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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)
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
dos Santos, César Frederico – Journal of Numerical Cognition, 2023
In the literature on numerical cognition, the presence of the capacity to distinguish between numerosities by attending to the number of items, rather than continuous properties of stimuli that correlate with it, is commonly taken as sufficient indication of numerical abilities in cognitive agents. However, this literature does not take into…
Descriptors: Number Concepts, Numeracy, Cognitive Ability, Mathematical Concepts
Phillips, Matthew; Robb, Kayla; Shipman, Barbara A. – PRIMUS, 2023
In an interplay between the Fundamental Theorem of Arithmetic and topology, this paper presents material for a capstone seminar that expands on ideas from number theory, analysis, and linear algebra. It is designed to generate an immersive way of learning in which students discover new connections between familiar concepts, create definitions, and…
Descriptors: Capstone Experiences, Algebra, Mathematics Education, Mathematics Instruction
Rojo, Megan; Wakim, Noura – Intervention in School and Clinic, 2023
Students with learning disabilities in mathematics often struggle with the underlying concepts of multidigit addition and subtraction. To help students build a conceptual understanding of these computations, teachers can utilize evidence-based practices such as the concrete-semi-concrete-abstract framework and the use of multiple visual…
Descriptors: Addition, Subtraction, Mathematics, Learning Disabilities
Peyser, Elizabeth E.; Bobo, Jessica – Mathematics Teacher: Learning and Teaching PK-12, 2022
Early mathematics plays an important role in introducing foundational concepts for number sense in children. One of the critical areas of learning is the establishment of a linear view of numbers. It is essential to create opportunities for young children to understand that numbers are equally spaced on the number line and that they increase in…
Descriptors: Number Concepts, Elementary School Mathematics, Arithmetic, Computation
Olkun, Sinan – International Electronic Journal of Elementary Education, 2022
The purpose of this paper is to propose an effective learning environment for the initial stages of mathematical learning. Basic numerical skills and the objects and actions that trigger those skills are conceptualized as a mathematics-learning environment. We discuss numerical learning mechanism and the basic skills and environments we use to…
Descriptors: Mathematics Education, Learning Processes, Mathematics Skills, Number Concepts
Sofia Tancredi – Digital Experiences in Mathematics Education, 2024
Can math concepts be experienced through the sensory modality of balance? Balance Board Math (BBM) is a set of pedagogical math activities designed to instantiate mathematical concepts through stimulation to the vestibular sense: an organ in the inner ear that detects our bodily balance and orientation. BBM establishes the different ways children…
Descriptors: Mathematical Concepts, Learning Modalities, Mathematics Activities, Stimulation
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
Alexandra M. Flathers; Lauren R. Paternostro; Sydney B. Resnick; Tutita M. Casa – Mathematics Teacher: Learning and Teaching PK-12, 2024
Over three decades ago, National Council of Teachers of Mathematics (NCTM, 1991) proposed that "for students to develop the ability to formulate problems, to explore, conjecture, and reason logically, to evaluate whether something makes sense, classroom discourse must be founded on mathematical evidence" (p. 34). As math interventionists…
Descriptors: Mathematics Instruction, Teaching Methods, Classroom Techniques, Number Concepts
Marah Sutherland; David Furjanic; Joanna Hermida; Ben Clarke – Intervention in School and Clinic, 2024
This article illustrates how teachers can use number lines to support students with or at risk for learning disabilities (LD) in mathematics. Number lines can be strategically used to help students understand relations among numbers, approach number combinations (i.e., basic facts), as well as represent and solve addition and subtraction problems.…
Descriptors: Number Concepts, Arithmetic, Mathematics Instruction, Teaching Methods
Norton, Anderson; Flanagan, Kyle – North American Chapter of the International Group for the Psychology of Mathematics Education, 2022
This paper frames children's mathematics as mathematics. Specifically, it draws upon our knowledge of children's mathematics and applies it to understanding the prime number theorem. Elementary school arithmetic emphasizes two principal operations: addition and multiplication. Through their units coordination activity, children construct two…
Descriptors: Mathematics Instruction, Arithmetic, Elementary School Students, Addition
Taylor Lesner; Marah Sutherland; Cayla Lussier; Ben Clarke – Intervention in School and Clinic, 2024
Building proficiency with fraction arithmetic poses a consistent challenge for students with learning difficulties or disabilities in mathematics. This article illustrates how teachers can use the number line model to support struggling learners in making sense of fraction arithmetic. Number lines are a powerful tool that can be used to help…
Descriptors: Number Concepts, Fractions, Arithmetic, Mathematics Skills