Abstract: This thesis reports on a longitudinal study of students' understanding of decimal notation. Over 3000 students, from a volunteer sample of 12 schools in Victoria, Australia, completed nearly 10000 tests over a 4-year period. The number of tests completed by individual students varied from 1 to 7 and the average inter-test time was 8 months. The diagnostic test used in this study, (Decimal Comparison Test), was created by extending and refining tests in the literature to identify students with one of 12 misconceptions about decimal notation.
Particular longitudinal measures and definitions of the prevalence of misconceptions were adapted from the medical literature. These measures were further refined to overcome the effect of repeated testing (which resulted in a 10% improvement) as well as various sampling issues. Analysis was conducted at both the coarse level (4 behaviours) and fine level (12 ways of thinking).
Improved estimates of the prevalence of expertise as well as for the various misconceptions are provided. Only 30% of Grade 6 students and 70% of Grade 10 students demonstrate expertise on this test and about 25% of students between Grades 7 and 10 completed tests by choosing the decimal with the fewest digits as the largest number, a behaviour which results from several different ways of thinking. Despite its high prevalence, this particular behaviour is not well known amongst teachers.
Three phenomena were investigated: persistence, hierarchy and regression. The misconceptions which are most persistent are those that involve the treatment of the decimal portion of a number as a whole number. A hierarchy of the misconceptions was determined by considering the relative rate to expertise on the next test: the hierarchy is different for primary and secondary students. About 20% of students were involved in regression, that is, they completed one test as an expert, but were unable to do so on a later test. This analysis provides additional evidence that many students are receiving teaching that covers over rather than overcomes their misconceptions. For example, some students appear to be following algorithms for comparing decimal numbers (such as rounding to two decimal places), but revert to a latent misconception when their incomplete algorithm fails. Furthermore, support is provided for the hypothesis that some misconceptions are due to the interference of new teaching.
Steinle, V. (2004). Changes with Age in Students’ Misconceptions of Decimal Numbers. Unpublished doctoral thesis, University of Melbourne.