This paper articulates a stance on the study of students' meaningful mathematics understanding with multiple representations. We integrate Thompson's theory of quantitative reasoning and Dreyfus' theory of multiple representations in our approach to frame and conduct empirical investigations of the study of meaningful understanding of function. We…

While spatial reasoning skills have been found to predict mathematical achievement, little is known about how primary students' conceptual understanding of three-dimensional objects develops. This paper reports insights into a Year 4 student's spatial reasoning when constructing and describing the properties of rectangular prisms, using Froebel's…

We investigate teacher beliefs about discourses for equation solving and the challenges these beliefs might pose for the implementation of instructional practices that promote deductive reasoning in algebra. To uncover these beliefs, we recorded three video explanations of solutions to the same linear equation with distinct discursive…

Probability and independence are difficult concepts, as they require the coordination of multiple ideas. This qualitative research study used clinical interviews to understand how three undergraduate students conceptualize probability and probabilistic independence within the theoretical framework of APOS theory. One student's reasoning was…

This report documents how one undergraduate student used set-based reasoning to reinvent logical principles related to conditional statements and their proofs. This learning occurred in a teaching experiment intended to foster abstraction of these logical relationships by comparing the predicate and inference structures among various proofs (in…

The aim of the research was the analysis of the problem dealing with solving of six Force concept inventory tasks by first-year university students using obtained eye-tracking data. Some characteristics like attention maps and sequences of fixations provide a deeper insight into the students' approaches to the tasks verifying their conceptual…

We have developed assessment tasks aligned to NGSS that require students to use practices along with disciplinary core ideas to make sense of energy-related phenomena. In this paper, we present an analysis of field test data and feedback from expert reviewers on the validity and reliability of a set of elementary school tasks. These tasks focused…

This article presents the results of a qualitative research with a group of 15 university students of social sciences on informal inferential reasoning developed in a computer environment on concepts involved in the confidence intervals. The results indicate that students developed a correct reasoning about sampling variability and visualized…

This paper explores the nature of students' quantitative reasoning and conceptions of functions supporting their ability to symbolize quadratic function rules, and the meanings students make of these rules. We analyzed middle school students' problem solving activity during a small group teaching experiment (n = 6) emphasizing quadratic growth…

This case study examines the salient features of two individuals' reasoning when confronted with a task concerning the cardinality and associated cardinal number of equinumerous infinite sets. The APOS Theory was used as a framework to interpret their efforts to resolve the "infinite balls paradox" and one of its variants. These cases…

We explore ways that university students handle proving statements that have the overall structure of a conditional implies a conditional, i.e., (p [right arrow] q) [implies] (r [right arrow] s). We structure our analysis using the theory of conceptual blending. We find conceptual blending useful for describing the creation of powerful new ideas…

This paper reports on the different gesture types employed by twenty-three Year 10 students as they endeavoured to explain their understanding of rate of change associated with the functions resulting from two different computer simulations. These gestures also have application to revealing students' understanding of functions. However,…

In the current study, it was examined whether successful learning is related to using different types of media. We compared the comprehension of an economic concept in novices (N = 82) under three conditions: a Wikipedia article, a funny, and a serious YouTube video. The media were presented in English which is a foreign language to most of the…

This study characterized and compared the arguments eighth grade mathematics students created while working in technological and non-technological environments. Toulmin's (1958/2003) argumentation model was used to analyze the content and structure of the arguments, including the ways in which the students used the tools (technological and…

To test the hypotheses that gifted Ss will be better able to supply missing conclusions to conditional syllogisms as a factor of maturation and that the hierarchy of difficulty in supplying conclusions will be from affirming the antecedent and denying the consequent, to linear transitivity to nonlinear transitivity, and finally to conditional…