An Exploratory Study of College Students' Views of Mathematical Thinking in a Historical Approach Calculus Course

Heuristic training alone is not enough for developing one's mathematical thinking. One missing component is a mathematical point of view. This study reports findings regarding outcomes of a historical approach calculus course to foster Taiwanese college students' views of mathematical thinking. This study consisted of 3 stages. During the initial phase, 44 engineering majors' views on mathematical thinking were tabulated by an open-ended questionnaire, and 9 randomly selected students were invited to participate in follow-up interviews. Students then received an 18-week historical approach calculus course in which mathematical concepts were problematized to challenge their intuition-based empirical beliefs about doing mathematics. Near the end of the semester, all participants answered the identical questionnaire, and we interviewed the same students to pinpoint any shifts in their views on mathematical thinking. We found that participants were more likely to value logical sense, creativity, and imagination in doing mathematics. Further, students were leaning toward a conservative attitude toward certainty of mathematical knowledge. Participants' focus seemingly shifted from mathematics as a product to mathematics as a process.     

The impact of instructor pedagogy on college calculus students’ attitude toward mathematics

College calculus teaches students important mathematical concepts and skills. The course also has a substantial impact on students’ attitude toward mathematics, affecting their career aspirations and desires to take more mathematics. This national US study of 3103 students at 123 colleges and universities tracks changes in students’ attitudes toward mathematics during a ‘mainstream’ calculus course while controlling for student backgrounds. The attitude measure combines students’ self-ratings of their mathematics confidence, interest in, and enjoyment of mathematics. Three major kinds of instructor pedagogy, identified through the factor analysis of 61 student-reported variables, are investigated for impact on student attitude as follows: (1) instructors who employ generally accepted ‘good teaching’ practices (e.g. clarity in presentation and answering questions, useful homework, fair exams, help outside of class) are found to have the most positive impact, particularly with students who began with a weaker initial attitude. (2) Use of educational ‘technology’ (e.g. graphing calculators, for demonstrations, in homework), on average, is found to have no impact on attitudes, except when used by graduate student instructors, which negatively affects students’ attitudes towards mathematics. (3) ‘Ambitious teaching’ (e.g. group work, word problems, ‘flipped’ reading, student explanations of thinking) has a small negative impact on student attitudes, while being a relatively more constructive influence only on students who already enjoyed a positive attitude toward mathematics and in classrooms with a large number of students. This study provides support for efforts to improve calculus teaching through the training of faculty and graduate students to use traditional ‘good teaching’ practices through professional development workshops and courses. As currently implemented, technology and ambitious pedagogical practices, while no doubt effective in certain classrooms, do not appear to have a reliable, positive impact on student attitudes toward mathematics.     

Using Sociocultural Theory to Teach Mathematics: A Vygotskian Perspective

This study describes an elementary teacher's implementation of sociocultural theory in practice. Communication is central to teaching with a sociocultural approach and to the understanding of students; teachers who use this theory involve students in explaining and justifying their thinking. In this study ethnographic research methods were used to collect data for 4 1/2 months in order to understand the mathematical culture of this fourth‐grade class and to portray how the teacher used a sociocultural approach to teach mathematics. To portray this teaching approach, teaching episodes from the teacher's mathematics lessons are described, and these episodes are analyzed to demonstrate how students created taken‐as‐shared meanings of mathematics. Excerpts from interviews with the teacher are also used to describe this teacher's thinking about her teaching.     

The role of mathematics in engineering education: an engineer's view

Students of mathematics in the first term studied calculus in two parallel courses. One course was in the standard approach, while the other was based on the non‐standard (Robinson) approach. The students participated in tests of the brain's hemispheres. The scores of both mathematical courses and the results of the hemispheric tests were correlated. The results show that the standard approach is positively correlated to both hemispheres. The n.s. approach is positively correlated to the right hemisphere and negatively correlated to the left hemisphere. It is possible to predict relative success in the two approaches to calculus.

     

Placenticeras: evolution of a task

We study a sequence of concrete case examples, anchored to a single image (of a fossil shell) to explore the complexity of building mathematical models in compelling realistic situations. The case examples are drawn from work with calculus undergraduates, and also dance majors from an experimental mathematics course who built evolving spiral forms in choreographic projects. The exposition is narrative, in the sense that it emphasizes, case by case, the evolution of the authors’ thinking in the course of extended conversations and collaborations with students over several years.     

The Mathematics Writer's Checklist: The Development of a Preliminary Assessment Tool for Writing in Mathematics

The use of writing as a pedagogical tool to help students learn mathematics is receiving increased attention at the college level ( Meier & Rishel, 1998 ), and the Principles and Standards for School Mathematics (NCTM, 2000) built a strong case for including writing in school mathematics, suggesting that writing enhances students' mathematical thinking. Yet, classroom experience indicates that not all students are able to write well about mathematics. This study examines the writing of a two groups of students in a college‐level calculus class in order to identify criteria that discriminate “;successful” vs. “;unsuccessful” writers in mathematics. Results indicate that “;successful” writers are more likely than “;unsuccessful” writers to use appropriate mathematical language, build a context for their writing, use a variety of examples for elaboration, include multiple modes of representation (algebraic, graphical, numeric) for their ideas, use appropriate mathematical notation, and address all topics specified in the assignment. These six criteria result in The Mathematics Writer's Checklist, and methods for its use as an instructional and assessment tool in the mathematics classroom are discussed.     

Engineering students’ conceptions of the derivative and some implications for their mathematical education

This paper explores Mechanical Engineering students’ conceptions of and preferences for conceptions of the derivative, and their views on mathematics. Data comes from pre-, post- and delayed post-tests, a preference test, interviews with students and an analysis of calculus courses. Data from Mathematics students is used to make comparisons with Mechanical Engineering students. The results show that Mechanical Engineering students’ conceptions of and preferences for the derivative develop in the direction of the rate of change aspects while those of Mathematics students develop in the direction of tangent aspects, and that Mechanical Engineering students view mathematics as a tool and want the application aspects in their course. Students’ developing conceptions, preferences and views with regard to teaching and departmental affiliation are considered and educational implications are suggested for the mathematical education of engineering students.     

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.     

How to make mathematics relevant to first-year engineering students: perceptions of students on student-produced resources

Many approaches to make mathematics relevant to first-year engineering students have been described. These include teaching practical engineering applications, or a close collaboration between engineering and mathematics teaching staff on unit design and teaching. In this paper, we report on a novel approach where we gave higher year engineering and multimedia students the task to ‘make maths relevant’ for first-year students. This approach is novel as we moved away from the traditional thinking that staff should produce these resources to students producing the same. These students have more recently undertaken first-year mathematical study themselves and can also provide a more mature student perspective to the task than first-year students. Two final-year engineering students and three final-year multimedia students worked on this project over the Australian summer term and produced two animated videos showing where concepts taught in first-year mathematics are applied by professional engineers. It is this student perspective on how to make mathematics relevant to first-year students that we investigate in this paper. We analyse interviews with higher year students as well as focus groups with first-year students who had been shown the videos in class, with a focus on answering the following three research questions: (1) How would students demonstrate the relevance of mathematics in engineering? (2) What are first-year students' views on the resources produced for them? (3) Who should produce resources to demonstrate the relevance of mathematics? There seemed to be some disagreement between first- and final-year students as to how the importance of mathematics should be demonstrated in a video. We therefore argue that it should ideally be a collaboration between higher year students and first-year students, with advice from lecturers, to produce such resources.     

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.     

The Tabular Mode: Not Just Another Way to Represent a Function

Understanding mathematical functions as systematic processes involving the covariation of related variables is foundational in learning mathematics. In this article, findings are reported from two investigations examining students' thinking processes with functions. The first study focused on seven middle school students' explorations with a dynamic physical model. Students were videotaped during the 20‐ to 45‐minute sessions occurring two or three times per week over a period of 2 months, and students' written work was collected. The second investigation included 19 preservice elementary and middle school teachers enrolled in a course focusing on a combination of mathematical content and pedagogy. Participants' written problem‐solving work and reflective writing were collected, and participants were individually interviewed in 50‐minute videotaped sessions. Results from both investigations indicated that students often relied on a table, or some variation of a table, as a cognitive link advancing the development of their reasoning about underlying function relationships.     

Preservice Elementary Teachers' Use of Mathematics in a Project‐Based Science Approach

The use of a project‐based science (PBS) approach to teaching encourages students to integrate mathematics and science in meaningful ways as they create projects. As a beginning study of how students use mathematics in such an approach, an analysis of 23 projects developed by preservice elementary teachers enrolled in an elementary science course was conducted. Findings showed that students made a number of different types of mathematical errors and underutilized data representation and summary forms. Implications included the importance of developing methods for supporting the use of mathematical tools in utilizing a project‐based approach and considering ways that such tools mediate scientific thinking.     

Collaborative Workshops and Student Academic Performance in Introductory College Mathematics Courses: A Study of a Treisman Model Math Excel Program

High failure rates in introductory college mathematics courses, particularly among underrepresented groups of students, have been of concern for many years. One approach to the problem experiencing some success has been Treisman's Emerging Scholars workshop model. The model involves supplemental workshops in which students solve problems in collaborative learning groups. This study reports on the effectiveness of Math Excel, an implementation of the Treisman model for introductory mathematics courses (college algebra, precalculus, differential calculus, and integral calculus) at Oregon State University over five academic terms. Regression analyses revealed a significant effect on achievement (.671 grade points on a 4‐point scale) favoring Math Excel students. Even after adjusting for prior mathematics achievement using linear regression with SAT‐M as predictor, Math Excel groups' grade averages were over half a grade point better than predicted (significant at the .001 level). This study provides supporting evidence that programs like Math Excel can help students in making a successful transition to college mathematics study.     

Researching young students’ mathematical world views

As an alternative to questionnaires suitable for young students, pictures, texts and interviews are used as data sources for studying mathematical world views of fifth and sixth graders in a several-step design. The project was developed in three successive studies. In the first study, the approach of using pictures, texts and interviews for researching young students’ mathematical world views was investigated. Object of the second study was the development of an interrater-method for determining mathematical world views which delivered a satisfactory degree of reliability. The empirical results in the second study indicated as well that quite often mathematics courses were dominated by a view on mathematics emphasizing numbers or calculations. An analysis of students’ utterances suggests that some young students might have mixed world views. This motivates a modified rating approach in a third study in which raters can give weights to several world views. The procedure indicates that various mixed forms of the world views can be observed. This brings up the question as to whether this phenomenon is due to the methodology or whether it describes the formation of mathematical world views at that age.     

Student Attitudes and Calculus Reform

This paper compares the attitudes about mathematics of students from traditionally taught calculus classes and those from a “reformed” calculus course. The paper is based on three studies, which together present a consistent picture of student attitudes about calculus reform. The reformed course appeared to violate students' deeply held beliefs about the nature of mathematics and how it should be learned. Although during their first months in the reformed course most students disliked it, their attitudes gradually changed. One and 2 years after, reform students felt significantly more than the traditionally taught students that they better understood how math was used and that they had been required to understand math rather than memorize formulas.     

‘I’m worried about the correctness’: undergraduate students as producers of screencasts of mathematical explanations for their peers – lecturer and student perceptions

Undergraduate mathematics is traditionally designed and taught by content experts with little contribution from students. Indeed, there are signs that there is resistance from mathematics lecturers to involve students in the creation of material to support their peers – notwithstanding the fact that students have been successfully engaged as co-creators of material in other disciplines. There appears to be little research into what issues may lead to reservations to using student-created content in mathematics learning. This paper takes a case study approach to investigate the reasons for lecturers’ resistance to undergraduate student contributions to learning material, in particular with a view to the production of screencasts of mathematical explanations. It also investigates the views of students producing mathematical screencasts. This study is part of a larger research project investigating undergraduate involvement in mathematics module design. Four second-year students, who were producing mathematics screencasts as part of an internship, and five academics, were interviewed to gain an understanding of their views of the value of student screencasts. The interviews focused on the particular contributions students make to screencasts, outcomes for the students and level of lecturer acceptance of these resources. We argue that students benefit from creating screencasts for their peers by gaining deeper mathematical understanding, improved technological skills and developing other generic skills required of today's graduates. In contrast, we confirm lecturer resistance to using student-generated screencasts in their teaching materials. Lecturer reservations pertain to students’ lack of mathematical maturity and concerns over the mathematical integrity of the content that students produce. We conclude that close collaboration between students and lecturers during the design and production phases of screencasts may help lecturers overcome reservations, whilst preserving the benefits for students. In addition, we provide evidence that the process is a valuable professional development opportunity for the lecturers themselves.