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Arithmetic is an elementary branch of mathematics that studies numerical operations like addition, subtraction, multiplication, and division. In a wider sense, it also includes exponentiation, extraction of roots, and taking logarithms.
Counting is the process of determining the number of elements of a finite set of objects; that is, determining the size of a set. The traditional way of counting consists of continually increasing a counter by a unit for every element of the set, in some order, while marking those elements to avoid visiting the same element more than once, until no unmarked elements are left; if the counter was set to one after the first object, the value after visiting the final object gives the desired number of elements. The related term enumeration refers to uniquely identifying the elements of a finite (combinatorial) set or infinite set by assigning a number to each element.
Numeracy is the ability to understand, reason with, and apply simple numerical concepts. The charity National Numeracy states: "Numeracy means understanding how mathematics is used in the real world and being able to apply it to make the best possible decisions...It's as much about thinking and reasoning as about 'doing sums'". Basic numeracy skills consist of comprehending fundamental arithmetical operations like addition, subtraction, multiplication, and division. For example, if one can understand simple mathematical equations such as 2 + 2 = 4, then one would be considered to possess at least basic numeric knowledge. Substantial aspects of numeracy also include number sense, operation sense, computation, measurement, geometry, probability and statistics. A numerically literate person can manage and respond to the mathematical demands of life.
Dyscalculia is a learning disability resulting in difficulty learning or comprehending arithmetic, such as difficulty in understanding numbers, numeracy, learning how to manipulate numbers, performing mathematical calculations, and learning facts in mathematics. It is sometimes colloquially referred to as "math dyslexia", though this analogy can be misleading as they are distinct syndromes.
Quantity or amount is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value multiple of a unit of measurement. Mass, time, distance, heat, and angle are among the familiar examples of quantitative properties.
Principles and Standards for School Mathematics (PSSM) are guidelines produced by the National Council of Teachers of Mathematics (NCTM) in 2000, setting forth recommendations for mathematics educators. They form a national vision for preschool through twelfth grade mathematics education in the US and Canada. It is the primary model for standards-based mathematics.
Mathematical software is software used to model, analyze or calculate numeric, symbolic or geometric data.
The intraparietal sulcus (IPS) is located on the lateral surface of the parietal lobe, and consists of an oblique and a horizontal portion. The IPS contains a series of functionally distinct subregions that have been intensively investigated using both single cell neurophysiology in primates and human functional neuroimaging. Its principal functions are related to perceptual-motor coordination and visual attention, which allows for visually-guided pointing, grasping, and object manipulation that can produce a desired effect.
Numerical cognition is a subdiscipline of cognitive science that studies the cognitive, developmental and neural bases of numbers and mathematics. As with many cognitive science endeavors, this is a highly interdisciplinary topic, and includes researchers in cognitive psychology, developmental psychology, neuroscience and cognitive linguistics. This discipline, although it may interact with questions in the philosophy of mathematics, is primarily concerned with empirical questions.
Number systems have progressed from the use of fingers and tally marks, perhaps more than 40,000 years ago, to the use of sets of glyphs able to represent any conceivable number efficiently. The earliest known unambiguous notations for numbers emerged in Mesopotamia about 5000 or 6000 years ago.
The following outline is provided as an overview of and topical guide to thought (thinking):
In human developmental psychology or non-human primate experiments, ordinal numerical competence or ordinal numerical knowledge is the ability to count objects in order and to understand the greater than and less than relationships between numbers. It has been shown that children as young as two can make some ordinal numerical decisions. There are studies indicating that some non-human primates, like chimpanzees and rhesus monkeys have some ordinal numerical competence.
Educational neuroscience is an emerging scientific field that brings together researchers in cognitive neuroscience, developmental cognitive neuroscience, educational psychology, educational technology, education theory and other related disciplines to explore the interactions between biological processes and education. Researchers in educational neuroscience investigate the neural mechanisms of reading, numerical cognition, attention and their attendant difficulties including dyslexia, dyscalculia and ADHD as they relate to education. Researchers in this area may link basic findings in cognitive neuroscience with educational technology to help in curriculum implementation for mathematics education and reading education. The aim of educational neuroscience is to generate basic and applied research that will provide a new transdisciplinary account of learning and teaching, which is capable of informing education. A major goal of educational neuroscience is to bridge the gap between the two fields through a direct dialogue between researchers and educators, avoiding the "middlemen of the brain-based learning industry". These middlemen have a vested commercial interest in the selling of "neuromyths" and their supposed remedies.
The approximate number system (ANS) is a cognitive system that supports the estimation of the magnitude of a group without relying on language or symbols. The ANS is credited with the non-symbolic representation of all numbers greater than four, with lesser values being carried out by the parallel individuation system, or object tracking system. Beginning in early infancy, the ANS allows an individual to detect differences in magnitude between groups. The precision of the ANS improves throughout childhood development and reaches a final adult level of approximately 15% accuracy, meaning an adult could distinguish 100 items versus 115 items without counting. The ANS plays a crucial role in development of other numerical abilities, such as the concept of exact number and simple arithmetic. The precision level of a child's ANS has been shown to predict subsequent mathematical achievement in school. The ANS has been linked to the intraparietal sulcus of the brain.
Number sense in animals is the ability of creatures to represent and discriminate quantities of relative sizes by number sense. It has been observed in various species, from fish to primates. Animals are believed to have an approximate number system, the same system for number representation demonstrated by humans, which is more precise for smaller quantities and less so for larger values. An exact representation of numbers higher than three has not been attested in wild animals, but can be demonstrated after a period of training in captive animals.
Spatial ability or visuo-spatial ability is the capacity to understand, reason, and remember the visual and spatial relations among objects or space.
Hypercalculia is "a specific developmental condition in which the ability to perform mathematical calculations is significantly superior to general learning ability and to school attainment in maths." A 2002 neuroimaging study of a child with hypercalculia suggested greater brain volume in the right temporal lobe. Serial SPECT scans revealed hyperperfusion over right parietal areas during performance of arithmetic tasks.
Kelly S. Mix is an American developmental psychologist known for her research on the development of numerical concepts and their origins in infancy and toddlerhood. She is professor and chair of the Department of Human Development and Quantitative Methodology at the University of Maryland. Mix was awarded the Boyd McCandless Early Career Award in 2002 for her innovative research on the early emergence of numerocity. Her co-authored book Quantitative Development in Infancy and Early Childhood, with Janellen Huttenlocher and Susan Cohen Levine, provides an overview of the early development of quantitative reasoning and mathematical concepts. Her co-edited book The Spatial Foundations of Language and Cognition, with Linda B. Smith and Michael Gasser, examines the role of space in structuring human cognition.
Lisa Feigenson is Professor of Psychological and Brain Sciences at Johns Hopkins University and co-director of the Johns Hopkins University Laboratory for Child Development. Feigenson is known for her research on the development of numerical abilities, working memory, and early learning. She has served on the editorial board of Cognition and the Journal of Experimental Psychology: General.
Arthur "Art" J. Baroody is an educational psychologist, academic, and an expert in mathematics education research. He is a Professor Emeritus of Curriculum and Instruction at the University of Illinois at Urbana-Champaign, and a Senior Research Fellow in Morgridge College of Education (COE) at the University of Denver.
References
- ↑ Piazza, M. (2010). "Neurocognitive start-up tools for symbolic number representations". Trends in Cognitive Sciences. 14 (12): 542–551. doi:10.1016/j.tics.2010.09.008. PMID 21055996. S2CID 13229498.
- ↑ "Number Sense: Rethinking Arithmetic Instruction for Students with Mathematical Disabilities".
- ↑ Berch, Daniel B. (2005). "Making Sense of Number Sense: Implications for Children With Mathematical Disabilities". Journal of Learning Disabilities. 38 (4): 333–339. doi:10.1177/00222194050380040901. PMID 16122065. S2CID 1657049.
- ↑ "Stages in Development of Number Sense - Harvard Education Letter".
- ↑ "Number Systems". www.math.twsu.edu.
- ↑ Dantzig, Tobias. Number: The Language of Science. New York: Macmillan Company, 1930.
- ↑ "Understanding a Child's Development of Number Sense".
- ↑ "UIL Number Sense".
- 1 2 "Unit 1: Number and Number Sense" (20-day lesson), STPSB.org, St. Tammany Parish School Board, Covington, LA (USA), 2009, overview webpage: ST-MathGrade7Unit-topics.
- ↑ "Freakonomics.com".