UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Conditional syllogistic reasoning and working memory capacity Instance, Stewart T. 1984

You don't seem to have a PDF reader installed, try download the pdf

Item Metadata

Download

Media
UBC_1984_A8 I57.pdf [ 1.79MB ]
Metadata
JSON: 1.0054511.json
JSON-LD: 1.0054511+ld.json
RDF/XML (Pretty): 1.0054511.xml
RDF/JSON: 1.0054511+rdf.json
Turtle: 1.0054511+rdf-turtle.txt
N-Triples: 1.0054511+rdf-ntriples.txt
Original Record: 1.0054511 +original-record.json
Full Text
1.0054511.txt
Citation
1.0054511.ris

Full Text

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  <g)  Stewart Instance, 1984  In 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  ABSTRACT  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 & 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 & 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.  iii.  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 & 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  iv.  TABLE OF CONTENTS (Cont'd) Page FIGURES 1 - Marcus & 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  1. 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 & 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 & 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->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 :  2. 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.  3.  The f i n d i n g s of T a p l i n , Staudenmayer & 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<->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  4. 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 & 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  5. According to Marcus & 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 & 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»-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 & 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  6. processing a f t e r any of the f i r s t three stages indicated above.  As  pointed out by Marcus & 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 & 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  7. 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).  8. 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.  9. 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  10. 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 «i + 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 .  11. 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 & 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 & 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  12. 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 & 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 & 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.  13. With the exception of RT measurement, the current study r e p l i c a t e d Experiment 2 of Marcus & 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  14. 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 & 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.  15. 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.  16. 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.  17. 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  18. 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 .  19. 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  •  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 .  20. 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 & 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 & 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  21. 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. •  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 & 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.  22. 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 &  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 & Staudenmayer (1973) and T a p l i n , Staudenmayer & 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.  INSERT TABLE 4 ABOUT HERE __5ee_pa£e__"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.  24. 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 & 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, £ < 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  25. 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 & 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 & 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.  26. 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 & 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 & 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.  27. REFERENCES Atkinson, R.C. & 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 & 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. & 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, & 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., & 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.  FOOTNOTES 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.  29.  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-•p]  1.20 (1.29)  2.24 (1.39)  2.59 (1.37)  [p. q]  Mean S.D.  30.  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-•P]  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.  31.  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» 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» q]  [Not-q, Not-p] A l l Arguments  TABLE 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) •  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  33. 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 & Staudenmayer Grade College 99.1 99.2 88.4 18.0 16.2 91.2 90.3 86.8  34.  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» p] [q, Not-p] [Not-q, Not-p]  YES  6  5  Respond Always True  Figure 1.  6  Respond Sometimes True  Respond Never True"  Marcus & 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.  APPENDIX 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  APPENDIX 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  37. 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.  38. 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 •***' 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.  39.  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.  

Cite

Citation Scheme:

    

Usage Statistics

Country Views Downloads
United States 14 0
China 8 5
Germany 6 1
France 5 1
Russia 3 0
Thailand 1 0
Serbia 1 0
City Views Downloads
Unknown 16 1
Ashburn 5 0
Beijing 4 0
Shenzhen 4 5
University Park 2 0
Mountain View 2 0
Sunnyvale 1 0
Chicago 1 0
Seattle 1 0
Redmond 1 0
Phoenix 1 0

{[{ mDataHeader[type] }]} {[{ month[type] }]} {[{ tData[type] }]}
Download Stats

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0054511/manifest

Comment

Related Items