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# Chapter

• ### Some key comparisons between statistics and mathematics and why teachers should care.

This article highlights some of the differences between statistics and mathematics and suggests some implications of these differences for teachers and students. Our aim is to provoke thought and to presents ideas that will guide classroom practice.

• ### The statistical education of grades Pre-K-2 teachers: A shared responsibility.

This article describes a collaborative effort at the university level between statisticians and mathematics educators as well as two initiatives of the American Statistical Association (ASA): Statistics Teacher Education: Assessment, Methods, Strategies (TEAMS) conference, and Guidelines in Assessment and Instruction for Statistical Education (GAISE).

• ### The GAISE project: Developing statistics education guidelines for grades Pre-K-12 and college courses.

This article presents brief summaries (introductions) for the two reports that came from the Guidelines for Instruction and Assessment in Statistical Education (GAISE) initiative of American Statistical Association (ASA). The reports focus on Pre-K-12 and college statistics education respectively.

• ### Why variances add - and why it matters.

We present examples in which the addition of variances is key to gaining insight when dealing with binomial probabilities, inference and the central limit theorem.

• ### Bootstrapping students' understanding of statistical concepts.

This article shows K-12 teachers how the bootstrap and permutations tests can be used to make abstract concepts such as sampling distributions and standard error concrete. Some background information on the bootstrap is provided before a demonstration on the mechanics of the bootstrap, including the calculation of confidence intervals and standard errors is provided. Finally a comparison of bootstrap confidence intervals with classical intervals is presented.

• ### Interpreting probabilities and teaching the subjective viewpoint.

On the basis of the literature on children's probabilistic conceptions and on the results of a survey of introductory statistics college students, this article argues that subjective probability should play a larger role in the probability curriculum in the schools. Also, a simple method for teaching conditional probability and illustrate Bayesian thinking is described. Other activities that are useful for applying subjective and alternative probability interpretations are suggested.

• ### Assessments, change, and exploratory data analysis.

This article examines some techniques that teachers and other individuals interested in school-related data may find helpful and informative in examining tabular displays of data that are frequently found in local newspapers and school publications. Such techniques include looking for tabular patterns, median polish, and examining trifold percents in a table of cross-categorized values.

• ### Fish 'n' chips: A pedagogical path for using an in-class sampling experiment.

This article is based on the mark-recapture activity (capture-recapture). We will (1) document how considering solutions approaches led students (and instructors) to discover a problem inherent in this approach, (2) examine approaches to pooling results from multiple cases, (3) highlight the differences between the framed mathematical problem and the actual practice of ecologists, and (4) propose two sampling activity formats that teachers can choose on the basis of their goals for students. These two formats allow students to mathematically contrast two approaches to handling data or to provide a real-world simulation.

• ### Research in the statistics classroom: Learning from teaching experiments.

This article describes two teaching experiments that focused on the development of statistical ideas and reasoning and illustrates different approaches (e.g. different rationale, activities, and technological tools). We focus on one aspect of each experiment: distribution and sample. Details of the experiments are provided to allow readers to appreciate some of the rich information that was gathered. Finally, suggestions for teachers are discussed.

• ### Representations of reversal: An exploration of Simpson's Paradox

To support NCTM's newest process standard, the potential of multiple representations for teaching repertoire is explored through a real-world phenomenon for which full understanding is elusive using only the most common representation (a table of numbers). The phenomenon of "reversal of a comparison when data are grouped" is explored in surprisingly many ways, each with their own insights: table, circle graph, slope &amp; correlation coefficients, platform scale, trapezoidal representation, unit square model, probability (balls in urns), matrix determinants, linear transformations, vector addition, and verbal form. For such a mathematically-rich phenomenon, the number of distinct representations may be too large to expect a teacher to have time to use all of them. Therefore, it is necessary to learn which representations might be more effective than others, and then form a sequence from those selected. Pilot studies were done with pre-service secondary teachers (n1 = 7 at a public research university and n2 = 3 at a public comprehensive university) on exploring a sequence of 7 different representations of Simpson's Paradox. Students tended to want to stay with the most concrete and visual representations (note: a concrete-visual-analytic progression may not be expected to apply in the usual manner in the particular case of Simpson's Paradox).