In order to learn more about student understanding of the structure of proofs, we generated a novel genre of tasks called "Proof Without Claim" (PWC). Our work can be viewed as an extension of Selden and Selden's (1995) construct of "proof framework"; while Selden and Selden discuss how the structure of a proof can be discerned…

We investigate how students make sense of irrational exponents. The data comprise 32 interviews with university students, which revolved around a task designed to examine students' sensemaking processes involved in predicting and subsequently interpreting the shape of the graph of f(x)=x[superscript square root of 2]. The task design and data…

One of the goals of science education is to help students make sense of the world around them. To that end, it is critical that students understand the central ideas in each discipline like, in chemistry, energy and interactions. These ideas are of particular importance because they are directly related to one another and are relevant across other…

Purpose: This paper aims to compare two types of prompts, encouraging participants to think about real-world examples or engineering principles to show how these two approaches can result in vastly different design practices. Design/methodology/approach: Two studies (N = 20, N = 40) examine the impact of two different prompts. Non-expert students,…

Conventional coordinate systems are often considered representational tools for reasoning about mathematical concepts. However, researchers have shown that students experience persistent difficulties as they engage in graphing activity. Using examples from research and textbooks, we present a framework based on a conceptual analysis of the use of…

The aim of this research is to analyse students' sense-making regarding matrix representation of geometric transformations in a dynamic geometry environment (DGE) within the perspective of semiotic mediation. In particular, the focus is on students' reasoning on the transition from the notion of function to transformation and to matrix…

In this paper, we investigate children's learning of reflectional symmetry in a dynamic geometry environment. Through a classroom-based intervention involving two 1-h lessons, we analyse the changes in the children's thinking about reflectional symmetry: first, they developed dynamic and embodied ways of thinking about symmetry after working with…

Post-problem reflective tutorial dialogues between human tutors and students are examined to predict when the tutor changed the level of abstraction from the student's preceding turn (i.e., used more general terms or more specific terms); such changes correlate with learning. Prior work examined lexical changes in abstraction. In this work, we…

College students learned artificial grammar under two conditions: paired associate learning (PA), and observation of exemplars (OBS). OBS induced abstract representation of the rules of grammar. PA produced very different learning--subjects knew some whole items but detected little structure. Grammar was learned largely by analogy rather than…

This paper takes the position that logical knowledge is distinct from conceptual and procedural knowledge and can make a unique contribution to the understanding of knowledge acquisition. This view of logical knowledge departs from the traditional Piagetian view of stages and the overriding view of logic as the sole means of constructing new…

The major results and conclusions of a program of research concerned primarily with the relationship of cognitive abilities to learning are summarized. The major purpose of this research was to develop theorems of instruction related to the interaction of task variables and individual difference variables and to develop them in a manner relevant…

The work of a three-year series of experimental studies of human cognition is summarized in this report. Proglem solving and learning in man-machine interaction was investigated, as well as relevant variables and processes. The work included four separate projects: (1) computer-aided problem solving, (2) computer-aided instruction techniques, (3)…