Cl THE EFFECT OF INSTRUCTION IN MODULAR ARITHMETIC ON THE ABILITY OF GRADE 6 STUDENTS TO DIVIDE FRACTIONS AND GIVE A RATIONAL EXPLAN-ATION OF THE PROCESS by ALEXANDER DAVID MACDONALD B.A.,University of B r i t i s h Columbia,1956 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF EDUCATION i n the D i v i s i o n of Elementary Education We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA August, 1973 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make i t freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. D i v i s i o n _ . _ Department of Elementary Education The University of British Columbia Vancouver 8, Canada Date August 27,1973 Abstract The problem under investigation i n t h i s study was to f i n d out what r e l a t i o n s h i p a unit i n modular arithmetic might have to Grade 6 pupils' s k i l l i n computing the d i v i s i o n of f r a c t i o n s and to t h e i r understanding of the mathematical basis of the algorithm. It was hypothesized that a unit i n modular arithmetic would aid i n developing s k i l l i n computing and understanding of the algorithm. The study was conducted with a sample of 58 Grade 6 students from the same school. The subjects were assigned to two treatment groups. Both groups received a review of f r a c t i o n concepts at the begin-ning of the study. Following t h i s , one group was taught modular arithmetic while the other group reviewed adding and subtracting of f r a c t i o n s . Then both groups were taught multi-p l i c a t i o n and d i v i s i o n of f r a c t i o n s . Following the i n s t r u c t i o n period, both groups were tested for a b i l i t y to compute d i v i s i o n of f r a c t i o n s . To test understanding of the d i v i s i o n of fract i o n s algorithm, an interview inventory test was administered to a l l subjects i n both groups. A s t a t i s t i c a l analysis of the data from these tests revealed no support f o r the hypotheses. The conclusion was that teaching modular arithmetic to the Grade 6 pupils p a r t i c i p a t i n g i n the study did not appear to improve t h e i r a b i l i t y to compute d i v i s i o n of f r a c t i o n s nor t h e i r understanding of the mathematical basis of the d i v i s i o n of f r a c t i o n s . i i i TABLE OF CONTENTS CHAPTER PAGE I. NATURE OF THE STUDY Introduction 1 Statement of the Problem 1 Uses of the Term "Fraction" 2 S u i t a b i l i t y of the Grade Level 3 Importance of Fractions i n the Elementary School 4 The Divi s i o n Algorithm 7 Modular Arithmetic 11 Related Studies 15 Statement of the Hypotheses 16 I I . DESIGN AND PROCEDURE Design 17 Sample 17 Measures of Population Variables 18 Formation of Treatment Groups 19 Instructional Procedure 23 Control of Variables 26 Development of Tests 27 S t a t i s t i c a l Treatment 32 I I I . ANALYSIS OF DATA Findings From the Division of Fractions Test 34 Findings From the Interview Test 38 Findings From the Modular Arithmetic Test 43 Additional Findings 44 i v CHAPTER PAGE IV. DISCUSSION AND CONCLUSIONS Discussion 49 Conclusions 53 BIBLIOGRAPHY 54 APPENDICES A Outline of Fraction Concepts Lessons 57 B Outline of Adding and Subtracting Fractions Lessons 62 C Outline of Lessons and Practice Booklet For Modular Arithmetic 67 D Outline of Lessons For M u l t i p l i c a t i o n of Fractions and Outline of Lessons and Selected Practice Pages For Divis i o n of Fractions 91 E Tests Constructed For and Used i n This Study 104 V LIST OF TABLES TABLE PAGE I Means, and Standard Deviations of I.Q. Scores For Treatment Groups 20 II Means and Standard Deviations of Arithmetic Computation Scores For Treatment Groups 21 III Mean Ages and Ranges For Treatment Groups. 22 IV Composition of the Treatment Groups 22 V Means, Adjusted Means, and Standard Errors of the Div i s i o n of Fractions Test (Correct Answers Only) 34 VI Summary of the Analysis of Covariance of Di v i s i o n of Fractions (Correct Answers Only) Scores With I.Q., Arithmetic Computation, Arithmetic Reasoning, and Fraction Concepts Held Constant 35 VII Means, Adjusted Means, and Standard Errors of the Div i s i o n of Fractions Test (Correct and P a r t i a l Answers) 36 VIII Summary of the Analysis of Covariance of Di v i s i o n of Fractions (Correct and P a r t i a l Answers) Scores With I.Q., Arithmetic Compu-tation, Arithmetic Reasoning, and Fraction Concepts Held Constant 37 IX The Number of T - l and T-2 Subjects i n C,P,W, and N Categories For Each of the Four Interview Test Items 38 X Comparison of the Treatment Groups i n the Number of Subjects Giving Correct Answers to Each Item i n the Interview Test 39 XI Comparison of the Treatment Groups i n the Number of Subjects Giving Correct and P a r t i a l Answers to Each Item i n the Interview Test 40 XII The Number of T - l and T-2 Subjects Giving Rational and Rote Responses to Each Item i n the Interview Test 41 v i TABLE PAGE XIII Comparison of the Treatment Groups i n the Number of Subjects Giving Rational Responses to Each Item i n the Interview Test 42 XIV Means and Standard Deviations of the Modular Arithmetic Test For the Total T-2 Group and the T-2 Group Divided by Sex and Intelligence 43 XV Means, Adjusted Means, and Standard Errors of the M u l t i p l i c a t i o n of Fractions Test 44 XVI Summary of the Analysis of Covariance of the M u l t i p l i c a t i o n of Fractions Test Scores With I.Q., Arithmetic Computation, Arithmetic Reasoning, and Fraction Concepts Held Constant 45 XVII Means, Adjusted Means, and Standard Errors of the Adding and Subtracting of Fractions Test.. 46 XVIII Summary of the Analysis of Covariance of the Adding and Subtracting of Fractions Test Scores With I.Q., Arithmetic Computation, Arithmetic Reasoning, and Fraction Concepts Held Constant 47 XIX Adjusted Mean Adding and Subtracting of Fractions Test Scores f o r T - l and T-2 Groups C l a s s i f i e d as High arid Low I.Q. Scores Divided at the Median (104.5) 48 ACKNOWLEDGEMENTS I wish to express my sincere thanks to Dr. Roland F. Gray for his advice and assistance during the course of t h i s study. Thanks are also extended to Dr. Donald E. A l l i s o n and Dr. Douglas T. Owens for th e i r help. I am very grat e f u l to my wife, E l v i e , for her support and encouragement. CHAPTER I NATURE OF THE STUDY Introduction Elementary school mathematics educators, whether they be classroom teachers or curriculum designers, face a common problem i n deciding what balance to maintain between comput-ational e f f i c i e n c y and an understanding of mathematical con-cepts. Emphasis has varied i n the past few decades. P r i o r to 1960, arithmetic programs tended to stress computational e f f i c i e n c y and the s o c i a l uses of arithmetic. During the 1960's, new elementary school programs l a i d greater stress on the understanding of mathematical concepts, with less emphasis on computational s k i l l s . At the present time the search continues for an accept-able balance between s k i l l s and understanding as some educa-tors question the rigorous approach of the 1960*s. Elemen-tary school mathematics programs continue to be revised i n attempts to maximize both s k i l l s and understanding. 1 This study was prompted by the apparent need to investigate ways of achieving t h i s balance. Statement of the Problem The problem under investigation i n t h i s study i s stated 1 B r i t i s h Columbia Department of Education, Mathematics, Primary - Years 1-3, Years 7-8, 1972, pp.1-2. 2 as follows: W i l l Grade 6 students who are taught a unit on modular arithmetic show a greater understanding of the mathematical basis of d i v i s i o n of f r a c t i o n s and a greater s k i l l i n perform-ing the computation of d i v i s i o n of f r a c t i o n s than students who are not taught modular arithmetic? Uses of the Term "Fraction" The term " f r a c t i o n " i s a potential source of misunder-standing i n elementary mathematics. In introducing the concept of f r a c t i o n a l numbers, Marks, Purdy, and Kenny point out that, "Accurate use of language dealing with f r a c t i o n a l numbers and f r a c t i o n s frequently requires so much verbiage 2 that i t may stand i n the way of c l a r i t y . " This section w i l l attempt to indicate how the term " f r a c t i o n " i s used i n t h i s study. There are three common usages of the term " f r a c t i o n " : f r a c t i o n as a symbol, f r a c t i o n as a number pair, and f r a c t i o n 3 as a number. A l l of these three usages appear at one point or another i n t h i s study. The f r a c t i o n as a symbol consists of three parts, a top numeral c a l l e d a numerator, a bottom numeral c a l l e d a denom-inator, and a bar between them. Students review the f r a c t i o n 2 John L. Marks, C. Richard Purdy, Lucien B. Kenny, Teaching Elementary School Mathematics for Understanding, 1970, p.191. 3 Truman Botts, "Fractions i n the New Elementary Curricula", The Arithmetic Teacher, 15:219, March 1968. 3 as a symbol i n the introductory lessons of t h i s study. The idea of the f r a c t i o n as a number pair i s introduced i n the primary grades and the f r a c t i o n symbol i s eventually introduced as a convenient notation for recording the number pa i r . The number pair can represent part of a whole or a subset of a set. Students review f r a c t i o n as a number pair i n the introductory lessons of t h i s study. The idea of f r a c t i o n as a number i s introduced i n the early intermediate grades. A choice i s usually made at t h i s point whether to introduce the term "rat i o n a l number", the term " f r a c t i o n a l number", or to continue to use the term " f r a c t i o n " . In t h i s study, the choice was to use " f r a c t i o n " rather than to introduce other terms. The term "rat i o n a l number" was not f a m i l i a r to the students nor was the term " f r a c t i o n a l number". The term " f r a c t i o n " as used i n the statement of the problem i n an e a r l i e r section of t h i s chapter ref e r s to f r a c t i o n as a number. Fractions as numbers are most often associated with the r a t i o n a l numbers. However, i n the elementary school the term " f r a c t i o n s " generally refers to the set of non-negative r a t i o n a l numbers. This i s the meaning of " f r a c t i o n s " i n t h i s study. S u i t a b i l i t y of the Grade Level Grade 6 i s often considered to be an appropriate grade l e v e l i n which to introduce d i v i s i o n of f r a c t i o n s . In the 4 primary grades students learn that a f r a c t i o n i s a symbol f o r an ordered pair of natural numbers with some physical r e f e r -ents. Students then learn to generate sets of equivalent f r a c t i o n s and from these sets to develop an i n t u i t i v e notion of r a t i o n a l numbers. Next the comparison of r a t i o n a l numbers i s introduced and students are taught to f i n d i f one f r a c t i o n i s greater than, less than, or equal to another f r a c t i o n . The order i n which students usually learn operations with r a t i o n a l numbers i s addition, subtraction, m u l t i p l i c a t i o n , and d i v i s i o n . Thus students at the Grade 6 l e v e l , the l e v e l chosen for t h i s study, are generally considered able to learn d i v i s i o n of f r a c t i o n s . Importance of Fractions i n the Elementary School Opinions vary as to the p r i o r i t y which should be given to i n s t r u c t i o n i n fra c t i o n s i n the elementary school mathe-matics program. Price, for example, suggests that there i s diminishing value i n teaching f r a c t i o n s because the opera-tions (especially, division) have l i t t l e p r a c t i c a l value i n da i l y l i f e and, furthermore, the metric system w i l l soon make the f r a c t i o n an anachronism. He asks, "Can you think of one s p e c i f i c example i n the r e a l world i n which d i v i s i o n of one f r a c t i o n by another i s necessary?" and suggests the answer, "The f r a c t i o n s that we teach are ju s t not fra c t i o n s of the 4 r e a l world." I f we convert to the metric system, he 4 Jack Price, "Why Teach Di v i s i o n of Common Fractions?" The Arithmetic Teacher, 16:111-112, February 1969. 5 argues, we s h a l l have to be concerned only with tenths and decimal computation takes care of these. Brueckner and Grossnickle, i n the i r 1953 ed i t i o n of 5 Making Arithmetic Meaningful , advised teachers to teach the pupil to invert the d i v i s o r and multiply even though he may not understand the mathematical basis of the operation. The teacher should on no account attempt to r a t i o n a l i z e the process because i t has no s o c i a l s i g n i f i c a n c e . Despite e f f o r t s to improve the mathematical correctness of elementary programs i n the past decade, the foregoing viewpoint may ex i s t today i n more sophisticated forms. Commenting on t h i s , Botts says, . . . we may be fostering a new kind of rote learning when we prompt teachers to i n s i s t that students always use the approved terms and notations - not because they make the subject any clearer, but because the teacher has been led to believe that t h i s i s what's " r i g h t " and important to mathematicians. 6 Marks, Purdy, and Kenny point out that f r a c t i o n s are valuable i n th e i r own r i g h t and that there are many everyday problems that are impossible to solve i f only whole numbers are used. They i l l u s t r a t e with the case of three boys wish-ing to share two apples equally. This problem can be solved with a knife, ". . . but to express the answer mathematically 7 demands using numbers other than whole numbers." Decimal 5 Leo J . Brueckner and Foster E. Grossnickle, Making A r i t h - metic Meaningful, 1953, p.338. 6 Botts, op.cit., p.220. 7 Marks, Purdy, Kenny, op.cit., p.189. 6 notation may be used to express these numbers but t h i s may not be the i d e a l way to introduce the concept to children. Even though metrication i s imminent i n Canada, there w i l l s t i l l be a need to use fract i o n s other than tenths, p a r t i -c u l a r l y halves, thirds , and fourths. Riess argues that a u t i l i t a r i a n viewpoint i s not s u f f i c i e n t . In A New Rationale f o r the Teaching of Fractions she states, Today we know that hardly any other aspect of a r i t h -metic has so much to contribute to the c h i l d ' s capacity for abstract reasoning and r e l a t i o n a l thinking as a thorough understanding of the concept of f r a c t i o n s . She contrasts t h i s with the r e a l i t y that, . . . no other phase i n elementary mathematics pres-ently contributes so l i t t l e to the c h i l d ' s appreciation of number and i s so unpopular with pupils and teachers a l i k e as the study of f r a c t i o n s . 8 Riess goes on to point out that attempts have been made to r e l i e v e t h i s drabness by teaching "useful" f r a c t i o n s but she questions whether i t i s r i g h t to carry pragmatic teaching beyond the f i r s t few grades. She suggests that, A program based on undistorted l i f e experiences of the c h i l d has i t s place i n the f i r s t grades, provided i t does not block a fundamentally d i f f e r e n t phase of dev-elopment by being car r i e d on too long. In other words, one of the c r u c i a l problems i n teaching f r a c t i o n s i s to s t r i k e the r i g h t balance between teaching them as a useful tool and helping the c h i l d extend h i s concept of number i n a mathematical sense, without losing con-tact with the c h i l d ' s own l e v e l of thought and i n t e r e s t . 8 Anita P. Riess, "A New Rationale for the Teaching of Fractions", Harvard Educational Review, XXV,No.2:105, Spring 1955. 9 Ibid., p.107. 7 The Division Algorithm There are three d i v i s i o n of fra c t i o n s algorithms i n common use; the common denominator method, the complex f r a c t i o n method, and the inverse operations method. The three methods are i l l u s t r a t e d here. Common denominator method 4 " * " 7 ~ 2 8 " J " 2 8 ~ 8 Complex f r a c t i o n method 2 3 7 21 3 2 4_ _ 4 * 2 8 _ 21 4 "*• 7 ~ 2 " 2 2 1 ~ 8 7 7 ' 2 Inverse operations method 2 2 _ 2 1 n * 7 " 2 ~ ~ 4 " 2 i 3 7 n • 1 = 4 • 2 21 n = — Some studies have been done to evaluate the three methods. Bidwell examines the three methods i n the l i g h t of modern learning theory and xn pa r t i c u l a r the theorxes of Gagne and Ausubel. By combining Gagne's suggestion of the need f o r hierarchies of required prior concepts with Ausubel's idea of advanced organizers, which are pri o r learned materials designed to maximize e f f i c i e n t learning, Bidwell attempts to esta b l i s h 8 the degree of meaningfulness i n each method. He concludes that, The inverse operations approach c l e a r l y u t i l i z e s a dom-inant generalization i n mathematics, the concept of i n -verse operations. S p e c i f i c a l l y , the method uses the pro-perty that m u l t i p l i c a t i o n and d i v i s i o n are inverse op-erations. The student has already used t h i s property with addition and subtraction of whole numbers. 10 Bidwell conducted t h i s study to compare the three methods of learning d i v i s i o n of f r a c t i o n a l numbers and concluded that the students were able to perform the inverse operations method with less error and better understanding and integration of concepts than with the other two methods. Ingersoll conducted an experiment to tes t the e f f i c i e n c y of two methods of d i v i s i o n which employ the m u l t i p l i c a t i v e inverse. These correspond to the two methods which Bidwell l a b e l l e d the "complex f r a c t i o n method" and the "inverse oper-ations method". Ingersoll omitted the common denominator method from his study because he f a i l e d to f i n d evidence that i t i s superior to other methods. Ingersoll found that, T y p i c a l l y , arithmetic texts which have emphasized the inversion p r i n c i p l e . . . have tended to employ one of two methods. The f i r s t uses complex fr a c t i o n s and the mu l t i p l i c a t i v e i d e n t i t y element and the other uses the associative property of m u l t i p l i c a t i o n and the multi-p l i c a t i v e i d e n t i t y element . . . . 11 It should be noted that Ingersoll uses the term "inversion" 10 James K. Bidwell, "Some Consequences of Learning Theory Applied to Di v i s i o n of Fractions", School Science and Mathe- matics, 71:429, June 1971. 11 Gary M. Ingersoll, "An Experimental Study of Two Methods of Presenting the Inversion Algorithm i n Div i s i o n of Frac-tions", C a l i f o r n i a Journal of Educational Research, XXII:18, January 1971. 9 to r e f e r to the use of the re c i p r o c a l , not to inverse operations. Ingersoll's study was conducted along the following l i n e s . A l l subjects were given an introductory program which stressed the development of the concept of the r e c i p r o c a l . Then one group was placed on Program A (Associative) while the second group was placed on Program CF (Complex Fra c t i o n s ) . Program A developed the re c i p r o c a l p r i n c i p l e i n d i v i s i o n using the associative property of m u l t i p l i c a t i o n and the m u l t i p l i c a t i v e i d e n t i t y element. I t emphasized the operation allowing r e l o c a t i o n of parentheses i n the expression (A x B) x C = A x (B x C). Program CF developed the recipro-c a l p r i n c i p l e i n d i v i s i o n by emphasizing the i d e n t i t y elements of m u l t i p l i c a t i o n and d i v i s i o n and complex f r a c t i o n s . A t h i r d program, Program R, was made up of a random selection of items from the other two programs. On a test of achievement o v e r a l l r e s u l t s favoured the complex f r a c t i o n approach. There i s some support f o r the view that none of the three methods of d i v i s i o n of fra c t i o n s referred to previously can be r e a l l y meaningful to children of t h i s age group. Bates believes that, Most of the concepts needed for an understanding of d i v i s i o n using f r a c t i o n s are . . . 'secondary* con-cepts - generalizations which are based upon previous abstractions, that i s , on 'primary 1 concepts. 12 12 Thomas Bates, "The Road to Inverse and Multiply", The Arithmetic Teacher, 15:348, A p r i l 1968. 10 He suggests that there i s a r e a l need to bridge the gap between the physical world and the abstract secondary concepts of r e c i p r o c a l and inverse operations. Furthermore, he i s not convinced that the gap can be bridged. In reaching the rote process of inv e r t and multiply, . . . the intermediate thought of tr a n s l a t i n g d i v i s i o n by n into m u l t i p l i c a t i o n by the m u l t i p l i c a t i v e inverse of n depends on a concept which I suspect i s not acquirable, at present, by many children of elementary school age. 13 It was the purpose of t h i s study to see i f modular arithmetic can help to bridge the gap. A search of the l i t e r a t u r e reveals that there are arguments i n favour of each of several methods of di v i d i n g f r a c t i o n s . Thus the choice of algorithm must be made on the basis of s u i t a b i l i t y for t h i s study. The method chosen was the inverse operation method. An outline of t h i s method showing the mathematical properties involved i s given here. Question: a. c_ _ _e b d ~ f cl C & Step 1: b = "d x "f D e f i n i t i o n of d i v i s i o n Step 2: ~ X b " = c " x ^ " d X " f ^ Cancellation property „. ~ d a /d C v e Associative property of Step 3: — x r- = (— x -r)x .. , . , . , . ^ ^ c b c d f m u l t i p l i c a t i o n „, „ a d , e Property of recip r o c a l s : Step 4: T - x — = 1 x — r i f J b c f d c . c X d = 1 , a d _ e_ M u l t i p l i c a t i v e i d e n t i t y ; btep b: b X c ~ f , e e 1 X T = T f f Step 6: -^-^^=-^x— Transitive property of the r e l a t i o n 13 Bates, op.cit., p.354. 11 I t was important to ensure that the d i v i s i o n of f r a c t i o n s method used i n t h i s study did not contain any concepts which could not be developed r e a d i l y during the course of i n s t r u c t i o n i n d i v i s i o n of f r a c t i o n s . Many of the subjects i n t h i s study had used the Seeing Through Arithmetic (1st. Canadian edition) 1 program i n previous school years, as f a r as was known from a study of t h e i r school records. This might suggest the use of the complex f r a c t i o n method inasmuch as i t i s the one used i n t h i s e d i t i o n of Seeing Through Arithmetic. However, an exam-ination of the algorithm yielded no compelling evidence that i t was the only one that could follow from the concepts developed i n the Seeing Through Arithmetic (1st. Canadian edition) program. Modular Arithmetic One group of subjects studied a unit on modular arithmetic i n t h i s study. Since modular arithmetic i s not a f a m i l i a r topic i n elementary school mathematics, i t i s discussed i n some d e t a i l i n t h i s section. There seems to be a f a i r l y wide divergence of opinion on the value of modular arithmetic i n the elementary school. Spitzer argues that because few elementary mathematics pro-grams include modular arithmetic, t h i s i s an i n d i c a t i o n of i t s lack of worth. Of the few programs that include a section 14 Maurice L. Hartung, et. a l . , Seeing Through Arithmetic (1st. Canadian e d i t i o n ) , 1965. 12 on modular arithmetic he says, "The producers . . . may have concluded that i n c l u s i o n of the topic w i l l be b e n e f i c i a l either to th e i r prestige or sales." On the other hand, Howard and Dumas f e e l that the topic has importance i n i n t r o -ducing c e r t a i n c h a r a c t e r i s t i c s of the mathematical system such as the properties of closure, a s s o c i a t i v i t y , commutativity, and p a r t i c u l a r l y inverse elements. Westcott and Smith believe that ". . . because a modular system operates within l i m i t s , i t makes basic mathematical p r i n c i p l e s less complex to demon-17 strate to c h i l d r e n . " Mueller states that, Properly slanted, work with modular arithmetic with an emphasis upon pattern can arouse the elementary student's mathematical imagination and i n t u i t i o n and, i n the process, provide a possible foretaste of the t h r i l l of scholarly discovery.. . . . at an elementary l e v e l , modular arithmetic provides f i n i t e , miniature number systems which may be explored rather thoroughly. 18 The term modular arithmetic re f e r s to number systems which use a f i n i t e number of units and then repeat themselves. I t i s sometimes c a l l e d "clock" arithmetic because i t i s c y c l i c i n nature and because the ordinary clock provides a physical representation of such a system. The analogy can 15 Herbert F. Spitzer, Teaching Elementary School Mathe- matics, 1967, p.358. 16 Charles F. Howard and Enoch Dumas, Teaching Contempor- ary Mathematics i n the Elementary School, 1966, p.80-81. 17 A l v i n M. Westcott and James A. Smith, Creative Teaching i n the Elementary School, 1967, p.136. 18 Francis J . Mueller, "Modular Arithmetic" i n the Twenty- Seventh Yearbook of the National Council of the Teachers of Mathematics, 1963, p.73. 13 be extended to include modular systems with any number of elements so that for i n s t r u c t i o n a l purposes the expressions "modulo seven clock" or "seven hour clock", for example, are used. While the regular 12-hour clock i s useful f o r introduc-ing the idea of modular arithmetic, i t i s not suitable for i l l u s t r a t i n g the mathematical properties of d i v i s i o n which are central to t h i s study. The properties of modular number systems vary depending upon the number of units i n the system. In systems having prime-numbered moduli there i s a unique quotient f o r every number i n the system for any non-zero d i v i s o r . This property i s ess e n t i a l i n order to demonstrate the inverse r e l a t i o n s h i p between d i v i s i o n and m u l t i p l i c a t i o n using the table of m u l t i p l i c a t i o n for the system. Non-prime moduli do not have unique quotients for every d i v i s o r ; there i s more than one quotient for some d i v i s i o n s . Although m u l t i p l i c a t i o n and d i v i s i o n are the operations used i n t h i s study, i t was reasoned that students should be introduced to addition and subtraction as well. Using the clock analogy, operations i n modular arithmetic are defined as follows. Addition i s defined as a clockwise movement of the hand, sta r t i n g from the numeral representing the f i r s t addend and moving through the number of points representing each success-ive addend. The sum i s represented by the f i n a l numeral to which the hand points. 14 S u b t r a c t i o n i s d e f i n e d as a co u n t e r - c l o c k w i s e movement of the hand, s t a r t i n g from the numeral r e p r e s e n t i n g the minuend and moving through a number of p o i n t s r e p r e s e n t i n g the subtrahend. The d i f f e r e n c e i s r e p r e s e n t e d by the f i n a l numeral to which the hand p o i n t s . M u l t i p l i c a t i o n i s d e f i n e d i n terms of s u c c e s s i v e a d d i -t i o n . The hand i s s t a r t e d a t 0 and moved i n a cl o c k w i s e d i r e c t i o n . One f a c t o r r e p r e s e n t s the s i z e of each move and the other f a c t o r r e p r e s e n t s the number of moves. The product i s r e p r e s e n t e d by the f i n a l numeral to which the hand p o i n t s . D i v i s i o n i s d e f i n e d on the c l o c k as repeated s u b t r a c t i o n . The hand i s pl a c e d a t the numeral r e p r e s e n t i n g the d i v i d e n d and moved i n a co u n t e r - c l o c k w i s e d i r e c t i o n . The s i z e o f each move i s determined by the d i v i s o r and the number of moves r e q u i r e d i n order f o r the hand to r e s t a t 0 r e p r e s e n t s the q u o t i e n t . By performing these f o u r o p e r a t i o n s i n a system having a prime-number modulus, complete t a b l e s of a d d i t i o n , sub-t r a c t i o n , m u l t i p l i c a t i o n , and d i v i s i o n can be c o n s t r u c t e d . These t a b l e s i n t u r n can be used to show the i n v e r s e r e l a -t i o n s h i p between a d d i t i o n and s u b t r a c t i o n and between m u l t i -p l i c a t i o n and d i v i s i o n . Other p r o p e r t i e s which can be developed once the o p e r a t i o n s are lear n e d i n c l u d e the m u l t i p l i c a t i v e i n v e r s e , the a s s o c i a t i v e and commutative p r o p e r t i e s of m u l t i -p l i c a t i o n , and the i d e n t i t y element f o r m u l t i p l i c a t i o n . 15 Most of the properties needed to explain d i v i s i o n of fr a c t i o n s are developed through the elementary school grades by the use of whole numbers. However, the m u l t i p l i c a t i v e inverse, or r e c i p r o c a l , i s not a property of whole numbers and i s usually introduced to students i n terms of f r a c t i o n s immediately pr i o r to i t s use i n d i v i s i o n of f r a c t i o n s . Mod-ular arithmetic provides an opportunity to introduce t h i s property to students i n another system. The table below i l l u s t r a t e s some properties i n terms of fr a c t i o n s and modular arithmetic. Reciprocal Fractions 2 5 1 5 x 2 = 1 M u l t i p l i c a t i v e ^ 7 i d e n t i t y ^ x 1 = ^ element Commutative property Associative property 1 1 - 1 A 4 X 3 ~ 3 X 4 1 ,S 3. ( 1 5, _3 4 x ^6X4 ~ 4 X6 X 4 Modular Arithmetic 3 x 2 = 1 (mod 5) 4 x 1 = 4 (mod 5) 3 x 4 = 4 x 3 (mod 5) (3x2) x 4 = 3 x (2x4) (mod 5) Related Studies Standard b i b l i o g r a p h i c a l sources were studied and no research studies could be located which had attempted to r e l a t e a study of modular arithmetic to operations with f r a c -t i o n s . Lyda and Tayler made a study of the e f f e c t of modular 16 arithmetic on student's understanding of the decimal numera-tion system. No s t a t i s t i c a l l y s i g n i f i c a n t evidence was ob-tained although the authors concluded that, "When pupils are given i n s t r u c t i o n i n modular arithmetic some growth occurs 19 i n t h e i r understanding of the decimal numeration system." Statement of the Hypotheses The following hypotheses were tested i n t h i s study. 1. Students who are taught modular arithmetic w i l l show a s i g n i f i c a n t l y better a b i l i t y to divide f r a c t i o n s , as meas-ured by a tes t of computation, than students who are not taught modular arithmetic. 2. Students who are taught modular arithmetic w i l l show a s i g n i f i c a n t l y better understanding of d i v i s i o n of fra c t i o n s , as measured by an interview test, than students who are not taught modular arithmetic. 19 W.J. Lyda and M.D. Taylor, " F a c i l i t a t i n g an Understand-ing of the Decimal Numeration System Through Modular A r i t h -metic", The Arithmetic Teacher, 11:10 3, February 1964. 17 CHAPTER II DESIGN AND,PROCEDURE Design The basic design of t h i s i n v e s t i g a t i o n i s that of a two' group study. The groups are i d e n t i f i e d and defined as follows. T - l (Review) Group - Grade 6 students who received i n s t r -uction based on the B r i t i s h Columbia curriculum guide f o r arithmetic"'", including a review of addition and subtraction of f r a c t i o n s , p r i o r to being taught m u l t i p l i c a t i o n and d i v i s i o n of f r a c t i o n s . T-2 (Modular Arithmetic) Group - Grade 6 students who received i n s t r u c t i o n based on the B r i t i s h Columbia c u r r i c u -lum guide f o r arithmetic, with the exception that a review of addition and subtraction of fr a c t i o n s was replaced by a unit on modular arithmetic i n which s p e c i f i c mathematical properties r e l a t i n g to m u l t i p l i c a t i o n and d i v i s i o n were emphasized. Sample The sample consisted of a l l Grade 6 students i n Queen's Park School, Penticton, B r i t i s h Columbia. This i s an urban public school of about 475 students which i s situated within School D i s t r i c t #15. The pupils are drawn from an area of 1 B r i t i s h Columbia Department of Education, Programme of Studies f o r the Intermediate Grades, Arithmetic 1965, Grades IV,V,VI, 1965. 18 lower socio-economic s t a t u s w i t h i n the community. However the community as a whole does not have areas of extremely h i g h or low socio-economic s t a t u s . A l l the students were white. There were 68 p u p i l s , 36 boys and 32 g i r l s , i n Grade 6 at the b e g i n n i n g of the school term. The mathematical back-ground of the p u p i l s v a r i e d s i n c e 25 of the p u p i l s had attended the same school s i n c e Grade 1 while 4 3 had attended one or more other s c h o o l s . No attempt was made to t r a c e the mathematical experience of t h i s l a t t e r group. Measures of P o p u l a t i o n V a r i a b l e s I n t e l l i g e n c e A l l students i n Grade 6 i n September, 1972 were adminis-t e r e d the Otis-Lennon Mental A b i l i t y Test, Elementary I I , 2 Form A d u r i n g the second week of s c h o o l . T h i s i s a widely used t e s t of g e n e r a l a b i l i t y i n which, Emphasis i s p l a c e d upon measuring the p u p i l ' s f a c i l i t y i n r e a s o n i n g and i n d e a l i n g a b s t r a c t l y with v e r b a l , symbolic, and c o n f i g u r a l content, sampling a broad range of c o g n i t i v e a b i l i t i e s . 3 A r i t h m e t i c computation A l l students i n Grade 6 i n September, 1972 were adminis-t e r e d the B r i t i s h Columbia Test, A r i t h m e t i c Computation, V-VI, 4 Form A d u r i n g the second week of s c h o o l . T h i s i s e s s e n t i a l l y 2 A r t h u r S. O t i s and Roger T. Lennon, Otis-Lennon Mental A b i l i t y Test, Elementary I I , Form J , 1967. 3 , Manual For A d m i n i s t r a t i o n , Otis-Lennon Mental A b i l i t y Test, Elementary, Form J , 1967, p.4. 4 B r i t i s h Columbia Department of Education, D i v i s i o n of T e s t s , Standards, and Research, B r i t i s h Columbia Test, A r i t h - metic Computation, V-VI, Form A, 1951. 19 a d u p l i c a t i o n o f the S t a n f o r d Achievement T e s t . The items on the t e s t are based on the work f o r the grades preceding the ones gi v e n i n the t i t l e . A r i t h m e t i c r e a s o n i n g A l l students i n Grade 6 i n December, 1972 were adminis-t e r e d the B r i t i s h Columbia Test, A r i t h m e t i c Reasoning, VI-VII, 5 Form A . T h i s t e s t i s a l s o based on the S t a n f o r d Achievement T e s t . Again the problems are based on the work f o r the grade preceding the ones gi v e n i n the t i t l e . Formation of Groups The T-l(Review) and T-2 (Modular A r i t h m e t i c ) Groups were s e l e c t e d i n the f o l l o w i n g manner. The r e s u l t s of the Otis-Lennon Mental A b i l i t y T e s t and the B r i t i s h Columbia Test, A r i t h m e t i c Computation were t a b u l a t e d and these steps taken. 1. Students were l i s t e d i n rank order of I.Q. as o b t a i n e d from the Otis-Lennon T e s t . S t a r t i n g with the f i r s t p a i r o f students i n t h i s l i s t , the f i r s t student i n each p a i r was assigned to A group i f a c o i n t o s s produced a head and to B group i f a c o i n t o s s produced a t a i l . The second student was assigned to the other group. The same procedure was f o l l o w e d w i t h each p a i r of students i n the l i s t u n t i l a l l were p l a c e d . 2. The students w i t h i n each group, A and B, were then l i s t e d i n rank order of scores on the A r i t h m e t i c Computation 5 , B r i t i s h Columbia Test', A r i t h m e t i c Reasoning, VI-VII, Form A, 1951. 20 T e s t . Each group was f u r t h e r d i v i d e d t o form A - l , A-2, B - l , and B—2 groups as f o l l o w s . S t a r t i n g w i t h the f i r s t p a i r o f s t u d e n t s i n A group, t h e f i r s t s t u d e n t i n the p a i r was a s s i g n e d t o A - l i f a c o i n t o s s produced a head and t o A-2 i f a c o i n t o s s produced a t a i l . The second s t u d e n t i n the p a i r was a s s i g n e d t o the o t h e r group. The same pr o c e d u r e was f o l l o w e d w i t h each p a i r o f s t u d e n t s i n the A group. B - l and B-2 groups were formed i n a s i m i l a r manner from B group. 3. The f o u r groups t h u s o b t a i n e d were r e - f o r m e d i n t o two groups as f o l l o w s . A - l was t o be combined w i t h B - l i f a c o i n t o s s produced a head; A - l was t o be combined w i t h B-2 i f a c o i n t o s s produced a t a i l . S i n c e the t o s s produced a head, the A - l and B - l groups were combined t o form one i n s t r u c t i o n a l group and A-2 and B-2 were combined t o form the o t h e r group. 4 . The two t r e a t m e n t groups were formed by a s i m i l a r method o f c o i n t o s s i n g . The A - l B - l group was a s s i g n e d the Modular A r i t h m e t i c Treatment (T -2 ) and t h e A-2 B-2 group was a s s i g n e d the F r a c t i o n Review Treatment ( T - l ) . The means and s t a n d a r d d e v i a t i o n s o f the I.Q. s c o r e s f o r each group a r e shown i n Ta b l e I . TABLE I MEANS AND STANDARD DEVIATIONS OF I.Q. SCORES FOR TREATMENT GROUPS Mean S t d . D e v i a t i o n T - l Group 10 3.6 T-2 Group 106.7 9.2 10.2 The means and standard deviations of the Arithmetic Computation Test scores are shown i n Table I I . TABLE II MEANS AND STANDARD DEVIATIONS OF ARITHMETIC COMPUTATION SCORES FOR TREATMENT GROUPS Mean Std. Deviation T - l Group 14.4 4.9 T-2 Group 13.2 5.9 During the period of the study, two members of the T - l Group and f i v e members of the T-2 Group l e f t the school. Five students who entered the school at various times during the study were included i n the i n s t r u c t i o n but not i n the study. Attendance was judged to be s a t i s f a c t o r y i f a student missed no more than an average of one period out of s i x . One member of the T-2 Group missed the entire period of i n s t r u c t i o n on modular arithmetic and had to be omitted from the study. Otherwise, one student was omitted from the T - l Group and no students were omitted from the T-2 Group because of poor attendance. The ages of a l l students taking part i n the study were v e r i f i e d from school records. The normal September age range for students i n Grade 6 i s 10 years 9 months to 11 years 8 months. One year beyond t h i s range was not considered unusual f o r Grade 6. One c h i l d aged 13-9 was omitted from 22 the f i n a l sample because of her age. Two other students aged 13-2 and 13-4 were retained i n the f i n a l sample because no record was available of when they f i r s t had entered school. Both had a non-English background. The mean age and range of each group i s shown i n Table I I I . TABLE III MEAN AGES AND RANGES FOR TREATMENT GROUPS Mean age (months) Range (months) T - l Group 137.3 32 (10-9 to 13-4) T-2 Group 135.5 25 (10-9 to 12-9) No attempt was made to control the sex variable i n the formation of the treatment groups. Composition of the f i n a l sample i s shown i n Table IV. TABLE IV COMPOSITION OF THE TREATMENT GROUPS Boys G i r l s . Total T - l Group 13 18 31 T-2 Group _17 10 27 Total 30 28 58 23 Instructional Procedure The following schedule of i n s t r u c t i o n was used i n t h i s study. During the f i r s t three days of the 1972-73 school year (September 8,9,10) the two Grade 6 classes remained i n t a c t and the time was used as an orientation period. Classroom teachers r e s t r i c t e d arithmetic i n s t r u c t i o n to a small amount of review of basic f a c t s and operations with whole numbers. On the next two school days (September 13 and 14) measures of population variables were administered and the i n s t r u c t i o n a l groups were formed. During the period September 15 to December 1, both groups were taught by the same classroom teacher (not the experimenter). Topics covered i n t h i s period were geometry, measures of perimeter, area and volume, d i v i s i o n with two and three d i g i t d i v i s o r s , signs of inequality, and the d i s -t r i b u t i v e law. Problem solving involved whole number compu-tatio n only. No work or review was done with f r a c t i o n s . Commencing December 4, i n s t r u c t i o n of both groups, T - l (Review) Group and T-2 (Modular Arithmetic) Group, was under-taken by the experimenter. The following schedule was followed. December 4 - 22 : Both groups reviewed these topics -concept of f r a c t i o n as a part of a whole and a subset of a set; review of terms such as proper, improper, mixed numeral, denominator, numerator; equivalent f r a c t i o n s , including lower terms, higher -terms, lowest terms; finding common denomin-ators by several methods. An outline of these lessons i s contained i n Appendix A. 24 Instruction i n t h i s section and i n succeeding sections was by various v i s u a l and manipulative devices such as a magnet board and an overhead projector. Practice was supplied by a 24-page booklet of pencil and paper exercises. Also each lesson commenced with a five-minute review of basic number facts and questions selected from measurement, geometry, and whole number computation. At the end of the unit the Fraction Concepts Test was administered to both groups. A copy of the Fraction Concepts Test i s included i n Appendix E. This three week period served to ensure a common background of knowledge of f r a c -tions for students i n both groups and also served to fami-l i a r i z e students with a d i f f e r e n t i n s t r u c t o r . One class period during t h i s time was used to administer the B.C. Arithmetic Test (Reasoning) to both groups. Following a Christmas vacation period of eleven days, T - l Group studied a unit on addition and subtraction of f r a c t i o n s for a period of twelve lessons. This was e s s e n t i a l l y a review of Grade 5 i n s t r u c t i o n although no assumption was made that the students remembered how to add or subtract f r a c t i o n s . No pre-test on these s k i l l s was given to the group. An outline of lessons i s included i n Appendix B. Practice was supplied by a 14-page booklet of pencil and paper exercises. During t h i s same period of January 3-19, 1973 T-2 Group studied a unit on modular arithmetic designed to develop an understanding of some mathematical properties which have an application to the d i v i s i o n of f r a c t i o n s . An outline of the 25 lessons and a copy of the 17-page booklet of practice exercises i s included i n Appendix C. In t h i s unit, i n s t r u c t i o n was supplemented by commercially prepared teaching tapes.^ Both groups continued to receive an i d e n t i c a l five-minute d r i l l and review session at the beginning of each lesson. The d r i l l s included review of f r a c t i o n a l ideas developed i n the review uni t but did not include any of the work that either group was currently doing. At the end of t h i s twelve-lesson period, T - l group was given the Adding and Subtracting Fractions Test. This was a computation test containing f i f t y items prepared by the experimenter. A copy of the Adding and Subtracting Fractions Test i s included i n Appendix E. At the same time, T-2 Group was given the Modular A r i t h -metic Test. This was a test on the a b i l i t y to compute with modular arithmetic and also on the understanding of some math-ematical properties of t h i s system. The t e s t was prepared by the experimenter and contained 49 items. A copy of the Modular Arithmetic Test i s included i n Appendix E. Following these units, both groups received i n s t r u c t i o n i n m u l t i p l i c a t i o n and d i v i s i o n of f r a c t i o n s for a period of seventeen lessons. An outline of these lessons i s included i n Appendix D. After ten lessons on m u l t i p l i c a t i o n , the Multi-p l i c a t i o n of Fractions Test was given to both groups. This was a computation test prepared by the experimenter and con-tained 34 items. A copy of the t e s t i s included i n Appendix E. 6 Wollensak Teaching Tape, Mathematics,Clock Arithmetic: Problems C-3504, 1968. 26 At the end of a further seven lessons, the D i v i s i o n of Frac-tions Test was given to both groups. This computation te s t was a c r i t e r i o n measure i n t h i s study. A description of how the D i v i s i o n of Fractions Test was developed i s contained i n a l a t e r section and a copy of the test i s included i n Appendix E. After the completion of the above schedule of i n s t r u c t i o n , the two groups were returned to the o r i g i n a l classroom teacher for i n s t r u c t i o n i n other topics i n the Grade 6 program. In the f i r s t three school days following the end of i n s t r u c t i o n by the experimenter, an i n d i v i d u a l interview was conducted with each c h i l d by the experimenter to determine the c h i l d ' s understanding of d i v i s i o n of f r a c t i o n s . A copy of the Interview Test and an explanation of how i t was administered i s included i n Appendix E. Also on the f i r s t day following the end of i n s t r u c t i o n by the experimenter, the T - l Group was re-tested with the Adding and Subtracting Fractions Test. At the same time, the T-2 Group was given the Adding and Subtracting Fractions Test. The T - l Group had received i n s t r u c t i o n i n adding and sub-t r a c t i n g f r a c t i o n s while the T-2 Group received i n s t r u c t i o n i n modular arithmetic. The T-2 Group had no review of adding or subtracting f r a c t i o n s since Grade 5. Control of Variables The experimental treatment i n t h i s study was the u n i t of i n s t r u c t i o n on modular arithmetic. The lessons i n t h i s t r e a t -ment were designed to a s s i s t the students i n the T-2 Group to 27 understand the mathematical properties which they would l a t e r need to understand d i v i s i o n of f r a c t i o n s . None of the subjects taking part i n the study were to l d the experimental nature of the study. I t was f e l t that the teaching arrangements were close enough to normal to be accepted by the students without query. The experimenter was known to the students and had taught arithmetic to a Grade 6 group i n the school i n the previous year. Thus a precedent had been set f o r him to teach Grade 6 students. The teacher variable was controlled by both groups being taught by the same teacher. However, t h i s meant that the time of day f o r the arithmetic lesson had to be d i f f e r e n t for the two groups. The times chosen for i n s t r u c t i o n i n arithmetic were 9:05 - 9:55 a.m. and 10:45 - 11:35 a.m. The T - l and T-2 Groups were assigned to a time f o r i n s t r u c t i o n by the toss of a coin. The T-2 Group was instructed from 9:05 to 9:55 each day of the week and the T - l Group was instructed from 10:45 to 11:35 each day of the week. The 9:05 time was immediately afte r school opening and the 10:45 time followed a 15-minute recess so that neither lesson followed immediately upon a lesson i n some other subject. No attempt was made to counter-balance the e f f e c t of t h i s variable by reversing the times of i n s t r u c t i o n for each group. Development of Tests The c r i t e r i o n measures i n t h i s study consisted of two tests, the Div i s i o n of Fractions Test f o r computation and 28 the I n t e r v i e w T e s t f o r understanding. Both t e s t s were designed by the experimenter. D i v i s i o n of F r a c t i o n s T e s t In order to develop a pool of q u e s t i o n s which would be r e p r e s e n t a t i v e of the computational s k i l l s r e q u i r e d f o r the d i v i s i o n of f r a c t i o n s , the f o l l o w i n g combinations of d i v i -dends, d i v i s o r s , and q u o t i e n t s were c o n s i d e r e d . F r a c t i o n "\ F r a c t i o n ~\ F r a c t i o n Whole number r + Whole number > = Whole number Mixed numeral^ Mixed numeralJ Mixed numeral There are 17 p o s s i b l e combinations o f the above i f whole number d i v i d e d by whole number i s omitted. In each case, the q u o t i e n t s c o u l d be a r r i v e d a t d i r e c t l y i n lowest terms or i t c o u l d be necessary f o r the students to r e - w r i t e the q u o t i e n t i n lowest terms. Using the above c l a s s i f i c a t i o n , i t was p o s s i b l e to i d e n t i f y 34 types of q u e s t i o n s . A d e c i s i o n was made to use no denomin-a t o r l a r g e r than 12 i n any d i v i d e n d or d i v i s o r . One q u e s t i o n of each type was w r i t t e n by the experimenter to c r e a t e a pool of 34 q u e s t i o n s . T h i s was judged, i n the experience of the exper-imenter, to be too many qu e s t i o n s f o r Grade 6 students to complete i n one s i t t i n g . From t h i s p o ol, 24 q u e s t i o n s were s e l e c t e d a t random. Students i n Grade 6 are u s u a l l y expected to express f r a c t i o n a l answers i n lowest terms. I t was reasoned t h a t students who had not reviewed a d d i t i o n and s u b t r a c t i o n of f r a c t i o n s might understand d i v i s i o n of f r a c t i o n s but not be 29 adept at writing answers i n lowest terms. For t h i s reason two sets of scores were co l l e c t e d for the Div i s i o n of Frac-tions Test. One set was designated Correct answers, that i s , correct answers expressed i n lowest terms. The other set was designated P a r t i a l answers, that i s , answers which were correct i n every respect except not being expressed i n lowest terms. Using the test scores of students included i n the two treatment groups, a r e l i a b i l i t y c o e f f i c i e n t of .90 was 7 obtained using the K-R 21 formula. Content v a l i d i t y was established for t h i s test by sub-mitting i t to a panel of f i v e experienced elementary teachers. Each agreed that i n his opinion the test was a v a l i d measure of d i v i s i o n of f r a c t i o n s for Grade 6. Interview Test An i n d i v i d u a l interview inventory was devised by the experimenter aft e r Gray's model. The purpose of such an instrument i s to evaluate pupil progress toward acquiring mathematical understanding. . . . the technique consists of facing a c h i l d with a problem or example, l e t t i n g him f i n d a solution, then challenging or questioning him to e l i c i t h is highest l e v e l of understanding of the process. 8 The s p e c i f i c purpose of the interview test i n t h i s study was to determine whether students performed the computation 7 Robert L. Thorndike and Elizabeth Hagen, Measurement and Evaluation i n Psychology and Education, 1961, p.181. 8 Roland F. Gray, "An Approach to Evaluating Arithmetic Understandings", The Arithmetic Teacher, 13:187-188, March 1966. 30 of d i v i s i o n of f r a c t i o n s solely by rote memory or whether they displayed a r a t i o n a l understanding of the process. In determining r a t i o n a l understanding, Brownell's d e f i n -9 i t i o n of "meaningful habituation" was used as a guide. Brownell l i s t s these c r i t e r i a for meaningful habituation: 1) a correct answer given at once and with apparent confidence, 2) i n a b i l i t y to convince the c h i l d that his answer was i n -correct by mentioning other p o s s i b i l i t i e s , and 3) the c h i l d ' s success i n defending the chosen answer. There were four items on the t e s t . Three of these items were chosen from the 24 items on the Div i s i o n of Fractions Test. The error count on a p i l o t form of the test adminis-tered to a group of Grade 7 students was used to select items of varying d i f f i c u l t y . A fourth item which was not included i n the Div i s i o n of Fractions Test was chosen to be the f i r s t item on the Interview Test. A questioning and scoring technique was developed and tested with Grade 7 students. The subjects were each given a test blank containing the questions and were allowed to use a pencil to f i n d the answers. Upon the completion of each question, the examiner noted the r e s u l t s on a separate scoring blank. A copy of the test blank, scoring blank, and d i r e c -tions for administering are included i n Appendix E. The t e s t was administered to 58 students and required an average of nine minutes to administer. 9 William A. Brownell, "Arithmetic Abstractions - Progress Toward Maturity of Concepts Under D i f f e r i n g Programs of Instruction", The Arithmetic Teacher, 10:324, October 1963. 31 A s c o r i n g technique was d e v i s e d to r e c o r d the d e s i r e d i n f o r m a t i o n . Student responses were c l a s s i f i e d as C o r r e c t , P a r t i a l (not i n lowest terms), or Wrong. C o r r e c t and P a r t i a l answers were c l a s s i f i e d as R a t i o n a l or Rote. A R a t i o n a l response was one i n which the student used the i n v e r s e r e l a -t i o n s h i p of m u l t i p l i c a t i o n and d i v i s i o n and the r e c i p r o c a l of the d i v i s o r to e x p l a i n the d i v i s i o n p r o c e s s . I t was not necessary f o r the student to use the term " r e c i p r o c a l " . A Rote response was one giv e n by a student who c o u l d g i v e no reason f o r the process except, perhaps, t h a t t h i s i s what he had been t o l d to do. I f the student made no attempt to perform the o p e r a t i o n , t h i s f a c t was noted on the score sheet. A l s o , i f the student used any other a l g o r i t h m except the one taught, t h i s f a c t was noted. I t was not p o s s i b l e to make any s t a t i s t i c a l t e s t s o f the r e l i a b i l i t y of the Interview T e s t i n t h i s study. A d d i t i o n a l t e s t s Four a d d i t i o n a l t e s t s were c o n s t r u c t e d by the experimenter. The F r a c t i o n Concepts T e s t was a t e s t of understanding which r e q u i r e d no computation. The items i n the t e s t were c o n s t r u c t e d to measure the student's understanding of the f r a c t i o n a l concepts taught i n the review u n i t a t the begin n i n g of the study. The Adding and S u b t r a c t i n g F r a c t i o n s T e s t was a computation t e s t c o n t a i n i n g items r e p r e s e n t a t i v e of the 32 types of adding and s u b t r a c t i n g q u e s t i o n s reviewed with the T - l Group. The Modular A r i t h m e t i c T e s t c o n t a i n e d items intended to measure computation w i t h modular a r i t h m e t i c and a l s o the student's understanding of the mathematical p r i n c i p l e s chosen to be i l l u s t r a t e d d u r i n g the u n i t of i n s t r u c t i o n . The M u l t i p l i c a t i o n of F r a c t i o n s T e s t was a computation t e s t c o n t a i n i n g items r e p r e s e n t a t i v e of the types of m u l t i -p l i c a t i o n q u e s t i o n s taught d u r i n g the u n i t of i n s t r u c t i o n . These t e s t s were judged to have content v a l i d i t y because they c o n t a i n e d items s i m i l a r to m a t e r i a l taught. Using the t e s t scores of the students i n the treatment groups t a k i n g the t e s t s , the f o l l o w i n g r e l i a b i l i t y c o e f f i c i -ents were obtained u s i n g the K-R 21 formula: F r a c t i o n Concepts T e s t .876 Adding and S u b t r a c t i n g F r a c t i o n s T e s t .880 M u l t i p l i c a t i o n of F r a c t i o n s T e s t .958 Modular A r i t h m e t i c T e s t .867 S t a t i s t i c a l Treatment Procedure The data from the t e s t s were t r e a t e d as f o l l o w s . The means, ad j u s t e d means, and the standard e r r o r s were c a l c u -l a t e d u s i n g the BMDX82 computer program 1 1 and the a n a l y s i s 10 Thorndike and Hagen, l o c . c i t . 11 W.J. Dixon,ed., BMP - Biomedical Computer Programs, 1972, p.81-85. 33 12 of c o v a r i a n c e was done u s i n g the BMD0 5V computer program at the U n i v e r s i t y of B r i t i s h Columbia Computing Centre. In t h i s study, I.Q. scores, A r i t h m e t i c Computation scores, A r i t h m e t i c Reasoning scores, and F r a c t i o n Concepts scores were used as c o v a r i a t e s . Subjects were d i v i d e d i n t o h i g h and low I.Q. groups on the b a s i s of I.Q. scores d i v i d e d a t the median. C h i square t e s t s were used to t e s t f o r s i g n i f i c a n c e of d i f f e r e n c e s of numbers of s u b j e c t s f a l l i n g i n the d i f f e r e n t response c a t e g o r i e s f o r the items on the Interview T e s t . In cases where c e l l s i z e s are s m a l l , the u s u a l method of compu-t i n g the c h i square would g i v e an over-estimate of the t r u e v a l u e and, as a r e s u l t , some hypotheses might be r e j e c t e d which i n f a c t should not have been. Therefore to c o r r e c t f o r t h i s p o s s i b i l i t y , where c e l l s i z e s were l e s s than 5, the Yates c o r r e c t i o n f o r small sample s i z e s was employed. Whenever t h i s c o r r e c t i o n was used, the f a c t was noted i n the t a b l e . D e c i s i o n r u l e In the s t a t i s t i c a l t e s t i n g procedure, the n u l l h y p o t h e s i s was r e j e c t e d whenever the p r o b a b i l i t y of committing a Type I e r r o r was equal to or l e s s than .05 . 12 I b i d . , p.543 - 555. 34 CHAPTER III ANALYSIS OF THE DATA The data were analyzed to tes t the hypotheses set fo r t h i n Chapter I and re-stated here. 1) Students who are taught modular arithmetic w i l l show a s i g n i f i c a n t l y better a b i l i t y to divide f r a c t i o n s , as mea-sured by a test of computation, than students who are not taught modular arithmetic. 2) Students who are taught modular arithmetic w i l l show a s i g n i f i c a n t l y better understanding of d i v i s i o n of fra c t i o n s , as measured by an interview test, than students who are not taught modular arithmetic. Findings From the Division of Fractions Test The means, adjusted means, and standard errors for the Divisi o n of Fractions Test (Correct answers only) are presented i n Table V. TABLE V MEANS, ADJUSTED MEANS, AND STANDARD ERRORS OF THE DIVISION OF FRACTIONS TEST (CORRECT ANSWERS ONLY) Mean Adj. Mean Std. Error Boys 15.97 15.59 1.31 Sex G i r l s 18.64 19.05 1.35 High 18.33 16.53 1.53 I.Q. Low 16.11 18.04 1.61 T - l 18.84 18.32 1.49 Treatment T-2 15.45 16.04 1.66 35 The significance of the differences of the adjusted mean scores for Correct responses only was tested by means of a 2(Treatment) by 2(Sex) by 2(I.Q.) analysis of covariance. A summary of the analysis of covariance i s presented i n Table VI. TABLE VI SUMMARY OF THE ANALYSIS OF COVARIANCE OF DIVISION OF FRACTIONS (CORRECT ANSWERS ONLY) SCORES WITH I.Q.,ARITHMETIC COMPUTATION, ARITHMETIC REASONING, AND FRACTION CONCEPTS HELD CONSTANT Source df Mean Square F Treatment 1 25.30 1.13 Sex 1 133.20 * 5.93 I.Q. 1 2.64: 0.12 Treatment x Sex 1 5.03 0.22 Treatment x I.Q. 1 34.83 1.55 Sex x I.Q. 1 44.22 1.97 Treatment x Sex x I.Q. 1 0.59 0.03 Error (within) 1 22.45 Total 53 * p <.05 36 The means, adjusted means, and standard errors for the Divisi o n of Fractions Test (Correct and P a r t i a l l y correct answers) are presented i n Table VII. TABLE VII MEANS, ADJUSTED MEANS, AND STANDARD ERRORS OF THE DIVISION OF FRACTIONS TEST (CORRECT AND PARTIAL ANSWERS) Mean Adj. Mean Std. Error Sex Boys 16.86 16.30 1.33 G i r l s 19.43 20.03 1.38 I.Q. High 18.80 17.21 1.57 Low 17.36 19.07 1.65 T - l 19.45 19.08 1.53 Treatment T-2 16.55 16.99 1.70 The significance of the differences of the adjusted mean scores for the Correct and P a r t i a l l y correct responses was tested by means of a 2(Treatment) by 2(Sex) by 2(1.Q.) analysis of covariance. A summary of the analysis of covariance i s presented i n Table VIII. 37 TABLE VIII SUMMARY OF THE ANALYSIS OF COVARIANCE OF DIVISION OF FRACTIONS (CORRECT AND PARTIAL ANSWERS) SCORES WITH I.Q.,ARITHMETIC COMPUTATION,ARITHMETIC REASONING,AND FRACTION CONCEPTS HELD CONSTANT Source df Mean Square F Treatment 1 15.64 0.65 Sex 1 148.21 6.19 I.Q. 1 1.98 0.08 Treatment x Sex 1 4.13 0.17 Treatment x I.Q. 1 33.93 1.42 Sex x I.Q. 1 16.10 0.67 Treatment x Sex x I.Q. 1 8.85 0.37 Error (within) 46 23.96 Total 53 p < .05 From Table VI and VIII i t can be seen that i n both cases, using Correct answers and Correct and P a r t i a l l y correct ans-wers, the only s i g n i f i c a n t difference was that due to sex, favouring g i r l s over boys. Since the difference due to t r e a t -ments ( T - l compared with T-2) was not s i g n i f i c a n t , the f i r s t hypothesis was not supported. Since i n t e l l i g e n c e test scores were included as one of the covariates, no s i g n i f i c a n t d i f f e r -ence due to i n t e l l i g e n c e was expected. Intelligence was included as part of the analysis so that possible i n t e r a c t i o n between I.Q. and Treatment or Sex might be revealed. 38 Findings From the Interview Test The response categories i n the Interview Test are abbrev-iated throughout the analysis. A l i s t of abbreviations used i n the analysis i s given here for convenience. C - Correct answer P — P a r t i a l l y correct answer (not i n lowest terms) W — Wrong answer N — No attempt to answer R - Rational response Ro - Rote response The data from the Interview Test were grouped according to the number of students i n each treatment group giving d i f f e r e n t types of answers to each of the four test items. Table IX shows the number of students giving C,P,W, and N answers. TABLE IX THE NUMBER OF T - l AND T-2 SUBJECTS IN C,P,W, AND N CATEGORIES FOR EACH OF THE FOUR INTERVIEW TEST ITEMS Test item Test item Test item Test item 1 2 3 4 Response T - l T-2 T - l T-2 T - l T-2 T - l T-2 category (N=31) (N=27) (N=31)(N=27) (N=31)(N=27) (N=31)(N=27) C 30 26 29 18 29 17 24 14 p 0 1 0 4 0 1 0 2 w 1 0 2 5 2 8 7 10 N 0 0 0 0 0 1 0 1 39 From the d i s t r i b u t i o n of scores i n Table IX, the differences between treatment groups i n the number of students obtaining Correct answers for each te s t item were tested for significance by means of the chi square t e s t . Table X shows the r e s u l t s of the ch i square t e s t s . (To be s i g n i f i c a n t at the . 0 5 l e v e l , chi square i n these analyses was equal to or greater than 3 . 8 4 1 . ) TABLE X COMPARISON OF THE TREATMENT GROUPS IN THE NUMBER OF SUBJECTS GIVING CORRECT ANSWERS TO EACH ITEM IN THE INTERVIEW TEST T - l Group T -2 Group Test item (N=31) (N=27) ^S . Number of correct Number of correct answers answers 1 . ( | i- j = ) 30 " 26 1 ? ( 9 3 -s- •§• = ) 29 18 6 . 8 6 1 l b o 3 . (4-i- t- 5 = ) 29 17 8 . 1 7 4 b 4. (3-| -s- 1-| = ) 24 14 4 . 1 9 8 1 f - f too small to apply Yates correction. o e rv -i . 40 Also from the d i s t r i b u t i o n of scores i n Table IX, the differences between the treatment groups i n the t o t a l number of students obtaining Correct and P a r t i a l answers for each tes t item were tested for significance by means of the chi square t e s t . Table XI shows the re s u l t s of the chi square t e s t s . TABLE XI COMPARISON OF THE TREATMENT GROUPS IN THE NUMBER OF SUBJECTS GIVING CORRECT AND PARTIAL ANSWERS TO EACH ITEM IN THE INTERVIEW TEST T - l Group T-2 Group " \ r * Test item (N=31) (N=27) J\ Number of C and P Number of C and P answers answers 1. (-| -r- ~ = ) 30 27 0 1 d ' K ^ -j- 4 = ) 29 22 .927 16 • 8*4 " 3 3. (4-^ -j- 5 = ) 29 18 6.861 D 4. (3r -e- l i = ) 24 16 2.188 1 Yates correction applied. From Table X i t can be seen that there were s i g n i f i c a n t differences i n the number of students scoring Correct answers i n Test Items 2, 3, and 4. From Table XI i t can be seen that there was a s i g n i f i c a n t difference i n Item 3 for students scoring Correct and P a r t i a l answers. In each case 41 the difference favoured the T—1 (Review) Group. Since there were no differences favouring the T—2 (Modular Arithmetic) Group i n any test item regardless of how i t was scored, the f i r s t hypothesis was not supported. However, the chief purpose of the Interview Test was to obtain a measure of understanding. For t h i s purpose, each Correct and P a r t i a l answer was categorized during the scoring of the test as being a Rational or a Rote response. Table XII shows the number of Rational and Rote responses i n each treatment group for each test item. TABLE XII THE NUMBER OF T - l AND T-2 SUBJECTS GIVING RATIONAL AND ROTE RESPONSES TO EACH ITEM IN THE INTERVIEW TEST Test item Test item Test item Test item 1 2 3 4 Response category R Ro T - l T-2 (N=30)(N=27) 3 4 27 23 T - l T-2 (N=29)(N=22) 4 4 25 18 T - l T-2 (N=29)(N=18) 4 2 25 16 T - l T-2 (N=24)(N=16) 2 2 22 14 From the d i s t r i b u t i o n of scores i n Table XII the d i f f e r -ences between treatment groups i n the number of students scoring i n the Rational category were tested by means of the chi square t e s t . Table XIII shows the r e s u l t s of the chi square t e s t s . 42 TABLE XIII COMPARISON OF THE TREATMENT GROUPS IN THE NUMBER OF SUBJECTS GIVING RATIONAL RESPONSES TO EACH ITEM IN THE INTERVIEW TEST Test item T - l Group T-2 Group 1 £ - ± - ) (N=30) 3 (N=27) 4 .027 1 (N=29) (N=22) 4 4 .006 (N=29) (N=18) 3 . ( 4 i t- 5~ = ) 4 2 2 • — (N=24)- (N=16) 4. C 3 § ^ i f = ) 2 2 • ^ — ™ 1 Yates correction used. 2 f - f too small i n Items 3 and 4 to apply Yates o e correction. From t h i s table i t can be seen that there were no s i g n i f i c a n t differences i n the number of subjects giving Rational responses to any test item. Since there was no observed difference due to treatment, the second hypothesis was not supported. 43 Findings From the Modular Arithmetic Test The T-2 (Modular Arithmetic) Group was the only group to receive the Modular Arithmetic Test and therefore no com-parison of Groups ( T - l with T-2) could be made. However, as a matter of i n t e r e s t the means and standard deviations of the Modular Arithmetic Test for the t o t a l T-2 Group and the T-2 Group divided by sex and i n t e l l i g e n c e are presented i n Table XIV. TABLE XIV MEANS AND STANDARD DEVIATIONS OF THE MODULAR ARITHMETIC TEST SCORES FOR THE TOTAL T-2 GROUP AND THE T-2 GROUP DIVIDED BY SEX AND INTELLIGENCE N Mean Std. Deviation Total T-2 Group 27 33.5 8.4 Boys 17 Sex G i r l s 10 35.1 30.9 7.2 10.0 jHlqh 15 Intelligence Low 12 35.9 30.5 9.7 5.3 1 Intelligence c l a s s i f i e d as High or Low I.Q. scores div-ided at the median of the combined T - l and T-2 Groups (104.5). 44 A d d i t i o n a l F i n d i n g s Since i t i s p o s s i b l e t h a t success i n computation of d i v i s i o n of f r a c t i o n s c o u l d be r e l a t e d to the student's a b i l i t y to compute m u l t i p l i c a t i o n of f r a c t i o n s , the scores from the M u l t i p l i c a t i o n of F r a c t i o n s Test were a l s o examined. The means, a d j u s t e d means, and standard e r r o r s f o r the M u l t i -p l i c a t i o n o f F r a c t i o n s Test are presented i n Table XV. TABLE XV MEANS, ADJUSTED MEANS, AND STANDARD ERRORS OF THE MULTIPLICATION OF FRACTIONS TEST Mean A d j . Mean Std. E r r o r Boys 23.17 22.21 2.05 Sex G i r l s 26.00 27.21 2.12 High I.Q. Low 25.97 23.01 21.32 27.30 2.29 2.37 T - l Treatment T-2 27.87 20.70 27.21 21.66 1.97 2.22 The s i g n i f i c a n c e of the d i f f e r e n c e s of the ad j u s t e d mean scores was t e s t e d by means of a 2(Treatment) by 2(Sex) by 2(I.Q.) a n a l y s i s of c o v a r i a n c e . A summary of the a n a l y s i s of c o v a r i a n c e i s presented i n Table XVI. 45 TABLE XVI SUMMARY OF THE ANALYSIS OF COVARIANCE OF THE MULTIPLICATION OF FRACTIONS TEST SCORES WITH I.Q., ARITHMETIC COMPUTATION, ARITHMETIC REASONING, AND FRACTION CONCEPTS HELD CONSTANT Source df Mean Square F Treatment 1 320.14 6.08 Sex 1 109.32 2.08 I.Q. 1 137.46 2.61 Treatment x Sex 1 54.06 1.03 Treatment x I.Q. 1 117.62 2.23 Sex x I.Q. 1 44.32 0.84 Treatment x Sex x I.Q. 1 0.28 0.01 Error (within) 45 52.64 Total 53 p <\05 From Table XVI i t can be seen that there was a s i g n i -f i c a n t difference between the treatment groups, favouring the T-l(Review) Group over the T-2(Modular Arithmetic) Group. I t would appear from t h i s finding that at the time i n s t r u c t i o n i n d i v i s i o n of fr a c t i o n s was begun, students i n the T - l Group were better able to compute m u l t i p l i c a t i o n of fr a c t i o n s than students i n T-2 Group. 46 The a b i l i t y of the two treatment groups to add and subtract f r a c t i o n s was examined also. Although the T-2 Group had received no review of adding and subtracting f r a c t i o n s since the previous school year, they were given the Adding and Subtracting Fractions Test aft e r completing the program of m u l t i p l i c a t i o n and d i v i s i o n of f r a c t i o n s . At the same time, the T - l Group were re-tested with t h i s same te s t . The means, adjusted means, and standard errors of the Adding and Subtracting Fractions Test are presented i n Table XVII. TABLE XVII MEANS, ADJUSTED MEANS, AND STANDARD ERRORS OF THE ADDING AND SUBTRACTING OF FRACTIONS TEST Mean Adj. Mean Std. Error Boys 13.16 12.35 2.68 Sex G i r l s 14.46 18.20 2.77 I.Q. High Low 18.87 11.96 16.69 13.55 2.83 2.98 Treatment T - l T-2 25.06 3.81 24.62 4.31 2.57 2.89 The significance of the differences of the adjusted mean scores was tested by means of a 2(Treatment) by 2(Sex) by 2(1.Q.) analysis of covariance. A summary of the analysis of covariance i s presented i n Table XVIII. 47 TABLE XVIII SUMMARY OF THE ANALYSIS OF COVARIANCE OF THE ADDING AND SUBTRACTING OF FRACTIONS TEST SCORES WITH I.Q. , ARITHMETIC COMPUTATION, ARITHMETIC REASONING, AND FRACTION CONCEPTS HELD CONSTANT Source df Mean Square . F. . Treatment 1 4887.84 57.09 * Sex 1 1.29 0.01 I.Q. 1 72.72 0.85 Treatment x Sex 1 78.93 0.92 Treatment x I.Q. 1 506.52 5.92 * Sex x I.Q. 1 75.57 0.88 Treatment x Sex x I.Q. 1 3.04 0.04 Error (within) 46 85.62 Total 53 p < .05 From Table XVIII i t can be seen that there was a s i g n i -f i c a n t difference between the treatment groups (T—1 compared with T-2) i n favour of the T - l Group, as might be expected. It can also be seen that there was a s i g n i f i c a n t difference due to in t e r a c t i o n of method with I.Q.. Subjects had been divided into high and low I.Q. groups on the basis of in t e l l i g e n c e test scores divided at the median (104.5). The adjusted mean scores of the Adding and Subtracting of 48 Fractions Test for the high and low i n t e l l i g e n c e groups within each treatment group are presented i n Table XIX. TABLE XIX ADJUSTED MEAN ADDING AND SUBTRACTING OF FRACTIONS TEST SCORES FOR T - l AND T-2 GROUPS CLASSIFIED AS HIGH AND LOW I.Q. SCORES DIVIDED AT THE MEDIAN, 104.5. Intelligence Grouping High Low T - l Group 30.1 19.5 Treatment T-2 Group 3.2 5.7 From Table XIX i t can be seen that the difference due to i n t e r a c t i o n of treatment with I.Q. favoured the low I.Q. group within T-2 Group. However the scores f o r the T-2 Group were so low that i t i s not l i k e l y that any inference can be drawn from the i n t e r a c t i o n . 4 9 CHAPTER IV DISCUSSION AND CONCLUSIONS The problem under discussion i n t h i s study was whether Grade 6 students who were taught modular arithmetic would show greater s k i l l i n performing the computation of d i v i s i o n of fra c t i o n s and greater understanding of the mathematical basis of the d i v i s i o n of fractio n s than students who were not taught modular arithmetic. The research hypotheses were based on the b e l i e f that Grade 6 students who were taught modular arithmetic would show greater s k i l l and understanding. An analysis of the re s u l t s of the Divis i o n of Fractions Test which was used as a measure of computational s k i l l revealed no support for the f i r s t hypothesis, which was that modular arithmetic would improve s k i l l i n computation of d i v i s i o n of f r a c t i o n s . An analysis of the re s u l t s of the Interview Test which was used as a measure of understanding revealed no support for the second hypothesis, which was that modular arithmetic would improve understanding of the mathematical basis of d i v i s i o n of f r a c t i o n s . Discussion Difference due to sex Although difference due to sex was not under i n v e s t i -gation i n t h i s study, the difference between the adjusted 50 mean scores for boys and g i r l s i n the D i v i s i o n of Fractions Test seems worth noting. The adjusted mean score for g i r l s was higher than the adjusted mean score for boys. The difference was s i g n i f i c a n t at the .0 5 l e v e l . Teaching d i v i s i o n of f r a c t i o n s The small number of r a t i o n a l responses from both groups i n the Interview Test may be supporting evidence that the concepts involved i n the d i v i s i o n of f r a c t i o n s algorithm may be too abstract for children of t h i s age group, thereby supporting Bates' contention. 1 Otherwise the unit on modular arithmetic seems to be unrelated to student's a b i l i t y to compute d i v i s i o n of f r a c t i o n s . M u l t i p l i c a t i o n of f r a c t i o n s The analysis of covariance of the M u l t i p l i c a t i o n of Fractions Test revealed a s i g n i f i c a n t difference between the adjusted mean scores of the treatment groups, favouring the T - l (Review) Group over the T-2 (Modular Arithmetic) Group. No reason can be given at t h i s point for t h i s difference. However, the r e l a t i o n s h i p between m u l t i p l i c a t i o n and d i v i s i o n of f r a c t i o n s might be a topic for further study. Adding and subtracting f r a c t i o n s The s i g n i f i c a n t difference i n adding and subtracting f r a c t i o n s between the T - l and T—2 Groups i n favour of the T - l Group was to be expected since the T-2 Group had had no review of the topic since the previous school year. However, the poor retention of s k i l l i n adding and subtracting 1 Thomas Bates, "The Road to Inverse and Multiply", The Arithmetic Teacher, 15:348, A p r i l 1968. 51 fr a c t i o n s on the part of the T-2 Group would seem to indicate that the substitution of an alternative program (modular arithmetic for adding and subtracting fractions) causes students to suffer i n s k i l l development. There was a s i g n i f i -cant i n t e r a c t i o n between Treatment and I.Q. favouring the low I.Q. group within the T-2 Group. However, the low scores of the T-2 Group i n adding and subtracting f r a c t i o n s makes i t un l i k e l y that an inference can be drawn from t h i s r e s u l t as f a r as the T—2 Group i s concerned. Instructional time A t o t a l of forty-three i n s t r u c t i o n a l periods were used i n t h i s study. This i s probably more time than i s normally devoted to the review and teaching of f r a c t i o n s i n Grade 6. Thus any implications that t h i s study might have for the teaching of d i v i s i o n of f r a c t i o n s i n Grade 6 might be influenced by the extensive nature of the i n s t r u c t i o n . However, not enough time may have been devoted to i n s t r u c t i o n i n modular arithmetic. A greater e f f e c t on the T-2 Group might have been observed i f more periods of i n s t r u c -tion had been given. Interview Test There i s one aspect of the Interview Test that seems to be of s u f f i c i e n t importance to mention here. This experimenter came to the conclusion that knowing the students as well as he did may have been more of a disadvantage than an advantage. While i t helped to put students at ease, i t also seemed to put the interviewer i n the position of sub-consciously 52 a n t i c i p a t i n g responses. I t might be a better technique for the interviewer to be someone unknown to the student. Limitations of the study Several l i m i t a t i o n s of t h i s study should be noted. The students a l l came from the same school and the sample size was small. This l i m i t a t i o n was o f f s e t to some extent by the experimenter being free to create two i n s t r u c t i o n a l groups at random from the sample. The conclusions from t h i s study w i l l r e l a t e only to other groups with the same character-i s t i c s . The time of day of i n s t r u c t i o n may have had some e f f e c t on the r e s u l t s of the study. The T-2 Group was taught f i r s t period i n the day while the T - l Group was taught aft e r a morning recess. I t may be that t h i s recreational period served to stimulate the T - l Group mentally. Suggestions for further study I t would be of i n t e r e s t to r e p l i c a t e t h i s study with a group of students who had a stronger background of mathe-matical understanding. The mathematics program which many of these students had followed i n t h e i r elementary school years did not stress mathematical properties such as a s s o c i a t i v i t y , commutativity, and the i d e n t i t y elements. Had the students i n the T-2 Group been more f a m i l i a r with these concepts, modular arithmetic may have been of more help to them, p a r t i c u l a r l y i n understanding the concept of the r e c i p r o c a l . 5 3 I t i s suggested that with further refinement, the interview test technique could reveal useful information about the l e v e l of mathematical understanding which children possess i n other areas of elementary school mathematics. The poor retention of adding and subtracting f r a c t i o n s s k i l l s on the part of the T-2 Group suggests the need for further studies to determine i f substituting modular a r i t h -metic for a review of adding and subtracting f r a c t i o n s does, i n f a c t , i n h i b i t pupil growth i n the l a t t e r s k i l l . Conclusions 1. Teaching modular arithmetic to the Grade 6 students p a r t i c i p a t i n g i n t h i s study did not appear to improve th e i r a b i l i t y to compute d i v i s i o n of f r a c t i o n s . 2. Teaching modular arithmetic to the Grade 6 students p a r t i c i p a t i n g i n t h i s study did not appear to improve th e i r understanding of the mathematical basis of the d i v i s i o n of f r a c t i o n s . BIBLIOGRAPHY 55 Bates, Thomas. "The Road to Inverse and Multiply. "The Arithmetic Teacher, 15:347-354, A p r i l 1968. Bidwell, James K. "Some Consequences of Learning Theory Applied to Divis i o n of Fractions". School Science and Mathematics, 71: 501-7, June 1971. Botts, Truman. "Fractions i n the New Elementary C u r r i c u l a " . The Arithmetic Teacher, 15: 216-20, March 1968. B r i t i s h Columbia. Department of Education, Div i s i o n of Curriculum. Programme of Studies f o r the Intermediate Grades, Arithmetic 1965, Grades IV,V,VI. V i c t o r i a , 1965. B r i t i s h Columbia* Department of Education, Div i s i o n of Instructional Services. Mathematics, Primary - Years 1-3, Years 7-8. V i c t o r i a , 1972. B r i t i s h Columbia, Department of Education, Div i s i o n of Tests, Standards, and Research. B r i t i s h Columbia Tests, A r i t h - metic Computation, V-VI, Form A. V i c t o r i a , 1951. . B r i t i s h Columbia Tests,Arithmetic Reasoning, VI-VII, Form A. V i c t o r i a , 1951. Brownell, William A. "Arithmetic Abstractions - Progress Toward Maturity of Concepts Under D i f f e r i n g Programs of Instruction." The Arithmetic Teacher, 10:321-9, October 1963. Brueckner, Leo J . and Foster E. Grossnickle. Making A r i t h - metic Meaningful. Philadelphia, The John C. Winston Company, 1953. Dixon, W.J. ed BMP - Biomedical Computer Programs. Los Angeles, University of C a l i f o r n i a , Health Service Computing F a c i l i t y , 1965 (Revised 1972). Gray, Roland F. "An Approach to Evaluating Arithmetic Under-standings ." Th^_ArJL_ttjme_tJ1c_Teac^ 13:187-91 March 1966. Hartung, Maurice L. e t . a l . Seeing Through Arithmetic, Grades 1-6. Toronto, W.J. Gage Ltd., 1965. Howard, Charles F. and Enoch Dumas. Teaching Contemporary Mathematics i n the Elementary School. New York, Harper and Row, 1966. Ingersoll, Gary M. "An Experimental Study of Two Methods of Presenting the Inversion Algorithm i n Division of Frac-t i o n s . " C a l i f o r n i a Journal of Educational Research, 22:17-25, January 1971. 56 Lyda, W.J. and M.D. Taylor. " F a c i l i t a t i n g an Understanding of the Decimal Numeration System Through Modular Arithmetic. The Arithmetic Teacher, 11:101-3, February 1964. Marks, John L., C. Richard Purdy, Lucien B. Kenny. Teaching Elementary School Mathematics f o r Understanding. New York McGraw-Hill, 1965. Mueller, Francis J . . Arithmetic, Its Structure and Concepts. Englewood C l i f f s , N.J., Prentice-Hall Inc. 1964 (Second edition) . . "Modular Arithmetic" i n the 27th. Yearbook of the National Council of the Teachers of Mathematics. Washing ton, D.C., 1963. Otis, Arthur S. and Roger T. Lennon. Otis Lennon Mental A b i l i t y Test, Elementary II, Form J . New York, Harcour.t, Brace, and World, Inc. 1967. ; . Manual For Administration, Otis-Lennon Test of Mental A b i l i t y , Elementary II, Form J . New York, Harcourt Brace, and World, Inc. 1967. Price, Jack. "Why Teach Div i s i o n of Common Fractions?" The Arithmetic Teacher, 16:111-2, February 1969. Riess, Anita P." A New Rationale f o r the Teaching of Fractions. Harvard Educational Review, 25:105-25, Spring 1955. Spitzer, Herbert F. Teaching Elementary School Mathematics. Boston, Houghton M i f f l i n Company, 1967. Thorndike, Robert L. and Elizabeth Hagen. Measurement and Evaluation i n Psychology and Education, (Second e d i t i o n ) . New York, John Wiley and Sons, Inc., 1961. Westcott, A l v i n M. and James A. Smith. Creative Teaching i n the Elementary School. Boston, A l l y n and Bacon, 1967. Willerding, Margaret F. Mathematics Around the Clock. Pasadena, C a l i f o r n i a , Franklin Publications, 1968. Wollensak Teaching Tape. Mathematics, Clock Arithmetic: Problems C-3504. St. Paul, Minnesota, Minnesota Mining and Manufacturing Company, 1968. APPENDIX A OUTLINE OF LESSONS FOR REVIEW OF FRACTION CONCEPTS T - l AND T-2 GROUPS 58 LESSONS FOR REVIEW OF FRACTION CONCEPTS Appropriate practice exercises were provided for these lessons by means of a 24-page booklet of exercises prepared by the experimenter. Lesson 1: Obj ective : To review the concept of f r a c t i o n as part of a whole and subset of a set. Lesson : Overhead transparencies of geometric shapes separated into halves, thirds , fourths, f i f t h s , sixths, and eighths. Counters on overhead projector to show sub-sets of a set. Students to have an opportunity to man-ipulate devices. Stress equal parts and equal subsets. Vocabulary - numerator, denominator, proper, improper, mixed numeral. Lesson 2: Objective : To continue review of concept of f r a c t i o n . Lesson : Continue a c t i v i t i e s of Lesson 1. Use practice exercises from Lesson 1 for discussion. Lesson 3: Obj ective : To review equivalent f r a c t i o n s . Lesson : Use paper f o l d i n g to show equivalent f r a c t i o n s which are part of a whole. Use multi-coloured counters on overhead projector to show equivalent fract i o n s which are subsets of a set. Vocabulary - lower terms, higher terms, lowest terms, equivalent. 59 Review of Fraction Concepts Lesson 4: Objective : To review equivalent f r a c t i o n s ; building sets of equivalent f r a c t i o n s ; changing to lower and higher terms. Lesson : Practice with f o l d i n g paper and magnet board 1 x 2 2 using f r a c t i o n a l parts. Also i l l u s t r a t e - 2 x 2 = "4 =, ^ 2 f 2 1 a n d 4~TT = 2 ' Lesson 5: Objective : To review equivalent f r a c t i o n s j writing f r a c t i o n s i n lowest terms. Lesson : Use blocks on overhead projector to i l l u s t r a t e 4 black » E 5 E a J - ! i black ••DQ 2 Give similar examples and practice on blackboard. Lesson 6: Objective : To review fi n d i n g the missing term i n a pair of equivalent f r a c t i o n s . Lesson : Learn f i r s t to f i n d by inspection what number both numerator and denominator of the complete f r a c t i o n i n a pair of f r a c t i o n s can be m u l t i p l i e d or divided by. Lesson 7: Objective : To review equivalent f r a c t i o n s ; f i n d i f a pair of f r a c t i o n s are equivalent. Lesson : Use concrete materials ( f r a c t i o n a l parts) to 3 6 f i n d i f , f o r example, = -g. Review Lesson 6 f o r inspec-ti o n method. 60 Review of Fraction Concepts Lesson 8: Objective : Review cross multiply t e s t to f i n d missing term i n a pair of equivalent f r a c t i o n s . Lesson : I l l u s t r a t e cross multiply t e s t . Give black-board pr a c t i c e . Lesson 9: Obj ective : To review comparison of size of f r a c t i o n s . Lesson : Build several l i n e s of a f r a c t i o n chart as a class exercise. Show on an overhead transparency how to use the chart to compare size of f r a c t i o n s . I 3 j 4 5 8 10 3 4 5 9 12 15 Build charts i n the practice booklet. Lesson 10: Obj ective : To review fi n d i n g the denominator common to two or more fr a c t i o n s , using the chart. Lesson : Use an overhead transparency of the f r a c t i o n chart to show how to locate a denominator common to two or more f r a c t i o n s . Lesson 11: Ob]ective : To review f i n d i n g a common denominator f o r two or more fr a c t i o n s by multiplying denominators. 61 Review of Fraction Concepts Lesson 11 (continued): Lesson : Find a common denominator f o r ^ , -|; . Show by an array that 2 x 3 gives a product that can be divided by both 2 and 3. Thus the product can be a common denominator for these two f r a c t i o n s . Lesson 12: Objective : To review lowest common denominator f o r two or more f r a c t i o n s . Lesson : Discuss methods which can be used: a) by inspec-ti o n - Can the denominator of one f r a c t i o n be a common denominator f o r the pair? b) use a chart, c) use the product of denominators. Lesson 13: Objective : To review writing a mixed numeral i n the form of an improper f r a c t i o n and v i c a versa. Lesson : Give students practice on the magnet board with situations such as ^m., !••>. f\ APPENDIX B OUTLINE OF LESSONS FOR ADDING AND SUBTRACTING FRACTIONS T - l GROUP ONLY 63 LESSONS FOR ADDING AND SUBTRACTING FRACTIONS Appropriate practice exercises were provided for these lessons by means of a 14-page booklet of exercises prepared by the experimenter. Lesson 1: Objective : To review pre-vacation study of f r a c t i o n concepts. Lesson : Use an overhead projector and transparencies and a magnet board to review b r i e f l y - f r a c t i o n s as a part of a whole and a sub-set of a set; equivalent fr a c t i o n s ; missing terms i n a pair of equivalent f r a c t i o n s . Lesson 2: Objective : Review mixed numerals and improper f r a c -t i o n s . Lesson : Give students an opportunity to perform a num-ber of manipulations on the magnet board, such as: a a 5 fourths Lesson 3: and Objective : Pupils learn to add and subtract f r a c t i o n s with l i k e denominators. 64 Lessons For Adding and Subtracting Fractions Lesson 3 (continued): Lesson : Use the overhead projector and opaque f r a c -t i o n a l parts to i l l u s t r a t e the following. 1 1 - 1 1 1 - 1 3 + 3 ~ 3 4 " 4 " 4 Use other s i m i l a r examples. Lesson 4: Objective : To provide practice f o r pupils i n adding and subtracting f r a c t i o n s with l i k e denominators. Lesson : Have pupils i l l u s t r a t e addition and subtract-ion with the overhead projector and magnet board with f r a c t i o n a l parts. Use questions such as: 1 1 - 1 1 - 1 1 -8 + 8 " 3 3 " 8 ~ 8 ~ Lesson 5: Objective : To review f i n d i n g a common denominator f o r two or more f r a c t i o n s . Lesson : Use f r a c t i o n charts from Fraction Concepts review booklet. Review cross multiply t e s t . Lesson 6: Objective : To use the concept of common denominator to add two fr a c t i o n s with unlike denominators. Lesson : Use the magnet board to i l l u s t r a t e several 1 1 3 examples of simple addition such as : + = Lesson 7: Objective : To use the concept of common denominator to subtract two f r a c t i o n s with unlike denominators. Lessons For Adding and Subtracting Fractions Lesson 7 (continued): Lesson : Use the magnet board to i l l u s t r a t e several 3 1 3 2 1 examples of subtraction such as: ~^~~2=~^~~^-~^ , 2 1 4 3 1 A N D 3 ~ 2 = 6 " 6 = 6 * Lesson 8 : Objective : To give students practice i n adding and subtracting f r a c t i o n s with unlike denominators. Lesson : Have pupils i l l u s t r a t e on magnet board with f r a c t i o n a l parts how to perform several sample ques-t i o n s . Lesson 9: Obj ective : Pupils learn to add mixed numerals. Lesson : Review a) fi n d i n g common denominators, b) re-writing mixed numerals where the f r a c t i o n a l part i s 7 3 improper, 5jr = 6 ^ . Use the magnet board to demonstrate questions such as : 2-j + 1 ^ = 3 ~ - = 4 ^ - = 4 - | - . Lesson 1 0 : Objective : Pupils learn to subtract mixed numerals. Lesson : Review: a) fin d i n g common denominators, b) re-3 1 1 writing mixed numerals i n the subtrahend, 2-g = 1-g- . I l l u s t r a t e on the magnet board, such as: 2 1 _ -LZ 1 3 7 6 _ 3 8 8 8 8 8 4 * Lesson 1 1 : Objective : To give pupils practice i n adding and sub-t r a c t i n g mixed numerals. 66 Lessons For Adding and Subtracting Fractions Lesson 11 (continued): Lesson : Have pupils i l l u s t r a t e several examples on the overhead projector and magnet board using f r a c t i o n a l parts. Lesson 12: Objective : Pupils learn to add three f r a c t i o n s with unlike denominators. Lesson : Use the magnet board to i l l u s t r a t e a simple example such as: 1 i + i _ _ 6 _ J_ _3_ _ -_1_ 2 + 3 4 ' 12 + 12 + 12 " 'A'12 APPENDIX C OUTLINE OF LESSONS AND PRACTICE EXERCISES FOR MODULAR ARITHMETIC T-2 GROUP ONLY 68 LESSONS IN MODULAR ARITHMETIC Lesson 1: Objective : To introduce to students the idea of a r i t h -metic operations on a clock, s t a r t i n g with addition. Lesson : Introduce clock arithmetic with a problem such as: a) I t i s 9 o'clock now. In 2 hours time we w i l l have a f i l m . What time w i l l we have a film? 9 + 2 = 11 b) I t i s 9 o'clock now. In 6 hours time you w i l l be going home. What time w i l l you go home? 9 + 6 = 3 Use an overhead transparency of a clock. Show that 12 on the clock acts the same a 0 i n ordinary arithmetic when adding. Vocabulary - modular, mod 12 . Equipment : Give students materials necessary to con-struct a small cardboard clock f o r computation. Practice : Exercises 1,2,3, and 4 on page 1 of prac-t i c e booklet. Complete the mod 12 adding table on page 2. Use t h i s table to do exercise 5 on page 3. Lesson 2: Objective : To introduce m u l t i p l i c a t i o n i n clock a r i t h -metic. Lesson : Introduce m u l t i p l i c a t i o n with a problem such as: A truck d r i v e r takes 2 hours to d e l i v e r a load and return to the warehouse. I f he sta r t s at noon and has 3 loads to delive r , what time w i l l he f i n i s h ? 3 x 2 = 6 Equipment : Individual clocks. 69 Modular Arithmetic Lesson 2 (continued): Practice : Exercises 6,7, and 8 on page 3 of practice booklet. Lesson 3: Objective : To show 12 as 0 i n mod 12 m u l t i p l i c a t i o n . Lesson : Use exercises 7 and 8 on page 3 of the prac-t i c e booklet as examples. What number i n each question i n exercise 8 acts the same as 12 i n each question i n exercise 7? E l i c i t 0. Equipment : Make sets of cards to use as aids. Practice : F i l l i n the m u l t i p l i c a t i o n table on page 4 using the cards as aids. Do exercise 9 on page 5. Lesson 4: Objective : To show addition i n another module besides 12, namely 7; to show subtraction. Lesson : Demonstrate the 7-clock on the overhead proj -ector. Use week as an example. Equipment : Make small i n d i v i d u a l clocks. Practice : Complete adding (mod 7) table on page 5 of the practice booklet. Exercises 10,11,12,and 13 on page 6. Project a l l exercises on the overhead projector for cor-rections . Lesson 5: Obj ective : To show m u l t i p l i c a t i o n i n another module besides 12, namely 7. 70 Modular Arithmetic Lesson 5 (continued) : Lesson : Use 7-clock transparency to demonstrate. Have students practice with i n d i v i d u a l clocks. Use cards f o r pra c t i c e . Practice : F i l l i n m u l t i p l i c a t i o n table (mod 7) on page 7. Exercise 14 on page 7, exercises 15,16,17, and 18 on page 8 of the practice booklet. Lesson 6: Obj ective : To show addition i n another module, namely 5. Lesson : Use a 5-clock transparency to demonstrate. Give students materials to construct a hand model of 5-clock. Use cards f o r practi c e . Practice : Complete the mod 5 addition table on page 9. Exercises 19 and 20 on page 9. Lesson 7: Obj ective : Review addition i n modular arithmetic. Lesson : Wollensak Teaching Tape R-3501 1 . Practice : Wollensak Worksheet for Clock Arithmetic, R-3501. Lesson 8: Obj ective : Review subtraction i n modular arithmetic. 2 Lesson : Wollensak Teaching Tape R-350 2 Practice : Wollensak Worksheet f o r Clock Arithmetic, R-3502. 1 Wollensak Teaching Tape, Mathematics, Clock Arithmetic: Problems C-3504, Tape R-3501, 1968. 2 , Mathematics, Clock Arithmetic: Problems C-3504, Tape R-3502, 1968. 71 Modular Arithmetic Lesson 9: Obj ective : Review m u l t i p l i c a t i o n i n modular arithmetic. 3 Lesson : Wollensak Teaching Tape R-3503 Practice : Wollensak worksheet f o r Clock Arithmetic, R-3503. Lesson 10: Obj ective : To show d i v i s i o n i n clock arithmetic. Lesson : a) Introduce d i v i s i o n on a clock by re-wording the problem i n Lesson 2. A truck driver takes 2 hours to de l i v e r a load and return to the warehouse. If he starts work at noon and stops work at 6 p.m., how many loads can he deliver? 6*-s- 2 = 3 . Show t h i s on the clock by st a r t i n g at 6 and moving back 2 hours at a time u n t i l a r r i v i n g at 12. b) D i v i s i o n on the m u l t i p l i c a t i o n table - show that 3 -r 2 = n means 3= 2 x n . On the table of mod 5 multi-p l i c a t i o n , the product 3 can be 2 x 4 or 4 x 2 . Thus, n = 4. Practice : Exercise 21 on page 16. Lesson 11: Obj ective : To show some properties of m u l t i p l i c a t i o n i n modular arithmetic and compare them with ordinary a r i t h -metic . Lesson : Use exercises 15,16,17, and 18 on page 8 of the practice booklet as examples for discussion. Have students locate a pattern and make a statement about each. 3 Wollensak Teaching Tape, Mathematics,Clock Arithmetic: Problems C-3504, Tape R-3503, 1968. Modular Arithmetic Lesson 11: Ex. 15 - You notice that when you multiply a number by 1, the answer i s the number you m u l t i p l i e d . Is t h i s true i n ordinary arithmetic (except f o r 0) ? (yes) Ex. 16 - Does i t matter how you group three numbers when you multiply? Is t h i s true i n ordinary arithmetic? (yes) Ex. 17 - Does i t matter i n what order you multiply num-bers i n mod 7? Is t h i s true i n ordinary arithmetic? (yes) Ex. 18 - Is there some other number i n the mod 7 system that you can multiply any number (except 0) i n the mod 7 system by to get the product 1? (yes) Is t h i s true i n ordinary arithmetic? (no) Practice : No written exercise i n t h i s lesson. Lesson 12: Objective : To apply properties discussed i n Lesson 11 to another module, mod 5. Lesson : Review the commutative and associative proper-t i e s of m u l t i p l i c a t i o n , the i d e n t i t y element fo r multi-p l i c a t i o n , and the r e c i p r o c a l . (These terms not used with the students; they are not f a m i l i a r with them.) Practice : Exercises 22, 23, 24, and 25 on page 17. PRACTICE EXERCISES MODULAR ARITHMETIC Practice Exercises - Modular Arithmetic 74 1. a) What time i s 5 hours a f t e r 1 o'clock? b) What time i s 4 hours a f t e r 3 o'clock? c) What time i s 4 hours a f t e r 11 o'clock? d) What time i s 8 hours a f t e r 9 o'clock? e) What time i s 11 hours a f t e r 4 o'clock? 2. Name the following a) 3 + 5 b) 7 + 2 c) 9 + 9 d) 10 + 12 e) 11 + 11 sums on the "12 clock". f ) 8 + 9 g) 1 1 + 5 h) 1 0 + 9 i ) 1 2 + 9 j) 1 0 + 1 1 3. Find the sum on the 12 clock f o r each question. Write the sum on the l i n e . a) 12 + 12 = e) 9 + 12 = b) 5 + 12 = f) 8 + 12 = c) 10 + 12 = g) 7 + 12 = d) 6 + 12 = h) 1 + 12 = 4. Answer these questions. a) When you add 12 to a number i n clock arithmetic, the sum i s . b) In clock arithmetic, 12 i s the same as i n ordinary arithmetic. Practice Exercises - Modular Arithmetic MOD 12 ADDING TABLE 0 I 2 3 4- 5 6 7 8 <? <0 o 1 2 3 4-£> 7 to <7 76 Prac t i ce Exercises - Modular Ari thmetic 5. Use your add i t ion table to f i n d the answers to the fo l lowing questions i n mod 12. a) 4 + = 10 f) + 8 = 2 b) 9 + = 11 g) + 7 = 3 c) 5 + = 0 h) + 5 = 5 d) 9 + = 2 i ) + 11 = 2 e) 4 + _____ = 1 j ) + 8 = 6 Use your 12 c lock to do the fo l lowing m u l t i p l i c a t i o n . a) 6 x 2 f) 7 x 9 b) 3 x 3 g) 11 x 4 c) 4 x 3 h) 7 x 10 d) 7 x 7 i ) 6 x 10 e) 8 x 6 j ) 11 X 7 7. F ind the answer to each question by doing m u l t i p l i c -a t ion on the 12 c l o c k . 4 X 12 = 9 X 12 = 2 X 12 = 1 X 12 = 3 X 12 = 12 X 12 = 8 X 12 = 10 X 12 = 5 X 12 = 6 X 12 = 8. Use ordinary m u l t i p l i c a t i o n to f i n d the answer to each question below. 4 x 0 = 9 x 0 = 2 x 0 = 1 x 0 = 3 x 0 = 12 x 0 = 8 x 0 = 1 0 x 0 = 5 x 0 = P r a c t i c e E x e r c i s e s - Modular A r i t h m e t i c MOD 12 MULTIPLICATION TABLE 77 X 0 ( z 3 Ar S 6 7 ? 9 0 1 z 3 Co 7 ? 9 /o l\ Practice Exercises - Modular Arithmetic 9. Use your mod 12 m u l t i p l i c a t i o n table to answer the questions. a) 3 X = 3 f ) 7 x = 1 b) 2 X 4 = g) 9 x 5 = c) 11 X = 4 h) 5 x = 11 d) 6 X 7 = ' ' i ) 11 X = 3 e) 10 X 11 = j ) 7 X = 9 MOD 7 ADDING TABLE + 0 1 2 3 4 S 0 1 2 3 4-5 to 79 Practice Exercises - Modular Arithmetic 10. Add these i n the mod 7 system. a) 3 + 4 = e) 2 + 0 = _ i ) 5 + 4 = b) 2 + 0 = f) 1 + 1 = _ j ) 6 + 3 = c) 1 + 6 = g) 4 + 4 = k) 5 + 5 = d) 6 + 1 - h) 0 + 6 = _ 1) 6 + 6 = Find n i n each question (mod 7). a) 3 + n = 6 e) n + 5 = 3 b) 5 + n = 2 f ) 4 + n = 0 c) 4 + n = 3 g) n + 6 = 0 d) n + 6 = 4 h) 3 + n = 0 12. Look at the mod 7 addition table and answer the follow-ing questions. Is 4 + 5 equal to 5 + 4 ? Is 3 + 4 equal to 4 + 3 ? Is 5 + 6 equal ; to 6 + 5 ? Is 2 + 4 equal to 4 + 2 ? Is 5 + 0 equal to 0 + 5 ? Is 5 + 3 equal to 3 + 5 ? 13. Use your mod 7 addition table to f i n d answers to these questions. a) 6 - 2 = f ) 4 - 6 = k) 0 - 4 = b) 5 - 3 = g) 5 - 6 = 1) 2 - 3 = c) 4 - 1 = h) 1 - 5 = m) 3 - 5 = d) 6 - 0 = i ) 3 - 6 = n) 0 - 5 = e) 4 - 3 = j ) 4 - 2 = o ) 0 - 6 = Practice Exercises - Modular Arithmetic 80 X 0 1 2 3 4 5" 6 0 2 3 4 5 14. Use your mod 7 m u l t i p l i c a t i o n table to f i n d the answers to these questions. 3 x 4 6 x 5 4 x 5 2 x 3 3 x 6 4 x 4 6 x 5 2 x 6 81 Practice Exercises - Modular Arithmetic 15. Use your mod 7 m u l t i p l i c a t i o n table to f i n d the answer to each question. 3 x 1 = 6 x 1 = _ 4 x 1 = 5 x 1 = 1 X 1 = _ 2 x 1 = Use your table, or your • clock, or your cards to f i n d the answers to these questions i n mod 7. ( 3 x 4 ) x 2 = 3 x ( 4 x 2 ) = ( 4 x 5 ) x 2 = 4 x ( 5 x 2 ) = ( 3 x 6 ) x 4 = 3 x ( 6 x 4 ) = 4 4, x 3 ) x 5 = 4 x ( 3 x 5 ) = ( 6 x 5 ) x 4 = 6 x ( 5 x 4 ) = ( 2 x 5 ) x 6 = 2 x ( 5 x 6 ) = 17. Use your mod 7 m u l t i p l i c a t i o n table to answer these questions. Is 3 x 4 equal to 4 x 3 ? Is 5 x 6 equal to 6 x 5 ? Is 3 x 6 equal to 6 x 3 ? Is 2 x 4 equal to 4 x 2 ? Is 4 x 5 equal to 5 x 4 ? 18. Use your mod 7 m u l t i p l i c a t i o n table to answer these questions. 4 x = 1 5 x = 1 6 x = 1 82 WOLLENSAK TEACHING TAPE WORKSHEET CLOCK ARITHMETIC: ADDITION R-3501 Name ADDITION TABLE (MOD 8) 0 t 2 3 4- s 6 7 0 i I j Z 3 > 7 opyright 1968 Minnesota Mining and Manufacturing Company 83 WOLLENSAK TEACHING TAPE WORKSHEET CLOCK ARITHMETIC: ADDITION R-3501 G. 3 + 4 = (mod 5) H. 4 + 4 + 3 = (mod 5) 1. 3 + 3 = (mod 4) 4. 4 + = 7 (mod 8) 2. 3 + 3 = (mod 5) 5. 4 + = 0 (mod 8) 3. 3 + 3 = (mod 6) 3 + = 2 (mod 8) 7. 5 + = 3 (mod 8) 8. 7 + = 4 (mod 8) 2 + 6 + = 7 (mod 8) 10. 4 + 2 = (mod 5) 14. 2 + = 0 (mod 7) 11. 3 + 5 = (mod 6) 15. + 4 = 3 (mod 5) 12. 8 + 7 = (mod 9) 16. + = 2(mod 6) 13. 3 + 2 = (mod 4) 17. + = (mod 5) 84 WOLLENSAK TEACHING TAPE WORKSHEET CLOCK ARITHMETIC: SUBTRACTION R-3502 Name - 0 t 2 j 4 5- 6 7 c Z 3 4- •7 s c 7 Copyright 1968 Minnesota Mining and Manufacturing Company WOLLENSAK TEACHING TAPE WORKSHEET CLOCK ARITHMETIC: SUBTRACTION R-350 2 K. 3 - 2 = (mod 4) L. 2 - 3 = (mod 4) 1. 0 - 3 = 1 - 2 = 3. 3 - 3 = (mod 4) (mod 4) (mod 4) 4 . 5. 6 -6. 2 -= 4 (mod 8) = 7 (mod 8) = 3 (mod 4) 5 -8. 0 -= 3(mod 8) = Kmod 4) 10. - 1 = 3 (mod 4) - 7 = 4 (mod 8) 11. - 2 = 3(mod 4) 12. - 6 = 5 (mod 8) 13. 3 - 0 = 14. 0 - 4 = 15. 1 - 3 = 16. 3 - 4 = 17. 4 - 5 = SUPPLEMENTARY PROBLEMS (mod 5) 18. - 4 = 1 (mod 5) (mod 5) 19. (mod 4) 20. (mod 6) (mod 6) 21. 3 -22 • 1 - 3 = 5 (mod 6) - 4 = 4 (mod 8) ___ = 1 (mod 4) = 3 (mod 4) Name 86 WOLLENSAK TEACHING TAPE WORKSHEET CLOCK ARITHMETIC: MULTIPLICATION R-350 3 MULTIPLICATION TABLE(MOD 5) X o 1 2 3 4 0 o 0 o o c » c 1 c 3 c o •MULTIPLICATION TABLE (MOD 8) Copyright 1968 Minnesota Mining and Manufacturing Company WOLLENSAK TEACHING TAPE WORKSHEET CLOCK ARITHMETIC: MULTIPLICATION R-350 3 B. 7 x 6 = (mod 8) D. 3 x 5 = (mod 8) C. 5 x 3 = (mod 8) E. 6 x 4 = (mod 8) P. 4 x 5 = (mod 3) I. 2 x 8 = (mod 4) G. 4 x 5 = (mod 4) J . 5 x 3 = (mod 6) H. 4 x 5 = (mod 15) K. 8 x 3 = (mod 7) SUPPLEMENTARY PROBLEMS 1 . 8 x 5 = (mod 6) 2. 7 x 6 = (mod 9) 3 . 9 x 3 = (mod 12) 4 . 4 x 7 = (mod 11) 5. 3 x 6 = (mod 7) 6 . 8 x 7 = (mod 9) 7. 2 x 3 = (mod 4) 88 Prac t i ce Exercises - Modular Ari thmetic 19. Use your mod 5 addi t ion table to answer these questions, 3 + 2 = 1 + 3 = 3 + 1 = 2 + 3 = 4 + 3 = 3 + 4 = 4 + 1 = 1 + 4 = 20. Use the mod 5 add i t ion table to f i n d the answers to these quest ions . 4 - 2 = 0 - 4 = 3 - 1 = 2 - 3 = 2 - 2 = 3 - 4 = 4 - 3 = 1 - 2 = 2 - 4 = 1 - 4 = -r 0 I 2 3 4-0 1 2 3 1-89 Prac t i ce Exercises - Modular Ari thmetic 21. Use your mod 7 m u l t i p l i c a t i o n table to answer these quest ions. 6 -s- 3 = N 4 • 5 = N 4 2 = N 3 * 6 = N 5 -t- 1 = N 4 6 = N 6 4 = N 3 f 2 = N 2 -j. 3 = N 6 5 = N X 0 i* 2 3 4-0 / 2 3 4-Practice Exercises - Modular Arithmetic 22. Use your mod 5 m u l t i p l i c a t i o n table to answer these questions. 1 X =1 2 x _ _ = 2 3 x Use your mod 5 m u l t i p l i c a t i o n table. ( 2 x 3 ) x 4 = 2 x ( 3 x 4 ) = ( 3 x 4 ) x 1 = 3 x ( 4 x 1 ) = ( 0 x 3 ) x 4 = 0 x ( 3 x 4 ) = ( 1 x 3 ) x 2 = 1 x ( 3 x 2 ) = Use your mod 5 m u l t i p l i c a t i o n table. 3 x 4 = 4 x 3 = II 2 x 3 = 3 x 2 = -4 x 1 = 1 x 4 = 0 x 2 = 2 x 0 = 25. Use your mod 5 m u l t i p l i c a t i o n table to answer these questions. 91 APPENDIX D OUTLINE OF LESSONS FOR MULTIPLICATION OF FRACTIONS OUTLINE OF LESSONS AND SELECTED PRACTICE PAGES FOR DIVISION OF FRACTIONS T - l AND T-2 GROUPS LESSONS FOR MULTIPLICATION OF FRACTIONS 92 Appropriate practice exercises were provided f o r these lessons by means of a 12-page booklet of exercises prepared Lesson 1: Objective : To introduce m u l t i p l i c a t i o n of f r a c t i o n s using regions as models (unit numerators only). Lesson : Use overhead transparencies with overlays to explain samples i n practice booklet. Example: Lesson 2: y *'f\ * ' 1 Objective : To i 11 usfri^.fcej^how_a number l i n e can be used as a model fo r m u l t i p l i c a t i o n of f r a c t i o n s . Lesson : Use the overhead transparency of a number l i n e to i l l u s t r a t e an example such as : 5 x Lesson 3: Objective : To i l l u s t r a t e m u l t i p l i c a t i o n of f r a c t i o n s using f r a c t i o n a l parts and using regions as models, (numerators greater than 1) Lesson : Use a measuring cup and l i q u i d to i l l u s t r a t e the example i n the practice booklet; use a magnet board 2 to i l l u s t r a t e examples such as 5 x . Use an overhead 1 3 transparency with an overlay to i l l u s t r a t e j x ^ using regions. by the experimenter. 1 4 _1_ 12 93 Lessons For M u l t i p l i c a t i o n of Fractions Lesson 4: Objective : To further i l l u s t r a t e m u l t i p l i c a t i o n of a f r a c t i o n by a whole number. Lesson : Use an overhead transparency to i l l u s t r a t e 1 3 1 — of 4 and ^ of 3. Show that, f o r example, — of 6 can be 6 halves or 3 wholes. Lesson 5: Objective : To r e l a t e m u l t i p l i c a t i o n of f r a c t i o n s to fi n d i n g area of a rectangle. Lesson : Review fi n d i n g the area of a rectangle, using whole numbers. Re-name whole numbers with f r a c t i o n a l name s. Lesson 6: Objective : To show that m u l t i p l i c a t i o n of f r a c t i o n s i s commutative and associative. (These terms not used.) Lesson : Give pairs of questions such as: and have students compare the answers i n each p a i r . Lesson 7: Obj ective : To show students a short cut i n multiply-ing f r a c t i o n s . 2 3 Lesson : I l l u s t r a t e that i n — x — , the denominator 3 2 3 i n — and the numerator 3 i n — can each be divided by 3 before m u l t i p l i c a t i o n takes place. 1 4 1 1 3 X 4 ~ and 94 Lessons For M u l t i p l i c a t i o n of Fractions Lesson 8: Objective : To give students practice i n multiplying f r a c t i o n s . Lesson : Review several examples b r i e f l y and i l l u s t r a t e with concrete aids. Lesson 9: Objective : To i l l u s t r a t e m u l t i p l i c a t i o n of mixed numerals. Lesson : Review re-writing mixed numerals as improper f r a c t i o n s . Stress that t h i s must be done before mult-i p l y i n g two mixed numerals. (At least, at t h i s i n t r o -ductory stage.) Lesson 10: Objective : To i l l u s t r a t e the r e c i p r o c a l . 1 4 Lesson : Use examples such as — x — = — = 1 and have students supply the missing numerator and denominator. Review that a whole number, such as 6, can be written — and that the r e c i p r o c a l of 6 i s — . 95 LESSONS FOR DIVISION OF FRACTIONS Lesson 1: Objective : To i l l u s t r a t e the d i v i s i o n of groups and of objects. Lesson : Use overhead transparencies and a magnet board with f r a c t i o n a l parts to i l l u s t r a t e examples A to I on pages 1 and 2 of the practice booklet. Have students cut and f o l d s t r i p of paper from Exercise 3 on page 3 to f i n d the answer. Lesson 2: Objective : To i l l u s t r a t e the inverse r e l a t i o n s h i p of m u l t i p l i c a t i o n and d i v i s i o n . Lesson : Use examples similar to the questions on page 4 of the practice booklet, s t a r t i n g with whole number examples. Show that i f 7 x 5 = 35, then 35 -r- 5 = 7 and 35 -r 7 = 5; s i m i l a r l y , i f —• x = — > then y^- -|- = = 1 1 _ 1 ana Y2 * 4 " 3 * Lesson 3: Objective : To review the r e c i p r o c a l , the commutative and associative properties of m u l t i p l i c a t i o n , and the i d e n t i t y element f o r m u l t i p l i c a t i o n . (These terms are not used with the students.) Lesson : Review examples of each property and i l l u s -t r a t e . Give the exercises on page 5 and 6 i n the booklet. Lessons For D i v i s i o n of Fractions Lesson 4: Objective : To teach the d i v i s i o n of f r a c t i o n s algorithm. Lesson : Explain to the students that they should look for mathematical properties within the algorithm that they have reviewed. Use an overhead transparency to colour code the r e c i p r o c a l throughout the algorithm. Remind students that mixed numerals must be written as improper f r a c t i o n s . Lesson 5: Objective : To review d i v i s i o n of f r a c t i o n s . Lesson : Review the algorithm completely. Stress that mixed numerals must be written i n the form of improper f r a c t i o n s . Review the r e c i p r o c a l of whole numbers. Lesson 6: Objective : To further review d i v i s i o n of f r a c t i o n s . Lesson : Review algorithm completely. Give practice exercises. Lesson 7 : Objective : To further review d i v i s i o n of f r a c t i o n s . Lesson : Review algorithm completely. Give further p r a c t i c e . Discuss previous exercises. SELECTED PAGES FROM DIVISION OF FRACTIONS PRACTICE BOOKLET A. (Page 1) In numerals; 6 + 3 * £~J In words: How many 3*s i n 6? Answer: .^L OOO ooo 98 OOO OOO B . In numerals: 1 . 1 r I : ?-i * U In words: How many 1 s i n ^ ? Answer: C» In numerals: 1? In words: How many - j 's i n 1 ? Answer: D. ~ *b i n 1<| ? In numerals: 1^ «f * In words: How many -| Answer: . E . In numerals: 3-r-1~ = ^ ^ In words: How many l | 's i n 3 ? Answer: In numerals: j ~ -| « | ] 1 3 In words: How many ^ »s i n ^ ? Answer: 5 o o o OO DO ii n (Page 2) Those were easy. Try these. G * In numerals: 4 T j = |" ^ ' In words: How many | «s i n 4? Answer: H « In numerals: 4~ -4- 3 « '£3 In words: How many 3»s i n 4^ ? Answer: 1 * In numerals : ~ 4- ~ » [ ] In words: How many | 's i n | ? or What p a r t o f ^ i s ? Answer : (Page 3) 100 Cut out t h i s portion* Fold i n two lengthwise. Fold In two again. The double shaded part i s what f r a c t i o n of the shaded part? S . 5 Look at these equations. Can you figure out a way of d i v i d i n g two fracti o n s to get the correct answer? (Page 4) 101 E* K <X rv-i p ic E * a m p I « • 3 4 r - 7 i - -r 4- * X /2 3 * 35 ±7 * * r% + 3 ± 7- X * s s ? x 2x / i » 3 * / r • 3 3 2^ 3 I I A A ^ = 15 ' ? x 3 s Z 8 /4> * 71 * * ±3i° zi loo 7+ IOO * 7-f- -2 » (Page 5) 102 A. A x » i L * / 3 Z 4> ' i x i , — / 1 X II / & 3 ' 77 * ^ A « , * 4-6 * . 2 « £_ „ / / 6 * ' * * * X — -,-•/ /**- S-k ^ — -7 / i x -•/ -/ — x 3 i * / T * mmmm 35 f x £ = 77 * z f x£ • 4 * I K 5"0 X J - a tX x / • » loo x f * /2 / / * 7 « x * - /OO * 75.* 4 * & r APPENDIX E TESTS CONSTRUCTED FOR AND USED IN THIS STUDY 1. FRACTION CONCEPTS 2. ADDING AND SUBTRACTING FRACTIONS 3. MODULAR ARITHMETIC 4. MULTIPLICATION OF FRACTIONS 5. DIVISION OF FRACTIONS 6. INTERVIEW TEST FRACTION CONCEPTS TEST a) What f r a c t i o n of A i s shaded? b) What f r a c t i o n of A i s not shaded? c) What f r a c t i o n of the objects i s shaded? \ d) What f r a c t i o n of the objects i s not shaded? e) What f r a c t i o n of C i s shaded? f) What f r a c t i o n of C i s not shaded? g) What f r a c t i o n of the objects i s shaded? h) What f r a c t i o n of the objects i s not shaded? a) Is r of t h i s object shaded? b) Are "o of these objects shaded? a) C i r c l e the proper f r a c t i o n s . 3 16 7 9 14 8 5 7 10 18 b) C i r c l e the improper f r a c t i o n s . 9 10 2 6 99 7 10 1 12 100 c) Write a mixed numeral. _^ F r a c t i o n Concepts Test 4. Look at the f r a c t i o n chart and write a f r a c t i o n t e l l i n g what part of each bar i s shaded. W//X///Am\WsA'/s/AW/M//AW/A I I I " •v I I I i i 5. For each f r a c t i o n , write an equivalent f r a c t i o n i n lowest terms. 12 21_ 42 _9_ = 75 16 ~ 24 ~ 14 = 21 100 3 6. C i r c l e the f r ac t ions that are equivalent to -g . 9 13 15 8 39 24 18 40 3 104 7. C i r c l e the numbers which can be common denominators 2 3 for — and — . 8 12 18 24 36 107 Frac t ion Concepts Test 8 . Show how to use the cross mul t ip ly te s t to f i n d i f each pa i r of f r ac t ions below i s an equivalent p a i r . a) - | , ~2 Y e s n o (check one) 7 6 b) "8 ' "7 Y e s n o (check one) 9. F ind a common denominator for each set of f r a c t i o n s . 3. 3 1 1 1 J _ . _8_ 7 ' 4 2 ' 3 ' 4 50 ' 75 10. Find the lowest common denominator for each set of f r a c t i o n s . 7 1 1 2 5 12' 3 5 ' 3 ' 6 11. Change each improper f r a c t i o n to a mixed numeral. Change each mixed numeral to an improper f r a c t i o n . 19 7 2 = 2_3_ = 9 7 3 - '5 " -16 5 -12. Put the cor rec t s ign between each pa i r of f r ac t ions . Use the signs = , ^ , ^ 2 1 9 7 3 6 10 8 ADDING AND SUBTRACTING FRACTIONS TEST Part I A 1 3 _ 5 5 7 7 " B * 6 + 6 C * 12 + 3 ~ D ' 16 + 8 = E l + 1 - P 3 4. IL t j m 7 + 3 ~ *' 4 + 10 G. | + 9| = H. - i i + 111 = I 2-2- + I - J i i - + 5-2-l . ^ 1 6 g - * 15 + ^IO K. 21-| + 13Jr = L. 6 3 — + 11-1 = 6 6 16 8 M. 6r~ + 1-g- = N. 3^ + &| = Adding and Subtracting Fractions Test S * I + i + I " T * 44 + 12 U. 40 + V. i s | + 6 2 3 W. 4 X. 6# 5 v * w8 7 2 I + ' 5 + 3 Part II C. Y. | Z. 1 6 ^ I x i 9 + 10 1. + 9 8 1 - R 2 I 9 " 9 " • # 8 8 15 i _ n 1 2 1 16 " 4 ~ * 20 " 5 p JL i _ P i l A. t i m 10 " 4 ~ 16 " 12 1 ? 1 5 G. 2 0 | - | - H. 3 3 i - f " i b12 8 " 5 4 Adding and Subtracting Fract ions Test K. 1 3 | - 4| = L . . M. 1 & ~ - 3-| = N. 20-^ iSrj 0. 2 0 | - l f i i = P. 12-| - 3 § I l l MODULAR ARITHMETIC TEST Use your mod 7 clock to f i n d the answers to these questions. Watch the signs. 3 + 3 = 0 + 4 = 5 + 6 = 6 - 4 = 0 - 3 = 2 - 6 = 2 x 3 = 4 x 4 = 6 x 5 = True or False ? a) 3 x 1 = 3 (mod 5) b) 4 x 5 = 5 x 4 (mod 8) c) 2 x 0 = 0 (mod 12) d) 5 x 6 = 3 x 3 (mod 7) e) 4 x 2 = 8 (mod 7) f ) (2 x 3) X 4 = 2 x (3 x 4) (mod 5) g) 4 x 2 = 1 (ordinary arithmetic) h) 11 x 3 = 3 x 11 (mod 12) i ) 6 x 9 = 9 x 6 (ordinary arithmetic) j ) 7 x 7 = 1 (mod 8) a) In ordinary arithmetic, what i s the answer when you multiply a number by 0 ? b) In mod 5 arithmetic, what i s the answer when you multiply a number by 0 ? c) In ordinary arithmetic, what i s the answer when you multiply a number by 1 ? 112 Modular Arithmetic Test 3. d) In mod 5 arithmetic, what i s the answer when you multiply a number by 1 ? e) In ordinary arithmetic, can you multiply any whole number (except 1) by another whole number and get the product 1 ? f) In mod 5 arithmetic, can you multiply every number (except 0 ) by some other whole number to get the product "1" ? Prove your answer to (f) by tr y i n g to f i n d the missing numbers i n each question below. 1 x = 1 (mod 5) 3 x = 1 (mod 5) 2 x = 1 (mod 5) 4 x = 1 (mod 5) 4. Use your cards to f i n d the missing numbers. a) 2 x 5 = (mod 6) g) 6 + 4 + 5 = (mod 8) b) 3 + 4 + 3 = (mod 5) h) 2 - 4 = (mod 5) c) 9 - 6 = (mod 12) i ) 7 x 0 = (mod 11) d) 11 + 11 = (mod 12) j ) 4 x = 2 (mod 5) e) 4 x 6 = (mod 8) k) 6 + = 0 (mod 12) f) 5 x 5 = (mod 6) 5. Find the missing numbers. 4 x 3 = (mod 5) 3 x 2 = (mod 5) 4 x 5 = (mod 7) 2 -*- 4 = (mod 5) 1 -J- 3 = (mod 5) 6 4 = (mod 7) 2 -t- 3 = (mod 5) 1 * 2 = (mod 5) 6 ^ 5 = (mod 7) MULTIPLICATION OF FRACTIONS TEST 1 3 . , 3 2 4 X 5 = ^* 4 X " 3 T6 X 3 = 4 ' To" X 4 " 1 x 1 4 = 6. § x 8 | x 10 = 8. i x l i x 2 = 10. - | x 2-| | x 2| = 12. | x e| 5 x Te = 1 4 ' 8 x Te 12 x I = 16. 11 x - | 17, 16 x r = 6 18. 12 x 2 j 19. 10 x 20, 15 x 2f 21. 2f x 8 = 22. 4 x ^ = 23. l r x 16 24. 4| x i = 25. 4-2- x 1 ^3 7 26. 2f x 15 « 27, I f x 7 - 28. 2f x 12 = 29. -.1 i l x I3 30. 4*4 31. ^3 ,J7_ Jrg X ± 1 5 32, = 1 114 33. X 16 = 1 34. x 2-j = 1 DIVISION OF FRACTIONS TEST D i v i s i o n of Fract ions Test 13. 4 _ ~ 2 14 5 ' ^3 " 15. Q l ,_5 16 8 2 ' *6 = 17. ~3 ,2 _ 18 4^ * X 3 ~ 19. 1 2 _ v l = 20 2 1 « 1 4 § - M 2 « 2 2 23. 1 - 5 , 24 1 ' 6 = INTERVIEW TEST 117 Directions f o r Administering and Scoring 1. Present the c h i l d with a copy of the Student's Sheet and a p e n c i l . Explain that the examiner would l i k e to f i n d out how well he understands d i v i s i o n of f r a c t i o n s . Assure him that t h i s i s not f o r h i s report card and t r y to put him at ease. T e l l him that you w i l l ask him to do some examples and then w i l l ask him a few questions about what he does. 2. F i l l i n the data at the top of the interview blank and the scoring sheet. Be sure that students write both f i r s t and l a s t names. En c i r c l e h i s group on the scoring sheet, T - l or T-2 . 3. Point to the f i r s t example and ask the c h i l d to f i n d the answer. T e l l him that he may do any necessary work on the paper. 4. When the c h i l d has completed each example, challenge him with these questions. a) Are you sure that t h i s i s the correct answer? What i f 3 1 I said that the answer was ( Ques. 1- •—=• , Ques. 2- 2—> 5 Ques. 3 - 20-g- , Ques. 4 - 2) ? How could you prove me wrong? b) Why did you change the divide sign to a multiply sign? 4 8 1 3 1 3 Why did you write ( — , — , — , -jr) instead of ( — , — , 2 5 , 1— )? Do you know why i t i s important to do t h i s ? 118 c) Can you give any reason f o r doing these l a s t two steps together? d) Do you know any other way to get the answer to t h i s question? I f the student does not understand the question as f i r s t put, re-phrase i t without i n d i c a t i n g the answer sought. Questioning should continue u n t i l each c h i l d has had f u l l opportunity to demonstrate his understanding. The examiner should be c a r e f u l not to use a leading question that might suggest the correct answer. 5. Record each response f o r each example on the interviewer's blank according to the code. Explain any of the subject's answers about which there i s uncertainty. I f the subject appears to know how to work an example but gets the wrong answer, urge him to check his work. Response categories and scoring procedure 1. C i r c l e the appropriate code l e t t e r s . C - Correct W - Wrong P - P a r t i a l l y correct " P a r t i a l l y c o rrect" - the subject gets the correct answer but does not write i t i n lowest terms. For example, 12 72 3 25 Ques. 1 - , Ques. 2 - jg- or , Ques. 3 - 30" > Ques. 4-45 9 • ^ j or ^ , or similar answers. 2. R ~ Rational response. This response i s recorded only with a C or P response. A Rational response requires an explanation of why the "invert and multiply" procedure i s used. This includes an understanding of the concept 119 of the r e c i p r o c a l but not neces sar i ly the use of the term "reciprocal 1 , 1 and the idea that d i v i s i o n and m u l t i -p l i c a t i o n are inverse operat ions . Ro - Rote response using the procedure " i n v e r t the d i v i s o r and m u l t i p l y " . This response could occur with a C or P response. No further explanation i s required from the student. 0 - Other. The subject can t e l l some other way of f i n d i n g the answer. This might be i n t u i t i v e l y or by some other algorithm such as the common denominator method. N - No attempt to do a quest ion. 120 INTERVIEW TEST Student's Sheet Name Date Group: T - l T-2 3 1 4 ' 4 _9_ 16 3 8 4 t 5 b 4. 3 2 J4 ' X3 121 INTERVIEW TEST Scoring Sheet Name Date Group: T - l T-2 1. _3 4 _1 4 1. C R Ro 0 W N 2. 9 16 3 8 2. C R Ro 0 W N 3- 4*5-6 3. C R Ro 0 W N 3 2 3-^ 1— = ' 3 C R Ro 0 W N
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The effect of instruction in modular arithmetic on the ability of grade 6 students to divide fractions… MacDonald, Alexander David 1973
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Title | The effect of instruction in modular arithmetic on the ability of grade 6 students to divide fractions and give a rational explanation of the process |
Creator |
MacDonald, Alexander David |
Publisher | University of British Columbia |
Date Issued | 1973 |
Description | The problem under investigation in this study was to find out what relationship a unit in modular arithmetic might have to Grade 6 pupils' skill in computing the division of fractions and to their understanding of the mathematical basis of the algorithm. It was hypothesized that a unit in modular arithmetic would aid in developing skill in computing and understanding of the algorithm. The study was conducted with a sample of 58 Grade 6 students from the same school. The subjects were assigned to two treatment groups. Both groups received a review of fraction concepts at the beginning of the study. Following this, one group was taught modular arithmetic while the other group reviewed adding and subtracting of fractions. Then both groups were taught multiplication and division of fractions. Following the instruction period, both groups were tested for ability to compute division of fractions. To test understanding of the division of fractions algorithm, an interview inventory test was administered to all subjects in both groups. A statistical analysis of the data from these tests revealed no support for the hypotheses. The conclusion was that teaching modular arithmetic to the Grade 6 pupils participating in the study did not appear to improve their ability to compute division of fractions nor their understanding of the mathematical basis of the division of fractions. |
Subject |
Mathematics -- Study and teaching (Elementary) |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2011-03-12 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0101049 |
URI | http://hdl.handle.net/2429/32395 |
Degree |
Master of Education - MEd |
Program |
Education |
Affiliation |
Education, Faculty of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
AggregatedSourceRepository | DSpace |
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