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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
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)
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
Jedediyah Williams – Mathematics Teacher: Learning and Teaching PK-12, 2024
Email filters classify new messages as either spam or not spam based on word frequency, syntax, and metadata. A "classifier" is an algorithm that maps input data into categories based on distinguishing characteristics, or "features." Features can be raw data or attributes derived from that data. "Feature engineering"…
Descriptors: Classification, Engineering, Numbers, Algorithms
Wha-Suck Lee – International Journal of Mathematical Education in Science and Technology, 2024
We view the (real) Laplace transform through the lens of linear algebra as a continuous analogue of the power series by a negative exponential transformation that switches the basis of power functions to the basis of exponential functions. This approach immediately points to how the complex Laplace transform is a generalisation of the Fourier…
Descriptors: Numbers, Algebra, Equations (Mathematics), Mathematical Concepts
Christy Pettis; Aran Glancy – Mathematics Teacher: Learning and Teaching PK-12, 2024
As students have struggled to use the "chip model" (i.e., red and yellow chips representing positive and negative numbers) to model integer addition and subtraction and have found it confusing, the authors developed a series of activities based on adding and removing opposite objects to and from a boat to better help students in this…
Descriptors: Mathematics Instruction, Numbers, Addition, Subtraction
Jeffrey Ehme – PRIMUS, 2024
The Miller-Rabin test is a useful probabilistic method for finding large primes. In this paper, we explain the method in detail and give three variations on this test. These variations were originally developed as student projects to supplement a course in error correcting codes and cryptography.
Descriptors: Probability, Numbers, Coding, Algorithms
Rafi' Safadi; Nadera Hawa – Mathematics Teacher: Learning and Teaching PK-12, 2025
Graded Troubleshooting (GTS) is a powerful routine that teachers can use easily to engender students' metacognitive thinking and boost their understanding of mathematics concepts and procedures. This article describes a new GTS activity designed to prompt students to efficiently exploit worked examples when asked to diagnose erroneous examples…
Descriptors: Mathematics Education, Mathematics Instruction, Problem Solving, Troubleshooting
Sean Chorney – International Journal of Mathematical Education in Science and Technology, 2024
In a pre-service mathematics methods class I taught, in which we mathematized political districting (first horizontally, then vertically), student questions led to engaging mathematics, in particular, the development of a new number sequence.
Descriptors: Elections, Mathematics Instruction, Numbers, Political Attitudes
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
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
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
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