Comparisons of a Physical Model and Computer Representationsin Reasoning and Learning About Linear Functions

拟议的研究将调查学生在不同情况下对定量变量和函数的推理，以支持理解：数量与线性函数相关的物理设备，包含数值信息的设备的计算机模拟，以及仅呈现数值信息的基于计算机的“函数机”。研究将集中在这些模型和表示为推理和学习函数提供资源的不同方式上。学生将以两人一组的形式接受面试，并针对这三个系统提出一系列平行的问题和问题。参加面试的学生包括没有学过代数的初中生和学过一年代数的高中生。教师将被要求推荐数学成绩处于全国平均水平的学生。在一项研究中，学生们将用他们选择使用的任何额外表示来回答问题和解决问题。在随后的两个研究中，学生将学习构造(1)公式和(2)图来表示情景中数量之间的函数关系。第四项研究将在本赠款期间开始，将把学习公式和图表作为系统的表示结合起来。关于学生推理和学习的结论将基于他们在解决问题和回答问题方面的成功，他们对他们的解决方案和答案的解释，以及他们用来指代定量性质和关系的语言术语。最后，这项工作将集中在学生推理和理解的特定方面，这些方面是相似的和不同的情况，并将差异与情况提供的不同类型的信息和活动联系起来。

The proposed research will investigate students' reasoning about quantitative variables and functions in different situations that support understanding: a physical device in which quantitities are related by linear functions, a computer simulation of the device that includes numerical information, and a computer-based "functions machine" that presents only numerical information. The research will focus on the different ways that these models and representations present resources for reasoning and learning about functions. Students will be interviewed in pairs, with parallel series of problems and questions for the three systems. Students who will be interviewed will include middle-school students who have not studied algebra and high-school students who have studied algebra for one year. Teachers will be asked to recommend students whose performance in mathematics is about at the average of national norms. In one study, students will answer questions and solve problems with the systems with whatever additional representations they choose to use. In two subsequent studies, students will learn to construct (l) formulas and (2) graphs to represent functional relations among quantitities in the situations. A fourth study, to be begun in this grant period, will combine learning of formulas and graphs as representations of the systems. Conclusions about the students' reasoning and learning will be based on their success in solving problems and answering questions, the explanations they give of their solutions and answers, and the linguistic terms that they use to refer to quantitative properties and relations. Finally, the work will focus on specific aspects of students' reasoning and understanding that are similar and different situations, relating differences to the different kinds of information and activity that the situations provide.