Andreas J. Stylianides

A Type of Parental Involvement With an Isomorphic Effect on Urban Children’s Mathematics, Reading, Science, and Social Studies Achievement at Kindergarten Entry

Research showed that children’s school-entry academic skills are strong predictors of their later achievement, thereby highlighting the importance of children’s achievement at kindergarten entry. This article defines a particular type of parental involvement in children’s education and uses a representative sample of American urban kindergarteners to examine its effect on urban children’s mathematics, reading, science, and social studies achievement at kindergarten entry. The findings in this article are isomorphic in the different subject areas and show that children with more access to this particular type of parental involvement tend to have higher academic achievement than their peers.

Proof and argumentation in Mathematics education research

In the chapter on proof in the previous PME Research Handbook, Mariotti (2006) observed that there had seemed to be “a general consensus on the fact that the development of a sense of proof constitutes an important objective of mathematics education” and also “a general trend towards including the theme of proof in the curriculum” (p. 173).

Introducing Young Children to the Role of Assumptions in Proving

The notion of assumptions permeates school mathematics, but instruction tends to highlight this notion only in the advanced grades. In this article, I argue that it is important for even young children to develop a sense of the role of assumptions in proving, and I investigate what it might mean and look like for instruction to promote this goal. Toward this end, I study an episode from third grade that describes the first time that the students in the class were introduced in a deliberate and explicit way to the role of assumptions in proving. The central role of the mathematical task in the episode is identified, and features of mathematical tasks that can generate rich mathematical activity in the intersection of assumptions and proving are discussed. In addition, issues of the role of teachers in fostering productive interactions between students and mathematical tasks that have those features are considered.

Validation of Solutions of Construction Problems in Dynamic Geometry Environments

This paper discusses issues concerning the validation of solutions of construction problems in Dynamic Geometry Environments (DGEs) as compared to classic paper-and-pencil Euclidean geometry settings. We begin by comparing the validation criteria usually associated with solutions of construction problems in the two geometry worlds – the ‘drag test’ in DGEs and the use of only straightedge and compass in classic Euclidean geometry. We then demonstrate that the drag test criterion may permit constructions created using measurement tools to be considered valid; however, these constructions prove inconsistent with classical geometry. This inconsistency raises the question of whether dragging is an adequate test of validity, and the issue of measurement versus straightedge-and-compass. Without claiming that the inconsistency between what counts as valid solution of a construction problem in the two geometry worlds is necessarily problematic, we examine what would constitute the analogue of the straightedge-and-compass criterion in the domain of DGEs. Discovery of this analogue would enrich our understanding of DGEs with a mathematical idea that has been the distinguishing feature of Euclidean geometry since its genesis. To advance our goal, we introduce the compatibility criterion, a new but not necessarily superior criterion to the drag test criterion of validation of solutions of construction problems in DGEs. The discussion of the two criteria anatomizes the complexity characteristic of the relationship between DGEs and the paper-and-pencil Euclidean geometry environment, advances our understanding of the notion of geometrical constructions in DGEs, and raises the issue of validation practice maintaining the pace of ever-changing software.

The Need for Proof and Proving: Mathematical and Pedagogical Perspectives

This chapter first examines why mathematics educators need to teach proof, as reflected in the needs that propelled proof to develop historically. We analyse the interconnections between the functions of proof within the discipline of mathematics and the needs for proof. We then take a learner’s perspective and discuss learners’ difficulties in understanding and appreciating proof, as well as a number of intellectual needs that may drive learners to prove (for certitude, for causality, for quantification, for communication, and for structure and connection). We conclude by examining pedagogical issues involved in teachers’ attempts to foster necessity-based learning that motivates the need to prove, in particular the use of tasks and activities that elicit uncertainty, cognitive conflict and inquiry-based learning.

The Role of Instructional Engineering in Reducing the Uncertainties of Ambitious Teaching

Ambitious teaching is a form of teaching that requires a high level of teacher responsiveness to what students do as they actively engage with the subject matter. Thus, a teacher enacting ambitious teaching is often confronted with uncertainties about how to advance students’ learning while also building on students’ contributions. In this article we propose a framework that aims to deepen understanding about the role of instructional engineering in helping reduce the uncertainties of ambitious teaching, particularly with regard to the design and implementation of task sequences that target academically important but difficult-to-achieve learning goals. To illustrate the framework, we consider how instructional engineering helped reduce the uncertainties in enacting ambitious teaching to advance university and secondary students’ understanding of what counts as “proof” in mathematics.

The Cultural Dimension of Teachers’ Mathematical Knowledge

In this chapter, we make a case for considering culture in research on teachers’ mathematical knowledge, and we review Chapters 7-10 with a focus on the interplay between the cultural context and mathematical knowledge for/in teaching. Our review illuminates three different, but complementary, aspects of the cultural embedding of mathematical knowledge for/in teaching. The first aspect, which is represented by the chapters of Andrews and Pepin, situates mathematical knowledge in teaching in the context of different national educational systems. The second aspect, which is represented by the chapter of Adler and Davis, situates mathematical knowledge for teaching in the context of diverse teacher education programmes. The final aspect, which is represented by the chapter of Williams, situates mathematical knowledge for teaching in the culture of a ‘knowledge economy’. We conclude by considering implications of the four chapters for teacher education research and practice.

An examination of the roles of the teacher and students during a problem-based learning intervention: lessons learned from a study in a Taiwanese primary mathematics classroom

ABSTRACT The benefits of problem-based learning (PBL) to student learning have prompted researchers to investigate this pedagogical approach over the past few decades. However, little research has examined how PBL can be applied to mathematics learning and teaching, especially in countries like Taiwan, where the majority of teachers are accustomed to lecture methods and students are used to this style of teaching. This study examines the actions of a teacher and her class of 35 fifth-grade students (10–11-year-olds) as they tried to take on and respond to the demands of their new roles as “facilitator” and “constructors”, respectively, during a one-year PBL intervention in a Taiwanese mathematics classroom. Our findings provide insights into classroom participants’ role transition, from a customary role to a new role, when engaging with PBL. We identify an interrelationship between the teacher and student roles and discuss implications for the implementation of PBL at the primary education level.

Teachers’ Selection of Resources in an Era of Plenty: An Interview Study with Secondary Mathematics Teachers in England

The proliferation of instructional resources and the potential impact of teachers’ resource selection on students’ learning opportunities create a need for research on teachers’ selection of resources. We report results from an interview study with 36 secondary mathematics teachers in England, designed to find out (1) what instructional resources teachers choose for their everyday practice, thus beginning to document what we call a resource “pool of possibilities” to represent teachers’ resource options, and (2) the reasons for teachers’ choices, thus beginning to construct a taxonomy of what we call teachers’ “resource pre-disposition” to schematize teachers’ selection decisions. Our results show a large pool of possibilities and a complex taxonomy of teachers’ resource pre-disposition.

Pre-service Mathematics Teachers’ Knowledge and Beliefs

The notions of mathematics teachers’ knowledge and beliefs have been conceptualized in manifold ways in the literature. Notwithstanding these different conceptualizations, however, the point stands that mathematics teachers’ knowledge and beliefs are important factors to consider both in the study of classroom instruction in mathematics and in thinking about the goals, curriculum, or organization of the education of pre-service mathematics teachers. In this commentary we discuss how the four preceding chapters in this section of the book contribute to this body of research. Specifically, the four chapters contribute, collectively, to the broad issue of describing, elaborating, or conceptualizing kinds of mathematical knowledge and beliefs that are important for the education of pre-service elementary teachers. In doing so, they raise interesting challenges for the curriculum of teacher education and research in this area.