Ferdinand Rivera

Formation of Pattern Generalization Involving Linear Figural Patterns Among Middle School Students: Results of a Three-Year Study

This chapter provides an empirical account of the formation of pattern generalization among a group of middle school students who participated in a three-year longitudinal study. Using pre-and post-interviews and videos of intervening teaching experiments, we document shifts in students’ ability to pattern generalize from figural to numeric and then back to figural, including how and why they occurred and consequences. The following six findings are discussed in some detail: development of constructive and deconstructive generalizations at the middle school level; operations needed in developing a pattern generalization; factors affecting students’ ability to develop constructive generalizations; emergence of classroom mathematical practices on pattern generalization; middle school students’ justification of constructive standard generalizations, and; their justification of constructive nonstandard generalizations and deconstructive generalizations. The longitudinal study also highlights the conceptual significance of multiplicative thinking in pattern generalization and the important role of sociocultural mediation in fostering growth in generalization practices.

Ethnomathematics in the Global Episteme: Quo Vadis?

This chapter discusses scholarly work in the field of ethnomathematics from three perspectives that seem to encompass much of the current work in the field: challenging Eurocentrism in mathematics; ethnomathematics praxis in the curriculum; and ethnomathematics as a field of research. We identify what we perceive to be strengths and weaknesses of these three perspectives for today’s learners who are faced with forces of a global nature. We propose a less traditional view of ethnomathematics that is compatible with postnational, global identities, and exemplify this approach through a professional development program in California. Finally, we raise several issues for future discussions relative to ethnomathematical theory and practice

Abduction and the Emergence of Necessary Mathematical Knowledge

The prevailing epistemological perspective on school mathematical knowledge values the central role of induction and deduction in the development of necessary mathematical knowledge with a rather taken-for-granted view of abduction. This chapter will present empirical evidence that illustrates the relationship between abductive action and the emergence of necessary mathematical knowledge.

Topic study group no. 20: Visualization in the teaching and learning of mathematics

This paper was published in the Proceedings of the 13th International Congress on Mathematical Education held at Hamburg, Germany from 24 - 31 July 2016

Patterns and Graded Algebraic Thinking

In this concluding chapter, we discuss ways in which algebra can be grounded in patterning activity. As a consequence, the development of algebraic generalization is also graded from nonsymbolic, to pre-symbolic, and finally to symbolic, reflective of the conceptual changes that occurred in the history of the subject. Over the course of four sections, we clarify the following points: (1) the different contexts of patterning activity and the kinds of algebraic generalizations they generate; (2) the relationship between arithmetical thinking and context-based structural thinking; (3) the grounding of algebra, functions, and models in nonsymbolic and pre-symbolic algebraic contexts; and (4) the graded vs. transitional nature of pattern generalization.

Types and Levels of Pattern Generalization

In this chapter, we synthesize at least 20 years of research studies on pattern generalization that have been conducted with younger and older students in different parts of the globe. Central to pattern generalization are the inferential processes of abduction, induction, and deduction that we discussed in some detail in Chaps. 1 and 2 and now take as given in this chapter. Here we explore the other equally important (and overlapping) dimensions of pattern generalization, namely: natures and sources of generalization; types of structures; ways of attending to structures; and modes of representing and understanding generalizations. In this chapter we remain consistent as before in articulating the complexity of pattern generalization due to differences in, and the simultaneous layering of, processes relevant to constructing, expressing, and justifying interpreted structures.

A Theory of Graded Representations in Pattern Generalization

In this chapter, we initially clarify what we mean by an emergent structure from a parallel distributed processing (PDP) point of view. Then we contrast an emergent structure from other well-known points of view of structures in cognitive science, in particular, symbol structures, theory-theory structures, and probabilistic structures. We also expound on the theory of PDP in semantic cognition in some detail and close the chapter with a discussion of the implications of the PDP theory on pattern generalization processes that matter to mathematical learning. In the closing discussion we discuss the need to modify some of the elements in the original PDP model based on cognitive factors that bear on pattern generalization processes involving school mathematical patterns. Further, we demonstrate the usefulness of a PDP network structure primarily as a thinking model that enables us to describe the complexity of students’ pattern generalization processes not in terms of transitions from, say, arithmetical to algebraic generalizations, but as parallel and graded, adaptive, and fundamentally distributed among, and dependent on, a variety of cognitive and extracognitive sources.

Graded Pattern Generalization Processing of Elementary Students (Ages 6 Through 10 Years)

In this chapter, we focus on pattern generalization studies that have been conducted with elementary school children from Grades 1 through 5 (ages 6 through 10 years) in different contexts. Our contribution to the current research based on elementary students’ understanding of patterns involves extrapolating the graded nature of their pattern generalization schemes on the basis of their constructed structures, incipient generalizations, and the use of various representational forms such as gestures, words, and arithmetical symbols in conveying their expressions of generality. The gradedness condition foregrounds the dynamic emergence of parallel types of pattern generalization processing that is sensitive to a complex of factors (cognitive, sociocultural, neural, constraints in curriculum content, nature and type of tasks, etc.), where progression is seen not in linear terms but as states that continually evolve based on more learning. In a graded pattern generalization processing view, there are no prescribed stages or fixed rules but only states of conceptual coalescences and coherent covariations that change with more experiences. The chapter addresses different aspects of pattern generalization processing that matter to elementary school children. We also explore approximate and exact pattern generalizations along three dimensions, namely: whole number knowledge, shape sensitivity, and figural competence. We further discuss the representational modes that elementary students oftentimes use to capture their emergent structures and incipient generalizations. These modes include gestural, pictorial, verbal, and numerical. In another section, we address grade-level appropriate use and understanding of variables via the notions of intuited and tacit variables. We close the section with an analysis of the relationship between elementary children’s structural incipient generalizations and the natural emergence of their understanding of functions.

Graded Pattern Generalization Processing of Older Students and Adults (Ages 11 and Up)

In this chapter, we explore the graded pattern generalization processing of older children and adults. Graded patterning processing occurs along several routes depending on the nature and complexity of a task being analyzed. That is, students’ graded pattern generalization processing and conversion can change in emphasis from manipulating objects to relationships (and possibly back to objects) in numerical or figural contexts (and possibly both). Toward the end of the chapter, we discuss how older students’ understanding of (linear) functions as an instance of generalizing extensions emerged from their experiences in pattern generalization activity.

Neural Correlates of Gender, Culture, and Race and Implications to Embodied Thinking in Mathematics

In this chapter, I discuss neuroscience research and selected findings that are relevant to mathematics education. What does it mean, for example, to engage in a neuroscientific analysis of symbol reference? I also discuss various research programs in neuroscience that have useful implications in mathematics education research. Further, I provide samples of studies conducted within and outside mathematics education that provide a neural grounding of gender, culture, and race. The chapter closes with three brief implications of neuroscientific work in mathematics education research, in general, and in individual- and intentional-embodied cognition in mathematical thinking and learning, in particular.