Generalization is critical to mathematical thought and to learning mathematics. However, students at all levels struggle to generalize. In this paper, I present a theoretical analysis connecting Piaget's assimilation and accommodation constructs to Harel and Tall's (1991) framework for generalization in advanced mathematics. I offer a theoretical…

In the most general sense, mathematical thinking can be defined as using mathematical techniques, concepts, and methods, directly or indirectly, in the problem-solving process. In this study, efforts were made to include the Graph Theory of mathematics, which is found abundantly in physics, chemistry, computer networks, economics, administrative…

Mental models are representations of students' minds concepts to explain a situation or an on-going process. The purpose of this study is to describe students' mental model in solving mathematical patterns of generalization problem. Subjects in this study were the VII grade students of junior high school in Situbondo, East Java, Indonesia. This…

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…

This aim of this study is to describe the process of local conjecturing in generalizing patterns based on Action, Process, Object, Schema (APOS) theory. The subjects were 16 grade 8 students from a junior high school. Data collection used Pattern Generalization Problem (PGP) and interviews. In the first stage, students completed PGP; in the second…

The study of functions is a critical route into teaching and learning algebra in the elementary grades, yet important questions remain regarding the nature of young children's understanding of functions. This article reports an empirically developed learning trajectory in first-grade children's (6-year-olds') thinking about generalizing functional…

The notions of "abstract" and "concrete" are central to the conceptualization of mathematical knowing and learning. Much of the literature takes a dualist approach, leading to the privileging of the former term at the expense of the latter. In this article, we provide a concrete analysis of a scientist interpreting an unfamiliar graph to show how…

The impact of prior learning on new learning is highlighted by the case of Dean, a Year 8 student who developed his own method to find the sum of the interior angles of a polygon without knowing why his method worked. Enriched transcripts and visual displays of the cognitive, social (Dreyfus, Hershkowitz, & Schwarz, 2001) and affective elements…

This research report presents a study of the work of agronomy majors in which an extension of linear models to non-linear contexts can be observed. By linear models we mean the model y=a.x+b, some particular representations of direct proportionality and the diagram for the rule of three. Its presence and persistence in different types of problems…

Investigated was student inability to understand mathematics across concepts, focusing on the processes by which students solve problems. In-depth clinical interviews were used with students during ninth-grade general mathematics. Mathematical abilities investigated were information gathering, generalization, reversibility, flexibility, and…

The effects of negative instances on the acquisition of the mathematical concepts of commutativity and associativity were examined. Also investigated were possible transfer effects that might result from the use of negative instances. For 64 ninth-grade subjects, results favored the treatments containing mixed instances and supported the transfer…

Mathematical tasks calling for relational thinking were given to 60 children of Indian, Pakistani, or Bangladeshi origin who had spent their school lives in England, 60 who had arrived within the past 3 years, and 60 British children. Differences in the ability to abstract, hypothesize, and generalize were studied. (KC)

Examines operations of generalization, identification, discrimination, and synthesis in mathematical concept development from early childhood to late adolescence according to Vygotsky's theory of development. (MDH)

This volume of the 27th International Group for the Psychology of Mathematics Education Conference includes the following research reports: (1) Improving Decimal Number Conception by Transfer from Fractions to Decimals (Irita Peled and Juhaina Awawdy Shahbari); (2) The Development of Student Teachers' Efficacy Beliefs in Mathematics during…