Numeracy is a key competency for living in our modern knowledge society. Therefore, it is essential to support numerical learning from basic to more advanced competency levels. From educational psychology it is known that learning is most effective when the respective content is neither too easy nor too demanding in relation to learners'…

Common fractions have been found to be processed intentionally but not automatically, which led to the conclusion that they are not represented holistically in long-term memory. However, decimals are more similar to natural numbers in their form and thus might be better candidates to be holistically represented by educated adults. To test this…

Teaching experiments have generated several hypotheses concerning the construction of fraction schemes and operations and relationships among them. In particular, researchers have hypothesized that children's construction of splitting operations is crucial to their construction of more advanced fractions concepts (Steffe, 2002). The authors…

A simple view, which dates back to Turing, proposes that complex cognitive operations are composed of serially arranged elementary operations, each passing intermediate results to the next. However, whether and how such serial processing is achieved with a brain composed of massively parallel processors, remains an open question. Here, we study…

In a previous article of the same journal, we have discussed the interrelations of students' beliefs and self-efficacy beliefs for the use of representations and their respective cognitive performance on the learning of fraction addition. In the present paper, we confirm a similar structure of cognitive and affective factors on using…

Educated adults solve simple addition problems primarily by direct memory retrieval, as opposed to by counting or other procedural strategies, but they report using retrieval substantially less often with problems in written-word format (four = eight) compared with digit format (4 = 8). It was hypothesized that retrieval efficiency is relatively…

The field of mathematics is complex, areas such as arithmetic, and geometry each consist of several sub-domains and encompass many cognitive processes. Achievement tests for elementary pupils assess a wide range of arithmetic skills such as number sense, procedural knowledge, and using problem solving strategies. Although mathematical tests often…

This quasi-experimental study aims to streamline cognitive models on fraction multiplication and division that contain the most worthwhile features of other existing models. Its exploratory nature and its approach to proof elicitation can be used to help establish its effectiveness in building students' understanding of fractions as compared to…

The purpose of this technical report is to describe a research study exploring students' cognitive processing while solving problems related to division of fractions. Four students were interviewed using verbal protocol techniques with the intention of better understanding their cognitive processing when dividing fractions. Information gained from…

The mixed method sequential nested study examines whether and how the cognitive type of teachers' content knowledge is associated with student achievement, and correlated with teaching practice. In the context of this study, the "cognitive type" refers to the kind of teacher content knowledge and thinking processes required to accomplish…

How do children learn math--and why do some children struggle with it? The answers are in "Number Sense and Number Nonsense," a straightforward, reader-friendly book for education professionals and an invaluable multidisciplinary resource for researchers. More than a first-ever research synthesis, this highly accessible book brings math…

The purpose of this study was to examine preservice mathematics teachers' knowledge about common (mis)conceptions and difficulties of elementary students. In addition, it was aimed to investigate preservice teachers' knowledge about the possible sources of these misconception/difficulties, and their suggested strategies to overcome those…

The aim of this research is to devise a cognitive tool for meeting the diverse needs of learners for comprehending new procedural knowledge. A model of affordances on teaching fraction equivalence for developing procedural knowledge for adding/subtracting fractions with unlike denominators was derived from the results of a case study of an initial…

Two samples of kindergarten children's representation and understanding of written number symbols were examined in two time points in one academic year. About 85% of Chinese five year olds (mean = 5 years 10 months) were able to use conventional number symbols to represent the quantity of 30 or larger. At the end of the kindergarten year, 94% of…

Within the large range of potential theoretical perspectives on fractions, this paper considers one particular interpretation: fractions' duality as process and object. By considering the number-fractionbar-number composite symbol as simultaneously representing division and rational, some process-object theories imply that fraction-as-process and…