Using Teacher Portfolios to Enrich the Methods Course Experiences of Prospective Mathematics Teachers

This paper illustrates ways to employ teacher portfolios to improve the quality of methods course experiences for prospective mathematics teachers. Based upon research conducted in an undergraduate teacher preparation program, this case study describes how the author used teacher portfolios to mentor prospective teachers in new ways. The case describes the author's experiences through a case study of his assessment of and response to one prospective teacher's portfolio. This portfolio illustrated themes that were present in other teachers' portfolios, but did so in ways that highlighted strategies for change to the methods course. Through the lens of this teacher's portfolio the author identified specific ways that the prospective teacher's beliefs were impacting her teaching practice, a result that enabled him to better help all of the teachers in the methods course reflect on their teaching. By providing a detailed account of the feedback process that led to this result, this paper illustrates how mathematics teacher educators can use prospective teachers' portfolios to enrich the quality of their methods courses.     

Advanced mathematics communication beyond modality of sight

This study illustrates how mathematical communication and learning are inherently multimodal and embodied; hence, sight-disabled students are also able to conceptualize visuospatial information and mathematical concepts through tactile and auditory activities. Adapting a perceptuomotor integration approach, the study shows that the lack of access to visual fields in an advanced mathematics course does not obstruct a blind student's ability to visualize, but transforms it. The goal of this study is not to compare the visually impaired student with non-visually impaired students to address the ‘differences’ in understanding; instead, I discuss the challenges that a blind student, named Anthony, has encountered and the ways that we tackled those problems. I also demonstrate how the proper and precisely crafted tactile materials empowered Anthony to learn mathematical functions.     

Beliefs,Practical Knowledge,and Context: A Longitudinal Study of a Beginning Biology Teacher's 5E Unit

The purpose of this three‐year case study was to understand how a beginning biology teacher (Alice) designed and taught a 5E unit on natural selection, how the unit changed when she took a position in a different school district, and why the changes occurred. We examined Alice's developing beliefs about science teaching and learning, practical knowledge, and perceptions of school context in relation to the 5E unit. Data sources consisted of interviews, classroom observations, and lesson materials. We found that Alice placed more emphasis on the explore phase, less emphasis on the engage and explain phases, and removed the elaborate phase over time. Alice's beliefs about science teaching and learning acted as a filter for making sense of practical knowledge and perceptions of context. Although her beliefs were student centered, they aligned with discovery learning in which little intervention from the teacher is required. We discuss how her beliefs, practical knowledge, and perceptions of context explained the changes in her practice. This study sheds insight into the nature of beliefs and how they relate to the 5E lesson phases, as well as the different lenses for viewing the 5E instructional model. Implications for science teacher preparation and induction programs are discussed.     

NEEDING TO USE ALGEBRA

We have completed a Teacher Training Agency (TTA) funded project looking at the teaching and learning of algebra with one mixed-ability year 7 class (11–12 year-olds in the UK). We take Kieran's (quoted in Sutherland, 1997) definition of algebra and see ourselves as developing a'community of practice’ (Lave and Wenger, 1991) in the classroom where the practice is not that of mathematician but of inquirer (Schoenfeld, 1996) into mathematics. The teacher in this community acts as role model for inquirer and metacomments (Bateson, 1972) on the practice of inquiry. In this paper we present evidence for how such metacommenting supports the development of a'community of inquiry’ which leads to the'need’ for algebraic activity. We conclude with a case study, illustrating one student's need for algebra.     

On the teaching and learing of Dienes' principles

Zoltan Dienes' principles of mathematical learning have been an integral part of mathematics education literature and applied both to the teaching and learning of mathematics as well as research on processes such as abstraction and generalization of mathematical structures. Most extant textbooks of cognitive learning theories in mathematics education include a treatment of Dienes' seminal contributions. Yet, there are no available studies at the tertiary level on how students internalise the meaning of Dienes' principles. This paper explores post-graduate mathematics education student's understanding of Dienes' principles and their ability to reflexively apply the principles to their own thinking on structurally similar problems. Some implications are offered for university educators engaged in the training of future researchers in the field.     

The State of Readiness of Initial Level Preservice Middle Grades Science and Mathematics Teachers and Its Implications on Teacher Education Programs

The purpose of this study was to document through interview and videotaped data the current state of readiness of 10 preservice middle grade teachers, regarding their ability to plan, implement, and reflect on an integrated mathematics and science lesson. The results showed that only one student was successful in implementing a lesson that compared favorably to national standards. This student's lesson plan contained minimal pedagogical considerations and consisted primarily of notes emphasizing fine detail of distinction about the content of the lesson using her own examples. The lesson plan and post-lesson-plan interview data of the remaining students indicated an adherence to algorithmic learning, rote memorization, and procedural knowledge. There were numerous content errors in the plans, and these students orally described a lack of self-confidence in their ability to teach this lesson successfully. The most successful student demonstrated her competence in meeting standards of pedagogical content knowledge and was most successful in analyzing her own teaching. The results showed that most subjects of this study needed extensive training in content and pedagogy and in synthesizing these in a way consistent with modern learning theory.     

SCHOOL OR COLLEGE MATHEMATICS AND WORKPLACE PRACTICE: AN ACTIVITY THEORY PERSPECTIVE

This paper reports on a case study in which we detail how a college mathematics and chemistry student struggles to make sense of the graphical output of an experiment in an industrial chemistry laboratory. The student's attempts to interpret unfamiliar graphical conventions are described and contrasted with those of college mathematics. Our analysis of this draws on activity theory to assist in understanding the position of the student in both the college and the workplace. This highlights the limitations of the experience of the student at college and we question how the mathematics curriculum might be adapted to assist students in making sense of workplace graphical output.     

MATHEMATICS STUDENT TEACHERS’ RESPONSES TO INFLUENCES AND BELIEFS

This article forms part of an ongoing study of student teachers of secondary mathematics. The aspect reported on in this article is an analysis of the effects of the influences brought to bear upon four individual student teachers of secondary mathematics as they progress through a one-year postgraduate course of teacher training (PGCE) based at a British University. The students have differing initial beliefs about teaching, learning and mathematics. As anticipated in the literature, the student's initial beliefs survive virtually intact throughout the year. However, the study suggests that the link between initial beliefs and teaching approach is not deterministic. The study suggests ways of encouraging student teachers to employ a range of pupil activities in their teaching.     

Metaphor as a possible pathway to more formal understanding of the definition of sequence convergence

This study presents how the introduction of a metaphor for sequence convergence constituted an experientially real context in which an undergraduate real analysis student developed a property-based definition of sequence convergence. I use elements from Zandieh and Rasmussen's (2010) Defining as a Mathematical Activity framework to trace the transformation of the student's conception from a non-standard, personal concept definition rooted in the metaphor to a concept definition for sequence convergence compatible with the standard definition. This account of the development of the definition of sequence convergence differs from prior research in the sense that it began neither with examples or visual notions, nor with the statement of the formal definition. This study contributes to the Realistic Mathematics Education literature as it documents a student's progression through the definition-of and definition-for stages of mathematical activity in an interactive lecture classroom context.     

How Preservice Teachers Interpret and Respond to Student Geometric Errors

Recognizing and responding to students' thinking is essential in teaching mathematics, especially when students provide incorrect solutions. This study examined, through a teaching scenario task, elementary preservice teachers' interpretations of and responses to a student's work on a task involving reflective symmetry. Findings revealed that a majority of preservice teachers identified the student's errors from conceptual aspects of reflection rather than from procedural aspects. However, when they responded to the student's errors, preservice teachers tried to cope with them by invoking procedural knowledge. This study also revealed the three types of responses and two different forms of address by preservice teachers to student errors; these categories might provide insight into the difficulties arising in communication between students and teachers.     

Attitudes and perceived usefulness of statistics among health sciences students

A questionnaire was completed at the cessation of semester 2 in November 1989 by 102 postgraduate and 58 undergraduate health sciences students studying introductory statistics units. The questionnaire measured; (i) the student's attitude towards statistics, (ii) the way in which they learned statistics, (iii) the student's intention to pursue further statistics training, and (iv) the perceived usefulness of statistics in their professions. It was found that the learning of statistics would be enhanced by smaller tutorial groups, and more exposure to computer printouts to assist in interpretation of results. An emphasis on the understanding of statistics presented in journal articles should be a priority. It was also apparent that the more computer and research experience the student had prior to commencing the course, the greater the likelihood of a positive attitude towards statistics. However, both undergraduate and postgraduate students indicated that they would not enrol in an advanced biostatistics course, but would rather consult a statistician when necessary. Suggestions for more effective statistical teaching for health sciences students are also discussed.     

Learning How Amanda,a High School Cerebral Palsy Student,Understands Angles

Little is known about how well cerebral palsy students learn high school geometry. A case study was used to better understand how one student, Amanda, understood angles. Three major accommodations were made to assist her in learning: a) a self-paced curriculum, b) The Geometer's Sketchpad, and c) nontraditional assessment (portfolio, interviews, observations). It was found that Amanda needed a lot of time to process visual information. The orientation of angles, the complexity of the diagram, and the length of the side of an angle all had an impact on her understanding. The software was beneficial for Amanda, because she could hide unnecessary and distracting information, she could make her drawings legible, and she could measure the angles without relying on her own visual perception.     

Factors affecting student success in a first-year mathematics course: a South African experience

In spite of sustained efforts tertiary institutions implement to try and improve student academic performance, the number of students succeeding in first-year mathematics courses remains disturbingly low. For most students, the gap between their mathematical capability and the competencies they are expected and need to develop to function effectively in these courses persists even after course instruction. In this study, an instrument for identifying and examining factors affecting student performance and success in a first-year Mathematics university course was developed and administered to 86 students. The overall Cronbach's Alpha coefficient for the questionnaire was found to be 0.916. Having identified variables from prior research known to affect student performance, factor analysis was used to identify variables exhibiting the greatest impact on student performance. The variables included prior academic knowledge, workload, student approaches to learning, assessment, student support teaching quality, methods and resources. From the analysis, students' perceptions of their workload emerged as the factor having the greatest impact on student's performance, followed by the matriculation examination score. The findings are discussed and strategies that can be used to improve teaching and contribute to student success in a first-year mathematics course in a South African context are presented.     

Formative Use of Intuitive Analysis of Variance

Students’ informal inferential reasoning (IIR) is often inconsistent with the normative logic underlying formal statistical methods such as Analysis of Variance (ANOVA), even after instruction. In two experiments reported here, student's IIR was assessed using an intuitive ANOVA task at the beginning and end of a statistics course. In both experiments, students were provided feedback regarding the normative logic underlying ANOVA and how their reasoning compared with it. Additionally, students in Experiment 2 were given an assignment in which they analyzed and interpreted other students’ performance on the intuitive ANOVA task. Results indicate that the feedback combined with the assignment (which required active explanation of both normative and non-normative reasoning applied to the task) led to more normative inferential reasoning at the end of the course, whereas providing feedback alone did not. Implications are discussed for using the intuitive ANOVA task as a formative classroom tool to help students improve their conceptual understanding of ANOVA.     

Conflicting Beliefs: A Story of a Chemistry Teacher's Struggle with Change

One teacher's struggle to develop and implement a curriculum focused on student understanding of chemistry is explored in this case study of a high school chemistry teacher. Conflicting beliefs about her roles as a teacher in the classroom and her professional responsibilities are addressed. Three primary conflicts that emerged from data collected over a two year period include, (a) conflicts between state curriculum mandates and individual student understanding; (b) conflicts between theoretical and applicable chemistry content knowledge, and (c) conflicts between the students' goals and the teachers' goals for the course. The impact of the research process on the teacher's change process included reconceptualization of constraints and development of confidence in her professional judgment. The case study provides insights into contextual problems teachers face as they attempt to change practices.     

Prospective Teachers' Perspectives on Mathematics Teaching and Learning: Lens for Interpreting Experiences in a Standards‐Based Mathematics Course

In a mathematics course for prospective elementary teachers, we strove to model standards‐based pedagogy. However, an end‐of‐class reflection revealed the prospective teachers were considering incorporating standards‐based strategies in their future classrooms in ways different from our intent. Thus, we drew upon the framework presented by Simon, Tzur, Heinz, Kinzel, and Smith to examine the prospective teachers' perspectives on mathematics teaching and learning and to address two research questions. What perspectives on the learning and teaching of mathematics do prospective elementary teachers hold? How do their perspectives impact their perception of standards‐based instruction in a mathematics course and their future teaching plans? Qualitative analyses of reflections from 106 prospective teachers revealed that they viewed mathematics as a logical domain representative of an objective reality. Their instructional preferences included providing firsthand opportunities for elementary students to perceive mathematics. They did not take into account the impact of a student's conceptions upon what is learned. Thus, the prospective teachers plan to incorporate standards‐based strategies to provide active experiences for their future elementary students, but they fail to base such strategies upon students' current mathematical conceptions. Throughout, the need to address prospective teachers' underlying perspectives of mathematics teaching and learning is stressed.     

Locally logical mathematics: An emerging teacher honoring both students and mathematics

This case study explores the mathematics engagement and teaching practice of a beginning secondary school teacher. The focus is on the mathematical opportunities available to her students (the classroom mathematics) and how they relate to the teacher's personal capacity and tendencies for mathematical engagement (her personal mathematics). We use a mathematical process-and-action approach to analyze mathematical engagement and then employ the teaching triad—mathematical challenge, sensitivity to students, and management of learning—to situate mathematical engagement within the larger context of teaching practice. The article develops the construct of locally logical mathematics to underscore the cogency of mathematical engagement in the classroom as part of a coherent mathematical system that is embedded within a teaching practice. Contributions of the study include the process-and-action approach, especially in tandem with the teaching triad, as a tool to understand nuances of mathematical engagement and differences in demand between written and implemented tasks.     

Understandings of Solutions to Differential Equations Through Contexts,Web‐Based Simulations,and Student Discussion

As in the case of elementary mathematics, the instruction of high‐level mathematical concepts can often be sacrificed at the expense of a focus on algorithmic procedures. Computer‐based simulations can expand an undergraduate mathematics instructor's opportunity to explore high‐level mathematical concepts in an applied environment. This study describes one instructor's approach to incorporating simulations and classroom discussions in a differential equations course and the subsequent effects on student learning attitudes and outcomes. Students made modest gains in the area of conceptualizing and applying ideas regarding solutions to differential equations in this learning environment. Implications of the study include the identification of specific gains relative to computer‐mediated learning environments and recommendations for using simulations to support conceptual development.     

Understanding a Teacher's Reflections: A Case Study of a Middle School Mathematics Teacher

The purpose of this research was to understand how one teacher reflected on different classroom situations and to understand whether the teacher's approach to these reflections changed over time. For the purposes of this study, we considered reflection as the teacher's act of interpreting her own practices and students' thinking to make sense of student understanding and how teaching might relate to that understanding. We investigated a middle school mathematics teacher's reflection on her students while watching videotapes of her classroom and categorized the reflection as Assess, Interpret, Describe, Justify, and Extend. The results show a higher percentage of Extend instances in later interviews than in earlier ones indicating the teacher's increasing attention to her own teaching in how her students developed their understanding. In addition, her reflection became clearer and better integrated as defined by the Cohen and Ball's triangle of interactions.     

Inquiry Teaching and Learning: The Best Math Class Study

This research reports on prospective middle school teachers' perceptions of a “best mathematics class” during their involvement in an inquiry‐designed mathematics content course. Grounded in the prestigious Glenn Commission report ( U.S. Department of Education, 2000 ), the study examined the prospective teachers' perceptions of effective mathematics instruction both prior to and after completing the inquiry course. Pre‐essay analysis revealed that students could be grouped into one of two categories: the Watch‐Learn‐Practice view and the Self as Initiator view. Post‐essay analysis indicated that over two thirds of all students involved in the study changed their views of a best math class after the inquiry courses. The Watch‐Learn‐Practice group's changes focused on developing reasoning skills and learning how one “knows” in mathematics. The Self as Initiator group noted expanded roles for the students, particularly emphasizing the importance of going beyond basic requirements to think deeply about the why and how of mathematics and expanded views of the benefits of group learning.