Functions are an essential concept in mathematics. The studies that have examined functions in advanced contexts have primarily focused on students' reasoning about specific types of functions (such as binary operations and isomorphisms) but not on the core characteristics of well-definedness and everywhere-definedness. Here, we report on a study…

Many students persistently misinterpret histograms. This calls for closer inspection of students' strategies when interpreting histograms and case-value plots (which look similar but are different). Using students' gaze data, we ask: "How and how well do upper secondary pre-university school students estimate and compare arithmetic means of…

Background/purpose: This study analyzes the commognition of students with low Working Memory Capacity (WMC) when solving calculus problems, particularly in integral material. Commognition, which merges cognition and communication, is explored through four indicators: keywords (stating knowns and unknowns), visual mediators (graphical…

The translation between mathematical representations was one of the indicators in understanding mathematical concepts. Understanding the things related to the process of student representation translation was very important in learning mathematics. One of the activities that plays a role in translation was unpacking the source. This study aimed to…

Generalization is critical to mathematical thought and to learning mathematics. However, students at all levels struggle to generalize. In this paper, I present a theoretical analysis connecting Piaget's assimilation and accommodation constructs to Harel and Tall's (1991) framework for generalization in advanced mathematics. I offer a theoretical…

We introduce a methodology for diagramming the ways students use sequences of mental actions to solve mathematical tasks. We studied 12 pre-service teachers as they solved a set of fractions tasks, ranked by cognitive demand. We present the unit transformation graphs for one of those pre-service teachers, to illustrate how she experienced and met…

Theorising STEM Education in the 21st Century is a book that captures the essence of Science, Technology, Engineering and Mathematics and the intricacies of STEM education in the contemporary society. It explores STEM as an interdisciplinary field as well as the individual disciplines that make up STEM. This ensures the field of STEM as a whole is…

Textbooks in applied mathematics often use graphs to explain the meaning of formulae, even though their benefit is still not fully explored. To test processes underlying this assumed multimedia effect we collected performance scores, eye movements, and think-aloud protocols from students solving problems in vector calculus with and without graphs.…

The ability to use numerical information in different contexts is a major goal of mathematics education. In health risk communication, outcomes of a medical condition are frequently expressed in probabilities. Difficulties to accurately represent probability information can result in unfavourable medical decisions. To support individuals with…

This study investigates cognitive difficulties and mathematical ideas that are related to making connections among representations. A three-week intervention was designed and implemented to help prospective secondary mathematics teachers develop understanding of big ideas that are critical to connection of representations in algebra. This study…

Verbal reports are a common method of data collection in studies of mathematics learning, often in studies with a longitudinal component or those employing microgenetic methods where several observations of problem-solving are made over a short period of time. Whilst there is a fairly substantial literature on reactivity to verbal reports,…

Reports on how students managed some technical aspects of graphics calculators as they used them to study graphs of straight lines and parabolas. Identifies three common student difficulties: (1) tendency to be unduly influenced by the jagged appearance of graphs; (2) poor understanding of the zoom operation; and (3) limited grasp of the processes…

Documents what high school students showed when asked to work on tasks that involved the use of various representations. Indicates that few students made proper connections between representations and the initial situation or problem on their own and exhibited competence in procedures such as expressing an area or solving quadratic equations, but…

Investigates adults' long-term memory of mathematics learned in high school. Reports findings regarding the subjects' attempts to draw graphs of simple linear functions. Categorizes diverse responses to the task of drawing a graph of a linear function. Analyzes in detail three cases based on recall theories that explain the mechanism of recalling…

This paper reports on the development and evaluation of cognitive tools used by learners to interpret graphs and tables. The development of these tools was informed by research about how learners interpreted graphs and tables. A prototype of the cognitive tools was trialed with a small sample of preservice teachers. This prototype was then…