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.     

Guiding Teachers in the Use of a Standards‐Based Mathematics Curriculum: Teacher Perceptions and Subsequent Instructional Practices After an Intensive Professional Development Program

Reforms in mathematics education call for K‐12 teachers to employ standards‐based pedagogies, which embody the National Council for Teachers of Mathematics' principles and standards. In order to effectively support teachers' implementation of standards‐based curricula, professional development must be provided that meets teachers' needs. The professional development program in this study focused on the implementation of a standards‐based mathematics curriculum entitled Investigations in Number, Data, and Space (Investigations). This study uses Guskey's framework as a guide to examining teachers' perceptions of the impact of the professional development that they received; their perceptions of mathematics teaching and learning; and how elements of the professional development translated into practice. Twenty‐two participants were randomly selected from the 53 professional development participants to be interviewed and observed during their mathematics teaching. Using a constant comparison method, the data sources in this study highlighted themes surrounding teachers' experiences with professional development and the implementation of the curricula. The analysis of the data sources in this study highlighted themes surrounding teachers' experiences with professional development: teachers as learners, teachers as self‐evaluators, shifting paradigms, enactment of professional development content into practice, and the influence of the state standardized mathematics test. The results of this study have several implications for future professional development and also highlight some of the more general issues that teachers face when attempting to enact new knowledge and skills into their practice.     

An accessible,justifiable and transferable pedagogical approach for the differential equations of simple harmonic motion

Recently, Gauthier introduced a method to construct solutions to the equations of motion associated with oscillating systems into the mathematics education research literature. In particular, Gauthier's approach involved certain manipulations of the differential equations; and drew on the theory of complex variables.

Motivated by the work of Gauthier, we construct an alternative pedagogical approach for the learning and teaching of solution methods to these equations. The innovation lies in drawing on factorization techniques of differential equations and harmonizing them with Gauthier's approach of the theory of complex variables. When blended together to form a new approach, the significance lies in its accessibility, justifiability and transferrability to other problems.

We pedagogically ground our approach in the educational development theory of Piaget, with the results informing the learning and teaching of solution methods to differential equations for lecturers, teachers and learners within universities, colleges, polytechnics and schools around the world.     

Research into teacher knowledge: a stimulus for development in mathematics teacher education practice

In this paper, we document some developments in teacher education practice at one university, brought about by reflection on research into mathematics teacher knowledge. The authors are three members of the Cambridge-based research team who developed the Knowledge Quartet (KQ), a theory of mathematics teacher knowledge, with a focus on classroom situations in which this knowledge is applied. At the same time as being researchers, the authors were elementary mathematics teacher education instructors. They found that the KQ research brought about new awareness of the importance of some components of mathematics didactics, as well as providing new tools for undertaking some aspects of their teacher educator role. The paper explores some of these awarenesses and tools in detail.     

Applying NAEP to Improve Mathematics Content and Methods Courses for Preservice Elementary and Middle School Teachers

This study explored how mathematics content and methods courses for preservice elementary and middle school teachers could be improved through the integration of a set of instructional materials based on the National Assessment of Educational Progress (NAEP). A set of eight instructional modules was developed and tested. The study involved 7 university instructors and 542 preservice teachers (PSTs) from three different universities. A quasi‐experimental nonequivalent groups design was used for this study in which the following data sources were collected and analyzed. Three versions of a Learning Mathematics for Teaching test were given to assess PSTs‘ mathematical content knowledge for teaching: (a) Elementary Number Concepts and Operations—Content Knowledge; (b) Elementary Geometry—Content Knowledge; and (c) Middle School Number Concepts and Operations—Content Knowledge. In addition, the Mathematics Teacher Efficacy Beliefs Instrument was given to assess PSTs’ teacher efficacy beliefs. Test results were analyzed using paired samples t‐tests. Findings suggest that use of instructional materials, based on NAEP, with PSTs results in increases in their mathematical content knowledge for teaching and in their teaching efficacy beliefs.     

The Effects on Students' Cognitive Achievement When Using the Cooperative Learning Method in Earth Science Classrooms

This study investigated the effects of cooperative learning instruction versus traditional teaching methods on students' earth science achievement in secondary schools. A total of 770 ninth-grade students enrolled in 20 sections of a required earth science course participated in this nonequivalent control group quasi-experiment. The control groups (n= 10) received a traditional approach, while the experimental groups (n= 10) used cooperative strategies. Study results include (a) no significant differences were found between the experimental groups and the control groups when overall achievement (F= 0.13, p > .05), knowledge-level (F= 0.12, p > .05), and comprehension-level (F= 0.34, p > .05) test items were considered; and (b) students who worked cooperatively performed significantly better than students who worked alone on the application-level test items (F= 4.63, p < .05). These findings suggest that cooperative-learning strategies favor students' earth science performance at higher but not lower levels of cognitive domains in the secondary schools.     

Computability of Solutions of the Korteweg‐de Vries Equation

In this paper we study computability of the solutions of the Korteweg‐de Vries (KdV) equation u_t + uu_x + u_xxx = 0. This is one of the open problems posted by Pour‐El and Richards [25]. Based on Bourgain's new approach to the initial value problem for the KdV equation in the periodic case, we show that the periodic solution u (x, t) of the KdV equation is computable if the initial data is computable.     

A note on localized directional weak lower semi-continuity

In minimization problems for functionals f : M → R, M ? E a subset of some infinite dimensional Banach space E, we typically have to rely on weak (sequential) lower semi-continuity of f on the whole space E even if M is a proper subset of E. The main reason for this lack of 'localized' weak lower semi-continuity seems to be that it is not known how to get and/or to characterize weak sequential lower semi-continuity on a subset M without knowing it on the whole space. As a first step to overcome this difficulty we propose the concept of 'localized directional weak sequential lower semi-continuity' and offer a way to implement it, namely in terms of conditions on the Gateaux derivative f′ of f (weak K-monotonicity). This allows to formulate a criterium and new sufficient conditions for the existence of a minimizer.

We conclude with a discussion of applications to the variational approach to the solution of (systems of) nonlinear partial differential equations where we focus on the case of integral functionals of vector fields for which the integrand is not assumed to be quasi-convex.     

Markovian representation of a bilinear time series model and maximum likelihood estimation of the parameters

A bilinear time series (BLTS) model is expressed in the form of Akaike's Markovian representation in order to use the Kalman recursive estimation approach. It is shown that Akaike's Markovian representation of autoregressive moving average models of orderp and q (ARMA(p,q)) and that of the bilinear model are equivalent. This equivalence facilitates the maximum likelihood estimation of the parameters involved in the bilinear model, which otherwise is an unwieldy problem. The present approach can easily be extended to take into account missing observations     

The Role of Teachers' Beliefs and Knowledge in the Adoption of a Reform‐Based Curriculum

Science as inquiry is a key content standard in the National Science Education Standards; however, few secondary science teachers successfully and consistently implement inquiry‐based instruction in their classrooms. This research examines the role of reform‐based curricular materials in influencing the classroom practices of 12 high school chemistry teachers and investigates the role of the teachers' knowledge and beliefs in their implementation of the reform‐based chemistry curriculum. Qualitative and quantitative data were collected in the form of beliefs interviews and classroom observations. The teachers' classroom practices were measured prior to and during the field test of the reform‐based chemistry curriculum. Analysis of the data revealed that teachers' classroom practice became more reform‐based in the presence of the new curriculum; however, the degree of change is related to the teachers' beliefs about teaching and learning, depth of chemistry knowledge, and years of teaching experience. Experienced, out‐of‐discipline teachers with transitional or student‐centered teaching beliefs demonstrated the most growth in reform‐based teaching practices. This study reinforces the need for reform‐based curriculum to assist teachers in implementing the intent of the National Science Education Standards.     

The beliefs of ‘Tomorrow's Teachers’ about mathematics: precipitating change in beliefs as a result of participation in an Initial Teacher Education programme

Mathematics education research has given increasing attention to the role of affective factors in the learning process. While 'affect' is used to refer to a variety of aspects including feelings, emotions, beliefs, attitudes and conceptions, this paper focuses on 'beliefs' of elementary pre-service teachers. In particular, the study evaluates the effect of participation in a reform-based elementary pre-service teacher education (referred to as Initial Teacher Education (ITE)) programme on participants' 'beliefs about the nature of mathematics'. This was completed using two (sub)scales of the Aiken's Revised Mathematics Scale measuring Enjoyment of Mathematics (E) and belief in the Value of Mathematics (V). Both scales were administered before and after participants completed the mathematics education programme, which consisted of 5 compulsory and consecutive modules. This study reveals that entry-level pre-service teachers report generally positive beliefs about the value of and enjoyment in doing mathematics. The findings challenge previous research, which report the tendency of teachers' beliefs to be resistant to change while in teacher education and suggest that it is possible for ITE mathematics education programmes to stimulate improvement in beliefs and attitudes among participants. Particular programme features are identified as instrumental in this positive change in beliefs about mathematics.     

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.     

The Development of an Instrument to Measure Preservice Teachers' Attitudes about Discourse in the Mathematics Classroom

This study explored the themes that comprise preservice teachers' attitudes regarding discourse in the K‐12 mathematics classroom. The initial development of the theory underlying preservice teachers' attitudes regarding mathematical discourse is documented through the development of a 5‐point Likert instrument. Analysis of the Preservice Teachers' Attitudes About Discourse in the Mathematics Classroom (PADM) Instrument (N – 277) resulted in three reliable factors: Promoting Mathematical Reasoning (α₁= .85), Examining Complex Mathematical Concepts (α₂= .81), and Valuing Students' Mathematical Ideas (α₃= .85). These results suggest a framework that mathematics educators can employ to address preservice teachers' attitudes regarding discourse in an effort to support their implementation of reform‐based discourse in the teaching of mathematics in their future classrooms.     

A Variational Principle for the Ackerberg-O'Malley Resonance Problem

A new variational principle is proposed for determining the asymptotic expansion of the solution of the Ackerberg-O'Malley resonance problem [Stud. Appl. Math. 49:277–295 (1970)] to any order in ε. The method used yields new higher-order results not permitted by the technique of Grasman and Matkowsky [SIAM J. Appl. Math. 32:588–597 (1977)]. Explicit results using the method are reported to O(ε) and confirmed with asymptotic expansions of the exact solution; the O(1) results agree with those reported in the literature. In the case where the coefficient functions are analytic, an exact solution is presented. It is not difficult to perform the higher-order calculations using the proposed variational approach, in contrast to the current methods in use.     

Cotorsion pair extensions

Assume that S is an almost excellent extension of R. Using functors HomR（S,-） and -×R S, we establish some connections between classes of modules lR and lS, cotorsion pairs （AR, BR） and （AS, BS）. If lS is a T-extension or （and） H-extension of lR, we show that lS is a （resp., monomorphic, epimorphic, special） preenveloping class if and only if so is lR. If （AS, BS） is a TH- extension of （AR, BR）, we obtain that （AS, BS） is complete （resp., of finite type, of cofinite type, hereditary, perfect, n-tilting） if and only if so is （AR, BR）.     

Development and Initial Validation of the Beliefs About Reformed Science Teaching and Learning (BARSTL) Questionnaire

This article describes the development and the initial validation of an instrument that can be used to assess teachers' beliefs about science teaching and learning. The instrument, which is called the Beliefs About Reformed Science Teaching and Learning (BARSTL) questionnaire, draws on the current national science education reform efforts in order to define a traditional‐reformed teaching and learning belief continuum that can be used to map teachers' beliefs. The reliability and validity of the instrument were examined using a multiple perspective approach. The psychometric properties of the BARSTL suggest that it is a valid and reliable instrument for measuring prospective elementary teachers' beliefs about science education.     

The Science Teaching Self‐Efficacy of Prospective Elementary Education Majors Enrolled in Introductory Geology Lab Sections

This study examined prospective elementary education majors' science teaching self‐efficacy while they were enrolled in an introductory geology lab course for elementary education majors. The Science Teaching Efficacy Belief Instrument Form B (STEBI‐B) was administered during the first and last lab class sessions. Additionally, students were asked an open‐ended question to describe their experience in the education majors' geology lab. The results of the STEBI‐B were analyzed using paired t‐tests to determine whether the students changed their personal science teaching efficacy (PSTE) and science teaching outcome expectancy (STOE). Results of this study indicate a significant increase in PSTE. No significant differences were found in STOE. This study suggests that science content courses designed for education majors may lead to a positive change in science teaching self‐efficacy and has implications for teacher educators in preparing science content courses for their teacher preparation program.