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Two different instructional procedures for a multiplication algorithm and their transfer effects to a.. 1972
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Title | Two different instructional procedures for a multiplication algorithm and their transfer effects to a higher-order algorithm |
Creator |
Hope, John Alfred |
Publisher | University of British Columbia |
Date Created | 2011-04-12 |
Date Issued | 2011-04-12 |
Date | 1972 |
Description | This was a study to determine the effects of two instructional procedures for a multiplication algorithm on the ability of elementary school children to extend this algorithm to the solving of computational tasks involving the use of a higher-order algorithm. Each of two groups was given preliminary instruction in solving multiplication problems via the application of the distributive law. After this readiness phase was completed, students were randomly assigned to either a Tl or T2 treatment group. The Tl subjects were taught a rote-type standard multiplication algorithm for determining the solution of 2 x 1 and 3 x 1 products. No explicit instruction was given to indicate the relationships between the two learning tasks, viz. the acquisition of the distributive law and the standard multiplication algorithm. Unlike the Tl instructional sequence, the T2 instructional sequence was designed to promote the learning of the relationships between the series of learning tasks. That is, the T2 subjects were taught a standard multiplication algorithm that required the explicit use of the distributive law and other acquired algebraic skills. It was hypothesised that this continual integration of learning tasks would enable the T2 subjects to exhibit superiority over the Tl subjects in extending their standard multiplication algorithm to computational tasks requiring the use of an untaught higher-order algorithm. A total of 238 subjects and 8 teachers were used in all phases of the experiment. A mixed model of analysis of variance was used to validate the performance hypothesis. It was found that the Tl subjects were significantly better than the T2 subjects in the performance of the standard multiplication algorithm. An analysis of covariance was performed to determine the validity of the transfer hypothesis. A subject's score on the performance test was used as a covariate in order to equate the disparate computational abilities of the Tl and T2 subjects. Although the mean score of the T2 subjects was higher than that of the Tl subjects on the transfer test, this difference was not statistically significant. |
Subject |
Mathematics -- Study And Teaching (Elementary) |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | Eng |
Collection |
Retrospective Theses and Dissertations, 1919-2007 |
Series | UBC Retrospective Theses Digitization Project |
Date Available | 2011-04-12 |
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 | 10.14288/1.0101695 |
Degree |
Master of Arts - MA |
Program |
Mathematics Education |
Affiliation |
Education, Faculty of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
URI | http://hdl.handle.net/2429/33581 |
Aggregated Source Repository | DSpace |
Digital Resource Original Record | https://open.library.ubc.ca/collections/831/items/1.0101695/source |
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TOO DIFFERENT INSTRUCTIONAL PROCEDURES FOR A MULTIPLICATION ALGORITHM AND THEIR TRANSFER EFFECTS TO A HIGHER-ORDER ALGORITHM. by John A. Hope B . S c , U n i v e r s i t y of B r i t i s h Columbia, 1965 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS i n the Department of Mathematics Education We accept t h i s t h e s i s as conforming to the re q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA May, 1972. In present ing th is thes is in p a r t i a l fu l f i lment o f the requirements for an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y sha l l make i t f r e e l y a v a i l a b l e for reference and study. I fu r ther agree that permission for extensive copying of th is t h e s i s fo r s c h o l a r l y purposes may be granted by the Head of my Department or by h is representa t ives . It i s understood that copying or p u b l i c a t i o n of th is thes is fo r f i n a n c i a l gain s h a l l not be allowed without my wr i t ten permiss ion . Department .of The Un ive rs i t y of B r i t i s h Columbia Vancouver 8, Canada Date ^Xt^vJLlX^ / Abstract This was a study to determine the e f f e c t s of two i n s t r u c t i o n a l procedures f o r a m u l t i p l i c a t i o n p a l g o r i t h m on the a b i l i t y of elementary school c h i l d r e n to extend t h i s a l g o r i t h m to the s o l v i n g of computational tasks i n v o l v i n g the use of a higher-order algorithm. Each of two groups was given p r e l i m i n a r y i n s t r u c t i o n i n s o l v i n g m u l t i p l i c a t i o n problems v i a the a p p l i c a t i o n of the d i s t r i b u t i v e law. A f t e r t h i s readiness phase was completed, students were randomly assigned to e i t h e r a T l or T2 treatment group. The T l subjects were taught a rote-type standard m u l t i p l i c a t i o n a l g o r i t h m f o r determining the s o l u t i o n of 2 x 1 and 3 x 1 products. No e x p l i c i t i n s t r u c t i o n was given to i n d i c a t e the r e l a t i o n s h i p s between the two l e a r n i n g t a s k s , v i z . the a c q u i s i t i o n of the d i s t r i b u t i v e law and the standard m u l t i p l i c a t i o n a l gorithm. U n l i k e the T l i n s t r u c t i o n a l sequence, the T2 i n s t r u c t i o n a l sequence was designed to promote the l e a r n i n g of the r e l a t i o n s h i p s between the s e r i e s of l e a r n i n g t a s k s . That i s , the T2 subjects were taught a standard m u l t i p l i c a t i o n a l g o r i t h m that required the e x p l i c i t use of the d i s t r i b u t i v e law and other acquired a l g e b r a i c s k i l l s . I t was hypothesised that t h i s c o n t i n u a l i n t e g r a t i o n of l e a r n i n g tasks would enable the T2 subjects to e x h i b i t s u p e r i o r i t y over the T l subjects i n extending t h e i r standard m u l t i p l i c a t i o n a l g o r i t h m to computational tasks r e q u i r i n g the use of an untaught higher-order algorithm. A t o t a l of 238 subjects and 8 teachers were used i n a l l phases of the experiment. A mixed model :-6f.analysis of variance was used to validate the performance hypothesis. It was found that the Tl subjects were significantly better than the T2 subjects in the performance of the standard multiplication algorithm. An analysis of covariance was performed to determine the validity of the transfer hypothesis. A subject's score on the performance test was used as a covariate in order to equate the disparate computational a b i l i t i e s of the Tl and T2 subjects. Although the mean score of the T2 subjects was higher than that of the T l subjects on the transfer test, this difference was not s t a t i s t i c a l l y significant. i TABLE OF CONTENTS Page LIST OF TABLES i i Chapter I OUTLINE OF,'-THE PROBLEM . . . . . . . . . . . 1 I n t r o d u c t i o n ' 1 Genera l Statement of the Problem 3 D e f i n i t i o n of Terms 4 D i s c u s s i o n and S i g n i f i c a n c e of the Problem 6 Hypotheses 15 Chapter I I SURVEY OF THE LITERATURE 16 Chapter I I I DESIGN OF THE EXPERIMENT 20 The Sample . 2 0 The I n s t r u c t i o n a l Sequences 20 The Measuring Instruments . . 23 Chapter IV ANALYSIS OF THE DATA 27 The Performance Hypothes is 27 The T r a n s f e r Hypothes is . . . . . . . . 29 Chapter V CONCLUSIONS AND IMPLICATIONS FOR FURTHER RESEARCH . . . . 32 Performance Hypothes is 32 T r a n s f e r Hypothes is 32 Problems f o r F u r t h e r Study 33 BIBLIOGRAPHY 34 APPENDICES 35 A . The I d e n t i f i c a t i o n of Another I n t e r n a l A l g o r i t h m . . . 35 B . Readiness Phase Lesson P lans . 38 C. Treatment Phase Lesson Plans . . . . . . . . . . . . . 49 D. The Measuring Instruments •. . 67 E . The E x p e r i m e n t a l Data 71 i i LIST OF TABLES Page Table I A n a l y s i s of the Performance Test: P o i n t , • • B i s e r i a l r f o r each item 26 Table I I A n a l y s i s of the Transfer Test: Point . , ,, B i s e r i a l r f o r each item 26 Table I I I A n a l y s i s of Variance; Performance Hypothesis 29 Table IV A n a l y s i s of Covariance: Transfer Hypothesis . . . . . . 31 CHAPTER I OUTLINE OF THE PROBLEM IN:T-R0D%:GTION> Most modern a r i t h m e t i c programs are i n agreement that the f i e l d p o s t u l a t e s f o r the system of a r i t h m e t i c should form an i n t e g r a l part of a r i t h m e t i c content. Both mathematicians and p s y c h o l o g i s t s have advised that the understanding of many of these p o s t u l a t e s be included as elementary school o b j e c t i v e s . P a r t i c i p a t i n g mathematicians at the Cambridge Conference on School Mathematics s t r e s s e d that students be f a m i l i a r w i t h part of the " g l o b a l s t r u c t u r e " of mathematics."'' They f e l t that a very s o l i d mathematical super s t r u c t u r e can be erected which w i l l help p u p i l s i n more advanced mathematical f i e l d s . Although the idea o f . " g l o b a l s t r u c t u r e " was never c l e a r l y defined there i s l i t t l e doubt, a f t e r examining t h e i r recommendations f o r c u r r i c u l u m content, 3 that the f i e l d p o s t u l a t e s formed part of i t . Jerome Bruner, again a v o i d i n g the knotty problem of d e f i n i t i o n , s t a t e d "there are at l e a s t four general claims that can be made f o r teaching the fundamental s t r u c t u r e of a su b j e c t , claims i n need of Goals f o r School Mathematics, (New York: Houghton M i f f l i n , 1963), p. 8. 2 I b i d . , p. 8. 3 I b i d . , p. 36. 2 d e t a i l e d study". He l i s t e d the f o l l o w i n g as supportive c l a i m s ; 1. Understanding fundamentals makes a subject more com- prehe n s i b l e . 2. Unless d e t a i l i s placed i n t o a s t r u c t u r e d p a t t e r n i t i s r a p i d l y f o r g o t t e n . 3. Understanding of fundamental p r i n c i p l e s and ideas leads to t r a n s f e r of t r a i n i n g . 4. By c o n s t a n t l y reexamining m a t e r i a l taught i n elementary and secondary schools f o r i t s fundamental c h a r a c t e r , one i s able to 4 narrow the gap between "advanced" knowledge and "elementary" knowledge. David Ausubel,claims that " p r e c i s e and i n t e g r a t e d understandings are, presumably, more l i k e l y to develop i f the c e n t r a l , u n i f y i n g ideas of d i s c i p l i n e are learned before more p e r i p h e r a l concepts and in f o r m a t i o n are introduced".^ In h i s o p i n i o n , "the most s i g n i f i c a n t advances that have occurred i n recent years i n the teaching of such subjects as mathematics, chemistry, physics and b i o l o g y are pr e d i c a t e d on the assumption that e f f i c i e n t l e a r n i n g and f u n c t i o n a l r e t e n t i o n of ideas and i n f o r m a t i o n are l a r g e l y dependent upon the adequacy of c o g n i t i v e s t r u c t u r e , i . e . upon the adequacy of an i n d i v i d u a l ' s e x i s t i n g o r g a n i z a t i o n , s t a b i l i t y and c l a r i t y of knowledge i n a p a r t i c u l a r subject-matter f i e l d " . Jerome S. Bruner, The Process of Education, (New York: Vintage Books, 1963), p. 23-26. ^David P. Ausubel, The Psychology of Meaningful Verbal Learning, (New York: Grune and S t r a t t o n , 1963), p. 21. I b i d . , p. 26. 3 In this writer's opinion Ausubel supports the early understanding of the f i e l d postulates when he claims that:'' "the acquisition of adequate cognitive structure, in turn, has been shown to depend upon both substantive and programmatic factors using for organizational and integrative purposes those substantive concepts and principles in a given discipline that have the widest explanatory power, inclusiveness, generalizability, and rela t a b i l i t y to the subject-matter content of that discipline".^ Although much has been hypothesised about the pedagogical benefits of subject-matter structure, l i t t l e validation has been attempted. Moreover, those studies that have been concerned with such issues have rarely attempted to offer suitable psychological explanations of the role of subject-matter structure in arithmetic understanding. Assuming that the f i e l d postulates form part of mathematical structure, the intent of this study is to provide both plausible psychological explanations and empirical data related to the role of the understanding of these f i e l d postulates in promoting arithmetic understandings. GENERAL STATEMENT OF THE PROBLEM; 1 Since computational algorithms are commonly given logical justification by using the f i e l d postulates, i t is hypothesised that the learning of the f i e l d postulates w i l l f a c i l i t a t e understanding, and through understanding, the learning of such algorithms. More specifically, this study w i l l attempt to determine under what instructional conditions Ibid., p. 26. 4 the understanding of the f i e l d p o s t u l a t e s promotes ease of extension to untaught computational algorithms. Moreover, an attempt w i l l be made to provide a p s y c h o l o g i c a l r a t i o n a l e f o r the i n c l u s i o n of these p o s t u l a t e s i n a contemporary a r i t h m e t i c program. 'DEFINITION OF TERMS In order to avoid an ambiguous and lengthy statement of hypotheses i t was f e l t necessary to d e f i n e the f o l l o w i n g terms: A l g e b r a i c p r i n c i p l e s . These are a l s o r e f e r r e d to as f i e l d axioms, f i e l d p r i n c i p l e s , and f i e l d p o s t u l a t e s . In t h i s study the subset of f i e l d p o s t u l a t e s w i t h which we are concerned i s the set of p o s t u l a t e s that apply to the whole numbers. Algorithm. Any r u l e or ordered set of procedures that can be used to produce a c o r r e c t s o l u t i o n to a computational task independent of the i n d i v i d u a l using that a l g o r i t h m ; f o r example, the usual column a d d i t i o n a l g o r i t h m . I n t e r n a l algorithm. Any a l g o r i t h m whose primary f u n c t i o n i s that i t i s used i n the generation of other algorithms. I t i s i n t e r n a l i n the sense that i t i s considered a means to an end r a t h e r than an end i n i t s e l f . That i s , i t s prime i n s t r u c t i o n a l purpose i s to serve as an a l g e b r a i c p r e r e q u i s i t e f o r more complex computational algorithms. The writer«will use the term f o r mainly r e f e r e n t i a l purposes and w i l l not a t t r i b u t e any s p e c i a l p s y c h o l o g i c a l p r o p e r t i e s to i n t e r n a l algorithms. The i n t e r n a l a l g o r i t h m used i n t h i s study i s the annexation a l g o r i t h m ; the reader should examine the T2 I n s t r u c t i o n a l Sequence on page 9 f o r an e x p lanation of t h i s algorithm. Appendix A describes another 5 i n t e r n a l a l g orithm. Standard m u l t i p l i c a t i o n a l g o r i t h m . For the purposes of t h i s study the standard m u l t i p l i c a t i o n a l g o r i t h m w i l l r e f e r to those procedures used to. compute products such as axb where e i t h e r a or b has a one d i g i t numeral and the other has a two or three d i g i t numeral. For example: 12 132 7 6 x 9 x 9 x 18 x 132 Hereafter such products w i l l be r e f e r r e d to as 2x1 and 3x1 products. Higher-order a l g o r i t h m . For the purposes of t h i s study a "higher-order a l g o r i t h m " w i l l r e f e r to an a l g o r i t h m used to compute products such as axb where.neither a nor b hase> a one d i g i t numeral and where e i t h e r a or b may have more than two d i g i t s i n the numeral. For example: 1001 132 12 x :7 x 111 x 1002 These algorithms are "higher" i n the sense that the standard m u l t i p l i c a t i o n a l g o r i t h m must be conc e p t u a l l y modified i n order to compute nov e l products. Further e l a b o r a t i o n i s given l a t e r i n the chapter. Performance tasks. This r e f e r s to those tasks r e q u i r i n g the a p p l i c a t i o n of the standard m u l t i p l i c a t i o n a l g o r i t h m . L e v e l of per- formance was measured by a w r i t t e n t e s t described i n Chapter I I I . . Transfer t a s k s . A s o l u t i o n of a t r a n s f e r task r e q u i r e d the s u c c e s s f u l extension of the p r e v i o u s l y taught standard m u l t i p l i c a t i o n a l g o r i t h m . L e v e l of t r a n s f e r was measured by a w r i t t e n t e s t described i n Chapter I I I . 6 Tl group. Those students who completed the Tl Instructional Sequence. The reader is referred to page 9 for details of this sequence. T2 group. Those students who completed the T2 Instructional Sequence. The reader is referred to page 9 for details of the sequence. The role of algorithms in arithmetic programs has changed considerably over the past twenty years. Previously, considerable instructional time was devoted to increasing a student's proficiency with an algorithm rather than his understanding of that algorithm; Arithmetic content was treated as i f i t were a series of logically unrelated algorithmic tasks rather than an integrated set of relation- ships between relatively simple concepts. With advances in technology less stress has been placed on mere performance of computational algorithms, although computational algorithms, s t i l l form the main substance of most modern arithmetic programs. Thus,the modern curriculum developer has been primarily concerned that children understand the rationale of an algorithm; i.e. concerned about the a b i l i t y of.children to explain the relationships between the algorithm and other previously acquired algebraic principles. t • Since computational algorithms are logically related to the properties of place value systems and the f i e l d principles, i t has frequently been claimed by some mathematics educators that these logical 7 r e l a t i o n s h i p s enhance the understanding of computational algorithms. E r i c MacPherson expresses t h i s view when he s t a t e s , "the c h i l d who understands a r i t h m e t i c i s the c h i l d who sees how each a l g o r i t h m g f o l l o w s from t h e s e ^ p r i n c i p l e s " . I t would be erroneous to conclude from such statements that c h i l d r e n who understand the f i e l d p r i n c i p l e s are able to d e r i v e spontaneously the usual standardized computational algorithms. Rather such views imply that when a c h i l d understands the r o l e of the p r i n c i p l e i n an a l g o r i t h m , (e.g. recognizes an instance of the p r i n c i p l e i n an algorithm,^demonstrates that a 'step' i n an a l g o r i t h m i s another a p p l i c a t i o n of some p r e v i o u s l y learned p r i n c i p l e , etc.) he i s more l i k e l y to understand the r a t i o n a l e of other r e l a t e d algorithms. However, what seems to be l a c k i n g i n the arguments of " s t r u c t u r e advocates" i s a reasonable p s y c h o l o g i c a l i n t e r p r e t a t i o n of the r o l e of subject-matter s t r u c t u r e i n e f f e c t i n g understanding. More s p e c i f i c a l l y , i n what sense does understanding of the r o l e of the f i e l d p o s t u l a t e s i n s p e c i f i c algorithms promote ease of extension to.untaught r e l a t e d algorithms? For the purposes of t h i s study, i t would seem that of the many l e a r n i n g p s y c h o l o g i s t s , David P. Ausubel and Robert M. Gagne1 are two whose views seem p a r t i c u l a r l y r e l e v a n t . In order to demonstrate the relevance of these p s y c h o l o g i c a l views to t h i s study, i t i s necessary to r e f e r c o n s t a n t l y to s p e c i f i c i n s t r u c t i o n a l sequences used i n t h i s 1 s t u d y . Hence i t seems a p p r o p r i a t e , f i r s t , to e x p l a i n the nature of these i n s t r u c t i o n a l sequences. The E r i c D. MacPherson, "The Foundations of Elementary School Mathematics", Modern I n s t r u c t o r , Volume 33 (October 1964), p. 70. 8 reader is referred to Figure 1. on page 9 for a diagrammatic explanation of these sequences. A. The Tl Instructional Sequence This sequence is typical of many that occur in modern textbooks. The f i r s t s k i l l taught in this sequence i s use of the distributive law. A child is assumed to understand the distributive law when he can: a) use the distributive law to solve such algebraic expressions as 9 x 5 = (9 x 3) + (9 x 1) + (9 xQ) 8 x 6 = (2 x 6) + (2 x 6) .<+"' (&x 6) 9 x 7 = (6 + 3) x (5 +2) = (6 x 5) + (6 x A) + (3 xD) + ( 3 x 2 ) (2 x 6) + (2 x 6) + (Qx 6) = 8 x 6 (8 x 3) + (8 x 5) + (8 x 1) = 8 x C_ b) compute products such as 9 x 8 by application of the distributive law: 9 x 8 = 9 x (2 + 6) = 18 + 54 = 72 9 x 8 = 9 x |2r+::5;^l)= 18 + 45 + 9 = 72 9 x 8 = (4 + 5) x (2 + 6) = 8 + 24 +10 + 30 = 72 The next objective i n the sequence is the acquisition of a rote-type standard multiplication-,;algorithm, the algorithm i s being considered to be rote-type in the sense that no attempt is made explic i t l y to indicate the relationships between the previously mastered s k i l l and this algorithm. FIGURE 1. THE INSTRUCTIONAL SEQUENCES ,9 Tl Instructional Sequence Distributive Law - Standard Multiplication Algorithm L- i * ' T2 Instructional Sequence Distributive Law Annexation Algorithm Standard Multiplication Algorithm Associative Law » i t Transfer Task: Higher-Order Algorithm TFT ,10 B. The T2 Instructional Sequence As with the T l sequence, the T2 instructionalfeseqaience incorporates the understanding of the distributive law as an i n i t i a l learning objective. However, additional algebraic s k i l l s are also considered necessary. These s k i l l s involve the use of the associative law and an internal algorithm, in this case the annexation algorithm. The child is taught to compute products in which 10 is a factor by "annexing the zeros". For example the product of 7 x 200 is i n i t i a l l y computed by using the associative law in the following manner: 7 x 200 = 7 x (2 x 100) = (7 x 2) x 100 = 14 x 100 = 1400 or 7 x 2 hundreds = ( 7 x 2 ) hundreds = 14 hundreds = 1400 Later computation simply involves direct annexation. For example, 7_ x 2 00 = 14 00 The standard multiplication algorithm utilized in this sequence validates procedural "steps" by expli c i t l y pointing out instances of the prior learned s k i l l s . This writer is primarily interested in the effects of each instructional sequences on the amount of transfer to computational tasks that involve an untaught higher-order algorithm. As mentioned earlier in this chapter, the views of Gagne and Ausubel would seem to provide possible explanations of these transfer differences. Gagne has developed what he considers a hierarchy-of-learning 9 • model. Before a specified learning task can be mastered, Gagne would Robert M. Gagne, The Conditions of Learning, (New York: Holt, Rinehart and Winston, 1970). 1.1 claim that a numbernof subordinate concepts must also be mastered. These concepts in turn depend upon other subordinate concepts so that i t can be argued that Gagne*'s model ultimately resembles that of S— R learning. As Gagne explains, when such an4'.analysis (selecting appropriate prerequisite tasks) is continued progressively to the point of delineating am .entire set of capabilities having an order relation to each other (in the sense that in each case prerequisite capabilities are represented as subordinate in position, indicating they need to be previously learned), one has a learning hierarchy. The analytic process may be carried out i f desired, u n t i l the simplest kinds of learnings (Ss-^R's, chains, dis- criminations) are reached and i d e n t i f i e d . ^ Thus:, once the terminal task is clearly specified, the problem is to select hypothesised prerequisites and arrange these in a hierarchical manner. Although i n i t i a l l y these prerequisites are selected logically on an a p r i o r i basis, a hypothesised prerequisite is concluded to be pedagogically necessary only after empirical investigation. As Gagne explains: "a subordinate s k i l l is determined to be pedagogically necessary i f i t facilitates the learning of the higher-level s k i l l to which i t is related. In contrast, i f the subordinate s k i l l has not been previously mastered, there w i l l be no f a c i l i t a t i o n of the higher- level s k i l l . This latter condition does not mean that the higher- level skill^cannot be learned — only that, on the average, in the group of students for whom a topic sequence has been designed, learning w i l l not be accomplished readily"."'"''" Thus i f transfer differences between the T l and T2 groups were observed, Gagne, rather than trying 10Gagne, op. c i t . , p. 2̂ 8'- 1 : LIbid. , p. 239-240. 12 to explain the differences in terms of any particular learning theory, would probably attribute these differences to the selections and arrangement of prerequisites, since he seems to be more concerned with the development of empirically validated,-hierarchies than the validation of contemporary psychological theories. Hence this study could prove to be valuable for the curriculum designer i f i t produced a more effective instructional sequence for teaching i n i t i a l multiplication s k i l l s . Ausubel would view the potential efficiency of each instructional sequence for promoting transfer in quite a different sense than would Gagne. For Ausubel, the amount of transfer brought to a learning task depends on an individual's cognitive structure, where "cognitive structure" means an individual's organization, s t a b i l i t y , and clarity ' 12 of knowledge in a particular subject-matter f i e l d at any given time. That is existing cognitive structure is regarded as the major factor influencing the learning and retention of potentially meaningful new material in the same f i e l d . According to Ausubel, a major criterion determining whether learning material is potentially meaningful is i t s relatability to the particular cognitive structure of a particular learner. As Ausubel states: for meaningful learning to occur in fact, i t is not sufficient that the new material simply be relatable to relevant ideas . in the abstract sense of the term. The cognitive structure of the particular learner must include a requisite intellectual . capacities, ideational content and experientfa^, background. Ausubel, pp. c i t . , p. 26. 'ibid. , p . 23. 13 The key concern of this study i s the effect of these instructional sequences on cognitive structure. That i s , which of the Tl and T2 sequences might be best integrated by the learner and in what sense this act of integration promotes greater transfer to tasks requiring the use of an untaught higher-order algorithm. According*to Ausubel, new learning is sometimes incorporated 14 into cognitive structure by correlative subsumption. This psychological phenomenon occurs when a learner somehow determines that new learning material i s related to relevant cognitive subsumers via some general principle. Thus new learning material may be best incorporated into an individual's cognitive structure i f those principles which require the least extension act as subsumers. In Ausubel's terms one might suppose that the learning of the algebraic principles of arithmetic may affect the learning of logically related computational algorithms in the same sense as 'advance organizers'. Thus i t i s hypothesised that the T2 instructional tasks might form relatively stronger subsumers than the Tl tasks, for future transfer tasks requiring the use of a higher-order algorithm. For example, consider the possible differing complexity of extension from.the standard multiplication algorithm to the higher- order algorithm that each treatment group must make for successful solution of such a transfer task as 107 x 11. Ausubel, op. c i t . , p. 77. 14 A typical solution that might be exhibited by the T2 group could be as follows: 11 1 0 + 1 (partitioning both factors into x 107 ^ 100 + .7 (binary sums involving powers of ten 10 + 1 100 + 7 (application of distributive 7 /law and annexation algorithm 70 100 1000 It i s assumed that no 'new'1 concept or s k i l l is required for successful extension from the standard multiplication algorithm to the higher- order algorithm. (The s k i l l of partitioning both factors, rather than just one factor, before application of the distributive principle was included in both instructional sequences.) The extension of the rote-type standard multiplication algorithm to the standardized higher order algorithm by the Tl procedure seems a very remote possibility: 11 11 f"move over one space to the le f t x 107 " ^ x 107 ^when multiplying by a factor of ten" 107 107 Suppose a Tl group member attempts to compute such products as 107 x 11 by considering the 11 as 'one di g i t ' and proceeds^as with the standard multiplication algorithm: 107 x 1 = 107 as with standard multiplication x d-0 7J algorithm, place 'seven'.and 'carry 7 ten' I 1 Q l l (107 x 1) + 10 = 117 place 'seven' and 'carry 11' x (L0_7J 1177 15 ' Although such partial products as 7 x 11 could be computed by using the Tl standard multiplication algorithm,this extended procedure.' seems much more d i f f i c u l t than the hypothesised T2 procedure. Transfer to tasks involving 3 x 3 and 4 x 3 products would seem even more unlikely considering the complexity of extending the Tl standard algorithm. HYPOTHESES Most textbooks andypractitioners are being urged by curriculum specialists to promote the understanding of algebraic principles. The arguments for the inclusion of such principles are based on the belief that much of arithmetic, and especially com- putational algorithms, may be better understood through the learning of algebraic principles. Hence from both a practical and'ahtheoretical point of view, i t seems worthwhile to investigate the validity of the following hypotheses: • Hypothesis One — T h e T l group w i l l score significantly higher than the T2 group in the performance of the standard multiplication algorithm, as measured by the performance test. Hypothesis Two — The T2.group w i l l score significantly higher than the T l group on the test of transfer from the standard multiplication algorithm to a higher-order algorithm. CHAPTER II SURVEY OF THE LITERATURE In reviewing the literature, one soon realizes that very few studies have been concerned with children's acquisition or use of the f i e l d of postulates to generate algorithms. Children's understanding of the f i e l d postulates without formal instruction was studied by Crawford in 1964.^ Using a multiple choice test of 45 items, he tested each of the eleven f i e l d axioms once at each level of Bloom's taxonomy. He found that the mean'scores increased significantly, from one even numbered grade to the next, ..in. a linear manner. Students exposed to 'modern mathematics' content in grades 9 and 10 had scores significantly superior to those of students in a l l other programs at the same level. This study seems important in that i t provides data on developmental processes which were occuring without explicit teaching. A study by Hall attempted to determine whether the rote learning of certain multiplication combinations could be accomplished more effectively through teaching procedures emphasising the commutative and ordered pair :approach in conjunction with practice on related 16' combinations. This procedure was compared to teaching procedures ."^Douglas Crawford, "An Investigation of Age-Grade Trends in Understanding.the Field Axioms," Dissertation Abstracts, Syracuse University, 1964. "^Kenneth D. Hall, "An Experimental Study of Two Methods of Instruction.for Mastering Multiplication Facts at the Third-Grade Level," Doctoral Dissertation, Duke University, 1967. 17 employing the traditional approach with practice on commuted combinations. He found no significant difference between the groups on both arithmetic computation and achievement in multiplication. This result lends support to the notion that there is no advantage in the mere acquisition of a f i e l d postulate. Gray.,, in 1964, tried to determine how a method of teaching introductory multiplication which stressed development of an under- standing of the distributive law would relate to pupil development as measured in terms of achievement, transfer, retention and progress toward maturity of understanding of multiplication."*"^ He used two treatment groups. cOne group, Tl was taught according to what was judged to be. the ..best of current .methods. The other group, T2,.was provided with introductory multiplication using an understanding of the distributive principle. Pre-experimental achievement and I.Q. were covaried. He constructed written pre-test, post-test, retention, and transfer.tests. Individual interviews of 110 random subjects measured maturity of understanding. His results warranted the follow- ing conclusions: 1. A program of arithmetic instruction which introduces multiplication by a method which stressed understanding of the dis- tributive property produced results superior to those of current methods. 2. Understanding of. the distributive property enables children to proceed independently to the finding of products of Roland.F. Gray, "An Experimental Study of Introductory Multiplication,"Doctoral Dissertation, University of California, Berkeleyi 1964. - • 18 novel m u l t i p l i c a t i o n combinations, to a greater extent than those c h i l d r e n not introduced to the d i s t r i b u t i v e p r i n c i p l e . 3. These c h i l d r e n appeared not to develop an understanding of the d i s t r i b u t i v e property unless i t was s p e c i f i c a l l y taught. There have been r e l a t i v e l y Jf;ew s t u d i e s which have been concerned w i t h the r e l a t i o n s h i p between understanding of the f i e l d p o s t u l a t e s and l e a r n i n g of computational algorithms. In most s t u d i e s the algorithms were i l l u s t r a t e d using p h y s i c a l devices. However, Schrankler t r i e d to evaluate the e f f e c t i v e n e s s of two p r e - a l g o r i t h m treatments.in. combination w i t h two algorithms f o r teaching the 18 m u l t i p l i c a t i o n of whole numbers at three i n t e l l i g e n c e l e v e l s . E f f e c t i v e n e s s was evaluated i n terms of computational s k i l l s , speed i n computation, understanding of the m u l t i p l i c a t i o n process, problem s o l v i n g and r e t e n t i o n of the four previous c r i t e r i a . The readiness phase placed emphasis on the 100 m u l t i p l i c a t i o n f a c t s f o r group B^. Emphasis was placed on the commutative, a s s o c i a t i v e and d i s t r i b u t i v e p r o p e r t i e s f o r group H^. F o l l o w i n g t h i s p e r i o d , these groups were subdivided i n t o a l g o r i t h m i c groups. Group subjects were taught the indent u n i t - s k i l l s a l g orithm: 57 x 28 . 456 114 1596 • W i l l i a m .Schrankler, "A Study of the E f f e c t i v e n e s s of Four Methods of Teaching M u l t i p l i c a t i o n of-.Whole Numbers i n Grade Four," D i s s e r t a t i o n A b s t r a c t s , U n i v e r s i t y of Minnesota, 1966. 19 Group subjects were taught the partial products algorithm: 57 x 28 56 400 140 1000 1596 No mention was made of the use of the annexation algorithm in the parti a l products algorithm. Students in each of the treatment groups, A1 B1' A2 B1' A1 B2' A2 B2' w e r e identified at one of three levels of intelligence. Schrankler found that the A^B^ group tested higher on the test of understanding than the other groups. This same group also tested higher on the retention test of understanding. The fact that theiA^B^ group was found to be superior to the A^B^ group on the test of understanding of the multiplication algorithm is of particular.interest. This result suggests that the understanding of computational algorithms is best promoted by the explicit application of.previously acquired algebraic principles. Studies such as Schrankler's have b een restricted to examining the use of algebraic principles in promoting understanding of already acquired computational algorithms. No:studies were found which examined the use of algebraic principles in promoting transfer to untaught higher-order algorithms. CHAPTER I I I DESIGN OF THE EXPERIMENT THE SAMPLE The experimenter decided to use grade three students as subjects i n the study s i n c e they had.had some experience w i t h m u l t i p l i c a t i o n but had not as yet been taught the standard m u l t i p l i c a t i o n a l g o r i t h m . Eight grade three c l a s s e s were s e l e c t e d from s i x B r i t i s h Columbia schools. A l l e i g h t of the teachers i n v o l v e d i n the study were v o l u n t e e r s . A f t e r the readiness phase, which w i l l be described i n the next s e c t i o n , students i n each classroom were randomly assigned to e i t h e r the Tl.'or T2 group. A student's t e s t scores were omitted from the study i f more than one treatment lesson was missed. A t o t a l of 238 su b j e c t s were used, to obt a i n the f i n a l s e t of data; 44 subjects were used f o r t e s t a n a l y s i s , and the remaining 194 subjects f o r t e s t i n g the hypotheses. THE INSTRUCTIONAL SEQUENCES A. The Readiness Phase.• During t h i s phase, a l l the subjects were taught the s k i l l s which were considered to be p r e r e q u i s i t e s f o r the treatment phase. 21 A set of lesson plans was provided for each teacher involved in the study. Briefly, these lessons stressed: - the relationship between multiplication and arrays. For example, 3 x 4 means a "3 by 4" array - the distributive law; both the right hand and the l e f t hand. This was to be accomplished by breaking arrays into the "sums" of smaller arrays. For example: 4 x 5 4 x (2 + 3) (4 x 2) + (4 x 3) X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X the application of the distributive law to multiplication problems. Only the techniques of breaking a product into the sums of smaller products was stressed and no attempt was made to have children provide a f i n a l numerical answer. For example, 28 2 0 + 8 2 0 + 8 x 19 x 19 x 19 19 x 8 + 19 x 20 For a f u l l description of these lessons, the reader i s referred to Appendix B. In order to parallel typical teaching practices and thus increase the generalizability of this study, the writer did not demand a fixed criterion of mastery of the distributive law. Rather, a l l teachers were instructed to terminate this phase when, in their judgement, the students indicated a mastery of the distributive law. 22 The teachers reported t h a t t h i s phase g e n e r a l l y took about f i v e hours of classroom i n s t r u c t i o n . B. The Treatment Phase. Every teacher was provided w i t h a set of w r i t t e n lesson plans s u i t a b l e f o r each treatment lesson. The lessons contained the general, dialogue, examples and seatwork to be used. The teachers met w i t h the w r i t e r twice during t h i s phase to ensure that they understood the lesson m a t e r i a l s . To minimize the e f f e c t of teacher d i f f e r e n c e s each teacher taught both groups w i t h i n her c l a s s . To minimize p u p i l i n t e r a c t i o n , i t was arranged to have the groups separated during a treatment l e s s o n . A l l p u p i l s were t o l d by t h e i r teacher they they were i n v o l v e d i n an experiment. To minimize outside influences,, teachers were i n s t r u c t e d to give no;;homework during t h i s phase. Both the T l and T2 groups had approximately four hours of treatment time. A b r i e f d e s c r i p t i o n of both treatments i s provided i n the f o l l o w i n g s e c t i o n but the reader i s r e f e r r e d to Appendix C f o r the lesson plans used. The T l I n s t r u c t i o n a l Sequence The T l group was taught the rote-type a l g o r i t h m described i n Chapter I. The a l g o r i t h m was r e s t r i c t e d to 2 x 1 and 3 x 1 products. To convince the students of t h e - l e g i t i m a c y of t h i s . - , a l g o r i t h m , a l l answers were i n i t i a l l y checked .using the d i s t r i b u t i v e law. For example^ the check might be made as f o l l o w s : 23 12 C 2 + 2 + 2 + 3 + 3 18 + 18 + 18 + 27 + 27 = 108 x 9 * x 9 * In contrast to the T2 Instructional Sequence, no explicit application of the distributive principle was stressed. Once the students were convinced that this rote-type algorithm yielded correct products, the objective of the succeeding lessons was merely to provide further practice. The T2 Instruction Sequence The T2 subjects were f i r s t taught the annexation algorithm. A l l computation of 2 x 1 and 3 x 1 products were accomplished by using the distributive principle.in conjunction with the annexation algorithm. The teachers were instructed to use the same examples and seatwork with both groups. THE MEASURING INSTRUMENTS Both the performance and transfer tests were written tests constructed by the experimenter. The teachers knew the general nature of each test prior to the treatment phase but did not see the actual test items un t i l the test administration date. Teachers were instructed to give students ample time to complete both tests. Any solution by repeated addition was disregarded for both tests. The reader is referred to Appendix D for the actual tests used. 24 A. .Performance Test. This test consisted of twenty items that required the use of the standard multiplication algorithm. The total number of correct responses was considered a measure of an individual's performance. In order to delete items that were either excessively d i f f i c u l t or easy, a point b i s e r i a l correlation was calculated for every item. It was decided to reject an item i f the point b i s e r i a l r was less than .20 19 in magnitude. As a result of this analysis, a l l items of the original test were retained. Since this test was designed to measure a very specific t r a i t , (viz. the a b i l i t y to use the standard multiplication algorithm), i t was f e l t that a measure of item homogeneity should be determined. Thus a KR20 was calculated for the twenty item test and was found to be .93. This value indicated that the performance test was high in item homogeneity. The results of the items analysis can be found in Table I. B> Transfer Test. This test consisted of fourteen items which were intended to measure the abi l i t y to compute novel products requiring the use of a higher-order algorithm. Neither the Tl group nor the T2 group had been previously exposed to any of these items. The total number of correct responses was considered a measure of the abi l i t y to extend the standard multiplication algorithm. As with the performance test, Nunnally, J.C., Psychometric Theory, (New York: McGraw- H i l l Book Company, 1967), p. 242. 25 a KR20 was c a l c u l a t e d ' t o evaluate item homogeneity. The KR20 of the f i n a l fourteen item t e s t was found to be .78. I t i s p o s s i b l e that the KR20 might have been increased i n magnitude by i n c l u d i n g a d d i t i o n a l t e s t items. However, t h i s lengthening procedure was f e l t to be in a p p r o p r i a t e s i n c e a very lengthy t e s t might have had the undesirable e f f e c t of i n c r e a s i n g t e s t anxiety of such young and 't e s t immature' students. The r e s u l t s of the item a n a l y s i s can be found i n Table I I . 26 TABLE I ANALYSIS OF THE PERFORMANCE TEST Item Point Item Point Number Biserial Number. Biserial 1 .45 11 .53 2 .61 12 .73 3 .50 13 .74 4 .77 14 .72 5 .64 15 .49 6 .75 16 .67 7 .74 17 .72 8 .70 18 .64 9 .54 19 .74 10 .64 20 .67 TABLE II ANALYSIS OF THE TRANSFER TEST Item Point Item Point Number Biserial Number Biserial 1 .50 13 0.0 * 2 .59 14 0.0 * 3 .43 15 0.0 * 4 - .59 16 0.0 * 5 .65 17 0.0 * 6 .65 18 0.0 * 7 0.0 * ' 19 0.0 * 8 .56 20 .24^ 9 .46 21 .47 10 .43 22 0.0 * 11 .65 23 .69 12 .51 Deleted items. CHAPTER IV ANALYSIS OF THE DATA EXPERIMENTAL RUN A;. The Performance Hypothesis. The s t a t i s t i c a l , h y p o t h e s e s to be t e s t e d were: H c: There w i l l be no s i g n i f i c a n t d i f f e r e n c e s between the means of the T l and T2 groups as measured by the performance t e s t . That i s : ̂ T = / ( T 2 H^: The mean of the T l group w i l l be s i g n i f i c a n t l y greater than the mean of the 1^ group as measured by the performance t e s t . That i s : T± >̂ J4^ T 2 Each classroom teacher taught both the T l and T2 groups i n her classroom. Thus a subject i n a classroom was given e i t h e r the T l or T2 i n s t r u c t i o n by h i s or her usual classroom teacher. The experimenter considered the d i f f e r e n c e s i n teacher performance to be a random e f f e c t , w h i l e d i f f e r e n c e i n treatment were considered to be a f i x e d e f f e c t . In other words, a mixed a n a l y s i s of vari a n c e model was f e l t to be the most appropriate s t a t i s t i c a l model to t e s t the hypothesis. The l i n e a r model chosen was: i n d i c a t e s the i t h teacher (random e f f e c t ) and represents the j t h treatment l e v e l ( f i x e d e f f e c t ) . 28 The experimenter made the usual assumptions underlying an ANOVA but did not test for these as the F test is reasonably robust to violations 20 of these assumptions. The assumptions made were: a) the teachers, used in the experiment, were randomly selected from a normal population, i.e. ± are NID (0, ̂ 2 ) and W i j a r e N I D C ° ' ^ > b) the£"^_.^ are normally distributed, i.e. £ i j k a r e N I D Co, 6£), c) the treatment variances are homogeneous, i.e. C f T 1 = G , n The reader is reminded that the denominator in the test for treatment (fixed) effects in a mixed model is the interaction term 21 and not the usual error term. The null hypothesis was considered to be rejected i f the probability of obtaining an F value, under the null hypothesis, was less than or equal to ̂ = .05. A l l calculations were done at the University of British Columbia Computer Centre using the BMD-X64 program. This program allows for differing numbers of subjects in a c e l l by using the least squares estimate technique. The results of this analysis are summarized in Table III. 20 Lindquist, E.F., Design and Analysis of Experiments in Psychology and Education (Boston: Houghton, M i f f l i n Company, 1953), pp. 78-90. and Winston^ o i f ? S ^ 3 - 4 4 t B t * B t * n p ( N e W Y ° r k : H o l t ' R i n e h a r t 29 TABLE I I I ANALYSIS OF VARIANCE: PERFORMANCE HYPOTHESIS Source of V a r i a t i o n df Sum of Squares Mean Squares Teacher 7 Treatment 1 I n t e r a c t i o n 7 E r r o r 178 812.197 1775.630 57.679 3927.752 116.028 1775.630 8.240 22.066 5.258 215.491 0.373 .0000083 Mean f o r T l group was 14.418 Mean f o r T2 group was 8.330 Since the p r o b a b i l i t y of o b t a i n i n g an F-value of 215.491 was c a l c u l a t e d to be f a r le s s * t h a n .05, the n u l l hypothesis H c was r e j e c t e d and the a l t e r n a t e hypothesis H^ was accepted. Bu, The Transfer Hypothesis. The s t a t i s t i c a l hypotheses to be teste d were: H Q: There w i l l be no s i g n i f i c a n t d i f f e r e n c e s between the means of the T l and T2 group as measured by the t r a n s f e r t e s t . That i s : J^T± = J L ( T 2 H^: The mean of the T2 group w i l l be greater than the mean of the T l group as measured by the t r a n s f e r t e s t . That i s : y 6 ^ T 2 > / < T 1 O r i g i n a l l y the experimenter had hoped to terminate the treatment phase only when both groups had reached a s p e c i f i e d per- formance c r i t e r i o n . That i s , u n t i l there were no s i g n i f i c a n t d i f f e r e n c e s between the two groups on the performance of the standard m u l t i p l i c a t i o n algorithm. Thus, i f any degree o f . c o r r e l a t i o n e x i s t e d between the - performance and t r a n s f e r t a s k s , t h i s p r e l i m i n a r y equating would minimize any d i f f e r e n c e s between the groups on the t r a n s f e r t e s t that might be a r e s u l t of d i f f e r e n c e s between the means on the performance tasks. However, to b r i n g about the e q u a l i t y of the groups on the performance t e s t , performance scores were covaried w i t h t r a n s f e r scores. Thus the l i n e a r model used to t e s t the t r a n s f e r hypothesis was: Y i j k + T 3 + <xv±i a... - x..k) +e..k where T'j 1 a n d £ „ k were p r e v i o u s l y d e f i n e d ; € ' i s an estimate of the common population r e g r e s s i o n w c o e f f i c i e n t ; X... i s a subject's performance score and X... i s the grand mean of the t o t a l sample on the performance t e s t . In a d d i t i o n to the necessary assumptions un d e r l y i n g an ANOVA that were discussed i n the previous s e c t i o n , the use of t h i s model n e c e s s i t a t e s the f o l l o w i n g a d d i t i o n a l assumptions: a) the population w i t h i n - c e l l r e g r e s s i o n c o e f f i c i e n t s are homogeneous, i . e . 8 w = £ w . . f 0 r 3 1 1 Because l i t t l e i s known about the F t e s t w i t h respect to v i o l a t i o n of the foregoing assumption, i t was decided to t e s t t h i s assumption at a l e v e l of s i g n i f i c a n c e equal to .10. Using the BMD-X82 computer program, which adjusts f o r d i f f e r i n g numbers of subjects 31 i n a c e l l , an F of 1.06, w i t h a numerator and denominator of 15 and 162 degrees of freedom r e s p e c t i v e l y , was obtained. Since the p r o b a b i l i t y of o b t a i n i n g such an F, under the n u l l hypothesis i s .398, homogeneity of the re g r e s s i o n c o e f f i c i e n t s was assumed. b) the pooled e s t i m a t e ^ ' i s not zero. w In t e s t i n g t h i s assumption at t h e . 0 5 l e v e l of s i g n i f i c a n c e , an F of 99.14, w i t h a numerator and denominator of 1 and 177 degrees of freedom r e s p e c t i v e l y , was obtained. Since the p r o b a b i l i t y of o b t a i n i n g — 8 such an F, under the n u l l hypothesis, i s l e s s than 10 the hypothesis of zero slope was e a s i l y r e j e c t e d . The r e s u l t s of the s t a t i s t i c a l a n a l y s i s of the t r a n s f e r hypothesis are summarized i n Table IV. TABLE IV ANALYSIS OF COVARIANCE: TRANSFER HYPOTHESIS Source of ^ Adjusted V a r i a t i o n Sum Square Teacher 7 179.984 25.712 3.961 Treatment 1 24.363 24.363 2.725 .141 I n t e r a c t i o n 7 62.570 8.939 1.377 E r r o r 177 1149.077 6.492 Adjusted mean f o r T l group was 3.776 Adjusted mean f o r T2 group was 4.544 Since the p r o b a b i l i t y of o b t a i n i n g an F of 2.725 i s .141, the n u l l hypothesis was accepted. That i s , the: mean of the T2 group was hi g h e r , but not s i g n i f i c a n t l y h i g h e r , than the mean of the T l group. Mean Square 1 CHAPTER V CONCLUSIONS AND IMPLICATIONS FOR FURTHER STUDY DISCUSSION OF CONCLUSIONS A./ Performance Hypothesis. With respect to the performance hypothesis, i t was found that subjects taught a rote-type algorithm did significantly better on tasks requiring the use of a standard multiplication algorithm than did the subjects taught a standard multiplication algorithm using previously learned algebraic principles. In fact, the performance level of the T2 group was so inferior to that of the Tl group that this researcher suspected thafc.t^ei'dfaihep'teacliers hadlnot-followed the " recommended treatment procedures. It was quite possible that, since most teachers had never used an instructional sequence like the T2 sequence, they may have had an experimental bias towards the rote Tl sequence. Perhaps more frequent observations of teacher performance would have eliminated such a bias towards treatment. B, Transfer Hypothesis. With respect to the transfer hypothesis, i t was found that subjects taught a standard multiplication algorithm using algebraic principles appeared to exhibit superior positive transfer to tasks requiring the use of a higher-order algorithm. However, this difference 33 in the amount of transfer was not statistically significant at the oC = level of significance. Because of the nature of the treatments, a T2 subject needed more time fo format correctly a computational problem than did a Tl subject. Thus, teachers were instructed to give students at least one hour to attempt a l l fourteen items of the transfer test. However, after a brief discussion with the teachers, i t was noted that some had allowed students about thirty minutes to complete this test. In fact, one teacher who obviously misunderstood the intent of the transfer test, stated that she gave children about fifteen minutes on this test because "the students weren't taught to compute such large products". This^situation could not be remedied by another test administration because school holidays immediately followed the test administration date. PROBLEMS FOR FURTHER STUDY Since the results of this study must remain inconclusive because of important uncontrolled factors, a replicate study employing controls to minimize teacher misunderstandings should be conducted. This writer also suggests that a study be conducted to examine the effect of instructional sequences that use algebraic principles to teach computational algorithms on a student's attitude toward arithmetic. It is postulated that instructional sequences that maximize the use of previously learned algebraic principles may enable a student to view arithmetic as a series of integrated tasks. This inte- grated view of arithmetic, might, in turn, have a positive effect on a student's attitude towards arithmetic. BIBLIOGRAPHY Ausubel, David P. The Psychology of Meaningful Verbal Learning. New »>York: Grune and St r a t ton, 1963. Bruner, Jerome S. The Process of Education. New York: Vintage Books, 1963. The Cambridge Conference on School Mathematics. Goals f o r School Mathematics. New York: Houghton M i f f l i n Company, 1963. Crawford, Douglas Houston. "An I n v e s t i g a t i o n of Age-Grade Trends i n Understanding the F i e l d Axioms." D i s s e r t a t i o n A b s t r a c t s , Syracuse U n i v e r s i t y , 1964. E i c h o l z , Robert E., et a l . Elementary School Mathematics, Book 3. Don M i l l s , Ontario: Addison-Wesley (Canada) L t d . , 1966. Gagne, Robert M. The Conditions of Learning. New York: H o l t , Rinehart and Winston, 1970. Gray, Roland F r a n c i s . "An Experimental Study of Introductory M u l t i p l i c a t i o n . " D o c t o r a l D i s s e r t a t i o n , U n i v e r s i t y of C a l i f o r n i a , Berkeley, 1964. H a l l , Kenneth Dwight. "An Experimental Study of Two Methods of I n s t r u c t i o n f o r Mastering M u l t i p l i c a t i o n Facts at the T h i r d Grade L e v e l . " D i s s e r t a t i o n A b s t r a c t s , Duke U n i v e r s i t y , 1967. Hays, W i l l i a m L. S t a t i s t i c s . New York: H o l t , Rinehart and Winston, 1963. L i n d q u i s t , E.F. Design and A n a l y s i s of Experiments i n Psychology and Education. Boston: Houghton M i f f l i n Company, 1953. MacPherson, Eric.D. "The Foundations of•Elementary School Mathematics." The Modern I n s t r u c t o r , V o l . 33, September 1964-April 1965. MacShell, Leo. "Two Aspects of Introductory M u l t i p l i c a t i o n : The Array and the D i s t r i b u t i v e P r o p e r t y . " D i s s e r t a t i o n A b s t r a c t s , State U n i v e r s i t y of Iowa, 1964. Nunnally, J.C. Psychometric Theory. New York: McGraw-Hill Book Company, 1967. Schrankler, W i l l i a m Jean. "A Study of the E f f e c t i v e n e s s of Four Methods f o r Teaching M u l t i p l i c a t i o n of Whole Numbers i n Grade Four." D i s s e r t a t i o n A b s t r a c t s , U n i v e r s i t y of Minnesota, 1966. APPENDIX A THE IDENTIFICATION OF ANOTHER INTERNAL ALGORITHM THE DIVISION OF FRACTIONS ALGORITHM AND THE EQUAL FACTORS ALGORITHM In addition to the mere rote, performance of an algorithm, most modern programs attempt to provide some rationale of that algorithm. Perhaps the most d i f f i c u l t algorithm to explain reasonably to the average elementary school child is the division of fractions algbarithm. In an attempt to provide this rationale, 22 a typical approach is as follows: 1 - 3 Step 1 5 y T--^ = W The work below shows how to 1 3 11 divide 5TT by —. Use — as another 2 4 2 a name for oy. Step 2 ^ Express the "divisiori-vr. in this way. 3. .'4 Step 3 11 First you need to get 1 for the 9 3 4 _J 4_ divisor, you multiply — x — to get 1. 4 X 3 ' 4 J 11 4 3 4 Step 4 — x — has been multiplied by —. So you x — must also multiply — x — • 11 4 —— x — You do not need to write the divisor when i t i s 1. 11 4 — x - So now you can write the computation in this way 7j You found 7̂- by multiplying -j by 2 2Maurice L. Hartung, et al.„ Seeing Through Arithmetic 6, Scott, Foresman and Co., Chxcago-7-p-r-198^ Step 5 11 '' 2 4 X 3 = 44 6 S I i 3 7^ 5 2 * 4 •73 37 One apparent assumption that has been made i s that the procedures taken i n Steps 3 and 4 can be followed by the elementary school c h i l d . However, the v a l i d i t y of these two steps must be b l i n d l y accepted by the c h i l d s i n c e no p r e l i m i n a r y work has been done that could be used to j u s t i f y these steps. One wonders what advantages t h i s modern treatment has over the rote " i n v e r t and m u l t i p l y " a l g o r i t h m because apparently we have merely s u b s t i t u t e d a long r o t e a l g o r i t h m f o r a short r o t e algorithm. What i s needed to v a l i d a t e steps 3 and 4 i s an i n t e r n a l a l g o r i t h m ; the equal f a c t o r s algorithm. This a l g o r i t h m s t a t e s that i f the d i v i s o r and dividend are m u l t i p l i e d or d i v i d e d by any non- zero r a t i o n a l number, the quotient remains unchanged. For example: (8 4- 4) = (2 x 8) 4- (2 x 4) I f t h i s i n t e r n a l algorithm i s mastered, the d i v i s i o n of f r a c t i o n s becomes much more reasonable to the elementary school c h i l d . Step 1 11 . 3 ,11 4N . ,3 4N ~2 T ~k =' ~2~ X "J 4" X 3" equal f a c t o r s a l g o r i t h m Step 2 (— x —; T (•£• x —) = (— x —) T 1 m u l t i p l i c a t i o n of r e c i p r o c a l s Step 3 (— x —; T 1 = — x — property of one Step 4 11 3 2 11 _ 2 ' 3 ~~ 2 X 3 - APPENDIX B READINESS PHASE LESSON PLANS THE READINESS PHASE These three lessons should; enable most students to acquire the necessary prerequisite s k i l l s before the actual experimental treatment begins. The teacher w i l l find that a l l lesson plans are quite detailed including examples to, use, questions to ask, answers one can expect, and seatwork problems to be used after each lesson. In order to minimize any misunderstanding that may result, w i l l the teachers please observe closely the following instructions: 1. Carefully read the lesson plans at least a day before the presentation. If you have any questions or suggestions, please don't hesitate to contact me. The phone number is 736-0595. 2. Try to give the answers to a l l seatwork questions before the students leave school for that day. Give NO HOMEWORK as outside influences must be discouraged. 3. Record any absenteeisms on the l i s t provided. 4. If more examples are needed to i l l u s t r a t e any concept before the seatwork is attempted, please feel free to do more. 5. If you feel that another period may be necessary, then extend this phase for another period. 40 LESSON 1: MULTIPLICATION AND ARRAYS The basic objectives of the lesson are:. A. To introduce the concepts of an array and i t s relationship to multiplication. B. To ill u s t r a t e the commutative principle for multiplication; a x b = b x a (in this case, an a x b array, though drawn differently, has the same number of elements as a b x a array). 1. Introduction of an Array "Today we w i l l see how we can multiply using an array." (Write the word array on the board). "Here i s an example of an array." x x x x x x "This array i s called a 2 x 3 array since i t has 2 rows of 3 crosses." „ 0 x x x row 1 2 x 3 x x x row 2 "We usually write the words '2 by 3' as '2 x 3'." Draw a 4 x 3 array on the board; ask children to give reasons for their responses. X X X x x x x x x X X X "This is a 4 x 3 array because i t has 4 rows of 3." 41 4 x 3 array x x x x x x x x x x x x row 1 row 2 row 3 row 4 Draw the following examples on the board (one at a time) and ask the children to name each. Ask children to give reasons for their responses. x x x x x x x x x x x x x x x 5 x 3 array (5 rows of 3) x x x x x x x x x x x x. 2 x 6 array (2 rows of 6) x x x x x x 1 x 6 array (1 row of 6) "Here i s the name of an array." (Put 3 x 6 on the board). "This time, try to draw what this array would look l i k e . " (Give children a few moments and then check individual pupil's work) Answer: 3 x 6 array x x x x x x row 1 x x x x x x row 2 x x x x x x row 3 Ask the children to draw the following arrays: 1 x 4 2 x 5 3 x 7 Check pupils' work and ask reasons for their responses. 2. The Commutative Principle for Multiplication; a x b = b x a "Can^anyone come up to the'board and draw a 2 x 4 array?" (Have a pupil come to the board and draw the array; ask the child how many crosses are in this array). 42 "Can̂ y anyone come up to the board and draw us a 4 x 2 o n array? Draw attention to the fact that a 2 x 4 array and a 4 x 2'- array have the same number of elements but are drawn differently. Repeat the same procedure using the 5 x 4 array and a 4 x 5 array. Draw an 8 x 4 array on the board. x x x x x x x x x x x x x x x x x x x x x x x x x x x x X X X . x Ask the children i f they can find another array which would have the same number of crosses, but would be drawn differently. Note: several answers are possible, but draw attention to the fact that i f we rotate the array we end up with a 4 x 8 array. x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x 8 x 4 array 4 x 8 array "Here is a very large array (13 x 8). Can*'anyone t e l l me another array that would be drawn differently but would have the same number of crosses?" (Answer: 8 x 13). 43 If needed, do other examples to emphasize the point that an a x b array has the same number of elements as a b x a array. 3. Seatwork These series of questions are to provide additional practice with the concepts covered in Lesson 1. Please allow enough time for marking the seatwork as this w i l l enable you to determine i f most of your class w i l l be ready for Lesson 2. A. Name the following arrays. 1. 2. 3. X X X X X X X X X X X X X X X : x x x x X X X X X X X X X X X X X X X B. Draw the following a-rrays. 1. 2. 3. 4. 5 x X X X X X X 2 x 6 8 x 2 10 x 4 1 x 11 11 x 1 (Answer: 5 x 3 ) (Answer: 1 x 7 ) (Answer: 3 x 5 ) (Answer: 7 x 1 ) 44 C. Name or draw another array which would have the same number of x's but would look different. 1. 2 x 6 2. x x x x x x x x x x x x x x x 45 LESSON 2: THE DISTRIBUTIVE LAW The basic objective of this lesson i s : To introduce both the l e f t hand and the right hand distributive law. The l e f t hand law states that a x (b + c) = (a x b) + (a x c). For example: 4 x 7-= 4 x (4 + 3) = (4 x 4) + (4 x 3). The right hand law states that (b + c) x a = (b x a) + (c x a). For example: 8 x 6 = (3+5) x 6 = (3 x 6) + (5 x 6). The teaching of both principles w i l l be accomplished by- dividing an array into smaller arrays. Please do not use the terms right hand and lef t hand d i s t r i - butive laws with the children as this only leads to confusion. 1. Review a) Draw a 6 x 7 array on the board and ask the children the name of this array.. Children should give reasons for their answers. Example: there are six rows of seven x's. b) Have a child come to the board and draw a 4 x-2 array. 2. "Let us look at the following array." x x x x x x x K't - X X X X X X X X X X - X X X X X X X X X X X X X X X X X "What is the name of this array?" ( 5 x 7 ) "How could we find out how many crosses there are in that array?" 46 (Children w i l l probably offer suggestions such as counting the individual elements, adding 5 seven's etc.). " A l l of these methods are fine, but here is another interesting way. Let's break up the 5 x 7 array into smaller arrays like this." Step 1 Step 2 Step 3 5 X 7 5 x (4 + 3) (5 X 4) + (5 x 3 X X X X X X X X X X X X X X X X X X X X X X X X X X X X x x x X X X X X X X X x x x X X X X X X X x x x X X X X X X X X x x x X X X X X X X x x x X v.X X X X X X X x x x X X X X X X X X X X X X X X X X X X x x x "Notice that the 5 X 7 array equals a 5 X 4 array 5 x 3 array. Can any of you think of other ways of breaking up this array?" (Let children suggest other p o s s i b i l i t i e s ) . For example: Step 1 Step 2 Step 3 5 x 7 5 x. (2 + 2 + 3) (5 x 2) + (5 x 2) + (5 x 3) X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X Allow children to break up a 7 x 8 array. Try to emphasize Step 1, 2 and 3. For example: 7 x 8 = 7 x (3 + 5) = (7 x 3) + (7 x 5). 47 3. The Right Hand Distributive Law "Here is another array." x x x x x x x x x x x x 5 x 4 x x x x x x x x "We have been breaking these arrays up by renaming the second number." For example: 5 x 4 = 5 x (5. x 2) + (5 x 2) "We can also/break an array into smaller arrays by renaming t^ i e ^ i r s t number." For example: Step 1 Step 2 Step 3 3 x 4 x x x x 3 x x x x x x x x x x x x 2 ^ ̂ ^ 2 x 4 x x x x 5 x 4 Ask children for further ways of breaking up this array by renaming the f i r s t number. For example: 5 x 4 = (1 + 1 + 2 + 1) x 4 = (1 x 4) + (1 x 4) + (2 x 4) + (1 x 4) . "Now we should be able to break up any>array into smaller arrays be renaming the second number or renaming the f i r s t number." Note: Several more examples w i l l probably be needed at this stage. The teacher .should emphasize the techh^iques or renaming both the f i r s t number and the second number. 48 4. Seatwork A. Break up each of these arrays by renaming the' second number. 1) . 6 x 7 = 2) 3 x 8 = B. Break mpseach of the following arrays by renaming the f i r s t number. 1) :6 x 7 = 2) 8 x.4 = C. Provide the numeral which makes the sentence true. 1) 6 X 7 = 6 x (4 + 3) = (6 x ?) + (6 x 3) 2) 4 X 8 = • : (4 x 2) + 0? x 6). 3) 3 X . 8 = ( 3 x 2 ) + (3 x 2) + (3 x ?: ) 4) 7 X 5 = (4 + ?) x 5 = (4 x 5) + (? x 5) 5) 7 X 9 = (7 x 8) + (7 x ? ) APPENDIX C TREATMENT PHASE LESSON PLANS TREATMENT ONE LESSON 1: MULTIPLICATION ,;OF TWO BY ONE PRODUCTS The objective of this less.on i s to teach the rote multiplication algorithm for 2 by 1 products; both with and without carrying. A. Without Carrying "Let us look at the following multiplication problem" 11 x 6 , "Can anyone suggest a way of solving this problem by renaming the top number?" (One possible answer might be): 11 5 + 5 + 1 5 + 5 + 1 x 6 -: x 6 x 6 6 x 1 = 6 + 6 x 5 =30 + 6 x 5 =30 66 The teacher should leave the work for 11 x 6 on the board and write down 11 x 6 somewhere else. "Today we w i l l learn another way that is probably faster than breaking up a multiplication problem. We merely have to work in the following way." 11 x 6 ; "We f i r s t ask ourselves what i s 6 x 1? Then we place the 6 ones in (the ones position." 11 x 6 51 "Then we ask ourselves again what is 6 x 1? This time we have 6 tens and must place the 6 tens in the tens position." 11 x 6 66 To confirm the answer, the teacher should refer to the problem 11 x. 6. done by the distributive principle ( f i r s t example) . At this point the teacher should ask one part of the T l group to try the problem, 11 x n by renaming the top number. The other half should try the new algorithm. When both groups have finished, the answers should be compared. If needed, try the problem of 11 x 9 in the same suggested manner. B. With Carrying 12 Write the problem. , on the board. x 6_ Ask,for suggestions as to how to solve this, problem by renaming the top number. One suggestion might be: 12 6 + 6 6 + 6 x 6_ x 6 x 6 6 x 6 = 36 + 6 x 6 =+36 72 "We can solve this problem using our new way." 12- x 6 "What is 6 x 2? This time we have 12 ones. Let's break this up into 1 ten and 2 ones. Now we can place the 2 in the one's place as before." 52 12 x 6 'We should place the 1 ten in the ten's place." i : 12 x 6 _2 "Now we ask ourselves—what is 6 x 1? This time we get 6 tens. But since we have another group of ten under- neath, we must add i t to the 6. Then we place the 7 tens in the ten's place." 12 x 6 I2 The teacher should then try a problem like 6 x 22 which invoices thegplacement of a 1 in the hundreds place. Use the same steps as before. Using the new algorithm, the pupils should attempt the following: 21 11 14 x 6 x 9 x 5 During this time help can be given to individuals as needed. C. One by Two Products This involves the handling of a problem such as: 6 x 23 Since in the readiness phase the commutative law for multiplication was taught, i t should be easy to convince the child that 53 w i t h t h i s type of problem we merely "turn i t upside down." 6 23 x 23 x 6 Now the c h i l d should be able to solve t h i s type of problem. D. Seatwork I t must be emphasized again that only these l i s t e d problems should be attempted. I t i s a l s o important that the answers be given to the c h i l d r e n before they leave school f o r that day. M u l t i p l y : 1. 65 2. 7 3. 89 4. 15 5. 49 6. 99 x 3 x 13 x 5 x 8 x 4 x 2 7. 6 x 41 54 LESSON 2: MULTIPLICATION OF THREE BY ONE PRODUCTS The objective of this lesson is to teach the rote multiplication algorithm for 3 x 1 products; both with and without carrying. Since the procedures for 3 x 1 products are very simple extensions of those for 2 by 1 products, a detailed lesson would be redundant. However, the teacher is urged to restrict a l l computation to only the examples given. A. Review Examples to use: 3 11 78 x 37 x 8 x 3 Emphasize the steps taken to get the f i n a l answer. B. Without Carrying Ask for suggestions to solve the problem 132 x 3 Most children w i l l probably suggest extending the pro- cedures used to solve 2 x 1 products. A typical explanation of the procedures to use might go as follows: 132 x 3 "Multiply the 3 x 2; we get 6 ones and have to place this 6 in the one's position." 132 x 3 55 "Multiply the 3 x 3; we get 9 tens and place this 9 in the ten's position." 132 x 3 96 "Finally, multiply the 3 x 1; we get 3 hundreds and place the 3 in the hundred's position." 132 x 3 396 Children should attempt: 1Q2 412 210 x 3 x 2 x 4. After sufficient time, ask the children to explain the procedure in addition to the fi n a l answer. C. With Carrying Example to use: 213 x 7 Again children w i l l probably extend procedures for 2 x 1 products. Go through steps as in problem.without carrying, but stress breaking up 7 x 3 = 21 ones = 2 tens + 1 one. Children should attempt: 120 108 223 x 7 x 5 x 6 D. Solution of 6 x 142 Again, as in 2 x 1 products, children should be urged to turn problem "upside down" and then solve. 6 142 x 142 x 6 56 E. Seatwork 222 x 7 2. 107 x 7 3. 24 x 101 27 x 550 5. 636 6. 101 7. 2 x 5 x 4 x 191 57 LESSON 3:. A REVIEW OF THE ROTE TYPE ALGORITHM FOR 3 x 1 AND 2 x 1 PRODUCTS This l e s s o n i s needed to a l l o w the T2 group to f i n i s h t h e i r treatment. Since most of the students i n the T l group w i l l have mastered the rote-type a l g o r i t h m , t h i s l e s s o n i s probably unnecessary f o r t h i s group. However, i t i s e s s e n t i a l f o r the purposes of t h i s study and can be used as merely a p r a c t i c e l e s s o n . The teacher should use only the problems given i n t h i s l e s s o n . Please do not give e x t r a problems to those who f i n i s h e a r l y . The teacher should have ample time to give i n d i v i d u a l help during t h i s p e r i o d . In a d d i t i o n to g i v i n g answers to the problems, the teacher should e x p l a i n the pro- cedures used to get the f i n a l answer i n 3 or 4 problems. M u l t i p l y : 62 6 1 x 49 23 x 6 40 x 2 425 x 5 8 x 280 3 x 949 108 x 2 39 x 3 10. 4 x 49 11. 57 x 7 12. 523 x 4 13. 730' x 5 14. 9 x 424 15. 61 x 5 16. 99 x 4 17. 45 x 9 18. 9 x 30 19. 253 x 4 20. 208 x 8 58 TREATMENT TWO LESSON 1: THE BEGINNINGS OF THE ANNEXATION ALGORITHM The o b j e c t i v e of t h i s l e s s o n i s to teach c h i l d r e n a technique f o r m u l t i p l y i n g any number by 10, 100, or 1000. The teacher should f i r s t q u i c k l y review m u l t i p l i c a t i o n as repeated a d d i t i o n . e.g. 3 x 8 = 8 + 8 + 8 = 24 L i s t the f o l l o w i n g s e r i e s of questions somewhere on the board. 2 x 10 = ? 6 x.10 = ? 3 x 10 = ? 7 x 10 = ? 4 x 10 = ? 8 x 10 = ? 5 x 10 = ? 9 x 10 = ? S t a r t i n g w i t h 2 x 10 ='? ask c h i l d r e n how to solve by adding (10 + 10). Solve each problem by adding. I t should befpfflinted out to the students that i n each problem the one d i g i t number has changed p l a c e s . For example 2_ x 10 = 20 "The 2 was o r i g i n a l l y i n the one's place but a f t e r m u l t i p l i c a t i o n by ten i t s h i f t e d to the ten's place and a zero was placed to the r i g h t . " 59 The children should quickly realize that to multiply by 10 we merely place a zero to the right of the other multiplier. The following series of questions should then be placed on the board. 10 x 11 = ? 10 x 21 = ? 12 x 10 = ? , 18 x 10 = ? 13 x 10 = ? Solve at least 2 or 3 problems by adding. Again have the students note that when multiplying by 10 the digits of the other multiplier a l l shift to the l e f t and a zero is placed to the right. 2. ClQtoeuMh'dVê dVCas a Factor Again the teacher should l i s t a series of questions such as: 2 x 100 = ? 4 x 100 = ? 3 x 100 = ? 5 x 100 = ? Solve each by adding. This time i t should be noted that students should recognize that the digits have shifted two places (from the one's place to the hundred's place) and two zero's are then placed to the right. The following series of questions should then be placed on the board. 11 x 100 = ? 13 x 100 = ? 12 x 100 = ? 26 x 100 = ? After ..solving the f i r s t problem or so by adding, the children should be able to quickly generalize that 26 x 100 = ? can be solved by ''placing two zeros to the right of the 26." 60 (26 x 100 = 2600) I t i s probably a d v i s a b l e to show the p u p i l s how the 2 d i g i t and 6 d i g i t of 26 have s h i f t e d two places to the l e f t . 3- ^oWIriSlas^Fact or By now the students should be able to g e n e r a l i z e to problems such as: 6 x 1000 = ? - - - 6000 12 x 1000 = ? 12000 To convince some p u p i l s of the l e g i t i m a c y of t h i s technique i t may be necessary to solve a problem or two by adding. Again the p u p i l s . s h o u l d r e a l i z e that the d i g i t s have s h i f t e d three places to the l e f t and three zeros have been placed to the r i g h t . Seatwork 1. 8 x 10 = ? 3. 15 x 100 = ? 5. 10 x 1000 = ? 7. 1000 x 8 9. 10 x 12 11. 72 x 10 = ? 13. 100 x 13 15. 28 x 10 2. 10 x ' 12 = ? 4. 12 x 100 = ? 6. 100 x 9 = ? 8. 12 x 100 10, 14. 100 x 9 12. 98 x .100 = ? 1000 x 3 When marking the teacher should have students e x p l a i n how they determined t h e i r f i n a l answers. 61 LESSON 2: THE ANNEXATION ALGORITHM The objectives of this lesson are: 1) to complete the annexation algorithm 5.x 80 = 240 2) to begin applying the annexation algorithm and dis- tributive principle to solve 2 x 1 products. 1. Review a) ask children to multiply the following: 3 x 10 = ? 100 x 11 = ? 18 x 100 = ? 1000 x 19 = ? Explain the procedures used to get f i n a l answer. Example: (multiply by 100; we place two zeros to the right of the other >fae.t.6.r.';ier';' etc.). b) review breaking up a productxinto the sum of smaller products by renaming the top or bottom number. Final answer not important. 12 10 + 2 7 7 x 7 x 7 and x 13 x 10 + 3 7 x 2 3 x 7 + 7 xlO + 10 x 7 2. Put the following series of questions on the board. 3 x 20 = • ? 110 x ' 3 = ? 300 x 4 = ? 70 x 5 = ? 20 x 3 = ? 110 x 2 = ? 3 x 200 = ? Show the children how to solve any of the above in the following manner. For example: 3 x 20 = ? 62 a) "How many tens are there in 20? (Ans. 2) "we can rewrite 20 as 2 x 10 3 x 20 = 3 x 2 x 10 b) Now the order in which we multiply in a question does not matter, so; 3 x 20 = 3 x 2 x 10 = 6_ x 10. c) We have already learned how to multiply a problem such as this" (place a zero to the right). 3 x 20 = 3 x 2 x 10 = 6 x 10 = 60 d) To have the children see the emerging pattern for the series of questions, the teacher should underline the following: 3 x 20 = 60 If the teacher does a few more examples in the above manner i t i s hoped that the.child w i l l see.how to multiply 3 x 200 = ? (Simply multiply 3 x 2 and place 2 zeros to the right 3_ x 200 = 600). Note: In solving 70 x 5 = ? the teacher should re- write 70 as 10 x 7 rather than 7 x 10 since: 70 x 5 = 7 x 10 x 5 (have to commute 7 and 10 to solve) 70 x 5 = 10 x 7 x 5 = 10 x 35 = 350 The next series of questions should be assigned to the pupils. This w i l l enable the teacher to quickly determine whether or not the class is ready to continue. If not, more example should be used to increase the competency with the annexation algorithm. 30 x 6 = ? 200 x 8 = ? 9 x 20 = ? 100 x 10 = ? 110 x 3 = ? 63 Multiplication <bf>, 2 by 1 products "Now we are ready to do some d i f f i c u l t multiplication problems like , x 6 a) rename "top" number as 12 10+2 x 6 x 6 b) we know how to multiply this type; 12 10+2 x 6 x 6 6 x 2 + 6 xlO c) Now i t becomes easy since 6 x 2 = 12 and we know that 6. x 10 = 60 12 10+2 x 6_ x 6 72 6 x 2 = 12 + 6 xlO = +60 72 The teacher should demonstrate: 21 2 0 + 1 x 6 x 6 6 x 1 = 6 + 6 x 20 = 120_ 126 Allow children to try 13 28 x :„A x 3 If class appears to be acquiring some mastery and i f time s t i l l permits, continue to the next section. 4. Renaming bottom numbers Problems such as ^ ̂ should be attacked in the following manner. 64 6 6 x 23 20+3 3 x 6 = 18 + 20 x 6 = 120 138 Other examples to use might be: 4 8 x 31 x 51 5. Seatwork Multiply the following: 1. 2 x 120 = ? 2. 3 x 70 = ? 3. 2 x 600 = ? 4. 100 x 100 = ? 5. 11 x 300 = ? 6. 65 x 3 7. 7 8. 89 x 13 _x 5 9. 15 .10. 6 x 8 x 41 The next series of questions should be assigned i f some students appear to have mastered the 2 x 1 products. 11. 6 .12. 100 + 1 0 + J 13. 140 x 102 x 7 x 2 65 LESSON 3: SOLUTION OF 3 BY 1 PRODUCTS Hopefully, this should be the last period of treatment for the T2 group. The objective of this lesson is to teach the techniques for solving several types of 3 by 1 products. 1. Review Pupils should be assigned the following: 8 x 20 = ? 34 9 x 6_ x 23 In addition to fin a l answers,ftthe procedures used to solve each should be re reviewed. For example: 8 x 20 = 160 34 30 +4 x 6 x 6 6 x 4 = 24 + 6 x30 = +180 9 9 x 23 20+3 3 x 9 = 27 +20 x 9 = +180 2. Renaming the Top Number Put the problem 102 on the board. Ask for suggestions for x 3 possible solution. Rename the top number in the followingsway. The rationale for each step should be explained in detail. 102 100 +2 100+2 x 3 x 3 '• x 3 3 x 2 = 6 multiplication fact + 3 x.100 = +300 multiplication by 100 306 Other examples that should be demonstrated by the • teacher are: 66 a) 134 100 + 20 + 4 100 + 30 + '-4 x 3 x 3 x 3 3 x 4 = 12 + 3 x 30 = +190 + 3 x 100 = +300 402 b) 240 240 + 40 x 4 x 4 4. x 40 -= 160 + 4 x 200 = +800 960 3. Renaming the Bottom Number The pupils should realize that sometimes i t i s advantageous to rename the bottom number. These examples should i l l u s t r a t e the techniques to be used. a) 8 8 x 101 x 100 + 1 1 x 8 = 8 + 100 x 8 = 800 808 b) 3 3 3 x 246 x 200 + 40 x 6 x 200 + 40+6 6 x 3 = 18 + 40 x 3 = +120 + 200 x 3 = +600 738 c) 7 7 7 x 120 100 + 20 100 + 20 20 x 7 = 140 +&100 x 7 = +.700 840 The preceding examples should be enough to enable most students to acquire some proficiency for solving 3 x 1 products. The seatwork to be assigned w i l l be good practice for a l l students and is lengthy enough to allow the teacher to give individual help. 67 Seatwork Multiply : 3 2. 730 3. 201 4. 485 5. 9 x 280 x 5 x 6 x 5 x.424 6. 61 7. 208 8. 9 9. 45 10. 8 x 5 x 5 x 14 x 3 x 101 APPENDIX D THE MEASURING INSTRUMENTS Performance Test Name School First Last Part A - Multiply the following Please show a l l work 1. 87 2. 53 3. 60 4. 93 x 5 x 6 x 2 , x 2 6. 3 7. .4 8. 7 9. 4 x;:41 x 15 x 23 x 68 11. 732 12. 623 13. 201 14. 840 x 6 x 2 x 4 x 2 16. 4 17. 4 18. 6 19. 5 x 606 x 433 x 218 x*330 70 Transfer Test Name F i r s t Last School Part B - M u l t i p l y the f o l l o w i n g Please show a l l work 1001 x 6 3 x 1234 6 :x. 1100 3461 x 3 12 x 11 11 x 26 13 x 64 25 x 12 1001 x 11 10. 12 x 2010 11. 1111 x 15 12. 16 x 1100 13. I l l 14. 203 15. 120 x 101 x 122 x 102 16. 113 17. 1001 18. 1200 x 201 x 101 x 110 19. 101 20. 122 21. 101 x 1111 x 12 x 18 22. 16 x 211 23. 11 xyl03 APPENDIX E THE EXPERIMENTAL DATA 73 TABLE V EXPERIMENTAL DATA TREATMENT TEACHER 1 Tl T2 Sub- Perfor- Trans- Sub- Perfor- Trans- ject mance fer ject mance fer 1 15 4 1 8 2 2 16 3 2 8 5 3 18 2 . 3 9 0 4 11 1 4 14 8 5 11 8 5 8 0 6 16 4 6 14 0 7 16 4 7 6 0 8 16 4 7 1 0 9 15 3 9 4 0 10 11 2 10 9 2 11 5 2 11 1 0 12 6 1 12 0 0 13 12 3 13 7 3 14 19 4 14 0 0 74 TABLE V EXPERIMENTAL DATA TREATMENT TEACHER 2 Tl T2 Sub- Perfor- Trans- Sub- Perfor- Trans- ject mance fer ject mance fer 1 20 11 1 5 0 2 10 1 2 14 0 3 18 8 3 14 0 4 10 AH 4 10 3 5 3 2 5 14 9 6 13 3 6 0 0 7 3 1 7 0 0 8 13 3 8 7 0 9 9 2 9 0 1 10 5 1 10 0 0 75 TABLE V EXPERIMENTAL DATA TREATMENT TEACHER 3 T l T2 Sub- P e r f o r - Trans- Sub- P e r f o r - Trans- j e c t mance f e r j e c t mance f e r 1 18 4 1 17 4 2 15 9 2 15 7 3 14 12 3 7 5 4 11 Of 4 8 5 5 18 4 5 2 2 6 15 6 6 0 1 7 12 5 7 0 0 8 13 2 8 7 0 76 TABLE V EXPERIMENTAL DATA TREATMENT TEACHER 4 Tl T2 Sub- Perfor- Trans- Sub- Perfor- Trans- ject mance fer ject mance fer 1 19 5 1 17 8 2 17 7 2 17 8 3 19 10 3 15 9 4 20 4 4 16 3 5 12 5 5 14 6. 6 16 3 6 11 12 7 17 4 7 6 3 8 19 12 8 9 1 9 19 4 9 6 3 10 15 4 10 10 3 11 14 3 11 2 3 12 14 5 ' 12 8 4 77 TABLE V EXPERIMENTAL DATA TREATMENT TEACHER 5 Sub- P e r f o r - Trans- Sub- P e r f o r - Trans- j e c t mance f e r . j e c t mance f e r ;i, l 20 12 1 19 12 2 19 14 2 12 11 3 18 8 3 19 10 4 18 7 4 17 12 5 17 12 5 14 8 6 16 9 6 6 5 7 19 12 7 13 7 8 19 10 8 15 6 9 19 9 9 14 7 10 18 12 10 13 5 11 19 8 11 20 6 12 16 11 12 10 4 13 12 6 13 8 2 14 17 11 14 6 1 15 16 4 15 4 1 16 12 1 16 4 0 17 16 1 17 1 0 78 TABLE V EXPERIMENTAL DATA TREATMENT TEACHER 6 T l T2 Sub- Perfor- Trans- Sub- Perfor- Trans- ject mance fer ; ject mance fer 1 15 6 1 8 4 2 19 14 2 12 7 3 19 9 3 15 9 4 11 4 4 7 2 5 18 17 5 9 3 6 14 4 6 9 1 7 4 3 7 11 5 8 16 7 8 6 0 9 16 3 9 13 4 10 9 4 10 3 0 11 16 3 11 0 0 12 17 3 12 3 0 13 9 2 13 0 0 14 0 0 14 0 0 79 TABLE V EXPERIMENTAL DATA TREATMENT TEACHER 7 Tl T2 Sub- Perfor- Trans- Sub- Perfor- Trans- ject mance fer ject mance fer 1 18 5 1 5 2 2 19 8 2 5 6 3 12 3 3 11 5 4 17 2 4 14 7 5 18 4 5 0 0 6 12 4 6 0 0 7 8 2 7 9 2 8 6 0 8 0 1 80 TABLE V EXPERIMENTAL DATA TREATMENT TEACHER 8 Tl T2 Sub- Perfor- Trans- Sub- Perfor- Trans- ject mance fer ject mance fer 1 18 13 1 15 10 2 15 4 -2 12 7 3 19 4 3 16 8 4 13 4 4 3 0 5 17 4 5 7 7 6 12 3 6 7 2 7 15 1 7 11 0 8 17 3 8 15 1 9 19 4 9 9 3 10 18 2 10 9 7 11 12 3 11 11 . 0 12 12 3 12 7 0 13 14 4 13 4 0 14 11 2 14 7 0
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