Since statistical inference is a probabilistic generalization about a population analyzed on the basis of a sample, inferential reasoning demands producing reasons ("statistical" and "contextual") to substantiate and validate generalizations. To convey an understanding of students' inferential reasoning, we present a…

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…

A common feature of many assessment plans is the use of multiple-choice questions. Although there are criticisms of multiple-choice questions, this assessment format is here to stay--multiple-choice questions are effective means of evaluation in large classes, central to many licensing and entry exams, used in most adaptive learning platforms, and…

This work is the result of a pre-experiment carried out as part of a master's course, dealing with the study of the parabola through different mathematical views. It aims to recognize possible didactic obstacles in its teaching, based on intuitive manifestations in the resolution of a didactic situation based on GeoGebra software. The methodology…

Flipped classrooms invert the traditional (lecture-based) classroom such that students review materials before class, and then participate in active learning techniques during class. While flipped classrooms have generally been shown to be beneficial, recent research indicates that there are numerous moderators that make an impact on learning…

Mass balance (MB) reasoning offers a rich topic for examination of students' scientific thinking and skills, as it requires students to account for multiple inputs and outputs within a system and apply covariational reasoning. Using previously validated constructed response prompts for MB, we examined 1,920 student-constructed responses (CRs)…

Game-based learning has gained significant attention from educational researchers because of its ability to create an engaging and enjoyable learning environment for students. However, there was a research gap regarding the design of game mechanics that specifically helped students understand abstract scientific concepts. Also, the impact of…

The goal of our research program is to explicate the learning of mathematical concepts in ways that are useful for instructional design and to develop design principles based on those explications. I review one type of concept and our elaboration of reflective abstraction, coordination of actions (COA) that accounts for its construction. I then…

This paper presents data from the first of three iterations of teaching experiments conducted with secondary teachers. The purpose of the experiments was to investigate how teachers' combinatorial reasoning could support their development of algebraic structure, specifically structural relationships between the roots and coefficients of…

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…

Literature suggests that "current characterizations of the terms procedural knowledge and conceptual knowledge are limiting and are, in fact, impediments to careful investigation of these constructs" (Star, 2005, p. 405). We examined secondary mathematics teachers' understanding of procedural and conceptual knowledge at superficial and…

Children often learn abstract mathematics concepts with concrete manipulatives. The current study compared different ways of using specific manipulatives -- base-ten blocks -- to support children's place value knowledge. Children (N = 112, M age = 6.88 years) engaged in place value learning activities in one of four randomly assigned conditions in…

Human thought can be characterised as being situated in the 'space of reasons'. That is to say that human thought is guided by the norms of theoretical and practical rationality which, in turn, enable autonomous thinking. But how do children learn to navigate the space of reasons? Building on the work of Tugendhat and Bakhurst, among others, we…

We articulate a framework for delineating student thinking in active, STEM-rich learning environments. Researchers have identified ways of reasoning that relate to specific content areas and practices within each of the STEM disciplines. However, attempts at characterizing student thinking in transdisciplinary STEM environments remains in its…

The basic design lesson is an introductory course that teaches students the fundamental principles of design, including elements such as form, space, scale, proportion, and composition. It is also an introduction to both concrete thinking and abstract one, which is rather about ideas and general thought. Abstraction is one of the steps ap- plied…