Ann Kajander

Transition from secondary to tertiary mathematics: McMaster University experience

The transition (‘gap’) between secondary and tertiary education in mathematics is a complex phenomenon covering a vast array of problems and issues. The aim of this paper is to present the ways in which the issues of mathematics transition have been dealt with at McMaster University. Roughly, the process of transition has been broken into three stages: students’ voluntary preparation for university mathematics courses facilitated by the Mathematics Review Manual; administration of Mathematics Background Survey; and redesign of the first-year Calculus (and, subsequently, other mathematics courses).

Mathematics Textbooks and Their Potential Role in Supporting Misconceptions.

As a fundamental resource, textbooks shape the way we teach and learn mathematics. Based on examination of secondary school and university textbooks, we describe to what extent, and how, the presentation of mathematics material–in our case study, the concept of the line tangent to the graph of a function–could contribute to creation and strengthening of students’ misconceptions. Our findings, roughly classified in several categories, raise awareness of non-obvious problems that need to be addressed when teaching calculus.

Interconnections of Knowledge and Beliefs in Teaching Mathematics

As part of a 5-year project that examined teachers’ knowledge and beliefs about math¬ematics and teaching mathematics, interviews were conducted with preservice elementary teachers during their certification year. The transcripts of five of these sets of pretest/posttest interviews were chosen as illustrative of the significant challenges that preservice teachers continue to face as they strive to deepen their subject-specific teaching knowledge and begin to address their beliefs about teaching and learning mathematics.RésuméDans le cadre d’un projet de 5 ans visant à analyser les connaissances en mathématiques des enseignants, ainsi que leurs points de vue sur cette discipline et son enseignement, des entrevues ont été menées auprès de futurs enseignants au primaire pendant leur année de certification. La tran¬scription de cinq des séries d’entrevues (pré-test et post-test) a été choisie pour illustrer l’importance des défis que doivent continuellement affronter les enseignants en formation alors qu’ils visent à approfondir leur niveau de connaissances spécifiques à l’enseignement des mathématiques et com¬mencent à réfléchir sur leurs propres points de vue concernant l’enseignement et l’apprentissage de leur discipline.

Examining teacher growth in professional learning groups for in‐service teachers of mathematics

Teachers may face important challenges when encouraged to improve their mathematics teaching. Their personal beliefs, knowledge, confidence and personal intentions towards growth and change are all complex factors which may influence teachers’ capacity, and their decisions about personal change in their teaching. In this study, intermediate teachers and the conversations that took place during their monthly Professional Learning Group meetings over a one-year period were examined in order to better understand issues teachers face in their growth and development as teachers of mathematics. We critically examine the notion and meaning of success to different stakeholders.RésuméLes enseignants sont susceptibles d’affronter des défis importants lorsqu’on les encourage à améliorer leur enseignemenl des mathématiques. Leurs idées personnelles, leur niveau de connaissances, leur assurance et leurs intentions devant la perspective de changer et de s’améliorer constituent des facteurs complexes qui peuvent influencer leur capacité de changer leur façon d’enseigner et les decisions qu’ils peuvent prendre à l’égard de ces changements. Dans cette étude, nous avons observé des enseignants de niveau intermédiare et analysé les discours qu’ils ont tenus pendant une année lors de rencontres mensuelles au sein d’un groupe de formation professionnelle, afin de mieux cerner les questions auxquelles font face les enseignants qui cherchent à améliorer leur enseignement des mathématiques. Nous proposons une analyse critique de la notion de « succès » et de ce que celle-ci représente aux yeux des différentes parties prenantes.

‘I Finally Get It!’: developing mathematical understanding during teacher education

A deep conceptual understanding of elementary mathematics as appropriate for teaching is increasingly thought to be an important aspect of elementary teacher capacity. This study explores preservice teachers’ initial mathematical understandings and how these understandings developed during a mathematics methods course for upper elementary teachers. The methods course was supplemented by a newly designed optional course in mathematics for teaching. Teacher candidates choosing the optional course were initially weaker in terms of mathematical understanding than their peers, yet showed stronger mathematical development after engaging in the extra hours the optional course provided.

Uncertainty and the Reform of Elementary Math Education

This paper investigates the notion of uncertainty as elementary teachers engage in conversations intended to develop their understanding and implementation of reform-based mathematics teaching. Using a narrative methodology, several sources of teacher uncertainty are investigated: teaching and learning, the subject, and improving one’s own teaching. The data analysis indicates two important findings. The first is the importance of substantive and syntactic subject knowledge as a necessary foundation for teachers to understand uncertainty in terms that renew their classroom practice. The second is the need to develop and sustain communities in which teachers value opportunities to critique their classroom practices.

Teachers Constructing Concepts of Mathematics for Teaching and Learning: “It's like the roots beneath the surface, not a bigger garden”

Conceptually rich classroom learning environments can only be supported by teachers with appropriate mathematical knowledge. A lack of clarity exists as to whether or how such teacher knowledge might go beyond knowledge of the relevant curriculum. This study contributes to the field by investigating further examples of what appropriate teacher mathematical knowledge might be, as rooted and contextualized in teachers’ daily classroom practices. Teacher journaling, individual meetings, and teacher focus-group discussions were used to identify relevant examples, and ultimately continue to collectively describe, in a specific, contextually based and practitioner-developed manner, the mathematical knowledge required for elementary teaching.RésuméUn environnement d’apprentissage riche en concepts doit se fonder avant tout sur des enseignants qui possèdent des connaissances mathématiques appropriées. Cependant, il n’est pas clair si ou comment ce niveau de connaissance peut aller au-delà des connaissances pertinentes pour le curriculum. Cette étude analyse certains exemples de ce qui pourrait constituer des connaissances mathématiques appropriées, telles que contextualisées dans les pratiques quotidiennes des enseignants dans leur classe. Des comptes-rendus sous forme de journal, des rencontres individuelles et des discussions de groupe ont permis de définir nombre d’exemples pertinents, et continuent de contribuer à une description collective, fondée sur le contexte et l’expérience dans la classe, des connaissances mathématiques requises pour l’enseignement au primaire.

STRIVING FOR REFORM BASED PRACTICE IN UNIVERSITY SETTINGS: USING GROUPS IN LARGE MATHEMATICS CLASSES

ABSTRACT As school curricula undergo changes in both content and pedagogy, pressure is placed on teacher preparation programs and undergraduate mathematics departments to model learning environments in mathematics appropriate for education students. Yet the reality of many post-secondary mathematics courses includes traditional style classes containing a large number of students. This article describes specific structures used within large post-secondary mathematics classes for teachers in which reform based pedagogy such as use of small group hands-on learning and problem solving is facilitated where possible. The suggestions should be of particular interest to post secondary mathematics faculty charged with teaching mathematics to education students.

Measuring Mathematical Aptitude in Exploratory Computer Environments

Mathematical aptitude may be considered a special trait which is present in varying degrees in academically gifted students. The ability to think in the abstract seems to be one of the most important characteristics of the mathematical gifted. To be truly gifted in mathematics, however, creative ability is abo required. In a recent study of gifted tenth graders, mathematical creativity appeared to be a special kind of creativity not necessarily related to divergent thinking ability. In this article, measures of abstract thinking preference and divergent thinking are compared to standard mathematics ability measures such as the SAT‐M.

Understanding and supporting teacher horizon knowledge around limits: a framework for evaluating textbooks for teachers

ABSTRACT As part of recent scrutiny of teacher capacity, the question of teachers’ content knowledge of higher level mathematics emerges as important to the field of mathematics education. Elementary teachers in North America and some other countries tend to be subject generalists, yet it appears that some higher level mathematics background may be appropriate for teachers. An initial examination of a small sample of textbooks for teachers suggested the existence of a wide array of treatments and depth and quality of mathematics coverage. Based on the literature, a new framework was created to assess the mathematical quality of treatments for both specialized knowledge and horizon knowledge in mathematics textbooks for teachers. The framework was tested on a sample topic of the circle area formula derivation, chosen because it draws heavily on both specialized and horizon knowledge. The framework may contribute to similar analyses of other topics in a broader range of resources, in the overall quest to better describe the details of what constitutes appropriate mathematics horizon knowledge for teachers.