: Even today, many complex and important skills, such as those required for language use and social interaction, are learned informally through apprenticeshiplike methods -- i.e., methods involving not didactic teaching, but observation, coaching, and successive approximation while carrying out a variety of tasks and activities. The differences between formal schooling and apprenticeship methods are many, but for our purposes, one is most important. Perhaps as a by-product of the specialization of learning in schools, skills and knowledge taught in schools have become abstracted from their uses in the world. In apprenticeship learning, on the other hand, target skills are not only continually in use by skilled practitioners, but are instrumental to the accomplishment of meaningful tasks. Said differently, apprenticeship embeds the learning of skills and knowledge in the social and functional context of their use. This difference is not academic, but has serious implications for the nature of the knowledge that students acquire. This paper attempts to elucidate some of those implications through a proposal for the retooling of apprenticeship methods for the teaching and learning of cognitive skills. Specifically, we propose the development of a new cognitive apprenticeship to teach students the thinking and problem-solving skills involved in school subjects such as reading, writing and mathematics.

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Cognitive Apprenticeship: Teaching the Craft of Reading, Writing and Mathematics

Brooks, R.A., Intelligence without representation, Artificial Intelligence 47 (1991) 139159. Artificial intelligence research has foundered on the issue of representation. When intelligence is approached in an incremental manner, with strict reliance on interfacing to the real world through perception and action, reliance on representation disappears. In this paper we outline our approach to incrementally building complete intelligent Creatures. The fundamental decomposition of the intelligent system is not into independent information processing units which must interface with each other via representations. Instead, the intelligent system is decomposed into independent and parallel activity producers which all interface directly to the world through perception and action, rather than interface to each other particularly much. The notions of central and peripheral systems evaporateeverything is both central and peripheral. Based on these principles we have built a very successful series of mobile robots which operate without supervision as Creatures in standard office environments.

Intelligence without Representation

http://people.csail.mit.edu/brooks/papers/representation.pdf

Intelligence without representation

Cognitive psychology, artificial intelligence, and computer technology have advanced to the point where it is feasible to build computer systems that are as effective as intelligent human tutors Computer tutors based on a set of pedagogical principles derived from the ACT theory of cognition have been developed for teaching students to do proofs in geometry and to write computer programs in the language LISP

https://science.sciencemag.org/content/sci/228/4698/456.full.pdf

Intelligent tutoring systems

The goals of this chapter are (1) to outline and substantiate a broad conceptualization of what it means to think mathematically, (2) to summarize the literature relevant to understanding mathematical thinking and problem solving, and (3) to point to new directions in research, development, and assessment consonant with an emerging understanding of mathematical thinking and the goals for instruction outlined here. The use of the phrase “learning to think mathematically” in this chapter’s title is deliberately broad. Although the original charter for this chapter was to review the literature on problem solving and metacognition, the literature itself is somewhat ill defined and poorly grounded. As the literature summary will make clear, problem solving has been used with multiple meanings that range from “working rote exercises” to “doing mathematics as a professional”; metacognition has multiple and almost disjoint meanings (from knowledge about one’s thought processes to self-regulation during problem solving) that make it difficult to use as a concept. This chapter outlines the various meanings that have been ascribed to these terms and discusses their role in mathematical thinking. The discussion will not have the character of a classic literature review, which is typically encyclopedic in its references and telegraphic in its discussions of individual papers or results. It will, instead, be selective and illustrative, with main points illustrated by extended discussions of pertinent examples. Problem solving has, as predicted in the 1980 Yearbook of the National Council of Teachers of Mathematics (Krulik, 1980, p. xiv), been the theme of the 1980s. The decade began with NCTM’s widely heralded statement, in its Agenda for Action, that “problem solving must be the focus of school mathematics” (NCTM, 1980, p. 1). It concluded with the publication of Everybody Counts (National Research Council, 1989) and the Curriculum and Evaluation Standards for School Mathematics (NCTM, 1989), both of which emphasize problem solving. One might infer, then, that there is general acceptance of the idea that the primary goal of mathematics instruction should be to have students become competent problem solvers. Yet, given the multiple interpretations of the term, the goal is hardly clear. Equally unclear is the role that problem solving, once adequately characterized, should play in the larger context of school mathematics. What are the goals for mathematics instruction, and how does problem solving fit within those goals? Such questions are complex. Goals for mathematics instruction depend on one’s conceptualization of what mathematics is, and what it means to understand mathematics. Such conceptualizations vary widely. At one end of the spectrum, mathematical knowledge is seen as a body of facts and procedures dealing with quantities, magnitudes, and forms, and the relationships among them; knowing mathematics is seen as having mastered these facts and procedures. At the other end of the spectrum, mathematics is conceptualized as the “science of patterns,” an (almost) empirical discipline closely akin to the sciences in its emphasis on pattern-seeking on the basis of empirical evidence. The author’s view is that the former perspective trivializes mathematics; that a curriculum based on mastering a corpus of mathematical facts and procedures is severely impoverished—in much the same way that an English curriculum would be considered impoverished if it focused largely, if not exclusively, on issues of grammar. The author characterizes the mathematical enterprise as follows:

Learning to Think Mathematically: Problem Solving, Metacognition, and Sense Making in Mathematics (Reprint):

https://files.eric.ed.gov/fulltext/ED159036.pdf

Diagnostic Models for Procedural Bugs in Basic Mathematical Skills

Computer-based tutoring/coaching systems have the promise of enhancing the educational value of gaming environments by guiding a student's discovery learning This paper provides an in-depth view of (i) the philosophy behind such systems, (ii) the kinds of diagnostic modeling strategies required to infer a student's shortcomings from observing his behavior and (iii) the range of explicit tutorial strategies needed for directing the Tutor to say the right thing at the right time Examples of these issues are drawn for a computer-based coaching system for a simple game-How the West was Won Our intention in writing this paper is to make explicit the vast amounts of tutorial knowledge required to construct a coaching system that is robust, friendly and intelligent enough to survive in home or classroom use During the past three years, we have witnessed how subtle the computer-based coaching problem really is We hope this paper conveys some of these subtleties—many of which continue to resist general solution

/pdf/an-investigation-of-computer-coaching-for-informal-learning-lvaafwrajn.pdf

An Investigation of Computer Coaching for Informal Learning Activities.

Publisher Summary This chapter provides an overview of SOPHIE, an intelligent instructional system that reflects a major attempt to extend Carbonell's notion of mixed-initiative Computer Aided Instruction (CAI) for the purpose of encouraging a wider range of student initiatives. Unlike previous AI-CAI systems that attempt to mimic the roles of a human teacher, SOPHIE tries to create a reactive environment in which the student learns by trying out his ideas rather than by instruction. SOPHIE incorporates a strong model of its knowledge domain along with numerous heuristic strategies for answering a student's questions, providing him with critiques of his current solution paths, and generating alternative theories to his current hypotheses. In essence, SOPHIE enables a student to have a one-to-one relationship with an expert who helps the student create, experiment with, and debug his own ideas. SOPHIE's expertise is derived from an efficient and powerful inferencing scheme that uses multiple representations of knowledge including (1) simulation models of its microcosm, (2) procedural specialists that contain logical skills and heuristic strategies for using these models, and (3) semantic nets for encoding time-invariant factual knowledge.

Multiple representations of knowledge for tutorial reasoning

This paper describes a fully operational AI-CAI system which incorporates artificial intelligence techniques to perform question answering, hypothesis verification and theory formation activities in the domain of electronic troubleshooting. Much of its logical or inferencing capabilities are derived from uses of simulation models in conjunction with numerous procedural specialists. The system also includes a highly tuned structural parser for allowing the student to communicate in natural language. Although the system is extremely large it is sufficiently fast to be thoroughly exercised in a training or classroom environment.

SOPHIE: A Step Toward Creating A Reactive Learning Environment.

This paper describes a paradigm for tutorial systems capable of automatically providing feedback and hints in a game environment. The paradigm is illustrated by a tutoring system for the PLATO game “How the West Was Won”. The system uses a computer-based “Expert” player to evaluate a student's moves and construct a “differential model” of the student's behavior with respect to the Expert's. The essential aspects of the student's behavior are analyzed with respect to a set of “issues”, which are addressed to the basic conceptual constraints that might prevent the student's full utilization of the environment. Issues are viewed as procedural specialists that “wake-up” or become active when an instance of an issue manifests itself in a move. These issue specialists help the Tutor isolate what to comment on. The intent of the system is to transform a “fun” game into a productive learning environment without altering the student's enjoyment.

https://files.eric.ed.gov/fulltext/ED151013.pdf

John Seely Brown

Papers

Diagnostic Models for Procedural Bugs in Basic Mathematical Skills

An Investigation of Computer Coaching for Informal Learning Activities.

Multiple representations of knowledge for tutorial reasoning

SOPHIE: A Step Toward Creating A Reactive Learning Environment.

A tutoring and student modelling paradigm for gaming environments