Children's understanding and application of the mathematical concepts of inversion and associativity

算术概念的发展是儿童算术知识发展的一个重要方面。在儿童的数学认知领域，重点一直放在加法概念的发展上，而不是乘法概念，但乘法概念更复杂，更难以为儿童所理解。此外，研究儿童的概念发展如何与算术的程序性和事实性知识结合起来，对于理解儿童的数学知识是至关重要的。我的研究计划旨在通过调查反转和关联性概念的加法和乘法版本如何随着时间的推移与程序和事实知识一起发展来解决这两个问题。反演的概念，即成对的操作是相互反向相关的，似乎是掌握，甚至在正规学校教育之前，当概念的加法方面是评估的问题，如a + B - B。然而，当出现乘法问题，如d x e，儿童似乎对反演概念的理解要弱得多。结合性的概念，即任何一对数都可以在a + B - c和d x e ∈ f的形式的问题中首先得到解决，在加法和乘法问题类型上似乎都很弱。有证据表明，对一个概念的理解与对另一个概念的理解有关。此外，有一些证据表明，对这些概念的理解和应用与对调查中的行动的事实和程序知识有关。本研究计划的目标是跟踪儿童对这两个概念的理解发展，并分离和确定如何增强或促进概念发展。孩子们对算术运算如何相互关联的扎实理解是他们在以后的正规学校教育中对数学问题和概念的进一步和更复杂理解的关键。

The development of arithmetic concepts is an essential aspect of children's arithmetic knowledge. In the area of children's mathematical cognition, much of the emphasis has been on the development of additive concepts rather than on multiplicative concepts and yet multiplicative concepts are more complex and difficult for children to understand. Further, research on how children's conceptual development occurs in conjunction with procedural and factual knowledge of arithmetic is critical to understanding children's mathematical knowledge as a whole. My research program aims to address both of these issues by investigating how both additive and multiplicative versions of the concepts of inversion and associativity develop across time in conjunction with procedural and factual knowledge. The concept of inversion, that pairs of operations are inversely related to each other, appears to be grasped, even before formal schooling when the additive aspect of the concept is assessed on problems such as a + b - b. However, when multiplicative problems such as d x e ÷ e are presented, children appear to have a much weaker understanding of the inversion concept. The concept of associativity, that any pair of numbers can be solved first in problems of the form a + b - c and d x e ÷ f, appears to be weak on both additive and multiplicative problem types. There is some evidence that understanding of one concept is related to understanding of the other. Further, there is some evidence that the understanding and application of the concepts is related to factual and procedural knowledge of the operations under investigation. The goal of this research program is to follow children's developing understanding of both concepts and to isolate and identify how concept development can be enhanced or promoted. Children's solid understanding of how the arithmetic operations relate to one another is key to their further and more complex understanding of the mathematic problems and concepts that they will be exposed to in their later years of formal schooling.