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 condition led 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 activ...

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Inventing to Prepare for Future Learning: The Hidden Efficiency of Encouraging Original Student Production in Statistics Instruction.

Statistical Literacy, Reasoning, and Thinking: Goals, Definitions, and Challenges.- Towards an Understanding of Statistical Thinking.- Statistical Literacy.- A Comparison of Mathematical and Statistical Reasoning.- Models of Development in Statistical Reasoning.- Reasoning about Data Analysis.- Learning to Reason About Distribution.- Conceptualizing an Average as a Stable Feature of a Noisy Process.- Reasoning About Variation.- Reasoning about Covariation.- Students' Reasoning about the Normal Distribution.- Developing Reasoning about Samples.- Reasoning about Sampling Distribitions.- Primary Teachers' Statistical Reasoning about Data.- Secondary Teachers' Statistical Reasoning in Comparing Two Groups.- Principles of Instructional Design for Supporting the Development of Students' Statistical Reasoning.- Research on Statistical Literacy, Reasoning, and Thinking: Issues, Challenges, and Implications.

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The challenge of developing statistical literacy, reasoning and thinking

Summary
This paper provides an overview of current research on teaching and learning statistics, summarizing studies that have been conducted by researchers from different disciplines and focused on students at all levels. The review is organized by general research questions addressed, and suggests what can be learned from the results of each of these questions. The implications of the research are described in terms of eight principles for learning statistics from Garfield (1995) which are revisited in the light of results from current studies.

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How Students Learn Statistics Revisited: A Current Review of Research on Teaching and Learning Statistics

This case study analyzes ways in which an experienced physics teacher uses questioning to guide student thinking during a benchmark discussion about measurement. Interactional issues involve ways of speaking: Why the teacher decided to ask what he did, when he did, of whom, in what way, and for what purpose. Conceptual issues involves ways of thinking: How students seemed to understand measurement concepts such as calculating an average value. We define a particular kind of question, a reflective toss, that the teacher uses to try to give students responsibility for thinking. A reflective toss sequence typically consists of a student statement, teacher question, and additional student statements. This unit of analysis directs attention to ways in which a teacher question influences student thinking. We analyze reflective tosses in terms of the immediate action plans they instantiated, the emergent goals they served, anal underlying beliefs they embodied during an episode that involved the public refinemen...

Using Questioning to Guide Student Thinking

This study reports on 2 upper-elementary teachers' learning through their use of potentially educative mathematics curriculum materials without additional professional development. 41 observations of the teachers' mathematics lessons and 28 interviews of the teachers were collected from October to May of an academic year. The case study analyses indicated that curriculum materials can be an effective professional development tool, but perhaps not for all teachers. 1 teacher's instructional focus and rationale for instructional practices remained stable throughout the school year, whereas the other's changed dramatically. The cases illustrated the teachers' dynamic and divergent nature of opportunities to learn through reading materials and enacting lessons. Findings also indicated that consideration of the interaction between beliefs integral to teachers' identity and those that are targets for change may illuminate responses to potentially educative curriculum materials.

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Curriculum Materials as a Professional Development Tool: How a Mathematics Textbook Affected Two Teachers' Learning

Whenever the need arises to describe a set of data in a succinct way, the issue of mathematical representativeness arises. The goal of this research is to understand the characteristics of fourth through eighth graders' constructions of "average" as a representative number summarizing a data set. Twenty-one students were interviewed, using a series of open-ended problems that called on children to construct their own notion of representativeness. Five basic constructions of representativeness are identified and analyzed. These approaches illustrate the ways in which students are (or are not) developing useful, general definitions for the statistical concept of average. One objective of statistics is to reduce large, unmanageable, and disordered collections of information to summary representations. The need to summarize data is present even among young children. For example, in the surveys conducted by primary-grade students, we see movement from focusing on individual pieces of data ("I have one brother") to highlighting and summarizing the data in some manageable form ("Most of the class members have only one brother or sister"). As soon as there is the need to describe a set of data in a more succinct way, the notion of representativeness arises: What is typical of these data? How can we capture their range and distribution?

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Children's Concepts of Average and Representativeness.

Beyond Arithmetic: Changing Mathematics in the Elementary Classroom

This study reports the results of a survey of fifty school districts. It assesses the extent to which special educators are moving beyond drill and practice software with learning disabled and emotionally disturbed students. It also describes factors that are preventing or discouraging applications beyond drill and practice. It was briefly described in a previous Computers in the Schools article by Hummel (1985).-J.W.H., Associate Editor.

Learner-centered Software: A Survey of Microcomputer Use with Special Needs Students

Student-Generated Conjecture about the Behavior of the Operations: Four Dimensions Supporting a Structural Understanding of Arithmetic

Abstract This article addresses the nature of student-generated representations that support students’ early algebraic reasoning in the realm of generalized arithmetic. We analyzed representations created by students for the following qualities: representations that distinguish the behavior of one operation from another, that support an explanation of a specific case of a generalization, and that support justification of a generalization. One key finding is that representations in the form of pictures, diagrams, arrangements of manipulatives, or story contexts that embody the meaning of the operation(s) allow students to distinguish between operations. Such representations can be used by young students to illustrate relationships conveyed in specific instances of a general claim. Further, extending these representations to class of numbers is a mechanism for proving a general claim. 

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Susan Jo Russell

Papers

Children's Concepts of Average and Representativeness.

Beyond Arithmetic: Changing Mathematics in the Elementary Classroom

Learner-centered Software: A Survey of Microcomputer Use with Special Needs Students

Student-Generated Conjecture about the Behavior of the Operations: Four Dimensions Supporting a Structural Understanding of Arithmetic

The centrality of student-generated representation in investigating generalizations about the operations