Student Understanding of Linear Scale in Mathematics: Exploring What Year 7 and 8 Students Know

Abstract

This thesis set out to undertake a curriculum review. Scale was chosen as the focus of the review because the Mathematics and Science Taskforce (1997) indicated that measurement was an area of the curriculum that needed priority attention, however comparatively little had been done to provide this by 2005. Scales are a common mathematical object. A search of many (Western) homes is likely to find a variety of scales being used for different purposes. They can be found on car dashboards, kitchen stoves, tape measures, and clocks. They will be found in graphs, newspapers, and magazines. Scales also underpin important mathematical learning over and above their everyday application to measurement and graphing. For example, concepts like gradient, rate of change, functions, and limits all rely on an understanding of scale. But how much do we know about how students learn to use scales? What does learning about scale involve? The thesis approaches the review through an exploration of student understanding of scale in the context of mathematics. It focuses on answering the question "what understanding of scales do students in Year 7 and 8 at the case study schools have?" A definition of scale is developed and is based on the mathematics to which the curriculum document indicates Year 7 and 8 students should have been exposed. This could be identified as the notion of a linear scale. Students in Years 7 and 8 were chosen because by that age the mathematics curriculum document implies students should have a good understanding of scale; at the same time, their errors and misconceptions are likely to indicate learning barriers that need to be addressed. The literature and the New Zealand mathematics curriculum were used to define a construct of scale appropriate to explore with Year 7 and 8 students. Two tests were then developed to measure understanding of that construct. Where possible, items were initially developed or adapted from the literature then, when early findings suggested new avenues for exploration, new items were developed to investigate further issues of understanding. Both tests were used at different schools in the format of a cognitive interview; Test 1 was also used at another school as a written test. Additional items were developed to use with groups of teachers in an attempt to challenge early findings; these teacher trials used a third assessment format requiring both a written answer and a written explanation of method. In Test 1 students were assessed with pairs of items. In each pair one item used a decontextualised number line, the other a measurement or graphing context promoted by the curriculum. During the cognitive interviews, the verbal responses of students were recorded on audiotape, while field notes were used to capture non-verbal data. Follow-up probe questions were used to clarify solution strategies and the understanding underlying these strategies. The written test was then used to identify if the interpretations made could be transferred. Test 2 repeated the data collection methodology from Test 1 but used a different structure within the test. In total, 45 cognitive interviews and 81 written tests were undertaken with students, while 32 teachers participated in the teacher trials. Facilities, point bi-serial correlation coefficients, and levels of significance were used to ascertain the fitness for purpose of the developed test items. Data collected during the cognitive interviews were also analysed using both qualitative and quantitative methods to ascertain patterns of response. The mathematics curriculum in New Zealand had assumed that students would develop understanding about scale from exposure to scales in measurement and graphing situations. This approach might have been appropriate if a scale was a simple (or single) object to be mastered but it is not. A scale is a mathematical tool of vast flexibility that can be applied in numerous forms to a wide range of situations. The results suggest that teaching of scale needs to be more deliberate, and also needs to be considered when curricula are designed. A high proportion of the students in the study had not developed the expected understanding of scale by Years 7 and 8. A complex series of factors were identified that impacted on how the students worked with scale. These included: their understanding of number and number symbols; their understanding of the measurement conventions that are foundational to scale; the strategies they had developed to partition unmarked intervals; their strategies to decide on the value of a partition in marked intervals; their understanding of the role of the marks and spaces on the scale; and their ability to iterate a unit. These different bodies of understanding needed to be integrated and used in a coordinated manner for the students to become effective users of scale.