Many children fail to master fraction arithmetic even after years of instruction. A recent theory of fraction arithmetic (Braithwaite, Pyke, & Siegler, 2017) hypothesized that this poor learning of fraction arithmetic procedures reflects poor conceptual understanding of them. To test this hypothesis, we performed three experiments examining…

Many children fail to master fraction arithmetic even after years of instruction. A recent theory of fraction arithmetic (Braithwaite, Pyke, & Siegler, in press) hypothesized that this poor learning of fraction arithmetic procedures reflects poor conceptual understanding of them. To test this hypothesis, we performed three experiments…

This paper focuses on the changes in thinking involved in the transition from school mathematics to formal proof in pure mathematics at university. School mathematics is seen as a combination of visual representations, including geometry and graphs, together with symbolic calculations and manipulations. Pure mathematics in university shifts…

Examined whether representational changes in digit memory are functions of children's expertise in mental abacus operation when abacus operators reproduced series of digits forward or backward. Found skilled operators equally facile with forward and backward reproduction, but novices slower going backward. Suggests advanced operators apply their…

Examined role of spontaneous gesture in 2- to 4-year-olds' counting and assessment of counting accuracy. Found that correspondence of children's speech and gesture varied systematically across age. Children adhered to one-to-one correspondence principle in gesture prior to speech. Counting accuracy related to correspondence of speech and gesture,…

Investigates numerical competence in five-month-old infants using a violation-of-expectation paradigm. Supports previous findings that young children possess not only the competence for limited numerical abstraction, but also the ability to carry out addition and subtraction operations. An alternative explanation, that infants' responses are based…

Examined whether preschoolers could recognize numerical equivalence for comparisons involving sequentially presented sets. Found that children recognized numerical equivalence for static sets earlier than for sequential sets. Memory of the number of sequentially presented objects emerged earlier than memory for the number of sequential events.…

Mathematical concepts presented within disciplines outside mathematics are either assumed to be already familiar to the students, or else, they are regarded as being peripheral to the appreciation of the content of the nonmathematical lesson. Because it is routinely included without regard for possible students' interpretations, the mathematical…

Examined whether 5- and 6-year-olds understand that addition and subtraction cancel each other and whether this understanding is based on identity or quantity of addend and subtrahend. Found that children used inversion principle. Six- to eight-year-olds also used inversion and decomposition to solve a + b - (B+1) problems. Concluded that…

This report presents the results of a teaching experiment which investigated (1) the role of mathematical experiences on the development of counting, addition, subtraction, mental arithmetic, classification, and other arithmetical topics and (2) the role of quantitative comparisons and class inclusion as readiness variables for learning the…

The major idea in this paper is the formulation of a theory of three distinct but interrelated worlds of mathematical thinking each with its own sequence of development of sophistication, and its own sequence of developing warrants for truth, that in total spans the range of growth from the mathematics of new-born babies to the mathematics of…

Arguing that cognitive development involves both conceptual and semiotic achievements, the authors emphasize the distinctness of semiotic issues and develop a differentiated appreciation of semiotic aspects of cognition, particularly in elementary mathematics cognition. Semiotic analyses are provided of differences between counting, adding, and…

Explores the mathematical abilities of human infants compared with various species of animals. Studies indicate that human infants enter the world capable of doing simple mathematical operations. Nonhuman animals can discriminate among sets of objects based on the number of items in each set. Further studies may pinpoint the age at which children…

Using R. S. Siegler's retrieval-required task, 19 male and 22 female third graders were examined before they had been introduced to multiplication in school. Examination of error patterns suggests that the basic assumptions of the distribution-of-associations model need to be tested directly and that the retrieval-required task confounds retrieved…

Two experiments investigated infants' sensitivity to amount of continuous quantity and to changes in amount of continuous quantity. Found that 6-month-olds looked significantly longer at a novel quantity than at the familiar quantity. Nine-month-olds looked significantly longer at an impossible event than at a possible event. Findings question…