What is the Mathematics in Mathematics Education?

In this paper I tackle the question What is the mathematics in mathematics education? By providing three different frames for the word mathematics. 1. Frame 1: Mathematics as an abstract body of knowledge/ideas, the organization of that into systems and structures, and a set of methods for reaching conclusions. 2. Frame 2: Mathematics as contextual, ever present, as a lens or language to make sense of the world. 3. Frame 3: Mathematics as a verb (not a noun), a human activity, part of one’s identity. After introducing the frames and examining their distinction and their overlap, I discuss their implication with respect to student-centered classroom, context, and culture

Preface

Bharath Sriraman noted in his Editorial for Vol. 10, nos. 1–2 that the first issue of The Mathematics Enthusiast (known then as The Montana Mathematics Enthusiast) published in April 2004 was “the result of four idealistic elementary school teachers believing in the mission of this journal and writing about their attempts to reconcile the mathematics content they were learning in a mathematics for elementary school teachers course with existing mathematics education research found in practitioners’ journals as well as standards imposed by institutions’ framing policy” (p. 2). Ten years later we return to a similar focus

Supporting novice mathematics teacher educators teaching elementary mathematics content courses for the first time

In order to be effectively prepared by a teacher education program, prospective elementary teachers (PTs) need to experience high quality mathematics instruction in their mathematics content courses. The instructors of these courses typically consist of individuals (mathematicians and mathematics educators) with ranging experiences, from tenured faculty members to first-year assistant professors or graduate students. This paper explores how to support novice mathematics teacher educators (MTEs) who are teaching elementary content coursework for PTs for the first time. We detail and describe how to implement three systems for supporting novice MTEs: working with a mentor, being provided with educative curriculum materials, and working in a collaborative teaching environment. We close by discussing specific challenges associated with these supports, and call for more institutions to share how they have successfully implemented systems to support novice MTEs

Networking Frameworks: a Method for Analyzing the Complexities of Classroom Cultures Focusing on Justifying

In this paper, we network five frameworks (cognitive demand, lesson cohesion, cognitive engagement, collective argumentation, and student contribution) for an analytic approach that allows us to present a more holistic picture of classrooms which engage students in justifying. We network these frameworks around the edges of the instructional triangle as a means to coordinate them to illustrate the observable relationships among teacher, students(s), and content. We illustrate the potential of integrating these frameworks via analysis of two lessons that, while sharing surface level similarities, are profoundly different when considering the complexities of a classroom focused on justifying. We found that this integrated comparison across all dimensions (rather than focusing on just one or two) was a useful way to compare lessons with respect to a classroom culture that is characterized by students engaging in justifying

Developing Mathematical Content Knowledge for Teaching Elementary School Mathematics

In this paper the authors present three design principles they use to develop preservice teachers\u27 mathematical content knowledge for teaching in their mathematics content and/or methods courses: (1) building on currently held conceptions, (2) modeling teaching for understanding, (3) focusing on connections between content knowledge and other types of knowledge. The authors share results of individual research projects and teaching approaches focusing on helping preservice elementary teachers develop such knowledge. Specific examples from different content areas (whole number, fractions, angle, and area) are discussed

Prospective Elementary Teacher Mathematics Content Knowledge: An Introduction

This Special Issue on the mathematical content knowledge of prospective elementary teachers (PTs) provides summaries of the extant peer-­‐reviewed research literature from 1978 to 2012 on PTs’ content knowledge across several mathematical topics, specifically whole number and operations, fractions, decimals, geometry and measurement, and algebra. Each topic-­‐specific summary of the literature is presented in a self-­‐contained paper, written by a subgroup of a larger Working Group that has collaborated across several years, resulting in this Special Issue sharing the final work. The authors hope this summative look at prospective teacher content knowledge will be of interest to the mathematics education community and will be a useful resource when considering future research as well as designing mathematics content courses for prospective elementary teachers

Prospective Elementary Mathematics Teacher Content Knowledge: What Do We Know, What Do We Not Know, and Where Do We Go?

In this Special Issue, the authors reviewed 112 research studies from 1978 to 2012 on prospective elementary teachers’ content knowledge in five content areas: whole numbers and operations, fractions, decimals, geometry and measurement, and algebra. Looking across these studies, this final paper identifies the trends and common themes in terms of the counts and types of studies and commonalities among findings. Analyses of the counts show that the number of articles published each year focusing on prospective teacher (PT) content knowledge is increasing. Most articles across the content areas show that PTs tend to rely on procedures rather than concepts. However, the focus of most articles is identifying PTs’ misconceptions rather than understanding PTs’ conceptions and the development thereof. Both the limitations of the reviews and the directions for future research studies are elaborated

New Working Group: Teaching Mathematics for Social Justice in the Context of University Mathematics Content and Methods Courses

There are three goals for this new working group: 1) To create a community of mathematics teacher educators (MTEs) who are (or are interested in) collaboratively teaching mathematics for social justice (TMfSJ) in their university content and/or methods classes. 2) To collaboratively select/develop/modify TMfSJ tasks and implement those in mathematics content/methods classes. 3) To research the implementation of TMfSJ tasks in content and methods classes

Prospective Elementary Mathematics Teacher Content Knowledge: What Do We Know, What Do We Not Know, and Where Do We Go?

The authors reviewed 112 research studies from 1978 to 2012 on prospective elementary teachers\u27 content knowledge in five content areas: whole numbers and operations, fractions, decimals, geometry and measurement, and algebra. Looking across these studies, this final paper identifies the trends and common themes in terms of the counts and types of studies and commonalities among findings. Analyses of the counts show that the number of articles published each year focusing on prospective teacher (PT) content knowledge is increasing. Most articles across the content areas show that PTs tend to rely on procedures rather than concepts. However, the focus of most articles is identifying PTs\u27 misconceptions rather than understanding PTs\u27 conceptions and the development thereof. Both the limitations of the reviews and the directions for future research studies are elaborated

Investigating Further Preservice Teachers’ Conceptions of Multidigit Whole Numbers: Refining a Framework

This study was designed to investigate preservice elementary school teachers’ (PSTs’) responses to written standard place-value-operation tasks (addition and subtraction). Previous research established that PSTs can often perform but not explain algorithms and provided a four-category framework for PSTs’ conceptions, two correct and two incorrect. Previous findings are replicated for PSTs toward the end of their college careers, and two conceptions are further analyzed to yield three categories of incorrect views of regrouped digits: (a) consistently as 1 value (all as 1 or all as 10), (b) consistently within but not across contexts (i.e., all as 10 in addition but all as 1 in subtraction), and (c) inconsistently (depending on the task)