Based on the relevant research in the context of constructivist principles, I have formulated some general principles of learning statistics:

*Students learn by constructing knowledge.*

Many research studies conducted both in education and in psychology support the theory that students learn by constructing their own knowledge, not by passive absorption of information (Resnick, 1987, von Glasersfeld, 1987) Regardless of how clearly a teacher or book tells them something, students will understand the material only after they have constructed their own meaning for what they are learning. Moreover, ignoring, dismissing, or merely ``disproving" the students' current ideas will leave them intact - and they will outlast the thin veneer of course content.

Students do not come to class as ``blank slates"or ``empty vessels" waiting to be filled, but instead approach learning activities with significant prior knowledge. In learning something new, they interpret the new information in terms of the knowledge they already have, constructing their own meanings by connecting the new information to what they already believe. Students tend to accept new ideas only when their old ideas do not work, or are shown to be inefficient for purposes they think are important.

*Students learn by active involvement in learning activities.*

Research shows that students learn better if they are engaged in, and motivated to struggle with, their own learning. For this reason, if no other, students appear to learn better if they work cooperatively in small groups to solve problems and learn to argue convincingly for their approach among conflicting ideas and methods (National Research Council, 1989). Small-group activites may involve groups of three or four students working in class to solve a problem, discuss a procedure, or analyze a set of data. Groups may also be used to work on an in-depth project outside of class. Group activities provide opportunities for students to express their ideas both orally and in writing, helping them become more involved in their own learning. For suggestions on how to develop and use cooperative learning activities see Johnson, Johnson, and Smith, (1991) or Goodsell et al. (1992).

*Students learn to do well only what they practice doing.*

Practice may mean hands-on activities, activities using cooperative small groups, or work on the computer. Students also learn better if they have experience applying ideas in new situations. If they practice only calculating answers to familiar, well-defined problems, then that is all they are likely to learn. Students cannot learn to think critically, analyze information, communicate ideas, make arguments, tackle novel situations, unless they are permitted and encouraged to do those things over and over in many contexts. Merely repeating and reviewing tasks is unlikely to lead to improved skills or deeper understanding (American Association for the Advancement of Science, 1989).

*Teachers should not underestimate the difficulty students have in
understanding basic concepts of probability and statistics.*

Many research studies have shown that ideas of probability and statistics are very difficult for students to learn and often conflict with many of their own beliefs and intuitions about data and chance (Shaughnessy, 1992; Garfield &Ahlgren, 1988).

*Teachers often overestimate how well their students understand
basic concepts.*

A few studies have shown that although students may be able to answer some test items correctly or perform calculations correctly, they may still misunderstand basic ideas and concepts. For example, Garfield and delMas (1991) found that when students were asked whether a sample of 10 tosses or 100 tosses of a fair coin was more likely to have exactly 70%heads, students tended to correctly choose the small sample, which seemed to indicate that they understood that small samples are more likely to deviate from the population than are large samples. When asked the same questions about whether a large, urban hospital or a small, rural hospital is more likely to have 70%boys born on a particular day, students responded that both hospitals were equally likely to have 70%boys born on that day, indicating that students could not transfer their understanding to a more real-world context.

*Learning is enhanced by having students become aware of and confront
their misconceptions.*

Students learn better when activities are structured to help students evaluate the difference between their own beliefs about chance events and actual empirical results (delMas and Bart, 1989; Shaughnessy, 1977). If students are first asked to make guesses or predictions about data and random events, they are more likely to care about the actual results. When experimental evidence explicitly contradicts their predictions, they should be helped to evaluate this difference. In fact, unless students are forced to record and then compare their predictions with actual results, they tend to see in their data confirming evidence for their misconceptions of probability. Research in physics instruction also points to this method of testing beliefs against empirical evidence (e.g., Clement, 1987).

*Calculators and computers should be used to help students visualize
and explore data, not just to follow algorithms to predetermined ends.*

Computer-based instruction appears to help students learn basic statistics concepts by providing different ways to represent the same data set (e.g., going from tables of data to histograms to boxplots) or by allowing students to manipulate different aspects of a particular representation in exploring a data set (e.g., changing the shape of a histogram to see what happens to the relative positions of the mean and median) (Rubin, Rosebery &Bruce, 1988). Instructional software may be used to help students understand abstract ideas. For example, students may develop an understanding of the Central Limit Theorem by constructing various populations and observing the distributions of statistics computed from samples drawn from these populations. The computer can also be used to improve students' understanding of probability by allowing them to explore and represent models, change assumptions and parameters for these models, and analyze data generated by applying these models (Biehler, 1991).

*Students learn better if they receive consistent and helpful
feedback on their performance.*

Learning is enhanced if students have opportunities to express ideas and and get feedback on their ideas. Feedback should be analytical, and come at a time when students are interested in it. There must be time for students to reflect on the feedback they receive, make adjustments, and try again (AAAS, 1989). For example, evaluation of student projects may be used as a way to give feedback to students while they work on a problem during a course, not just as a final judgment when they are finished with the course (Garfield, in press). Since statistical expertise typically involves more than mastering facts and calculations, assessment should capture students' ability to reason, communicate, and apply, their statistical knowledge. A variety of assessment methods should be used to capture the full range of students' learning (e.g., written and oral reports on projects, minute papers reflecting students' understanding of material from one class session, or essay questions included on exams). Teachers should become proficient in developing and choosing appropriate methods that are aligned with instruction, and should be skilled in communicating assessment results to students (Webb &Romberg, 1992). For a variety of classroom assessment techniques designed to help instructors better understand and improve their students' learning, see Angelo and Cross (in press).

*Students learn to value what they know will be assessed.*

Another reason to expand assessment beyond the exclusive use of traditional tests, is that students will only apply themselves to those skills and activities on which they know they will be evaluated. If students know they will be evaluated on their ability to critique and communicate statistical information, or work collaboratively on a group project, they will be more willing to invest themselves in improving skills required by these activities.

*Use of the suggested methods of teaching will not ensure that
all students will learn the material.*

No method is perfect and will work with all students. Several research studies in statistics as well as in other disciplines show that students' misconceptions are often strong and resilient - they are slow to change, even when students are confronted with evidence that their beliefs are incorrect. And this is only part of the problem. Another is whether students are engaged enough to struggle with learning new ideas.

Wed Jun 29 13:57:26 EDT 1994