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…

This paper contains interesting facts regarding the powers of odd ordered special circulant magic squares along with their magic constants. It is shown that we always obtain circulant semi-magic square and special circulant magic square in the case of even and odd positive integer powers of these magic squares respectively. These magic squares…

This study sought to understand how students activate number sense in determining the position of fractions on a number line and identify how the natural number bias and number sense influences students' thinking processes. The study utilized the Cognitive Task Analysis (CTA), involving four fifth-grade elementary students as the research…

It is widely assumed within developmental psychology that spontaneously-arising conceptualizations of natural number--those that develop without explicit mathematics instruction--match the formal characterization of natural number given in the Dedekind-Peano Axioms (e.g., Carey, 2004; Leslie et al., 2008; Rips et al., 2008). Specifically,…

The authors describe a lesson--"You Decide"--which challenges students but also provides opportunities for success for those who may struggle. They show how this lesson has been helpful for teachers in revealing some misconceptions that often exist in primary students' thinking. In this article, they share data on the apparent relative…

In this study, we investigated prospective secondary mathematics teachers' development of a meaning for the Cartesian form of complex numbers by examining the roots of quadratic equations through quantitative reasoning. Data included transcripts of the two sessions of classroom teaching experiments prospective teachers participated in, written…

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…

This article highlights how the Mathematics Teaching Practice of "facilitating meaningful mathematical discourse" (NCTM 2014, p. 10) was used to support a class of eighth-grade students as they engaged in recognizing patterns in which inputs are related to an output. As Carpenter, Franke, and Levi (2003) recommended, the authors chose…

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.

Extensive research has documented students' difficulty understanding and applying the Empirical Law of Large Numbers, the statistical principle that larger random samples result in more precise estimation. However, existing interventions appear to have had limited success, perhaps because they merely demonstrate the Empirical Law of Large Numbers…

Even though algebraic ideas are addressed across a number of grades, algebra continues to serve as a gatekeeper to upper mathematics and degree attainment because of the high percentage of students that fail algebra classes and become halted in their educational progress. One reason for this is students not having the opportunity to build on their…

Children's understanding of fractions, including their symbols, concepts, and arithmetic procedures, is an important facet of both developmental research on mathematics cognition and mathematics education. Research on infants', children's, and adults' fraction and ratio reasoning allows us to test a range of proposals about the development of…

Background: While it has been widely demonstrated that children's and adolescents' understanding of rational number plays an important role in their mathematics achievement, we have limited knowledge about the cognitive correlates of this understanding. Aims: The current study aimed at examining whether children's non-symbolic ratio processing 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…

Teachers often use number talks at the beginning of class to hear the ways that students understand mathematical topics so that they can base future instruction on student thinking. Students who engage in number talks become accustomed to thinking about mathematics in multiple ways, sharing their thoughts and critiquing the thoughts of others. The…