Component Analysis of Hutchings' Low-Stress Addition Algorithm

Transcription

1 Western Michigan University ScholarWorks at WMU Master's Theses Graduate College Component Analysis of Hutchings' Low-Stress Addition Algorithm Daniel V. McCallum Western Michigan University Follow this and additional works at: Part of the Educational Psychology Commons, and the Science and Mathematics Education Commons Recommended Citation McCallum, Daniel V., "Component Analysis of Hutchings' Low-Stress Addition Algorithm" (1981). Master's Theses This Masters Thesis-Open Access is brought to you for free and open access by the Graduate College at ScholarWorks at WMU. It has been accepted for inclusion in Master's Theses by an authorized administrator of ScholarWorks at WMU. For more information, please contact

2 A COMPONENT ANALYSIS OF HUTCHINGS' LOW-STRESS ADDITION ALGORITHM by D aniel V. McCallum A T hesis Subm itted to th e F acu lty of The Graduate College in p a r t i a l f u lf illm e n t of the requirem ent fo r the Degree of M aster of A rts Department of Psychology W estern Michigan U n iv ersity Kalamazoo, Michigan A p ril 1981

3 A COMPONENT ANALYSIS OF HUTCHINGS' LOW-STRESS ADDITION ALGORITHM D aniel V. McCallum, M.A. W estern M ichigan U n iv e rsity, 1981 An A-B-A-C counterbalanced r e v e rs a l design was used w ith two groups to analyze th e two components of H utching's Low -Stress additio n algorithm. Fourth grade stu d e n ts achiev in g 96% accuracy on a p r e te s t of b a s ic math f a c ts were su b je c ts of th is study. Subjects were taught two new methods of com putation; H utchings' Low -Stress and th e Conventional alg o rith m w ith a w ritte n record,, along w ith review ing the Conventional a lg o rith m. Students were given w orksheets conta in in g fix ed siz e a d d itio n problem s and asked to com plete as many as p o ssib le w ith a fiv e-m in u te timed se ssio n. Accuracy and speed were m onitored acro ss th e th re e methods of com putation. The r e s u lts showed su p e rio r perform ance w ith the alg o rith m using a f u l l record versus no reco rd. A s im ila r b u t le s s profound e f f e c t was seen w ith the algorithm th a t u t i l i z e d only b a sic math f a c ts versus comp lex f a c ts. A most im portant fin d in g was th a t a l l su b je c ts had h ig h er perform ances w ith th e Low -Stress alg o rith m when compared w ith the Conventional alg o rith m.

4 ACKNOWLEDGEMENTS This study would n o t have been p o ssib le w ithout th e help of many people. F i r s t, I would lik e to thank the fa c u lty and s ta f f of th e Department of Psychology of Western Michigan U n iv ersity fo r t h e i r guidance and tra in in g. S pecial thanks go to D rs. Galen A le s s i, Wayne Fuqua, and Cheryl Poche fo r serv in g as committee members fo r th is th e s is. An e x tra s p e c ia l thanks goes to my major a d v iso r, Galen. His knowledge, sugg e s tio n s, and a s s is ta n c e have helped to improve t h i s paper and my ed u catio n. I would lik e to thank the elem entary school p r in c ip a l, Beverly K lienhaus, fo r her in t e r e s t and coop eratio n ; and Dave Linton and Ed Huth fo r recommending stu d e n ts fo r th is study. I would a lso lik e to thank Dave Snyder, Steve Wong, Dave Keenan, Sue Dickerman and the members of G alen s recearch group, F a ll Most im p o rtan tly, I wish to express my a p p re c ia tio n to my p are n ts fo r th e ir support and help throughout my c o lle g e y e a rs. And f i n a l l y, an e x tra s p e c ia l thanks to my w ife, J a n e t, fo r h er p a tie n c e, su p p o rt, and understanding. Her encouragement was in v a lu ab le and g re a tly ap p re c ia te d. D aniel V. McCallum

5 INFORMATION TO USERS This was produced from a copy of a document sent to us for microfilming. While the most advanced technological means to photograph and reproduce this document have been used, the quality is heavily dependent upon the quality of the material submitted. The following explanation of techniques is provided to help you understand markings or notations which may appear on this reproduction. 1. The sign or target for pages apparently lacking from the document photographed is Missing Page(s). If it was possible to obtain the missing page(s) or section, they are spliced into the Him along with adjacent pages. This may have necessitated cutting through an image and duplicating adjacent pages to assure you of complete continuity. 2. When an image on the Him is obliterated with a round black mark it is an indication that the film inspector noticed either blurred copy because of movement during exposure, or duplicate copy. Unless we meant to delete copyrighted materials that should not have been filmed, you will find a good image of the page in the adjacent frame. 3. When a map, drawing or chart, etc., is part of the material being photographed the photographer has followed a definite method in sectioning the material. It is customary to begin filming at the upper left hand comer of a large sheet and to continue from left to right in equal sections with small overlaps. If necessary, sectioning is continued again beginning below the first row and continuing on until complete. 4. For any illustrations that cannot be reproduced satisfactorily by xerography, photographic prints can be purchased at additional cost and tipped into your xerographic copy. Requests can be made to our Dissertations Customer Services Department. 5. Some pages in any document may have indistinct print. In all cases we have filmed the best available copy. University Mkrc)ri!ms International 300 N. ZEEB ROAD, ANN ARBOR, Ml BEDFORD ROW, LONDON WC1 R 4EJ, ENGLAND

6 MCCALL LM» DANIEL VERM A COMPONENT ANALYSIS OF HUTCHINGS' LGW-STRESS ADD I 11DN ALGORITHM. WESTERN MICHIGAN UNIVERSITY, M.A., 1901 University Microfilms International 300 N. ZEEB RD., ANN ARBOR, Ml 48106

7 TABLE OF CONTENTS ACKNOWLEDGEMENTS... i i LIST OF TABLES AND FIGURES... v Chapter I. INTRODUCTION... 1 I I. METHOD... 9 S ubjects... 9 S e ttin g... 9 I n str u m e n ts Independent v a ria b le Dependent v a r ia b le s In te rs c o re r agreem ent D e s i g n T raining procedures P r o c e d u r e s I I I. RESULTS In te rs c o re r agreem ents O rganization of d a t a Comparison of f u l l reco rd vs. no r e c o r d Comparisons of b a sic f a c ts vs. complex f a c ts Comparisons of H utchings' Low -Stress alg o rith m (HA) and the Conventional alg o rith m w ithout a f u l l w ritte n record ( C A ) Student perform ance on stan d ard ized probes Sequence e f f e c ts i i i

8 TABLE OF CONTENTS (Continued) IV. DISCUSSION APPENDICES BIBLIOGRAPHY iv

9 LIST OF TABLES AND FIGURES Table I. In d iv id u al stu d en t means and stan d ard d e v ia tio n s fo r se ssio n perform ance a cro ss com putational methods Table I I. In d iv id u al stu d en t means and stan d a rd d e v iatio n s fo r sessio n perform ance acro ss com putational methods Table I I I. In d iv id u al stu d en t perform ance on place value and a d d itio n com putation p r o b e s...39 F igure 1. F igure 2. F igure 3. F igure 4. F ig u re 5. F igure 6. F ig u re 7. F igure 8. Columns/minute c o r r e c t and in c o rr e c t a l l com putational methpds fo r S ubject P ercentage of columns added c o r r e c tly fo r a l l com putational methods fo r S u b ject Columns/minute c o r r e c t and in c o rr e c t fo r a l l com putational methods fo r S u b ject Percentage of columns added c o r r e c tly fo r a l l com putational methods fo r S ubject Columns/minute c o rre c t and in c o rr e c t fo r a l l com putational methods fo r S ubject P ercentage of columns added c o r r e c tly fo r a l l com putational methods f o r S ubject Columns/minute c o rre c t and in c o rr e c t fo r a l l com putational methods fo r S ubject Percentage of columns added c o r r e c tly fo r a l l com putational methods fo r S ubject v

10 CHAPTER I Introduction A number of stu d ie s have been conducted w ith th e H utchings' alg o rith m sin ce i t s in c e p tio n (Hadden, 1981; G ille s p ie, 1976; Rudolf, 1976; Z oref, 1976; and A le s s i, 1974). This i n i t i a l rese a rc h has conc e n tra te d on id e n tify in g th e fu n c tio n a l r e la tio n s h ip between the a l gorithm and perform ance. Since th i s re la tio n s h ip has been r e lia b ly dem onstrated, i t is time fo r rese a rc h in th i s area to move on to a component a n a ly s is of th i s alg o rith m. This was th e undertaking of the p re se n t study. There has been a growing concern in education over the p a st few y e a rs about h te competency of the stu d en ts being graduated. The knowledge th a t th e re have been d ecreasing scores on the s c h o la s tic a p p titu d e t e s t given to high school s e n io rs w anting to e n te r co lle g e has added p ressu re to p a re n ts, te ac h e rs and a d m in ista to rs to c o rre c t th e d ecreasing s k i l l s le v e ls of th e ir stu d e n ts. R e-ev alu atio n of c u rre n t teaching techniques has occurred. This has lead some i n s t i tu tio n s to adopt minimal competency exams th a t must be passed in o rd er fo r a stu d en t to graduate from high school, and even between grad es. In m athem atics th e re has been a developing tre n d over the y ears toward focusing on concepts and le s s emphasis on a c tu a l com putationa l s k i l l s (A lessi, 1979). The N atio n al Council of Teachers of M athematics (NCTM) has expressed i t s support fo r a balanced math 1

11 curriculum th a t s tr e s s e s b a s ic s in the context of t o t a l m athem atics in s tru c tio n (A rith m etic T eacher, 1976). They a lso p o in t out the importance of sim ple com putational s k i l l s in everyday s itu a tio n s. Cox (1975) p o in ted out th a t the development of b a sic com putational s k i l l s should not be n e g le c te d. Yet many te ach ers co ntinue to see o ld e r stu d e n ts who have n o t m astered these b asic o p e ra tio n s. This lack of m astery may be p a r t i a l l y due to the stu d e n ts' n e g a tiv e re a c tio n to m athem atics in g e n era l. This re a c tio n by the stu d e n t o ften r e s u lts in te a c h e rs spending as l i t t l e time as p o s s ib le on d r i l l and more on m a te ria l of g re a te r in h eren t in te r e s t (as d efin e d by the tr a d i ti o n a l educatio n system ). Wheatley (1976) su g g ests th a t com putational p rocedures th a t contin u a lly s tr e s s some degree of understanding may r e s u l t in le s s comp u ta tio n a l f a c i l i t y. King (1972) seems to agree w ith t h i s when he s ta te s th a t e x p lan atio n s of algorithm s o ften confuse and f r u s t r a t e stu d e n ts. I t should be noted th a t K ing's statem ent i s based upon p erso n al experience and those of o th er teach ers he has ta lk e d w ith r a th e r than em p irica l d ata. Rem ediation of com putation s k i l l s is u su a lly accom plished through d r i l l and p r a c tic e which has a number of d isad v an tag es. One disadvantage is th a t p r a c tic e often consumes v alu a b le classroom time th a t could be used fo r o th e r s k i l l s and in o th e r a re a s. Teachers o ften have only a few techniques a t th e ir d isp o sa l fo r use in d r i l l. A nother d isadvantage is the monotony of the ta sk s. H utchings (1972) and A le s s i (1974) share th is view point th a t monotony and boredom generated by rep eated p ra c tic e of math fa c ts may r e s u l t in a poor

12 a t tit u d e toward th e s u b je c t being p ra c tic e d. Skinner s ta te s th a t "Figures and symbols of m athem atics have become em otional s t i m u l i... the glimpse of a column of fig u r e s, not to say an a lg e rb ra ic symbol or an in te g ra l sig n, is lik e ly to s e t o f f, not m athem atical b eh av io r, b u t a re a c tio n of a n x ie ty, g u i l t, or fe a r" (Skinner, 1968, p. 18). A lte rn a tiv e tech n iq u es fo r in s tru c tio n and p ra c tic e can be v alu ab le to the te a c h e r and stu d en t. Wheatley (1976) compared the methods of d ir e c t a d d itio n (summing each d ig it w ith the follow ing d ig it) and the ten s method (th e stu d en t looks fo r com binations of d ig i ts th a t sum to te n ). He found the d ir e c t method to be 15-18% f a s te r and ju s t as a c c u ra te as th e te n s method. He follow ed th is up in 1977 (Wheatley and McHough) and found th a t th is e f f e c t was seen a c ro ss grade le v e ls and stu d e n t a b i l i t i e s. Wheatley and McHough suggest th a t the most e f f i c i e n t alg o rith m may be one th a t co n ta in s the le a s t number of d e c is io n s. The H utchings' alg o rith m seems to f i t th i s d e s c rip tio n. Nichol (1978) d iscu ssed the use of p alin d ro n es w ith stu d e n ts in the p ra c tic e of b a s ic s k i l l s. A palindrone is a word or number th a t can be read the same b o th forw ards and backwards. Examples include noon, 212, 797, and The p alin d ro n es were found to be h e lp fu l in m otivating stu d e n ts (by tu rn in g the ta sk in to a game) when the ta sk involved p r a c tic e of b a sic com putational o p e ra tio n s. Other methods of a d d itio n com putation have been suggested. One such method was suggested by Sanders in This a d d itio n procedure was developed fo r stu d e n ts who tended to get l o s t and fo rg e t where they were in a d d itio n problem s. This method has the stu d en t

13 s i l e n t l y compute th e sum of two d ig it s and then hold up fin g e r s to rep resen t the ten s p la c e. This method i s continued down the column u n t il a l l binary a d d itio n s have been com pleted. This method does reduce the number o f responses required by the stu d en t. I t should be noted th a t Sanders pointed out th a t t h is method should only be used a fte r the studen t i s competent in a d d itio n, because th ere i s no permanent product which could be used to id e n tify error p a tte rn s. O M alley (1969) presented another com putational method. Here, in stea d of hold in g up fin g e r s fo r the ten s p la ce as done w ith the Sanders method, the student w r ite s down the te n s, hundreds, e t c., p ortion of th e sum of added p a irs to the l e f t of each column. This procedure thus provides the teacher w ith a p a r tia l w r itte n record of the stu d en t's work. An example looks lik e th is : 268 C D Another p a r tia l record algorithm was developed by Fulkerson in This method i s s lig h t ly d iffe r e n t from O M a lley 's (1969) method. Fulkerson has the student draw a lin e through th e la s t d ig it used in obtain in g a sum of ten or grea ter when adding two addends. When the student has fin ish e d adding a column, a l l th at needs to be completed i s adding the lin e s drawn through th e d ig it s

14 to o b ta in the sum to be c a rrie d to the next column. This method does provide th e te ach er w ith a p a r t i a l record of th e s tu d e n t's work. An example looks lik e th is : a * / 0b yp 1473 In 1972, H utchings developed an a lte r n a te s e t of algorithm s fo r the four b a sic m athem atical o p e ra tio n s. He s ta t e s th a t they w ill "o p e ra te w ith much le s s s tr e s s on the u ser than Conventional a lg o rith m s" (H utchings, 1976, p. 219). This method has some advan tag es over th e alg o rith m s mentioned above. I t p ro v id es a f u l l w ritte n record of a l l component o p e ra tio n s, in ste a d of the p a r t i a l reco rd provided by the O'Malley (1969) and Fulkerson (1963) methods. S anders' (1971) alg o rith m provides fo r no permanent record of the s tu d e n t's work. Since th e in tro d u c tio n of the H utchings; L ow -Stress alg o rith m, re c e n t rese a rc h has been conducted to compare stu d e n t perform ance using the H utchings' Low -Stress ad d itio n alg o rith m to th a t w ith the C onventional alg o rith m (Hadden, 1981; G ille s p ie, 1976; R udolf, 1976; Z o re f, 1976; A le s s i, 1974). The r e s u lts of the above s tu d ie s in d ic a te b e t t e r perform ance w ith the Low -Stress alg o rith m. The G ille s p ie (1976) study a ls o in d icate d stu d en t p re fe re n c e fo r the Low -Stress alg o rith m. Since these s tu d ie s in d ic a te p re fe re n c e s fo r,

15 and increased perform ance w ith the H utchings' alg o rith m, a q u estio n a r is e s concerning which components of the alg o rith m a re c r i t i c a l to these r e s u lt s. Research w ith the alg o rith m has co n cen trated on the issu e s of p re fe re n c e, perform ance, a c q u is itio n r a te s, and r e p lic a tio n of the fin d in g s w ith th e a d d itio n alg o rith m. As of now, an em pirical component a n a ly s is of the H utchings' a d d itio n alg o rith m has not been udertaken. The purpose of th i s study was to conduct such an a n a ly s is. There a re two c r i t i c a l d iffe re n c e s between H utchings' Low -Stress alg o rith m and the C onventional alg o rith m. These a re : a) binary a d d itio n w ith the Low -Stress alg o rith m versus complex a d d itio n w ith the Conventional alg o rith m, and b) the form at of th e Low -Stress alg o rith m re q u ire s th a t the stu d en t com plete a f u l l com putational reco rd. The Conv e n tio n a l a lg o rith m re q u ire s no such reco rd. This study is concerned w ith the a n a ly s is of these two components. H utchings' Low -Stress f u l l reco rd algorithm has many advantages: a) i t is easy to lo c a te problem a re a s and s p e c ific e rro rs made by the stu d e n ts, b) i t re q u ire s le s s in s tr u c tio n a l time to meet a m astery le v e l (Z oref, 1976), and c) only a knowledge of b a sic math f a c ts vs. complex f a c ts is needed fo r stu d e n ts to use the alg o rith m e ff e c tiv e ly. This l a s t advantage reduces fa c t knowledge by 90% sin ce th e re a re only 100 b a sic f a c ts and 900 complex f a c t s. Note th a t complex f a c ts involve the a d d itio n of a tw o -d ig it number w ith a s in g le - d ig it number, e. g., , and th a t b a sic f a c ts involve the a d d itio n of of two s in g le - d ig it numbers, e.g., 4+7. One

16 7 disadvantage of the f u l l record alg o rith m may be th e in creased amount of space needed on a sheet of paper per problem. This occurs because H utchings' Low -Stress alg o rith m uses h a lf - s te p n o ta tio n, which re q u ire s e x tra space between columns and rows fo r the s tu d e n t's p e n c il work. To provide th is e x tra space, te a c h e rs would need to make up s p e c ia l w orksheets fo r each lesso n. Could a f u l l record Conventional alg o rith m re q u irin g complex f a c ts be as e ffe c tiv e in a c q u is itio n and perform ance as the H utchings' f u l l record u t i l i z i n g only b a sic fa c ts? This q u estio n can only be answered by analyzing the components of the H utchings' alg o rith m to determ ine the e x te n t in flu en ce the f u l l record alo n e has on p e rfo r- mace. H utchings (1976) d e fin e s the alg o rith m as fo llo w s: The Low -Stress alg o rith m uses a new n o ta tio n, c a lle d h a lf - s te p n o ta tio n, to record in d iv id u a l s te p s. H a lf-ste p n o ta tio n uses num erals of no more than a h a lf-s p a c e in h e ig h t to record the sums of two d ig i ts. With h a lf-sp a c e n o ta tio n, the u n its p o rtio n of the sum of two d ig i ts is w ritte n a t the lower r ig h t of the b o tto m...we add the f i r s t two d i g i t s...a n d record the sum in the new n o ta tio n...t h e complete sum of each tw o -d ig it a d d itio n is recorded in h a lf-s p a c e n o ta tio n, but only th e ones portio n of the columns sum is always the same as the ones p o rtio n of the l a s t tw o -d ig it sum...the tens p o rtio n of the column sum is always the same as the number of tens recorded a t the l e f t of the column. These a re simply counted. For a column in some m ulti-colum n e x e rc is e s then, the l a s t step - th a t i s, counting the tens a t th e l e f t of the columns - would be s lig h tly changed. The counting i t s e l f is not changed in any way, but the answ er, the to t a l number of te n s, is no longer w ritte n in the tens place of the f i r s t colum n's sum but in ste a d a t the top of th e next column a t the le ft...w o rk co n tin u es in th is manner u n ti l the ex e rc ise is com pleted. N ote, however, th a t the column sum fo r the l a s t column in a m u lticolumn example is recorded in e x a c tly the same way as the sum of a single-colum n e x e rc ise (pp ).

17 8 Examples: X >.c I 5 b i il % i P- i >& 3-3. i 1 c a A ll a d d itio n o p e ratio n s a re performed in each column, follow ed by a l l n ecessary regrouping. There is no need to a l te r n a t e between a d d itio n and regrouping o p eratio n s as w ith the C onventional alg o rith m.

18 CHAPTER I I Method S ubjects The su b je c ts of th i s study were f if te e n fo u rth grade stu d e n ts atte n d in g the G alesburg-a ugusta Elementary School. G alesburg and Augusta a re two sm all ru r a l communities lo c ated approxim ately t h i r teen m iles from Kalamazoo, Michigan. The popu latio n of th e se two communities is p rim a rily Caucasian w ith the prim ary in d u stry of farm ing. A ll stu d e n ts of the fo u rth grade c la s s were given a p r e te s t to help s e le c t the su b je c ts of th is study, sin ce a knowledge of b a sic math f a c t s is considered a p re re q u is ite fo r e f f e c tiv e in s tru c tio n in the Low -Stress alg o rith m. The p r e te s t was comprised of 56 b in ary a d d itio n problem s (used by A le s s i, 1974). F ifte e n stu d e n ts who com pleted the p r e te s t in a fiv e-m in u te p erio d and achieved a 96% accuracy or b e t te r, became su b je c ts of th is study. Informed consent form s, ex p lain in g the purpose, advantages and r is k s of the stu d y, were signed and retu rn e d b efo re the stu d e n ts could p a r tic ip a te in th is study. There were no p a re n ts who chose not to have th e ir c h ild p a r tic ip a te in th is study. S ubjects were randomly assig n ed in to two groups. S ettin g The s e ttin g of th is study was a re g u la r classroom not being 9

19 10 used during the morning hours of each day. This room had no open space to o ther classroom s and was divided in to th re e se c tio n s v ia a p a r titio n and a b o o k sh elf. The room had a blackboard on each of two w alls w ith c h a irs and ta b le s throughout the room. Sessions were held once a day between 9:10-9:30 a.m. each school day. Sessions were a lso run in th e aftern o o n fo r sev eral su b je c ts during the l a s t week of the study. These a d d itio n a l sessio n s were held in the s u b je c ts ' re g u la r classroom between 2:30-3:00 p.m. Instrum ents Requirements fo r a measurement instrum ent fo r use in s tu d ie s of com putational speed and accuracy were suggested by Hutchings (1972). One of these is pro v id in g a range of examples th a t might occur in lesso n s b u t in a form which would n o t load fo r reading or eye movement s k i l l s. He a ls o s ta te s " i t is req u ire d th a t a p p lic a tio n s of the id e n tity elem ent (0) be avoided, as these a re consid ered to load fo r a d i s t i n c t p e rip h e ra l concept w hile c o n trib u tin g very l i t t l e to demands upon m em ory-retrieval fu n ctio n s" (p. 51). To conform w ith one of th ese req u irem en ts, problems were s e t on standard 8% by 11 inch pap er, fo u r per page, in two rows. Fixed siz e a d d itio n problems were used in th is study. Follow ing A le s s i's suggestion (1974), an IBM S e le c tric ty p e w riter u t i l i z i n g an o ra to r element was used to make up the w orksheets. These problems were c o n stru cted by com pleting tr ip le - s p a c in g between columns and doublespacing between rows. There was a lso double-spacing between the l a s t row of d ig its and the sum lin e. This c o n stru c tio n allow ed fo r

20 11 s u f f ic ie n t space fo r a s tu d e n t's w ritte n responses when u ti liz in g a f u l l w ritte n record algorithm. The problems were g enerated by a computer program. This program se le c te d d ig its a t random and placed them in th e proper a rray form at (5 x 7) used in th is study. Follow ing an o th er requirem ent suggested by Hutchings (1972), zeros were not included in the problem s. As mentioned above, the problem s iz e a rra y fo r th is study was fiv e columns by seven rows, (or 34 b in a ry o p e ra tio n s ). This a rray siz e was chosen because param etric s tu d ie s in d ic a te a se p a ra tio n in accuracy between the Low -Stress and C onventional a l gorithm begins a t le v e ls g re a te r than 15 b in a r ie s. P ast research is ex ten siv e w ith 7 rows by 2 columns (13 b i n a r i e s ), 7 rows by 3 columns (20 b in a r ie s ), and 7 rows by 5 columns (34 b in a r ie s ). g r e a te s t sep aratio n has occurred w ith the 7 x 5 a rra y s iz e. The The number of b in a rie s per problem i s c a lc u la te d by m u ltip ly in g the number of rows by the number of columns, minus one. This assumes th a t a l l rows and columns a re f i l l e d. A Huer-Leonidas Trackm aster model stopw atch was used to time each sessio n. S c ie n tif ic Research A sso ciatio n probes (M-l and M-13) (S c ie n tif i c Research A sso ciatio n, 1972) were used throughout the study to a sse s s any changes in the s u b je c t's com putational o r place value s k i l l s. The num erical v alu es on the probes were changed to help prevent any p ra c tic e e f f e c ts. However, th e co n ten t of the m a te ria l covered in the measurement instrum ent was c o n s is te n t a c ro ss p rese n ta tio n s.

21 12 Independent v a ria b le s Independent v a ria b le s involved the alg o rith m used by a p a r t i c u la r stu d en t to com plete th e d a ily w orksheets. These algorithm s included: a) the C onventional alg o rith m, b) H utchings' Low -Stress alg o rith m, and c) the C onventional alg o rith m w ith a f u l l w ritte n reco rd. Dependent v a ria b le s There were th re e dependent v a ria b le s involved in th is study. 1. P ercen t c o rre c t: the number of colums computed c o r r e c tly divided by the to t a l number of columns atte m p t ed, and expressed as a p e rc e n t. 2. C orrect r a te : c a lc u la te d by adding the number of columns c o rre c tly added, divided by the sessio n len g th and expressed as columns per m inute. 3. In c o rre c t r a te : c a lc u la te d by adding the number of columns in c o rre c tly added, divided by the sessio n len g th and expressed as columns in c o rre c t p er m inute. In te rs c o re r agreement Agreement data were taken on c o rre c tio n of th e s tu d e n ts' papers fo r the number of columns c o rre c t and in c o rre c t. Approxim ately 30% of the s tu d e n ts' w orksheets were c o lle c te d in each phase of the study and scored by independent sc o re rs. This agreement was c a lc u la te d by d iv id in g the to t a l number of agreem ents by the t o t a l

22 number of agreem ents p lu s disagreem ents, m u ltip lie d by 100 to y ie ld a p ercen t m easure. In a l l in s ta n ts where th e re were d is agreem ents between the two sc o re rs, the w orksheets were again scored to determ ine the tru e m easure. This in no way a ffe c te d the rep o rted agreement m easures but did allow fo r th e author to re p o rt the a c c u ra te measure of the dependent v a ria b le s. Design A re v e rs a l design is commonly used to dem onstrate r e l i a b l e cont r o l of an im portant b e h a v io ra l change (Baer, Wolf and R is le y, 1968). V a ria tio n s on th i s b a sic re v e rs a l design a re o fte n needed to overcome some of th e problem s of th is design. One problem th a t a r i s e s i f more than one treatm en t c o n d itio n is introduced is s e q u e n tia l e f f e c ts. This e f f e c t can be d e te c te d if two groups of su b je c ts a re used w ith each group re c e iv in g both trea tm e n ts b u t in d if f e r e n t o rd e rs. Since th is study was u t i l i z i n g more than one treatm en t c o n d itio n, thus producing the p o s s i b ili ty of sequence e f f e c t s, an A-B-A-C counterbalanced re v e rs a l design was used (B ailey and Bostow, 1979). The sequences used were A-B-A-C and A-C-A-B. This design is s e t up to d e te c t an d /o r co u n terb alan ce any sequence e f f e c ts th a t might occur due to the order of p re s e n ta tio n of the alg o rith m s. T raining procedures A ll stu d en ts receiv ed s im ila r in s tru c tio n s on a l l th re e a lg o r ithm s. A tw enty-m inute tr a in in g session was conducted b e fo re the

23 14 i n i t i a l p re se n ta tio n of a new alg o rith m. This tra in in g was based on w ritte n lesson plans adopted from H utchings (1972) and p rev io u sly used by o th er in v e s tig a to rs of the H utchings' Low -Stress alg o rith m (A lessi, 1974; Boyle, 1975; G ille s p ie, 1976; Rudolph, 1976; Z oref, 1976). This lesson plan was a ls o adapted fo r use in tra in in g the Cone n tio n a l algorithm w ith a f u l l w ritte n record and the Conventional algorithm w ith no w ritte n reco rd. Before r e i n s t i t u t i n g a p rev io u sly taught alg o rith m, a 10-m inute review was conducted. T his was done to help ensure the stu d en t u ti liz e d the c o rre c t com putational alg o rith m. The review lesso n plan wass adapted from H utchings' (1972) review lesso n plan. Procedures Before the a c tu a l study began, stu d e n ts were given the S c ie n tif i c Research A sso ciatio n (SRA) probe to a sse ss th e ir c u rre n t s k i l l s in a d d itio n u t i l i z i n g th e C onventional alg o rith m, and th e ir knowledge of place value concepts. This probe was a lso given b efo re the in t r o duction of each new a lg o rith m. These probes were then used to determ ine if and when any changes in s k i l l le v e ls occurred during th is stu d y. Before the beginning of each com putational se ssio n, the stu d e n ts were given a v erb al prompt th a t s p e c ifie d the algorithm c u rre n tly being used. Students were then given w orksheets w ith a d d itio n problems and asked to w rite th e ir name and the date on the space provided a t the top of th e paper. Once th is was com pleted, they

24 were asked to com plete as many problems as they could in the a l lo tt e d tim e. Each timed sessio n began w ith the v erb al prompt, "Ready, s ta r t " and ended w ith the prompt "S top". At th is p o in t the stu d en ts were in s tru c te d to b rin g th e ir papers up to the in v e s tig a to r and lin e up to re tu rn to th e ir re g u la r classroom. Each stu d en t remained on a given alg o rith m u n ti l a steady s ta te of responding was achieved. The stu d en t was then sw itched to th e next alg o rith m in the sequence. The performance of two stu d e n ts (S ubjects 3 and 7) was not cons is te n t w ith th a t of th e ir fello w stu d e n ts. Subject 3 had a very low c o rre c t ra te w ith high accu racy, and Subject 7 had e r r a t ic p erformance. A sim ple reinforcem ent procedure was se le c te d to achieve experim ental c o n tro l. Under th i s c o n d itio n, the stu d e n ts were r e in fo rced fo r e ith e r a h ig h er c o rre c t r a te (Subject 3) or fo r s ta b le perform ance (s ta te d as p lu s or minus 10 percentage p o in ts w ith Subject 7). The re in fo rc e r was one th a t the su b je c ts s e le c te d themselv es (use of the stopw atch to time ev en ts, e.g. going to and from th e ir re g u la r classro o m ). the rem ainder of the study. These co n tin g en cies were in e ff e c t fo r This procedure was n o t used w ith any o th e r su b je c ts.

25 CHAPTER I I I R esults In te rs c o re r agreem ents There were a to ta l of fo u rteen agreement checks made ap p ro x i m ately every th ird sessio n throughout th is study. These checks y ie ld e d sco res ranging from 91.4% to 100% w ith a mean of 96.7%. O rganization of data The r e s u lt s of th is study a re p resen ted in th re e main s e c tio n s: a) com parisons between the record vs. no reco rd component. This in volves s tu d e n ts ' perform ances on the C onventional alg o rith m (CA) and on the C onventional alg o rith m w ith a f u l l w ritte n reco rd ; b) com parisons between the complex a d d itio n f a c ts v s. b a sic f a c ts component. This involves s tu d e n ts' perform ances on th e H utchings' L ow -Stress alg o rith m (HA) and on the C onventional alg o rith m w ith a f u l l w ritte n record (WR); and f in a l ly, c) stu d en t perform ance on the stan d ard ized probes throughout th i s study. Comparisons w ill a ls o be made between the H utchings' alg o rith m (HA) and the Conventio n a l alg o rith m (CA). The comparisons w ill f i r s t be made in terms of g en eral e f f e c ts acro ss stu d en ts and then by s p e c ific in t r a - and in te r s u b je c t d iffe re n c e s. Perform ances in each of the f i r s t th re e s e c tio n s w ill be made in term s of the th ree dependent v a ria b le s : a) p erce n t accuracy, b) c o rre c t r a te, and c) in c o rre c t r a te. D aily sco res fo r each su b ject 16

26 . 17 are presented in Appendix D. Summary data fo r these com parisons are presented in th re e ta b le s and four fig u re s. The four fig u re s are re p re s e n ta tiv e of th e d if f e r e n t general e f f e c ts seen ac ro ss a l l su b je c ts. Comparison of f u l l reco rd v s. no record Percent of columns c o r r e c t. T ables I and I I p re se n t the means and standard d e v ia tio n s fo r the d a ily session sco res ac ro ss a l l comp u ta tio n a l methods. th e ir p re s e n ta tio n.) (Note th a t the methods a re l i s t e d in o rd er of As shown in the ta b le s, 12 of 15 su b je c ts had g re a te r accuracy when u t i l i z i n g the Conventional alg o rith m w ith the f u l l w ritte n re c o rd, when compared to the same alg o rith m w ith no reco rd. S u b jects 3, 4 and 8 showed no observable d iffe re n c e s between the two methods. Subject 4 's data in d ic a te th a t the C onventional alg o rith m w ithout the reco rd i s more a c c u ra te when compared w ith th e f i r s t co n d itio n (WR). F ig u res 1 and 2 show th e in d iv id u a l scores fo r Subject 4 through- our the study. These data a re re p re s e n ta tiv e of those su b je c ts who had no o b servable d iffe re n c e s between th ese two alg o rith m s. The excep tio n to th i s is the low mean and high standard d e v ia tio n fo r Subject 4 in th e f i r s t c o n d itio n (WR). A c lo se r look a t the d a ily raw scores and w orksheets in d ic a te s th a t the su b je c t made a number of system atic e r r o r s in the f i r s t co n d itio n (S essions 1, 3 and 4) which re s u lte d in a low mean of 63.5 and a high standard d e v ia tio n of There is a sharp upward tren d in the beginning of the f i r s t co n d itio n (see F igure 1) because of these system atic e r r o r s

27 T able I. In d iv id u a l stu d e n t means and stan d ard d e v ia tio n s fo r sessio n perform ance acro ss com putational methods.

28 TABLE I In d iv id u a l Student Means and Standard D eviations fo r Session Performance Across Com putational Methods < X RC RI # c I RC RI # X SD X SD X SD X SD X SD X SD WR S4 WR CA CA WR WR HA HA WR WR WR S6 WR CA CA WR WR HA HA WR WR WR S8 WR CA CA WR WR HA HA WR WR

29 TABLE I (Continued) % RC RI // % RC RI # X SD X SD X SD X SD X SD X SD WR S CA CA WR WR HA HA WR WR WR CA WR HA WR Note: WR r e f e r s to C onventional alg o rith m w ith a w ritte n record CA r e f e r s to Conventional alg o rith m HA r e f e r s to H utchings' alg o rith m % in d ic a te s mean se ssio n p erce n t accuracy RC in d ic a te s mean sessio n r a te of columns c o rre c t/m in u te RI in d ic a te s mean sessio n r a te of columns in c o rre c t/m in u te # in d ic a te s th e number of sessio n s in each co n d itio n ho o

30 Table I I. In d iv id u a l stu d e n t means and stan d ard d e v ia tio n s fo r sessio n perform ance acro ss com putational methods.

31 TABLE I I In d iv id u a l Student Means and Standard D eviations fo r Session Performance A cross Com putational Methods % RC RI # % RC RI # X SD X SD X SD X SD X SD X SD WR S2 WR CA 76^ CA WR WR HA HA CA CA WR S13 ^ CA HA WR WR HA CA CA WR N3 ro

32 TABLE I I (Continued) < Z RC RI # % RC RI // X SD X SD X SD X SD X SD X SD WR S15 WR HA HA WR WR CA CA WR r e f e r s to Conventional alg o rith m w ith a w ritte n record CA r e f e r s to Conventional alg o rith m HA r e f e r s to H utchings' alg o rith m % in d ic a te s mean sessio n p ercen t accuracy RC in d ic a te s mean sessio n r a te of columns c o rre c t/m in u te RI in d ic a te s mean sessio n r a te of columns in c o rre c t/m in u te // in d ic a te s the number of se ssio n s in each co n d itio n

33 24 F igure 1 F igure 2 Columns/minute c o rre c t and in c o rre c t fo r a l l com putational methods fo r Subject 4. P ercentage of columns added c o rre c tly fo r a l l com putational methods fo r Subject 4.

34 WR CA WR HA WR h- O UJ tr cc o o i- 2: U J o cr CORRECT U J t Z> z 2 \ c/) _l O o INCORRECT 10 5ESSI0N S 40

35 26 and hence a la rg e stan d ard d e v ia tio n score. If th ese system atic e rro rs a re l e f t o u t, the mean in c re ases to 91.5 and the stan d ard d ev ia tio n drops to 6.9. Figure 3 shows the in d iv id u a l sco res fo r Subject 10 throughout the study. These data a re re p re s e n ta tiv e of those su b je c ts who had g re a te r accuracy when u t i l i z i n g the w ritte n record alg o rith m. There is some overlap among the raw scores between th e w ritte n record (WR) and th e C onventional alg o rith m (CA). But the range of c o rre c t responding is in c re ased when th e su b ject u t i l i z e s the a lg o r ithm w ith the f u l l w ritte n reco rd. F igure 5 shows th e in d iv id u a l sco res fo r Subject 9. Again, h ig h er accuracy is seen when a f u l l w ritte n record is used w ith the Conventional a lg o rith m. This su b je c t a lso showed a decrease in v a r ia b ili ty when u t i l i z i n g the w ritte n record alg o rith m. This same e ff e c t was seen w ith a number of o th e r su b je c ts who had g re a te r g re a te r accuracy w ith the w ritte n reco rd vs. no reco rd. Rate of columns c o r r e c t. The mean ra te s of columns c o rre c t w ith standard d e v ia tio n s fo r a l l su b je c ts a re p resen ted in T ables I and I I. Ten s u b je c ts had mean c o rre c t r a te s th a t were cons is te n t ly h ig h er w ith th e w ritte n record alg o rith m (WR). S ubjects 2, 4 and 15 had no o b serv ab le d iffe re n c e between means w ith th ese two methods. S u b jects 8 and 14 had s lig h tly h ig h er c o rre c t r a te s fo r the Conventional alg o rith m w ith no record (CA). F igure 2 shows no observable d iffe re n c e w ith Subject 4, There were upward tre n d s fo r th is su b je c t in C onditions 3 and 5. F igures 6 and 8 show th e higher c o rre c t r a te s w ith the w ritte n

36 27 Figure 3 F igure 4 Columns/minute c o rre c t and in c o rre c t fo r a l l com putational methods fo r S ubject 10. P ercentage of columns added c o rre c tly fo r a l l com putational methods fo r S ubject 10.

37 WR CA HA WR WR H 75- o LlI or Cf o o H* z Ll I O or U J 25- CORRECT UJ H 3 Z 2 \ to o o INCORRECT 1 0 SESSIONS 40

38 29 F ig u re 5 F ig u re 6 Columns/minute c o rre c t and in c o rre c t fo r a l l com putational methods fo r Subject 3. P ercentage of columns added c o rre c tly f o r a l l com putational methods fo r Subject 3.

39 c o l u m n s / minute percent c o r r e c t c o r r e c t INCORRECT i

40 31 F igure 7 F igure 8 Columns/minute c o rre c t and in c o rre c t fo r a l l com putational methods fo r S ubject 9. P ercen tag e of columns added c o rre c tly fo r a l l com putational methods fo r S ubject 9.

41 32 WR CA WR c o l u m n s / minute percent c o r r e c t 4 - CORRECT INCORRECT i -w\av SESSIONS 40

42 33 record (WR) versus no reco rd (CA) fo r S u b jects 3 and 9. There is an upward trend in the c o rre c t r a te fo r Subject 9 in the l a s t cond itio n. There a re a lso upward tre n d s in C onditions 3 and 5 fo r Subject 3. F igure 4 shows a s lig h tly h ig h er c o rre c t r a te fo r Subject 10 when using the w ritte n record alg o rith m (WR) versus the Conventiona l alg o rith m (CA). Here a g a in, th e re is an upward trend in the c o rre c t r a te in the l a s t c o n d itio n fo r th is s u b je c t. Rate of columns in c o r r e c t. The mean r a te of columns in c o rre c t and the standard d e v ia tio n s fo r a l l su b je c ts a re p resen ted in Tables I and I I. In every ca se, except one, th e mean r a te of columns in c o rre c t was lower when the su b je c t u ti liz e d th e w ritte n record (WR) versus no record (CA) a lg o rith m. The one exception is in the f i r s t c o n d itio n of Subject 4. Again, looking a t the d a ily raw scores and w orksheets, system atic e r ro rs a re d e te c te d in Sessions 1, 3 and 4. W ithout th ese se s sio n s, the e rro r r a te drops to.24 and is c o n s is t en t w ith the data fo r th i s su b je c t in subsequent co n d itio n s. The C onventional algorithm w ith the w ritte n record c o n s is te n tly shows lower e rro r r a te s when compared w ith the C onventional algorithm w ith no record. Comparisons of b a sic f a c ts vs. complex f a c ts P ercent of columns c o r r e c t. The means and standard d e v ia tio n s fo r the p ercen t accuracy a c ro ss a l l com putational methods a re p resented in Tables I and I I. Eleven of the 15 su b je c ts had g re a te r mean a c c u ra c ie s when u t i l i z i n g the H utchings' Low -Stress algorithm

43 34 (HA) when compared w ith the C onventional alg o rith m w ith the f u l l w ritte n record. No observable mean d iffe re n c e s were seen w ith Subj e c t s 2, 4, 5 and 9. Not one su b ject had o b serv ab le mean a c c u ra c ie s th a t were cons is te n t ly higher w ith the C onventional alg o rith m w ith the w ritte n record (WR) when compared w ith the H utchings' Low -Stress algorithm. However, these su b je c ts did have one co n d itio n each where the Conv e n tio n a l algorithm w ith th e reco rd did have a s lig h tly higher mean than the Low -Stress a lg o rith m. S u b jects 2 and 9 had p o s itiv e mean d iffe re n c e s fo r the C onventional alg o rith m of.1 and 1.1%, resp ectiv e ly. S ubjects 4 and 5 had p o s itiv e mean d iffe re n c e s fo r the C onventional w ritte n record alg o rith m of 1.9 and 2.4%, re sp e c tiv e ly. S ubjects 12 and 15 had only s lig h t mean p ercentage d iffe re n c e s in fav o r of the Low -Stress alg o rith m. Though th e re were no observable mean d iffe re n c e s fo r S ubject 9 looking a t F igure 7 (which shows d a ily sco res) th ere does appear to be a decrease in the v a r ia b ili ty of accuracy when the su b je c t u t i l i z e d th e Low -Stress alg o rith m. This same e ffe c t was seen w ith a number of s u b je c ts, exem plified by Subj e c t 3 (see Figure 5). Use of the Low -Stress alg o rith m e ith e r dec re a se s the v a r ia b ili ty (seen w ith S ubjects 3 and 9), in creases the mean accuracy (Subject 10, see F igure 3 ), or shows no observable d iffe re n c e s (Subject 4, F ig u re 1). Rate of columns c o r r e c t. A ll su b je c ts except Subject 9 had cons is te n tly higher mean c o rre c t r a te s w ith the Low -Stress algorithm than the means obtained w ith th e Conventional alg o rith m w ith the w ritte n record (see T ables I and I I ). S ubjects 2, 4 and 5, who had

44 35 no m easurable d iffe re n c e s in mean a c c u ra c ie s, did show increased mean r a te s when using the Low -Stress a lg o rith m. Subject 9 showed no mean d iffe re n c e s between th ese two alg o rith m s (HA and WR) in the l a s t two c o n d itio n s (see F igure 8). A number of su b je c ts showed in c re a sin g tre n d s in c o rre c t r a te s when using the Low -Stress alg o rith m. This is shown in F igures 2, 4, 6 and 8. There g en e ra lly was a gradual in c re a se in c o rre c t r a te s over a l l experim ental c o n d itio n s irre s p e c tiv e of the sequence in which the c o n d itio n s were p rese n te d. When th e re was a decrease ac ro ss experim ental c o n d itio n s, i t g e n e ra lly occurred in th e l a s t co n d itio n (see F ig u res 2, 4 and 6). Rate of columns in c o r r e c t. There a re seven su b je c ts who had lower mean e rro r r a te s w ith the Low -Stress alg o rith m when compared w ith the w ritte n record alg o rith m (see T ables I and I I ). There were e ig h t su b je c ts fo r whom one comparison (acro ss co n d itio n s) between the C onventional alg o rith m w ith a record and th e H utchings' alg o rith m showed the w ritte n record having a s lig h t ly lower mean e rro r r a te. In th e o th e r comparison the Low -Stress alg o rith m had the lower mean e rro r r a te. F igures 2, 4 and 6 show how c lo se the a c tu a l e rro r r a te s were between these two methods. Mean d iffe re n c e s in e rro r r a te s were u su a lly in the te n th s value w ith sometimes the only d i f feren ce being seen in a hundredths of a p o in t. Comparisons of the H utchings' Low -Stress alg o rith m (HA) and the C onventional algorithm w ithout a f u l l w ritte n reco rd (CA) P ercen t of columns c o r r e c t. T ables I and I I show th a t w ith every su b je c t the Low -Stress alg o rith m y ie ld ed h ig h e r mean percent

45 c o rre c t when compared w ith th e Conventional alg o rith m (CA). The sm a lle st d iffe re n c e is seen w ith Subject 4 (2.4%) to the la rg e s t d iffe re n c e w ith Subject 13 (43.6%). Subject 10 showed a d iffe re n c e between means of 41.4%. F igure 3 g ra p h ic a lly shows th a t th e re is only one data p o in t of overlap between these two methods (Session 14). F igure 7 shows the same e ffe c t w ith Subject 9: th e re is only one data p o in t of overlap between the two methods (Session 15). H utchings' Low -Stress alg o rith m o ften decreased the v a r i a b i l i t y in accuracy which can be seen by looking a t F igure 5 fo r S ubject 3. Subject 4 has the sm alle st d iffe re n c e between the means. F igure 1 shows the d a ily sco res fo r t h i s s u b je c t. I t is d i f f i c u l t to d e te r mine through v is u a l in sp e c tio n whether th e re is a d iffe re n c e in accuracy between the two methods w ith Subject 4. This graph is ty p ic a l of those s u b je c ts who had mean d iffe re n c e s of le s s than 10% (S ubjects 1, 2, 4, 5 and 8). Rate of columns c o r r e c t. Fourteen out of the 15 s u b je c ts had h ig h er mean c o rre c t r a te s when u t i l i z i n g th e Low -Stress alg o rith m a s compared w ith the C onventional algorithm (CA). Subject 14 had a mean c o rre c t r a te of 2.5 w ith the Conventional alg o rith m compared to a mean of 2.4 c o rre c t r a te w ith the Low -Stress a lg o rith m (Conditio n s 2 and 4 in Table I ). This is a r e la tiv e ly sm all d iffe re n c e between the two means. S ubjects 4, 10, 3 and 9 a re ty p ic a l of those su b je c ts who had h ig h e r mean c o rre c t ra te s w ith th e Low -Stress alg o rith m. F igures 2, 4, 6 and 8 show th a t the d a ily sco res were h ig h e r in alm ost a l l in sta n c e s. I t should be noted th a t th e re a re in c re a sin g

46 37 tren d s in the c o rre c t r a t e fo r a l l of th ese su b je c ts when they used the Low -Stress a lg o rith m. This e f f e c t was seen w ith 11 of the 15 su b je c ts in th is stu d y. The rem aining four had r e la tiv e ly s ta b le c o rre c t ra te s when using th i s method of com putation. Subject 9 (see F igure 8) showed an upward tren d in the c o rre c t r a te w ith the C onventional alg o rith m a ls o. This same e f f e c t was seen w ith th ree o th e r su b je c ts. Student perform ance on sta n d a rd iz e d probes A ddition com putation p ro b e s. Table I I I p re s e n ts the data fo r a l l su b je c ts on a l l pro b es. A ddition com putation scores a re in d i cated in the ta b le by th e l e t t e r "N". Ten su b je c ts had no changes in th e ir p ercen t of item s scored as c o rre c t acro ss the e n tir e study. S ubjects 5, 6 and 15 had in c re a s e s of 7.3%, 18.5% and 14.7%, re s p e c tiv e ly. This in c re ase was n o t noted to occur w ith any p a r tic u la r cond itio n of the study and was u s u a lly a gradual in c re ase over the course of th is in v e s tig a tio n. Two s u b je c ts, 12 and 13, had decreases of 11.0% and 7.5%, re s p e c tiv e ly. Place value p ro b e s. These data a re p resen ted in Table I I I. P lace value scores a re in d ic a te d in the ta b le by the l e t t e r "P". The data here a re somewhat d if f e r e n t than the a d d itio n probes. Only fo u r su b je c ts showed no changes in p erce n t of item s scored c o rre c t (S ubjects 3, 5, 8 and 12). Ten s u b je c ts showed some in c re a se in sc o res. This ranged from 3.4% (one more item c o rre c tly answered) to 41.4% (12 more item s c o r r e c tly answ ered). Four su b je c ts showed in c re ases le s s than 10%, two had in c re a se s in the te e n s, and four