Student performance and attitudes in a collaborative and flipped linear algebra course

Flipped learning is gaining traction in K-12 for enhancing students’ problem-solving skills at an early age; however, there is relatively little large-scale research showing its effectiveness in promoting better learning outcomes in higher education, especially in mathematics classes. In this study, we examined the data compiled from both quantitative and qualitative measures such as item scores on a common final and attitude survey results between a flipped and a traditional Introductory Linear Algebra class taught by two individual instructors at a state university in California in Fall 2013. Students in the flipped class were asked to watch short video lectures made by the instructor and complete a short online quiz prior to each class attendance. The class time was completely devoted to problem solving in group settings where students were prompted to communicate their reasoning with proper mathematical terms and structured sentences verbally and in writing. Examination of the quality and depth of student responses from the common final exam showed that students in the flipped class produced more comprehensive and well-explained responses to the questions that required reasoning, creating examples, and more complex use of mathematical objects. Furthermore, students in the flipped class performed superiorly in the overall comprehension of the content with a 21% increase in the median final exam score. Overall, students felt more confident about their ability to learn mathematics independently, showed better retention of materials over time, and enjoyed the flipped experience.     

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.     

An evaluation of the Supplemental Instruction programme in a first year calculus course

Supplemental Instruction (SI) incorporates collaborative learning in small, peer-led, group settings in order to integrate instruction in learning and reasoning skills with course content. Several meta-analyses speak to the efficacy of SI but fail to address selection bias due to ability/motivation and gender. In this study, SI was paired with a first year calculus for non-majors course. An ANCOVA indicated that: ability/motivation, as measured by prior grade point average, was a useful predictor of course letter grade; gender differences were statistically significant but trivial; and, SI participation was statistically and practically significant, a 1.8 letter grade improvement after correction for selection bias. For the pass/fail analysis, a sequential binary logistic regression indicated there was a sizable statistically significant improvement with SI participation after accounting for gender and ability/motivation selection biases. The odds of success were 2.7 times greater for the SI participants. No gender differences of any significance were found.     

Integrating supplementary application-based tutorials in the multivariable calculus course

This article presents a study in which applications were integrated in the Multivariable Calculus course at the Technion in the framework of supplementary tutorials. The purpose of the study was to test the opportunity of extending the conventional curriculum by optional applied problem-solving activities and get initial evidence on the possible impact of the tutorials on students’ beliefs about the value of learning mathematics with applications. The study lasted three semesters and consisted of three experiments in which supplementary tutorials were offered in different forms: a weekly evening meeting for interested students, a weekly extra hour added to the conventional calculus class, and a workshop which introduces mathematics concepts from the application perspective. The study reveals: (1) significant positive effect of the tutorials on the students’ beliefs (in all three experiments); (2) statistically significant advantage of the group involved in the tutorials in relation to the group that learned in the conventional way (second experiment); (3) students’ positive evaluation of the workshops for better understanding the course lectures (third experiment). Grounding on the study experience, we propose for further discussion a stage model of the applied problem solving cycle.     

Mathematics learning support and engagement in first year engineering

This paper considers the effects of both free optional mathematics learning support and engagement on the mathematics performance in a foundation mathematics subject of a cohort of engineering students entering university with poor mathematical skills. New engineering students were directed to either a foundation or standard mathematics subject based on the results of a placement test. For students in the foundation subject, it was found that high levels of learning support were associated with greater improvement over the semester. Some form of learning support was used by 57.9% of the students, a reasonably high proportion of the cohort. Some factors for this high level of use of learning support are considered. One possible factor, the placement test, appears to have had a positive effect. Engagement in the subject activities as measured by tutorial attendance and learning management system use was found to have a positive association with final mark. Students who utilized a high level of learning support were more likely to be more engaged with the subject, making it impossible to draw conclusions about improvements being solely due to the use of learning support.     

Implementing a flipped classroom approach in a university numerical methods mathematics course

This paper describes and analyses the implementation of a ‘flipped classroom’ approach, in an undergraduate mathematics course on numerical methods. The approach replaced all the lecture contents by instructor-made videos and was implemented in the consecutive years 2014 and 2015. The sequential case study presented here begins with an examination of the attitudes of the 2014 cohort to the approach in general as well as analysing their use of the videos. Based on these responses, the instructor makes a number of changes (for example, the use of ‘cloze’ summary notes and the introduction of an extra, optional tutorial class) before repeating the ‘flipped classroom’ approach the following year. The attitudes to the approach and the video usage of the 2015 cohort are then compared with the 2014 cohort and further changes that could be implemented for the next cohort are suggested.     

Mathematical tasks,study approaches,and course grades in undergraduate mathematics: a year-by-year analysis

Students approach learning in different ways, depending on the experienced learning situation. A deep approach is geared toward long-term retention and conceptual change while a surface approach focuses on quickly acquiring knowledge for immediate use. These approaches ultimately affect the students’ academic outcomes. This study takes a cross-sectional look at the approaches to learning used by students from courses across all four years of undergraduate mathematics and analyses how these relate to the students’ grades. We find that deep learning correlates with grade in the first year and not in the upper years. Surficial learning has no correlation with grades in the first year and a strong negative correlation with grades in the upper years. Using Bloom's taxonomy, we argue that the nature of the tasks given to students is fundamentally different in lower and upper year courses. We find that first-year courses emphasize tasks that require only low-level cognitive processes. Upper year courses require higher level processes but, surprisingly, have a simultaneous greater emphasis on recall and understanding. These observations explain the differences in correlations between approaches to learning and course grades. We conclude with some concerns about the disconnect between first year and upper year mathematics courses and the effect this may have on students.     

A report on the introduction of a multiple choice examination for a first year university engineering mathematics course

The paper is an account of the experience of introducing a multiple choice mathematics examination paper for the first year engineering mathematics course at Nottingham University. The form of the paper is described, as is a system for providing the candidates with feedback on their performance without the publication of the examination paper.     

Sources of engineering teaching self-efficacy in a STEAM methods course for elementary preservice teachers

This study investigated the effect of a STEAM (science, technology, engineering, arts, and mathematics) methods course on elementary preservice teachers’ (PTs’) perceptions of self-efficacy to teach engineering practices. The course positioned engineering as the primary content area from which to integrate other subjects. To enhance PT’s perception of engineering self-efficacy, the course provided instruction that leveraged the following sources of self-efficacy: cognitive content mastery, cognitive pedagogical mastery, vicarious experience, verbal persuasion, and emotional state. The study also examined to what extent the various sources of self-efficacy contributed to changes in self-efficacy. Data was collected from 14 participants that included a self-efficacy survey and focus group interview. After completing the course, elementary PTs’ self-efficacy to teach engineering practices increased significantly. Qualitative data analysis revealed cognitive pedagogical mastery, vicarious experience (specifically simulated modeling), and emotional state were the most influential sources linked to positive changes in self-efficacy, with cognitive content mastery, and other forms of vicarious experience contributing, but to a lesser degree. These results suggest that teacher preparation programs can better support elementary PTs to teach engineering practices by offering additional methods courses focused on engineering, rather than providing short-term exposure to engineering practices and pedagogy in overloaded science methods courses.     

Integrated STEM in practice: Learning from Montessori philosophies and practices

In theory, STEM (interdisciplinary science, technology, engineering and mathematics) is cross‐disciplinary and situated in real‐world problem‐solving contexts. In practice, STEM disciplines are often implemented separately using contrived contexts. This paper examines theoretical and empirical aspects of Montessori middle school science in the United States, and its alignment with the conceptual framework of integrated STEM. We selected Montessori adolescent environments because the Montessori philosophy involves interdisciplinary application contextualized in purposeful work and learning. Our research sought to investigate how Montessori middle schools have designed their science programs, and to situate these findings within the current landscape of STEM education and reform‐based science. Based on the results of our survey of 96 U.S. Montessori middle schools, we argue Montessori offers an integrated educational approach that meaningfully situates academic disciplines to mirror local and global challenges, well supported by theory and literature on STEM and situated learning theories. We assert that integrated STEM happens organically in many Montessori middle schools, and takes place through authentic work in communities of practice. Our research communicates the value of looking outside traditional school settings to examine alternative formal education spaces, like Montessori classrooms where integrated STEM happens organically.     

PBL as a pedagogical approach for integrated STEM: Evidence from prospective teachers

Problem-based learning (PBL) and science, technology, engineering, and mathematics (STEM) are two acronyms widely visible in education literature today. However, few studies have explored these in connection with one another, specifically with regard to teacher preparation. This study investigated how 47 prospective elementary teachers developed PBL units and how they integrated STEM and other disciplines into those units. It also addressed the affordances and constraints of integrated STEM as perceived by the prospective elementary teachers. Data sources in this multimethod study included PBL units and interviews. Findings revealed that all of the units integrated at least two of the STEM disciplines, as well as literacy, in a variety of ways. The prospective teachers articulated perceived benefits of integrated STEM, such as: making connections across content areas, preparing students for the real world, teaching students that failure is not a bad thing, and providing future opportunities. They also addressed perceived barriers of integrated STEM, such as: having limited experience with the content, diminishing the effect of individual content areas, and needing better curriculum alignment. Overall, this study provides evidence that PBL can be a pedagogical approach to integrate STEM. Implications for teachers, teacher educators, and curriculum specialists are discussed.     

Incentivizing advanced mathematics study at upper secondary level: the case of bonus points in Ireland

Secondary level mathematics education in Ireland has recently experienced a period of significant change with the introduction of new curricula and the addition of an incentive to study upper secondary mathematics at the most advanced level (Higher Level). This incentive, typically referred to as ‘bonus points’, appears to have aided a significant increase in the number of students studying upper secondary mathematics at Higher Level. However, thematic analysis of interviews with experienced upper secondary mathematics examiners and exploration of mathematics diagnostic test data outlined in this paper suggest that the difficulty of the Higher Level upper secondary mathematics final examination in Ireland has reduced since the introduction of the bonus points initiative. The sharp increase in students attempting this examination coupled with a policy of maintaining a consistent proportion of students achieving passing grades was identified as a key reason for this possible reduction in standards.     

Driving student-centred calculus: results of a comprehensive case study for Kaizen learning in the Sultanate of Oman

The art of teaching freshmen students is undergoing a rapid paradigm change. Classical forms of teaching are not applicable any more and an unmanageable offer of new multimedia tools and concepts is glutting the market. Moreover, compared to previous courses, the class size triples. In view of these challenges, we implemented a new teaching concept best described as Kaizen learning. By Kaizen learning, we define a teaching philosophy that is based on a concise mix of short learning units (with feedback loops and tests) and of carefully chosen repetitions (also with feedback loops and tests) to calibrate a course for the students. Here, this intensive blended, student-centred learning paradigm is analysed together with its direct impact on the students’ performance. This case study leads to easy-to-implement key drivers for successfully teaching science in Oman, such as (1) human–human interaction, (2) clearly communicated expectations, (3) avoidance of a short-term learning attitude, (4) a no-calculator policy, (5) continuous Kaizen learning, and (6) balanced combination of traditional teaching and e-learning.     

Models are a “metaphor in your brain”: How potential and preservice teachers understand the science and engineering practice of modeling

We investigated beginning secondary science teachers’ understandings of the science and engineering practice of developing and using models. Our study was situated in a scholarship program that served two groups: undergraduate STEM majors interested in teaching, or potential teachers, and graduate students enrolled in a teacher education program to earn their credentials, or preservice teachers. The two groups completed intensive practicum experiences in STEM‐focused academies within two public high schools. We conducted a series of interviews with each participant and used grade‐level competencies outlined in the Next Generation Science Standards to analyze their understanding of the practice of developing and using models. We found that potential and preservice teachers understood this practice in ways that both aligned and did not align with the NGSS and that their understandings varied across the two groups and the two practicum contexts. In our implications, we recommend that teacher educators recognize and build from the various ways potential and preservice teachers understand this complex practice to improve its implementation in science classrooms. Further, we recommend that a variety of practicum contexts may help beginning teachers develop a greater breadth of understanding about the practice of developing and using models.     

Retention of differential and integral calculus: a case study of a university student in physical chemistry

This paper reports a study on retention of differential and integral calculus concepts of a second-year student of physical chemistry at a Danish university. The focus was on what knowledge the student retained 14 months after the course and on what effect beliefs about mathematics had on the retention. We argue that if a student can quickly reconstruct the knowledge, given a few hints, this is just as good as retention. The study was conducted using a mixed method approach investigating students’ knowledge in three worlds of mathematics. The results showed that the student had a very low retention of concepts, even after hints. However, after completing the calculus course, the student had successfully used calculus in a physical chemistry study programme. Hence, using calculus in new contexts does not in itself strengthen the original calculus learnt; they appeared as disjoint bodies of knowledge.     

Opportunities for learning: the use of variation to analyse examples of a paradigm shift in teaching primary mathematics in England

We demonstrate the power of Variation Theory as an analytical tool used to understand the underlying conceptual structure of mathematics lessons taught by English primary school teachers. We study excerpts of three lessons that are posted on a professional website. We show how lesson analysis using variation allows us to focus on what is made available to be learnt in the lesson excerpts. We identify some differences in their use of dimensions of variation and the associated ranges of change and discuss how suitable patterns of variation and invariance might differ according to the nature of the learning focus. We reflect on the value of our analytical approach.     

Seeking mathematics success for college students: a randomized field trial of an adapted approach

Many students enter the Canadian college system with insufficient mathematical ability and leave the system with little improvement. Those students who enter with poor mathematics ability typically take a developmental mathematics course as their first and possibly only mathematics course. The educational experiences that comprise a developmental mathematics course vary widely and are, too often, ineffective at improving students’ ability. This trend is concerning, since low mathematics ability is known to be related to lower rates of success in subsequent courses. To date, little attention has been paid to the selection of an instructional approach to consistently apply across developmental mathematics courses. Prior research suggests that an appropriate instructional method would involve explicit instruction and practising mathematical procedures linked to a mathematical concept. This study reports on a randomized field trial of a developmental mathematics approach at a college in Ontario, Canada. The new approach is an adaptation of the JUMP Math program, an explicit instruction method designed for primary and secondary school curriculae, to the college learning environment. In this study, a subset of courses was assigned to JUMP Math and the remainder was taught in the same style as in the previous years. We found consistent, modest improvement in the JUMP Math sections compared to the non-JUMP sections, after accounting for potential covariates. The findings from this randomized field trial, along with prior research on effective education for developmental mathematics students, suggest that JUMP Math is a promising way to improve college student outcomes.     

Traditional instruction in advanced mathematics courses: a case study of one professor’s lectures and proofs in an introductory real analysis course

It is widely accepted by mathematics educators and mathematicians that most proof-oriented university mathematics courses are taught in a “definition-theorem-proof” format. However, there are relatively few empirical studies on what takes place during this instruction, why this instruction is used, and how it affects students’ learning. In this paper, I investigate these issues by examining a case study of one professor using this type of instruction in an introductory real analysis course. I first describe the professor’s actions in the classroom and argue that these actions are the result of the professor’s beliefs about mathematics, students, and education, as well as his knowledge of the material being covered. I then illustrate how the professor’s teaching style influenced the way that his students attempted to learn the material. Finally, I discuss the implications that the reported data have on mathematics education research.     

Exploring MLD in mathematics education: Ten years of research

Mathematical learning disabilities or difficulties (MLD) are an increasing source of educational inequalities. This article explores research about MLD in Mathematics Education over the past ten years. The methodology focuses on specific, validated keywords. These keywords are used to identify articles in leading journals in mathematics education. Our work makes several new contributions to the field of mathematics education, notably: a reusable methodology and keywords for a literature review; an exhaustive list of articles about MLD in leading mathematics education journals; a discussion of the definitions and features of MLD used in these articles; and a tool to classify research dealing with MLD (categories that characterize students with MLD). We also highlight some unexplored dimensions regarding MLD in Mathematics Education research.     

Animating an equation: a guide to using FLASH in mathematics education

Macromedia's FLASH development system can be a great tool for mathematics education. This article presents detailed Flash tutorials that were developed and taught by the author to a group of mathematics professors in a summer course in 2005. The objective was to educate the teachers in the techniques of animating equations and mathematical concepts in Flash. The course was followed by a 2-year study to assess the acceptance of the technology by the teachers and to gauge its effectiveness in improving the quality of mathematics education. The results of that 2-year study are also reported here.