Knowing more than their students: Characterizing secondary science teachers’ subject matter knowledge

Despite agreement among teacher educators, scholars, and policymakers on the importance of teachers’ subject matter knowledge (SMK), existing models provide limited information about the nature of this foundational component of teacher knowledge. The common assumption is that teachers need to know more about the science subject matter than their students are expected to learn, but what and how much more is underspecified. In order to more characterize science teachers’ SMK, we present the science knowledge for teaching (SKT) model, which has been adapted from the mathematics education literature to apply to science education. The SKT model includes three domains: core content knowledge, specialized content knowledge, and linked content knowledge. We used this model to explore the SMK new secondary chemistry teachers in South Africa and the United States drew on when they explained the conservation of mass and analyzed a related teaching scenario, two important tasks of teaching. Findings indicated these new teachers drew on knowledge from all three SKT domains in order to engage in these tasks of teaching. This result suggests the potential of the SKT model to characterize the nature of science teachers’ SMK and thereby better inform teacher preparation and professional development programs.     

Chinese elementary mathematics teachers’ knowledge in mathematics and pedagogy for teaching: the case of fraction division

In this study, we investigated the extent of knowledge in mathematics and pedagogy that Chinese practicing elementary mathematics teachers have and what changes teaching experience may bring to their knowledge. With a sample of 18 mathematics teachers from two elementary schools, we focused on both practicing teachers’ beliefs and perceptions about their own knowledge in mathematics and pedagogy and the extent of their knowledge on the topic of fraction division. The results revealed a gap between these teachers’ limited knowledge about the curriculum they teach and their solid mathematics knowledge for teaching, as an example, fraction division. Moreover, senior teachers used more diverse strategies that are concrete in nature than junior teachers in providing procedural justifications. The results suggested that Chinese practicing teachers benefit from teaching and in-service professional development for the improvement of their mathematics knowledge for teaching but not their knowledge about mathematics curriculum.     

Special Issue Article: A Look at Technology's Role in Professional Development of Mathematics Teachers at the Middle School Level

In last month's issue of School Science and Mathematics, Glenda Lappan addressed the dilemma of supporting teachers in continuing to grow professionally in four domains: Learning more mathematics content; improving their pedagogical skills; improving assessment skills; and addapting the curriculum to the needs of their students. Though it is clear that this kind of learning is best done in the context of what is going on in the classroom, teachers' schedules and other problems make it difficult to implement a coherent classroom based, professional development program. In this article, two more knowledge domains are added to this mix: the ability to (a) use technology and (b) teach with technology effectively. Since the process of learning and teaching is a dynamic one, the author describes his vision of the classroom as a laboratory where teachers get to practice and improve in these six areas and get feedback from an audience of their peers. Reflections are based on a current project in Paterson, New Jersey where the author helps middle school teachers use computer software to improve their mathematics teaching and learning.     

Classification of stability-like concepts and their study using vector Lyapunov functions

In the first section, stability-like definitions for ordinary differential equations are derived from a general qualitative concept. It is shown that the classical definitions of stability in the sense of Lyapunov, and their extensions can easily be deduced from this general formulation. A classification of all the definitions which may be derived is proposed.The second section contains the main results of this paper. It deals with the “comparison method” based upon one of T. Wazewski's theorems on differential inequalities. Several authors have used this method in order to investigate stability-like properties. We display the structure of this method, in order to state and prove some general comparison principles. These apply to the class of concepts considered earlier.In the last section some new results about stability and attractivity of sets are obtained as examples for the comparison principles. A theorem on stability in tube-like domains is proved in order to emphasize the generality and the flexibility of the comparison method.     

On the content-dependence of prospective teachers’ knowledge: a case of exemplifying definitions

In this article, we demonstrate that prospective teachers’ content knowledge related to defining mathematical concepts is dependent on content area. We use the example of generation (a research tool we developed in a previous study) to investigate prospective teachers’ knowledge. We asked prospective secondary mathematics teachers to provide multiple examples of definitions of concepts from different areas of mathematics. We examined teacher-generated examples of concept definition and analysed individual and collective example spaces, focusing on their correctness and richness. We demonstrate differences in prospective teachers’ knowledge associated with defining mathematical concepts in geometry, algebra and calculus.     

Engineering design and the development of knowledge for teaching among preservice science teachers

The goal of this article is to inform professional understanding regarding preservice science teachers’ knowledge of engineering and the engineering design process. Originating as a conceptual study of the appropriateness of “knowledge as design” as a framework for conducting science teacher education to support learning related to engineering design, the findings are informed by an ongoing research project. Perkins’s theory encapsulates knowledge as design within four complementary components of the nature of design. When using the structure of Perkins’s theory as a framework for analysis of data gathered from preservice teachers conducting engineering activities within an instructional methods course for secondary science, a concurrence between teacher knowledge development and the theory emerged. Initially, the individuals, who were participants in the research, were unfamiliar with engineering as a component of science teaching and expressed a lack of knowledge of engineering. The emergence of connections between Perkins’s theory of knowledge as design and knowledge development for teaching were found when examining preservice teachers’ development of creative and systematic thinking skills within the context of engineering design activities as well as examination of their knowledge of the application of science to problem‐solving situations.     

Scepticism,context and modal reasoning

I analyze some classical solutions of the skeptical argument and some of their week points (especially the contextualist solution). First I have proposed some possible improvement of the contextualist solution (the introduction of the explicit-implicit belief and knowledge distinction beside the differences in the relevance of some counter-factual alternatives). However, this solution does not block too fast jumps of the everyday context (where empirical knowledge is possible) into skeptical context (where empirical knowledge is impossible). Then I analyze some formal analogies between some modal arguments on the contingency of empirical facts (and the world as whole) and the skeptical arguments against empirical knowledge. I try to show that the skeptical conclusion “Empirical knowledge does not exist” is logically coherent with the thesis that they are empirical facts and that we have true belief on them. In order to do that without contradictions I have to accept a non-classical definition of knowledge: S knows that p:= S is not justified to allow that non-p. Knowledge and justified allowance function here as some pseudo-theoretical concepts which allow only some partial and conditional definitions by some “empirical” terms and logical conditions.     

Towards curricular coherence in integers and fractions: a study of the efficacy of a lesson sequence that uses the number line as the principal representational context

This paper reports a study of the efficacy of Learning Mathematics through Representations (LMR), an innovative curriculum unit designed to support upper elementary students’ understandings of integers and fractions. The unit supports an integrated treatment of integers and fractions through (a) the use of the number line as a cross-domain representational context, and (b) the building of mathematical definitions in classroom communities that become resources to support student argumentation, generalization, and problem solving. In the efficacy study, fourth and fifth grade teachers employing the same district curriculum (Everyday Mathematics) were matched on background indicators and then assigned to either the LMR experimental classrooms (n = 11) or the comparison group (n = 8 with 10 classrooms). During the fall semester, LMR teachers implemented the LMR unit on 19 days and district curriculum on other days of mathematics instruction. HLM analyses documented greater achievement for LMR students than Comparison students on both the end-of-unit and the end-of year assessments of integers and fractions knowledge; the growth rates of LMR students were similar regardless of entering ability level, and gains for LMR students occurred on item types that included number line representations and those that did not. The findings point to the efficacy of the LMR sequence in supporting teaching and learning in the domains of integers and fractions.     

Which is more fundamental: Outward peace or being true to our humanity?

How we perceive and define concepts should be a major concern for teachers and educators in general and for math teachers and educators in particular. This issue, however, is usually ignored or marginalised in curricula and classrooms, where textbooks form the major source of information, knowledge, meanings and definitions. The two words in the theme of this ZDM-issue—math and peace—are themselves good examples to explore. “What do we include in the math curriculum and what do we exclude, and why?” and “What do we mean by peace, and whose peace, and at what human and environmental cost?” are questions that the article raises and tries to point out how we should go about dealing with them. The article challenges the absolute meanings of concepts and stresses the importance of discussing meanings only within context. This concern requires giving teachers and students a greater say in the curriculum, to explore and discuss issues that affect their lives profoundly. The article challenges current values and goals, and suggests what it considers more fundamental values. In particular, it suggests that when there is conflict between peace and being true to our humanity, the latter should be given priority as a guiding value.     

Elementary Teachers' Progressive Understanding of Inquiry through the Process of Reflection

What is inquiry? Although many teachers are using inquiry based curricula, often they have not engaged in answering or personalizing this question. This study examined teachers' changing definitions of inquiry over a semester using the process of guided reflection. Through inquiry experiences and reflection, these teachers developed and communicated a more sophisticated understanding of inquiry. The findings suggest that conscious consideration of what inquiry means assisted teachers in broadening their perceptions of inquiry in four distinct aspects: 1) inquiry is a coherent process consisting of particular actions, 2) inquiry exists on a continuum, 3) the goal of inquiry is science conceptual development, and 4) inquiry provides a context for building connections between those engaged in inquiry, science and other content areas, and science and life.     

Elementary school teachers’ growth in inquiry-based teaching of mathematics

This article reports on a self-directed, school-based, practice-based professional development (PD) experience aimed at helping elementary school teachers to develop knowledge and expertise in inquiry-based teaching of mathematics. It discusses the characteristics of the self-directed orientation of this PD that supported the teachers’ learning, the nature of the inquiry-based knowledge they constructed, and the impact on their teaching. It highlights the centrality of agency, practical knowledge, and situated learning in this PD approach. The findings suggest that this approach can help mathematics teachers who want to be the architect of their own learning to transform their classrooms in meaningful and desirable ways.     

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.     

Examining novice teacher leaders’ facilitation of mathematics professional development

This paper reports on novice teacher leaders’ efforts to enact mathematics PD through an analysis of their facilitation in workshops conducted at their schools. We consider the extent to which teacher leaders facilitated the Problem-Solving Cycle model of PD with integrity to its key characteristics. We examine the characteristics they enacted particularly well and those that were the most problematic to enact. Facilitators were generally successful with respect to workshop culture and selecting video clips for use in the PD workshops. They had more difficulty supporting discussions to foster aspects of mathematics teachers’ specialized content knowledge and pedagogical content knowledge. We suggest a number of activities that may help to better prepare novice PD leaders to hold effective workshops. Furthermore, we conjecture that leaders of mathematics PD draw from a construct we have labeled Mathematical Knowledge for Professional Development (MKPD), and we posit some domains that may comprise this construct.     

A Comparison of TIMSS Items Using Cognitive Domains

The results of international assessments such as the Trends in International Mathematics and Science Study (TIMSS) are often reported as rankings of nations. Focusing solely on national rank can result in invalid inferences about the relative quality of educational systems that can, in turn, lead to negative consequences for teachers and students. This study seeks an alternative data analysis method that allows for improved inferences about international performance on the TIMSS. In this study, four classroom teachers categorized a sample of TIMSS items by the cognitive domains of knowing and applying using the definitions provided by the TIMSS 2011 Assessment Frameworks. Items of different cognitive domains were analyzed separately. This disaggregation allowed for more valid inferences to be made about student performance. Results showed almost no significant difference between the performance of U.S. students and the students of five other nations. Additionally, no differences were observed in U.S. students' performance on knowing items and applying items, although students from some sample nations performed significantly better on knowing items. These results suggest that policy makers, educators, and citizens should be cautious when interpreting the results of TIMSS rank tables.     

How do domains of knowledge integrate into mathematics teachers’ practice?

This study is a part of a research project that seeks to characterize the relationship between mathematics teachers’ knowledge and their practice. In this paper, we focus on identifying the characteristics of subject matter knowledge and pedagogical content knowledge that two teachers integrate in decisions they make about the introduction of specific mathematical content. Then, we examine the changes that arise in their classrooms as their plans are put in action. Data were obtained through audiotapes of several semi-structured interviews, through observations, and through videotapes. Although the two teachers in this study had similar backgrounds and experiences, our analysis shows differences in the characteristics of the domains of knowledge they integrated in their planning as well as differences in the adaptations that each made in the classroom. In this sense, this study contributes to better understanding the complexity of teachers’ professional practice.     

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

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.     

The Winds Are Variable: Student Intuitions About Variation

This study uses the context of the weather to explore the development of students' intuitive ideas of variation from pre‐Grade 1 to Grade 9. Three aspects of understanding these intuitions associated with variation are explored in individual videotaped interviews with 73 students: explanations, suggestions of data, and graphing. The development of these three aspects across grades is explored, as well as the associations among them. Fifty‐eight of the students also answered a general question on the definitions of “variation” and “variable,” and these responses are discussed and compared with responses to the weather task. The interview protocol may prove useful for teachers, particularly with younger children, to appreciate students' developing understanding of variation and provide starting points for classroom work of a more specific nature, either with respect to weather or other contextual topics.     

An analysis of the form and content of quadrilateral definitions composed by novice pre-service teachers

Mathematical definitions are an important mathematical construct which has been noted as a challenging topic for both teachers and students. This study provides an analysis of the form and content of a set of 308 definitions of quadrilateral types provided by 44 novice pre-service elementary teachers (NPSTs) who had not yet studied geometry or definitions on the college level. Analysis of definition structure including necessary, sufficient, and minimal conditions, as well as hierarchical and partitional structure, provides insight into what NPSTs may think about the form of mathematical definitions in general. Analysis of the properties used in definitions of each shape type and the frequency of those properties reveals the shape types with which NPSTs are most and least familiar. These results are presented alongside possible applications and implications for instruction.     

Topological Properties of Neumann Domains

A Laplacian eigenfunction on a two-dimensional manifold dictates some natural partitions of the manifold; the most apparent one being the well studied nodal domain partition. An alternative partition is revealed by considering a set of distinguished gradient flow lines of the eigenfunction—those which are connected to saddle points. These give rise to Neumann domains. We establish complementary definitions for Neumann domains and Neumann lines and use basic Morse homology to prove their fundamental topological properties. We study the eigenfunction restrictions to these domains. Their zero set, critical points and spectral properties allow to discuss some aspects of counting the number of Neumann domains and estimating their geometry.     

Building teachers’ expertise in understanding, assessing and developing children’s mathematical thinking: the power of task-based, one-to-one assessment interviews

In this paper, we outline the benefits to teachers’ expertise of the use of research-based, one-to-one assessment interviews in mathematics. Drawing upon our research and professional development work with teachers and students in primary and middle years in Australia and the research of others, we argue that the use of the interviews builds teacher expertise through enhancing teachers’ knowledge of individual and group understanding of mathematics, and also provides an understanding of typical learning paths in various mathematical domains. The use of such interviews also provides a model for teachers’ interactions and discussions with children, building both their pedagogical content knowledge and their subject matter knowledge.