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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
Viviane Durand-Guerrier – ZDM: Mathematics Education, 2024
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…
Descriptors: College Mathematics, Numbers, Number Concepts, Number Systems
Kristyn Sartin – Mathematics Teacher: Learning and Teaching PK-12, 2024
Integer operations are typically introduced in sixth grade, but they are a consistent area of struggle among these students. This struggle also inhibits their understanding of algebraic computations involving positive and negative terms. In this article, the author provides introductory tasks and models for adding and subtracting integers that can…
Descriptors: Grade 6, Middle School Students, Mathematics Instruction, Numbers
Jillian M. Cavanna; Byungeun Pak; Brent E. Jackson – Mathematics Teacher: Learning and Teaching PK-12, 2024
Teachers can make number talks more ambitious and increase learning opportunities for students by debugging errors, promoting multi-student thinking, and orienting to make connections. This article describes the basic number talks routine and examines some common tendencies, such as "Serial Sharing." It concludes with a vignette to…
Descriptors: Number Concepts, Numbers, Mathematics Teachers, Mathematics Instruction
Jungic, Veselin; Yan, Xiaoheng – For the Learning of Mathematics, 2020
The aim of this article is to advise readers that natural numbers may be introduced as ordinal numbers or cardinal numbers and that there is an ongoing discussion about which come first. In addition, through several examples, the authors demonstrate that in the process of answering the question "How many?" one may, if convenient, use…
Descriptors: Number Concepts, Mathematics Instruction, Cognitive Processes, Numbers
Sella, Francesco; Slusser, Emily; Odic, Darko; Krajcsi, Attila – Child Development Perspectives, 2021
Learning the meaning of number words is a lengthy and error-prone process. In this review, we highlight outstanding issues related to current accounts of children's acquisition of symbolic number knowledge. We maintain that, despite the ability to identify and label small numerical quantities, children do not understand initially that number words…
Descriptors: Numbers, Knowledge Level, Vocabulary, Number Concepts
Wetherell, Chris – Australian Mathematics Teacher, 2017
This is an edited extract from the keynote address given by Dr. Chris Wetherell at the 26th Biennial Conference of the Australian Association of Mathematics Teachers Inc. The author investigates the surprisingly rich structure that exists within a simple arrangement of numbers: the times tables.
Descriptors: Numbers, Mathematics Teachers, Professional Associations, Number Concepts
Howe, Roger – ZDM: The International Journal on Mathematics Education, 2019
This paper makes a proposal, from the perspective of a research mathematician interested in mathematics education, for broadening and deepening whole number arithmetic instruction, to make it more relevant for the twenty-first century, in particular, to enable students to deal with large numbers, arguably an essential skill for modern citizenship.…
Descriptors: Number Concepts, Numbers, Error of Measurement, Computation
Farrugia, Marie Therese – For the Learning of Mathematics, 2017
In this article, I describe a research/teaching experience I undertook with a class of 5-year-old children in Malta. The topic was subtraction on the number line. I interpret the teaching/learning process through a semiotic perspective. In particular, I highlight the role played by the gesture of forming "frog jumps" on the number line.…
Descriptors: Mathematics Instruction, Subtraction, Foreign Countries, Young Children
Vozzo, Enzo – Australian Senior Mathematics Journal, 2017
Ever since their serendipitous discovery by Italian mathematicians trying to solve cubic equations in the 16th century, imaginary and complex numbers have been difficult topics to understand. Here the word complex is used to describe something consisting of a number of interconnecting parts. The different parts of a complex number are the…
Descriptors: Mathematics Instruction, Mathematics, Professional Personnel, Numbers
Costello, Pat – PRIMUS, 2018
In 1981 Dixon introduced a clever idea for factoring large numbers. This idea has become the basis for many current factoring techniques. In this paper, we show how to implement the idea on the computer in the classroom. Additionally, pseudocode is given for finding examples suitable for demonstrating Dixon factorization.
Descriptors: Number Concepts, Numbers, Theories, Educational Technology
Woods, Dawn Marie; Ketterlin Geller, Leanne; Basaraba, Deni – Intervention in School and Clinic, 2018
A strong foundation in early number concepts is critical for students' future success in mathematics. Research suggests that visual representations, like a number line, support students' development of number sense by helping them create a mental representation of the order and magnitude of numbers. In addition, explicitly sequencing instruction…
Descriptors: Number Concepts, Numbers, Numeracy, Visual Aids
Throndsen, Jennifer; MacDonald, Beth; Hunt, Jessica – Australian Primary Mathematics Classroom, 2017
Building students' understanding of cardinality is fundamental for working with numbers and operations. Without these early mathematical foundations in place, students will fall behind. Consequently, it is imperative to build on students' strengths to address their weaknesses with the notion of cardinality.
Descriptors: Mathematics, Mathematics Instruction, Kindergarten, Numbers
Herrera, Christine A.; McCabe, Terrance; Strictland, Sharon; White, Alexander – PRIMUS, 2018
In an undergraduate analysis course taught by one of the authors, three prompts are regularly given: (i) What do we know? (ii) What do we need to show? (iii) Let's draw a picture. We focus on the third prompt and its role in helping students develop their confidence in learning how to construct proofs. Specific examples of visual models and their…
Descriptors: Mathematics Instruction, Mathematical Logic, Validity, Mathematics Skills
Firozzaman, Firoz; Firoz, Fahim – International Journal of Mathematical Education in Science and Technology, 2017
Understanding the solution of a problem may require the reader to have background knowledge on the subject. For instance, finding an integer which, when divided by a nonzero integer leaves a remainder; but when divided by another nonzero integer may leave a different remainder. To find a smallest positive integer or a set of integers following the…
Descriptors: Mathematics Instruction, Numbers, Mathematical Concepts, Equations (Mathematics)