years to determine students' conceptions of mathematics, and orientations they follow in learning the subject. Respondents were 459 first year mathematics students from four universities and one teacher college. Results indicated that more than half the sample reported mathematics to be a subject made of numbers and formulae that could be memorized. This suggests a shallow emphasis when learning the subject, with no intention to understand. However, most students passed their examinations. It was concluded that there was no statistically significant relationship between examinations results and students' learning orientations. It is recommended that lecturers should foster students' meta-learning capabilities and an awareness of different learning strategies.     

Students’ experiences with mathematics teaching and learning: listening to unheard voices

This study documents students’ views about the nature of mathematics, the mathematics learning process and factors within the classroom that are perceived to impact upon the learning of mathematics. The participants were senior secondary school students. Qualitative and quantitative methods were used to understand the students’ views about their experiences with mathematics learning and mathematics classroom environment. Interviews of students and mathematics lesson observations were analysed to understand how students view their mathematics classes. A questionnaire was used to solicit students’ views with regards to teaching approaches in mathematics classes. The results suggest that students consider learning and understanding mathematics to mean being successful in getting the correct answers. Students reported that in the majority of cases, the teaching of mathematics was lecture-oriented. Mathematics language was considered a barrier in learning some topics in mathematics. The use of informal language was also evident during mathematics class lessons.