A common theme of models of conceptual growth is to establish the hierarchical structures of abilities that can be interpreted along developmental lines. Integrating the literature on the development of mathematical concepts and skills in children, a comprehensive 6 level model for describing, explaining and predicting the development of key…

This article reports on work undertaken by schools as part of Qualifications and Curriculum Authority's (QCA's) "Engaging mathematics for all learners" project. The goal was to use in the classroom, materials and approaches from a Royal Institution (Ri) Year 10 master-class, "Number Sense", which was inspired by examples from…

Reflecting concerns about student attainment and participation in mathematics and science, the Effecting Principled Improvement in STEM Education ("epiSTEMe") project attempted to support pedagogical advancement in these two disciplines. Using principles identified as effective in the research literature (and combining these in a novel…

This study evaluates the efficacy of a mathematics intervention devised by Essex Educational Psychology Service (EPS), UK. The intervention was designed to develop understanding and skills across four key domains within arithmetical development, by applying the principles of errorless learning, distributed practice and teaching to mastery. A…

We present a calculation involving a wide class of real integrals by means of integration in the complex plane. Some particular cases where Euler numbers and Bell polynomials appear are discussed and a generalisation of some previous results is also provided. (Contains 1 table and 1 figure.)

This article describes the development of number concepts between infancy and early childhood. It is based on a diary study that tracked number word use in a child from 12 to 38 months of age. Number words appeared early in the child's vocabulary, but accurate reference to specific numerosities evolved gradually over the entire 27-month period.…

At grade 7, Common Core's content standards call for the use of long division to find the decimal representation of a rational number. With an eye to reconciling this requirement with Common Core's call for "a balanced combination of procedure and understanding," a more transparent form of long division is developed. This leads to the…

As students contribute to social and sociomathematical norms, they will often reorganize their own understandings. As such, the purpose of this article is to describe the ways in which prospective elementary teachers contributed to the norms that were established and "re"-established throughout an entire semester in an undergraduate…

This paper examines spatial metaphors in the English language associated with the number line, in particular metaphors of direction and motion, and how these are manifested in actual spatial practices associated with number. It considers how these metaphors are culturally influenced, and how the influences of other cultures, such as Arabic,…

The study used Linguistic Inquiry and Word Count and Coh-Metrix software to examine linguistic differences with deception in an ultimatum game. In the game, the Allocator was given an amount of money to divide with the Receiver. The Receiver did not know the precise amount the Allocator had to divide, and the Allocator could use deception.…

This article confronts the issue of why secondary and post-secondary students resist accepting the equality of 0.999... and 1, even after they have seen and understood logical arguments for the equality. In some sense, we might say that the equality holds by definition of 0.999..., but this definition depends upon accepting properties of the real…

In this article, we present an integer sequence approach to solve the classic water jugs problem. The solution steps can be obtained easily by additions and subtractions only, which is suitable for manual calculation or programming by computer. This approach can be introduced to secondary and undergraduate students, and also to teachers and…

Although number sense is difficult to define, it involves having a good intuition about numbers and their relationships, including the ability to have a "feel" for the relative size of numbers and to make reasonable estimations. Students with good number sense typically recognise the relative magnitude of numbers, appreciate the effect…

Many students have trouble grasping algebra. In this book, bestselling authors Judith, Gary, and Erin Muschla offer help for math teachers who must instruct their students (even those who are struggling) about the complexities of algebra. In simple terms, the authors outline 150 classroom-tested lessons, focused on those concepts often most…

This paper examines Roy Harris' discussion of number and the language of science in his book "History, Science and the Limits of Language" in light of recent work in cognitive linguistics. While many of his points are valid, linguistic theory has since addressed some of these concerns by formulating usage-based, cognitive frameworks to explore and…