CALIFORNIA STATE UNIVERSITY, NORTHRIDGE AN ANNOTATED BIBLIOGRAPHY OF HEURISTICS FOR THE SECONDARY LEVEL (7-12) A graduate project submitted in partial satisfaction of the requirements for the degree of Master of Arts in Secondary Education by Jennifer Irene Conner June, 1980 Project Irene Conner is approved: H. Heimler, Advisor California State University, Northridge ii TABLE OF CONTENTS Chapter Page 1. INTRODUCTION 1 Purpose of Heuristics 1 History 2 Terminology 4 Strategies 7 Studies 10 Advantages and Disadvantages 13 References 18 2 . ANNOTATED BIBLIOGRAPHY 22 Algebra 23 Arithmetic 24 Calculator 31 Calculus 31 Geometry 32 Logic 39 Theory 39 iii ABSTRACT AN ANNOTATED BIBLIOGRAPHY OF HEURISTICS FOR THE SECONDARY LEVEL (7-12) by Jennifer Irene Conner Master of Arts in Secondary Education This graduate project is an annotated bibliography of articles written from 1975 through 1979 concerning the heuristic method of teaching and its applications in the classroom. The articles are from these professional Journals: Arithmetic Teacher, Mathematics Teacher, Mathematics Teaching, and School Science and Mathematics. To introduce the reader to the subject of heuristics, the following topics are presented: purpose, history, termi­ nology, strategies, studies, and advantages and disadvan­ tages. iv ' ' INTRODUCTION The use of heuristics, or simply heuristics, is a well- known method for the teaching and learning of mathematics. Often called "discovery learning," it challenges the student to think one's way through the problem at hand. Apart from a growing book-literature on the subject, numerous articles have appeared in professional journals. This graduate project is an annotated bibliography of such articles, for the period 1975-1979. To set the stage for the use of the bibliography, the science of heuristics is discussed in the following details: purpose, history, terminology, strate- gies, studies, and advantages and disadvantages. Purpose of Heuristics A major goal in mathematics ecucation is teaching stu- dents problem-solving skills. They need to know how to attack a problem and think logically through to its comple- tion. Problem-solving skills may not only be applicable in the area of mathematics, but in other subject areas as well. When you are reasoning in a particular branch of mathematics (e.g. geometry or arithmetic) you are generally thinking about the subject matter of that branch--although, as Hilbert, Boole, and others pointed out, if your reasoning is correct, the subject matter is irrelevant and the reasoning would a~ply equally well to any other subject area. l, , 3 1 2 The process of attacking and thinking logically is also necessary, for example, in areas of the humanities, where sentence structure is vital for communication, or in the physical, behavioral, or social science, where following through on a particular hypothesis demands logical reason­ ing. The heuristic method of teaching mathematics is one method that strives to develop problem-solving skills in students. History The heuristic method of teaching is not new. One of the earliest references to this method was Plato's mention of the conversation between Socrates and the slave boy. He quotes Socrates, "Do you observe, Meno, that I am not teaching the boy anything, but only asking him questions?" 4 The word "heuristic" seems to have been first applied to teaching mathematics by A. W. Grube in his book, Leitfaden f~r das Rechnen in der Elementarschule nach den Grundsatzen einer heuristischen Methode, published in 1842. 5 Other early advocates of the heuristic method of teaching were David P. Page, Herbert Spencer, William Bagley, and Frank and Charles McMurry. In 1847, David P. Page, the first principal of New York's first normal school wrote, "There is great satisfaction in discovering a difficult thing for one's self--and the teacher does the scholar a lasting injury who takes this pleasure from him." 6 In 1860, Herbert 3 Spencer, a scientist-philosopher, wrote, "Children should be led to make their own inferences. They should be told as little as possible and induced to discover as much as possi­ ble."7 The McMurry brothers, Frank and Charles, authors of The Method of the Recitation, published in 1897, in describ- ing their method, state The questions are put to the child before their answers have been presented. More than that, the child is expected to conceive these answers himself; he is systematically required to make discoveries, to judge what might reason­ ably follow from a given situation, to put two and two together and declare the result. 8 In 1905, William Bagley, a member of the faculty of Teachers College, Columbia University, wrote, "The pupil is not to be told but led to see. Whatever the pupil gains must be gained with the consciouness that he is, in a sense, the discoverer." 9 One of the more recent advocates of this method is George Polya. One of his books, How to Solve It, first published in 1945, is devoted entirely to problem­ solving. In this book Polya states, "One of the most im- portant tasks of the teacher is to help students. But if the student is left alone without any help, he may make no progress at all. If the teacher helps too much, nothing is left to the student." 10 Polya expounds upon this help by advising a teacher to challenge the curiosity of his stu- dents by setting them problems proportionate to their know- ledge, and helping them to solve their problems with stimulating questions. 11 4 Terminology The term "heuristic" comes from the Greek heuriskein which means, "to discover." 12 Different people have various ideas when it comes to defining heuristics. According to Jasper Hassler and Rolland Smith, the heuristic method of teaching is similar to the method used by Socrates. The knowledge that the student gains comes through his own efforts with the assistance of the teacher who asks correct questions designed to provoke thought. As Hassler and Smith explain, . The heuristic method is the process of leading the pupil by skillful questions to find the desired knowledge himself. It in­ volves very little telling of facts .... It is the aim of the pure heuristic method to tell a pupil nothing that he can discover for himself. 13 Jon L. Higgins further emphasizes, "Heuristics is not simply asking questions but the logical strategy behind the questions asked is important." 14 Higgins continues with a more complete definition of heuristic teaching when he states, "Heuristic teaching in mathematics is a category of instructional methods that make primary use of one or more problem-solving techniques in mathematics." 15 In addition to this, he lists four characteristics of heuristic teaching: it approaches content of material by assuming everything is a problem to be solved, reflects problem- solving techniques as the logical construction of instruc- 5 tional procedures, demands the flexibility for uncertain and alternate approaches, and, seeks to maximize student action and participation in the teaching-learning process. 16 Most mathematicians view the heuristic method as using problem-solving techniques. Hugh Ouellette goes beyond only solving the problem at hand when he states, "Heuristic teaching is the art of seeking generalizations from problem situations. 1117 Once a problem is solved, it should not be forgotten, but used to formulate generalizations regarding similar problems. According to George Polya, when we speak about modern heuristics, we not only focus attention on solving problems, but also on the thought process involved. Modern heuristic endeavors to understand the process of solving problems, especially the mental operations typically useful in this process. 18 Another recent author agreeing with Polya of the importance of the thinking process is Barnabas Hughes. He states The primary purpose of heuristic teaching is to teach mathematical thinking, not mathematical thought. Heuristic clarifies the critical dis­ tinction between the logical and the psychologi­ cal approach to mathematics. The former convinces doubters; the latter brings understanding. Before one can be certain of anything, he must have some plausible reason for his thinking. 19 A term frequently interchanged with "heuristic" is "discovery." Whereas heuristics is linked with problem- solving, the discovery method can involve any aspect of mathematics. A discovery method is any activity through which the learner creates, invents, finds, or gains under- 6 standing of some mathematical principle, concept, or pro­ cedure through his own efforts. 20 The discovery method is similar to the heuristic method in that it involves no direct information given to the stu­ dent. Discovery teaching is a teaching process through which a student is persuaded to feel that he 11 sees 11 some­ thing--a mathematical fact or a relationship--that he had not previously seen, and that this perception was not di­ rectly told, given, or displayed to him by another person. 21 Discovery teaching also requires that the student be able to substantiate any discoveries. In teaching by discovery, students are guided into making their own conjectures and trying to prove or disprove them. 22 Besides 11 heuristic 11 and 11 discovery," three other terms which are also interchangeably used are the socratic method, the inquiry method, and the process method. The socratic method takes its name from one of the famous originators of the heuristic method--Socrates--who engaged in a question and answer strategy for learning. The inquiry method sug­ gests that questions are to be asked by both the teacher and the student; the teacher uses questioning to aid the student in understanding the method of solving a particular problem and then the student can also use the same method of inquiry when he attacks a problem on his own. The process method is used to denote certain step-by-step procedures that are to be followed in order to solve the problems encountered.