The Learning Trajectories of Similarity in Mathematics Curriculum: An Epistemological Analysis of Hong Kong Secondary Mathematics Textbooks in the Past Half Century

Qiao-Ping Zhang,Ngai-Ying Wong

doi:10.3390/math9182310

Abstract

The topic of similarity plays an essential role in developing students’ deductive reasoning. However, knowing how to teach similarity and understanding how to incorporate deductive reasoning and proof along with plane geometry remain a challenge to both school curriculum creators and teachers. This study identified the problems and characteristics regarding how similarity is treated in secondary mathematics textbooks in Hong Kong in the past half century. The content analysis method was used to analyze six secondary mathematics textbook series published in different periods. From the epistemological perspective of the textbook contents, our analysis shows the historical context and learning trajectories of how similarity was treated in school curriculum. The natural axiomatic geometry paradigm is not emphasized too much at different stages and most of the textbooks did not provide formal proofs of similarity. The intuitive idea was gradually consolidated into a formal definition of similarity. Furthermore, the way that rigorous geometric deduction can be performed from intuitive concepts and experimental geometry to the idea of proofs and formal proofs is also discussed.

Highlights

Debates and changes in the goals of geometry instruction have arisen during the twentieth century [1]
Before we explore which of the above definitions should be used as a starting point to introduce similarity in school mathematics, we shall keep in mind that whatever is defined in geometry is not always the same as that adopted in textbooks
The analysis and results in this paper contribute to a greater understanding of how deductive reasoning and the topic of similarity is arranged in mathematics textbooks

Summary

Introduction

Debates and changes in the goals of geometry instruction have arisen during the twentieth century [1]. Even though deductive reasoning plays a central position in school geometry, how to teach deductive reasoning does not always have a clear process It is still not clear which concepts or definitions should be taught at the beginning stage before teaching students how to write proofs in school teaching, as well as which statement should be used as a starting point to deduce another theorem. It is discussed whether teachers shall emphasize the rigid logic reasoning process or students’ intuitive and hands-on experience. To understand the nature of similarity, we should first define the concept

Methods

Results

Discussion

Conclusion