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Abstract
This paper considers the experiences of two sets of students who attended schools that taught mathematics in completely different
ways. One of the schools used a traditional, textbook approach, and the other used an open, project-based approach. The latter
approach produced equity between girls and boys whereas the textbook approach prompted many of the girls to under achieve.
This paper will consider the experiences of girls and boys who followed the project-based approach, reflect upon the sources
of equity within this approach and relate the differences between the two approaches to Gilligan’s notions of “separate” and
“connected” knowing.
... In other words, they are receiving a form of mathematics education that is inferior to what they could be receiving under reform. Boaler's (1997aBoaler's ( , 1997bBoaler's ( , 1997c research with high-performing students suggests that students in traditional mathematics classrooms neither retain for very long the information they have learned nor can they explain in real-world or conceptual terms what they are doing in mathematics. In her 3-year study, she found that students who learned in reform-oriented classrooms did as well on tests of basic skills and better on tests of conceptual skills than their peers who learned in traditional mathematics classrooms. ...
... This level of analysis would take into consideration such uncertainties as how students will respond to a given practice. For example, Boaler (1997a) suggested that students who tend to do well in traditional (basic) school mathematics tend to be the most opposed to their teachers changing their practices. So, understanding how teachers might anticipate or respond to student opposition would be included in a focus on teacher practice. ...
In this article, I address the need for a more clearly articulated research agenda around equity issues by proposing a working definition of equity and a focal point for research. More specifically, I assert that rather than pitting them against each other, we must coordinate (a) efforts to get marginalized students to master what currently counts as "dominant" mathematics with (b) efforts to develop a critical perspective among all students about knowledge and society in ways that ultimately facilitate (c) a positive relationship between mathematics, people, and equity on the planet. I make this argument partly by reviewing the literature on (school) contexts that engage marginalized students in mathematics. Then, I argue that the place that holds the most promise for addressing equity is a research agenda that emphasizes enabling the practice of teachers and that draws more heavily on design-based and action research, thereby redefining what the practice of mathematics means along the way. Specific research questions are offered.
... This sense of a collective in terms of the generation of the mathematical arguments and the distributed authority that comes along with it is in contrast to the pervasive perception of the field of mathematics. This perception of mathematics as a body of knowledge passed from the teacher to students is often a barrier to students experiencing a sense of mathematical agency (Boaler, 1997). ...
... As discussed in detail in Goodell (1998), several recent studies have shown the effectiveness of connected teaching (e.g. Boaler (1997) in two UK schools; Silver, Smith and Nelson (1995) in the QUASAR project in the US; Knapp, Shields and Turnbull (1995) in a study of 140 teachers in high-poverty US elementary classrooms; Rogers (1990;) in relation to college-level mathematics; and Tate (1995) in a study of one teacher's success in providing meaning to mathematics through students' exploration of a problem of immediate relevance to them (viz., the close proximity of their school to a number of liquor stores). ...
In this chapter we develop, from the literature on reform in mathematics education, a detailed operational definition of the essential features of an equitable mathematics classroom. We examine the documented reasons for gender differences in enrolment and performance in mathematics courses, discuss research into explanations for these differences, then critique initiatives which have been designed to encourage females to continue with the study of mathematics. Our analysis focuses on three main elements: students, classroom practices, and the curriculum. From the above analysis, we distil a definition of what, given contemporary evidence from practice and research, appears to be the ideal "Connected Equitable Mathematics Classroom" (CEMC). Following this definition, we explore, through data from a large-scale reform project in the USA, the possibilities and limits of implementation of the CEMC by mathematics teachers. We conclude the chapter with a short discussion of the implications of this work for reformers and equity advocates.
... In the process, several qualitative data analysis (QDA) techniques were employed. Although the use of grounded theory in mathematics education research is not novel (see, for example, Boaler, 1997a;Galbraith, 1995;Nardi, 1996;Shield, 1995), this usually follows the classical method of Glaser and Strauss (1967) which is a strongly inductive emergent approach which produces "a theoretical formulation or integrated set of conceptual hypotheses" to be tested or verified by followup quantitative research (Glaser, 1992, p. 16). With the reformulated method, the research product is constructed rather than emergent from the data, and verification of hypotheses occurs throughout the construction process, without the substantial use of quantitative techniques. ...
The current study was conducted with the goal of helping to address the research gap of improving preservice teachers' pedagogical knowledge about probability by investigating how preservice teachers explored concepts of probability during aleatoric music composition, as well as their contemplation process during follow-up reflections focused on using music-centred activities for developing elementary students’ probabilistic reasoning. Preservice teachers (n = 71) were guided to compose a piece of aleatoric music, with coin tossing utilized as the technique to generate the tonal patterns. A sum of 421 pieces of qualitative written data was collected. A content analysis was conducted to find the recurring themes and subthemes across the entire set of writing pieces. The overall findings indicated that most preservice teachers’ probabilistic reasoning was influenced (for better or worse) by the observed data obtained when composing their music, rather than a conceptual understanding of the Law of Large Numbers and the relationship between theoretical and experimental data. The conceptions – and misconceptions – held by the preservice teachers after aleatoric music composition were found to be related to their expressed teaching strategies.
This article describes a study that investigated pre-service teachers’ awareness of the interdisciplinary connections among mathematics, personal finance, and social justice issues. When pre-service teachers understand these connections, they can prepare their students to be responsible, participatory, and justice-oriented citizens in a global economy. Possessing and articulating this awareness are integral components of providing children and youth with the knowledge and skills to improve their life quality and bring about social change. The 68 participants in the study were pre-service teachers at two public universities in the United States, one in the Midwest and one in the Southeast. The participants at the Midwestern university were enrolled in general education classes, and the participants at the Southeastern university were enrolled in elementary, middle school, and high school mathematics methods classes in the Fall 2010 and Spring 2011 semesters. They completed surveys mid-semester, in which they were prompted to discuss mathematics literacy, financial literacy, and their social applications. The researchers found the majority of the pre-service teachers had narrow or intermediate conceptions of mathematics and financial literacy, discussing the need to understand mathematics and financial terminology, as well as their social connections. However, the participants were largely unable to articulate related social justice issues, seeing mathematics as an isolated discipline.
Schools classified as marginalized exhibit a complex cluster of factors, including parents who have low socio-economic status and low levels of education, and contexts where social networks are weak, there are few role models and in general there is a lack of opportunity. In the Australian context these schools tend to be in isolated geographical locations, and have large cohorts of students who either have English as their second language or are Indigenous. The problems that these schools face are universal. Teachers often struggle to work in these contexts, and students are at the greatest risk of not succeeding at school let alone mathematics. The focus of this book is to share the findings from a four-year longitudinal study Representations, Oral Language and Engagement in Mathematics (RoleM) that was situated in the most marginalized schools in Queensland, Australia. The participating students were in their first four years of school. The overall aim of the book is to share the journey of these teachers and students, and to draw out the dimensions that assisted these students to become successful learners of mathematics.
Learning environments are never identical. Research findings from the Learner’s Perspective Study (LPS) affirm just how “culturally-situated are the practices of classrooms around the world and the extent to which students are collaborators with the teacher, complicit in the development and enactment of patterns of participation that reflect individual, societal and cultural priorities and associated value systems” (Clarke, Emanuelsson, Jablonka, & Mok, 2006, p. 1). In this book we attend closely to this collaboration with our focus on the voice of the student.
The results of international large-scale assessments have revealed the emergence of gender disparities in attitudes to mathematics, with girls generally demonstrating lower levels of interest in and enjoyment of mathematics than boys. Given that attitudes to mathematics are key determinants of future STEM participation, collaborative cognitive-activation teaching strategies, which harmonise with the core tenets of feminist mathematical pedagogy, are proposed as a possible approach to improving girls’ relationships with mathematics. The results of a small-scale cross-national case study that incorporated this approach through a six week intervention are reported. The findings show a significant increase in girls’ enjoyment of mathematics but there was no significant change in boys’ attitudes. Potential implications for mathematics education policy and practice are elucidated.
This critical ethnographic study of an after-school mathematics club for elementary-aged Latina/o youth focuses on connecting critical, community, and mathematical knowledge in the context of authentic, community-based investigations. We present cases of two extended projects to highlight tensions and dilemmas that emerged, particularly tensions related to ensuring rich mathematics in the contexts of projects that were personally and socially meaningful to the students. Our analysis offers insights into critical mathematics education with elementary aged students, and has the potential to counter dominant deficit perspectives of Latina/o youth. Additionally, the findings of this study inform critical approaches to teaching mathematics in schools attended by marginalized students in order to reverse prevalent trends of our educational system failing these students.
This paper reports on data from seventy-eight 9-11 year old children attending four schools that had participated in a national numeracy initiative in New Zealand. Children's responses to a multi-digit addition problem, including alternative solution strategies, and their ideas about
the importance of getting the ‘right answer’ were analysed. Separate analysis of the responses of girls and boys revealed interesting gender differences. Girls in the study were more likely to give an alternative strategy for solving a multi-digit problem than boys, but more boys
than girls reported that effort put into a problem was more important than getting a correct solution.
In this article, I deconstruct the concept of understanding in mathematics education, examining how it is spoken into being and what work it does for primary school student teachers. I use poststructural analysis to unpack interviews with a student teacher, Jane, drawn from a larger longitudinal study. I show how she negotiates tensions between “romantic” discourses of understanding within mathematics education research and “functional” discourses of understanding within neoliberal mathematics education policy. A romantic discourse constructs understanding as an aspect of being resulting from the natural curiosity of the child. A functional discourse constructs understanding as performances within which the child is indistinguishable from automata. I argue that Jane takes on both functionality and romanticism, but they collide creating a disorderly discourse of understanding that reproduces inequity.
The analytical stance taken by equity researchers in education, the methodologies employed, and the interpretations that are drawn from data, all have an enormous impact upon the knowledge that is produced about sources of inequality. In the 1970's and 1980's, a great deal of interest was given to the issue of women's and girls' underachievement in mathematics. This prompted numerous different research projects that investigated the extent and nature of the differences between girls' and boys' achievement and offered reasons why such disparities occurred. This work contributed towards a discourse on gender and mathematics that flowed through the media channels and into schools, homes and the workplace. In this article I will consider some of the scholarship on gender and mathematics, critically examining the findings that were produced and the influence they had. In the process, I will propose a fundamental tension in research on equity, as scholars walk a fine and precarious line between lack of concern on the one hand, and essentialism on the other. I will argue in this article that negotiating that tension may be the most critical role for equity researchers as we move into the future. Abstract
This study explores opportunities to learn mathematics problem solving for Latina/o students in 3 kindergarten classrooms in the southwest. Mixed methods were used to examine teaching practices that engaged Latina/o students in problem solving and supported their learning. Findings indicate that although students in all 3 classrooms showed growth on pre-/postassessment measures, students in Ms. Arenas's classroom outperformed students in the 2 other classrooms. More time spent on problem solving; exposure to a broader range of problems involving multiplication, division, and multiple steps; and consistent access to students' native language, Spanish, distinguished Ms. Arenas's class.
This chapter acknowledges that access to mathematics plays a role in potential earnings and that this access is closed to
English language learners (ELLs) and to students living in poverty. We provided evidence that poverty, academic language development,
and discourse diversities are intertwined and cannot be easily separated when it comes to mathematical achievement. In fact,
evidence is provided to suggest that English-speaking students living in poverty may be more severely impacted by discourse
diversities and academic language development than ELLs. Even the discourse practices espoused by reform curricula can have
a negative influence on the achievement of students in poverty. The authors suggest a pedagogical approach, such as Cognitively
Guided Instruction, to aid both ELLs and students in poverty because of its focus on identifying and using what students already
know to plan instruction and its focus on developing mathematical language through negotiation and discussion of students’
thinking.
KeywordsAcademic discourse-Academic language development-Achievement gap-Cognitively guided instruction (CGI)-English language learners (ELLs)-Equity-Socio-economic status (SES)-Poverty and mathematics
In the course of a program of research into how students respond to typical word problems, it quickly became clear that patterns
in their responses showing an apparent willingness to suspend sense making could not be explained in cognitive terms alone.
Rather, it is necessary to consider the culture of the mathematics classroom and, in particular, the set of beliefs underlying
the “Word Problem Game” that, largely implicitly, governs classroom practice. Findings from systematic research studies with
students and teachers-in-training are reported that cohere with others in the literature and anecdotal evidence to elucidate
the nature of practices surrounding word problems. Further, initial teaching experiments are reported which suggest that it
is possible to change beliefs about word problems. However, these beliefs cannot be considered in isolation. Rather, they
form part of more general beliefs about the nature of mathematics and its relation to the real world. Moreover, beliefs about
word problems shaped by classroom culture are embedded within the nested and interacting contexts of school culture, the educational
system, and society in general. We argue for the reconceptualization of word problems as a vehicle for promoting early awareness
of the relationship between mathematics and aspects of reality that it models. This proposal reflects our own beliefs about
the nature of mathematics and the proper goals of mathematics education.
This paper presents case study data from two schools which taught mathematics in completely different ways. One of the schools followed a traditional, procedural approach which caused many girls to underachieve. The girls in this school related their underachievement to the closed nature of their working environment. The second school followed an open, project based approach which appealed to more girls than boys, yet produced parity of attainment. Interviews with disaffected girls and boys from both schools are used to inform perspectives on motivation and learning styles. It is argued that the performance of girls is generally increasing, relative to boys, because school approaches are becoming more equitable and girls are being allowed to achieve at levels that are consonant with their interest and motivation. Also, that it is wrong to blame schools, teachers or girls for the low attainment of boys, whose problems need to be located within a broader social perspective.
ABSTRACT In this paper I consider what appears to be an emerging feminist perspective within mathematics education that suggests that theories such as 'attribution theory' lay too much 'blame' upon girls and women for their underachievement in mathematics and not enough blame upon the wider school system. I attempt to extend this theoretical position further through the use of case study data from two schools. Interviews with underachieving girls are used to show the way in which girls link their underachievement, not to themselves, but to the type of mathematics that is widely taught in the UK, which they believe denies them access to understanding. An alternative model of mathematics teaching is described that is open and project-based and that may be able to eradicate underachievement and disaffection amongst girls.
The teaching of qualitative analysis in the social sciences is rarely undertaken in a structured way. This handbook is designed to remedy that and to present students and researchers with a systematic method for interpreting qualitative data', whether derived from interviews, field notes, or documentary materials. The special emphasis of the book is on how to develop theory through qualitative analysis. The reader is provided with the tools for doing qualitative analysis, such as codes, memos, memo sequences, theoretical sampling and comparative analysis, and diagrams, all of which are abundantly illustrated by actual examples drawn from the author's own varied qualitative research and research consultations, as well as from his research seminars. Many of the procedural discussions are concluded with rules of thumb that can usefully guide the researchers' analytic operations. The difficulties that beginners encounter when doing qualitative analysis and the kinds of persistent questions they raise are also discussed, as is the problem of how to integrate analyses. In addition, there is a chapter on the teaching of qualitative analysis and the giving of useful advice during research consultations, and there is a discussion of the preparation of material for publication. The book has been written not only for sociologists but for all researchers in the social sciences and in such fields as education, public health, nursing, and administration who employ qualitative methods in their work.
This paper reports on 3-year case studies of 2 schools with alternative mathematical teaching approaches. One school used a traditional, textbook approach; the other used open-ended activities at all times. Using various forms of case study data, including observations, questionnaires, interviews, and quantitative assessments, I will show the ways in which the 2 approaches encouraged different forms of knowledge. Students who followed a traditional approach developed a procedural knowledge that was of limited use to them in unfamiliar situations. Students who learned mathematics in an open, project-based environment developed a conceptual understanding that provided them with advantages in a range of assessments and situations. The project students had been "apprenticed" into a system of thinking and using mathematics that helped them in both school and nonschool settings.
In this paper I present some of the results of a three-year case study of a mathematics department in a UK school that taught in setted' groups. Interviews, observations, questionnaires, and assessment data are used to show the way in which high ability students, particularly girls, underachieved and became disaffected as a result of being in the top set'. Various qualitative and quantitative results of the case study are used to show the way in which top sets can diminish, rather than enhance, achievement for high ability students. It is also suggested that the gender inequalities in mathematics achievement that continue to prevail among the top five per cent of students may partly be caused by features of top set' learning.
Gender differences in mathematics learning outcomes persist and several explanatory models incorporate affective variables. Current understandings of how children learn mathematics seem inconsistent with traditional mathematics instruction. The literature reveals that little is known about the relationship between classroom factors and students' beliefs about themselves as learners of mathematics. The study reported here explored this relationship at two levels: a large scale survey and in-depth studies of two classrooms. Classroom factors which might influence beliefs were identified, and provided partial explanations for some of the gender differences noted. There were clear implications for the teaching of mathematics. © 1996 Mathematics Education Research Group of Australasia Inc.
A year-long study in 40 eighth grade mathematics classrooms compared the effect on student achievement of teaching with meaning with algorithmic-practice teaching. In general, teaching with meaning was found to increase student achievement. However, differences in response to the methods was seen in the top, middle, and lower thirds of the students. (Author/MDH)
Reports on an analysis of standardized test score data covering 9 yrs (1982–1991) to study the effects of a major long-term intervention (INT) in the teaching of mathematics in a K–8 elementary school in New Jersey. The concern was whether a more thoughtful approach to elementary mathematics would have the undesirable side-effect of producing a decline in scores on standardized tests. Student scores were derived from results of student performance on the math component of the Iowa Tests of Basic Skills (ITBS). Student national percentile rank (NPR) scores prior to 1986 were adjusted to be in line with the present ITBS 1985–1986 norms. The analysis examines the distribution of ITBS NPR scores for Math Concepts Score, Math Problem Solving, and Math Computation in Grades 4–8 and separately for each grade level year-to-year variability in mean NPR scores for pre-INT, INT, and post-INT periods. The effect on test scores was generally positive. (PsycINFO Database Record (c) 2012 APA, all rights reserved)
There is, now, an extensive critical literature on gender and the nature of science three aspects of which, philosophy, pedagogy and epistemology, seem to be pertinent to a discussion of gender and mathematics. Although untangling the inter-relationships between these three is no simple matter, they make effective starting points in order to ask similar questions of mathematics to those asked by our colleagues in science. In the process of asking such questions, a major difference between the empirical approach of the sciences, and the analytic nature of mathematics, is exposed and leads towards the definition of a new epistemological position in mathematics.
Various research and writings are used to frame a major issue in U.S. schooling: Our schools apparently do little to change
power relations between the students of our elite families and those of the minorities and of low socioeconomic status (SES).
The authors use situated cognition as an instructional strategy; curriculum designed for depth—not coverage; and student responsibility
and choice as a theoretic rationale for a collaborative curriculum design in which power is shared by teachers and students
to provide equal educational opportunity. After setting the organizational context at a School Without Walls (SWW), a public,
urban high school in Rochester, New York, the authors describe how such a collaborative curriculum works. Data both from SWW
student outcomes and from a national study on school restructuring are employed to support the efficacy of this collaborative
curriculum in providing a schoolwide, student-centered context and equal opportunity for students.
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