F. D. Rivera

• PhD
• Professor at San Jose State University

4036 Background: In the phase 3KEYNOTE-859 study (NCT03675737), first-linepembrolizumab (pembro) + chemotherapy (chemo) continued to provide longer OS (HR, 0.79; 95% CI, 0.71-0.88) and PFS (HR, 0.76; 95% CI, 0.68-0.85), and a higher ORR (51.0% vs 42.0%) vs placebo + chemo in participants (pts) with HER2-negative G/GEJ adenocarcinoma, after a median...

4045 Background: At the protocol-specified interim analysis of the KEYNOTE-859 study (NCT03675737), first-line pembro + chemo significantly improved OS (HR, 0.78; 95% CI, 0.70-0.87; P<0.0001), PFS (HR, 0.76; 95% CI, 0.67-0.85; P<0.0001), and ORR (51.3% vs 42.0%; P=0.00009) vs placebo (pbo) + chemo in patients (pts) with HER2-negative G/GEJ cancer....

The fundamental aim in this article is to elucidate cognitive factors that influence the development of mathematical structures and incipient generalizations in elementary school children on the basis of their work on patterns, including how they use various representational forms such as gestures, words, and arithmetical symbols to convey their ex...

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 emerg...

This longitudinal study empirically addresses the issue of structure construction and justification among a class of US seventh and eighth-grade Algebra 1 students (mean age of 12.5 years) in the context of novel semi-free pattern generalization (PG) tasks before and after a teaching experiment that emphasized a multiplicative thinking approach to...

Drawing on a review of recent work conducted in the area of pattern generalization (PG), this paper makes a case for a distributed view of PG, which basically situates processing ability in terms of convergences among several different factors that influence PG. Consequently, the distributed nature leads to different types of PG that depend on the...

In this chapter you will deal with content-practice, teaching, and learning issues relevant to the Number System (NS) domain of the CCSSM from Grade 6 to Grade 8 and Algebra 1. In particular, this chapter will help you establish connections between whole numbers and fractions in the earlier grades and integers, rational numbers, and irrational numb...

In this chapter you will deal with issues relevant to middle school students’ content practice learning of mathematics. In section 13.1 you will establish and articulate a general definition of learning that will help guide the manner in which you expect middle school students to learn the CCSSM.

In this chapter you will deal with content-practice, teaching, and learning issues relevant to the domains of geometry and measurement from Grade 5 to Grade 8 of the CCSSM. Table 10.1 lists the appropriate pages for your convenience. The development of fifth-grade students’ mathematical understanding of measurement follows the same multiplicative s...

This methods book takes a very practical approach to learning to teach middle school school mathematics in the Age of the Common Core State Standards (CCSS). The Kindergarten through Grade 12 CCSS in Mathematics (i.e., CCSSM) was officially released on June 2, 2010 with 45 of the 50 US states in agreement to adopt it. Consequently, that action also...

In this chapter you will deal with content-practice, teaching, and learning issues relevant to constructing and manipulating expressions from Grade 5 to Grade 8 and Algebra 1 of the CCSSM. Table 7.1 lists the specific domains and relevant content standards that are covered in this chapter.

In this chapter you will deal with various issues relevant to setting up and running an effective content-practice driven mathematics classroom. Such classrooms typically provide a climate that is conducive for teaching and learning mathematics, that is, a sociocultural condition that supports meaningful engagement with the content and practice sta...

In this chapter you will deal with content-practice, teaching, and learning issues relevant to the equations and inequalities domain of the CCSSM from Grades 6 to 8 and Algebra 1. Table 8.1 lists the appropriate pages for your convenience. To better appreciate the clusters of content standards that deal with equations and inequalities in school mat...

In this chapter you will deal with content-practice, teaching, and learning issues relevant to the following CCSSM number-related domains: Numbers and Operations in Base Ten (NBT) in Grade 5; Numbers and Operations – Fractions (NF) in Grade 5; and portions of Number System (NS) in Grade 6. In Chapter 4, you will reconceptualize them in terms of the...

In this chapter you will explore different content-practice assessment strategies that will help you measure middle school students’ understanding of the CCSSM. The term measure should not be conceptualized merely in terms of scaled values or proficiency labels relevant to students’ performances on a content practice standard being assessed.

The practice standards explicitly articulate how middle school students should engage with the Common Core State Standards for mathematical content (or “content standards”) “as they grow in mathematical maturity and expertise throughout the [middle school] years” (NGACBP &CCSSO, 2012, p. 8). When you teach to the content standards, the practice sta...

Like physical and concrete manipulatives, technology-mediated tools for teaching and learning middle school mathematical concepts and processes provide students with an opportunity to actively engage in thinking while they tinker with the relevant objects in a virtual context.

In this chapter you will deal with content-practice, teaching, and learning issues relevant to the domains of data, statistics, and probability in the CCSSM from Grade 5 to Grade 8 and Algebra 1. Table 11.1 lists the appropriate pages for your convenience. Since data, statistics, and probability involve relationships between quantities, middle scho...

In this chapter you will deal with content-practice, teaching, and learning issues relevant to the domains of functions and models in the CCSSM from Grade 5 to Grade 8 and Algebra 1. Table 9.1 lists the appropriate pages for your convenience. Functions in the CCSSM are formally defined in eighth grade – that is, as rules that assign to each input e...

In this chapter you will deal with content-practice, teaching, and learning issues relevant to the Ratio and Proportion (RP) and Quantities (N-Q).domains of the CCSSM from Grades 6 to 8 and Algebra 1.

In this chapter you will deal with content-practice issues relevant to teaching middle school mathematics. You will learn different teaching models and write unit plans and lesson plans. The component of teaching completes the alignment mindset that you have been asked to frequently bear in mind in your developing professional understanding of teac...

This is a methods book for preservice middle level majors and beginning middle school teachers. It takes a very practical approach to learning to teach middle school mathematics in an emerging Age of the Common Core State Standards. The Common Core State Standards in Mathematics (CCSSM) is not meant to be “the” official mathematics curriculum; it w...

In this chapter, you will deal with content-practice, teaching, and learning issues relevant to fractions and operations in the Numbers and Operations –Fractions (NF) domain from Grades 1 through 5 of the CCSSM.

This methods book takes a very practical approach to learning to teach elementary school mathematics in an emerging Age of the Common Core State Standards (CCSS). The Kindergarten through Grade 12 CCSS in Mathematics (i.e., CCSSM) was officially released on June 2, 2010 with 45 of the 50 US states in agreement to adopt it.

In this chapter, you will deal with content-practice, teaching, and learning issues relevant to the place value structure of whole numbers and decimal numbers up to the thousandths place in the Number and Operations in Base Ten (NBT) domain from kindergarten (K) through Grade 5 of the CCSSM. Table 4.1 lists the appropriate pages for your convenienc...

In this chapter, you will explore content-practice, teaching, and learning issues relevant to the Counting and Cardinality domain in the kindergarten (K) CCSSM. Table 3.1 shows the three clusters that comprise this domain, representing the seven individual content standards that every K student needs to learn by the end of the school year.

In this chapter, you will deal with content-practice, teaching, and learning issues relevant to the four arithmetical operations of addition, subtraction, multiplication, and division involving whole numbers greater than 10 and decimal numbers up to the hundredths place.

In this chapter, you will deal with content-practice, teaching, and learning issues relevant to the Measurement and Data (MD) domain from kindergarten (K) through grade 5 of the CCSSM. The measurement clusters build on multiplicative structures, while the data clusters focus on ways in which data can be organized and reasonably interpreted around w...

In this chapter, you will deal with content-practice, teaching, and learning issues relevant to the Operations and Algebraic Thinking (OA) domain from kindergarten (K) through Grade 5 of the CCSSM. Table 5.1 lists the appropriate pages for your convenience.

In this chapter, you will deal with content-practice, teaching, and learning issues relevant to elementary geometry in the Geometry (G) domain from kindergarten (K) through grade 5 of the CCSSM.

This is a methods book for elementary majors and preservice/beginning elementary teachers. It takes a very practical approach to learning to teach elementary school mathematics in an emerging Age of the Common Core State Standards. The Common Core State Standards in Mathematics (CCSSM) is not meant to be “the” official mathematics curriculum; it wa...

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 gener...

In this chapter, we discuss issues of depth that are relevant to the concept and process of generalization. We clarify the following useful terms that are now commonly used in patterns research: abduction; induction; near generalization and far generalization; and deduction. We also explore nuances in the meaning of generalization that have been us...

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 tak...

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 expo...

This paper provides a longitudinal account of the emergence of whole number operations in second- and third-grade students (ages 7–8 years) from the initial visual processing phase to the converted final phase in numeric form. Results of a notational documentation and analysis drawn from a series of classroom teaching experiments implemented over t...

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 t...

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 sectio...

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 s...

In Chapter 4, we drew on a few examples from my own classroom work in discussing a progressive account of symbol formation in school mathematics. A visually grounded approach provides an alternative and effective route that could assist students in understanding mathematics better. Otte (2007) notes how mathematical knowledge seems to be already “e...

In Chapters 4 and 5, we specifically addressed contexts in which visuoalphanumeric symbols in school algebra and number sense could be interpreted as symbolic entities that have roots in structured visual experiences. We also discussed the significance of progressive symbolization relative to intra- and inter-semiotic transitions that occur from ic...

In this chapter, we explore cultural and blind-specific issues and their implications to visual thinking in mathematics. Issues on culture refer to those patterns of knowledge and skills, tool use, thinking, acting, and interacting that are favored by, and specific to, groups that support them. It is interesting to consider the possibility that stu...

In writing this book, I have definitely stood on the shoulders of giants whose works on various aspects of mathematical visualization have enriched our understanding of how students actually learn mathematics. The impressive critical syntheses of research studies on visualization in mathematics by Presmeg (2006) and Owens and Outhred (2006), which...

This book synthesizes research findings on patterns in the last twenty years or so in order to argue for a theory of graded representations in pattern generalization. While research results drawn from investigations conducted with different age-level groups have sufficiently demonstrated varying shifts in structural awareness and competence, which...

In this research report, I explore the implications of parallel distributed processing in explaining differences in figural patterning competence among 19 Grade 2 students after a classroom teaching experiment.

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 the...

In my Algebra 1 class, solving for the unknown in a linear equation occurred when they had to deal with reversal tasks in patterning situations such as item 4 in Fig. 4.1. Figure 4.2 shows the written work of Dung (eighth grader, Cohort 1), who understood the process of solving for the unknown in the context of finding a particular stage number p w...

Jackie (Cohort 2) was in seventh grade when she joined my Algebra 1 class to participate in a yearlong teaching experiment involving various aspects of algebraic thinking at the middle school level. Her earlier mathematical experiences had solidified for her the impression that mathematics was something that she merely followed on the basis of rule...

In this closing chapter, I should point out that we certainly have come a long way since the time Klotz (1991) asserted that “visualization has a more important role to play in mathematics education” and the need to be “willing to ask hard questions about this approach” (p. 103). Research data from a variety of sources at least in the last 20 years...

Fischbein (1977) in the opening epigraph is certainly correct in pointing out our natural predisposition toward constructing images in order to make sense of some knowledge that appears to us perhaps initially in either linguistic or alphanumeric form. In the case of Gemiliano, he liked mathematics despite his many struggles with its symbolic aspec...

An analysis of the combinatorics problems in many algebra textbooks for high school students reveals that time-honored classic problems are valued. These problems often involve finding combinations and arrangements of numbers and letters on license plates for fictitious states (with and without repetition); digits in an n -digit number that is eith...

In this research article, I present evidence of the existence of visual templates in pattern generalization activity. Such templates initially emerged from a 3-week design-driven classroom teaching experiment on pattern generalization involving linear figural patterns and were assessed for existence in a clinical interview that was conducted four a...

Findings, insights, and issues drawn from a three-year study on patterns are intended to help teach prealgebra and algebra.

This paper discusses the content and structure of generalization involving figural patterns of middle school students, focusing on the extent to which they are capable of establishing and justifying complicated generalizations that entail possible overlap of aspects of the figures. Findings from an ongoing 3-year longitudinal study of middle school...

The article deals with issues concerning the abductive–inductive reasoning of 42 preservice elementary majors on patterns that consist of figural and numerical cues. We discuss: ways in which the participants develop generalizations about classes of abstract objects; abductive processes they exhibit which support their induction leading to a genera...

This paper provides an instrumental account of precalculus students’ graphical process for solving polynomial inequalities. It is carried out in terms of the students’ instrumental schemes as mediated by handheld graphing calculators and in cooperation with their classmates in a classroom setting. The ethnographic narrative relays an instrumental s...

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 t...

Induction Plays A Central Role In Performing generalization and abstraction, two important processes that are necessary and highly valued in all areas of mathematics (Kaput 1999; Mason 1996; Romberg and Kaput 1999; Schoenfeld and Arcavi 1988). From 2000 to 2004, at least 30,000 eighth-grade students in northern California were tested on algebra tas...

involving linear patterns. Our research questions were: What enables/hinders students' abilities to generalize a linear pattern? What strategies do successful students use to develop an explicit generalization? How do students make use of visual and numerical cues in developing a generalization? Do students use different representations equally? Ca...

In this report, we address the following questions: What aspects of information do preservice elementary teachers rely on when performing inductive reasoning? What contexts enable them to perceive the inherent invariant relationships from a finite sample and, thus, formulate viable generalizations? To what extent are they able to justify inductive...

In this paper we explain generalization of patterns in algebra in terms of a combined abduction-induction process. We theorize and provide evidence of the role abduction plays in pattern formation and generalization and distinguish it from induction.

This study reports on findings we obtained from pre-and post-interviews of twelve 6th grade students. We address the following questions: What abilities do they have that influence the manner in which they express and justify generalizations in algebra? How, and to what extent, are they capable of extending finite samples of objects in a larger and...