Future teachers' professional knowledge on argumentation and proof : a case study from universities in three countries

In this paper, qualitative results of a case study about the professional knowledge in the area of argumentation and proof of future teachers from universities in three countries are described. Based on results of open questionnaires, data about the competencies these future teachers have in the areas of mathematical knowledge and knowledge of mathematics pedagogy are presented. The study shows that the majority of the future teachers at the participating universities situated in Germany, Hong Kong and Australia, were not able to execute formal proofs, requiring only lower secondary mathematical content, in an adequate and mathematically correct way. In contrast, in all samples there was evidence of at least average competencies of pedagogical content reflection about formal and pre-formal proving in mathematics teaching. However, it appears that possessing a mathematical background as mandated for teaching and having a high affinity with proving in mathematics teaching at the lower secondary level are not a sufficient preparation for teaching proof.

A framework for primary teachers’ perceptions of mathematical reasoning

Mathematical reasoning has been emphasised as one of the key proficiencies for mathematics in the Australian curriculum since 2011 and in the Canadian curriculum since 2007. This study explores primary teachers’ perceptions of mathematical reasoning at a time of further curriculum change. Twenty-four primary teachers from Canada and Australia were interviewed after engagement in the first stage of the Mathematical Reasoning Professional Learning Program incorporating demonstration lessons focused on reasoning conducted in their schools. Phenomenographic analysis of interview transcripts exploring variation in the perceptions of mathematical reasoning held by these teachers revealed seven categories of description based on four dimensions of variation. The categories delineate the different perceptions of mathematical reasoning expressed by the participants of this study. The resulting outcome space establishes a framework that facilitates tracking of growth in primary teachers’ awareness of aspects of mathematical reasoning.

Analyzing Elementary Preservice Teachers’ Development of Content and Pedagogical Content Knowledge in Mathematics through Microteaching Lesson Study

This study examines the effect of Microteaching Lesson Study participation on the mathematics content and pedagogical content knowledge of 52 elementary preservice teachers. Preliminary findings, which are positive, are discussed.

The Effect of Authentic Literacy Experiences as Book Buddies with Hispanic Fourth Graders on Preservice Teachers’ Literacy Content Knowledge and Reading Maturity

From a sociocultural perspective, individuals learn best from contextualized experiences. In preservice teacher education, contextualized experiences include authentic literacy experiences, which include a real reader and writer and replicate real life communication. To be prepared to teach well, preservice teachers need to gain literacy content knowledge and possess reading maturity. The purpose of this study was to examine the effect of authentic literacy experiences as Book Buddies with Hispanic fourth graders on preservice teachers’ literacy content knowledge and reading maturity. The study was a pretest/posttest design conducted over 12 weeks. Preservice teacher participants, the focus of the study, were elementary education majors taking the third of four required reading courses in non-probabilistic convenience groups, 43 (n = 33 experimental, n = 10 comparison) Elementary Education majors. The Survey of Preservice Teachers’ Knowledge of Teaching and Technology (SPTKTT), specifically designed for preservice teachers majoring in elementary or early childhood education and the Reading Maturity Survey (RMS) were used in this study. Preservice teachers chose either the experimental or comparison group based on the opportunity to earn extra credit points (experimental = 30 points, comparison = 15). After exchanging introductory letters preservice teachers and Hispanic fourth graders each read four books. After reading each book preservice teachers wrote letters to their student asking higher order thinking questions. Preservice teachers received scanned copies of their student’s unedited letters via email which enabled them to see their student’s authentic answers and writing levels. A series of analyses of covariance were used to determine whether there were significant differences in the dependent variables between the experimental and comparison groups. This quasi-experimental study tested two hypotheses. Using the appropriate pretest scores as covariates for adjusting the posttest means of the subcategory Literacy Content Knowledge (LCK), of the SPTKTT and the RMS, the mean adjusted posttest scores from the experimental group and comparison group were compared. No significant differences were found on the LCK dependent variable using the .05 level of significance, which may be due to Type II error caused by the small sample size. Significant differences were found on RMS using the .05 level of significance.

The pedagogical content knowledge of Danish geography teachers in a changing schooling context

This study examines the self-reported, topic-specific professional knowledge (TSPK) of Danish geography teachers seen as an aspect of their pedagogical content knowledge (PCK) when teaching weather formation and climate change. This topic is considered representative of geography teaching in Denmark. In the last ten years Danish primary and lower-secondary schooling has undergone several significant changes, including the introduction of a final multiple-choice exam in geography in 2007, and a fundamental reconstruction of the curriculum in 2014. These changes are expected to influence the TSPK of geography teachers in ways that potentially have an impact on their classroom practice. Teachers´ responses to specific questions relating to their choice of learning goals and the content and organisation of their lessons show that geography teachers take into account not only the knowledge aspects which point to the final multiple-choice exam, but also the ‘bildung’ perspectives of the subject equipping students to develop their own opinions when dealing with socio-scientific issues (SSI).

Second-year pre-service teachers’ responses to proportional reasoning test items

A recent international study of pre-service teachers identified that proportional reasoning was problematic for pre-service teachers. Proportional reasoning is an important topic in the middle years of schooling and therefore it is critical that teachers understand this topic and can rely on their Mathematical Content Knowledge (MCK) when teaching. The focus of this paper is second-year Australian primary pre-service teachers’ MCK of real number items related to ratio, rate, proportion and proportional reasoning. This paper reports on strengths and weakness of pre-service teachers’ MCK when responding to test items; including a method suitable for analysing responses to five items and ranked by three levels of difficulty. The results revealed insights into their correct methods of solutions and common incorrect responses, identifying difficulty, where multiplication and division were required. The method of coding test items by difficulty ranking may assist with developing an appropriate learning trajectory, which will assist pre-service teachers develop their MCK of this and other difficult topics.

Conocimiento de matemáticas especializado de los estudiantes para maestro de primaria en relación al razonamiento proporcional

Un dominio particular del conocimiento matemático para la enseñanza es el conocimiento de matemáticas especializado. Este estudio se centra en examinar el conocimiento de matemáticas especializado en el ámbito del razonamiento proporcional de un grupo de estudiantes para maestro de Educación Primaria. Los resultados muestran que los estudiantes para maestro tienen un conocimiento especializado sobre el razonamiento proporcional limitado puesto de manifiesto por la dificultad en identificar situaciones no proporcionales, en desarrollar formas de razonar en relación a la construcción de la unidad y en manejar el significado multiplicativo de la idea de operador.

Primary teachers’ algebraic thinking: example from lesson study

Previous studies have reported on primary children’s algebraic thinking and generalising in a range of problem settings but there is little evidence of primary teachers’ knowledge of algebraic thinking. In this paper the development in algebraic thinking of one primary teacher who taught a research lesson in a Japanese Lesson Study project involving teachers from three primary schools is presented. The findings suggest the need for professional learning in algebra and reasoning and indicate the value of Lesson Study.

Mathematical knowledge of some entrants to a pre-service education course

Standards for teaching emphasise the need for teachers to have deep content knowledge. To assess the mathematical knowledge of students enrolling in its B.Ed. program, the University of New England has introduced a mathematics diagnostic test. This work is the first stage of an ongoing research project into the numeracy needs of students entering the B.Ed. program. The test is a pen-andpaper test that replaces previous on-line, multiple-choice tests. This paper reports on the test results, discusses some common errors made by students and outlines the future direction of the research.

Deepening the mathematical knowledge of secondary mathematics teachers who lack tertiary mathematics qualifications

A professional learning program for unqualified practising secondary mathematics teachers regarding senior secondary mathematics teaching is described in this paper. The VCE (Victorian Certificate of Education) mathematics professional learning program for senior secondary mathematics was designed for practising secondary teachers of mathematics who had no experience of teaching advanced senior secondary mathematics and who had not completed the recommended qualifications. Professional learning episodes, artefacts and reflections of three teachers who participated in the program are analysed to identify the development of these teachers' mathematical and pedagogical content knowledge (PCK). The PCK framework developed by Chick et al. was used to analyse teachers' knowledge, and the cases of teachers' knowledge presented in the paper illustrate the entwining of knowledge of mathematics and knowledge of teaching and learning The findings indicate that a program designed for senior secondary mathematics can enable practising teachers to deepen and broaden their understanding of junior secondary mathematical pedagogy.

No Royal Road but at Least a Gateway to the Mathematical Knowledge: The EuDML Project

Радослав Павлов - Представен е проектът EuDML – Европейската цифрова библиотека по математика (http://www.eudml.eu), който цели: • да създаде обща инфраструктура за безпроблемна навигация, търсене и взаимодействие в рамките на плътна мрежа от разпределено валидирано многоезично математическо съдържание в цифрова форма, което да е достъпно в цяла Европа, и така да направи математиката лесно достъпна за всички потребители; • да задоволи изискването за надежден и дългосрочен достъп до математическите изследвания. Представен е и българският принос в проекта – BulDML – цифрово хранилище за математическа литература на Института по математика и информатика на БАН (http://sci-gems.math.bas.bg).

Connecting with the horizon : developing teachers' appreciation of mathematical structure

A professional learning program for teachers of junior secondary mathematics regarding the content and pedagogy of senior secondary mathematics is the context for this study of teachers’ mathematical and pedagogical knowledge. The analysis of teachers’ reflections on their learning explored teachers’ understanding of mathematical connections and their appreciation of mathematical structure. The findings indicate that a professional learning program about senior secondary mathematics can enable practicing teachers to deepen and broaden their knowledge for teaching junior secondary mathematics and develop their practice to support their students’ present and future learning of mathematics. Further research is needed about professional learning approaches and tasks that may enable teachers to imbed and develop awareness of structure in their practice.