Fuzzy sets have experienced multiple expansions since their conception to enhance their capacity to convey complex information. Intuitionistic fuzzy sets, image fuzzy sets, q-rung orthopair fuzzy sets, and neutrosophic sets are a few of these extensions. Researchers and academics have acquired a lot of information about their theories and methods…

In this paper we examine the application of Tzur's (2007) fine-grained assessment to the design of an assessment measure of a particular multiplicative scheme so that non-interview, good enough data can be obtained (on a large scale) to infer into elementary students' reasoning. We outline three design principles that surfaced through our recent…

This article shows how to modify classroom evaluation items to avoid four potential difficulties that limit a teacher's insight into students' mathematical understanding by addressing these issues: (1) poor choice of numbers; (2) implausible or inappropriate contexts; (3) inclusion of graphics that do not help make learning visible; and…

We used both student interviews and diagnostic testing to reveal students' misconceptions about number representations in computing systems. This article reveals that students who have passed an undergraduate level computer organization course still possess surprising misconceptions about positional notations, two's complement representation, and…

Lee and Sarnecka (2010) developed a Bayesian model of young children's behavior on the Give-N test of number knowledge. This paper presents two new extensions of the model, and applies the model to new data. In the first extension, the model is used to evaluate competing theories about the conceptual knowledge underlying children's behavior. One,…

Specific Language Impairment (SLI) is understood to be a disorder that predominantly affects phonology, morphosyntax and/or lexical semantics. There is little conclusive evidence on whether children with SLI are challenged with regard to Gricean pragmatic maxims and on whether children with SLI are competent with the logical meaning of quantifying…

Previous evidence has shown that 11-month-olds represent ordinal relations between purely numerical values, whereas younger infants require a confluence of numerical and non-numerical cues. In this study, we show that when multiple featural cues (i.e., color and shape) are provided, 7-month-olds detect reversals in the ordinal direction of…

To implement current reforms regarding proof and reasoning in secondary school mathematics successfully, pre-service secondary mathematics teachers must have adequate understandings of concepts across mathematical domains for their future teaching. This paper examined the strategies that pre-service secondary mathematics teachers had for…

While there is convincing evidence that preverbal human infants and non-human primates can spontaneously represent number, considerable debate surrounds the possibility that such capacity is also present in other animals. Fish show a remarkable ability to discriminate between different numbers of social companions. Previous work has demonstrated…

Now entering into its 20th year of data collection, Trends in International Mathematics and Science Study, (TIMSS) is an international assessment of mathematics and science at the fourth and eighth grades. TIMSS 2015 is the most recent in the TIMSS series, which began with the first assessments in 1995 and has continued every four years--1999,…

Useful for small groups or one-on-one instruction, this book offers successful math interventions and response to intervention (RTI) connections. Teachers will learn to target math instruction to struggling students by: (1) Diagnosing weaknesses; (2) Providing specific, differentiated instruction; (3) Using formative assessments; (4) Offering…

A tight correspondence has been postulated between the representations of number and space. The spatial numerical association of response codes (SNARC) effect, which reflects the observation that people respond faster with the left-hand side to small numbers and with the right-hand side to large numbers, is regarded as strong evidence for this…

Developmental dyscalculia (DD) is a deficit in number processing and arithmetic that affects 3-6% of schoolchildren. The goal of the present study was to analyze cerebral bases of DD related to symbolic number processing. Children with DD aged 9-11 years and matched children with no learning disability history were investigated using fMRI. The two…

The balance between the preservation of early cognitive functions and serious transformations on these functions shifts across time. Piaget's writings, which favored transformations, are being replaced by writings that emphasize continuities between select cognitive functions of infants and older children. The claim that young infants possess…

Dyscalculia, a poor understanding of the number concept and the number system, is a learning problem affecting many individuals. However, less is known about this disability than about the reading disability, dyslexia, because society accepts learning problems in mathematics as quite normal. This article provides a summary of the research on…