Investigates students' understanding of the idea of average. Relationship between data and the average of the data; Confusion of the term with mode and median.
Investigates students' understanding of the idea of average. Relationship between data and the average of the data; Confusion of the term with mode and median.
We distinguish two conceptions of sample and sampling that emerged in the context of a teaching experiment conducted in a high school statistics class. In one conception 'sample as a quasi-proportional, small-scale version of the population' is the encompassing image. This conception entails images of repeating the sampling process and an image of variability among its outcomes that supports reasoning about distributions. In contrast, a sample may be viewed simply as 'a subset of a population' - an encompassing image devoid of repeated sampling, and of ideas of variability that extend to distribution. We argue that the former conception is a powerful one to target for instruction.
In this chapter, I provide a discussion of selected discoveries that have been made about students' conceptions of probability. I also include a discussion of some trends in student performace on probability items on the 1996 National Assessment of Educational Progress (NAEP). In light of what is known about student thinking and performance in probability, I present some suggestions for the teaching of probability, I point out that a separation of research discussions of probability and statistics is artificial, just as artificial as the separation of data and chance when teaching. Principles and Standards for School Mathematics (NCTM, 2000) aptly places probability and statistics under one shared heading. I believe the most interesting research questions for the future reside in the joint realm of the areas of probability and statistics, just as the most interesting teaching challenges for the future lie in making interconnections between these two areas.
As part of a research project on students' understanding of variability in statistics, 272 students, (84 middle school and 188 secondary school, grades 6 - 12) were surveyed on a series of tasks involving repeated sampling. Students' reasoning on the tasks predominantly fell into three types: additive, proportional, or distributional, depending on whether their explanations were driven by frequencies, by relative frequencies, or by both expected proportions and spreads. A high percentage of students' predominant form of reasoning was additive on these tasks. When secondary students were presented with a second series of sampling tasks involving a larger mixture and a larger sample size, they were more likely to predict extreme values than for the smaller mixture and sample size. In order for students to develop their intuition for what to expect in dichotomous sampling experiments, teachers and curriculum developers need to draw explicit attention to the power of proportional reasoning in sampling tasks. Likewise, in order for students to develop their sense of expected variation in a sampling experiment, they need a lot of experience in predicting outcomes, and then comparing their predictions to actual data.
This paper summarizes the thinking of 84 middle school mathematics students' about variability in three stochastics tasks that involve repeated trial. Differences in students' acknowledgement of variability were found, depending on whether the task was from a sampling environment, or a probability environment. Students' tended to neglect variability in the probability environment. We conjecture that the way that probability is normally introduced to students is part of the cause of this phenomenon.
Are you increasing your emphasis on probability and statistics with students? Are more of your students studying statistics or probability during secondary school? Are your students improving in their performance in data and chance? If you answered yes to any of these three questions, you are not alone, according to the most recent National Assessment of Educational Progress (NAEP). The NAEP is administered approximately every four years to students attending a representative sample of schools across the United States.
Although research has been done on students' conceptions of centers (averages, means, medians), there has not been a corresponding line of research into students' conceptions of variability or spread. In this paper we describe several exploratory studies designed to investigate school students' conceptions of variability. A sampling task that was a variation of an item on the 1996 National Assessment of Educational Progress (NAEP) was given to 324 students in Grades 4 - 6, 9, and 12 from Oregon, Tasmania, and New South Wales. Three different versions of the task were presented in a Before, and in an After setting. The Before and After students did the task both before and after carrying out a simulation of the task. Responses to the sampling task were categorized according to their centers (Low, Five, High) and spreads (Narrow, Reasonable, Wide). Results show a steady growth across grades on the center criteria but no clear corresponding improvement on the spread criteria. There was considerable improvement on the task among the students who repeated it after the simulation. Students' growth on the center criteria may be due to the emphasis that instruction places on centers in school mathematics. Similarly, the lack of clear growth on spreads and variability, and the inability of many students to integrate the two concepts (centers and spreads) on this task, may be due to instructional neglect of variability concepts.
An appreciation of variation is central to statistical thinking. In this study, four students from each of grades 4,6,8 and 10 were interviewed individually on aspects of variation present in three settings. The first setting was an isolated random sampling situation, whereas the other two settings were real worlds sampling situations. Four levels of responding were identified and described in relation to developing concepts of variation. Implications for teaching and future research on variation are considered.
This study follows two earlier studies of school students' abilities to draw inferences when comparing two data sets presented in graphical form (Watson and Moritz, 1999; Watson, 2001). Using the same interview protocol with a new sample of 60 students, 20 from each of grades 3, 6 and 9, cognitive conflict was introduced in the form of video clips of reasoning expressed by students in the earlier studies. This methodology was intended to mimic the type of argumentation that might take place in the classroom but in a controlled setting where identical arguments could be presented to different students. Interviews were videotaped and analysed in a similar fashion to the earlier studies in order to document change associated with the presentation of cognitive conflict. Change was documented with respect to the levels of observed response for two parts of the protocol and for the use of displayed variation in the graphs. Implications of the methodology for future research and teaching are discussed.
This study follows an earlier study of school students' abilities to draw inferences when comparing two data sets presented in graphical form (Watson and Moritz, 1999). Forty-two students who were originally interviewed in grades 3 to 9, were subsequently interviewed either three or four years later. The results for individual student development add to the credibility of the cross-age observations, as well as support the hierarchical framework suggested by the original study. Changes in levels of performance and strategies for drawing conclusions are documented. A further step from the original study is the consideration of how students used the variation displayed in the graphical presentation of the data sets as a basis for decision-making. Implications for teaching and for further research are discussed.