California State University, Northridge an Annotated

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

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)

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, strategies, studies, and advantages and disadvantages.

Purpose of Heuristics

A major goal in mathematics ecucation is teaching students 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 important 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 students by setting them problems proportionate to their knowledge, 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. In 7

this way the learner must determine what the proper se

quence of operations must be in order to arrive at the pre

ferred result.

Strategies

There are several teaching strategies used as part of

the discovery method. The first of these strategies is the

inductive method, the second is the nonverbal awareness method, the third is the incidental learning method, the

fourth is the variation method, the fifth is the deductive method, and the sixth is the analogy method.

In the inductive method the student is led into the

knowledge of a generalization by various examples. This method is characterized by verbalization of the generaliza tion.23 For example, when teaching addition of integers,

students are able to formulate the different rules for addi tion of "like" and "unlike" signs by various examples pre

sented.

The nonverbal awareness method is similar to the induc tive method in that a generalization is sought from the examples given. The difference is the student may learn the generalization being taught without being able to express verbally what he is doing. 2 ~ The student may be able to add

integers correctly, but may not be able to state the rules that he is following.

A characteristic of the incidental learning method 8

occurs when a student learns new information as a result or by-product of something else which is being taught. 25

Pauker (1979) illustrates this method in his algebra class

6 26 where they are factoring x - 64. Some factor this using the difference-of-squares technique, while others use the difference-of-cubes approach. The two factorizations ap pear to be different but are, in fact, equal. This leads to the unplanned discovery of a new factor form for the

2 2 trinomials w + wz + z •

The variation method is a process of changing part of the data or conclusion, or both, of a mathematical state ment accepted or proven true, resulting in a new statement to prove or disprove. 27 This is especially true in proof exercises where the student might be asked, "Could we still prove these triangles congruent if we were given that sides a and a' are congruent instead of sides b and b' being con gruent?" The problem posed might result in two corres ponding sides and a nonincluded pair of angles congruent, instead of two corresponding sides and an included pair of angles congruent.

Many writers consider discovery as only inductive rea soning. Marvin Bittinger feels the deductive method is also used in discovery teaching. 28 Deduction can be described as manipulating sentences by logical rules. 29 John Corcoran states deductive reasoning is important because

When a set of known sentences are considered , 9

deductive reasoning is used to extend knowledge. When a set of hypotheses is considered, deductive reasoning can be used to refute the hypotheses by deducing them from a known falsehood. 30

This is seen in indirect proofs. For example, a student might be asked to prove there can be no more than one right angle in a triangle. The student starts by asking, "What would happen if we assumed there was more than one right angle?" He knows this could only happen if the measures of the angles in the triangle summed to be more than 180°.

This is a known falsehood, so the student knows his assump- tion is also false.

The analogy method is used when the problem put before the student is difficult but there exists a similar problem which is easier. 31 One example of this is the similarity between the problems, "Find the diagonal of a rectangular solid in which all sides are given," and, "Find the diagonal of a rectangle with all sides given." The first problem is reduced from a three-dimensional problem to a two- dimensional problem.

With respect to problem-solving, P6lya lists four steps for students to follow when presented with a problem. The students must understand the problem, determining the un- known and any information given; devise a plan for solving the problem; carry out the plan checking each step; and examine the solution obtained, checking the results. 32 One goal that P6lya has for students is to learn to ask themselves questions. If the same question posed by the 10

teacher is repeatedly helpful, the student will notice it

and will be motivated to ask the question by himself in a

similar situation. 33

Whatever method is used to inspire students to dis

cover, there are certain steps which the students need to be

involved in actively. This process includes gathering

information, searching for patterns, making generalizations,

and verifying or proving the generalizations made. 34

Studies

A number of studies have been made at different levels

regarding the discovery method of teaching. The following

studies have been designed using students from the elemen

tary to the college level, in an attempt to study the

effects of the heuristic method. These studies result in

positive effects on learning, although this might be attrib

utable to the notion that those making the studies are

looking for positive results. There are, however, no known

studies which show the heuristic method as having negative

effects on learning. The major studies of the heuristic method will be presented in the following paragraphs.

A study by Scandura (1974) using elementary students

proposed the hypothesis that problem-solving ability is

linked to the presence or absence of higher order capabil

ities which modify or combine simple rules. The students were divided into two groups. Each group was taught several 11

simple rules, however, one of the groups was taught how to combine the rules. Each group was then presented with a problem which required combining the rules. The group that had been taught how to combine the rules was able to solve the problem, while the other group was not. 35

Kenneth Henderson and James Rollins (1967) designed a study investigating three stratagems of the discovery method used in teaching. The stratagems studied were the method of agreement whereby examples are presented which agree with the generalization desired, the method of difference whereby examples are presented which are different than the general ization desired, and the joint method of agreement and difference. They reached the conclusion that whatever stratagem is chosen, students will learn no matter what their ability. 36

Jack Price (1967) used tenth grade students in his study. He divided them into three groups: Group One was taught using a traditional textbook, Group Two was taught using materials prepared to promote student discovery, and

Group Three was taught using the materials prepared for

Group Two in addition to materials designed to aid in the transfer of mathematical thinking to the real world. Price indicated Groups Two and Three showed a slight but nonsig nificant gain in achievement, Groups Two and Three showed a great increase in mathematical reasoning, and Groups Two and

Three showed a positive attitude change towards mathe- .»l,· - ·------

matics. 37

Another study using elementary students was done by

Worthen (1968). In this study, 536 fifth and sixth grade

pupils in the Salt Lake City School District were chosen.

The control group consisted of 106 students (3 sixth grade

classes). The teachers in the study, having been taught the

discovery method of teaching, taught one class using the

discovery method and one class using the expository method.

The results indicated that classes taught using the discov-

ery method exhibited more retention and had better transfer

ability in using the discovery problem approaches in new

situations. 38

Kenneth Cummins (1960) designed a study using calculus

students. He divided them into two groups: one group was

taught using the discovery method, and the other was taught

using the traditional method. Each group was given two

tests: one test was designed for the discovery group and

the other test for the traditional group. The results

showed the discovery group scored higher on the discovery

test, but there was no significant difference in scores on

the traditional test. 39 An observation made by Marvin

Bittinger concerning this study was that, "This may indicate

not that discovery learning is better but only that

discovery-trained students will do better on a discovery

type test. 40 13

Advantages and Disadvantages

The heuristic method affords students basic advantages in learning. Keeping in mind the concept that students develop more of an interest when actively involved in the learning process, it tends to follow that since this is the principle of the heuristic method, then, in general, stu dents should learn more effectively. It is a tenet of the heuristic method that the students' active involvement and participation consequently leads to a deeper understanding of the subject. The following authors have pinpointed specific advantages to this principle. The advantages Arthur Schultze (1912) lists are that students think for themselves rather than merely listening for information; students acquire a real understanding of the subject and are, therefore, better able to remember it; the interest of the students and the resulting willingness to work are greater when they are taught heuristically than when they are taught by the information method; teachers are in complete touch with their class; and home study is not nearly so heavy or tedious as when the informational methods are used. 41 The results of the study made by Price (1967) support the statement that the students' interest is increased. William Lowry (1967) agrees with the advantages listed by Schultze and adds, "The student learns a way of going about learning on his own; in mathematics he learns some techniques, or strategies, for discovering other ~.· ------

things."~t 2 Bruner (1961) continues the list of advantages by adding the following: There is an increase in intellectual potency, which is the ability to assemble material sensibly; the student learns the heuristics of discovery, which are the attitudes and activities that go with inquiry; there is an intrinsic reward in finding out for one's self; and there is conservation of memory, which is the effect of making material more readily accessible in memory.~t 3

Many authors have been critical and have expressed disadvantages in using the heuristic method. Their criticisms apply equally to both students and teachers. Most criticize the heuristic method because educators should realize that more than one method might be preferable in teaching strategies and the fear follows that educators will forget this in their zeal to develop the heuristic method.

As a result, there is believed to be a tendency for students not to receive the fullest possible learning experiences and, indirectly, to cause teacher frustration based on this.

In an attempt to show these disadvantages some authors have used the following arguments.

Arthur Schultze (1912) states the heuristic method is slow and time-consuming, especially in the beginning; it is sometimes difficult to make students discover certain facts; the method does not work well in the hands of every teacher; and the method is difficult for the teacher, for he cannot simply follow a textbook.~t~t David Newton (1968) listed 15

several criticisms of the discovery method of teaching. His criticisms are that it is not consonant with the demonstra ted needs of adolescents--teenagers need the security of authority; it is not an honest preparation for college bound students--the method is seldom used in colleges; it does not honestly reflect the nature of science--science should be presented, in part, as an organized body of know ledge; and it has not been analyzed adequately so that it is commonly an ineffective and inefficient technique of teach ing.45 Marvin Bittinger (1968) feels some of the research has not been quite fair. "The didactic method (teacher sets forth knowledge) has not been given a fair chance in the control group. The nondiscovery group should be given the same well-tuned effort as the discovery group. ""6 Lee

Cronbach (1966) agrees that the studies made have rarely given didactic instruction a "fair shake." The control group suffers in the amount of time it receives in teaching.

Since the discovery method takes longer to reach the speci fied criterion, more time must be spent with the group.

When more time is spent learning a task, that task becomes easier to remember. 47 David Ausubel has done extensive work on the negative aspects of the discovery method. One par ticular criticism he (1964) finds is that, "Children tend to jump to conclusions, to generalize on the basis of limited experience, and to consider only one aspect of a problem at a time." "8 The conclusions reached by students may not 16

always be the right ones and it then becomes difficult for

them to accept a different conclusion. Ausubel also states

On developmental grounds the discovery method is generally unnecessary and inappropriate for teaching subject-matter content, except when the pupils are in the concrete stage of cognitive development. During the abstract stage of cognitive development, pupils need no longer depend on current or recently prior concrete empirical experience, and hence are able to by pass completely the intuitive type of under standing reflective of such dependence. Through proper expository teaching they can proceed directly to a level of abstract understanding that is qualitatively superior to the intuitive level in terms of generality, clarity, precision, and explicitness. 49

Ausubel finds positive and negative aspects in both the discovery method and the traditional method of teaching.

Both expository and problem-solving techniques can be either rote or meaningful depending on the conditions under which learning occurs. In both instances meaningful learning takes place if the learning task can be related in nonarbitrary, substantive fashion to what the learner already knows, and if the learner adopts a corresponding learning set to do so. However, it should seem rather self-evident that per forming laboratory experiments in cookbook fashion, without understanding the underlying substantive and methodological principles involved, confers precious little meaningful understanding, and that many students studying mathematics and science find it relatively simple to discover correct answers to problems without really understanding what they are doing. They accomplish the latter feat merely by rotely memorizing "type problems" and procedures manipulating symbols. 50

People continuously talk and ask questions about the heuristic method of teaching: Does it work? Is it a superior method of teaching? A variety of ways to implement 17

the heuristic method have been demonstrated. As these methods become more fully developed by teachers, then stu dents will definitely have more effective learning exper iences. More research should be done relative to this method by unbiased educators to determine the validity of heuristics as a teaching strategy. It is true that when a new concept is presented, more interest is aroused. The interest which is shown by enthusiastic teachers can be transferred to the student. In this way, as the students become more interested and enthusiastic, more learning will take place. This is the goal of all educators whether their field is methematics or another discipline. It is to that ideal of student learning and understanding, rather than merely teaching concepts to students, that forms the basis of the method called Heuristic. 18

REFERENCES

1 Constance Reid, Hilbert (New York, Heidelberg-Berlin, 1970)' pp. 57ff. 2 George Boole, Studies in Logic and Probability, ed. by R. Rhees (LaSalle, Illinois, 1952):-Pp. 235ff.

3 John Corcoran, "Discourse grammars and the structure of reasoning; II: the nature of a correct theory of proof and its value," Journal of Structural Learning, III (1971), pp. 1-16.

~Plato, Meno (Library of Liberal Arts, Indianapolis: Bobbs-Merrill~49), p. 39. 5 Phillip S. Jones, "Discovery teaching-from Socrates to modernity," Arithmetic Teacher, XVII (1970), pp. 503-510.

6 David P. Page, Theory and Practice of Teaching (Syracuse: Hall and Dickson:-1847), p. 8~

7 Herbert Spencer, Education: Intellectual, Moral, and Physical (London: Hurst and Co., n.d.), p. 126.

8 Charles A. and Frank M. McMurry, The Method of the Recitation (New York: Macmillan, 1897):-P. 67.

9 William Bagley, The Educative Process (New York: Macmillan, 1905), p. 2~

wGeorge P6lya, How to Solve It (Princeton, New Jersey: Princeton University Press, 1957)-,-p. 1.

11 Ibid. , p. v.

12 Mauritz Johnson, "Who discovered discovery," Phi Delta Kappan, XLVIII (1966), pp. 120-123.

13 Jasper Hassler and Rolland Smith, The Teaching of Secondary Mathematics (New York: Macmillan Co., 1930~ pp. 141-14 .

1~Jon L. Higgins, "A new look at heuristics," Mathe matics Teacher, LXIV (1971), pp. 487-495. 19

15 Ibid.

16 Ibid.

17 Hugh Ouellette, "The even triangle: a heuristic approach," Mathematics Teacher, LXXI (1978), pp. 684-688.

18 George Polya, How to Solve It (Princeton, New Jersey: Princeton University Press, 1957)-,-p. 129.

19 Barnabas B. Hughes, "Heuristic teaching in mathe matics," Educational Studies in Mathematics, V (1974), pp. 291-299. --

20 William Lowry, "Approaches to discovery learning in mathematics," High School Journal, L (1967), pp. 254-260.

21 Phillip S. Jones, "Discovery teaching-from Socrates to modernity,".Arithmetic Teacher, XVII (1970), pp. 503-510.

~Steve Bergen, "A discovery approach for the y intercept," Mathematics Teacher, LXX (1977), pp. 675-676.

23 Gertrude Hendrix, "A new clue to transfer of train ing," Elementary School Journal, XLVIII (1947), pp. 197-208.

~Ibid., pp. 197-208.

25 Ibid.

26 Andy Pauker, "Discovering a new factor form," Mathe matics Teacher, LXXII (1979), p. 275.

27 Clarence Heinke, "Variation, a process of discovery in Geometry," Mathematics Teacher, L (1957), pp. 146-154.

28 Marvin L. Bittinger, "A review of discovery," Mathe matics Teacher, LXI (1968), pp. 140-146.

29 Gertrude Hendrix, "A new clue to transfer of train ing," Elementary School Journal, XLVIII (1947), pp. 197-208.

30 John Corcoran, "Discourse grammars and the structure of mathematical reasoning; I: mathematical reasoning and the stratification of language," Journal of Structural Learning, III (1971), pp. 55-74.

31 William Lowry, "Approaches to discovery learning in mathe.matics," High School Journal, L ( 1967), pp. 254-260. ~.· ------

~George P6lya, How to Solve It (Princeton, New Jersey: Princeton University Press, 1957)-,-pp. 5-6.

33 Ibid., p. 4.

34 Hugh Ouellette, "The even triangle: a heuristic approach," Mathematics Teacher, LXXI (1978), pp. 684-688.

35 J. M. Scandura, "Mathematical problem solving," American Mathematical .Monthly, LXXXI (1974), pp. 273-280.

36 Kerineth B. Henderson and James H. Rollins, "A com parison of three stratagems for teaching mathematical con cepts and generalizations by guided discovery," Arithmetic Teacher, XIV (1967), pp. 583-588.

~Jack Price, "Discovery: its effect on critical thinking and achievement in mathematics," Mathematics Teacher, LX (1967), pp. 874-876.

38 Blaine R. Worthen, "A study of discovery and exposi tory presentation: implications for teaching," Journal of Teacher Education, XIX (1968), pp. 223-242.

39 Kenneth Cummins, "A student experience-discovery approach to the teaching of Calculus," Mathematics Teacher, LIII (1960), pp. 162-170.

lj()Marvin L. Bittinger, "A review of discovery," Mathe matics Teacher, LXI (1968), pp. 140-146.

41 Arthur Schultze, The Teaching of Mathematics in Secondary Schools (New York: Macmillan Co., 1912), pp. 44- 46.

42 William Lowry, "Approaches to discovery learning in mathematics," High School Journal, L (1967), pp. 254-260.

43 Jerome S. Bruner, "The act of discovery," Harvard Educational Review, XXXI (1961), pp. 21-32.

~Arthur Schultze, The Teaching of Mathematics in Secondary Schools (New York: Macmillan Co., 1912),-pp. 44- 46.

45 David E. Newton, "The dishonesty of inquiry teaching," School Science and Mathematics, LXVIII (1968), pp. 807-810.

116 . Marvin L. Bittinger, "A review of discovery," Mathe- matics Teacher, LXI (1968), pp. 140-146. -~··

~7 Lee Cronbach, Learning by Discovery: A Critical Appraisal ed. by Lee S. Shulman and Evan R. Keislar (Chicago: Rand McNally and Co., 1966), p. 80.

~David P. Ausubel, "Some psychological and educational limitations of learning by discovery," Arithmetic Teacher, XI (1964), pp. 290-302.

~9 Ibid.

50 Ibid. 22

ANNOTATED BIBLIOGRAPHY

Fifty-nine articles are annotated in the pages that follow. Each article was read in one of four journals:

Arithmetic Teacher, Mathematics Teacher, Mathematics

Teaching, or School Science and Mathematics. The years consulted are from 1975 through 1979. The articles are grouped, alphabetically by author, under the topics:

Algebra, Arithmetic, Calculator, Calculus, Geometry, Logic, and Theory. These topics are sufficient for grade-level classification of the articles. 23

Algebra

Bergen, Steve, "A discovery approach for the y

intercept," Mathematics Teacher 1977, 1..Q_:675-676. The question posed in this article was, "Find the y intercept given two points." After the standard method using slope was examined, a student suggested finding the difference between the inner and outer products of the ordered pairs. This was done and a new method of finding the slope was found.

Borenson, Henry, "Promoting discovery in Algebra,"

Mathematics Teacher 1978, 71:751-752. This article is about twelfth grade algebra students who were examining whether a binary operation * is associative on the set K = {a,b,c,d} as defined in a table. The ques tion was then asked, "What is the minimum number of triplets of K that need to be verified for associativity to conclude that * is associative on K?" Lamon, Richard A., "Dense, denser, densest,"

Mathematics Teacher 1976, ~:547-548. This article expands three methods which are used to find a fraction between two given fractions.

Oliver, Charlene, "The twelve days of Christmas," Mathematics Teacher 1977, 70:752-754. The problem, "Find the total number of gifts given in 'The twelve days of Christmas,'" was given to a second year 24

algebra class. They started by constructing a table, and searched for patterns. Soon a student discovered the general formula.

Pauker, Andy, "Discovering a new factor form,"

Mathematics Teacher 1979, ~:275.

6 An algebra class was factoring x - 64. Some in the class treated the problem of factoring by using the difference-of squares technique, while others used the difference-of-cubes approach. The answers were different in appearance but equal. A new factor form was generalized for trinomials

2 2 w + wz + z •

Ursell, John H., "Guess my formula," Mathematics

Teacher 1978, 71:32-35. This article describes a game which can be played to figure out a formula or equation. The game uses class teamwork.

In the sample game, the students used the methods of plotting points, differences, derivatives, and integrals.

Arithmetic

Brown, Stephen I., "A new multiplication algorithm: on

the complexity of simplicity," Arithmetic Teacher 1975,

~:546-554. Factors which differ by 2, 4, and 6 are considered and generalizations are formed. Squares of numbers which end in

5 are also examined. 25

Crawford, John A. and Calvin T. Long, "Guessing,

mathematical induction, and a remarkable Fibonacci

result," Mathematics Teacher 1979, 11._:613-616. This lesson is entirely teacher-directed. An example of a proof by induction is given using Fibonacci numbers. The number 1/89 is written in decimal form and found to be a Fibonacci sequence.

Curtis, Timothy, "Two sixth form investigations," Mathematics Teaching 1975, n. 73:40-43. In the first investigation, a pattern is found for finding intersections between the set of square numbers and the set of triangular numbers. In the second investigation, an algorithm is found for producing Pythagorean triples.

DiDominico, AngeloS., "Discovery of a property of

consecutive integers," Mathematics Teacher 1979, 11._:285-286. Students begin by writing a natural number c as the sum of nand c- n. Next they multiply n(c- n). Trying this with the first dozen positive integers shows that some consecu tive integers yield the same result [ 3(7 - 3) = 2(8 - 2)]. A generalization relating to prime numbers is formed.

Ehrlich, Amos, "Columns of difference," Mathematics

Teaching 1977, n. 79:42-45. Four numbers are chosen at random. These are put in a column and by taking the absolute value of the differences, a new column of four numbers is found. This is continued 26

and the result is generalized for columns of an even amount of numbers and an odd amount of numbers in the columns.

Ehrlich, Amos, "Doubling-diagrams," Mathematics

Teaching 1978, n. 84~35-39. Several discoveries are made based upon doubling numbers and writing them in modulos. The doubled numbers form a cycle in the various modulos.

Erb, Clinton A., "What do you see?-A discovery approach to prime numbers," Arithmetic Teacher 1975,

~:272-273.

Patterns are found in two tables. One of the tables lists the multiples of 2, 3, 4 ... The other table lists the number of times each number is found in the frist table.

The findings are related to prime numbers.

Francis, Richard L., "A search for root-multiples,"

Mathematics Teacher 1976, 69:554-556. The digital root of a positive integer is the digit obtained by summing all the digits in the number's representation. A root-multiple is a number which is a multiple of the digital root. A pattern is found with the root-multiples of seven.

This is extended to form a generalization for finding the root-multiples of any digit. Several problems are presented at the end of the article.

Hampton, Homer F., "A fresh approach to prime numbers,"

School Science and Mathematics 1979, 79:59-60. Before prime numbers are defined the teacher asks students 27

to "rename" the natural numbers using multiplication. They are not allowed to use the number one as a factor, and the smallest factors possible must be used. Soon the students find that some of the numbers cannot be renamed and the others are always renamed using the "un-nameables."

Fraser. Majory, "Prospecting on Pascal's mountain,"

School Science and Mathematics 1979, 12:54-58. Triangular and pyramidal (tetrahedral) numbers are discov ered in Pascal's triangle as well as many other properties and patterns.

Hartman, Janet, "Figurate numbers," Mathematics

Teacher 1976, §2:47-50. A set of three worksheets are presented dealing with square, triangular, and pentagonal numbers. The method of con structing each number is shown as well as the relationships between the different types of figures. Patterns involving differences are also found.

Keller, Clifton, "Using tables to teach mathematics,"

Mathematics Teacher 1978, 71:655-656. This article uses tables of: the list of decimal equiva lents of the reciprocals of integers, and the integers put into six columns. Each table is examined for patterns. In the first, a pattern of powers is found and in the second, a pattern of primes is found.

Killian, C. Rodney and HenryS. Kepner, Jr., "Pascal's

triangle and the binomial probability distribution," 28

Mathematics Teacher 1976, 69:561-563. A triangle is formed by taking each element of Pascal's triangle and dividing it by the corresponding row sum. The resulting triangle has elements which are decimals less than or equal to one. The decimals represent the expected proba bility values resulting from tossing a fair coin. This new triangle has many characteristics similar to Pascal's triangle.

Ouellette, Hugh, "Discovery with number triangles,"

Mathematics Teacher 1978, 71:678-682. This is an activity which leads the student to find patterns in the first few rows of number triangles, and then leads the student to predict what the outcomes would be for the lOOth row. There are three triangles given: the even triangle, the multiple triangle, and the mixed up triangle.

Ouellette, Hugh, "Fibonacci's triangle," School Science

and Mathematics 1979, 79:248-254. The numbers 1, 3, 5, ... are put in a triangular array which is called Fibonacci's triangle. Six patterns are found and the generalizations proven.

Ouellette, Hugh, "Number triangles-A discovery Lesson," Mathematics Teacher 1975, 68:671-674. This is a set of three worksheets dealing with number tri angles. The triangles used are the integer triangle,

Fibonacci's triangle, and Pascal's triangle. The student is led to make discoveries regarding the lOOth row in each. 29

Ouellette, Hugh, "The even triangle: a heuristic

approach," Mathematics Teacher 1978, 71:684-688. This article presents five activities using information gathered from the even triangle. The activities are designed to help the student search for patterns. Then seven properties are listed and proven.

Palagi, George H., "Polya's triangular array problems,"

Mathematics Teacher 1976, ~:564-566. This article is a dialogue between a student and a teacher.

First an even triangle is formed. The elements are divided by two and patterns are found. The method of differences is used to find the general formula to relate the row number to the sum of that row.

Pauker, Andy, "Pattern discovery with binary trees,"

Mathematics Teacher 1979, ~:337-340. A binary tree, similar to a family tree except each position has exactly two "children," and each "child" has only one

"parent," is explored. Several patterns are found and generalizations are formed. The numbers are then changed into the binary system and more patterns are found.

Prielipp, Robert W., and Norbert H. Kuenzi, "Sums of

consecutive positive integers," Mathematics Teacher 1975, .§.§_:18-21. Positive integers are written as a sum of two or more consecutive positive integers. Problems which are presented and proven include, "Which positive integers can be 30

represented as the sum of two or more positive integers?"

"Which cannot be represented in this way?" "Which have a unique representation?" and "Which have more than one representation?"

Richart, Ronald W., "Something different for middle

school mathematics," Mathematics Teacher 1977,

70:674-675.

This article looks at the patterns formed by listing mul tiples of the numbers 9, 7, 3, and 31. It is suggested that a pattern appears with the multiples of every number. The writer encourages students to find other patterns.

Sicklick, Francine P., "Patterns in integers,"

Mathematics Teacher 1975, 68:290-292.

Students discover their own rules for adding, subtracting, and multiplying integers using the patterns found in the problems presented by the teacher. A different subtraction rule is presented.

Stanley, Francis v.J., "Serendipi taus discovery of

Pascal's triangle," Mathematics Teacher 1975, .§.§.:95-98.

The class lesson started with the rules for divisibility by

2, 3, 4, 5, 6, 8, 9, and 10. One student asked, "What is the smallest positive integer that is divisible by l, 2, 3,

4, 5, 6, 7, 8, 9, and 10?" The class found 2,520 was the number they were searching for. Using the differences be tween the quotients of 2,520 and the numbers 1 through 10, several patterns relating to Pascal's triangle were found. 31

Woodburn, Douglas, "Can you predict the repetend?"

Mathematics Teacher 1976, 69:675-678.

Using worksheets the students find the repetend for numbers divided by 9, 99, and 999. Hints are given at the bottom of each worksheet to help the student find the pattern.

Calculator

Maor, Eli, "The pocket calculator as a teaching aid,"

Mathematics Teacher 1976, §2:471-475.

Using a pocket calculator students can make discoveries and

TI e then show why they happen. For example, e ~ TI , because

TI ~ e ln TI. Another use for calculators is to verify generalizations. Trignometric identities are usually verified using angles which measure 30°, 45°, or 60°. With a calculator, any angle measure is easy to use.

Schmalz, S. P., Rosemary, "Calculator capers,"

Mathematics Teacher 1978, 71:439-442.

First problems are worked on a calculator. Then by looking at the patterns formed, the students try to work out similar problems without a calculator.

Calculus

Shilgalis, Thomas W., "Using discovery in the Calculus

class," Mathematics Teacher 1975, 68:144-147.

This article shows how to present discoverable theorems from elementary calculus. The theorems presented are the mean 32

value theorems, one for derivatives and one for integrals.

Geometry

Aichele, Douglas B., "Using constructions to discover

mathematical relationships," School Science and

Mathematics 1975, 75:466-470.

Three examples are given on how to use constructions to discover mathematical relationships. The constructions involve the incircles and excircles of triangles, and

Fermat's point.

Bader, William A., "Problem solving via soap bubbles,"

School Science and Mathematics 1975, ~:343-353. The first problem involves plotting the shortest distance between three points. After the students make some guesses, the teacher introduces the soap bubble idea. The problem is expanded to four, five, and six points. The author ex plains that the students must follow Polya's process for solving their problems.

Botsler, L. Carey and Evan M. Maletsky, "Tangram

mathematics," Mathematics Teacher 1977, I..Q_:l43-146.

This is a set of three worksheets with activities designed for tangrams. The first worksheet gives the student the tangram pieces, the second gives shapes for the student to form, and the third lists questions regarding the tangram pieces.

Bowling, J. Michael, "Word chains: a simulation of ~.· - ---

Proof," Mathematics Teacher 1977, 70:506-508.

This article relates the thinking process used in solving

word-chains (the first and last words are given with the

object of getting to the last word by changing one letter at

a time and making another word) to the thinking process used

in writing proofs.

Bright, George W., "Learning to count in Geometry,"

Mathematics Teacher 1977, IQ:l5-19.

The problems suggested in this article are solved by

creating a series of simpler similar problems or simple

cases of the problem. The problems asked are, "What is the

maximum number of regions determined by five circles of

arbitrary radii lying in a plane?" "What if the circles

must be congruent?" "What if congruent squares are used?"

and "What if the squares are not congruent?"

Bright, George W., "Using tables to solve Geometry

problems," Arithmetic Teacher 1978, 25:39-42.

First considered are the number of rectangles in a chain of

squares. Second, the number of squares in an n x n set of

unit squares are counted. Third, the number of squares

which can be formed using an n x n set of dots are counted.

Fourth examined are the number of regions formed by drawing

n chords through a circle in which each chord intersects

every other chord. Fifth, the number of triangles in n rows

of n unit triangles are counted. In each of these problems,

tables are used to form generalizations. _¥~,·

Comella, James J. and James D. Watson, "Sum squares on

a Geoboard," Mathematics Teacher 1977, l.Q_:150-153.

The problem, "Generate a formula that gives the number of squares having vertices at pegs that may be constructed on an n x n geoboard," is considered. First, the squares which have their sides parallel to the edges of the board are counted and listed in a table. The formula is found by using differences.

Dolan, Daniel T., "Right or not: a triangle investiga

tion," Mathematics Teacher 1979, 11_:279-282.

Students are led by manipulative exercises with squares to find rela,tionships between the squares formed on the shorter sides of a triangle, and the square on the longest side. On the basis of these relationships, generalizations are formed as to whether the triangle is acute, right, or obtuse. Fielker, David, "Five sticks, " Mathematics Teaching 1975, n. 72:12-16.

At first five sticks are given to the students. The students are led into making observations about the various placements of the sticks. Then the students take one stick at a time and add to it, continuously looking at the dif ferent ways the sticks can be arranged. Then they concen trate on the maximum number of intersections and regions found. A pattern emerges. Grignon, Jean, "J'ai Choisi L'espace .. " Mathematics Teaching 1977, n. 79:18-21. Triangular, square, and hexagonal grids are examined to find the number of ending points to paths "traveled." Patterns are found and class activities are suggested.

Hale, David, "One thing leads to another," Mathematics

Teaching 1975, n. 72:18-21.

The question was, "How many triangles with sides of integer lengths can be drawn having a longest side (or sides) of length n units." A table was produced and several patterns seemed to appear but after checking them,they were not true patterns. To find the general rule, the perimeters were included in the table.

Henry, Loren L., "Discovering the Euler line by paper

folding," School Science and Mathematics 1978,

1..§_:665-668.

A paper-folding activity is described in which the student can "discover" the Euler line. After the exercise is

Completed using an acute scalene triangle, it is suggested that the students try the activity with other triangles.

Kuper, Marie and Marion Walter, "From edges to solids,"

Mathematics Teaching 1976, n. 74:20-23.

The students are first asked to find all possible rectangles with sides 3, 5, or 8 units. Then they are given the pos sible rectangles and asked to find the number of boxes which can be made. A table shows there is a pattern between the number of sides and possible rectangles and boxes. Another table lists the volumes compared with the sides. 36

Lichtenberg, Donovan R., "From the geoboard to number

theory to complex numbers," Mathematics Teacher 1975, §.§.:370-375. Starting with the geoboard, students are asked to find squares of various areas. The areas which can be found are square numbers or numbers which are the sum of two squares.

Next it was discovered that any number that is the product of two sums of squares is itself the sum of squares. The identities are then shown to relate to the complex number system.

Mathews, John, and William Leonard, "A discovery

activity in Geometry," Mathematics Teacher 1977, 70:126. A different approach is shown in this activity. In triangle

ABC, let D and E be the midpoints of BC and AB, respectively. Let CE intersect AD at P; then AP:PD = 2:1. The problem is varied to CD:CB = 1:3, rather than CD:CB = 1:2. Olson, Alton T., "Exploring skewsquares," Mathematics

Teacher 1976, Q2:570-573. Skewsquares are quadrilaterals in which the diagonals are equal and perpendicular to each other. Seven activities exploring skewsquares are given. These activities use rotations to show the diagonals are perpendicular and equal.

Ranucci, Ernest R., "Isosceles," Mathematics Teacher

1976, ~:289-294. This article solves the problem of finding the conditions 37

under which an isosceles triangle can be separated into other noncongruent, nonoverlapping, isosceles triangles. A triangle is found where two interior triangles are isosce

les, one where three interior triangles are isosceles, one with five interior isosceles triangles, and one with eleven.

A general formula is given to find the angle measures.

Regan, Michael, "Painted cubes and cuboids,"

Mathematics Teaching 1977, n. 78:49-51. Cubes and cuboids are painted white on the outside and then cut up into unit cubes in order to see how many faces of each cube are painted. The information is listed in a table and generalizations are made.

Sconyers, James M., "From polygons to pi," Mathematics

Teacher 1978, 71:514. This article suggest activities which help the student to understand the meaning of n. At first students are given copies of regular polygons; they are to determine the ratio between the perimeter and the diameter (longest diagonal).

Then students are given circles of various sizes to measure the circumference and diameter, and find the ratio of their measures.

Sharlow, John F., "Visualizing mathematics with

rectangles and rectangular solids," Mathematics

Teacher 1977, 70:60-63. A rectangle is divided into 2, 3, and 6 parts and the total number of rectangles are counted. A pattern and general 38

formula are found for generating the total number of rectangles in any n x m rectangle. The formula is modified for use with rectangular solids.

Smith, Lyle R., "Discovery in one, two, and three

dimensions," Mathematics Teacher 1977, ]_Q:733-738. This article starts with the idea of joining unit line segments and looking for patterns in the number of segments which result. After the pattern and generalization is formed, the article next considers n x n squares to identify patterns in the number of smaller squares inside. Next cubes are examined to find patterns in the number of smaller cubes included.

Spitler, Gail and Marion Weinstein, "Congruence

extended: a setting for activity in Geometry,"

Mathematics Teacher 1976, ~:18-21. The question posed is, "What are the minimum conditions necessary to guarantee the congruence on two polygons?"

First, special triangles are considered (right, isosceles, and equilateral), then triangles in general. Second, quadrilaterals are examined, each type individually. A table of the data and a general formula is developed.

Walter, Marion I., and Stephen I. Brown, "Problem

posing and problem solving: an illustration of their

interdependence," Mathematics Teacher 1977, I.Q:4-13.

In the process of solving the problem, "Given two equilat eral triangles find a third one whose area is the sum of 39

the areas of the other two," other problems are suggested.

The problem is related to the Pythagorean theorem which has

been varied to include other shapes besides squares.

Logic Bookman, Jack, "Why 'False +False' is true-A

discovery explanation," Mathematics Teacher 1978, 71:675-676. Many students will not accept F + F as a true statement.

This article leads students through an exercise, ([(A+ B)

A ~ + B), to discover the definition of F + F.

Theory

Feinstein, Irwin K., "Dare we discover or the dangers

of over-generalization," School Science and Mathematics 1979, 1_2_:22-33- Problems in which quick generalizations are found but not

always right are listed. In these problems, several

"patterns" appear that turn out not to be continued on

indefinitely. The article also lists four reasons why

students over generalize or generalize too quickly.

Lawson, Anton, E., Alfred Devito, and Floyd H.

Nordland, "How's your I.Q. (Inquiry Quotient)?: An

instrument to measure inquiry teaching in the science classroom," School Science and Mathematics 1976, 1..§_:139-151. - Although this article was written basically for the science 40

classroom, most of what is written can be applied to the mathematics classroom, too. The article lists several questions designed to measure the amount of inquiry teaching being accomplished in the classroom.