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Xerox University Microfilms 300 North Zaab Road Ann Arbor, Michigan 40100 77-2383 DAMARIN, Suzanne Kidd, 1941- AN INQUIRY INTO THE USE OF LOGIC IN MATHEMATICAL CONTEXTS BY PRESERVICE ELEMENTARY TEACHERS.

The Ohio State University, Ph.D., 1976 Education, mathematics

Xerox University MicrofilmsAnn , Arbor, Michigan 48106

© Copyright by

Suzanne Kidd Damarin

1976 AN INQUIRY INTO THE USE OF LOGIC IN MATHEMATICAL CONTEXTS

BY PRESERVICE ELEMENTARY TEACHERS

DISSERTATION

Presented in Partial Fulfillment of the Requirements for

the Degree Doctor of Philosophy in the Graduate

School of The Ohio State University

By Suzanne Kidd Damarin, A.B., M.A.

*■* * *

The Ohio State University

1976

Reading Committee: Approved By

Richard J. Shumway Joseph Ferrar Alan Osborne Arthur White Advisor Faculty of Science and Mathematics Education ACKNOWLEDGMENTS

Many individuals have contributed to the success of this research. More than 300 elementary education students served as research subjects. They were quite cooperative and several of them volunteered their thoughts and concerns about the tests.

Professors Joan Leitzel, T. Ralley, P. Rolfe, J.

Schultz and L. Stull generously permitted the testing of these students during class time. The Undergraduate Commit­ tee of the Department of Mathematics and several of the professors named above raised probing questions concerning the nature and purpose of the research, thereby helping to shape it.

Each member of the reading committee had a unique influence on the development of the dissertation. Professor

Joseph Ferrar raised questions which led to the clarifica­ tion of relationships among several parts of the manuscript.

Discussions with Dr. Alan Osborne concerning the relation­ ships between this research and other studies and points of view were quite helpful, as was his critical reading of the manuscript. Dr. Arthur White*s keen scientific intuition and concern with experimental validity influenced the research design and development from the outset. Dr. Richard J. Shumway served conscientiously and well in the many roles of advisor. Concerning himself with all aspects of the dissertation he raised many perti­ nent questions especially with regard to the implications of the research. His spirit of inquiry and keen insight will be a lasting resource. VITA

February 19, 1941. . . . Born - Easton, Pennsylvania

1962 ...... A.B., Wilson College, Chambers- burg, Pennsylvania

1964 ...... M.A., Bryn Mawr College, Bryn Mawr, Pennsylvania

FIELD OF STUDY

Mathematics Education

v TABLE OF CONTENTS

Page ACKNOWLEDGMENTS...... iii VITA ...... v

LIST OF TABLES ...... viii

LIST OF FIGURES...... xi

Chapter I. STATEMENT OF THE PROBLEM...... 1

II. REVIEW OF THE LITERATURE...... 14

Studies in the Tradition of Experi­ mental Psychology...... Studies and Findings in the Genetic Tradition...... 26 Studies in the Tradition of Educa­ tional Research...... 33 Synthesis of Three Traditions ...... 37

III. THE CONSTRUCTION OF A MODEL 41

A Formal Relationship Between Mathe­ matical and Logical Statements. . . . The Operation of the Expanded Model . . 50 Testing the Model I: Delineation of Contexts ...... 56 Testing the Model II: Research Hypotheses...... 57

IV. PROCEDURES FOR TESTING THE MODEL...... 68

Segment 1. Finding Sortings for Statements...... 74 Segment 2. Comparing Content D o m a i n s...... 76 Segment 3. Selecting Statements for Sortings...... 77 Segment 4. Finding Examples fcr which Statements Differ ...... 82

vi Page Segment 5. Bringing Six Tasks Together...... 89 Segment 6. The Class Room Logic T e s t s ...... 91

V. R E S U L T S ...... 94

Segment 1. Finding Sortings for Statements...... 96 Segment 2. Comparing Content D o m a i n s ...... 100 Segment 3. Selecting Statements for Sortings...... 102 Segment 4. Finding Examples for which Statements Differ ...... 121 Segment 5. Bringing Six Tasks Together...... 133 Segment 6. The Classroom Logic T e s t s ...... 150

VI. CONCLUSIONS...... 155

APPENDIX A. A Preliminary Investigation: Interpretations of Statements ...... 167

B. Instruments...... 179

BIBLIOGRAPHY...... 259

vii LIST OF TABLES

Page 1. Presentations of equivalent information in 3 m o d e s ...... 11

2. Principles used in inference tests...... 19

3. Equivalence of two sample statements...... 46

4. Standard statements...... 69

5. Brief descriptions of research segments .... 70

6. Pairing of tests for segment 3 ...... 80 7. Classification of pairs available for test LM . 86

8. Classification of pairs available for test ML . 87

9. Test administration data...... 94

10. An overview of results as related to hypoth­ eses...... 95

11. Basic test statistics for segment 1 tests . . . 97

12. Basic test statistics for segment 2 tests . . . 101

13. Mean percentages correct for subsets of MA. . . 102

14. Basic test data for segment 3 tests ...... 105

15. Correlations between test scores...... 106 16. Repeated measures ANOVA for ordered tests . . . 106

17. Basic test data for segment 3 subscales .... 107

18. ANOVAs subscales of four tests {segment 3) . . . H I

19. Repeated measures ANOVAs. for L and M test f o r m s ...... H 2

20. Analysis of Variance: Comparing L with M . . . 115

viii Page 21. Subscale means for L and M ...... 116 22. Distribution of errors in segment 3 ...... 120

23. Speededness data for segment 4 tests...... 122

24. Basic test data for segment 4 t e s t s ...... 123

25. Summary of ANOVA for statement types...... 124

26. Basic test data for segment 4 subscales .... 125

27. Summary of ANOVA: LM vs. LL (segment 4). . . . 127

28. Summary of ANOVA: ML vs. MM (segment 4) . . . . 129

29. Comparison of true key and conjunctive key scores...... 132

30. Basic test data for segment 5 t e s t s ...... 134

31. Differences between means for segment 5 t e s t s ...... 135 32. Means and reliability estimates for test L (segment 5 ) ...... 137

33. Means and reliabilities when scored for conjunctive interpretation...... 138

34. Means and intercorrelations for selected subscales of test M (segment 5 ) ...... 140

35. Drawing power of options in L*M (segment 5) . . 141

36. Distribution of errors on test M*L (segment 5 ) ...... 142

37. Correlations among scale scores ...... 144

38. Correlations corrected for attenuation (segment 5 ) ...... 146 39. Multiple correlations among tests (segment 5 ) ...... 147

40. Tetrachoric correlations (high vs. low group)...... 148

ix Page 41. Reliabilities of three-test units ...... 149

42. Basic test data for segment 6 t e s t s ...... 150

43. Errors on correct and incorrect arguments...... 151

44. Data for hypothetical tests ...... 152

45. Classification of "feedback" responses...... 153

x LIST OF FIGURES

Page 1. A schematic representation of the Piagetian Model ...... 42

2. The expanded model...... 45

3. Model operation on a mathematical statement . . 52

4. Model operation on a logical statement...... 53

5. Expansion of operation 6 ...... 54

6. Integration of the research base...... 55 7. Hypotheses 1-4 concern single operations. . . . 58

8. Hypotheses 5 and 6 each compare two operations...... 60 9. Hypothesis 7 compares operations...... 61

10. Hypothesis 8 concerns non-equivalent statements...... 62

11. Hypothesis 9 ...... 64

12. Hypothesis 1 0 ...... 65

13. Two sample items from segment 3 ...... 78

14. Design for comparing order effects...... 80

15. Sample items from Statement Differences T e s t s ...... 83 16. Pairs of truth tables forms used in Statement Differences Tests ...... 85

17. Error patterns on four tests (segment 1). . . . 99

18. Means for segment 3 subscales ...... 108

19. Comparing subscales of L and M (segment 3 ) ...... 114

xi Page 20. Subscale means for L and M ...... 117

21. Design of ANOVA: mathematical vs. logical statements (segment 4) ...... 124

22. Design for two ANOVAs: segment 4 subscales. . 126

23. Means for subscales of LL and L M ...... 128

24. Means for subscales of ML and M M ...... 130

25. Significant differences between means (segment 5)...... 136

26. Operations tested in segment 5 ...... 143

27. Two clusters of tasks by difficulty. . . . . 145 CHAPTER I

STATEMENT OF THE PROBLEM

Pure Mathematics is the class of all propo­ sitions of the form "p implies q" where p and q are propositions containing one or more variables...... Elementary Arithmetic might be thought to form an exception: 1+1-2 appears neither to contain variables nor to assert an implication. But as a matter of fact, as will be shown in part II, the true mean­ ing of this proposition is "If x is one and y is one, and x differs from y, then x and y are two.M Bertrand Russell, 1903

Like many mathematicians and philosophers of the turn of the century, Bertrand Russell argued that mathematics, and, in particular, arithmetic, was nothing more than applied logic. The decades prior to 1903 when Russell wrote the passage cited above had seen fundamental changes in the fields of logic and mathematics. The study of mathematical logic was born; the axiomatic method, developed centuries earlier by the Greek geometers, was successfully applied to arithmetic (Peano Postulates); and a host of other contri­ butions to the logical foundations of mathematics were made by Hilbert, Cantor, Boole, Dedekind, Whitehead, Frege, and others. 2

Since that time the frontiers of both logic and mathe­ matics, as well as their interrelations, have expanded and changed. Godel's proof that any axiomatically defined sys­ tem contains undecidable propositions and subsequent related developments, the application of model theory to diverse branches of mathematics, and the discovery of nonstandard logics are among the many events related to the changing relationship between logic and mathematics.

Today we find ourselves in an era when logic itself has been given mathematical foundations, and in which the use of probabilistic methods of mathematical proof is being debated.

It is probably a rare mathematician who would wholeheartedly endorse Russell's definition of pure mathematics. Yet if interpreted not as a definition of the domain of mathemati­ cal endeavor, but rather as a description of the mathematics which we teach, Russell's point of view would seem to be a valid one.

In order to examine that validity we must assume a point of view concerning the nature of logic and a position with regard to the mathematics taught and the teaching of mathe­ matics. The point of view taken concerning the nature of logic is that expressed in 1962 by the Swedish logician

Jurgen Jorgensen: ...logic has to do neither with relations between words and sentences, nor between calculatory shapes or sounds, but with relations between meanings or concepts... It is, however, possible to express logically related concepts (proofs. 3

argumentations, inferences) in everyday language, and when this is done everyday language will contain small selections of linguistic expressions that are expressions of logically related concepts. By formali­ zation of such selections of everyday language it is possible to reach what may be called a "logical language" with a "logical syntax."... there are many different linguistic syntaxes, but one logic common to all intelligent people..."

The concepts and meanings of concern here are mathemati­ cal concepts and meanings. The relations between them are formal mathematical relations. These relations are frequent­ ly expressed in two different languages: the mathematical language of equations, inequalities, and set membership; and a spoken language which is closely related to (a small selec­ tion of) the natural language of the speaker. The elements and rules of formation of the second of these languages are described in elementary logic textbooks.

In the teaching and discussion of elementary mathematics we must move freely from one of these languages to the other, implicitly translating symbolic mathematical statements into the natural language, and conversely, in much the same way

Russell discusses the "true meaning" of "1 + 1 * 2." While it is, perhaps, possible to "teach" the basic facts of arith­ metic (e.g., 1+1-2) by relying solely on the use of symbolic expressions from the mathematical language, the application of the facts to problems would seem to depend upon the use of valid and well-constructed arguments. Such arguments must use the propositional language. 4

Moreover, it is these arguments which form the basis for generalization from simpler to more complex mathemati­ cal problems and systems. Although this generalization is often reflected in a parallelism of expression forms or algorithms, reliance by teachers on parallel treatment of symbolic expressions as a means of generalization would seem to invite inappropriate but common generalizations (such as

"a + (bxc) = (a+b) x (a + c)). While the appropriate use of propositional language in the mathematics classroom cannot guarantee the elimination of errors of this type, habitual use of propositional language and thought struc­ tures provides a tool for recognizing, criticizing, and elucidating such errors. It might be argued that young students do not under­ stand logic, and that teachers' use of logical language and structures is therefore of minimal importance. Several rejoinders to this argument are available: there is some evidence that children do understand logic (Hill 1961), and that they understand some statement forms as well as their teachers (Eisenberg and McGinty 1975a). Even if this evidence were unavailable, it would be reasonable to expect that children taught in an environment in which logical forms were used consistently might well learn to use them appropriately, much as they learn to use complex or special­ ized vocabulary, grammatical structures, or even foreign languages. Even if they did not learn to use the language and methods of propositional logic in the early grades,

children would certainly not be harmed by them. The accumulation of models of correct usage might aid them in

the later development of sound logical thinking. Finally,

children deserve clear and correct discussion of situations

involving necessary and sufficient conditions; these dis­ cussions cannot avoid use of propositional forms.

The importance of elementary teachers* understanding and use of logic in mathematical contexts has been stressed recently by Eisenberg and McGinty (1975b) and others. An­ other testimony, if somewhat muted, to this importance is the increasing inclusion of chapters on logic in many mathe­ matics texts designed for elementary teachers.

In some respects the current trend in elementary teacher education seems to be away from emphasis on the ability to think and argue clearly or rigorously. Even though text­ books include chapters on logic they avoid integration of the logic with other material; some even place the chapter on logic at the end of the book. Many textbook units on logic stress algorithmic procedures for completing truth tables and applying rules of inference. In the absence of applications of these algorithms to problems in other areas of the text, such units are, at best, a questionable use of the limited time available for the mathematics instructions of future teachers. 6

The current popularity of teacher education curricula

which rely heavily on the use of manipulative materials is,

in the opinion of the writer, further evidence of the trend

away from expecting or demanding the use of logical (or

abstract) reasoning by preservice teachers. Manipulative

materials are by their nature finite. They invite the pre­

service teacher to "prove" hypotheses by example or by

exhaustive (of the materials) case by case testing. Since

the "negation" of one hypothesis with respect to the mater­ ials is tantamount to the creation of a new state of the

materials, the possibility of developing the ability to

reason indirectly is virtually non-existent. Moreover,

even as these materials "embody" the mathematical principles

they are intended to convey, they also embody material-based

explanations for observations made.

Manipulative materials can be extremely useful in help­

ing to build concepts among children and adults alike.

Teachers should certainly be familiar with them and with

their uses. However, if teachers are to use these materials

effectively, their understanding of the mathematical prin­

ciples illustrated by them must go well beyond the illustra­

tions themselves. The purpose of the current study is to examine some

aspects of the understanding of the logic underlying mathe­ matical principles ; in particular it is directed toward preservice elementary teachers' use and interpretation of propositional statement forms in mathematical contexts. In­

formal evidence gleaned from conversations with the instruc­

tors of preservice elementary teachers, as well as from per­

sonal experience, provides sufficient motivation for the

study of this general problem: confusion of statements

with their converses, of proof with example, and of explana­

tion with restatement. A series of researches (Eisenberg and McGinty 1975,

Jansson 1975, Juraschek 1976) has effectively established

that preservice elementary teachers make consistent errors

on tests of logical inference. However, it is difficult to

assess the importance of these studies to the issues regard­

ing mathematics education. The reasoning tests from which

these findings emerge consist of items dealing with "real

world situations" rather than with relations between con­

cepts. There is some evidence (Henle 1962, Wason and John-

son-Laird 1972) that subjects tend to misinterpret the

tasks given them on such tests.

The scope of the study. The current study is the beginning of an examination of preservice elementary teachers' under­

standing and use of statements in propositional forms when

these statements deal with elementary mathematical concepts.

The vehicles for examining this understanding are tests re­

quiring the declaration of whether statements are true under given conditions, the identification of differences between 8

statements, and the selection of statements descriptive of

situations presented.

The statements used are of two sorts: (1) statements which contain one logical connective (and, or, if...then,

if and only if), but do not explicitly contain a mathematical operation., and (2) statements which contain arithmetical

operations (+, x) but do not explicitly contain a logical

connective. Statements of type (1) are referred to as

logical statements, while statements of type (2) are called

mathematical statements. Thus the distinction between log­

ical and mathematical statements is dependent upon the for­

mal properties of the statements themselves. While it is

recognized that this distinction between the logical and the mathematical is specialized and, in a sense, naive, this

usage would seem to be in the spirit of the research on

logical thinking, and appropriate to the goals of the study.

As one means of limiting the scope of the study, it was decided that explicit use of the logical operation of nega­ tion would be avoided in the posing of statements. Research

(see Wason and Johnson-Laird, 1972) has shown that use of

the word "not" increases the difficulty of problems in a variety of tests of logic. While negation is implicitly

involved in the many aspects of the study, it is always handled without using the word Mnot;" this is accomplished

through the selection of item content so as to have affirma­

tive names for negations (e.g., "p is not even" is expressed as np is odd"). 9

Other restrictions on the scope include the use of stan­ dard sets of statements, and the selection of test items.

These will be discussed in the course of the exposition.

A Preliminary Investigation. In the process of defining the nature of the research a preliminary study was conducted.

The purpose of this initial research was to examine subjects' behavior on tasks modeled on truth tables. Subjects were presented with logical statements referring to sets pictured at the top of the page, and asked to indicate whether the statements were true or false. The tests were designed to examine subjects' assignment of truth values to compound statements in situations where they had to determine the truth values of constituent simple statements by examining pictures. Responses to these questions were found to be consistent with subjects' treatment of conditional and bi­ conditional statements as if they were conjunctive.

At the outset the underlying hypothesis of the researcher had been that forcing subjects to examine a situation in order to determine the truth or falsity of constituent parts of compound statements, rather than simply to accept the test-writer's word, would result in better performance.

Such was not the case. In retrospective examination of the test and the research literature, the similarity of this test to the instruments used by other researchers was apparent: in all cases the simple statements concerned sin­ gle events (F is in set A; The dog is brown). The 10 appropriateness of using statement forms other than the conjunctive to link two discrete events would seem to be at issue.

On the basis of this and other considerations it was decided that further investigation of the problem would use statements about variables. The set theory context was re­ jected for subsequent research phases because of the diffi­ culty of establishing situations to which statements might refer without using statements of the sorts being tested.

A report on the preliminary study is presented in Appendix A.

A Model for the Use of Logic in Mathematical Contexts.

In order to interpret the implications of the complex body of psychological theories and research results concern­ ing logic for the use of logical forms in mathematical con­ texts an organizational framework is useful. Such a frame­ work was derived by extending certain aspects of Piagetian theory to mathematical domains. The vehicle for this exten­ sion is the observation that the use of the language of propositional logic is but one of several ways of stating logical relationships among mathematical variables. At least three modes of presenting information concerning dichotomous concepts, such as odd/even, are identifiable.

Consider the context of odd and even integers: If M and N are two variables defined over the positive integers, information concerning the parities (oddness or evenness) of M and N can be presented in at least three ways: (1) a 11

logical statement, that is, a simple statement or one in­

volving a logical connective; (2) a mathematical statement

involving an operation defined ovdr the integers; or (3) a

sorting of the set of possible combinations of parities of

M and N into a "truth set" and its complement. Examples

of these presentations are given in Table 1.

Table 1. Presentations of equivalent information in 3 modes

Mathematical Logical Sorting of the Statement Statement "replacement set"

M + N is odd M is even if and Truth set = {(M odd, N only if N is odd even), (M even, N odd)|; complement = {(M odd, N odd), (M even, N even)|

M x N is even M is even or N is Truth set: {(M odd, N even even), (M even, N odd), (or) (M even, N even)| If M is odd then N is even

M x N is odd M is odd and N Truth set: {(M odd, N is odd odd)}

The importance of logic to the elementary teacher lies chiefly in the ability to use each of these presentation modes, and to move freely from one to another in classroom discourse.

A model integrating these three presentation forms was developed. This model is essentially an algorithmic descrip­ tion of the process of translating from one presentation mode 12

to another. The translation of a mathematical statement to

a logical statement (or conversely) is viewed as a sequence of five mental operations, one of which results in the sort­

ing of the replacement set. In each direction (logical to mathematical, mathematical to logical), the final step in

the translation process yields an element of a set of equiv­ alent statements. For the purposes of the current research

single elements of these sets of equivalent statements were

identified as "standard statements" to be used throughout.

The research literature on logical thinking was found to apply to certain steps of the model. The literature and the model together were used to formulate several hypotheses concerning the interpretation and use of logical statements

in mathematical contexts by preservice elementary teachers.

These hypotheses concerned the ability of subjects to per­ form the mental operations which are identified as steps in the model, and the relationships between performance on different operations.

Testing the model. Standard mathematical and logical state­ ments were used in the development of tests designed to as­ sess the performance of preservice elementary teachers on the operations which provide the steps of the model.

In a sequence of research segments these tests were used in a variety of research designs to examine the hypotheses.

As initial tests of the viability of the model the abilities of subjects to perform the basic operations of the model on 13 standard statements was investigated (Segments 1, 2, 3) .

After the behavior of preservice teachers with respect to the individual operations was examined, their performance on the operations hypothesized to be composite was compared with performance on the (assumed) constituent parts (Segment

5) . The sorting of the set of possible combinations plays a key role in the model developed. In order to assess the potential functioning of the model in a task related to, but not constituent parts of, the modeled process, preservice teachers were asked to determine the element of the set of possibilities for which two statements differed (Segment 4).

A sixth research segment was designed to investigate preservice teachers' use of the set of possibilities, and, in particular, counter-examples in judging arguments in a simulated situation. CHAPTER II

REVIEW OF THE LITERATURE

In 1846 when George Boole wrote An Investigation into the Laws of Thought, and for almost half a century there­ after, philosophers, including those who contributed to the founding of scientific psychology, believed that formal log­ ic represented the laws of thought (Bourne, Ekstrand, and

Dominowski 1971) . Near the turn of the century this view was rejected by theorists with a variety of concerns, and in this fertile period which fostered major developments in logic and the logical foundations of mathematics, three major approaches to the study of the use of logic by mature and maturing individuals began to take form.

Oswald Kulpe, a student of Wilhelm Wundt, the founder of experimental psychology, began to study thinking proces­ ses in a laboratory setting. He and his collaborators at

Wurzburg conducted experiments involving simple thought prob­ lems in which they found that subjects' introspective and retrospective solution methods were decidedly not based on rational or logical relations among problem components

(Bourne et al., 1971). These experiments formed the basis for what has become a prolific research into the use of logic from the point of view of experimental psychology. 15

Less than ten years after the founding of the Wurzburg

school, James Mark Baldwin published his four volumes opus

Thought and Things; A Study of the Development and Meaning

of Thought, subtitled Genetic Logic (1906) in which he de­

tailed "an inductive, psychological, genetic research into the actual movement of the function of knowledge (v.l, p.

viii)." This work, an often neglected but very thorough

attempt to provide a genetic basis for reasoning at all

levels of complexity, from the simplest to the most sophis­

ticated forms, and other works by Baldwin influenced the

early work of Piaget (see, e.g., Piaget 1928). Piaget and

his collaborators at Geneva have continued and expanded the

study of logical usage in the tradition of genetic studies.

Among the works cited by Baldwin as important to the

issues of Genetic logic is John Dewey*s Studies in Logic

Theory (1903). Today Dewey’s influence on the study of

logical usage is derived from his influence on education in

general, and mathematics education in particular. Dewey's

importance lies, not in the specification of a research

theory or method, but rather in that he identified roles of

logical thought and logical usage in the domain of education

(Dewey 1933, 1936). Much of the concern with logic on the part of educators, and consequent research activity, has reflected or grown out of concerns expressed by Dewey and

the Progressive Education Movement. 16

Thus three distinct empirical traditions for the study

of the use of logic have emerged and flourished during the

twentieth century: experimental, genetic, educational. Although these traditions are not entirely disjoint, each

provides a view of the problem which is unique.

Studies in the Tradition of Experimental Psychology

Experimental studies of the relationships between logic

and human behavior are almost as diverse as the situations

in which logic is or can be used. Although no comprehensive

review of all the literature in this field seems to be

available, analyses and syntheses of findings related to

classes of problems within this area are numerous (e.g.,

Wason and Johnson-Laird 1972, Bourne and Dominowski 1972,

Clark 1971, Woodworth and Schlossberg 1960). Three areas of experimental research are of special interest:

1. Studies of inference from statements in standard

logical forms. The major experimental task given the

subject is to assess the truth, falsity, or indeter­

minacy of one statement given the truth of one or

more different statements. The given statements involve

at least one concept or phenomenon which is not mentioned

in the statement to be judged. 2. Studies of interpretations of statements in stan­

dard logical forms. Subjects are required to determine the truth or falsity of a compound statement from infor­

mation about the truth or falsity of its constituent

parts. 17

3. Studies of the formation of concepts defined by

statements in standard logical forms. Subjects learn concepts by examination of examples and nonexamples.

The concepts learned involve two (relevant) variables

related by a logical connective; the subject may or

may not be instructed on the logical connective in

advance. Although these categories of research have been operational- ally defined as disjoint, research in one classification of­ ten has implications for problems more closely related to another category.

Studies of inference. The most thoroughly researched ques­ tion in the area of the use of logic is "what inferences do subjects draw from pairs of statements in the four funda­ mental forms of class or conditional logic?” It was dis­ covered quite early (Bourne et al. 1971) that inference pat­ terns used by the general population are not identical with those mandated by formal logic. In 1935 Woodworth and Sells postulated the "atmosphere effect" as an explanation for this discrepancy in the area of class logic. According to their theory, affirmatively worded premises create an atmosphere favorable for affirma­ tive conclusions; similarly negative premises induce nega­ tive conclusions. Quantifiers have similar atmosphere ef­ fects; and according to the theory negative premises and 18 existential quantifiers are stronger in their effect than affirmative premises and universal .quantifiers.

Since 1935, many studies of class logic have been car­ ried out; the general trends of these studies have been to investigate the effects of varying statement content (abstract/concrete; familiar/symbolic, etc.), and to build probability models which might refine or replace the atmos­ phere effect as the principal theory (Ceraso and Provitera

1971, Chapman and Chapman 1959, Ericksen 1974).

Studies of conditional logic have not yet yielded a theory with the explanatory power of the atmosphere effect hypothesis. None-the-less many findings in this area have been well-established by manifold replication. The order of difficulty of four problem types (Table 2) is clear: modus ponens is easiest, followed by modus tollens, reject­ ing the fallacy of affirming the consequent, and, finally, rejecting the fallacy of denying the antecedent.

Effects of statement content have been explored in laboratory settings, and effects of developmental level of subjects have been explored in educational settings. In both of these areas fairly consistent data have been ob­ tained (Jansson 1975, Roberge 1972, 1975, Wason and Johnson

Laird 1972). In an effort to explain their developmental findings O'Brien, Shapiro, and Reali (1971) identified "Child Logic" as a "logical system" in which conditional statements (if Table 2. Principles used in Inference Tests

Modus Ponens:

Given: If p then q.

p is true

Is q true? (Yes) No Maybe

Modus Tollens

Given: If p then q.

q is false.

Is p true?

Yes ^No) Maybe

Affirming the Consequent

Given: If p then q

q is true

Is p true?

Yes No Correct No Maybe Fallacy

Denying the antecedent

Given: If p then q

p is false

Is q true? Yes No (jlaybe) Correct

Yes Maybe Fallacy 20

p then q) are consistently used as if they were bicondition­

al (p if and only if q). In later work Shapiro and O'Brien

(1973) identified five forms of "Quasi-Child Logic." All of

these are response patterns which were used consistently

(at least 8 of 12 times) by subjects participating in a

series of studies by these experimenters.

The notions of child logic and quasi-child logics en­

able researchers to classify subjects according to perform­ ance on propositional tasks, but they lack the explanatory

power and the hypothesis suggestion of the atmosphere effect

and probabilistic theories of class logic. A more fruitful

way of examining the data reported by Shapiro and O'Brien

would be to examine the types of errors consistently made

by subjects. Although the full data set is not available, restruc­

turing the published data from the Shapiro and O'Brien study

indicates that only a few types of errors are made. Some

subjects make each of these errors consistently, and develop­ mental trends in error reduction do occur. Shapiro and O'­

Brien accept 8 of 12 consistent responses on each of four

tasks as the criterion for assigning a subject to a cate­ gory. Using this criterion they find that their adult pop­ ulation (medical school students) is more inconsistent than their school-age populations. Under these conditions it would seem that systematic analysis and classification of errors (rather than subjects) would be very enlightening. 21

It has been asserted that errors in tests of logic

result from subjects* misunderstanding of the tasks assigned

them rather than from logical errors {Henle 1962, Wason

1969). Henle*s argument on this point suggests that sub­

jects simply do not understand that the logic to be used is

other than the useful thought patterns associated with day-

to-day discourse.

Wason*s argument is derived from his experience with the

"four card problem," a task related to inference in which subjects are required to check instances in order to decide whether certain inferences are appropriate. In discussing these problems he notes that subjects check unnecessary instances in order to reassure themselves of the correctness of their reasoning. Although the arguments of Henle and Wason are quite different they share the position that the subjects' inter­ pretations of the tasks and the statements with which they are presented is an issue.

Studies of Interpretations. Although the question of whether subjects understand statements involving logical connectives in a manner consistent with the truth tables associated with those statements has been discussed for some time, this question does not seem to have been formally studied until recently. In 1973 Paris asked subjects at five educational levels to indicate whether compound statements were true or false under the four true/false combinations for simple 22

statements involved. Paris' data indicate that his older

subjects interpreted conditional and biconditional state­

ments equivalently as biconditional (a finding consistent with O'Brien's Child Logic). Developmental trends are

apparent in Paris* data.

Among the trends observed by Paris is the tendency of

older subjects to treat the disjunctive as an "exclusive

or." This finding is comparable to results of Neimark (1970, Neimark and Slotnick 1970) and Jurashcek (1976). The latter

researcher has found that adults (preservice elementary

teachers) interpret the "or" connective in this manner.

Studies by Neimark in the area of class logic indicate that

the correct formation of set union requires formal opera­

tions, and that the interpretation of the "or" as exclusive reflects the improper formation of such unions.

The area of subject interpretations of statements is far from being completely understood. In their book Psychology of Reasoningi Structure and Content (1972) Wason and John- son-Laird provide several amusing instances in which rigor­ ous interpretations of logical connectives would be mis­ leading. Salmon (1973) has shown that use of ordinary logic in some scientific contexts would lead to paradoxes. These writings indicate that the context in which statements occur should effect the interpretation of them.

Moreover, there is ample evidence that the verbal con­ struction of statements has an effect on subjects' 23 interpretations of them. Several researchers (O'Brien 1972,

Jansson 1974, Shipman 1975, Neimark 1970) found that the subject matter of items (e.g., symbols, nonsense words, causal relationships, concrete objects, etc.) effects the difficulty of the items. Others (Suppes and Feldman 1971,

Johnson-Laird and Tagart 1969, Johnson-Laird 1969) have found that the semantic structure of statements tends to determine (or contribute to the determination of) subjects' interpretations of them. In this connection Johnson-Laird studied deliberately ambiguous statements and discussed a

"semantic atmosphere effect." Suppes and Feldman (1971) examined the effects of different idiomatic expression of logical connectives when presented to young children, and found differences attributable to change in idiom. Several researchers (Eiferman 1961, Wason 1961, Wason and Jones 1963, Johnson-Laird and Tridgell 1972, Wason and

Johnson-Laird 1972, Feldman 1972) have studied the effects of the presence of negations in the statements used In tests of inference, and have found that negation increases the difficulty of items. It is clear that the content, context, and linguistic structure of statements all affect the ability of subjects at diverse developmental levels to interpret and use the statements in tests of logic. 24

Studies of Concept Formation. Several findings reported in studies of concept learning have important implications for studies of the use of propositional logic. Beginning with Bruner, Goodnow, and Austin's A Study of Thinking (1956) many researchers have investigated subjects' learning of concepts which are defined by logical relationships between attributes of stimuli presented by the experimenter. A great deal of work has been done in this area, and several compilations and reviews are available (Clark 1971, Bourne et al. 1971, Bourne and Dominowski 1972) .

Among the findings reported in this area are a number of studies (Bruner et al. 1956, Haygood and Bourne 1968, and others; see Clark 1971) which have established an order of difficulty for tasks involving different logical connec­ tives. These findings are generally consistent with each other and indicate that conjunctive concepts are easiest, followed by disjunctive, conditional, and finally bicondi­ tional concepts. This ordering differs from the ordering of difficulty found by Paris (1973) for truth table tasks in that the order of conditional and the biconditional is reversed.

Thus it would appear that, although subjects find it easier to learn relationships of implication than equivalence when they are learning through the trial-and-error or hypothesis- testing experiences of a concept learning experiment, when these relationships are presented to them through verbal 25

expression of the logical relationships the equivalence relationship is more easily understood and learned than

the implicative one.

A finding related to this phenomenon is Bourne's (1970)

observation that subjects who learn concepts defined by

logical relations do not use the language of propositional

logic to report the concepts learned. Rather they tend to

list the combinations which are consistent with the concept. By comparison with the study of concept learning, the

study of the use of propositional logic is in its infancy. As more becomes understood about the use of logic at various developmental levels and in various contexts, many of the results of concept learning research will be available as sources of information, hypotheses, and questions concerning the use of propositional statements. For example, work on the effects of relevant and irrelevant information on con­ cept formation may have implications for problems involving statements such as "If p or q, then r."

General considerations. Concept formation research is not alone among the topics of experimental psychology which have the potential for informing the investigation of the use of propositional logic. The investigations of psycholin­ guistics and of information processing both have a priori relationships to the study of logic. Summarizing the research in the tradition of experimental psychology as it applies to the problem of the use of 26 propositional logic in mathematic contexts we find that the most directly related research concerns patterns of infer­ ence. None of the psychological studies reported involve

inference in the context of mathematics, or even in domains in which the concepts or phenomena related by the pro- positional statements have structural or conceptual relation­ ships to each other independently of the statement presented.

Certain findings (e.g., difficulty ordering of problems, developmental trends) have been replicated over a variety of contexts, and are likely to apply to the mathematical context as well.

Research on the construction of truth tables and forma­ tion of concepts, when integrated with the results concern­ ing inference suggest that subject difficulties in drawing appropriate inferences and rejecting inappropriate ones may arise from erroneous interpretations of statements or from difficulties in the formation of concepts which the experimenter intends the statements to convey.

Studies and Findings in the Genetic Tradition

Although the genetic tradition for the study of logical development was begun at Johns Hopkins by J. H. Baldwin, the importance of that tradition in studies of logic today is due principally to Jean Piaget and his collaborators, notably Barbel Inhelder, in Geneva.

Piaget* s theory of the stages of cognitive development has been stated, interpreted, and analyzed in many volumes 27

{e.g., Piaget 1928 and elsewhere, Flavell 1963, Furth 1970,

Ginsberg and Opper 1969). As is well known, in Piaget's

view, cognitive development culminates in the stage of

formal operations. In his view one characteristic of

persons who have attained the level of formal reasoning is

their ability to engage in combinatorial thought.

As described by Piaget in both theoretical discussions

{Piaget 1957, Beth and Piaget 1966) and in reports of sub­

ject performance (Inhelder and Piaget 1958), combinatorial

thought in the solution of problems involves consideration

of all potentially relevant variables and systematic test­

ing of the effects of the full domain (lattice) of combin­

ations of relevant variables. According to Piaget the

formally operational individual generates the lattice of

combinations by repeated mental applications of the INRC

group, a group of transformations on the lattice which is

(isomorphic to) the Klein 4-group. When a conclusion is

reached, whether it be the full solution to the problem, a partial solution, or a decision to reject an hypothesis,

the formal subject expresses his conclusion using the relationships and languages of propositional logic.

Criticisms of Piagetian theory concerning the development and nature of logical thought have developed along several dimensions. Over the years since the publication of The

Growth of Logical Thinking from Childhood to Adolescence

(Inhelder and Piaget 1958), various writers have attacked 28

Piaget's notions on the grounds of their theoretical valid­

ity (internal consistency, independence, completeness), the paucity of the empirical base, the generality claimed for

the formal stage, and the overall usefulness of the theory

as a tool for understanding the logical processes of individ­ uals (Ennis 1975).

Questions of Theoretical Validity. Perhaps because his theories deal explicitly with logic, or perhaps because

Piaget has claimed for them an affinity with Bourbakian structures (Beth and Piaget 1966, Piaget 1957) and in par­ ticular algebraic structures such as the Klein 4-group, group theory, and lattice theory, the theories of Piaget have been subjected to greater scrutiny by logicians than is ordinarily accorded to psychological theories. As early as 1940 Quine

(1940a, b), in reviewing papers of Piaget, criticized his mathematization of the groupement. Parsons, in his review of The Growth of Logical Thinking

(Parsons 1960) expanded Quine's criticisms, presenting a detailed technical analysis of Piaget's logic. Parsons' critique states in part: Although Piaget uses the standard notation of the propositional calculus, it turns out that he vacillates somewhat in his inter­ pretation of the notation, and his use of it obscures some features of inference undoubtedly made by subjects which logicians call logical. This vitiates the claim that propositional logic provides the essential structure of the final stage of logical development. For the same reason it is 29

uncertain that Piaget's model completely describes the difference between his stages II and III.

The problem of multiple interpretations of mathematical

and logical symbols seems to be endemic to Piagetian studies

of logic. The logician Jean Grize has provided a system of

axioms for Piagetian theory (see Beth and Piaget, pp. 267-

272; Osherson, volume 1, pp. 18-19), but attempts by Piaget

(Beth and Piaget), Osherson (volume 1, pp. 23ff.) and the

current writer to produce models for these axioms which are in the form of Piagetian problem solving tasks seem to

lead invariably to multiple interpretations of single events. A related issue has been addressed by Bynum, Thomas, and Weitz (1972). This team of logicians and a psychologist was motivated by its analysis of a protocol presented by

Inhelder and Piaget (1958, p. 102) to try to devise every­ day expressions for the logical forms Inhelder and Piaget claim are used by their subject. After a year's effort these researchers concluded that colloquial forms expressing

6 of the 16 logical relationships do not exist.

The Paucity of the Empirical Base. The criticism most fre­ quently leveled against the entire fabric of Piaget's research concerns the small number of subjects used in

Genevan studies and the lack of experimental control. While the force of this criticism has been lessened in some areas by manifold successful replications of Piagetian experiments 30

(Flavell 1963, Lovell 1971 a,b# Sigel and Hooper 1968), replications of experiments in the areas of formal thought are not numerous, and in some cases have been unsuccessful.

Although Inhelder and Piaget offer protocols from sub­

jects in fourteen separate experiments to illustrate the general developmental trends in subjects1 approaches to explanation of phenomena, only one protocol (GOU) from one experiment (invisible magnetism, Inhelder and Piaget 1958, p. 102) is offered as evidence that individuals eventually use the full lattice of 16 binary operations in problem solving.

In their reanalysis of GOU's protocol Bynum, Thomas, and Weitz (1972) found only ten of the sixteen operations present. These researchers attempted to replicate the invisible magnetism experiment (Weitz, Bynum, Thomas, and

Steger 1973) using a larger pool of subjects, and found that their subjects used only 5 of the 16 logical operations.

While this study certainly does not support the Piagetian theory, due to internal problems of its own, it is not sufficient to refute Piaget's position (Damarin and Shumway

1976). On the other hand, some evidence in support of the notion that subjects do use the Klein 4-group (INRC) in the solu­ tion of logical problems has been reported by Markel (1973, 1974). Although Markel's research, based on a card game modeled on Piagetian theory and a task related directly to 31

the 4-group, reveals a possible role for algebraic struc­

tures in models of formal thought, the 4-group itself is

too simple a structure to describe adequately the logical

problem solving process (Markel 1974, Bart 1971).

Piagetian tasks for formal thought have been used in

several research situations in which the main objective

has been to classify subjects as formal or non-formal (e.g.,

Juraschek 1974, Schwebel 1975), but these surveys shed lit­

tle light on Piagetian Theory itself. Other studies involv­

ing these tasks have compared success on the tasks with

other forms of intellectual success; again, these studies

neither support nor refute Piaget's claims.

In the absence of further attempts to replicate Piaget­

ian experiments, the research base for the theory of com­

binatorial thought remains meager. Not only are the exper­

imental findings and replications small in number, but also

research has been limited to reasoning almost exclusively in

the realm of physical science. Thus the question of the

generality of the findings, even if replicated, must also be addressed.

The Generality of the Stage of Formal Operations. Early works of the Piagetian School expressed, or often assumed,

the notion that the acquisition of formal operations is a unitary process. Under this assumption Inhelder and Piaget

in their preface to The Growth of Logical Thinking speak of their purpose as "to describe the formal structures that mark 32

the completion of the operational development of intelli­

gence." In the last chapter they address what they viewed

as a parallelism between intellectual and moral development

of the adolescent, this parallelism based on the view that

all intellectual and moral problems were approached by the

formally operational individual using the formal logical

structures. Recent research has revealed that the formal stage,

whether it be characterized by Piagetian structures or not,

cannot be considered as a unitary phenomenon (Bart 1972 a,b,

Berzonsky, Weiner and Raphael n.d., 1975, Higgins-Trenk and

Gaite 1971, Stome and Ausubel 1969). Each of these research­ ers has reported at least one pair of measures of the use of

formal operations between which there was observed a zero-

order correlation.

In one of his more recent statements on this matter,

Piaget (1972) partially abandons the notion that formal

thought is a single acquisition which is uniformly applied by individuals who have attained it:

. . . a formal structure seems in contrast (to a concrete structure) generalizable as it deals with hypotheses. However, it is one thing to dissociate the form from the content in a field which is of interest to the subject and within which he can apply his curiosity and initiative, and it is another to be able to generalize this same spontaniety of research and comprehension to a field foreign to the subject's career and interests. We now ask the following critical question: Can one demonstrate at this level of 33

development (15 to 20 years) as at previous levels, cognitive structures common to all individuals which will, however, be applied or used differently by each person accord­ ing to his particular activities?

The reply will probably be positive but must be established by the experimental methods used in psychology and sociology."

Thus without denying the lattice theoretic/4-group model for formal operations which he developed in conjunction with studies of persons performing scientific tasks (and reitera­ ted in the 1972 paper cited), Piaget has conceded that this model may not be universal in its applicability, and raised the question of whether such a universal model can be devel­ oped.

A more fundamental question might have been posed, name­ ly, whether a single model is appropriate to all areas of intellectual investigation. Perhaps the structures of the content of different subjects and endeavors require diver­ sified cognitive structures.

Studies in the Tradition of Education Research

The distinction between an educational study of the use of propositional logic and one from the other traditions is not a distinction in research methodology or conceptual framework as much as it is a difference in purpose. Indeed many of the studies discussed in regard to the experimental and genetic traditions were conducted by educational re­ searchers or in educational settings. 34

Other types of studies by educators focus, not on the development or nature of logical ability as a psychological phenomenon, but rather on the relationships between use of propositional logic and other educational variables. Stud­ ies in this category are extremely varied; they seem to share only the underlying assumption that logical thinking is a valid and important goal of the educational process.

One important category of studies and discursive papers in the educational tradition is directed toward the further specification of the importance of logic in the educational process.

The development of logical abilities has long been a goal of general education and at various times writers on the goals of mathematics education in particular have cited objectives such as the ability to think clearly as related to the rationale for mathematics instruction (Osborne and

Crosswhite 1970).

Moreover, with the ascendant popularity of discovery, problem solving, and laboratory based instruction, as well as the use of computers in mathematics classrooms, the importance of logical reasoning as a tool for learning is enhanced. Many mathematics educators predict increased adoption of these instructional approaches as use of the hand calculator becomes the accepted algorithm for complex computation, thus increasing the amount of classroom time available for "higher level” activities (Suydam 1976). 35 Strike (1975) reviewed and analyzed studies of the many

differing approaches to discovery based instruction in an

attempt to find their common core. Concluding that the concept of verification is the "logical center" of the idea of discovery, he went on to observe that "a reasonable degree of sophistication concerning the logic of verifi­ cation and concept formation is prerequisite for an ade­ quate conceptualization of the variables involved in dis­ covery teaching and learning."

Some authorities on mathematics education (Vinner 1976,

Sernadeni 1975) have argued recently against the teaching of mathematics at pre-college levels from an axiomatic or definitional point of view. None-the-less these educa­ tional theorists advocate the teachers' use of "valid and convincing" arguments in forms which generalize to more abstract situations. It is apparent that such forms must be grounded in the sound use of logic.

Research by Gregory and Osborne provides correlational evidence in support of teachers* ability to use proposi­ tional logic. In a series of researches (1975, Gregory 1972,

1975) teachers' use of conditional language during mathe­ matics instruction was coded, and this usage was correlated with students' scores on the Cornell Critical Thinking Test. The results indicate that students instructed by teachers who frequently use the language of propositional logic show greater gains in critical thinking than students whose teach­ ers rarely or never use conditional statements. 36 Against this background ascribing several areas of importance to teachers' ability to use logic, there stand only a few studies of elementary teachers' logical ability. Juraschek (1974) tested 141 preservice elementary teach­ ers using three Piagetian tasks; his findings indicated that 52.5% of these subjects were still in the stage of concrete operations, 42.6% were in the transitional stage preceding full formal operations, and only 5% had fully attained the formal operational period. Two studies (Eisenberg and McGinty 1974, Jansson 1975) have investigated logical inference among preservice ele­ mentary teachers. Both studies showed rather poor per­

formance over the full range of tasks. Overall Eisenberg and McGinty's population performed somewhat better (an average of 54.5% correct over 5 problem types) than Jansson's (an average of 36.8% correct over 4 problem types). The order of difficulty of the three conditional problem types common to both investigations was the same in both studies. Eisenberg and McGinty compared performance of preservice teachers with that of second and third graders and found no significant differences on their two disjunctive problems. These researchers also analyzed performance on the basis of affirmative vs. negative and mixed wording of statements, and found that wording effects were roughly the same for all populations tested. In a follow-up of part of Eisenberg and McGinty's study Juraschek investigated performance of pre-service elementary 37 teachers on disjunctive items when these items were present­ ed under different sets of instructions. On the basis of this experiment he concluded that the results of Eisenberg and McGinty's experiment were due to teachers' interpre­ tation of disjunctive statements using the "exclusive or."

Both Jansson and Eisenberg and McGinty observe that pre­ service elementary teachers are insufficiently trained in the area of conditional reasoning, and advise that develop­ ment and implementation of instruction in logic for pre­ service teachers is long overdue. Jansson suggests that efficient instruction can be accomplished only after further understanding of the problem is acquired.

Synthesis of Three Traditions

The values and goals of twentieth century education pro­ vide adequate motivation for the study of teachers' use of propositional logic. However, the logical demands of current curricular and instructional developments when con­ trasted with the findings of recent studies of the use of logic by teachers lend a note of urgency to this problem.

Studies of logical usage by preservice elementary teach­ ers conducted by Eisenberg and McGinty, Jansson, and Juras­ chek produced results not very different from those of similar studies of the general college population conducted by researchers in the experimental and genetic traditions. In the analysis of research in each of these latter traditions several common issues emerge. Findings in both 38

traditions seem to vary to some extent with the conceptual context from which the problems are drawn. Familiarity of the experimental subject with the subject matter or level

of abstraction of items has a facilitating effect on per­ formance in both cases. The issue of the meaning of statements in standard log­ ical forms is raised in connection with research in both traditions. In experimental research the interpretations made by subjects are at issue: do subjects faced with inference problems interpret the premises as conveying the information intended? Piagetian theories and research, on the other hand, have been criticized on the grounds that the experimenter makes inconsistent (Parsons) and inappro­ priate (Bynum et al.) interpretations of subjects* state­ ments. In both cases the effectiveness of communication

between experimenter and subject is questioned. The listing and testing of possible combinations of

events defined by variables relevant to a problem also emerges as an aspect of subject behavior in both research traditions. Inhelder and Piaget (1958) discuss in some detail the development of the capacity to construct (mentally) the combinations to be tested during problem solving, and assert that the ability to construct the full lattice is a necessary condition for formal operations. In the tradition of experimental psychology, on the other hand. Bourne (1970) finds that subjects use just such a listing to report (logically defined) concepts learned. Even as research in these two traditions shares some

salient characteristics and findings, from the point of view

of extending the findings to the use of logic in mathematical contexts they share a deficiency. In neither tradition has

an experiment been reported in which problem content is

itself dependent upon logical relationships. The typical

problems of experimental research involve artificial rela­

tionships between "real life” phenomena such as blue balls

and brown dogs. Piaget's work involves observed relation­

ships between physical states of apparatus; the relation­

ships between these states are certainly not artificial, but they are empirically, not logically, determined.

Summary

Mathematics educators are concerned with the use of logic by preservice elementary teachers for a variety of reasons. Salient among these is the accumulation of evi­ dence that future elementary teachers do not perform well on tests of inference. However, these tests are based on rather artificial situations, and performance on them may not generalize to situations such as mathematics in which logical considerations are natural and important.

The use of logic in mathematical contexts appears not to have been investigated by educators, experimentalists, nor Piagetians. None-the-less, research in all of these traditions provides direction for study in this area. The

Piagetian analysis of combinatorial thought as systematic consideration of variables provides a basis for the develop­ ment of an algorithmic model for the interpretation of logical and mathematical statements. The investigations and findings of experimental psychologists are useful in suggesting issues on which to focus; taken together they mandate the initial approach: analyze errors, not sub­ jects . CHAPTER III THE CONSTRUCTION OF A MODEL

In an area in which the theories, questions, and research techniques are as varied and complex as they are in the area of logical thinking, a hypothetical model of the process be­ ing studied serves a variety of useful purposes. Not only does a good model enhance the integration of information from diverse sources, but it also facilitates the formula­ tion of research problems for further studies.

The model developed here has its basis in certain ele­ ments of the Piagetian theory of combinatorial thought.

Analysis of the interface of Piaget's ideas and research with other types of psychological studies of logic, and of the problems inherent in attempts to apply either Piagetian or experimental ideas to mathematical problems, leads rather directly to this model for the use of logic in elementary mathematical contexts. For the purposes of exposition the terms logic and logi­ cal refer to elementary traditional logic such as might be included in a first logic textbook. A logical statement is an English language sentence which is either a simple declar­ ative statement or a compound statement in one of the forms 42 of elementary logic (p and q; p or q; if p, then q; p if and only if q; where p and q are simple declarative state­ ments) . A mathematical statement is a simple declarative statement which explicitly involves mathematical operations or relations. Mathematical statements are frequently in the forms of equations, inequalities, or equivalence class membership relationships.

Underlying Piaget's discussions of logical thinking, or combinatorial thought, is a three stage model for the process of solving problems by logical methods. During the first stage the problem is presented. The second stage of the process consists of the systematic hypothesizing and testing of possible solutions or partial solutions. During this stage the INRC group operates, according to Piaget, to monitor the order in which possibilities are tested. Final­ ly, the problem solution is expressed in a logical statement. This model is illustrated schematically in Figure 1 below.

INCR Group Monitors

Figure 1. A schematic representation of the Piagetian Model 43

Piaget's model was developed with reference to problems

drawn primarily from the empirical sciences, and has been

subject to test only in that domain. Efforts to define methods of verifying this model for problems in the domain

of mathematics lead immediately to two difficulties.

The first of these difficulties is one of directionality.

It seems clear that the direction of problem solving in the physical sciences is one of moving from a question posed in empirical terms to a determination of logical or structural relationships among variables. In mathematics, on the other hand, problems frequently occur (or are posed) in terms of

logical relationships and are solved by determination of mathematical relationships among the problem variables.

Therefore, if the Piagetian model is to be made applicable to the full use of logic in elementary mathematical situa­ tions, it is necessary to add to that model reversals of the operations.

The second difficulty lies in the process of systematic consideration and testing of possibilities. In the problems used by Inhelder and Piaget consideration and testing can be viewed as a unitary process because the actual testing of possibilities considered is a trivial matter. Once a sub­ ject has determined to combine three colorless liquids, for example, the actual test involves only the simple act of pouring. The consideration and testing procedure necessary for solution of problems involving mathematical relationships between variables is frequently more complex than it is in Piaget's experimental tasks. After a subject determines a possibility for consideration, his test of that possibility will generally require the use of an algorithm which he may or may not execute correctly. If the algorithm is incorrect­ ly performed the subject may erroneously accept or reject the possibility under consideration.

In order to accommodate these differences, the consider­ ation and testing phase of the original model must be expand­ ed so as to separate the non-trivial aspects of the solution process in mathematical problems.

By making these adjustments a five-stage symmetric model for the use of elementary logic in mathematical contexts was developed. A static representation of this model is pre­ sented in Figure 2 below. In order to examine the hypothe­ sized operation of this model on simple mathematical prob­ lems a formal analysis of the relationships between elemen­ tary mathematical and logical statements is necessary. 45

Jenerate rGenerate [Possibil­ Possibil­ ities ities {L>. Mathe- Sort Logical [matical Possibil­ Statement State­ ities ment ^Evaluatey» Evaluate 1 Possibil-J [Possibil-1 V ities J V ities / V )

Figure 2. The expanded model

A formal relationship between mathematical and logical state­ ments . Consider the pair of statements: xy * 0

x = 0 or y = 0 The equivalence between the mathematical statements ”xy * 0H and the logical statement "x = 0 or y = 0" can be demon­ strated in tabular form very much like a truth table as shown in Table 3. 46

Table 3. Equivalence of two sample statements.

Cases Is xy = 0? x=0 y=0 x=0 or y-0 x = 0 y = 0 yes T T T x = 0 y / 0 yes T F T x 7* 0 y = 0 yes P T T x 7* 0 y ^ 0 no F F F

compare' J

The purpose of the following formal discussion is to describe a procedure which generalizes that used in building Table 3, and which provides a formal method of determining whether two statements concerning two variables are equivalent. The method described consists of systematic assignment of codes called "truth table forms" to statements; two statements assigned the same code are equivalent. The truth tables of elementary logic provide the vehicle for formal description of logical statements; this descrip­ tion is independent of the content of the statements. A symbolic code representing truth tables is easily described. A truth table form is a four digit binary expression. The truth table form of a truth table (organized in the usual manner) for a compound statement can be obtained from the right hand column of the table by replacing T's and F's with l's and 0's, respectively, and transposing. For example, the truth table form of "if p, then q" is 1011; it is obtained as follows; 47

p :> q T T T - T F F 1011 F T T F F T

Truth table forms can also be determined for many mathe­ matical statements. If D is a domain (e.g. a set of numbers) and C a dichotomous concept defined over the domain D, then C partitions D into two subdomains and Dg? these subdo­ mains are arbitrarily ordered (D^, Dg). The characteristic function C* of maps D onto the set |l, oj , and the pro­ duct map C* x C* maps D x D to {llr 10, 01, OoJ . Sympolically this process is represented as follows:

C: D ------> (D1# Dq) ; D1 UDg=D; Din D0 ® 0

C* 5 D ------> {o, l} if d£ D, C* (d) - £ if dtD, D x D ( n , io, oi, oo} 'll if d^ Dx, d2£ Dx 10 if dxt D1# d 2€ Dg C* x C* (d1, d2) = 01 if dj^C Dg, d2 t d x 00 if dxe Dg, d2e Dg

For an example of these mappings consider D as the domain of integers, and C the dichotomous concept odd/even. C partitions D into D^ the set of odd integers and Dg the set of even integers. C* then assigns to any odd integer the value 1 and to any even integer the value 0. For any ordered pair of integers C* x C* assigns a two digit code reflecting 48 the parities of the two members of the pair. C* x C*

(2, 5) - 01; C* x C* (5, 7) = 11, and so on.

By the convention described above a mathematical state­ ment concerning two variables contains an operation 0, Not all mathematical statements have unique truth table forms for the partition D^, Dg. In order that a truth table form be well-defined for a statement concerning the operation 0 it is necessary and sufficient that the condition (**) de­ fined below be satisfied.

Now, suppose 0 is an operation of the arbitrary domain

D:

0 : D x D ------> D which satisfies the property (**) .

<**) If C*(d1) = C*(d2) and C*(d3) = C*(d4), then

C* (0 (dv d3)) - C M ${&2, d4>>

Such an operation induces a map 0' which maps {ll, 10, 01, oo) to (l, 0j , and for which the following diagram commutes: 0 D

C* x C*

11,

The truth table form associated with a statement pf(x, y)6 D1 where x and y are variables is defined to be the ordered 4-tuple {0' (11) , 0* (10) , 0* (01), 0 % (00)) written without commas and parentheses. The truth table form of 49

0(x,y)£ Dq is obtained from the above by interchanging the

l's and 0's.1

Returning to the example in which D is the set of inte­

gers and C is the odd/even concept consider the operation of

addition of integers. Since the sum of two odd or two even

integers is always even while the sum of an odd and an even

integer is odd, property (**) is satisfied, and the map jrf*

is induced. The truth table form of the statement "M + N is odd" is 0110; that of "M + N is even" is 1001.

If the input statements, p and q, of a truth table are

P: x^ belongs to q: X2 belongs to D1 then the truth table form of a mathematical statement re­ lating variables x^ and x2 is identical with the truth table form of the logical statement describing the solution set for the mathematical statement. For example, the truth table form of "M x (N + 1) is even" is 1011; considering the statement p and q as "M is odd" and "N is odd" respectively, the truth table form of the statement "If M is odd then N is odd” is also 1011. This statement describes the solution set for the mathematical statement.

As noted above some statements do not have truth table forms for a given partition. An example of such a statement is "max(M,N) is even." For this statement condition {**) is not satisfied; max (5,6) is even, but max (7,6) is odd although 5 and 7 are both odd (and 6 is constant). 50

Although each logical or mathematical statement has a

unique truth table form, there are many statements having the same truth table form.

This formulation of equivalence between pairs of statements forms the basis for the following definitions: Two statements are LD-equivalent provided they have the same truth table form and refer to the same partition defined over the same ordered partition of the same domain. In the development of the model used and tested in this study the principal concern has been with LD-equivalence of state­ ments. The logical statements considered were restricted to those which did not explicitly involve negation. The effects of this restriction were mitigated in part by the selection of domains and partitions for which both sets

(D^, Dq) have positive identities (e.g. odd/even). However the restriction did exclude such statements as "It is not the case that M is odd and N is odd." from consideration al­ though both the negation of this statement (M is odd and N is odd), and LD-equivalent forms (M is even or N is even) were included.

The Operation of the Expanded Model

The basic problems to be considered are problems which are stated in logical or mathematical forms and solved by choosing the mathematical or logical statement having the 51

same truth table form. The connection between these two

statements (problem and solution) is assumed to be the solu­

tion set of the first statement, or, more precisely, a sort­

ing of the set D X D into the set of solutions and its complement. Where the domain D is infinite this sorting can

not be done by exhaustive case by case testing, and it is assumed to be done by applying the characteristic

function C* and sorting the set {11, 10, 01, 00).

Under these assumptions a problem solution is obtained

in five steps:

1. The initial (problem) statement is posed.

2. The set { 11, 10, 01, 00 } is generated in the

appropriate context. 3. The four cases 11, 10, 01, and 00 are indivi­

dually tested

4. The set { 11, 10, 01, 00 > is sorted into a "true" or solution set and its complement.

5. The final (solution) statement is provided.

It should be noted that since several statements have the same truth table form, the fifth step of this procedure does not have a unique result. For each truth table form there exist classes of equivalent logical statements and mathematical statements; correct performance of step 5 52 is evidenced by selection of a member of the appropriate 2 equivalence class.

Figures 3 and 4 illustrate the operation of the model on a mathematically stated problem and a logically stated problem, respectively.

Sr*rt tiihikhi M t44, N ^44 M «44, N M even, A/ 944

M * N a r u t . even , 'M 944, *Jo44 M tv if t , AAevett F t l i t rt4d, 0 4 4 } o44,Ntue* (•44, tVtn}*o44 turn# tytn, e44)*o44i

Da/it

Figure 3: Model Operation on a Mathematical Statement

6 and e" are not operations in the sense of mathematical operations since they are not well-defined. For the empirical testing of the model, however, standard state­ ments representing the equivalence classes associated with truth table forms were identified. When restricted sets of statements are considered to be the ranges of 6 and 0~3-, these operations are well defined. 53

/H i i O f N /*

Figure 4: Model Operation on a Logical Statement

The operations identified as $ and (J ^ in the general model- (figure 2) and in the problem solutions identified in figure 3 and 4 above are undoubtedly more complex than the single arrows indicating them show. In each case they seem to involve a hypothesis testing procedure which is based on iteration of certain operations in the general model.

The expanded hypothesis testing process which could account for the operation is illustrated in figure 5 below. The operation of (J-1 can be explained by a procedure symmetric to that for 8. \ iSf&ttmenf

Figure 5: Expansion of Operation 6

Integration of the Research Base with the Model

The hypothesized model for logical usage in mathema­

tical contexts has emerged from consideration of the elements

common to the problems and results of research in several

traditions. The model having been posed, it is possible and

appropriate to re-examine the research base in the light of

the model.

Examination of the literature of experimental psy­

chology from this point of view reveals that problems

studied experimentally are chiefly concerned with two aspects

of the model: the transition from logical statements to evaluation of possibilities (inference studies), and the

relationships between sorting of possibilities and evaluation of possibilities (concept learning research). One paper

(Bourne 1970) concerns the transition from sorting of possibilities to logical statements. 55

Piagetian research, on the other hand, has been devoted

primarily to the generation of possibilities for problem

statements and the use of logical statements in describ­

ing sortings. Implications of this research for problems

in mathematical contexts would therefore concern the arrows

a and 6 of the model. The major criticisms of Piagetian work on logical thinking concern the conclusions drawn concerning logical statements. A diagrammatic summary of the research literature as

it applies to the model is provided in figure 6.

Figure 6: Integration of the Research Base 56

Testing the Model I; Delineation of Contexts

A thorough testing of the model, even in one mathe­

matical context generalizable to one population group,

would demand a huge investment in subject tine, and the

making of more than 100 comparisons of subject performance

on specific pairs of tasks. Efficiency in selection of mathematical contexts and key questions to be investigated

initially is of obvious importance. This selection roust

take into account the nature of the subject population. For the current research this population consists of preservice

elementary teachers enrolled in courses on mathematics or methods of teaching mathematics. With this population in mind, mathematical contexts to be available for use in the testing of the model were chosen to meet the following requirements:

- The concepts related by the logic are familiar to persons with high school education (college preparat­

ory program); moreover, the conceptual content of

items used with any subpopulation has been a part of

required college mathematics coursework covered prior

to the experimental testing.

- The domain (D) of a concept is large enough to

necessitate use of abstract reasoning rather than

exhaustive case by case testing. 57

- The concept imposes a natural dichotomization of the

domain; this dichotomization allows the construction

of formal relationships between mathematical and logical

statements as described above. -Relatively simple mathematical statements are avai­

lable for each of the 16 possible truth table forms.

Three mathematical contexts were selected as fulfilling these

requirements. These contexts are defined by a domain and

a concept as follows:

Code Domain Concept

OE Integers Odd/even

ZG Counting Numbers Zero/nonzero

PN Nonzero rationals Positive/negative

Testing the Model II; Research Hypotheses

Three categories of questions were identified for initial testing of the viability of the model. Questions in the first category concerned the abilities of subjects to perform the single operations which form the elements of the model. In connection with these first order questions the following research hypotheses were formulated.

Hi: Given a mathematical statement subjects can generate

the 3et of (relevant) possibilities: 58

H2. Given a mathematical (resp. logical) statement

subjects can evaluate the possibilities given in

a list, thus forming a sorting of the list. H3. Given a sorting of the set of possibilities in

a proscribed context subjects can select appro­

priate mathematical (resp. logical) statements

from a list provided, H4. Given a mathematical (resp. logical) statement,

subjects can select an equivalent logical (resp,

mathematical) statement from a list provided.

Figure 7 below indicates the operations of the model which are to be tested under these hypotheses.

Figure 7: Hypotheses 1 - 4 concern the ability of subjects to perform single operations 59

The second category of questions identified concerns

comparative performance of subjects on pairs of single

operations. Within this category two types of comparisons

are considered: (1) the relationship between a subject's

performance on one operation and his performance on another,

and (2) subjects' ability to identify differences between

non-equivalent statements. Stated in terms of research

hypotheses, the questions posed are: H5. Subjects who can select logical statements

descriptive of sortings can also select mathe­

matical statements describing those sortings,

and conversely. H6. Subjects who can select logical statements

equivalent to given mathematical statements can

also select mathematical statements equivalent

to logical statements, and conversely.

H7. Subjects who can evaluate the possibilities for mathematical (resp. logical) statements can also

select mathematical (resp. logical) statements

descriptive of the sortings, and conversely.

H8. When asked to compare two non-equivalent state­

ments subjects can select an element of the set

of possibilities for which the statements differ.

Figures 8, 9, and 10 provide a schematic summary of the pairs of operations compared under these hypotheses. 60

6 e o e ra tt Gt re m it Possibilities) Pi35ibtl*ti (*) M

tts comparts

Ms & r % tls \ PbSSibhhrs)

H i caitparts <2

Figure 8i Hypotheses 5 and 6 aach coapara two operations 61

Ev&fotbc PtsvMitHea) fbs&bif/ M

Figure 9i Hypothesis 7 compare■ operations of finding sortings for statenants with finding state­ nants for sortings. 62

((yenernf-e. PoiHbiUfiti\ PaiSibilih (») a)

ntarnervTi

rigura 101 Hypothesis 8 concerns ability to distin­ guish sortings for aon-aguiralent state­ ments. 63

The final pair of questions concerns the relationship

of the ability to translate mathematical statements into

logical ones (and conversely) to sorting ability and the ability to select statements descriptive of sortings.

According to the model, the latter two processes are consti­ tuent parts of the former. Stated as research hypotheses, the two questions investigated in this regard are:

H9. Subjects can select logical statements equivalent to given mathematical statements if and only if

they can evaluate the possibilities for mathe­

matical statements, and select logical state­ ments descriptive of sortings of the set of

possibilities.

H10. Subjects can select mathematical statements

equivalent to given logical statements if and only

if they can evaluate the possibilities for logical

statements and select mathematical statements descriptive of sortings of the set of possibi­

lities .

The hypothesized compositions of operations tested under each of these hypotheses is shown in Figures 11 and 12 respectively. 64

Figur* lit Hypothesis 9 compares translation froa mathematical to logical statements with two other operations. 65

Flgura 121 Hypothesis 10 coatparas translation from logical to aathamatical stateaents with two othar oparations. Of overriding concern in the consideration of each of

these hypotheses is the role and importance of the logical

connective used in the logical statements, and, to a somewhat

lesser extent, of the operations involved in, and the comple­

xity of, mathematical statements. Therefore, the hypotheses

stated above must be considered not only in a global sense,

but more importantly in each of several instances defined by

the logical (or mathematical) form of the statements

presented in problems or required in problem solutions. In

the execution of the model testing, each of the hypotheses

listed above must be considered as a set of five hypotheses;

these five are specified as follows:

• • the global hypothesis as stated Hil

♦ Hi2 • Hi restricted to conjunctive logical statements

■ • Hi to disjunctive logical Hi3 restricted statements

* • Hi restricted to conditional logical statements Hi4

• Hi_ * Hi biconditional logical statements D restricted to

These considerations, and the results reported in the litera­ ture review dictate the posing of two further hypotheses concerning the operation of the model:

Hll. Performance of subjects on each phase of the

model testing will vary as a function of the

logical connectives used in statements. In all phases of model testing error patterns will be consistent with conjunctive interpre­ tations of all logical connectives. CHAPTER IV

PROCEDURES FOR TESTING THE MODEL

The experimental approach taken to the testing of the model was basically an exploratory one. A number of key hypotheses were identified on a priori grounds, and the major objective of the procedures was to test these hypo­ theses in an efficient manner. At the same time, post hoc examination of data, especially of error patterns, was viewed as essential. The testing of the model was conducted in five research segments. For each segment a question or set of hypotheses was identified for testing, and a brief instrument package directed toward the specific question was developed.

The questions identified for a given segment were inter­ preted in terms of relationships between truth table forms and their logical, mathematical, and sorting equivalents.

Once these formal relationships had been established, stan­ dard mathematical and logical statements were selected from predetermined lists for use in items. Thus the linguistic structure of statements, as well as the content, was subject to fairly strict control. The standard statements used are provided in Table 4 (some statements used in segment 1 differed from these; see Appendix B). 69

Table 4. Standard Statements

Truth table form Logical Statements Mathematical statements

0000 2(M + N) is odd

Conjunctive 0001 M is even and N is even M + N + MxN is even 0010 M is even and N is odd N(M + N) is odd 0100 M is odd and N is even M(M + N) is odd 1000 M is odd and N is odd M x N is odd

Simple 0011 M is even 2N + M is even 0101 N is even N is even 1010 N is odd N is odd 1100 M is odd 2N + M is odd

Biconditional 1001 M Is odd if and only if N is odd M + N is even M is even if and only if N is even 0110 H is odd if and only if N is evenM + N is odd M is even if and only if N is odd

Disjunctive 0111 H is even or N is even M x N is even 1011 M is even or N is odd M(M + N) is even 1101 M is odd or N is even N (M + N) is even 1110 M is odd or N is odd MxN + M + N is odd3

Conditional 0111 If M is odd then N is even M x N is even 1011 If M is odd then N is odd M(M + N) is even 1101 If N is odd then M is odd N(M + N) is even 1110 If N is even then M is odd MxN + M + N is odd3 aAlternate used in test M ( segment 5: (M + 1) (N + 1) Is even 70

Table 5. Brief descriptions of research segments

Segment 1. Finding Sortings for State­ ments . The purpose of segment 1 was to gather information concerning subjects' sortings of sets of possibilities for mathematical and logical statements. The effects of abstract vs. concrete possibilities were compared. Hypotheses H2a and H2b were tested.

Segment 2. Comparing Content Domains. The purpose of segment 2 was to compare the difficulty of finding sortings for mathematical statements from three con­ tent domains: odd/even, positive/ negative, zero/nonzero.

Segment 3. Selecting Statements for © © Sortings. The purpose of segment 3 was to explore the abilities of subjects to select logical and mathematical state­ ments descriptive of the various sortings of the set of possibilities. Hypotheses H3a, H3b, H5, Hll, and H12 were tested.

Segment 4. Finding examples for which 0 statements differ. The purpose of segment 4 was to investigate subjects' ability to identify elements of the set of possi­ bilities for which one statement is true and another is false. Hypotheses H8 and Hll were tested.

Segment 5. Bringing six tasks together. 6 The purpose of segment 5 was to compare and contrast subjects' abilities to translate from one to another of the three inodes: logical statements, mathe­ matical statements, sortings of the set of possibilities. Hypotheses H4a, H4b, H6, H7a, H7b, H9 and H10 were tested. 71

Segment 6. The Classroom Logic Tests. The purpose of segment 6 was to examine the use of the set of possibilities in a The model in simulated classroom situation involving a classroom both logical and mathematical statements. context 72

The purposes of individual segments were related to the

testing of direct translation of statements and of (counter) example selection. In some segments the effects of content domain were examined, while in others only a fixed domain was considered. The purposes of individual segments, and their relationships to the model and hypotheses are indi­ cated in Table 5. A sixth research segment was designed to examine sub­ jects' use of the set of possibilities in a simulated teach­ ing context. This segment required subjects to judge and critique arguments involving both mathematical and logical statements. It was hypothesized on the basis of the model that subjects would use counterexamples in the criticism of invalid arguments.

Subjects. Each research segment involved the testing of all students enrolled in a section or sections of a required course for preservice elementary teachers. The courses involved in the study were (I) Mathematics for Elementary Teachers I (Number Theory), (II) Mathematics for Elementary Teachers II (Geometry), and (III) Elementary Education: Arithmetic (Methods). These courses are essentially sequential although some students take course III prior to taking course II. It was expected that overall performance of subjects would improve with each course, not only because of learning and intellectual maturation, but also because of dropout of 73

less able subjects. However, as the course syllabi and texts

do not call for or offer specific consideration of any of

the model components, it was assumed that either the model

components would not be differentially effected throughout

the course sequence, or that if differential effects did

exist they would be such as to increase differences in sub­

ject performance on different components but not change the direction of those differences.

Preliminary Informal Research Prior to the formal research segments data were collected informally on several aspects of the study. Inform&j. data collection procedures were as follows:

1. Verification that subjects could generate sets of possibilities when explicitly told to do so. Two homework problems and one quiz problem assigned during the first week of course I required students to list possible combinations of two, three, and four elements. Listings provided by 32 students were coded as complete or incomplete. Thirty-one students (96.88%) provided complete listings for all three problems. 2. Verification that subjects respond to mathematics problems stated in propositional language, and use that language in their responses. Fifteen students' responses to each of 21 homework and test problems requiring proof or disproof of statements involving logical connectives were retained and coded. Response and success rates for these 74

problems were comparable with those rates for all problems

in the course (course II). All students used the words "if"

and "or" in responding to some of the problems.

3. Determination of Statements to be used in the

instruments. Nine graduate students in mathematics (5),

mathematics education (3), and computer science (1) were

asked to produce logical and mathematical statements describ­

ing the 16 sortings of the set of possibilities for two

variables in the odd/even context. The statements generated

were compared with those previously written by the exper­

imenter and generally found to agree except for order of

clauses and similar minor variations.

0 0 Segment 1. Finding Sortings for Statements. The purpose of segment 1 was to gather infor- >— •— • mation concerning subjects* sortings of sets of possibilities for mathematical and logical 0 0 statements. The effects of abstract vs. con­ crete possibilities were compared. Errors were analyzed. Hypotheses H2a and H2b were tested.

Four tests were constructed using the odd/even dichotomy.

Two tests contained one mathematical statement for each of

the 16 truth table forms; two tests contained one logical

statement for each of 14 truth table forms (0000 and 1111 were omitted due to the awkwardness of the logical state­ ments) . Two sets of possibilities were used in conjunction with the statements. One test containing mathematical state­ ments (form HA) and one containing logical statements (LA) 75 offered the following set of "abstract" possibilities:

A: M is odd, N is odd

B: M is odd, N is even

C: M is even, N is odd

D: M is even, N is even

The other two forms (MC and LC) offered these "concrete" pos­ sibilities:

A: M = 1, N = 3

B: M = 1, N » 4

C: M = 2, N ■= 3

D: M = 2, N = 4

These tests were administered to students enrolled in a section of course I shortly after a unit on prime numbers and modular systems was completed. Tests were paired and test pairs were spiraled for administration; fourteen subjects were given forms LC and MA and 15 were given forms

MC and LA.

It was hypothesized that there would be no difference in total scores attributable to logical vs. mathematical state­ ments or to abstract vs. concrete elements in the set of possibilities. To test these hypotheses means, standard deviations, and reliabilities of tests were computed; differ­ ences in means were submitted to t-test (two-tailed). In post hoc analyses errors were classified as belonging to one of seven types reflecting the sorting of too many possibil­ ities as true, too many as false, or sorting errors in both 76 directions. Error patterns were compared with expected fre­ quencies of errors of these types. Rank orders of diffi­ culty of statements having the same truth table form were compared across test forms using Friedman*s rank test for correlated samples.

0 0 Segment 2. Comparing Content Domains. The purpose of segment 2 was to compare the •— ^— • 0 difficulty of finding sortings for mathemat­ ical statements from three content domains: 0 0 odd/even, positive/negative, zero/nonzero.

In order to determine whether preservice elementary teachers would handle other dichotomies more easily than the odd/even dichotomy in tasks involving mathematical state­ ments and abstract possibilities, a 20 item test was con­ structed and administered to students enrolled in a section of course II. The test consisted of 4 five-item subscales, one for each of the dichotomies odd/even (OE), zero/nonzero

(ZG), and positive/negative (PN), and one with (course- related) geometric content (this test appears in Appendix

B). All items presented a fact and four possibilities; subjects were to select all possibilities consistent with the fact. The same truth table forms were represented on each of the subscales. The test was administered to all students enrolled in a section of the second of two mathematics courses for pre­ service elementary teachers (course II). One paper was de­ leted from the data set because it was not completed, leav­ ing 20 subjects. 77

Means, standard deviations, and reliabilities (KR20) were computed for the three five item subscales (OE, ZG,

PN). Differences between means were tested using the t- statistic. Results of the OE subscale were compared with segment 1 results on the same subscale.

0 0 Segment 3. Selecting statements for sort- xngs. The purpose of segment 3 was to •— < I >— • explore the abilities of subjects to select logical and mathematical statements descrip- 0 0 tive of the various sortings of the set of possibilities. Hypotheses H3a, H3b, H5, Hll, and HI2 were tested.

In order to examine the ability of subjects to select statements equivalent to sortings four tests were developed using items of the form shown in Figure 13. Test direc­ tions instructed the subject to examine the true set and the false set and select the statement which was true for all possibilities in the true set and false for all possibil ities in the false set. All item stems presented sortings of the possibilities:

M - 1 M = 1 M = 2 M « 2 N is odd N is even N is odd N is even 73

N is odd N is even N is even N is odd

True set False set

A. M is odd if and only if N is odd

B. M is odd or N is even

C. M is odd or N is odd

D. N is even

N i3 even N is odd N is odd N is even

True set False set

A. M + N is odd

B. M x N is even

C. N is even

D. N(M + N) is odd

Figure 13. Two sample items from segment 3. Item 1 is from Ll; item 2 is from M2. 79

Two eighteen item tests (LI and L2) contained logical statements; correct answers were simple statements (4 items), conjunctive statements (4), disjunctive statements (4), conditional statements (4), or biconditional statements (2).

Each sorting for which both the true and false sets were non­ empty was included, with the 3 true-1 false sortings ap­ pearing twice, once with disjunctive answer and once with a conditional answer. Conditional and disjunctive state­ ments were never offered as options in the same item. Two fourteen item tests (Ml and M2) contained mathematical state­ ments; in these tests each sorting except the "all true" and

"all false" sortings appeared once.

Distractors for items were selected by systematic mani­ pulation of the truth table forms in order to insure that the options presented in LI and Ml were logically equiva­ lent; likewise options in L2 and M2 were logically equiva­ lent. Ll and L2 differed only in the order in which sort­ ings were presented and in the distractors used with each item; the same was true of Ml and M2. Tests were paired as indicated in Table 6; test pairs were spiraled for administration to 56 students enrolled in a mathematics methods course (course III). The number of usable returns varied with the analysis being performed; for each analysis all returns on which the tests being considered had been completed were included in the data base. 80

Table 6. Pairing of tests for segment 3

First Test

Second Test LI Ml

L2 L1-L2 M1-L2 36 items 32 items

M2 L1-M2 M1-M2 32 items 28 items (+4 items from Ll not scored)

Means, standard deviations, and reliabilities (KR-20)

were computed for Ml, M2, Ll, L2, M1-M2 combined, and Ll-

L2 combined. Correlations were computed for the four test

pairs (Ml, L2), (Ml, M2), (Ll, L2), and (Ll, M2).

Several analyses of variance were performed. The pur­

pose of the first analysis was to establish that the order

in which mathematically and logically stated test items were

presented did not effect performance; since the tests were

of unequal length percent correct scores were used in this

analysis. Mo significant differences were found and data

were pooled for subsequent analyses. The design used in

this analysis is illustrated in Figure 14.

Test Statement form (repeated measures) form Mathematical Logical

M1-L2 - — - - 12 subjects L1-M2 - - - - 12 subjects - - - - -

Figure 14. Design for comparing order effects (segment 3) 81

The major objective of data analysis was to compare per­

formance on subscales of the logical statement tests {Ll and

L2) with performance of parallel subscales of the tests (Ml

and M2) using mathematical statements. This analysis was

accomplished through a sequence of steps. Means, standard

deviations, and reliabilities were computed for five sub­

scales of the L tests and four subscales of the M tests.

A one factor repeated measures ANOVA was performed on each

of the four tests, and the results were examined to ascertain

effects for subscales within forms. Secondly, two analyses of variance were performed, one

on the L tests and one on the M tests. These analyses

treated subjects within test form, and subscales as repeated

measures; the purpose of these ANOVAs was to determine

whether there were effects for test forms, and whether there

was a significant form by subscales interaction.

Finally L test data were pooled, and M test data were

pooled and 2 factor repeated measures analyses of variance were performed to compare L data with M data; three such analyses were done using different methods of accommodating

to the imbalance in number of items between L tests and M tests.

In all analyses significant effects were identified using the Geiser-Greenhouse Conservative F-test; differ­ ences between means were tested using Tukey's "Honestly

Significant Difference." 82

For each item in the tests all errors were coded to

reflect the difference between the truth table form of the

sorting presented and that of the endorsed statement.

Error codes were tallied for items and for subscales.

Observed differences in error patterns were tested for

significance using the Chi-square statistic.

Segment 4. Finding examples for which statements differ. The purpose of segment 4 was to investigate subjects' ability to identify elements of the set of possibil­ ities for which one statement is true and another is false. Hypotheses H8 and Hll were tested.

In order to examine subjects' ability to distinguish

between two statements by identifying instances for which they differ, four 28-item Statement Differences Tests were

constructed; all item content concerned the odd/even dichot­

omy. Each test item contained one statement declared to be

true and one statement declared to be false. Subjects were

instructed to select a possibility for which the first state­ ment was true and the second false; if a subject could not

find a possibility meeting these criteria he was instructed

to circle option "X". Sample items are provided in Figure

15. The four test forms differed in the nature of the true and false statements presented. Items in form MM presented two mathematical statements; in form ML the true statement 83

Possibilities: A - M = 1, N is odd

B - M = 1, N is even

C - M - 2, N is odd

D-M— 2, N is even

An item from LL:

True statement: M is even or N is odd

False statement: M is even if and only if N is even A B C D X

An item from LM: True statement: M is even or N is odd

False statement: M + K is even A B C D X

Tin item from ML

True statement: M + N is odd

False statement: M is even and N is odd A B C D X

An item from MM:

True statement: MxN + M + N is odd

False statement: 2N + M is odd A B C D X

Figure 15. Sample items from statements differences tests 84 was mathematical and the false statement was logical; in form LM the true statement was logical and the false state­ ment mathematical; in form LL both statements were logical.

(In order to equalize test length four LM items were included in MM and ML.)

The two statements presented in each item differed for exactly one element of the set of possibilities. The pairs of truth table forms used in construction of items for the four tests are presented in Figure 16. In this figure each arrow represents a potential item; the source of the arrow is the truth table form of the true statement and the terminus of the arrow is the truth table form of the false statement. Because the forms 0111, 1011, 1101, and 1110 are expressed by standard disjunctive and conditional statements, each arrow originating in one of these forms was represented by two items in LL and LM. In order to equalize test length four items comparing 1111 with 0111, 1011, 1101, and 1110 and four items selected from LM were included in MM and ML. The standard statements were used in each of these tests.

Standard statements were classified according to logical connective or mathematical operation(s) involved. These classifications were as follows:

Logical: Simple statements: 1100, 1010, 0101, 0011

Conjunctive statements: 0001, 0010, 0100, 1000

Disjunctive statements: 0111, 1011, 1101, 1110 1111

1110 1101 1011 0111

0110 1100 1010 1001 0011

1000 0100 00010010

0000

Figure 16. Pairs of truth table forms used in

Statement Differences Tests 86

Table 7. Classification of pairs available for test LM

Classification Classification of Mathematical Statement of Logical Statement + only + and coeff x only + and x

Simple statement 1100— 1000 1100— 0100 1010— 1000 1010— 0010 0101— 0100 0101— 0001 0011— 0010 0011— 0001

Conjunction No arrows originate at conjunctive statement forms

Disjunction 0111— 0110 0111— 0011 1011— 1001 1011-0011 1101— 1001 1101— 1100 1110— 0110 1110— 1010 0111— 0101 1011— 1010 1101— 0101 1110— 1100

Conditional Same as Same as above above

Bicond i tlonal 1001— 1000 1001— 0001 0110— 0100 0110— 0010 87

Table 8. Classification of pairs available for test ML

Classification Classification of Mathematical Statement of Logical Statement + only + and coeff. x only + and x

Simple Statement 0111— 0101 1011— 1010 0111— 0011 1011— 0011 1101— 0101 1101— 1100 1110— 1100 1110— 1010

Conjunction 1001— 0001 0011— 0001 1001— 1000 0011— 0010 0110— 0010 0101— 0001 0110— 0100 0101— 0100 1010— 1000 1010— 0010 1100— 0100 1100— 1000

Disjunction No arrows terminate in disjunctive statement forms

Conditional No arrows terminate in conditional statement forms

Biconditional 0111— 0110 1011— 1001 1101— 1001 1110— 0110 89

Conditional statements: 0111, 1011, 1101, 1110

Biconditional statements: 1001, 0110

Mathematical:

+ only: 1001, 0110

+ and a coefficient: 0011, 0101, 1010, 1100

x only: 1000, 0111

+ and x: 0001, 0010, 0100, 1011, 1101, 1110

These classifications were applied to the arrows in Figure to yield distributions of items available for inclusion in forms LM and ML; these distributions are provided in Tables

7 and 8. These tables were used in the construction of ML and LM according to the following rules:

1. If a cell (of the table) contains 4 pairs, all will

be used in the test.

2. If a cell contains more than four pairs, then four

will be randomly selected for the test.

3. In the case of "simple, x only" each pair will be

represented by two items.

4. Items involving the biconditional with x only or

+ and x will be considered together.

The four tests are attached in Appendix B.

The four test forms constructed were spiraled for admin­ istration to 121 students enrolled in a large lecture section of course II; 25 minutes of testing time were available.

Means, standard deviations, and reliabilities (KR-20) were computed for all forms and for all subscales of each of 89 the forms. Subscale scores on ML and MM were submitted to a repeated measures analysis of variance; a similar analysis was done on subscale scroes from LL and LM.

Errors were examined by computing frequency counts for specific errors and error types for individual items and for subscales. Post hoc testing of observed differences was accomplished using the chi-square statistic.

0 0 Segment 5. Bringing six tasks together. The purpose of segment 5 was to compare and con­ _ ■ > — p- fl V trast subjects' abilities to translate from one to another of the three modes: logical 0 0 statements, mathematical statements, sortings v . .. of the set of possibilities. Hypotheses H4a, _- - > <■- H4b, H6, H7a, H9 and H10 were tested.

Six 8 item tests were developed for this segment, one devoted to each of the following tasks:

L: Given a logical statement, sort the set of possi­

bilities by indicating for which possibilities

the statement is true.

M: Given a mathematical statement, sort the set of

possibilities by indicating for which possibili­

ties the statement is true.

L*M: Given a logical statement, select an equivalent

mathematical statement.

M*L: Given a mathematical statement, select an equiv­

alent logical statement.

L': Given a sorting of the set of possibilities,

select a logical statement describing the truth

set. 90

M ' : Given a sorting of the set of possibilities,

select a mathematical statement describing it.

Each of the eight item tests contained two conjunctive,

two disjunctive, two conditional, and two biconditional

items. In tests M and M* there were four items which were

equivalent to both disjunctive and conditional forms.

Each student enrolled in a large lecture section of

course I (N = 127) was given three of these tests to

complete. Six ordered combinations of tests were used:

1. L, L*M, M ’

2. M, M*L, L'

3. L, M*L, L f 4. M, I»*M, M'

5. L, M*L, L*M

6. M, L*M, M*I»

These tests were spiraled for administration during a regular lecture session. The class had previously completed a unit on modular arithmetic. One-half hour was allowed for the testing.

Means, standard deviations, and reliabilities (KR-20) were computed for each test using the full population taking each instrument. Correlation coefficients were computed for each pair of tests across the population taking the test pair (thus different subsamples were used for different test pairs). Multiple correlations were computed for test combinations 1 and 2. In addition performance on test pairs 91

was examined by investigating the extent to which high scor­

ing subjects for one test also belonged to the high scoring

group for the second test in a pair. For the purposes of this analysis scores were considered to be in the high

group provided they exceeded the mean by at least one stan­

dard deviation (rounded to the nearest integer). Comparison was accomplished by means of the tetrachoric correlation.

For each of the three-test sequences KR-20 reliabilities

were computed for the set of 24 items as additional "internal

consistency" indices for the test sequences.

Each of the six individual tests was examined for error

patterns.

Segment 6 . The Classroom Logic Tests. The purpose of segment 6 was to examine the use The model in of the set of possibilities in a simulated a classroom classroom situation involving both logical context. and mathematical statements.

Segment 6 was designed to examine subjects' use of the

set of possibilities, and of counterexamples, in particular,

in the judgment of arguments. Two forms of the Classroom Logic Test (CLT-A and CLT-B) were developed. Each item contained an argument concerning

a point in elementary mathematics. Subjects were asked to imagine themselves as teachers, and the arguments as given by

their students. Under this assumption they were to judge

the arguments as correct or incorrect and, if incorrect,

provide feedback for the student. In form A items included 92 lists of four examples (or example types — essentially the set of possibilities) from which feedback was to be selected.

Form B items required subjects to write out their own feed­ back. The same arguments were presented in both forms; 5 arguments were correct, and 13 incorrect.

The two forms were spiraled for administration to 50 students enrolled in two sections of a mathematics methods course (course III). Mathematics courses covering the con­ tent of the items were prerequisite to the methods courses. Items on both tests were first scored for the judgment of arguments as correct or incorrect. Means, standard deviations, and reliabilities (KR-20) were computed for the two forms. The difference between means was submitted to a t-test (two tailed, corrected for unequal variances).

For each test form error frequencies were computed for items containing valid arguments and for items containing invalid arguments. These data were submitted to a chi- square analysis in order to determine whether the presence of options had a differential effect on judgment of valid and invalid arguments.

In order to determine whether performance on items from different content areas was differentially effected by the presence of examples, items on each form were ranked accord­ ing to difficulty and Spearman*s rank order correlation was computed and tested for significance. Following this analysis responses to the "feedback"

part of each item were coded. For each of the 13 invalid

arguments subjects who marked the item as incorrect were

classified according to whether they chose a counterexample

as feedback or not. The average percentages of subjects

providing counterexamples to invalid arguments were computed.

Again the items were ranked according to difficulty and

Spearman's rank order correlation was computed and tested

for significance. CHAPTER V

RESULTS

The six research segments were carried out sequentially

as planned (see Table 5, p.70 ). in general subjects were

cooperative, although somewhat frustrated by the tasks given

them. Table 9 provides an overview of the test administra­ tion.

Table 9. Test administration data

Segment N Tests Items per S Time Speededness

1 30 4 30 No time limit 2 17 1 20 - No time limit 3 56 4 32-36 30 min. Math forms moder­ ately speeded 4 104 4 28 25 min.* Math forms highly speeded 5 50 2 18 (2 part) 30 min. Not speeded 6 127 6 24 30 min. Not speeded

* Subject confusion with format and other administrative

problems occurred *

Test data were submitted to standard test and item analyses and to detailed analysis of errors. On the whole the tests were difficult for the population. Estimates of test reliability were computed using Kuder-Richardson formula 20. These ranged from .009 to .932; in many cases involving two item tests the KR-20 was found to be misleading due to

94 96 small marginals (McNemar, 1969) and tetrachoric correlations gave truer pictures of subject-itern consistency.

Table 10 provides an overview of the relationship of the results obtained to the hypotheses posed (pp.57-67 ).

0 0 Segment 1. Finding Sortings for Statements. The purpose of segment 1 was to gather infor- •— >— •— 4-- • mat ion concerning subjects' sortings of sets of possibilities for mathematical and logical 0 0 statements. The effects of abstract vs. con­ crete possibilities were compared. Hypotheses H2a and H2b were tested.

The tasks of segment 1 required subjects to sort the set of possibilities by indicating which of the possibilities could be true when given statements were true. Four test forms were used; two of these (MA and LC) contained logical statements, and two (MA and MC) contained mathematical state­ ments. One logical (LA) and one mathematical form (MA) contained abstract possibilities (e.g., M is even, N is odd) while the others used concrete possibilities (e.g.,

M = 2, N - 3) . The most salient single finding of segment 1 was the magnitude of the differences between the MA (mathematical statements, abstract possibilities) mean, and the means of the other three tests. These means are provided in Table

11, together with the standard deviations and reliability estimates for each form. 97

Table 11. Basic test statistics for segment 1

Test Form X s.d. KR-20 N

MA 5.64 2.91 .704 11

MC 11.13 1.73 .348 15

LA 10.13 .99 .009 15

LC 10.20 1.32 .691 10

The difference between the means of MA and the other

tests were found to be significant (p <.001# two-tailed t-

test) suggesting that although subjects could perform com­

putations and reason with many of the logical statements

using the words "odd” and "even," they have difficulty

using these concepts in conjunction with algebraic expres­

sions. The low reliabilities of MC and LA, especially in com­

parison with the reliabilities of the other tests, are some­

what puzzling; they may be due, at least in part, to a

selection factor. Test pairs MA-LC and LA-MC were spiraled

for administration; the same number of subjects (15) was

given each pair. However, only 11 of the subjects given

the MA-LC pair completed the first test, and one of these did not attempt LC. Perhaps the least consistent subjects were "weeded out" of the MA-LC population by the difficulty of MA, while similar subjects remained in the LA-MC popu­

lation. A further explanation of the unreliability is

suggested by the types of errors made. 98

Analysis of overall error patterns shows marked differ­ ences in the types of errors made on the different tests.

For each item sorting errors were scored according to the following system: +k: classifying k elements of the false set as belonging to the true set; no other errors

-k: classifying k elements of the true set as

belonging to the false set; no other errors

x: errors in both directions The number of errors of each type was tabulated for each test

(14 items): the distributions obtained are graphed in Figure

17. When these distributions are examined in comparison with the distribution of errors expected by chance they lead to the following hypotheses: • Subjects taking MC seem to be computing values case

by case. Most errors are of the +1 and -1 types,

suggesting computational errors for individual cases.

m Subjects taking MA seem to be guessing. On the basis

of chance alone a larger proportion of their errors

would be expected to be of type x, but the symmetry of

the response pattern suggests that errors are due, at

least in part, to guessing. • Most of the errors made by subjects taking forms LA

and LC are errors in the negative direction, that is,

errors of not declaring elements of the true set.

The majority of these errors occur on conditional Figure 17, Error Patterns on Four Tests (segment 1)

30

20

10

3 -2 1X 1 2

Test MA Test MC

30

- 20

10

-3 -2 -1 1 2 3x 3 2 1 x

Test LA Test LC

1 1 0 -

28- 18- 4- -3 2 1 x 1 2 3

Distribution of Possible Errors on each of the four tests 100

statements for which many subjects declare only the

possibility "both parts true."

The percentage of subjects answering each item correctly was computed and these percentages were compared. The four test forms were ranked according to the relative difficulty of items representing each truth table form, and Friedman's rank test was performed. Although there large absolute differences in subject performance on different items, the value obtained was not significant (6.1926, .10>p>.05).

On the basis of these results it was decided that testing in later segments would use possibilities in which one vari­ able ranged over a finite set and the other over an infinite set. Thus for items involving the odd/even dichotomy the set of possibilities to be used was determined to be:

A: M = 1, N is odd

B: H * 1, N is even

C: M - 2, N is odd D: M = 2, N is even

This decision was made in the expectation that the use of guessing in response to items containing mathematical state­ ments would be diminished.

0 0 Segment 2. Comparing Content Domains. The purpose of segment 2 was to compare the •----- • 0 difficulty of finding sortings for mathemat­ ical statements from three content domains: 0 0 odd/even, positive/negative, zero/nonzero. 101

The second research segment compared the difficulty and

functioning of items sampled from three mathematics content

areas: odd and even integers (OE), positive and negative

integers (PN), and zero and nonzero (positive) rationale

(ZG). One purpose of this segment was to determine which content area or areas would be used in segments 4 and 5.

Means, standard deviations, and KR-20's were compared

for the three 5-item scales concerned with the OE, PN, and

ZG dichotomies, and for the 15 item scale composed of these

three. These data appear in Table 12.

Table 12. Basic test statistics for segment 2 scales

Scale Number of items Mean S.D. KR-20

OE 5 4.35 .7456 .6297

ZG 5 3.40 1.3917 .4012

PN 5 3.15 1.3717 .4185

Composite 15 10.70 2.2266 .7812

The differences between the OE mean and the ZG and PN means were tested with the t-statistic (correlated samples),

and the OE mean was found to be significantly higher than

each of the others (t « 2.7622 (ZG), t = 3.7130 (PN), df»19, pc.Ol, two-tailed test). The standard deviation of the OE

scores is lower, and the reliability higher than the com­ parable statistics for the other two scales, indicating that subjects are more homogeneous and consistent with respect to OE than PN or ZG. 102

Table 13. Mean percentages correct for subsets of MA

Item subset %+

Complete test (MA) (16 items) 43.16

Items used in segment 1 analysis (14 items excluding 0000 and 1111) 40.86

Items selected for use in segment 2 (5 items) 36.20

Thus the differences between the findings of segments 1

2 must be attributed to subject maturity or other differences

in the subject populations.

0 0 Segment 3. Selecting Statements for Sortings. The purpose of segment 3 was to explore the •— <--- • • abilities of subjects to select logical and mathematical statements descriptive of the 0 0 various sortings of the set of possibilities. Hypotheses H3a, H3b, H5, Hll, and H12 were tested.

The objective of segment 3 was to examine subjects'

performance on the reversals of the operations tested in

segments 1 and 2. The problems in earlier segments asked

subjects to sort possibilities in a manner consistent with given statements; in segment 3 test items presented the sort­

ing and asked that a statement descriptive of the sorting be

selected from among four options. Four tests were developed and administered using items of the form: 103

Under the assumption that the most efficient testing of the model would occur in the content area for which sorting of possibilities is least difficult for subjects, the results of this segment clearly indicated that the odd/even dichotomy should be used. The subjects* performance on this scale was better and more consistent (within subjects across items, as well as between subjects) than their performance on other scales.

The very high mean for the odd/even dichotomy (87% correct was not expected on the basis of testing in segment

1. In the earlier segment the mean score on a test of 16 items in similar format (test MA) was 43.16% correct. Since segment 1 testing was done in a course which is prerequisite for the course in which segment 2 testing was done, three explanations for this discrepancy are possible.

1. The population used in segment 2 is more able than

the population used in segment 1 (due to a weeding

effect).

2. Learning related to OE item content took place

after the administration of MA in course I .

3. There was bias in the selection of items for segment

2. The selection of items does not appear to have been biased. Table 13 displays the average %-pass for subsets of items used in test MA (segment 1). These data show that the five items selected from the segment 1 test for the seg~ ment 2 test were not easier than the test as a whole. 104

N is odd N is odd N is even N is even

True set False set A. M is odd B. M is even

C. N is odd

D. N is even

These tests are described more fully on pages Pairs of tests were administered to each of 56 students enrolled in course III (methods). Means, standard deviations, and reliabilities (KR-20) were computed for each test, and for the test pairs having both tests in the same mode (i.e., both logical or both mathematical). For each of the tests subjects were dropped from the statistical sample if they had omitted all items beyond the midpoint in the test. These data are presented in Table 14. Although LI and L2 were constructed as parallel forms containing identical item stems and correct answers, dif­ fering only in distractors used, when the L1/L2 combination was considered as a test the reliability was less than that of either of the shorter tests of which it was composed.

This effect, contrary to the logic of the Spearman-Brown prophecy formula may be due to idiosyncrasies of the small population (N ** 14} given the combination test. However, 105

Table 14. Basic test data for segment 3 tests

Test n (items) X s.d. KR-20 N Logical Statements

Ll 18 13.64 3.51 .818 28

L2 18 13.44 3.24 .754 27

L1/L2 36 28.00 4.05 .704 14

Mathematical Statements

Ml 14 10.00 3.41 .826 29

M2 14 10.27 3.54 .854 22

M1/M2 28 21.00 7.07 .937 10 it also suggests that distractors play an especially impor­ tant role in tests of this sort. Since test pairs were randomly assigned and the L1/L2 sample accounted for half of the subjects given each of the 18 item tests, and thus contributed to their reliabilities as well as to that of L1/L2, the latter explanation cannot be dismissed without further investigation. Pearson product moment correlations (Table 15) between test scores were computed over subjects completing each test pair. These coefficients reveal that the correlations for pairs (Ll, M2)f (Ml, L2), and (Ml, M2) are significant­ ly greater than zero (p .01) , but that the correlation between Ll and L2 is not. Again this may be attributed to the sample or to effects due to differences in distractors. 106

Table 15. Correlations between test scores

Second Test First Test

Ml Ll

M2 .8928* .8207*

L2 .7976* .5102 (N=13) (N=14)

* p .01

In order to determine whether the overall performance of subjects on mathematical and logical statement items was the same for test pairs M1/L2 and L1/M2, a repeated measures analysis of variance was performed. Subjects were considered within the order of the M and L tests; percent correct on each test provided the repeated measures. No significant effects were observed in the analysis which is summarized in Table 16.

Table 16. Repeated measures ANOVA for ordered tests

Source df MS F

Order (0) 1 385.333 .34

Content (C) 1 142.000 1.55

0 x C 1 48.000 .39

Subjects (within O) 21 1119.665

S x C (within 0) 21 124.223

Five subscales of the L tests (logical statements) were defined by the logical connectives used in the X07

Table 17. Basic test data for Segment 3 subscales

Test Mean s.d. KR-20

Ll Subscales

Conjunctive 3.32 .92 .475

Simple 3.29 .96 .628

Biconditional .96(1.92) .68 .064

Disjunctive 3.04 .94 .381

Conditional 3.04 1.15 .645

L2 Subscales

Conjunctive 3.11 .92 .391

Simple 3.41 1.10 .780

Biconditional 1.33(2.66) .72 .197

Disjunctive 2.67 1.22 .626

Conditional 2.93 1.12 .604

HI Subscales

Conjunctive 2.69 1.29 .670

Simple 3.03 1.27 .757

Biconditional 1.34(2.68) .66 .357

Di s junctive/conditional 2.93 1.08 .677

M2 Subscales

Conjunctive 2.68 1.22 .621

Simple 3.27 1.17 .754 Biconditional 1.50(3.00) .72 .478

Disjunctive/conditional 2.82 1.23 .638 108 -1- * • Biconditional ■ Conditional -■Disjunctive - - simple *’Conjunctive (projected)

109

answers; comparable subscales of the M tests were identified by equating truth table forms. Means, standard deviations, and reliability estimates were computed for each subscale of each test. These statistics are presented in Table 17, and the relationships between means are presented graphically in Figure 18.

Subjects were assigned scores for each scale they had taken. Since the biconditional scales were represented by only half as many items as the other scales on each test, biconditional scores were doubled for purposes of comparison with other scales.

A one-factor repeated measurements analysis of variance was performed on each of the tests Ll, L2, Ml, and M2, in order to examine differential performance on subscales with­ in a test. Summary tables for these analyses are presented in Table 18. When the values of F obtained were tested for significance using the Geiser-Greenhouse conservative F test, the effect of subscales was found to be significant

(p < .01) only for test Ll. The F value obtained for test L2 was tested using a less conservative approach (df^-4, df2=104), but was not found to be significant.

Subscales on Ll were compared using Tukey's statistic for "honestly significant difference" (critical value » .623, p < .01). Using this criterion the (projected) biconditional score was found to be significantly lower than each of the other scores for Ll. 110

The goal of analysis was to compare subscale scores on

M tests with those on L tests. In light of the findings of differences between Ll and L2 (correlational and role of biconditional), it was necessary to examine the interaction of test forms with subscales before pooling data for the comparison of M tests with L tests.

Repeated measures analyses of variance were performed on two pairs of test forms: Ll and L2f Ml and M2. For each analysis subjects were nested within forms which were crossed with subscales. Overlap between samples was not considered; for each analysis subjects using one form were randomly deleted from the sample in order to equalize (and maximize) cell size. The summary tables for these analyses are presented in Table 19.

The values of F obtained were tested for significance using both the conservative and liberal criteria; the only significant value (p<.01) observed was due to subscales of the logical statements tests. In both analyses the values of F for forms and for the forms by subscales interactions failed to reach significance at the 5% level under the liberal criterion. The inability of the variance analysis to detect differ­ ences between forms of the L and M tests respectively per­ mitted the comparison of L with M using data pooled from the two forms of each test. Ill

Table 18. ANOVAs: subscales of four tests (segment 3)

Source df MS F

Ll Subscales (A) 4 9.1321 12.60*

Subjects (S) 27 3.1606

Error (S x A) 108 .7247

L2 Subscales (A) 4 2.6741 2.32

Subjects (S) 26 2.4974

Error (S x A) 104 1.1548

Ml Subscales (A) 3 .8821 1.00

Subjects (S) 28 3.7549

Error (S x A) 84 .8821

M2 Subscales (A) 3 1.4356 1.76

Subjects (S) 21 4.3317

Error (S x A) 63 .8166

* p < .0 1 112

Table 19. Repeated Measures ANOVAs for L and M test forms

Source df MS F

Ll versus L2

Test Forms (F) 1 .0593 0.02

Subscales (A) 4 8.3094 8.95* F x A 4 2.1241 2.29

Subjects (within F) 52 2.8883

S x A (within F) 208 .9281

Ml versus M2

Test Forms (F) 1 .2045 0.05

Subscales (A) 3 1.7727 2.21

F x A 3 .1894 0.24

Subjects (within F) 42 3.9843

S x A (within F) 126 .8024

* p < .0 1 113

Data on tests Ml and M2 were pooled to form data set M;

Ll and L2 formed data set L. Four subjects were randomly

selected for elimination from set L in order to equalize the

number of subjects per data set at N = 51. Again, overlap

between samples was not considered.

An imbalance built into M and L by duplicating 4 truth

table forms (item stems) in the L tests for use with both

disjunctive and conditional items had to be accommodated

in the analyses. The relationships between subscales and

truth table forms is displayed in Figure 19. As no best decision on the handling of this imbalance was obvious,

three analyses were performed. For the first analysis an artificial subscale was added to M by assigning to each

subject a fifth score equal to his score on subscale four.

In the second analysis subscale 4 was deleted from L, and

in the third subscale 5 was deleted from L. Summaries of

these analyses appear in Table 20. For each analysis the

Geiser-Greenhouse conservative F test was used in determining

the significance of F values. Critical values of Tukey's honestly signficant difference were computed for each case.

The three analyses yielded very similar results. In each case there was no effect directly attributable to whether statements were logical or mathematical in nature. The pos­

sibility of a "mild" interaction between forms and subscales

is apparent in each analysis, and each produced a significant effect for subscales. 114

L subscales LI L2 L3 L4 L5 A 1 T I Truth 0001f 0011 1001 0111 0111 Table 0010 0101 0110 1011 1011 Forms 0100 1010 1101 1101 1000 1100 1110 1110

M subscales fi. M3 V M4

Figure 19. Comparing Subscales of L and M

Cell means were computed (Table 21) and plotted (Figure

20). Comparison of differences between means using Tukey's HSD yielded identical results for the three analyses. These comparisons revealed that for conjunctive truth table forms

M was significantly (p<.01) more difficult than L, while for biconditional forms L was significantly more difficult than M. Only one subscale (M- conjunctive) was not signifi­ cantly more difficult than L~ biconditional. The difference between M-conjunctive and M-simple was also significant

(p^ .05) . Despite the significance of the subscale effect and the potential significance of the interaction, taken together forms (L vs. M),subscales, and their interactions contribute less than 1% of the variance in the system. When the data are considered from the point of view of variance accounted for they indicate that by and large subjects were consistent in responding correctly to about 75 percent of items independent of subscale. Despite the findings of significant differences, the real import of segment three results seem 115

Table 20. Analysis of Variance: Comparing L with M

Source df MS F

Analysis 1: 5 subscales

L vs. M (F) 1 .1255 .03

Subscales (A) 4 5.3216 6.18****

F x A 4 3.1941 3.71*

Subjects (within F) 100 3.8977

S x A (within F) 400 .8617

Critical values for HSD: .484 (P < .0 1 ), .413 (p < .05)

Analysis 2: 4 subscales, disjunctive deleted

L vs. M (F) 1 .4142 .12

Subscales (A) 3 6.9175 7.75****

F x A 3 4.0809 4.57**

Subjects (within F) 100 3.3682

S x A (within F) 300 .8923

Critical values for HSD: .472 Cp < .0 1 ), .399 (p < .05)

Analysis 3: 4 subscales, conditional deleted

L vs. M (F) 1 .0612 .02

Subscales (A) 3 7.0874 7 .43****

F x A 3 4.2508 4.46**

Subjects (within F) 100 3.1215

S x A (within F) 300 .9539

Critical values for HSD: .488 (p< .0 1 ), .412 (p < .05)

**** p < .01 *** p < .025 ** p < .05 * p < .06 1X6

Table 21. Subscale Means for L and M

Subscale L M Total

Conjunctive 3.176 2.686 2.931

Simple 3.314 3.137 3.225

Biconditional 2.353 2.824 2.588

Disjunctive 2.784 2.882 2.833

Conditional 2.941 2.882 2.912

Total 2.914 2.882 2.898 117

3.0.

2,5..

2 ,0.

01

Figure 20. Subscale means for L and M 118

to lie in this consistency, which stands in contrast to the

results of other segments (e.g. 1 and 5) where the logical

connective involved in statements had a profound effect

on performance.

Several nonstandard assumptions were made in the course

of data analysis. Of special note in this regard is the

assumption that subjects' scores on a four item biconditional

scale would be twice their scores on the 2 item scale ad­

ministered. (This assumption is equivalent to the use of

percent correct scores.) Several counterarguments to this

assumption are available, and, in any case, the relationship

of the assumption to the finding of significant differences

between performance on biconditional and other items is an

open question. However, the analysis of errors, which does

not depend on this assumption, also reveals a special role

for the biconditional.

Errors on all test items were coded according to the

difference between the truth table form of the sorting pre­

sented and that of the statement endorsed. All responses were coded according to the scheme:

+1 - selection of a response for which one or more element should be in the true set (e.g. for item

1001, the response 1011 would be coded +1)

-1 - selection of a response for which one more

element should be in the false set (e.g. for item

1001, the response 1000 would be coded -1) 119

x - selection of a response which differs from the

true key in both directions (e.g. for item 1001,

the response 1010 would be coded x).

Error types were tallied for all subscales for all tests; the

resulting distribution is presented in Table 22. When this

table was examined it appeared that -1 errors were more

prevalent on L-items while +1 errors were more frequent on M-items. This observation was tested with chi-square, but found to be non-significant (Of1 - 4.947, df = 2, .10> p >.05).

The distribution of errors on logical items and mathe­ matical items were examined separately. Errors on L-items were found to be disproportionately frequent for bicondition­ al sortings, and infrequent for sortings described by simple statements ( y 1 = 14.246, df = 4, p ^ .01). For mathematical items the conjunctive and disjunctive-conditional items evoked a greater than expected proportion of all errors

(X2 = 8.866, df « 3, p<.01); in general these items involve more complicated mathematical expressions than the other items. Examination of the actual statements erroneously se­ lected in response to items revealed that 72 of the 196 errors on L-items involved inappropriate selection of bicon­ ditional statements. The distribution of biconditional statements among distractors was such that, on the basis of chance alone, only 33 errors would be expected to be of 120 Table 22. Distribution of Errors in Segment 3

Truth Table Logical Items Mathematical Items forms -1 x +1 -1 x +1

Conjunctive 0001 0010 NA 7 34 NA 15 49 0100 1000

Simple 1100 1010 6 6 17 13 4 20 0101 0011

Biconditional 1001 19 5 20 13 3 14 0110

Disjunctive 0111 1011 26 17 NA 30 25 NA 1110 1101 Conditional 0111 1011 29 10 NA Same as above 1101 1110

Total 80 45 71 56 47 83 196 errors 186 errors 121 this type. The difference between expected and observed 2 frequencies yielded x = 92.18 (df = 1, p <.001).

It is clear that the biconditional statement was the greatest single source of errors on L-tests. Not only did subjects fail to select this form when it was appropriate, but they also tended to select it in a diversity of situ­ ations for which it was inappropriate. There was no analo­ gous statement or sorting in the mathematical tests.

Segment 4. Finding Examples for which State­ ments Differ" the purpose of segment 4 was to investigate subjects* ability to identify elements of the set of possibilities for which one statement is true and another is false. Hypotheses H8 and Hll were tested.

The task of subjects in segment 4 was to determine the element of the set of possibilities for which one statement was true and a second statement was false. Statement pairs were of four types: (1,1), (1, m), (m,l), and (m,m), where

1 indicates a logical statement and m a mathematical state­ ment. Each of the four Statement Differences Tests contained

28 items of one type. Each test contained seven subscales defined by the logical connectives or mathematical operations used in the statements. These tests are described more fully on pages 82-87 and appear in Appendix B.

The most salient differences in the functioning of the four test forms was their unequal speededness. Although the underlying logical structure of statement pairs was the 122 same for all tests, the two tests in which the first state­

ment was a mathematical statement were considerably more

speeded than the others. Speededness data for the four test

forms are presented in table 23.

Table 23. Speededness Data for segment 4 tests

Test % of subjects completing Last item completed Form N 28 items 21 items 14 items by 75%

LL 29 76 90 97 28 LM 31 61 81 94 23

ML 32 34 63 94 20

MM 29 27 48 83 15

A clear implication of the speededness data is that it

took subjects longer to process the mathematical statements

than the logical statements. To help in the determination

of how the differential speededness of test forms should

effect the selection of data analysis procedures, the mean

percentage of subjects reaching an item who solved it correct­

ly (%+) was computed for each test. The mean percentages were compared with mean scores on test forms. These data,

together with standard deviations and reliabilities for

each form are reported in table 24. 123 Table 24. Basic test data for segment 4 tests

___ * Test Form X s.d. KR-20 %+

LL 14.52 5.24 .830 57.86

LM 10.71 6.12 .875 44.18

ML 11.16 5.29 .854 46.36

MM 8.76 3.82 .695 41.14

♦Average percentage of items reached which were answered correctly for Segment 4 tests.

It was observed that the rank orders of X and %+ were identical, and that the relative magnitudes of differences of these statistics for pairs of tests were comparable. It was noted that the major analyses to be performed would com­ pare subscales of LL with subscales of LM, and subscales of

ML with subscales of MM. Moreover, the distribution of items from comparable subscales was identical for each pair of tests to be compared; therefore effects of speededness on a subscale would be expected to be roughly the same for both tests in a pair. On the basis of these considerations (and the desire to maintain N as large as possible) it was decided to use the full set of data in analyses and comparisons of the tests.

In order to determine whether there was an interaction between the form (mathematical vs. logical) of the first statement and that of the second statement an analysis of variance was performed. Two levels of the true statement 124 were crossed with two levels of the false statement in the

factorial design shown in Figure 20. Table 25 provides a

summary of this analysis.

False Mathe­ Statement matical Logical

Mathe­ Subjects Subjects matical given MM given LM

Subjects Logical Subjects given ML given LL

Figure 21. Design of ANOVA: mathematical vs. logical statements (segment 4).

Table 25. Summary of ANOVA for statement types

Source df MS F

True Statement (T) 1 178.76 7.28** False Statement (F) 1 311.21 12.67** T X F 1 0.55 0.02 Error 112 24.57

** p < .01

Results of the analysis of variance indicated that there was no effect due to interaction of the forms of the first (true) and second (false) statements. If the first 125

Table 26. Basic test data for segment 4 subscales

Scale Description

Subscales of LL A. Simple statement vs. conjunction 2.83 1.23 .621 B. Simple vs. Conjunction (2) 2.69 1.32 .664 C. Disjunctive vs. Biconditional 1.97 1.33 .604 D. Conditional vs. Biconditional .38 .76 .548 E. Disjunctive vs. Simple 2.72 1.36 .713 F. Conditional vs. Simple 1.66 1.18 .406 G. Biconditional vs. Disjunctive 2.28 1.36 .754 Subscales of LM A. Simple vs. MxN is odd 1.87 1.13 .364 B. Simple vs. complex algebraic 1.74 1.22 .502 C. Disjunctive vs. M+N odd/even 1.94 1.29 .584 D. Conditional vs. M+N odd/even .68 1.17 .788 E. Disjunctive vs. 2M+N (2N+M) 1.52 1.43 .728 F. Conditional vs. 2M+N (2N+M) 1.26 .84 .016 G. Biconditional vs. complex alg. 1.71 1.25 .582 Subscales of ML A. (+,x) expression vs. simple 1.59 .86 .229 B. M+N vs. conjunction 2.63 1.24 .668 C. 2M+N(2N+M) v s . conjunction 1.97 1.53 .769 D. x or (+,x) vs. biconditional 1.16 1.23 .784 E. MxN even vs. simple 2.28 1.18 .622 F. 2 (M+N) vs. conditional/disjunction .13 .93 .192 G. Items from LM .41 .65 .350 Subscales of MM A. ( + / X ) expression vs. 2N+M 1.07 .83 .183 B. M+N vs. (+,x) expression 1.59 1.03 .323 C. 2M+N vs. (+,x) expression 1.55 1.13 .360 D. x, (+,x) vs. M+N 1.55 .97 .127 E. MxN even vs. 2N+M 1.55 .93 .212 F. 2 (M+N) even vs. complex algebraic 1.17 .91 .269 G. items from LM .28 .69 .634 126

statements in pairs were mathematical the pairs were more

difficult than if the first statements were logical. Simi­

larly, tests in which the second statement was mathematical were more difficult that similar tests in which the second

statement was logical.

Two repeated measures analyses of variance were executed. The design for these analyses is shown in figure 21.

Test Subscales (Repeated Measures) form

1 2 3 4 5 6 (7)

LL Subjects ujifhtn test form (MM)

LM Svbji c>fs within test form (ML)

Figure 22. Design for two ANOVAs on Segment 4 subscales 127

The first was designed to compare performance of sub­ jects on the seven subscales of LM. In this analysis the subscales were considered as repeated measurements of the ability to distinguish a logical statement from a second statement; this ability was compared for two conditions defined by whether the second statement was logical or mathematical. The results of the analysis of variance are presented in table 27. Obtained F-values were tested for significance using the Geiser-Greenhouse Conservative F- test, and the effects of test form, subscales, and their interaction were all found to be significant.

Table 27. Summary O f ANOVA: LM VS • LL (segment 4}

Source df MS F

Test form 1 20.3966 4.55*

Subscales 6 23.9523 23.67***

Subjects (within form) 56 4.4848

Form x Subscales 6 4.0804 4.03**

Subjects x Subscales 336 1.0120 (within forms)

*p < .05 **p < .01 ***p < .001 128

Means (table 26) were plotted for the cells involved in the interaction (figure 22). The critical value of

Tukey's HSD was computed to be .898 (df2 - 56, steps =13, p-^.05) and used to test differences between means for significance. The principal findings indicated that LL was easier than LM overall, and that on both tests scale D was more difficult than the other scales from either form.

Scale F was found to be more difficult than scales A, B, and E in test LL; no comparable difference was evident in test LM.

4

A X \ \ / \

V

A B c O BT F

Figure 23. Means for subscales of LL and LM. 129

A similar analysis of variance was performed on ML and

MM. Scale G (taken from LM) was omitted from this analysis.

A summary of the ANOVA results is presented in table 28.

Table 28. Summary of ANOVA: ML vs. MM (segment 4}

•

Source df MS F

Test forms 1 25.3908 8.54*

Subjects (within forms) 56 2.9733

Subscales 5 1 0 . 1 0 1 1 13.74*

Forms x Subscales 5 4.9900 6.80*

Subjects x Subscales 280 ,7350 (within forms)

*p < .01

Again, both variables (forms and subscales) were signi­ ficant, together with their interaction. The critical value of Tukey's HSD was found to be 1.09 (p<,05, 11 steps, df =56).

This figure was used to test the significance of differences between means (table 26) as plotted in figure 23. As was previously determined, ML was easier overall than MM. Within

ML scale B was found to be significantly easier than scales

D and F. 130

Figure 24. Means for subscales of ML and MM,

Analysis of Errors. When errors on individual subscales were examined several interesting patterns emerged. Of particular interest in forms LL and LM were scales C, D, and F. Scale D, by far the most difficult in both forms asked subjects to find a possibility for which a conditional state­ ment was true and a biconditional statement (LL) or statement of the form "M + N is odd (even)” (LM) was false. In both forms the most popular option for each item was the choice for which each of the simple statements joined by the conditional was true.

Example:

True Statement: If M is odd then N is odd.

False Statement (LL) M is even if and only if N is even.

False Statement (LM): M + N is even 131

Attractive option: M = 1, N is odd (LL-19, IM-14Ss) Correct: M = 2, N is odd (LL-3, LM-6 Ss) "X" (no difference) (LL-2, IM-7 Ss)

The response pattern exhibited in scale D, as well as in scales C and F (LL), F (LM), and D and F (ML) suggested that many subjects were interpreting conditional and bicon­ ditional connectives as if they were conjunctive. In order to examine thi3 possibility tests LL, LM, and ML were rekeyed using conjunctive interpretations of all logical statements.

For each test the conjunctive key was compared with the true key for each subscale in order to determine the expected

(true key) score profile of a subject who interpreted all connectives as conjunctions (and processed the erroneous information perfectly). This profile was then correlated with the profile of mean subscale scores for the group taking the test. The correlations obtained were ,8570

(LL), .8596 (LM), and .8213 (ML); these add credence to the hypothesis that subjects were interpreting all connectives as conjunctions.

In order to examine this hypothesis more closely, forms

LL and LM were rescored for all subjects using the conjunctive

Interpretation key and submitted to a standard test and item analysis.1 Selected data from this analysis are presented

^Form ML was omitted from this analysis because, under the conjunctive interpretation of connectives, 8 items had multiple "correct" answers. 132 in comparison with data from the original analysis in

Table 29.

This analysis revealed that mean scores and variances under the conjunctive keys were almost identical with those obtained with the true key. However, there was some shift in subscale scores. In particular, scale D (conditional vs. biconditional) scores on LL showed a marked increase under the new system; this increase did not occur on the parallel subscale of test LM. Examining items from each scale D, it is clear that in form LL a conjunctive inter­ pretation of the false statement (e.g. M is even if and only if N is odd) leads directly to the response; direction is not provided by the false statement (e.g. M + N is odd) of form LM. Comparison of error patterns on scales D of forms

ML and MM is in agreement with these.

Table 29. Comparison of true key and conjunctive key scores

Statistic Correct Key Conjunctive Key

Test LL Mean 14.52 14.76 S.D, 5.24 6.04 KR-20 .830 .874 Scale D Mean (s.d.) .38 (.76) 2.14 (1.31) Scale E Mean (s.d.) 2.72 (1.36) ,79 ( .61)

Test LM Mean 10.71 9.77 S.D. 6.12 6.05 KR-20 .875 .883 Scale D Mean (s.d.) .68 (1.17) .97 (1.56) Scale E Mean (s.d.) 2.28 (1.18) .35 ( .8 6 ) 133

Scores on subscale E of both forms LL and LM were

greatly decreased by conjunctive rescoring; it appears that

on this type of test, at least, subjects do not make con­

junctive interpretations of the connective "or".

Aside from scale D mentioned above, no discernible

patterns of errors or error differences were evident for

tests ML and MM.

® 6 Segment 5 . Bringing six tasks together. The purpose of segment 5 was to compare and contrast subjects* abilities to translate from one to another of the three modes: logical statements, mathematical statements, sortings of the set of possibilities. Hypotheses H4a, H4b, H6 , H7a, H7b, H9 and H10 were tested.

The purpose of segment 5 was to compare subject perfor­ mance on six tasks. These tasks were sorting on the basis of

logical and mathematical statements (tests L and M) , selecting mathematical translations of logical statements (test L*M), selecting logical translations of mathematical statements

(test M*L), and selecting logical and mathematical statements descriptive of sortings of the set of possibilities (tests L* and M'). An eight-item test was constructed for each of these tasks. One hundred twenty seven students enrolled in Course X were each given three of the tests to complete. The tests, and the grouping of tests for administration are discussed on pages 89-91 . Test copies appear in Appendix B. 134

Means, standard deviations, and reliability estimates

for the eight-item tests were computed. These are reported

in table 30. Differences between means were submitted

to t-test (table 31), and a schematic description of the

difficulty relationships among tests was developed (figure

24). Before the relationships between the tests was

examined further, each test was analyzed on its own merits.

Test L (sorting possibilities given logical statements) The item pairs representing each logical connective within

Test L were considered as separate tests. Tetrachoric correlations were computed as reliability estimates for these two item tests, and means for each pair were computed.

These data appear in table 32,

Table 30 Basic test data for segment 5 tests

Test Mean S.D. KR-20

L 2.569 1.311 .638

M 4.548 2.274 .764

L*M 2.512 1.579 ,311

M*L 3.694 1.725 .485

h* 3.512 1.856 .505

M* 4.024 2.343 .609 135

Table 31: Differences (column-row) between means for segment 5 tests

Test M L*M L'

L d=l.979 d— 0.057 d=l.121 d=.941 d=l.451 (N=65) t=5.995 t=4.327 t=3.054 t=4.062 p < . 0 0 1 p <. 001 p- < .01 p < . 0 0 1

M d=2.036 d=-.858 d=-l.038 d*-.528 (N=62) t=6 .343 tas2.580 t= 2.449 tal.137 p <.001 p < .02 p <. 02

L*M d=l.181 d= .998 d=l.508 (N=84) t=4.609 t-3.144 t=*4.248 p <.0 0 1 p <. 01 p < . 0 0 1

M*L d=-0.18 d=0.33 (N=85)

L' daO.51 (N=43

M' (N=42)

(Note on p values: The probability that one or more of the six differences labeled "p < *0 0 1 " was obtained by chance is less than .007; the probability that one or more of the 10 significant differences indicated was obtained by chance is less than .067). 136

M #L A

kr

L

Figure 25: Significant Differences Between Means (Segment 5)

p < . 0 0 1

-I “ I P p <. 02 137

Table 32: Means and reliability estimates for Test L (Segment 5)

Connective Mean Tetrachoric r

Conjunction 1.935 .9542

Disjuntion .600 .8004

Conditional .046 .6446

Biconditional .077 .9322

When errors on individual items were examined together with these means it appeared that subjects were treating all connectives as if they were conjunctive. In order to examine this possibility further, all items were rescored according to the scheme:

1 - the response would be correct if the connective

had been replaced with "and"

0 - the response would be incorrect if the

connective had been replaced with "and"

Again means were computed, and correlation coefficients used as estimates of the reliability with which the revised scoring system assigns scores. These are reported in table 33. 138

Table 33 Means and reliabilities when scored for conjunctive interpretation

Connective Mean Tetrachoric r

Conjunction 1.935 ,9542

Disjunction .784 .9054

Conditional 1.708 .9370

Biconditional 1.800 .9659

These data indicate that most subjects treated the

conditional and biconditional connectives as if they were

conjunctions, and that they did so with great consistency.

When interpreting the disjunctive "or" 17 of the 65 sub-

jucts given test L treated both items in agreement with the

conjunctive interpretation of "or;" 15 treated the items in

a manner consistent with the logical "inclusive or;" the

remaining subjects were inconsistent in their usage, or did not use either of these interpretations. No subject used

the "exclusive or" in both cases although 4 subjects did use it in one case. Test M (sorting possibilities given mathematical state­ ments) Several subscales of test M were identified using the connective involved in the equivalent logical statement, or the presence of common mathematical expressions. The four statements involving only one mathematical operation 139 were found to be significantly easier for subjects to sort

than the statements involving more than one operation(t -

4.846, df = 61, p<.001). Data for other item pairs and

subscales are provided in table 34; taken together, these data indicate that subjects were fairly consistent in

their ability to sort statements and their reversals;

the one exception to this consistency lies in the consider­ ation of the parity of M x N. The ability of subjects to sort statements having conjunctive truth table forms res­ ted heavily on the complexity of the statement.

Errors were coded -, or X” depending upon whether they reflected erroneous identification of at least one possibility as true, failure to identify at least one true possibility correctly, or errors in both directions. The distribution of errors was tallied and compared with the 2 expected distribution by chance; a significant (X -59.466, df - 2 , pc.0 0 1 ) tendency to commit errors of type was found. 140

Table 34: Means an intercorrelation for selected subscales of test M (Segment 5)

Subscale n Mean r

Conjunctive (1 0 0 0 ,0 0 1 0 ) 2 1.393 .1331

Disjunctive Conditional (1 1 1 0 , 1 1 0 1 , 1 0 1 1 , 0 1 1 1 ) 4 1.803 .7161

M x N 2 1.492 .3258

M + N 2 1.475 .8034 Biconditional

N (M + N) 2 .918 .8624

Test L*M (Selecting mathematical translations of logical statements) L*M was (with test L) one of the most diffi­ cult tests in the series, and the least reliable. When item pairs defined by logical connective were examined, zero order correlations between success on the elements of each pair were observed. Examination of the four fold tables for pairs of items revealed that in each case one item was considerably more difficult than the other.

When errors on individual items were examined two res­ ponse trends were observed, both related to the possibility of subjects making conjunctive interpretations of the compound logical statements presented. For two items 141

(one conditional, one biconditional), the mathematical

equivalent of the conjunctive interpretation was available

as an option; when available this option was a powerful

distractor. Options were classified as true or false under

a conjunctive interpretation of the stem, and the average

drawing power of "true" and "false" options computed. The

"true" options were considerably more attractive to subjects

than the "false" options, as shown in table 35.

Table 35: Drawing power of options in L*M (Segment 5)

Truth Table Correct Conj. Average Average correct Trans. "true" "false" Form

1 0 0 0 36 NA 30.5 10.5 0 1 0 0 9 NA 23.33 9.0 1 1 1 0 (con.) 35 NA 27.00 13.5 1 0 1 1 (con.) 5 30 26.33 3.0 110 1 (dis) 34 NA 24.67 8.0 0 1 1 1 (dis) 36 NA 20.25 NA 1 0 0 1 16 38 20.75 NA 0 1 1 0 37 NA 21.33 19.0

Test M*L (Selecting logical transaltions of mathematical statements) Test M*L was of moderate difficulty. Except for the pair of disjunctive items whose correlation was negative, the correlations for success on a pair of items using the same connective in the correct answer was around

.3, like the overall reliability, somewhat higher than esti­ mates for L*M, 142

Errors on items were classified according to the con­

nective used in the option chosen. Table 36 shows the

distribution of responses for the 8 items. Although the

test was not balanced with respect to availability of op­

tions, the number of selections of simple statements is

somewhat surprising, in the five items for which the

number of subjects selecting a simple statement exceeded

7, this choice reflected translation of a statement of the form

Factor1 x Factor2 is even to one of the form

Factor^ is even.

Table 36 Distribution of errors on test M*L (segment 5)

Correct Error Connective Connective conj. disj. cond. bicond. simple

Conjunction 17------7 17 ---- 10 10 Disjunction -- 17 14 31 18 11 Conditional 9 12 19 30 18 20 Biconditional 23 20 -- -— 3 9 10 7 Totals 49 82 39 54 108 143

Tests L 1 and M 1 (Selecting statements describing sortings)

These tests consisted of items selected from tests LI and Ml

used in segment 3. The items functioned in much the same

way they did in the longer tests; no particular error patterns

other than those observed in segment 3 were noted.

The Six Tests Together; Application to the Model

The underlying purpose of segment six was to examine the relationships between and among six elements of the model

proposed. These elements are isolated in figure 25 below.

Figure 26. Operations tested in segment 5.

Although the tests in segment 6 were neither universal enough in their coverage of problems, nor (consequently) reliable enough to provide a definitive test of the model, the results of this segment do support the model. This support is evidenced in several approaches to the data.

Means, standard deviations, and reliabilities (KR-20) for each of the six tests are provided in table 30 (p. 134) differences between means were tested for significance (table

31). Comparison of test difficulties revealed two clusters 144

of tests as shown in figure 26, Lines in this diagram indicate

that the means of the tests at the end points are not signifi­

cantly different at the 1% level. Thus, under the difficulty

analysis L and L*M (5 ^ and 3 *) belong together, with the

remaining four tests forming a separate clique.

Pearson product-moment correlations were computed for

each pair of tests given to a common population. These co­

efficients appear in table 37. Of the eleven correlations

computed only 5 were found to be significantly (p<.05) differ­

ent from 0 when tested with Fisher's z-test. Each of the

significant correlations associated one of the tests L' and M* with another test.

Table 37. Correlations among scale scores.

L M L*M M*L L* M'

L mmwm — .1812 -.0721 .2006 .4891* (n = 43) (n = 43) (n = 2 2 )

* P < ,05 ** pc.Ol

Under the assumption that true scores are not correlated with error, the correlations between tests can be corrected for attenuation. The adjusted coefficients obtained under this correction are presented in table 38. 145

Figure 27. Two clusters of tasks by difficulty (p<.0 1 ) 146

Table 38. Segment 5 correlations corrected for attenuation.

L*M M*L L' M'

L .4068* -.1296 .3534 .7847*

M .4060* .4682* .8249* .7725*

L*M — .6265* — .7603*

M*L — — — .8776* — —

*p< .01

When the corrected correlation coefficients are considered all are significant except correlations of U S *“) with L* (6 ) and M*I. ($); the lack of correlation in these cases strongly suggests that the ability to use (select) logical statements in describing situations is quite different from the ability to determine the sortings or truth tables of logical statements.

Using the original (uncorrected) correlation coefficients —1 it is possible to compare scores on tests of 4 and 4 . with the best (linear) estimates of these scores from scores on other tests. The multiple correlations for prediction of (M*L) from performance on tests M(0) and L* (6 ), and for prediction of

4~* (L*M) from performance on L (5-*) and M f ($”*) were compu­ ted and tested for significance with an F-test. The results are presented in table 39. 147

Table 39. Multiple correlations among tests (segment 5)

Comparison Multiple r F

4 *06 .733 11.06 (p<.0 1 )

4 -1 -0~1 6 " 1 .332 1.17

These comparisons indicate that overall performance on operation 4 can be predicted from performance on 6 and 6 .

Performance of is not predicted by performance on 6 1 and 0**^; however, it should be noted that tests L and L*M were very difficult for the population, with scores quite close to the chance level. Thus the failure to find a signi­ ficant multiple r does not preclude the possibility of a relationship between the inverse operations in a population more able with respect to their performance.

A somewhat different picture of the relationships between tests emerged when subjects were classified into two groups and tetrachoric correlations computed. For each test a sub­ ject was assigned to class 1 provided his score was at least one standard deviation above the mean (X + sd was rounded to the nearest integer), and to class 0 otherwise. Tetrachoric correlations were computed for each pair of tests given to a common population, and tested by comparison with tabled values. These correlations appear in table 40. 148

Table 40. Tetrachoric correlations (High vs. low group)

L M L*M M*L L* *1 S l l .3043* -.2653 .2770 .1834

not com­ .6216** .4728* .0000 putable

L*M .4386** .4899**

M*L .3214*

L*

* p < .05 ** p < . 0 1

Under this analysis we find that M*L is correlated with each

of M and L 1, the elements hypothesized to be its constituent

parts. Similarly L*M is correlated with I* and M'. L*M and

M*L are correlated with each other. Each of the preceding analyses depend upon the use

of scores which are less reliable than desirable. A third

approach to the comparison of these tests was designed to

avoid dependence on these scores. Each set of three tests which had been administered to a common population was con­

sidered as a unit. Kuder-Richardson formula twenty relia­ bilities were computed for each of these units, as well as

for subscales defined as sets of items related to the same

logical connective. Reliabilities of these units, being measures of internal consistency, were then interpreted as 149

indices of the extent to which the unit measured a single phenomenon. Table 41 presents these indices.

Table 41. Reliabilities for three--test units

Unit Total Con j. Disj. Cond. Bicond

L/L*M/M' .470 -.150 .453 .171 .174

M/M*L/L' .767 .581 .548 .476 .310

L/M*L/L' .309 .208 .734 .208 .381

M/L*M/M' .522 .555 .700 .746 .450

L/M*L/L*M .588 .639 .379 .225 .317

.693 .542 .515 .522 .170

Examination of these data reveals that M/M*L/L’ is the most consistent of the 6 instruments overall; this finding supports the notion that there is an intrinsic relationship among the three constituent tests. Several other interesting relationships are also revealed. None of the three test com­ binations provides a reliable measure of subjects performance on biconditional items. For conditional items the M/L*M/M' combination is clearly the most consistent; this unit is the most heavily "mathematical" and least "logical" of the six.

For the disjunctive items the two units which involve sorting and describing sortings in the same mode are the most reliable.

Taken together these analyses provide considerable support for the model, at least in the direction of translation 150

of mathematical statements into a logical mode. In each

analysis the group of tests M, M*L, L 1 emerges as a coherent

unit, augmented in some cases by M 1.

Segment 6 . The Classroom Logic Tests. The purpose of segment 6 was to examine The model in the use of the set of possibilities in a a classroom simulated classroom situation involving context. both logical and mathematical statements,

The Classroom Logic Tests extended the investigation from abstract exercises to simulated classroom problems. The research questions of particular interest in segment 5 were whether the presence of a list of possibilities would effect subjects' abilities to judge arguments as valid or invalid, and whether subjects would use counterexamples to refute invalid arguments. Items on both tests (CLT-A and CLT-B) were first scored for the judgement of arguments as correct or incorrect. Means, standard deviations, and reliabilities (KR-20) for the two forms are reported in table 42 below.

Table 42. Basic test data for segment 6 tests.

Test form Mean S.D. KR20

CLT-A 12.80 2.19 .3221

CLT-B 11.28 2.91 .6583 151

The difference between means was submitted to a t-test

(corrected for unequal variances, two-tailed), and performance

on CLT-A was found to be significantly better (p<.05) than

performance on CLT-B. Any potential importance of this finding

is obliterated, however, by the analysis of errors.

For each of the test forms error frequencies were com­ puted for items containing valid arguments and for items containing invalid arguments. These data (presented in table

43) were submitted to a Chi-square analysis and a significant 2 difference (x ** 10.75, p<.01) in error patterns was detected.

Subjects given form A made a greater proportion of errors by rejecting valid arguments, while form B subjects tended to accept invalid arguments.

Table 43, Errors on correct and incorrect arguments.

Argument is CLT-A CLT-B

correct 28 errors 14 errors

incorrect 101 errors 154 errors

In order to determine whether the difference in test means was due entirely to the imbalances in argument validity and error tendencies, a theoretical test including 13 (rather than

5) valid arguments and 13 invalid arguments was considered.

It was assumed that errors would occur on the 8 additional items at the same rate that they occurred on the 5 correct 152

arguments tested. Under this assumption the ’’da t a ’’ in table

44 were calculated. For these data there is no significant

difference between means.

Table 44. Data for hypothetical tests.

Hypothetical Hypothetical Statistic CLT-A CLT-B

Errors (correct arguments) 73 36.4

Mean scores 19.04 18.36

In order to determine whether performance on (content)

items was differentially effected by the presence of examples,

items on each form were rank ordered according to difficulty,

and Spearman's rank order correlation was computed and tested

for significance. A significant correlation was found (t » 2.7462, p<.02, two tailed), suggesting that items were not differentially effected.

This analysis indicates that the presence of a list of possible examples had two effects: it enhanced the ability of subjects to identify invalid arguments, but it also debi-

lited their ability to identify valid arguments.

Following the analysis of argument judgements the res­ ponses to the "feedback" parts of the 13 items using invalid arguments were considered. Answers to CLT-A and CLT-B were classified using the following scheme: 153

A B c non-counterexamples presented in CLT-A D

* counterexample

E example provided or discussed; not clearly one of the above, but mathematically sound

F faulty example (e.g. arithmetic error)

O other response

X no response (although argument labelled incorrect)

N argument labelled correct

Item omitted completely For each form the total number of responses of each type was tallied. These data are presented in Table 45.

Table 45. Classification of "feedback" responses.

Response type CLT-ACLT-B

Number of "incorrect" judgements 222 171

Feedback Counterexample(*) 179 111

A, B, C, or D (not counterexample) 38 7

E --- 18

F --- 15 0 --- 12

X 5 8

N 96 135

Item omitted entirely 7 19 154

On both forms the majority of subjects who judged in­ valid arguments to be incorrect provided counterexamples as feedback to their hypothetical students. Moreover, even in cases where counterexamples were not used subjects did use examples as feedback for form B items.

It is clear from these data that subjects who were asked to provide feedback {form B) showed a marked tendency to use examples: 88.30% of all feedback responses used examples of one sort or another. Of these, a large majority were counterexamples (73.51%).

The average percentages of subjects providing counter­ examples to invalid arguments were 78.55 for CLT-A and 56.64 for CLT-B. Again items were rank ordered according to diffi­ culty and the Spearman rank order correlation was computed.

This correlation was not significant (t = ,9475) suggesting that item content was important in determining whether counter­ examples would be provided or selected if provided. Examination of individual item responses tended to bear out this observa­ tion; several items in form A seemed to lure subjects to responses which were not produced at all by subjects using form B. When considered in light of the informal finding that subjects can generate lists of combinations (see p.73 ), these data seem to indicate that the generation of such lists is not an effective part of the process used by subjects in judging arguments; however, elements of the listing are provided by these subjects as counterexamples to invalid arguments. CHAPTER VI

CONCLUSIONS

This study explored the question of how preservice ele­ mentary teachers interpret and use statements involving four logical connectives when those statements refer to mathe­ matical concepts. Toward this end an hypothetical model for the process of translation between "logical" and "mathemati­ cal" statements was developed, and a sequence of tests was designed and administered to examine various aspects of the model.

An Implicit Hypothesis and Related Limitations

The overriding implicit hypothesis of the research was that the ability of the subject population to use or inter­ pret statements in propositional forms correctly is depen­ dent upon the context in which the statement forms occur.

This context has at least three aspects: (1) the concepts or phenomena with which the statements deal, (2) the, avail­ ability of additional information concerning the concepts or phenomena, and (3) the nature of the interpretive task.

This study was concerned only with elementary mathemat­ ical contexts; these contexts were limited in their general­ ity by the imposition of criteria for selection of items.

155 156

and by varying the amount of information available jointly with the nature of the task in the definition of tests to be used. An early decision to avoid explicit use of negation was prompted by reports of other researchers that use of the word ''not" tends to increase the difficulty of problems.

This decision led to the use of dichotomous concepts which split the domain into two parts each having positive identi­ ties (e.g., odd/even/ positive/negative). The odd/even dichotomy was used exclusively in most research segments.

One effect of the avoidance of negation was to limit the statements (although not the underlying relationships) which could be used in the study. For example, the state­ ment "Neither M nor N is even" was not included in the study although both the equivalent statement "M is odd and N is odd" and the implicit negation "M is even or N is even" occur throughout the research segments.

Other effects of the decision to avoid explicit nega­ tion are unknown. It remains for further research to deter­ mine whether the phrases "even" and "not odd" are handled equivalently by subjects. The tasks used were essentially tasks of translation (or selection of translations) from one of three modes to another. The modes were logical statements, mathematical statements, and sortings of the set of possibilities. The four basic tasks in which logical statements occured were: 157

1. Given a logical statement tell for which possibil­

ities it is true.

2. Given a logical statement select a mathematical

statement which defines the same relationship be­

tween variables. 3. Given a mathematical statement select a logical

statement which defines the same relationship

between variables.

4. Given a sorting of the set of possibilities into

true and false sets select a logical statement

describing the true set. The amount of information readily available to the subject

would seem to increase from task 1 through task 4. In

task 1 the logical statement is the sole source of informa­

tion; in 2 the nature of the limited set of options provides

some additional information. In task 3 the mathematical statement provides information concerning the relationship

between variables, but this information must be decoded before a logical statement describing the relationship

between variables can be selected. In task 4 the informa­

tion presented is a sorting of the set of possibilities

in conformity with the logical relationship. One hypothesis

of the study is that the decoding of information in mathe­ matical statements (task 3) results in a sorting as pre­

sented in task 4. This hypothesis was used in building the model for translation between logical and mathematical state­ ments . 158

A Model for the Use of Logic in Mathematical Contexts

A model for the use of logic in mathematical contexts

was hypothesized in Chapter III, and aspects of the model

were subjected to test throughout the course of the study.

This model was derived by seeking a best a priori fit for

Piagetian theory concerning logical thought with the find­

ings of experimental psychology and the observed use of

logical statement forms in discourse concerning elementary

mathematics. The elements of the model are operations whereby infor­ mation is transformed from one of three representation

states to another. These states are: Cl) logical state­ ment, (2) mathematical statement, and (3) partition of the

"replacement set" into the solution set and its complement. The results of the tests designed to elicit operations of the model supported the model in the direction of trans­ lating from mathematical to logical statements. No evidence was obtained in the reverse direction due to the extreme difficulty of two tasks (1 and 2 above) for the subject pop­ ulation.

The Role of the "Set of Possibilities" A key component of the model is the "set of possibil­ ities," which is comparable to the "replacement set" for an equation, in a sense the model asserts that the set of possibilities is used as a tool for mediation between equiv­ alent logical and mathematical statements. The tests 159

designed to investigate the operation of the model required

subjects to examine and use the set of possibilities in

responding to items. The question arises concerning the

validity of results derived in situations where this usage

is forced. As a partial test of the assumption that the subject

population uses examples and counterexamples (i.e./ accepted

and rejected possibilities) naturally in dealing with mathe­ matical problems subjects were given one of two forms of

the Classroom Logic Test. Form A provided a set of possi­

bilities for use in judging arguments and required the

selection of a counterexample from this set; form B

requested free response answers. Subjects using form B were found to use examples (and particularly counterexamples)

in their responses almost as often and effectively as subj-

jects using form A. The salient difference between perfor­ mance on the two forms was that form A subjects tended to err by rejecting valid arguments, while those using form B

tended to err by accepting invalid arguments. Thus the presence of the set of possibilities seemed to make subjects more critical of arguments presented to them.

Conjunctive Interpretations of Statements

This research has produced massive evidence that pre­ service elementary teachers tend to treat all compound state­ ments as conjunctive — or, simply, as lists of "true facts."

The preliminary study demonstrated this phenomenon in a situation in which subjects were asked to determine whether

statements about sets were true or false. A later research segment (segment 5, task L) required subjects to select

instances for which given statements were true. Despite the considerable difference between these tasks, the results were almost indistinguishable; in both cases more than 90% of subjects responded to all conditional and biconditional statements as if they were conjunctions; a little less than half of both populations treated disjunctions, also, as if they were conjunctive. Moreover, this phenomenon does not seem to have been limited to the two types of tasks in which it was directly tested. In virtually all forms of the six other types of problems used in this research prevalent errors could be attributed directly to subjects' treatment of nonconjunctive statements as if they were conjunctive.

In several of the research segments parallel test forms were used to compare subjects' interpretations and uses of equivalent logical and mathematical statements. An un­ expected, but omnipresent, result of this comparison was the finding that subjects require much more time to process the mathematical statements than to process the logical statements intended to convey the same information.

Unfortunately, this time differential does not always seem to result in more accurate processing; however, the evidence on this point is not uniform. The effects of test speeded- ness may have depressed most scores on tests using 161

mathematical statements. In the single unspeeded test

situation (segment 5) in which use of mathematical and

logical statements was compared scores on tests presenting

mathematical statements were significantly (p < .001) higher

than scores on tests presenting equivalent logical state­ ments . Unlike the logical statements, mathematical statements were not subject to consistent misinterpretation. Errors

in interpretation of mathematical statements did tend to occur in direct proportion to the complexity (number of operations) of the mathematical expressions involved, but the specific errors made were generally quite diffuse.

A Note on Reliability and Related Issues

Throughout the research a trade-off among theee impor­ tant variables was necessary: test speededness, test relia­ bility, and subject motivation (more precisely, subject bore­ dom and frustration). To seme extent these factors were related to the limited availability of subject time, but this managerial issue was not the sole source of concern.

Several subjects simply terminated their participation in the experiment after completing a few items; others expressed a great deal of frustration with the tasks, especially when they found them to be repetitive. This repetitiveness could result from at least two sources: inclusion of very similar items to enhance instrument reliability, or subjects'

(erroneous) interpretation of diverse statements as 162

identical. The effects on test scores, reliability, and

speededness of subjects' feelings that they are being asked

the same questions repeatedly are unknown.

Moreover, in the absence of duplicate items, it is not

clear how reliabilities of these tests should be estimated.

The techniques used in this research were two: the Kuder-

Richardson Formula 20 and the tetrachoric correlation coef­

ficient. In tests which had subscales clearly defined by

one variable the tetrachoric correlation often reflected

the consistency of subject responses much better than the

KR-20. For tests in which items are defined by more complex

principles the choice of estimates is not obvious.

In any case, some support for the model has been gener­ ated, and the direction for further study of the model seems clear. Tightly clustered set of items should be developed to examine the model across one problem type (for which higher reliabilities should obtain). Considerations of differential speededness and subject motivation almost dictate that items or pairs of items be validated in connec­ tion with the model before comparisons among problem types

(e.g. disjunctive, etc.) can be made effectively.

Problems for Further Research A major contribution of this research has been to posit a model for the use of propositional logic in the contexts of elementary mathematics and to establish the tenability of that model in the context of odd and even integers. 163

Several questions related to the model have been answered

within this context, but many more remain open. Among these

two stand out as being of primary important: (1) validation

of the model for a specific logical connective (not con­

junction) by exhaustive testing of all components of the

model using several replications of each item form; (2) test­

ing of the model in a context other than odd and even inte­

gers .

A host of problems arise from attempts to integrate this

research with the findings of other investigators. Some

results seem to contradict published findings. For example,

Juraschek (1976) seems to have found that preservice ele­ mentary teachers use the "exclusive or" when solving dis­

junctive inference problems. By contrast, not one of the 424 preservice elementary teachers participating in various segments of this study used the exclusive interpretation of the disjunction with any consistency. O'Brien and his collaborators (1971) claim that interpretation of conditional statements as if they were biconditional is the principal cause of errors in inference tasks. Yet the current research reveals not only that both conditional and biconditional statements are interpreted as conjunctions, but also that, when the context precludes this interpretation, subjects are more puzzled by the biconditional than they are by the conditional. The resolution of these conflicts must be a priority area for research. 164

The roots of this research lie in problems of class­ room communication, and in particular in the problem of

teaching preservice elementary teachers to use the language of propositional logic effectively in their reading of mathematics texts, in the solution of mathematical problems, and in their own classroom discourse. A third area of re­

search involves the effectiveness of strategies for teacher training which can be derived from or suggested by the results of the study.

Implications for Teacher Education

Teacher educators concerned with the use of logic by elementary teachers have often advocated the incorporation of special units on elementary logic in the mathematics component of the teacher education program. Evidence that this is an effective approach has not been obtained, and among some teacher educators at least, there is a belief that such units do not serve the intended purpose. Exten­ sion of the findings of this study into the preservice mathematics classroom leads to several recommendations.

The study of logic should be integrated with other mathematical units, and not segregated into a special unit.

The structured contexts of mathematical problems facilitate the appropriate use of logical statements. In many studies of logic, as well as in units designed to teach logic, the statements used create artificial relationships which have no meaning to the reader. The objective of teaching logic 165 is to enable students to use propositional modes in dealing, not with contrived problems, but with empirically or logi­ cally determined relationships. By integrating the teaching of logic with the development of these relationships the propositional forms can be made more meaningful to the student population.

A potential advantage of teaching logic in the context of elementary mathematics may be the ability to postpone the explicit consideration of negation until the connectives are well established. Research has shown that negation of statements is quite difficult for subjects (Wason and

Johnson-Laird 1972). Many elementary mathematical concepts divide the universe into two sets, both of which have identities of their own (e.g., odd/even# prime/composite, whole number/proper fraction, etc.). In these contexts the connectives can be established, and implicit negation used before the formal operation of negation is introduced. Research on the effectiveness of such an approach could provide considerable insight into the nature of the diffi­ culty associated with negation. In most treatments of elementary logic negation is used in the definition of connectives (e.g., poq=H(pA-«q) , and in most studies of negation tests use statements involving logical connectives.

Thus the effects of negation and connectives are currently confounded in both research and instruction. 166

Within the contexts of elementary mathematics the dis­

junction is the most easily understood of the non-conjunc­

tive connectives. The disjunction arises naturally in the

solution of equations, in the determination of least common multiples, and in other areas of elementary mathematics.

Instructional strategies which initially rely more heavily on the disjunction (and less on the conditional), and then use the disjunction to introduce the implicative form should be investigated. Finally, the effects of instruction designed to high­

light the sorting of the set of possibilities should be examined. This study has shown that the sorting of this set is related to the ability of subjects to select logical statements descriptive of mathematical relationships. More­ over, it was shown that in the presence of a listing of the elements of the set of possibilities preservice teachers were more inclined to be critical (rather than accepting) of both valid and invalid arguments. Taken together these results suggest that mathematics courses which deal explicit' ly with use of the full set of possibilities might be effective in improving teachers* use of logical connectives. 167

APPENDIX A

A PRELIMINARY INVESTIGATION: INTERPRETATIONS OP STATEMENTS

Most studies of the understanding of statements in the standard forms of propositional logic have been concerned with the drawing of inferences from pairs of statements purported to be true. The consistent findings can be attri­ buted to errors in reasoning or errors in statement interpre­ tation. They lead to the question: under what conditions of truth (or falsity) of the constituent parts of a compound statement do individuals find that (compound) statements are true?

The purpose of the initial research phase was to investi­ gate the interpretation of statements through the use of tests modelled on truth tables. Two tests were used: one required subjects to assign one of the values "true" and

"false" to each compound statement; the second asked subjects to indicate whether a statement was "true", "false", or

"neither".

Procedures

The Truth Value Inventory (TVI), a thirty-two item instrument was devised by the experimenter. Each item of form A (TVI—A) contains a single statement referring to a pair of sets pictured at the top of the page, and requires the subject to declare whether the statement is true or 168 false. A second part of each item requires the subject to indicate minimum changes in set membership which would change the truth value of the statement. Items in TVI-B were identical with the first 16 items in

TVI-A except that they allowed subjects to declare statements to be true, false, or neither. The second part of an item was to be done only if the statement was not declared true.

Abbreviated instructions and sample items are provided in

Figure A-l.

Two items were constructed for each cell in the 4 by 4 matrix shown in Table A-l. The order of items within the test was determined by applying a Graeco-Roman square to the array.

The TVI-A was administered to all students in attendance on an unannounced date for a lecture session of a mathematics course for pre-service elementary teachers. (Logic was not discussed in the course.) One student's paper was deleted as incomplete, leaving a sample size of 70.

Responses to the true-false parts of items were scored by the systems true - 3, omit - 2, false = 1. The pair of items in each cell of the matrix was considered as a two-item test; tetrachoric correlations were computed s b reliability estimates for each pair.

Item scores were submitted to a three factor analysis of of variance. Factors were defined by connectives (4 levels), truth values of p and q (4 levels), and subjects (70 levels); 169

The four statements on this page refer to these sets:

A L f

AN ITEM FROM TVI-A

Read each statement carefully; then circle its truth value (T if true, F if false). Then circle the objects you would move to the other set in order to change the truth value of the statement. Remember to move as few as you can and still change the truth value!

1. Ais in set A or iis in set B. Move: L?

THE PARALLEL ITEM FROM TVI-B

Read each statement carefully; then circle its truth value

If you marked F or N circle the objects you would move to the other set to make the statement true. Remember to move as few as you can and still make the statement true.

1. A is in set A or ifis in set B. T F N Move: a • # ■ A *******

Figure A-l. Abbreviated directions and sanple items from TVX 170

two item scores appeared in each of the 1120 cells. Pairwise

comparison of means was accomplished by Tukey*s "honestly

significant difference" method.

This analysis was used to subject the following research

hypotheses to statistical test:

HI: The proportion of compound statements declared

to be true does not vary as a function of the

connective.

H2: The proportion of compound statements declared to

be true does not vary as a function of the combin­

ation of truth values of the constituent parts.

H3: When the truth values of the constituent parts are

held constant, the proportion of compound statements

declared to be true does not vary as a function of

the connective.

H4: When the connective is held constant, the proportion

of compound statements declared to be true does not

vary as a function of the truth values of the con­

stituent parts.

Responses to the second parts of items or "moves" were scored according to the system: 0 = no move made, 1 ** move only the object in the first simple statement, 2 « move only the object in the second simple statement, 3 =* move both objects mentioned in the compound statement, and 4 » any deviation from the above. These scores were combined with 171

true-false scores to yield a 15 point nominal scale. Frequency distributions were computed for each item. Examination of

these distributions suggested post hoc hypotheses which were 2 tested with the x statistic. Observed effects were considered to be significant only

if the probability of obtaining them by chance was less than .01. After TVI-A data were collected and analyzed, TVI-B was

administered in a pilot test situation in order to determine whether sizable differences in subject responses would occur when subjects were given the option of asserting that a state­ ment was neither true nor false. TVI-B was administered to

17 students during the first meeting of a section of the second of two required mathematics courses for elementary teachers.

One subject answered only two items and was deleted from the sample. Responses to the first parts of items were scored accor­ ding to the system: true = 3 , neither = 2, false « 1. Mean scores for items were computed and compared with mean scores for items in TVI-A. Data were examined to determine usage patterns for the "neither option and patterns in the responses to the second parts of items.

Results

Item pair reliabilities for TVI-A are provided in Table

A-l. Fourteen of the 16 tetrachoric correlations were above

.8; one was .6578, and the final one was incomputable due to 172

a 0 in one cell of the four-fold table. These correlations

indicate that subjects were quite consistent in their responses

to the two items testing the '.same problem.

Table A-l. Matrix used in constructing TVI with item pair reliabilities for TVI-A entered in cells.

Truth values of Statement Form

P <3 p and q p or q if p then q p iff q

T T .8496 .9104 .6578 .9030

T F .8135 .9564 not com­ .8935 putable F T .8496 .8083 .9023 .8847

F F .8418 .8258 .9650 .8121

Analysis of variance, summarized in Table A-2, revealed significant (p ^ .001) effects for connectives, truth values, and the interaction of these variables. As expected, subjects also provided a significant source of variance, and the inter­ actions of subjects with all other variables were significant.

When Tukey’s HSD was used to test the 120 differences between means determined for levels of truth values combined with levels of connectives, 83 were shown to be significant at the 1% level of significance (critical value = .31). The means subjected to these comparisons are shown in Table A-3.

On the basis of this analysis hypotheses 1 and 2 were rejected. The assignment of truth values to statements did vary with both connectives and truth values of constituent 173

Table A-2. Analysis of Variance for TVI-A Scores

Source df MS F

Truth Values (T) 3 353.56 2302.30***

Connectives (C) 3 16.18 105.33***

Subjects (S) 69 1.13 7.37***

T x C 9 11.19 72.86*** T x S 207 0.81 5.31***

C X S 207 0.51 3.32*** T x C x S 621 0.37 2.38***

Error 1120 0.15

*** p<.001

Table A-3. Mean scores for items in TVI-A

Truth Values Statement Form p q_ p and g P or q if p then q P if Q T T 2.95 2.84 2.92 2.80 (3.00)a (3.00) (3.00) (3.00)

T F 1.05 1.78 1.04 1.04 (1.00) (3.00) (1.00) (1.00)

F T 1.08 1.98 1.19 1.17 (1.00) (3.00) (3.00) (1.00)

F F 1.13 1.14 1.43 1.50 (1.00) (1.00) (3.00) (3.00) a Figures in parentheses indicate values which would be obtained by perfect use of logic. 174

statements. Hypotheses 3 and 4 each invlove four sub­

hypotheses, one for each pair of truth values (H3) or con­

nective (H4). Table A-4 displays the significant differences

in mean scores when comparisons are made among connectives

with truth values constant for each connective. There is

only one case (false-false) for which the conjunction differs

from the biconditional, and in no case does the conditional

differ from either the conjunction or the biconditional. The

disjunction differs from all other forms in the true-false

and false-true cases and from the biconditional when both

simple statements are false.

Table A-5 provides a similar display of differences when

comparisons are made among input truth values with connectives held constant. There is no significant difference between

true-false and false-true statements. For conjunctions there is no difference between false-false and false-true or true- false; for implications false-true and false-false are not significantly different. All other differences exceed the critical value. Examination of the data generated by subjects' responses to the second part of each item led to two hypotheses; H5: The "move" part of an item will be omitted more

often when the statement is declared true than when

it is declared false.

H6: In balanced statements (conjunctions, disjunctions,

and biconditionals with both parts true or both 17S

Tables A-4 and A-5. Differences between means (TVI-A)

Table A-4. Comparison by connective

Disjunction Conditional Biconditional TT TF FT FF TT TF FT FF TT TF jFT FF

Conjunction TT 11 .03 .15 TF .73* .01 .01 FT .90* .11 .09 FF •01 ,30 ,37* Disjunction TT .08 .04 TF .74* .74* FT .79* .81* FF .29 .36* Conditional TT .12 TF .00 FT .02 FF .07 * p< . 01

a Table A-5. Comparison by truth values

true-false false-true false-false c d i e cdiecdie

true-true c 1.90* 1 .87* 1.82* .86* 1.70* d■ 1.06* i 1.88* 1.73* 1.49* e 1.76* 1.63* 1.30* true-false c .03 .08 d .20 .64* i .15 .39* e .13 .46* false-true c .05 d .84* i .24 e .33* * ------p < . 0 1 - ~ . ** is a4 oiiinnf ■{rtn . i m f?nnd i t iona. 176

parts false) the move made will involve the second

simple statement more often than the first. 2 These hypotheses were subjected to x analysis, and hypotheses

5 and 6 were retained (H5: x2 “ 859.84, p < .001; H6: x2 ** 26.95, p < .001).

Mean scores for items in TVI-B are presented in

Table A-6. The pattern of means was found to be quite similar

to that for TVI-A (Table 4).

Table A-6. Mean scores for items in TVI-B.

Truth values Statement form

P q p and q p or q if p then q P q

T T 3.00 2.47 2.64 2.68

T F 1.06 2.31 1.00 1.06

F T 1.00 2.17 1.19 1.44

F F 1.00 1.00 1.26 1.40

The "neither" option was found to be used infrequently; the most frequent usage occurred in disjunctive items having the first simple statement false and the second true (4 subjects)

Two subjects used the "neither" option 3 or more times, and

11 did not use it at all.

Discussion

The major finding of this initial study is that participatir 177 preservice elementary teachers tended to treat conjunctive, conditional, and biconditional statements in the same way, declaring them true only if both parts were true. Roughly half of the population also tended to treat the disjunction in this manner.

Overall, subject responses to the part of each item requiring a move in order to change the truth value appeared to be consistent with their endorsements of statements as true or false.

Moves were omitted more frequently when the statement was declared true (86.32% of omissions). Of the cases in which subjects failed to answer the move part of the item,

63.25% involved statements in which both simple statements were true. Omissions were distributed approximately equally over connectives.

A hypothesis that the truth value of the second simple statement might have a greater effect on subject behavior than the first grew out of the observation of (nonsignificant) differences between responses to true-false and false-truei combinations, especially in the disjunctive case. In the current study and in a study reported by Paris (1973) the direction of the differences in adolescent subjects' treatment of disjunction agreed with the hypothesis. The significant tendency for subjects to move the object mentioned second also lends credence to this hypothesis. 178

The possibility that subjects responding to items in

TVI-A might have been using a 3-valued logic to classify statements as true, false, or neither true nor false was considered. Administration of a shortened form of the test which allowed three, rather than two, options did not support this hypothesis. 179

APPENDIX B

The instruments used in the six research segments are provided in this appendix as follows.

Page

Segment 1 Test MA 180 Test MC 182 Test LA 184 Test LC 186 Segment 2 Test 188 Segment 3 Directions 192 Form LI 193 Form L2 199 Form Ml 205 Form M2 210 Segment 4 Directions 215 Form LL 216 Form LM 219 Form ML 222 Form MM 225 Segment 5 Directions 228 Test M 229 Test L 230 Test M * L 231 Test L,* M 233 Test L' 235 Test M* 239 Segment 6 CLT -■ form A 243 CLT - form B 251 180

Segment 1: Test MA (Mathematical Statements, Abstract Po s s i'bi iit'ie s}

In the following puzzles M and N are whole numbers. M is either odd or even, and N is either odd or even. Thus there are four possibilities for M and N as shown below. Possibilities: A: M odd, N odd B: M odd, N even C: M even, N odd D: M even, N even

In each of the problems read the statement carefully. Then look at each possibility and decide whether it could be true when the statement is true.

Circle all possibilities that could be true when the ment is true.

1. M x N is odd. A BC D 2. M + N is odd. A B C D

3. 2M + N is odd. A BC 0

4. M x N is even. A BC D

5. M x (M + N) is even. AB C D 6. 2M + N is even, M + N is odd. A B C D

7. M x N and M + N are even • ABC D

8. 2M + N is even. AB C D

9. M + 2N is even. A B CD 10. M + 2N is odd. ABC D

11. 2 x (M + N) is even. A B C D

12. N x (M + N) is even. A B C D

13. M + 2N is even, M + N is odd. A B C D 181

Possibilities: A M odd, N odd B M odd, N even C M even, N odd D M even, N even

14. M + N is even. AB C D

15. M + N + (M x N) is odd. A B C D

16. 2 X (M + N) is odd. A B CD 182

Segment 1; Test MC (Mathematical Statements, Concrete Possibilities)

In the following puzzles M and N are whole numbers. M is either 1 or 2, and N is either 3 or 4. Thus there are four possibilities for the exact values of M and N as shown below.

Possibilities: A: M - 1, N = 3 B: M = 1, N - 4 C: M = 2, N ■ 3 D: M - 2, N = 4

In each of the problems read the statement carefully. Then look at each possibility and decide whether it could be true when the statement is true. Circle all possibilities which could be true when the state­ ment is true.

1. M x N is odd. A B C D

2. M + N is odd. A B C D

3. 2M + N is odd. A B c D

4. M x N is even. A B c D 5. M x (M + N) is even. A B c D

6. 2M + N is even, M + N is odd. A B c D 7. M x N and M + N are even. A B c D 8. 2M + N is even. A B c D

9. M + 2N is even. A B c D

10. M + 2N is odd. A B c D

11. 2 x (M + N) is even. A B c D 12. N x (M + N) is even. AB c D 183

Possibilities: A: M = 1, N = 3 B: M = 1. N = 4 C: M = 2, N = 3 D: M = 2, N = 4

13. M + 2N is even, M + N is odd. AB C D

14. M + N is even. A B C D

15. M + N + (M x N) is odd. A B c D

16. 2 X (M + N) is odd. A B c D 184

Segment 1: Test LA'(Logical Statements, Abstract Possibi1ities)

In the following puzzles M and N are whole numbers. M is

either odd or even, and N is either odd or even. Thus there

are four possibilities for M and N as shown below.

Possibilities: A: M odd, N odd B: M odd, N even C: M even, N odd D: M even, N even

In each of the problems read the statement carefully. Then look at each possibility and decide whether it could be true when the statement is true.

Circle all possibilities that could be true when the state­ ment is true.

1. If H is even, then N is even. AB C D

2. Only one of the numbers is even. AB C D

3. H is even and N is odd. AB c D

4. N is odd. AB c D

5. At least one of the numbers is even. A B c D

6. M is even. A B c D

7. Neither M nor N is even. A B c D

8. M and N are not both even. AB c D

9. H is even and N is even. AB c D

10. N is even. AB c D

11. M is odd. AB c D

12. M is odd and N is even. AB c D 185

Possibilities: A M odd, N odd B M odd, N even C M even, N odd D M even, N even

13. If one number if even, so is the other. A B C D

14. If M is not even, then N is not even. A B C D 186

Segment 1: Test LC (Logical Statements, Concrete Possibilities)

In the following puz2les M and N are whole numbers. M is either 1 or 2, and N is either 3 or 4. Thus there are four

possibilities for the exact values of M and N as shown

below.

Possibilities: A: M = 1, N -3 B: M = 1, N = 4 C: M = 2, N = 3 D; M = 2, N = 4 In each of the problems read the statement carefully. Then

look at each possibility and decide whether it could be

true when the statement is true. Circle all possibilities which could be true when the

statement is true.

1. If M is even, then N is even. A B C D

2. Only one of the numbers is even. A B C D

3. M is even and N is odd. AB c D

4. N is odd. A B c D

5. At least one of the numbers is even. A B c D

6. M is even. A B c D 7. Neither M nor N is even. AB c D

8. M and N are not both even. AB c D

9. M is even and N is even. A B c D

10. N is even. A B c D

11. M is odd. A B c D

12. H is odd and N is even. A B c D X87

Possibilities: A: M 1, N = 3 B: M s: 1, N = 4 C: M ss 2, N = 3 D; M = 2, N = 4

13. If one number is even, so is the A B C D other.

14. If M is not even, then N is not A B C D even. 188

Segment 2; Sorting in Four Content Areas

Directions. In each of the following problems you are given one fact and four options. Some of the options are possible when the fact is known and others are not. Read the fact and each option carefully. Circle the letters indicating all options which are possible when the fact is known.

Example: Fact: x - y = 10 00 CM >» If 1 Options: A) X a *• B) X * - 8 , y = 2 C) X * 0 D) X < Y Answer: You should have circled A and C

A is possible because 8-(-2) =*10 B is not possible because -8-2=10 C is possible because 0-(-10)=10 D is not possible because x - y>0, so x > y.

1. Fact: M and N are counting numbers and M + N is even

A) M is even, N is odd B) M is even, N is even C) M is odd, N is odd D) M is odd, N is even

2. Fact: M and N are counting numbers and 2M + N is odd

A) M is even, N is odd B) M is even, N is even C) M is odd, N is odd D) M is odd, N is even

3. Fact: M and N are counting numbers and H(M + N) is even

A) M is even, N is odd B) M is even, N is even C) M is odd, N is odd D) M is odd, N is even 189

4. Fact: M and N are counting numbers and 2 (M + N) is even

A) M is even, N is odd B) M is even, N is even C) M is odd, N is odd D) M is odd, N is even

5. Fact: M and N are counting numbers and M x N + M + N is odd

A) M is even, N is odd B) M is even. N is even C) M is odd, N is odd D) H is odd, N is even

6. Fact; y are rationals and xy = 0

A) x = 0, y = 0 B) x - 0, y j* 0 C) x 0, y = 0 D) x ^ 0, y ^ 0

7. y are rationals and x + y > A) x - 0, y = 0 B) x - 0, y 0 C) x ^ 0, y = 0 D) xj< 0, y / 0

8. Fact: y are rationals and x(y - 1) A) x = 0, y = 0 B) x = 0, y j* 0 C) x ^ 0, y * 0 D) x j* 0, y ft 0 2 9. Fact: x and y are rationals and xy > x > 0 A) x = 0, y = 0 B) x = 0, y i* 0 C) x y 0, y •* 0 D) x M i y j 1 0 10. Fact: are rationals and 2xy - 2yx

A) x *» 0, y » 0 B) x = 0, y / 0 C) x f 0, y * 0 D) x / 0, y / 0 Fact: P and Q are integers and P x Q > 0

A) P > 0, Q > 0 B) P > 0, Q < 0 C) P < 0, Q > 0 D) P < 0, Q < 0

ire integers and

A) P > 0, Q > 0 B) P > 0, Q < 0 c) P < 0, Q > 0 D) P < 0, Q < 0 Fact: are integers and = P4 A) P > 0, Q > 0 B) P > 0, Q < 0 C) P < 0, Q > 0 D) P < 0, Q < 0

Fact: P and Q are integers and (P x Q) > 0

A) P > 0, Q > 0 B) P > 0, Q < 0 c) P < 0, Q > 0 D) P < 0, Q < G

Fact: are integers and

A) P > 0, Q > 0 5 B) P > 0, Q < 0 C) P < 0, Q > 0 D) P < 0, Q < 0

Fact: In triangle ABC above AC is 10 cm. long.

A) AB is 6 cm. long and BC is 8 cm. long B) AB is 6 cm. long and BC is 3 cm. long C) AB is 15 cm. long and BC is 8 cm. long D) AB is 15 cm. long and BC is 3 cm. long 191

17. Pact: Figure S is a square

A) S has area 4 and perimeter 4 B) S has area 4 and perimeter 1 C) S has area 1 and perimeter 4 D) S has area 1 and perimeter 1

18. Fact: Figure R is a rectangle

A) R has area 4 and perimeter 8 B) R has area 4 and perimeter 10 C) R has area 3 and perimeter 8 D) R has area 3 and perimeter 10

19. Fact: A and B are rectangles with the same area

A) A and B have the same base and same height B) A and B have the same base and different heights C) A and B have different bases and the same height D) A and B have different bases and different heights 20 . Fact: r

A B

In triangle ABC the measure of angle A is a ” and the measure of angle B is b*

A) a > 90, b > 90 B) a > 90, b < 90 C) a < 90, b > 90 D) a < 90, b < 90 192

Segment 3i Instructions for tests

In the following puzzles M and If are whole number*. H la either 1 or 2. IT la either odd or area* Thtte there are four possibilities for M and N ae shown below:

M - 1 K - 1 I H - 2 H - 2 B is Odd B is eves | K la odd B is even

In each problem these possibilities are divided Into two sets, a "true set" and a "false set,* and you are given four state* manta. Examine the true set and false set carefully. Then select the statement which la true for all the possibilities In the true set, and false for all the possibilities in the false set. Circle the letter which indicates your choice.

Example: M ■ 1 M « 2 ^ (H - 1 M - 2 N is odd H is odd ) \B Is even B is evenD True set False set

A. M is odd B. M Is even C. B is odd 2>. N is even

The correot answer for this problem is C. Segment 3; Selecting Logical Statements 193 for Sortings {Form LI) II

1. f , M « 1 H » 2 J v w 18 N 18 / It is 'Pro* set Falsa sat

A. N la odd B* M la avan or V Is odd C. Hla odd If and only If H is aran D. M la odd and It is odd

2. M - 2 N la odd B la odd V la <

A. K la odd B. V la odd C* 7f la odd or V la odd D. M la odd or It la swan

! H la odd It la

A. M la odd if and only If It la odd B* Mla odd or V la avan C. H la odd or H is odd D* V la aran 194

LI

4. M - 1 H is odd H Is R la IT ia odd

1. VIs odd B* M is odd sad K is odd

C* K is v t s b or H ia odd D* M is odd if and only if S is odd

5. M - 1 1 R is odd N is odd B la i

Talas sat

A. K is srsn and R is odd B. M is odd If and only if R is a m C« M is srsn and R is trrsn D. R is odd

6. « - 1 R is odd R is R is odd

A* M is a m or R is aran B. M is a m C. M is odd or R ia soon D* K is srsn if and only if R is rrsn 195

LI

7. N Is odd If Is odd

False sst

A. K Is even and If is srsn B* Mis srsn C* H Is srsn or K Is even D* If Is even

8. is odd ( W is 5 is odd

A. M is odd if and only if B is odd B* If H la odd than If is srsn C. If K is odd than K is odd D. N is odd r — 9. K - 1 ; ■ M « 2 M - j If is even: ! N la odd H is srsnran j/ C \ J” mis lodd ^ ) /

True sst False set

km H is even B. If K la odd tbsn If is srsn 0. If « Is odd then H is odd D. K is odd if and only if K is odd 196

XI

1 0 . B is odd IT is odd

A. M Is odd sad B is odd B. M Is odd If and onlj If IT is srsn C. Mis odd and B is «r«a D. H is odd

U. 5 Is odd b is odd

A. M is even or IT is avan B. M is svsa C. M is odd sad B is arsn D. B is srsn

12 K - 2 M - 1 V la i B Is odd B Is < B is odd '

A. K is «TOB B. M is srsn sod B ia odd C. B is srsn D. M la srsn and B is « w a 197

15. n «1 K - 2 Is odd a9 is onra 9 is odd is srsn / Balsa sst

i. K is odd if and only if 9 is assn

B. M is s t s n and 9 is odd C. M is odd and V is odd D. H is odd

14. IT is odd ■ 9 is odd

d. If K is odd than 9 is odd B. H is odd C. If 9 ia v r m than M is odd D. H is odd if and only If N la odd

H - 2 ' If is odd V is odd H I S

i. K la odd B. If:.* is'odd than 9 is odd C. B Is odd D. If 9 is srsn than H is odd 198

V is odd B is is odd S is «T*& :

A. N is Odd B. M is odd C. M is odd and B ia srsn D. M is odd or If is srsn

M * 2 A f M-l M-2 ^ B is B is odd y V B is Odd B is ersa J

true sst Fslss sst

A. M is odd sad B is srsn B* K is odd or B is odd C. M is odd if sad only if B is srsa D* B is odd

' B is odd

Trus sst

A. M is srsa or H is odd B. M is srsn C. K is srsn or B is srsn D» H is odd if sad only if B is odd 199 Segment 3: Selecting Logical Statements for Sortings (Form Li) 12

1 6 . H « 2 I J I K - 1 M - 2 j 9 Is odd ' / I N Is odd R Is srsn /

Trne sat False set

A. 9 Is even B. M is odd If and only If 9 Is area C. M is even and 9 is odd D. M Is even or 9 la even

17. 9 is odd ‘la odd! 9 is odd ‘la

A. M is even B. If K is odd then 9 is odd C. M is odd if and only if 9 is odd D. If M is odd then 9 is even

18. 9 is odd 9 is odd ,9 iais

A* 9 la odd

b ; K is odd and 9 is even c. K is odd if and only If 9 ia odd D.M is odd and 9 is odd 200

12

19. 9 is ; 9 is odd

A. MIs area and 9 is srsn B. H is srsn 0. K is odd and 9 is eesn D. 9 is area

20. H - 1 9 is odd ; 9 is 9 is odd 9 IS

True set Falss sst

A. K is odd and 9 is odd B. M is odd C. M is odd or 9 is odd D. 9 is avsa

21. 9 is Odd 9 is add j is 9 is STOD ;

True set

A. K is odd if and only if 9 is s t s b

B. His odd and 9 Is odd 0. Xla even and 9 is odd D. X is srsn 201

L2

22 . IT is odd ; H is odd -

A. IT !■ srsn B. M ia ersn or N ia srsn C. M ia srsn or B ia odd D. M is odd if and only if N is odd

25. B is odd IT is IT is odd

A. If IT is odd than M la odd B. N ia odd if and only if H is odd C* If Bia srsn than M is odd D. M is odd

24. : » is odd IT is odd n ia

A. M la odd if and only if B ia odd B; M is odd or B la odd C* K ia srsn or B la odd D. B ia odd 202

L2

25. N is 9 is odd, IT ia odd

A. M is odd if and only if N ia sron B. H is odd and 9 is odd C. K la odd and 9 is srsn D. M is eran

26. [ ! M - 1 M « 1 J 9 ia I V is odd 9 is «r*n J

Talas sot

A. M ia sren and 9 ia odd B. M ia srsn or 9 is odd 0. 9 ia odd D. M is srsn

27. 9 is

A. 9 is a m B. M la srsn and 9 la srsn 0. K is odd or 9 is srsn D. M la odd 203

L2

28. oddIT is odd K Is R Lb oddIT

A. If R Is odd than « Is odd B. If R la area than H la odd C. M la odd S. N la odd

29. M « 1 ; R la area R la odd

A. K la aven B. If R la odd than K la odd C. M la odd If and only If R la odd D. If H la odd than R la arran

30. R la oddR la odd R la oddR

A. M la srsn and R ia odd B* M la odd or R la odd C. K la odd If and only if R la odd D. R la odd 204

L2

51. K - 1 is odd IT la «T«nN is area IT is odd

A. K is m n sad V is srsn 3. M is odd or V is srsn C. N is srsn 0. H is odd if sad only if 3 is odd

52. ! 3 is odd i

A* M is odd if sad only if H is odd B. M Is ovsn or 3 is rron C. H is odd or V is even S. K is odd

f \ M - 1 M-l M » 2 J I N is odd V is srsn If is odd J

Tras sst

A. H is odd or If is odd 3. H is odd C. H is srott or K is odd 0. If is odd 205 Segment 3i Selecting Mathematical Statements ^°r Sortings (Form m TT Ml

1 N la odd ; V is odd

V ia odd B. M i N is odd C. M(M + V) is area 0. M + H is odd

2. H is FT is odd is odd V is svsn i

True sot

A* fficK + K + N is odd B. K + B is odd C. K(M + V) it odd D. N is odd

Iff is odd ,N i s o d d If is o w n If is

A* M + II is oven B. M(M + S) is oven C. 21T + M is even D. M x V is even 206

Ml

4. V la odd N la odd

Troa aat

A. MxR + If + IT Is odd B. M + IT la aran C. IT(M + V) la eras D. R la « m

5. V Is odd H Is odd IT la aaan: 1 R is aaan'

A. N(M +V) Is odd B. K + H ia odd C. MxN + M + R Is avan B. H is odd

6. M - 1 R Is odd is odd V isH

Raisa aat

A. M(M + R) Is odd B. R(M + R) la aaaa 0. R la odd D. 2R + M la odd 207

HI

7. H » IT ia aran B 19 Odd

Trna aat

A. M i H is area B. K(H + IT) is odd C. H is aran D. 25 + M is aran

8. B is odd B is odd

A. ZB ♦ M is odd B. K (17 + H) la odd C. M + IT ia odd D. K x IT is odd

B Is odd H is odd

A. MxH. + M + 5 is odd B. B is odd C. B(K + B) Is aran B. 25 + H is odd 208

HI

1 0 . 9 Is odd v is odd

A. M + B IS odd B. M X IT is odd c. H is odd D. M(M + H) is odd

11. - IT Is odd B is i ir is odd1 N is rrsnj

A. MxN + H + H is s t s b B* H x H is svsn C* 29 + M Ia srsn D. V is even

12. is odd B is

A. 29 4- H is srsn

B. Hllil STSB

C. 9(K + V) is «*s b D. M + S is srsn 209

13. - IT is avan N is odd |

A. H X IT 1* Odd B. M(M + IT) is awn C. N la odd D. M + IT ia evan

14. V is IT Is odd N is odd

A. IT ia even B. IT(K + K) is odd C# MxB + M + N ia aran P. 2N + M ia aran 210

Segment 3; Selecting Mathematical Statements for Sortings (Form

15. B la odd

True eat

A* M x IT ia odd B. MxB + H + H ia odd C. 2N + M la odd D. B la even

1 6 . N is odd iff la even H ia odd

True aet

A. B(K + N) la area B. M + F ia even C. HiNli even D. 2N + K ia odd

K - 2 17. 5 la odd V la B la odd B la <

A. B ia odd B. K(K + B) la odd C. K + B ia odd D. M x B ia odd 211

IS. j I M - 2 : ;M - 2 j ] ( IT la odd 5 1* w i n J IT is odd

I t s * sat

k. B(W + IT) is odd B. 2B + H is aran C. S is odd D. *f(K + B) la avan

19. B Is odd b is odd

km H + B is even B. H z B is even C. M(M + B)ia even D. B la aran

2 0 . |H - 1 777 ! B la < B ia odd odd 5 la a r m J

falsa sat

km M + B la odd B; N x B ia aran C. B is aran B. B(M + B) la odd 212

21 * V is odd is odd if is

Trus sst

A* 2H + M is s t s o B. M x H is odd 0. N(K + K) is odd D, M + Iff is odd

22 K » 1 !* is odd is V is < S is odd

A. KxtT + M + H is srsn B. B(M + 5) is srsn C. IT is s m D. K + K Is srsn

23. : K is odd V Is If la rr«i :

A. M(M + S) Is srsn B. V is odd G. KZH + M + V is odd s D. 2S 4- H is odd 213

24. 9 is odd is odd

A. K is odd B. H(M + N) Is srsn C. 29 + H is odd D. HxN + M + 9 ia odd

25. ^ C t ^ T 777“ ^ 9 is areai j I 9 is odd H la odd J

Bals* sst

A. 29 + H Is odd B. 9 is even C. 9(M + 9) is srsn D. Ml9 + M + 9 is srsn

26. 9 is ,9 is 9 is odd

A. K(H + 9) is odd B. 29 + K is « m

C. M d + 9 + 9 Is srsn D. 9 is srsn 214

27. IT is odd N is odd

A. K is «r«n B. H + N la odd C. M(M + n) is odd D. Hsdf + K + IT is «T«n

28. H is odd N is Odd i H is H is even '

Trus sat

A. H(H + IT) is odd B» N is odd C. M + H Is avan D. MxN + M + H is odd 215

Segment 4: Directions for Statement Differences Tests

Statement Differences Test

In the following puzzles M and N are whole numbers. M is either 1 or 2, and N is either odd or even. Thus there are four possi­

bilities for M and N as shown below: Possibilities: A - M ■ 1, N is odd B - M - 1, N is even C - M * 2i N is odd + * D - M - 2, N is even In each of the problems there are two statements. Read both of the statements carefully: then find a possibility in the list for which the first statement is true and the second statement is false.

Circle the letter which indicates your choice. If you cannot find a possibility for which the first state­ ment is true and the second statement is false, circle "X"

Examples: I. True statement: M - 1 False statement: M - 1 and N is odd A B C D X II. True statement: M * 1 or H is even False statement: M * 1 ABCDX

The correct answer to problem I is B (H ■ 1, H Is even). The correct answer to problem II is D (H a 2, N is even). 216

Segment 4: Statement Differences Test - Form LL

LL - 2 Possibilities: A - H * 1, N is odd B - M ■ 1, N is even C - M - 2, N is odd D - M ■ 2, N is even

X. True statement: K is even or N is even Palse statement: H is even if and only if N is odd A B C D X

2. True statement: If H is odd then N is even Palse statement: N is even A B C D X

3. True statement: H is even or N is odd False statement: N is odd A B C D X

4. True statement: If M is odd then N is odd False statement: H is even if and only if N is even A B C D X

5. True statement: H is odd or N is even False statement: M is even if and only if N is even A B C 0 X

6. True statement: M is even False statement: M is even and » is even A B C D X

7. True statement: M is odd False statement: H is odd and N is odd A B C D X

8. True statement: M is even or N is even False statement: M is even A B C D X

9. True statement: If M is odd then N is even False statement: M is even if and only if N is odd A B C D X

10. True statement: M is even or Iff is odd False statasmnt: M is even if and only if II is even A B C 0 X 217

LL - 3 Possibilities: A - M - 1 . N is odd B - M - 1 , N is even C - M - 2 . K is odd D - M - 2 1 N is even

11. True Statementi If M Is even then N is even False statement: H is even if and only if N is even A B c D X

12. True statement: If M is even then N is odd False statement: M is odd if and only if W is even A B C D X

13. True statement: N is odd False statement; N is odd and H is even A B C 0 X

14. True statement; N is odd False statement: H is odd and N is odd A B C D X

15. True statement: M is odd if and only if N is even False statement: K is odd and tt is even A 8 C D X

16. True statement: M is odd or N is odd False statement; H is odd A B C D X

17. True statement: M is odd if and only if K is even False statement: M is even and N is odd A B C D X

18. True statement: H is odd if and only if N is odd False statement: H is even and N is even A B C D X

19. True statement: M is odd or N is odd False statement: M is odd if and only if N is even A B C 0 X

10. True statement: N is odd False statement: N is odd and M is odd A 8 C 0 X 218

IX - 4 Possibilities: A - M - 1: N is odd B - K m 1, N is even C - M - 2: N is odd D - M - 2, N is even

21. True statement* N is even or M is odd False statement* N is even A B C D X

22. True statement: If M is even then N is odd False statement: N is odd A B C D X

23. True statement* M is even if and only if N is even False statement* M is odd and N is odd A B C D X

24* True statement: If M is even then N is even False statement: H is odd A B C D X

25* True statement: H is odd False statement: M is odd and N is even A B C D X

26. True statement: If H is odd then N is odd False statement: M is even A B C D X

27. True statement: H is odd False statement: N is odd and M is odd A B C 0 X

28. True statement: N is even False statement: M is odd and M is even A B C D X 219

Segment 4: Statement Differences Test - Form LM

lm - 2 Possibilities: A - M 1, N it odd B - M 1, N is even C - M 2, N is odd D - M 2, N is even

1. True statement: H is even or N is even False statement: M + M is odd A B C D X

2. True statement: If M is odd then N is even False statement: 2M + N is even A B C D X

3. True statement: M is even or N is odd False statement: 2M + N is odd A B C D X

4. True statement: If M is odd then K is odd False statement: M + N is even A B C D X

5. True Statement: M is odd or N is even False statement: It + If la even A B C D X

6. True statement: M is even False statement: MXN + M + N is even A B C D X

7. True statement: M is odd False statement: MxN is odd A B C D X

8. True statement: M is even or N is even False statement: 2N + M is even A B C D X

9. True statement: If H is odd then N is even False statement: M + » is odd A B C D X

10* True statement: M is even or N is odd False statement: H + N it even A B C 0 X 220

LM - 3 Possibilities: A - M W 1. N is odd B - M - 1. N is even C - M - 2, N is odd D - M - 2, N is even

11. True statement: If M Is even then N is even False statement: M + N is even A B C D X

12. True statement: If H is even then N is odd False statement: M + N is odd A B C D X

13. True statement; N is odd False statement: N(N+M) is odd A B C D X

14. True statement: N is odd False statement: M x N is orfJ A B C □ X

15. True statement: H is odd if and only if N is even False statement: M(M + N) is odd A B C D X

16. True statement: M is odd or N is odd False statement: 2N + M is odd A B C 0 X

17. True statement: H is odd if and only if N is even False statement: N{M + N) is odd A B C D X

IB. True statement: M is odd if and only if N is odd False statement: MXN + « + N it even A B C D X

19. True statement: M is odd or N is odd False statement: M +■ N is odd A B C D X

20. True statement: N is odd False statesient: M X N is odd A B C D X 221

LM - 4 Possibilities: A - M - i. N is odd B - M a i. N is even C - M m 2, N is odd D - M m 2. N is even

21. True statement: M is odd or N is even False statement: 2M + N is even A B C D X

22. True statement: If M is even then N is odd False statement: 2H + H is odd A B C D X

23. True statement: M is even if and only if N is even False statement: H x N is odd A 8 C 0 X

24. True statement: If H is even then N is even False statement: 25 •*> M is odd A B C D X

25. True statement: If M is odd then N is odd False statement: 2N + M is even A B C 0 X

26. True statement: M is odd False statement; M{M + N) is odd A B C D X

27. True statement: M is odd False statement; M x N is odd A B C 0 X

28. True statement: N is even False statement: M(M + H) is odd A B C 0 X 222 Segment 4: Statement Differences Test - Form ML

ML - 2 Possibilities: A - M “ i. H is odd B - a m i. N is even C - M m 2, N is odd *D - M - 2, N is even

1. True statement: M(M + N) is even False statement: M is even A B C D X

2. True statement: N(M + N) is even False statement: M is even if and only if N is even A B C D X

3. True statement: 2M + N is odd False statement: H is even and N is odd A B C D X

4. True statement: 2 (M + N) is even False statement: M is odd or N is even A B C O X

5. True statement: M x N is even False statement: H is even A B C D X

6. True statement: 2 (M + N) is even False statement: If M is even then N is odd A B C D X

7- True Statement: M(M + N) is even False statement: M is even if and only if N is even A B C D X

8. True statement; 2M + 0 is even False statement: M is odd and N is even A B C D X

9. True statement: M + W is even False statement; M is odd and N is odd A B C D X

10. True statement: M + N is odd False statement: H is even and N is odd A B C D X 223

HI. - 3 possibilities: A - M *• 1, N is odd B - M *■ 1, Mi s even C - H ■ 2. N is Odd D - H ■ 2, N is even

11. True statement: MxN + M + N is odd False statement: M is odd A B C D X

12. True statement: H x N is even False statement: N is even A B C D X

13. True statement: M x N is even False statement: H is even if and only if N is odd A B C D X

14. True statement: 2N + M is even False statement: H is even and H is even A B C D X

15. True statement: M + N is even False statement; H is even and N is even A B C D X

16. True statement: 2N + H is odd False statement M is odd and N is odd A B C D X

17. True statement: M x H is even False statement: N is even A B C D X

18. True statement: H is even or N is odd False statement: 2M + N is odd A B C 0 X

19. True statement: 2(M + N) is even False statement: If H is odd then N is odd A B C D X

20. True statement: If H is odd then N is even False statement: M + H is odd A B C D X 224

ML - 4 Possibilities: A - M ■ 1, N is odd B - M - 1, N is even C - M » 2, N is odd D - H - 1, N is even

21. True statement: MXN + M + N is odd False statement: M is even if and only if N is odd A B C D X

22. True statement: If M is even then N is odd False statement: 2M + N is odd A B C D X

23. True statement: M + N is odd False statement: M is odd and N is even A B C 0 X

24. True statement: M x N is even False statement: M is even A B C 0 X

25. True statement: N(M + N) is even False statement: M is odd A B C D X

26. True statement: If M is even then N is even False statement: M + N is even A B C 0 X

27. True statement: 2{M + N) is even False statement: M is even or N is even A B C D X

28. True statement: N(M + N) is even False statement: N is even A B C D X 225

Segment 4: Statement Differences Test - Form MM

MM - 2 Possibilities: A - M * 1, N is odd B • M ■ It N is even C - H ■ 2, N is odd D - M - 2, N is even

1. True statement: M(M + N) is even. False statement: 2N + M is even A B C D X

2. True statement: N{M + N) is even False statement: M + N is even A B C 0 X

3. True statement: 2M + N is odd False statement: N(M + N) is odd A B C D X

4. True statement: 2 (M + N) is even False statement: N(M + N) is even A B C D X

5. True statement: M x N is even False statement: N is even A B C D X

6. True statement: 2{M + N) is even False statement: MxN + M + N is odd ^ A B C D X

7. True statement: M(M + N) is even False statement: M + N is even A B C 0 X

8. True starement: 2M + n is even False statement: MxN + M + N is even A B C D X

9. True statement: M + N is even False statement: MxN is odd A B C 0 X

10. True statement: M + N is odd False statement: N{M + N) is odd A B C O X

11. True statement: MxN + M + N is odd False statementi 2N + M is odd A B C D X 226

MM - 3 Possibilities: A • K * 1, B i» odd B - M * 1, N 1« even C - M - 2, N is odd D - H ■ 2, N is even

12. True statement: M x II is even False statement: 2M + M is even A B C D X

13. True statement: ' + H + 3 Is odd False statement: M + H is odd A B C D X

14. True statement: 2N + M is even False statement: MxN + M + N is even A B C D X

15. True statement: M + N is even False statement: MxN + M + N is even A B C D X

15. True statement: 2N + M is odd False statement: M x N is odd A B C D X

17. True statement: M x N is even False statement: 2N + M is even A B C 0 X

IS. True statement: M is even or N is odd False statement: 2M + N is odd A B C D X

19. True statement: 2(M + N) is even False statement; M(M + H) is even A B C D X

20. True statement: if H is odd then N is even False statement; M + N is odd A B C D X

21. True statement: M x N is even False statement: M + N is odd A B C 0 X 227

MM - 4 Possibilities: A - M - 1, N is odd B - M - 1* N is even C - M - 2, N is odd D - M - 2, N is even

22. True statement: If H is even then N is odd False statement: 2M + N is odd A B C D X

23. True statement: M + H is odd False statement: M(M + N) is odd A B C D X

24. True statement: MxN is even False statement: M is even A B C 0 X

25. True statement: M(M + N) is even False statement: 2M + N is odd A B C D X

26. True statement: If M is even then N is even False statement: M + N is even A B C 0 X

27. True statement: 2(M + N) is even False statement; M x N is even A B C D X

18. True statement: M(H + N) is even False statement: 2M + N is even A B C D X 228

Segment 5: Directions for completing test packages

The purpose of this questionnaire is to help determine which of several methods of presenting the same information about even and odd numbers is most easily understood.

Please work as rapidly as you can and try to answer all the questions. There are Zty> problems.

Tou will be asked to do several different types of problems. In the problems If and N are counting numbers* M is either 1 or 2, and S is either odd or even. Thus there are four possibilities for H and S as shown below,

Possibilities: A: K =* 1. N is odd B: M ■ 1, R is even C: H * 2, K la odd D* M =* 2, N is even

Ton should use this information for all problems.

THABK TOT FOR TOUR HELP! 229 Segment 5: Sorting for Mathematical Statements (Test M)

Possibilities: A: PI - 1, !f ia odd B: M ■ 1, FT is even Ci K ■ 2. N is odd D: M m 2, If is ewen

In each of the problems below read the statement carefully. Then look at each possibility and decide whether it could be true when the statement is true. Circle all possibilities which could be true when the statement is true.

1. M x 9 la odd. A B C D

2. M + 9 is even. A B CD

5. M(M + 9) ia even. A B c 0

4. (M + 1)(5 + 1) is even. A B c D

5. N(K + 9) is odd. A B c 0

6. M + 9 is Odd. A B c D

7. H z N ia even. A B c D 8. 9(H + 9) la even* A B c D 230

Segment 5 Sorting for Logical Statements (Test L)

Possibilities: A: M 1, Jf is odd B: H 1, H is t m m C: M 2, If is odd D: X 2, H is « m

In each of the problems below reed the statement carefully. Then look at each possibility and decide whether it could be true when the statement is true. Circle all possibilities which could be true when the statement is true.

1. H is odd or IF is even. B 0 D

2. M is odd if and only if V is even. B CB

3. X is even and N is odd. B C D

4* If H is odd then N is odd. B C 2>

5. K is odd or IF is odd. B CD

6. If 5 la odd then If is even. B CD

7. N is odd and N is odd. B C D

8. X is even if and only if X is even. B 0 V 231 Segment 5 i Translation from Mathematical to Logical Statements (Test M*L)

Possibilities: M * 1, H Is odd. M ~ 1, If la even M - 2, N is odd H = 2, ft ia even

In each of these problems you are givan a major statement and four choices. Read the major statement carefully; then read the choices. Circle the oholce vfaieh is the best description of the solutions to the major statement.

1. M + If is odd. A) K is odd and R is even. B) M is odd or R is odd. C) M is odd If and only if ft is even. D) ft is odd.

2. M(M + ft) is even* A) Kis even or H is odd. B) Mis even. C) Mis even or ft is even. D) Mis odd if and only if ft ia odd.

3. H x ft is even. k) ft is even. B) If K is odd then ft is even. C) If K is odd then ft is odd. D) M is odd if and only if ft is odd.

4. WXft + M + ft is even. A) K is even. B) K is even and ft la odd. C) ft is even. D) M is even and ft is even. 232

Possibilities* M - 1, N la odd H * 1, V ia even M - 2, N Is odd X * 2, N li even

5. N(M + N) is odd. A) M is even end H is odd. B) M is odd 1 f sad only if V is even. C) M is odd or N is odd. D) N is odd.

6. (M + 1)(N + 1) is area. A) K Is odd. B) N ia odd. C) M is odd or B is odd. D) H is odd or N is even.

7. M +■ K is even. A) 9 is odd. B) M is odd and IT is even. C) H is even or V Is odd. D) H is odd if sad only if B is odd.

8. JT(M + N) is even. A) H is odd if sad only if W is odd. B) If M is odd then H is even. C) If H is odd then M isodd. D) IT is even. 233 Segment 5: Translation from Logical to Mathematical Statements (Test L*M)

Possibilities: Mol, N is odd W m Xt N Is even H » 2, S’ Is odd H = 2, N is even

In each of thsss problems you are given a major statement and four choices. Read the major statement carefully; then read the choices* Circle the choice vhich Is the best mathematical translation of the major statement.

1. M Is odd and N is odd. A) 2N + M is odd. B) IT CM + S) is odd. . C) M + N is odd. S) M i N le odd.

2. M iseven If and only if K Is odd. A) KxH + M + H is odd. B) M + R is odd. C) M(M + N) is odd. D) 5 la odd.

5. If M ia even then IT Is odd. A) MxN + M + N la odd. B) K is odd. C) N(M ♦ H) is even. D) 2IT + M la odd.

A. M la odd or H ia even. A) MxN + M + N ia odd. B) K + IT is even. C) Jf(H + If) la even. D) V ia even. 234

5. If M ia odd than IT la odd. A) H + K la avail. B) M(M + IT) la even. C) 27 +■ M la even. D) Hllla odd.

6. M is odd if and only if 7 la odd, A) H x V is odd. B) K(M + N) ia even. C) 7 is odd. D) M + 7 is even.

7. M is even or IT is even. A) 27 + K la even. B) H x K ia even. C) V(M + 7) is even. D) H + 7 ia even.

8. K is odd and 7 is even. A) K + 7 Is odd. B) It X 7 is odd. C) H la odd. D) W<7 + M) la odd. 235

Segment 5: Selecting Logical Statements £ ° r Sortings (Test L~rT

Follow these directions for the next B problems. The four possibilities for M and IT are shown belovi

M - 1 H is odd N is even H is Odd

In each problem these possibilities are divided Into two sets, s “true set** sad a "false set,* and you are given four state­ ments. Examine the true set sad false set carefully* Then select the statement which 1s true for all the possibilities In the true set, and false for all the possibilities In the false set. Circle the letter which indicates your choice.

Is odd S la odd IT is even

True set False set

A. M is odd B* N is even C. H is odd D. H is even

The correct answer for this problem is C. 236

A. N is even B. M isodd if and only if K is even C. H is even and N is odd D. H is evsn or N is even

K > 1 M - 2 M - 2 ia odd F is even F is odd K is even

Trae set False set

A , N is odd B; K is odd and N is erven C. K is odd if and only if N Is odd D. K is odd and F is odd

F la odd N is F Is odd

A. M Is even or F is even B, K la even 0 . M is odd or 1 is even V. H is even If sad only if V is even 237

N is odd IT is odd

True set False set

A. M is oddif and only if K is even B. M is odd end IT ia odd C* M is odd and IT Is even D. K is even

Is odd N is even n—V H is odd nJ

False set

A* If N is odd then M is odd B* V is odd if and only if His odd C* If V is even then M is odd D* M is odd

K isH is odd K isH IT is

A. V is odd B. K Is odd and H ia odd 0* II la even or V la odd D* K la odd If andonly if If is odd 238

ft is odd is odd I ft is even

True set

A* M is odd if and only if ft is odd B; M ia odd or ft is odd C. Mis even or ft ia odd D. ft is odd

ft is odd ft is odd

True set

A. If ft ia odd then M is odd B. If ft is erwn then ft is odd C. M is odd D. ft is odd

/ / / 239

Segment 5; Selecting Mathematical Statements ?or Sortings (Test M M

Follow these directions for the next 8 problems* The four possibilities for M and IT are shown below:

H - 1 M * 1 M - 2 M *■ 2 B is odd N is even H is Odd K is even

In each problem these possibilities are divided Into two seta, & "true set** and a "false set," and you are given four state­ ments. Examine the true set and false aet carefully. Then select the statement which is true for all the possibilities in the true set. and false for all the possibilities In the false set. Circle the letter which indicates your choice.

Example: f |m - i M « 2 I ir is odd V is odd is

True set

A. K is odd B. K is even C. V is Odd P. B Is erres

The correct answer for this problem is C. 240

V 1b area IT Is Odd N la odd

'Pro# sot Falsa sat

A. Mxtf + M + N Is odd B. M + N Is odd C« M(M + tr) Is odd B* H Is odd

M - 1 5 Is odd V Is evan IT is odd

Trua set Falsa sat

A. 21f 4- H Is odd B. N(IT + M) Is odd C. M + H la odd P. H x N la odd

M m 1 V Is i Is odd H Is

Falsa sat

Km 2H + M is area B. Hxlla sran C. N(M + V) is sran P. M + If is sran 241

H is odd

A* X is even B. H + U ia odd C. M(H + N) is odd D. MxN + M + X is even

M - 1 IT is < X is odd

Trus sat Falsa set

A* X(M + X) is area B. If + H is even C. K x N is even V, 2JT + M is odd

isX is odd X is isX

A. MxH + M + X is m s B. X(K + V) is svstt 0* X is avsn D» R + X is even 242

K ia odd ST ia odd N ia

Falaa sat

JU H + If is aaan B. K(M + If) la even C. 2R + M la eran D. M x If is s v «

X la odd N ia even H is odd

Tru* sot Falsa sat

A. MxR + M + V ia odd B. H la odd C. N(H + N) is araa 9. 2K + M is odd 243

Segment 6: Classroom Logic Test (Form A) c l t - A

CIASSROOM LOGIC TEST

Directions. Imagine that your class has just taken a special test for a state-wide math contest, and that you are now leading a class discussion of the problems on the test. As each student presents his interpretations and ideas concerning various problems you want to provide appropriate feedback to him or her. For each item Read the student's statement carefully and decide whether the argument is correct or incorrect. Place a check (y / l which indicates your judgment concerning the student's argument. If you check correct, go on to the next question. If you check incorrect, read the four options, and place a check (y) beside the one you would show tne student in order to point out his error to him. Work the Sample items below: A. Jackie: If a number is even, then the remainder is 0 when you divide by 2. In this problem the number is even, and we're dividing by 2, so the remainder is 0. Jackie is correct Jackie is incorrect; I would show her _____ An even number with remainder 0 when divided by 2. An even number with remainder 1 when divided by 2. An odd number with remainder 0 when divided by 2. An odd number with remainder 1 when divided by 2. B. Tim: in this problem the number A is either 0, 1, or 2, and B is bigger than C. Therefore AB is bigger than AC. _ _ _ Tim is correct. Tim is incorrect. I would show him A, B, and C with B greater than C, AB greater than AC B greater than C, AB equal to AC _____ B less than or equal to C, AB greater than AC. B less than or equal to C, AB equal to AC

Now turn to the next page and check your answers to the sample items. 244

CLT - A - 2

Answers to the sample items:

A. Jackie: If a number is even, then the remainder is 0 when you divide by 2. In this problem the number is even, and we’re dividing by 2 , so the remainder is 0. ✓ Jackie is correct _____ Jackie is incorrect; I would show her _____ An even number with remainder 0 when divided by 2. _____ An even number with remainder 1 when divided by 2. _____ An odd number with remainder 0 when divided by 2. An odd number with remainder 1 when divided by 2.

B. Tim: In this problem the number A is either 0, 1, or 2, and B is bigger than C. Therefore AB is bigger than AC. _____ Tim is correct ^ Tim is incorrect; I would show him A, B, and C with B greater than C, AB greater than AC B greater than C, AB equal to AC B less than or equal to C, AB greater than AC _____ B less than or equal to C, AB equal to AC

(Note that Tim is incorrect because 0 times any number is 0)

Now turn to the next page and do the rest of the problems. Remember that you are to decide whether the student's arguments are correct or incorrect. 245

CLT - A - 3

1. Betsy: If p is even or q is even, then p times q is even. In this problem is says p times q is an odd number. So we know p is not even and q is not even. Betsy is correct Betsy is incorrect; I would show her numbers p and q with _____ p even, q even p even, q odd p odd, q even p odd, q odd

2. Bill: It says A and B are different sets. I think A is not a subset of B and B is not a subset of A. Because if A is a subset of B and B is a subset of A, then A and B have to be equal. Bill is correct Bill is incorrect; I would show him sets A and B with A a subset of B, A unequal to B A a subset of B, A equal to B A not a subset of B, A unequal to B A not a subset of B, A equal to B

3. Sarah: You told us that if a computation is correct, we will get 0 when we cast our nines. Look— >1 get 0 when I cast out nines, so my answer is correct. Sarah is correct Sarah is incorrect; I would show her A correct caegrutatlon with 0 for casting out 9's A correct computation without 0 for casting out 9‘s _____ An incorrect computation with 0 for casting out 9's _ _ _ An incorrect computation without 0 for casting out 9*s 246

CLT - A - 4

4. Johnny: You told us that if a 2-digit number is prime it has to be odd. There are 45 odd 2-digit numbers* so there have to be 45 2-digit primes. Johnny is correct Johnny is incorrect; I would show him A two-digit prime that isn't odd _ _ _ _ A two-digit prime that is odd ______A two-digit odd number that isn't prime A two-digit odd number that is prime

5. Frank: He know that if b is positive, then -10-b is negative. The test says -10-b i£ negative, so b has to be positive. Frank is correct . Frank is incorrect; I would show him a number b with b positive and -10-b positive b positive and -10-b negative b negative and -10-b positive b negative and -10-b negative

6. Joan: If two squares have the same area, then their bases have the same length. So if you have two squares and they have different bases they can’t have the same area. Joan is correct Joan is incorrect; I would show her two squares with same base and same area same base and different areas different bases and same area different bases and different areas 247

CLT - A - 5

7. Lisa: In this problem we know that c times b is less than 1. So we must have c less than 1 or b less than 1. The problem says b is less than 1. That means c is either 1 or greater than 1. Lisa is correct Lisa is incorrect; I would show her b (less than 1) and c with _ _ _ _ c less than 1, be less than 1 c less than 1, be greater than 1 e greater than 1, be less than 1 c greater than 1, be greater than 1

8. Annei A and B are finite sets. A is a proper subset of B means that B has more elements than A. In this problem B does have more elements than A, so A is a subset of B. Anne is correct Anne is incorrect; I would show her sets A and B with A a subset of B, B has more elements than A _ _ _ _ A a subset of B, A has more elements than B A not a subset of B, 8 has more elements than A _ _ _ A not a subset of B, A has more elements than B

9. Kevin: If the product of two integers p and q is prime, then p ■ 1 or q ■ 1. In this problem q is not 1, so pq is not prime. — Kevin is correct. Xevin is incorrect; I would show him p and q with q ■ 1, pq prime q ^ 1, pq prime q * 1, pq not prime

q j 1 1, pq not prime 248

CLT - A - 6

10. Sam; If a Is even, then a+2b has to be even. In this prob­ lem a+2b is odd, so a has to be odd. _____ Sam is correct

Sam is incorrect; I would show him numbers a and b with a even, a+2b even a even, 4.+2b odd a odd, a+2b even a odd, 2+2b odd

11. Mary: if two triangles are congruent they have the same area, so to tell if these two triangles are congruent we can check the areas and see if they are the same. Mary is correct Mary is incorrect; I would show her _____ Two congruent - triangles with the same area _ _ _ _ Two congruent triangles with different areas Two noncongruent triangles with the samearea Two noncongruent triangles with different areas

12. Tom; Since p time* q is a multiple of 3, we have p is a multiple of 3 or q is a multiple of 3. I just showed that p is not a multiple of 3, so q has to be a multiple of 3. _____ Torn is correct Tom is incorrect; I would show him p and q, with pxq a multiple of 3 and p a multiple of 3, q a multiple of 3 p a multiple of 3, q not a multiple of 3 p not a multiple of 3, q a multiple of 3 p not a multiple of 3, q not a multiple of 3 249

CLT - A - 7

13. Martha: If two rectangles have the sane base and height they have to have the same areas. But these rectangles have different bases and different heights, so they have to have different areas. Martha is correct Martha is incorrect; I would show her rectangles with _____ equal bases, equal heights, equal areas _____ equal bases, equal heights, unequal areas unequal bases, unequal heights, equal areas unequal bases, unequal heights, unequal areas

14. Josh; The cancellation property says that if a jt 0 and ab - ac, then b - c. In this problem, 4b ■ 4c - 0, so b - c Josh is correct Josh is incorrect; I would show him b and c with 4b • 4c, b ■ c

4b “ 4c, b / c

_____ 4b 7* 4c, b ** c

_____ 4b jt 4c, b j* c

15. Jim: I figured out that if you take two numbers, m and n,

both bigger than 1 ,0 0 0 , and you multiply them your answer will be more than a million. So I say thatsince m times

n is more than 1 ,0 0 0 , 0 0 0 in this problem, m and n have to

both be over 1 0 0 0 . Jim is correct Jim is incorrect; I would show him m and n with

m 1 0 0 0 , n 1 0 0 0 , mxn 1 ,0 0 0 , 0 0 0

m 1 0 0 0 , n 1 0 0 0 , mxn 1 ,0 0 0 , 0 0 0

_____ m 1 0 0 0 , n 1 0 0 0 , mxn 1 ,0 0 0 , 0 0 0

m 1 0 0 0 , n 1 0 0 0 , mxn 1 ,0 0 0 , 0 0 0 250

CLT - A - 8

16. Rhonda: We know that if a ia less than b and c less than d, then ac is less than bd. In this problem ac is 4SO and bd is 500. So we have to have a less than b or c less than d. Rhonda is correct Rhonda is incorrect; I would show her a,b,c,d with ac • 450, bd " 500, and a greater than b, c greater than d _____ a greater than b, c less than d _____ a les3 than b, c greater than d _____ a less than b, c less than d

17. Herb: If p divides q, then p divides q2. In this problem 2 P is 12 and p does not divide q— so it can't divide q . Herb is correct _ Herb is incorrect; I would show him a number q with

12 divides q., 12 divides q 2

12 divides q, 12 does not divide q 2

12 does not divide q, 12 does not divide q2 2 12 does not divide q, 12 divides q

18. Pete: You calculate both the area and the perimeter of a rectangle from the same numbers. So if two rectangles have the same area they also have the same perimeter. Pete is correct Pete is incorrect; I would show him two rectangles with Santarea, same perimeter Same area, different perimeters Different areas, same perimeter Different areas, different perimeters 251

Segment 6; Classroom Logic Test (Form B) CLT - B

CLASSROOM LOGIC TEST

Directions. Imagine that your class has just taken a special test for a statewide math contest, and that you are now leading a class discussion cf the problems on the test. As each student presents his interpretations and ideas concerning various prob­ lems you want to provide appropriate feedback to hin or her. For each item: Read the student*s argument carefully and decide whether the argument is correct or incorrect. Place a check (vO to indicate your judgment concerning the student'sargument. If you check correct go on to the next question. If you check incorrect, write a brief statement outlining what you would tell or show the student in order to point out his error to him.

Work the sample items below:

A. Jackie: If a number is even, then the remainder-is 0 when you divide by 2. In this problem the number is even, and we're dividing by 2 , so the remainder is 0 . Jackie is correct. Jackie is incorrect. I would show her ______

B. Tim: In this problem the number A is either 0, 1, or 2, and B is bigger than C. Therefore AB is bigger than AC. Tim is correct. _ Tim is incorrect; I would show him numbers A, B, and C with ______

Now turn to the next page and check your answers to the sample items. 252

CLT - B - 2

Answers to sample items:

A. Jackie: If a number is even, then the remainder is 0 when you divide by 2. In this problem the number is even, and we're dividing by 2 , so the remainder is 0 .

\/ Jackie is correct Jackie is incorrect; I would show her ______

B. Tim: In this problem the number A is either 0, 1, or 2, and B is bigger than C. Therefore AB is bigger than AC. ____ Tim is correct Tim is incorrect; I would show him numbers A, B, and C with A - O. R > C . r AB~AC*o

Now turn to the next page and do the rest of the problems. Kemember that you want to decide whether the student's arguments are correct or incorrect. 253

CLT - B - 3

1. Betsy: If p is even or q is even, then p times q is even. In this problem it says that p times q is an odd number. So we know p is not even and q is not even. Betsy is correct _ _ _ _ Betsy is incorrect; I would show her numbers p and q with ______

2. Bill: It says A and B are different sets. I think A is not a subset of B and B is not a subset of A. Because if A is a subset of B and B is a subset of A, then A and B have to be equal. _ _ _ _ Bill is correct Bill is incorrect; I would show him sets A and B with______

3. Sarah: You told us that if a computation is correct, we will get 0 when we cast out nines. Look — I get 0 when I cast out nines, so my answer is correct. Sarah is correct Sarah is incorrect; I would show her ______254

CLT - B - 4

4. Johnny: You told us that if a 2-digit number is prime it has to be odd. There are 45 odd 2-digit numbers, so there have to be 45 2-digit primes. Johnny is correct _ _ _ Johnny is incorrect; I would show him ______

5. Frank: We know that if b is positive, then -10-b is negative. The test says -10-b is negative, so b has to be positive. Frank is correct Frank is incorrect; I would show him ______

6 . Joan: If two squares have the same area, then their bases have the same length. So if you have two squares and they have different bases, they can't have the same area. Joan is correct ____ Joan is incorrect. I would show her two squares with ______255

CLT - B - 5

7. Lisa: In this problem we know that c times b is less than 1. So we must have c less than 1 or b less than 1. The

problem says b is less than 1 ; that means e is either 1

or greater than 1 . Lisa is correct Lisa is incorrect; I would show her a number b

{less than 1 ) and c with ______

S. Anne; A and B are finite sets. A is a proper subset of B means that B has more elements than A. In this problem B does have more elements than A, so A is a subset of B. Anne is correct Anne is incorrect; I would show her sets A and B

with ______

9. Kevin: If the product of two integers p and q Is prime, than p <• 1 or q ■ 1. In this problem q is not 1, so pq is not prime. Kevin is correct Kevin is incorrect; I would show him ______256

CLT - B - 6

10. Sam: If a Is even, then a+2b has to be even. In this

problem a+2b is odd, so a has to be odd. Sam is correct Sam is incorrect; I would show him numbers a and b with

11. Mary: If two triangles are congruent they have the same area, so to tell if these two triangles are conavuent we can check the areas and see if they are the same. Mary is correct Mary is incorrect; I would show her ^ ___

12. Tom; Since p x q is a multiple of 3, we have p is a multiple of 3 or q is a multiple of 3. I just showed that p is not a multiple of 3, so q has to be a multiple of 3. Tom is correct Tom is incorrect; I would show him p and q with

p x q a multiple of 3 and . 257

CLT - B - 7

13. Martha: If two rectangles have the sane base and height they have to have the sane area. But these rectangles so have different bases and different heights, andAthey have to have different areas. Martha is correct Martha is incorrect; 1 would show her rectangles with

14. Josh: The cancellation property says that if a j* 0 and ab ■ ac, then b ■ c. In this problem 4b and 4c are both

0 , so b ■ c. Josh is correct Josh is incorrect; I would show him b and c with

IS. Jim: I figured out that if you take two numbers, m and n,

both bigger than 1 0 0 0 , and you multiply them your answer will be more than a million. So I say that since m time

n is more than 1 ,0 0 0 , 0 0 0 in this problem, m and a have

to both be over 1 0 0 0 . Jim is correct Jim is incorrect; I would show him m and n with 258

CLT - B - 8

16. Rhondai We know that if a is lass than b and c is leas than d, then ac is less than bd. in this problem ac is 450 and bd is 500. So we have a less than b or c less than d. _ _ _ _ Rhonda is correct _____ Rhonda is incorrect} X would show her a, b, c, d with ac « 450, bd - 500, and ______

2 17. Herbt If p divides q, then p divides q . In this problem p is 12 and p does not divide q — so it can't divide q2. Herb is correct ____ Herb is incorrect; I would show him a number q with.

18. Pete; You calculate both the area and the perimeter of a rectangle from the same numbers. So if two rectangles have the same area they also have the same perimeter. Pete is correct Pete is incorrect; I would show him two rectangles with ______259

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