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Series:

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

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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.

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Dianne Siemon and Rosemary Callingham

Abstract This chapter traces the origins of the Reframing Mathematical Futures II (rmfii) research project before describing how rich assessment tasks and Rasch analysis techniques were used to develop an evidence-based resource to support the teaching and learning of algebraic, statistical and geometric reasoning in Years (i.e., Grades) 7 to 10. Hypothesised learning progressions were developed for each of the three target domains based on prior research. Potential assessment tasks, together with detailed scoring rubrics, that addressed the learning progressions were developed and trialed. A detailed examination of the Rasch analysis output indicated modifications that were needed and the modified questions were retrialed. Finally, the resulting variable was interpreted and segmented into Zones that were used as the basis for developing teaching advice. In describing this process, we comment on the application of learning trajectories more generally and suggest further avenues for research in this field.

Series:

Tasos Barkatsas and Claudia Orellana

Abstract

The aims of this study were to explore the factorial structure of the goals items from a student survey used as part of the Australian Reframing Mathematical Futures II (rmfii) project, and to examine whether statistically significant differences were evident between the derived factors and the variable Year Level. Three factors were derived using an Exploratory Factor Analysis (efa). The three factors were consistent with prior studies based on Goal Achievement Theory: Performance Approach Goal Orientation (F1), Mastery Goal Orientation (F2), and Performance Avoidance Goal Orientation (F3). Statistically significant differences were also found between Year Level and one of the factors (F2). The findings have highlighted potential avenues for further research with respect to students’ goal orientations across different Year Levels.