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.     

Undergraduate mathematics achievement in the emerging ethnic engineers programme

This paper addresses the issue of collegiate mathematics achievement of underrepresented minority students as it investigates the impact of a cooperative learning calculus programme on the first-year calculus experience of non-Asian ethnic minority engineering students. The Emerging Ethnic Engineers Programme in the College of Engineering at the University of Cincinnati is a successful, comprehensive programme that focuses on the recruitment, retention, academic success, professional development, and timely graduation of underrepresented coloured students. The objectives of the programme are accomplished through three interrelated phases: pre-college science and mathematics programmes, first-year collegiate programmes, and upper-division programmes. The underlying principles of the first-year programme include academic achievement and establishing a strong sense of community among the cohort. This report will focus on the cooperative learning calculus programme that has been successful in improving retention and academic success rates for coloured freshmen engineering students.     

Predicting performance in a first engineering calculus course: implications for interventions

At the University of Louisville, a large, urban institution in the south-east United States, undergraduate engineering students take their mathematics courses from the school of engineering. In the fall of their freshman year, engineering students take Engineering Analysis I, a calculus-based engineering analysis course. After the first two weeks of the semester, many students end up leaving Engineering Analysis I and moving to a mathematics intervention course. In an effort to retain more students in Engineering Analysis I, the department collaborated with university academic support services to create a summer intervention programme. Students were targeted for the summer programme based on their score on an algebra readiness exam (ARE). In a previous study, the ARE scores were found to be a significant predictor of retention and performance in Engineering Analysis I. This study continues that work, analysing data from students who entered the engineering school in the fall of 2012. The predictive validity of the ARE was verified, and a hierarchical linear regression model was created using math American College Testing (ACT) scores, ARE scores, summer intervention participation, and several metacognitive and motivational factors as measured by subscales of the Motivated Strategies for Learning Questionnaire. In the regression model, ARE score explained an additional 5.1% of the variation in exam performance in Engineering Analysis I beyond math ACT score. Students took the ARE before and after the summer interventions and scores were significantly higher following the intervention. However, intervention participants nonetheless had lower exam scores in Engineering Analysis I. The following factors related to motivation and learning strategies were found to significantly predict exam scores in Engineering Analysis I: time and study environment management, internal goal orientation, and test anxiety. The adjusted R² for the full model was 0.42, meaning that the model could explain 42% of the variation in Engineering Analysis I exam scores.     

Relationships between emotional dispositions and mathematics achievement moderated by instructional practices: analysis of TIMSS 2015

ABSTRACT

This research is a secondary analysis with Korean students’ data collected in the TIMSS 2015 to describe the moderation effects of instructional practices on the relationships between students’ emotional dispositions toward mathematics and mathematics achievement. From the TIMSS 2015 database, we collected mathematics achievement scores, a student-level contextual scale for students’ emotional disposition, and teacher-level contextual scales representing teachers’ instructional practices. We applied hierarchical linear modelling to construct multilevel models. The findings showed that the achievement gap between emotional dispositions – like and dislike – became smaller when teachers more frequently implemented certain instructional practices like asking students to complete challenging exercises, decide their own problem-solving procedures, and express their ideas in class. Students who disliked mathematics were likely to have higher scores as their teachers implemented each of those practices more frequently. Findings provide important implications to teachers regarding: It is important to encourage students to reason through instructional practices like asking them to decide their own problem-solving procedures and to solve challenging problems.     

The interaction between dropout,graduation rates and quality ratings in universities

This paper investigates in a non-parametric framework whether academic programmes maximize their student graduation rates and programme quality ratings given the first-year student dropout rates. In addition, it explores what institutional and programme characteristics explain this interaction. The results show a large variation in how academic programmes are able to deal with the selective nature of first-year dropout. Nevertheless, we can accurately explain the variation among programmes by programme and institutional characteristics. It seems that universities can maximize the relation between first-year dropout, graduation rates and quality ratings in several ways: (1) by improving student programme satisfaction, (2) by better preparing certain groups of students for higher education, (3) by supporting male students, (4) by supporting ethnic minority students, (5) by attracting older staff, and (6) by strengthening the selective nature of the first year (ie, increasing the academic dismissal policy threshold).     

Student enrichment in mathematics: a case study with first year university students

ABSTRACT

This paper presents an enrichment case study to showcase a possible avenue for attending to the needs of academically strong mathematics students. We report on a group of university students who were presented with the opportunity of exploring a specific first year mathematics topic deeper, using an inquiry-based learning approach as part of an enrichment programme. Following the intervention, students completed a questionnaire and a few were interviewed to establish their experiences of the enrichment programme. We discuss the successes and pitfalls of the intervention and report on the impact it had on the participants.     

The Effects of Project‐Based Learning on Students' Academic Achievement,Attitude, and Retention of Knowledge: The Subject of “Electricity in Our Lives”

The aim of this study is to analyze the effects of project‐based learning on students' academic achievement, attitude, and retention of knowledge in relation to the subject of “Electricity in Our Lives” in a fourth‐grade science course. The study was conducted in a quasi‐experimental design as a “pre‐test, post‐test with control group.” In the experimental group, the unit was taught through the project‐based learning method. The measuring tools were administered to both groups before and after the applications. To perfectly analyze the “process” of the method, seven different learning assessment “forms” were administered to the students. The findings of the forms indicated that the students learn to construct their own learning and to evaluate changes in their own behavior through the application of the method. The application of different methods between both groups had a statistically significant effect in terms of academic achievement, (F(1,112) = 46.78, p = .000) and of retention of knowledge (F(1,112) = 35.24, p = .000). However, there were no statistically significant effects from being in different groups for the attitudes of students (F(1,112) = .99, p = .321). For the students, being in the project‐based learning groups resulted in better academic achievement and retention of knowledge than being in the traditional teaching group.     

Ways of talking and ways of positioning: Students’ beliefs in an inquiry-oriented differential equations class

As part of developmental research for an inquiry-oriented differential equations course, this study investigates the change in students’ beliefs about mathematics. The discourse analysis has identified two different types of perspective modes - i.e., discourse of the third-person perspective and discourse of the first-person perspective - in the students’ mathematical narratives, depending on their ways of positioning themselves with respect to mathematics. In the third-person perspective discourse, the students positioned themselves as passive recipients of mathematics that has been established by some external authority. In the first-person perspective discourse, the students positioned themselves as active mathematical inquirers and produced mathematics by interweaving their own mathematical ideas and experiences. Over the semester, students’ mathematical discourse changed from third-person perspective narratives to first-person perspective narratives. This change in their discourse pattern is interpreted as an indication of change in their beliefs about mathematics. Finally, this article discusses the instructional features that promote the change.     

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.     

The effectiveness of resources created by students as partners in explaining the relevance of mathematics in engineering education

First-year engineering students often struggle to see the relevance of theoretical mathematical concepts for their future studies and professional careers. This is an issue, as students who do not see relevance in fundamental parts of their studies may disengage from these parts and focus their efforts on other subjects they think will be more useful to them. In this study, we surveyed engineering students enrolled in a first-year mathematics subject on their perceptions of the relevance of the individual mathematical topics taught. Surveys were administered at the start of semester when some of these topics were unknown to them, and again at the end of semester when students had not only studied all these topics but also watched a set of animated videos. These videos had been produced by higher-year students to explain where they had seen applications of the mathematical concepts presented in the first year. We notice differences between the perceived relevance of topics for future study and for professional careers, with relevance to study rated higher than relevance to careers. We also find that the animations are seen as helpful in understanding the relevance of first-year mathematics. The majority of students indicated that lecturers with students as partners should work collaboratively to produce future videos.     

Mathematics in engineering programs: what teachers with different academic and professional backgrounds bring to the table. An institutional analysis

ABSTRACT

In this paper, we examine how differences in the academic and professional backgrounds of engineering teachers shape their personal relationship to the use of mathematics in engineering practices, and whether these differences affect some of their practices. The analyses herein are based on an institutional perspective and employ Chevallard's anthropological theory of the didactic (ATD). We interviewed two teachers in an engineering programme to identify specific elements of their practice that could be attributable to the mobilisation of knowledge and skills derived from their distinct academic backgrounds and experience. The results indicate that the teachers mobilise different tasks, techniques, and technologies in many of their practices, and that they take different approaches to using mathematics and applying rigour.     

Exercising QS: quantitative skills in an exercise science course

This study seeks to bring the discipline of exercise science into the discussion of Quantitative Skills (QS) in science. The author's experiences of providing learning support to students and working with educators in the field are described, demonstrating the difficulty of encouraging students to address their skills deficit. A survey of students’ perceptions of their own QS and of that required for their course, demonstrates the difficulties faced by students who do not have the prescribed assumed knowledge for the course. Limited results from academics suggest that their perceptions of students’ QS deficits are even direr than those of the under-prepared students.     

Using Primary Source Projects to teach undergraduate mathematics content: analysis of instructor implementation and perceptions

ABSTRACT

This study investigates changes in instructor teaching tendencies, instructor’s perception of impact on student learning and dispositions, and methods of implementation of Primary Source Projects (PSPs). PSPs are curricular modules designed to teach core mathematical topics from primary historical sources rather than from standard textbooks. In essence, they are a form of inquiry-based learning that incorporates the history of mathematics through original source texts. We provide an overview of results from two semesters of implementation reports and surveys administered at the beginning and end of the semester by instructors who implemented PSPs in their undergraduate mathematics classes.     

Factors contributing to academic achievement: a Bayesian structure equation modelling study

In Iran, high school graduates enter university after taking a very difficult entrance exam called the Konkoor. Therefore, only the top-performing students are admitted by universities to continue their bachelor's education in statistics. Surprisingly, statistically, most of such students fall into the following categories: (1) do not succeed in their education despite their excellent performance on the Konkoor and in high school; (2) graduate with a grade point average (GPA) that is considerably lower than their high school GPA; (3) continue their master's education in majors other than statistics and (4) try to find jobs unrelated to statistics. This article employs the well-known and powerful statistical technique, the Bayesian structural equation modelling (SEM), to study the academic success of recent graduates who have studied statistics at Shahid Beheshti University in Iran. This research: (i) considered academic success as a latent variable, which was measured by GPA and other academic success (see below) of students in the target population; (ii) employed the Bayesian SEM, which works properly for small sample sizes and ordinal variables; (iii), which is taken from the literature, developed five main factors that affected academic success and (iv) considered several standard psychological tests and measured characteristics such as ‘self-esteem’ and ‘anxiety’. We then study the impact of such factors on the academic success of the target population. Six factors that positively impact student academic success were identified in the following order of relative impact (from greatest to least): ‘Teaching–Evaluation’, ‘Learner’, ‘Environment’, ‘Family’, ‘Curriculum’ and ‘Teaching Knowledge’. Particularly, influential variables within each factor have also been noted.     

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.     

Pre-service teachers' point of views about learning history of mathematics: a case study in Turkey

This study examines pre-service teachers’ points of view about learning history of mathematics during their undergraduate education. An open-ended questionnaire was administered to one hundred and twenty pre-service teachers, during the fall semester of the 2013–14 academic year. The participants indicated that learning history of mathematics could increase their content knowledge as they understand how formulas, theories and relations were developed over time. In addition, it could develop them intellectually as they learn life stories of mathematicians. Also, it could help them to hold the attention of students, and answer some of the why questions. Particularly, they reported using history of mathematics knowledge while teaching Geometry and Numbers.     

Examining epistemic beliefs across conceptual and procedural knowledge in statistics

The purpose of this paper was to examine whether students’ epistemic beliefs differed as a function of variations in procedural versus conceptual knowledge in statistics. Students completed Hofer’s (Contem Edu Psychol 25:378–405, 2000) Discipline-Focused Epistemological Beliefs Questionnaire five times over the course of a semester. Differences were explored between students’ initial beliefs about statistics knowledge and their specific beliefs about conceptual knowledge and procedural knowledge in statistics. Results revealed differences across these contexts; students’ beliefs differed between procedural versus conceptual knowledge. Moreover, students’ initial beliefs about statistics knowledge were more similar to their beliefs about conceptual knowledge rather than procedural knowledge. Finally, regression analyses revealed that students’ beliefs about the justification of knowledge, attainability of truth and source of knowledge were significant predictors of examination performance, depending on the examination. These results have important theoretical, methodological and pedagogical implications.     

Adolescent girls’ construction of moral discourses and appropriation of primary identity in a mathematics classroom

This qualitative study examines the way three American young adolescent girls who come from different class and racial backgrounds construct their social and academic identities in the context of their traditional mathematics classroom. The overall analysis shows an interesting dynamic among each participant’s class and racial background, their social/academic identity and its collective foundation, the types of ideologies they repudiate and subscribe to, the implicit and explicit strategies they adopt in order to support the legitimacy of their own position, and the ways they manifest their position and identity in their use of language referring to their mathematics classroom. Detailed analysis of their use of particular terms, such as “I,” “we,” “they,” and “should/shouldn’t” elucidates that each participant has a unique view of her mathematics classroom, developing a different type of collective identity associated with a particular group of students. Most importantly, this study reveals that the girls actively construct a social and ideological web that helps them articulate their ethical and moral standpoint to support their positions. Throughout the complicated appropriation process of their own identity and ideological standpoint, the three girls made different choices of actions in mathematics learning, which in turn led them to a different math track the following year largely constraining their possibility of access to higher level mathematical knowledge in the subsequent schooling process.     

Identification and definition of lexically ambiguous words in statistics by tutors and students

Lexical ambiguity arises when a word from everyday English is used differently in a particular discipline, such as statistics. This paper reports on a project that begins by identifying tutors’ perceptions of words that are potentially lexically ambiguous to students, in two different ways. Students’ definitions of nine lexically ambiguous words are also collected at the beginning and end of a semester of introductory statistics study, in a complex design taking account of multiple tutors and multiple words in multiple contexts. Tutor perceptions and actual student difficulties at the beginning of a semester are compared. The lexical ambiguity associated with the word ‘significance’ is shown to be evident in students even after completing an introductory statistics course.     

Undergraduate Engineering Majors' Beliefs About Mathematics

This study examined the mathematics beliefs of college students in 10 undergraduate mathematics classes at a large engineering school in the Midwest. The beliefs of 254 engineering majors were measured by the Indiana Mathematics Belief Scales and compared to the beliefs of elementary education majors and remedial college mathematics students obtained from earlier studies using the same instrument. The results were interpreted in terms of the students' daily attitudes towards their mathematics classes and corresponding academic and demographic parameters. The study showed that in many respects, the beliefs of the engineering majors were not that different from the other populations. The correlations among beliefs for the engineering group tended to be higher although there were relatively few significant correlations between belief and background variables. Attitude data were collected across a full semester for the engineering majors. The relatively modest day-to-day variation in those attitudes suggests that they are based on deeply seated beliefs.