Research

  • This article describes aspects of the statistical content knowledge of 46 preservice elementary school teachers. The preservice teachers responded to a written item designed to assess their knowledge of mean, median, and mode. The data produced in response to the written item were examined in light of the Structure of the Observed Learning Outcome (SOLO) taxonomy (Biggs & Collis, 1982, 1991) and Ma's (1999) conception of Profound Understanding of Fundamental Mathematics (PUFM). The article describes 4 levels of thinking in regard to comparing and contrasting mean, median, and mode. Several different categories of written definitions for each measure of central tendency are also described. Connections to previous statistical thinking literature are discussed, implications for teacher education are given, and directions for further research are suggested.

  • The study describes students' patterns of thinking for statistical problems set in two different contexts. Fifteen students representing a wide range of experiences with high school mathematics participated in problem-solving clinical interview sessions. At one point during the interviews, each solved a problem that involved determining the typical value within a set of incomes. At another point, they solved a problem set in a signal-versus-noise context [Konold, C., & Pollatsek, A. (2002). Data analysis as the search for signals in noisy processes. Journal for Research in Mathematics Education, 33, 259-289]. Several patterns of thinking emerged in the responses to each task. In responding to the two tasks, some students attempted to incorporate formal measures, while others used informal estimating strategies. The different types of thinking employed in using formal measures and informal estimates are described. The types of thinking exhibited in the signal-versus-noise context are then compared against those in the typical value context. Students displayed varying amounts of attention to both data and context in formulating responses to both problems. Suggestions for teachers in regard to helping students attend to both data and context when analyzing statistical data are given.

  • Professional probabilists have long argued over what probability means, with, for example, Bayesians arguing that probabilities refer to subjective degrees of confidence and frequentists arguing that probabilities refer to the frequencies of events in the world. Recently, Gigerenzer and his colleagues have argued that these same distinctions are made by untutored subjects, and that, for many domains, the human mind represents probabilistic information as frequencies. We analyze several reasons why, from an ecological and evolutionary perspective, certain classes of problem-solving mechanism in the human mind should be expected to represent probabilistic information as frequencies. Then, using a problem famous in the "heuristics and biases" literature for eliciting base rate neglect, we show that correct Bayesian reasoning can be elicited in 76% of subjects - indeed, 92% in the most ecologically valid condition - simply by expressing the problem in frequentist terms. This result adds to the growing body of literature showing that frequentist representations cause various cognitive biases to disappear, including overconfidence, the conjunction fallacy, and base-rate neglect. Taken together, these new findings indicate that the conclusion most common in the literature on judgment under uncertainty - that our inductive reasoning mechanisms do not embody a calculus of probability - will have to be re-examined. From an ecological and evolutionary perspective, humans may turn out to be good intuitive statisticians after all.

  • Activities that promote student invention can appear inefficient, because students do not generate canonical solutions, and therefore the students may perform badly on standard assessments. Two studies on teaching descriptive statistics to 9th-grade students examined whether invention activities may prepare students to learn. Study 1 found that invention activities, when coupled with subsequent learning resources like lectures, led to strong gains in procedural skills, insight into formulas, and abilities to evaluate data from an argument. Additionally, an embedded assessment experiment crossed the factors of instructional method by type of transfer test, with 1 test including resources for learning and 1 not. A "tell-and-practice" instructional conditioned to the same transfer results as an invention condition when there was no learning resource, but the invention condition did better than the tell-and-practice condition when there was a learning resource. This demonstrates the value of invention activities for future learning from resources, and the value of assessments that include opportunities to learn during a test. In Study 2, classroom teachers implemented the instruction and replicated the results. The studies demonstrate that intuitively compelling student-centered activities can be both pedagogically tractable and effective at preparing students learn.

  • A sample of 134 sixth-grade students who were using the Connected Mathematics curriculum were administered an open-ended item entitled, Vet Club (Balanced Assessment, 2000). This paper explores the role of misconceptions and naive conceptions in the acquisition of statistical thinking for middle grades students. Students exhibited misconceptions and naive conceptions regarding representing data graphically, interpreting the meaning of typicality, and plotting 0 above the x-axis.

  • The influences on adult quantitative literacy were studied using information from the National Adult Literacy Survey, 1,800,000 individuals between 25 and 35 years of age and not in school. The major influences on quantitative literacy were educational background (t=123-; df=1; p[less than].0001), daily television usage (t=1538; df=1; p[less than].0001), and disability (t=713; df=a;p[less than].0001). Education impacted television usage (t=691; df=1;p[less than].0001) and personal yearly income (t=991; df=1;p[less than].0001). Ethnicity affected income levels (t=898; df=1 p[less than].0001), which in turn influenced television viewing (t=1514; df=1; p[less than].0001). The results indicated that education seemed the key to increasing adult levels of quantitative literacy. Library usage, parents' education, and gender did not exhibit any relationship with quantitative literacy

  • In order to investigate the impact of simulation software on students' understanding of sampling distributions, the Sampling Sim program (delMas, 2001) was developed. The use of this software with students has been the subject of several classroom research studies conducted in a variety of settings (see Chance, delMas, and Garfield, in press). This paper examines the effect of several versions of a structured activity on students' understanding of sampling distributions. The first version of the activity was created to guide the students' interaction with the simulation software based on ideas from previous studies as well as the research literature. Two subsequent versions introduced a sticker and scrapbook activity that allowed students to keep a visual record of the effects of change in population shape and sample size on the resulting distributions of sample means. Four questions guided the studies reported in this paper: how can the simulations be utilized most effectively, how can we best integrate the technology into instruction, which particular techniques appear to be most effective, and how is student reasoning of sampling distributions impacted by use of the program and activities. A variety of assessment tasks were used to determine the extent of students' conceptual understanding of sampling distributions. As classroom researchers, a main goal was to document student learning while providing feedback for further development and improvement of the software and the learning activity. Ongoing collection and analysis of assessment data indicated that despite students' engagement in the activity and apparent understanding of sampling distributions and the Central Limit Theorem,<br>they were unable to apply this knowledge to solve novel problems. In particular, they had<br>difficulty solving graphical items that resembled tasks in the activity as well as well as applying the Central Limit Theorem to different situations. We comment on the lessons we have learned from this research, our explanations for why so many students continue to have difficulties, and our plans for revising the activity.

  • To support the development of statistical inquiry, teachers must know more than statistical procedures; they must learn to investigate, reason, and argue with statistical concepts and techniques. For mathematics teachers, this is often unfamiliar territory. This study investigates the process of teachers' statistical inquiry and documents its development during a 6-month professional development sequence that immerses teachers in learning statistics while analyzing their students' high-stakes state assessment results. We will report, based on empirical research, how an immersion model in statistics can influence that development.

  • This study developed a graphicacy assessment instrument which can (a) assess and measure pupils' progress in graphical understanding for the National Curriculum, (b) identify significant errors and misconceptions in graphing held by Year 10 Mathematics pupils and so (c) help raise teachers' awareness of their pupils' mathematical thinking.<br>A carefully constructed set of graphical problems related to the research literature and located in the National Curriculum was administered to a pilot group of pupils. The problems were deliberately posed in such a way as to encourage relevant isconceptions to come to the surface. The test was 'scaled' using Rasch methodology and a graphicacy measure results, with the main misconceptions plotted with the items on the same scale. The pupils producing the most common errors in the test were selected to take part in some group interviews to get an insight into their way of thinking and to collect evidence for discussion with teachers. Data from teachers' interviews and questionnaires were also collected to ascertain the curriculum validity of the ssessment and to probe their knowledge of their pupils' understanding.

  • Attempts to teach statistical thinking using corrective feedback or a ``rule-training'' approach have been only moderately successful. A new training approach is proposed which relies on the assumption that the human mind is naturally equipped to solve many statistical tasks in which the relevant information is presented in terms of absolute f requencies instead of probabilities. In an investigation of this approach, people were rained to solve tasks involving conjunctive and conditional probabilities using a requency grid to represent probability information. It is suggested that learning by doing, whose mportance was largely neglected in prior training studies, has played a major role in the current training. Study 1 showed that training that combines external pictorial epresentations and learning by doing has a large and lasting effect on how well people an solve conjunctive probability tasks. A ceiling effect prevented comparison of the requency grid and a conventional pictorial representation (Venn diagrams) with respect o effectiveness. However, the grid representation was found to be more effective in Study 2, which dealt with the more difficult topic of conditional probabilities. These results uggest methods to optimize the teaching of statistical thinking and the presentation of statistical information in the media.

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