Why it is important for in-service elementary mathematics teachers to understand the equality .999… = 1

Researchers conducted semi-structured interviews with in-service fifth grade teachers. The purpose of these interviews was to examine teachers’ reactions to arguments that .999… = 1. Previously reported results indicate that some pre-service elementary school teachers possess misunderstandings about mathematical issues related to decimals with single repeating digits. This research investigates whether some in-service teachers possess misunderstandings about mathematical issues related to .999…. This paper reports on one instance of a teacher whose responses indicate that the teacher's sense of number and sense of measurement are intertwined, resulting in fragile understanding of repeating decimals. These data present evidence that teachers continue to develop repeated decimal understandings and misunderstandings throughout their careers, and that the curriculum, everyday experience, and perceptions of student learning combine to form or reinforce these understandings. Because decimals with a single repeating digit (e.g. .333… and .666…) are an integral part of the elementary mathematics curriculum, we argue that it is important that in-service elementary mathematics teachers have a clear understanding of concepts related to the concept of infinity as they emerge through the study of the equality .999… = 1.

► Cognitive disequilibrium about the relationship between .999… and 1 can reveal misconceptions about repeating decimals.
► Some teachers continue to develop repeated decimal understandings and misunderstandings throughout their careers.
► Curriculum, experience, and perceptions of student learning combine to form or reinforce repeated decimals understandings.
► A sense of number and a sense of measurement can intertwine to form a fragile understanding of repeating decimals.
► Less than accurate elementary teaching can develop from teacher misunderstandings about repeated decimals.