Abstract
In this chapter, we paint in broad strokes issues on pattern generalization in school mathematical contexts that we explore in this book over several chapters. We clarify what we mean by patterns, structures, and pattern generalization activity. Readers are also introduced to the basic characteristic features of our proposed theory of graded pattern generalization processing. We end the chapter with an overview of the remaining chapters.
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Notes
- 1.
This characterization of structure in generalization is consistent with Davydov’s (2008) definition of generalization, “(g)eneralization involves searching for some invariant property within a class of objects …. The general, as something recurring or stable, is a definite invariant of the various properties of a given type of object—i.e. it is essential. In many works, the terms ‘general’ and ‘essential’ are used interchangeably: ‘To identify attributes as essential, they must be found to be general or common to a certain set of objects but not to some other set of objects’” (p. 74).
- 2.
In this book, we share Davydov’s (2008) view regarding the strong link between generalization and abstraction, as follows: “Generalization is regarded as inseparably linked with the operation of abstraction. Singling out some essential quality as a general or common one means abstracting it from other qualities. This enables the child to transform the general quality into an independent and special object of subsequent actions (the general quality is labeled by some word). Knowledge of the general, since it is the result of making a comparison and recording the result using a word, is always something abstract, extracted, thinkable” (p. 75; italics added for emphasis).
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Acknowledgment
This work has been supported by a Career Grant from the National Science Foundation under Grant Number DRL 0448649 awarded to me between 2005 and 2012. I take full responsibility for all the views and opinions expressed in this book. My sincere thanks to all my program officers who allowed me to pursue work in this research area. A warm thanks as well to Joanne Rossi Becker, SJSU colleague, who collaborated with me in the early foundational stages of this work. I am grateful to all my students and their teachers, from first to eighth grade, who generously shared their thinking on patterns.
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Rivera, F. (2013). Introduction. In: Teaching and Learning Patterns in School Mathematics. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-2712-0_1
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