This article discusses a pedagogical technique for utilizing breakout rooms for small group discussions and synchronous drawing features in Zoom to create crowdsourced solutions to mathematical problems that involve graphing or visualization. Students begin with specific examples and then move quickly to the development of a more general…

Although there is much research exploring students' covariational reasoning, there is less research exploring the ways students can leverage such reasoning to coordinate more than two quantities. In this paper, we describe a system of covariational relationships as a comprehensive image of how two varying quantities, having the same attribute…

Inferential reasoning is an integral part of science and civic society, but research shows that it is a problematic domain for many people. One possibility for a more accessible approach to inferential reasoning is to use randomization tests via computer simulations. A case study was conducted with primary preservice teachers after they had passed…

We are all fascinated by loops and their formation in space. When a line cuts itself, it forms an intersection point and creates a space. This is an experimental study done by analyzing several loops, forming a concrete formulation by visualizing the patterns observed, and then proving the formulations proposed using the known standard…

At the undergraduate level, the composite motion of simple harmonic vibrations has always been the main content of physics as well as several other scientific disciplines. Many textbooks tell us clearly that when the frequency ratio [omega][subscript 1]/[omega][subscript 2] of two perpendicular vibrations is simple integer ratio n[subscript…

The concept of derivative is used in many areas including applied problems and requiring mathematical modelling in different disciplines. One of the most important approaches for teaching the derivative is to support students in visualizing the concept. Also, it is necessary to shift researchers and teachers' focuses to students' dynamic mental…

This study focuses on connections between linear functions and their graphs that were made by tertiary remedial algebra students. In particular, we describe students' work on a Task designed to examine the connection between points on a graph and the equation of a line. The data consist of 63 responses to a written questionnaire and individual…

For misguided reasons, social scientists have long been reluctant to use gain scores for estimating causal effects. This article develops graphical models and graph-based arguments to show that gain score methods are a viable strategy for identifying causal treatment effects in observational studies. The proposed graphical models reveal that gain…

Research shows that though some students can easily find the correct solutions to the problem(s) at hand, the algorithms that these students create are not always ones that would, when implemented, produce the correct solution(s). Towards shedding light on this phenomenon, the present study explicates a mechanism--accounting for features of the…

Professionals working with individuals at risk for, or with, disabilities are required by federal legislation to select, implement, and formatively and summatively evaluate the effectiveness of evidence-based practices in typical settings. Although multiple resources related to graph construction and visual analysis are available in the…

Single-case research designs (SCRDs) are rigorous methods used to investigate the effects of interventions on socially important outcomes. A hallmark of SCRDs is repeated measurement of observable behavior presented in time-series graphs. The construction of time-series graphs can inflate Type I or Type II errors made via visual analysis. In the…

In this paper, we describe a framework for characterizing students' graphical reasoning, focusing on providing an empirically-based list of students' graphical resources. The graphical forms framework builds on the knowledge-in-pieces perspective of cognitive structure to describe the intuitive ideas, called "graphical forms", that are…

This article recounts the life of mathematician Leonhard Euler and discusses the use of Graph Theory in student learning.

Visual representations of data are widely used for communication and understanding, particularly in science, technology, engineering, and mathematics (STEM). However, despite their importance, many people have difficulty understanding data-based visualizations. This work presents a series of three studies that examine how understanding time-based…

This study aimed to examine the quality of the open-access GeoGebra applets for function transformations by attending to the opportunity to learn function transformations afforded by them. Guided by the variation theory, this study conceptualized the opportunity to learn function transformations as the opportunity to discern critical features of…