Functional thinking is an established route into algebra. However, the learning mechanisms that support the transition from arithmetic to functional thinking remain unclear. In the current study we explored children's pre-instructional intuitive reactions to functional thinking content, relying on a conceptual change perspective and using mixed…

Teaching and learning Calculus concepts and procedures, particularly the definite integral concept, is a challenge for teachers and students in their academic careers. In this research, we supplement the analysis made by different authors, applying the theoretical and methodological tools of the Onto-Semiotic Approach to mathematical knowledge and…

Insightful problem solving is a vital part of human thinking, yet very difficult to grasp. Traditionally, insight has been investigated by using a set of established "insight tasks," assuming that insight has taken place if these problems are solved. Instead of assuming that insight takes place during every solution of the 9 Dot, 8 Coin,…

Intuitive conceptions in mathematics guide the interpretation of mathematical concepts. We investigated if they bias teachers' conceptions of student arithmetic word problem solving strategies, which should be part of their pedagogical content knowledge (PCK). In individual interviews, teachers and non-teaching adults were asked to describe…

There have been mixed results in studies investigating proportional reasoning in young children. The current study aimed to examine whether providing visual scaling cues and structuring the reasoning process can improve proportional reasoning in 5- to 6-year-old children. In a series of computerized tasks, children compared the sweetness of 2…

Higher working memory capacity (WMC) improves performance on a range of cognitive and academic tasks. However, a greater ability to control attention sometimes leads individuals with higher WMC to persist in using complex, attention-demanding approaches that are suboptimal for a given task. We examined whether higher WMC would hinder insight…

This study tested the hypothesis that intuitions about the effect of operations, e.g., "addition makes bigger" and "division makes smaller", are still present in educated adults, even after years of instruction. To establish the intuitive character, we applied a reaction time methodology, grounded in dual process theories of reasoning. Educated…

Much research supports the existence of an Approximate Number System (ANS) that is recruited by infants, children, adults, and non-human animals to generate coarse, non-symbolic representations of number. This system supports simple arithmetic operations such as addition, subtraction, and ordering of amounts. The current study tests whether an…

Individual differences in working memory capacity (WMC) increase the ability and tendency to devote greater attentional control to a task--improving performance on a wide range of skills. In addition, recent research on enclothed cognition demonstrates that the situational influence of wearing a white lab coat increases controlled attention, due…

Gestalt psychologists proposed two distinct learning mechanisms. Associative learning occurs gradually through the repeated co-occurrence of external stimuli or memories. Insight learning occurs suddenly when people discover new relationships within their prior knowledge as a result of reasoning or problem solving processes that re-organize or…

The present research deals with the perception of matter in young children, inquiring if it is perceived as discrete or continuous. It investigates the existence of the intuitive rule "everything can be divided in two" in elementary school children (K-6) by using questions on the subdivision of mathematical and material objects. In addition, we…

Explores different types of Bingo games in which students can work on mental arithmetic, solve equations, and recognize geometric shapes. (KHR)

An influential proposal about aquiring mathematical knowledge is that it entails linking instruction-based concepts to intuitions derived from informal activities. In the case of non-positive numbers, informal knowledge is unlikely to emanate from observing physical objects, because non-positive objects or sets of objects do not exist. However, it…

This paper discusses a radically different set of assumptions to improve educational outcomes for disadvantaged students. It is argued that disadvantaged children, when exposed to carefully organized thinking-oriented instruction, can acquire the traditional basic skills in the process of reasoning and solving problems. The paper is presented in…