Using the basic properties of the base-b representation of rational numbers, we will give an elementary proof of Gauss's lemma: "Every real root of a monic polynomial with integer coefficients is either an integer or irrational." The paper offers a new perspective in understanding the meaning of 'irrational numbers' from a deeper…

Imaginary latent variables are variables with negative variances and have been used to implement constraints in measurement models. This article aimed to advance this practice and rationalize the imaginary latent variables as a method to detect possible latent deficiencies in measurement models. This rationale is based on the theory of complex…

Children's mathematical knowledge at school entry varies considerably and predicts long-term achievement outcomes. Differences in children's exposure to math and number talk at home may help to explain variations in school-entry math ability. However, nearly all research on exposure to math and number talk has been conducted with parents and…

We explore the connection between the notion of critical point for a function of one variable and various inequalities for iterated exponentials defined on the positive semiline of real numbers.

Metaphors are recognized as bridging scientists' challenges in communication with the public. In mathematics, however, which often involves abstract concepts and relations, less attention has been paid to the role of metaphors in public communication. In this article we examine the metaphors appearing in the discourse of 10 mathematicians in five…

We present an algorithm for a matrix-based Enigma-type encoder based on a variation of the Hill Cipher as an application of 2 × 2 matrices. In particular, students will use vector addition and 2 × 2 matrix multiplication by column vectors to simulate a matrix version of the German Enigma Encoding Machine as a basic example of cryptography. The…

A study conducted with 25 Year 6 primary school students investigated the potential for a short classroom intervention to begin the development of a "Modelling" conception of mathematics on the way to developing a sense of mathematics as a way of thinking about life. The study documents the developmental roots of the cognitive activity,…

Research suggests that people use a variety of strategies for comparing the numerical values of two fractions. They use holistic strategies that rely on the fraction magnitudes, componential strategies that rely on the fraction numerators or denominators, or a combination of both. We investigated how mathematically skilled adults adapt their…

It is not yet understood how children acquire the meaning of numerical symbols and most existing research has focused on the role of approximate non-symbolic representations of number in this process (see Piazza, ("Trends in Cognitive" 14(12):542-551, 2010). However, numerical symbols necessitate an understanding of both order and…

Many studies have shown the difficulties of learning and teaching the decimal number system for whole numbers. In the case of numbers bigger than one hundred, complexity is partly due to the multitude of possible relationships between units. This study was aimed to develop conditions of a resource which can help teachers to enhance their teaching…

Using a dog's paw as a basis for numerical representation, sixth grade students explored how to count and regroup using the dog's four digital pads. Teachers can connect these base-4 explorations to the conceptual meaning of place value and regrouping using base-10.

Children and animals of all kinds are said to develop some degree of number sense. The search for "number neurons" and neural correlates of computational thinking aims to identify biological primitives to explain the emergence of number sense. This work typically looks for the sources of number sense in organisms, but one might extend…

Babies and infants love mathematics. Give babies a set of blocks, and they will build and order them, fascinated by the ways the edges line up. Children will look up at the sky and be delighted by the V formations in which birds fly. Count a set of objects with a young child and then move the objects and count them again, and they will be…

This article introduces the formative assessment probe--a powerful tool for collecting focused, actionable information about student thinking and potential misconceptions--along with a process for targeting instruction in response to probe results. Drawing on research about common student mathematical misconceptions as well as the former work of…

The sample mean is sometimes depicted as a fulcrum placed under the Dot plot. We provide an alternative geometric visualization of the sample mean using the empirical cumulative distribution function or the cumulative histogram data.