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Giovanni Vincenzi – International Journal of Mathematical Education in Science and Technology, 2025
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…
Descriptors: Mathematical Logic, Validity, Numbers, Mathematics
F. M. S. Lima – International Journal of Mathematical Education in Science and Technology, 2025
In this short note I present an elementary proof of irrationality for the number "e," the base of the natural logarithm. It is simpler than other known proofs as it does not use comparisons with geometric series, nor Beukers' integrals, and it does not assume that "e" is a rational number from the beginning.
Descriptors: Mathematical Logic, Number Concepts, Geometry, Equations (Mathematics)
Yuan, Lei; Prather, Richard; Mix, Kelly S.; Smith, Linda B. – Child Development, 2020
The number-line task has been extensively used to study the mental representation of numbers in children. However, studies suggest that proportional reasoning provides a better account of children's performance. Ninety 4- to 6-year-olds were given a number-line task with symbolic numbers, with clustered dot arrays that resembled a perceptual…
Descriptors: Number Concepts, Numbers, Young Children, Visual Stimuli
Carola Ruiz; Saskia Kohnen; Rebecca Bull – European Journal of Psychology of Education, 2024
Number line estimation has been found to be strongly related to mathematical reasoning concurrently and longitudinally. However, the relationship between number line estimation and mathematical reasoning might differ according to children's level of performance. This study investigates whether findings from previous studies that show number line…
Descriptors: Number Concepts, Computation, Mathematics Skills, Mathematical Logic
Rani, Narbda; Mishra, Vinod – International Journal of Mathematical Education in Science and Technology, 2022
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…
Descriptors: Numbers, Mathematical Logic, Mathematics Education, Mathematical Concepts
Karaali, Gizem; Yih, Samuel – PRIMUS, 2020
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…
Descriptors: Mathematics Instruction, College Mathematics, Algebra, Numbers
Maria Blanton; Angela Murphy Gardiner; Ingrid Ristroph; Ana Stephens; Eric Knuth; Rena Stroud – Mathematical Thinking and Learning: An International Journal, 2024
Understanding how young learners come to construct viable mathematical arguments about general claims is a critical objective in early algebra research. The qualitative study reported here characterizes empirically developed progressions in Grades K-1 students' thinking about parity arguments for sums of evens and odds, as well as underlying…
Descriptors: Persuasive Discourse, Algebra, Learning Processes, Elementary School Students
Estela A. Vallejo-Vargas; David A. Reid – International Journal of Science and Mathematics Education, 2024
This article presents a case study of two Grade 5 boys' argumentation concerning addition and subtraction of negative numbers while using an interactive tablet-based application simulating positive and negative tiles. We examine the properties of integers they conjectured, and the kinds of evidence and arguments they used to support their…
Descriptors: Grade 5, Persuasive Discourse, Addition, Subtraction
Johanna Schoenherr; Stanislaw Schukajlow – ZDM: Mathematics Education, 2024
External visualization (i.e., physically embodied visualization) is central to the teaching and learning of mathematics. As external visualization is an important part of mathematics at all levels of education, it is diverse, and research on external visualization has become a wide and complex field. The aim of this scoping review is to…
Descriptors: Visualization, Mathematics Education, Educational Research, Pictorial Stimuli
Utomo, Dwi Priyo – Journal of Research and Advances in Mathematics Education, 2020
Relational understanding constitutes students' awareness of appropriate procedures to solve problems along with logical reasoning. It is pivotal to help students solve problems in mathematics. It is necessary that the teaching of mathematics be directed to achieve relational understanding. Accordingly, students are capable of solving complicated…
Descriptors: Numbers, Problem Solving, Elementary School Students, Grade 5
Yu, Shuyuan; Kim, Dan; Fitzsimmons, Charles J.; Mielicki, Marta K.; Thompson, Clarissa A.; Opfer, John E. – Developmental Psychology, 2022
Children display an early sensitivity to implicit proportions (e.g., 1 of 5 apples vs. 3 of 4 apples), but have considerable difficulty in learning the explicit, symbolic proportions denoted by fractions (e.g., "1/5" vs. "3/4"). Theoretically, reducing the gap between representations of implicit versus explicit proportions…
Descriptors: Elementary School Students, Mathematics Skills, Fractions, Number Concepts
Hino, Keiko; Kato, Hisae – ZDM: The International Journal on Mathematics Education, 2019
Whole-number arithmetic is a core content area of primary mathematics, which lays the foundation for children's later conceptual development. This paper focuses on teaching whole-number multiplication (WNM) to build a stepping stone for children's proportional reasoning. Our intention in writing this paper is to obtain a practice-based perspective…
Descriptors: Mathematics Instruction, Numbers, Multiplication, Children
Shuyuan Yu – ProQuest LLC, 2022
Analogy is a powerful learning mechanism for children to learn novel, abstract concepts from only limited input, yet also requires cognitive supports. My dissertation sought to propose and examine number lines as a mathematical schema of the number system to facilitate both the development of rational number understanding and analogical reasoning.…
Descriptors: Logical Thinking, Mathematical Logic, Mathematics Instruction, Visual Aids
Marmur, Ofer; Moutinho, Ion; Zazkis, Rina – International Journal of Mathematical Education in Science and Technology, 2022
This study aims to explore the notion of the density of the set of rational numbers in the set of real numbers, as interpreted by undergraduate mathematics students. The data comprise 95 responses to a scripting task, in which participants were asked to extend a hypothetical dialog between two student characters, who argue about the existence of…
Descriptors: Undergraduate Students, College Mathematics, Number Concepts, Mathematics Skills
Josephine Relaford-Doyle – ProQuest LLC, 2022
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,…
Descriptors: Mathematical Concepts, Number Concepts, Mathematical Logic, Undergraduate Students