Dream Realizations Math Research

"When children are acquiring the symbolic system for representing numbers and learning about math in school, they're tapping into this primitive number sense," says Elizabeth Brannon, a professor of psychology and neuroscience at Duke University, who led the study. "It's the conceptual building block upon which mathematical ability is built." Understanding how infants and young children conceptualize and understand number can lead to the development of new mathematics education strategies, says psychology and neuroscience graduate student Ariel Starr. In particular, this knowledge can be used to design interventions for young children who have trouble learning mathematics symbols and basic methodologies. "Our study shows that infant number sense is a predictor of symbolic math," Brannon says. "We believe that when children learn the meaning of number words and symbols, they're likely mapping those meanings onto pre-verbal representations of number that they already have in infancy," she says. "In fact our infant task only explains a small percentage of the variance in young children's math performance. But our findings suggest that there is cognitive overlap between primitive number sense and symbolic math. These are fundamental building blocks."

A Proposed Framework for Examining Basic Number Sense Two scenarios: Story of a boy adding two-digit numbers and clerk taking 50% off. JSTOR: For the Learning of Mathematics, Vol. 12, No. 3 (Nov., 1992), pp. 2-8, 44

These findings provide evidence that infants have robust abilities to represent large numerosities. In contrast, infants may fail to represent small numerosities in visual-spatial arrays with continuous quantity controls, consistent with the thesis that separate systems serve to represent large versus small numerosities.

Abstract Four experiments investigated infants' sensitivity to large, approximate numerosities in auditory sequences. Prior studies provided evidence that 6-month-old infants discriminate large numerosities that differ by a ratio of 2.0, but not 1.5, when presented with arrays of visual forms in which many continuous variables are controlled. The present studies used a head-turn preference procedure to test for infants' numerosity discrimination with auditory sequences designed to control for element duration, sequence duration, interelement interval, and amount of acoustic energy. Six-month-old infants discriminated 16 from 8 sounds but failed to discriminate 12 from 8 sounds, providing evidence that the same 2.0 ratio limits numerosity discrimination in auditory-temporal sequences and visual-spatial arrays. Nine-month-old infants, in contrast, successfully discriminated 12 from 8 sounds, but not 10 from 8 sounds, providing evidence that numerosity discrimination increases in precision over development, prior to the emergence of language or symbolic counting.

Some monkeys are smarter than college freshman. A quote

Abstract Assessed the impact of linguistic differences in ordinal number names on children's acquisition and use of ordinal numbers and their understanding of ordinal concepts. Elementary school children (aged 5.4-10.6 yrs) in China and the US performed a series of tasks assessing understanding of ordinal numbers and concepts. The results show differences in the acquisition and use of ordinal numbers corresponding to linguistic differences in ordinal names in their native languages. On tasks assessing children's conceptual knowledge of ordinal relations, a more complicated picture emerged. These results suggest that (1) children induce their language's set of ordinal number names by generalization based on rules sanctioned by early examples, and (2) the relation between ordinal names and ordinal concepts is a complex one, with language only one source of difficulty in understanding ordinal relations. Implications for studies of the relation between linguistic structure and cognitive development are discussed, in particular the possibility that effects of linguistic differences may vary for different levels of development and for different aspects of cognition. (PsycINFO Database Record (c) 2012 APA, all rights reserved)

Abstract Three- to 5-year-old children participated in one of 4 counting experiments. On the assumption that performance demands can mask the young child's implicit knowledge of the counting principles, 3 separate experiments assessed a child's ability to detect errors in a puppet's application of the one-one, stable-order and cardinal count principles. In a fourth experiment children counted in different conditions designed to vary performance demands. Since children in the errror-detection experiments did not have to do the counting, we predicted excellent performance even on set sizes beyond the range a young child counts accurately. That they did well on these experiments supports the view that errors in counting-at least for set sizes up to 20-reflect performance demands and not the absence of implicit knowledge of the counting principles. In the final experiment, where children did the counting themselves, set size did affect their success. So did some variations in conditions, the most difficult of which was the one where children had to count 3-dimensional objects which were under a plexiglass cover. We expected that this condition would interfere with the child's tendency to point and touch objects in order to keep separate items which have been counted from those which have not been counted.

...aspects of children's counting and numerical thinking and with some notion of the overall kinds of changes children's thinking and performance undergo over the years 2 through 8.

We describe the preverbal system of counting and arithmetic reasoning revealed by experiments on numerical representations in animals. In this system, numerosities are represented by magnitudes, which are rapidly by inaccurately generated by the Meck and Church (1983) preverbal counting mechanism. We suggest the following. (1) The preverbal counting mechanisms is the source of the implicit principles that guide the acquisition of verbal counting. (2) The preverbal system of arithmetic computation provides the framework for the assimilation of the verbal system. (3) Learning to count involves, in part, learning a mapping from the preverbal numerical magnitudes to the verbal and written number symbols and the inverse mappings from these symbols to the preverbal magnitudes. (4) Subitizings is the use of the preverbal counting process and the mapping from the resulting magnitudes to number words in order to generate rapidly the number words for small numerosities. (5) The retrieval of the number facts, which plays a central role in verbal computation, is mediated via the inverse mappings from verbal and written numbers to the preverbal magnitudes and the use of these magnitudes to find the appropriate cells in tabular arrangements of the answers. (6) This model of the fact retrieval process accounts for the salient features of the reaction time differences and error patterns revealed by expriments on mental arithmetic. (7) The application of verbal and written computational algorithms goes on in parallel with, and is to some extent guided by, preverbal computations, both in the child and in the adult.

Critical Mathematics in the classroom. RL has this journal Vol50 Num5 Oct13 pg1050

The understanding is that mathematics is universal, but in reality is it culturally defined: ie, measurement & conversions.

Abstract: Numerical cognition encompasses the concepts of quantity ('how many?') and serial order ('which position?'). Yet, although numbers can convey different meanings, a recent imaging study by Fias and coworkers showed that ranking letters in the alphabet is subserved by a cortical network highly similar to that involved in judging magnitudes. In terms of neural processing, quantity and rank might just be two sides of the same coin.

The authors show that Ss can make a fast "countability" judgment indicating whether or not they could, if requested, give an accurate numerosity response. These judgments were fast and produced a "yes" response within the subitizing range and a "no" response thereafter. It is concluded, on the basis of 5 studies with 48 adults, that the RT function found in subitizing consisted of 3 processes: a response to arrays of 1-3 that was fast and accurate and was based on acquired canonical patterns; a response to arrays 4 to 6 or 7 that was based on mental counting; and an estimating response for arrays larger than 6 that could be held in consciousness for mental counting. (36 ref)

Mental rotation is a spatial skill regarded as essential for science and math achievement. The average percentage of correct choices increased from 54 percent at age 3 to 69 percent at age 4 and 83 percent at age 5.

..Whole number arithmetic, fractions, simple algebra, and measurement, with performance on these tests predicting employability and wages in adulthood, controlling for other factors.