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

儿童对反演和结合性数学概念的理解和应用

• 批准号: 261626-2008
• 负责人: Robinson, Katherine
• 金额: $ 1.09万
• 依托单位: University of Regina
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2008
• 资助国家: 加拿大
• 起止时间: 2008-01-01 至 2009-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=395138
• 关键词: Children; understanding; application; mathematical; concepts

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.

算术概念的发展是儿童算术知识的重要方面。在儿童的数学认知领域，大部分的重点一直是发展加法概念，而不是乘法概念，然而乘法概念对儿童来说更加复杂和难以理解。此外，研究儿童的概念发展是如何与算术的程序和事实知识一起发生的，对于从整体上理解儿童的数学知识至关重要。我的研究项目旨在通过调查反转和关联性概念的加法和乘法版本如何结合程序性和事实知识随时间发展来解决这两个问题。甚至在正规学校之前，当对a+b-b等问题的加法方面进行评估时，人们似乎就已经掌握了倒置的概念，即运算对彼此成反比。然而，当提出乘法问题时，如d x e？e，孩子们对倒置概念的理解似乎要弱得多。结合性的概念，即在a+b-c和d xe？f形式的问题中，任何一对数字都可以首先求解，在加法和乘法问题类型中似乎都是弱的。有一些证据表明，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，对概念的理解和适用与所调查行动的事实和程序知识有关。这个研究项目的目标是跟踪儿童对这两个概念的发展理解，并分离和确定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扎实理解是他们进一步和更复杂地理解数学问题和概念的关键，他们将在以后的正规学校学习中接触到这些问题和概念。