It is now possible to build machine learning systems whose behavior is consistent with data from human students. How can education use such simulated students? Applications that help three user groups are discussed. Teachers can practice the art of tutoring byhaving them teach a simulated student. Using a simulation instead of a real student allows teachers to see how their actions affect that student's knowledge, to undo their actions, and to try their skills on students with varying prior knowledge and learning strategies. Students can learn in collaboration with a simulated student. Because the simulated student can be simultaneously an expert and a colearner, it can scaffold and guide the human's learning in subtle ways. Instructional developers can test their instruction on simulated students. Unlike formativeevaluations with real students, a simulation-based evaluation can indicate exactly what piece of the instruction caused which pieces of knowledge, and thus help developers troubleshoot their instructional designs early in the design process. For each of these three areas of application, inherent technical limitations, existing systems and prospective systems are discussed.

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The opinions and positions expressed in this practice guide are the authors ’ and do not necessarily represent the opinions and positions of the Institute of Education Sciences or the U.S. Department of Education. This practice guide should be reviewed and applied according to the specific needs of the educators and education agencies using it and with full realization that it represents only one approach that might be taken, based on the research that was available at the time of publication. This practice guide should be used as a tool to assist in decision-making rather than as a “cookbook.” Any references within the document to specific education products are illustrative and do not imply endorsement of these products to the exclusion of other products that are not referenced. U.S. Department of Education

As I write this sentence, I am glancing over at the color printer sitting beside my screen. In the popular jargon of the computer industry, that printer is called a "peripheral"—which, upon reflection, is a rather odd way to describe it. What, precisely, is it peripheral to? If the ultimate

student achievement

Abstract Exploration can be an effective learning experience. if suitably constrained and guided. Moreover. it can provide this benefit for specifically targeted formal skills in the arithmetic curriculum. This paper presents two complementary techniques for promoting success in computer-based exploration environments. Semantic constraints on eploration cut out meaningless options. and heuristic guidance facilitates search on the basis ofa heuristic function with both cognitive and problem-solving components. fJi> have implemented an environment that permits semantically constrained exploration for subtraction, as well as a related environment that facilitates the transition to paper-and-pencil subtraction. The authors tested the system in individual hour-long sessions with more than twenty children in grades 1-3.

The opinions and positions expressed in this practice guide are the authors ’ and do not necessarily represent the opinions and positions of the Institute of Education Sciences or the U.S. Department of Education. This practice guide should be reviewed and applied according to the specific needs of the educators and education agencies using it and with full realization that it represents only one approach that might be taken, based on the research that was available at the time of publication. This practice guide should be used as a tool to assist in decision-making rather than as a “cookbook.” Any references within the document to specific education products are illustrative and do not imply endorsement of these products to the exclusion of other products that are not referenced. U.S. Department of Education

The term "learning " has always covered a wide range of phenomena-from schoolchildren learning their multiplication tables to deans learning that their budgets have been cut. Suddenly, In just the last few years, the range has gotten much wider-by orders or magnitude, it seems. We now have learning corporations, even learning societies, institutional memory. and distributed expertise. Marshall McLuhan's catchy phrase, "learning a living, " has been resurrected as a characterization of what life will be like in the fiber optics age. Traditional conceptions of learning do not comfortably embrace these newer notions. The fact that people are able to talk about them Is mainly a testimony to the remarkable flexibility of language. The term "learning' * has been extended metaphorically, but it does not follow that our basic understanding of learning has changed. The conception of learning that has guided education through recent millennia is grounded in what has come to be called "folk psychology " (Bruner. 1990; Stich, 1983). To say this Is no particular aspersion on educators, for the same could be said about all the other professions and disciplines In which considerations of human mental activity are Involved-law. psychiatry, theology, even contemporary work in artificial Intelligence.

This article offers a new perspective on the use of concrete objects to teach mathematics. It is commonly assumed that concrete manipulatives are effective because they allow children to perform mathematics without understanding arbitrary, written mathematical symbols. We argue that the sharp distinction between concrete and abstract forms of mathematical expression may not be justified. We believe instead that manipulatives are also symbols; teachers intend for them to stand for or represent a concept or written symbol. Consequently, research on how young children comprehend symbolic relations is relevant to studying their comprehension of manipulatives. We review evidence that many of the problems that children encounter when using manipulatives are very similar to problems that they have using other symbol systems such as scale models. Successful use of manipulatives depends on treating them as symbols rather than as substitutes for symbols. A persistent dilemma for teachers of mathematics concerns how to help children understand abstract concepts, such as addition and multiplication, and the symbols that are used to represent these concepts (Hiebert & Carpenter, 1992; Resnick & Ford, 1984). Teachers face a double challenge. Symbols may be difficult to teach to children who have not yet grasped the concepts that they represent. At the same time, the concepts may be difficult to teach to children who have not yet mastered the symbols. Not surprisingly, both teachers and mathematics researchers have called for better techniques to help children learn mathematical concepts and symbols.

A controversy has arisen concerning the relative merits of conceptually-oriented teaching versus calculation-centred teaching. Marks (1989) maintains that concepts are far more important than computations, and that they can be successfully taught without the related computations. In contrast, Khamis (1989) claims that students cannot truly