The ability to generalize previous knowledge to new contexts is a key aspect of human cognition and relational learning. A well-known learning maxim is that breadth of training predicts "breadth of training predicts breadth of transfer." When examples vary in their surface features, this provides evidence that only the common relational…

This paper will elaborate five levels of algebraic generalisation based on an analysis of students' responses to Reframing Mathematical Futures II (RMFII) tasks designed to assess algebraic reasoning. The five levels of algebraic generalisation will be elaborated and illustrated using selected tasks from the RMFII study. The five levels will be…

Learning progressions have become an important construct in educational research, in part because of their ability to inform the design of coherent standards, curricula, assessments, and instruction. In this paper, I discuss how a learning progressions approach has guided our development of an early algebra innovation for the elementary grades and…

In this paper, we explore how students' algebraic noticing's and explanations changed across a two-year period with the introduction of designed instructional material. The data in this report is drawn from n=53 Year 7-8 students' responses to a free-response assessment task across two different years. Analysis focused on how students noticed and…

Students who take an online course, such as a MOOC, use the course's discussion forum to ask questions or reach out to instructors when encountering an issue. However, reading and responding to students' questions is difficult to scale because of the time needed to consider each message. As a result, critical issues may be left unresolved, and…

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…

The transition from preformal and propaedeutic generalization-actions to a symbolically explicit use of the concept of variable has been a matter of significant attention in mathematics education, for example in the context of generalization processes on a preformal level and regarding the specific nature of algebraic concepts. This contribution…

The purpose of this study is to clarify for what students do generalize something in learning mathematics. In this study, we make a distinction between generalization and extension, and focus on the function of generalization in terms of its meaning, purpose, and usefulness. Through reviewing literature on generalization and philosophical…

The general purpose of this study is to increase, in a classroom environment, formal communication by using reading and writing. Our research focuses on reading as a means to develop the writing of a good précis, which in its turn contributes to improve: 1) memory, vocabulary and grammatical-syntactical structures, in brief, the organization of a…

The purpose of this paper is to describe (a) multivariable calculus students' meanings for the domain and range of single and multivariable functions and (b) how they generalize their meanings for domain and range from single-variable to multivariable functions. We first describe how students think about domain and range of multivariable functions…

In this paper, we describe our learning progressions approach to early algebra research that involves the coordination of a curricular framework, an instructional sequence, written assessments, and levels of sophistication describing the development of students' thinking. We focus in particular on what we have learning through this approach about…

The theory that language acquisition is guided and constrained by inborn linguistic knowledge is assessed. Specifically, the "no negative evidence" view, the belief that linguistic theory should be restricted in such a way that the grammars it allows can be learned by children on the basis of positive evidence only, is explored. Child language…

An alternative to the standard theory that language learners always formulate the simplest rule to accommodate data is proposed. This new position states that the system of formulating rules and the generalizations made from it by children and adults in the stages of language learning needs to be more specific. The present theory excludes the use…

This paper demonstrates a method for constructing long variables using items that elicit partically correct responses across ages. Long variables may be defined by students at different ages (year levels) attempting common items within a test containing other items considered to be appropriate for each age or year level. A developmental model of…

Research on Vygotsky's zone of proximal development theory was used as a foundation to investigate differences in the learning and transfer ability of 30 students with learning disabilities (LD) and 30 students with matched low achievement (LA), from grades 7 and 8. The two groups were assessed on their problem-solving ability on a balance scale…