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The Gold and the Dross

Althusser for Educators

Series:

David I. Backer

In the last decade, there has been an international resurgence of interest in the philosophy of Louis Althusser. New essays, journalism, collections, secondary literature, and even manuscripts by Althusser himself are emerging, speaking in fresh ways to audiences of theorists and activists. Althusser is especially important in educational thought, as he famously claimed that school is the most impactful ideological state apparatus in modern society. This insight inspired a generation of educational researchers, but Althusser’s philosophy—unique in a number of ways, one of which was its emphasis on education—largely lost popularity.

Despite this resurgence of interest, and while Althusser’s philosophy is important for educators and activists to know about, it remains difficult to understand. The Gold and the Dross: Althusser for Educators, with succinct prose and a creative organization, introduces readers to Althusser’s thinking. Intended for those who have never encountered Althusser’s theory before, and even those who are new to philosophy and critical theory in general, the book elaborates the basic tenets of Althusser’s philosophy using examples and personal stories juxtaposed with selected passages of Althusser’s writing. Starting with a beginner’s guide to interpellation and Althusser’s concept of ideology, the book continues by elaborating the epistemology and ontology Althusser produced, and concludes with his concepts of society and science. The Gold and the Dross makes Althusser’s philosophy more available to contemporary audiences of educators and activists.

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Edited by Dianne Siemon, Tasos Barkatsas and Rebecca Seah

The relationship between research and practice has long been an area of interest for researchers, policy makers, and practitioners alike. One obvious arena where mathematics education research can contribute to practice is the design and implementation of school mathematics curricula. This observation holds whether we are talking about curriculum as a set of broad, measurable competencies (i.e., standards) or as a comprehensive set of resources for teaching and learning mathematics. Impacting practice in this way requires fine-grained research that is focused on individual student learning trajectories and intimate analyses of classroom pedagogical practices as well as large-scale research that explores how student populations typically engage with the big ideas of mathematics over time. Both types of research provide an empirical basis for identifying what aspects of mathematics are important and how they develop over time.

This book has its origins in independent but parallel work in Australia and the United States over the last 10 to 15 years. It was prompted by a research seminar at the 2017 PME Conference in Singapore that brought the contributors to this volume together to consider the development and use of evidence-based learning progressions/trajectories in mathematics education, their basis in theory, their focus and scale, and the methods used to identify and validate them. In this volume they elaborate on their work to consider what is meant by learning progressions/trajectories and explore a range of issues associated with their development, implementation, evaluation, and on-going review. Implications for curriculum design and future research in this field are also considered.

Contributors are: Michael Askew, Tasos Barkatsas, Michael Belcher, Rosemary Callingham, Doug Clements, Jere Confrey, Lorraine Day, Margaret Hennessey, Marj Horne, Alan Maloney, William McGowan, Greg Oates, Claudia Orellana, Julie Sarama, Rebecca Seah, Meetal Shah, Dianne Siemon, Max Stephens, Ron Tzur, and Jane Watson.

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Ron Tzur

Abstract

In this chapter I introduce a distinction between two research types, marker studies and transition studies, that I find useful in developing and categorizing hypothetical learning trajectories (hlts). I stress the latter type as one closely related with the third element in Simon’s (1995) definition of hlt. To further depict the linkage between hlt and transition studies, I discuss the starting points and development of hlts, including a model of learning as cognitive change – reflection on activity-effect relationship. I follow those sections with a description of four ways in which I find hlts that specify conceptual transitions to be useful.

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Dianne Siemon, Tasos Barkatsas and Rebecca Seah

Series:

Tasos Barkatsas and Claudia Orellana

Abstract

The purpose of this study was to explore the factorial structure of motivation and perception items from a student survey utilised as part of the Reframing Mathematical Futures II (rmfii) Project. Data were collected in 2016 from students in Years 7 to 10 from various different states and territories across Australia. An exploratory factor analysis identified four factors which were consistent with the studies the items were adapted from: Intrinsic and Cognitive Value of Mathematics, Instrumental Value of Mathematics, Mathematics Effort, and Social Impact of School Mathematics. A multivariate analysis of variance (manova) also revealed that there were statistically significant differences between Year Level for some of these factors. The results from the study have confirmed that the survey items continue to be valid and reliable in the mathematics context. The findings also highlight the need for further investigations to examine how students’ motivations and perceptions of mathematics develop and differ across the different states and territories in Australia.

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Dianne Siemon

Abstract

Identifying and building on what students know in relation to important mathematics is widely regarded as essential to success in school mathematics. However, determining what is important and identifying what students actually understand in relation to what is deemed to be important are by no means uncontested or straightforward endeavours. In recent years attention has turned to the development of evidenced-based learning trajectories (or progressions) as a means of identifying what mathematics is important and how it is understood over time. But for this information to be useful to practitioners, it needs to be accompanied by accurate forms of assessment that locate where learners are in their learning journey and evidenced-based advice about where to go to next. This chapter traces the origins of the Scaffolding Numeracy in the Middle Years (snmy) research project that used rich assessment tasks and Rasch analysis techniques to develop an evidence-based framework to support the teaching and learning of multiplicative thinking in Years (Grades) 4 to 9.

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Rebecca Seah and Marj Horne

Abstract

This chapter reports the development of a learning progression for geometric reasoning. Geometric reasoning is the ability to critically analyse axiomatic properties, formulate logical arguments, identify new relationships and prove propositions. All types of geometric concepts develop over time, becoming increasingly integrated and synthesized as individuals learn to visualise beyond the physical images, and participate in ‘taken-as-shared’ mathematical discourse to describe, analyse, infer and deduce geometric relationships, leading to engaging in formal proof. By analysing data collected through a series of assessment items we have designed and verified an eight-zone learning progression. Examples will be provided to show how the assessment items, activities and teaching advice can be used to help develop and nurture geometric reasoning.

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Julie Sarama and Douglas H. Clements

Abstract

Approaches to standards, curriculum development, and teaching are diverse. However, increasingly learning trajectories are being used as a basis for each of these. In this chapter, we present our own definition and use of the construct, positing that learning trajectories can serve as one effective foundation for scientifically-validated mathematics standards, curricula, and pedagogy. We discuss the current state of learning trajectories in early childhood mathematics and present the development and expansion of the concept in our work, which began with studies of individuals and progressed to large scale-up implementations and evaluations. Throughout this work, learning trajectories served as the structure that maintained coherence.

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Lorraine Day, Marj Horne and Max Stephens

Abstract

Mathematical reasoning is an important component of any mathematics curriculum. This chapter focuses on algebraic reasoning in the middle years of schooling, often termed as a predictor of later success in school mathematics. It describes the design research process used in the Reframing Mathematical Futures (RMFII) project to develop an evidence-based Learning Progression for Algebraic Reasoning framed by three big ideas: Equivalence, Pattern and Function, and Generalisation. The Learning Progression for Algebraic Reasoning may be used to identify where students are in their learning journey and where they need to go next. Once the Learning Progression for Algebraic Reasoning was developed it was used to design Teaching Advice to help teachers to provide appropriate activities and challenges to support student learning. Two implied recommendations of this chapter are that algebraic reasoning based on these three key ideas should precede symbol use; and, that algebraic reasoning as described here needs to be cultivated in the primary school years.