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An analysis of two approaches to teaching used by grade 8 mathematics teachers Tam, Rosita Tseng 1983

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Cry ( AN ANALYSIS OF TWO APPROACHES TO TEACHING . USED BY GRADE 8 MATHEMATICS TEACHERS By ROSITA TSENG TAM B.Sc, The University of B r i t i s h Columbia, 1973 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS in THE FACULTY OF GRADUATE STUDIES Department of Mathematics and Science Education Department of Educational Psychology and Special Education Faculty of Education We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA August 1983 © Rosita Tseng Tarn, 1983 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. I t i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of Mathematics and Science Education Department of Educational Psychology The University of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date August, 1983 DE-6 (.3/81) ABSTRACT The purpose o f the s t u d y was t o p e r f o r m an a n a l y s i s o f two approaches t o t e a c h i n g used by Grade 8 mathematics t e a c h e r s i n B r i t i s h Columbia. A comparison was made i n t h e o r i e n t a t i o n t o c o n c r e t e and a b s t r a c t approaches. The major q u e s t i o n a d d r e s s e d i n t h e s t u d y was: Were t e a c h e r p r a c t i c e s i n accordance w i t h t h e t h e o r y o f i n s t r u c t i o n proposed by s e v e r a l a u t h o r s , i n which i t i s recommended t h a t c o n c e p t s be p r e s e n t e d u s i n g m a n i p u l a t i v e m a t e r i a l s , p i c t u r e s , o r diagrams, and t h e n t h r o u g h a b s t r a c t p r e s e n t a t i o n s w i t h symbols? To address t h i s q u e s t i o n , an a n a l y s i s was made o f t h e d a t a c o l l e c t e d from a p p r o x i m a t e l y 100 mathematics t e a c h e r s who took p a r t i n t h e Second I n t e r n a t i o n a l Mathematics Study (SIMS) d u r i n g t h e 1980-81 s c h o o l y e a r . The i n s t r u m e n t s used t o c o l l e c t t h e d a t a were a s e t o f f i v e t o p i c - s p e c i f i c q u e s t i o n n a i r e s on c l a s s r o o m p r o c e s s e s . The i t e m s i n t h e s e q u e s t i o n n a i r e s r e l a t e d t o many a s p e c t s o f c l a s s r o o m p r a c t i c e . T h i s s t u d y was concerned w i t h t h o s e i t e m s t h a t were c o n s t r u c t e d t o c o l l e c t i n f o r m a t i o n on methods used i n p r e s e n t i n g c e r t a i n c o n c e p t s and s k i l l s i n t h e Grade 8 mathematics c u r r i c u l u m . S i n c e t h e n a t u r e o f t h e d a t a was not known p r i o r t o a n a l y s i s , t h e s t u d y was d e s i g n e d t o e x p l o r e t h e n a t u r e o f t h e d a t a so t h a t v a l i d comparisons c o u l d be made on t h e c o n c r e t e - a b s t r a c t v a r i a b l e . S e v e r a l d a t a p r o c e s s i n g p r o c e d u r e s were used. These p r o c e d u r e s i n c l u d e d t h e c a t e g o r i z a t i o n o f i t e m s by a p a n e l o f e x p e r t s ; t h e s c o r i n g o f i t e m s ; the p r e l i m i n a r y a n a l y s i s t o i n v e s t i g a t e t h e j u s t i f i a b i l i t y o f a g g r e g a t i o n o f i t e m s c o r e s ; and t h e c a l c u l a t i o n o f c o n c r e t e , a b s t r a c t , and d i f f e r e n c e s c o r e s f o r a l l t h e t e a c h e r s . E x p l o r a t o r y d a t a a n a l y s i s t e c h n i q u e s were t h e n used t o compare t h e t e a c h e r s ' c h o i c e between c o n c r e t e and a b s t r a c t approaches. The r e s u l t s o f t h e a n a l y s i s seemed t o i n d i c a t e t h a t t e a c h e r s were a b s t r a c t l y - o r i e n t e d i n t e a c h i n g most o f the s k i l l s and c o n c e p t s i n c l u d e d i n t h e s t u d y . I n t h e i r t e a c h i n g o f t h r e e new c o n c e p t s and s k i l l s i n t h e Grade 8 c u r r i c u l u m , mathematics t e a c h e r s were c o n c r e t e l y - o r i e n t e d i n t h e i r t e a c h i n g o f one concept o n l y . T h i s f i n d i n g l e d t o t h e c o n c l u s i o n t h a t t e a c h e r p r a c t i c e s were not i n accordance w i t h t h e t h e o r y o f i n s t r u c t i o n which s t a t e s t h a t new c o n c e p t s and s k i l l s i n mathematics s h o u l d be i n i t i a l l y t a u g h t u s i n g c o n c r e t e l y - o r i e n t e d approaches. i v TABLE OF CONTENTS Chapter Page ABSTRACT i i ACKNOWLEDGEMENTS i x 1 THE PROBLEM 1 Background 1 Framework f o r the SIMS 6 D e f i n i t i o n of Terms 8 Purpose of the Study 11 2 REVIEW OF THE LITERATURE 13 Theories of I n s t r u c t i o n and Related Research 13 Classroom P r a c t i c e s and Related Research 18 R e l i a b i l i t y and V a l i d i t y of Q u e s t i o n n a i r e s 24 3 DESIGN AND PROCEDURE 33 Po p u l a t i o n and Sample S e l e c t i o n . . . . 33 The Instruments 36 Data C o l l e c t i o n 39 Data P r o c e s s i n g Procedure 39 C a t e g o r i z a t i o n of Items 40 Sco r i n g of Items 44 P r e l i m i n a r y A n a l y s i s 44 Scor i n g of Subtopics 47 D i f f e r e n c e Score f o r Each Subtopic 48 Method of A n a l y s i s 49 4 RESULTS OF THE STUDY 52 Concrete and A b s t r a c t Scores 52 Common and Decimal F r a c t i o n s . . 53 Algebra 57 Geometry 61 Measurement 61 D i f f e r e n c e Scores 65 Summary 68 V T a b l e of Contents - continued Chapter Page 5 CONCLUSIONS AND IMPLICATIONS 6 9 Answer to Research Question 69 Limitation of the Study 71 Implications 72 Suggestions for Further Research.. 75 Summary 78 REFERENCES 80 APPENDICES A Common and Decimal Fractions Questionnaire 86 B Ratio, Proportion and Percent Questionnaire 104 C Algebra Questionnaire 120 D Geometry Questionnaire 144 E Measurement Questionnaire 168 F The I n i t i a l L i s t of Subtopics and Items as Categorized by the Panel of Experts 187 G P r o f i l e s of Responses to the Concrete and Abstract Items for Each Subtopic for the Sample of Ten Teachers 189 v i LIST OF TABLES Table Page 1 C o r r e l a t i o n s o f St u d e n t s and Teacher w i t h Observer 29 2 D e s c r i p t i o n o f P o p u l a t i o n A T e a c h e r s . . . . 35 3 The L i s t o f S u b t o p i c s and C o r r e s p o n d i n g Items f o r the C o n c r e t e and A b s t r a c t C a t e g o r i e s as V a l i d a t e d by the P a n e l of E x p e r t s 42 4 The L i s t o f S u b t o p i c s and C o r r e s p o n d i n g Items I n c l u d e d f o r A n a l y s i s i n T h i s Study 43 5 The Mean V a r i a n c e s of Item Scores from C o r r e s p o n d i n g S u b t o p i c Scores f o r the Sample o f Ten Teachers 46 v i i L I S T OF FIGURES F i g u r e Page 1 Framework f o r the i n t e r n a t i o n a l s tudy 7 2 An a b s t r a c t t r e a t m e n t o f the concept o f n e g a t i v e i n t e g e r s 8 3 An a b s t r a c t t r e a t m e n t o f the concept o f f r a c t i o n s 9 4 A c o n c r e t e t r e a t m e n t o f the concept o f n e g a t i v e i n t e g e r s 10 5 A c o n c r e t e t r e a t m e n t o f t h e co n c e p t o f f r a c t i o n s 10 6 Items d e a l i n g w i t h methods o f i n s t r u c t i o n f o r the co n c e p t o f i n t e g e r s 38 7 D i s t r i b u t i o n o f a b s t r a c t s c o r e s and c o n c r e t e s c o r e s f o r the concept o f f r a c t i o n s 54 8 D i s t r i b u t i o n o f a b s t r a c t s c o r e s and c o n c r e t e s c o r e s f o r the co n c e p t o f d e c i m a l f r a c t i o n s 54 9 D i s t r i b u t i o n o f a b s t r a c t s c o r e s and c o n c r e t e s c o r e s f o r the a d d i t i o n o f f r a c t i o n s 56 10 D i s t r i b u t i o n o f a b s t r a c t s c o r e s and c o n c r e t e s c o r e s f o r the c o n c e p t o f i n t e g e r s 58 11 D i s t r i b u t i o n o f a b s t r a c t s c o r e s and c o n c r e t e s c o r e s f o r t h e s u b t r a c t i o n o f i n t e g e r s 60 12 D i s t r i b u t i o n o f a b s t r a c t s c o r e s and c o n c r e t e s c o r e s f o r t h e co n c e p t Pythagorean Theorem 6 2 v i i i L i s t o f F i g u r e s - c o n t i n u e d F i g u r e Page 13 D i s t r i b u t i o n o f a b s t r a c t s c o r e s and c o n c r e t e s c o r e s f o r the concept number fr 64 14 D i s t r i b u t i o n o f a b s t r a c t s c o r e s and c o n c r e t e s c o r e s f o r the r e l a t i o n s h i p among v a r i o u s m e t r i c u n i t s 64 15 D i s t r i b u t i o n o f d i f f e r e n c e s c o r e s f o r the e i g h t s u b t o p i c s 66 ACKNOWLEDGEMENTS The a u t h o r would l i k e t o e x p r e s s her g r a t i t u d e t o members o f her t h e s i s committee -- Dr. David R o b i t a i l l e Dr. Todd Rogers, and Dr. Doug Edge -- f o r t h e i r a d v i c e and encouragement; t o members o f her p a n e l o f e x p e r t s -Dr. G. Bluman, Dr. A. A d l e r , Dr. W. S z e t e l a , and Mr. M. D i r k s — f o r c o n s u l t a t i o n s e r v i c e s r e g a r d i n g d a t a p r o c e s s i n g ; and t o members o f her f a m i l y , Mr. & Mrs. J o r g e Tseng, Mr. Theodore Tam, and M i s s Genevieve Tam f o r t h e i r s u p p o r t and c o o p e r a t i o n . C h a p t e r 1 THE PROBLEM Background I t i s a p p a r e n t from t h e l i t e r a t u r e t h a t i n t h e l a s t 20 t o 30 y e a r s many changes i n t h e m a t h e m a t i c s c u r r i c u l a have been p r o p o s e d i n Canada and i n c o u n t r i e s t h r o u g h o u t t h e w o r l d (House, 1979; Howosn, 1970; R o b i t a i l l e , 1980; and Van d e r B l i j , H i l d i n g and W e i n z w e i g , 1980). N i c h o l s (1968) d e s c r i b e d t h i s p e r i o d as "a p e r i o d o f r e v o l u t i o n " Cp. 1 6 ) . New t o p i c s s uch as p r o b a b i l i t y , a l g e b r a and number t h e o r y were added t o t h e e l e m e n t a r y s c h o o l c u r r i c u l u m . T o p i c s such as s t a t i s t i c s , l o g i c and computer s c i e n c e became a p a r t o f t h e s e c o n d a r y s c h o o l c u r r i c u l u m (Van d e r B l i j , H i l d i n g and Weinzweig, 198 0 ) . New m a t h e m a t i c s programs p r o d u c e d by g r o u p s s u c h as t h e U n i v e r s i t y o f I l l i n o i s Committee on S c h o o l M a t h e m a t i c s (UICSM) (Howson, K e i t e l and K i l p a t r i c k , 1981) and t h e S c h o o l M a t h e m a t i c s Study Group (SMSG) (Howson, K e i t e l and K i l p a t r i c k , 19 81) were i n t r o d u c e d i n t o t h e s c h o o l s . Nev; t h e o r i e s i n d e v e l o p m e n t a l p s y c h o l o g y ( P i a g e t , 1963) and i n i n s t r u c t i o n ( B r u n e r , 1966; D i e n e s , 1971) were d e v e l o p e d . B r u n e r (1966) p r e s e n t e d h i s t h e o r y o f i n s t r u c t i o n when he e x t e n d e d h i s d e v e l o p m e n t a l work i n t o p r e s c r i p t i o n s f o r c l a s s r o o m i n s t r u c t i o n . He was i n f l u e n c e d by P i a g e t ' s work 1 and d e f i n e d t h r e e "modes of r e p r e s e n t a t i o n " (p. 46) i n concept development. An e n a c t i v e r e p r e s e n t a t i o n i s a mode of r e p r e s e n t i n g a concept t h r o u g h a p p r o p r i a t e motor response o r p h y s i c a l m a n i p u l a t i o n o f r e l e v a n t c o n c r e t e o b j e c t s . An i c o n i c r e p r e s e n t a t i o n i n v o l v e s the use o f p i c t u r e s , maps o r o t h e r forms of mental imagery. A s y m b o l i c r e p r e s e n t a t i o n i s a word o r mark t h a t stands f o r some concept o r event but i n no way resembles t h a t c oncept o r e v e n t . Bruner s t a t e d t h a t new c o n c e p t s s h o u l d be t a u g h t i n a s p e c i f i c sequence: The o b j e c t was t o b e g i n w i t h e n a c t i v e r e p r e s e n t a t i o n . . . --something t h a t c o u l d l i t e r a l l y be done or b u i l t - -and t o move from t h e r e t o an i c o n i c r e p r e s e n t a t i o n . . . . A l o n g the way, n o t a t i o n was d e v e l o p e d . . . i n t o a p r o p e r l y s y m b o l i c system (p. 64, 6 5 ) . Dienes (1971), i n h i s p r i n c i p l e o f m u l t i p l e embodiments emphasized the use o f c o n c r e t e r e p r e s e n t a t i o n s i n the i n t r o d u c t i o n o f new c o n c e p t s . H i s argument was t h a t a c q u a i n t a n c e w i t h a c o n c e p t t h r o u g h a v a r i e t y o f c o n c r e t e m a t e r i a l s h e l p s the l e a r n e r w i t h the a b s t r a c t i o n o f the c o n c e p t . Prominent mathematics e d u c a t o r s such as A s h l o c k (1967), Fennema (1972), Suydam and D e s s a r t (1976), R a t h m e l l (1978), and R e s n i c k and F o r d (1981) a l s o urged t h a t new c o n c e p t s be t a u g h t t h r o u g h th e use o f c o n c r e t e m a t e r i a l s . Suydam (1976) summarized the o p i n i o n s o f numerous mathematics e d u c a t o r s t o the e f f e c t t h a t "mathematical i d e a s and s k i l l s be d e v e l o p e d from a c o n c r e t e , p h y s i c a l base" (p. 6 ) . T h i s summary i s c o n s i s t e n t w i t h recommendations from c u r r e n t r e s e a r c h . A f t e c o m p l e t i n g a l i t e r a t u r e r e v i e w o f s t u d i e s on the use o f c o n c r e t e methods i n mathematics i n s t r u c t i o n , R e s n i c k and 3 Ford (1981) found that most researchers suggested that new concepts be introduced using concrete material, followed by semi-concrete material such as pictures and f i n a l l y through abstract presentation with symbols. There i s evidence that certain proposed changes in the curriculum were in accordance with Piaget's developmental theory and Bruner's theory of in s t r u c t i o n . For example, programs produced by SMSG r e f l e c t various features that are in harmony with the results of Piaget's work. K i l p a t r i c k (1970) wrote: Piaget's evidence that the c h i l d of nine or ten can handle many of the basic concepts of Euclidean s p a t i a l representation and measurement i s mirrored i n SMSG1s placement of such topics in the elementary curriculum (p. 249). Regarding teaching strategies, he also wrote that "Piaget's emphasis on the pa r t i c i p a t i o n of the learner as he performs re a l actions on the learning materials... i s shown in SMSG's use of discovery exercises in a l l of i t s text materials" (p. 249). In the SMSG curriculum, other advanced topics such as pr o b a b i l i t y , s t a t i s t i c s and number theory were also introduced at lower grade l e v e l s . Van der B l i j , Hilding and Weinzweig (1980) contended that t h i s s h i f t of topics was inspired by Bruner who stated that "any subject can be taught e f f e c t i v e l y in some i n t e l l e c t u a l l y honest form to any c h i l d at any stage of development" (Bruner, 1963; p. 33). Unfortunately, although "teachers found the demonstrations [of new programs ] persuasive, they did not adopt the new programs on returning to the i r schools" (House, 1979; p. 2) . 4 Howson (1979) attributed the f a i l u r e o f d i f f u s i o n o f these new programs into regular classrooms in part to the composition of writing teams of these new products: In the United Kingdom such teams usually contained a preponderance of teachers, whereas in the United States the balance was more even or was weighted towards the university academic. The dangers of placing r e s p o n s i b i l i t y for writing in the hands of university mathematicians are exemplified i n the work of many projects whose materials have long since been cast aside as impracticable. Yet teams which lack professional mathematical support produce material lacking in coherence, purpose and balance. Indeed the achievement of 'balance' i s notoriously d i f f i c u l t (p. 144). While the l i t e r a t u r e (House, 1979; Howson, 1979; Howson, K e i t e l and K i l p a t r i c k , 1981) i s in agreement that dissemination of most of these new programs has f a i l e d and that t h e i r content was not adopted in mathematics classrooms, i t i s s i l e n t on the subject of whether or not theories proposed by developmental psychologists, mathematicians and mathematics educators have influenced teacher behaviour and methods of i n s t r u c t i o n in the classroom. In p a r t i c u l a r , i t i s not known i f teacher practices are in accordance with the theory that concrete approaches be used in the instruction of mathematical concepts. An assumption held by some educators has been that once a theory of i n s t r u c t i o n i s developed, i t would subsequently be implemented in educational practice (Common, 1980). Un-fortunately, there has been l i t t l e research to v e r i f y t h i s assumption. The few studies (Goodlad and Klein, 1970; Price, Kelley and Kelley, 1977; Fey, 1979) that have investigated c l a s s r o o m p r a c t i c e s have c o n c l u d e d t h a t many of the changes b e l i e v e d by e d u c a t o r s to be t a k i n g p l a c e have not been, a c c e p t e d and implemented by c l a s s r o o m t e a c h e r s . However, such a s m a l l sample of s t u d i e s cannot p r o v i d e v a l i d , g e n e r a l i z a b l e i n f o r m a t i o n . S e v e r a l mathematics e d u c a t o r s (Westbury, 1972; L a n i e r , 1978; R o b i t a i l l e , 1980; McKnight, 1980; and Fulla.n and P a r k , 1981) have w r i t t e n t h a t d e s c r i p t i v e a n a l y s e s o f the methods o f i n s t r u c t i o n used by teacheps in, c l a s s r o o m s ajre absent i n mathematics e d u c a t i o n l i t e r a t u r e . The a u t h o r s o f the NACOME Report (1975) s t a t e d t h a t " a p a l l i n g l y l i t t l e i s known about t e a c h i n g i n any l a r g e f r a c t i o n o f U.S. c l a s s r o o m s . The vacuum o f d a t a on c l a s s r o o m p r a c t i c e s s h o u l d g i v e pause t o t h o s e who p r e s e n t s i m p l i f i e d c a u s e - a n d - e f f e c t e x p l a n a t i o n s " p. 68). Thus, th e problem i s t h a t t h e r e i s a l a c k o f i n f o r m a -t i o n on c l a s s r o o m p r a c t i c e s . L i t t l e i s known about the r e a l i t y o f i m p l e m e n t a t i o n o f i n n o v a t i v e i d e a s i n mathematics c l a s s r o o m s . The Second I n t e r n a t i o n a l Mathematics Study (SIMS) ( R o b i t a i l l e , O'Shea and D i r k s , 1982) was d e s i g n e d i n p a r t t o p r o v i d e h i g h l y s p e c i f i c i n f o r m a t i o n on c l a s s r o o m p r a c t i c e s . F o r example, i n f o r m a t i o n on methods used by t e a c h e r s when t e a c h i n g s p e c i f i c t o p i c s i n the c u r r i c u l u m was c o l l e c t e d . The SIMS, t h e n , p r o v i d e d a d a t a base f o r r e s e a r c h e r s t o examine the methods adopted by t e a c h e r s . I n p a r t i c u l a r , the d a t a i n c l u d e d i n f o r m a t i o n on t e a c h e r s ' o r i e n t a t i o n t o the c o n c r e t e o r a b s t r a c t approaches i n t e a c h i n g s e l e c t e d c o n c e p t s and s k i l l s . A d e s c r i p t i v e a n a l y s i s o f t h i s p o r t i o n o f the 6 d a t a would p r o v i d e the much needed i n f o r m a t i o n r e g a r d i n g the t r a n s l a t i o n o f t h e o r y i n t o p r a c t i c e . T h i s p r e s e n t study was d e s i g n e d t o p e r f o r m such an a n a l y s i s . The d a t a used f o r the a n a l y s i s were c o l l e c t e d from the SIMS and r e l a t e d t o the c o n c r e t e - a b s t r a c t v a r i a b l e . The framework f o r the i n t e r n a t i o n a l study i s p r e s e n t e d i n the n e x t s e c t i o n t o p r o v i d e the c o n t e x t i n which the p r e s e n t study was co n d u c t e d . Framework f o r the SIMS A c c o r d i n g t o R o b i t a i l l e , O'Shea and D i r k s (1982), the i n t e r n a t i o n a l s t u d y i s "a b r o a d l y - b a s e d , c o m p a r a t i v e i n v e s t i g a t i o n o f t h e mathematics c u r r i c u l u m as p r e s c r i b e d , as t a u g h t , and as l e a r n e d " (p. 5 ) . In t h i s s t u d y , the mathematics c u r r i c u l u m i s viewed from t h r e e p e r s p e c t i v e s : the i n t e n d e d c u r r i c u l u m , t h e implemented c u r r i c u l u m , and the a t t a i n e d c u r r i c u l u m . The i n t e n d e d c u r r i c u l u m i s the mathematics c u r r i c u l u m as p r e s c r i b e d by the M i n i s t r y o f E d u c a t i o n i n each o f the p a r t i c i p a t i n g c o u n t r i e s . The implemented c u r r i c u l u m i s the c u r r i c u l u m as t a u g h t by t e a c h e r s i n t h e i r c l a s s r o o m s . The a t t a i n e d c u r r i c u l u m i s the c u r r i c u l u m as l e a r n e d by the s t u d e n t s as e v i d e n c e d by t h e i r knowledge o f and a t t i t u d e s toward mathematics. F i g u r e 1 g i v e s a view o f the framework f o r the s t u d y . 7 ComDonent: Level of Focus: I. Curriculum Analysis Intended Curriculum Educational System II. Classroom Processes III. Student Outcomes Implemented Curriculum Attained Curriculum School and Classroom Student F i g u r e 1. Framework f o r t h e i n t e r n a t i o n a l s t u d y . As shown i n F i g u r e 1, t h e t h r e e main components o f t h e SIMS c o r r e s p o n d t o t h e t h r e e p e r s p e c t i v e s o f t h e m a t h e m a t i c s c u r r i c u l u m : a c u r r i c u l u m a n a l y s i s ; an a n a l y s i s o f c l a s s r o o m p r o c e s s e s ; and an a n a l y s i s o f s t u d e n t outcomes, b o t h c o g n i t i v e and a f f e c t i v e , i n t h e l i g h t o f t h e n a t u r e o f t h e c u r r i c u l u m and t h e k i n d o f i n s t r u c t i o n r e c e i v e d . F o r t h e a n a l y s i s o f t h e i n t e n d e d c u r r i c u l u m , c u r r i c u l u m g u i d e s and r e l a t e d documents were e x a m i n e d . F o r t h e s t u d y o f t h e i m p l e m e n t e d c u r r i c u l u m , v a r i o u s q u e s t i o n n a i r e s on c l a s s r o o m p r o c e s s e s , e a c h d e a l i n g w i t h a s p e c i f i c t o p i c , were d e v e l o p e d T h e s e i n s t r u m e n t s were d e s i g n e d t o c o l l e c t h i g h l y s p e c i f i c i n f o r m a t i o n f r o m t e a c h e r s r e g a r d i n g t h e methods t h e y used i n t e a c h i n g t h e s e t o p i c s i n t h e c u r r i c u l u m . The f i v e s p e c i f i c t o p i c s s e l e c t e d f o r one p o p u l a t i o n o f t h e s t u d y were: common and d e c i m a l f r a c t i o n s ; r a t i o , p r o p o r t i o n and p e r c e n t ; a l g e b r a ; geometry; and measurement. The a t t a i n e d c u r r i c u l u m was a s s e s s e d by t e s t i t e m s , q u e s t i o n n a i r e s and a t t i t u d e s c a l e s . D e f i n i t i o n o f Terms The f o l l o w i n g d e f i n i t i o n s o f terms were adopted f o r use i n t h i s s t u d y : An a b s t r a c t r e p r e s e n t a t i o n i s a r e p r e s e n t a t i o n which uses a word o r mark t h a t s t a n d s f o r some concept o r event but i n no way resembles t h a t concept o r e v e n t . An a b s t r a c t t r e a t m e n t o f c o n t e n t r e l i e s p r i m a r i l y on an e x p l a n a t i o n which i s s y m b o l i c i n n a t u r e and d e r i v e s i t s meanings from o t h e r m a t h e m a t i c a l c o n t e n t . Examples of a b s t r a c t t r e a t m e n t of c o n t e n t a r e g i v e n i n F i g u r e s 2 and 3. F i g u r e 2 i s an a b s t r a c t t r e a t m e n t of the c oncept o f n e g a t i v e i n t e g e r s found i n t h e a l g e b r a q u e s t i o n n a i r e (see Appendix C, i t e m 2 1 ) . The need t o extend the number system t o i n c l u d e the n e g a t i v e i n t e g e r s i s d i s c u s s e d i n o r d e r t o f i n d a s o l u t i o n t o e q u a t i o n s l i k e : + 7 - 5 F i g u r e 2. An a b s t r a c t t r e a t m e n t o f the concept o f n e g a t i v e i n t e g e r s . F i g u r e 3 i s an a b s t r a c t t r e a t m e n t o f the concept of f r a c t i o n s found i n the common and d e c i m a l f r a c t i o n s q u e s t i o n n a i r e (see Appendix A, i t e m 24). F r a c t i o n s are d e f i n e d as q u o t i e n t s : 3/4 means "3 d i v i d e d by 4" F i g u r e 3. An a b s t r a c t t r e a t m e n t o f the concept o f f r a c t i o n s . A c o n c r e t e r e p r e s e n t a t i o n i s a r e p r e s e n t a t i o n t h a t i n v o l v e s the use o f m a n i p u l a t i v e m a t e r i a l s , diagrams o r p i c t u r e s . A c o n c r e t e t r e a t m e n t o f c o n t e n t r e l i e s p r i m a r i l y on m a n i p u l a t i v e m a t e r i a l s , diagrams o r e x p e r i e n c e s from the environment. I t s h o u l d be noted t h a t c o n c r e t e r e p r e s e n t a t i o n s i n t h i s s tudy i n c l u d e d r e p r e s e n t a t i o n s which were, i n B r u n e r ' s d e f i n i t i o n s , e i t h e r e n a c t i v e o r i c o n i c i n n a t u r e . Examples o f c o n c r e t e t r e a t m e n t o f c o n t e n t are g i v e n i n F i g u r e s 4 and 5. F i g u r e 4 i s an example of a c o n c r e t e t r e a t m e n t o f the concept o f n e g a t i v e i n t e g e r s found i n the a l g e b r a q u e s t i o n n a i r e (see Appendix C, i t e m 2 0 ) . 10 N e g a t i v e i n t e g e r s are i n t r o d u c e d by e x t e n d i n g the number ray (0 and p o s i t i v e numbers) t o the l e f t . In t h i s way, d i r e c t i o n as w e l l as magnitude are shown i n the diagram. <r 1 1 1 1 1 1 1 > - 3 - 2 - 1 0 1 2 3 -3 means 3 u n i t s t o the l e f t o f 0. F i g u r e 4. A c o n c r e t e t r e a t m e n t o f the concept of n e g a t i v e i n t e g e r s . F i g u r e 5 i s an example of a c o n c r e t e t r e a t m e n t o f the c oncept of f r a c t i o n s found i n the common and d e c i m a l f r a c t i o n s q u e s t i o n n a i r e (see Appendix A, i t e m 2 1 ) . F r a c t i o n s are d e f i n e d as p a r t s o f r e g i o n s : 3/4 means F i g u r e 5. A c o n c r e t e t r e a t m e n t o f t h e c oncept of f r a c t i o n s . A t o p i c i s d e f i n e d by the c o n t e n t c o v e r e d by each c l a s s r o o m p r o c e s s q u e s t i o n n a i r e . 11 A s u b t o p i c i s a concept o r s k i l l w i t h i n a s e l e c t e d t o p i c . For example, f o u r s u b t o p i c s w i t h i n the t o p i c a l g e b r a were: concept of i n t e g e r s , a d d i t i o n of i n t e g e r s , s u b t r a c t i o n o f i n t e g e r s , m u l t i p l i c a t i o n o f i n t e g e r s and f o r m u l a . A new s u b t o p i c i s a s u b t o p i c t h a t i s i n t r o d u c e d f o r the f i r s t time i n Grade 8 a c c o r d i n g t o the c u r r i c u l u m g u ide used i n B r i t i s h Columbia. The new s u b t o p i c s i n t h i s study were: concept of i n t e g e r s , a d d i t i o n o f i n t e g e r s , s u b t r a c t i o n o f i n t e g e r s , m u l t i p l i c a t i o n o f i n t e g e r s , f o r m u l a e , and Pythagorean Theorem. A r e v i e w s u b t o p i c i s a s u b t o p i c t h a t has been i n t r o d u c e d i n t h e mathematics c u r r i c u l u m b e f o r e Grade 8 a c c o r d i n g t o t h e c u r r i c u l u m g u i d e used i n B r i t i s h Columbia. The r e v i e w s u b t o p i c s i n t h i s s tudy were: concept of f r a c t i o n s , a d d i t i o n o f f r a c t i o n s , c oncept of d e c i m a l f r a c t i o n s , o p e r a t i o n s w i t h d e c i m a l f r a c t i o n s , r a t i o , p r o p o r t i o n , number% and r e l a t i o n s h i p among v a r i o u s m e t r i c u n i t s . Purpose o f t h e Study The p r e s e n t s t u d y was f o c u s e d on t h e p r a c t i c e o f t e a c h i n g mathematics t h r o u g h the use o f c o n c r e t e m a t e r i a l s . The purpose o f the study was t o d e s c r i b e and compare the t e a c h e r s ' c h o i c e o f c o n c r e t e o r a b s t r a c t approaches i n t e a c h i n c r c e r t a i n c o n c e p t s and s k i l l s w i t h i n s e l e c t e d t o p i c s i n G r a d e 8. The majo r q u e s t i o n a d d r e s s e d i n t h e s t u d y was: Were t e a c h e r p r a c t i c e s i n a c c o r d a n c e w i t h t h e t h e o r y o f i n s t r u c t i o n p r o p o s e d by s e v e r a l a u t h o r s , i n w h i c h i t i s recommended t h a t c o n c e p t s be p r e s e n t e d u s i n g m a n i p u l a t i v e m a t e r i a l s , p i c t u r e s , o r d i a g r a m s , and t h e n t h r o u g h a b s t r a c t p r e s e n t a t i o n s w i t h s y m b o l s ? To a d d r e s s t h i s q u e s t i o n , t h e f o l l o w i n g a s p e c t s o f i n s t r u c t i o n a l b e h a v i o u r were e x a m i n e d : 1) The c o n s i s t e n c y w i t h i n i n d i v i d u a l t e a c h e r s i n t h e i r o r i e n t a t i o n t o t h e c o n c r e t e and a b s t r a c t a p p r o a c h e s i n t e a c h i n g e a c h s u b t o p i c . 2) T e a c h e r s ' o r i e n t a t i o n t o t h e a b s t r a c t o r c o n c r e t e a p p r o a c h e s i n t e a c h i n g v a r i o u s new and r e v i e w s u b t o p i c s . Chapter 2 REVIEW OF THE LITERATURE The l i t e r a t u r e r e l a t e d t o t h r e e major t o p i c s r e l e v a n t t o t he purpose o f the study i s reviewed i n t h i s c h a p t e r . F i r s t , t he v i e w p o i n t s o f v a r i o u s p s y c h o l o g i s t s , m a t h e m a t i c i a n s and mathematics e d u c a t o r s r e g a r d i n g the use o f c o n c r e t e approaches a r e r e v i e w e d . T h i s i s f o l l o w e d by a r e v i e w o f the r e s e a r c h on c l a s s r o o m p r a c t i c e s . S i n c e c l a s s r o o m p r o c e s s r e s e a r c h o f t e n i s based upon d a t a and i n f o r m a t i o n c o l l e c t e d u s i n g q u e s t i o n n a i r e s , the r e l i a b i l i t y and v a l i d i t y o f such i n s t r u m e n t s i s examined i n t h e l a s t s e c t i o n . T h e o r i e s o f I n s t r u c t i o n and R e l a t e d Research Bruner (1966) s p e c i f i e d t h a t the most e f f e c t i v e sequence i n w hich t o i n t r o d u c e new c o n c e p t s t o s t u d e n t s i s t o p r o g r e s s i n t he o r d e r e n a c t i v e , i c o n i c t o s y m b o l i c . In h i s c l a s s r o o m s t u d i e s , B r uner worked c l o s e l y w i t h i n d i v i d u a l s t u d e n t s on ex p e r i m e n t s t h a t were m a i n l y concerned w i t h mathematics l e a r n i n g . F o r example, i n one st u d y (Bruner and Kenney, 1965), q u a d r a t i c e q u a t i o n s were i n t r o d u c e d t o Grade t h r e e s t u d e n t s u s i n g base t e n b l o c k s . Another procedure i n v o l v i n g the m a n i p u l a t i o n o f w e i g h t s on a b a l a n c e beam was used t o r e i n f o r c e the same con c e p t . D u r i n g the c o u r s e o f i n s t r u c t i o n , B runer (1966) o b s e r v e d the sequence i n which concept development o c c u r r e d : 13 The c h i l d r e n always began by c o n s t r u c t i n g an embodiment of some c o n c e p t , b u i l d i n g a c o n c r e t e model....The f r u i t of the c o n s t r u c t i o n was an image...that stood f o r the c o n c e p t . From t h e r e on, the t a s k was t o p r o v i d e [ s y m b o l i c ] r e p r e s e n t a t i o n s t h a t were f r e e o f p a r t i c u l a r m a n i p u l a t i o n s and s p e c i f i c images (p. 65). Bruner emphasized the importance o f t e a c h i n g i n t h i s sequence as he wrote: But what s t r u c k us about t h e c h i l d r e n as we observ e d them i s t h a t they not o n l y u n d e r s t o o d the a b s t r a c t i o n s they had l e a r n e d but a l s o had a s t o r e o f c o n c r e t e images t h a t s e r v e d t o e x e m p l i f y the a b s t r a c t i o n s . When they s e a r c h e d f o r a way t o d e a l w i t h new problems, the t a s k was u s u a l l y c a r r i e d out not s i m p l y by a b s t r a c t means but a l s o by m a tching up images (p. 6 5 ) . On the b a s i s o f h i s o b s e r v a t i o n , Bruner argued t h a t even though some s t u d e n t s might be q u i t e ready f o r a p u r e l y s y m b o l i c r e p r e s e n t a t i o n , i t seemed r e a s o n a b l e and wise t o p r e s e n t a t l e a s t the i c o n i c r e p r e s e n t a t i o n f i r s t . He d e s c r i b e d t h i s t h e o r y as c o n s e r v a t i v e , and s t a t e d : When the l e a r n e r has a w e l l - d e v e l o p e d s y m b o l i c system, i t may be p o s s i b l e t o by-pass t h e f i r s t two s t a g e s . But one does so w i t h the r i s k t h a t the l e a r n e r may not p o s s e s s the imagery t o f a l l back on when h i s s y m b o l i c t r a n s f o r m a t i o n s f a i l t o a c h i e v e a g o a l i n problem s o l v i n g (p. 49). K i l p a t r i c k (1970) p o i n t e d o u t t h a t "much o f B r u n e r 1 s i n t e r p r e t a t i o n o f new co n c e p t s and methods i n e d u c a t i o n i s made i n l i g h t o f what P i a g e t has s a i d about i n t e l l e c t u a l development" (p. 249). For t h a t r e a s o n , P i a g e t ' s views on c o g n i t i v e development and the i m p l i c a t i o n o f h i s views f o r mathematics e d u c a t i o n are summarized i n the f o l l o w i n g p a r agraphs i n o r d e r t h a t the f o u n d a t i o n o f Bruner's t h e o r y can be b e t t e r u n d e r s t o o d . 15 P i a g e t (1963) summarized the f o u r s t a g e s o f c o g n i t i v e development as f o l l o w s : s e n s o r i - m o t o r i n t e l l i g e n c e , from b i r t h t o two y e a r s ; p r e o p e r a t i o n a l thought, from two t o seven y e a r s ; c o n c r e t e o p e r a t i o n s , from seven t o 11 y e a r s ; and f o r m a l o p e r a t i o n s , from 11 t o 15 y e a r s . Each stage o f development i s c h a r a c t e r i z e d by s p e c i f i c b e h a v i o u r . The p e r i o d o f s e n s o r i - m o t o r i n t e l l i g e n c e i s c h a r a c t e r i z e d p r i m a r i l y by motor b e h a v i o u r , whereas the p e r i o d o f p r e o p e r a t i o n a l thought i s c h a r a c t e r i z e d p r e d o m i n a n t l y by language development. D u r i n g the p e r i o d o f c o n c r e t e o p e r a t i o n s , the c h i l d d e v e l o p s the a b i l i t y t o a p p l y l o g i c a l thought t o problems t h a t can be p r e s e n t e d i n a c o n c r e t e manner. The c h i l d then becomes a b l e t o a p p l y l o g i c t o problems t h a t are p r e s e n t e d i n an a b s t r a c t manner when he r e a c h e s the p e r i o d o f f o r m a l o p e r a t i o n s . R e g a r d i n g the i m p l i c a t i o n s o f P i a g e t ' s views f o r mathematics e d u c a t i o n , E a s l e y (1979) wrote: P i a g e t ' s t h e o r y emphasized r e f l e c t i v e a b s t r a c t i o n from one's own p h y s i c a l a c t i o n i n the f o r m a t i o n o f l o g i c o - m a t h e m a t i c a l mental s t r u c t u r e s . . . . W h a t he c a l l s c o n c r e t e o p e r a t i o n s i s the a p p l i c a t i o n o f i n t e r n a l i z e d l o g i c o - m a t h e m a t i c a l t r a n s f o r m a t i o n s o r c o r r e s p o n d e n c e s , o n e - a t - a - t i m e , t o o b j e c t s o r t h e i r i n t e r n a l r e p r e s e n t a -t i o n s . . . and may i n v o l v e the m a n i p u l a t i o n o f p h y s i c a l o b j e c t s o r n o t , as c i r c u m s t a n c e s p e r m i t (p. 1 0 ) . Thus, p h y s i c a l a c t i o n and m a n i p u l a t i o n o f o b j e c t s a r e i m p o r t a n t i n t h e p r o c e s s o f a c q u i r i n g m a t h e m a t i c a l knowledge. The m a n i p u l a t i o n o f o b j e c t s or t h e i r i n t e r n a l r e p r e s e n t a t i o n s l e a d s t o u n d e r s t a n d i n g and a p p l i c a t i o n o f l o g i c a l thought t o m a t h e m a t i c a l problems. H e r e i n l i e s the f o u n d a t i o n o f 16 B r u n e r ' s t h e o r y t h a t e n a c t i v e o r i c o n i c p r e s e n t a t i o n s be used b e f o r e s y m b o l i c p r e s e n t a t i o n i n the t e a c h i n g o f any ma t h e m a t i c a l c o n c e p t . T h i s sequence of i n s t r u c t i o n i s recommended so t h a t the l e a r n e r w i l l have "not o n l y a f i r m sense o f the a b s t r a c t i o n u n d e r l y i n g what he was working on, but a l s o a good s t o c k o f v i s u a l images f o r embodying them" (Bruner, 1966; p. 66). Dienes (1973) a l s o advocated t h e use o f c o n c r e t e m a t e r i a l s i n mathematics i n s t r u c t i o n . In h i s p r i n c i p l e o f m u l t i p l e embodiments, Dienes emphasized the use o f c o n c r e t e m a t e r i a l s i n the i n t r o d u c t i o n o f m a t h e m a t i c a l c o n c e p t s . He s t a t e d t h a t a c q u a i n t a n c e w i t h a con c e p t through a v a r i e t y o f c o n c r e t e r e p r e s e n t a t i o n s h e l p s the l e a r n e r w i t h the a b s t r a c t i o n o f the con c e p t . In one o f h i s e a r l i e r s t u d i e s w i t h young c h i l d r e n , Dienes (1963) used m u l t i b a s e a r i t h m e t i c b l o c k s (a s e t o f mathematics m a t e r i a l s which he designed) t o p r e s e n t the f a c t o r i n g p r i n c i p l e u n d e r l y i n g q u a d r a t i c e q u a t i o n s . In a l a t e r s t u d y , Dienes and G o l d i n g (1971) used a t t r i b u t e b l o c k s t o d i s p l a y p r i n c i p l e s o f c l a s s i f i c a t i o n , s e t t h e o r y and l o g i c . I n b o t h c a s e s a c o n c r e t e approach was used when i n t r o d u c i n g new c o n c e p t s . Suydam (1976) has summarized t h e views o f t h e o r i s t s such as P i a g e t , Bruner and Di e n e s . She s t a t e d : G e n e r a l l y , r e s e a r c h e r s have c o n c l u d e d t h a t u n d e r s t a n d i n g i s b e s t f a c i l i t a t e d by the use o f c o n c r e t e m a t e r i a l s , f o l l o w e d by s e m i - c o n c r e t e m a t e r i a l s (such as p i c t u r e s ) , and f i n a l l y by the a b s t r a c t p r e s e n t a t i o n w i t h symbols.... The c h i l d s h o u l d have many e x p e r i e n c e s i n which r e a l o b j e c t s a re m a n i p u l a t e d . Only a f t e r an i d e a has been developed w i t h r e a l m a t e r i a l s s h o u l d p i c t u r e s , c h a r t s , and o t h e r l e s s c o n c r e t e m a t e r i a l s be used -- and the use of symbols a l o n e s h o u l d be d e l a y e d u n t i l t he c h i l d has a b a s i c u n d e r s t a n d i n g . . . . These s t e p s s h o u l d not be l i m i t e d t o the p r i m a r y l e v e l , t h e y a re a l s o i m p o r t a n t a t l a t e r grade l e v e l s when new m a t h e m a t i c a l i d e a s a re i n t r o d u c e d (p. 6, 7 ) . T h i s same endorsement was a l s o suggested by R a t h m e l l (1978) who contended t h a t c o n c r e t e m a t e r i a l s s h o u l d p r o b a b l y be used d u r i n g the i n i t i a l i n s t r u c t i o n f o r an o p e r a t i o n . C o n c r e t e m a t e r i a l p l a y an i m p o r t a n t r o l e i n con c e p t development. The m a t e r i a l s become a r e f e r e n t f o r work i n v o l v i n g the o p e r a t i o n . They p r o v i d e a l i n k t o conn e c t the o p e r a t i o n t o r e a l - w o r l d p r o b l e m - s o l v i n g s i t u a t i o n s (p. 1 6 ) . And, more r e c e n t l y , R e s n i c k and F o r d (1981) advocated the t e a c h i n g o f new c o n c e p t s u s i n g the c o n c r e t e approach: The s t r u c t u r e s o f mathematics may be t a u g h t i n an i n t e l l e c t u a l l y honest way a t an e a r l y age by p r e s e n t i n g them i n c o n c r e t e form, e s p e c i a l l y i n the form o f math m a t e r i a l s t h a t p h y s i c a l l y embody t h o s e s t r u c t u r e s (p. 126). In summary, d e v e l o p m e n t a l p s y c h o l o g i s t s , m a t h e m a t i c i a n s and mathematics e d u c a t o r s s t r o n g l y s u p p o r t i n i t i a l l y t e a c h i n g m a t h e m a t i c a l c o n c e p t s t h r o u g h c o n c r e t e l y - o r i e n t e d approaches, w i t h the g r a d u a l i n t r o d u c t i o n o f s y m b o l i c r e p r e s e n t a t i o n s . 18 C l a s s r o o m P r a c t i c e s and R e l a t e d Research I t i s d i f f i c u l t t o o b t a i n an a c c u r a t e p i c t u r e o f what i s happening i n o r d i n a r y c l a s s r o o m s (NACOME Report, 1975). The a u t h o r s of the NACOME Report (1975) p o i n t e d out t h a t when what i s " c u r r e n t " i s d i s c u s s e d i n c o n f e r e n c e s , i t u s u a l l y means t h a t " c u r r e n t t r e n d s a t the c u t t i n g edge of i n n o v a t i o n " (p. 67) are b e i n g d i s c u s s e d . However, the i m p r e s s i o n i s l e f t t h a t t h e p r e s e n t time " i s a time of g r e a t change and ferment" (p. 6 7 ) , when i n r e a l i t y , l i t t l e i s known about what i s happening i n o r d i n a r y c l a s s r o o m s . What seems t o be known i s t h a t c l a s s r o o m s which are p a r t o f some w e l l - p u b l i c i z e d , w e l l - f u n d e d p r o j e c t appear t o be i m p l e m e n t i n g the i n n o v a t i v e i d e a s suggested by t h e o r i s t s . But one o f t h e c o n c l u s i o n s o f the r e p o r t i s t h a t l i t t l e i s known about methods of i n s t r u c t i o n i n the r e m a i n i n g l a r g e r number of c l a s s r o o m s . L a n i e r (1978) summarized the essence o f t h i s d e f i c i e n c y w i t h the s t atement: " D e s c r i p t i v e a n a l y s e s o f t e a c h e r s p l a n n i n g f o r and i n s t r u c t i n g groups o f l e a r n e r s i n c l a s s r o o m s are o b v i o u s l y absent i n mathematics l i t e r a t u r e " (p. 7 ) . Very few r e s e a r c h s t u d i e s have been conducted t o i n v e s t i g a t e c l a s s r o o m p r a c t i c e s . The few s t u d i e s i n t h e l i t e r a t u r e (Goodlad and K l e i n , 1970; P r i c e , K e l l e y and K e l l e y , 1977; R o b i t a i l l e and S h e r r i l l , 1977; R o b i t a i l l e , 1981) which were d e s i g n e d t o examine c l a s s r o o m p r a c t i c e s p r o v i d e g e n e r a l i n f o r m a t i o n such as the c o n t e n t t h a t was t a u g h t , r e s o u r c e s 19 used, c l a s s r o o m o r g a n i z a t i o n and methods of e v a l u a t i o n . These s t u d i e s do not p r o v i d e s p e c i f i c and d e t a i l e d i n f o r m a t i o n on i n s t r u c t i o n a l methods. In p a r t i c u l a r , q u e s t i o n s r e l a t e d t o the t e a c h e r s ' o r i e n t a t i o n t o c o n c r e t e o r a b s t r a c t approaches were not a d d r e s s e d . Goodlad and K l e i n (1970) conducted a study t o f i n d out whether the i n n o v a t i v e e d u c a t i o n a l p r a c t i c e s recommended by e d u c a t o r s were a c t u a l l y f i n d i n g t h e i r way i n t o the s c h o o l s . They s e l e c t e d t h r e e t y p e s o f s c h o o l s f o r c l a s s r o o m o b s e r v a t i o n : s c h o o l s e n r o l l i n g a h i g h p r o p o r t i o n o f c u l t u r a l l y d i s a d v a n t a g e d c h i l d r e n , s c h o o l s c o n s i d e r e d t o be t y p i c a l , and s c h o o l s c o n s i d e r e d t o be i n n o v a t i v e . T h e i r f i n d i n g s were based on o b s e r v a t i o n s and i n t e r v i e w s conducted i n 150 c l a s s r o o m s r a n g i n g from k i n d e r g a r t e n t o t h i r d grade. They c o n c l u d e d from t h e i r f i n d i n g s t h a t many o f the most noted and recommended c u r r i c u l u m i n n o v a t i o n s were " d i m l y c o n c e i v e d and, a t b e s t , p a r t i a l l y implemented" (p. 72) i n s c h o o l s c l a i m i n g t h e i r use. I t was t h e i r o p i n i o n t h a t n o v e l f e a t u r e s seemed t o be b l u n t e d i n the e f f o r t t o t w i s t t h e i n n o v a t i o n i n t o f a m i l i a r c o n c e p t u a l frames o r e s t a b l i s h e d p a t t e r n s o f schooling....Many o f the changes b e l i e v e d by e d u c a t o r s t o be t a k i n g p l a c e i n s c h o o l i n g have not been g e t t i n g i n t o c l a s s r o o m s (p. 72). P r i c e , K e l l e y and K e l l e y (1977) conducted a study t o d e t e r m i n e what a c t u a l c l a s s r o o m p r a c t i c e s o c c u r i n second and f i f t h grade mathematics c l a s s e s i n t h e U n i t e d S t a t e s . A q u e s t i o n n a i r e d e s i g n e d t o g a t h e r i n f o r m a t i o n 20 about p r a c t i c e s i n mathematics c l a s s r o o m s was sent t o a sample of second and f i f t h grade t e a c h e r s a c r o s s the U n i t e d S t a t e s . T h i s sample o f 1200 t e a c h e r s was s e l e c t e d u s i n g t h e f o l l o w i n g p r o c e d u r e : D u r i n g the s p r i n g of 1975, 300 s u p e r v i s o r s from a l i s t o f more than 800 p r o v i d e d by the N a t i o n a l C o u n c i l o f Teachers o f Mathematics (NCTM) were randomly s e l e c t e d and asked t o d i s t r i b u t e 10 q u e s t i o n n a i r e s each. The q u e s t i o n n a i r e s were t o be d i s t r i b u t e d randomly, 5 of them t o second-grade t e a c h e r s and 5 t o f i f t h - g r a d e t e a c h e r s , i n the a r e a s e r v e d by each s u p e r v i s o r . The s u p e r v i s o r s were p r o v i d e d w i t h d e t a i l e d p r o c e d u r e s f o r the random s e l e c t i o n o f the t e a c h e r s (p. 323). In the s t u d y , the r e s e a r c h e r s i n v e s t i g a t e d the i m p l e m e n t a t i o n of i n n o v a t i v e p r a c t i c e s i n t h e s e c l a s s r o o m s . The i n n o v a t i v e p r a c t i c e s examined were: i n d i v i d u a l i z e d and s m a l l group i n s t r u c t i o n , t h e use of m u l t i p l e sources o f mathematics i n f o r m a t i o n f o r i n s t r u c t i o n , and the use o f c o n c r e t e m a t e r i a l s . The f i n d i n g s i n d i c a t e d t h a t 40% o f the t e a c h e r s used whole-c l a s s i n s t r u c t i o n most o f the t i m e ; 82% o f t h e t e a c h e r s used one o r p r e d o m i n a n t l y one t e x t ; and 72% o f the t e a c h e r s used c o n c r e t e l y - o r i e n t e d approaches l e s s than 10% o f t h e i r i n s t r u c t i o n a l t i m e . R e g a r d i n g the g e n e r a l i z a b i l i t y o f t h e s e f i n d i n g s , the r e s e a r c h e r s w r o t e : I t i s c o n c e i v a b l e t h a t t h e method of s e l e c t i n g r e s p o n d e n t s t h r o u g h s u p e r v i s o r s i n t r o d u c e d a b i a s i n t h e r e s u l t s , s i n c e not a l l s c h o o l d i s t r i c t s have mathematics s u p e r v i s o r s . I t might be assumed t h a t the t e a c h e r s were more p r o f e s s i o n a l , b e t t e r p r e p a r e d , and more l i k e l y t o t r y i n n o v a t i v e methods i n mathematics t e a c h i n g t h a n the average t e a c h e r (p. 324). The a u t h o r s c o n c l u d e d t h a t mathematics t e a c h e r s and c l a s s r o o m s have changed f a r l e s s i n the p a s t f i f t e e n 21 years than had been supposed: "Teachers are es s e n t i a l l y teaching the same way that they were taught in school. Almost none of the concepts, methods, or big ideas of modern mathematics programs have appeared in t h i s median classroom" (p. 330). The Ministry of Education in the province of B r i t i s h Columbia (B.C.) conducted an assessment of mathematics classrooms in Grades 4, 8 and 12 i n 1976-77. In this study ( R o b i t a i l l e and S h e r r i l l , 1977), 3500 mathematics teachers completed a comprehensive questionnaire dealing with numerous aspects of the methods and materials used in the teaching of mathematics in the province. Each questionnaire consisted of fi v e parts: background and general information, learning outcomes, classroom organization, classroom instruction, and use of textbooks. The part on classroom instruction was designed to c o l l e c t information on the resources, aids and methods teachers used when teaching mathematics. The s p e c i f i c i n s t r u c t i o n a l methods included for investigation were: use of concretely-oriented approaches such as learning centers and laboratories, individualized i n s t r u c t i o n , t o t a l class i n s t r u c t i o n , team teaching, and computer-aided instr u c t i o n . The results of the assessment indicated that less than 25% of the secondary mathematics teachers used concretely-oriented approaches in their i n s t r u c t i o n . In the report, the recommendation was made to teachers at a l l levels that they should "vary t h e i r teaching approaches to include 22 [ c o n c r e t e l y - o r i e n t e d methods ] ...such as the use o f l e a r n i n g c e n t e r s and mathematics l a b o r a t o r y a c t i v i t i e s " (p. 91). A recommendation was a l s o made t o t e a c h e r e d u c a t o r s t h a t they s h o u l d "encourage t h e i r s t u d e n t t e a c h e r s t o d e v e l o p the s k i l l s r e q u i r e d t o use such t e c h n i q u e s " (p. 9 1 ) . The second B.C. Mathematics Assessment ( R o b i t a i l l e , 1981) was c a r r i e d o ut d u r i n g the 1980-81 s c h o o l y e a r . A p p r o x i m a t e l y 1600 t e a c h e r s o f mathematics completed q u e s t i o n n a i r e s which d e a l t w i t h a number of i m p o r t a n t a s p e c t s o f the t e a c h i n g and l e a r n i n g o f mathematics. Much i n f o r m a t i o n was c o l l e c t e d on v a r i o u s c l a s s r o o m a c t i v i t i e s i n c l u d i n g t h e use o f o r a l , i n d i v i d u a l and group work; the use o f o b s e r v a t i o n , t e a c h e r - p r e p a r e d t e s t s and c o m m e r c i a l l y -made t e s t s f o r e v a l u a t i o n ; the n a t u r e o f the c u r r i c u l u m ; and the use o f c a l c u l a t o r s . However, q u e s t i o n s r e l a t e d t o the o r i e n t a t i o n t o c o n c r e t e o r a b s t r a c t approaches i n i n s t r u c t i o n were not i n c l u d e d i n the second assessment. A l t h o u g h t h e few s t u d i e s r e p o r t e d i n the p r e c e d i n g paragraphs i n v o l v e d i n v e s t i g a t i o n i n t o c l a s s r o o m p r a c t i c e s , such a s m a l l sample cannot p r o v i d e v a l i d , g e n e r a l i z a b l e i n f o r m a t i o n on c l a s s r o o m p r o c e s s e s . Once a g a i n i t becomes apparent t h a t t h e r e i s a l a c k o f d e s c r i p t i o n o f c l a s s r o o m p r a c t i c e s . T h i s o p i n i o n i s s u p p o r t e d by s e v e r a l mathematics e d u c a t o r s . F o r example, R o b i t a i l l e (1980), i n a d d r e s s i n g the i s s u e o f t h e r e a l i t y o f c u r r i c u l u m r e f o r m i n s c h o o l mathematics s i n c e 1960, r a i s e d the q u e s t i o n o f how much 23 anyone r e a l l y knows about the a c t u a l i m p l e m e n t a t i o n p r o c e s s e s i n mathematics c l a s s r o o m s . He d e s c r i b e d the s i t u a t i o n as one i n which "the e x t e n t t o which the w i d e l y p u b l i c i z e d changes of the p a s t two decades i n methods o f t e a c h i n g mathematics have r e s u l t e d i n o b s e r v a b l e changes i n the c l a s s r o o m b e h a v i o u r s o f t e a c h e r s " (p. 90, 91) i s not known. In B u l l e t i n No. 4 o f the SIMS the same message i s echoed: Very l i t t l e d e t a i l e d i n f o r m a t i o n i s a v a i l a b l e on what i n s t r u c t i o n a l s t r a t e g i e s are employed by t e a c h e r s as they go about t e a c h i n g . . . . I t i s e s s e n t i a l t h a t we have more i n f o r m a t i o n about what s t u d e n t s e n c o u n t e r as they study i n the mathematics c l a s s r o o m (p. 1 3 ) . T h i s same o p i n i o n was a l s o e x p r e s s e d by McKnight (1980) who contended t h a t t h e r e a r e "no adequate p o r t r a i t s o f what goes on i n s c h o o l s and c l a s s r o o m s as mathematics c u r r i c u l a a re implemented" (p. 2 4 1 ) . He p o i n t e d out t h a t "a d e s c r i p t i v e mosaic [ t h a t ] c o v e r s a v a r i e t y o f a s p e c t s r e l a t e d t o a v a r i e t y o f s p e c i f i c i n s t r u c t i o n a l s i t u a t i o n s and i n s t r u c t i o n a l d e c i s i o n - m a k i n g " (p. 257) i s much needed. However, he suggested t h a t t h e items o f t h e f i v e t o p i c - s p e c i f i c q u e s t i o n -n a i r e s used i n t h e SIMS t o d e s c r i b e c l a s s r o o m p r o c e s s e s do p r o v i d e s p e c i f i c , d e t a i l e d p i e c e s o f i n f o r m a t i o n t h a t make the c o n s t r u c t i o n o f such a d e s c r i p t i v e mosaic p o s s i b l e . Another group o f s t u d i e s t h a t i n v o l v e d the i n v e s t i g a t i o n o f t e a c h i n g p r a c t i c e s i s e x p e r i m e n t s on t e a c h i n g methods. These s t u d i e s were d e s i g n e d t o compare th e e f f e c t i v e n e s s o f c e r t a i n i n s t r u c t i o n a l methods. Medley (1979) completed a l i t e r a t u r e r e v i e w o f t h e s e methods e x p e r i m e n t s and c o n c l u d e d : Almost every methods experiment that I have found in the l i t e r a t u r e was designed to use the pupil (rather than the teacher) as the unit of analysis. As a r e s u l t , no v a l i d generalization to teachers other than those who actually took part in the experiment could be made. In order to make such generalization possible, many teachers (the more the better) would have to teach by each method, so that an accurate estimate could be made of the consistency of the results obtained by d i f f e r e n t teachers using the same method. This has rarely been done (p. 14) . In summary, i t i s evident that there i s a lack of description of classroom practices. This i s p a r t i c u l a r l y evident when focusing on the concrete-abstract variable. In spite of the strong endorsement by theorists and mathematics educators for concretely-oriented approaches, l i t t l e i s known about the r e a l i t y of the translation of this theory of inst r u c t i o n into practice. R e l i a b i l i t y and V a l i d i t y of Questionnaires In t h i s section, research on the r e l i a b i l i t y and v a l i d i t y of questionnaires i s reviewed. The studies considered seem to provide contradictory results on the v a l i d i t y of teacher self-report data. A possible reason for the contradiction might be that they d i f f e r in design, purpose, variable in question and instrument used. In t h e i r study on classroom processes, Goodlad and Klein (1970) interviewed 150 teachers to obtain the teachers' opinion of classroom a c t i v i t i e s such as i n d i v i d u a l i z a t i o n of instruction, use of a wide range of i n s t r u c t i o n a l materials, group processes, and inductive or discovery methods in the i r classrooms. Trained observers also made anecdotal records of the same a c t i v i t i e s in these classrooms. In comparing the teacher data with the observer records, the researchers found a discrepancy between the reports of observers of classroom a c t i v i t i e s and the reports of the teachers who were interviewed. When interpreting t h i s finding, Goodlad and Klein pointed out that a major l i m i t a t i o n of the study was that the observation records "varied in t h e i r comprehensiveness and in the degree to which observational data were separated from evaluations without data" (p. 34). Therefore, no recommendation was made regarding the use of teacher self-report data in research. Ehman (1970) reported on the varying degree of consistency among three sources of data with respect to the students' freedom to express themselves. The three sources of data were teacher, student and observer. Observations of discussions on controversial issues were made by trained observers in 14 s o c i a l studies classrooms. At the end of the i n s t r u c t i o n a l unit, the teachers completed a paper and pencil self-administered questionnaire about students' freedom to express t h e i r opinion during these discussions. A sample of the i r students also responded to a similar questionnaire. When the data from the three sources were compared, i t was found that the teacher data 26 disagreed with the other two sources. Although Ehman concluded that teacher self-report i s an unsatisfactory source of data for the "freedom of students to express opinions" (p. 4) variable, he also pointed out two major limit a t i o n s to the generalization of the r e s u l t s : This study i s based on a limited number of teachers from a single school, and represents only one subject area -- secondary school s o c i a l studies i n s t r u c t i o n . Generalizations from the findings, therefore, cannot be broad, although they can suggest cautions which should be heeded by researchers interested in studying classroom phenomena in general (p. 4). House and Steele (1971) investigated the v a l i d i t y of two sources of information on classroom c h a r a c t e r i s t i c s in a state-wide evaluation program in I l l i n o i s . These two sources were student observation and teacher se l f - r e p o r t information. A study was conducted in 32 classes comparing the student and teacher data to records obtained by trained observers. These observation records were used as external c r i t e r i a for the "percentage of i n s t r u c t i o n a l time the teacher spent i n talking" variable. The finding indicated that there was a high discrepancy between teacher s e l f -report data and data collected from students or observers. For example, i n a case where the observed teacher talk was 73%, the teacher estimate was 25% and the median student estimate was 75%. Thus, the decision was taken to process the student data for the evaluation of the g i f t e d program. In comparing teachers and trained observers as alternative sources of i n s t r u c t i o n a l time data, Marliave, Fisher and 27 F i l b y (1977) compared the r e c o r d s kept by t e a c h e r s and d a t a c o l l e c t e d by t r a i n e d o b s e r v e r s when measuring a l l o c a t e d and engaged i n s t r u c t i o n a l time i n s i x grade two c l a s s e s . A n a l y s e s showed.that t e a c h e r r e c o r d s o f a l l o c a t e d i n s t r u c -t i o n a l time i n each c l a s s were p o s i t i v e l y c o r r e l a t e d w i t h both a l l o c a t e d and engaged time o b t a i n e d by d i r e c t o b s e r v a -t i o n . The r e s e a r c h e r s c o n c l u d e d t h a t t e a c h e r and o b s e r v e r are comparable s o u r c e s i n i n s t r u c t i o n a l time d a t a . Hook and Rosenshine (1979), i n t h e i r r e v i e w o f the l i t e r a t u r e on t h e a c c u r a c y o f t e a c h e r r e p o r t s o f c l a s s r o o m b e h a v i o u r , c o n c l u d e d t h a t t e a c h e r r e p o r t s o f t h e i r own s p e c i f i c b e h a v i o u r might be i n a c c u r a t e . They recommended t h a t i f p o s s i b l e , d a t a o t h e r than s e l f - r e p o r t d a t a s h o u l d be used i n r e s e a r c h . I n l i g h t o f t h e s e c o n t r a d i c t o r y f i n d i n g s and recom-mendations r e g a r d i n g t h e use o f q u e s t i o n n a i r e s , F l e x e r (1980) conducted a s t u d y t o a s s e s s the v a l i d i t y o f t h e t o p i c -s p e c i f i c q u e s t i o n n a i r e s used i n the SIMS. The purpose o f the F l e x e r s t u d y was t o i n v e s t i g a t e the v a l i d i t y o f t h e s e t e a c h e r s e l f - r e p o r t i n s t r u m e n t s f o r g a t h e r i n g d a t a about c l a s s r o o m p r a c t i c e s . The two t o p i c - s p e c i f i c q u e s t i o n n a i r e s used were i n t e g e r s ; and r a t i o , p r o p o r t i o n and p e r c e n t . Data were a n a l y z e d o n l y on the s e c t i o n s o f each q u e s t i o n n a i r e t h a t d e a l t w i t h i n s t r u c t i o n a l methods and a l l o c a t i o n o f i n s t r u c t i o n a l t i m e . The q u e s t i o n s i n t h e s e two s e c t i o n s 28 were d e s c r i b e d as "very s p e c i f i c , low i n f e r e n c e q u e s t i o n s " ( B u l l e t i n No. 4, p. 2 5 ) , meaning t h a t the q u e s t i o n s were unambiguous and c l e a r l y worded, and t h a t they r e q u i r e d m i n i m a l s u b j e c t i v e i n t e r p r e t a t i o n on the p a r t o f the r e s p o n d e n t s . F l e x e r d i d not i n c l u d e the s e c t i o n s on reasons f o r i n c l u s i o n and e x c l u s i o n o f t o p i c s , on s o u r c e s o f m a t e r i a l s , and on p r o v i d i n g f o r i n d i v i d u a l d i f f e r e n c e s . She reasoned t h a t on t h e s e v a r i a b l e s , "many t e a c h e r s w i l l shade the t r u t h , w a n t i n g t o appear as good t e a c h e r s t o the r e s e a r c h s t a f f " (p. 9 ) . The sample c o n s i s t e d o f t h r e e e i g h t h grade mathematics c l a s s e s shared by two t e a c h e r s . These t e a c h e r s were observed by t r a i n e d o b s e r v e r s d u r i n g a p e r i o d o f an i n s t r u c t i o n a l u n i t . A t the end o f the i n s t r u c t i o n a l u n i t , t h e t e a c h e r s , t h e s t u d e n t s o f t h e c l a s s e s and t h e o b s e r v e r s were a l l g i v e n t h e a p p r o p r i a t e t o p i c - s p e c i f i c q u e s t i o n n a i r e s t o f i l l o u t . Responses on the q u e s t i o n n a i r e s f i l l e d out by s t u d e n t s were c o m p i l e d and means f o r each i t e m c a l c u l a t e d . C o r r e l a t i o n c o e f f i c i e n t s were c a l c u l a t e d among s t u d e n t means, t e a c h e r s ' r e s p o n s e s , and o b s e r v e r s ' r e s p o n s e s . These r e s u l t s a r e shown i n Table 1. From the f i n d i n g s , F l e x e r c o n c l u d e d t h a t the " s t r o n g - p o s i t i v e c o r r e l a t i o n o f t e a c h e r s ' and o b s e r v e r ' s response on items r e l a t i n g t o t o p i c s p r e s e n t e d i n the c l a s s r o o m s seem t o l e g i t i m i z e t h e use o f t h e s e q u e s t i o n n a i r e s f o r g a t h e r i n g such d a t a on c l a s s r o o m p r a c t i c e " (p. 6 ) . Table 1 Correlations of Students and Teacher with Observer Subgroups n of items Students Teacher Algebra Introduction to integers and ordering 7 .92** .73 (berating with integers (+, x) 16 .60* .65** Introduction to integers, ordering, operating with integers @ 23 .67** .69** Time spent on each activity 7 .96** .94** Importance of topics to teacher 6 .39 .53 Ratio, Proportion and Percent Ratio and proportion 17 .47 .89** Percent 25 .61** .93** Ratio, proportion and percent @ 42 .38* .91** Time spent on each activity 5 .33 .86 Importance of topics to teacher 8 .12 .48 Note. F l e x e r , 1980 *p .05 **p .01 @ Combination of above two The r e s e a r c h on v a l i d i t y c o n c e r n i n g the use o f q u e s t i o n n a i r e s seems t o p r o v i d e c o n t r a d i c t o r y r e s u l t s . Some s t u d i e s (Ehman, 1970; House and S t e e l e , 1971) recommended the use of s t u d e n t d a t a o r o b s e r v e r d a t a over the use o f t e a c h e r s e l f - r e p o r t d a t a . Other s t u d i e s ( M a r l i a v e , F i s h e r and F i l b y , 1977; F l e x e r , 1980) co n c l u d e d t h a t t e a c h e r s e l f - r e p o r t d a t a were r e l i a b l e when s p e c i f i c , l o w - i n f e r e n c e q u e s t i o n s were used. B e r d i e and Anderson (1974) attempted t o e x p l a i n t h i s phenomenon by a t t r i b u t i n g the c o n t r a d i c t i o n t o the f a c t t h a t t h e s e r e s u l t s u s u a l l y have been based on e x p e r i m e n t a l d e s i g n s t h a t were not chosen e x c l u s i v e l y t o t e s t q u e s t i o n n a i r e v a l i d i t y . R a t h e r , the r e s u l t s were o f t e n b y - p r o d u c t s of s u r v e y s d e s i g n e d f o r o t h e r purposes. T h e r e f o r e , they c o n c l u d e d t h a t , "...the c o n t r a d i c t o r y r e p o r t s c o n c e r n i n g q u e s t i o n n a i r e methods a r e not s u r p r i s i n g , as t h e y are based on r e s u l t s from d i f f e r e n t q u e s t i o n n a i r e s used f o r d i f f e r e n t r e a s o n s w i t h d i f f e r e n t p e o p l e a t d i f f e r e n t t i m e s " (p. 1 2 ) . The e x p l a n a t i o n o f f e r e d by B e r d i e and Anderson seems t o be a l o g i c a l and s u i t a b l e e x p l a n a t i o n f o r the c o n t r a d i c -t o r y r e s u l t s found i n the s t u d i e s reviewed i n t h i s s e c t i o n . These s t u d i e s i n v e s t i g a t e d d i f f e r e n t v a r i a b l e s such as i n d i v i d u a l i z a t i o n o f i n s t r u c t i o n , use o f m a t e r i a l s , the s t u d e n t s ' freedom t o e x p r e s s t h e m s e l v e s , a l l o c a t e d and engaged i n s t r u c t i o n a l time and i n s t r u c t i o n a l methods. D i f f e r e n t q u e s t i o n n a i r e s were a d m i n i s t e r e d t o d i f f e r e n t 31 grade l e v e l s r e g a r d i n g these v a r i a b l e s . The r e s u l t o f the House and S t e e l e (1971) study was a b y - p r o d u c t of an e v a l u a t i o n s t u d y f o r a g i f t e d program. T h e r e f o r e , the c o n c l u s i o n s o f t h e s e s t u d i e s can o n l y be a p p l i c a b l e t o t h e s p e c i f i c i n s t r u m e n t s used, and t o the s i t u a t i o n s i n which they were used. Thus, the r e s u l t s o f s t u d i e s reviewed i n t h i s s e c t i o n cannot g i v e v a l i d , g e n e r a l i z a b l e i n f o r m a t i o n on t h e v a l i d i t y o f the q u e s t i o n n a i r e s used i n the SIMS because of t h e i r d i f f e r e n c e s i n purpose, d e s i g n , v a r i a b l e i n q u e s t i o n and i n s t r u m e n t used. S i n c e the s m a l l - s c a l e F l e x e r s t u d y i s the o n l y s o u r c e o f i n f o r m a t i o n on the v a l i d i t y o f t h e s e t o p i c - s p e c i f i c q u e s t i o n n a i r e s , an i n v e s t i g a t i o n i n t o the development o f t h e s e q u e s t i o n n a i r e s was conducted i n o r d e r t h a t c o n c l u s i o n s r e g a r d i n g t h e i r v a l i d i t y c o u l d be drawn. B u l l e t i n No. 5 o f the SIMS r e p o r t e d on how v a l i d i t y c o n c e r n s have been t a k e n i n t o account d u r i n g v a r i o u s s t a g e s o f development o f the q u e s t i o n n a i r e s . The i n i t i a l s t a g e c o n s i s t e d o f g a t h e r i n g i n p u t from mathematics s p e c i a l i s t s from Canada, West Germany and the U.S.A. t o map out t h e g e n e r a l framework f o r the q u e s t i o n n a i r e s . D r a f t s o f the q u e s t i o n n a i r e s were then reviewed by the I n t e r n a t i o n a l Mathematics Committee which c o n s i s t e d o f prominent members of the mathematics e d u c a t i o n community. D e t a i l e d r a t i n g s , a t the i t e m l e v e l , as t o a p p r o p r i a t e n e s s , f e a s i b i l i t y and s u i t a b i l i t y were performed by the Committee i n the second s t a g e . On the b a s i s o f t h i s i n f o r m a t i o n , r e v i s i o n s o f the q u e s t i o n n a i r e s were made. Care was ta k e n a t each stage t o ensure t h a t the q u e s t i o n s were unambiguous, c l e a r and l o w - i n f e r e n c e i n n a t u r e . F o r example, q u e s t i o n s t h a t r e q u i r e d s u b j e c t i v e i n t e r p r e t a t i o n by the t e a c h e r s , such as whether o r not t e a c h e r s are open t o s t u d e n t o p i n i o n s , were not i n c l u d e d i n t h e q u e s t i o n n a i r e s . The f i n a l s t a g e c o n s i s t e d o f p i l o t - t e s t i n g o f the i n s t r u m e n t s . E x p e r i e n c e d r e s e a r c h e r s i n mathematics e d u c a t i o n conducted i n - d e p t h i n t e r v i e w s w i t h c l a s s r o o m t e a c h e r s c o n c e r n i n g the c l a r i t y and i n t e n t i o n o f the q u e s t i o n s , the coverage of the q u e s t i o n n a i r e s w i t h r e s p e c t t o c o n t e n t and method, and the time demands o f the i n s t r u m e n t s . S u g g e s t i o n s c o l l e c t e d from t e a c h e r s d u r i n g the p i l o t - t e s t i n g were then i n c o r p o r a t e d i n subsequent r e v i s i o n s . Thus, the q u e s t i o n n a i r e s were deve l o p e d by mathematics s p e c i a l i s t s , r e v i e w e d by a committee c o n s i s t i n g o f prominent members o f the mathematics e d u c a t i o n community, and p i l o t - t e s t e d by e x p e r i e n c e d r e s e a r c h e r s . T h i s l e n g t h y p r o c e s s s u b s t a n t i a t e s t h e c l a i m t h a t "concerns f o r the v a l i d i t y o f d a t a on c l a s s r o o m p r o c e s s e s . . . have been c e n t r a l t o the development o f the i n s t r u m e n t s " ( B u l l e t i n No. 5, p. 30). T h i s f a c t , s u p p o r t e d by the f i n d i n g s from the F l e x e r s t u d y l e a d one t o co n c l u d e t h a t the s e c t i o n s of the q u e s t i o n n a i r e s i n which low-i n f e r e n c e q u e s t i o n s were used are r e l i a b l e and v a l i d . Chapter 3 DESIGN AND PROCEDURE The present study was designed to compare teachers' use of concrete and abstract approaches i n teaching c e r t a i n subtopics i n Grade 8. The comparison was made on the data c o l l e c t e d from approximately 100 mathematics teachers who took part i n the SIMS. The instruments used to c o l l e c t t h i s data were a set of f i v e t o p i c - s p e c i f i c question-naires on classroom processes. Since the nature of the data was not known p r i o r to a n a l y s i s , c e r t a i n data processing procedures and analysis techniques were used i n order that v a l i d comparisons could be made on the concrete-abstract v a r i a b l e . This chapter out l i n e s the data processing procedures used, and presents the r a t i o n a l e for choosing exploratory data analysis techniques to compare the teachers' choice between concrete and abstract approaches. Population and Sample Selection Two populations of students were i d e n t i f i e d for i n v e s t i g a t i o n i n the i n t e r n a t i o n a l study. The f i r s t , Population A, was defined as co n s i s t i n g of a l l those students e n r o l l e d i n the grade where the majority of 33 34 s t u d e n t s have reached the age o f 13 by the m i d d l e of the s c h o o l y e a r . In B.C., t h i s p o p u l a t i o n was d e f i n e d t o i n c l u d e a l l s t u d e n t s e n r o l l e d i n r e g u l a r Grade 8 c l a s s e s i n the p u b l i c s c h o o l s as o f September, 1980. The second, P o p u l a t i o n B, was i n t e n d e d t o encompass a l l s t u d e n t s i n the l a s t y e a r o f secondary s c h o o l who were s t u d y i n g mathematics as a s i g n i f i c a n t p a r t o f an academic program. I n B.C., t h i s p o p u l a t i o n was d e f i n e d t o i n c l u d e a l l s t u d e n t s e n r o l l e d i n A l g e b r a 12 c l a s s e s i n the p u b l i c s c h o o l s as o f September, 1980. The t e a c h e r s who t a u g h t the s t u d e n t s b e l o n g i n g t o p o p u l a t i o n s A and B formed the p o p u l a t i o n o f t e a c h e r s i n the s t u d y . A sample s i z e o f a p p r o x i m a t e l y 100 c l a s s e s and t h e i r t e a c h e r s from each p o p u l a t i o n was e s t a b l i s h e d i n B.C. T h i s p r e s e n t s t u d y i s concerned w i t h P o p u l a t i o n A o n l y . R o b i t a i l l e , O'Shea and D i r k s (1982) gave a d e s c r i p t i o n o f the sample s e l e c t i o n f o r the SIMS i n B.C.: In o r d e r t o a c h i e v e a sample s i z e o f a p p r o x i m a t e l y 100 Grade 8 and 100 A l g e b r a 12 c l a s s e s s t r a t i f i e d a c c o r d i n g t o g e o g r a p h i c zone o f the p r o v i n c e and by s c h o o l s i z e , i n i t i a l samples o f 125 c l a s s e s a t each l e v e l were drawn. In most c a s e s t h i s r e s u l t e d i n the s e l e c t i o n o f no more than one c l a s s per s c h o o l . Of t h e 125 c l a s s e s , 105 were s e l e c t e d f o r i n i t i a l c o n t a c t and the remainder r e s e r v e d t o be used as needed. In the s p r i n g o f 1980, l e t t e r s were sen t from t h e M i n i s t r y o f E d u c a t i o n t o a l l o f the p r i n c i p a l s o f the s c h o o l s s e l e c t e d , s o l i c i t i n g t h e i r c o o p e r a t i o n i n the s t u d y and a s k i n g them t o s e l e c t a Mathematics 8 o r A l g e b r a 12 t e a c h e r o r t e a c h e r s a t random from among the t e a c h e r s a v a i l a b l e . In c a s e s where i t was not p o s s i b l e t o make a random s e l e c t i o n , the p r i n c i p a l s were asked t o e x e r c i s e t h e i r b e s t judgment about which t e a c h e r o r t e a c h e r s t o s e l e c t (p. 9 ) . R e g a r d i n g t h e r e p r e s e n t a t i v e n e s s o f t h e a c h i e v e d s a m p l e s , they noted: On the whole, i t appears that, at both levels, the geographic d i s t r i b u t i o n of classes partic i p a t i n g in the international study i s s u f f i c i e n t l y close to that of the design sample to be representative of the province as a whole. Moreover, the gender and age s t a t i s t i c s for the students in the two samples compare favorably with those for the populations. In the case of the teachers selected for p a r t i c i p a t i o n in the study, there are some s i g n i f i c a n t differences between them and the population of teachers of mathematics. In p a r t i c u l a r , the [SIMS] teachers are more experienced and more l i k e l y to have specialized in the teaching of mathematics than their colleagues Table 2 l i s t s descriptive information that was collected from teachers in the Population A sample. (p. 13). Table 2 Description of Population A Teachers Variables Mean Values Age i n years 38.6 Years of teaching experience 13.8 Years spent in teaching mathematics to Grade 8 students 8.5 Percentage of time spent in teaching mathematics during the 1980-81 school year 80.0 Number of mathematics courses included in teachers' post-secondary education 9.1 36 As shown i n Table 2, few inexperienced teachers or teachers of s u b j e c t s other than mathematics who teach one or more mathematics c l a s s e s i n order to complete t h e i r t e a c h i n g schedules were s e l e c t e d f o r p a r t i c i p a t i o n i n the i n t e r n a t i o n a l study. The f a c t t h a t experienced and s p e c i a l i z e d t e a c h e r s were s e l e c t e d f o r p a r t i c i p a t i o n might c r e a t e a b i a s i n the r e s u l t s of t h i s study and might l i m i t the g e n e r a l i z a t i o n of the r e s u l t s to the p o p u l a t i o n of mathematics teachers i n B r i t i s h Columbia. The Instruments Each P o p u l a t i o n A teacher was asked to complete f i v e q u e s t i o n n a i r e s on classroom processes f o r the f o l l o w i n g t o p i c s : - Common and Decimal F r a c t i o n s - R a t i o , P r o p o r t i o n and Percent - A l g e b r a (Integers, Formulae and Equations) - Geometry - Measurement The classroom process q u e s t i o n n a i r e s f o r these f i v e t o p i c s are i n Appendices A to E. Each t o p i c - s p e c i f i c q u e s t i o n n a i r e was designed to c o l l e c t i n f o r m a t i o n on the f o l l o w i n g aspects of classroom p r a c t i c e : 1) Resources used i n t e a c h i n g the t o p i c 2) S p e c i f i c s u b t o p i c s t a u g h t 3) I n s t r u c t i o n a l methods used t o p r e s e n t s p e c i f i c c o n c e p t s such as the concept o f n e g a t i v e i n t e g e r s and c e r t a i n o p e r a t i o n s such as m u l t i p l i c a t i o n o f i n t e g e r s . 4) F a c t o r s t e a c h e r s p e r c e i v e d as i n f l u e n c i n g t h e i r c h o i c e o f approach o r proce d u r e 5) Time a l l o c a t e d t o the t o p i c and s u b t o p i c s w i t h i n the same s u b j e c t 6) Types o f a p p l i c a t i o n problems u t i l i z e d by the t e a c h e r 7) Teachers' o p i n i o n s r e g a r d i n g i s s u e s such as t h e use of c a l c u l a t o r s The p r e s e n t s t u d y was concerned w i t h the items t h a t were c o n s t r u c t e d t o c o l l e c t i n f o r m a t i o n on i n s t r u c t i o n a l methods used i n p r e s e n t i n g s p e c i f i c c o n c e p t s and s k i l l s ( c a t e g o r y 3 ) . These items were d e s c r i b e d by B u l l e t i n No. 4 o f the SIMS as b e i n g " v e r y s p e c i f i c , [and] low-i n f e r e n c e " (p. 25) i n n a t u r e . Examples o f t h e s e items a r e g i v e n i n F i g u r e 6. A l i s t o f the items (20 -24) d e a l i n g w i t h the t e a c h i n g o f t h e c o n c e p t o f i n t e g e r s i s shown i n t h i s f i g u r e . The inlt^icXaXiotu o f inltgt<4 given b c l t w M i y be included i n y<mt ifilCAjJC I^wfuU f j J i o g u a . CHECC Uit ^ u ^ o i u t cude lUiicJi d c i c t i b e a (hi VttaXr<nt o{ each Ojpic in youA tJjxxk. nsrcusE covtsi I. tmfiiatiitd ( u i c d ai a p^imiAy tiplaiialton, *t(cAAtd to ttttntivttij 0\ i\tt)utnllij\ I. Uttd Lui nal LnpliAiiitd i. Hut uttd Iht inteApn.tati.oni o£ integcAl given iciooj may be included i n youA in&tAuc-tioruil pxojlflffl . CHECK Cht MAponit code. uViicli dtACAibtl tin (ie.aUn\en(. o j each topic in yoa\ c l a i i . KLSrOHSE C O W S i I. tmpha&iztd \uAtd cu a pvuna/iy Lanation, Kt^tAA.td to cxtcniiveJy 01 intqurntttj) I. Used but not tir^hmiLztd i. Hot tutd 20. Extending the nuuber ray to tlie nuulier 1 Ine: . I entended the nuofcer ray (0 and positive nuuiiers) to the left by Introducing direction ts well is cagnllude. E«: — -4-3-2-1 0 1 2 J i -J M a n s i units to the left of 0. 21. Eitendlng the nunaier system to find solutions to equations: I discussed the need to eitend the positive integers in order to find a solution to equations like • 7 - 5. 22. Using vectors or directed seg-nentl on the outlier line: I defined an Integer as a set of vectors (directed line seg-ments) on the nmber line. £«: -2 can be represented by any of: E«: -'id " " - L " ' 6 & • 1 • '10 •2 can be represented by any of: -Vo" 2 3 . D e f i n i n g i n t e g e r s as e q u i v a -l e n c e c l a s s e s o f who le numbers: 1 d e v e l o p e d the I n t e g e r s as e q u i v a l e n c e c l a s s e s o f o r d e r e d p a i r s o f who le numbers. E x : ( ( 0 . 2 ) . ( 1 . 3 ) . ( 2 . « - ) . . . . ) - "2 o r ( ( a . b ) c WXW: b • a • 2}« "2 ?4. U s i n g examples o f p h y s i c a l -s i t u a t i o n s : I d e v e l o p e d i n t e g e r s by r e f e r r i n g to d i f f e r e n t p h y s i c a l s i t u a t i o n s w h i c h can be d e s c r i b e d w i t h I n t e g e r s . E x : thermometer , e l e v a t i o n , money ( c r e d i t / d e b i t ) , s p o r t s ( s c o r i n g ) , t ime ( b e f o r e / a f t e r ) , e t c . F i g u r e 6. Items d e a l i n g w i t h methods o f i n s t r u c t i o n f o r the concept o f i n t e g e r s . Data C o l l e c t i o n Data c o l l e c t i o n took place during the 1980-81 school year. Classroom process questionnaires were sent out to teachers i n sealed envelopes at the beginning of that school year. Teachers were in s t r u c t e d to open the envelope f o r a given questionnaire only a f t e r they had f i n i s h e d teaching that p a r t i c u l a r t o p i c . This procedure was used to minimize the influence of the questionnaires on teaching p r a c t i c e s ( R o b i t a i l l e , O'Shea and'Dirks, 1982). The teachers were also asked to f i l l out the t o p i c — s p e c i f i c questionnaires as soon as the unit of i n s t r u c t i o n on a given to p i c was f i n i s h e d . In t h i s way, errors on the information gathered by these question-naires were kept to a minimum. Data Processing Procedure The data processing procedure consisted of various steps which were performed i n sequence. The i n i t i a l step involved the c a t e g o r i z a t i o n of items into subtopics within each t o p i c . Then these items were categorized into the concrete and abstract categories by a panel of experts. Only those items which were agreed upon by four out of f i v e experts as belonging to a category were included i n the a n a l y s i s . Preliminary analysis 40 was performed on the responses t o items t o determine the j u s t i f i a b i l i t y of aggregation of item scores to form s u b t o p i c s c o r e s . F i n a l l y , s u b topic scores were obtained by aggregating the item s c o r e s . C a t e g o r i z a t i o n of Items In each q u e s t i o n n a i r e , the items t h a t were i d e n t i f i e d by the r e s e a r c h e r as p e r t a i n i n g to v a r i o u s s u b t o p i c s were c a t e g o r i z e d i n t o the concrete and a b s t r a c t approaches by a panel of e x p e r t s . The panel c o n s i s t e d of two mathematicians from the F a c u l t y of Science at the U n i v e r s i t y of B r i t i s h Columbia (U.B.C), two mathematics educators from the F a c u l t y of Education a t U.B.C. and a d o c t o r a l student who was an experienced teacher of Grade 8 mathematics. The panel was not given the terms "concrete" and " a b s t r a c t " to a s s i s t i n t h e i r c a t e g o r i z a t i o n of items. Instead they were g i v e n d e f i n i t i o n s as o u t l i n e d below and were asked to c l a s s i f y the items as e i t h e r A, B or C. A The items i n t h i s category i n v o l v e the use of ma n i p u l a t i v e m a t e r i a l s , diagrams or p i c t u r e s . A major c h a r a c t e r i s t i c of the items i n t h i s c ategory i s t h a t treatment of the mathematical content r e l i e s p r i m a r i l y on m a n i p u l a t i v e m a t e r i a l s , diagrams or experiences from the environment. B The items i n t h i s category i n v o l v e the use of symbols to stand f o r some concept or event but the symbols may not resemble t h a t concept or event. A major c h a r a c t e r i s t i c of the items i n t h i s category i s t h a t the treatment of the content r e l i e s p r i m a r i l y on e x p l a n a t i o n which d e r i v e s i t s meaning from other mathematical content. C The items i n t h i s category do not belong to A or B. Table 3 shows the l i s t o f s u b t o p i c s and c l a s s i f i c a t i o n of items as v a l i d a t e d by the p a n e l o f e x p e r t s a c c o r d i n g t o the d e f i n i t i o n s o u t l i n e d above. Only the c a t e g o r i z a t i o n s t h a t were agreed upon by f o u r out o f the f i v e e x p e r t s were i n c l u d e d i n t h i s l i s t . A c c o r d i n g t o the p a n e l , some s u b t o p i c s were d e f i n e d by items i n one c a t e g o r y o n l y , and t h e s e s u b t o p i c s were e x c l u d e d from subsequent a n a l y s e s . For example, the s u b t o p i c a n g l e s o f t r i a n g l e was e x c l u d e d from the Geometry s e c t i o n because the items i n the q u e s t i o n n a i r e c o r r e s p o n d i n g t o t h i s s u b t o p i c do not c o n s i s t o f b o t h the c o n c r e t e and a b s t r a c t approaches. (See Appendix F f o r the o r i g i n a l l i s t o f s u b t o p i c s and items.) S i x more s u b t o p i c s were th e n e x c l u d e d from f u r t h e r a n a l y s e s because t h e y were d e f i n e d by o n l y one i t e m i n e i t h e r o r b o t h c a t e g o r i e s . These s u b t o p i c s were: o p e r a t i o n s w i t h d e c i m a l f r a c t i o n s , r a t i o , p r o p o r t i o n , a d d i t i o n o f i n t e g e r s , m u l t i p l i c a t i o n o f i n t e g e r s , and f o r m u l a . The r a t i o n a l e f o r e x c l u d i n g t h e s e s u b t o p i c s i s t h a t " r a r e l y i s one i t e m s u f f i c i e n t l y r e l i a b l e and v a l i d t o make i t w o r t h w h i l e t o r e p o r t i t s s c o r e a l o n e " ( A l l e n and Yen, 1979; p. 130). The e i g h t s u b t o p i c s t h a t remained were a l l d e f i n e d by two o r more items i n b o t h c a t e g o r i e s . These s u b t o p i c s w i t h t h e i r c o r r e s p o n d i n g items a r e l i s t e d i n T a b l e 4. 42 Table 3 The L i s t of Subtopics and Corresponding Items for the Concrete and Abstract Cateaories as Validated by the Panel of Experts Topic / Subtopic Category Concrete Items Abstract Items Common and Decimal Fractions Concept of Fractions 21, 28, 22, 30 23, 27, 24, 25 Addition of Fractions 31, 38 32, 33, 37, 34, 35, Concept of Decimal Fractions 51, 53, 56 52, 54, Operations with Decimal Fractions 59 57, 58 Algebra Concept of Integers 20, 22, 24 21, 23 Addition of Integers 25, 27 26 Subtraction of Integers 28, 32 29 30 M u l t i p l i c a t i o n of Integers 36 34, 38 Formula 45, 47, 48 44 Ratio, Proportion and Percent Ratio 21 23 Proportion 27 29 Geometry Pythagorean Theorem 67, 68, 71 69, 72 Measurement Number If 48, 53, 54 49, 51 Relationship Among Various 69, 70 66, 67, Metric Units 43 Table 4 The L i s t o f S u b t o p i c s and C o r r e s p o n d i n g Items I n c l u d e d f o r A n a l y s i s i n T h i s Study Ca t e g o r y T o p i c / S u b t o p i c C o n c r e t e Items A b s t r a c t Items Common and D e c i m a l F r a c t i o n s Concept o f F r a c t i o n s 21, 22, 23, 27, 24, 25 28, 30 Concept of Decimal F r a c t i o n s 51, 53, 56 52, 54, 55 A d d i t i o n o f F r a c t i o n s 31, 32, 33, 37, 34, 35, 36 38 A l g e b r a Concept of I n t e g e r s 20, 22, 24 21, 23 S u b t r a c t i o n o f I n t e g e r s 28, 32 29, 30 Geometry Pythagorean Theorem 67, 68, 71 69, 72 Measurement Number TT 48, 53, 54 49, 51 R e l a t i o n s h i p Among V a r i o u s 69, 70 66, 67, 68 M e t r i c U n i t s Scoring of Items As shown i n Figure 6 (p. 38) , three response options were given for each item i n the questionnaire. The options were: used as a primary method of explanation, used but not as a primary means of explanation, and not used. These options f o r t h i s p a r t i c u l a r section of the question-naire remained uniform across a l l f i v e t o p i c s . For the purposes of t h i s study, the responses to the items were assigned values 0, 1 or 2 according to the option chosen. A response that was "used as a primary method of explanation" was assigned a value of 2. A response that was "used but not as a primary means of explanation" was assigned a value of 1. A response that was "not used" was assigned 0. Preliminary Analysis The purpose of the preliminary analysis was to investigate the j u s t i f i a b i l i t y of aggregation of item scores to obtain a corresponding subtopic score. An analysis of response patterns of a randomly selected sample of teachers was performed. This analysis was used to determine the differences within each teacher i n h i s or her response to the items i n the concrete or abstract category for each subtopic considered. Teacher responses to both the concrete items and the abstract items for each subtopic were graphed. Since the items i n each c a t e g o r y were few i n number, and t h e r e were o n l y t h r e e response o p t i o n s t o each i t e m , the number of d i f f e r e n t response p a t t e r n s was l i m i t e d . A d e c i s i o n was t a k e n t h a t the sample s i z e , n, s h o u l d be determined by the p a t t e r n s d i s p l a y e d by the graphs t h e m s e l v e s . I f d e f i n i t e p a t t e r n s appeared and t h e s e p a t t e r n s s t a r t e d t o r e p e a t , then the random s e l e c t i o n o f more t e a c h e r s f o r the a n a l y s i s would s t o p . The r e s u l t o f t h i s p r o c e d u r e showed t h a t d e f i n i t e p a t t e r n s were found and t h a t t h e s e p a t t e r n s s t a r t e d t o r e p e a t when the re s p o n s e s o f the 6 t h o r 7th t e a c h e r were graphed. T h e r e f o r e , i t was r e a s o n a b l e t o c o n c l u d e t h a t an i n c r e a s e i n sample s i z e beyond t e n would not y i e l d more i n f o r m a t i o n r e g a r d i n g the t e a c h e r s ' p a t t e r n o f r e s p o n s e s . The graphs showing t h e response p a t t e r n s o f 10 randomly s e l e c t e d t e a c h e r s a re d i s p l a y e d i n Appendix G. The mean and v a r i a n c e f o r each t e a c h e r were c a l c u l a t e d f o r each c a t e g o r y f o r a l l s u b t o p i c s . The mean v a r i a n c e s were a l s o c a l c u l a t e d f o r t h e 10 t e a c h e r s f o r each c a t e g o r y f o r a l l s u b t o p i c s . These means a r e l i s t e d i n T a b l e 5. T a b l e 5 shows t h a t t h e mean v a r i a n c e s o b t a i n e d f o r the s u b t o p i c s range from .15 t o .55 f o r t h e c o n c r e t e items and from .13 t o .55 f o r the a b s t r a c t i t e m s . With the e x c e p t i o n o f two v a l u e s , a l l the means are l e s s t h a n h a l f a s c o r e p o i n t on a s c a l e t h a t ranges from z e r o t o two, thus s u g g e s t i n g t h a t the i t e m s c o r e s f o r each Table 5 The Mean V a r i a n c e s o f Item S c o r e s from C o r r e s p o n d i n g S u b t o p i c Scores f o r t h e Sample of Ten Teachers Categ o r y S u b t o p i c s C o n c r e t e Items A b s t r a c t Items Concept o f F r a c t i o n s . 37 .13 Concept o f Decimal F r a c t i o n s .35 .33 A d d i t i o n o f F r a c t i o n s .25 .17 Concept o f I n t e g e r s .55 .40 S u b t r a c t i o n o f I n t e g e r s .22 .40 Pythagorean Theorem . 44 .55 Number IT .15 .40 R e l a t i o n s h i p Among V a r i o u s M e t r i c U n i t s .30 .35 s u b t o p i c do not d i f f e r s i g n i f i c a n t l y from each o t h e r w i t h i n each c a t e g o r y . Hence, i t i s p o s s i b l e t o aggregate i t e m s c o r e s t o o b t a i n s u b t o p i c s c o r e s f o r the 10 t e a c h e r s w i t h o u t much l o s s o f i n f o r m a t i o n . The p r o f i l e s o f the s e t e n t e a c h e r s can be c o n s i d e r e d t o be t y p i c a l o f the response p a t t e r n s o f a l l t e a c h e r s i n v o l v e d i n the study f o r the f o l l o w i n g r e a s o n s : the 47 ten teachers were randomly selected from the t o t a l sample of teachers, and a sample s i z e of 10 was s u f f i c i e n t because of the f a c t that d e f i n i t e patterns were found, and these patterns had started to repeat within the 10 teachers that were chosen. Therefore, since the item scores for the 10 teachers can be aggregated, the item scores f o r a l l teachers can also be j u s t i f i a b l y aggregated to a r r i v e at a s i n g l e concrete score as well as a sing l e abstract score f o r each subtopic. Scoring of Subtopics Figure 6 (p. 38) shows a l i s t of the items (20 - 24) dealing with the teaching of the concept of integers. Items 21 and 23 represent abstract approaches to teaching t h i s subtopic whereas items 20, 22 and 24 represent concrete approaches to teaching the same concept. Obtaining the sum of assigned values to the responses to these items made i t possible to create a score for each teacher for the concrete and abstract categories i n teaching the concept of integers. The sum for each category was divided by the number of items involved i n that p a r t i c u l a r category to provide equal weighting and theh m u l t i p l i e d by ten to give whole number values. Thus, each teacher had two scores for each subtopic: an abstract score and a concrete score. Both scores range from 0 to 20. A score of 0 means that the teacher chose the option "not used" f o r a l l the items 48 i n that category. A score of 20 means that the teacher chose the option "used as a primary explanation" for a l l items i n that category. Difference Score for Each Subtopic The concrete and abstract scores, when considered f o r any teacher, give information as to whether or not most concrete or abstract items were emphasized or used i n the i n s t r u c t i o n of a c e r t a i n subtopic. These scores, however, do not give information on the teacher's preference fo r e i t h e r approach. This information can be obtained by combining the concrete and abstract scores to obtain a s i n g l e score that r e f l e c t s the preference for or o r i e n t a t i o n to e i t h e r approach f o r each teacher. This opinion was also expressed by Cooney (1980) who suggested that teachers be reassigned d i s c r e t e scores on a scale of 0 to 5 according to the differences between t h e i r concrete and abstract scores. However, an examination of the data showed that the differences between the concrete and abstract scores for the teachers were continuous i n nature and, therefore, could not be c l a s s i f i e d into d i s c r e t e categories. On the other hand, d i f f e r e n c e scores obtained by subtracting the abstract scores from the concrete scores do r e f l e c t accurately the teacher's o r i e n t a t i o n to e i t h e r approach. Hence, the f i n a l step i n the data processing procedure was to c a l c u l a t e the di f f e r e n c e scores for a l l teachers. The d i f f e r e n c e s c o r e s have unique p r o p e r t i e s t h a t are d i f f e r e n t from the c o n c r e t e and a b s t r a c t s c o r e s . The d i f f e r e n c e s c o r e s range from -20 t o 20. A p o s i t i v e d i f f e r e n c e s c o r e shows t h a t the t e a c h e r i s c o n c r e t e l y -o r i e n t e d i n t e a c h i n g t h a t s u b t o p i c and a n e g a t i v e d i f f e r e n c e s c o r e shows t h a t the t e a c h e r i s a b s t r a c t l y -o r i e n t e d i n t e a c h i n g t h a t s u b t o p i c . The a b s o l u t e v a l u e of the d i f f e r e n c e s c o r e i n d i c a t e s the degree o f o r i e n t a t i o n f o r t h a t p a r t i c u l a r approach. F o r example, a t e a c h e r who s c o r e d -15 was more s t r o n g l y - o r i e n t e d t o t h e a b s t r a c t approaches t h a n a t e a c h e r who s c o r e d -5. S i m i l a r l y , a t e a c h e r who s c o r e d 15 was more s t r o n g l y - o r i e n t e d t o t h e c o n c r e t e approaches than a t e a c h e r who s c o r e d 5. I t i s t h i s unique p r o p e r t y o f the d i f f e r e n c e s c o r e t h a t p e r m i t s t h e comparison among t e a c h e r s i n t h e i r c h o i c e o r emphasis o f approach. Method o f A n a l y s i s E x p l o r a t o r y d a t a a n a l y s i s t e c h n i q u e s were used i n t h i s s t u d y . These t e c h n i q u e s were more a p p r o p r i a t e t o use f o r the purposes o f t h i s s tudy than c o n v e n t i o n a l s t a t i s t i c a l p r o c e d u r e s f o r two r e a s o n s : 1) E x p l o r a t o r y d a t a a n a l y s i s t e c h n i q u e s c o n c e n t r a t e on " s i m p l e a r i t h m e t i c and easy-to-draw p i c t u r e s " t o p r o v i d e a d e s c r i p t i o n o f "what [the data] seem t o say" (Tukey, 1977; p. v ) . S i n c e the purpose o f t h i s s tudy was t o compare the t e a c h e r s ' c h o i c e o f i n s t r u c t i o n a l methods, i t f o l l o w s t h a t the method o f a n a l y s i s s h o u l d f i r s t o f a l l p r o v i d e an a c c u r a t e p i c t u r e o f the d a t a b e f o r e v a l i d comparisons can be made. For t h i s purpose, e x p l o r a t o r y t e c h n i q u e s seem t o be most a p p r o p r i a t e . 2) The use o f c o n v e n t i o n a l s t a t i s t i c a l p r o c e d u r e s r e q u i r e s t h a t c e r t a i n assumptions be s a t i s f i e d by the d a t a so t h a t r e s u l t s from the a n a l y s i s can be m e a n i n g f u l l y i n t e r p r e t e d . S i n c e the n a t u r e o f the d a t a i n t h i s s t u d y was not known b e f o r e the a n a l y s i s , i t i s i n a p p r o p r i a t e t o use t h e s e p r o c e d u r e s . Two s p e c i f i c t e c h n i q u e s f o r d i s p l a y i n g t h e d a t a were used. These were st e m - a n d - l e a f d i s p l a y s and box-and-whisker p l o t s . Tukey emphasized t h a t t h e s e t e c h n i q u e s are " f o r g i v i n g " t e c h n i q u e s (p. v i i i ) , meaning t h a t t h e y do not produce " d i g i t - p e r f e c t " r e s u l t s (p. v i i i ) , r a t h e r they a l l o w us " t o l o o k a t the g e n e r a l p a t t e r n o f the [ d a t a ] " (p. 2 7 ) . F o r each s u b t o p i c , the c o n c r e t e , a b s t r a c t , and d i f f e r e n c e s c o r e d i s t r i b u t i o n s were graphed i n s i d e -b y - s i d e and ba c k - t o - b a c k stem-and-leaf d i s p l a y s . The c h o i c e o f d i s p l a y depended on the number o f d i s t r i b u t i o n s i n v o l v e d i n a p a r t i c u l a r comparison (Tukey, 1977; p. 6 5 ) . The back-to-back d i s p l a y was used t o compare c o n c r e t e and a b s t r a c t s c o r e d i s t r i b u t i o n s f o r each s u b t o p i c because i t i s more s u i t a b l e f o r comparisons i n v o l v i n g two d i s t r i b u t i o n s . The s i d e - b y - s i d e d i s p l a y was used t o compare d i f f e r e n c e s c o r e d i s t r i b u t i o n s f o r a l l the sub-t o p i c s because i t i s more s u i t a b l e f o r comparisons i n v o l v i n g s e v e r a l d i s t r i b u t i o n s . Chapter 4 RESULTS OF THE STUDY Stem-and-leaf d i s p l a y s and box-and-whisker p l o t s used t o examine the d i s t r i b u t i o n s o f c o n c r e t e , a b s t r a c t , and d i f f e r e n c e s c o r e s a re p r e s e n t e d and d i s c u s s e d i n t h i s c h a p t e r . Both stem-and-leaf d i s p l a y s and box-and-w h i s k e r p l o t s were employed i n o r d e r t o maximize the i n f o r m a t i o n p r o v i d e d by the d a t a . Each x on the stem-a n d - l e a f d i s p l a y s r e p r e s e n t s the s c o r e o f one t e a c h e r . I n the box-and-whisker p l o t s , t he d o t s show the extreme v a l u e s and the h o r i z o n t a l l i n e s show the rounded whole number v a l u e s o f t h e median, the 25th and the 75th p e r c e n t i l e s o f the d i s t r i b u t i o n (Tukey, 1977; p. 33-39). C o n c r e t e and A b s t r a c t Scores The c o n c r e t e and a b s t r a c t s c o r e d i s t r i b u t i o n s f o r the e i g h t s u b t o p i c s a r e d i s p l a y e d i n F i g u r e s 7 t o 14. Each f i g u r e c o n s i s t s o f two p a r t s : P a r t a) i s the stem-and-leaf d i s p l a y and P a r t b) i s the box-and-whisker p l o t . S i n c e t h e s u b t o p i c s r e p r e s e n t e d by thes e f i g u r e s can be grouped i n t o c o r r e s p o n d i n g t o p i c s , d i s c u s s i o n and i n t e r p r e t a t i o n o f t h e s e f i g u r e s are made under each t o p i c h e a d i n g . 52 Common and Decimal F r a c t i o n s The t h r e e s u b t o p i c s under t h i s t o p i c heading a r e : f r a c t i o n s , d e c i m a l f r a c t i o n s and a d d i t i o n o f f r a c t i o n s . A l l t h r e e s u b t o p i c s are c o n s i d e r e d t o be r e v i e w m a t e r i a l f o r grade 8 s t u d e n t s . Two p a t t e r n s o f t e a c h e r b e h a v i o u r can be found i n the i n s t r u c t i o n o f the s e s u b t o p i c s . The f i r s t p a t t e r n i s a s t r o n g o r i e n t a t i o n t o the a b s t r a c t approaches when t e a c h i n g f r a c t i o n s and d e c i m a l f r a c t i o n s . The second p a t t e r n does not g i v e any c o n c l u s i v e e v i d e n c e as t o the p r e f e r e n c e o f approach i n the i n s t r u c t i o n o f a d d i t i o n o f f r a c t i o n s . The f i r s t p a t t e r n i s r e p r e s e n t e d by F i g u r e s 7 and 8. These f i g u r e s a re c h a r a c t e r i z e d by skewed a b s t r a c t s c o r e d i s t r i b u t i o n s , w i t h medians t h a t a r e above 15. T h i s means t h a t more than h a l f o f the t e a c h e r s used the a b s t r a c t i t e m s when t e a c h i n g t h e s e c o n c e p t s . F o r example, i n F i g u r e 8, an extreme v a l u e o f 10 i n the a b s t r a c t s c o r e d i s t r i b u t i o n means t h a t a l l the t e a c h e r s t a u g h t d e c i m a l f r a c t i o n s as another way o f w r i t i n g f r a c t i o n s ( i t e m 52), as a s e r i e s ( i t e m 55) or as an e x t e n s i o n o f p l a c e v a l u e ( i t e m 5 4 ) . F u r t h e r m o r e , a median o f 17 and a 25th p e r c e n t i l e o f 13 i n d i c a t e t h a t more than 75% o f the t e a c h e r s emphasized th e s e items i n t e a c h i n g the con c e p t . A n o t h e r c h a r a c t e r i z t i c o f t h e s e f i g u r e s i s the skewed c o n c r e t e s c o r e d i s t r i b u t i o n s , w i t h medians t h a t a re Abstract Concrete S xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxx X Scores 20 - - |—< 15 -- U-10 - -5 - • 0 J -I - 20 x xxxxxxx • - 15 xxx xxxxxx xxxxxxxxxx - - 10 xxxxxxxxxx xxxxxxxxxxxxxxxx xxxxxx 5 xxxxxxxx xxxxxxx xxxx - - 0 x F i g u r e 7 . D i s t r i b u t i o n of a b s t r a c t s c o r e s and c o n c r e t e s c o r e s f o r the concept o f f r a c t i o n s . Abstract Concrete a) Scores xxxxxxxxxxxxxx -f 20 x xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxx 4 15 4- 10 x x x x x x x x x x x x x 4 5 x x x x x x x x x x x x x x x x x x x x i x X X X X X X X X X X X X X X X X X X X l 4 0 x x x x Scores b) f 20 4 15 4 io 4 5 4 Figure 8. D is t r ibu t ion of abstract scores and concrete scores for the concept of decimal f r a c t i o n s . below 10. T h i s means t h a t the c o n c r e t e items were not emphasized by t e a c h e r s when t e a c h i n g t h e s e c o n c e p t s . For example, i n F i g u r e 8, a 75th p e r c e n t i l e f o r the c o n c r e t e s c o r e s i s 10 and t h e median i s 7. T h i s means t h a t l e s s than 25% o f the t e a c h e r s emphasized the use of the number l i n e ( i t e m 5 1 ) , diagrams ( i t e m 53) o r rods ( i t e m 56) when t e a c h i n g d e c i m a l f r a c t i o n s . The t h i r d c h a r a c t e r i s t i c o f F i g u r e s 7 and 8 i s t h a t the 25th p e r c e n t i l e o f t h e a b s t r a c t s c o r e s i s h i g h e r t h a n t h e 75th p e r c e n t i l e o f the c o n c r e t e s c o r e s , showing t h a t t e a c h e r s s c o r e d much h i g h e r w i t h the a b s t r a c t items than the c o n c r e t e i t e m s . A l l t h e s e c h a r a c t e r i s t i c s c o n s t i t u t e a p a t t e r n which s u g g e s t s t h a t t e a c h e r s showed a d e f i n i t e and s t r o n g p r e f e r e n c e f o r the a b s t r a c t approaches i n t e a c h i n g the r e v i e w c o n c e p t s f r a c t i o n s and d e c i m a l f r a c t i o n s . The second p a t t e r n i s r e p r e s e n t e d by F i g u r e 9. The c o n c r e t e and a b s t r a c t d i s t r i b u t i o n s f o r t h i s s k i l l a r e v e r y s i m i l a r i n shape and range. The c o n c r e t e s c o r e s range from 0 t o 16 whereas the a b s t r a c t s c o r e s range from 0 t o 17. The box-and-whisker p l o t s a re s i m i l a r i n l e n g t h and have the same median. T h i s i n d i c a t e s t h a t the v a r i a b i l i t y o f s c o r e s i s s i m i l a r f o r bot h d i s t r i b u t i o n s . W h i l e the graphs i n d i c a t e t h a t t e a c h e r s used b o t h c o n c r e t e and a b s t r a c t approaches i n Abstract Scores Concrete xxx xxxxxxxxx xxxxxxxx xxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxx 20 15 X XX xxxxxx 4- 10 xxxxxxx xxxxx xxxxxxxxxxxxxxxx 5 xxxxxxxxxxxxxxxxxx xxxxxxxxxx 0 xxxxxxxxxxxxxx 5*' ores 20 15 + 10 + 5 + Figure <j. Distribution of abstract scores and concrete scores for addition of fractions. t e a c h i n g a d d i t i o n o f f r a c t i o n s , the s i m i l a r i t y g i v e s no c o n c l u s i v e e v i d e n c e as t o t e a c h e r s ' p r e f e r e n c e o f approach. A l g e b r a The two s u b t o p i c s under t h i s t o p i c h e a d i n g a r e : i n t e g e r s and s u b t r a c t i o n o f i n t e g e r s . B o t h s u b t o p i c s a r e c o n s i d e r e d t o be new m a t e r i a l f o r s t u d e n t s i n Grade 8 Two p a t t e r n s o f t e a c h e r b e h a v i o u r can be found i n the i n s t r u c t i o n o f the s e s u b t o p i c s . The f i r s t p a t t e r n i s a s t r o n g o r i e n t a t i o n t o t h e c o n c r e t e approaches when t e a c h i n g i n t e g e r s . The second p a t t e r n i s an o r i e n t a t i o n t o t h e a b s t r a c t approaches when t e a c h i n g s u b t r a c t i o n o f i n t e g e r s . The f i r s t p a t t e r n i s r e p r e s e n t e d by F i g u r e 10. T h i s f i g u r e i s almost the m i r r o r image o f F i g u r e 8, and i s c h a r a c t e r i z e d by skewed d i s t r i b u t i o n s f o r b o t h the c o n c r e t e and a b s t r a c t s c o r e s . A median a t 5 and a 75th p e r c e n t i l e a t 10 f o r t h e a b s t r a c t s c o r e s i n d i c a t e t h a t l e s s t h a n 25% o f t h e t e a c h e r s emphasized a l g e b r a i c approaches ( i t e m 21) o r e q u i v a l e n c e c l a s s e s ( i t e m 23) when t e a c h i n g i n t e g e r s . A median a t 13 and a 25th p e r c e n t i l e a t 10 f o r the c o n c r e t e s c o r e s i n d i c a t e t h a t o v e r 75% o f the t e a c h e r s used o r emphasized t h e number l i n e ( items 20 and 22) and p h y s i c a l s i t u a t i o n s ( i t e m 24) SB Abstract Scores Concrete a) 4- 20 xxxx xxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx 4- 5 xxxxxxxxxxxxxxx 4- 0 1 5 xxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxx -4- 10 xxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxx b) Scores 20 15 10 + 5 + Figure lo. Distribution of abstract scores and concrete scores for the concept of integers. when t e a c h i n g t h i s c o n c e p t . In t h i s f i g u r e , t h e 75th p e r c e n t i l e of the a b s t r a c t s c o r e d i s t r i b u t i o n i s the 25th p e r c e n t i l e o f the c o n c r e t e s c o r e d i s t r i b u t i o n s , showing t h a t t e a c h e r s s c o r e d much h i g h e r w i t h t h e c o n c r e t e items than the a b s t r a c t i t e m s . T h i s d i f f e r e n c e i n the d i s t r i b u t i o n s s u g g e s t s t h a t t e a c h e r s showed a s t r o n g p r e f e r e n c e f o r t h e c o n c r e t e approaches i n t e a c h i n g t h e new concept i n t e g e r s . The second p a t t e r n i s r e p r e s e n t e d by F i g u r e 11. The c o n c r e t e and a b s t r a c t s c o r e d i s t r i b u t i o n s i n t h i s f i g u r e showed t h a t t e a c h e r s used o r emphasized t h e a b s t r a c t approaches more t h a n the c o n c r e t e approaches. A 25th p e r c e n t i l e a t 10 f o r t h e a b s t r a c t s c o r e s showed t h a t more th a n 75% o f the t e a c h e r s emphasized o r used r u l e s (items 29 and 30) i n t e a c h i n g s u b t r a c t i o n o f i n t e g e r s . The c o n c r e t e s c o r e s are more e v e n l y d i s t r i b u t e d on the s c a l e . A median a t 10 showed t h a t 50% o f t h e t e a c h e r s used o r emphasized the number l i n e (items 28 and 32) when t e a c h i n g t h i s s k i l l . Comparing the two d i s t r i b u t i o n s , a p r e f e r e n c e f o r t h e a b s t r a c t approaches i s i n d i c a t e d i n t h e i n s t r u c t i o n o f t h i s new Abstract Concrete Scores xxxxxxxxxxxxx 1- 20 xxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx 4- 15 xxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxx 10 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx x x x x _L 5 xxxxxxxxxxxxxx 4- 0 xxxxxxxxxxx b) Scores 20 4 15 4-10 4 5 -4-Figure II. Distribution of abstract scores and concrete scores for the subtraction of integers. 61 Geometry P y t h a g o r e a n Theorem i s t h e o n l y s u b t o p i c u n d e r t h i s h e a d i n g . T h i s s u b t o p i c i s c o n s i d e r e d t o be new m a t e r i a l f o r s t u d e n t s i n G r a d e 8. F i g u r e 12 i s t h e c o n c r e t e and a b s t r a c t s c o r e d i s t r i b u t i o n s f o r t h i s c o n c e p t . As shown i n t h i s f i g u r e , t h e c o n c r e t e and a b s t r a c t d i s t r i b u t i o n s , a l t h o u g h n o t m i r r o r images, a r e q u i t e s i m i l a r i n shape and r a n g e . The s i m i l a r l e n g t h s o f t h e b o x - a n d - w h i s k e r p l o t s i n d i c a t e s t h a t t h e v a r i a b i l i t y o f t h e two s c o r e s a r e s i m i l a r . The g r a p h i n d i c a t e s t h a t t e a c h e r s u s e d c o n c r e t e a p p r o a c h e s s u c h as m e a s u r i n g d e v i c e s ( i t e m 6 7 ) , g e o b o a r d s ( i t e m 71) and d i a g r a m s ( i t e m 68) and a b s t r a c t a p p r o a c h e s s u c h as f o r m u l a ( i t e m 69) and a l g e b r a i c d e d u c t i o n s ( i t e m 72) i n t e a c h i n g t h i s c o n c e p t . The s i m i l a r i t y i n t h e two d i s t r i b u t i o n s g i v e s no c o n c l u s i v e e v i d e n c e as t o t h e t e a c h e r s ' p r e f e r e n c e o f a p p r o a c h . Measurement The two s u b t o p i c s u n d e r t h i s h e a d i n g a r e : number IT and r e l a t i o n s h i p among v a r i o u s m e t r i c u n i t s . B o t h s u b t o p i c s a r e c o n s i d e r e d t o be r e v i e w m a t e r i a l f o r s t u d e n t s i n G r a d e 8. A c o n s i s t e n t p a t t e r n i s f o u n d i n t h e i n s t r u c t i o n o f t h e s e two s u b t o p i c s : t e a c h e r s were a b s t r a c t l y - o r i e n t e d when t e a c h i n g t h e s e c o n c e p t s . 62 a) Abstract Concrete Scores xxxx 4- 15 xxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxx 4- 5 20 xxxxxx xxxxxxxxx 10 xxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxx xxxxxxx xxxxxxxx + 0 xxxxx b) Scores 20 15 10 4 5 A Figure 12. Distribution of abstract scores and concrete scores for the concept Pythagorean Theorem. The two s u b t o p i c s a re r e p r e s e n t e d by F i g u r e s 13 and 14. In bo t h f i g u r e s , the c o n c r e t e s c o r e s showed skewed d i s t r i b u t i o n s . T h i s i s an i n d i c a t i o n t h a t most t e a c h e r s s c o r e d low on the c o n c r e t e i t e m s . I n F i g u r e 13, the median i s 3 and the 75th p e r c e n t i l e i s 7 f o r t h e c o n c r e t e s c o r e s , thus s u g g e s t i n g t h a t few t e a c h e r s emphasized the use o f measuring d e v i c e s ( i t e m 4 8 ) , p o l y g o n s ( i t e m 53) or g r i d s ( i t e m 54) when t e a c h i n g the c o n c e p t f r . F u r t h e r m o r e , 4 3% o f the t e a c h e r s s c o r e d 0 on t h e s e c o n c r e t e i t e m s , i n d i c a t i n g t h a t t h e s e approaches were not used. In F i g u r e 14, the median i s 5 and t h e 75th p e r c e n t i l e i s 10 f o r the c o n c r e t e s c o r e s . T h i s s u g g e s t s t h a t few t e a c h e r s emphasized the use o f t h e metre s t i c k ( i t e m 69) o r c e n t i m e t r e and d e c i m e t r e cubes ( i t e m 70) t o e s t a b l i s h r e l a t i o n s h i p s among v a r i o u s m e t r i c u n i t s . A n o t h e r c h a r a c t e r i s t i c o f t h e s e f i g u r e s i s t h a t t h e a b s t r a c t s c o r e d i s t r i b u t i o n s a r e q u i t e evenLy spr e a d o u t on t h e s c a l e , s u g g e s t i n g t h a t most t e a c h e r s used o r emphasized t h e a b s t r a c t approaches when t e a c h i n g t h e s e c o n c e p t s . F o r example, i n F i g u r e 13, t h e median i s 10 and t h e 25th and the 75th p e r c e n t i l e s a re 5 and 15 r e s p e c t i v e l y . T h i s i n d i c a t e s t h a t a l t h o u g h t h e a b s t r a c t approaches such as f o r m u l a ( i t e m 49) and c h a r t s ( i t e m 51) were used, t e a c h e r s d i f f e r i n t h e i r degree o f emphasis i n u s i n g t h e s e approaches. In both f i g u r e s , the d i f f e r e n c e s a) Abstract Concrete Scores xxx % 20 xxxxxxxxxxxxxxxxxx 4- 15 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx 4- 10 xxxx xxxxxxxxx 4- 5 xxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxx xxxxxxxxxxx 4- 0 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx b) Scores 20 15 4-10 + Figure 13. Dist r ibut ion of abstract scores and concrete scores for the concept number IT. a) b) Abstract Concrete Scores X X X X X X X X X X X X X £ 2 n X X X xxxxxxxxxx 4- 15 xxxxxx xxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxx 4- 10 xxxx X X X X X X X X X X X X X H I > . X X X X X X X X X X X X X Scores 20 - -15 - -10 - -5 - -4- 5 X X X X X X X X X X X X X X X X X X X X X X 4" 0 X X X X X X X X X X X X X Figure It. D is t r ibu t ion of abstract scores and concrete scores for the re la t ionship among various metric un i ts . 4> between the c o n c r e t e and a b s t r a c t s c o r e d i s t r i b u t i o n s s u g g e s t s t h a t t e a c h e r s showed a p r e f e r e n c e f o r the a b s t r a c t approaches i n t e a c h i n g t h e s e two r e v i e w c o n c e p t s . D i f f e r e n c e Scores The comparison o f d i f f e r e n c e s c o r e d i s t r i b u t i o n s c o n f i r m s t h e c o n c l u s i o n s made about t e a c h e r p r e f e r e n c e s as shown i n t h e stem-and-leaf c o n c r e t e and a b s t r a c t s c o r e d i s t r i b u t i o n s . I t was c o n f i r m e d by t h i s p a r t i c u l a r comparison t h a t most t e a c h e r s were c o n c r e t e l y - o r i e n t e d i n t h e i r i n s t r u c t i o n o f i n t e g e r s ; and a b s t r a c t l y - o r i e n t e d i n t h e i r i n s t r u c t i o n o f f r a c t i o n s , d e c i m a l f r a c t i o n s , s u b t r a c t i o n o f i n t e g e r s , number Ti" and r e l a t i o n s h i p among v a r i o u s m e t r i c u n i t s . F o r t h e r e m a i n i n g two s u b t o p i c s , a d d i t i o n o f f r a c t i o n s and Pythagorean Theorem, t h i s p a r t i c u l a r c omparison i n d i c a t e d t h a t perhaps most t e a c h e r s showed no s i g n i f i c a n t p r e f e r e n c e f o r e i t h e r approach i n t h e i r i n s t r u c t i o n . The d i s t r i b u t i o n s o f t h e d i f f e r e n c e s c o r e s f o r the e i g h t s u b t o p i c s were graphed i n s i d e - b y - s i d e stem-and-l e a f d i s p l a y s i n F i g u r e 15. Each d i s p l a y i s accompanied by the c o r r e s p o n d i n g box-and-whisker p l o t . S e v e r a l p a t t e r n s can be found i n t h i s f i g u r e . The f i r s t p a t t e r n i s c h a r a c t e r i z e d by t h e f a c t t h a t the box i n the box-and-whisker p l o t i s c o m p l e t e l y below t h e z e r o l i n e . l u l l l l l l , . I l l " " J l l l l 1 1 11 A d d i t i o n 5T lliit--ffii H Pythtqorun TV»fT« I I M H U 1 1 * > ) l t l O « l M p *nO*vq V i r i o n H e t r U UwIM i l l . H i l l " 1 H U H . T T J L l i l t " ' H H H I " I" Figure 15. D i s t r i b u t i o n of difference scores for the eight subtopics: f r a c t i o n s , addition of f r a c t i o n s , decimal fractions, integers, subtraction of integers, Pythagorean Theorem, number TT , rela t i o n s h i p among various metric u n i t s . 6 7 The s e c o n d p a t t e r n by t h e box i n t h e b o x - a n d - w h i s k e r p l o t e x t e n d i n g f r o m t h e z e r o l i n e t o b e l o w . The t h i r d by t h e box i n t h e b o x - a n d - w h i s k e r p l o t b e i n g c o m p l e t e l y a b o v e t h e z e r o l i n e , and t h e f o u r t h by t h e box i n t h e b o x - a n d - w h i s k e r p l o t s t r a d d l i n g t h e z e r o l i n e . I n t h e f i r s t p a t t e r n , t h e b o x e s a r e c o m p l e t e l y b e l o w t h e z e r o l i n e . T h i s i n d i c a t e s t h a t more t h a n 75% o f t h e t e a c h e r s a r e a b s t r a c t l y - o r i e n t e d i n t h e i r i n s t r u c t i o n o f t h e f o l l o w i n g s u b t o p i c s : f r a c t i o n s , d e c i m a l f r a c t i o n s , and r e l a t i o n s h i p among v a r i o u s m e t r i c u n i t s . I n t h e s e c o n d p a t t e r n , t h e b o x e s e x t e n d f r o m t h e z e r o l i n e t o b e l o w , m e a n i n g t h a t 75% o f t h e t e a c h e r s s c o r e d z e r o o r n e g a t i v e l y i n t h e i r d i f f e r e n c e s c o r e s . T h i s p a t t e r n i s f o u n d i n two d i s t r i b u t i o n s : s u b t r a c t i o n o f i n t e g e r s a n d number TT . The f a c t t h a t t h e m e d i a n s a r e b e l o w z e r o i n d i c a t e s t h a t more t h a n 50% o f t h e t e a c h e r s w e r e a b s t r a c t l y - o r i e n t e d i n t h e i r i n s t r u c t i o n o f s u b t r a c t i o n o f i n t e g e r s a n d number Tr I n t h e t h i r d p a t t e r n , t h e box i s a b o v e t h e z e r o l i n e , i n d i c a t i n g t h a t more t h a n 75% o f t h e t e a c h e r s a r e c o n c r e t e l y -o r i e n t e d i n t h e i r i n s t r u c t i o n . T h i s p a t t e r n c a n be f o u n d i n t h e i n s t r u c t i o n o f i n t e g e r s o n l y . I n t h e f o u r t h p a t t e r n , t h e b o x e s s t r a d d l e t h e z e r o l i n e . T h i s p a t t e r n i s f o u n d i n two d i s t r i b u t i o n s : a d d i t i o n o f f r a c t i o n s and P y t h a g o r e a n Theorem. The f a c t 68 t h a t the boxes s t r a d d l e the z e r o l i n e might be an i n d i c a t i o n t h a t the t e a c h e r s used both c o n c r e t e and a b s t r a c t approaches i n t h e i r i n s t r u c t i o n w i t h o u t showing any s i g n i f i c a n t p r e f e r e n c e f o r e i t h e r approach. Summary Three p a t t e r n s o f t e a c h e r b e h a v i o u r seemed t o emerge from the d i s t r i b u t i o n s o f c o n c r e t e , a b s t r a c t and d i f f e r e n c e s c o r e s . The p a t t e r n s were: a b s t r a c t - o r i e n t a t i o n , c o n c r e t e - o r i e n t a t i o n and no p r e f e r e n c e f o r e i t h e r approach. Teachers tended t o be a b s t r a c t l y - o r i e n t e d i n t h e i r i n s t r u c t i o n o f f i v e o f t h e e i g h t s u b t o p i c s i n c l u d e d f o r i n v e s t i g a t i o n i n t h i s s t u d y . These f i v e s u b t o p i c s were: f r a c t i o n s , d e c i m a l f r a c t i o n s , s u b t r a c t i o n o f i n t e g e r s , number ^  and r e l a t i o n s h i p among v a r i o u s m e t r i c u n i t s . I n t e g e r s was the o n l y s u b t o p i c t h a t was t a u g h t w i t h a c o n c r e t e - o r i e n t a t i o n . F o r the r e m a i n i n g two s u b t o p i c s , a d d i t i o n o f f r a c t i o n s and Pythagorean Theorem, most t e a c h e r s might have showed no s i g n i f i c a n t p r e f e r e n c e f o r e i t h e r approach i n t h e i r i n s t r u c t i o n . C h a p t e r 5 CONCLUSIONS AND IMPLICATIONS The p urpose o f t h i s s t u d y v/as t o p r o v i d e an a n a l y s i s o f t h e ap p r o a c h e s used by ma t h e m a t i c s t e a c h e r s i n t e a c h i n g c e r t a i n c o n c e p t s and s k i l l s i n Grade 8. I n p a r t i c u l a r , a c o m p a r i s o n was made i n t h e o r i e n t a t i o n t o two a p p r o a c h e s : c o n c r e t e and a b s t r a c t . The r e s u l t s o f t h i s a n a l y s i s r e v e a l f i n d i n g s w h i c h have c e r t a i n i m p l i c a t i o n s f o r f u r t h e r r e s e a r c h i n m a t h e m a t i c s e d u c a t i o n . Answer t o R e s e a r c h Q u e s t i o n The m a j o r q u e s t i o n a d d r e s s e d i n t h e s t u d y was; VJere t e a c h e r p r a c t i c e s i n a c c o r d a n c e w i t h t h e t h e o r y o f i n s t r u c t i o n p r o p o s e d by s e v e r a l a u t h o r s , i n w h i c h i t i s recommended t h a t c o n c e p t s be p r e s e n t e d u s i n g m a n i p u l a t i v e m a t e r i a l s , p i c t u r e s , o r d i a g r a m s , and t h e n t h r o u g h a b s t r a c t p r e s e n t a t i o n s w i t h symbols? To a d d r e s s t h i s q u e s t i o n , t h e f o l l o w i n g a s p e c t s o f i n s t r u c t i o n a l b e h a v i o u r were examined: 1) The c o n s i s t e n c y w i t h i n i n d i v i d u a l t e a c h e r s i n t h e i r o r i e n t a t i o n t o t h e c o n c r e t e and a b s t r a c t a p p r o a c h e s i n t e a c h i n g each s u b t o p i c . 2) T e a c h e r s ' o r i e n t a t i o n t o t h e a b s t r a c t o r c o n c r e t e a pproaches i n t e a c h i n g v a r i o u s new and r e v i e w s u b t o p i c s . 69 When teachers' choice o f methods for the instruction o f i n d i v i d u a l subtopics were examined, i t was found that teachers were consistent in their emphasis or choice among concrete and abstract approaches for most concepts or s k i l l s . Preliminary analysis procedures showed that teachers' responses to the concrete and abstract items do not d i f f e r s i g n i f i c a n t l y within each category. This means that teachers tended to use with equal emphasis either none or most of the items within each category for the in s t r u c t i o n of each subtopic. When i n s t r u c t i o n a l behaviours for the three new subtopics were examined, i t was found that teachers were concretely-oriented in th e i r teaching of one subtopic only — integers. For the two remaining new subtopics, teachers were abstractly-oriented in the i r i n s t r u c t i o n of subtraction of integers, and showed no s i g n i f i c a n t preference of approach in th e i r instruction of Pythagorean Theorem. When i n s t r u c t i o n a l behaviours for the review subtopics were examined, i t was found that teachers were abstractly-oriented in the i r i n s t r u c t i o n of four of the f i v e subtopics. These subtopics were fractions, decimal f r a c t i o n s , number "» and relationship among various metric units. For the remaining s k i l l , addition of fractions, teachers used both concrete and abstract 71 approaches and showed no s i g n i f i c a n t preference toward either approach The findings from the investigation of the d i f f e r e n t aspects of i n s t r u c t i o n a l behaviour lead to the conclusion that teacher practices related to the use of concrete and abstract approaches were not in accordance with the theory of i n s t r u c t i o n proposed by Bruner, Dienes and many mathematics educators. According to t h i s theory, new concepts should be introduced through concretely-oriented approaches, followed by the introduction of symbolic representations. This i n s t r u c t i o n a l behaviour has not been found among the Grade 8 mathematics teachers in t h i s study. In the i r teaching of three new concepts and s k i l l s i n the Grade 8 curriculum, mathematics teachers were concretely-oriented in t h e i r teaching of one subtopic only. If teacher practices were in accordance with t h i s theory of in s t r u c t i o n , the finding of the study should indicate that teachers were concretely-oriented in teaching a l l the new concepts and s k i l l s . Limitation of the Study A l i m i t a t i o n of the study i s that while the questionnaires were constructed i n order to provide information on classroom practices, they were not constructed for investigation into the concrete-72 abstract v a r i a b l e alone. Many subtopics were eliminated from analysis i n t h i s study because of i n s u f f i c i e n t numbers of items i n the questionnaires that r e l a t e d to t h i s v a r i a b l e . The questionnaires also did not provide other information that would be valuable such as the teachers' reasons f o r p r e f e r r i n g abstract approaches to concrete approaches i n i n s t r u c t i n g the concepts and s k i l l s considered f o r t h i s study. Another l i m i t a t i o n i s that the r e s u l t s of t h i s study might be biased due to the f a c t that the teachers i n the sample were more experienced and s p e c i a l i z e d i n the teaching of mathematics than t h e i r colleagues. The findings might not provide accurate, generalizable information on the population of mathematics teachers i n Grade 8. Implications According to Bruner's "conservative doctrine" (Bruner, 1966; p. 49), a l l concepts and s k i l l s , e s p e c i a l l y new concepts and s k i l l s , should be taught with concretely-oriented approaches f i r s t , with gradual introduction of abstract representations. This p r a c t i c e has not been found among the teachers i n the study. The f a c t that t h e i r behaviours were not i n accordance v/ith the suggestions made by t h e o r i s t s leads one to question the r e a l i t y of the i m p l e m e n t a t i o n of the t h e o r y i n a l l mathematics c l a s s r o o m s There can be s e v e r a l r e a s o n s f o r the f a i l u r e o f the implementation p r o c e s s . One o b s t a c l e t o the i m p l e m e n t a t i o n p r o c e s s might be t h a t t e a c h e r s were not c o n v i n c e d by t h e t h e o r i s t s t h a t a c o n c r e t e - o r i e n t a t i o n t o i n s t r u c t i o n i s a b e t t e r approach and l e a d s t o b e t t e r u n d e r s t a n d i n g o f m a t h e m a t i c a l c o n c e p t s W h i l e the t h e o r i s t s have based t h e i r c o n c l u s i o n s on c l a s s r o o m s t u d i e s , the numbers o f s t u d e n t s i n v o l v e d i n t h e s e s t u d i e s were few, and achievement r e s u l t s between s t u d e n t s t h a t were t a u g h t w i t h a c o n c r e t e - o r i e n t a t i o n and t h o s e t h a t were t a u g h t w i t h an a b s t r a c t - o r i e n t a t i o n were not compared. A s i g n i f i c a n t d i f f e r e n c e i n achievement r e s u l t s i n f a v o u r o f the c o n c r e t e l y - o r i e n t e d approaches w i l l s u g gest t o t e a c h e r s t h a t s t u d e n t s t a u g h t w i t h a c o n c r e t e - o r i e n t a t i o n do r e f l e c t b e t t e r u n d e r s t a n d i n g o f the m a t h e m a t i c a l c o n c e p t s i n v o l v e d . The p r a c t i c a l a s p e c t o f i m p l e m e n t a t i o n might be an o b s t a c l e . Teachers might have found t h a t the r e l e v a n t m a t e r i a l s needed f o r t e a c h i n g c e r t a i n s u b t o p i c s were not e a s i l y a c c e s s i b l e . Time might a l s o have been an i m p o r t a n t f a c t o r i n a f f e c t i n g t e a c h e r s ' d e c i s i o n s r e g a r d i n g i n s t r u c t i o n a l methods. When under the p r e s s u r e of t e a c h i n g a s e t number of p r e s c r i b e d t o p i c s i n a y e a r a c c o r d i n g t o t h e c u r r i c u l u m , t e a c h e r s might have found c o n c r e t e approaches t o be t o o time-consuming. T h e r e f o r e , 74 they might have chosen t o use t h e i r time e c o n o m i c a l l y and t a u g h t most c o n c e p t s and s k i l l s a b s t r a c t l y . A t h i r d o b s t a c l e t o i m p l e m e n t a t i o n might be the i n a d e q u a t e p r e - s e r v i c e t r a i n i n g o f t e a c h e r s i n u s i n g c o n c r e t e approaches i n mathematics i n s t r u c t i o n . T h i s o b s t a c l e might be overcome by changes i n t e a c h e r -t r a i n i n g programs. One p o s s i b l e change might be an i n c r e a s e i n emphasis i n the i n s t r u c t i o n o f c o n c r e t e l y - o r i e n t e d approaches i n s p e c i f i c t o p i c s i n mathematics. Another change might be a s i g n i f i c a n t i n c r e a s e i n time a l l o c a t e d t o the i n s t r u c t i o n o f methods i n t e a c h i n g s p e c i f i c c o n c e p t s and t o p i c s i n mathematics. The f a i l u r e i n the i m p l e m e n t a t i o n p r o c e s s o f the t h e o r y can a l s o i m p l y t h a t the t h e o r y i s not a v a l i d one. The f a c t t h a t e x p e r i e n c e d t e a c h e r s i n mathematics have not a p p l i e d t h i s t h e o r y i n t h e i r i n s t r u c t i o n o f a wide v a r i e t y o f c o n c e p t s c o u l d c a s t doubt on the v a l i d i t y o f the t h e o r y i t s e l f . The s m a l l - s c a l e s t u d i e s on which Bruner and Dienes based t h e i r c o n c l u s i o n s about mathematics l e a r n i n g might be i n s u f f i c i e n t t o prove t h a t c o n c r e t e approaches a r e i n d e e d n e c e s s a r y o r advantageous f o r the u n d e r s t a n d i n g o f m a t h e m a t i c a l c o n c e p t s . 75 S u g g e s t i o n s f o r F u r t h e r Research A major f i n d i n g o f the study i s t h a t t e a c h e r s t y p i c a l l y p r e f e r r e d a b s t r a c t approaches t o c o n c r e t e approaches i n t e a c h i n g mathematics. However, one l i m i t a t i o n o f t h i s s t u d y i s t h a t t h e q u e s t i o n n a i r e s d i d not p r o v i d e d e t a i l e d i n f o r m a t i o n on t h e r e a s o n s f o r t e a c h e r s ' p r e f e r e n c e f o r a b s t r a c t approaches. T h e r e f o r e , a f u r t h e r s t u d y c o u l d be done u s i n g an i n s t r u m e n t d e s i g n e d s p e c i f i c a l l y t o i n v e s t i g a t e the c o n c r e t e - a b s t r a c t v a r i a b l e . A r e p r e s e n t a t i v e sample o f Grade 8 mathematics t e a c h e r s s h o u l d be used i n o r d e r t o produce more g e n e r a l i z a b l e r e s u l t s . The f i n d i n g o f t h i s f u r t h e r s t u d y might r e v e a l t e a c h e r s ' reasons f o r p r e f e r r i n g a b s t r a c t approaches t o c o n c r e t e approaches, t h u s c a s t i n g l i g h t on t h e o b s t a c l e s t o the i m p l e m e n t a t i o n o f t h e t h e o r y . The v a l i d i t y o f the t h e o r y can a l s o be examined i n f u r t h e r r e s e a r c h . Achievement r e s u l t s can be compared between s t u d e n t s t h a t a r e t a u g h t w i t h a c o n c r e t e - o r i e n t a t i o n and t h o s e t h a t a r e t a u g h t w i t h an a b s t r a c t - o r i e n t a t i o n . I n o r d e r t o make v a l i d g e n e r a l i z a t i o n s , a r e p r e s e n t a t i v e sample o f c l a s s e s s h o u l d be used, and t e a c h e r s s h o u l d be t r a i n e d s p e c i f i c a l l y f o r the c o n c r e t e o r a b s t r a c t approaches t h e y use. A wide v a r i e t y o f t o p i c s s h o u l d a l s o be used i n o r d e r t h a t p a t t e r n s can be d e t e c t e d a c r o s s t o p i c s . Another f i n d i n g of t h i s study i s t h a t Grade 8 mathematics t e a c h e r s were not c o n s i s t e n t i n t h e i r o r i e n t a t i o n t o e i t h e r the c o n c r e t e o r a b s t r a c t approaches a c r o s s t h e e i g h t s u b t o p i c s . For example, t e a c h e r s were more a b s t r a c t l y - o r i e n t e d i n d e a l i n g w i t h r e v i e w s u b t o p i c s t h a n they were i n d e a l i n g w i t h new s u b t o p i c s . I n f o r m a t i o n c o l l e c t e d from a n o t h e r p o r t i o n o f the q u e s t i o n n a i r e s r e v e a l s t h a t t e a c h e r s assumed t h a t s t u d e n t s had p r e v i o u s knowledge o f the r e v i e w s u b t o p i c s . T h e r e f o r e , t h e i m p l i c a t i o n i s t h a t the t e a c h e r s made the assumption t h a t t h i s knowledge was both c o r r e c t and s u f f i c i e n t f o r t h e s t u d e n t s t o handle f u r t h e r work i n t h e s e s u b t o p i c s i n a b s t r a c t r e p r e s e n t a t i o n s . I t would be i n t e r e s t i n g t o i n v e s t i g a t e the b a s i s on which t h i s a ssumption was made, whether t e a c h e r s used v a l i d i n s t r u m e n t s t o a s s e s s s t u d e n t s ' knowledge o f t h e r e v i e w s u b t o p i c s , o r whether the d e c i s i o n t o assume t h i s knowledg was made s u b j e c t i v e l y . An i m p o r t a n t f i n d i n g o f t h i s s t u d y c o ncerns t h e n a t u r e o f t h e SIMS d a t a r e g a r d i n g the c o n c r e t e - a b s t r a c t v a r i a b l e . The e x p l o r a t o r y d a t a a n a l y s i s r e v e a l s t h a t the d i s t r i b u t i o n o f s c o r e s a re uni-modal and approximate t h e normal d i s t r i b u t i o n . T h i s f i n d i n g s u g g e s t s t h a t the b a s i c a s s u m p t i o n t o c o n v e n t i o n a l d a t a a n a l y s i s p r o c e d u r e s i s s a t i s f i e d by t h i s d a t a . T h e r e f o r e , t h e s e s t a t i s t i c a l p r o c e d u r e s can be a p p l i e d t o the d a t a t o f i n d out 77 r e l a t i o n s h i p s between t e a c h e r p r a c t i c e s and o t h e r v a r i a b l e s d e a l t w i t h i n the i n t e r n a t i o n a l s t u d y . S i n c e the u l t i m a t e aim o f the SIMS was t o " r e l a t e s t u d e n t achievement t o t e a c h i n g p r a c t i c e " ( R o b i t a i l l e , O'Shea and D i r k s , 1982; p. 9 3 ) , r e l a t i o n s h i p s between t e a c h e r d i f f e r e n c e s on the c o n c r e t e - a b s t r a c t v a r i a b l e and s t u d e n t outcomes can be examined u s i n g c o n v e n t i o n a l s t a t i s t i c a l p r o c e d u r e s . A problem t h a t d e s e r v e s f u r t h e r s t u d y i s the i n v e s t i g a t i o n o f the e f f e c t o f t e a c h e r s ' c h o i c e o f c o n c r e t e o r a b s t r a c t approaches on s t u d e n t s ' achievement i n and a t t i t u d e toward mathematics. Another problem i s the i n v e s t i g a t i o n o f the impact o f t e a c h e r s ' c h o i c e o f approach on the v a r i a b i l i t y o f s t u d e n t r e s u l t s . T h i s i n f o r m a t i o n can be found i n t h e r e l a t i o n s h i p between the t e a c h e r s ' o r i e n t a t i o n t o c o n c r e t e o r a b s t r a c t approaches and t h e v a r i a b i l i t y o f s t u d e n t achievement and a t t i t u d e w i t h i n each c l a s s . R e l a t i o n s h i p s between t e a c h e r p r a c t i c e s and o t h e r v a r i a b l e s c o n c e r n i n g t e a c h e r d a t a can a l s o be examined. For example, t e a c h e r p r a c t i c e s might be r e l a t e d t o t e a c h e r s ' e x p e r i e n c e i n t e a c h i n g mathematics, amount o f u n i v e r s i t y e d u c a t i o n , the a t t i t u d e toward mathematics o r how mathematics i s p e r c e i v e d as a d i s c i p l i n e by the t e a c h e r . I n f o r m a t i o n on t h e s e v a r i a b l e s was c o l l e c t e d by o t h e r i n s t r u m e n t s i n the SIMS. Hence, i t i s p o s s i b l e t o 78 i n v e s t i g a t e and compare the r e l a t i o n s h i p s among thes e v a r i a b l e s . F i n d i n g s from t h e s e i n v e s t i g a t i o n s might c r e a t e a p r o f i l e of a c o n c r e t e l y - o r i e n t e d t e a c h e r and p r o v i d e v a l u a b l e i n f o r m a t i o n t o the mathematics e d u c a t i o n community. Summary The t h e o r y o f i n i t i a l l y t e a c h i n g m a t h e m a t i c a l c o n c e p t s t h r o u g h c o n c r e t e l y - o r i e n t e d approaches, f o l l o w e d by the i n t r o d u c t i o n o f a b s t r a c t r e p r e s e n t a t i o n s , has been proposed by t h e o r i s t s such as Bruner and D i e n e s , and advocated by many mathematics e d u c a t o r s . However, i n s p i t e o f the s t r o n g endorsement f o r t h i s - t h e o r y , i t was found i n t h i s s t u d y t h a t t h i s t h e o r y has not been implemented t o the e x t e n t t h a t e d u c a t o r s had i n t e n d e d . S e v e r a l r e a s o n s have been suggested f o r t h e f a i l u r e o f i m p l e m e n t a t i o n . These s u g g e s t i o n s have y e t t o be v e r i f i e d by f u r t h e r r e s e a r c h . The f a c t i s t h a t the i m p l e m e n t a t i o n o f the t h e o r y has not o c c u r r e d . T h i s f i n d i n g i s i n agreement w i t h f i n d i n g s from some o t h e r s t u d i e s on c l a s s r o o m p r o c e s s e s . These f i n d i n g s " p r o v i d e r e a s o n t o q u e s t i o n the e x t e n t t o which any o f t h e s e p r o p o s a l s f o r i n n o v a t i v e pedagogy have i n f l u e n c e d predominant i n s t r u c t i o n a l p a t t e r n s " (Fey, 1979; p. 493). I t appears t h a t t h e many changes proposed i n t h e mathematics c u r r i c u l a i n the l a s t twenty t o t h i r t y y e a r s have not r e s u l t e d i n s i g n i f i c a n t changes i n t e a c h e r b e h a v i o u r i n mathematics c l a s s r o o m s . REFERENCES A l l e n , M. J . , & Yen, W. M. I n t r o d u c t i o n t o Measurement  Theory. Monterey, C a l i f o r n i a : B r o o k s / C o l e P u b l i s h i n g Company, 1979. A s h l o c k , R. B., & West, T. A. P h y s i c a l r e p r e s e n t a t i o n s f o r signed-number o p e r a t i o n s . 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"New Math" i m p l e m e n t a t i o n : a l o o k i n s i d e the c l a s s r o o m . J o u r n a l  f o r R e search i n Mathematics E d u c a t i o n , 1977, 8_, 323-331. R a t h m e l l , E. C. U s i n g t h i n k i n g s t r a t e g i e s t o t e a c h the b a s i c f a c t s . I n M. N. Suydam & R. E. Reys ( E d s . ) , D e v e l o p i n g C o m p u t a t i o n a l S k i l l s . R e ston, V i r g i n i a : The N a t i o n a l C o u n c i l o f Teachers o f Mathematics, I n c . , 1978. R e s n i c k , L. B., & F o r d , W. W. The P s y c h o l o g y o f Mathematics f o r I n s t r u c t i o n . New J e r s e y : Lawrence Erlbaum A s s o c i a t e s , P u b l i s h e r s , 1981. R o b i t a i l l e , D. F. I n t e n t i o n , i m p l e m e n t a t i o n , r e a l i z a t i o n : c ase s t u d i e s o f the impact o f c u r r i c u l u m r e f o r m . In Comparative S t u d i e s o f Mathematics C u r r i c u l a -Change  and S t a b i l i t y 1960-1980-, B i e l e f e l d : I n s t i t u t f u r D i d a k t i k d e r Mathematik der U n i v e r s i t a t B i e l e f e l d , 1980. R o b i t a i l l e , D. F. ( E d . ) . The 1981 B.C. Mathematics  Assessment: G e n e r a l R e p o r t . B.C.: M i n i s t r y o f E d u c a t i o n , 1981. R o b i t a i l l e , D. F., O'Shea, T. J . , & D i r k s , M. K. The  T e a c h i n g and L e a r n i n g o f Mathematics i n B.C. B.C.: M i n i s t r y o f E d u c a t i o n , 1982. R o b i t a i l l e , D. F., & S h e r r i l l , J . M. The B.C. Mathematics  Assessment Report Number 2. B.C.: M i n i s t r y o f E d u c a t i o n , 1977. Suydam, M. R e f l e c t i o n s from r e s e a r c h : f o c u s i n g on t e a c h i n g s t r a t e g i e s from v a r i o u s d i r e c t i o n s . T e a c h i n g  S t r a t e g i e s . Ohio, Columbus: ERIC Center f o r S c i e n c e , M a t h e m a t i c s , and E n v i r o n m e n t a l E d u c a t i o n , 1976. Suydam, M. N., & D e s s a r t , D. J . Classroom Ideas from  R e s e a r c h on C o m p u t a t i o n a l S k i l l s . Reston, V i r g i n i a : The N a t i o n a l C o u n c i l o f Teachers o f Mathematics, I n c . , 1976 . Suydam, M. N., & Reys, R. E. (Eds.) . D e v e l o p i n g  C o m p u t a t i o n a l S k i l l s . R e ston, V i r g i n i a : The N a t i o n a l C o u n c i l o f Teachers o f Mathematics, I n c . , 1978. The C o n t i n u i n g R e v o l u t i o n i n Math e m a t i c s . Washington, D.C: N a t i o n a l C o u n c i l o f Teachers o f Mathematics, 1968 . Tukey, J . W. E x p l o r a t o r y Data A n a l y s i s . Don M i l l s , O n t a r i o : Addison-Wesley P u b l i s h i n g Company, 1977. Van der B l i f , F., H i l d i n g , S., & Weinzweig, A. I . A s y n t h e s i s o f n a t i o n a l r e p o r t s on changes i n c u r r i c u l a . I n Comparative S t u d i e s on Mathematics C u r r i c u l a  -Change and S t a b i l i t y 1960-1980-. B i e l e f e l d : I n s t i t u t f u r D i d a k t i k der Mathematik d e r U n i v e r s i t a t B i e l e f e l d , 1980 . Westbury, I . C o n v e n t i o n a l c l a s s r o o m s , "open" c l a s s r o o m s and t h e t e c h n o l o g y o f t e a c h i n g . J o u r n a l o f C u r r i c u l u m S t u d i e s , 1973, 5, 99-121. APPENDIX A Common and Decimal F r a c t i o n s Q u e s t i o n n a i r e 86 INTERNATIONAL ASSOCIATION for the EVALUATION of EDUCATIONAL ACHIEVEMENT — S E C O N D — Study of MATHEMATICS G R A D E 8 T O P I C S P E C I F I C Q U E S T I O N N A I R E C O M M O N A N D D E C I M A L F R A C T I O N S { Booklet 10 L ) FOR NATIONAL CENTRE USE ONLY PROVINCE OF BRITISH COLUMBIA MINISTRY OF EDUCATION DIVISION OF PUBLIC INSTRUCTION LEARNING ASSESSMENT BRANCH 87, Check here i f neither common fractions nor decimal fractions are included in your program. In that case, disregard the rest of the questionnaire and return i t to B.C. Research in the envelope provided. 1 CHECK tkt nzApon&e. mkick boj>£ dzAcnibeA the. UA<L you. made, o^ zach ofa the. 6oZlouti.ng mcutQJujoULi> -in. youA A.YU>inaction on common and/on. d&CAjnat inwctAjom,. RESPONSE COVES: 1. pnMn<viy 4ouAce, aied finzqucntty 2. 6zcondaA.y &oancz, uAdd occcu><Lonatty 3. not LLizd on, naxaty uAzd 1. School Mathematics IT (Addison-Wesley) 1 2. Mathematics II (Ginn) 3. Essentials of Mathematics II (Ginn) 4. Other published text materials (e.g., textbooks, workbooks, and worksheets) 5. 6. Commercially or locally produced in-dividualized materials (e.g., program-med instruction or computer assisted instruction) 7. 8. Commercially or locally produced lab-oratory materials for student use (e.g., games or manipulatives) 1 2 3 22 1 2 3 23 1 2 3 2i+ 1 2 3 25 1 2 3 26 1 2 3 27 1 2 3 28 Locally produced text materials (e.g., textbooks, workbooks, or worksheets) Commercially or locally produced films, fi lm-strips, or teacher demonstration models. TEACHING TOPICS The. topic* given beZow may be. indiude.d in youA in&tnjuctionaJL pnogKam. CHECK the. ?ieApont>e. which deACAibeA the. tx.ejcutme.wt e.ach topic in LJOUA cZa&A. RESPONSE COVES: 7. taught ai new content 1. fieviewe-d and then extende.d 3. fie.\)ime.d only 4. a&Aume.d ai pn.2Ae.qui(>i£e. knouilcdge. and nzitheA taught non. fieviewzd 5. not taught and not ai-i tuned at> pne.-n.e.quii>i£e. knowledge Fractions 9. Developing the concept 1 10. Finding equivalent fractions -including reducing fractions 1 11. Adding and subtracting - including finding common denominators 1 12. Multiplying 1 13. Dividing 1 14. Ordering 1 Decimals 15. Developing the concept 1 16. Converting decimals to fractions or vice versa 1 17. Adding and subtracting 1 18. Multiplying 1 19. Dividing 1 20. Ordering 1 IF YOU DID NOT TEACH COMMON FRACTIONS, PROCEED 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 2 3 4 DIRECTLY TO ITEM 51. - 4 - 89 5 — .a S"S i 2-3 IS •a * U J O u J si a v j i •t?«s> q -o E s o -o v e 3 • _ SlV 3 3 3 V oo a>_<5 a g J : • • • • • • • • • • • • • • • • 01 o >- z s •o 0) u -a o 3 o» ai -c -c C -W o c o 3 .er •o - 5 -90 < a i ^ a i -a " V O i a l 3 * S a i 3 f a i E w * s 3 a i ~ <=> -a 3 *r Z W I— E S ^ - § 3 3 W — 3 V 3 — S C §>= is =o o c i— V <J E <5 V a l a o -o w v -a o o <w co aJ <i * 3 al ai « - o V a a ) « ? - w avi a is a a v c a) c -a a ) 1 V S S • f l V a a ) -O a i aJ «J O E o a ) * O H o 5 3 < ! • o v > o a « • s o e • * J a t a a i - a < > a v 3 a i « 3 i •3-5 a i § a l « r -a "o >*> a. • w a l c a 1! s i u j a l j e <5 r « J - C a ) a i * a a i 9 _ 1 * '•3 3 o •T> v -o a o •• uu *K ai a ai t— tl l -a a l < W V > a l ~ o ~ <1 S a a " a - a + J •« < j a i a v J; c i a o a o a i " a UJ S o -O V C = a) 3 -? J; a N 3 —,"W al 3 s « a o a a i < U 3 » U I : aJ a l ! a o o a , a * 2 a ) a 3 S •§ . a ) v a « o S < a l a i 3 - a . e * o o W 3 •V » o • w « < > s > d a o a ) a a l o o § " 3 • 3 a a a i 3 V ' • V a « r - a • V <J> o - a a i o - w - = a i < ; - a v - - o a s s O O « » a 3 - o "a "a S a i s> E x a i , 5 o u j a i V 3 z • • • • • • • • • • • • I O o 1 o 2 o + II < < <l The inttrpAeXatiom given below may be Included in your initruetional program. CHECK the rtiponie code which deicribei the treatment oi each topic in your claii. RESPONSE COPES: I . Emphaiized [uied ai a primary explanation, reieAAed to ex-teniively ox irequently) Uied bat not emphaiized Not uied b!ai thii interpretation in the itudenti' text? li you EMPHASIZED the. given inteipneXation [i.e. it you checked I in the iirit column) WRITE the mimbehi ot the reiponie coda which ihow the turn primary neaiom ior iti emphaiii. RESPONSE COPES: I . Welt, known to me In B.C. oi local cuAAXcutum guide Eaiy ion itudenti to underitand Enjoyed by itudenti Related to math oi prior glade* Hie fail in math oi later gradei I wai taught it wai appropriate Emphaiized in itudent text li you PIP NOT USE the given interpreta-tion [i.e. ii you checked 3 in the iiAit column) WHITE the numben oi the reipome c dei which ihow the. two primary reaioni ior not. uiing iX. RESPONSE COPES: I. Not well known to me Not in 6.C. or local curriculum guide 2. 3. 4. 5. 6. 1. Hard lor itudenti to underitand Pit Liked by itudenti Not related to math oi prior gradei Not uieiul in math oi later gradei I wai taught it wai inappropriate Not emphaiized in itudent. text 28. F r a c t i o n s as measurements: t h i s c o n t a i n e r ho lds t h i s s t i c k Is 3_ cm 4 Yes No 1s t Pr imary Reason 2nd P r imary Reason 1s t P r i m a r y Reason 2nd P r imary Reason • • • • 26-31 29. F r a c t i o n s as o p e r a t o r s : Yes No • • • • 30. F r a c t i o n s as comparisons M< /.////////A u n i t rod 1^  rod 3 Yes No i r: i 1 2 2 rod 3" • • • • - 7 - 92 A d d i t i o n o f F r a c t i o n s The iMtzAp^ztatiom oi the. addition o{ $Aactian& givzn bzioui may be, indudzd in youA inttfuicXionaJL pnnqnjm. CHECK thz n.upotu>e. which, dziaUbzi the. txeatmexiX. otf zach topic in youA. CZA&A. RESPONSE COOES: I. Smpkmizzd (uaed <u a. piimivuj vt.ptanation, -te^evied to extzniiviZy on. ^ejqatntly) I. Uizd by not emphuizzd. 3. Hot ai<id. 3 1 . The sum o f two f r a c t i o n s as the un ion o f two r e g i o n s E x . 2 + 1 as 7 T 3 32 . The sum o f two f r a c t i o n s as combina t ion o f f r a c t i o n a l p a r t s o f a c o l l e c t i o n E x . 2 t 1 as 7 T (Mote : the c o l l e c t i o n c o n s i s t s o f 20 d o t s ) 2 T 3 3 . The sum o f two f r a c t i o n s on the number 1 ine E x . 2 7 3 a s : , 34 . The sum o f f r a c t i o n s as the sum o f two q u o t i e n t s Ex.. 2 + 3 as (2 • 31 + C3 • 4) 7 T 35 . The sum o f two f r a c t i o n s as the sum o f two d e c i m a l s . E x . 3 + 2 T 7 0.75 + 0 .40 1.15 3 36 . The sum o f two f r a c t i o n s u s i n g f r a c t i o n s as repea ted a d d i t i o n o f the u n i t f r a c t i o n s Ex . 2 + 4 7 7 l 7 ? ( 7 5 7 ' ? " l + l + l + l + l + l 7 7 7 7 7 7 » 6 7 37 . The sum o f two f r a c t i o n s as a c o m b i n a t i o n o f two measurements E x . 2 + 1 as 7 7 i*9 -3" SO 38 . The sum o f two f r a c t i o n s as j o i n i n g two segments E x . Z + 3 as 3 4 <*6 X. SI S i n c e 2 * 3 = 8 + 12 And 3 • 4 » 9 * 12 C8 • 121 + (9 + 121 * (3 + 9) * 12, = 17 * 12 - 8 - 93 Procedures for Adding Fractions Thz pnjoczdun.Zi> fan adding finactionA givzn bzlow may be included in youn instructional pnognam. CHECK tkz nzbponbz which. dzAcnibzA thz tnzatmznt zach topic in youn cla6&. RESPONSE COVES: 7 . Emphasizzd [u&zd ai a pnimany pnoczdunz, nz^znnzd to extensively on. £n.equently) 2. tided by not emphaiized 3. Mot uied 39. Using the least common denominator in a horizontal format 42. 40. 4 + 1 9 6 4 x 2 + 1 x 3 9 I 6" T 8 + 3 18 18 11 Using the least common denominator in a vertical format 4 = 8 9 IS" + 1 = A 6 18 TT T8" 1 2 3 41. Using the "formula" i + "BT a + c = ad + be b I i + 1 9 6 (4 x 6) + (1 x 9) TT~6 = 24 + 9 54 33 54 11 T8" Using any common denominator in a horizontal format 4_ + j_=4x6_ + ]_x9_ 9 6 9 6 6 9 = 2 4 + 9 = 33 54 =" Ii 18 52 55 43. Using any common denominator in a vertical format 4 = 24 9 54 1 = 9 + 6 54 33 54 l =u 18 53 56 44. Using decimals 1 + 5 = 0.2 + 0.625 = 0.825 .= 825 • 54 - 9 - 94 Techniques for Adding Fractions 45. Which one of the following best describes the technique you used in teaching the addition of fractions? CHECK tkz ewe mo&t ap--pfiopnlajtz itiponAz. a) I presented only numerical examples demonstrating the procedure(s). Ex. 3 = 21_ 4 28 + 4=16_ i 7 28 37 = ,9_ 2"8~ '28 b) I f irst presented the procedure symbolically and then illustrated it with numerical examples. Ex. Symbolically: a_ + £ = ad + cb b d bd Numerically: 3 + 4 = 3 x 7 + 4 x 5 f 7 TT7 = 21 x 20 35 = 4J_ 35 c) I f irst used numerical examples and then presented the procedure symbolically. 3 58-- 10 - 95 Finding Common Denominators The pK.oce.duLn.es!> fan finding common denominators given below may be included in youn instructional program. CHECK the response which describes'the treatment of each topic in youn class. RESPONSE COVES: I. Emphasized {used as a pnimany explanation, referred to extensively or frequently) I. Used but not emphasized 3. Not used 46. 47. Using the product of the denominators Ex. To find a common denominator of 1 and 1, find the product 6 8" of 6 and 8. Using common multiples of the denominators Ex. To find, a common denominator of 1_ and 1_, 1 ist the 6 8 multiples of 6 and the multiples of 8, and find common multiples 49. Multiples of 6 Mulitples of 8 {6, 12, 18, 24,, {8, 16, 24, 32,, 3 ' 50. 48. Using prime factorization Ex. To find a common denominator of 1 and 1, factor each 6 8" denominator and take the product of the prime factors of each denominator. Listing the multiples of the greater of the two denominators until one that is divisible by the other denominator is found. Ex. 1_ and 1 6 8 8, 6 does not divide 8 16, 6 does not divide 16 24, 6 divides 24 Hence, 24 is L.C.D. 1 2 3 Using the trial multiples of the denominators until equal products are obtained. Ex. 1_ and J_ 6 8 59 62 18 and 2 24 and 3 x 8 x 8 16 6 0 24 63 6 = 2 x 3 and 8 = 2 x 2 x 2 Take 2 x 2 x 2 x 3 . This product contains the prime factors of 8 and those of 6. 1 2 3 61 Decimals The inteApxetations given below may be included in yowi instAuctional pxogxam. CHECK the xesponse code which descxibes tiiflAea.tme.nt oi each topic in youx clai6. RESPOUSE cooes: /. Emphasized [used ai a pximaxy explanation, xeiexxed to ex-tensively ox ixeauently) 2. Used bat not emphasized 3. Not used 51. A dec imal as the coo rd ina t e o f a p o i n t on the number l i n e . .28 bias this intexpxetation in the student A ' text? Yes No .28 < 52. A dec imal as another way o f w r i t i n g a f r a c t i o n . 0.17 = 17 TuU 0 . 8 = 8 TO" 53. A dec imal as p a r t o f a r e g i o n . Yes No 0. 38 0.7 Yes No 1 2 3 1^ you EHPHAS11EV the given inteipfie.tati.on U.e, ii you checked 1 in the iixst. column) WRITE the numbexs oi the Aesponse codeJS which show the two pximaxy xeasons ioi its emphasis. RESPOUSE COVES: 1. Hell known to me 2. In B . C . on local cuxxiculum guide 3. Easy ioA students to undeAstand 4. Enjoyed by students 5. Related to math oi pxiox glades 6. Useiut in math oi latex gxadiu 1. 1 M I 4 taught it was appxopxiate t. Emphasized in student text 1st Pr imary Reason 2nd P r i m a r y Reason 1^ you P IP NOT USE the given intexpxeta-ti n, li.e, H you checked 3 in the iiAst column) WRITE the nimbeAS pi the xesponse codes which show the two pximaxy xeasons iox not using it, RESPONSE COPESi 1. Not well known to me 2. Not in B . C . o i local cuxxiculum guide i. HaAd iox students 4. Qisliked by students 5. Wot xetated to math oi pxiox gxades 6. Not use in I in math oi latex gxades 7. I tttu taught, it aus inappxopxiate t. Not emphasized in student text 1s t P r imary Reason 2nd P r imary Reason • • • • — • • • • • 19 C@ • • • • 2025 C3> The interpretatiom given below may be included in your initructional program. CHECK the reipome code wliich deicribei the treatment oi each topic in your clan. RESPONSE COPES: 1. Emphaiized [uied ai a primary explanation, referred to ex-teniively or irequently) 2. Uied but not emphaiized 3. Not uied 54. A decimal as an e x t e n s i o n o f p l a c e v a l u e . Wai tliii interpretation in the itudenti' text? li you EUPHAS11EP the given interpretation I t . e . it you checked 1 in the iint column) WRITE the numbeM oi the reipome eodei wliich ihow the two primary reaioiu ior iti emphaiij>. RESPONSE COPES: I. 2. 3. 4. 5. 6. 1. Well known to me In B.C. or local curriculum guide Eaaj ior itudenti to undentand Enjoyed by itudenti Related to math oi prior gradei Uieiul in math «i later gradei I wai taught it wai appropriate Emphaiized in itudent text li you PIP NOT USE the given interpreta-tion li.e. ii you che.cked 3 in the iin.it column) WRITE the. numbeAi oi the nesponAe codei which ihow the. two primary reaioni ior not uiing it.. RESPONSE COPES: 1. Not welt known to me 2. Not in B.C. or local curriculum guide 3. Hard lor itudenti 4. Viitiked by itudenti 5. Not related to math oi prior gradei 6. Not uieiul in math ofr later grade.i 7. 1 wai taught it wai inappropriate i. Not emphaiized in itudent. text. 1st Pr imary Reason 2nd P r imary Reason 1s t P r imary Reason 2nd P r imary Reason Yes No 55. A dec imal as a s e r i e s 0 .243 = 2 + 4 + 3 ITT TOO" TUDfJ Yes No • • • • 26-31 ro I • • • • 32- 37 56. A dec imal as a comparison 11 i 1111 i r n u n i t rod rxrrrxj 0 - 6 Yes No 0 . 4 5 • • n • - 13 - 98 Operations with Decimals The tz.chvu.qu.eM for teaching operations with decimals given below may be indluded in youA instructional program. CHECK the response which describes the treatment of each topic in your class. RESPONSE COVES: 1. Emphasized {used as a primary explanation, referred to extensively or frequently). 2. Used but not emphasized 3. Not used 57. 58. Related operations with decimals to operations with fractions. Ex. 0.7 x 0.6 = But 0.7 = 7 and 0.6 = 6 TO TO So 0.7 x 0.6= _7 x _6 10 10 = 42 100 Therefore 0.7 x 0.6 = 0.42 1 2 3 Related operations with decimals to operations with whole numbers, teaching rules for placing the decimal point. Ex. 1.38 x 5.2 = Since 138 x 52 276 690 59. Used concrete materials to illustrate operations with decimals. Ex. 3.47 + 2.13 = Using rods I demonstrated that 3.47 m and 2.13 m makes 5.60 m 1 2 3 44 46 7176 1.38 x 5.2 = 7.176 Iplaces 1 place 3 places l 2 3 45 - 14 - 99 TIME ALLOCATIONS 60. What was the average length (in minutes) of each class period? | ' [ "| i+7-»+8 Fractions 61. How many total class periods did you spend on teaching fractions? (Combine partial lessons when necessary.) [ | | 49-so Indizcutz tkz numbzn. o£ claAi pzhXadh &pznt on zack o£ thz bottoming actlvitizA [incut u>, dzmon4tsia£Lon&, zxplancutioni, AtudzntA doing computational ZX.ZA.CJJ> z6, uiing manipuZativzA, ztc. ) uiitk youn. clat>A. Round youn. OYIAWZA. to thz nzanzt>t whoZz numbzn.. 62. Activities related to developing the concept of fraction | | I 5.1-52 63. Activities related to finding equivalent fractions including reducing fractions | | | 53-51+ 64. Activities related to adding and subtracting fractions including finding common denominators | | | 55-56 65. Activities related to multiplying fractions , , 1 | [ 57-58 66. Activities related to dividing fractions ). | " | 59-60 67. Activities related to ordering fractions | | | 61-62 68. Problem-solving activities related to fractions * (textbook word probems, problems arising from real l i fe situations, recreational problems, _ _ _ _ challenging problems, etc.) | [ | 63-6"+ NOTE: THE SUM OF THE PERIODS GIVEN FOR ITEMS 62 TO 68 SHOULD NOT EXCEED THE NUMBER GIVEN FOR ITEM 61. - 15 - 100 Decimals 69. How many total class periods did you spend on teaching decimals? (Combine partial lessons when necessary.) 1 | 6 5 - 6 6 Indicate, the. number of class periods spent on tack of the following activities [that is, demonstrations, explanations, students doing computational exercises, using manipulatives, etc.) with your class. Round youn. answer to the nearest whole number. 70. Activities related to developing the concept of decimal.. 1 1 6 7 - 6 8 71. Activities related to converting decimals to fractions or vice versa I 1 69-70 72. Activities related to adding and subtracting decimals.... 1 I 71-72 73. Activities related to multiplying decimals I I 7 3 - 7 4 74. Activities related to dividing decimals | 1 7 5 - 7 6 75. Activities related to ordering decimals | 1 7 7 - 7 8 76. Problem solving activities related to decimals (textbook word problems, problems arising from real l i fe situations, recreational problems, challenging problems, etc.) | | 7 9 - 8 0 NOTE: THE SUM OF THE PERIODS GIVEN FOR ITEMS 70 - 76 SHOULD NOT EXCEED THE NUMBER GIVEN FOR ITEM 69. - 16 - 101 19 cGD OPINIONS Indicate the. extent to which you agncz on dLUagmz with zach o/j the {ottowing AtcutcmcntA $on youn clcu>A. CIRCLE the. choice, which bej>t dc6cni.bej> youA faceLing*. 77. Computation with common fractions should be taught. Strongly1 Disagree2 Undecided3 Agree1* Strongly5 Disagree Agree 2 0 78. The degree to which the students are skilled at computing is an indicator of their understanding of fractions and/or decimals. Strongly1 Disagree2 Undecided3 Agree4 Strongly5 Disagree Agree 2 1 79. Computations with common fractions should be delayed until students are at least 12-13 years of age. Strongly1 Disagree2 Undecided3 Agree4 Strongly5 Disagree Agree 2 2 80. Computation with decimals and common fractions should be done with hand-held calculators. Strongly1 Disagree2 Undecided3 Agree4 Strongly5 Disagree Agree 2 3 81 Only common fractions with small denominators should be taught (e.g., 1/2, 1/3, etc.) . Strongly1 Disagree2 Undecided3 Agree4 Strongly5 Disagree Agree 82. It is important to dri l l on computation with common fractions and decimals until students are very good at computing. Strongly1 Disagree2 Undecided3 Agree4 Strongly5 Disagree Agree 2 5 83. Rules for operations with common fractions and decimals should be memorized. Strongly1 Disagree2 Undecided3 Agree4 Strongly5 Disagree Agree 2 6 84. Emphasis should be placed on teaching applications involving common fractions and decimals. Strongly1 Disagree2 Undecided3 Agree Strongly5 Disagree Agree 2 7 85. Problem solving activities and applications with common fractions and decimals should be emphasized more than computations with fractions and decimals. Strongly1 Disagree2 Undecided3 Agree Strongly5 Disagree " Agree 2 8 2<t - 17 -102 86. In teaching common fractions it is important that structural properties (distributivity, associativity, commutativity, identity, inverse elements) be emphasized. Strongly1 Disagree2 Undecided3 Agree*4 Strongly5 Disagree Agree 87. Estimation, approximation, and checking the reasonableness of an answer are more important than becoming skilled in computing with common fractions and decimals. Strongly1 Disagree2 Undecided3 Agree4 Strongly5 Disagree Agree 88. Decimals and their operations should be emphasized more than common fractions and their operations. Strongly1 Disagree2 Undecided3 Agree4 Strongly5 Disagree Agree 89. Mental calculation should be emphasized with common fractions and decimals. Strongly1 Disagree2 Undecided3 Agree4 Strongly5 Disagree Agree 90. Instruction on common fractions should precede instruction on decimals. Strongly1 Disagree2 Undecided3 Agree4 Strongly5 Disagree Agree 91. Instruction on addition of common fractions (like and unlike denominators)should precede instruction on multiplication of fractions. Strongly1 Disagree2 Undecided3 Agree4 Strongly5 Disagree Agree 92. It is important for students to know how to find the least common multiple of two whole numbers. Strongly1 Disagree2 Undecided3 Agree4 Strongly5 Disagree Agree 93. It is important for students to know how to find the greatest common factor of two whole numbers. Strongly^ Disagree2 Undecided3 Agree4 Strongly5 Disagree Agree 94. When reducing fractions, students should first find the greatest common factor (GCF). of the numerator and denominator and then divide the numerator and the denominator by the GCF. Strongly1 Disagree2 Undecided3 Agree4 Strongly5 Disagree Agree APPENDIX B R a t i o , P r o p o r t i o n and P e r c e n t Q u e s t i o n n a i 104 INTERNATIONAL ASSOCIATION for the EVALUATION of EDUCATIONAL ACHIEVEMENT — S E C O N D — Study of MATHEMATICS GRADE 8 TOPIC SPECIFIC QUESTIONNAIRE RATIO, PROPORTION AND PERCENT ( Booklet II L ) FOR NATIONAL CENTRE USE ONLY PROVINCE OF BRITISH COLUMBIA MINISTRY OF EDUCATION DIVISION OF PUBLIC INSTRUCTION LEARNING ASSESSMENT BRANCH Check here i f none of ratio, proportion, or percent are included in your program. In that case, disregard the rest of the questionnaire and return i t to B.C. Research in. the" envelope provided. CHECK tke response wkick best describes tkz use you madz of eack of tkz following materials in your i n s t r u c t i o n on ratio, proportion, or percent. RESPONSE COVES: 1. primary source, used frequently I. secondary source, used occasionally 3. not used or rarely used 1. School Mathematics II (Addison-Wesley) 2. Mathematics IT (Ginn) 3. Essentials of Mathematics II (Ginn) 4. Other published text materials (e.g., textbooks, workbooks, and worksheets) 5. Locally produced text materials (e.g., textbooks, workbooks, or worksheets) 6. Commercially or locally produced in-dividualized materials (e.g., program-med instruction or computer assisted instruction) 7. Commercially or locally produced films, film-strips, or teacher demonstration models. 8. Commercially or locally produced lab-oratory materials for student use (e.g., games or manipulatives) - 3 - 106 TEACHING TOPICS Thz topic* given bzlow may be included in youA instructional pnagKam. CHECK thz AZipoMz which. dz&cAibz* thz tsizatment oi zach topic in yoan. CIOAA. RESPONSE COVES: J. taught at new contznt nzvizwzd and thzn zxtzndzd A,&v<iwzd only 2. 3. 4. 5. a&Aumzd a& pfiznzquiAitz knowlzdgz and nzitkzn. taught noA. xzvizwzd not taught and not a&Aumzd ai, ptiz-AzquiAitz knowledge 9. The concept of ratio" 10. The concept of proportion 11. Solving proportional equations 12. The concept of percent 13. Computing percents: Find a percent of a given number or find what percent one number is of another 14. Changing percents to common fractions 15. Changing percents to decimal fractions 16. Changing common fractions to percents 17. Changing decimal fractions to percents 18. Percents greater than 100% 19. Percents less"than 1% .... 2 2 29 30 31 32 33 3h 35 36 37 38 39 Tkz interpretations below may bz included in your instructional program. CHECK tkz response wkick describes thz trzatmznt of zack topic In your class. RESPONSE COVES: 1. Emphasized [used as a primary zxplanation, referred to extensively or frequently). 2. Uszd, but not extensively. 3. Hot uszd. Ratio 20. Ratio as a rate Exs: i) 13 km in one hour i i ) 72 heartbeats per minute 21. Ratio as comparison Exs: i) One part cleaner to 10 parts water i i ) Three pencils per student 22. Ratio as a fraction Ex: 3:5 means 3/5 (three fifths) 23. Ratio as a quotient of two whole numbers Ex: 3:5 means 3 * 5 1 2 3 Percent 24. Percent as a fraction ( i .e . , a synonym for hundredths) Ex: 83% means 83/1Q0 or 0.83 1 2 3 25. Percent as a ratio with a second term of 100 Ex: 832 means 83:100 - b - 108 a l 3 3 •s as a. « a i * » V a l S i 5 a l a. s a i w > a v* a l • a l o - = « - c a l 6 3 a ) L U V 3 — S O S i s 3 3 >^*5 S 3 ^  O a l 5 - -5 i i ft* «o « I ? : * ft) "<c ft! 5* a l e -a "«o>3 = a s a l a. a l 5 - s - « 3 a i tt u a l . SUA 1-3 S > a l 3 a l * c a "5 <5 3 " 3 -.5 ft) <T. * r O a l O K " * * c n C X * ' y a . a a a «J «• s ^* a a i « a a t r O < 3 3 g e 4 t i j " 4 s £ 3 E 5 __ o " 3 < *r s 3 e> v<s «f t" 1 c i - * cj a « - « c - « i a « s t o V V « * W < « V co U J - • i t 3 S - a l 3. < e n * N a ca-. « a. -^ *5 5 '«3 3 o 1 a -5 S i § <w a l 3 = 0 © 9 O t J * "° 3 B si s.1 * ••<*.. & y ^ * a s u a a a l pa a / V t o a i U l N *4 2 3 "2 v I o. • • • • • • • • • • • o • • • • I S cr QJ 1_ VI o o O * J i- 10 o. u -15 L a g ai ut a. w <u at M £ a +4 a vi O) IA*I .2 — «3 * J a * a t e o a i « 3 cr 01 VI VI v . i a vi 3 a-. 01 O 1/1 -^« n X C M N | e o m X e l c n X CM O l | t O CM X r> CM X CM <o(xr <"5|«M m r M W W o» • o a §•£ . U O X a. u uj a 4 a. < es CM 3 01 VI c •• O VI 4 J * J e U 01 o ~ a.«-> o o U 3 c w e r 01 * J o • c e 01 ^ 01 — -o u 5— I/) c <o *c» m-to VI M I O I S U " 3 r» • c ~ o" •- • O 01 _ 01 cno - o II * J VI 01 - o n o u c CM 3 — 1 0 « » -^ cr^  « CM W * * fll 3 » --C cr •-r i n o i * ' 01 o - & - 109 Procedures for Solving Proportions Thz pnjDC.zduA.oj> 'fan solving pnaponjtions givzn bzlow may bz included in youn. in&tnjucjtlonal pnognam. CHECK thz nzspomz which dzAcnibzs thz treatment o£ zach topi.c i n youn class. RESPONSE COVES: J . Emphasized (used as a pnJjnany pnocedune, nz{znxzd ta extensively on, ^Azquzntly) 2. Used, bat not zmphasizzd. 3. Not used. 30. Finding the cross products, and then solving the resulting equation. Ex: Given 3 3 6 T4" x 3 • x * 14 • 6 or 3 • x = 84, etc. 1 2 __3 • 70 37. Using multiplication or division to equate numerators and denominators. Ex: Given 3 = 6 TT x 3'Z = 6_ or 6 = 6_ T F ? x IB" x Since the numerators are equal and the ratios are equivalent, the denominators must be equal. Hence x = 28. 1 2 3 71 32. Dividing the terms of one ratio and then solving the resulting equation. Ex: Given x = 17 9 4 x••• 4.25, so x = 9 x 4.25, or 38.25 • 1 2 3 1 1 0 Methods for Solving Problems Involving Proportions Several methods of solving problems involving proportions are Listed below. CHECK the response which describes the treatment of each topic in your class. RESPONSE COVES: 1. Emphasized [used as a primary method, Kefenred to extensively or frequently) 2. Used, but not emphasized 3. Not used EXAMPLE PROBLEM: Three neckties cost $20.00. How much do 12 neckties cost? 33. Solved i t using proportional reasoning without an equation. For example: 12 neckties are four times as much as three neckties, so they would cost four times as much, or $80.00. 34. Solved i t using a proportional equation For example 3 = 12 where x is the cost of 12 neckties. 2o~ x 73 Solve for x. 1 2 35. Solved i t using the unit method without an equation. For example: one necktie cost $20/3 or $6.67, therefore, 12 neckties cost 12>($6.67)=$80.00. 7h 75 - 8 -111 Techniques for Teaching Solving Proportional Equations Thz following statements describe, technlquzs a tza.ck.2X might use when teaching a pnocedune fon solving pnopontlonal equations. CHECK thz nesponse which descnlbes thz tnzatmznt of zach topic In youn. class. RESPONSE COVES: 7. Emphaslzzd [uszd as a pnlsnany pnoczdunz, nefenxed to extensively on. frequently) 1. Uszd, bat not zmphaslzzd 3. Not uszd 36. I presented only numerical examples demonstrating the procedure(s). Ex: 3 = 6 5 n 37. I f i rs t used numerical examples and then presented the procedure symbolically ( i .e . , the general case). Ex: Numerically Symbolically 3 » 6_ a_ = c t n b n 76 77 38. I f i rst presented the procedure symbolically ( i .e . , the general case) and then illustrated i t with numerical examples). Ex: Symbolically Numerically a_ = c_ 3 = 6_ b n ST n 78 - 9 -112 Applications and Problems Several applications of ratio and pn.opzAJU.zA one. listed below. CHECK the response which, describes the treatment of each topic in your class, RESPONSE COVES: 1. Emphasized [used frequently) 2. Used, but not emphasized 3. Not used 39. Scale Models (airplanes, automobiles) 1 2 3 40. Finding distances from maps 1 2 3 41. Scale drawings 1 2 42. Calculating the size of a population from a sample estimate 1 2 3 43. Problems involving buying decisions based on cost rates Ex: Pay $1.00 for 3 items or 35<t for each? 1 2 3 44. Mixture or recipe problems 1 2 3 45. Real world problems using similar triangles Ex: A 12 foot tree casts a shadow of 4 feet. A building has a shadow of 25 feet. How tall is the building? 79 80 19 C. fJJ 20 21 22 23 2i+ - 10 Applications and Problems Several applications of percents are. listed, below. CHECK the response which describes the treatment of each topic In your class. RESPONSE COVES: 1. Emphasized (used frequently) 2. Used, but not emphasized 3. Not used 46. Commission 1 2 3 25 48. General Word Problems Ex: John bought 25 toys. 40% were defective. How many were defective? 113 47. Discount 1 2 3 26 27 49. Simple or compound interest 1 2 3 28 50. Percent of increase or decrease 1. 2 3 29 51. Circle or bar graphs 1 2 3 30 - 11 -1 1 4 Sources of Applications and Problems Several source* of apptLc.aXJ.OYii of ratio, proportion, and percent are l i s t e d below. CHECK the response which describes the treatment of each topic In your class. RESPONSE COVES: 1. uied frequently I. uied occasionally 3. not used at a l l 52. Students' textbooks 1 2 3 31 53. Supplementary textbooks or workbooks 1. 2 3 32 54. Worksheets or exercises designed by myself or local teachers 1 2 3 33 55. The B.C. curriculum guide or a local guide 1 2 3 34 56. Articles or papers published by professional assocations 1 2 3 3 5 57. Applications or problems suggested by my students 1 2 3 36 Methods o f S o l v i n g P e r c e n t Problems four methodi oi iolvlng percent problem* are luted below ion each o{ three type* oi percent probtemt, CHECK the reipome which deicKtbei the treatment oi each topic in your clan. RESPONSE CODES: I. Emphaiized (tued ai primary procedure, ior thii type, oi problem] I, Taught, but not ai a primary procedure, ior thii type, oi problem 3. Not taught Type I : G i v e n the base and p e r c e n t f i n d the p e r c e n t a g e : E X : Sa ra bought a new d r e s s p r i c e d a t $150 .00 . The s a l e s tax was 3 t o f the p r i c e . What was the s a l e s t ax? 58. The e q u a t i o n method: E x : 0 .03 x 150 = x S o l v e f o r x . 59 . The p r o p o r t i o n method: E x : L e t x be the s a l e s t a x . Then: x 150 3_ 00 1 S o l v e f o r x . 2 3 60 . The a r i t h m e t i c method: Ex M u l t i p l y the p e r c e n t ( I n dec ima l o r f r a c t i o n a l form) t imes the base t o ge t the p e r c e n t a g e , u s i n g o n l y a r i t h m e t i c . 150 x . 0 3 6 1 . The u n i t method: E x : 3X o f $1 I s $ ^ J J 3X o f $150 I s t x 150 Type II: G i v e n the base and percen tage f i n d the pe r cen t : E X : The Mathemat ics C l u b has 40 members. Twen ty -e igh t o f the members were a t a meet ing t o e l e c t o f f i c e r s . What pe rcen t o f the members a t t ended the meet ing? 62 . The e q u a t i o n method: Ex 100 (28 • 40) = x S o l v e f o r X . 6 3 . The p r o p o r t i o n method: E x : 28 40 x TOO" Then s o l v e f o r x . 2 3 64. The a r i t h m e t i c method: E x ; D i v i d e the base I n t o the percen tage and m u l t i p l y by 100 t o f i n d the pe rcen t u s i n g o n l y a r i t h m e t i c . 40 A t x 10*0 65 . The u n i t method: E x : 1 member Is 1 o f 100S TO 28 members 1s 28 o f 100* M Type III: E X : 6 6 . G i v e n p e r c e n t and p e r c e n t a g e , f i n d the base : On a c e r t a i n s c h o o l d a y , t h e r e were 30 s t u d e n t s a b s e n t . That was 51 o f the t o t a l . How many s t u d e n t s were t he r e? The e q u a t i o n method: E x : 0 . 0 5 x • 30 S o l v e f o r x l 6 7 . The p r o p o r t i o n method: E x : 30 x 5 100 I S o l v e f o r x . 2 6 8 . The a r i t h m e t i c method: Ex D i v i d e the p e r c e n t ( I n dec ima l o r f r a c t i o n a l form) I n t o the pe rcen tage t o ge t the base . i .05 2 7 3 0 6 9 . The u n i t method E x : 5 s t u d e n t s a r e St o f 100 s t u d e n t s . 1 s t u d e n t i s 6* o f 100 o r 20 s t u d e n t s . T T h e r e f o r e , 30 s t u d e n t s a r e 5X o f 30 x 20 s t u d e n t s . TIME ALLOCATIONS 70. What was the average length (in minutes) of each class period? | 1 | minutes 71. How many total class periods did you spend on teaching ratio, proportion, and percent? , — t — , (Combine partial lessons when necessary.) I I I periods Indicate, the. number of class periods spent on each of the following activities (that Is, demonstrations, explanations, students doing, computational exercises, using manipulative^, etc.) with your class. Round your answers to the nearest whole number. 72. Activities related to developing the concept of ratio. I I 1 73. Activities related to developing the concept of proportion. I I I 74. Activities related to solving proportional . . . equations ••• . . . . . . . |_J [ 75. Applications/problem solving activities related to ratio and proportion — (textbook word problems, problems arising from real l i fe situations, recreational problems, challenging problems,, etc.) I I I 76. Activities related to developing the concept of percent... I I I 77. Activities related to computing with percents.... I I I 78. Activities related to changing percents to r—r—i common fractions I I I 79. Activities related to changing percents to decimal fractions I I I 80. Activities related to changing common fractions to percents I I I 81. Activities related to changing decimal . fractions to percents I I I 82. Applications/problem solving activities related to percents—(textbook word problems, problems arising from real l i fe situations, recreational problems, challenging problems, etc. l I I I NOTE: THE SUM OF THE PERIODS GIVEN FOR ITEMS 72 TO 82 SHOULD NOT EXCEED THE NUMBER GIVEN FOR ITEM 71. - 44 117 OPINIONS Indicate the extent to which you agree or disagree with, each of the following statements r e l a t i v e to your, class. CIRCLE the choice which best describes your feelings. 83. The study of percent should be related to the study of proportion. Strongly1 Disagree2 Undecided3 Agree1* Strongly5 Disagree Agree 84. The study of Dercent should precede the study of ratio and proportion. Strongly1 Disagree2 Undecided3 Agree1* Strongly5 Disagree Agree 85. The study of proportion should be delayed until the students learn how to solve linear equations. Strongly1 Disagree2 Undecided3 Agreed Strongly5 Disagree Agree 86. Students should be taught to identify each of the three types of percent problems before solving them. Strongly1 -Disagree2 Undecided3 Agree1* Strongly5 Disagree Agree 87. The study of proportion should be delayed beyond this grade level. Strongly1 Disagree2 Undecided3 Agree1* Strongly5 Disagree Agree 88. The students should init ial ly learn how to solve proportional problems using arithmetical methods (without setting up proportional equations), Strongly1 Disagree2 Undecided3 Agree1* Strongly5 Disagree Agree 89. Students should be given a specific procedure for each type of per-cent problem. Strongly1 Disagree2 Undecided3 Agree1* Strongly5 Disagree Agree 90. The degree to which the students are skilled at computing when solving proportions is an indicator of their understanding of proportions. Strongly1 Disagree2 Undecided3 Agree1* Strongly5 Disagree • Agree 75 76 7 7 78 79 80 21 - 15 -118 Strongly1 Disagree2 Undecided3 Agree4 Strongly5 Disagree Agree 22 91. Computation with percent should be done with hand-held calculators. Strongly1 Disagree2 Undecided3 Agree4 Strongly5 Disaqree Agree 92. Applications of proportion should be emphasized more than solving proportional equations. Strongly1 Disagree2 Undecided3 Agree4 Strongly5 Disagree Agree 93. Applications involving consumer arithmetic (discount, interest, etc.) should be emphasized when students study percent. Strongly1 Disagree2 Undecided3 Agree4 Strongly5 Disagree Agree 2 4 94. Ratio should be taught as fractions or quotients rather than as rates or comparisons of collections. 23 25 APPENDIX C Alg e b r a Q u e s t i o n n a i r e 120 INTERNATIONAL ASSOCIATION for the EVALUATION of EDUCATIONAL ACHIEVEMENT — S E C O N D — Study of •MATHEMATICS G R A D E 8 T O P I C S P E C I F I C Q U E S T I O N N A I R E A L G E B R A ( F O R M U L A S A N D E Q U A T I O N S ) A N D I N T E G E R S ( Booklets I4L and I5L Combined) FOR NATIONAL CENTRE USE ONLY PROVINCE OF BRITISH COLUMBIA MINISTRY OF EDUCATION DIVISION OF PUBLIC INSTRUCTION LEARNING ASSESSMENT BRANCH - 2 - 121 Check here i f none of integers (positive and negative whole numbers), formulae or equations are included in your program. In that case, disregard the remainder of the questionnaire and return i t to B.C. Research in the envelope provided. 1 CHECK the response which, best describes the use you made of each of the following materials In your I n s t r u c t i o n on Integers, formulae and equations. RESPONSE COVES: 1. primary source, used frequently 2. secondary source, used occasionally 3. not used or rarely used 1. School Mathematics II (Addison-Wesley) 2. Mathematics II (Ginn) 3. Essentials of Mathematics II (Ginn) 4. Other published text materials (e.g., textbooks, workbooks, and worksheets) 5. Locally produced text materials (e.g., textbooks, workbooks, or worksheets) 6. Commercially or locally produced individualized materials (e.g., programmed instruction or computer assisted instruction) 7. Commercially or locally produced films, filmstrips, or teacher demonstration models 21 22 23 2k 25 26 27 8. Commercially or locally produced laboratory materials for student use (e.g., games or manipulatives) 28 - 3 - 122 TEACHING TOPICS The topics given below may bz included In youn. I n s t r u c t i o n a l pnognam. CHECK thz nzsponsz codz which dzscnibzs thz tnzatmznt of zach topic in youn class. RESPONSE COVES: /. taught as new content 1. reviewed and then extended * 3. reviewed only 4. assumed as p r e r e q u i s i t e knowledge and neither taught nor reviewed 5. not taught and not assumed as pre-r e q u i s i t e knowledge Integers 9. The concept of positive and negative integers. 10. Addition of integers (+ and -) . 11. Subtraction of integers (+ and -) . 12. Multiplication of integers (+ and -) . 13. Division of integers (+ and -) . 14. Structural properties of the set of integers (e.g., commutativity, associativity, distributivity, etc.) 15. Order relations in the set of integers. Formulas and Equations 16. Evaluation of formulas for given values of the variables. EX: Given A = L x W. If L = 4 and W = 5, substitute for L and W and find the value of A. 17. Deriving formulas or equations. EX: Each weight stretches a spring 3 cm. What formula gives the stretch (total) for n weights? 18. Solving literal equations. EX: Solve Y = 2x + r for r. z 19. Solving linear equations. EX: Solve 4x - 3 = 19 1 2 3 <• 5 29 1 2 3 it 5 30 1 2 3 1+ 5 31 1 2 3 4 5 32 1 2 3 1+ 5 33 1 2 3 1+ 5 3i+ 1 2 3 1+ 5 35 1 2 3 4 5 36 1 2 3 4 5 .37 1 2 3 1+ 5 38 1 2 3 1 + 5 4 - 123 5 o a -s 2 a <y <y-a -w «i-s-= •» o -s s V -B _ a -a « a a v o a 5* i l "34 ^ M al a - a u * a . si 1-3 3 3'-3 ^ •» V S al "3-1 *m23 •3 3 •3 31*' i * "3**f 3 -» CJ 3 = 3 3 3 £ 2 -3 e 15 vi • • • • • • • • • • • im VI Ol 3 3 e a ~ * 1 c U « — e <u a w u S <U > <= 2 J£^ Sw*- VI _ w vi .a c ^ -a a "o — a e ^ — 3 •a a oi t j 4i * * *«» C C * - C >— U — 41 O b CI o -o u C C 41 < 3 " 4 I - 0 -V* W im * C X O n * * a a* _ : ui e c O — a 3 ut C L CT ^ U * 41 41 41 41 4 ai o* e S O C 41 C J3 * 41 41 U — 41 « — — 41 U 41 41 41 <a 41 O l * * d 41 C 3 <— n wl a: —. tu :S1 > o e *•* CM — <J <—» i Ol v* 4) VI S * * 4> > — c -o c -VI 41 4J X 3 a — o s ui 'T s T Wl > !r3 The intexpxetations oi integtxs given below may be included in youn instxuc-tional pxogxan. CHECK the xesponse code which describes the treatment oi each topic in youx class. RESPONSE COPES! I . Emphasized {used as a pximaxy explanation, xeiexxed to extensively on frequently] I. Used but not emphasized 3. Not used tlon [i.e. ii you c/ieeEed I in the iixst column) WRITE the mwhexs the xesponse codes wliich show the ttoo pximaxy xeasons iox its emphasis. RESPONSE COVES: Was this inteApxetation in the students' textt Well known to me In B . C . oh local cuMiculum guide Easy ion students to understand Enjoyed by students Related to math oi pxiox gxades Useiul in math oi latex gxades I was taught it was appxopxiate Emphasized in Student text tlon {i.e. ii you cheeked 3 in the iixst cotunui) WRITE the numbexs 0(j the Xesponse codes uiiich show the two px.imaxy xeasons iox not using it. RESPONSE COVES: 1. Not well knoion to me. 2. Not in B . C . ox local cuX-iicutum guide i. Haxd iox students to undexstand 4. Vistiked by students 5. Hot xelated to math oi pxiox gxades 6. Not useiul in math oi latex gnades 7. I MU taught it was inappxopxiate i. Not emphasized in student text 23 . D e f i n i n g I n t e g e r s as e q u l v a - 1s t P r i m a r y Reason 2nd P r i m a r y Reason 1 s t P r i m a r y Reason 2nd_Pr | inary Reason l e n c e c l a s s e s o f whole numbers: I deve loped the In t ege r s as e q u i v a l e n c e c l a s s e s o f y e s l o r d e r e d p a i r s o f whole numbers. No E x : { ( 0 , 2 ) . ( 1 . 3 ) , ( 2 , 4 ) . . . . ) • - "2 o r l ( a . b ) c WXM: b = a + 2}= "2 • • • • 50"63 24. U s i n g examples o f p h y s i c a l s i t u a t i o n s : I deve loped In t ege r s by r e f e r r i n g t o d i f f e r e n t Yes p h y s i c a l s i t u a t i o n s which N o 2 can be d e s c r i b e d w i t h I n t e g e r s . E x : thermometer , e l e v a t i o n , money ( c r e d i t / d e b i t ) , s p o r t s ( s c o r i n g ) , t ime ( b e f o r e / a f t e r ) , e t c . • • • • a i © * a. SI «/ ^ *f J V 9 a * S U a l a i - c S3 -•5 s = o o O - a 3 OS B t ! 3 = 3 S a i 3 a > « j « S a *r a o c < v « J « 3 "a a ) a i * « O * a. a l 3 LU 2 « 0 3 ' I CO * s o u a. q s. s •3 < S a , « o r-a S - w J : s a i g * » 3 s " B B S "* 3 E • w v o N " o O V S a l • o - W * J 3 V I-ai * J CJ --•8 33 u u s to 125 o OS <M • • • a I Q. • • • • • • • • • i / l « 01 e 3 i. >. 01 -O. 3 • e l / l o U • a OI w 01 a i wi 01 3 • a c < VO ca at o* •a p- 01 o 3 U T3 <0 i— e c a u i s o i o i o > — > J= ^ i n . a o > ^ - » i s B a i o i I I T3 3 4 - > c . e £ c c 3 — •!-> 10 3 C O i / > O t - O — • ^ V - «l o c 01 01 —) Cn » - - C vt JZ IO -O — —• ~J -»- I Q V I I O I - — W 44 OJ o c < n M . - I O « r « >, VO f J < 3 fliu i n O — fl * r »— i*. O l f l ' D 0 * 0 c > cj aj -u U O I ^ I O S - O S * - ^ 0 ^ « fl^1. > > 0 . 01 * U 1 * i « J f ->,*J p— ul IO O f £ c u - a > . a . * - 1 0 t - o i i s c o i 1 0 e "3 TJ — O U 01 IO 01 "O 7 a> a o 5-1 - ai % • - c - c ( V j ai . 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Ol U M Ol Ul w u Ul w o ul • M i . *J J-" Ol .3 3 * .3 t . ul 3 <*- 3 W C C 3 — 1/) o Ul Ul (0 3 Ul "O c !. o 41 I k . <n Ol o I S -w i m Ct 01 01 ^- u ••• ie e • ul 01 <a — Ol !_ Ul Ol e •<-> Ol c — > Ol IO -O 3 ID Ul c U l 3 3 Ul O w e ui 41 o n U O *i a . u a Ul > o n ID o 10 e m + Ul u *» Ul s •o c i c Ol o . ai -o o 4) o p—" 3 — £ i > 3 .Ul c Ul w I . Ul u o c u u O u I . ID — •a oi ta 01 + to I S 41 i . * J Ol 41 w Ol u 01 Ol — ui s 41 ul 01 3 ja * J 3 *J • • 3 -C 41 3 c X 3 C X t / l IS M ^ U| UJ U l - a 127 © 2 ai a. © S -o a> « "C V «r a. al -Q u al o —= a a. S ai S s J : « ai *» e1 •2 c ? * ©V* ai © aj "1 It •» TS £ ai ai - S WJJ oiv =2 3 «- r cs, s \3 o 3 P— ^ 3 3 s © . <w -a < »«•.» a © © co o u LU co i CO © © al 0 . 3 * - O B 3 V al al TJ -fl * « »* < © -«»-= « l a | V i r v a al a. •> s fl, -V -V < 9 JB'TS 1 al S Q o al LU a l - S I»I v CO LU fl SL\ 3lOC u 2 ? r i al V S-3 •o © •si > O © o cj 3 2 LU -a v al al -•W fl o u 2 co S3 >• -C «J a lTS Q. Q.+J al <u E X V -o © Lu al al S 2 — CM ' a Si 01 3 <0 C o 3 LO • • • • • • • • • • • VI e o 01 > c — 01 e — x «*- •"-» M o o e «> 1-5 ' u - s * >, oi o 01 u. *J •= O • S — *> 0) 3<»"» «•! C • C >i I 13 O at -*"~ — oi o i -a -w -c •»-» c o » « , >i oi <o i. X + p L aj ui X 4U ^ C S 01 .n c fl o 3 3 " —i <n C 0) o •8** 3 r i si 01 k X E»- LU 01 3 0) O 01 01 3 e e v— jr. c ic - * o u -w — _ S vi we » - L. ul 01 O 01 -O g L j L £ * - i o i l - -IO 3 11 C ^ O s <W 01 —> e coi u o o) — f c ^ — ^ A C ,|U 4U 4U 4U 01 (J U OI VI . £ IO *o « c — fl W 41 W ^ - "D L. •J V I M ' S <w 3 3 - ~ O 3 CO — (ft < ^ ^ » VI o o o C I 0) VI VI o •o J= c -o » • O) * • fl 01 •u -o a) ™ u "O *> 6 -o *J fl fl 0) . 1 0 fl U OI 1 • £Z •u 01 013 X La fl-3 4U •»• VI ^* *> VI VI i IO T3 3 C 3 01 01 E OI CO E OI c •M J= 1 , _ o •* 0) IO L fl L 1 fl *> •fa* ^3 .c U OI «• X ro « T5 OI s 3 c + - i ^» V) OI 10 c c 3 ^ fl OI c 3 01 0J — X LO J3 — £ E LU - 9 - 128 The following statements describe method* by which, a teacher might develop the concept of the product of integers. CHECK the response code to i n d i c a t e the extent to which that method of developing the concept was used with your class. RESPONSE COVES: 7. Emphasized [used as a primary method of - development, referred to extensively or frequently) 2. Used, but not emphasized 3. Not used. 34. Development by use of repeated addition: I developed the concept of multiplication by appealing to repeated addition, e .g. , 4 x "3 = "3 + "3 + "3 + "3 = -12 1 2 3 35. Development by the extension of properties of the whole number system: I developed the concept of multiplication of integers by using the commutative, associative, and distributive properties to justify the products, e .g . , "4 x "3 = But 0 = ("4 + +4) x "3 = ("4 x "3) + (+4 x ~3) = ("4 x "3) + "12 Hence "4 x '3 = +12 36. Development by use of physical situations: I developed the concept of multiplication of integers by appealing to physical situations that might illustrate the product of positive and negative numbers, e .g. , A refrigerator is cooling at a rate of 4° per minute. Its thermometer is at 0°. What wilT be its temperature 4 minutes from now? - 10 - 129 37. Development by use of patterns: I developed the concept of multiplication of integers by appealing to patterns of products, e .g. , U x '3 = '12 T3 x "3 = "9 +2 x '3 = "6 +1 x '3 = "3 0 x "3 = 0 "1 x "2 x 38. No development— students were given rules: I did not develop the facts for multiplica-tion of integers by using any of the above methods. I instead gave them rules similar to the following. If the signs are alike, the answer is positive. If they are different the answer is negative. If one factor is zero, the answer is zero. - i I - 130 I 2 o a. o 55 ° 5 S a i a i - = WJ! 91V S _, t - O O J - S 3 > - C _ a O S Jt W a a > 3 _o -| — • § *•>• 3 5 a a » a t 3 a i • -we * a a a •a 2 C O • t A >o T s . a o _ , i " B - C * « _ S J a l 3 - M * V a o-g'W 8 ^ a o « » 3 » a a -a v s • a-3-a 3 a l d 5 u j « « J o e » a 3 5 0 a i < ? • — t » j a l 3 * 5 s 3 s a l < p 3 » u u K 3 y> UJ a • • • • • • • >» ut -w 1 . ••- 01 •=•§ 3 3 cr e 0 1 S 2 01 II I tn m 9 1 . a. o « — tn 2: LOU~. 01 L U I n r n| *a . 0 . - 0 n — 11 > 3 — t. — 11 is> ut x « 3 x*» -a _ r-.| •—* x 1 •— x ) X U J J E O +J 4 J > > J 3 X in in 1 V I f " » V I Ol e + 0 • 0 • < -0 m — 0 a ai C O V I • V V I I O L . • J = <a u II Ol *J U 0 1 > - — . 0 0 a. 0 m c 1/1 J 3 L . 0 V I ai 11 a. • 0 0 1 u 1 0 1 • — 0 V I J = V I tn X CL Ol L . m w ro -1- L B Ol j= •*J > + -y ^ « J 11 1— 0 1 e •• •a 0 • 3 J = V I x x a J S X S •*-> L . 1 — jro Ol Ol C _Q - 1 vi 5 X = c U J a -12 - 131 a o a s a. a -o>r a* •V 5 § "»"I 4 " O S a ) ai -a 4 « V u - a _ O LU-S 1- •« Ol 3 » a * ° ~« "* S o ai 5 ai ; a j"3 4 * a a a — U u « e <3i s -= 3 S i i 3 3 a ) a j o - 3 al al 3 " 3 * r * 3 3 « a. al a . J a v • • ( i f a 5 a. a •5 a ai 3 2 0 a -a V «W *J s .-> 3 f «"a 1 .-3 •a W -» •«> a -t? £ v - c 3 ** a --5-0 •o V _a ai a>"B •a ' 3 " S1 i-5 , - -a-v ^ 3 * = g 5 a a 3 a a a al - o - o S 3 a a *s V a) ai s * * al ' S - - a v • 3 § al al a-e -a , -v <J a ) . ai-a 2 -o u -= -o . . . _ a. 3 I a ~ -c SlCS U £ 3 al • 3 * a a I O al 3 * J -a a a " 3 -w 3 a S O 31TJ a -a *-> a a ai a • a " B"3 *?-y -a o «o ai ai » J 3 I * J 3 3 n w i u a . 3 » i u 4 | J al 3 a. 3 S a , •a S S V o iS oi s * -O -J 3 al S a i < ai a V 3 •< . a . * u a UJ 3 •B a . al-e -5 ai V a a al a 3 3 al a 5 * a U •a ai ' u a O " 3 * O O 3 •8*3 a a. *J al E al ! Uj al al 3 2 §1 I 1.1 a. • • • • • • • • • • • n to V I — <u c JZ o in *> IA f 9s 01 U 3 -a o — — — c ** ° ^ V I oi to r*» m v» ro oi n e j= oi c- oi v» ^ - oi u * J o oi <o *> 3 vi I -2 < J o •»-> e •» x -•-i in in &. CO L, CO C Ol UO Ol oi -a vi JS II - s * r = i ts c u c Ol 4 J —» J= * J x UO x £ 0 «i ui 3 *a- E o + Ol o C Ol . >4-u •r Ol -3" ro 01 — ' 14— o — e •U J= s 01 b_ II • » II V I 4 J 01 L . o X c — Ol <J Ol "S it) * f*» uo V I « « + e O J 3 U O I O 4-1 Ol 01 J= s 3 C I II - — 01 u uo J* 4 J I S X 0 0 « V I >— ^ rO • uo uo uo 01 I O — > u 1 -~ >l **•• V I Ol J 3 « e • It f~. . + 3 <-l — J= o Ol Ol ce u f j - f *r e C O Ol > X X Ol c « o •a 4 - 1 II • II C H Ol •— o II J 3 S . II o >l L. — "O 01 > Ol X L> 01 X — e "a o e c j = X o X c <•» Ol 01 E U O IO 4-1 u •o > > . ! > • > ia V I X u o O X 1 i • L U H- to •— to L U 1 i • - 13 - 132 TEACHING TECHNIQUES Thz following statements describz tzcM.ru.qu.Z6 a tza.dn.zfi might UAZ tn teaching formulas. CHECK tkz response code which describes the treatment of each topic in your class. RESPONSE COVES: 7. Emphasized [used as a primary technique, referred to extensively or frequently) 2. Uszd, but not zmphasizzd. 3. Not used 44. Presenting formulas and explaining the meaning of the terms in the formulas: Ex: Formula: A = % bh A stands for the area of a triangle J) stands for the base of a triangle h, stands for the height of a triangle 45. Having the students inspect graphs and find formulas to express the relation-ships portrayed by the graph: Ex: 4 3 2 4 1 s 1 1 1 I I I I I 1 2 3 4 5 6 7 8 A = 2 x L i+5 46. Providing data from which formulas or equations are developed: 0 1 2 3 4 5 0 3 5 7 9 11 Hence y = 2x + 1 i+6 - 14 - 133 Teaching Techniques (Con't.) RESPONSE COVES: 1. Empln.asi.zzd [uszd as a pnJjnany technique,, n.e.fejm.zd to zxtznsivzly on. fnzquzntty) 2. Uszd, but not zmpkaslzzd 3. Not uszd 47. Having students collect data on related variables and formulate the relationship between the variables: ''~~\ one revolution ' \ ' 15.6 cm Ratio: 15.6 - 3 12 5 one revolution / N " ' v /' 40.9 cm Ratio: 40.9 _ , 1 C nrr "3 , 1 5 Hence | a 3 < l f S o c = 3 l d 4 7 - 15 - 134 Teaching Techniques (Con't) RESPONSE COVES: 1. Emphasized [used as a primary technique., referred to extensively on. frequently) 2. Used, but not emphasized 3. Hot used 48. Having students create new formulas based on known, simpler formulas: Ex. Create formula for surface area of a cylinder based on formulas for area of the rectangle and the circle. So, surface area = 2irrh + 2wr SA = 2irr (h + r) l 2 3 i+8 - 16 - 135 APPLICATIONS AND PROBLEMS Szvznal typzs of pnoblzms anz ll&tzd bzlow which may havz bzzn Included -in youn. Instructional prognam. CHECK thz nzsponsz code to Indicate the. degA.ee to which, a paKXA.cjjJiaA.typz. of problem was studlzd by youn. etas*. RESPONSE COVES: 1. Emphaslzzd [uszd as a primary typz of pnobZzm, uszd zxtznsivzly on. frzquzntly) 2. Uszd, but not zmphaslzzd 3. Not uszd 49. Age problems Roberta is now 15 years older than Stan. In 3 more years Roberta will be 3 times as old as Stan was 4 years ago. How old is Roberta now? 50. Digit problems If 4/5 of a number is added to 3/5 of that number, the result is the same as i f 10 is added to the number. What is the number? 51. Mixture problems A feed dealer plans to mix corn (at $1.12 a bushel) with wheat (at $1.74 a bushel) to get a mixture that sells at $1.43 per bushel. How many bushels of corn are needed to make 200 bushels of the mixture? 52. Percent problems In 1980 about 4/7 of the telephones in Georgia had direct distance dialing capabilities. What percent was this? 53. Distance-Rate-Time problems How long does i t take a rainstorm to travel 360 km at a rate of 45 km per hour? - 17 - 136 Applications and Problems (Con't) RESPONSE COVES: 7. Emphasized [used as a primary type of problem, used extensively on. frequently) 2. Used, but not emphasized 3. Not used 54. Interest problems Les borrowed $3000 from the bank at 11% interest per year. How much interest would he have to pay at the end of 9 months? 55. Area-Volume problems The Great Pyramid in Eqypt has a square base measuring 240 m on a side. Its altitude is 160 m. What is its volume? 56. Physical-Natural Science problems (lever problems, Hooke's law, ...) If Sue has a mass of 56 kg and Sara has a mass of 42 kg, how far will Sue have to s i t from the middle of the teeter-totter to balance with Sara, i f Sara is 1.2 m from the middle? 57. Energy or Ecological problems 2 An adult guppy requires 60 cm of air surface to live in an aquarium. How many adult guppies can live in a rectangular aquarium that is 45 cm long and 30 cm wide? - 18 - 137 SOURCES OF APPLICATIONS AND PROBLEMS SZVZAJOI souA.cz* of appJU.ccuLi.OYii,I'pn.obl.zms of Intzqzns, fonmulas, and zquaZions anz l i s t z d bzloui. CHECK tkz xzsponsz code to skow kow fn.zquzntly zack SOUACZ IMS uszd. RESPONSE COVES: 7. Uszd fn.zquzntly 1.. Uszd occasionally 3. Hot uszd at a l l 58. Students' textbooks 1 2 3 59. Supplementary textbooks or workbooks 60. Worksheets or exercises designed by myself or local teachers 61. The curriculum guide or syllabus 62. Articles or papers published by professional associations 63. Applications or problems suggested by my students 64. Applications or problems from real world sources such as newspapers or individuals involved in the use of mathematics l 2 3 - 19 - 138 TIME ALLOCATIONS 65. What was the average length (in minutes) of each class period? 1 | INTEGERS 66. How many total class periods did you spend on the development of the integers and operations with integers? [Combine. partial periods wken necessary) | 1 Indicate, the. number of class periods spent on each of the following activities (that is, demonstrations, explanations, students doing computational exercises, using manlpulatlves, etc.) with your class. Round your answer to the nearest whole number. 67. Activities related to the development of the concept of positive and negative integers 68. Activities related to the addition of integers (positive and negative) I I I 69. Activities related to the subtraction of integers (positive and negative) I I I 70. Activities related to the multiplication of integers (positive and negative) 71. Activities related to the division of integers (positive and negative) 72. Activities related to the structural properties of the set of integers (commutativity, associativity, distributivity, etc.) 73. Activities related to order relations with the set of integers 74. Application/problem solving activities related to integers (textbook word problems, problems related to real world problems, recreational problems, challenging problems, etc 75-76 77-78 79-80 19 C. L U 20-21 22-23 NOTE: THE SUM OF THE VER10VS GIVEN FOR ITEMS 67 TO 74 SHOULD NOT EXCEEV THE NUMBER GIVEN FOR ITEM 66. - 20 -Time Allocations (Con't.) FORMULAS AND EQUATIONS 75. How many total class periods did you spend on teaching formulas and equations? [Combine. p a r t i a l lessons when necessary.) | Indicate the. number of class periods spent on each, of the following activities (that Is, demonstrations, explanations, students doing computational exercises using rnanlpulatlves, etc.) with your class. Hound your answer to the nearest whole number. 76. Activities related to evaluation of formulas (for given values of the variables) 77. Activities related to deriving formulas or equations [ 78. Application/problem solving activities related to use of formulas (textbook word problems, problems related to.real world problems, recreational problems, challenging problems, etc.) 79. Activities related to solving literal equations 80. Activities related to solving linear equations 81. Application/problem solving activities related to the use of equations (textbook word problems, problems related to real world problems, recreational problems, challenging problems, etc.) NOTE: THE SUM OF THE PERIODS GIVEN FOR ITEMS 76 TO 81 SHOULD NOT EXCEED THE NUMBER GIVEN FOR ITEM 75. - 21 -OPINIONS Indicate the extent to which, you. agree, or disagree, with each of the following statements for your class. CIRCLE the choice which best describes your feelings. 82. The use of the number line adds a lot to the teaching of integers. Strongly Strongly Disagree1 Disagree2 Undecided3 Agree1* Agree5 83. It is very important to justify the rules for multiplying integers. Strongly Strongly Disagree1 Disagree2 Undecided3 Agree1* Agree5 84. A great deal of practice is required in order for students to acquire competence in performing operations with directed numbers. Strongly Strongly Disagree1 Disagree2 Undecided3 Agree1* Agree5 85. It is important for students to understand how integers obey general laws like the distributive law, the associative law, etc. Strongly Strongly Disagree1 Disagree2 Undecided3 Agree1* Agree5 86. Average students are usually not satisfied with knowing only the rules for performing operations with integers; they want to know why the rules work. Strongly Strongly Disagree1 Disagree2 Undecided3 Agree1* Agree5 87. Most students find i t diff icult to appreciate the significance of studying the structural properties (additive inverse, order relation, distributive law, etc.) of the set of integers. Strongly Strongly Disagree1 Disagree2 Undecided3 Agree1* Agree5 88. Most students cannot be expected to master the use of letters for unknowns quickly; they have to become accustomed to this usage slowly over a long period of time. Strongly Strongly Disagree1 Disagree2 Undecided3 Agree1* Agree5 - 22 -Opinions (Con't.) 89. Linear equations whose solution is a fraction (like 5x - 2 = 1) are generally more diff icult for students to solve than linear equations whose solution is an integer (like 6x - 3 = 15). Strongly Strongly Disagree1 Disagree2 Undecided3 Agree1* Agree5 90. In solving equations, i t is important that students be able to justify each step in their solution procedure. Strongly Strongly Disagree1 Disagree2 Undecided3 Agree1* Agree5 91. Solving linear equations by trial and error helps students understand the meaning of a solution. Strongly Strongly Disagree1 Disagree2 Undecided3 Agreek Agree5 92. The notion "solution set" (those values of the unknown which make the relation true) aids the students' comprehension of linear equations. Strongly Strongly Disagree1 Disagree2 Undecided3 Agree1* Agree5 93. Average students have difficulty in solving word problems involving linear equations. Strongly Strongly Disagree1 Disagree2 Undecided3 Agree1* Agree5 94. Average students have difficulty in translating verbal and written sentences into mathematical sentences, and vice versa. Strongly Strongly Disagree1 Disagree2 Undecided3 Agree1* Agree5 95. Average students have difficulty with applications involving linear equations. Strongly Strongly Disagree1 Disagree2 Undecided3 Agree1* Agree5 - 2 3 - 142 Opinions (Con't.) 96. When solving problems, i t is important for students to f irst identify the type of problem (age, digit, mixture, etc.) being solved. 52 Strongly Strongly Disagree1 Disagree2 Undecided3 Agree14 Agree5 97. Solving equations requiring students to justify the steps in the solution procedure has a detrimental effect on learning how to solve equations. Strongly Strongly Disagree1 Disagree2 Undecided3 Agree1* Agree5 5 3 98. The notion of equivalent equations is useful in helping students under-stand solutions. 55 Strongly Strongly Disagree1 Disagree2 Undecided3 Agree1* Agree5 99. Formulas taught should be memorized by students. Strongly Strongly Disagree1 Disagree2 Undecided3 Agree1* Agree5 100. Formulas should be used mainly to aid students in solving classes of story problems. Strongly Strongly Disagree1 Disagree2 Undecided3 Agree1* Agree5 101. Formulas should be used mainly to find volumes, areas, and perimeters of geometric figures. Strongly Strongly Disagree1 Disagree2 Undecided3 Agree1* Agree5 102. Formulas should be used mainly in applications to practical situations. Strongly Strongly Disagree1 Disagree2 U n d e c i d e d 3 Agree1* Agree5 5 8 56 57 APPENDIX D Geometry Q u e s t i o n n a i r e 144 INTERNATIONAL ASSOCIATION for the EVALUATION of EDUCATIONAL ACHIEVEMENT —SECOND— Study of MATHEMATICS G R A D E 8 T O P I C S P E C I F I C Q U E S T I O N N A I R E G E O M E T R Y (Booklet 13 L ) FOR NATIONAL CENTRE USE ONLY /""TL- 'N PROVINCE OF BRITISH COLUMBIA I V J L / l M , N , S T R Y OP EDUCATION lejSp/ D l v , s , 0 N 0 F PUBLIC INSTRUCTION \^ —' J LEARNING ASSESSMENT BRANCH - 2 - 145 Check here i f no geometry content is included in your program for your class. In that case, disregard the remainder of the questionnaire and return it to B.C. Research in the envelope provided. 1 2 0 CHECK tkz AZAponAZ which, best dzAcnibzA thz UAZ you madz of za.dk of thz fottowing matznialA in youn. instruction on gzomztny. RESPONSE COVES: 7. pAAMLfuj AOUKCZ, uszd fnzquzntZy 2. Azcondany AOUA.CZ, uAzd occoAionaJLly 3. not uAzd on. nojizLy uAzd 1. School (Mathematics II (Addison-Wesley) 1 2 3 2 1 2. Mathematics II (Ginn) 1 2 3 2 2 3 . Essentials of Mathematics II (Ginn) 1 2 3 2 3 4. Other published text materials (e.g., textbooks, workbooks, and worksheets) 1 1 3 Z h 5. Locally produced text materials (e.g., textbooks, workbooks, or worksheets) 1 2 3 2 5 6. Commercially or locally produced in-dividualized materials (e.g., programmed instruction or computer assisted instruction) 1 2 3 2 6 7. Commercially or locally produced films, film-strips, or teacher demonstration models 1 2 3 2 7 8.. Commercially or locally produced labora-tory materials for student use (e.g., games or manipulatives) 1 2 3 2 8 TEACHING TOPICS Tke topics given below may be included in your instructional program. CHECK tkz rzsponsz wkick describes tkz trzatmznt of zack topic in your class. RESPONSE COVES: I . taugkt as new contznt 1. rzviewzd and tkzn zxtzndzd 3. rzviewzd only 4. assumed as prerequisite knowledge and neither taugkt nor reviewed 5. not taugkt and not assumed as prerequisite knowledge 9. Angles (acute, right, supplementary, etc.) 10. Transformations (translations, rotations, reflections) 11. Vectors 12. The Pythagorean Theorem 13. Triangles and their properties (excluding congruent triangles) 14. Polygons and their properties (exeluding properties related to congruent or similar polygons) 15. Circles and their properties 16. Congruence of geometric figures (including congruent triangles) 17. Similarity of geometric figures (including similar triangles) 18. Parallel lines 19. Spatial relations 20. Geometric solids and their properties 21. Geometric constructions with ruler and compass 22. Proofs (formal deductive demonstrations) 23. Tessellations 24. Coordinate geometry 2 3 <t 2 3 it 2 3 <t 2 3 it 2 3 k 2 3 k 2 3 k 2 3 4 2 3 4 2 3 k 2 3 4 2. 3 it 2 3 k 2 3 k 2 3 k 2 3 k - 4 - 147 INSTRUCTIONAL APPROACHES Several approaches to teaching geometry are given below, which describe* the teaching approach In your class. RESPONSE COVES: CHECK the response code emphasized [used as a primary means of developing geometric content, used extensively or frequently) used but not emphasized not used 25. An informal Euclidean approach based on inductive reasoning, measurement, or students' intuitions 26. A formal Euclidean approach based on an axiomatic system used to prove theorems 27. An informal transformational approach based on inductive reasoning or students' intuitions 28. A formal transformational approach based on an axiomatic system used to prove theorems 29. A coordinate approach (either informal or formal) using coordinates of points, equations, etc. 30. A vector approach (either informal or formal) using addition of ordered pairs, a scalar times an ordered pair, etc. - 5 - 148 INSTRUCTIONAL AIDS For each of the following aids to the teaching and learning of geometry, CHECK the response code which indicates the degree to which you. and your students used the aid. RESPONSE COVES: 7. used extensively or frequently 1. used occasionally 3. not used 31. Ruler and compass 32. Protractor 33. Set squares (draftsman's triangles) 34. Geoboards 35. Paper cutouts or patterns 36. Models of solids (cones, pyramids, cylinders, etc.) 37. Paper folding 38. Tracing paper 39. Graph paper 40. Mirrors or translucent reflectors 41. Filmstrips and films 42. Computer graphics 43. Kits for constructing plane or solid figures 2 3 51 2 3 . 52 2 3 53 2 3 51+ 2 3 55 2 3 56 2 3 57 2 3 58 2 3 . 59 2 3 60 2 3 61 2 3 62 2 3 63 - 6 -1 4 9 TEACHING TRANSLATIONS If you did not teach translations, check here and proceed directly to Item 50. l 64 Szvzral iutzA.pKzXaXX.oru, of translations arz givzn bzlow. CHECK tkz rzsponsz codz wkick dzscAibzs tkz trzatmznt of zack interpretation in your class. RESPONSE COVES: 7 . Empkasizzd [uszd as a primary zxplanation, referred to zxtznsivzly or frzqazntly) 1. Uszd but not zmpkasizzd 3. Not uszd —* 44. I defined the vector AB as the set of equivalent pairs of points: AB = {(M,N) | MeP, NeP, (M,N) ^(A,B)} where (M,N) -v. (A,B)^(A,N) and (B,M) have the same midpoint. Then the translation along the vector V was defined as the map of P onto P which associates to each point M a point N such that MN* = V (or (M,N) elf). 1 2 3 6 5 45. Given (A,B) a pair of points on the plane P, I defined the translation associated with the pair as the map of P onto itself which makes each point M correspond- to a point N such that ABNM is a parallelogram. 1 2 3 6 6 46. I defined a translation as the composition of two central symmetries. 1 2 3 6 7 47. I presented translations by a physical approach involving displacements determined by their direction, orientation and magnitude. 1 2 3 68 Teaching Translations (Con't.) 48. Using graph paper we studied the mappings t^a ^ from Z 2 onto Z 2 such that t^a b)(x>y) = U',y') where x1 = x + a y' = y + b Then a translation of the plane P was defined as the map T , , . :P-*-P' (a,b) which associates to each point M with coordinates (x,y) a point M with coordinates (x1,y') such that x' = x + a y' = y + b 49. I presented the axioms of incidence and defined the translation on the plane P as a bisection of P satisfying the following axioms: 1. the identity map I of P is a translation. 2. the image of any line i under a translation, is a line V parallel to i. 3. for every translation (other than the identity), there exist one and only one direction d_, such that any line % with orientation d_ has itself for an image. 4. for every A and for every B, there exists one and only one translation t such that t(A) = B. TEACHING VECTORS If you did not teach vectors, check here and proceed directly to Item 59 Several Interpretations of vectors are given below. CHECK the response code which describes the treatment of each Interpretation In your etas*. RESPONSE COVES: 7. Emphasized (used as a primary explanation, referred to extensively or frequently) I. Used but not emphasized 3. Not used 50. After choosing the axes, the vector t associated with the translation xf* K\ 1 S defined as the pair (a,b). va,D| Addition of vectors is then defined in terms of the composition of translations. 51. A vector t is defined as the set of pairs (M,T (M)) where M is a point and x is given translation. 52. A vector is defined as an equivalence class of pairs of points. The pairs — » — » AB and MN are equivalent i f there exists a translation that transforms A into B and M into N. 53. A vector AB is defined by: 1. its orientation (that of the <-» line AB) 2. its direction (from A to B) 3. its length (distance from A to B) 54. A vector is defined as an equivalence class of pairs of points. The pairs AB and MN are equivalent i f and only i f M and BM have the same midpoint. 55. I do not define vectors since a defini-tion is not necessary. It suffices that students know how to work with them. - 9 - 152 TEACHING ADDITION OF VECTORS Several interpretations of addition of vectors are given below. CHECK the response code which describes the treatment of each interpretation in your class. RESPONSE COVES: 1. Emphasized [used as a primary explanation, referred to extensively or frequently] 2. Used but not emphasized 3. Not used 56. Given that a vector is associated with a translation, I presented the addition of two vectors as the vector associated with the composition of their transla-tions. Ex: AB + BC = AC Since. T.gj* o T g . ^ 57. Given (A,B) and (A,D) as the pairs of points representative of the vectors — » — » U and V, respectively, I defined the sum of U and V as the vector W representative of the pair (A,C) where ABCD is a parallelogram. 58. Since the equivalence relation is defined as "having the same midpoint" (see 54 above), I introduced trans-lations before defining the addition of vectors. i - 10 -3 al 3-= o — o V fl CTl 'V p- -S "J .3 3 «J 2 -2 13 3 "a UJ Q CJ CO CJ 3 a l •3 ,, 8.3 i fl a) c v si s a) a 1? 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OJ £ M . 3 Q. rg X M M IQ UJ - 14 - 157 TECHNIQUES FOR TEACHING CONGRUENT TRIANGLES Several techniques for tea.cki.ng, congruent triangles are given below. CHECK tke response code wkick describes tke tecknique used in your class. RESPONSE COVES: 1. used extensively or frequently 2. used occasionally 3. not used 74. State definition and properties Students were given a definition and conditions under which two triangles are congruent, e.g. , SSS, SAS, or ASA. 75. Graph paper or tracing paper Congruent triangles were constructed using graph paper or tracing paper. 76. Measurement Measurement activities were used to study properties of congruent triangles, e.g. , congruence of corresponding sides and angles, 77. Constructions with ruler and compass Students constructed congruent triangles using a ruler and compass. 78. Geoboard Students used the geoboard to make con-gruent triangles and study their properties. 79. Environment Examples of congruent triangles from the environment were discussed. Ex. Scaffolding 50 51 52 53 54 A ABC ~A.BAD 55 80. Transformations Students formed congruent triangles by finding images of triangles using reflections, rotations or translations. - 15 158 TECHNIQUES FOR TEACHING SIMILAR TRIANGLES Several techniques for teaching similar triangles are given below, response code which describes the technique used In your class. RESPONSE COVES: CHECK the 1. used extensively or frequently 1. used occasionally 3. not used 81. State definition and properties Students were given a definition and conditions under which two triangles are similar, e.g. , AAA, SAS or SSS. 82. Graph paper or tracing paper Similar triangles were constructed using graph paper or tracing paper. 83. 84. 85. 86. Measurement Measurement activities were used to study properties of similar triangles, e.g. , proportionality of sides. Constructions with ruler and compass Students constructed similar triangles using a ruler and compass. Geoboard Students used the geoboard to make similar triangles and study their properties. Environment Examples of similar triangles from the environment were discussed. Ex. # 87. Dilations (stretching or shrinking) Students constructed the image of triangles under an enlargement or dilation (stretching or shrinking). 57 58 59 60 61 62 63 - 16 - 159 TECHNIQUES FOR TEACHING PARALLEL LINES Szvzral ttc.hyu.qu.zs for teaching parallel lines arz given below. CHECK tkz response codz wklck describes tkz technique uszd tn your class. RESPONSE COVES: 1. uszd extensively or frequently 2. uszd occasionally 3. not uszd 88. Definition and examples Students were given a definition and examples of parallel and nonparallel lines were illustrated. l 2 3 64 89. Paper folding Paper folding activities were used to present and study parallel lines. 1 2 3 6 5 90. Measurement Measurement activities were used to study such properties as: parallel lines are everywhere equidistant, parallel lines form congruent corresponding angles with a transversal, etc. l 2 3 66 91. Constructions with ruler and compass Ex. Given a line i, and a point A not on the line, students constructed a line a' through the point parallel to the given line. 92. Tessellations Given tessellations of the plane such as floor or ceiling t i les, students inspected the tessellations for parallel lines and their properties. 93. Geoboards Given a geoboard, students inspected lines on the board to determine parallel lines and study their properties. ' 1 2 3 6 9 - 17 - 160 Techniques for Teaching Parallel Lines (Con't.) RESPONSE COVES: 7 . used extensively on frequently 2. uszd occasionally 3. not used 94. Constructions with straightedge and set squares (draftsman's triangles) Ex. Students constructed i' parallel to i T v > — • V 95. 96. Environment Examples of "parallel lines" from the environment, e.g. , railroad tracks or telephone lines, were discussed. Translations Parallel lines were studied through the use of translations. Ex. Given the translation T , points A and B and their image points A1 and B' then AA1 11 B? . 97. 98. BB1 Reflections Parallel lines were studied through the use of reflections. Ex. Given two lines, students used translucent materials (e.g. miras) to determine whether the lines are parallel. Rotations Parallel lines were studied through the use of rotations. Ex. Given a line, students determined its image under a half-turn (180° rotation). 70 71 72 73 - 18 - 161 TEACHING SPATIAL RELATIONS Several techniques for teaching s p a t i a l relations are given below, response code which describes the technique used In your class. RESPONSE COVES: I . used extensively or frequently 1. used occasionally 3. not used CHECK the 99. Using ready-made two-dimensional patterns (nets) to build three dimensional figures. Ex. 75 100. 101 Designing a two-dimensional pattern for a given three-dimensional object. Ex. Making a two-dimensional drawing for a given three-dimensional object. Ex. SOUP 76 77 102. Drawing plans and elevation (orthogonal projections) of geometric solids. Ex. 78 Teaching Spatial Relations (Con't.) RESPONSE COVES: 103. 7. used extensively on frequently 2. used occasionally 3. not used Representing the intersection of a plane and a solid by a two-dimensional drawing. Ex. 104. Finding numerical or algebraic expressions that describe relationships among the parts of a geometric figure. Ex. }B = 60° because A ABC is equilateral since its sides are the diagonals of the faces of the cube. 105. Building models of intersecting planes in space. 106. Predicting the shape of the shadows cast by various objects under a fixed source of light. - 20 -TIME ALLOCATIONS 107. What was the average length (in minutes) of each mathematics period? Minutes 108. How many total class periods did you spend on geometry? (Combine partial periods when necessary) Periods Indicate the amount, of tune spent on za.cn of tkz following actlv-lties [tkat Is, demonstrations, explanations, students doing computational. exercises on. proofs, using manlpulatlves, etc.) with your class. Round your answer to tkz nearest wkole number. 109. Activities related to the development of the concept of angles (acute, right, supplementary, etc.) Periods 110. Activities related to transformations (transla-tions, rotations, reflections) Periods 111. Activities related to vectors Periods 112. Activities related to The Pythagorean Theorem Periods 113. Activities related to triangles and their properties (excluding congruent triangles) Periods 114. Activities related to polygons and their properties (excluding properties related to congruent or similar polygons) Periods 1T5. Activities related to circles and their properties Periods 116. Activities related to congruence of geometric figures (including congruent triangles) Periods 117. Activities related to similarity of geometric figures (including similar triangles) Periods - 21 -Time Allocations (Con't.) 118. Activities related to parallel lines Periods 119. Activities related to spatial relations Periods 120. Activities related to geometric solids and their properties Periods 121. Activities related to geometric constructions with ruler and compass Periods 122. Activities related to proofs (formal deductive demonstrations) Periods 123. Activities related to tessellations Periods 124. Activities related to coordinate geometry Periods 125. Application/problem solving activities related to geometry (textbook word problems, problems related to real world problems, recreational problems, challenging problems, etc.) Periods NOTE: THE SUM 0F~THE PERIODS GIVEN FOR ITEMS 109 TO 125 SHOULD NOT EXCEED THE NUMBER GIVEN FOR ITEM 108. - 22 - 165 OPINIONS Indicate, thz zxtznt to which you agxzz ox disagxzz with zach of thz following statzmznts fox youA class. CIRCLE thz choice which bzst dzscxibzs yoax fzzlings. 126. The main objective of teaching geometry at this grade level is that of constructing a mathematical model of real situations. Strongly1 Disagree2 Undecided3 Agree1* Strongly5 60 Disagree Agree 127. Mastery of deductive procedures (e.g. proving theorems) is the goal of teaching geometry at this grade level. Strongly1 Disagree2 Undecided3 Agree1* Strongly5 61 Disagree Agree 128. The objective of teaching geometry at this grade level is to present the students with situations in which they have to formally demonstrate something about which they have an intuitive notion. Strongly1 Disagree2 Undecided3 Agree1* Strongly5 62 Disagree Agree 129. It is desirable that the presentation of geometric concepts follow an order determined by an axiomatic approach. Strongly1 Disagree2 Undecided3 Agree4 Strongly5 63 Disagree Agree 130. An intuitive approach to geometry is more meaningful to students at this grade level than a formal approach. Strongly1 Disagree2 Undecided3 Agree1* Strongly5 6h Disagree Agree 431. Geometry should be taught mainly through transformations (flips, turns, stretches). Strongly1 Disagree2 Undecided3 Agree1* Strongly5 65 Disagree Agree 132. The use of concrete models and instructional aids is essential in teaching geometry. Strongly1 Disagree2 Undecided3 Agree1* Strongly5 66 Disagree Agree 133. Three dimensional geometry should be taught only in the context of measurement (volume, surface area, etc.) for these students. Strong!yi Disagree Disagree2 Undecided3 Agree1* Strongly5 Agree 67 - 23 - 166 68 Opinions (Con't.) 134. The concept' of translation should be part of the knowledge of students at this grade level. Strongly1 Disagree2 ' Undecided3 Agree4 Strongly5 Disagree Agree 135. The concept of vector should be part of the knowledge of students at this grade level. Strongly i Disagree2 Undecided3 Agree4 Strongly 5 6 9 Disagree Agree 136. It is preferable to delay the study of vectors to a later time. Strongly1 Disagree2 Undecided3 Agree4 Strongly5 7 0 Disagree Agree 137. Activities to improve students' ability to visualize spatial figures should be included in the instructional program. Strongly1 Disagree2 Undecided3 Agree4 Strongly5 7 1 Disagree Agree 138. The study of polygons and their properties should be limited only to triangles and quadrilaterals. Strongly1 Disagree2 Undecided3 Agree4 Strongly5 7 2 Disagree Agree 139. The students should be skilled in geometric constructions using ruler and compass. Strongly1 Disagree2 Undecided3 Agree4 Strongly5 7 3 Disagree Agree 140. Teachers' demonstration of proofs of theorems should be an essential part of an instructional program in geometry for these students. Strongly1 Disagree2 Undecided3 Agree4 Strongly5 7 4 Disagree Agree 141. Geometric topics should be taught only to those students who will pursue higher education. Strongly1 Disagree2 Undecided3 Agree4 Strongly5 7 5 Disagree Agree 142. Proofs of theorems should be delayed until these students are at least 15 years of age. Strongly1 Disagree2 Undecided3 Agree4 Strongly5 7 6 Disagree Agree APPENDIX E Measurement Q u e s t i o n n a i r e 168 INTERNATIONAL ASSOCIATION for the EVALUATION of EDUCATIONAL ACHIEVEMENT —SECOND— Study of •MATHEMATICS G R A D E 8 T O P I C S P E C I F I C Q U E S T I O N N A I R E M E A S U R E M E N T (Booklet 12 L ) PROVINCE OF 8RITU5H OTLUMBIA MINISTRY OF EDUCATION DIVISION OF PUBLIC INSTRUCTION LEARNING ASSESSMENT BRANCH FOR NATIONAL CENTRE USE ONLY Check here i f measurement is not included in your program for your class. In that case, disregard the remainder of the questionnaire and return it to B.C. Research in the envelope provided. 1 CHECK tke response wkick best describes tkz use you made of zack of tkz following materials tn youn instruction on mzaAun.zm.znt. RESPONSE COVES: 7. primary Aoun.cz, used frequently 1. secondary source, uszd occasionally 3. not uszd or rarely used 1. School Mathematics II (Addison-Wesley) 1 2 2. Mathematics II (Ginn) 3. Essentials of Mathematics II (Ginn) 4. Other published text materials (e.g., textbooks, workbooks, and worksheets) 5. Locally produced text materials (e.g., textbooks, workbooks, or worksheets) 6. Commercially or locally produced in-dividualized materials (e.g. programmed instruction or computer assisted instruction) 7. Commercially or locally produced films, film-strips, or teacher demonstration models 8. Commercially or locally produced labora-tory materials for student use (e.g., games or manipulatives) - 3 - 170 TEACHING TOPICS Thz topics givzn below may bz included in youn. instructional pKoqKam. CHECK thz Kzsponsz zodz which dzscKlbzs thz tizatmznt of zach topic in youn. etas*. RESPONSE COVES: 7. taught as nzw content 2. izvizwzd and thzn zxtzndzd 3. KZvlzwzd only 4. 044tuned cu, pneAzquisitz knowledgz and nzithzn. taught no A. nzvlzwzd 5. not taught and. not assumzd as pneAequlsltz knowtzdgz 9. Concept of measurement (including the selection of appropriate units) 1 2 3 k 5 10. Names of units of measure in the metric system (SI) 1 2 3 h 5 11. Names of units of measure in the English system 1 2 3 ** 5 12. Conversion of units within a system Ex. 5 centimetres = 50 millimetres 24 inches =- 2 feet 1 2 3 ~* 5 13. Conversion of units between systems Ex. Convert 5 inches to centimetres How many miles in 60 kilometres? 1 2 3 ** 5 14. Estimating measurements Ex. Find a stick 15 cm long. How many metres high is the ceiling? 15. Operations with measurements Ex. 4 yards 2 feet 8 inches + 2 yards 1 foot 10 inches 2.5 m + 67 cm = 16. Precision, accuracy, percent error, or relative error 29 30 31 32 33 34 35 36 - 4 -Teaching Topics (Con't.) 17. Concept of TT 18. Linear measurement Ex. Find the length of segment AB 19. Perimeter of a polygon (including triangles, quadrilaterals and other polygons) 20. Circumference of a circle 21. Area of a triangle 22. Area of a rectangle (including squares) 23. Area of a parallelogram other than a rectangle 24. Area of a trapezoid 25. Area of a circle 26. Surface area of a rectangular solid (including a cube) 27. Surface area of a cylinder 28. Surface area of a sphere 29. Volume of a rectangular solid (including a cube) 30. Volume of a cylinder and prism 31. Volume of a sphere 32. Volume of a cone and pyramid - 5 - 172 INSTRUCTIONAL AIDS fou zack of tkz following aids to tkz teaching and learning of mzasuAzmznt, CHECK tkz response code which indicates tkz degree to which you. and youn. students uszd tkz aid. RESPONSE COVES: 1. uszd extensively on frequently I. uszd occasionally 3. not uszd 33. Rulers (metrestick, yardstick, 12" ruler, etc.) 34. Measuring tape 35. Trundle wheel 36. Examples of non-standard units (paper clips, hand span, foot length, popsicle sticks, sugar cubes, matchboxes, etc.) 37. Geoboards, graph paper, or grids 38. Models for standard units of area (cm squares, cm cubes, cm rods, etc.) 39. Graduated cylinders 40. Containers (litre, gallon, etc.) 41. 'Tillable" models of geometric solids 53 54 55 56 57 58 59 60 61 TEACHING METHODS Thz mzthods uszd to intxodu.cz thz usz of units of mzasuxzmznt givzn below may havz been included in youx Instxuctional pxogxam. CHECK thz xzsponsz code which describes thz treatment of zach intzxpxztation in youx class. RESVOMSE COVES: 1. zmphasizzd {uszd as a pximaxy method, ^ .e^ eAAed to zxtznslvzly ox fxzquzntly) 1. uszd, but not zmphasizzd 3. not uszd 42. I have my students use non-standard units of measurement. Ex. Measure the length of a desk using paper clips. 1 43. I have my students use standard units in measuring objects. Ex. Measure the length of the room in metres. 44. I have my students estimate the size of real world objects. Ex. Estimate how many sugar cubes will f i t into a given container. Estimate the length of the hallway. 45. I have my students identify objects whose measurement is as close as possible to a given number of units. Ex. Which of these four containers has a capacity closest to two litres? Cut a long piece of string about 10 centimetres long without using a ruler. 46. I have my students measure a given object using different units of measure. Ex. Measure the width of the paper in mm and cm. Find the height of the table in cm and inches. 47. I have my students increase the precision of their measurements by using smaller units. Ex. The length of the stick is between 5 cm and 6 cm. More precisely, it is between 53 mm and 54 mm. @ 174 3 uj a ~ f — - a . ~ <j ai c* •3-= -If-5 cs. 3 •» co a a) 3 ai "3 V o a k u u -2 *• -I *• al 4? a a i a. - 3 • ^ - o » <i t - al a 3 T O 41 S • 3 a» "V ai a< w a t o 3 3 - a S • y ai o . a l j l H 3 5-3 "3 al ai 3 1^ -W *r 3 3 i i ai 3 c . o r_i 2 CO o u r x a. a) a a v 4? -w a.-" a. 3 a. a «<3<v 2 a x»-5 3 i *j a - * » o » - > a v « 4 . 5 «j al si *4 -3 3 > a • » » ai 3 -w 3 x • »4 V "3 S 3 a s c v , : *? ai V a "3 a "3 o -v a a> > 3 4 J N : • a > - 3 - a - v I U • » - a ai ay cn -o • _ . V 3 3 3 i c o 4- - a « » = 3 - a - - •33 * ' a-^ ^ 4 B l I 4 J 4F> 4) V V 3 V > a 3 a 2 a a a ai • • • t ' i S * ' « — M "i f ui < f - « 3 8.. = a ai o 3 -r * 3 • a oi < -3 U J 3 3 ol ol 3 "<= 3 - a " B <_l al a ay al 6 a a ai c r i a 3 a-a a a ai a '•5 2 f a 3 ' a •w g «j 4- al a a) _ 3 - a - y g 3 a ai » y •y S • J 3 al a 44 a «e 4-al V < CJ al ti J < a « v co - a ' « J a a..ai a uj u. o v y 3 • « c n — a V _ a a > a •3 a aj ai al V , N v > 3 -3 • v 3 - y - a ai -a a -a <i Ial 5 - S * 3 § . g . * { S x * ^ o 4> ^ C £ <o (O 3 -U (ft 0) t- O •*-» >> U C 3 (J W tU <0 crt OJ s 5 I ° £ ro i - , Ul U » -<- ra U1TO 4 J O "O •— c a « CJ a a i u i 4-» • a - J - - e u g w 13 u — l / l w o X 414- O 2» 2 S - c o i a . • 4-) al -. O. Ol ^ C 01 IO *— in ^  o i - a u J I C - s ' . U — 01 3 C — c o g VI PO & 3 ^ c \ l | r — •5~' l — s I w l Ol I 01 — I 3 - 3 — Ol Cfl Ol c u — c j a -4 J 14. O IQ S - Ol #-— 3 Ui JZ 01 U Ol 4 U W W 4 4 » 01 Ol IO Ol — f — - c - c - a U 4-1 U l Ol Ol (O * - ^ -— O 4-1 u • a u a ; ui a - — » .. 4-* o i o ai c n e 3 c Ol »— 4-1 U l U l 10 IO 3 X o i > - e o o a . oi ro O •a c e oi 4-J — W ro <4- 01 -C 4- j - 4J 0 S u _ 4-> 3 ro *0 U 01 Ol rn , Ul 1— r-1 4J -I- U 4-s o o < — 01 t- ro | I d 01 I-3 - = S- Ol 4-1 4-1 o i a Ul 4-1 Ol C4 a — CJ c n o i o 3 43 (O -O *J o c • ~ ^ ro « r -O 4J _ Ol rO 01 Ol -X - . J = -— « — Ul 4-1 U >l ? 01 J . U l 01 -C C 4 J 01 H ro Ol (J 4 J C W 4 J a — — ro O "O ul 4 1 U > > Ol 4 - C ul J g ^ ro c 3 a . a -01 c o o 3 -— 4 J Ol • 4 / Qf- U U U l C 3 C CJ O ui al 4-1 >>— Ol U Ol 0 4 J i. 01 S io 'tr io ^ i - 01 £5 — -— J = 3 ^ O — 4 J o — t^3 t 'J — 8 -• or al 3 - B 2 «( ai ! o *« s: • S a i 3 v a < -w -s: A ^ W V ai a «? : •w r * ai a) < V 5-3 •w at a a -« w S3 J =1 v a s a cj 3 S e. O at CO S a v CJ a a A V aj +4 3 * at 3 o -a 1 .3 i t CJ fl ai a S w , • C at at en * •>» v CJ)*> X a, al a *r 3 V •-y at a. a a. = •v e al «o = T5 a a s ** -* i | 1 | V a -t) T J A V = at =|T3 JtW "° -w ^  3 3 •B a - c i j a al »5 al V a, 3 * 3 fl a. fl W 2 V •O a a a & 2 2 >- 2 •y v fl V se — <M < ^ w " i ^ tn >o N 1 al a fl a. 175 2 \ •o o 2 ca lm t- COICM C3I U 01 id CJ * - ro O 01 i — v IQ U 41 U * S_ IQ 'CJ V IO C J — 41 S-V » - I S - C J p ui eofin • i/i J Z ca o Sevexat methods ion teaching the frxmula frx the axea oi a paxaltelogxam axe given below. CHECK die appxopxiate xesponse code which describes the treatment oi eacli method hi youn dass. RESPONSE COPES: 1. Emphasized I d l e d as a pximaxy explanation, xeiexxed to extensively ox frequently) 2. Used, but not emphasized 3. Hot used Was this method in the students' tex-tt ton. tlwse methods used as I'KJMARV explanations ( i . e . , ii you checked I in the iixst column) WRITE the nuuibexs oi the response codes which show the two pxiimiy xeasons fri using them. RESPONSE COVES: 1. Welt known to me 2. In B . C . ox local cuxxicutum guide 3. Easy fri students to undexstand 4. Enjoyed by students i. Related to math oi pxiox gxades 6. Usefrt in math oi latex gxades 7. I was taught it was appxopxiate t. Emphasized in student text Evx those methods HOT USEV, {i.e., ii you checked 3 in the iixst column) WRITE the numbers oi the xesponse codes which show the two pximaxy xeaions frx not using them. RESPONSE COPES: 1. Hot well known to me 2. Hot in B . C . ox local cuxxicutum guide 3. Itaxd frx Students to undexstand 4. Vistiked by students 5. Hot xetated to math oi pxiox gxades 6. Hot usefrt in math oi latex gnades 1. I was, taught it was inappxopxiate $. Hot emphasized in student text \ 55. I presented the formula A = b x h and demonstrated how to app ly I t by means ° f examples. 4 r m Ex. ' 1.7 cm 1st Pr imary Reason 2nd Pr imary Reason 1st P r imary Reason 2nd Pr imary Reason 56. A = 4 cm x 1.7 cm » 6 .8 cm 2 i 2 1 I presented a p a r a l l e l o g r a m on a g r i d (o r a geoboard) l i k e the one below ( p a r a l l e l o g r a m ABCD), and had the s tudents r e l a t e the number o f square u n i t s i n s i d e ABCD to the base and a l t i t u d e o f the p a r a l l e l o g r a m . Yes No Yes No 50-55 I 57. I presented a p a r a l l e l o g r a m on a g r i d (o r a geoboard) l i k e the one shown above and had the s tudents count the square u n i t s i n s i d e t r i a n g l e s ABE and CDF. Then I had them r e l a t e the area o f ABCD to that o f r ec t ang l e BEFC based on the congruence o f A ABE and A B C F . Yes No ON Sevexal methods frx teaching the frxmuta frx the axea oi a paxatlelogxam ale given below. CHECK the appxopxiate xesponse code which descxibes the txeatment oi each method in youx etait. RESPONSE COVES: I . Emptiasized [used as a pximaxy explanation, xeiexxed to extensively ox frequently) Used, but not emphasized Hot used Was tills method in the students' texil fox those methods used as PRIMARY explanations [i.e., ii you checked I in the iixst eotuam) WRITE the numbexs oi the Xesponse codes which show the two pximaxy xeasons frx uiing them. RESPOHSE COVES: Well known to me In B . C . ox local cuxxicutum guide Eauj frx students to undexstand Enjoyed by students Related to math oi pxiox gxades Usefrt in math oi latex, gxades I was taught it was appxopxiate Emphasized in student text. fox those methods HOT USEV, [i.e., H you checked 3 in the iixst column) WRITE the inunbeAl oi the xesponse codes wliich shoio the two pximaxy xeasons frx not using them. RESPOHSE COVES: 1. Hot well known to me 2. Hot in B . C . ox local cuxxicutum guide 3. Haxd frx students to undexstand 4. Vistiked by students 5. Hot xetated to math oi pniox gxades 6. Hot usefrt in math oi latex gxades 7. I was taught it was inappxopxiate t. Hof. emphasized in student text 58. I d e r i v e d the formula A = b x h by comparing the area o f the p a r a l l e l o g r a m to that o f a r e l a t e d r ec t ang le o f equal d imens ions . Yes No 1s t Pr imary Reason 2nd Pr imary Reason 1st Pr imary Reason 2nd P r i n a r y Reason 68-73 59. I gave the student a p a r a l l e l o -gram l i k e the one below, and asked them to cu t o f f t r i a n g l e FDC and to use t h i s to form a r e c t a n g l e ( A F ' F D ) . The s tudents then r e l a t e d the formula fo r the area o f the r e c t a n g l e to the area o f the p a r a l l e l o g r a m . A " Yes No 60. I p a r t i t i o n e d the p a r a l l e l o g r a m by a d iagona l i n t o two congruent t r i a n g l e s . Then the area o f A ABO i s >s bh and the area o f the p a r a l l e l o -gram Is then bh . Yes No o i 7',-71 19 C. (JJ 2 0 - 2 5 — - n - 178 • y i - 3 o *— a . c a s - o • 3 a ) * ? a l -C c . — o x : • y a a i — E • « . 3 « S C l 4 | 4 Q u j o a zi s . a i K - S * - • - a x j a t .2 •* "T5 a l V o - a - a V •3 o B - y « a i ro 5 - a a* a • v - a a i l l S 4 § 3 5 © a j -5 <y a l -3 "3 cn s - 3 I I a l o 5 S l l a i a a i a © O o a l V a * V "3 a a §- * a l V O t ! -9 §•..*?* JX c j « j - a «u eo' «• "3 » i o a i o a ~ o . c s 3 - y •^3 5-"3 a l a l « - a V T O O c n « r - y < - > e n s — * a . a i o *• o V ••V i 4? - w a , « y x , " * 1-3 = - y a g *s 2 a 0 -a v a a i • a - a - y a i « y cn o • w 3 a a a ^ a < • w a i v § . 1 < E < 3 - g a l « J » J 4 C 5 s i ? * 3 I"5 UJ " 3 x a t - . a i * y i — • a at 3 - 3 3 3 o - a o c n • a a a i - y 13 3 '3s 5) 3 § 3 2-1 I 5 a al < • o . a l x j "3 O « o a o al »*3 "S a l a l O O "3 3 -q * J "3 ai 3 3 "3 - y 3)3 S.*-! c n o « * ? a l •q QI V « c J j - 3 a o x i 3 = o ** 4 I * 3 -« a . a l a l • o x - a - a u, s i y y 3 a o « * a • © Ji c j x i a l t d S i ? c 3 a n o c n * T 3 - a • > ? a 3 a i • y 3 IM - 3 - 3 , * } - y i a l a l a J o 3 > t y 3 -9 3 '•2.5 7 1 "S. a a i « uj o c 3 Uj ^3 s a a i 31 CO -3 O LU => a i - y o • y 3 • y - a a l u 3 § S .3 U J - a « u - a a a . C L W a l * u S x X - o o g u j a l « 3 CO e x — CM * n a : c e e n x CJ — Q 5t3< a . 2, O -M C - Q < « «3 3 _^ (9 > i « . o > _ <n a j u r o **- a i c a L . O L . O ^ c n to 01 o t n O l c <— « oi — • — a i CJ - C — L J E j a i o S * J O W Ol o u r o JT L . O l M Q . 4 J a - L . - 12 - 179 Several method* for teaching the formula, for tke volume of a rectangular prism are given below. CHECK the response code which describes the treatment of each interpretation in your class. RESPONSE COVES: 1. Emphasized [used as a primary explanation, referred to extensively or frequently] 2. Used, but not empkasized 3. Not used 63. I presented the formula V = 1 x W x h or V = (area of base) x height and demonstrated how to apply it by means of examples. Ex. 2 64. I presented a physical model of a right rectangular prism (box) with its faces marked off in square units, as illustrated below. I had students generate the formula by relating the number of cubic units contained in the prism to the dimensions of the box, giving hints only if necessary. 65. I provided my students with unit cubes and asked them to build rectangular prisms of specified dimensions. I asked them to relate the number of unit cubes used to the given dimensions, giving hints only if necessary. - 13 - 180 Several techniques a teacher, might use in teaching the relationship among various metric (SI) units are listed below. CHECK the response code which describes the treatment of each technique In your class. RESPONSE COVES: 7. Emphasized {used as a primary explanation, referred to extensively or frequently] I. used, but not emphasized 3. Not used 66. I established the analogy between decimal numer-ation system and the basic metric units of measurement. Ex. One kilolitre is 1,000 litres and 121 centimetres is 1.21 metres. 67. I taught my students rules to change from one metric unit to another. Ex. To convert from a unit to a smaller unit, multiply. To convert from a unit to a larger unit, divide. 68. I presented a table showing definitions and adjacent relationships. Ex. km = 1000 m hm = dam = m dm = cm = mm -100 m • 10 m 0.1 0.01 0.001 10 hm 10 dam m = 10 cm m m 10 mm 69. I used a number line or a metre stick (grad-uated in centimetres and millimetres) to illustrate interrelationships among units. 70. I used centimetre cubes and decimetre cubes to establish relationships among units. 71. I demonstrated the relationship between metric units of length, metric units of capacity and metric units of mass using instruments and units. Ex. 1,000 cubic centimetres = 1 litre 1 cubic centimetre of H20 - 1 gram Therefore, 1 litre of H20 = 1 kilogram 41 i*2 43 44 45 46 - 14 - 181 TIME ALLOCATIONS 72. What was the average length (in minutes) of each class period? Minutes 73. How many total class periods did you spend on teaching measurement? (Combine partial lessons when necessary.) Periods ^-so Indicate the amount of time spent on each, of the following activities {that is, demonstrations, explanations, students doing computations, using manipulative*, etc.) with youn. class. Round youn answer to the nearest whole number. 74. Activities related to the concept of measurement (including selection of units and use of unit to assign a number) Periods s i - 5 2 75. Teaching units in the metric system (SI) 76. Teaching units in the English system 77. Activities related to conversion of units within a system Ex. 5 cm - 50 mm 24 inches = 2 feet Periods 53-54 Periods ss-56 Periods 57-58 78. Activities related to conversion of units between systems Ex. Convert 5 inches to centimetres. How many miles in 60 kilometres? Periods 5 9 - 5 0 79. Activities related to estimating measurements Ex. Find a stick 15 cm long. How many metres high is the ceiling? Periods s i - 6 2 80. Activities related to determining precision, accuracy, percent error or relative error Periods 63-64 Time Allocations (Con't.) 81. Activities related to operations with measurements Ex. 4 yards 2 feet 8 inches + 2 yards 1 foot 12 inches 2.5 m + 67 cm = 82. Activities related to the concept of IT 83. Activities related to linear measurement Ex. Find the length of segment A"B. 84. Activities related to finding the perimeter of a polygon (including triangles, quadrilaterals, and other polygons) 85. Activities related to finding the circumference of a circle 86. Activities related to finding the area of a triangle 87. Activities related to finding the area of a rectangle (including spheres) 88. Activities related to finding the area of a parallelogram other than a rectangle 89. Activities related to finding the area of a trapezoid 90. Activities related to finding the area of a circle 91. Activities related to finding the surface area of a rectangular solid (including cubes) 92. Activities related to finding the surface area of a cylinder Periods Periods Periods Periods Periods Periods Periods Periods Periods Periods Periods Periods 16 - 183 Time Allocations (Con't.) 93. Activities related to finding the surface area of a sphere Periods 2 8 - 2 9 94. Activities related to finding the volume of a rectangular solid (including cubes) Periods 3 0 - 3 1 95. Activities related to finding the volume of a cylinder and prism Periods 32-3 3 96. Activities related to finding the volume of a sphere Periods 3<+-3 5 97. Activities related to finding the volume of a cone and pyramid Periods 3 6 - 3 7 98. Application/problem solving activities related to measurement (textbook word problems, problems arising from real l i fe situations, recreational problems, challenging problems, etc.) Periods 3 8 - 3 9 NOTE: THE SUM OF THE PERIODS GIVEN FOR ITEMS 74 TO 98 SHOULD NOT EXCEED THE NUMBER GIVEN FOR ITEM 73. - 17 -184 OPINIONS Indicate, the extent to which you agA.ee OK disagree with each of the following statements foA youx class. CIRCLE the choice which best describes your feelings. 99. Estimation and approximation should be emphasized in the teaching of measurement. Strongly1 Disagree2 Undecided3 Agree1* Strongly5 Disagree Agree 100. Students' use of standard instruments for measuring should be emphasized in the mathematics program. Strongly1 Disagree2 Undecided3 Agree1* Strongly5 Di sagree Agree 101. Measurements other than length, area, or volume should be taught as part of the school science program and not as a part of the school mathematics program. Strongly1 Disagree2 Undecided3 Agree1* Strongly5 1 + 2 Di sagree Agree 102. Work with non-standard units is essential for increasing students' understanding of the concept of measurement. Strongly1 Disagree2 Undecided3 Agree1* Strongly5 Di sagree Agree 103. Measurement of time, temperature, mass, and weight should be taught as part of the mathematics program at this grade level. Strongly1 Disagree2 Undecided3 Agree1* Strongly5 Disagree Agree 104. Work with formulae for finding the perimeter, area, and volume should be emphasized. Strongly1 Disagree2 Undecided3 Agree1* Strongly5 Di sagree Agree Opinions (Con1t.) 105. Computations involving measurement should be done with hand-held calculators. Strongly1 Disagree2 Undecided3 Agree1* Strongly5 Disagree Agree 106. The best way students learn about measurement is by actually measuring things. Strongly1 Disagree2 Undecided3 Agree1* Strongly5 Disagree Agree 107. Students should be expected to know and apply standard area and volume formulas. Strongly1 Disagree2 Undecided3 Agree1* Strongly5 Disagree Agree APPENDIX F I n i t i a l L i s t o f S u b t o p i c s and Items C a t e g o r i z e d by t h e P a n e l o f E x p e r t s 187 The I n i t i a l L i s t o f S u b t o p i c s a n d I t e m s a s C a t e g o r i z e d b y t h e P a n e l o f E x p e r t s C a t e g o r y T o p i c / S u b t o p i c Common and D e c i m a l F r a c t i o n s C oncept o f F r a c t i o n s 21, 22, 23, 27, 24, 25 28, 30 A d d i t i o n o f F r a c t i o n s 31, 32, 33, 37, 34, 35, 36 38 Concept o f D e c i m a l 51, 53, 56 52, 54, 55 F r a c t i o n s O p e r a t i o n s w i t h D e c i m a l 59 57, 58 F r a c t i o n s A l g e b r a C o n c e p t o f I n t e g e r s 20, 22, 24 21, 23 A d d i t i o n o f I n t e g e r s 25, 27 26 S u b t r a c t i o n o f I n t e g e r s 28, 32 29, 30 M u l t i p l i c a t i o n o f I n t e g e r s 36 34, 38 Formula 4 5 , 4 7 , 4 8 44 R a t i o , P r o p o r t i o n , and P e r c e n t R a t i o 21 23 P r o p o r t i o n 27 29 Geometry Measures o f A n g l e s o f 59, 60, 61, 62, T r i a n g l e 63, 64, 66 P y t h a g o r e a n Theorem 67, 68, 71 69, 72 Measurement Number ~ 48, 53, 54 49, 51 A r e a o f P a r a l l e l o g r a m 56, 57, 58, 59, 60, 61, 62 Volume o f r e c t a n g u l a r 64, 65 p r i s m 188 APPENDIX G P r o f i l e s o f Responses t o t h e C o n c r e t e and A b s t r a c t Items f o r Each S u b t o p i c f o r t h e Sample of Ten Teachers 189 Figure 1. Profiles of resoonses to the concrete items for the concept of fractions. 190 Figure 2. Profiles of responses to the abstract items for the concept of fractions." 191 Figure 3. Profiles of responses to the concrete items for the skil l addition of fractions. 1 9 2 Figure 4. Profiles of responses to the abstract items for the ski l l addition of fractions. Figure 5. Profiles of responses to the concrete items for the concept of decimal fractions. 194 Figure 6. Profi les of resoonses 'to the abstract items for the c o n c e D t of decimal fract ions. 1 9 5 Figure 7. Prof i les of resoonses to the concrete items for the conceDt of integers. 196 Fiaure 8. Profiles of resoonses to the abstract items for the conceDt of integers. 197 Figure 9. P r o f i l e s o f responses to the concre te items f o r the s k i l l s u b t r a c t i o n o f i n t e g e r s . 1 9 8 Figure 10. Profiles of resoonses to the abstract items for the skil l subtraction of integers. 199 Figure 11. Profiles of resDonses to the concrete items for the concept Pythagorean Theorem. 200 Figure 12. Profiles of responses to the abstract items for the c o n c e D t Pythagorean Theorem. 201 Figure 13. P r o f i l e s o f responses to the concre te items f o r the concept tr 202 Ficure 14. Profiles of resDonses to the abstract items for the concept TT . 203 Figure 15. Profiles of responses to the concrete items for the relationship amona various metric units. 204 Figure 16. Profiles of responses to the abstract items for the relationship among various metric units. 

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