Prompting people to estimate climate change numbers before showing them the true value can shift learners' attitudes and conceptions. Yet, interventions created for such a learning experience are not easily accessible to the general public. The purpose of this preregistered study was to address this research gap by developing and testing an openly…

Supporting students of all levels to move beyond empirical arguments, which employ example based reasoning to endorse universal truths and are thus mathematically invalid, remains a challenging goal in mathematics education. Arguments that make use of generic examples are both mathematically valid and accessible for even young learners. However,…

In this paper, we explore student solutions to a free response mathematical assessment task which had opportunities for students to notice structural properties in the context of number systems. In total, 308 students aged between 10 years to 13 years participated in the study. Their responses were analysed to determine whether they noticed…

This research aims to determine second-year university students' understanding in interpreting and representing fractions. A set of fraction tests was given to students through two direct learning interventions. An unstructured interview was used as an instrument to obtain explanations and confirmations from the purposive participants. A total of…

In Asia, some children are taught a calculation technique known as the 'mental abacus'. Previous research indicated that mental abacus experts can perform extraordinary feats of mental arithmetic, but it disagrees as to whether the technique improves working memory. The present study extended and clarified these findings by contrasting performance…

A second course in linear algebra that goes beyond the traditional lower-level curriculum is increasingly important for students of the mathematical sciences. Although many applications involve only real numbers, a solid understanding of complex arithmetic often sheds significant light. Many instructors are unaware of the opportunities afforded by…

The natural numbers may be our simplest, most useful, and best-studied abstract concepts, but their origins are debated. I consider this debate in the context of the proposal, by Gallistel and Gelman, that natural number system is a product of cognitive evolution and the proposal, by Carey, that it is a product of human cultural history. I offer a…

We review research literature concerning "number sense" from several related fields. Whereas other authors have pointed to difficulty defining "number sense" or to some degree of inconsistency in the literature, we argue instead that this is a case of polysemy: There are 3 different constructs that go by the same name. In this…

This report delineates the outcomes of an intervention conducted with in-service high school educators, focusing on elucidating three distinct scenarios within geometric and arithmetic domains: the infinitely large, infinitely numerous, and infinitesimally close. Grounded in the theoretical framework of conceptual change, it is posited that when…

The "Reasoning Algebraically about Operations Casebook" was developed as the key resource for participants' Developing Mathematical Ideas seminar experience. The thirty-four cases, written by teachers describing real situations and actual student thinking in their classrooms, provide the basis of each session's investigation into the…

Across two experiments, we investigated how verbal labels impact the way young children attend to proportional information, well before the introduction of formal fraction education. Five- to seven-year-old children were introduced to equivalent non-symbolic proportions labeled in one of three ways: (a) a single, categorical label for multiple…

Many of the applied mathematics courses we teach touch upon a range of rich and important ethical issues -- issues that, for a number of reasons, are rarely openly discussed in class. In this work, we describe a sequence of activities co-designed and co-taught by philosophy and mathematics faculty in the hopes of bridging the divide between the…

When Johann Bernoulli published his lectures on integrals in 1742, integral calculus had become very advanced since the time of their composition in 1692. Nevertheless, these lectures are of excellent clarity and simplicity even when the book deals with major problems of Mathematical Physics. Just to pique some interest, we offer a commented…

Children do not understand the meanings of count words like "two" and "three" until the preschool years. But even before knowing the meanings of these individual words, might they recognize that counting is "about" the dimension of number? Here in five experiments, we asked whether infants already associate counting…

We investigated the effect of conceptual transparency in the physical structure of manipulatives on place-value understanding in typically developing children and those at risk for mathematics learning disabilities. Second graders were randomly assigned to one of three manipulatives conditions: (a) attachable beads that did not make the…