In this article a new parameterization of magic squares of order three is presented. This parameterization permits an easy computation of their inverses, eigenvalues, eigenvectors and adjoints. Some attention is paid to the Luoshu, one of the oldest magic squares.

A visual proof of Ptolemy's theorem.

The year students take Algebra I historically determines how far they progress in secondary mathematics, creating complex equity issues around access to this course. By examining a case study of one large, urban school district adjusting to the Common Core State Standards in Mathematics (CCSS-M), we demonstrate how district leaders' interactions,…

Purpose: The purpose of this paper is to describe the design and delivery of a lesson study unit in mathematics to pre-service primary teachers and to identify the opportunities and challenges resulting from the need to deliver the unit wholly online due to the COVID-19 pandemic. Design/methodology/approach: Cross-case analysis, using a…

This paper presents an intelligent tutoring system (ITS) for the learning of arithmetical problem solving. This is based on an analysis of (a) the cognitive processes that take place during problem solving; and (b) the usual tasks performed by a human when supervising a student in a one-to-one tutoring situation. The ITS is able to identify the…

This study investigated middle school students' conceptual understanding of algebraic equations. 257 sixth- and seventh-grade students solved algebraic equations and generated story problems to correspond with given equations. Aspects of the equations' structures, including number of operations and position of the unknown, influenced students'…

The benefits of rich tasks, project-based learning, and other inquiry-based approaches in terms of student understanding and engagement with mathematics are well documented. Such pedagogies are consistent with the development of mathematical proficiencies as described in the "Australian Curriculum: Mathematics" (Australian Curriculum…

Developing expertise in physics problem solving requires the ability to use mathematics effectively in physical scenarios. Novices and experts often perceive the use of mathematics in physics differently. Students' perceptions and how they frame the use of mathematics in physics play an important role in their physics problem solving. In this…

The authors explored whether a testing effect occurs not only for retention of facts but also for application of principles and procedures. For that purpose, 38 high school students either repeatedly studied a text on probability calculations or studied the text, took a test on the content, restudied the text, and finally took the test a second…

In this article we want to point out as brief as possible the particularities of the transition from the pre-university education to the academic environment, by offering some examples from the backstage of the first evaluative meeting with future students. They were candidates for the Primary and Preschool Education Pedagogy specialization of the…

The National Council of Teachers of Mathematics (NCTM) has called for changes in mathematics teaching from a procedural to conceptual focus since 1980, yet the way mathematics is taught in many classrooms continues to contradict the recommended practices. The pervasiveness of this challenge has led some educators to suggest changes in university…

Adaptive curriculum sequencing (ACS) is still a challenge in the adaptive learning field. ACS is a NP-hard problem especially considering the several constraints of the student and the learning material when selecting a sequence from repositories where several sequences could be chosen. Therefore, this has stimulated several researchers to use…

Computational thinking can provide a basis for problem solving, for making evidence-based decisions, and for learning to code or create programs. Therefore, it is critical that all students across the K-12 continuum--including students in the early grades--have opportunities to begin developing problem solving and computational thinking skills.…

Problem solving is a necessary component of developing a strong mathematics curriculum that will help all students achieve their life goals, regardless of their specific academic plans. What day-to-day instructional decisions do teachers need to make if they believe that problem solving is a vehicle for learning mathematical content? In this…

While the value of "schematic representations" in problem solving requires no further demonstration, the way in which students should be taught how to construct these representations invariably gives rise to various debates. This study, conducted on 146 grade 4 students in Luxembourg, analyzes the effect of two types of "schematic…