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.

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.

The exploration of higher education (HE) teachers' approaches to teaching has mainly been done using quantitative instruments which have been criticised for being too narrow in examining the dimensions of teaching. "Higher education approaches to teaching" (HEAT) inventory is a recently developed instrument to capture the dimensions of…

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…

In recent years, professional organizations in the United States have suggested undergraduate mathematics shift away from pure lecture format. Transitioning to a student-centered class is a complex instructional undertaking especially in the proof-based context. In this paper, we share lessons learned from a design-based research project centering…

Professor Gordon argues for a significant reorientation in the focus and impact of assessment in education. For the types of assessment activities that he advocates to prosper and positively impact education, serious attention must be paid to two important topics: (1) the conceptual underpinnings of the assessment practices we develop and use to…

This study aims to find out: firstly, the qualifications for developing teaching materials to evaluate observation-based history learning and secondly the level of students' critical thinking skills. The results of this research contribute to improving students' critical thinking skills through the development of teaching materials. This research…

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…

Since 1983, the Instructional Theory Framework (ITF) (and its subsequent improvements) has guided instructional designers and researchers in designing and developing useful learner experiences (LX). For the past 40 years, the ITF was laser-focused on the selection of instructional methods, downplaying delivery methods (media) and management…

We propose the 'flow of a proof' as a construct that relates to the balance lecturers achieve in their presentation between the proof as a whole and its different parts, taking into account various aspects (informal, contextual) of proof classroom presentation. We conducted expert interviews in which we presented two different proofs of the same…

This study aims to measure the development of Islamic education modules integrated with "Maja Labo Dahu" culture that can improve the character of elementary school students. Module development was conducted at State Elementary School 45 Pane, Bima City, using the Borg and Gall model development research (R&D) method. The results of…

Recent usage of virtual reality (VR) technology in surgical training has emerged because of its cost-effectiveness, time savings, and cognition-based feedback generation. However, the quantitative evaluation of its effectiveness in training is still not thoroughly studied. This article demonstrates the effectiveness of a VR-based surgical training…

In this paper we develop a case for introducing a new teaching tool to undergraduate mathematics. Lean is an interactive theorem prover that instantly checks the correctness of every step and provides immediate feedback. Teaching with Lean might present a challenge, in that students must write their proofs in a formal way using a specific syntax.…

GeoGebra is a dynamic software that is frequently used and of increasing importance in mathematics teaching processes in our digital age. Accordingly, in this study a new perspective has been brought to the proofs of the "two square difference identity" expressed for the square, which is a flat polygon, made with different approaches.…

Although critical thinking has been widely accepted as an educational goal for decades, clear evidence for the development of students' critical thinking is sparse. Critical thinking has been conceptualised as skill, knowledge, and disposition respectively in psychology, philosophy and critical theory, but which applies to education is unclear. In…