<http://www.w3.org/2009/08/skos-reference/skos.html#note>
"CONDITIONAL SYLLOGISTIC REASONING AND WORKING MEMORY CAPACITY by STEWART T. INSTANCE B.S., Cornell U n i v e r s i t y , 1970 M. Arch., State U n i v e r s i t y of New York at B u f f a l o , 1974  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS in THE FACULTY OF GRADUATE STUDIES (Human Learning, Development and I n s t r u c t i o n Program, Department of Educational Psychology and S p e c i a l Education)  We accept t h i s thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA September 1984  &lt;g)  Stewart Instance, 1984  \u000CIn p r e s e n t i n g  t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of  requirements f o r an advanced degree at the  the  University  o f B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make it  f r e e l y a v a i l a b l e f o r reference  and  study.  I  further  agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying of t h i s t h e s i s f o r s c h o l a r l y purposes may  be granted by the head o f  department or by h i s or her r e p r e s e n t a t i v e s .  my  It i s  understood t h a t copying or p u b l i c a t i o n o f 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 allowed without my  permission.  Department of  EDUCATION  The U n i v e r s i t y of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 13th. OCTOBER. 1984  DE-6  (3/81)  written  \u000CABSTRACT  ii.  A r e l a t i o n s h i p between working memory capacity and p r o p o s i t i o n a l reasoning a b i l i t i e s i s examined within the framework of Marcus &amp; Rips (1979) verification  model of c o n d i t i o n a l syllogisms and the mental operator model of  c o g n i t i v e development proposed by Pascual-Leone (1970). verification  Using the four-stage  model to explain required c o g n i t i v e processes, i t i s argued that  development i n the a b i l i t y to solve c o n d i t i o n a l syllogisms can be a t t r i b u t e d , i n part, to an e p i g e n e t i c a l l y determined increase  i n working memory capacity.  With a sample composed of 77 pre-adolescent and u n i v e r s i t y students, micro-computers presented i n d i v i d u a l subjects with two 40-item c o n d i t i o n a l s y l l o g i s t i c reasoning (CSR) tasks and a backward d i g i t span (BDS) task, i n two sessions. The  r e s u l t s are not as predicted.  Indexing memory capacity by BDS, analyses  of covariance and polynomial regression a n a l y s i s , f a i l to i d e n t i f y a r e l a t i o n s h i p with correct CSR responses.  While grade i s shown to e x p l a i n a  major percentage of variance i n CSR scores, knowledge of the .conditional r u l e i s also i d e n t i f i e d  as an important f a c t o r .  Arguments are grouped according to  order of d i f f i c u l t y and v a l i d a t i n g response time, and the r e s u l t s of subjects identified  as knowing the c o n d i t i o n a l r u l e f a i l to agree with the groupings  predicted  by the Marcus &amp; Rips model while supporting development of a s i n g l e  operative  scheme f o r c o n d i t i o n a l s y l l o g i s t i c reasoning.  \u000Ciii.  TABLE OF CONTENTS Page I.  II.  STATEMENTS OF PROBLEMS AND HYPOTHESES  1  Introduction C o n d i t i o n a l S y l l o g i s t i c Reasoning Developmental Trends i n C o n d i t i o n a l Reasoning V e r i f i c a t i o n Model of C o n d i t i o n a l Syllogisms Theoretic Role of Working Memory Investigations i n t o Development i n Working Memory Capacity Working Memory Capacity and the Verf i f i c a t i o n Model Summary of Hypothesis  1 1 2 4 6 6 11 11  METHOD  14  Subjects and Design Equipment and M a t e r i a l s Procedure  14 15 16  I I I . RESULTS  19  A n a l y s i s of CSR Test Responses 19 A n a l y s i s of Performance on Eight Types of Argument by Mastery L e v e l . . 21 Comparison of Present Data w i t h the Reported Data 22 IV.  DISCUSSIONS AND CONCLUSIONS  24  REFERENCES  27  FOOTNOTES  28  TABLES 1 (a &amp; b) - Mean Backward D i g i t Spans and Mean Number of Correct Responses f o r Each C o n d i t i o n a l Argument Type by Grade 2 - Analyses of Covariance of the Number of Correct Responses f o r Each CSR Argument Type of Set 2 by Grade 3 - Mean Number of Correct Responses, Mean V a l i d a t i o n Time, and Rank Order f o r Each Argument Type of CSR Set 2 Items by Mastery Level 4 - Percent of Correct Responses f o r Each of Eight C o n d i t i o n a l S y l l o g i s t i c Arguments by Grade; Comparison with Previous Studies  29 31 32 33  \u000Civ.  TABLE OF CONTENTS (Cont'd) Page FIGURES 1 - Marcus &amp; Rips Four-Stage Model f o r V e r i f i c a t i o n of Conditional Syllogisms  34  APPENDICES A. B. C. D. E.  Eight Conditional S y l l o g i s t i c Arguments Selected Spans f o r Backward D i g i t Span Task I n i t i a l I n s t r u c t i o n s to Subjects I n s t r u c t i o n s f o r Backward D i g i t Span Task I n s t r u c t i o n s f o r the Conditional S y l l o g i s t i c Reasoning Task  35 36 37 38 39  \u000C1. I.  STATEMENTS OF PROBLEMS AND HYPOTHESES  Introduction The purpose of the study was t o i n v e s t i g a t e a r e l a t i o n s h i p between pre-adolescent and c o l l e g e students' p r o p o s i t i o n a l reasoning a b i l i t i e s and working memory capacity.  S p e c i f i c a l l y , the present study explored the  p o s s i b i l i t y of explaining developmental f i n d i n g s i n c o n d i t i o n a l s y l l o g i s t i c reasoning, i n d i c a t e d i n studies by P a r i s (1973), T a p l i n , Staudenmayer &amp; Taddonio (1974), and Sternberg (1979), i n terms of epigenetic growth i n the capacity of working memory.  Pascual-Leone's (1970) Mental Operator model of  c o g n i t i v e development, which had previously been demonstrated as having a r e l a t i o n s h i p between working memory capacity and Piagetian c o g n i t i v e substages, provided a t h e o r e t i c a l framework.  A t h e o r e t i c a l basis f o r the  processing of c o n d i t i o n a l syllogisms was found i n the four-stage model proposed by Marcus &amp; Rips (1979).  Conditional S y l l o g i s t i c Reasoning In a p r o p o s i t i o n a l statement, an antecedent premise such as \" I f you eat your dinner\", i s combined with a consequent premise, such as \"then you may have dessert\", to imply a u n i d i r e c t i o n a l causal r e l a t i o n s h i p between the two. To a c t i n a r a t i o n a l manner, according to what i s implied i n the p r o p o s i t i o n a l statement, a c h i l d must understand the meaning of the c o n d i t i o n a l i t y , and deduce an appropriate conclusion from the premises.  An e m p i r i c a l  i n v e s t i g a t i o n of a c o n d i t i o n a l s y l l o g i s t i c deductive reasoning problem f i r s t present an \" i f , then\" premise, expressed symbolically as P-&gt;Q.  A second  premise i s subsequently presented that a f f i r m s or denies e i t h e r the antecedent or the consequent.  Thus, four arguments can be defined, as f o l l o w s :  \u000C2. 1.  A f f i r m i n g the antecedent (Modus ponens)  2.  Denying the antecedent (Modus tollendo t o l l e n s )  3.  A f f i r m i n g the consequent  4.  Denying the consequent  For each s y l l o g i s t i c argument, the conclusion may be e i t h e r a f f i r m a t i v e or negative, r e s u l t i n g i n a t o t a l of eight d i f f e r e n t argument forms.  The  subject i s asked to evaluate the v a l i d i t y of the conclusion based on information derived from the f i r s t and second premise.  T y p i c a l l y , the  syllogisms are of the form''': I f there i s a p, then there i s a q There i s a p There i s a q Developmental Trends i n C o n d i t i o n a l Reasoning A developmental trend i n comprehension of language connectives has been c l e a r l y e s t a b l i s h e d .  In h i s study with grades two, f i v e , eight and  eleven, P a r i s (1973) i d e n t i f i e d two patterns of comprehension evident w i t h age, i n c l u d i n g the increased d i f f e r e n t i a t i o n of conjunctive from d i s j u n c t i v e p r o p o s i t i o n s , and the causal i n t e r p r e t a t i o n of b i c o n d i t i o n a l and c o n d i t i o n a l sentences.  He showed that i n i n t e r p r e t i n g c a u s a l i t y ,  younger subjects tended to judge a complete p r o p o s i t i o n as f a l s e I f any part of the s y l l o g i s m , antecedent or consequent, was f a l s e .  The  responses of subjects changed with age, however, suggesting search f o r a causal r e l a t i o n s h i p between premises and one event's dependence on a second event.  \u000C3.  The f i n d i n g s of T a p l i n , Staudenmayer &amp; Taddonio (1974) supported those of P a r i s i n suggesting that comprehension of the causal r e l a t i o n s h i p of the c o n d i t i o n a l connective \" i f p, then q\", was the most d i f f i c u l t , and involved an intermediary stage of b i c o n d i t i o n a l i t y .  In  t h i s t r a n s i t i o n stage, the c h i l d comprehended a r e l a t i o n s h i p with r e v e r s i b l e c a u s a l i t y between the two premises, whereby neither were e x c l u s i v e l y antecedent nor consequent.  As a r e s u l t , P implied Q and Q  implied P; t h i s r e l a t i o n s h i p may be s y m b o l i c a l l y represented as P&lt;-&gt;Q. T a p l i n et a l ' s (1974) i n v e s t i g a t i o n of developmental changes i n c o n d i t i o n a l reasoning, with subjects from grades three, f i v e , seven, nine and eleven, i n d i c a t e d that there was improved performance on a c o n d i t i o n a l reasoning task with age.  Their r e s u l t s a l s o indicated that  the degree of improved performance was more evident i n some argument forms than others. Sternberg (1979) studied developmental patterns f o r l o g i c a l connectives, and examined the r e l a t i v e importance of l o g i c a l and l i n g u i s t i c processes i n t h i s development.  Using a form of componential  a n a l y s i s , a technique d i f f e r e n t from that used by T a p l i n et a l , he compiled data that generally supported the previous f i n d i n g s .  Looking at  both l i n g u i s t i c encoding and l o g i c a l combination, he confirmed that the c o n d i t i o n a l l o g i c a l connective was the most d i f f i c u l t and supported a developmental trend.  His data i n d i c a t e d that In encoding tasks, grade  two c h i l d r e n evidenced conjunctive and d i s j u n c t i v e i n t e r p r e t a t i o n s of the c o n d i t i o n a l , while performance of those i n grades four, s i x and eight suggested a b i c o n d i t i o n a l i n t e r p r e t a t i o n .  With some evidence of the  \u000C4. c o r r e c t c o n d i t i o n a l i n t e r p r e t a t i o n beginning at grade e i g h t , i t was  not  u n t i l high school and c o l l e g e age that t h i s i n t e r p r e t a t i o n was strongly i n evidence.  Sternberg i n f e r r e d from the data that the l o g i c a l  combination of premises followed the same general developmental trend as l i n g u i s t i c encoding, but lagged by about two  years.  V e r i f i c a t i o n Model of C o n d i t i o n a l Syllogisms In developing a process model of c o n d i t i o n a l s y l l o g i s t i c  reasoning,  Marcus &amp; Rips (1979) appear to have assumed that i n d i v i d u a l s already knew an underlying c o n d i t i o n a l 'rule' or scheme.  The information processing  model developed by these authors consisted of four stages, and r e l i e d both ' s t r u c t u r a l ' and 'error' assumptions; the former comprising  on  an  information processing sequence underlying correct reasoning while the l a t t e r considered explanations of erroneous reasoning. The model makes a number of p r e d i c t i o n s about response l a t e n c i e s f o r v a l i d i t y decisions of each of the eight argument forms.  These  p r e d i c t i o n s assume that while some s y l l o g i s t i c arguments require two stages of processing, others require three and four.  The i n v e s t i g a t o r s  supported t h e i r p r e d i c t i o n that arguments formed three response latency (RT) groups, depending on the number of processing stages required. noted that RT increased as syllogisms increased i n the number of negations, a r e s u l t also reported by Lee  (1984).  They  \u000C5. According to Marcus &amp; Rips (1979), [p,q] and [p, Not-q] syllogisms are processed  through only the f i r s t two stages of the model, 'Encoding  Premises and Conclusions' and 'Does Second Premise Equal F i r s t Premise?', and form the f i r s t c l u s t e r of arguments.  The remaining s i x arguments  proceed f o r f u r t h e r processing i n the t h i r d stage, 'Is Conclusion Consistent with F i r s t and Second Premises?'  This stage uniquely  i d e n t i f i e s the [Not-q, p] s y l l o g i s m as r e q u i r i n g a Never True response. Requiring three processing stages, t h i s s i n g l e argument forms a second 'cluster'.  The remaining f i v e syllogisms a l l require the f o u r t h  processing stage and thereby form a t h i r d c l u s t e r .  I d e n t i f i e d as Always  True at t h i s stage i s the s y l l o g i s m [Not-q, Not-p], i n which the negating of the conclusion produces a doubly negated p r o p o s i t i o n i n c o n s i s t e n t with the c o n d i t i o n a l . The remaining syllogisms, despite negated conclusions, are consistent with the premises and are concluded to be Sometimes True. Figure 1, which schematically reproduces the Marcus &amp; Rips model, i d e n t i f i e s the three c l u s t e r s of syllogisms by the stage i n processing at which a conclusion can be drawn and the appropriate response prepared.  INSERT FIGURE 1 ABOUT HERE -see p\u00BB-ge 34 In a d d i t i o n to the above s t r u c t u r a l elements, the model takes i n t o account three p o t e n t i a l sources of e r r o r .  According to Marcus &amp; Rips,  the source of most probable e r r o r i s the premature termination of processing; hence an inference e r r o r could occur with termination of  \u000C6. processing a f t e r any of the f i r s t three stages indicated above.  As  pointed out by Marcus &amp; Rips, the model considers the psychological meaning of l o g i c a l connectives which does not correspond e n t i r e l y to the propositional logic.  Errors may a l s o r e s u l t from the processing of  negative premises or conclusions during stages three and four, or from the r e v e r s a l of the P-Q sequence.  Theoretic Role of Working Memory I m p l i c i t i n the Marcus &amp; Rips model i s the assumption that the c o g n i t i v e processes involved i n comprehending, analyzing and reaching a conclusion about a problem i n c o n d i t i o n a l reasoning occur i n working memory.  Viewed i n the information processing paradigm, the c o g n i t i v e  processes involved i n a t t a i n i n g a l o g i c a l l y v a l i d conclusion f o r a c o n d i t i o n a l p r o p o s i t i o n necessitate a minimum capacity In working memory, a capacity that appears to undergo developmental change (Pascual-Leone, 1970).  From t h i s i m p l i c a t i o n , a p o t e n t i a l explanation of the  developmental trend noted i n s y l l o g i s t i c reasoning may be advanced:  that  the a b i l i t y to s u c c e s s f u l l y solve a s y l l o g i s t i c reasoning problem i s contingent on the working memory requirements of the problem and the a v a i l a b l e working memory capacity I n the i n d i v i d u a l .  Investigations i n t o Development i n Working Memory Capacity The development of working memory capacity was explored by Pascual-Leone (1970) who proposed a neo-Piagetian model.  This model  postulated a q u a n t i t a t i v e parameter t o account f o r Piaget's q u a l i t a t i v e  \u000C7. d e s c r i p t i o n of i n t e l l e c t u a l development.  According to Piaget ( c f . 1958),  the i n t e g r a t i o n of information occurs i n a ' f i e l d of c e n t r a t i o n ' or ' f i e l d of e q u i l i b r i u m and that t h i s f i e l d increases i n s i z e with age. Pascual-Leone attempted to quantify the increasing s i z e of t h i s  'field'  and to r e l a t e i t to the P i a g e t i a n construct of i n t e l l e c t u a l development. In h i s subsequent i n v e s t i g a t i o n , Case (1972) demonstrated that the Pascual-Leone model could be v a l i d a t e d by a d i f f e r e n t measure, and added support to the Mental Operator as the set measure of ' f i e l d of c e n t r a t i o n ' or 'M-space'. In v a l i d a t i n g the neo-Piagetian construct, Case (1972) used a d i g i t i n s e r t i o n technique by which subjects were required to locate a target d i g i t w i t h i n a previously presented s e r i e s of d i g i t s .  He noted, however,  that the Backward D i g i t Span (BDS) task y i e l d e d i d e n t i c a l norms to those obtained w i t h the d i g i t i n s e r t i o n technique.  I t was suggested that the  transformation of d i g i t order required by the BDS task i n t e r f e r e d with rehearsal and 'chunking' s t r a t e g i e s , and thereby equated i t with the c o g n i t i v e processing requirements of the d i g i t i n s e r t i o n task. According t o Case (1972, p. 287), what was measured i n the d i g i t i n s e r t i o n task, and by i m p l i c a t i o n i n the BDS task, was the \"maximum number of a c t i v a t e d schemes which (could) be coordinated at any one time.\"  I t can be i n f e r r e d from t h i s d e f i n i t i o n that Piaget's ' f i e l d of  c e n t r a t i o n ' and Pascual-Leone's 'M-space' are f u n c t i o n a l l y synonymous w i t h working memory and s i m i l a r to the Short Term Store of Atkinson and S h i f f r e n (1968).  \u000C8. As i n d i c a t e d by Pascual-Leone (1970), the growth i n M-space, or working memory, i s considered to be l i n e a r , and determined p r i m a r i l y by epigenetic factors.  Occurring generally between the ages of three and s i x t e e n years, the  modal value of M-space increases from a + 1 to a + 7, and can be r e l a t e d to Piagetian  substages.  In the notation used to i n d i c a t e working memory c a p a c i t y , lc represents the number of a c t i v a t e d schemes that can be attended to and manipulated  at a  given developmental stage, while a_ represents the working memory capacity requirements of the schemes that d i r e c t and coordinate the manipulation.  By  'scheme' i s meant an \" o r i g i n a l set of reactions ... s u s c e p t i b l e to being transferred from one s i t u a t i o n to another by a s s i m i l a t i o n of the second to the f i r s t , \" (Pascual-Leone,  1970, p. 306). They share common features, such as  being r e c u r s i v e , definable by t h e i r content ( i . e . , perceptual, c o g n i t i v e , e t c . ) , and form three general groups, i d e n t i f i e d as superordinate, f i g u r a t i v e and operative. Superordinate schemes are the o v e r a l l plans a c t i v a t e d to consider a s p e c i f i c problem s i t u a t i o n .  S i m i l a r to a computer program that uses  subroutines, these 'executive' schemes are i n t e r n a l representations of procedures appropriate f o r a t t a i n i n g p a r t i c u l a r o b j e c t i v e s . The second type, f i g u r a t i v e schemes, are capable of r e l e a s i n g responses of superordinate schemes; that i s , they are i n t e r n a l representations of known or recognizable elements of information and correspond  to 'chunks' ( M i l l e r , 1956).  Finally,  there are operative schemes which are i n t e r n a l representations of functions or r u l e s applied to f i g u r a t i v e schemes to generate transformations.  \u000C9. According to these d e f i n i t i o n s , a includes the working memory requirement f o r the superordinate and operative schemes, and k includes the working memory requirement of the f i g u r a t i v e schemes to be manipulated.  I n view of the  a d d i t i v e r e l a t i o n s h i p between a and k i n d e r i v i n g memory c a p a c i t y , and the maximum representational capacity of working memory or Jc, at a given age, these f a c t o r s appear to be important i n e s t a b l i s h i n g the l e v e l of i n t e l l e c t u a l functioning. Underlying the Pascual-Leone construct I s the notion that a_ remains constant across age groups f o r a s p e c i f i c , well-learned task.  Here,  superordinate and operative schemes associated with a given task tend to become w e l l - e s t a b l i s h e d i n the i n d i v i d u a l .  At the point at which the task i s  thoroughly, or 'overly' learned, the working memory requirement of these schemes, a, a t t a i n s a t a s k - r e l a t e d minimum that remains constant i n any subsequent performance of the same task. the value of  I t should be noted, however, that  across d i f f e r e n t tasks can vary and w i l l depend upon the  complexity and amount of transformation and coordination required. I m p l i c i t i n the Pascual-Leone proposal i s that the working memory t h e o r e t i c a l l y a v a i l a b l e to r e t a i n the f i g u r a t i v e schemes i s the capacity remaining i n the e p i g e n e t i c a l l y determined M-space a f t e r the executive and operative schemes have been accommodated.  This suggests that working memory  assigned to a or k i s interchangeable, and i s governed by the demands of the task and o v e r a l l capacity.  The BDS task can be used to i l l u s t r a t e t h i s  i n t e r c h a n g e a b i l i t y and the t a s k - r e l a t e d constancy of j i . B r i e f l y , i n a BDS task the subject i s s e q u e n t i a l l y shown a s e r i e s of two-to-nine d i g i t s .  With no e x t e r n a l memory a i d , t h e i r task i s to r e c a l l the  d i g i t s e r i e s i n the reverse order of presentation.  To accomplish t h i s , the  \u000C10. subject must r e t a i n the i n d i v i d u a l d i g i t s i n memory, and then manipulate them i n t o the required reverse order and r e c i t e t h i s sequence back to the experimenter.  This manipulation and coordination f u n c t i o n , that i s , the  backward transformation, i s assumed to be governed by task appropriate superordinate and operative schemes which require a p o r t i o n of working memory; t h i s working memory requirement i s equivalent to _a, and assumed to be constant once the BDS task i s well-learned. I t i s a l s o assumed (Case, 1972) that the nature of the BDS task keeps the subjects from 'chunking' d i g i t s together and thereby implies that each d i g i t i s equivalent to a s i n g l e f i g u r a t i v e scheme.  The span of d i g i t s that a  subject i s capable of r e c a l l i n g i n a reverse order therefore indexes the number of f i g u r a t i v e schemes that they can manipulate f o r the BDS task; t h i s span i s considered to be an i n d i r e c t measure of k. I t becomes apparent that f o r any given overlearned task then, ja and It should have unique values, but i n no case may \u00ABi + lc exceed the e p i g e n e t i c a l l y determined maximum capacity of working memory.  This suggests t h a t , where a  task requires more capacity than an i n d i v i d u a l has a v a i l a b l e , that task should not be s u c c e s s f u l l y performed. As summarized by Case (1972), i t should be noted that a_ and _k do not account f o r a l l v a r i a b l e s of c o g n i t i v e performance i n terms of v a r i a t i o n i n working memory.  Also important i s the proportion of working memory devoted to  a p a r t i c u l a r task and the r e p e r t o i r e of schemes a v a i l a b l e to the i n d i v i d u a l , p a r t i c u l a r l y as influenced by learning f a c t o r s and f i e l d f a c t o r s that govern what schemes are to be a c t i v a t e d .  \u000C11. Working Memory Capacity and the V e r i f i c a t i o n Model The notion of f i g u r a t i v e and operative schemes may be applied to the model of Marcus &amp; Rips, where d i f f e r e n t working memory requirements may be inferred.  As i n d i c a t e d by the author's model, the number of processing  stages  through which each argument passes d i r e c t l y a f f e c t s the amount of processing time required.  The processing stages may be considered equivalent to three  operative schemes, implying three working memory l e v e l s (_a,jf,_a). Each l e v e l represents one of the three argument groupings as defined by the model and supported by RT observed by Marcus &amp; Rips (1979).  These researchers a l s o  suggested that processing stages were not the only determinant of response latency.  They also incorporated i n t o t h e i r model f a c t o r s f o r the negation of  premises, and the r e v e r s a l of premises i n four of the arguments (Types: [q,p], [q, Not-p],  [Not-q,p],  [Not-q, Not-p]).  Summary and Hypothesis There i s evidence t o suggest that working memory capacity i s not f i x e d , but increases as the c h i l d develops (Pascual-Leone,  1970; Case, 1972, 1974).  Hence, i t i s proposed that the a b i l i t y of a subject to s u c c e s s f u l l y solve a s y l l o g i s t i c reasoning problem should be a f f e c t e d by the working memory capacity a v a i l a b l e at a p a r t i c u l a r point i n development. hypothesized  S p e c i f i c a l l y , i t was  that the a b i l i t y of subjects to solve each of the eight  s y l l o g i s t i c arguments, according to the c o n d i t i o n a l t r u t h f u n c t i o n , w i l l depend on t h e i r working memory capacity as defined by backward d i g i t span. The present study attempted to I d e n t i f y r e l a t i v e working memory c a p a c i t y , as i n f e r r e d from BDS scores, required to s u c c e s s f u l l y solve s y l l o g i s t i c arguments, g i v i n g consideration to the a l l o c a t i o n of memory between operative  \u000C12. and f i g u r a t i v e schemes.  Subjects were presented with a series of concrete and  abstract syllogisms, and t h e i r working memory capacity determined using a BDS task.  I n the present study, the premises involved i n a s y l l o g i s t i c  p r o p o s i t i o n were considered to be equivalent to f i g u r a t i v e schemes. supplemented  This view  that of Marcus &amp; Rips (1979) and provided a d d i t i o n a l explanation  f o r the response l a t e n c i e s predicted.  By t h i s view, each p o s i t i v e l y stated  premise, or more s p e c i f i c a l l y , the subject of that premise, was assumed to represent one f i g u r a t i v e scheme and correspond to a k value of one. In the most common of arguments, [p,q], two p o s i t i v e l y stated premises are involved and represented the l e a s t number of f i g u r a t i v e schemes to be manipulated i n s o l v i n g a s y l l o g i s t i c problem.  I n t h i s case, i t was reasoned that k took on a  value of two. Negation or r e v e r s a l of premises required a d d i t i o n a l manipulation of f i g u r a t i v e schemes, and r e s u l t e d i n a d d i t i o n a l processing time. I t must be noted that the v e r i f i c a t i o n model i d e n t i f i e d the encoding of premises and conclusions as a separate stage from processing arguments. However, the experimental methodology of Marcus &amp; Rips d i d not make t h i s d i s t i n c t i o n ; RT was measured from onset of the complete syllogism to v a l i d a t i n g response.  I m p l i c i t i n t h i s methodology and the r e s u l t i n g a n a l y s i s ,  i s that encoding should be constant across argument types.  Such an assumption  i s open to question. There e x i s t s strong support f o r encoding being the source of d i f f e r e n t l e v e l s of d i f f i c u l t y i n s o l v i n g each of the eight syllogisms (Sternberg, 1979; T a p l i n et a l , 1974).  To emphasize the evaluating  process, the current study measured v a l i d a t i n g time (VT) from onset of the conclusion to v a l i d a t i n g response; t h i s procedure reduced, but d i d not eliminate measurement of encoding time, r e s t r i c t i n g i t to the encoding of the conclusion.  \u000C13. With the exception of RT measurement, the current study r e p l i c a t e d Experiment 2 of Marcus &amp; Rips (1979), and attempted to determine i f the v e r i f i c a t i o n model i s consistent with the performance of subjects c l a s s i f i e d as knowing the c o n d i t i o n a l r u l e .  I f the model were to be supported, VT of  subjects with a mastery of the c o n d i t i o n a l r u l e should increase according to the complexity of processing required.  In a d d i t i o n , the l e v e l of d i f f i c u l t y ,  as measured by the number of o v e r a l l c o r r e c t responses, should a l s o increase with the number of processing stages proposed by the model.  Both VT and  number of c o r r e c t responses should separate arguments i n t o three s i m i l a r groups corresponding to the three l e v e l s of operative scheme memory capacity, a_ _a', and y  logic rule.  at l e a s t f o r those subjects who can be regarded as knowing the  \u000C14. II.  METHOD  Subjects and Design I n i t i a l l y , a t o t a l of 92 subjects was i d e n t i f i e d , of whom 31 were drawn from each of grades f i v e and seven, and 30 from paid undergraduate and graduate u n i v e r s i t y students. Elementary students were selected from a school i n the Lower Mainland of B r i t i s h Columbia; the u n i v e r s i t y subjects were drawn from students at the U n i v e r s i t y of B r i t i s h Columbia.  Upon obtaining the  p a r t i c i p a n t s ' consents through the school and u n i v e r s i t y i n s t r u c t o r s , a t o t a l of 77 subjects remained i n the sample f o r the present experiment.  The sample  consisted of three groups, 25 grade f i v e , 27 grade seven and 27 c o l l e g e students. As r e s u l t s of previous studies ( T a p l i n , Staudenmayer &amp; Taddonio, 1974; Sternberg, 1979) had i d e n t i f i e d l i t t l e evidence of c o n d i t i o n a l s y l l o g i s t i c reasoning below grade f i v e , the youngest subjects f o r the current study were selected from t h i s grade l e v e l .  Based on the i n v e s t i g a t i o n of Pascual-Leone  (1970), i t was determined that these younger subjects could be expected to have a modal M-space value of a + 4 to a + 5. To provide subjects with a range i n modal M-space values t o a + 7, the maximum i d e n t i f i e d by Pascual-Leone, grade seven and u n i v e r s i t y students were also s e l e c t e d .  With  t e s t i n g occurring a t the end of the school year, grades f i v e and seven subjects were assumed to correspond to the Late Concrete and Early-Middle FormalPiagetian substages, r e s p e c t i v e l y ; college subjects were assumed to correspond to Late Formal and beyond.  \u000C15. Equipment and M a t e r i a l s The  two tasks, Backward D i g i t Span and Conditional S y l l o g i s t i c Reasoning,  were both presented i n d i v i d u a l l y to subjects using a micro-computer; t h i s equipment automatically recorded item responses and v a l i d a t i n g response latency.  S i x systems were used, each c o n s i s t i n g of an Apple H E  micro-computer, a 12-inch monochrome monitor, and two d i s k d r i v e s ; one d r i v e was used to run the program and the second to record the data.  Each system  was so arranged as to prevent subjects from seeing a screen other than t h e i r own. In the BDS task, eight d i g i t spans were evaluated twice; spans tested were from two through nine d i g i t s .  The sequences and order of d i g i t span  length were determined randomly from Random Number Tables (Edwards, 1968); consecutive eliminated.  d u p l i c a t e and s e q u e n t i a l l y ordered d i g i t s i n any span were The selected spans are presented i n Appendix B.  A random  presentation of target spans was selected to avoid a response set, t h e o r e t i c a l l y consistent with the established BDS t e s t i n g paradigm. The c o n d i t i o n a l s y l l o g i s m reasoning eighty items used by Lee (1984).  (CSR) task consisted of the same  B r i e f l y stated, these syllogisms were the  r e s u l t of ten semantic s i t u a t i o n s , two abstract and eight concrete, i n a f a c t o r i a l combination with the eight argument forms previously described and summarized i n Appendix A.  The syllogisms comprised three statements or  p r o p o s i t i o n s , i n c l u d i n g a major premise, a minor premise, and a conclusion.  \u000C16. Procedure Subjects were tested i n groups of s i x , i n two 30-45 minute periods.  The  same experimenter administered a l l sessions f o r grades f i v e and seven subjects, a s s i s t e d by a female graduate student; t h i s a s s i s t a n t tested a l l u n i v e r s i t y subjects.  During the f i r s t t e s t sessions, subjects received f o r t y  CSR problems; during the second session, they received the BDS task and the second set of f o r t y CSR problems.  Each subject's f i r s t t e s t session began  with a b r i e f i n t r o d u c t i o n t o acquaint p a r t i c i p a n t s w i t h the experiment and t o confirm that p a r t i c i p a n t s were s u f f i c i e n t l y f a m i l i a r with the computer keyboard t o accomplish the proposed tasks (Appendix C). The tasks were s e l f - t i m e d , with presentation of a l l materials computer-controlled according to d u p l i c a t e programs copied from a common master.  Each task was preceded by s p e c i f i c i n s t r u c t i o n s presented on the  computer screen p e r t a i n i n g to the task.  Presented f i r s t was the BDS task  (Appendix D) Following the i n s t r u c t i o n s , the f i r s t p r a c t i c e d i g i t span sequence s t a r t i n g with the word 'READY', shown f o r 1.2 seconds.  began,  The screen then went  blank f o r 1.2 seconds before the f i r s t d i g i t appeared; each d i g i t was presented i n d i v i d u a l l y i n the centre of the screen f o r 1.2 seconds.  At the  end of the f i r s t p r a c t i c e s e r i e s only, the subject was reminded of s p e c i f i c instructions. ... F i r s t p r a c t i c e s e r i e s ... Now, please i n d i c a t e the d i g i t s you have j u s t seen i n backwards order. Remember, i f you cannot think of a d i g i t , put an '-' i n i t s place.  \u000C17. These i n s t r u c t i o n s remained on the screen f o r f i v e seconds.  The subject  had a maximum time l i m i t f o r responding of f i f t e e n seconds; responses were not displayed on the screen. At the end of each d i g i t span sequence, the screen became blank f o r f i v e seconds and then the next sequence began w i t h the word, 'READY.' A f t e r the four pertaining problems, subjects were t o l d they had completed the four p r a c t i c e problems and to proceed to the a c t u a l task.  The  experimental task was i d e n t i c a l to the p r a c t i c e session, but excluded a l l instructions. At the conclusion of the BDS task, the screen became blank f o r f i f t e e n seconds while the computer loaded the CSR program; the i n s t r u c t i o n s f o r the next task were then displayed (Appendix E ) . Following the i n s t r u c t i o n s , the f i r s t argument appeared on the screen, beginning with the f i r s t p r o p o s i t i o n which appeared on the screen f o r f i v e seconds: Item 1: Suppose that you know that, ( f i r s t p r o p o s i t i o n ) , The screen then went blank f o r 1.2 seconds, u n t i l the second p r o p o s i t i o n was shown: and (second proposition) This p r o p o s i t i o n was also displayed f o r f i v e seconds, when the screen again went blank f o r 1.2 seconds and the conclusion was displayed! Then would t h i s be true?  (Conclusion)  A f t e r f i v e seconds, the m u l t i p l e choice answers appeared: Always true: Sometimes true: Never true:  A S N  \u000C18. Once answered, c o r r e c t i v e feedback was presented.  At the end of each problem  sequence, the screen went blank while the computer recorded the subject's response and v a l i d a t i o n response time (VT) recorded up to one m i l l i s e c o n d on the d i s k e t t e .  Subjects were presented w i t h the same i n s t r u c t i o n s before the  second set of f o r t y CSR problems when tested a few days l a t e r .  \u000C19. III.  RESULTS  I n i t i a l processing of data from i n d i v i d u a l subjects r e s u l t e d i n a set of three measurements, i n c l u d i n g backward d i g i t span and two c o n d i t i o n a l s y l l o g i s t i c reasoning scores, number of correct responses, and associated v a l i d a t i n g time f o r i n d i v i d u a l requirements.  BDS was established as the  longest span answered c o r r e c t l y by subjects i n both span r e p l i c a t i o n s .  Five  subjects f a i l e d to a t t a i n the c r i t e r i o n ; BDS was estimated f o r these subjects based on the o v e r a l l number of d i g i t s i n the c o r r e c t r e l a t i v e p o s i t i o n .  The  raw number of c o r r e c t CSR responses f o r each of the eight arguments was determined f o r each of the two presentations of the task.  Each task consisted  of eight arguments i n d i v i d u a l l y presented f i v e times, permitting a maximum score per task of f i v e f o r each argument.  These data are summarized i n  Table 1. A f u r t h e r CSR datum, VT, was measured from onset of the argument's conclusion to making the c o r r e c t response.  INSERT TABLE 1 ABOUT HERE A  See  \u2022  p a g e  2?  A n a l y s i s of CSR Test Responses To determine the e f f e c t of knowledge of the c o n d i t i o n a l r u l e on CSR performance, subjects were c l a s s i f i e d i n t o one of two groups, mastery and non-mastery.  Mastery-level subjects were determined from r e s u l t s of the f i r s t  CSR task, according to a method o r i g i n a l l y proposed by Lee (1984), i n which a score of four or greater was required on at l e a s t s i x of each eight arguments.  Sixteen of the 77 subjects met t h i s c r i t e r i o n , i n c l u d i n g two i n  Grade 5, two i n Grade 7, and twelve at c o l l e g e l e v e l .  \u000C20. Under the premise that c o n d i t i o n a l s y l l o g i s t i c reasoning resembled the v e r i f i c a t i o n model proposed by Marcus &amp; Rips (1979), the a n a l y s i s had two purposes:  to determine (1) the e f f e c t of working memory capacity on CSR  problems, and (2) the extent to which p r e d i c t i v e performance of the model could be explained by knowledge of the c o n d i t i o n a l r u l e i m p l i c i t l y assumed by Marcus &amp; Rips (1979). To determine the Influence of working memory capacity on c o n d i t i o n a l s y l l o g i s t i c problems, a s e r i e s of analyses of covariance were performed on the number of correct responses and VT of i n d i v i d u a l arguments. were used as covariants, with grade the grouping f a c t o r . as predicted.  BDS and mastery  The r e s u l t s were not  In no argument d i d BDS exceed the chance l e v e l , while mastery  was a s i g n i f i c a n t f a c t o r i n a l l arguments, and grade i n s i x .  BDS was a l s o  tested i n a polynomial regression a n a l y s i s , which a l s o f a i l e d to i d e n t i f y any e f f e c t of working memory capacity.  INSERT TABLE 2 ABOUT HERE See page 3' To examine f u r t h e r the f i n d i n g s of the a n a l y s i s of covariance, a determination was made of the percent of variance of correct argument responses accounted f o r by each of three p r e d i c t o r s : mastery of the c o n d i t i o n a l r u l e .  grade l e v e l , BDS, and  I n a l l but one argument, [Not-p, Not-q], the  majority of variance a t t r i b u t e d to the three f a c t o r s was explained by grade l e v e l , accounting f o r between 3.3% and 29.8% of variance, with a mean of 15%. While grade l e v e l explained only 5.4% of the variance on argument [Not-p, Not-q], mastery l e v e l accounted f o r 12.1%, the most variance explained f o r by t h i s f a c t o r on any of the arguments; with a mean of 6.8%, mastery accounted  \u000C21. for  2.1% to 12.1% of variance.  By contrast, BDS, with a range of 0% to 2.2%  and a mean of 0.9%, explained l i t t l e .  The a n a l y s i s was repeated on the t o t a l  score of a l l eight arguments from the second CSR task, to determine the e f f e c t of the  the three f a c t o r s on the o v e r a l l response pattern.  The a n a l y s i s supported  f i n d i n g s f o r i n d i v i d u a l arguments by i d e n t i f y i n g grade as explaining the  majority of variance at 32.1%, and mastery as the second f a c t o r , explaining 17.8% of variance. of  As i n the previous a n a l y s i s , BDS accounted f o r only 1.1%  variance. C l e a r l y , the r e s u l t s f a i l e d t o support the hypothesis by i d e n t i f y i n g no  argument i n which BDS accounted i n any s i g n i f i c a n t way f o r performance on the CSR task.  This f i n d i n g suggests that development i n working memory capacity  has l i t t l e e f f e c t on c o g n i t i v e a b i l i t i e s , as defined by the c o n d i t i o n a l reasoning problem.  I n view of the importance of grade l e v e l to CSR  performance, i t must be i n f e r r e d that other developmental f a c t o r s c o n t r i b u t i n g to improved performance on t h i s task with age, remain t o be I d e n t i f i e d .  In  a d d i t i o n , f a c t o r s other than grade and master l e v e l appear to be involved, as the  majority of variance on a l l arguments remains unexplained. \u2022  A n a l y s i s of Performance on Eight Types of Arguments by Mastery Level To examine whether the order of argument d i f f i c u l t y was consistent with that predicted by apparent working memory requirements of the Marcus &amp; Rips (1979) model, repeated measures analyses of variance were performed on combined r e s u l t s of the two CSR presentations.  As mastery of the c o n d i t i o n a l  r u l e was i m p l i c i t l y assumed i n the v e r i f i c a t i o n model, data f o r mastery and non-mastery subjects were analyzed separately.  \u000C22. I n i t i a l analyses across a l l eight arguments, f o r each group, found that arguments varied i n d i f f i c u l t y , according to the number of correct responses and i n VT.  Further a n a l y s i s , with repeated measures analyses of variance  across p a i r s of arguments, i d e n t i f i e d the rank order of arguments i n d i c a t e d i n Table 3.  INSERT TABLE 3 ABOUT HERE  _S_li_P.mJL?: The analyses revealed d i f f e r e n t orders of argument d i f f i c u l t y f o r each mastery l e v e l on each CSR datum.  Further, and as predicted by the Marcus &amp;  Rips (1979) model, arguments could be grouped i n t o c l u s t e r s of s i m i l a r difficulty.  However, the order of d i f f i c u l t y i n neither mastery l e v e l group  was as expected from the model.  Those subjects c l a s s i f i e d as possessing the  c o n d i t i o n a l r u l e evidenced fewer argument c l u s t e r s than non-mastery In both correct responses and VT.  subjects  This reduction i n the number of c l u s t e r s ,  from four to two, w i t h improved performance where one of the c l u s t e r s f o r mastery subjects represented seven of the eight arguments, suggests a developmental trend towards a s i n g l e VT c l u s t e r .  Comparison of Present Data with the Reported Data To test e x t e r n a l v a l i d i t y , an a n a l y s i s was conducted on data from the c o n d i t i o n a l s y l l o g i s t i c reasoning task to permit comparison w i t h f i n d i n g s reported by T a p l i n &amp; Staudenmayer (1973) and T a p l i n , Staudenmayer &amp; Taddonio (1974).  Results of a l l three experiments are summarized i n Table 4.  While  some d i f f e r e n c e s are noted on s p e c i f i c arguments, p a r t i c u l a r l y f o r grade 5 subjects, there appears a s i m i l a r o v e r a l l grade-related trend.  \u000CINSERT TABLE 4 ABOUT HERE __5ee_pa\u00A3e__\"5j A f u r t h e r comparison was made with r e s u l t s reported by Sternberg (1979) on the percent of correct c o n d i t i o n a l sets; that i s , the number of sets of eight consecutive arguments i n a s i n g l e s e r i e s as a percentage of the t o t a l number of sets.  The r e s u l t s are very s i m i l a r .  I n the current study, the  percent of c o r r e c t sets was 3.2%, 3.6%, and 12.6% f o r grades f i v e , seven and c o l l e g e , r e s p e c t i v e l y , compared to 3% f o r grade s i x and 19.0% f o r c o l l e g e subjects reported by Sternberg.  \u000C24. IV.  DISCUSSION AND CONCLUSION  The study found l i t t l e support f o r the c e n t r a l hypothesis, nor f o r the model of c o n d i t i o n a l s y l l o g i s t i c reasoning, as proposed by Marcus &amp; Rips (1979).  However, support was found f o r v a r i a t i o n i n argument d i f f i c u l t y  somewhat d i f f e r e n t from that predicted by the v e r i f i c a t i o n model. While i t was argued that development i n working memory capacity could contribute to the age-related improvement i n CSR performance noted by previous i n v e s t i g a t o r s such as P a r i s (1973), t h i s was not the case.  In the current  study, grade l e v e l was i d e n t i f i e d as a major f a c t o r In CSR performance with l i t t l e r e l a t i o n s h i p to BDS. As discussed by Lee (1984), the c o n d i t i o n a l t r u t h f u n c t i o n i s frequently i m p l i c i t l y assumed i n studies of c o n d i t i o n a l s y l l o g i s t i c reasoning.  I n the  current study, those subjects appearing t o know the c o n d i t i o n a l r u l e were e x p l i c i t l y i d e n t i f i e d by t h e i r r e s u l t s on one of the two CSR tasks. Performance on the CSR task by mastery and non-mastery groups v a r i e d very s i g n i f i c a n t l y , (F(l,75) = 124.5, \u00A3 &lt; 0.01). Once a v a i l a b l e to the subject, the c o n d i t i o n a l r u l e appears to be stable and a good p r e d i c t e r of performance on the second CSR task.  I t can be  i n f e r r e d , then, that knowledge of the c o n d i t i o n a l t r u t h f u n c t i o n , or the a b i l i t y to a c t i v a t e the appropriate operative schemes, may be a better explanation of success i n answering syllogisms than working memory capacity. Before dismissing the working memory capacity hypothesis, the v a l i d i t y of the Backward D i g i t Span task, as used i n the current study, must be questioned.  The computerized task v a r i e d from standard t e s t i n g approach i n  presenting a l l subjects with a random order of span lengths.  In the t y p i c a l  \u000C25. BDS  task, subjects are i n d i v i d u a l l y presented with spans of increasing length  u n t i l they f a i l to c o r r e c t l y respond to a span of s p e c i f i c length. computerized task, subjects could a t t a i n a high BDS,  In the  such as s i x , while  f a i l i n g shorter spans; t h i s s i t u a t i o n i s not possible with the standard t e s t i n g paradigm. Viewed from the information processing paradigm, the VT data of the mastery subjects suggests that a s i n g l e operative scheme may the CSR task.  be involved i n  This operative scheme appears to develop i n stages and  may  r e s u l t from the gradual i n t e g r a t i o n of at l e a s t one other scheme; a review of data from non-mastery subjects i n d i c a t e s that arguments may be separated  Into  four VT c l u s t e r s , while only two c l u s t e r s were evident f o r mastery subjects. The a v a i l a b i l i t y of a f u n c t i o n a l c o n d i t i o n a l r u l e , or c o n d i t i o n a l operative scheme, may help to account f o r the l a c k of agreement between the current data and the Marcus &amp; Rips (1979) model.  The three argument c l u s t e r s  predicted from the model f a i l e d to appear through e i t h e r the number of correct responses or VT, suggesting that argument d i f f i c u l t y may not be explained by processing f a i l u r e at selected stages w i t h i n the model, as i t s authors proposed.  Rather, the explanation may be a l a c k of a s i n g l e , integrated  process or c o n d i t i o n a l t r u t h f u n c t i o n , with the Marcus &amp; Rips (1979) f i n d i n g s r e s u l t i n g from a c o l l e c t i v e developmental trend i n the a c q u i s i t i o n of the c o n d i t i o n a l r u l e present i n t h e i r college-age subjects. Caution must be used i n i n t e r p r e t i n g the VT data f o r some non-mastery subjects where very low VT and high r i s k e r r o r rates on three arguments suggests that these i n d i v i d u a l s may have been guessing.  As these three  arguments, [q,p], [not-q, p ] , and [not-q, not-p], were a l s o found to be the most d i f f i c u l t by non-mastery subjects, guessing may have r e s u l t e d from minimal development of appropriate processing a b i l i t y f o r these syllogisms.  \u000C26. I n t e r e s t i n g l y , VT's representing c l u s t e r s 2 and 3 i n data from non-mastery subjects which correspond with two of the e a s i e r syllogisms f o r t h i s group, are somewhat lower than VT on s i m i l a r arguments by mastery subjects.  For syllogism [p,q], i t i s possible that non-mastery subjects  viewed the argument c o n j u n c t i v e l y .  Such an i n t e r p r e t a t i o n i s c o n s i s t e n t with  f i n d i n g s reported by such previous authors as P a r i s (1973), T a p l i n (1973), and Staudenmayer &amp; Bourne (1977), where frequent exposure to the conjunctive i n e a r l y development can be reasoned to r e s u l t i n a separate operative scheme (Marcus &amp; R i p s , 1979).  The data suggests that, i n the developmental process,  as the CSR operative scheme adapts to respond to more varied syllogisms, i t integrates the conjunctive scheme. While the study d i d not support working memory as being an important v a r i a b l e i n the development towards c o n d i t i o n a l s y l l o g i s t i c reasoning, i t d i d support development towards a s i n g l e CSR operative scheme.  The  encoding-evaluation issue i n v e s t i g a t e d by Sternberg (1979), c l e a r l y determined the primacy of encoding i n the o v e r a l l development of CSR processing.  The  current work a s s i s t s i n our understanding of the two year l a g i n attainment of the e v a l u a t i o n subprocess noted by Sternberg.  Further, i t provides evidence  that younger subjects are able to accomplish complex problem solving such as c o n d i t i o n a l s y l l o g i s t i c reasoning, once the appropriate scheme i s a v a i l a b l e to them.  The challenge f o r education i s to a s s i s t the c h i l d i n b u i l d i n g these  appropriate schemes.  \u000C27. REFERENCES Atkinson, R.C. &amp; S h i f f r e n , R.M. (1968) Human memory: A proposed system and i t s c o n t r o l processes. I n K.W. Spence &amp; J.T. Spence (Eds.) Advances i n the psychology of learning and m o t i v i a t i o n research and theory. V o l . 2 New York: Academic Press. Case, R. (1972) V a l i d a t i o n of a neo-Piagetian mental capacity construct. Journal of Experimental C h i l d Psychology, 14, 287-302. Case, R. (1974) Mental s t r a t e g i e s , mental c a p a c i t y , and i n s t r u c t i o n : A neo-Piagetian i n v e s t i g a t i o n . Journal of Experimental C h i l d Psychology, 18, 382-397. Edwards, A.L. (1972) Experimental Design i n Psychological Research, New York, H o l t , Rinehart and Winston, Inc. Lee, S.S. (1984) Children's a c q u i s i t i o n of c o n d i t i o n a l l o g i c s t r u c t u r e : Teachable? Contemporary Educational Psychology, 9, 419-483. Marcus, S.L. &amp; R i p s , L . J . (1974) C o n d i t i o n a l reasoning. Learning and Verbal Behavior, 18, 199-224.  Journal of Verbal  M i l l e r , G.A. (1956) The magical number seven, plus or minus two: Some l i m i t s on our capacity f o r processing information. P s y c h o l o g i c a l Review, 63, 81-97. P a r i s , S.G. (1973) Comprehension of Language connectives and p r o p o s i t i o n a l l o g i c a l r e l a t i o n s h i p s . Journal of Experimental C h i l d Psychology, 16, 278-291. Pascual-Leone, J . (1970) A mathematical model f o r the t r a n s i t i o n r u l e i n Piaget's development stages. Acta Psychologica, 32, 301-345. Piaget, J . (1958) A s s i m i l a t i o n et connaissance. I n A. Jonckheere, B. Mandelbrot, &amp; J . Piaget (Eds.), La Lecture de 1'experience. P a r i s : 42-108.  P.U.F.  Sternberg, R.J. (1979) Developmental patterns i n the encoding and combination of l o g i c a l connectives. Journal of Experimental C h i l d Psychology, 28, 469-498. T a p l i n , J.E., Staudenmayer, H. (1973) I n t e r p r e t a t i o n of a b s t r a c t c o n d i t i o n a l sentences i n deductive reasoning. Journal of Verbal Learning and Verbal Behavior, 12, 530-542. T a p l i n , J.E., Staudenmayer, H., &amp; Taddonio, J.L. (1974) Developmental Charge i n c o n d i t i o n a l reasoning: L i n g u i s t i c or l o g i c a l ? Journal of Experimental C h i l d Psychology, 17, 360-373.  \u000CFOOTNOTES As t h i s paper w i l l r e f e r to each of the eight arguments i n d i v i d u a l l y , i t w i l l be done by placing a square bracket around the second premise and conclusion; a negated premise or conclusion w i l l be i n d i c a t e d by \"Not-.\" As an example, the notation f o r the modus ponens argument above, i s [p,q]. A l l eight arguments are i l l u s t r a t e d i n Appendix A, which includes the notation used throughout t h i s paper.  \u000C29.  TABLE 1(a)  BDS  Mean Backward D i g i t Spans and Mean Number of Correct Responses f o r Each C o n d i t i o n a l Argument Type by Grades (N = 77) Grade 5 Grade 7 College (N = 25) (N = 25) (N = 27) Mean 4.40 4.68 4.93 S.D. (Range) 1.66 (2-8) 1.28 (2-7) 1.98 (2-9)  CSR Set 1 3.04 (1.27)  4.00 (1.00)  4.63 (1.01)  [p, Not--q]  2.00 (1.58)  2.40 (1.35)  4.48 (1.01)  [Not-p, q)  2.68 (1.28)  3.32 (1.18)  3.85 (1.20)  [Not-p, Not--q]  3.44 (1.23)  2.48 (1.50)  3.56 (1.34)  [q, p]  2.08 (1.44)  1.84 (1.52)  2.81 (1.62)  [q, Not--P]  3.00 (1.19)  2.88 (1.42)  3.81 (1.15)  [Not-q, P]  1.84 (1.28)  2.24 (1.39)  3.48 (1.34)  [Not-q, Not-\u2022p]  1.20 (1.29)  2.24 (1.39)  2.59 (1.37)  [p. q]  Mean S.D.  \u000C30.  TABLE 1(b)  BDS  Mean Backward D i g i t Spans and Mean Number of Correct Responses for Each C o n d i t i o n a l Argument Type by Grades (N = 77) Grade 5 Grade 7 College (N = 25) (N = 25) (N = 27) Mean 4.40 4.68 4.93 1.66 (2-8) 1.28 (2-7) S.D. (Range) 1.98 (2-9)  CSR Set 2 3.00 (1.23)  3.68 (1.11)  4.22 (0.85)  [p, Not--q]  2.28 (1.31)  3.28 (1.40)  4.11 (0.85)  [Not-p, q]  2.28 (1.40)  2.76 (1.17)  3.37 (1.33)  [Not-p, Not--q]  2.76 (1.36)  2.72 (1.43)  3.41 (1.31)  (q, p]  1.80 (1.15)  1.32 (1.11)  2.59 (1.42)  [q, Not-\u2022P]  2.48 (1.42)  2.20 (1.26)  2.85 (1.70)  [Not-q, P]  1.88 (0.88)  2.48 (1.19)  3.37 (1.42)  [Not-q, Not-p]  1.08 (1.22)  1.60 (1.32)  2.41 (1.55)  [p, q]  Mean S.D.  \u000C31.  TABLE 2 Analyses of Covariance of the Number of Correct Responses for Each CSR Argument Type of Set 2 by Grade, with Students' BDS and Mastery L e v e l of Conditional Rule Based on CSR Set 1 (N = 77; Mastery N = 16, Non-mastery N = 61)  Argument Type of CSR Set 2  E f f e c t of Grade F(2, 72) P  BDS Regression Coefficient  _t  P  Mastery L e v e l Regression Coefficient t  P  9.18  0.01  0.10  -1.30  0.20  0.95  2.73  0.01  [p, Not-q]  19.69  0.01  0.12  1.54  0.13  1.18  3.31  0.01  [Not-p, q]  5.17  0.06  0.94  0.10  0.92  1.29  3.08  0.01  [Not-p, Not-q]  2.34  0.10  0.80  -0.81  0.42  1.42  3.22  0.01  [q\u00BB p]  7.56  0.01  0.37  0.41  0.68  0.96  2.37  0.02  [q, Not-p]  1.34  0.26  0.15  1.34  0.18  0.64  1.32  0.19  [Not-q, p]  12.40  0.01  0.15  1.85  0.07  0.96  2.61  0.11  6.41  0.05  0.01  0.07  0.95  0.93  2.03  0.05  535.95  0.01  0.44  1.30  0.20  8.31  5.23  0.01  [p\u00BB q]  [Not-q, Not-p] A l l Arguments  \u000CTABLE 3  Mean Number of C o r r e c t Responses, Mean V a l i d a t i o n Time, and Rank. Order f o r Each Argument Type o f CSR Set 2 Items by M a s t e r y L e v e l Argument Type  Predicted Rank Order  [p. q]  1  [p, Not--q]  Mean S.D.  Non-mastery (N = 61) Rank VT Rank C o r r e c t s Order (Sec) (Drder  M a s t e r y (N = 16) Rank Rank VT Order (Sec) lDrder Corrects  7.07 (2.04)  1  1.28 (0.69)  3  .9.44 (0.81)  1  1.49 (0.59)  2  1  5.38 (2.45)  2  1.22 (0.87)  3  9.56 (0.63)  1  1.37 (0.61)  2  [Not-p, q]  3  5.51 (2.01)  2  1.06 (0.67)  2  8.44 (1.21)  2  1.26 (0.55)  2  [Not-p, Not-q]  3  5.56 (2.22)  2  1.35 (0.86)  3  8.38 (1.26)  2  1.45 (0.55)  2  [q, p]  3  3.38 (1.90)  4  0.84 (0.86)  1  7.31 (1.62)  2  1.05 (0.42)  1  [ q , Not-- P i  3  5.28 (2.27) \u2022  2  2.08 (1.16)  4  7.63 (1.45)  2  1.44 (0.66)  2  [Not-q, P]  2  4.33 (1.93)  3  0.88 (0.55)  1  8.25 (1.29)  2  1.25 (0.50)  2  [Not-q, Not-p]  3  3.12 (2.15)  4  0.74 (0.70)  1  6.13 (1.86)  3  1.12 (0.54)  2  \u000C33. TABLE 4 Percent of Correct Responses f o r Each of Eight Conditional S y l l o g i s t i c Arguments by Grade; Comparison with Previous Studies  Conditional Syllogistic Arguments [p,q] [p,Not-q] [Not-p,q] [Not-p,Not-q] [q,p] [q,Not-p] [Not-q,p] [Not-q,Not-p]  Current Study Grade 5 7 College 60.4 42.0 49.6 62.0 38.8 54.8 37.2 22.8  76.8 56.8 60.8 52.0 31.6 50.8 47.2 38.4  88.5 85.9 72.2 69.2 54.1 66.7 68.5 50.0  T a p l i n , Staudenmayer and Taddonio Grade 5 7 8 11 90.0 75.0 24.0 10.3 7.5 21.8 66.6 58.7  81.0 70.5 28.5 19.3 13.2 26.3 63.8 54.6  89.2 80.2 30.8 26.0 34.6 37.3 69.8 61.4  94.3 91.0 46.3 37.9 51.5 54.2 66.9 59.3  T a p l i n &amp; Staudenmayer Grade College 99.1 99.2 88.4 18.0 16.2 91.2 90.3 86.8  \u000C34.  Syllogism about which a conclusion can be drawn at model stage  Stage 1 Encode premises and conclusion YES; Conclusion = Q  Stage 2 Does second premise equal P?  YES; Conclusion  Not-Q  Cluster 1 [p,q] p, Not-q]  NO Stage 3 Is conclusion consistent with with f i r s t and second premises?  Cluster 2 [Not-q, p]  NO  Cluster 3  V Y E S  Stage 4 Is negation of conclusion consistent with f i r s t and second premises?  NO  K-  [Not-p, q] [Not-p, Not q] [q\u00BB p] [q, Not-p] [Not-q, Not-p]  YES  6  5  Respond Always True  Figure 1.  6  Respond Sometimes True  Respond Never True\"  Marcus &amp; Rips four-stage model f o r v e r i f i c a t i o n of c o n d i t i o n a l syllogisms; the point of which a conclusion can be c o r r e c t l y drawn f o r each s y l l o g i s m i s a l s o shown.  \u000CAPPENDIX A EIGHT CONDITIONAL SYLLOGISTIC ARGUMENTS ARGUMENT FORM AFFIRMING THE ANTECEDENT  DENYING THE ANTECEDENT  AFFIRMING THE CONSEQUENT  DENYING THE CONSEQUENT  RESPONSE PATTERN CONDITIONAL BICONDITIONAL  CONCLUSION  FIRST PREMISE  AFFIRMATIVE  I f P, then q  P  q  [p. q]  Always true  Always true  NEGATIVE  I f P. then q  P  q  [p, Not-q]  Never true  Never true  AFFIRMATIVE  I f P. then q  P  q  [Not-p, q]  Sometimes true  Never true  NEGATIVE  If P, then q  P  q  [Not-p, Not-q] Sometimes true  Always true  AFFIRMATIVE  I f P. then q  q  p  [q, p]  Sometimes true  Always true  NEGATIVE  I f P, then q  q  p  [q, Not-p]  Sometimes true  Never true  AFFIRMATIVE  I f P. then q  q  p  [Not-q, p]  NEGATIVE  I f P. then q  q  p  [Not-q, Not-p] Always true  SECOND PREMISE  CONCLUSION  NOTATION  Never true  Never true Always true  Ul  \u000CAPPENDIX B SELECTED SPANS FOR BACKWARD DIGIT SPAN TASK  n  Sequence  6 5  4,2,9,3,7,5, 9, 7, 4 , 1, 6  3  3,  8 5 2  8, 5, 5, 2, 8, 5  5, 2 3, 6,  6, 8,  4, 3  7,  9,  2  9 3 6 2  5, 7, 4, 8, 1, 4 8, 5 , 7, 3, 7  6,  1,  9,  3,  8,  2  3,  9,  1  7 9 7 4 8 4  7, 4 , 2 , 6, 9, 4, 8, 1, 9, 2 , 4, 1, 3, 9, 4, 6 5, 9, 3, 7, 2, 8, 1, 7  8, 6, 5,  3, 2, 3,  5 6, 8  7,  3  2,  6,  4,  8  \u000C37. APPENDIX C INITIAL INSTRUCTIONS TO SUBJECTS  H e l l o . My name i s and we're going to play two games on the computer i n f r o n t of you using the numbered keys. F i r s t , I would l i k e you to type i n your given names and press the RETURN key at the r i g h t of the keyboard. (Subject types i n c h r i s t i a n names...) Good! key.  Now,  enter your surnames and then, again, press the RETURN  (Subject types i n surname  ...)  OK. Now, the games you w i l l be playing w i l l require no more knowledge of the computer than that. You w i l l each have two quite d i f f e r e n t tasks; one w i l l be remembering a l i s t of numbers and the other w i l l be a t r u e / f a l s e quiz. Some of you w i l l have the numbers task f i r s t and others, the t r u e / f a l s e . In both cases, you w i l l have four p r a c t i c e problems f i r s t . Let's s t a r t with the f i r s t task by pressing the spacebar.  \u000C38. APPENDIX D INSTRUCTIONS FOR BACKWARD DIGIT SPAN TASK The computer w i l l show you a s e r i e s of d i g i t s , or numbers, one at a time. At the end of each s e r i e s , your job w i l l be to t r y to remember a l l the d i g i t s and to l i s t them back to the computer by using the appropriate number keys. However, you are to l i s t them i n the reverse, or backwards, order to which they were shown. For example, you may see a s e r i e s such as: 1 2  3  4  The computer w i l l then show you these symbols \u2022***' displayed i n the middle of the screen. Your job w i l l be l i s t the d i g i t s back to the computer i n the f o l l o w i n g order: 4  3  2  1  I f you cannot remember one or more of the d i g i t s , you should enter a *-' i n i t s place. Let's say that you forgot the d i g i t , 3, you should then have entered the f o l l o w i n g : 4  -  2  1  The computer w i l l give you time to remember each d i g i t and w i l l then go on to the next s e r i e s . The word 'READY' w i l l show just before a new s e r i e s i s about to s t a r t . There are 4 p r a c t i c e and 16 a c t u a l items. You should work as q u i c k l y as you can but without making mistakes. Now,  l e t ' s t r y the f i r s t 4 p r a c t i c e items.  \u000C39.  APPENDIX E INSTRUCTIONS FOR THE CONDITIONAL SYLLOGISTIC REASONING TASK  This task w i l l help your l o g i c a l thinking a b i l i t y improve, i f you f o l l o w the i n s t r u c t i o n s very c a r e f u l l y . Your job i s to determine the correct conclusion that can be drawn from two premises (or c l u e s ) . For example, here are two c l u e s . The f i r s t c l u e : I f P, then Q. The second c l u e : P From these c l u e s , you are to determine whether, The conclusion: Q, i s 'always true (A),' 'sometimes true ( S ) ' or 'never true (N).' The important thing i s to understand the meaning of each c l u e . To ensure your understanding, i t i s suggested that you read each clue c a r e f u l l y f o r up to 5 seconds and then think about what you have read for another 5 seconds. Of course, i f you f i n i s h reading and are sure of your answer, then you can immediately go to the next step by pressing the spacebar. I f you don't f i n i s h w i t h i n 10 seconds, the computer w i l l go on to the next step. The most important thing i s to see whether or not your answer to each conclusion i s c o r r e c t . I f your answer was i n c o r r e c t , the corrected answer w i l l f l a s h on the screen and you should t r y to understand why that answer i s r i g h t . Good luck.  "@en
<https://open.library.ubc.ca/cIRcle/collections/ubctheses/831/items/1.0054511>