This is an article about abstraction, generalization, and the beauty of mathematics. We claim that abstraction and generalization in of itself may very well be a beauty of the human mind. The fact that we humans continue to explore and expand mathematics is truly beautiful and remarkable. Many years ago, our ancestors understood that seven stones,…

This article examines whether geography can provide the powerful knowledge that is a key element of a Future 3 curriculum, and an important component of the GeoCapabilities Project's proposals for the teaching of geography. Powerful knowledge is knowledge that gives students the intellectual ability to analyse, explain, predict, evaluate and think…

In this study, we identify ways in which a sample of 18 graduates with mathematics-related first degrees found the nth term for quadratic sequences from the first values of a sequence of data, presented on a computer screen in various formats: tabular, scattered data pairs and sequential. Participants' approaches to identifying the nth term were…

Background: The reciprocal internal/external frame of reference (RI/E) combines two models of academic self-concept formation, namely the reciprocal effects model (REM) and the internal/external frame of reference (I/E) model. The REM assumes reciprocal relations between achievement and academic self-concept. The I/E model assumes contrast effects…

Transfer of learning is a fundamental goal of education but is challenging to achieve, especially where far transfer to remote contexts is at stake. How can we improve learners' flexible application of knowledge to distant domains? In a counterintuitive phenomenon termed the "derring effect," deliberately committing and correcting errors…

Mole, molar mass and the Avogadro number are used constantly to measure the amount of substance in chemical changes. However, students have difficulty in understanding these concepts because they are not familiar with measuring very small amounts of matter that cannot be seen and weighed in daily life, like atoms. The mole concept was developed to…

Understanding fraction as a quantity has been identified as a key developmental understanding. In this study, students in Grades 5, 8, and 11 were asked to compare the areas of two halves of the same square--a rectangle and a right triangle. Findings from this study suggest that students who understand fraction as a quantity use reasoning related…

This study investigated how 155 pre-service teachers solved three pattern generalization problems in a two-part written test and sequenced them for teaching purposes to demonstrate their curricular noticing. Participants' solutions were analyzed using inductive content analysis, which showed that only 8.4% of PSTs produced correct answers to all…

The purpose of this study is to establish the evolution and expose the trends of research in mathematics education between 1980 and 2019. The bibliometric analysis of the articles in Web of Science database indicated four-clustered structure. The first cluster covers the items related to the theoretical framework of mathematics education whereas…

As a multifaceted construct reflecting one's self-esteem, generalized self-efficacy, locus of control, and emotional stability, core self-evaluations (CSE) has become popular to measure in applied psychology research, especially given its conceptual importance and empirical usefulness for understanding the dispositional effects on employee…

Background: Competency models provide useful frameworks for organizing learning and assessment programs, but their construction is both time intensive and subject to perceptual biases. Some aspects of model development may be particularly well-suited to automation, specifically natural language processing (NLP), which could also help make them…

Background: One criterion of adaptive learning is appropriate generalization to new instances based on the original learning context and avoiding overgeneralization. Appropriate generalization requires understanding what features of a solution are applicable in a new context and whether the new context requires modifications or a new approach. In…

Purpose: The objective of the current study was to investigate the short-term test-retest reliability of contralateral suppression (CS) of click-evoked otoacoustic emissions (CEOAEs) using commercially available otoacoustic emission equipment. Method: Twenty-three young normal-hearing subjects were tested. An otoscopic evaluation, admittance…

We describe 24 third (8-9 years old) and 24 fifth (10-11 years old) graders' generalization working with the same problem involving a function. Generalizing and representing functional relationships are considered key elements in a functional approach to early algebra. Focusing on functional relationships can provide insights into how students…

Similarity is one of the most important relations humans perceive, arguably subserving category learning and categorization, generalization and discrimination, judgment and decision making, and other cognitive functions. Researchers have proposed a wide range of representations and metrics that could be at play in similarity judgment, yet have not…