The mathematics curricula of Australia (ACARA, 2019), Scotland, England and America all require an understanding of proof by contradiction. Specifically, proof by contradiction is included as a Geometry topic in Specialist Mathematics (Version 8.4). In Specialist Mathematics, it is expected that students construct proofs in a variety of contexts…

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…

In order to learn more about student understanding of the structure of proofs, we generated a novel genre of tasks called "Proof Without Claim" (PWC). Our work can be viewed as an extension of Selden and Selden's (1995) construct of "proof framework"; while Selden and Selden discuss how the structure of a proof can be discerned…

Proof validation plays a significant role in students' understanding and learning of mathematical proofs. Recent studies have shown that university students were lacking skills in proof validation and were challenged by the implementation of the appropriate acceptance criteria when validating proofs. Drawing on Habermas' construct and rational…

Tychonov's 1935 solution of the heat equation, exhibiting nontrivial heat fluxes spontaneously appearing along an isolated conducting rod initially held at zero degrees, has intrigued some specialists for almost a century. No doubt those practicing heat engineers who took mathematics seriously were initially relieved to learn that the construction…

PWW style are presented. The difference of evidence is based on various additional auxiliary constructions, which in itself is a "mathematical art" that the student can master as a result of much practice. The PWW are accompanied by GeoGebra applets containing HINT buttons that allow the student to get step-by-step help in completing the…

Clearinghouses are influential repositories of information on the effectiveness of social interventions. To identify which interventions are "evidence-based," clearinghouses review intervention evaluations using published standards of evidence that focus primarily on internal validity and causal inferences. Open science practices can…

The purpose of this paper is to demonstrate and describe why alternating treatments design (ATD) researchers should consider employing a no-treatment series during the alternating treatments phase. To identify limitations associated with excluding this design element, we obtained examples of ATD studies published in five school psychology journals…

Traditional cloze tests (such as the CBM-maze) may be poor measures of comprehension processes beyond the single sentence level. This paper presents an alternative, a deep cloze test with gaps that are strategically chosen to assess comprehension beyond the sentence level. To fill each gap, the reader has to draw global cohesion inferences during…

Purpose: Auditory feedback is thought to contribute to the online control of speech production. Yet, the standard method of estimating auditory feedback control (i.e., reflexive responses to auditory-motor perturbations), although sound, requires specialized instrumentation, meticulous calibration, unnatural tasks, and specific acoustic…

The purpose of this study is to discern and discuss issues with relevance to the tension between contextuality and generalisation, which recurrently are identified over time in research reviews of teaching methods. The 75 most cited reviews on teaching methods listed in the Web of Science from 1980 to 2017 were analysed. Since our interest is the…

Background: Much confusion exists about the underachievement of gifted students due to significant variations in how the phenomenon has been identified. From a review of the literature, five methods were found to be commonly used to identify gifted underachievement. Aims: The purpose of the study was to assess the equivalence of the commonly used…

Validity of performance standards is a key element for the defensibility of standard setting results, and validating performance standards requires collecting multiple pieces of evidence at every step during the standard setting process. This study employs a statistical procedure, latent class analysis, to set performance standards and compares…

Frequency of behaviors or amounts of variables of interest are essential topics in many surveys. The use of heuristics might cause rounded answers, resulting in the increased occurrence of end-digits (called heaping or digit-preference). For web surveys (or CASI), we propose using a conditional prompt as input validation if digits indicating…

Among other aspects, the quantitative researchers point out that defining the research sample is an extremely important question which contributes to the research validity and objectivity, thus raising dillemas on specifying the sample in the qualitative researches related to language phenomena and questions. Therefore, this paper aims to answer…