DOCUMENT RESUME 30'113 152. SE 019 635 AUTHOR Suydam, Marilyn N. Ed.; Osborne, Alan R., Ed. _TITLE . Algorithmic- Learning. INSTITUTION ERIC Information Analysis Center fOr Science, Mathematics, and Environmental Education, Columbus, Ohio.; Ohio State Univ., Columbus. Center for Science and Mathematics Education. SPONS AGENCY National Inst. of Education (DREW), Washington, D.C. Career Education Program. PUB DATE [75] NOTE 194p. AVAILABLE FROMOhio State University, Center for Science and Mathematics Education, 244;Arps Hall, Columbus, Ohio 43210 ($3.75) \ EDRS PRICE MF-$0.76 HC-$9.51 Plus Postage DESCRIPTORS *Algorithms; Cognitive Processes; Elementary Secondary Education; Instruction; *Learning; Learning Theories; *Literature Reviews; *Mathematics Education; Memory; *Research; Teaching Methods IDENTIFIERS *Algorithmic Learning ABSTRACT Thi volume contains a series of papers on algorithmic learning. Included are six reviews of research pertaining to various aspects of algorithmic learning, six reports of pilot experiments in this area, a theoretical discussion of "The Conditions for'Algol:ithmic Imagivation," and an annotated bibliography. All the papers assume a common definition of algorithmic learning as "the process,of developing and/or applying methods or procedures, i.e., algorithms, with the goal of learning-how-to-learn." 2 common definition of algorithm is also used. Topics covered by literature reviews include algorithmic processes for r,'gnition, algorithms and hierarchies, conceptual bases for the learning of algorithms, interference with the learning of algorithms, algorithmic problem solving, and algorithms and mental computations. Research papers report on studies related to lgebra (3), arithmetic (2) and the use of desk calculators (1). The authors conclude that there are many open researchable questions in tharea of algorithmic learning. 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These reports fall into three broad categories. Research reviews summarize and analyze recent research in specific areas of mathematics education. Resource guides identify and analyze materials and references for use by mathematics teachers at all levels. Special bibliographies announce the availability of documents and review the literature in selected interest areas of mathematics education. Reports in each of these categories may also be targeted for specific sub-populations of the mathematics education community. Priorities for the development of furture Mathematics Education Reports are estab- lished by the advisory board of the Center, in\cooperation with the National Council of Teachers of Mathematics, the Special Interest Group for Research in Mathematics Education, the Conference Board of the Mathematical Sciences, and other professional groups in mathematics education. Individual comments on past Reports and suggestions for future Reports are always welcomed. This publication was prepared pursuant to a contract with the National Institute of Education, U.S. Department of Health, Educa- tion and Welfare. Contractors undertaking such projects under Government sponsorship are encouraged to express freely their judgment in professional and technical matters. Points of view or opinions do not, therefore, necessarily represent official National Institute of Education position or policy ii CONTENTS Algorithmic Learning Section Page I Algorithmic Learning: Introduction 3 II Conditions for Algorithmic Imagination, by Alan R. Osborne 13 III Reviews of Previous,Research on Specific Aspects A. Algorithmic Processes for Cognition, by Jesse D. Parete 25 r B. Algorithmic Learning and Hierarchies, by Paul H. Wozniak C. Investigations of Conceptual Bases Under- lying the Learning of Algorithms, by .1, Diane Thomas ti 56 D. Algorithms: Interference, Facilitation, and Comparisons, by Brady Shafer 73 E. Algorithmic Problem Solving, by-Richard W. Corner 85 F. Algorithms and Mental Computation, by Raymond Zepp 9r( IV Report's of Some Exploratory Research A. Algorithmic Processes in Arithmetic and ,Logic, by Jesse D. Parete 107 B. A Comparison of Different Conceptual Bases for Teaching Subtraction of Integers, by Diane Thomas 119 C. Solving Quadratic Inequalities:,More Than One Algorithm? by Brady Shafer 137 D. A Comparison of Two Strategies for Teaching Algorithms for Finding Linear Equations, by 147 . Richard W. Corner E. Some Computational Strategies of Students Using Desk Calculators, by Raymond Zepp... 154 iii CONTENTS (continued) Section Page V Annotated List of Selected Research References Related to Computational Algorithms, by Marilyn N. Suydam 165 VI Summary 197A iv- \ I. Algorithmic Learning: Introduction 1 Algorithmic Learning: Introduction Marilyn N. Suydam To many people, "algorithmic.,learning" means "the learning of algorithms". They think of algorithms for addition, subtraction, multiplication, and division with whole numbers,such as: / 13 6, 54 86 183W +37 4 x7 18 11 27 774 74 80 602 54 91 6794 They think of algorithms for operationswith fractions and dedimals, of a square root algorithm,of procedures in the content of algebra and calculus and other mathematical areas. But algorithmic learning involves morethan just the learning of specific algorithms. It connotes having learners generalize from specific skills to broader process applications. It is related to learning-how-to-learn. As Simon (1975) pointed out, teaching the algo- rithm and teaching the characteristics of analgorithmic solution are / two different things. The importance of algorithmiclearning is being increasingly recognized, across other content areas aswell as within mathematics. In the past few years, it has beendeveloped as the approach in at least one textbook. The research interest in artificial intelligence is built on a foundation of algorithmiclearning. Several Russian psychologists, among others, have been Very muchconcerned with the implications of algorithmic learning(Gerlach and Brecke, 1974; Landa, 1974). The focus of much current writing isstill on algorithms, but the need to provide for algorithmiclearning is becoming increasingly more evident. The use of hand-held calculators atall levels from the ele- mentary years through life has raised newquestions about algorithms-- 3 and emphasizes the need to explore ways in which algorithmic learning can be .promoted, as calculators decrease the need to focus'so much of our attention on the algorithms for calculation. Explanation This document is not intended to be all-inclusive (although we had dreams of being comprehensive at one early point:).It is basically the reiport of a year of emphasis on algorithms and algorithmic learning in a seminar for mathematics education doctoral students at The Ohio State University. It dogsn't include all that the seminar encompassed. But it does present some results, both in the form ofresearch reviews and mini- research studies. It is hoped that it will serve to have others do more thinking.about what is known about algorithmic learning, and, even more important, to think about what still needs to be explored and learned, In proposing the seminar, it was noted that there is a tradition of concern for algorithms in the computational orientation of elementary school mathematics, but new information-processing models of learning seem to be stimulating a new body of research problems and studies. A more general interest is suggested, in broadly conceived algorithmic learning non-specific to the computational needs of young children. The relation of algorithmic learning to problem solving, logical ability, creativity, and the like have not been explored. And they should be. Our focus was indicated by this flow diagram for our initial discussions: What in an algorithm? Definitions from texts 4 Computational algorithms (examples, research, selected studies) Other elementary Secondary school algorithms Computers t Broad meaning/purpose of algorithms Learning hOw to learn ALGORITHMIC LEARNING: the evolving focus )4. Thus, this document attempts to: (1) review the status of some aspects of research related to algorithmic learning