An investigation of post-primary students' images of mathematics

This research study investigates the image of mathematics held by 5th-year post-primary students in Ireland. For this study, “image of mathematics” is conceptualized as a mental representation or view of mathematics, presumably constructed as a result of past experiences, mediated through school, parents, peers or society. It is also understood to include attitudes, beliefs, emotions, self-concept and motivation in relation to mathematics. This study explores the image of mathematics held by a sample of 356 5th-year students studying ordinary level mathematics. Students were aged between 15 and 18 years. In addition, this study examines the factors influencing students‟ images of mathematics and the possible reasons for students choosing not to study higher level mathematics for the Leaving Certificate. The design for this study is chiefly explorative. A questionnaire survey was created containing both quantitative and qualitative methods to investigate the research interest. The quantitative aspect incorporated eight pre-established scales to examine students‟ attitudes, beliefs, emotions, self-concept and motivation regarding mathematics. The qualitative element explored students‟ past experiences of mathematics, their causal attributions for success or failure in mathematics and their influences in mathematics. The quantitative and qualitative data was analysed for all students and also for students grouped by gender, prior achievement, type of post-primary school attending, co-educational status of the post-primary school and the attendance of a Project Maths pilot school. Students‟ images of mathematics were seen to be strongly indicated by their attitudes (enjoyment and value), beliefs, motivation, self-concept and anxiety, with each of these elements strongly correlated with each other, particularly self-concept and anxiety. Students‟ current images of mathematics were found to be influenced by their past experiences of mathematics, by their mathematics teachers, parents and peers, and by their prior mathematical achievement. Gender differences occur for students in their images of mathematics, with males having more positive images of mathematics than females and this is most noticeable with regards to anxiety about mathematics. Mathematics anxiety was identified as a possible reason for the low number of students continuing with higher level mathematics for the Leaving Certificate. Some students also expressed low mathematical self-concept with regards to higher level mathematics specifically. Students with low prior achievement in mathematics tended to believe that mathematics requires a natural ability which they do not possess. Rote-learning was found to be common among many students in the sample. The most positive image of mathematics held by students was the “problem-solving image”, with resulting implications for the new Project Maths syllabus in post-primary education. Findings from this research study provide important insights into the image of mathematics held by the sample of Irish post-primary students and make an innovative contribution to mathematics education research. In particular, findings contribute to the current national interest in Ireland in post-primary mathematics education, highlighting issues regarding the low uptake of higher level mathematics for the Leaving Certificate and also making a preliminary comparison between students who took part in the piloting of Project Maths and students who were more recently introduced to the new syllabus. This research study also holds implications for mathematics teachers, parents and the mathematics education community in Ireland, with some suggestions made on improving students‟ images of mathematics.

The effects of teaching a mathematics problem solving strategy on the problem solving ability of grade nine students

Forty grade 9 students were selected from a small rural board in southern Ontario. The students were in two classes and were treated as two groups. The treatment group received instruction in the Logical Numerical Problem Solving Strategy every day for 37 minutes over a 6 week period. The control group received instruction in problem solving without this strategy over the same time period. Then the control group received the treat~ent and the treatment group received the instruction without the strategy. Quite a large variance was found in the problem solving ability of students in grade 9. It was also found that the growth of the problem solving ability achievement of students could be measured using growth strands based upon the results of the pilot study. The analysis of the results of the study using t-tests and a MANOVA demonstrated that the teaching of the strategy did not significaritly (at p s 0.05) increase the problem solving achievement of the students. However, there was an encouraging trend seen in the data.

Teachers' professional learning within communities of mathematics-for-teaching practice

Ontario bansho is an emergent mathematics instructional strategy used by teachers working within communities of practice that has been deemed to have a transformational effect on teachers' professional learning of mathematics. This study sought to answer the following question: How does teachers' implementation of Ontario bansho within their communities of practice inform their professional learning process concerning mathematics-for-teaching? Two other key questions also guided the study: What processes support teachers' professional learning of content-for-teaching? What conditions support teachers' professional learning of content-for-teaching? The study followed an interpretive phenomenological approach to collect data using a purposive sampling of teachers as participants. The researcher conducted interviews and followed an interpretive approach to data analysis to investigate how teachers construct meaning and create interpretations through their social interactions. The study developed a model of professional learning made up of 3 processes, informing with resources, engaging with students, and visualizing and schematizing in which the participants engaged and 2 conditions, ownership and community that supported the 3 processes. The 3 processes occur in ways that are complex, recursive, nonpredictable, and contextual. This model provides a framework for facilitators and leaders to plan for effective, content-relevant professional learning by placing teachers, students, and their learning at the heart of professional learning.

Mathematics teacher education and technology

Various applications of electronic technology have contributed to the field of mathematics teacher education, but the array of tools now' available to teacher educators in developed countries can be somewhat daunting. In this chapter, the authors present an overview of some of these tools and the ways that they are being used. In addition, they address issues surrounding the development and applications of digital technology in teacher education programs in a number of countries. In particular, they present recent applications of videotapes, multimedia resources, internet-based communities, and tutorial conferencing facilities in pre-service and in-service education of mathematics teachers. As professionals who have used technology in teacher education, the authors blend their own insights into this review of technological tools and relevant issues.

Variables influencing the mathematics performance of first-year tertiary students : a case study

The need to understand which factors most strongly affect performance in first-year mathematics programs at Khon Kaen University (KKU), in North Eastern Thailand, provided the main focus of the study which is described. First-year mathematics students in the 1990-1991 academic year, from four KKU faculty groups (Medicine and Nursing, Agriculture, Science and Education, and Engineering) were involved in this study. Research literatures addressing variables which were likely to influence performance in early tertiary mathematical study, and variables associated with difficulties in learning mathematics at the transition from upper secondary school to tertiary studies, were reviewed. The first major aim of the study was to identify the variables which were good predictors of first-year mathematics performance at KKU. Results from stepwise multiple regression analyses indicated that the following predictor variables were statistically significant and entered the regression equations for most Faculty groups: School Mathematics Achievement, Self-Esteem, Study Habits in Mathematics, and Faculty of Study. Other predictor variables that sometimes entered regression equations (depending on the Faculty group) were Socio-Economic-Status, Mathematics Language Competence, Mathematics Confidence, Attitude Towards Mathematics, and Gender. Depending on Faculty group, the statistically significant variables accounted for between 11% and 74% of scores on fist-year KKU mathematics examinations. The predictor variables contributed much more to the variance of scores on first-semester mathematics examinations than to the variance of scores on second-semester mathematics examinations. It was also found that scores on the Direct Entry Examination Mathematics test (administered by KKU) and the School Mathematics Achievement test (developed and administered by the author) had stronger correlations with first-year KKU mathematics performance than did scores on the National Entry Examination Mathematics tests (administered by the Thai Ministry of University Affairs). Scores on the three pre-university mathematics achievement test instruments were better predictors of first-semester mathematics performance than of second-semester mathematics performance. It was found that the mean Mathematics Confidence of male students was statistically significantly higher than that of female students, but there were no statistically significant gender differences in Mathematics Misplaced Confidence. Only about 30% of the main sample ( 30% of the male and 30% of the female sample groups) had appropriate confidence in mathematics, that is, they thought their answers were correct when they were, in fact, correct, and they thought they were wrong when they were, in fact, incorrect. So far as Faculty performance differences were concerned, Engineering students had the highest Mathematics Confidence scores, followed by the Medicine and Nursing group of students and the Science and Education group students. Agriculture students had the lowest mean Mathematics Confidence score. No statistically significant differences occurred in Mathematics Misplaced Confidence between different Faculty groups. The second main aim of the study was to investigate why many first-year students experienced difficulties in coping with their mathematics units. A small group of senior secondary mathematics teachers, university mathematics lecturers, and first-year mathematics students were interviewed during the first semester of the 1990-1991 academic year. Interviews were conducted by the author according to a questionnaire format, and were aimed at identifying factors causing difficulty in the transition from senior secondary to university mathematical study. The analysis of the quantitative data together with the interview data indicated that the major sources of difficulty were associated with: (a) students' mathematical abilities; (b) curriculum content; (c) course organisation; (d) students' study habits; (e) instructional styles; and (f) assessment procedures. The results of the investigation are discussed in the light of the relevant literature and related research. The study concludes with recommendations which are addressed to mathematics teachers and education administrators in senior secondary schools in Thailand, to the Thai Ministry of Education, and to the KKU Department of Mathematics.

Investigation into the skill levels achieved by mathematics students in the V.C.E. and the H.S.C. mathematics courses

This study examines whether recent changes to the mathematics courses offered in the final year of secondary school (Year 12) in the state of Victoria, Australia have affected the learning outcomes of students in terms of then: skill levels in algebra, calculus and problem solving; and in terms of their preparation for a tertiary mathematics unit. The impact of these changes on the transition from secondary to tertiary mathematics is also considered. A comparison is made between students who attempted a first year mathematics unit at the University of Melbourne (U. of M.) having completed the new V.C.E. (Victorian Certificate of Education) mathematics courses and mathematics courses from the previous H.S.C. (Higher School Certificate) system. The comparison involves the use of tests administered upon entrance to a tertiary mathematics unit at the U. of M., and questionnaires. In 1991, V.C.E, students and H.S.C. students attempted the same mathematics test at the U. of M. and their results were compared. In 1992, the tests were attempted by V.C.E. students only. To compare new V.C.E. students and H.S.C. students, questions on the 1991 test were matched with similar questions on the 1992 tests and a panel of experts determined what the H.S.C. students who attempted the 1991 test would have been expected to average on these matched questions on the 1992 tests had they attempted them. These expected average scores were then compared with the actual scores of the new V.C.E. students. The scores of the groups were scaled when necessary. Questionnaires were administered to 1991 U. of M, mathematics students who were part of the V.C.E. pilot group in 1990, secondary mathematics educators, tertiary mathematics educators, and 1991 V.C.E. (1992 U. of M.) students. The mathematical misconceptions exhibited by new V.C.E. students are discussed and their frequencies stated. The research indicates that the new V.C.E. mathematics courses have provided the V.C.E. mathematics students in this study with significantly lower skill levels and a significantly poorer preparation for a tertiary mathematics unit than those which were previously provided by the H.S.C. mathematics courses.

Subject cultures and pedagogy : comparing mathematics and science

This research found that a teacher is both a member of a culture and an individual, building practice within parameters set by a dynamic and multifaceted subject culture. Feelings of competence and confidence grow as an aesthetic understanding of what it means to know, teach, and appreciate a subject.

Mathematical notebook, and commonplace book, circa 1689-1769

Manuscript notebook, possibly kept by Harvard students, containing 17th century English transcriptions of arithmetic and geometry texts, one of which is dated 1689-1690; 18th century transcriptions from John Ward’s “The Young Mathematician’s Guide”; and notes on physics lectures delivered by John Winthrop, the Hollis Professor of Mathematics and Natural Philosophy at Harvard from 1738 to 1779. The notebook also contains 18th century reading notes on Henry VIII, Tudor succession, and English history from Daniel Neal’s “The History of the Puritans” and David Hume’s “History of England,” and notes on Ancient history, taken mainly from Charles Rollin’s “The Ancient History of the Egyptians, Carthaginians, Assyrians, Babylonians, Medes and Persians, Macedonians and Grecians.” Additionally included are an excerpt from Plutarch’s “Lives” and transcriptions of three articles from “The Gentleman’s Magazine, and Historical Chronicle,” published in 1769: “A Critique on the Works of Ovid”; a book review of “A New Voyage to the West-Indies”; and “Genuine Anecdotes of Celebrated Writers, &.” The flyleaf contains the inscription “Semper boni aliquid operis facito ut diabolus te semper inveniat occupatum,” a variation on a quote of Saint Jerome that translates approximately as “Always good to do some work so that the devil may always find you occupied.” In the seventeenth and eighteenth centuries, Harvard College undergraduates often copied academic texts and lecture notes into personal notebooks in place of printed textbooks. Winthrop used Ward’s textbook in his class, while the books of Hume, Neal, and Rollin were used in history courses taught at Harvard in the 18th century.