This study describes how 24 third graders (8–9 years old) relate and represent the relationships between variables when working with a functional thinking problem. This aspect contributes to providing insights about how elementary school students attend properties and relationships between covarying quantities rather than isolated computations. From a functional approach to early algebra, we describe written students’ answers when working with a problem that involves a function, which includes questions for specific values and to generalize. Design research guidelines, specifically those set out for Classroom Teaching Experiment were followed. This study addresses the fourth and last Classroom Teaching Experiment session, which involved a function of the type y = ax + b and students had not previously worked it. Students primarily evidenced correspondence relationship, using natural language and numerical representation to express this functional relationship. Our findings let us to state that (a) although students were not used to working with these types of problems, eleven of them go beyond arithmetic computations, finding relationships that relate the variables; and (b) three students generalized using natural language as a useful vehicle, while there are other students who perceived the same regularity for different specific values but they are unable to represent generalization clearly.

这项研究描述了24名三年级学生（8-9岁）在处理功能性思维问题时如何相互联系并代表变量之间的关系。这方面有助于提供有关小学生如何参加可变变量之间的属性和关系的见解，而不是孤立的计算。从功能性方法到早期代数，我们描述书面学生在处理涉及功能的问题时所得到的答案，其中包括针对特定价值的问题并进行概括。遵循设计研究指南，特别是针对课堂教学实验的研究指南。本研究针对第四和最后一个课堂教学实验课程，其中涉及类型y = ax + b的函数而且学生以前没有工作过。学生主要通过自然语言和数字表示来表达这种功能关系，从而证明了对应关系。我们的发现让我们指出：（a）尽管学生不习惯处理这类问题，但其中有11个超出了算术计算的范围，找到了与变量相关的关系；（b）三名学生使用自然语言作为有用的工具进行了概括，而另一些学生则对不同的特定值感觉到相同的规律性，但无法清晰地表示概括。