Structural reasoning is a combined ability to "look for structures, recognize structures, probe into structures, act upon structures, reason in terms of general structures, and see how a piece of knowledge acquired resolves a perturbation experienced" (Harel and Soto, 2017). The purpose of this study was to explore the cognitive…

Rational number knowledge is critical for mathematical literacy and academic success. However, despite considerable research efforts, rational numbers present perennial difficulties for a large number of learners. These difficulties have led some to posit that rational numbers are not a natural fit for human cognition. In this chapter, we…

Mathematical proof is a logically formed argument based on students' thinking process. A mathematical proof is a formal process which needs the ability of analytical thinking to solve. However, researchers still find students who complete the mathematical proof process through intuitive thinking. Students who have studied mathematical proof in the…

This qualitative study aimed to examine: (1) the manner in which kindergarten children and first graders make sense of the term "area" regarding optimization problems; (2) how this manner is manifested in their decision-making and "STEAM" (science, technology, engineering, art and math) skills; and (3) how kindergarten children…

The cognitive relationship between intuition and proof is complex and often students struggle when they need to find mathematical justifications to explain what appears as self-evident. In this paper, we address this complexity in the specific case of open geometrical problems that ask for a conjecture and its proof. We analyze four meaningful…

Structure sense can be interpreted as an intuitive ability towards symbolic expressions, including skills to perceive, to interpret, and to manipulate symbols in different roles. This ability shows student algebraic proficiency in dealing with various symbolic expressions and is considered important to be mastered by secondary school students for…

Gestures are associated with powerful forms of understanding; however, their causative role in mathematics reasoning is less clear. We inhibit college students' gestures by restraining their hands, and examine the impact on language, recall, intuition, and mathematical justifications of geometric conjectures. We test four mutually exclusive…

Dual-process theories posit two distinct types of cognitive processing: Type 1, which does not use working memory making it fast and automatic, and Type 2, which does use working memory making it slow and effortful. Mathematics often relies on the inhibition of pervasive Type 1 processing to apply new skills or knowledge that require Type 2…

The observation that the human mind operates in two distinct thinking modes--intuitive and analytical- have occupied psychological and educational researchers for several decades now. Much of this research has focused on the explanatory power of intuitive thinking as source of errors and misconceptions, but in this article, in contrast, we view…

In this paper two studies are reported in which two contrasting claims concerning the intuitiveness of the law of large numbers are investigated. While Sedlmeier and Gigerenzer ("J Behav Decis Mak" 10:33-51, 1997) claim that people have an intuition that conforms to the law of large numbers, but that they can only employ this intuition…

The main goal of this research was to assess the effect of a dynamic environment on relationships between the three geneses (figural, instrumental, and discursive) of Spaces for Geometric Work. More specifically, it was to determine whether the interactive geometry program GeoGebra could play a specific role in the geometric work of future…

Understanding contingency table analysis is a facet of mathematical competence in the domain of data and probability. Previous studies have shown that even young children are able to solve specific contingency table problems, but apply a variety of strategies that are actually invalid. The purpose of this paper is to describe primary school…

Students' belief that a larger number has more factors is outlined as a particular example of 'the more of A, the more of B' intuitive rule. Discusses the robustness of this belief by demonstrating students' tendency to perceive conflicting evidence as an exception to the rule. Considers some pedagogical approaches. (Contains 12 references.)…

High school students normally encounter the study and use of formal proof in the context of Euclidean geometry. Professional mathematicians typically use an informal trial-and-error approach to a problem, guided by intuition, to arrive at the truth of an idea. Formal proof is pursued only after mathematicians are intuitively convinced about the…

One theoretical framework which addresses students' conceptions and reasoning processes in mathematics and science education is the intuitive rules theory. According to this theory, students' reasoning is affected by intuitive rules when they solve a wide variety of conceptually non-related mathematical and scientific tasks that share some common…