Development of Mathematical Knowledge in Young Children: Attentional Skill and the Use of Inversion

The principle of inversion, that a + b ? b must equal a, is a fundamental property of arithmetic, but many children fail to apply it in symbolic contexts through 10 years of age. We explore three hypotheses relating to the use of inversion that stem from a model proposed by Siegler and Araya (2005 Siegler , R. S. , &; Araya , R. ( 2005 ). A computational model of conscious and unconscious strategy discovery . In R. V. Kail (Ed.), Advances in child development and behavior (pp. 1 – 42 ). New York , NY : Elsevier .[Crossref] , [Google Scholar]). Hypothesis 1 is that greater calculational skill is related to greater use of inversion. Hypothesis 2 is that greater attentional skill is related to greater use of inversion. Hypothesis 3 is that the relation between attentional skill and the use of inversion is particularly strong among children with high skill in calculation. We found suggestive evidence for Hypothesis 2 and clear evidence for Hypothesis 3, indicating that for children who are strong at calculation, attentional flexibility is related to use of inversion.     

Knowledge of counting principles: How relevant is order irrelevance?

Most children who are older than 6 years of age apply essential counting principles when they enumerate a set of objects. Essential principles include (a) one-to-one correspondence between items and count words, (b) stable order of the count words, and (c) cardinality—that the last number refers to numerosity. We found that the acquisition of a fourth principle, that the order in which items are counted is irrelevant, follows a different trajectory. The majority of 5- to 11-year-olds indicated that the order in which objects were counted was relevant, favoring a left-to-right, top-to-bottom order of counting. Only some 10- and 11-year-olds applied the principle of order irrelevance, and this knowledge was unrelated to their numeration skill. We conclude that the order irrelevance principle might not play an important role in the development of children’s conceptual knowledge of counting.     

What counts as knowing? The development of conceptual and procedural knowledge of counting from kindergarten through Grade 2

The development of conceptual and procedural knowledge about counting was explored for children in kindergarten, Grade 1, and Grade 2 (N = 255). Conceptual knowledge was assessed by asking children to make judgments about three types of counts modeled by an animated frog: standard (correct) left-to-right counts, incorrect counts, and unusual counts. On incorrect counts, the frog violated the word-object correspondence principle. On unusual counts, the frog violated a conventional but inessential feature of counting, for example, starting in the middle of the array of objects. Procedural knowledge was assessed using speed and accuracy in counting objects. The patterns of change for procedural knowledge and conceptual knowledge were different. Counting speed and accuracy (procedural knowledge) improved with grade. In contrast, there was a curvilinear relation between conceptual knowledge and grade that was further moderated by children's numeration skills (as measured by a standardized test); the most skilled children gradually increased their acceptance of unusual counts over grade, whereas the least skilled children decreased their acceptance of these counts. These results have implications for studying conceptual and procedural knowledge about mathematics.     

Multiplication by eye and by ear for Chinese-speaking and English-speaking adults.

English-speaking (n = 32) and Chinese-speaking adults (n = 32) solved single-digit multiplication problems. In one condition, problems were presented as visual digits (e.g., 8 x 9). In the other condition, problems were presented as auditory number words in the participant's first language (e.g., /eit/ /taimz/ /nain/). Chinese-speaking adults made proportionately more operand-intrusion errors (e.g., 4 x 8 = 24) than English-speaking adults. Both groups made more operand-intrusion errors with auditory than with visual presentation. These findings are similar to those found when participants solve problems presented as visual number words (e.g., eight x nine), suggesting that in both cases the activation of phonological codes interferes with processing.