Using the MKT measures to reveal Indonesian teachers’ mathematical knowledge: challenges and potentials

The purpose of this study was to examine the adaptability of the US-based mathematical knowledge for teaching (MKT) geometry measures for use to study Indonesian elementary teachers’ MKT geometry. We selected the geometry scales Form A and Form B, and then adapted the items using a framework developed by Delaney et al. (J Math Teach Educ 11(3):171–197, 2008). We administrated the adapted learning mathematics for teaching measures to 210 elementary and middle school teachers. During translation and adaptation of the measures, issues arose regarding the mathematical substance of the items related to the use of inclusive and exclusive definitions of shapes. Psychometric analyses confirmed that these items were more difficult for the Indonesian elementary teachers compared to the US sample. Implications for future direction for item adaptation to measure Indonesia teachers’ MKT are presented.     

Quality assessment process for dynamic geometry resources in Intergeo project

One of the objectives of the Intergeo project was to provide European mathematics teachers with ??good quality?? pedagogical material supporting the use of dynamic geometry software in classrooms. With this aim in view, an online repository/platform was developed to gather various dynamic geometry resources, based on the idea of a teachers?? community feeding the repository, (re)using available resources and sharing practices related to the use of dynamic geometry in classrooms. The repository is thus open to contributions of any user who can deposit, browse, download and use resources, which naturally raises the question how to handle the resource quality issue in such an open environment. This paper reports on the way we tackled this issue in the Intergeo project. We first explain what we mean by a ??good quality?? dynamic geometry resource. We then provide a rationale behind the design of a questionnaire, the main tool for resource quality reviews, which are at the core of the quality assessment process implemented in the repository. Several experiments carried out with groups of teachers in order to confront our research-based view of the resource quality with the teachers?? one and to observe teachers?? usages of the quality assessment process are also reported in the paper. The outcomes of these experiments highlight strengths and limitations of the resource quality assessment process. They also tend to show that the idea of involving teachers into the resource quality assessment is a promising way of stimulating the use of dynamic geometry in classrooms, provided that teachers benefit from a support to make the quality process their own.     

Co-facilitation of study groups around animated scenes: the discourse of a moderator and a researcher

This paper examines the methodological issues pertaining to study group facilitation assisted by representations of teaching. The study groups, which comprised high school geometry teachers, aimed at achieving two goals: enhancing professional development among teachers and studying the practical rationality of geometry teaching. Therefore, session facilitation involved playing two roles that were often contradictory. A specific co-facilitation model was developed to achieve both goals. This paper examines the characteristics of the discourse of each of the facilitators. The results of this study are used to explain how the facilitators’ discourse created a relevant environment for achieving both goals.     

Assessing views of coaching via a video-based tool

Mathematics coaches represent a unique group of didacticians, or individuals who work with practicing teachers. Twenty-eight mathematics coaches participated in this exploratory study, which used video viewing to examine the coach–teacher dynamic. To gather data about participants’ views of effective coaching practices, we developed the Video Assessment of Coaching instrument, which provided coaches with opportunities to express their views of effective practice and implementation. The participants expressed views of effective coaching that often did not align with those of coaching authors. The significance of this research lies in its efforts to document the views that mathematics coaches develop as practitioners, as an early step in the examination of the relationships between the views of coaches and coaches’ effectiveness in improving teacher practice, knowledge, and attitudes.     

Toward Improving the Mathematics Preparation of Elementary Preservice Teachers

Research suggests the importance of mathematics knowledge for teaching (MKT) for enabling elementary school teachers to effectively teach mathematics. MKT involves both mathematical content knowledge (M‐CK) and mathematical pedagogical content knowledge (M‐PCK). However, there is no consensus on how best to prepare elementary preservice teachers (PSTs) to achieve M‐CK and M‐PCK. This study builds on research related to MKT by investigating influences of mathematics content courses designed specifically for elementary PSTs (IMPACT courses—Impact of Mathematics Pedagogy and Content on Teaching) on their attitudes (i.e., confidence and motivation) toward M‐CK and M‐PCK. Results suggest that the PSTs who participated in these IMPACT courses not only acquired high levels of confidence and motivation toward M‐CK, but also showed significant and greater gains in attitudes toward M‐PCK, after taking the required mathematics methods course, than their counterparts. Further, the findings suggest that these IMPACT courses provided a mathematical foundation that allowed the PSTs to engage in mathematics teaching methods better than those PSTs who did not have such a foundation. These results suggest potential course experiences that may enhance M‐CK and M‐PCK for elementary PSTs.     

Pre-constructed dynamic geometry materials in the classroom – how do they facilitate the learning of ‘Similar Triangles’?

The use of dynamic geometry software (DGS) is becoming increasingly familiar among teachers, but letting students conduct inquiries using computers is still not a welcome idea. In addition to logistics and discipline concerns, many teachers believe that mathematics at the lower secondary level can be learned efficiently through practice alone. Thus, the application of DGS remains limited to demonstration and explanation. This article discusses how a set of pre-constructed dynamic geometry (DG) materials was designed to teach the ‘similar triangles’ concept. The reactions and behaviour of students with relatively low levels of mathematic achievement are also analysed. Finally, the potential value of pre-constructed DG materials, with lab sheets and teacher intervention, in inquiry activities for junior-level students is discussed.     

Unpacking Online Asynchronous Collaboration in mathematics teacher education

We report on one aspect of an extended research and development project that was conducted to support teachers?? development of mathematical knowledge for teaching (MKT) algebra through participation and authentic engagement in online collaborative mathematical problem solving. This article expands on our recent work, which has succeeded in developing a model for supporting teachers?? mathematical development at a distance that has shown great promise for supporting significant gains in teachers?? MKT. Specifically, this ex-post-facto analysis consisted of qualitative, micro-level analysis of the content of teachers?? activity and generated artifacts and helps us understand how the various collaborative activities (specific combinations of interaction, instructor support and feedback, and technology) supported and/or constrained the development of MKT algebra in an online environment.     

Using the Knowledge Quartet to develop mathematics content knowledge: the role of reflection on professional development

In this paper I report findings from a four year study of beginning elementary school teachers which investigated development in their mathematical knowledge for teaching (MKT). The study took a developmental research approach, in that the teachers and the researcher collaborated to develop the mathematics teaching of the teachers, while also trying to understand how such development occurred and might be facilitated. The Knowledge Quartet (KQ) framework was used as a tool to support focused reflection on the mathematical content of teaching, with the aim of promoting development in mathematical content knowledge. Although I focused primarily on whether and how focused reflection using the KQ would promote development, it was impossible to separate this from other influences, and in this paper I discuss the ways in which reflection was found to interrelate with other areas of influence. I suggest that by helping the teachers to focus on the content of their mathematics teaching, within the context of their experience in classrooms and of working with others, the KQ framework supported development in the MKT of teachers in the study.     

Understanding kindergarten teachers’ perspectives of teaching basic geometric shapes: a phenomenographic research

Geometry is one of the disciplines children involve within early years of their lives. However, there is not much information about geometry education in Turkish kindergarten classes. The current study aims to examine teachers’ perspectives on teaching geometry in kindergarten classes. The researchers inquired about teachers’ in-class experiences in geometry and asked a series of questions such as “what are the benchmarks in your kindergarten class?”; “what kind of tools and materials you use to teach geometry in your class?”; “what shape do you teach first in your kindergarten class?”; “what do you expect to hear when you asked your students ‘what is square’?”; “how do you teach rectangular?”. The study utilized one of the qualitative research methods, namely phenomenography, to collect the data and analyze the data. The study involved with eight kindergarten teachers who work in different schools in central Kutahya, Turkey. The researchers collected data by conducting face-to-face half-structured interviews. The findings of this phenomenographic research showed that kindergarten teachers have some difficulties in teaching geometry and have lack of knowledge and skills in teaching geometry in kindergarten classes.     

The effectiveness of ‘what if not’ strategy coupled with dynamic geometry software in an inquiry-based geometry classroom

In this paper we present the results of a study which was carried out in an inquiry-based teaching and learning environment with the use of ‘what if not’ methodology coupled with the integration of dynamic geometry software. The vast majority of the students reported that they perceived themselves as participants rather than spectators. Most of the prospective teachers came to the conclusion that the implementation of the findings of this study in their future teachings was a good idea and that it will raise the students’ motivation and enhance and deepen the knowledge pool of the learners.     

Dynamic geometry as a context for exploring conjectures

The purpose of this paper is to provide examples of ‘non-traditional’ proof-related activities that can explored in a dynamic geometry environment by university and high school students of mathematics. These propositions were encountered in the dynamic geometry environment. The author believes that teachers can ask their students to construct proofs for these propositions.     

Using dynamic geometry to explore non-traditional theorems

The purpose of this article is to provide examples of ‘non-traditional’ theorems that can be explored in a dynamic geometry environment by university and high school students. These theorems were encountered in the dynamic geometry environment. The author believes that teachers can ask their students to construct proofs for these theorems. The Appendix contains proof outlines for each theorem.     

An epistemic model of task design in dynamic geometry environment

Dynamic geometry environment (DGE) has been a catalytic agent driving a paradigm shift in the teaching and learning of school geometry in the past two decades. It opens up a pedagogical space for teachers and students to engage in mathematical explorations that niche across the experimental and the theoretical. In particular, the drag-mode in DGE has been a unique pedagogical tool that can facilitate and empower students to experiment with dynamic geometrical objects which can lead to generation of mathematical conjectures. Furthermore, the drag-mode seems to open up a new methodology and even a new discourse to acquire geometrical knowledge alternative to the traditional Euclidean deductive reasoning paradigm. This discussion paper proposes an epistemic model of techno-pedagogic mathematic task design which serves as a theoretical combined-lens to view mathematics knowledge acquisition. Three epistemic modes for techno-pedagogic mathematical task design are proposed. They are used to conceptualize design of dynamic geometry tasks capitalizing the unique drag-mode nature in DGE that opens up an explorative space for learners to acquire mathematical knowledge.     

Examining Secondary Mathematics Teachers' Opportunities to Develop Mathematically in Professional Learning Communities

To make progress toward ambitious and equitable goals for students’ mathematical development, teachers need opportunities to develop specialized ways of knowing mathematics such as mathematical knowledge for teaching (MKT) for their work with students in the classroom. Professional learning communities (PLCs) are a common model used to support focused teacher collaboration and, in turn, foster teacher development, instructional improvement, and student outcomes. However, there is a lack of specificity in what is known about teachers’ work in PLCs and what teachers can gain from those experiences, despite broad claims of their benefit. We discuss an investigation of the work of secondary mathematics teachers in PLCs at two high schools to describe and explicate possible opportunities for teachers to develop the mathematical knowledge needed for the work of teaching and the ways in which these opportunities may be pursued or hindered. The findings show that, without pointed focus on mathematical content, opportunities to develop MKT can be rare, even among mathematics teachers. Two detailed images of teacher discussion are shared to highlight these claims. This article contributes to the ongoing discussion about the affordances and limitations of PLCs for mathematics teachers, considerations for their use, and how they can be supported.     

Analysis of psychometric properties as part of an iterative adaptation process of MKT items for use in other countries

Researchers at the University of Michigan have developed sets of items that can be used to analyze teachers’ mathematical knowledge for teaching (MKT). In this paper, we consider what is required in the adaptation of a set of these items for use in a Norwegian context. We discuss how analysis of item difficulty and point–biserial correlation can be applied in combination with qualitative approaches to ensure a high-quality process of piloting adapted MKT items. Findings indicate that researchers who attempt to adapt MKT items for use in cultural contexts other than those for which they were designed need to use different methods to analyze all aspects of the adaptation process. The results from the different analyses conducted might then be used to inform other parts of the process, and this will mean that the process of adapting and piloting items becomes cyclic and iterative.     

Game Design as an Interactive Learning Environment for Fostering Students'' and Teachers'' Mathematical Inquiry

Many learning environments, computer-based or not, have been developed for either students or teachers alone to engage them in mathematical inquiry. While some headway has been made in both directions, few efforts have concentrated on creating learning environments that bring both teachers and students together in their teaching and learning. In the following paper, we propose game design as such a learning environment for students and teachers to build on and challenge their existing understandings of mathematics, engage in relevant and meaningful learning contexts, and develop connections among their mathematical ideas and their real world contexts. To examine the potential of this approach, we conducted and analyzed two studies: Study I focused on a team of four elementary school students designing games to teach fractions to younger students, Study II focused on teams of pre-service teachers engaged in the same task. We analyzed the various games designed by the different teams to understand how teachers and students conceptualize the task of creating virtual game learning environment for others, in which ways they integrate their understanding of fractions and develop notions about students' thinking in fractions, and how conceptual design tools can provide a common platform to develop meaningful fraction contexts. In our analysis, we found that most teachers and students, when left to their own devices, create instructional games to teach fractions that incorporate little of their knowledge. We found that when we provided teachers and students with conceptual design tools such as game screens and design directives that facilitated an integration of content and game context, the games as well as teachers' and students' thinking increased in their sophistication. In the discussion, we elaborate on how the design activities helped to integrate rarely used informal knowledge of students and teachers, how the conceptual design tools improved the instructional design process, and how students and teachers benefit in their mathematical inquiry from each others' perspectives. In the outlook, we discuss features for computational design learning environments. This revised version was published online in August 2006 with corrections to the Cover Date.     

To change or not to change: adapting mathematical knowledge for teaching (MKT) measures for use in Korea

This paper examines challenges in adapting mathematical knowledge for teaching (MKT) measures developed in the United States for use in Korea. After an initial analysis of candidate issues regarding the “fit” of items to the Korean context—whether items were familiar, authentic, and realistic as characterized by Delaney et al. (J Math Teach Educ 11:171–197, 2008)—we adapted and administered an instrument developed by the Learning Mathematics for Teaching project with 93 Korean teachers and conducted follow-up interviews with nine teachers. Based on analysis of this data, we conducted a second round of revision and then administered the revised test to 101 Korean teachers. Results showed that small modifications that were made to increase fit often increased teachers’ performance on the items as expected, but the impact of changes was at times difficult to interpret. For several items, modifications introduced unanticipated validity issues. The paper discusses dynamics that arise in making changes to MKT items—in particular, the tension in modifying items to increase the fit to specific educational contexts while maintaining validity.     

To be or to become: how dynamic geometry changes discourse

In this article, we investigate the impact of the introduction of a dynamic geometry environment on mathematical thinking by identifying changes in discourse engendered by its introduction in a high school geometry class. Our focus is on the teacher, and we find significant differences between static and dynamic geometry in terms of the ways in which the teacher talks about geometric objects, makes use of visual artifacts and models geometric reasoning. Even though these changes have major implications for the geometry being studied, they are made only very implicitly in the classroom.     

Expanding in-service mathematics teachers’ horizons in creative work using technology

This article describes the experiences gained from a seminar in the teaching of mathematical reasoning and problem solving designed to prepare in-service high school mathematics teachers to teach genuine mathematical activity in a computer-based environment. Presented with a set of unfamiliar tasks and activities, the participants were encouraged to investigate each of them, using the Geometer's Sketchpad software, and then to justify their assertions accordingly. In the exploratory process the student teachers make the major mathematical contributions while the teacher plays the role of facilitator. The mathematics teachers began to realize the power of technology in teaching mathematics and were pleased to return to their classrooms with a great number of experiences and ideas for immediate use.