Jb Moritz

The Longitudinal Development of Understanding of Average

The development of the understanding of average was explored through interviews with 94 students from Grades 3 to 9, follow-up interviews with 22 of these students after 3 years, and follow-up interviews with 21 others after 4 years. Six levels of response were observed based on a hierarchical model of cognitive functioning. The first four levels described the development of the concept of average from colloquial ideas into procedural or conceptual descriptions to derive a central measure of a data set. The highest two levels represented transferring this understanding to one or more applications in problem-solving tasks to reverse the averaging process and to evaluate a weighted mean. Usage of ideas associated with the three standard measures of central tendency and with representation are documented, as are strategies for problem solving. Implications for mathematics educators are discussed.

Development of understanding of sampling for statistical literacy

Abstract The development of understanding sampling is explored through responses to four items in a longitudinal survey administered to over 3000 students from Grades 3 to 11. Responses are described with reference to a three-tiered framework for statistical literacy, including defining terminology, applying concepts in context, and questioning claims made without proper justification. Within each tier increasing complexity is observed as students respond with single, multiple, and integrated ideas to four different tasks. Implications for mathematics educators of the development of sampling concepts across the years of schooling are discussed.

The development of chance measurement

This paper presents an analysis of three questionnaire items which explore students’ understanding of chance measurement in relation to the development of ideas of formal probability. The items were administered to 1014 students in Grades 3,6 and 9 in Tasmanian schools. The analysis, using the NUD•IST text analysis software, was based on the multimodal functioning SOLO model. An analysis of the results and a developmental model for understanding chance measurement are presented, along with implications for curriculum and teaching practice.

A Model for Assessing Higher Order Thinking in Statistics

ABSTRACT As in other areas of the school curriculum, the teaching, learning and assessment of higher order thinking in statistics has become an issue for educators following the appearance of recent curriculum documents in many countries. These documents have included probability and statistics across all years of schooling and have stressed the importance of higher order thinking across all areas of the mathematics curriculum. This paper reports on a pilot project which applied the theoretical framework for cognitive development devised by Biggs and Collis to a higher order task in data handling in order to provide a model of student levels of response. The model will assist teachers, curriculum planners and other researchers interested in increasing levels of performance on more complex tasks. An interview protocol based on a set of 16 data cards was developed, trialed with Grade 6 and 9 students, and adapted for group work with two classes of Grade 6 students. The levels and types of cognitive function...

School Students' Reasoning about Conjunction and Conditional Events.

The objective of this study was to provide baseline data in an area of the mathematics curriculum that is beginning to receive greater attention than previously. Four survey items were completed by 2615 students in grades 5 to 11. Two survey items asked for estimates of probability or frequency for everyday events (A), (B), and their conjunction (A and B). Two survey items asked for estimates of probability or frequency for conditional events, (X|Y) and (Y|X). Cross-sectional and longitudinal analyses revealed improvement with grade in expressing probability numerically and in distinguishing conditional events, but no change in incidence of conjunction errors. The relationships of responses to conjunction items with those to conditional items, and of both with responses to other items of basic chance measurement were considered. Implications were related to interpretation of the results in terms of previous research and suggestions for educators.

Development of reasoning associated with pictographs: Representing, interpreting, and predicting

A developmental model involving four response levels is proposed concerning how students arrange pictures to represent data in a pictograph, how they interpret these pictographs, and how they make predictions based on these pictographs. The model is exemplified by responses from three related interview-based studies. In Study 1, examples of each response level are provided from 48 preparatory- to tenth-grade students. Students from higher grades were more likely to respond at higher levels. In Study 2, 22 students were interviewed longitudinally after a three-year interval; many improved in response level over time, although a few responded at lower levels. In Study 3, 20 third-grade students were interviewed and then prompted with conflicting responses of other students on video; many improved their initial responses to higher levels after exposure to the conflicting prompts. Associations among levels of representing, interpreting, and predicting were explored. Educational implications are discussed concerning reasonable expectations of students and suggestions to develop these skills in students at different grades.

Constructing Coordinate Graphs: Representing Corresponding Ordered Values with Variation in Two-Dimensional Space

Coordinate graphs of time-series data have been significant in the history of statistical graphing and in recent school mathematics curricula. A survey task to construct a graph to represent data about temperature change over time was administered to 133 students in Grades 3, 5, 7, and 9. Four response levels described the degree to which students transformed a table of data into a coordinate graph.Nonstatistical responses did not display the data, showing either the context or a graph form only.Single Aspect responses showed data along a single dimension, either in a table of corresponding values, or a graph of a single variable.Inadequate Coordinate responses showed bivariate data in two-dimensional space but inadequately showed either spatial variation or correspondence of values.Appropriate Coordinate graphs displayed both correspondence and variation of values along ordered axes, either as a bar graph of discrete values or as a line graph of continuous variation. These levels of coordinate graph production were then related to levels of response obtained by the same students on two other survey tasks: one involving speculative data generation from a verbal statement of covariation, and the other involving verbal and numerical graph interpretation from a coordinate scattergraph. Features of graphical representations that may prompt student development at different levels are discussed.