Teacher listening: The role of knowledge of content and students

In this research report we consider the kinds of knowledge needed by a mathematician as she implemented an inquiry-oriented abstract algebra curriculum. Specifically, we will explore instances in which the teacher was unable to make sense of students’ mathematical struggles in the moment. After describing each episode we will examine the instructor's efforts to listen to the students and the way that these efforts were supported or constrained by her mathematical knowledge for teaching. In particular, we will argue that in each case the instructor was ultimately constrained by her knowledge of how students were thinking about the mathematics.     

Teacher time out as a site for studying mathematical knowledge for teaching

The special mathematical knowledge that is needed for teaching has been studied for decades but the methods for studying it have challenges. Some methods, such as measurement and cognitive interviews, are removed from the dynamics of teaching. Other methods, such as observation, are closer to practice but mostly involve an outsider perspective. Moreover, few methods tap into the tacit and often invisible demands that teachers encounter in teaching. This article develops an argument that teacher time outs in rehearsals and enactments might be a productive site for studying mathematical knowledge for teaching. Teacher time outs constitute a site for professional deliberation, which 1) preserves the complexity and gets inside the dynamics of teaching, where 2) tacit and implicit challenges and demands are made explicit, and where 3) insider and outsider perspectives are combined.     

Using the Knowledge Quartet to quantify mathematical knowledge in teaching: the development of a protocol for Initial Teacher Education

This study examined trainee teachers' mathematical knowledge in teaching (MKiT) over their final year in a US Initial Teacher Education (ITE) programme. This paper reports on an exploratory methodological approach taken to use the Knowledge Quartet to quantify MKiT through the development of a new protocol to code trainees' teaching of mathematics lessons. This approach extends Rowland's et al. work on the Knowledge Quartet (KQ). Justification for using the KQ to quantify MKiT, and the potential benefits such an attempt might provide those involved with ITE, are discussed. It is suggested that quantified MKiT data based on the Knowledge Quartet can be used to consider MKiT development in novice teachers in order to inform ITE programmes and form new theoretical loops between theory and practice in teacher education.     

Mathematics university teachers' perception of pedagogical content knowledge (PCK)

Teaching mathematics in university levels is one of the most important fields of research in the area of mathematics education. Nevertheless, there is little information about teaching knowledge of mathematics university teachers. Pedagogical content knowledge (PCK) provides a suitable framework to study knowledge of teachers. The purpose of this paper is to make explicit the perception of mathematics university teachers about PCK. For this purpose, a phenomenological study was done. Data resources included semi-structured interviews with 10 mathematics university teachers who were in different places of the mathematics university teaching experience spectrum. Data analysis indicated a model consisting of four cognitive themes which are mathematics syntactic knowledge, knowledge about mathematics curriculum planning, knowledge about students' mathematics learning and knowledge about creating an influential mathematics teaching–learning environment. Besides, it was found out that three contextual themes influenced on PCK for teaching mathematics in university levels which were the nature of mathematics subjects, university teachers' features and terms of learning 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.     

Four (algorithms) in one (bag): an integrative framework of knowledge for teaching the standard algorithms of the basic arithmetic operations

In this article we present an integrative framework of knowledge for teaching the standard algorithms of the four basic arithmetic operations. The framework is based on a mathematical analysis of the algorithms, a connectionist perspective on teaching mathematics and an analogy with previous frameworks of knowledge for teaching arithmetic operations with rational numbers. In order to evaluate the potential applicability of the framework to task design, it was used for the design of mathematical learning tasks for teachers. The article includes examples of the tasks, their theoretical analysis, and empirical evidence of the sensitivity of the tasks to variations in teachers’ knowledge of the subject. This evidence is based on a study of 46 primary school teachers. The article concludes with remarks on the applicability of the framework to research and practice, highlighting its potential to encourage teaching the four algorithms with an emphasis on conceptual understanding.     

Preservice elementary teachers’ knowledge for teaching the associative property of multiplication: A preliminary analysis

This study examines preservice elementary teachers’ (PTs) knowledge for teaching the associative property (AP) of multiplication. Results reveal that PTs hold a common misconception between the AP and commutative property (CP). Most PTs in our sample were unable to use concrete contexts (e.g., pictorial representations and word problems) to illustrate AP of multiplication conceptually, particularly due to a fragile understanding of the meaning of multiplication. The study also revealed that the textbooks used by PTs at both the university and elementary levels do not provide conceptual support for teaching AP of multiplication. Implications of findings are discussed.     

Defining as a mathematical activity: A framework for characterizing progress from informal to more formal ways of reasoning

The purpose of this paper is to further the notion of defining as a mathematical activity by elaborating a framework that structures the role of defining in student progress from informal to more formal ways of reasoning. The framework is the result of a retrospective account of a significant learning experience that occurred in an undergraduate geometry course. The framework integrates the instructional design theory of Realistic Mathematics Education (RME) and distinctions between concept image and concept definition and offers other researchers and instructional designers a structured way to analyze or plan for the role of defining in students’ mathematical progress.     

A new Jacobi operational matrix: An application for solving fractional differential equations

In this paper, we derived the shifted Jacobi operational matrix (JOM) of fractional derivatives which is applied together with spectral tau method for numerical solution of general linear multi-term fractional differential equations (FDEs). A new approach implementing shifted Jacobi operational matrix in combination with the shifted Jacobi collocation technique is introduced for the numerical solution of nonlinear multi-term FDEs. The main characteristic behind this approach is that it reduces such problems to those of solving a system of algebraic equations which greatly simplifying the problem. The proposed methods are applied for solving linear and nonlinear multi-term FDEs subject to initial or boundary conditions, and the exact solutions are obtained for some tested problems. Special attention is given to the comparison of the numerical results obtained by the new algorithm with those found by other known methods.     

The impact of integrated STEM modeling on elementary preservice teachers’ self‐efficacy for integrated STEM instruction: A co‐teaching approach

The purpose of the current study was to evaluate the impact of co‐taught integrated STEM methods instruction on preservice elementary teachers’ self‐efficacy for teaching science and mathematics within an integrated STEM framework. Two instructional methods courses (Elementary Mathematics Methods and Elementary Science Methods) were redesigned to include STEM integration components, including STEM model lessons co‐taught by a mathematics and science educator, as well as a special education colleague. Quantitative data were gathered at three time points in the semester (beginning, middle, and end) from 55 preservice teachers examining teacher self‐efficacy for integrated STEM teaching. Qualitative data were gathered from a purposeful sample of seven preservice teachers to further understand preservice teachers’ perceptions on delivering integrated STEM instruction in an elementary setting. Quantitative results showed a significant increase in teacher self‐efficacy across all three time points. Item‐level analysis revealed that self‐efficacy for tasks involving engineering and assessment (both formative and summative) were low across time points, while self‐efficacy for tasks involving technology and flexibility were consistently high. Qualitative results revealed that the preservice teachers did not feel adequately prepared by university‐level science and mathematics courses, in terms of content knowledge and integration of science and mathematics for elementary students.     

Learning trajectory of a student with learning disabilities for the concept of length: A teaching experiment

This study aims to map the learning trajectory (LT) of a student with learning disabilities (LDs) regarding the unit concept in length measurement and the usage of rulers. The article draws on data from a teaching experiment with a 10-year-old student with LDs in Turkey. Data were analyzed in two stages, including microanalysis, where each successive teaching session was separately analyzed, and macroanalysis, where the teaching sessions regarding interrelated instructional goals were analyzed to construct the LT. The main findings of the study illustrate that this student with LDs eliminated her misconceptions about the unit concept and using a ruler, accomplished the determined instructional goals to a large extent, and reached a higher level of thinking with a 4-month teaching experiment designed based on her specific developmental capacity.     

Characterizing prospective secondary teachers’ foundation and contingency knowledge for definitions of transformations

One promising approach for connecting undergraduate content coursework to secondary teaching is using teacher-created representations of practice. Using these representations effectively requires seeing teachers' use of mathematical knowledge in the work of teaching. We argue that the dimensions of Rowland's (2013) Knowledge Quartet, especially Foundation and Contingency, form a fruitful framework for this purpose. We contribute an analytic framework to characterize the quality of mathematical knowledge observed in the Foundation and Contingency dimensions, developed using a purposive sampling from over 300 representations. These representations all featured geometry teaching. We showcase the framework with examples of "high" and "developing" Foundation and Contingency.Then, we compare our coding along these dimensions with performance on a measure of mathematical knowledge for teaching geometry. Finally, we describe the potential for generalizing this framework to other domains, such as algebra and mathematical modeling.     

A novel method for nonlinear fractional partial differential equations: Combination of DTM and generalized Taylor's formula

In this article, a novel numerical method is proposed for nonlinear partial differential equations with space- and time-fractional derivatives. This method is based on the two-dimensional differential transform method (DTM) and generalized Taylor's formula. The fractional derivatives are considered in the Caputo sense. Several illustrative examples are given to demonstrate the effectiveness of the present method. Results obtained using the scheme presented here agree well with the analytical solutions and the numerical results presented elsewhere. Results also show that the numerical scheme is very effective and convenient for solving nonlinear partial differential equations of fractional order.     

On an initial-value method for quickly solving Volterra integral equations: A review

A method of converting nonlinear Volterra equations to systems of ordinary differential equations is compared with a standard technique, themethod of moments, for linear Fredholm equations. The method amounts to constructing a Galerkin approximation when the kernel is either finitely decomposable or approximated by a certain Fourier sum. Numerical experiments from recent work by Bownds and Wood serve to compare several standard approximation methods as they apply to smooth kernels. It is shown that, if the original kernel decomposes exactly, then the method produces a numerical solution which is as accurate as the method used to solve the corresponding differential system. If the kernel requires an approximation, the error is greater, but in examples seems to be around 0.5% for a reasonably small number of approximating terms. In any case, the problem of excessive kernel evaluations is circumvented by the conversion to the system of ordinary differential equations.