1. Background Research into students' beginning intuitions about distribution has generally been associated with variation in single variable settings (e.g., Ben-Zvi & Sharett-Amir, 2005; Watson & Kelly, 2005) and has often focused on graphing attributes of students' created representations. Kelly and Watson (2002) for example found that students' graphs to represent the imagined outcomes of repeated sampling trials in a probability setting ranged from idiosyncratic drawings of the physical scenario, to time-series type graphs inconsistently justified by " more " of a certain characteristic, to informal graphing based on " middle, " to a conventional distribution recognizing variation and center. In sampling or measurement investigations focusing mainly on single variable distributions, Shaughnessy (2006) described six aspects of variation as (i) extremes or outliers, (ii) change over time, (iii) the whole range, (iv) the likely range, (v) distance or difference from some fixed point, or (vi) sums of residuals. These descriptors are not seen as hierarchical and inform this study to assist in characterizing the story that students attempt to tell with the graphs they produce to represent a verbal description of covariation. This paper, in moving from the consideration of the distribution associated with a single variable to that associated with two variables, builds on earlier research on correlational reasoning and its representation (e.g., Ross and Cousins, 1993a, 1993b). Shaughnessy's (2006) aspects of variation also apply when two variables are involved, seen as the trend in the relationship between the two variables, and seen as deviations from the trend. Various researchers asked students to create graphical distributions from data values. Brasell and Rowe (1993), for example, asked physics students to construct a graph of five paired values representing the heights from which a ball was dropped and the height to which it rebounded; they found students drew pictures, produced poorly labeled graphs, or plotted points, but rarely gave evidence of graphing to show a trend. Rather than starting with data values, Mevarech and Kramarsky (1997) asked grade 8 students to graph four different verbal claims about trend relationships of time spent studying and the marks received at school. Whereas slightly over half of students appropriately graphed a positive, negative, or no association, fewer than half graphed a curvilinear association. The most common errors observed related to graphing only a single point, to graphing only a single variable, or to graphing an increasing function regardless of the conditions set in the …
Comprehension of chance language, such as is found in newspapers, is a fundamental aspect of statistical literacy. In this study, students' understandings of chance language were explored through responses to two items in surveys administered to 2,726 students from grades 5 to 11. One item involved evaluating the chance expressed in phrases from newspaper headlines using a number line, and responses were described in four levels of chance language evaluation. The other item involved interpreting, in context, an expression of percent chance, and responses were described in four levels of chance language interpretation. Students in higher grades were more likely to demonstrate higher levels of both evaluation and interpretation. The association between levels of evaluation and interpretation was further explored generally and in relation to one of the headlines involving percent. Implications for mathematics educators in relation to chance language in the curriculum across the years of schooling are discussed.
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.
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.
Quantitative literacy involves not just basic number skills, often called numeracy, but also the ability to integrate basic skills in contexts that require high levels of literacy to interpret situations and make judgments. This paper discusses the development, implementation, and evaluation of a unit designed to assist preservice teachers of mathematics to see the relevance of quantitative literacy to daily life, to understand and discuss aspects of quantitative literacy themselves, and to prepare and deliver lessons in schools using an internet site as a teaching resource. Forty pre-service teachers (a) selected a newspaper article and developed student questions and brief comments for discussion with teachers, identifying issues for mathematics teaching; (b) developed a teaching unit for one or more class lessons and implemented it in the classroom; (c) prepared an evaluation of the experience; and (d) discussed their colleagues’ work in relation to these tasks online. Responses indicated most of the pre-service teachers could engage with these tasks and came to believe that they supported effective teaching in the classroom. Some difficulties are identified, however, related to the implementation of the project, and suggestions are made for future variations.
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.
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.
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.
Bar graphs are commonly used throughout society as a simple tool for representing data. How do students in early childhood interpret bar graphs in a familiar context? Thirty Grade 3 students were interviewed to explore their interpretations of a bar graph of how children travel to school and their predictions for various changed circumstances. Student responses to the task reflected three levels of graph comprehension: reading the data, reading between the data, and reading beyond the data.
Two survey items askingfor estimates of probability or frequency of everyday events (A), (B), and their conjunction,(A and B), were completed by 2719 school students in grades 5 to 11. Cross-sectional and longitudinal analyses revealed chance expression improved with grade, but no change in incidence of conjunction errors. Gender differences favouring males occurred for some grades. Comparisons with responses to other probability items indicated incidence of conjunction errors is independent of development of basic chance measurement. The conjunction fallacy arises in contexts where probabilities are considered for two events and their intersection (conjunction). According to classical probability theory, if sets A and B are defined, it is necessary that the conjunction set (A and B) is a subset of A and of B, thus peA and B) is necessarily less than or equal to both peA) and P(B). Previous research has found that people often violate this principle using reasoning based on a conjunction fallacy. Tversky and Kahneman (1983) and subsequent researchers used problems in social contexts, often variations of a problem that involved a character description ofLinda, with respondents asked to judge the likelihood that Linda has various occupations (A), hobbies (B), or both (A and B). Tversky and Kahneman (1983) found that around 90% of university students violated the conjunction rule under certain conditions. They considered that some respondents may be averaging peA) and P(B) to arrive at peA and B), or thinking of the causal relations between A and B, thus rating (A and B) as more typical than (A) and (B) for the character description given. Their major finding, however, was that fewer conjunction errors (only 11 %) occurred for simpler questions requesting frequency rather than probability estimates. They interpreted this as evidence that responses to probability items were often based on reasoning of typicality and intentional meaning of tenns, using a more general representative heuristic, whereas responses to frequency items were based on extensional referents, which are countable.
Two survey items asking for estimates of probability or frequency of conditional events, (AIR) and (BIA), were completed by 2719 school students in grades 5 to 11. Cross-sectional and longitudinal analyses revealed improvement with grade in expressing probability numerically and in distinguishing conditional events. Conditional events were better distinguished for the frequency item than the probability item. Comparisons with responses to other probability items indicated understanding of conditional probability was related to development of basic chance measurement. Historically understanding of conditional probability was explored with tertiary students. Not only were tertiary students readily available as participants in the researchers' classes but also there was a lack of emphasis on probability in the school curriculum generating little interest by mathematics education researchers. Over the last decade, however, mathematics curricula of most western countries have recognised the importance of chance and probability generally and included specific reference to conditional probability. In Mathematics - A Curriculum Profile for Australian Schools (Australian Education Council, 1994) secondary students are expected to assign conditional probabilities based on data in two-way tables (level 7.23, p. 124). In the Curriculum and Evaluation Standards for School Mathematics (National Council of Teachers of Mathematics, 1989) in the United States, recommendations for grades 9 to 12 include addressing probabilistic intuitions (Fischbein, 1975; Kahneman & Tversky, 1972). Concepts of probability, such as independent and dependent events, and their relationship to compound events and conditional probability should be taught intuitively. Fonnal definitions and properties should be developed only after a firm conceptual base is established ... (National Council of Teachers of Mathematics, 1989, p. 171) What it means to teach or learn intuitively however may still be open to debate following the research of Fischbein and others in probability generally (Fischbein, 1975; Fischbein & Gazit, 1984; Fischbein & Schnarch, 1997) and more specifically in relation to conditional probability (Bar-Hillel & Falk, 1982; Falk, 1986). Findings show mixed results in terms of the development of intuitive understanding; for example, Fischbein and Schnarch (1997) found diminished performance in relation to some concepts with increasing age. In discussing misconceptions associated with significance testing, Falk (1986) suggested that much of the difficulty is related to confusing conditional probabilities and equating the probability of rejecting a null hypothesis when it is true, P(RIHo), with the probability of it being true when it is rejected, P(HoIR). The contexts in which conditional problems have been set have varied widely. In questioning middle school students' understanding of replacement and non-replacement aspects of conditional probability, Fischbein and Gazit (1984) and Tarr and Jones (1997) employed events of drawing objects from bags or choosing among alternatives in games. In questioning tertiary students who had not undertaken a statistics course, Pollatsek, Well, Konold, Hardiman, and Cobb (1987) employed events from everyday social settings, such as having green eyes, for which some degree of topic knowledge was assumed. No studies were found that used everyday definitions of events in terms of social characteristics with school students.
The development of school students' understanding of comparing two data sets is explored through responses of students in individual interview settings. Eighty-eight students in grades 3 to 9 were presented with data sets in graphical form for comparison. Student responses were analysed according to a developmental cycle which was repeated in two contexts: one where the numbers of values in the data sets were the same and the other where they were different. Strategies observed within the developmental cycles were visual, numerical, or a combination of the two. The correctness of outcomes associated with using and combining these strategies varied depending upon the task and the developmental level of the response. Implications for teachers, educational planners and researchers are discussed in relation to the beginning of statistical inference during the school years.
ABSTRACT 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 functioning associated with the outcomes achieved by students completing the task in the two contexts will be discussed, as will the implications for classroom teaching and for further research. Additional informationNotes on contributorsJane M. WatsonThis material is based on work supported by the Australian Research Council, Grant Number A79231392, the National Center for Research in Mathematical Sciences Education ‐ Models of Authentic Assessment Working Group (University of Wisconsin) and the Department of Education and the Arts in Tasmania. Any opinions, findings and conclusions or recommendations expressed are those of the authors and do not necessarily reflect the views of the supporting organisations. An earlier version of this paper was presented at the annual meeting of the Australian Association for Research in Education, Newcastle, NSW, Australia, in November, 1994.
