Understanding the concept of completeness for an ordered field is known to be difficult for many university mathematics students. We hypothesise that the variety of possible axioms of completeness for the set of real numbers is one of the sources of difficulties as is the lack of understanding of the "raison d'être" of these axioms. In…

Young children with limited knowledge of formal mathematics can intuitively perform basic arithmetic-like operations over nonsymbolic, approximate representations of quantity. However, the algorithmic rules that guide such nonsymbolic operations are not entirely clear. We asked whether nonsymbolic arithmetic operations have a function-like…

What comes to mind when one thinks about building? One may envision constructions with blocks or engineering activities. Yet, constructing and building a number system requires the same sort of imagination, creativity, and perseverance as building a block city or engaging in engineering design. We know that children invent their own notation for…

Studies have demonstrated a significant correlation between the approximate number system (ANS) and early mathematical achievement. However, various explanations exist regarding the underlying cognitive mechanisms that underpin this association. The present study investigated whether the two hypotheses of inhibition control and visual form…

The interpretation of numerals has been traditionally debated across disciplines including, but not limited to, Linguistics, Developmental Psychology, and Philosophy. A big portion of the debate concentrates on the ontology of numerals, i.e., their basic meaning, and the mechanisms by which (non) exact interpretations (e.g., at least) are derived.…

Previous research suggests that the Approximate Number System (ANS) allows people to approximate the cardinality of a set. This ability to discern numerical quantities may explain how meaning becomes associated with number symbols. However, recently it has been argued that ANS representations are not directly numerical, but rather are formed by…

Although interest in mathematics anxiety in young children (6-11 years) has increased recently, the results of previous studies on the role of mathematics anxiety in the relationship between number sense and math performance are inconsistent. The study's main objective was to establish whether math anxiety is a mediator or a moderator in the…

In this paper, the researchers report an experimental study on conceptual knowledge learning and application through embodied interactions, expecting that bodily movements would facilitate learning in a positive manner. The intervention was enabled by Unity3D and Kinect V2, and taught novice adult learners the concepts of and conversions between…

This study investigates how the approximate number system (ANS) and young children's symbolic skills jointly develop and interact. Specifically, the study aims at disentangling the directionality of the association between ANS acuity and a wide range of symbolic skills that reflect 4- to 5-year-olds' symbolic quantitative knowledge (enumeration…

This article describes students learning to build their own numbering system by recognizing and identifying patterns with interlocking cubes in different place values. The students used the Egyptian hieroglyphic numeral system in conjunction with this activity to connect learning in other subjects. Students used prior knowledge of place value to…

Research over the past 20 years has suggested that our intuitive sense of number--the Approximate Number System (ANS)--is associated with individual differences in symbolic math performance. The mechanism supporting this relationship, however, remains unknown. Here, we test whether the ANS contributes to how well adult observers judge the…

When first learning how to write mathematical proofs, it is often easier for students to work with statements using the universal quantifier. Results that single out special cases might initially come across as more puzzling or even mysterious. In this article we explore three specific statements from abstract algebra that involve the number…

This paper reports on an experiment involving 108 sixth-grade French students in which the history of Chinese numeration was used in mathematics education. The aim of this study is to evaluate the relevance of using ancient number systems in mathematics education, which is common in French textbooks. In the experiment a pre-test with usual…

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…

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…