Whither social theory?

What is the place of social theory in mathematics education research, and what is it for? This special issue of Educational Studies in Mathematics offers insights on what could be the role of some sociological theories in a field that has historically privileged learning theories coming from psychology and mathematics as the main theoretical frames informing research. Although during the last 10 years the term “socio-cultural” has become part of the accepted and widespread trends of mathematics education research when addressing learning, this issue gathers a collection of papers that depart from a “socio-cultural” approach to learning and rather deploy sociological theories in the analysis of mathematics education practices. In this commentary paper, we will point to what we see to be the contributions of these papers to the field. We will do so by highlighting issues that run through the six papers. We will try to synthetize what we think are the benchmarks of the social approach to mathematics education that they propose. We will also take a critical stance and indicate some possible extensions of the use of social theory that are not addressed in this special issue but nonetheless are worth being explored for a fuller understanding of the “social” in mathematics education.     

中国数学教育研究方法调查研究--基于《数学教育学报》（2011-2015）的统计分析

运用内容分析法对《数学教育学报》（2011—2015）刊载的研究性论文的研究方法从5个维度进行了系统的量化分析。研究发现：中国数学教育研究最常用的研究方法是理论思辨、统计调查和内容分析,叙事研究、田野调查等质性研究方法较为罕见;实证研究正逐渐成为中国数学教育研究的主流,然而实证研究以量化研究为主,质性研究和混合研究较少;使用数学方法研究数学教育已经成为人们的共识,但使用的数学方法主要是简单统计;描述型研究在中国数学教育研究中占有绝对的主导地位,关联型研究和干预型研究严重不足。最后,针对中国数学教育研究方法中存在的问题提出了若干建议。     

Mathematics Teacher Educators' Perceptions and Use of Cognitive Research

Instructors (N = 204) of elementary mathematics methods courses completed a survey assessing the extent to which they value cognitive research and incorporate it into their courses. Instructors' responses indicated that they view cognitive research to be fairly important for mathematics education, particularly studies of domain‐specific topics, and that they emphasize topics prominent in psychology studies of mathematical thinking in their courses. However, instructors reported seldom accessing this research through primary or secondary sources. A mediation analysis indicated that mathematics methods instructors' perception of the importance of the research predicts their incorporation of it in their courses, and that this relation is partially mediated by their accessing of it. Implications for psychologists who have an interest in education and recommendations for facilitating the use of cognitive research in teacher preparation are discussed.     

研究性数学、教育数学与数学教育

数学可分为研究性数学和教育性数学。研究性数学主要注重科学逻辑的序,而教育性数学则注重科学逻辑的序与认知心理的序之间的完美结合,后者即“教育数学”。教育数学是在研究性数学基础上的再创造、再提高,教育数学对数学教育改革具有深远的意义。     

Constructivism,mathematics and mathematics education

Learning theories such as behaviourism, Piagetian theories and cognitive psychology, have been dominant influences in education this century. This article discusses and supports the recent claim that Constructivism is an alternative paradigm, that has rich and significant consequences for mathematics education. In the United States there is a growing body of published research that claims to demonstrate the distinct nature of the implications of this view. There are, however, many critics who maintain that this is not the case, and that the research is within the current paradigm of cognitive psychology. The nature and tone of the dispute certainly at times appears to describe a paradigm shift in the Kuhnian model. In an attempt to analyse the meaning of Constructivism as a learning theory, and its implications for mathematics education, the use of the term by the intuitionist philosophers of mathematics is compared and contrasted. In particular, it is proposed that Constructivism in learning theory does not bring with it the same ontological commitment as the Intuitionists' use of the term, and that it is in fact a relativist thesis. Some of the potential consequences for the teaching of mathematics of a relativist view of mathematical knowledge are discussed here.     

A cognitive analysis of Cauchy’s conceptions of function,continuity, limit and infinitesimal,with implications for teaching the calculus

In this paper, we use theoretical frameworks from mathematics education and cognitive psychology to analyse Cauchy’s ideas of function, continuity, limit and infinitesimal expressed in his Cours D’Analyse. Our analysis focuses on the development of mathematical thinking from human perception and action into more sophisticated forms of reasoning and proof, offering different insights from those afforded by historical or mathematical analyses. It highlights the conceptual power of Cauchy’s vision and the fundamental change involved in passing from the dynamic variability of the calculus to the modern set-theoretic formulation of mathematical analysis. This offers a re-evaluation of the relationship between the natural geometry and algebra of elementary calculus that continues to be used in applied mathematics, and the formal set theory of mathematical analysis that develops in pure mathematics and evolves into the logical development of non-standard analysis using infinitesimal concepts. It suggests that educational theories developed to evaluate student learning are themselves based on the conceptions of the experts who formulate them. It encourages us to reflect on the principles that we use to analyse the developing mathematical thinking of students, and to make an effort to understand the rationale of differing theoretical viewpoints.     

Unraveling the gap between natural and rational numbers

The foundations for more advanced mathematics involve a good sense of rational numbers. However, research in cognitive psychology and mathematics education has repeatedly shown that children and even adults struggle with understanding different aspects of rational numbers. One frequently raised explanation for these difficulties relates to the natural number bias, i.e., the tendency to inappropriately apply natural number properties to rational number tasks. This contribution reviews the four main areas where systematic errors due to the natural number bias can be found, i.e., their size, operations, representations and density. Next, we discuss the major theoretical frameworks from which rational number understanding is currently investigated. Finally, an overview of the various papers is provided.     

`A historical angle’, a survey of recent literature on the use and value of history in geometrical education

In this article we give a survey of recent literature on the use and value of the history of geometry in mathematics education. It turns out that many authors have contributed to the debate why we should apply history. Their arguments can be divided into conceptual, (multi-) cultural and motivational ones. The number of authors who concentrated on the methodological question “How could we introduce history into mathematics lessons?” is considerably smaller. Those who want to make history an integral part of mathematics education have to bridge the gap between theoretical arguments and practical ideas, and, connected with this,between historians and teachers, and ultimately between mathematicians from the past and present day students. We divided the various surveyed publications, mainly journal articles, into categories according to a framework in order to clarify the discussion on the role of the history of geometry in education. Details about content and purpose of the articles are presented in two appendices, which make the articles more accessible both for further research and for practical teacher purposes.     

Black Mathematics Educators: Researching Toward Racial Emancipation of Black Students

This article focuses on the scholarship of Black mathematics education researchers whose work focuses on Black students in P–20 mathematics spaces. We conducted a metasynthesis literature review of empirical studies by Black mathematics education researchers. The authors utilized critical theories of race and racism to aid in the synthesis of the literature. The Black researchers we reviewed challenged the perspective that failure and limited persistence in Black students who are learning and participating in mathematics is normative. As a critical defense, these scholars offer research that problematizes test score data, race and racism, opportunities to learn mathematics, identity considerations, and other constructs that produce unequal effects in mathematics learning. We found that Black mathematics education researchers strategically disrupt the deficit narrative about Black students. Black scholars select theoretical frameworks that allow them to focus on race and how racism operates in mathematics education. We present this research to incite dialogue among all mathematics educators about improving the mathematical context for Black students.     

数学教育学学科建设三十年：回顾与反思

我国数学教育学学科建设始于对“中学数学教材教法”相关问题的探讨,经历了作为一门课程的“数学教育学”、作为系列课程的“数学教育学”及作为学科群的“数学教育学”和“数学教育学”的主题研究繁荣等几个发展阶段,初步形成了具有中国特色的数学教育学学科。数学教育学是一门涉及数学、教育学、哲学、心理学、文化学、传播学、教育技术学、思维科学等有关内容的新兴交叉学科,在数学教育学的学科建设过程中,通过理论与实践两方面研究,形成了数学教育专门研究人员与一线教师组成的研究团队,发展、完善了有中国特色的数学教育学科体系。今后,数学教育学学科建设仍需关注理论体系建构、研究团队建设、研究视角拓展等问题。     

Personal and Social-Contextual Factors in K–12 Academic Performance: An Integrative Perspective on Student Learning

Our extensive literature review in the fields of educational, social, and cognitive psychology has led us to identify about a dozen variables that demonstrate direct empirical links to academic achievement at the K–12 level. Those variables are grouped into four major categories: student engagement, learning strategies, school climate, and social-familial influences. We then categorize the first two variables as personal factors and the latter two as social-contextual factors. We document empirical findings that have shown particular relationships between the reviewed personal and social-contextual factors and academic achievement, mainly in the areas of reading and mathematics. Based on our conceptualization, we propose an integrated perspective that students’ personal factors in the domains of behavior, affect, attitude, and cognition as well as their social-contextual environment have to work in concert to produce optimal school performance. We conclude with a discussion on educational implications and future research to be addressed.     

关于数学学习心理研究的几个问题

数学学习心理研究要处理好一般学习理论与数学学习心理研究之间的关系 ;要在学习、引进、借鉴西方先进学习理论的同时 ,主动将我国有特色的数学学习心理研究成果推向国际 ,并在理论框架与研究方式上实现与国际的“可公度性”;应充分关注建构主义对数学学习研究所产生的特殊影响 ;为促使数学学习心理研究走向成熟 ,仍需采取多学科的视角 ,并更加重视对数学学习真实具体的心理活动的探索     

Educational psychology and the reform of teacher education

Benton and Hoyt (1989) perform a service for educational psychology by obtaining empirical evidence about educational psychologists' reactions to changes in teacher education proposed by the Holmes Group and the Carnegie Commission. Their paper should serve to stimulate debate about the proper role of educational psychology in teacher education. In this paper, we challenge the empirical findings reported by Benton and Hoyt and suggest reasons why the findings may misrepresent educational psychologists' interest in teacher education. We argue that some of the Holmes and Carnegie recommendations contradict the emerging consensus among cognitive psychologists about the contextually bound nature of knowledge. Benton and Hoyt conceptualize educational psychology according to the traditional middleperson viewpoint and make recommendations with respect to educational psychology that presume this conceptualization. We counter that the middleperson viewpoint is being made obsolete by changes in psychology, and describe an alternative conception of educational psychology. This alternative conception leads to a set of alternative recommendations about the appropriate role of educational psychology in the reform of teacher education.     

Simplest Shapes First! But Let's Use Cognitive Science to Reconceive and Specify What “Simple” Means

Children's informal and formal learning experiences with geometric shapes currently result in misconceptions that persist into adulthood. Here, we combine research from mathematics education as well as cognitive science pertaining to concepts, categories, and learning strategies to propose a more optimal progression that is better specified and justified than the current standards. To do so, we reframed what constitutes a “simple” shape from perceptual simplicity to simplicity of properties. Our Property-Based Shape Sequence uses property-based criteria of what makes shapes “simple” and progresses in a way that affords opportunities for learners to develop hierarchical conceptions of two-dimensional and three-dimensional shapes. Our goals are threefold: (1) recommend an optimal, mathematically-correct shape learning sequence, (2) correct misconceptions that adults and children harbor about shapes, and (3) encourage cross-disciplinary collaborations between mathematics education and psychology researchers to validate the proposed learning sequence.     

跨文化数学教育研究综述

从我国民族地区数学课程教材改革、双语教学、学生数学学习心理研究、跨文化数学教育理论研究与方法等方面综述了我国近20年来跨文化数学教育研究取得的成就。目前,对我国少数民族学生数学学习心理研究已达到较高水准,数学课堂的跨文化教育实验研究——中小学＂数学情境与提出问题＂教学实验颇具中国特色并在实践层面产生重要影响。进一步还需研究构建符合少数民族学生思维特点的数学课程体系,编写民族文字数学教材,加强少数民族双语数学教学研究与高素质双语教师队伍建设。     

A systematic review of the role of learning games in fostering mathematics education in K-12 settings

In the recent decade, a number of literature reviews were conducted to examine the effectiveness of learning games. However, prior reviews typically focused on providing a synopsis of the overall research trends and the games’ impact on cognitive and non-cognitive learning, without providing critical and contextual information of learning-gameplay integration or the game design features. The current review focuses on recent empirical studies that implement learning games for K-12 mathematics education during the time period of 2008–2021 via a systematic search of the databases. Forty-three papers were identified as the result of a three-stage data extraction process. We identified the trends of implementing learning games in mathematics education and the ways to designing and integrating math content in gameplay. We propose that more research is needed to examine the design and use of learning games for math learning in K-12 settings. Recommendations for future game-based learning design and research are presented.     

Educational Psychology: some thoughts on the transformation of learning in social practices and its consequences for a field of research

The relevance of psychological knowledge to education and learning has been recognized for long time. Educational psychology was thus, one of the first subdisciplines of applied psychology to emerge on the academic scene, and most scholars in the area — internationally as well as in the Nordic countries — have had a firm grounding in psychology. For a couple of decades or more during the second half of this century, educational psychology held a central position within education in most of the Nordic countries. Research within this field has been varied and vigorous. However, although psychological theories should be seen as important intellectual resources, it must be recognized that the relationship between a general discipline such as psychology and a particular area such as education is complex. Important problematics inherent in this relationship are illuminated and discussed in the contribution by Säljö on Sweden. It is argued, that recent attempts to develop a cultural psychology, explicitly recognizing the socio‐cultural nature of learning and other psychological processes, promise to pave the way for conceptions of human activities that are relevant to human concerns in general. The second article in this section, on Finland, contrasts with the first by focusing on educational research performed during the last few decades. This article describes trends in the content and methodologies of this research. The developmental orientation has been very strong in Finnish educational psychology during the whole of the post‐war period. Early research was highly influenced by a differential‐psychology approach, whereas later research has adopted social, motivational and cognitive frameworks. This article discuss the influential theoretical trends and developments.     

对我国教学教育学研究的反思

我国数学教育学的研究以分化－综合－分化的模式流变，经历了从理论移植到自我理论开发、从理论思辨到理论与实践整合的发展历程，取得了丰富的成果，然而，数学教育学的研究还存在许多盲点，诸如：学科理论建构的逻辑起点、数学学习心理、学科性质与功能定位、学科研究的价值取向、学科研究内容的拓展等，都需要作更深入的研究。     

Student characteristics and learning environment interactions in mathematics and physics education: A resource perspective

In two studies, one on secondary mathematics education, the other on secondary physics education, data were collected on students' cognitive achievement and characteristics of students and their learning environment. In this chapter the findings of the two studies are brought together in secondary analyses. The purpose is to explore whether relations between cognitive achievement and characteristics of students and their learning environment can be interpreted from a resource perspective. Although the results only partially support the theoretical model, planning of further research from a resource perspective may be fruitful.     

Affective and Cognitive Responses to Poetry in the University Classroom

Abstract

In universities, as in mainstream education more widely, cognitive approaches to poetry are often dominant. Far from being irrelevant to the serious study of literature, we argue that eliciting students’ affective responses to poetry can deepen their cognitive understanding and analytical skills. Drawing on recent research in psychology on the relationship between cognition and affect, we show that poetry has particular potential to make us aware of the crucial interrelation of our cognitive and affective processes; and that bringing those responses into balance can deepen our understanding of poetry. Building on recent educational studies of typical student (and teacher) anxieties and assumptions about working with poetry, and on our observations from our own initial, exploratory seminars, we explore some of the obstacles to rebalancing the cognitive and affective dimensions of poetry in higher education, and point to the potential value of such an approach if such obstacles are overcome.