Promoting Student Collaboration in a Detracked, Heterogeneous Secondary Mathematics Classroom

Detracking and heterogeneous groupwork are two educational practices that have been shown to have promise for affording all students needed learning opportunities to develop mathematical proficiency. However, teachers face significant pedagogical challenges in organizing productive groupwork in these settings. This study offers an analysis of one teacher’s role in creating a classroom system that supported student collaboration within groups in a detracked, heterogeneous geometry classroom. The analysis focuses on four categories of the teacher’s work that created a set of affordances to support within group collaborative practices and links the teacher’s work with principles of complex systems.

Supporting Whole-Class Collaborative Inquiry in a Secondary Mathematics Classroom

Recent mathematics education reform efforts call for the instantiation of mathematics classroom environments where students have opportunities to reason and construct their understandings as part of a community of learners. Despite some successes, traditional models of instruction still dominate the educational landscape. This limited success can be attributed, in part, to an underdeveloped understanding of the roles teachers must enact to successfully organize and participate in collaborative classroom practices. Towards this end, an in-depth longitudinal case study of a collaborative high school mathematics classroom was undertaken guided by the following two questions: What roles do these collaborative practices require of teacher and students? How does the community’s capacity to engage in collaborative practices develop over time? The analyses produced two conceptual models: one of the teacher’s role, along with specific instructional strategies the teacher used to organize a collaborative learning environment, and the second of the process by which the class’s capacity to participate in collaborative inquiry practices developed over time.

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An online community of practice for pre-service and beginning teachers of secondary mathematics

The aim of this study was to investigate how a community of practice focused on learning to teach secondary mathematics was created and sustained by pre-service and beginning teachers. Bulletin board discussions of one pre-service cohort are analysed in terms of Wenger’s (1998) three defining features of a community of practice: mutual engagement, joint enterprise, and a shared repertoire. The study shows that the emergent design of the community contributed to its sustainability in allowing the pre-service teachers to define their own professional goals and values. Sustainability was also related to how the participants expanded, transformed, and maintained the community during the pre-service program and after graduation.

Understanding technology integration in secondary mathematics: Theorising the role of the teacher

Previous research on computers and graphics calculators in mathematics education has examined effects on curriculum content and students’ mathematical achievement and attitudes while less attention has been given to the relationship between technology use and issues of pedagogy, in particular the impact on teachers’ professional learning in specific classroom and school environments. This observation is critical in the current context of educational policy making, where it is assumed – often incorrectly – that supplying schools with hardware and software will increase teachers’ use of technology and encourage more innovative teaching approaches. This paper reports on a research program that aimed to develop better understanding of how and under what conditions Australian secondary school mathematics teachers learn to effectively integrate technology into their practice. The research adapted Valsiner’s concepts of the Zone of Proximal Development, Zone of Free Movement and Zone of Promoted Action to devise a theoretical framework for analysing relationships between factors influencing teachers’ use of technology in mathematics classrooms. This paper illustrates how the framework may be used by analysing case studies of a novice teacher and an experienced teacher in different school settings.

The relationship between the secondary mathematics curriculum, college persistence, and success at an urban community college

According to Venezia, Kirst, and Antonio (2003) and Barth’s 2002 Thinking K16 Ticket to Nowhere report, the disconnect between K-12 and postsecondary education was a contributing factor to high attrition rates. Since mathematics emerged as a primary concern for college readiness, Barth (2002) called for improving student transitions from K-12 to postsecondary institutions through the use of state or local data. The purpose of the present study was to analyze mathematics course-taking patterns of secondary students in a local context and to evaluate high school characteristics in order to explore their relationships with Associate degree attainment or continuous enrollment at an urban community college. Also, this study extended a national study conducted by Clifford Adelman (The Toolbox Revisited, 2006) as it specifically focused on community college students that were not included his study. Furthermore, this study used the theoretical framework that human capital, social capital, and cultural capital influence habitus—an individual’s or a group’s learned inclination to behave within the parameters of the imposed prevailing culture and norms. Specifically, the school embedded culture as it relates to tracking worked as a reproduction tool of ultimate benefit for the privileged group (Oakes, 1994). ^ Using multilevel analysis, this ex post facto study examined non-causal relationships between math course-taking patterns and college persistence of public high school graduates who enrolled at the local community college for up to 6 years. One school-level variable (percent of racial/ethnic minorities) and 7 student-level variables (community college math proportion, remedial math attempts, race, gender, first-year credits earned, socioeconomic status, and summer credits earned) emerged as predictors for college persistence. Study results indicated that students who enter higher education at the community college may have had lower opportunities to learn and therefore needed higher levels of remediation, which was shown to detract students from degree completion. Community college leaders are called to partner with local high schools with high percentages of racial/ethnic minorities to design academic programs aimed at improving the academic preparation of high school students in mathematics and promote student engagement during the first year and summers of college. ^

EXAMINING SECONDARY MATHEMATICS TEACHER CANDIDATES’ LEARNING AND ENACTMENT OF MATHEMATICS TEACHING PRACTICES: A MULTIPLE CASE STUDY

This qualitative case study explored three teacher candidates’ learning and enactment of discourse-focused mathematics teaching practices. Using audio and video recordings of their teaching practice this study aimed to identify the shifts in the way in which the teacher candidates enacted the following discourse practices: elicited and used evidence of student thinking, posed purposeful questions, and facilitated meaningful mathematical discourse. The teacher candidates’ written reflections from their practice-based coursework as well as interviews were examined to see how two mathematics methods courses influenced their learning and enactment of the three discourse focused mathematics teaching practices. These data sources were also used to identify tensions the teacher candidates encountered. All three candidates in the study were able to successfully enact and reflect on these discourse-focused mathematics teaching practices at various time points in their preparation programs. Consistency of use and areas of improvement differed, however, depending on various tensions experienced by each candidate. Access to quality curriculum materials as well as time to formulate and enact thoughtful lesson plans that supported classroom discourse were tensions for these teacher candidates. This study shows that teacher candidates are capable of enacting discourse-focused teaching practices early in their field placements and with the support of practice-based coursework they can analyze and reflect on their practice for improvement. This study also reveals the importance of assisting teacher candidates in accessing rich mathematical tasks and collaborating during lesson planning. More research needs to be explored to identify how specific aspects of the learning cycle impact individual teachers and how this can be used to improve practice-based teacher education courses.

Basic components in the scienctific didactical training of the secondary school mathematics teachers

Secondary mathematics teacher training in Spain is currently the subject of a heated revision debate. The speed of social, cultural, scientific and economic changes have left a hundred years old teacher training model well behind. However, academical inertia and professional interests are impeding a real new training of the mathematics teacher as an autonomous mathematical educator. Teachers of Didactic of Mathematics and the Spanish Associations of mathematics teachers have recently been discussing the issue. Their conclusions are included here.

Assessing the potential of mathematics textbooks to promote deep learning

Curriculum documents for mathematics emphasise the importance of promoting depth of knowledge rather than shallow coverage of the curriculum. In this paper, we report on a study that explored the analysis of junior secondary mathematics textbooks to assess their potential to assist in teaching and learning aimed at building and applying deep mathematical knowledge. The method of analysis involved the establishment of a set of specific curriculum goals and associated indicators, based on research into the teaching and learning of a particular field within the mathematics curriculum, namely proportion and proportional reasoning. Topic selection was due to its pervasive nature throughout the school mathematics curriculum at this level. As a result of this study, it was found that the five textbook series examined provided limited support for the development of multiplicative structures required for proportional reasoning, and hence would not serve well the development of deep learning of mathematics. The study demonstrated a method that could be applied to the analysis of junior secondary mathematics in many parts of the world.

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.

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.