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Aaron E. Naiman – International Journal of Mathematical Education in Science and Technology, 2024
We automate and randomize the building of linear systems with a parameter, appropriate for assigning to students. When the parameter takes on a specific value, the system has no solutions. When the parameter takes on a different value, the system has an infinite number of solutions.
Descriptors: Mathematics Instruction, Advanced Courses, Teaching Methods
Moshe Stupel; Jay M. Jahangiri – International Journal of Mathematical Education in Science and Technology, 2025
In this article, we state an interesting geometric conservation property between the three angle bisectors of three similar right triangles and provide a proof without words for its justification. A GeoGebra applet is also presented to help with the understanding of the progression of the proof from inductive to deductive stage.
Descriptors: Geometry, Mathematics Instruction, Computer Software, Teaching Methods
Detchat Samart – International Journal of Mathematical Education in Science and Technology, 2024
For a given rational number r, a classical theorem of Niven asserts that if cos(rp) is rational, then cos(rp) [element-of] {0,±1,±1/2}. In this note, we extend Niven's theorem to quadratic irrationalities and present an elementary proof of that.
Descriptors: Mathematics Instruction, Teaching Methods, Validity, Mathematical Logic
Joanna Mamona-Downs – International Journal of Mathematical Education in Science and Technology, 2024
This paper initiates a teaching sequence that focuses on building up equivalent definitions to the standard ones for the limit concept in Real Analysis. It comprises two parts: The first provides a classroom assignment where students, guided by Analysis lecturers, are led to develop an alternative definition to the [epsilon] - [delta] one for…
Descriptors: Mathematics Instruction, Teaching Methods, Definitions, Mathematical Concepts
Xiaoheng Yan; Gila Hanna – International Journal of Mathematical Education in Science and Technology, 2025
As new technological developments continue to change the educational landscape, it is not an exception in the area of proof and proving. This classroom note introduces the use of one of the trending proofs assistants -- the Lean theorem prover. We first provide a technical account of Lean, then exemplify Lean proofs in propositional logic, number…
Descriptors: Mathematics Instruction, Undergraduate Students, Mathematical Logic, Validity
Ribeiro, Ademir Alves; Barbosa, José Renato Ramos – International Journal of Mathematical Education in Science and Technology, 2022
This short note discusses how the optimality conditions for minimizing a multivariate function subject to equality constraints have been covered in some undergraduate Calculus courses. In particular, we will focus on the most common optimization problems in Calculus of several variables: the 2 and 3-dimensional cases. So, along with sufficient…
Descriptors: Calculus, Mathematics Instruction, Undergraduate Students, Teaching Methods
Abboud, Elias – International Journal of Mathematical Education in Science and Technology, 2023
In this article, we consider certain minimization problems. If d[subscript 1], d[subscript 2] and d[subscript 3] are the distances of a boundary or inner point to the sides of a given triangle, find the point which minimizes d[subscript 1][superscript n] + d[subscript 2][superscript n] + d[subscript 3][superscript n] for positive integer n. These…
Descriptors: Computer Software, Mathematics Instruction, Geometry, Calculus
Ismael Cabero; Carl Winsløw – International Journal of Mathematical Education in Science and Technology, 2025
The notion of function is central in all of the secondary curriculum, and indeed functional models appear in almost all higher education that is based on mathematics. However, in secondary education, functions usually appear in restricted and somewhat sterile forms. In this (mostly theoretical) paper, we present a proposal -- exemplified by a…
Descriptors: Mathematics Instruction, Mathematical Models, Teaching Methods, Secondary School Mathematics
Vorob'ev, Evgenii M. – International Journal of Mathematical Education in Science and Technology, 2023
This paper discusses the mathematical and didactical problems of teaching indefinite integral in the context of the ubiquitous availability of online integral calculators. The symbol of indefinite integral introduced by Leibniz, unfortunately, does not contain an indication of the interval on which the antiderivatives should be calculated. This…
Descriptors: Teaching Methods, Mathematics Instruction, Internet, Calculators
Samuel B. Allan; Peter K. Dunn; Robert G. McDougall – International Journal of Mathematical Education in Science and Technology, 2024
In this note we demonstrate two instances where matrix multiplication can be easily verified. In the first setting, the matrix product appears as matrix element concatenation, and in the second, the product coincides with matrix addition. General proofs for some results are provided with a more complete description for 2×2 matrices. Suggested for…
Descriptors: Mathematics Instruction, Teaching Methods, Multiplication, Addition
McCartney, Mark – International Journal of Mathematical Education in Science and Technology, 2021
An elementary method to calculate the area, centroid and volume of rotation of the Koch curve is presented. Classroom extensions are provided to allow students to investigate the method used.
Descriptors: Mathematics Instruction, Teaching Methods, Geometry, Computation
Alves, Alexandre – International Journal of Mathematical Education in Science and Technology, 2023
Taylor series play a ubiquitous role in calculus courses, and their applications as approximants to functions are widely taught and used everywhere. However, it is not common to present the students with other types of approximations besides Taylor polynomials. These notes show that polynomials construed to satisfy certain boundary conditions at…
Descriptors: Mathematics Instruction, Teaching Methods, Calculus, Error Patterns
Ullah, Mukhtar; Aman, Muhammad Naveed; Wolkenhauer, Olaf; Iqbal, Jamshed – International Journal of Mathematical Education in Science and Technology, 2022
The natural exponential and logarithm are typically introduced to undergraduate engineering students in a calculus course using the notion of limits. We here present an approach to introduce the natural exponential/logarithm through a novel interpretation of derivatives. This approach does not rely on limits, allowing an early and intuitive…
Descriptors: Engineering Education, Teaching Methods, Mathematics Instruction, Numbers
Lozada-Cruz, German – International Journal of Mathematical Education in Science and Technology, 2021
This note deals with some variants of the integral mean value theorem. Mainly a variant of Sahoo's theorem and a variant of Wayment's theorem were proved. Our approach is rather elementary and does not use advanced techniques from analysis. The simple auxiliary functions were used to prove the results.
Descriptors: Mathematics Instruction, Teaching Methods, Geometry, Problem Solving
Lou, Hongwei – International Journal of Mathematical Education in Science and Technology, 2023
In current teaching materials, when using Dedekind cuts to construct real numbers, the definition of a Dedekind cut is always involved in defining addition and multiplication. In this paper, as it is done in many current textbooks, Dedekind cuts are used to construct the set of real numbers. Then the order in it is defined, and the…
Descriptors: Mathematics Instruction, Addition, Multiplication, Arithmetic