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Grade placement of symbolic logic Grant, Douglas Robin 1961

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GRADE  PLACEMENT  SYMBOLIC  OF  LOGIC  by DOUGLAS ROBIN  GRANT  B.A., U n i v e r s i t y of B r i t i s h Columbia, 1948  A  THESIS THE  S U B M I T T E D IN P A R T I A L REQUIREMENTS  FOR  MASTER. OF  FULFILMENT OF  THE DEGREE  OF  ARTS  in t h e College of EDUCATION  We  accept this thesis as conforming t o t h e  required standard  THE  UNIVERSITY  OF  BRITISH  O c t o b e r , 1961  COLUMBIA  In p r e s e n t i n g  t h i s thesis i n p a r t i a l fulfilment- of  the requirements f o r an advanced degree a t t h e U n i v e r s i t y British  Columbia, I agree t h a t the  a v a i l a b l e f o r reference  and  study.  of  L i b r a r y s h a l l make i t f r e e l y I f u r t h e r agree t h a t  permission  f o r e x t e n s i v e copying o f t h i s t h e s i s f o r s c h o l a r l y purposes may g r a n t e d by  the  Head o f my  Department o r by h i s  be  representatives.  It i s understood t h a t copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain  s h a l l not  be  alloived w i t h o u t my  College Depapt»e*>t o f E d u c a t i o n The U n i v e r s i t y o f B r i t i s h Vancouver 8, Canada.  Date  October, 1961.  Columbia,  written  permission.  ABSTRACT  T h i s study was designed t o determine  the e f f e c t i v e -  ness o f teaching symbolic logic in t h e high school.  Three  hundred eighty—seven students enrolled on the U n i v e r s i t y  Pro-  gramme in grades nine t o t h i r t e e n a t Como L a k e High School, in School D i s t r i c t No. 43 (Coquitlam), took p a r t in the investigation.  T h e students were grouped according t o the  mathematics course they were studying. A n s w e r s were sought  t o two specific questions. :  Do  significant d i f f e r e n c e s exist between t h e means o f the final t e s t s c o r e s o f t h e students in each of t h e groups?  A t which  grade levels can this material be e f f e c t i v e l y mastered?  As a  c r i t e r i o n f o r determining t h i s , 75 per cent o f t h e students at a particular level were required t o obtain a s c o r e o f 50 per cent o r b e t t e r on the final t e s t .  In order t o answer  the f i r s t question, the r e s u l t s were studied by analysis o f covariance with scholastic aptitude being the variable controlled. The  answer t o t h e second  question was obtained by comparing  the performance o f each group with the standard outlined. O n the basis of this information, decisions were made r e g a r d ing the suitability of the material f o r t h e various grade levels. A l l o f the d i f f e r e n c e s between t h e means were found to be significant a t the one per cent level.  T h e highest mean  score was  obtained by the students in Mathematics  followed in order by those o f Mathematics 91, 30» 10.  The  101, 20, and  students of Mathematics 101, 91 and 30 s a t i s f i e d  the requirement  t h a t 75 per cent should obtain a score of  50 per cent o r b e t t e r on the final t e s t .  The students o f  Mathematics 20 and 10 failed t o s a t i s f y this  requirement.  T A B L E  OF  CONTENTS  CHAPTER I.  PAGE  THE  PROBLEM  1  Introduction Statement  1 of  the  Problem  Justification  for  Definition  Terms  of  the  2  Study.  1+ 9  Summary II. I I I .  SURVEY THE  12 OF  THE  EXPERIMENTAL  Content  of  the  Description Mastery  of  Selection  AND  T E S T S .  Unit the  of  of  25  Lessons  26 28  the  the  25  Final  Test  29  items  29  Reliability  30  Validity  31  METHOD The  OF  STUDY  Subjects  Experimental Statistical V.  UNIT  13  Questions  Construction  IV.  L I T E R A T U R E  A N A L Y S I S  . Procedure  Treatment OF  35  R E S U L T S  .  .  35 37 i+1 2+6  CHAPTER VI.  PAGE  SUMMARY  AND  CONCLUSIONS  52  Summary  52  Conclusions  5k  Major conclusions  5k  Discussion  5k  L i m i t a t i o n s o f the S t u d y  56  Problems f o r F u r t h e r S t u d y  57  BIBLIOGRAPHY  58  APPENDIX  A.  L e s s o n Plans  62  APPENDIX  B.  Exercises  75  APPENDIX  C.  M a s t e r y Questions .  83  APPENDIX  D.  Final T e s t  98  APPENDIX  E.  Reliability o f Final T e s t  APPENDIX  F.  Raw  Score Data  109 I l l  LIST  OF  TABLES  TABLE I. II. III.  PAGE Content of the Lessons C o n t e n t Analysis o f the T e s t Topics f r o m the  IV. V.  27  Symbolic Logic  Items . . .  Involved in  Solution o f E a c h Item  33  Grouping of S t u d e n t s f o r I n s t r u c t i o n  . .  Significance of D i f f e r e n c e s  37 between Means  f o r Intelligence VII. VIII. IX. X.  Methods f o r Assigning  i+2 L e t t e r Grades  . .  Analysis o f Covariance . A d j u s t e d C r i t e r i o n Means Significance of D i f f e r e n c e s  XII.  2+8 between Means 48  Distributions o f S c o r e s Intelligence Quotients and L o g i c  2+9 Test  S c o r e s o f Mathematics 101 S t u d e n t s . . XIII.  Intelligence Quotients and L o g i c  Intelligence Quotients and Logic  112  Test  S c o r e s o f Mathematics 91 S t u d e n t s XIV.  2+2+ 2+7  f o r Logic XI.  35  D e s c r i p t i o n o f S t u d e n t s Grouped A c c o r d i n g to Mathematics C o u r s e  VI.  32  . .  113  . .  112+  Test  S c o r e s o f Mathematics 30 S t u d e n t s  VII  TABLE XV.  PAGE Intelligence Quotients Scores  XVI .  Test  of Mathematics 20 S t u d e n t s  Intelligence Quotients Scores  and L o g i c  and L o g i c  OF  . .  116  FIGURES  FIGURE 1.  115  Test  o f Mathematics 10 S t u d e n t s  LIST  . .  PAGE Distribution of Scores  by G r a d e s  51  CHAPTER  THE  I.  I  PROBLEM  INTRODUCTION  In the past f e w years much has been w r i t t e n indicating d i s s a t i s f a c t i o n with the teaching o f high school mathematics  and with the content o f the traditional high school  mathematics program. and  While other subjects such as physics  chemistry have kept up with the developments in their  respective f i e l d s , no significant changes have taken place in the high school mathematics curriculum.  Although  great  pro-  gress has been made during the past sixty years in t h e development o f new mathematics, t h e mathematics curricula o f the 1950's contained essentially the same mathematics as did those o f the 1900's.  Some o f the newer t o p i c s ,  however,  have made p a r t s o f the traditional course less important they once were. far  Electronic  than—  computers, f o r instance, are  superior t o logarithms as a calculating aid and hence have  lessened the importance o f logarithms as such.  With  progress  in the behavioural sciences, newer and more e f f i c i e n t mathematical theories of s t a t i s t i c s and probability have been  developed.  Until about t h i r t y years ago the field o f mathematical ' logic  2 was generally of classical i n t e r e s t only.  Now,  t h e r e is an  A s s o c i a t i o n f o r Symbolic L o g i c which publishes a journal devoted t o r e s e a r c h in this f i e l d .  While i t is not t h e sugges-  tion t h a t all o f t h e new topics should be introduced into t h e high school, some modifications should be made in t h e c u r r i c ulum. Davis has noted The subjects which make up the whole mathematical knowledge of most people a r e a r i t h m e t i c , geometry and algebra. E v e n those who have some scientific training have o f t e n studied only these and t h e calculus. When these f o u r subjects were built into t h e c u r r i c ulum the subject, or method of analysis, called s e t theory had not y e t begun t o influence t h e development of mathematics. Thus what may have been a very sound curriculum f i f t y years ago no longer serves t h e purpose O f o r i e n t ing students in t h e use o f mathematical ideas. ^ These remarks could equally well be applied t o many o f t h e newer topics in mathematics.  II.  The  STATEMENT  OF  THE  PROBLEM  problem o f this study is t o determine t h e  e f f e c t i v e n e s s o f teaching a unit on symbolic  logic in t h e high  R o b e r t L . Davis ( e d . ) , E l e m e n t a r y Mathematics o f Sets (Ann A r b o r , Michigan: Mathematical A s s o c i a t i o n of A m e r i c a , 1958), p. 1.  school. For  p r a c t i c a l considerations, i t was necessary t o limit  the amount o f material t o be included in the experimental The  unit.  following topics were included: the basic operations o f  conjunction, disjunction, negation,  and implication, the use o f  t r u t h t a b l e s , algebra o f logic, applications t o electrical  net-  works, derived implications and the use o f q u a n t i f i e r s .  The  emphasis was on language and notation. chapters  o f t h r e e college t e x t s  ' '  A  study of the f i r s t  shows t h a t this i s , gen-  erally, the material which is included when symbolic taught t o students  logic is  in f i r s t year mathematics courses a t univ-  ersities . The  investigation included students  U n i v e r s i t y Programme Como L a k e High School The  students  enrolled on the  in each o f grades nine t o t h i r t e e n a t in School D i s t r i c t No. i+3  (Coquitlam).  were grouped according t o the mathematics  C a r l B . A l l e n d o e r f e r and C l e t u s O. Oakley, Principles o f Mathematics (New Y o r k : M c G r a w - H i l l Book Company, 1955). -'John G. Kemeny, U . L a u r i e Snell and G e r a l d L . Thompson, Introduction t o F i n i t e Mathematics (New Y o r k : P r e n t i c e - H a l l , I n c . , 1957). ^"Robert R. C h r i s t i a n , Introduction t o L o g i c and S e t s (preliminary edition; B o s t o n : Ginn and Co., 1958).  4 course they were studying. Using these materials, with these subjects, the f o l lowing specific problems were investigated: 1.  A r e t h e r e significant d i f f e r e n c e s in the means of the five groups (Math. 1 0 1 , 9 1 , 3 0 , 2 0 , 1 0 ) ?  2.  A t which grade levels can this material be e f f e c t i v e l y mastered?  A s a criterion f o r  determining t h i s , 7 5 per cent o f t h e students at a particular level were required t o obtain a score of 5 0 per cent or b e t t e r on a m a s t e r y t e s t of symbolic logic. III.  The  JUSTIFICATION  FOR  THE  STUDY  l i t e r a t u r e o f t h e past five years indicates t h a t  a major revision in our high school mathematics curriculum is needed.  Many groups in the United S t a t e s and Canada are  aware o f the problem and a r e attempting t o solve i t .  Although  many committees a r e working on experimental courses and p r o grams, t h e r e appear t o have been f e w , i f any, rigorous experiments  conducted  on the r e s u l t s of these new c o u r s e s .  Since  educational authorities in this province a r e contemplating a revision of the curriculum, some objective information should be available.  5 A s a guide in this undertaking, Rourke^ o f f e r s what he believes are f i v e good, sound reasons f o r using c e r t a i n ideas of  modern mathematics in building a high school mathematics  curriculum. ing,  He feels t h a t we  (2) simplify, (3) unify,  introduce important new One  need t o : (1) clarify our t e a c h -  (4) broaden old ideas, and  (5)  ideas.  must recognize t h a t with the increasing use being  made of devices such as electronic computers, less emphasis may  be placed on such topics as mental arithmetic or the solv-  ing of complicated numerical problems in t r i g o n o m e t r y by means of logarithms.  However, the students need t o know the p r i n -  ciples underlying the operation of these devices.  A s the fields  of probability and s t a t i s t i c a l inference are becoming increasingly important, so the need f o r high school students to have an understanding of the principles on which they are based i n c r e ses. to  T h e theory of probability, f o r instance, enables students  have some understanding o f the basis on which the large  insurance companies o p e r a t e .  The increasing misuse of s t a t i s -  t i c s by those t r y i n g t o s t r e n g t h e n their own  argument  makes  R o b e r t E.K.. Rourke, "Some implications o f t w e n t ieth ©enfeu-ry mathematics f o r high schools," The Mathematics T e a c h e r , L I ( F e b r u a r y , 1958), p. 74-75.  6 it important t h a t school students be i n s t r u c t e d in the principles of s t a t i s t i c a l i n f e r e n c e . Some of the newer developments in mathematics do not appeal t o the layman to the same degree t h a t probability does.  However, s e t t h e o r y , symbolic logic, geometries r a t h e r  than geometry and modern algebra all contribute t o a b e t t e r appreciation of the type of problem  suggested above.  These  are some of the topics t h a t could be introduced into a modified curriculum. Since the mathematics under discussion is intended f o r the student planning to enter university, i t is necessary t o consider the f u t u r e training he is t o receive. students who  will not in some way  matics or mathematicians.  There  are  few  require the use of mathe-  Richardson points out t h a t "mathe-  matics is the background of all scientific  (logical) subjects,  whether physical, biological, social or o t h e r w i s e . "  The  high  school must continue t o prepare students f o r university and t r y to f u l f i l the requirements  outlined by t h a t body.  Mathem-  atics departments of this and other universities are asking t h a t high school students be introduced t o modern mathematics  (rev.  Moses Richardson, Fundamentals of Mathematics, ed.; New Y o r k : The Macmillan Company, 1 9 5 8 ) , p. i+82.  7 so t h a t they will have a b e t t e r understanding of the logical s t r u c t u r e of the subject and  consequently be more adequately  prepared t o take advanced work. Indeed,  i t may  well be t h a t teaching mathematics as  a unified subject and by taking care t o develop an ing of these unifying concepts, the student may  understand-  m a s t e r the  7 manipulative skills more readily.  Beberman  r e p o r t e d t h a t the  students using the U n i v e r s i t y of Illinois Committee on School Mathematics experimental materials were making b e t t e r than average p r o g r e s s and seemed to be capable of completing f a r more subject m a t t e r than is ordinarily contained in the t r a d i t ional course.  While this is a subjective opinion, i t does  cause  one t o wonder whether the traditional course has done all t h a t it should f o r our s t u d e n t s . Robinson,  who  has expressed d i s s a t i s f a c t i o n with the  psychological shortcomings  of the traditional mathematics course,  seems t o f e e l t h a t the new  mathematics may  be a definite  improvement. I t appears t h a t we may be entering a new e r a in mathematical education and we should make every  'Max Beberman, D i r e c t o r of the U n i v e r s i t y of Illinois Committee on School Mathematics, address b e f o r e D o w e r Mainland Mathematics T e a c h e r s , Vancouver, B.C., F e b r u a r y , 1 9 5 9 .  8 attempt to follow the new outset. ®  developments f r o m  the  I f the objectives of any revision include teaching the s t r u c t u r e of mathematics or the mathematical  method of  deductive reasoning, then symbolic logic should be one of the units introduced.  Traditionally, geometry has been the main  subject by which logical reasoning has been taught.  Not all  mathematicians are of the opinion t h a t this subject is the most useful one f o r the  purpose.  Euclidean geometry is quite a complicated mathematical s y s t e m , and as presented in most textbooks is not even completely logical. ... The reason t h a t Euclidean geome t r y has been used as the prime example of logical reasoning is t h a t until r e c e n t times t h e r e was no other example to which t e a c h e r s could t u r n . 9 Symbolic turn.  logic is one of--the examples to which t e a c h e r s  A  can  series of lessons on this topic would give students  a good background f o r f u r t h e r study with other topics of modern mathematics. f e r and  The  This position is supported by Allendoer-  Oakley:  B.C.  E . G . Robinson, "Change Secondary Mathematics," T e a c h e r . X L , ( F e b r u a r y , 1961), p. 221.  C a r l B . A l l e n d o e r f e r , "Deductive Methods in M a t h ematics," Insights Into Modern Mathematics, T w e n t y - t h i r d Yearbook of The National Council of T e a c h e r s of Mathematics (Washington, D . C : N.C.T.M., 1957), p. 66. 9  9  S i n c e t h e l o g i c a l p r e e m i n e n c e o f m a t h e m a t i c s i s due to t h e peculiar n a t u r e of t h e s u b j e c t . . . i t is necess a r y t o begin t h e serious s t u d y o f m a t h e m a t i c s w i t h an e x a m i n a t i o n o f t h e logical principles which underly it.!0  A l o n g w i t h s e t t h e o r y , t h i s t o p i c helps t o e s t a b l i s h a g o o d foundation f o r a study of modern  mathematics.  S i n c e much o f t h e proposed m a t e r i a l is new  t o many  t e a c h e r s o f m a t h e m a t i c s , c u r r i c u l u m w o r k e r s have l i t t l e guidance r e g a r d i n g t h e grade p l a c e m e n t o f s p e c i f i c t o p i c s .  Research  is n e e d e d t o e s t a b l i s h t h e l o w e s t g r a d e a t w h i c h s t u d e n t s c a n master the topics  efficiently.  IV.  DEFINITION  Traditional.  OF  TERMS  F o r purposes of this study "traditional"  shall mean t h e m a t h e m a t i c s  c o u r s e in o u r high schools as i t  has e x i s t e d f o r t h e p a s t f i f t y t o one h u n d r e d y e a r s .  The  c o l l e g e p r e p a r a t o r y c u r r i c u l u m i n t h e U n i t e d S t a t e s and  Canada  has g e n e r a l l y c o n s i s t e d o f a l g e b r a in g r a d e nine, d e m o n s t r a t i v e geometry  i n g r a d e t e n , m o r e a l g e b r a i n g r a d e e l e v e n , and  onometry,  solid g e o m e t r y  - ^ A l l e n d o e r f er  trig-  and a d v a n c e d a l g e b r a i n g r a d e twelve.-'  and O a k l e y , o p . c i t . ,  p. 1 . .  l l R i c h a r d F . B r u n s and A l e x a n d e r F r a z i e r , " S e q u e n c e and R a n g e o f H i g h S c h o o l M a t h e m a t i c s , " T h e M a t h e m a t i c s T e a c h e r , L., ( D e c e m b e r , 1 9 5 7 ) , P P . 5 6 2 - 5 6 6 .  10 This is not identical, but essentially the same as t h e program followed by students in B r i t i s h Columbia,, namely: grade nine, algebra and intuitive geometry; grades t e n and eleven, the equivalent of one year each o f algebra and demonstrative geome t r y ; grade twelve, algebra and t r i g o n o m e t r y .  In the p r e s e n t  discussion, r e f e r e n c e t o the traditional course will mean a course similar t o the above. Modern.  The term  "modern" shall be used t o r e f e r t o  the new subject m a t t e r and t o a new approach.  T h e emphasis  in the modern approach is t o be placed on t h e fundamental concepts r a t h e r than on a dozen o r so isolated skills.  In  order t o do this e f f e c t i v e l y , some new topics must be t a u g h t . Meder points out the d i f f e r e n c e between modern and traditional mathematics,  and a t the same time their interdependence.  Modern mathematics is both a point o f view and new subject m a t t e r . These a r e , however, inextricably intertwined. T h e new point o f view is attained by looking a t the content o f elementa'ry mathematics in the light of the new subject m a t t e r . ^ Meder illustrates his s t a t e m e n t by saying t h a t a t e a c h e r o f modern mathematics would not leave students with the idea  ^ A l b e r t E . Meder, J r . - , "Modern Mathematics and its P l a c e in the Secondary S c h o o l , " T h e Mathematics T e a c h e r , L , ( O c t o b e r , 1 9 5 7 ) , p. 418'.  t h a t x^+y2 cannot be f a c t o r e d but would explain t h a t whereas x^-y2  is f a c t o r a b l e in the field o f rational numbers,  is not.  I t is f a c t o r a b l e , however, in the field of com-  plex numbers. The  R e p o r t of the Commission on Mathematics, in  describing the nature of contemporary mathematics s t a t e s t h a t is is characterized by (1) a tremendous development quantitatively; (2) the introduction of new content; (3) the reorganization and extension of older content; and (2+) renewed, increased, and conscious emphasis upon the view t h a t mathematics is concerned with a b s t r a c t p a t t e r n s of thought.  13'-  This may  be summed up by saying t h a t modern mathematics  r e f e r s not only to new matrices, new  material such as logic, s e t s , v e c t o r s ,  modern algebra and  number t h e o r y , but also t o a  approach to the traditional material so t h a t the  principles and  s t r u c t u r e of the subject may  underlying  become more evi-  dent . Symbolic L o g i c .  In this study "symbolic logic" r e f e r s  specifically to those aspects  of the topic which are outlined in  ^ R e p o r t of the Commission on Mathematics, P r o g r a m f o r College P r e p a r a t o r y Mathematics, ( P r i n c e t o n , N.J.: College E n t r a n c e Examination B o a r d , 1 9 5 9 ) , P^ 3»  12 the experimental u n i t . ^  No a t t e m p t is being made to'buud a  \ complete logical s y s t e m but r a t h e r the emphasis is on language and notation. V.  SUMMARY  I t is generally accepted by m; themati'cians t h a t while great progress has been made in t h e development of new mathematics during t h e past sixty y e a r s , t h e r e has been no real change in t h e content of high school mathematics courses.. During t h e past decade many suggestions have been made regarding ways t o introduce some\of these new ideas into t h e schools.  S i n c e , a t p r e s e n t , we have'little information r e g a r d -  ing t h e grade level a t which suggested  topics could be taught,  it is f e l t t h a t r e s e a r c h in this area is needed.  I t is t h e  purpose o f t h e present study t o provide some information concerning t h e grade placement o f a unit on symbolic  covered  logic.  L e s s o n plans indicating specifically t h e work t o be in t h e experimental unit are included in Appendix A .  CHAPTER  SURVEY The ematics  OF  THE  II  LITERATURE  l i t e r a t u r e dealing with curriculum revision in math-  contains much information regarding proposed  courses  in which certain topics of modern mathematics are included. Although  many groups are t r y i n g revised courses with experi-  mental classes, t h e r e is little evidence of r e s e a r c h related to the grade placement of these t o p i c s . type, however, have been conducted  Studies of the present in other fields and  grade placement of a topic has been determined  the  by objective  •means. In general, i t is not just a change in content t h a t is being sought in these revisions, although this does happen necessarily, but r a t h e r a change in the whole approach to mathematics.  M o s t of the proposed revisions a t t e m p t to i n -  culcate in the students an understanding mathematics.  of the principles of  In d i r e c t c o n t r a s t to our present  wide-spread  method of teaching mathematics as a group of isolated mechanical skills or even t r i c k s , t h e r e is now  an e f f o r t to a s s i s t  students to see some s o r t of organization in the subject. In contemporary mathematics, the emphasis is on unifying principles.  Ik The  course being developed  by the U n i v e r s i t y of  Illinois,. Committee on School Mathematics includes the following topics in the f o u r year program: G r a d e nine- principles of real numbers, inequalities, some concepts of s e t s , linear and Grade t e n -  quadratic equations,  graphing;  s e t s and relations, linear and quadratic f u n c t i o n s ,  properties of angles, polygons  and  circles;  Grade eleven- mathematical induction, exponents and  logarithms  (continuity and limit concept, binomial s e r i e s ) , complex numbers, polynomial functions; Grade twelve-- circular f u n c t i o n s , deductive t h e o r i e s . M c C o y points out one  of the main f e a t u r e s of the  method-, namely,, t h a t students are expected  to discover  generalizations on their own.  The  a way  exercises lead students to dis-  t h a t both exposition and  cover principles and  rules.  t e x t s are designed in such  T h i s is borne out in the following  remarks taken f r o m the T e a c h e r s  Commentary accompanying  The information' regarding the course content of the U I C S M courses was obtained f r o m mimeographed materials published by the committee. M. Eleanor M c C o y , "The Secondary School Mathematics P r o g r a m of the U n i v e r s i t y of Illinois Committee on School Mathematics," Bulletin of the National A s s o c i a t i o n of S e c o n d a r y School Principals, X L I I I (May, 1 9 5 9 ) , p. 12. 2  . U n i t One * We believe t h a t s t u d e n t s should be gi ven an o p p o r t u n i t y t o d i s c o v e r a g r e a t deal o f t h e m a t h e m a t i c s which t h e y are expected t o l e a r n . M a t h e m a t i c a l ideas which a s t u d e n t d i s c o v e r s make sense t o him. ... t h e s t u d e n t is expected t o c o n t r i b u t e ideas, p r i n c i p l e s , and. rules.3 The the  c o u r s e s planned and being used by t h e c o m m i t t e e f i r s t emphasis on c o n t e m p o r a r y  contemporary man^  was  content.  The  t h a t s t u d e n t s who  place  method r a t h e r t h a n on  impression r e c e i v e d f r o m s t a r t e d w i t h the  Beber-  committee's  method covered t h e t r a d i t i o n a l c o n t e n t more quickly and s t o o d i t much b e t t e r t h a n s t u d e n t s who ional method. mathematics,  The  studied by t h e t r a d i t -  "Illinois s t u d e n t s " developed a liking f o r  w a n t e d more m a t e r i a l and consequently as t h e  p r o g r a m evolves j more t o p i c s f r o m c o n t e m p o r a r y are  under-  mathematics  being f i t t e d i n . The; Commission on M a t h e m a t i c s  o f t h e College E n -  t r a n c e E x a m i n a t i o n B o a r d in i t s R e p o r t ^ has recommended t h e  ^ H i g h S c h o o l M a t h e m a t i c s , U n i t One, T e a c h e r s ' E d i t i o n , ( U r b a n a , I l l i n o i s : U n i v e r s i t y o f Illinois C o m m i t t e e on S c h o o l M a t h e m a t i c s , 1 9 5 8 - 5 9 ) , pp. I—ii ^Max B e b e r m a n , D i r e c t o r o f U n i v e r s i t y o f Illinois C o m m i t t e e on S c h o o l M a t h e m a t i c s , address b e f o r e L o w e r M a i n land M a t h e m a t i c s T e a c h e r s , V a n c o u v e r , B.C., F e b . , 1959. -'College E n t r a n c e E x a m i n a t i o n B o a r d , C o m m i s s i o n on M a t h e m a t i c s , P r o g r a m f o r College P r e p a r a t o r y M a t h e m a t i c s (New Y o r k : College E n t r a n c e E x a m i n a t i o n B o a r d , 1 9 5 9 ) . f  16 following sequence o f mathematics courses f o r students on the University  Program:  G r a d e nine- algebra  ( s e t s , variation, equations  and inequalities  in one and t w o unknowns, deduction in algebra, quadratic equations) ; Grade t e n - informal geometry, deductive reasoning, of theorems culminating in the Pythagorean theorem, geometry  sequence coordinate  (locus, s t r a i g h t line, c i r c l e ) , solid geometry;  Grade eleven- linear f u n c t i o n s , radicals, quadratic f u n c t i o n s , exponents and logarithms, s e r i e s , number fields, v e c t o r s , trigonometry; G r a d e twelve-  s e t s and combinations,  mathematical induction,  functions and relations, exponential and logarithmic f u n c t i o n s , circular functions and one-of—-the following: A . i n t r o d u c t o r y probability with s t a t i s t i c a l applications, B . modern algebra (introduction) , or C . selected topics f o r f u r t h e r  treatment.  In this outline the emphasis is on preparation both in understanding  o f concepts  and m a s t e r y o f skills f o r college  mathematics a t the level o f calculus and analytic geometry. The students  are t o be given an understanding  of t h e nature  and role of deductive reasoning in algebra as well as in geometry.  17 This is another of the major points of emphasis in the  various revisions. Up  until the p r e s e n t time, deductive reasoning has  usually been taught through Euclidean geometry  which exper-  ience has shown has not been too s a t i s f a c t o r y a model f o r this p u r p o s e . the  0  T h e feeling held by some mathematicians t h a t  deductive method can be taught b e t t e r by using a model  other than " E u c l i d " is borne out by the last item in the Illinois  plan, namely, deductive t h e o r i e s .  In f a c t , the Illinois  course makes little r e f e r e n c e t o Euclidean geometry The  U n i v e r s i t y o f Maryland study group^ has been  working a t the grade seven and eight level. of  as such.  A  large portion  the seventh grade course is devoted t o a study of number.  This course includes units on natural numbers, plane geometry, and graphs.  The  eighth grade course includes a section on  p r o o f s and equations in the s y s t e m of real numbers. probability and a considerable amount of geometry  Logic,  are also  C a r l B . A l l e n d o e r f e r , "Deductive Methods in M a t h ematics," Insights Into Modern Mathematics, T w e n t y - t h i r d Yearbook of the National Council of T e a c h e r s of Mathematics, (Washington, D.C.: N a t . Coun. T e a c h , o f Math., 1957), pp. 65-66. u  ? M . L . Keedy, "The University of Maryland Mathematics P r o j e c t , " The A m e r i c a n Mathematical Monthly, L X V I ( J a n u a r y , 1959), pp. 58-59.  18 included in the grade eight course.® The  School Mathematics S t u d y Group at Yale  organized in 1958  was  and plans t o prepare sample textbooks, to  t e s t experimental units, to prepare monographs and t o produce teacher-training materials.9  A t present,  experimental  seventh grade units are being t r i e d o u t , ninth t o t w e l f t h grade courses have been outlined, and some monographs have been published. two  The  high school courses are similar t o the  t h a t have been outlined. The  Teachers  curriculum committee of the National Council of  of Mathematics was  charged  with producing a r e p o r t  formulating proposals f o r strengthening and improving  mathem-  10 atics education in the secondary principles emerge f r o m  school.  the r e p o r t .  New  isolated but woven into the curriculum.  Two  important  topics should not For  be  example,  U n i v e r s i t y of Maryland Mathematics P r o j e c t , Mathematics f o r the J u n i o r High S c h o o l , F i r s t Book; Second Book, P a r t s I & I I and Teacher's Guide, (College P a r k , Md., College of E d u c a t i o n , U n i v e r s i t y of Maryland, 1 9 5 9 ) .  1 and 2,  ^School Mathematics S t u d y G r o u p , N e w s l e t t e r s (March and J u n e , 1 9 5 9 ) .  No.  -^The National Council of T e a c h e r s of Mathematics Secondary-School Curriculum Committee, "The S e c o n d a r y Mathematics Curriculum", The Mathematics T e a c h e r , L I I ,  (May, 1 9 5 9 ) , PP. 389-417.  19  equations and inequalities can be t r e a t e d t o g e t h e r using s e t s as a unifying concept.  Secondly, new approaches  used t o develop ideas and techniques r a t h e r than skill.  should be computational  T h e r e p o r t suggests t h a t such things as t r i c k y  factor-  ing, business a r i t h m e t i c and extensive computational problems should be eliminated. A  small group of students in Tallahassee have r e p o r -  t e d their reactions t o a unit on symbolic logic.  These  dents f i r s t studied symbolic logic and then proceeded a study o f permutations in gambling,  and combinations,  through  mathematical  topology and the theory o f s e t s .  They  stu-  odds  concluded  as follows: C h i e f among the benefits o f our study was the valuable experience in working in a new mathematics with strange symbols and new operations. H T h i s comment is most significant f r o m  the point o f view o f  the c u r r e n t discussion on mathematics revision.  Teachers  may feel t h a t a c e r t a i n topic is too d i f f i c u l t f o r a particular grade level but i t might be t h a t t h e novelty t o students o f "working  in a new mathematics" may well o f f s e t the d i f f i c u l t y  f e l t by the t e a c h e r s .  S t u d e n t s o f A l g e b r a I I , U n i v e r s i t y High S c h o o l , Tallahassee, F l o r i d a , "Symbolic L o g i c and logical c i r c u i t r y in the high school," T h e Mathematics T e a c h e r , L , ( J a n u a r y , 1957),  p.  26.  20 The  Ball S t a t e  experimental program  12  has been r e s 13  ponsible f o r the publishing of two  t e x t s , one'in-algebra  and  J  11  one  in geometry  .  The  material included in the t e x t s  had  been taught prior to publication "to normal classes in average high schools f o r five y e a r s . I n  each t e x t t h e r e  is a chap-  t e r on logic which serves  as a basis or foundation  work t h a t is to follow.  In the introductions to both t e x t s  the authors urge the teacher  to r e t u r n to these logical ideas  frequently in order to clarify the mathematics. and ples underlying deductive  f o r the  proof.  The  the  princi-  authors reveal t h a t t h e i r  work is in the spirit of modern mathematics. The principal d i f f i c u l t y t h a t t e a c h e r s . . .have found is t h a t our material is concept-centered r a t h e r than symbol-centered. I t is harder to talk to students about ideas than i t is to tell them how t o arrange marks on paper, but in the long run i t leaves one with a b e t t e r t a s t e in his m o u t h . ^  12 Charles B r u m f i e l , R o b e r t E i c h o l z , and Merrill Shanks "The Ball S t a t e experimental program," The Mathematics T e a c h e r , L I U ' , ( F e b r u a r y , I960), pp. 75-82+ ^3Charles B r u m f i e l , R o b e r t E i c h o l z , and Merrill Shanks A l g e b r a I , (Reading, Mass.: Addison-Wesley Publishing Company Inc., 1961). "^Charles B r u m f i e l , R o b e r t E i c h o l z , and Merrill Shanks G e o m e t r y , (Reading, Mass.: Addison-Wesley Publishing Company I n c . , I960). 1  5 l b i d . , p. ix.  Charles B r u m f i e l , Shanks, op. c i t . , p. 82+.  R o b e r t E i c h o l z , and  Merrill  21 Many o t h e r groups-'-? a r e working w i t h experimental units i n a manner similar t o t h a t being f o l l o w e d by t h e C o m mission" on M a t h e m a t i c s o f t h e College E n t r a n c e Board.  Examination  T h e R e p o r t o f t h e Commission includes a s e c t i o n on  t h e procedure used in determining course c o n t e n t .  T h e com-  m i t t e e recognizes t h a t although t h e mathematicians  m u s t pass  judgement on t h e d e s i r a b i l i t y o f including c e r t a i n t o p i c s i n t h e c u r r i c u l u m , t h e advice o f educational p s y c h o l o g i s t s is also required t o help determine  methodology, grade placement o f  m a t e r i a l , learning t h e o r y and t h e l i k e .  T h e r e p o r t goes on  t o point o u t t h a t i t s recommendations a r e "based on t h e needs of m a t h e m a t i c s , on i t s u s e s , and on t h e a n t i c i p a t e d needs o f t h e s t u d e n t s and society."-'-® T h e y a r e having experimental c o u r s e s t r i e d o u t and a r e t a k i n g a "middle course b e t w e e n detailed experimental s t u d i e s and pronouncements based on a p r i o r i .judgements . "  y  17  ' S t u d i e s in m a t h e m a t i c s e d u c a t i o n , (Chicago:. S c o t t , F o r e s m a n and Company, 1959) • T h i s pamphlet contains a b r i e f outline o f t h e w o r k o f t h e major groups working on m a t h e m a t i c s revision f r o m grades one t h r o u g h college. 1 P  College E n t r a n c e E x a m i n a t i o n B o a r d , op_. c i t . , pp. 13, li+. ^Ibid..  22  Although most of these groups are trying out proposed materials in experimental their experimentation  situations, i t is f e l t t h a t  is not yielding the maximum amount of  information t h a t might be obtained. in the case of the students level of d i f f i c u l t y of a new  As  has been pointed  out  in Tallahassee and in Illinois, the topic is something which i t is d i f f -  icult f o r t e a c h e r s to p r e d i c t .  The  various groups are  ing primarily on the basis of educators' and opinions and  the  operat-  mathematicians'  i t is possible t h a t c e r t a i n topics could well be  placed more e f f e c t i v e l y at a grade level other than the  one  suggested. Much of the information r e p o r t e d in the journals on curriculum revision, although r a t h e r subjective. revision, E v a n s  carefully worked out, is still  In summarizing a chapter  on  curriculum  says,  The contributions of the studies reviewed in this chapter t o the r e s e a r c h process or t o the s t o c k of r e s e a r c h findings leaves something to be d e s i r e d . . . T h e r e is no clean b r e a k - t h r u to a f i r m e r r e s e a r c h base f o r making decisions and planning organizational operations . 2 0 T h i s comment might be i n t e r p r e t e d as suggesting, t h a t the prevailing method of conducting  curriculum revision, that- i s ,  2 0 H . M » E v a n s , "Organization f o r Curriculum Development ," _Rej/iew_of_J3^^ X X V I I , (June, 1957),  PP.  291-292.  23  working primarily on the basis of educators' opinions, is not the most s a t i s f a c t o r y .  The  present study suggests  a more  rigid design. 21  Washburne and the Committee of S e v e n a study attempting curriculum.  conducted  to relate mental age and the arithmetic  Information was  gathered  f r o m a large number  of schools regarding the actual grade placement of a specific topic in a r i t h m e t i c . this topic one or two  The  experiment consisted of teaching  grades above and below the level at*  which the previous survey showed t h a t most schools placed i t . The  committee s e t , as an a r b i t r a r y standard, an eighty per  cent grade from, t h r e e - q u a r t e r s of the class.  In o t h e r  words, this particular group f e l t t h a t b e f o r e a topic as long division should be assigned to a specific grade,  such three-  quarters of the students in t h a t grade should have been able to obtain a t least an eighty per cent mark on the t e s t .  If  this degree of mastery could not be obtained a t the grade five level, say, then i t was  not considered e f f i c i e n t to t e a c h  the topic to grade five s t u d e n t s .  The  principle of readiness  which underlies the Washburne experiment is also the principle C a r l e t o n W. Washburne, "Mental Age and the A r i t h metic C u r r i c u l u m , " J o u r n a l of Educational R e s e a r c h , X X I I I , (March, 1 9 3 1 ) ,  PP.  210-231.  22+  which underlies t h e present study. In general, t h e revisions proposed by t h e various groups contain new material and t r e a t much of t h e traditional subject m a t t e r in a modern way."  T h e t e x t s produced by t h e  Ball S t a t e group illustrate this point v e r y clearly. the geometry  In both  and algebra t e x t s , a chapter is devoted t o  logic which in t u r n is used t o a s s i s t in t h e development traditional material.  of "  Although many o f these experimental  units have been t r i e d out in classroom situations, t h e r e have been f e w studies designed t o determine t h e grade placement of a specific topic. other connections.  S u c h studies have been conducted in Specifically, Washburne investigated t h e  grade placement of topics in arithmetic a t the elementary school level.  T h e p r e s e n t study is designed t o investigate  the grade placement of symbolic logic in a similar way.  CHAPTER  THE  Two  EXPERIMENTAL  III  UNIT AND  TESTS  questions have been raised in this study, namely;  are t h e r e significant d i f f e r e n c e s in the means of the five groups? A n d  at which grade levels can this material be  tively mastered?  effec-  In order to answer these questions, a series  of lessons, exercises and t e s t questions were prepared presented to the s t u d e n t s .  At  and  the conclusion of the unit,  the s c o r e s made on the final t e s t by each group were analyzed. Decisions were then made regarding the suitability of the material f o r the various grade levels. The  experimental unit in symbolic logic was  of ten lessons followed by a final t e s t . review  composed  E a c h lesson contained  material based on the preceding lesson, new  work, an  assignment and questions designed to t e s t the m a s t e r y of the lesson.  I.  CONTENT  OF  THE  UNIT  Since the purpose of the study was  to determine the  e f f e c t i v e n e s s of teaching a f i r s t year university unit on symbolic logic in the high school, i t was  necessary to r e f e r to  some of the newer college t e x t s to determine the content' of  26 such a unit.  A  study of the chapters on logic of the t h r e e  college f r e s h m a n t e x t s r e f e r r e d t o earlier core of subject m a t t e r .  A  revealed  a common  typical f i r s t year unit on symbolic  logic generally includes the basic operations  of -con junction, dis-  junction, negation and implication, the use o f t r u t h tables, algebra of logic, applications t o electrical n e t w o r k s , derived implications and the use of q u a n t i f i e r s . material t h a t the present  II. The  I t is upon this basic  unit has been developed.  DESCRIPTION  OF  THE  LESSONS  unit consisted of t e n instructional periods of  f i f t y minutes each, the t e n t h being devoted primarily t o review exercises.  T h e topics t h a t were taught in the various  lessons are outlined in Table I . (overleaf) At  the beginning o f t h e f i r s t lesson the students  were told t h a t they were participating in an experiment and t h a t i t was the subject m a t t e r t h a t was being t e s t e d  rather  C a r l B . A l l e n d o e r f e r and C l e t u s O. Oakley, P r i n c i ples o f Mathematics., (New Y o r k : M c G r a w - H i l l Book Company, 1955). 'Robert R. C h r i s t i a n , Introduction t o L o g i c and S e t s (Preliminary edition; B o s t o n : Ginn and Co., 1 9 5 8 ) . J o h n G . Kemeny, J * L a u r i e Snell and Gerald L . Thompson, Introduction t o F i n i t e Mathematics, (New Y o r k : Prentice-Hall, Inc., 1 9 5 7 ) .  27  TABLE CONTENT  OF  I  THE  L e s s o n Number  LESSONS  Topic ..  1  B a s i c operations  2-  T r u t h tables Implication and equivalence Derived implications A n algebra of logic "Black boxes" as a method o f illustrating logical s t r u c t u r e Switching networks Switching networks Q u a n t i f i e r s and verbal s t a t e m e n t s V e r b a l s t a t e m e n t s and review Final t e s t  3 4* 5 6 7 8 9 10 11  than the students themselves. a t t e m p t was  (conjunction, disjunction, negation)  I t was pointed out t h a t an  being made t o determine the e f f e c t i v e n e s s o f  teaching symbolic logic at various grades.  Students  were  urged t o do their best even though a t times the material might appear t o be d i f f i c u l t . p  The  lesson plans  was necessary  give an outline of each lesson.  It  f o r the teacher t o adhere closely t o the plan  f o r each lesson in order t o ensure uniformity of instruction f o r the various c l a s s e s . See  Appendix A .  T h e students  were provided with a  28 set  o f exercises based on t h e work of each lesson.3  Although  the amount o f time spent on each p a r t o f the lessons varied, in general t e n minutes was spent in working and discussing t h e review  questions, t w e n t y minutes on i n s t r u c t i o n , f i f t e e n  minutes on exercises based on the lesson and the" final five minutes on the mastery questions.  No homework was required  of the s t u d e n t s .  " i l l .  MASTERY  QUESTIONS  In order t o determine the extent t o which the lesson had been understood,  mastery questions were prepared and  given t o the students during the l a s t five minutes of each period.  T h e s e were marked by t h e t e a c h e r .  were informed The  The  students  o f the marks, but the papers were not r e t u r n e d .  major points of each lesson were t e s t e d by these  m a s t e r y questions.^  4-  T h e questions f o r a specific lesson were  designed t o t e s t only the material taught Since more subject m a t t e r was covered  in t h a t lesson. in some lessons than'in  o t h e r s , the number o f questions varied f r o m lesson t o lesson. In general, s t r a i g h t f o r w a r d questions were followed by more  T h e s e exercises are contained in Appendix The  B.  m a s t e r y questions are contained in Appendix  C.  29 complex the  ones.  In o t h e r words, an a t t e m p t was made t o t e s t  material taught and t o v a r y t h e complexity o f t h e ques-  tions. A s well as serving as a gauge o f t h e e f f e c t i v e n e s s of the teaching, t h e mastery questions served another purpose.  A n a t t e m p t was made t o arrange them in an approx-  imate o r d e r o f increasing d i f f i c u l t y by computing t h e p e r c e n tage passing each i t e m .  T h i s information was used in the  construction of t h e final t e s t , the items o f which were devised so t h a t they were comparable t o t h e m a s t e r y  ques-  tions in t e r m s o f content and form.. IV.  CONSTRUCTION OF  THE FINAL  TEST  T h e final t e s t ^ on symbolic logic was composed o f f o r t y multiple-choice i t e m s .  S e l e c t i o n o f items Since t h e t e s t was intended t o measure t h e m a s t e r y of the subject m a t t e r , i t was c o n s t r u c t e d in such a way t h a t , except f o r f o u r items, the final t e s t was similar t o t h e mast e r y questions used a t the conclusion o f each lesson.  5The final t e s t is contained in Appendix D.  30 I t was f e l t t h a t t h e final t e s t  must parallel the  emphasis of t h e m a s t e r y questions if an adequate measure o f the students' comprehension was t o be obtained. items  Alternate  were c o n s t r u c t e d t o match as closely as possible each  of t h e m a s t e r y questions in content and f o r m .  Since o n e  D  of the m a s t e r y questions t e s t e d memory o f a r a t h e r unimportant point r a t h e r than understanding  o f t h e material, a ques-  tion t o match this one was not put on t h e final t e s t . tions t h i r t y - e i g h t , thirty-nine original number o f items.  Ques-  and f o r t y were added t o t h e  T h e s e t h r e e questions were design- '  ed t o be a more complex measure of t h e students' ability t o apply t h e principles t o a slightly d i f f e r e n t situation than t h e others.. An  a t t e m p t was made t o arrange t h e questions in an  increasing order o f d i f f i c u l t y  on the basis of t h e responses  made on t h e m a s t e r y questions.  Reliability The  reliability of t h e t e s t was determined  a formula developed  Question  by H o y t .  7  by using  This c o e f f i c i e n t of reliability  1 of L e s s o n 5«  7 C J . H o y t , "Note on a simplified method of com- -• puting t e s t reliability," Educational and Psychological Measurement , I , ( J a n u a r y , 19^+1), pp. 93-95.  measures the internal consistency of a t e s t and hence will tend to  provide a minimum estimate of reliability i Since this formula is not applicable t o speed  it was  necessary to.ensure t h a t time was  tests,  not a f a c t o r .  Usirg the time required t o complete the m a s t e r y questions as a guide, the experimenter be adequate.  The  decided t h a t f i f t y minutes should  students were i n s t r u c t e d to take their  time, check their work i f they desired and t o t u r n in their papers  when they f e l t t h a t they were finished.  noted.  At  T h i s r e p r e s e n t s 2.08  which was  time  was  the conclusion of the f i f t y minutes eight students  f e l t t h a t they had not completed tion.  The  the t e s t t o t h e i r  satisfac-  per cent of the t o t a l group,  considered low enough to w a r r a n t  the use of this  procedure. Using this method, the reliability  c o e f f i c i e n t of the q  t e s t was  .87 with a standard e r r o r of 2.70  .  Validity The of  extent t o which this t e s t was  a valid measure  the material of the unit as outlined in the lesson plans.  Henry E , G a r r e t t , S t a t i s t i c s in Psychology and E d u c a t i o n , (New Y o r k : Longmans, G r e e n and Co., 1 9 4 9 ) , p. 386.. ^ D e t a i l s of the calculations may E.  be found in Appendix  32  could only be determined from, the point o f view o f content validity. .As f a r as can be determined, no comparable measuring instruments A in Table  exist*  content analysis o f t h e items  of the t e s t is shown  I I . I t will be noted t h a t only two questions involved  simple recall o f f a c t s . relate various f a c t s .  T h i r t y - o n e required the student t o Nine questions were of the s o r t  whose  solutions could be determined by applying the material studied. T A B L E II CONTENT ANALYSIS  Topic  Factual  B a s i c Operations T r u t h Tables Implication Equivalence Derived Implications A l g e b r a of Logic Black Boxes Switching N e t w o r k s Quantifiers Totals  1 1  2  OF  TEST  Relationships  ITEMS  Application  Total  5  5  2  2  2  2  3  1  h  3  3  7  6  7  2  2  3  1  3  4  31  9  h 7 2+0  In order t o give a more complete description of t h e items, an analysis showing the various p a r t s o f the material involved in each item is given in Table I I I .  33 T A B L E III T O P II C CS F R O M S Y M B O L I C L O G I C I INNVVCO L V E D IN T H E S O L U T I O N O F E A C H I T E M 3  I t e m B a s i c T r u t h Impli- Equiv- D e r . A l g . Black S w i t c h QuantNo. Op. Tables cation alence Imp. L o g i c Boxes Netw'ks i f i e r s  1 2 3 2+ 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36  • x x x x  x x x X  X  X  x  x x .x  x x x x x x x x x  x x  x x x x  X  x x x  x  x x  x x x x x x x  x x  x  X  x x  x  x  x  x x  x x x x x  x  x  x x x  x x x x x  X  3k TABLE  I I I (continued)  I t e m B a s i c T r u t h Impli- Equiv- D e r . A l g . Black S w i t c h QuantNo Op. Tables cation alence Imp. L o g i c Boxes Netw'ks i f i e r s  37 38 39  i+0  x  X X X  X  X X X  X  I t will be seen, f o r instance, t h a t according to this analysis t h e r e a r e six questions requiring the use of t r u t h tables whereas f r o m Table I I only two questions specifically t e s t this t o p i c .  In other words, many of the topics taught  t o w a r d the end of the unit involved applications of topics studied earlier.  CHAPTER  The  IV  METHOD  OF  I.  SUBJECTS  THE  STUDY  subjects chosen f o r this experiment  consisted o f  387 pupils in /-/grades" nine t o t h i r t e e n inclusive enrolled on t h e U n i v e r s i t y Programme a t Como L a k e High School in School D i s t r i c t No. 43 ( C o q u i t l a m ) . Since i t was intended t h a t all students should receive the same i n s t r u c t i o n , students f r o m courses were, in some cases, grouped pose.  T h i s enabled t h e experimenter  different  mathematics  t o g e t h e r f o r this purt o include a larger num-  ber of students in t h e study than would otherwise have been possible*  T h e grouping f o r instructional purposes  is shown in  Table I V .  T A B L E IV GROUPING OF STUDENTS FOR INSTRUCTION Group Math. Course No, o f Students  A  B  C  D  F  G  H  101  30  10  20  20  30  20  91  30  10  91  31  36  33  27  32  35  36  32  35  63  29  36  Groups A 19,  to G  i 9 6 0 to October  October  13,  were taught the unit f r o m  6,. i 9 6 0 .  Group H  i 9 6 0 until O c t o b e r  31,  was  September  taught  from  i960.  I t is to be noted t h a t the numbers indicated in Table I V r e f e r t o the net numbers of s t u d e n t s .  Students  who  were absent f o r a lesson r e p o r t e d t o the experimenter their r e t u r n to school and i f i t was  possible f o r them to  attend the missed lesson in another section, this was Otherwise,  the student was  upon  not counted in the  done.  experiment.  F o r t y - t h r e e of the original number, of students were not counted. Table V in the study f r o m  shows the number of students who each mathematics course*  took p a r t  In addition, the  mean and standard deviation of the O t i s Intelligence Quotients of these groups is indicated. F  showing the O t i s I.Q.  Tables are included in Appendix  and the s c o r e made on the logic  t e s t f o r each individual used in the study. In this study, students taking Mathematics 1 0 1 in high school .are being considered approximately equivalent t o those in f i r s t year university t o whom material similar t o t h a t of this study is being taught at some universities. grade t h i r t e e n students in B r i t i s h Columbia  Most  w r o t e the Domin-  ion Group T e s t of L e a r n i n g C a p a c i t y in November,  i960.  37 TABLE  V  DESCRIPTION OF STUDENTS GROUPED TO MATHEMATICS COURSE  No. of Subjects  Course  ' Mean I.Q.  S.D  ' 111.7 112.7 110.0  6.3 8,1 8.9 8.5 8.9  31  Ma 101 Ma 91 Ma 30 Ma 20 Ma 10  ACCORDING  61 106 93 96  113.0 113.3  .  2 The  mean I*Q. provincially was 1 1 5 . 7  group used in this study was 116.3.  while t h a t f o r t h e I t may be concluded t h a t  this group was f a i r l y typical, in t e r m s o f scholastic aptitude, of t h e grade t h i r t e e n students in the province. II,  The  EXPERIMENTAL  PROCEDURE  students were taught t h e unit a t t h e time when  they would normally receive t h e i r mathematics i n s t r u c t i o n . In certain cases t w o classes were combined in order t o involve  1  T h e intelligence quotients used in t h e preparation o f this table were derived f r o m t h e O t i s Quick S c o r i n g Gamma T e s t , given t o t h e Ma 10 and Ma 3 0 students in A p r i l , i 9 6 0 , and t o t h e others in A p r i l , 1 9 5 9 * 2 N o r m s published by t h e D e p a r t m e n t o f E d u c a t i o n , Victoria, B.C., March, I 9 6 I .  38 more students in the study.  In this way,  were needed in the school time-table.  no  adjustments  The ..details of the  grouping are shown in Table I V on page 3 5 . The  i n s t r u c t i o n was  directed a t the level of under-  standing of the students in Mathematics 1 0 1 . t e n t of each lesson was with the Mathematics 101 making adjustments  determined  The  actual  by t r y i n g the materials  students of the previous year^  where i t was  con-  and  f e l t advisable. A l l e n d o e r f e r  and Oakley^ suggest t h a t t e n lessons is adequate t o cover the section on L o g i c in their t e x t .  While the experimental unit  does not contain material identical with t h a t in the above t e x t , the t o t a l amount is roughly the same. reason t h a t the experimenter  I t was  f o r this  originally divided the work into  t e n lessons. In general, the routine followed in each lesson the same.  A  description of L e s s o n  Four  was  will serve t o  •^The mean I.Q. of the 1 9 5 9 - 6 0 group on the Dominion Group T e s t of Learning Capacity was 1 1 7 . 1 with a standare deviation of 9 « 5 8 , while the mean I.Q.. on the same t e s t f o r the experimental Ma 101 group was 116.3 with a standard deviation of 9 . 2 7 * A t e s t f o r the significance of the d i f f e r ence in these means gave a C.R. of . 0 6 which is not significant a t the 1% level. ^"Carl B, A l l e n d o e r f e r and C l e t u s O. Oakley, P r i n c i ples of Mathematics. (New Y o r k : M c G r a w - H i l l Book Company, 1 9 5 5 ) , P. v i i .  39 illustrate the method used. tained in Appendix  Outlines of all lessons are  con-  A.  B e f o r e the students entered the room, the t e a c h e r w r o t e the review exercises on the blackboard. t h e r e were t h r e e . began t o work.  As  In this case,  soon as the students entered they  When about 80 per cent- of the students  had  finished, the t e a c h e r solved the problems on the blackboard, explaining any d i f f i c u l t i e s .  Since L e s s o n  with derived implications, i t was the students knew how ation.  Four  was  t o deal  necessary t o ensure t h a t  t o c o n s t r u c t a t r u t h table f o r implic-  Hence, question one received p a r t i c u l a r a t t e n t i o n .  Question two,  which dealt - w i t h the c o n s t r u c t i o n of an ordin-  ary t r u t h table, presented little t r o u b l e . logical equivalence (*  The  symbol f o r  ») used in question t h r e e was  of most of the d i f f i c u l t y here.  the cause  Working and discussing the  review exercises took about t e n minutes of- the lesson. In order t o introduce the new  material on derived  implications, the t e a c h e r w r o t e "p—*q" and " i f i t rains then I shall get wet"  on the blackboard.  One  of these is w r i t t e n  in the symbolism of logic, the other in ordinary language. was  s t a t e d t h a t by interchanging the two  tence and by introducing negation, new constructed from  the original one.  p a r t s of the  sen-  implications could be  In this manner, the  It  converse, inverse and contrapositive were introduced, each o f the derived implications was discussed, was c o n s t r u c t e d i the  As  a t r u t h table  I t was noted t h a t the t r u t h table f o r  contrapositive agreed with t h a t f o r t h e original implication.  In other words, when the original implication was t r u e , so was the  contrapositive.  This was not necessarily t h e case with  the  converse and inverse. This particular point gave rise t o some discussion in  those groups where the converse o f a geometric theorem had been studied.  Although t h e students realized t h a t t h e con-  v e r s e o f a theorem was not necessarily t r u e , they were impressed with seeing a "logical" p r o o f f o r this f a c t .  These  students were equally impressed with the idea t h a t i t was possible t o prove t h a t t h e t r u t h value o f the contrapositive and  the original implication must always agree with each  other.  "When the symbolic f o r m s , the English s t a t e m e n t s and the t r u t h tables f o r each o f t h e new implications had been put on the blackboard, t h e students copied them into their notebooks.  T h e y then began t h e assignment.  About twenty  minutes was spent on i n s t r u c t i o n . The  assignment required t h a t the students w r i t e t h e  converse, inverse and contrapositive o f s t a t e m e n t s such as " I f a is an even integer, then 2_a is divisible by i+" and o f  i+1  s t a t e m e n t s expressed in symbols such as " ~p—»~q".  I n this  particular lesson, about f i f t e e n minutes were spent on t h e assignment. Following  t h i s , the sheets containing  questions were d i s t r i b u t e d .  the mastery  T h e students were able t o com-  plete these in approximately eight minutes. III. The  STATISTICAL  TREATMENT  students who participated in the experiment were  divided into five groups determined by the mathematics course they were studying.  Since t h e number o f students in each  group varied greatly and since the smallest  group originally con-  tained only t h i r t y - t h r e e , i t was considered impractical t o match individuals. subjects  Matching would have eliminated  t o o many  and would have r e s u l t e d in groups t h a t were unneces-  sarily small.  I t was f e l t t h a t analysis o f covariance would  provide t h e best method f o r analyzing  the d a t a .  Intelligence was used a t t h e control variable since it was conceivable t h a t this might have been a f a c t o r could have a f f e c t e d the r e s u l t s of the study.  that  In order t o  t e s t f o r significant d i f f e r e n c e s between t h e means o f the intelligence quotients,  B a r t l e t t ' s t e s t was f i r s t used t o  determine whether t h e variances were homogeneous.  The  42  result o f this t e s t was a X than t h e tabled e n t r y of results o f t h e t - t e s t s  2  value o f  3-3.277  5 . 5 9 2  Q"  which is less  a t t h e .01 level.  The  are summarized in Table V I . TABLE  VI  SIGNIFICANCE OF DIFFERENCES BETWEEN MEANS FOR INTELLIGENCE •  D i f f . in Means  Groups  Ma  101-91 101-30  101-20 101-10 91-30  91-20 91-10 30-20 30-10  20-10  The  1.1 1.7 2.7 1.6 2.7 ,7 .6 3.0 3.3 •3  t  C «R .  d.f.  ,22 .28 • 47 .26  90  2.63  135  2.62  122 125 165 155  2.62 2.62 2.61 2.61 2.61  197  2.60  200 187  2.60  .72  .02 .16 1.34 4.3 .13  152  .01  2.60  d i f f e r e n c e between t h e means o f t h e Mathematics  3 0 and 1 0 groups is t h e only one t h a t is significant a t t h e one per cent level. Reference  has been made t o t h e Washburne study^  on a r i t h m e t i c in which an a r b i t r a r y standard o f p e r f o r m a n c e  -'Carleton W. Washburne, "Mental A g e and t h e Arithmetic Curriculum," J o u r n a l o f Educational R e s e a r c h , X X I I I , (March, 1 9 3 1 ) , PP. 2 1 0 - 2 3 1 .  43 was  established.  I t was  noted t h a t t h r e e - q u a r t e r s of the  students in a specific grade were required to obtain a mark of eighty per cent b e f o r e the topic was f o r t h a t grade.  The  t e d in a similar way.  considered suitable  B i n e t scale of intelligence was B e f o r e a particular t e s t was  construcplaced  at the six-year level (say) , seventy-five per cent of the s i x year olds must have passed i t .  Although the percentage  successes f o r each age group was  not always seventy-five  per cent, the same general principle of a standard was  predetermined  employed at each level.  While a person may  or may  ard required by an experimenter,  not agree with the s t a n d -  the general principle involved  has been established as a r e s e a r c h technique.  When the data  are available, i t is always possible t o apply a standard ent f r o m  the one The  of  actually used by the  differ-  experimenter.  D e p a r t m e n t of Education of this Province  has,  by implication, s e t out a standard to be used as a guide in the awarding of l e t t e r grades t o students in the schools of the province.^  secondary  T h i s information is presented in  " P r o v i n c e of B r i t i s h Columbia, D e p a r t m e n t of E d u c ation, Division of Curriculum, A d m i n i s t r a t i v e Bulletin f o r Secondary Schools, i 9 6 0 , (Victoria, B.C.: Queen's. P r i n t e r , i 9 6 0 ) , p. 55.  Table V I I .  This table provides a rough relationship between  certain score intervals and t h e percentage  o f students t h a t  should obtain scores in t h e respective i n t e r v a l s .  Since various  methods based on these distributions and on information regarding t h e intelligence quotients o f t h e students are used in most secondary  schools throughout  t h e province, i t was  f e l t t h a t a standard o f performance related t o this system was  warranted  in this  study. TABLE VII  METHODS  Percentage of students  Top Next  Next  FOR ASSIGNING L E T T E R  Score  on t e s t  L e t t e r Grade  5 % 20  15 20 15  A B  C +  25 %  From  GRADES  C C Fail  86 - 100 % 73-85  66-72 58 - 65 50 - 57 below 50 %  Table V I I i t will be seen t h a t , generally, 75  per cent o f t h e students in a group a r e expected a mark o f 50 per cent or more on a t e s t .  t o obtain  In the present  study i t was decided t o examine t h e r e s u l t s o f each group f r o m this point o f view.  I f 75 per cent o f t h e students in  k5 any group obtained a mark o f 50 p e r cent o r m o r e , t h e n t h e p e r f o r m a n c e o f t h a t group would be considered  satisfactory.  CHAPTER  ANALYSIS  Before  data may  OF  V  RESULTS  properly be analyzed by the co-vari-  ance method, i t is necessary of the groups are similar.  to establish t h a t the variances An  overall t e s t f o r significant  d i f f e r e n c e s between variances of several groups is provided by B a r t l e t t ' s t e s t of homogeneity of variance. applied a value of —* i~.  t e s t was  tabled e n t r y f o r X^  Obtained  for  Since C  is always g r e a t e r  since 6 . 5 8 3 is already less than 1 3 . 2 7 7 , i t  unnecessary to calculate C. of variance was The  X  .  The  with f o u r degrees of freedom is 1 3 . 2 7 7  at the one per cent level. unity and  was  When this  The  than was  hypothesis of homogeneity  upheld.  procedure used in the analysis of covariance  t h a t outlined by W e r t et a l . ^ The sis are summarized in Table  data related t o the  was analy-  VIII.  1-The procedure to be followed f o r B a r t l e t t ' s t e s t may be found in Palmer O. Johnson, S t a t i s t i c a l Methods in R e s e a r c h (New Y o r k : P r e n t i c e Hall, Inc., 1 9 4 9 ) pp. 8 3 - 8 5 . J a m e s E . W e r t , Charles O. Neidt and J . S t a n l e y Ahmann, S t a t i s t i c a l Methods, in Educational and Psychological R e s e a r c h (New Y o r k : A p p l e t o n - C e n t u r y - C r o f t s , Inc., 1954),  PP.  343-363.  47 TABLE ANALYSIS  OF  S o u r c e of Variation  Degrees of Freedom  Total Within Difference  385  COVARIANCE  Sum of Squares  4 4,38i  Mean Square  19636 12539 7097  381  F  VIII  =  5 3  32.91 1774.25  :-9  T h e value of F obtained through the analysis  was  53«9 which, with 4 and 381 degrees of freedom is significant at the one per cent level.  Hence, significant d i f f e r e n c e s did  exist between the means. In order t o locate the specific areas where d i f f e r e n c e s existed, i t was necessary t o p e r f o r m of t - t e s t s .  Lindquist  3  these  a series  outlines a method f o r t e s t i n g f o r the  significance of d i f f e r e n c e s between adjusted means.  The  adjusted means are given in Table I X .  -^E.F. Lindquist, Design and Analysis of Experiments in Psychology and E d u c a t i o n ( B o s t o n : Houghton M i f f l i n Company, 1953), p. 327.  1+8 TABLE ADJUSTED  Group  CRITERION  MEANS  Original Mean  M a t h 101 91 30 20 10  A  IX  Adjusted Mean  30.7 27.9 23.0 20.3 17-8  31.7 27.7 2I4..4 20.1 1 7.4  series of mean d i f f e r e n c e  t e s t s was then  ducted using t h e adjusted means f r o m  Table I X .  con-  The results  of these t e s t s a r e shown in Table X .  TABLE  X  SIGNIFICANCE OF DIFFERENCES MEANS FOR LOGIC  Groups  101 91 30 20  -  D i f f . between A d j . Means  91 30 20 10  When t e s t e d  4.0 3*3 4.3 2.7  by Lindquist's  C R . 3,1 3-6 5.3  3.2  BETWEEN  d.f. 90  t  .01 2.63  165  2.61  197 187  2.60 2.60  method, all t h e d i f f e r -  ences were found t o be significant a t the one p e r cent level.  49  A n analysis of the d a t a according t o the standard outlined in C h a p t e r  I V , namely, t h a t 7 5 per cent of the  students should obtain a score of 5 0 per cent o r b e t t e r is given in Table X I .  Letter Grade A B C + C C Fail  TABLE  XI  DISTRIBUTIONS  OF  Percent of Students  Range of Scores  MA 101  SCORES  MA  MA  MA  MA  91  30  20  10  1% 5  5  34-40  33$  15%  20  29-33  39  34  20  2% 10  15  26-28  19  18  13  12  4  20  23-25 20-22  15 10  25 12  12  13  12  17  25%  52%  60%  15 25  '  below 2 0  I t will be seen f r o m Table  3 3  3%  8%  5%  X I t h a t just 7 5 per cent  of the Mathematics 3 0 students obtained a mark of 5 0 per cent or b e t t e r .  T h e Mathematics 1 0 1 and 9 1 students p e r -  f o r m e d a t a level superior t o t h a t of the Mathematics 30 s t u d e n t s , while t h e Mathematics 2 0 and 1 0 students  were  inferior. C u r v e s representing the distributions of t h e scores of the five groups a r e shown in F i g u r e 1 . only t h r e e "peaks" r a t h e r than f i v e .  T h e r e are really  T h e distributions of the  50 Mathematics 20 and 10 groups overlap each other t o a considerable e x t e n t . the  T h e same type o f overlapping exists between •  curves o f the Mathematics 101 and 91 groups. The  curve representing the Mathematics 3 0 d i s t r i b -  ution lies between the two extremes just discussed.  This  curve overlaps those o f both the Mathematics 1 0 1 - 9 1 and Mathematics 2 0 - 1 0 groups, while i t s maximum point falls between t h e two large groups.  T h e scores in Table X I show  t h a t the distribution f o r this group approximates t h e t h e o retical distribution suggested by the D e p a r t m e n t ation, very  closely.  of Educ-  F I G U R E Distribution  of  1  Scores  by  Grades  CHAPTER V i  SUMMARY I. Symbolic  AND  CONCLUSIONS  SUMMARY  logic is one o f several topics f r o m  contem-  porary mathematics t h a t , according t o some a u t h o r i t i e s , should be included in the high school mathematics curriculum. Since v e r y little r e s e a r c h related t o t h e grade placement o f these topics has been conducted  and since our provincial  education authorities a r e contemplating  a revision o f t h e  mathematics curriculum, i t was f e l t t h a t some information related t o t h e grade placement o f symbolic logic should be gathered.  T h i s study was designed t o determine t h e e f f e c -  tiveness o f teaching symbolic logic in t h e high school.  The  material comprising t h e unit was f a i r l y similar t o t h a t taught to  f r e s h m a n university students a t some universities. The  experimental unit consisted o f t e n lessons on  symbolic logic followed by a final t e s t . to  T h i s unit was taught  387 students enrolled on t h e university programme in  grades nine t o t h i r t e e n a t C o mo L a k e High S c h o o l . The  study was designed t o answer two specific  questions, namely, do significant d i f f e r e n c e s exist between the means o f the various groups? A n d a t which grade levels  53 can this material be e f f e c t i v e l y mastered?  A s a criterion  f o r determining t h i s , 75 per cent o f t h e students a t a p a r ticular level were required t o obtain a score o f 50 per cent or b e t t e r on a m a s t e r y t e s t o f symbolic  logic.  In order t o obtain an answer t o t h e f i r s t question, the r e s u l t s were studied by analysis o f co variance with scholastic aptitude being t h e variable controlled.  Since i t would  have been impractical t o match t h e groups, and since t h e r e was a significant d i f f e r e n c e between t h e mean intelligence quotients o f two o f t h e groups, this technique p e r m i t t e d t h e experimenter  t o obtain t h e information required.  B e f o r e t h e material was considered suitable f o r introduction t o a specific mathematics course, t w o additional f a c t o r s were investigated.  F i r s t , t h e p e r f o r m a n c e o f each  group was compared with t h e standard outlined in t h e second question above.  T h e groups in which 75 p e r cent o f the  students scored 50 per cent o r b e t t e r -on the final t e s t were considered t o have p e r f o r m e d s a t i s f a c t o r i l y . erial was considered unsuitable f o r o t h e r groups.  T h e matT h e curves  representing the distributions o f t h e scores were compared with each o t h e r .  Decisions were then made regarding t h e  suitability o f t h e material f o r t h e various c o u r s e s .  II.  Major  CONCLUSIONS  Conclusions The  conclusions drawn f r o m this study apply only t o  the subjects, materials and method o f teaching used in t h e experiment. the  Caution must be exercised in generalizing f r o m  results. 1.  A l l o f t h e d i f f e r e n c e s between the means o f  the groups were found level.  t o be significant a t t h e one per cent  No two groups p e r f o r m e d a t the same level.  The  highest mean score was obtained by t h e students o f M a t h ematics  101, followed in order by those o f Mathematics  91, 30, 20, and 10. 2.  T h e students o f Mathematics 101, 9 1 , and 30  s a t i s f i e d t h e requirement  t h a t 75 per cent should obtain a  score o f 50 per cent o r b e t t e r on t h e final t e s t .  The  students o f Mathematics 20 and 10 failed t o s a t i s f y this requirement.  Discussion An  examination  of t h e curves representing the dis-  tributions o f t h e logic t e s t  scores showed t h a t t h e r e was  a considerable amount of overlapping o f t h e Mathematics 101 and 91 groups, and also between the Mathematics 10 and 20  55 groups.  The  ution was  curve representing the Mathematics 30  located between these two  I t was  distrib-  larger groups.  noted t h a t just 75 per cent of the Mathema-  t i c s 30 students obtained a score of 50 per cent or b e t t e r on the t e s t and t h a t the curve representing the distribution of scores f o r this group fell between the two mentioned  above.  larger  groups  Since the lessons were designed f o r M a t h -  ematics 101 s t u d e n t s , more material was  covered in a class  period than is normally the case f o r Mathematics 30 s t u d e n t s . I t is possible t h a t they might have p e r f o r m e d  at a more s a t -  i s f a c t o r y level i f the material had been presented in a more leisurely manner thus allowing the students more time t o absorb the work. Although the material, presented in the f o r m a l mathematical language normally found at the college f r e s h m a n appears to be unsuitable f o r students  level,  in Mathematics 20  and  10, i t is interesting t o note t h a t i+8 per cent of those in Mathematics 20 and i+0 per cent of those in Mathematics  10  obtained a mark of 50 per cent or b e t t e r on the final t e s t . This might  suggest t h a t the material could be used with above  average students at these grade levels. T h a t the mean s c o r e s of the groups  were in descend-  ing order f r o m the highest mathematics course t o the lowest was  not entirely unexpected.  I t is quite possible t h a t this is  56  a d i r e c t r e f l e c t i o n of the mathematics background of the v a r ious groups.  Although this background was  group, the material was in Mathematics 30 III.  d i f f e r e n t f o r each  found to be suitable f o r the students  and 91 as well as those in Mathematics  L I M I T A T I O N S OF  T h i s study was  THE  STUDY  designed to investigate a relatively  small area of one topic in modern mathematics. iment was  101.  The  exper-  intended t o serve as a guide in determining the most  e f f e c t i v e grade placement of a unit on symbolic logic.  Infor-  mation concerning the best method of teaching this material and the possible e f f e c t of i t on the abilities or i n t e r e s t s of the students must be ascertained by o t h e r means.  Different  methods of presenting the material, including modifications in both language and concepts, would probably alter the findings of the study.  Since this study was  r e s t r i c t e d t o the topic  of symbolic logic, inferences regarding the possible grade placement of o t h e r topics f r o m justified.  modern mathematics would not  I t must also be noted t h a t conclusions drawn  be from  the study apply to the particular sample cf students used f o r the  experiment.  57  IV.  PROBLEMS  FOR  FURTHER  STUDY  This study has given rise t o several problems which require f u r t h e r study. 1.  "What adjustment should be made in t h e number  of lessons in order t h a t this material might be more e f f e c tively taught as a p a r t of t h e Mathematics 30  course?  Since the lessons were designed t o challenge the Mathematics 101 s t u d e n t s , i t is possible t h a t students in Mathematics 30 did not have s u f f i c i e n t time t o absorb the material. 2.  What is the possibility of adapting t h e conceptual  f r a m e w o r k t o the language o f students in Mathematics  20,  10 or even lower? 3. ial?  What is the best method o f teaching t h e m a t e r -  Since this is a new topic so f a r as the high schools  are concerned,  studies in methodology should be  undertaken.  This might be an area in which a teaching machine program could be used t o good advantage. i+. conducted  S t u d i e s similar t o the present one should be with o t h e r topics o f contemporary mathematics  in order t o gain information regarding t h e i r grade placement.  BIBLIOGRAPHY  59 BIBLIOGRAPHY  A l l e n d o e r f e r , C a r l B . "Deductive Methods in Mathematics," Insights Into Modern Mathematics, pp. 65-99. T w e n t y t h i r d Yearbook o f t h e National Council o f T e a c h e r s o f Mathematics. Washington, D . C : National Council o f T e a c h e r s o f Mathematics, 1957.  New  , and C l e t u s O. Oakley. Principles of M a t h e m a t i c s . Y o r k : M c G r a w - H i l l Book Company, 1955.  B r u m f i e l , C h a r l e s , R o b e r t E i c h o l z , and Merrill Shanks. A l g e b r a I . Reading, Mass.: Addison-Wesley Publishing Company, Inc., 1961. . G e o m e t r y . Reading, Mass.': Publishing Company, Inc., I 9 6 0 .  Addison-Wesley  . "The Ball S t a t e E x p e r i m e n t a l P r o g r a m , " The Mathematics T e a c h e r , L I I I ( F e b r u a r y , I 9 6 0 ) , 7 5 - 8 4 . B r u n s , Richard F . , and Alexander F r a z i e r . "Sequence and Range o f High School Mathematics," T h e Mathematics T e a c h e r , L (December, 1957), 562-566. C h r i s t i a n , R o b e r t R. Introduction t o L o g i c and S e t s . Preliminary edition. B o s t o n : Ginn and Co., 1958. Davis, R o b e r t L . (ed.). E l e m e n t a r y Mathematics o f S e t s . A n n A r b o r , Michigan: Mathematical A s s o c i a t i o n o f A m e r i c a , 1958. E v a n s , H.M. "Organization f o r Curriculum Development," Review o f Educational R e s e a r c h , X X V I I (.June, 1 9 5 7 ) ,  287-294.  G a r r e t t , Henry E . S t a t i s t i c s in Psychology and E d u c a t i o n . New Y o r k : Longmans, G r e e n and C o . , 1949. High School Mathematics, Unit One. T e a c h e r s edition. Urbana, Illinois: U n i v e r s i t y o f Illinois Committee on School Mathematics, 1 9 5 8 - 5 9 .  60 H o y t , C . J . "Note on a Simplified Method o f Computing T e s t Reliability," Educational and Psychological . Measurement, I (.January, 1941), 93-95. Johnson, Palmer O. S t a t i s t i c a l Methods in R e s e a r c h . New Y o r k : P r e n t i c e - H a l l Inc., 1949. Keedy, M . L . Project,"  " T h e U n i v e r s i t y of Maryland Mathematics T h e A m e r i c a n Mathematical Monthly, L X V I  (January, 1959), 58-59. Kemeny, J o h n G., J . L a u r i e S n e l l , and G e r a l d L . Thompson, Introduction t o F i n i t e Mathematics, New Y o r k : P r e n t i c e Hall Inc., 1957Lindquist, E . F . Design and Analysis o f E x p e r i m e n t s in Psychology and E d u c a t i o n . B o s t o n : Houghton M i f f l i n Co., 1953M c C o y , M. E l e a n o r . "The Secondary School Mathematics P r o g r a m of the U n i v e r s i t y o f Illinois Committee on School Mathematics," Bulletin o f the National A s s o c i a t i o n o f Secondary School Principals, X L I I I  (May, 1959), 12-18.  Meder, A l b e r t E . J r . "Modern Mathematics and i t s Place in the Secondary School," T h e Mathematics T e a c h e r .  L  ( O c t o b e r , 1957), 418-423-  National Council o f T e a c h e r s o f Mathematics, S e c o n d a r y School Curriculum Committee. "The Secondary Mathematics C u r r i c u l u m , " T h e Mathematics T e a c h e r . L I I  (May, 1959), 389-417. Province o f B r i t i s h Columbia, D e p a r t m e n t o f E d u c a t i o n , Division o f C u r r i c u l u m . , A d m i n i s t r a t i v e Bulletin • f o r Secondary Schools, I960. V i c t o r i a , B . C . : Queen's P r i n t e r , i960. R e p o r t o f the Commission on Mathematics. Program f o r College P r e p a r a t o r y Mathematics. Princeton, N . J . : College E n t r a n c e Examination B o a r d , 1959. Richardson, Moses. Fundamentals o f M a t h e m a t i c s . Revised edition. New Y o r k : T h e Macmillan Company, 1958.  61 Robinson, F . G . "Change Secondary Mathematics," The B.C. Teacher, X L ( F e b r u a r y , 1961), 219-221. Rourke, R o b e r t E . K . "Some Implications of T w e n t i e t h C e n t u r y Mathematics f o r High S c h o o l s , " The Mathematics T e a c h e r , L I ( F e b r u a r y , 1958), 74-86. School Mathematics S t u d y Group. (March and J u n e , 1959).  N e w s l e t t e r s No.  1 and  2.  S t u d e n t s of A l g e b r a I I , U n i v e r s i t y High S c h o o l , Tallahassee, Florida. "Symbolic L o g i c and Logical C i r c u i t r y in the High S c h o o l , " The Mathematics T e a c h e r , L ( J a n u a r y , 1957), 23-26. Studies in Mathematics E d u c a t i o n . and Company, 1959.  Chicago:  Scott-Foresman  U n i v e r s i t y of Maryland Mathematics P r o j e c t . ' Mathematics f o r the J u n i o r High S c h o o l , F i r s t Book; Second Book; P a r t s I and I I and Teacher's Guide. College P a r k , Md. : College o f E d u c a t i o n , U n i v e r s i t y of Maryland, 1959. Washburne, C a r l e t o n W. "Mental A g e and the A r i t h m e t i c Curriculum," J o u r n a l of Educational R e s e a r c h , X X I I I (March, 1 9 3 D , 210-231. W e r t , J a m e s E . , Charles O. Neidt, and J . S t a n l e y Ahmann. S t a t i s t i c a l Methods in Educational and Psychological Research. New Y o r k : A p p l e t o n - C e n t u r y - C r o f t s , Inc., 1954.  APPENDIX LESSON  A  PLANS  LESSON  Specific  ONE  Objectives  1.  T o explain the need f o r precise language and clear thinking in mathematics.  2.  T o introduce the concept o f a proposition.  3.  T o teach the basic operations disjunction, and negation.  o f conjunction,  Method 1.  ' Discuss  undefined words, " c i r c u l a r " definitions.  2.  A proposition - a sentence - i t s t r u t h o r falsity.  3.  Work out the conventions o f T r u e o r F a l s e f o r the t h r e e operations.  Assignment Work the exercises f o r lesson  1.  M a s t e r y Questions O n separate sheet R e f erences A l l e n d o e r f e r and Oakley, pp 6 - 7 , 1 5 - 1 6 . C h r i s t i a n , pp 1 - 6 . Kemeny e t a l , pp 1-3 N a t . Coun. T e a c h , o f M a t h . Y r b k 2 3 , pp 7 6 - 7 8 .  62+ LESSON  Specific 1.  TWO  Objective T o study t r u t h tables.  Revision E x e r c i s e s  ( w r i t t e n on blackboard)  1.  What is the t r u t h value o f T ( e v b ) ? ~C(~eAb)? (where e and b a r e known propositions)  2.  Which o f the following pairs o f f o r m s are essentially the same? a)  m v n  a v b  b)  rv(svt)  qv(pvm)  c) m v n  m  A  n  Method Work out t r u t h tables f o r the following. (q q), A  (qA-q), (pAq), (pvq),  ~ p , (pA~q) ,  (~p/\q)v(pA-~q) , (mv~n)A~(~m/\n) Assignment Work the exercises f o r lesson 2. M a s t e r y Questions On separate sheet. R e f erences A l l e n d o e r f e r and Oakley, pp 16-17. C h r i s t i a n , pp 7-9• Kemeny et a l , pp 5-9-  65 LESSON  THREE  S p e c i f i c Objectives T o t e a c h implication. T o t e a c h t h e concept o f logical equivalence. Exercises I f m is "2 + 3 = 5" and p is "2 + 3 = 7", determine t h e t r u t h value o f  2.  a) mAn  b) mvn  c )~mA~n  e)^(mvn)  f )~Umv«n)  d) m w i  C o n s t r u c t a t r u t h table f o r b) r A ( r v s )  a) r A ~ s Method  (~pAq)  v (pA~q) .  1.  Review  2.  T e a c h implication:  3-  Develop and explain t h e t r u t h table f o r implication.  4.  Work out the t r u t h table f o r (p -» q) A (Q. -> P)  5.  p*->q  6.  Show t h a t table.  i f - then.  if -f[p) = -<(q)  L,(pvq) U—^HP/wq —  by means o f a t r u t h  ent Work t h e exercises •• Questions . On separate sheet.  f o r lesson  3.  66 References A l l e n d o e r f e r and Oakley, pp 7-9 C h r i s t i a n , p 10. Kemeny e t a l , pp 10-11. NCTM.  23 Y r b k , pp 78-80.  LESSON  Specific 1. Revision  FOUR  Objective T o t e a c h converse, inverse, and contrapositive, Exercises  1.  Construct  a t r u t h table f o r p->„q.  2.  Construct  a t r u t h table f o r  3.  Show the following t o be t r u e :  (~rvs) PAq<—^ qAP  Method 1.  Give definitions o f converse, inverse, and contrapositive.  2.  Construct  the t r u t h tables f o r each.  Assignment Work the exercises f o r lesson 2+. M a s t e r y Questions On separate sheet. References A l l e n d o e r f e r and Oakley, pp 25-27. Kemeny e t a l , pp 39-40. NCTM.  23 Y r b k . , pp 83-84.  LESSON  Specific 1. Revision 1.  FIVE  Objective T o develop an algebra o f logic. Exercises Show the following t o be t r u e : a) ^ ( P A q ) <=—=• ~ p v ~ q b ) ^  (pvq)  <r—»  ^p -q A  c) p p<-vp A  Method 1.  U s e the questions above and show the similarity to algebra. A replaces • , vreplaces+, <—» replaces =.  2.  Develop the laws o f the s y s t e m . Explain t h a t these are merely logically equivalent expressions. T h e y can be proved by t r u t h t a b l e s . P r o v e a couple.  3.  Work the following examples: — (fAt),  (-fAt).'vf,  ~f*t,  XA(-~XVX),  X V (~xvx) . 4.  Solve the folloving equations f o r x. XVf = t (XAf ) V (XAt ) = t  Assignment Work the exercises f o r lesson 5. M a s t e r y Questions On separate  sheet.  References C h r i s t i a n , pp 11-17.  ~t A t ,  ,-,-XVf,  L E S S O N  Specific 1.  Revision 1.  S I X  Objective T o introduce t h e idea o f "black boxes" r e l a t e this t o logical s t r u c t u r e .  and t o  Exercises Write  the converse,  inverse  and contrapositive  of _p_>q. 2.  Make  the t r u t h table  with  that  f o r  f o r ^ r v s and compare i t  r—»s .  Method 1.  Work a)  t h e following  x  examples,  A N  yz_  b)  NOT  D  O R  ^(xAy) (DeMorgan's  c )  Rules)  x v—y  Assignment Work Mastery  the exercises  f o r lesson  Questions On  separate  sheet.  Reference Christian,  pp  20-22.  6.  LESSON  Specific 1. Revision  SEVEN  Objective T o develop "switching n e t w o r k s " and t o relate these t o logical s t r u c t u r e . Exercises XA (xAt) = x  1.  Solve f o r x:  2.  Write an equivalent proposition t o the following; " I t is false t h a t 3 + 7 = 5 o r i t is false t h a t Canada is in the U . S . A . "  3.  Show by means o f t r u t h tables t h a t t h e following propositions a r e equivalent: x-  and  'Xvy  Method Show analogy of series circuits and A parallel circuits and v one switch closed, other open and ~ . Explain t h e similarity between t h e following t w o networks. SiA  S  3  B  S.2R e l a t e this t o logical  operations.  Assignment Work the exercises f o r lesson 7. M a s t e r y Questions. On separate sheet.  51  S3.  5  S3.  A 2  71 Reference Christian,  pp  29-32.  72 LESSON  Specific 1. Revision 1.  EIGHT  Objective T o extend the work on "switching n e t w o r k s " . Exercises D r a w a diagram t o i l l u s t r a t e a) S v 1  (SgvS^)  b)  S  1  A S  2  )  V.  (S^Sg)  Method 1.  Develop an electrical voting machine. e.g. 3 man committee - t o r e c o r d simple majority. D e v e l o p this in such a way t h a t the principle underlying i t may become apparent t o t h e s t u d e n t s . Show o t h e r a r r a n g e m e n t s of v o t i n g m a c h i n e s .  Assignment Work exercises f o r lesson 8. M a s t e r y Questions O n separate  sheet.  References A l l e n d o e r f e r and Oakley, pp 115-118. C h r i s t i a n , pp 2 4 - 2 7 • Kemeny et a l , pp  49-52.  LESSON  Specific  NINE  Objectives  1.  T o teach the use o f quantifiers in propositions.  2.  T o show the use o f negation with q u a n t i f i e r s .  Revision  Exercises  1.  S t a t e the converse, inverse, and contrapositive " I f a quadrilateral has one right angle, i t is a rectangle". Which o f these derived implications true?  2.  Write the converse, inverse, and contrapositive of these implications: a) m-vr-m b) -~m-»~n  Method 1.  T h e following points are t o be emphasized: a) propositions sometimes r e f e r t o s e t s o r collections. b) "some", "all",, "none" r e f e r t o the number of individuals involved - called " q u a n t i f i e r s " . c) symbolism d) negation - the negation of a t r u e proposition must be f a l s e . e) discuss the axioms of negation.  Assignment Work the exercises f o r lesson 9« M a s t e r y Questions O n separate sheet. References A l l e n d o e r f e r and Oakley, pp 13-15, 20-21+. N C T M . 23 Y r b k . , pp 81+-88.  7k L E S S O N  Specific 1. Revision  T E N  Objective Review Exercises (t or f ) of  (~t Af) | f  1.  C a l c u l a t e t h e value  2.  Simplify:  3.  Write  t h e negation  of "All teachers  k.  Write  t h e negation  of  v  A  (tvf)  ~xv(x y) A  are lunatics"  p—»(~q).  Method 1.  E x t e n d t h e ideas o f negation, by w o r k i n g through t h e n e g a t i o n o f " I f s o m e people go s k i i n g , s o m e l e g s will be b r o k e n " .  2.  The the as  r e s t o f t h e l e s s o n is t o be s p e n t w o r k i n g e x e r c i s e s - w i t h t h e t e a c h e r taking up problems the lesson  progresses.  Assignment Work Mastery  t h e e x e r c i s e s f o r l e s s o n 10.  Questions On  separate  sheet.  APPENDIX  B  EXERCISES  76  EXERCISES  FOR  LESSON  1 /  1.  Write the disjunction and conjunction f o r each pair of propositions. a) R o s e s are r e d . V i o l e t s are blue. b) 6 x 3 = 18 3 + 7 = 10 c) A square is a rectangle. .A donkey has t h r e e legs. d) W a t e r is wet. Ice is cold.  2.  W r i t e the negation of the following. a) I am t i r e d . b) 3 is less than 4 c) T h e r e are eight days in a week.  3.  L e t d be "the dog is f r i s k y " and l e t c be "the cat is fast". Write in words the meaning of each of the following. a) ~ d b) dvc c)~d/\-~-c " d) ( ~ d A c ) v ( d A ~ c )  i+.  L e t g_ be "George is b r i g h t " and l e t rn be "Mary is slow", and l e t a be "Alec is s m a r t " . W r i t e each of the following in symbolic f o r m . a) I t is false t h a t G e o r g e is bright. b) George is bright or M a r y is slow. c) I t is false t h a t (George is bright and A l e c is s m a r t ) . d) I t is false t h a t A l e c is not s m a r t .  5.  L e t £ be " 1 + 1 = 2", and l e t £ be " 2 + 1 = 8". D e t e r mine the t r u t h value of each of the following. a) p A ~ q b) pv~q c ) — p*<~q d) ~(~'p/\~q) e) P A ( — p v q )  6.  Which of the following pairs r e p r e s e n t essentially the same f o r m ? a) p/\r s/\t b) p A r svt c) p v ( p A r ) qv(qA.s) d) (p/\~r)vr (q/\~s)vs  E X E R C I S E S  Construct  a truth  table  F O R  f o r each  L E S S O N  of the  1.  PAP  5.  2.  pv~>p  6.  (avb)A-(aAb)  3.  ~(pvq)  7.  (pA~q)v(~p/vq)  2+.  -pv-^q  8.  a truth  (rvs)v~r  table  (pAq)—>.p  3.  2.  p—*(pvq)  2+. Pn-'P  p  be  t h e following  1.  I f it rains,  then  2.  I f t h e wind  blow,  3.  I t is f a l s e t h a t  Show  by  truth  L E S S O N  of the  into  then  be  following:  form:  i t is f a l s e t h a t blows  t h e following  i t is raining.  then are  i t is raining) true:  (pvq)*->~|T-~p  1.  (p-»q) <—7>~[p/\('~q)]  3.  2.  (p,\q)  4. ~ ( p q ) ^ - ^ p v - q  A  is blowing".  blows.  ( i ft h e wind that  " t h e wind  symbolic  t h e wind  tables  (q p)  3  (pvq)-» (qvp)  " i t is raining" and £  Translate  F O R  f o r each  1.  Let  following:  ~(rA~s)  E X E R C I S E S  Construct  2  A  78 E X E R C I S E S  F O R L E S S O N  1+  1.  S t a t e t h e converse, inverse, and contrapositive o f t h e following t r u e implications. a ) I f a i s a n e v e n i n t e g e r , t h e n 2 a i s d i v i s i b l e b y i+. b) I f t h e angles o f a t r i a n g l e a r e all equal, t h e n t h e triangle is equilateral. c) I f a r h o m b u s h a s o n e r i g h t angle, i t i s a s q u a r e .  2.  L e tp—»q be t h e given implication. W r i t e in symbolic f o r m and simplify t h e r e s u l t s : a) t h e c o n t r a p o s i t i v e o f t h e c o n t r a p o s i t i v e o f p — » q . b) t h e c o n t r a p o s i t i v e o f t h e c o n v e r s e o f p — * q . c) t h e c o n t r a p o s i t i v e o f t h e i n v e r s e p—»q.  3.  Write these a) b) c)  t h e converse, implications:  inverse,  q—>P ~p->~q ,-q—>~p  d) e)  E X E R C I S E S  1.  2.  Calculate  t h e value  and contrapositive o f  p-*~q -p—>q  F O R L E S S O N  ( t o r f ) o f each  5  o f t h e following:  a)  (tAt)Af  c)  (tA-f)  V  b)  (tA~f)vt  d)  (~tAf)  v  (fA~t)  (fA(tvf)|  S o l v e each o f t h efollowing equations f o r x. That is, find all values ( t o r f ) f o r x which makes each equation a true statement. T h e r e m a y b e no s o l u t i o n s , o n e solution, o r t w o solutions. a) x v f= t d) = t b) x v f= f e) X A (xAt) = x  XAf  c) 3.  XVt  = t  Using t h e algebra a) X A ( X A X ) b) X A ( x A y ) c ) —XA(XAy)  f )  (XAf) V (XAt)  o f logic, simplify each d) ~ ( - X A y ) e) --'Xv(xAy) f ) '-(xvf) v  = t  o f t h e following:  (XAt)  79 EXERCISES  FOR  LESSON  5  2+.  Write t h e converses of t h e following s t a t e m e n t s : a) E v e r y man born.in B . C . is a Canadian citizen. b) I f plants a r e green, they must have had sunshine.  5.  Write the contrapositives of the following s t a t e m e n t s : a) I f t h e ground hog sees his shadow, then we shall have a cold spring. b) I f you are not s a t i s f i e d , then your money will be returned.  80 EXERCISES  a diagram AND Y) OR NOT AND Y)  FOR  LESSON  6  1.  Draw a) ( X b) ( X c) ( X  f o r each o f the following n e t w o r k s : OR X Y) A N D (X A N D Y) OR (X A N D 2 )  2.  F i n d an equivalent network (by means o f algebra o f logic) which uses f e w e r black boxes. a) ( X O R Y ) A N D ( N O T X O R Y ) b) N O T ( N O T X O R N O T Y ) c) ( N O T X ) O R ( N O T Y ) d) ( X O R Y ) A N D ( X O R 2 )  EXERCISES 1.  FOR  LESSON  7  D r a w .diagrams f o r the following n e t w o r k s : a) ( S S ) V S I c) ( S i v S ) A ( S i v S ) i A  b)  S  1  2  V ( S  2  2  A S  3  )  3  d) ( S ^ S g ) v ( S ^ S ^ )  Write an expression f o r each o f t h e following networks: b)  a)  •s  S^  2 S  l  i .  s3.  Show by t h e algebra o f logic t h a t the networks o f 2(a) and 2(b) a r e electrically equivalent.  81 EXERCISES  1.  What compound s t a t e m e n t  FOR  LESSON 8  r e p r e s e n t s this circuit? QQ  1  Q2.  Work out a t r u t h table o f t h e s t a t e m e n t  in exercise 1.  3.  What does this table tell about t h e circuit?  i+.  Using this information, design a simpler circuit having the same p r o p e r t i e s .  5.  Design a circuit f o r an electrical version o f of matching pennies: A t a given signal each two players either opens o r closes a s w i t c h control. I f they both do t h e same thing, A they do the opposite, then B wins. Design so t h a t a light goes on i f A wins.  6.  A committee has five members. I t takes a majority vote t o c a r r y a measure, except t h a t the chairman has a veto ( i . e . t h e measure carries only i f he votes f o r it). Design a circuit f o r t h e committee, so t h a t each member votes f o r a measure by pressing a b u t t o n and the light goes on i f and only i f the measure is c a r r i e d .  t h e game o f the under his wins; i f t h e circuit  E X E R C I S E S  FOR  L E S S O N  9  W r i t e in w o r d s t h e n e g a t i o n o f : 1. I f w h i t e l o o k s b l a c k , t h e n I am blind. 2. These two lines i n t e r s e c t , o r t h e y a r e p a r a l l e l . 3. I, w i l l g o t o t h e g a m e o r I w i l l g o t o t h e m o v i e . 2+. T o d a y is M o n d a y and t h e w e a t h e r is cold. 5. 6. 7. 8. 9. 10.  S o m e numbers are positive. A l l birds have wings. A l l squares have f o u r sides. A l l soldiers a r e "heroes. S o m e boys play f o o t b a l l . A l l s t u d e n t s in B . C . like  F o r m the negation 1. (~p) — * q 2. (pAq) v r  of:  E X E R C I S E S  Write  the negation  mathematics.  FOR  L E S S O N  10  of:  1. 2. 3.  Some Some  4. 5.  I f all chemicals a r e dangerous, t h e n w a t e r A l l mathematics teachers are brutes.  6.  Some  State 1. 2.  the  sailing is d a n g e r o u s , and all f i s h i n g h a n d w r i t i n g is good o r all s t u d e n t s  c a t s can  swim  i n v e r s e and  I f some I f some  people people  but  chickens  contrapositive  can.  of:  then  he  is  rich.  I f she s i n g s in t h e c h o i r t h e n I will n o t I f I am s l e e p i n g , t h e n I am b r e a t h i n g .  What is t h e green"?  is h a r m f u l .  go s k i i n g , s o m e l e g s will be broken. w o r k h a r d , t h e n a l l p e o p l e do n o t e a t .  S t a t e the converse of: 1. I f a man is f r o m Chicago 2. 3.  no  is t e d i o u s . are lazy.  negation of  "the  sky  is blue and  go  the  to  Church.  g r a s s is  APPENDIX MASTERY  C  QUESTIONS  84 Q U E S T I O N S  Place the letter to the right. 1.  T O  T E S T  M A S T E R Y  corresponding  O F  to the best  L E S S O N  answer  1  o n t h e line  L e t s_ b e " t h e s u n i s s h i n i n g " a n d l e t r b e " i t i s r a i n i n g " . G i v e t h e v e r b a l m e a n i n g o f '~s./\r. a) b) c) d) e)  t h e s u n is shining and i t i s raining. i t is f a l s e t h a t t h e s u n is shining o r i t i s raining. t h e s u n i s shining o r i t i s f a l s e t h a t i t i s r a i n i n g . i t i s f a l s e t h a t t h e s u n i s shining and i t i s r a i n i n g . i t i s f a l s e t h a t t h e s u n is shining and i t i s f a l s e t h a t i t is raining. 1.  2.  L e t s_ a n d r h a v e t h e m e a n i n g s a s a b o v e . verbal meaning o f ( ~ s v r ) v ( s/\~~r) .  Give the  a)  ( i t is f a l s e t h a t t h e s u n i s shining and i t is raining) and ( t h e s u n is shining o r i t i s f a l s e t h a t i t is raining).  b)  ( t h e s u n i s shining and i t is raining) o r ( t h e s u n is shining and i t is f a l s e t h a t i t is r a i n i n g ) . ( i t i s f a l s e t h a t t h e s u n is shining o r i t i s raining) o r ( t h e s u n i s shining and i t is r a i n i n g ) .  c) d) e)  ( t h e s u n i s shining o r i t i s raining) o r ( t h e s u ni s shining and i t is f a l s e t h a t i t is r a i n i n g ) . ( i t i s f a l s e t h a t t h e s u n i s shining o r i t i s raining) and ( t h e s u n is shining o r i t is raining) 2.  3.  L e t s_ a n d r b e a s a b o v e . G i v e t h e s y m b o l i c f o r m f o r ( I t i s f a l s e t h a t t h e s u n is shining and i t i s f a l s e t h a t i t i s raining) o r ( t h e s u n i s shining o r i t i s raining). a)  (~SA~r)  v  b) c)  (-sv~r) v ( s v r ) (-^s/\~r) A ( s v r )  d)  (-^s/\r  e)  none  ) v  (~~'svr)  (~SA~r)  of these 3.  85 4.  I f £ is "1+3=4" and i f £ is "2+1=7", determine t h e t r u t h value o f ~pv^q. a) T  b) F  c) impossible t o determine k.  5.  I f p_ is "1+3=4" and i f £ is "2+1=7", determine t h e t r u t h value o f p A ( p v ~ q ) . a) T  b) F  c) impossible t o determine 5.  86 Q U E S T I O N S  Place  TO  the letter  T E S T  corresponding  line  to the right.  1.  What  is t h e final T T  b)  column c)  T F  F F  M A S T E R Y  T F T T  F T  O F  to the best  of the truth d)  e)  T T T F  L E S S O N  answer  table  2  on t h e  o f ~ pvq?  T F F F 1.  2.  What  is t h e final  column  (rA~a)  a)  T F F T  b)  F T T F  Q U E S T I O N S  1.  What T T  v  •>c)  F F T F  TO  T E S T  is t h e final  column  b)  T  F T F  F  T  c)  of the truth  (—TAa)  T T  d)  F T F T  e)  of  T F T F  M A S T E R Y  O F  of the truth d)  table  ?  e)  T F  F  T  T  F  L E S S O N  table  o f p—>~q?  F T T T 1.  2.  What  is t h e final  column  (TAS)—»r a)  T  b)  T  c  )  T  F  F  <p  F  T  T  of the truth  table  of  ? d)  F F F F  e  )  3  T F  T p 2.  87 QUESTIONS  TO  TEST  Place the l e t t e r corresponding to the r i g h t . 1.  MASTERY  OF  L E S S O N h,  t o the best answer on t h e line  Which one of t h e following is logically equivalent t o (pvq)vr ? a)  (p/\q)v(pAr)  c) pA(qAr) e) none o f these  b) p v ( q v r )  d) (pvq)A(pvr) 1.  2.  Which one o f the following is logically equivalent t o  P-*q ?  a) p (~q) c) pv(~q) A  e) none of these  3.  b (~a)—»(~b) d (<~b )—* ("-a) 3.  I f an implication is t r u e , what may be said about i t s converse ? a) always t r u e c) never t r u e  5.  '2.  Given a—»b, which one of t h e following is the expression f o r t h e contrapositive? a)~a—*b c)-~b—*a e) b—^»a  i+.  b) •~[pA(~ql] d) _[p v (~q)j  b) not always t r u e d) none o f these  What would be t h e inverse o f the implication ~ a—»~b? a)  a—>b  c) b—»a e) ~a —>-b  b) r-b—>~a  d) ^ b — * a  ;  5.  88 6.  What is t h e final column in the t r u t h table converse o f an implication? a)  7.  T F T T  b) T T F T  c) T F T F  d) T T T T  o f the  e) none o f these  6.  W r i t e in symbolic f o r m and simplify t h e r e s u l t : "The inverse of the contrapositive o f t h e implication a—• b." a) ( ~ a ) ^ ( ~ b ) c) b—>a e) none o f these  b) d)  (~b)-*(~a) a—fb  89 QUESTIONS  P l a c e t o  1.  t h e  t h e  TEST  MASTERY  c o r r e s p o n d i n g  t o  t h e  OF LESSON  b e s t  a n s w e r  on  c o m p l e m e n t following  l a w s  m a y  b e  r e p r e s e n t e d  a)  pA—<p«-»f  b)  p A t*—»t  c)  pAp<->p  d)  p  e)  n o n e  o f  b y  w h i c h  e x p r e s s i o n s ?  A  q«-?q A  p  t h e s e  1.  Which one o f t h e following laws is r e p r e s e n t e d by the expression p A ( q v r ) * — » ( p A q ) v ( p A r ) ? a) A s s o c i a t i v e law c) D i s t r i b u t i v e law e) Idempotent law 3.  Solve f o r x:  Solve f o r x:  b) Commutative law d) DeMorgan's rules  xvt=f  a) t c) t , f e) none o f these  4.  b) f  d) no solution possible 3-  x—»f=f  a) t c) t , f e) none of these  b) f d) no solution possible  4.  Calculate t h e t r u t h value o f (tA-'t)vf. a) T  b) F  c) can't be calculated 6,  5  t h e  line  r i g h t .  T h e t h e  l e t t e r  TO  5.  Calculate t h e t r u t h value o f jf v( t v ~ ~ f )j A j ~ f A t a) T c) can't be calculated  b) F 6.  one  o f  90 QUESTIONS  TO  TEST  MASTERY  OF LESSON  6  Place t h e l e t t e r corresponding t o t h e best answer in t h e space t o t h e right. 1.  F i n d an equivalent n e t w o r k which uses f e w e r black boxes. ( X AND Y ) OR ( X A N D 2 ) a) X A N D ( Y O R 2 ) c) ( X A N D 2 ) O R Y e) none of these  b) d)  X OR (Y A N D NOT ( X AND  2) Y ) 1.  2.  Which of t h e logical designs below is a r e p r e s e n t a t i o n o f this system o f "black boxes"? X  > A N Y > D —H N O T  O R  a)  (xAy)vx  b) ( x v y ) A ~ X  c)  (XAV)V~X  d)  e) none o f these 3.  (xvy)  A X  2.  Which o f t h e following propositions is equivalent t o t h e proposition " ( P a t went t o a play) and (George stayed home o r M a r y went t o t o w n ) " ? a) ( P a t went t o a play) o r (George stayed home and M a r y went t o t o w n ) . b) ( P a t went t o a play o r George stayed home) and ( P a t went t o a play o r M a r y went t o t o w n ) . c) ( P a t went t o a play and George stayed home) or ( P a t went t o a play and M a r y went t o town) . d) ( P a t went t o a play o r George stayed home) o r ( P a t went t o a play o r M a r y went t o t o w n ) . e) none o f these 3'  i+.  Solve f o r x:  (^xAt)vf = f  a) t c) t , f e) none o f these  b) f d) no solution possible 1+,  Simplify the  following  a) x c) t e) none of these  expression: ^ ( x v f ) v ( x A t ) b) y d) f  • ;  QUESTIONS  TO  TEST  MASTERY  OF  LESSON  7  Place the l e t t e r corresponding t o the best answer in the space t o the right. 1.  Which one of t h e following designs would be equivalent to  (s^ A S g )  V  (S1AS3)?  a) (s^vsg) A (sivsgi)  b) ( s i v s p j A S3  c)  d)  S ^ A C S ^ V S ^ )  siv(s2AS3)  e) none of these 2.  1.  Which logical design r e p r e s e n t s the following n e t w o r k ? s  l -  • 2 s  s  a)  (S^AS2)VS3  b)  c)  ( s ^ V S g ) A S 3  d) (s^vsg) V S 3  e )  3.  3  n o n e  o f  t h e s e  ( S ^ S ^ J A S ^  2.  Which of t h e following diagrams r e p r e s e n t s this logical design? [pv (~p/\~q)j v [i>AqJ  93  e)  3.  94 QUESTIONS  TO TEST MASTERY O F LESSON 8  P l a c e t h e l e t t e r corresponding t o t h e b e s t a n s w e r in t h e space a t t h e r i g h t . 1.  The following r e p r e s e n t s a p o r t i o n o f a t r u t h t a b l e developed when t r y i n g t o design a c i r c u i t t o solve a problem. "Which one o f t h e logical expressions would be a p a r t o f t h e r e s u l t i n g design? Desired X Y 2 t r u t h value  a) c) e)  2.  T  F  T  T  T  T  T  T  F  T  F  F b) XVY^-'VS  XAYA2  d) X . V Y A Z  XAYVZ X W Y V Z  1  1.  1  W r i t e t h e logical design o f t h i s c i r c u i t  ^2  •Si-  s  S6-  5  s  a)  S i ^ fS  b)  S  c)  Si/\r ' ( s  1  /\^(S  2  V S )/\S 3  AS  2  2  7  v s  3  ) 3  V  k  V  s  ) A S  j 4  j  5  A |(S V  V  ( S /\S ) 6  V  5  |(S  5  AS  Sy  S )ASr 6  6  )  V S^j,  95  d)  V  |(S  e)  V  J(s  2  2  V  S ) AS |j>  A  s  3  3  4  v  V  s^lr A  |(S A S 5  (S  5  V  V  S  ?  S A  S  7 j  6  6  2.  QUESTIONS  TO T E S T  MASTERY  OF LESSON 9  Place t h e l e t t e r corresponding t o t h e b e s t answer in the space t o t h e right. What is the best s t a t e m e n t f o r t h e negation o f "Some people a r e f a t " ? a) c) d) c)  2.  A l l people a r e f a t . b) A l l people a r e not f a t . Some people a r e not f a t . No people a r e f a t . I t is false t h a t some people a r e f a t .  What is t h e negation o f a)  Vx  c) V x e) 3.  [(~p)—(~q)]  [ P A  (~q)|  Vx  ]x e) Vx c)  —  (~q)  (pAq)  2.  none o f these  What is the negation o f a)  Vx [ ( ~ p ) b) "]x [p-» d) ^ x  [(~p)  A  (~q)  [(~p) A (~q)  ^x  (pAq) ?  b)  '-Vx  d)  ^x [(~p)  (pvq ) v (~q)  [(~p) v (~q) 3-  97 QUESTIONS  TO  TEST  MASTERY  Place t h e l e t t e r corresponding space t o t h e right. 1.  OF  ^x  a) ~]x  pp^q)  b)Vx[~(pvq)  ]x  [(~p) v  (~ y q  10  t o t h e best answer in t h e  What is the negation of  c)  LESSON  {j '~ ^ )A  v<  ^  d) Vx [(~p) v (~q)]  e) none of these  1. 2.  What is t h e negation of t h e proposition "Some bears are bown and some dogs a r e white"? a) b) c) d) e)  3.  No bears a r e brown o r no dogs a r e white. No bears a r e brown and no dogs are white. Some bears a r e brown o r some dogs a r e white. Some bears are not brown and some dogs a r e not white, none o f these  What is t h e inverse of " I f some s c i e n t i s t s a r e subversive, all s c i e n t i s t s a r e subversive"? a) I f some s c i e n t i s t s a r e not subversive, then s c i e n t i s t s a r e subversive. b) I f no s c i e n t i s t s are subversive, then some are not subversive. c) I f all s c i e n t i s t s are subversive, then some are subversive. d) I f some s c i e n t i s t s a r e subversive, then no are subversive. e) none o f these  no scientists scientists scientists  3.  APPENDIX FINAL  D  TEST  99 F I N A L  Place  T E S T  the letter  ON  S Y M B O L I C  corresponding  L O G I C  to the best  answer  on t h e  line t o t h e r i g h t . In all w o r k with t r u t h tables, i t is assumed t w o c o l u m n s will a l w a y s be a s f o l l o w s : p  q  T T  T F T F  F F  1.  that  L e t s_ b e " t h e s u n i s s h i n i n g " a n d l e t r b e G i v e t h e verbal meaning o f s v ~ r . a) b) c) d) e)  t h e sun is i t is false t h e sun is i t is false t h e sun is  the  first  " i t is raining".  shining and i t i s r a i n i n g . t h a t t h e sun is shining o r i t is raining. shining o r i t is f a l s e t h a t i t is raining. t h a t t h e s u n is shining and i t is raining. shining and i t is f a l s e t h a t i t is raining. 1.  2.  Calculate a)  the truth  T  value  b ) F  c)  of can't  ( t A t ) A f .  be  calculated.  2. 3.  Which o f t h e logical designs below this s y s t e m o f "black boxes"?  is a  representation  A  Y  N NOT  a)  (XAY)VX  b)  c) ( X A Y ) V~X e)  none  of  these  >  D  (XVY)A-X  d ) (XVY')AX 3.  of  Which logical design r e p r e s e n t s the following network?  '1 52i 5  a)  ( S  c)  (SjASg) V S 3  e)  none of these  1  A S  2  ) A  3'  b)  S  d)  ( S 3 V S 2 )  1  V S  2  ) A s  3  V S 3  4,  What is the final column of the t r u t h table f o r — ' p — ? a)  T T T F  b)  F T F T  c)  T T F T  d) F  T  e)  F  T T  F  T T  Which one of the following diagrams r e p r e s e n t s this logical design? ±  a)  AJSJA (S  2  V S  3  )  V  Js^A  b)  I —  T-SlL - S  3  I—  - s — s 4  '3 c)  d)  5  101 e)  3"  5  -S  4  7.  G i v e n a—>b, of  8.  the  6.  which  one o f t h e following  is t h e  expression  inverse?  a ) '-a—>b c) b—> a •e) b ^ a  b ) -ja-»~b  I f an implication is t r u e ,  what  d) ~b-»'~a  7.  may  be said  about i t s  contrapositive? a) c)  always true never true  b) n o t always t r u e d) none o f t h e s e 8.  9.  I f £ i s "1+3=4" a n d i f £ i s "2+1=7", d e t e r m i n e t h e truth a)  value  T  of  A (~pv~q).  b) F  c) impossible  t o determine  . 9. 10.  What a)  is t h e final  column  b)  c)  T T F T  T F F  of the truth T F T  d)  e)  T T T  T  T  table  F  of  p/»—q?  T F F F  10. 11.  What a ) Vx c)  Vx  is t h e negation  of  Vx Ip-^q] ?  [~p-*~q|  b.) jx  [p  d)  A  ~ql  ]]x  [p—»~qj [p  A  q]  102 L  12.  11.  J  Which one of t h e following designs would be equivalent t o (S VS ) A (S VS ) 1  a)  2  ( S  1  1  A S  2  )  V  3  ( S  c)  SiACS VS )  e)  none of these  2  1  A S  3  )  b)  ( S  d)  3  1  V S  SiV  2  ) A S  3  (S AS ) 2  3  12. 13.  Solve f o r x:  x A f=f  a)t  c ) t , f  b ) f  d)  no solution possible  13.  12+.  Calculate the t r u t h value of ( ~ t v f ) A [t a)  T  b) F  c) Impossible  A  (tAf^J  to-determine  14. 15.  L e t s_ be "the sun is shining" and l e t r? be " i t is raining". Give the verbal meaning of the following. (  XAr) v  (sA^r)  a) (it is f a l s e t h a t the sun is shining and i t is raining) and (the sun is shining or i t is f a l s e t h a t i t is raining). b) (the sun is shining and i t is raining) or (the sun is shining and i t is f a l s e t h a t i t is raining). c) (it is f a l s e t h a t the sun is shining or i t is raining) or (the sun is shining and i t is raining). d) (the sun is shining or i t is raining) or (the sun is shining and i t is f a l s e t h a t is is raining). e) (it is f a l s e t h a t the sun is shining or i t is raining) and (the sun is shining o r i t is raining).  1  103 16.  I f £ is "1+3=2+" and i f £ is "2"+l=7" , determine the t r u t h value o f ^~pn~q. a)  T  b)  F  c)  impossible t o determine 16.  17.  What is t h e final column in the t r u t h table o f ( r A sO v ~> ( rv;s ) ? a)  T F F T  b) F T T F  c)  F F T F  d)  F T F T  e)  T F T F  17. 18.  What is the best s t a t e m e n t o f t h e negation o f " A l l dogs have f o u r legs"? a) b) c) d) e)  No dogs have f o u r legs. A l l dogs do not have f o u r legs. I t is false t h a t all dogs have f o u r legs. Some dogs do not have f o u r legs . None o f t h e s e . 18.  19.  What would be t h e converse o f t h e implication -~ba) c) e)  a —> b b —> a b—^-~a  b ) ~b —> a d) ~a—> ~ b  19. 20.  What is t h e negation o f a)Vx  j^pvq)]  c) V x  [J~(p-->q)J  e)  ^jx  [('--p)  b) ] x d)  ^x  > (~q)J  [H?)A(~q)] [^(pAqjJ  none of t h e s e .  20.  21.  Which one o f t h e following is logically equivalent t o (pvq)? a) ~ ( p v q ) c) pv^q e) none of t h e s e .  b) d)  (P*q) ~[T~p)  21, 22.  W r i t e in symbolic f o r m and simplify t h e r e s u l t : T h e converse o f t h e inverse o f &—>b. a) ( - a ) ^ ( - b ) c ) b—>a e) none o f t h e s e .  b) d)  -b)-*b  (~a)  22, 23.  What is t h e final column o f t h e t r u t h table o f (rvs) >r? T T T F  a)  21+.  b)  F T F T  c)  d)  T F T F  e)  F T T T  23.  L e t s be "the sun is shining" and l e t r be " i t is raining". Give t h e symbolic f o r m f o r : ( I t is f a l s e t h a t t h e sun is shining and i t is not raining) o r (the sun is shining and i t is raining). a) (~s/v~r) v (~-~svr) c) ( < ~ S A ~ r ) A ( s v r ) e) none o f t h e s e .  25.  T T F T  b) (~sv~r) v ( s v r ) d) h s / i r ) v (~s/i~r)  21+.  Which one o f t h e following is logically equivalent t o p (qvr) ? A  a) ( p q ) v (p«r) c) pv (qvr) e) none o f these A  b) p A ( q/\r) d) ( p v q ) A ( p v r )  25. 26.  Solve f o r x: a)  t  b) f  x—*t=f c ) t , f d ) n o solution possible. 26.  105 27-  Find an equivalent network which uses f e w e r black boxes; NOT ( NOT X AND NOT Y ) a) X O R Y c) N O T X A N D N O T e) none o f these  Y  b) N O T X O R d) X A N D Y  NOT  Y  27. 28.  Solve f o r x: a)t  b) f  (xAf)A(xvt)  =t  c) t , f d ) n o solution possible. 28.  29.  What is the final column in t h e t r u t h table o f the contrapositive o f an implication? a)  30.  b)  T T F T  c)  T F T F  d) T e) none o f these T T T 29.  Which one o f t h e following propositions is equivalent t o the proposition " I t is f a l s e t h a t ( P a r i s is in F r a n c e and 2+1 = 2+)"? a) b) c) d) e)  31.  T F F T  Paris Paris Paris Paris None  is not in F r a n c e and i t is false t h a t .2+1=2+. is not in F r a n c e o r i t is f a l s e t h a t 2+1=4. is in F r a n c e and 2+1=4. is in F r a n c e o r 2+1=4. of these. 30.  What is the inverse o f " I f all prices a r e falling, some businesses a r e failing"? a) I f no prices a r e falling, then some businesses are not failing. b) No prices a r e falling and no businesses are failing. c) I f some prices a r e falling, all businesses a r e failing. d) I f no prices a r e falling, then some businesses a r e f ailing. e) I f some prices a r e not falling, then no businesses are f ailing. 31.  106 32.  What is t h e negation o f Vx (pvq) ?  a) Vc(~p/\~q) c) Jx(<~p/\~q) e) Nx i (~p) v (~q) '  b) ~\/x (pvq) d) ]x (~pv~q)  32. 33-  Simplify t h e following expression: a)x  b)y  c ) t  d ) f  (xvy) A, (xv—y)  e) none o f t h e s e .  33. 34.  What is t h e negation o f the proposition; "Some people work hard, o r all people do not e a t " . a) Some people do eat. b) No people work c) Some people do d) No people work e) None o f t h e s e .  not work hard and all people do not hard o r some people e a t . not work hard o r all people e a t . hard and some people e a t .  34. 35.  Which o f t h e following laws is represented by t h e expression ^~(pvq)«—>~p/\~>q ? a) A s s o c i a t i v e law c) D i s t r i b u t i v e law e) None o f these  b) Commutative law d) DeMorgan's Rules  35. 36.  Write t h e logical design o f this c i r c u i t .  3  "4"  107  ( S I V S 2 V S 3 ) A  b)  (s^vs^vs^)  c)  (S^AS^AS-^) A [ ( s v s v s ) A s^j  (  s  S]_A  V  S  I  +  v [ ( s i v s g v s ^ ) vs^j  1  d) (s-^ASgAS^) e)  [( S I A S 2 A S 3 )  ~J  a)  A  j|(  2  3  S - ^ A S g A S ^  )  2 A 3 ) v JTS1VS2VS3) s  A  S^J  A .S4J S  36. The following r e p r e s e n t s a portion o f a t r u t h table developed when t r y i n g t o design a circuit t o solve a problem. Which one o f t h e logical expression would be p a r t o f t h e resulting design? Desired T r u t h value  A  B  C  T  F  - T  F  T  T  T  T  F  T  F  F  a) A A B A C  b) A V B  c) A A B A C  d)  VC  AVBVC  e) A l A B A C l 37. What is t h e converse o f : "You must be in grade 10 i f you a r e taking  geometry"?  a) Y o u must be in grade 10 and you a r e not taking geometry. b) I f you a r e n o t in grade 10, yo'u a r e not taking geometry. c) I f you a r e not taking geometry, you are not in grade 10. d) I f you a r e in grade 10, then you a r e taking geometry. e) None o f t h e s e . 38.  108 What is t h e negation of "The apple harvest is poor and everyone will a) I f everyone s t a r v e s , b) T h e apple harvest is not s t a r v e . c) T h e apple harvest is will not s t a r v e . d) T h e apple harvest is starve. e) None o f these.  starve"?  t h e apple harvest is poor, notpoor o r some people will notpoor and some people notpoor o r everyone will  39. What is t h e converse o f "The picnic will be cancelled in case o f rain"? a) b) c) d) e)  I f i t rains, the picnic will not be cancelled. I f t h e picnic is not cancelled, i t will not rain. I f t h e picnic is cancelled, i t will rain. T h e picnic will not be cancelled i f it does not rain. None o f these 40.  APPENDIX RELIABILITY  OF  E  FINAL  TEST  110 RELIABILITY  OF  FINAL  TEST  Hoyt's formula f o r t h e reliability coefficient is  tt where  KS  n  R  KS  n-1  + S i - T (T+K) - 'T* S  r ^ is the coefficient of reliability n is the number of items K is t h e number of subjects S  s  is the sum of t h e squares of t h e s c o r e s f o r all subjects  S j is obtained by squaring the number of c o r r e c t responses f o r each item. T The  is the sum of t h e scores f o r all subjects.  data related t o the computation n  K  2+0  387  T  8773  rjn 2  are as follows: i  S  2,133,763  219,863  s  76,965,529  The  standard deviation of t h e scores is  The  standard e r r o r is given by  S  m  7.365  = CJ~Jl-r^.  Using this formula, t h e standard e r r o r is 2.70 r  t t  = .87  s  APPENDIX RAW  SCORE  F DATA  112 T A B L E XII INTELLIGENCE QUOTIENTS AND LOGIC T E S T S C O R E S OF M A T H E M A T I C S 101 S T U D E N T S  Name  Otis- D G T C C I.Q. I.Q.  AA AB AC AD AE AF AG AH Al AU AK AL AM AN AO  106  117 115 108 108 •120  112+ 109 112+ 112+ 112+ 120  117  120 122+  107 121 122 125  109 118  123 112 127  132+  112 123 116 129 122+  Logic  Name  27  BA BB BC BD BE BF BG BH BI BJ BK BL BM BN BO BP  31 33 29 22  32+ 33 35 38  32 29 36  34 31 34  Otis D G T L C I.Q. I.Q.  115 108  103 112 106 102 •100 116  117 110 113 105 101 121 105 109  117 111 107 111 . 105 120 108  116 119 99 134 110 96 115 121 115  Logic  27 34 25 27 32 29 27 29 36 27 38 32 35 33 16 26  113 TABLE  XIII  INTELLIGENCE QUOTIENTS AND LOGIC T E S T S C O R E S O F M A T H E M A T I C S 91 S T U D E N T S  Name  Otis Logic I.Q. S c o r e  FA PB PC PD PE PF PG PH PI PJ PK PL PM PN PO PP PQ PR PS PT PU  108  120 108 118  111 101 111 104 101 135 117 120 103 103 120 115 112 133 127 99 116  25 32 19 27 34 25 28  30 21 31 23 21 27 30 34 24 28  37 39 30 28  Name  QA QB QC QD QE QF QG QH QI QJ QK QL QM QN QO QP QQ QR QS QT  Otis Logic I.Q. S c o r e 111  105 113 112 111 108  123 107  113  117  23 31 35  32 32  34 35  19  20 24  117 134  34  109  32  117 115 119  33 28 35 25 33  116 107 121  108  23 30 29  Name  RA RB RC RD RE RF RG RH RI RJ RK RL RM RN RO RP RQ RR RS RT  Otis Logic I.Q. S c o r e  100 106 106 105 123 107 121 112 103 117 120 109 112 111 110 116 113 108  109 102  28  25 20 20 31 33 32 28  15 27 28  27 30 17 29 30 30 21 19 31  lli+ TABLE XIV INTELLIGENCE QUOTIENTS AND LOGIC T E S T S C O R E S OF M A T H E M A T I C S 30 S T U D E N T S  Name  MA MB MC MD ME MF MG MH MI MU MK ML MM MN MO MP MQ MR MS MT MU MV MW MX MY MS JA JB JC JD UE J F  JG JH UI IQ  Otis I.Q. 103 110 112 107 ' 120 116 109 112 95 115 119 121 108 117 83 126 111 116 122 101 90 116 111 122 97 93 ;95 107 104 100 106 123 119 99 103 103  Logic Score  Name  25 18 11 20 18 24 26 31 32 22 26 24 27 28 20 12 22 3423 26 9 12 24 27 16 18 17 18 27 23 22 31 30 28 16 21  NA NB NO ND NE NF NG NH NI NJ NK NL NM NN NO NP NQ NR NS NT NU NV NW NX NY NZ UU UK UL  JM UN J O  UP UQ J R  Otis I.Q. 104 110 118 119 121 97 109 108 100 106 124 ' 101 97 109 112 124 126 103 112 115 107 105 111 108 109 114 114 107 117 115 110 123 107 126 104  Logic Score 14 23 20 25 '32 22 35 18 25 23 35 20 17 25 31 37 25 25 32 31 15 25 18 29 25 15 20 16 29 24 25 33 21 35 17  Name  Otis Logic S I .Q. c o r e  OA OB OC OD OE OF OG OH OI OU OK OL OM ON OO OP OQ OR OS OT OU OV OW OX OY OS US UT UU UV UW  116 118 96 114 94 113 110 113 115 113 107 96 112 107 118 117 113 109 105 124 116 111' 115 110 118 113 100 101 89 114 125 103 108 113 110  u x  UY US IP  10 32 31 25 23 30 28 31 27 31 30 17 27 11 33 16 25 25 24 17 24 28 19 28 17 22 ...11 24 21 27 33 29 23 25 32:  115 TABLE  XV  INTELLIGENCE QUOTIENTS AND LOGIC T E S T S C O R E S O F M A T H E M A T I C S 20 S T U D E N T S  Name GA GB GC GD GE GF GG GH GI GJ GK GL GM GN GO GP GQ GR GS GT GU GV GW GX GY GS IA IB IC ID IE  Otis Logic I.Q. S c o r e 122 101+ 119 127 111 114 122 120 117 115 107 111 120 113 108 109 110 113 99 116 118 96 117 96 115 115 111 118 111 108 111  27 11 18 20 25 10 24 19 16 13 16 20 32 20 25 24 15 18 21 28 24 15 9 18 28 27 15 20 15 19 22  Name  Otis I.Q.  HA HB HC HD HE HF HG HH HI HJ HK HL HM HN HO HP HQ HR HS HT HU HV HW HX HY HS IF IG IH II IJ  108 126 119 104 119 117 119 115 117 129 109 115 117 117 127 112 111 105 99 102 118 117 110 105 109 111 123 113 123 98 131  Logic Score 18 29 12 18 23 16 27 28 16 17 15 20 28 16 24 20 19 11 17 13 26 18 12 31 10 19 27 18 37 15 28,  Name  LA • LB LC LD LE LF LG LH LI LJ LK LL LM LN LO LP LQ LR LS LT LU LV LW LX LY LS . IK IL IM IN IO  Otis I.Q. 102 101 95 109 120 107 105 124 103 109 107 120 101 128 115 93 114 112 122 110 99 122 122 122 106 102 132 117 -122 111 116  Logic Score 14 33 23 30 18 14 19 31 15 18 20 13 10 10 32 25 9 : 27 25 18 15 29 22 21 19 18 31 24 17 14 35  116 TABLE  XVI  INTELLIGENCE QUOTIENTS AND LOGIC T E S T S C O R E S OF M A T H E M A T I C S 10 S T U D E N T S  Name CA CB CC CD CE C F  CG . CH CI CJ CK CL CM CN CO CP CQ CR CS CT CU CV CW  cx  CY CS FA FB' FC FD FE FF  Logic Otis I.Q. S c o r e 115 133 123  106 125 94  6 32  16 9  24 13  10 121 20 . 122 • 24 101 17 . 104 9 116 21 111 20 118 24 107 16 10 99 108 20 111 16 103  133  112 104 121 117 115  111 ' 113  110 120 102 107 117  112  25 15  11 30  21 18 24 17 13  15 12 11 21 23  Name  Otis Logic I.Q. S c o r e  Name  Otis Logic I.Q. S c o r e  DA DB DC DD DE DF DG DH DI DU DK DL DM DN DO DP DQ DR DS DT DU DV DW DX DY DZ FG FH FI F J FK FL  126 111 109 118  20  117  26  14  104 100 102 121  17  104  EA EB EC ED EE EF EG EH EI EJ EK EL EM EN EO EP EQ ER ES ET EU EV EW EX EY ES FM FN FO FP FQ FR  14 15  29  116 105  18 10 27 16  136  107 $02 1  99  115  118  104 119  92 117  118 131  125 119  122 122  N  14 17 24  11 12 8 10 16 18 31 32  10 25 24  104 124 117  17 20 21  107 121 116 124 100  '22  113  13  26 15 24  12  ,  113  112+ 108 112 116 118 100 110 102 111 115  106 127 117  112 113 119  106 131  106 119 113 113 113  110 • 127 110  8 10  23 13 15  20 24 19 14  18 22 11 17 13 15 31  16 19 15  16 15  22 20 27 17  10 20 11 36  6  

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