This paper's purpose is to investigate the attitude of students in mathematical discovery processes in terms of the handling of counterexamples. By understanding this attitude as a kind of scientific attitude, it consists of different aspects that become visible in the behaviour during a mathematical discovery process. Since such a process is…

In this paper we follow a hypothetical mathematician who is working on a problem that is eventually solved. We treat this problem as if it were difficult for the mathematician. In following the mathematician's work, we note both what she does and what she doesn't do in the process. By the latter, we consider the times when progress is not being…

This article is a discovery-based instructional resource for faculty to use as a capstone course or exploratory project for undergraduates who are familiar with (but not necessarily fluent in) calculus, linear algebra, complex variables, and geometry. The explorations flow from a simple change in sign: in the complex numbers a + bi, where…

Minimal guidance or maximal independence? How can teachers achieve that complex balance when teaching mathematics? In this pre-and post-test quasi-experimental research study, the authors explored the above broad questions in Grade 10 Financial Mathematics following guided discovery and mathematical modelling principles. Fifty-four (54) Grade 10…

Computer simulations are considered efficient in supporting exploratory learning. This paper highlights instruction challenges and undesirable consequences on student learning in exploratory learning with computer simulations in classroom situations. A classroom with students who explore using simulations contains ideas explicated on several…

The metaphors that students form and encounter have been shown to exert a powerful influence on how they think about mathematics. In this paper, we explore the linguistic metaphors about learning and doing mathematics that were prevalent in 11 advanced mathematics lectures. We present four metaphor clusters that were common in the corpus that we…

The pandemic period forces the teaching and learning process to be carried out online. However, most of the teaching materials are still in the form of those used in pre-pandemics and, therefore, are irrelevant to use. Therefore, educators need alternative teaching materials suitable to support online learning in a pandemic. For this reason, this…

Internal and external assessment instruments, such as the Vanuatu Standard Test of Achievements (VANSTA) and the Pacific Island Literacy and Numeracy Assessment (PILNA), reveal that mathematics achievements in the Republic of Vanuatu remain below the minimum standard. This study drew on the constructivist grounded theory approach to explore…

This article discusses the use of number talks to engage kindergarten children in regular joyful math opportunities in the classroom. As an educator of four- and five-year-old students in a full day kindergarten (FDK) program in Ontario, Canada, I embrace inquiry-based learning to guide children's activities. Inspired by the childcare centres in…

This conceptual paper considers what it would mean to take seriously Freudenthal's suggestion that mathematics should be taught like swimming. The general claim being made is that "direct instruction" and "discovery" are not opposite but complementary, linked by repetitive yet explorative practice. This claim is elaborated…

With limited prior research on young children's learning of the measurement of mass, the study reported in this paper provides insights that can usefully inform teaching. Close examination of the actions and conversations of 12 children of 5-7 years of age, as they experimented for the first time with suspended balance scales, led to the…

In this paper, we describe three practical exercises relating to algorithm education. The exercises are based on a learning support system that offers visualization of program behavior. Systems with the ability to visualize program behavior are effective to promote the understanding of algorithm behavior. The introduction of these systems into an…

This paper discusses the importance of the joy of learning in mathematics and some ways that teachers can foster this in mathematics education.

In reviewing the six articles within this "Instructional Science" special issue, we are reminded of Schoenfeld's ("Educ Res" 45(2):105-111, 2016) review of American Educational Research Association president-authored papers for the centennial celebration of AERA. There, he succinctly unveiled the content focus of AERA research…

An intervention experiment served to evaluate the efficacy of highly guided discovery learning of relations underlying add-1 and doubles combination families and to compare the impact of such instruction with minimally guided instruction. After a pretest, 78 first graders were randomly assigned to one of three intervention conditions: highly…