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的一个重要方面。在儿童的数学认知领域，许多重点一直放在加法概念的发展上，而不是乘法概念，然而乘法概念对儿童来说更复杂，更难理解。此外，研究儿童的概念发展如何与程序性和事实性算术知识相结合，对于整体理解儿童的数学知识至关重要。我的研究计划旨在通过研究反转和联想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。反转的概念，即成对的运算彼此成反比，似乎在正式上学之前就掌握了，当概念的加法方面被评估为a + b - b等问题时。然而，当提出乘法问题如d × e ÷ e时，孩子们对反转概念的理解似乎要弱得多。结合律的概念，即在a + b - c和d × e ÷ f的形式的问题中，任何一对数字都可以首先得到解，在加法和乘法问题类型上似乎都是弱的。有证据表明，对一个概念的理解与对另一个概念的理解是相关的。此外，有一些证据表明，对这些概念的理解和应用与对正在调查的行动的事实性和程序性知识有关。这个研究项目的目标是跟踪儿童对这两个概念的理解发展，并找出如何加强或促进概念发展。孩子们对算术运算如何相互关联的深刻理解是他们在以后的正规学校教育中将接触到的对数学问题和概念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.