Functions and graphs are fundamental mathematical concepts in mathematics and are vital to helping students comprehend the relationship between variables and other advanced topics in higher-level mathematics. Research has shown that students continually possess misconceptions and inaccurate thinking about functions and function representations.…

Supporting young students in developing meanings for the set-theoretic function definition is emphasized in 6th-12th grade curricula around the world. In prior work, we have highlighted how covariational reasoning supports college students in constructing relationships that afford considering mathematical properties important for a set-theoretic…

Studies on student learning of scientific concepts have shown that students often experience various difficulties in understanding basic physics concepts such as vector quantities, as a result of formal and informal learning. The purpose of the present study is to document Grade 11 students' difficulties in understanding vectors and their…

Previous studies have identified physics students' difficulties with graph slope and area under a graph in different contexts. In this study we compared physics and nonphysics (psychology) students' understanding of graphs; i.e., we evaluated the effects of concept (graph slope vs area under graph), type of question (qualitative vs quantitative),…

Students' graph comprehension may be affected by the background of the students who are the readers or interpreters of the graph, their knowledge of the context in which the graph is set, and the inferential processes required by the graph operation. This research study investigated these aspects of graph comprehension for 152 first year…

To understand the mathematical concept of function, students must understand certain subconcepts, such as domain and range. Many researchers have studied students' understanding of functions, but no study has focused on how students come to understand the domain and range for the graphs of functions. In this study, we identified four common…

This article describes the development of the Precalculus Concept Assessment (PCA) instrument, a 25-item multiple-choice exam. The reasoning abilities and understandings central to precalculus and foundational for beginning calculus were identified and characterized in a series of research studies and are articulated in the PCA Taxonomy. These…

The revolution of technology in education has heralded a new wave of learning styles and in some cases total dependence on technology. Yet, statistics reveal no significant improvement in student performance in Mathematics and a downward trend in basic algebra skills. The major impediment in the learning process is lack of understanding of the…

Since the dawn of civilization, pictorial representations and symbols have been used to communicate simple statistics. Efficient and effective, they are still used today in the form of pictures and graphs to record and present data. Who can tie their shoes? How many calories are in your favorite food? Make data and graphs relevant and interesting…

The power of mathematical inscriptions, such as graphs, is often attributed to the fact that they summarize a lot of information independent of their contextual particulars. There is evidence, however, that even quintessential experts and scientists have difficulties interpreting graphs when they are unfamiliar with the entities represented and…

A survey was administered to calculus students who had previously been exposed to a course on integral calculus. The purpose of the survey was to explore students' understanding of the definition of a definite integral, their abilities to evaluate definite integrals, and their graphical interpretations of definite integrals. The analysis of…

This present study investigated engineering students' conceptions and misconceptions related to derivative, particularly interpreting the graph of a function and constructing its derivative graph. Participants were 147 first year engineering students from four universities enrolled in first year undergraduate calculus courses with or without the…

In this paper, we discuss issues in planning and conducting research into mathematics learning. We emphasize two central themes: (1) the learners' mathematics (especially the issues and ideas, in given problem situations, that learners choose to think about and to present; and (2) the kinds of knowledge that learners may be building (including…

This book provides elementary and middle school teachers with practical ideas on the use and teaching of graphs. Five sections consist of: (1) "Graphs--What Are They and How Are They Used?"; (2) "Levels of Graph Comprehension"; (3) "Collecting, Organizing, and Analyzing Data"; (4) "Constructing, Interpreting, and Writing about Graphs"; and (5)…

This article describes an approach to teaching linear equations using multiple representations in a highly motivating context. Connections among representations are made explicit to enhance understanding and empower students. (Contains 4 figures.)