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A comparison of the inductive and the deductive methods in teaching two units of sequential mathematics… Holyoke, Frederick Vernon 1954

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A COMPARISON OF THE INDUCTIVE AND THE DEDUCTIVE METHODS IN TEACHING TWO UNITS OF SEQUENTIAL MATHEMATICS IN HETEROGENEOUS CLASSES OF THE SENIOR HIGH SCHOOL by FREDERICK VERNON HOLYOKE  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS In the School of EDUCATION  We accept this thesis as conforming t o the standard required from candidates f o r the degree of MASTER OF ARTS.  Members of the School of Education  THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 195k  ABSTRACT Problem:  Does the i n d u c t i v e method o f f e r advantages over the  deductive f o r heterogeneous c l a s s e s i n S e n i o r High School mathematics? A p r o p o s a l i s made t h a t a l l students i n such c l a s s e s s t a r t t o g e t h e r w i t h p r a c t i c a l a p p l i c a t i o n s and t h a t each p r o ceed as f a r i n t o theory as he i s a b l e .  There i s some q u e s t i o n ,  however, as t o whether the i n d u c t i v e order and s t y l e of p r e s e n t a t i o n would r e s u l t i n l o s s of l e a r n i n g , e s p e c i a l l y i n the t h e o r e t i c a l aspeets, as compared w i t h the deductive method* To h e l p answer t h i s q u e s t i o n a c o n t r o l l e d experiment was conducted  i n which two c l a s s e s , equated by mean and s t a n d -  a r d d e v i a t i o n on the bases o f I.Q. and p r e v i o u s mathematics marks, worked during e i g h t l\.0 minute p e r i o d s on elementary trigonometry and during seven s i m i l a r p e r i o d s on chords circle.  T h i s s u b j e c t matter,  ina  the same f o r both c l a s s e s ,  formed p a r t of t h e i r r e g u l a r course i n Grade X I mathematics. The i n d u c t i v e group began w i t h p r a c t i c a l a p p l i c a t i o n s and proceeded t o theory w h i l e the deductive group f o l l o w e d the r e v e r s e order; both c l a s s e s were h e l d t o the same l e n g t h of time f o r each type of work, however. p r o v i d e d t o p u p i l s f o r each l e s s o n . as t o method f o r the second u n i t .  Mimeographed sheets were The groups were r e v e r s e d  Teacheir-made t e s t s were  employed f o r measuring l e a r n i n g g a i n .  The f i r s t u n i t of the experiment was l a t e r c a r r i e d on w i t h sample c l a s s e s  i n two other s c h o o l s .  R e s u l t s showed no s t a t i s t i c a l l y  significant  differ-  ences i n g e n e r a l l e a r n i n g g a i n between the two methods. R e s u l t s i n the f i r s t u n i t by the o r i g i n a l sample i n d i c a t e d no l o s s i n the t h e o r e t i c a l aspects under the i n d u c t i v e method.  Information concerning t h i s f e a t u r e was  not  a v a i l a b l e from the other groups or from the second u n i t . In g e n e r a l , the evidence f a v o u r e d the n u l l hypothesis.  ii  TABLE OF CONTENTS Page CHAPTER I  BACKGROUND OF THE PROBLEM  T r a n s i t i o n and Trends i n Secondary Mathematics 1 L i m i t a t i o n s of the Double or M u l t i p l e Track System.... $ A Review of Attempts t o Provide f o r I n d i v i d u a l Differences.... 6 O r g a n i z a t i o n of S u b j e c t Matter f o r Heterogeneous C l a s s e s . . . . 11 The I n d u c t i v e vs the Deductive Approach 13 Summary If? Statement of the Problem i n the Present I n v e s t i g a t i o n . 16 CHAPTER I I  PLAN, SETTING, AND LIMITATIONS OF THE STUDY  General P l a n S e t t i n g f o r the Experiment Design of the Experimental Study S u b j e c t Matter Measurement and Comparison L i m i t a t i o n s of the Study CHAPTER I I I  PROCEDURE  P r e p a r a t i o n of M a t e r i a l Classroom Procedure. Measurement Administration CHAPTER IV  17 17 19 20 21 21  23 29 30 30  ANALYSIS OF RESULTS  E q u a l i t y of Groups. General Achievement of Groups Achievement In P r a c t i c a l and i n T h e o r e t i c a l Work C o r r e l a t i o n Between A b i l i t y and Achievement  3i| 36 39 lj.1  iii  Page CHAPTER V  SUMMARY AND CONCLUSIONS  Summary of the Problem and Its Background The Experiment Equality of Groups Measurement. Summary of Results Interpretation of Results Summary of Conclusions C r i t i c a l Note and Suggestions f o r Further Study  lj.3 • ij.6 1+6 Ij-7 lj.8 f>l f?2  BIBLIOGRAPHY APPENDIX A  MIMEOGRAPHED DETAIL OF LESSONS  -. £9  APPENDIX B  TESTS  6l  APPENDIX C  SAMPLE OF CALCULATIONS  62  iv  LIST OP TABLES TABLE  Page  I  O u t l i n e o f Lesson Sequences Elementary Trigonometry  II  Unit  21+  O u t l i n e of Lesson Sequences Chords i n a C i r c l e U n i t  III  P r e l i m i n a r y Equating of Kamloops Groups  IV V VI  25 31  P i n a l Equating of Kamloops Groups  3i|.  Equating of C h i l l i w a c k Groups.  35  Equating of Langley Groups  35  VII  General Achievement of Kamloops Groups i n U n i t I . . ; 36  VIII  General Achievement of Kamloops Groups I n U n i t I I . . 37  IX  Test and Gain D i f f e r e n c e s - U n i t I I  37  X  General Achievement of C h i l l i w a c k Groups i n U n i t I. 38  XI  General Achievement of Langley Groups I n U n i t I . . . . 38  XII  Comparison of Mean Gains I n General L e a r n i n g  XIII  I n T h e o r e t i c a l and P r a c t i c a l Work Shown S e p a r a t e l y Mean Score D i f f e r e n c e s Between P r a c t i c a l  39  Results  XIV  lj.0  and T h e o r e t i c a l . . . . lj.0 XV  C o r r e l a t i o n Between I.Q. and Teat Score Gain  XVI  1+1  C o r r e l a t i o n Between I.Q,. and Test Marks - P r a c t i c a l and T h e o r e t i c a l - U n i t I - Kamloops Groups 1+2 LIST OP ILLUSTRATIONS I  Lesson 1,  Sequence A  26  II  Lesson 1,  Sequence B  27  III  Lesson 5 , Sequence A..  28  V  ACKNOWLEEG-MENTS  The w r i t e r wishes t o acknowledge the guidance, encouragement a t p o i n t s o f d i f f i c u l t y , and i n s i s t e n c e upon h i g h standards both, i n r e s e a r c h and i n e x p r e s s i o n , g i v e n by Dr. J . Ranton Mcintosh of the S c h o o l of E d u c a t i o n , the U n i v e r s i t y of B r i t i s h Columbia, under whose d i r e c t i o n t h e s i s was p r e p a r e d .  The w r i t e r wishes t o acknowledge a l s o the a s s i s t a n c e of Miss M. ¥ . How of C h i l l i w a c k , B. C and Mr. James P. C l a r k of Langley, B. C. who conducted the work of U n i t I i n t h e i r r e s p e c t i v e High S c h o o l s ,  this  CHAPTER I BACKGROUND OP THE PROBLEM T r a n s i t i o n and Trends i n Secondary Mathematics The r e t r e a t of t r a d i t i o n a l mathematics b e f o r e the advance of modern psychology, the emphasis upon a c t u a l needs which forms p a r t of our p h i l o s o p h y of e d u c a t i o n to-day, and the  tremendous i n c r e a s e i n attendance a t High Schools as they  seek t o p r o v i d e f o r a l l y o u t h have combined t o b r i n g about a g r e a t change i n secondary mathematics during the p a s t s e v e r a l decades.  And the movement i s not y e t complete.  Our whole  secondary s c h o o l system appears t o be i n a t r a n s i t i o n a l stage and the mathematics phase of i t remains i n process of r e organization.  Current e d u c a t i o n a l l i t e r a t u r e c o n t a i n s many  expressions of o p i n i o n by l e a d e r s i n the mathematics t e a c h i n g field;  the m a j o r i t y of these p o i n t towards common goals but  remain c o n s i s t e n t l y g e n e r a l .  In development of programs and  r e b u i l d i n g of mathematics c u r r i c u l a trends have become f a i r l y w e l l marked, but d e t a i l e d surveys are few and c o n s t r u c t i o n i s almost  erratic. In a f a i r l y r e c e n t a r t i c l e E . R. Breslich" " has 1  t r a c e d the o u t s t a n d i n g changes of the p a s t f i f t y y e a r s , n o t i n g the  development of g e n e r a l mathematics w i t h emphasis upon ap-  plications to daily l i f e ,  1.  g e n e r a l c o r r e l a t i o n , the i n c r e a s e  B r e s l i c h , E . R., How Movements of Improvement Have A f f e c t e d Present l a y Teaching of Mathematics, Sch. S c i . & Math», Feb. 1951: 131-llp..  2  of emphasis on meaning and understanding, g r e a t e r s t r e s s concrete m a t e r i a l s , use of m u l t i - s e n s o r y  on  a i d s , and the p r o j e c t  and l a b o r a t o r y methods t o provide f o r i n d i v i d u a l d i f f e r e n c e s ,  Fehr^  H. P.  i n d i s c u s s i n g a modern program compares the  older  aim of c o l l e g e p r e p a r a t i o n w i t h t h a t of mathematics f o r a l l members of s o c i e t y , y e t s t r e s s e s In the l a t t e r r e s p e c t  that  p a l l are not a l i k e . trends  P. S. Jones  attempts to summarize such  as teaching f o r meaning and understanding (both  and mathematical), emphasis upon l o g i c , types of course, use  social  development of s e v e r a l  enrichment m a t e r i a l s , l a b o r a t o r y methods,  of Instruments and t e a c h i n g a i d s , source u n i t s , and  i z a t i o n of a p p l i c a t i o n s .  util-  He notes t h a t l i t t l e has been done  to improve the s e q u e n t i a l courses.  W.  D. Reeve-^ o u t l i n e s  such s i g n i f i c a n t trends as s t r e s s on meaning, g e n e r a l mathematics, m u l t i - s e n s o r y  a i d s , omissions and changed emphases i n  p a r t i c u l a r f e a t u r e s of s u b j e c t matter, and r e c o g n i t i o n of i n d i v i d u a l d i f f e r e n c e s ; a l t h o u g h he questions  whether much more  than l i p s e r v i c e has been p a i d t o the l a s t named. r e p o r t s t h a t a F l o r i d a workshop group studying  Wm  A. Gager'  improvement of  1.  Fehr, H. P., A Proposal f o r a Modern Program i n Mathemati c a l E d u c a t i o n i n the Secondary Schools, Sch. S c i . & Math., Dec. 1949: 723-730.  2.  Jones, P. S. and others, Report on Progress i n Mathematics Education, Sch. S c i . & Math., June 191+9: 1+65-1+71+.  3.  Reeve, • W. D., S i g n i f i c a n t Trends i n Secondary Mathematics, Sch. S c i . & Math., Mch. 19^9: 229-236.  1+. Gager, Wm  1951:  A.,  F u n c t i o n a l Mathematics, Math. Tch.,  297-301.  May  -  3  mathematics c u r r i c u l a favoured functional mathematics as a constant i n grades 7 to 10 and e l e c t i v e i n grades 11 and 12, with sequential algebra, geometry, and trigonometry from grades 9 to 12 also e l e c t i v e *  While this was advocated, r e -  ported indications were that few schools had come close to the plan i n actual p r a c t i c e . The Commission on Post War Flans of the National Council of Teachers of Mathematics survey*  1  conducted an extensive  In t h e i r second report i n 19l|lj. they set f o r t h the  r e s p o n s i b i l i t y of the High School as twofold: to provide sound mathematical t r a i n i n g f o r future leaders i n science, mathematics, and r e l a t e d f i e l d s , and to ensure mathematical competence i n ordinary a f f a i r s of l i f e f o r a l l *  To meet t h i s  r e s p o n s i b i l i t y they recommended a u n i f i e d program of general mathematics f o r grades 7 and 8, followed by a "double track" of sequential mathematics f o r those of higher a b i l i t y and of general mathematics f o r the remainder*  H* Schorling,  who  was a member of the commission, l a t e r published an interpret a t i o n of some of t h e i r data, summarizing that two-thirds of the schools which had reported d i d o f f e r the double track i n grade 9, and one-half c a r r i e d i t on through grade 10, but a r e l a t i v e l y small number provided any alternative to the single track of sequential mathematics courses i n grades 11 or 12* 1* Commission on Post War Plans - National Council of Teachers of. Mathematics, Second Report, Math* Teh*, May 19k$: 19^-221. 2. Schorling; R*, What's Going On i n Your School, Math. Tch*, Apr* 19P: llj.7-153.  Quite recently, a report on a special research project, sponsored by the Southern Section of the C a l i f o r n i a Mathematics Council but c a r r i e d on over t h i r t y - f i v e states, Indicated favour f o r a three track or a multiple track program,  This  1  report did not give a s p e c i f i c plan, and there does not appear to have been agreement upon organization d e t a i l s *  I t was r e -  commended that formal mathematics courses be strengthened and that there be "sequential ungraded courses In non-traditional mathematics.  11  Beyond these appeared to be simply the aim to'  expand general mathematics*  I t i s noteworthy that a consid-  erable number of the personnel engaged on t h i s project had also been leading members of the Commission on Post War Plans* A l l these reports show considerable general  accord,  and from them the writer attempts to summarize presently established trends as follows: 1* Emphasis on meaning and understanding: a* more concrete i l l u s t r a t i o n s and applications b* c l o s e r r e l a t i o n to r e a l l i f e situations c* use of multi-sensory aids 2* Re-emphasis on c r i t i c a l thinking: a* Inclusion of non-mathematical subject matter b. teaching f o r transfer 3* General mathematics: a* decorapartmentalization of arithmetic, algebra, and geometry b* s o c i a l value topics c. c o r r e l a t i o n with other subjects fy. Recognition of Individual differences 1* I r v i n , Lee, The Organization of Instruction i n Arithmetic and Basic Mathematics i n Selected Secondary Schools,  Math. Tch., Apr. 1953: 235-21+0.  5>. Curriculum reorganization: a* general mathematics - compulsory f o r grades 7 and 8 b. double track (expanding to multiple track) - f o r grades 9 to 12, one or two years compulsory. Limitations of the Double or Multiple Track System Looking more p a r t i c u l a r l y at the Senior High School or at grades 9 to 12, while the double or multiple track seems a working attempt to meet the needs of our greatly increased and v a r i e d population, i t appears to have l i m i t a t i o n s . Commission on Post War P l a n s  1  The  noted i n t h e i r survey that more  than two-thirds of a l l High Schools had less than 200 students and eight teachers.  To meet t h e i r s i t u a t i o n , I t was recom-  mended that two courses be handled simultaneously by one teacher, that c y c l i n g of courses be c a r r i e d on, and that use be 2  made of correspondence  courses.  Schorling  l a t e r pointed out,  however, that the response to the enquiry had been very weak from small High Schools.  There seems reason to think, then,  that the Commission's recommendations f o r them were not as adequately considered as was  the general question.  In order  to provide a single track many small schools have used the practices recommended by the Commission.  In these s i t u a t i o n s ,  where a teacher previously handled two courses simultaneously, the double or multiple track would require him to direct a variety of interests at once. 1. Commission on Post War Plans, op. c i t . 2. Schorling, R.,  op. c i t .  6  Even i n the large schools which can o f f e r a number of e l e c t i v e courses, guidance services are not perfect and s o c i a l pressures e x i s t , hence the composition of many classes appears l i k e l y to be quite heterogeneous  f o r some time to  come. There remains, then, a considerable problem of making reasonably adequate provision f o r the varied a b i l i t i e s , needs, and interests which occur w i t h i n single c l a s s e s . A Review of Attempts to Provide f o r Individual Differences A number of methods of providing f o r i n d i v i d u a l differences have been considered i n the past.  An i n v e s t i g -  ation of supervised study by Minniek as early as 1913  bas  been reviewed by Heed* together with l a t e r ones by Jones and 1,  Douglass.  Another by Johnson combined supervised study with  a project method and s o c i a l i z e d presentation.  Stokes achieved  unusual success with a low I.Q. class by i n d i v i d u a l i n s t r u c tion.  Reed'3 general summary of a l l these indicates some  value i n homogeneous grouping, supervised study, project method, d i f f e r e n t i a t e d assignments, i n d i v i d u a l Instruction, and special teaching f o r slow p u p i l s . 2 The University of Chicago High School  developed  over some years a general pattern of supervised study with 1, Reed, Homer B., Psychology and Teaching of Secondary School Subjects, New York, Prentice-Hall, 1939. 2. university of Chicago, Mathematics Instruction i n the University High School, Pub. No. 8, Nov. 1910.  7  added Instruction f o r the weak and enrichment f o r students of high a b i l i t y ; they also found sectional grouping worthwhile* Brownman i n a controlled study compared lecture-demonstration 1  with the individual-laboratory method f o r teaching experimenta l geometry, and he found the l a t t e r s i g n i f i c a n t l y superior 2 i n test scores and i n experimental concepts.  Duroll  consid-  ered three stages of mastery and advocated gradation of exercises; this plan i s used i n a number of modem text-books. Lane^ graded o r i g i n a l exercises i n plane geometry on three l e v e l s ; i n an experimental class students were allowed to make t h e i r own selection so that those completing d i f f i c u l t exercises did not have to attempt the easier ones, while i n the control class students simply were assigned a c e r t a i n number of exercises per day.  According to her report, com-  parison of test results gave indications of s u p e r i o r i t y f o r the experimental method except In the case of students of low ability.  She also reported that the majority of those who  had  choice of exercises appeared to select i n t e l l i g e n t l y rather than l a z i l y .  1.  Brownman,. David E», Measurable Outcomes of Two Methods of Teaching Experimental Geometry, J n l , Exp. Ed., Sept. 1938: 31-31+.  2* Durell, Fletcher, Mathematical Adventures, Boston, Bruce Humphries, 1938: 60-75. 3. Lane, Ruth, The Use of Graded Originals i n Plane Geometry, Math. Tch., Nov. 191+0: 291-300.  8  Several more recent studies d i f f e r i n t h e i r t r e a t ment of i n d i v i d u a l differences,  Albers and Seagoe i n a ninth 1  grade algebra class allowed f i f t e e n minutes of each daily period to students whose I.Q, was 12£ or above f o r explorative enrichment on a more or l e s s voluntary basis; a small l i b r a r y of enrichment material was provided and extra voluntary homework allowed.  In the control class the corresponding students  c a r r i e d on only regular work.  In a f i n a l t e s t on algebra  achievement the experimental group showed progress equal to that of the control group, and In a further t e s t on the enrichment material they showed good r e s u l t s . The conclusions were that superior students can a f f o r d time f o r enrichment, that such work i s self-motivating, and also that the procedure i s administratively possible i n small schools unable to use homogeneous grouping.  But the findings of t h i s study are 2  l i m i t e d to students of superior a b i l i t y ,  Lee  described a  plan used by a large High School where several general or f u n c t i o n a l courses, each including some t h e o r e t i c a l work, were c a r r i e d on simultaneously algebra and geometry.  with more formal offerings i n  A l l these were organized on a semester  basis, and a student showing Interest and a b i l i t y could move 1, Albers, M. E. and Seagoe, M, V., Enrichment f o r Superior Students i n Algebra Classes, J n l . Ed, Rsch,, Mch. 191+7: 2, Lee, Wm, Provision f o r Individual Differences i n High School Mathematics Courses, Math, Tch,, Oct, 19U7: 291*297.  9  from the general to the formal or vice versa a t the beginning of any term.  While the plan appears complicated and, i n the  form described, l i m i t e d to large schools, i t s mosaic pattern strikes the writer as uniquely apt f o r any heterogeneous group of developing youth. In a project by Fowler,  1  several features were  combined i n an experimental procedure f o r teaching geometry. A mimeographed syllabus was prepared containing d e f i n i t i o n s , axioms, postulates, constructions, and theorems, grouped around eight main t o p i c s .  Class sets of two texts were pro-  vided and a small l i b r a r y of supplementary .material.  Basic  concepts were developed through discussion with some teacher demonstration, apparently much l i k e the pattern described by 2  Fawcett  i n his c l a s s i c .  Formal proof was approached through  exercises and cooperative work with "the i n s t r u c t o r prodding and questioning."  Formal proof of only 21* out of 133 theorems  was required of the students, the remainder being e i t h e r i n formally demonstrated or discovered; the ideas were learned but the weight of proof was eliminated In favour of p r a c t i c e and application.  Homework "was of a standing v a r i e t y . " Four  groups of students were employed i n the Investigation, each under a d i f f e r e n t teacher; one followed the above plan f o r a whole year, two others c a r r i e d on the usual practice with text 1. Fowler, Wynette, An Experiment i n the Teaching of Geometry, . Math. Tch., Feb. 19ij.7: 8I4.-88. 2. Fawcett, H. P., The Nature of Proof, Thirteenth Yearbook, National Council of Teachers of Mathematics.  10  and homework during the f i r s t term but changed to the experimental procedure f o r the second term, and the f o u r t h followed the routine of text and homework over the whole year. A l l groups were tested by standardized plane geometry tests given p e r i o d i c a l l y throughout the year and the r e s u l t s showed supe r i o r achievement by those i n the experimental s i t u a t i o n . While the m u l t i p l i c i t y of variables would seem to render v a l i d conclusions somewhat questionable i n t h i s study, provision f o r i n d i v i d u a l differences appears to be inherent i n the experimental procedure.  Modification of subject matter  requirements  coupled with the placing of an outline syllabus i n the hands of each student and standing homework could provide f o r considerable d i f f e r e n t i a t i o n . resemblance,  There even seems to be. some  although vague, between t h i s procedure and Lee's  plan. Through these attempts to provide f o r student d i f ferences there appears to be a c e r t a i n progression of development.  Individual i n s t r u c t i o n i s the dominant i f not the only  feature of the early studies but, while i t remains common, i t s l i m i t a t i o n s i n the group s i t u a t i o n have become recognized. Homogeneous grouping within classes, with added Instruction f o r the weak and enrichment f o r the more able students, i s aided by the grading of exercises on a three-level b a s i s . More recent attempts lean towards p r o v i s i o n f o r greater vari a t i o n i n a b i l i t y and achievement, both i n amount and i n type* with the student p a r t i c i p a t i n g to some extent i n s e l e c t i o n of  11  material, and f i n d i n g his own l e v e l .  This plan f o r meeting  i n d i v i d u a l differences within a single class seems a promising attempt to achieve the advantages of the multiple track plan f o r groups. Organization of Subject Matter f o r Heterogeneous Classes The foregoing trends i n the teaching of secondary mathematics are by no means unique to that f i e l d but are part and parcel of general developments i n education.  They are  consistent with a general tendency to adjust a l l subject matter to the needs, a b i l i t i e s , and i n t e r e s t s of the  student.  Such an arrangement f o r a single heterogeneous class i s w e l l i l l u s t r a t e d by the " d i f f e r e n t i a t e d u n i t " advocated by B i l l e t t , i n which c e r t a i n minimum essentials are expected of a l l , but v a r i a t i o n i n both amount and type of f u r t h e r a c t i v i t y and achievement i s regarded as a natural occurrence to be provided f o r by f l e x i b i l i t y i n subject matter.  Eurell's proposal f o r  heterogeneous classes i n mathematics, that a l l pupils s t a r t at the same place but that some proceed further than others, may  o r i g i n a l l y have dealt with only three l e v e l s , but applied  to a continuum i t could provide the basis f o r a type of d i f ferentiated unit. Certain problems suggest themselves, however. What i s to be the common s t a r t i n g place? cises to be graded?  Upon what basis are exer-  Modern psychology has substantiated the  1. B i l l e t t , Roy 0., Fundamentals of Secondary-School Teaching, Cambridge, The Riverside Press, 19lj.O, esp. Chap. XVII.  12  p r i n c i p l e of leading from the concrete to the abstract, noted levels of learning, and pointed out that some pupils require longer periods of concrete work and are more l i m i t e d i n t h e i r a b i l i t y to generalise or deal with abstractions*  Organization  of subject matter In l i n e with this psychological approach now i s c a r r i e d generally from Elementary School arithmetic through Junior High to some of the general mathematics courses of the Senior High School, but l i t t l e change has been made i n the sequential courses which tend to continue at the upper l e v e l s i n the t r a d i t i o n a l style* "How  In this connection Wren  1  asks:  do we know that the t r a d i t i o n a l sequence and the t r a d i t -  ional treatment of subject matter i s the most s i g n i f i c a n t possible In the perspective of problems of modern education?" Perhaps organization and presentation along a progression from the concrete to the abstract, from the p r a c t i c a l a p p l i c a t i o n to the t h e o r e t i c a l background, can provide the common s t a r t i n g place and the basis of gradation even f o r the material of sequential courses i n the Senior High School*  1. Wren, P. Lynwood, What about the Structure of the Mathe matics Curriculum, Math. Tch., Mch. 19f>l: 166-167*  13  The Inductive vs the Deductive Approach Some studies of this inductive type of approach have been made i n addition to those previously mentioned as attempts t o provide f o r i n d i v i d u a l differences,  Luchins  1  t r i e d building the concept of areas through the use of concrete materials* proceeding to diagrams, and leading to deductive geometry.  He reported a clearer grasp and good  retention of the deductive proof as w e l l as proper and wide a p p l i c a t i o n of formulas, but t h i s was only a subjective view, 2 Michael,  with f i f t e e n classes i n ninth grade algebra, com-  pared an inductive method, i n which the class discovered rules through numerous exercises b u i l t around f a m i l i a r s i t u a t i o n s , with a deductive method where the teacher gave rules without reasons followed by extensive p r a c t i c e .  He found the deduc-  t i v e group s i g n i f i c a n t l y better i n generalizations, but otherwise no evidence to support preference f o r either method. I t seems important to note, however, that with the inductive group no attempt was made to state v e r b a l l y the discovered rules —  that these pupils had not had practice i n expressing  generalizations.  1, Luchins, A. S. & E, H,, A Structural Approaoh to the Teaching of the Concept of Areas i n Intuitive Geometry, J n l , Ed, Rseh., Mch, 1947: 528-533. 2, Michael, R. E,, The Relative Effectiveness of Two Methods of Teaching Certain Topics i n Ninth Grade Algebra, Math. Tch., Feb. 191+9: 83-87.  111.  Ebdes,  i n a review of experimental studies, has  noted that so f a r there i s no strong evidenoe i n favour of either method. Theoretically, then, the inductive approach offers a basis of progression which.could be used to advantage with heterogeneous classes whose study includes formal mathematics, and the evidence thus f a r indicates -that no general learning loss would r e s u l t .  But the amount of that evidence i s r e l -  a t i v e l y small; the e x i s t i n g t o t a l of systematically gathered data concerning p u p i l achievement under the inductive method as compared with the deductive i s i n s u f f i c i e n t to j u s t i f y any conclusion.  Moreover, the inductive approach might be sus-  pected of emphasizing  the concrete, and a loss i n comprehen-  sion of theory coupled with a gain i n p r a c t i c a l achievement could appear as no loss i n general learning.  Serious consid-  eration should be given, therefore, to the two aspects viewed separately. With the aim of securing f u r t h e r information as to the r e l a t i v e merits of each method, the w r i t e r proposes to undertake an experimental study, comparing results under progression from the concrete or the p r a c t i c a l application to the underlying theory with those where students proceed i n the t r a d i t i o n a l style from theorems or rules to applications.  1.  Dodes, Irving A l l e n , The Science of Teaching Mathematics, Math. Tch., Mch. 1953: 159.  15  Summary Secondary s c h o o l mathematics Is I n a s t a t e o f transition.  Emphasis on meaning and understanding  w i t h a p p l i c a t i o n s t o everyday l i f e marked t r e n d s .  together  s i t u a t i o n s have become w e l l  "General mathematics," compulsory through the  J u n i o r High S c h o o l , i s o r g a n i z e d around these p r i n c i p l e s . In the S e n i o r High S c h o o l , t o a s s i s t i n p r o v i d i n g for  the g r e a t l y I n c r e a s e d and v a r i e d p o p u l a t i o n , s o c i a l  u t i l i t y mathematics courses have been added t o the t r a d i t i o n a l o f f e r i n g s of f o r m a l geometry and a l g e b r a , and a l l of these made e l e c t i v e .  T h i s "double  (or m u l t i p l e ) t r a c k " can h a r d l y  be c a r r i e d on i n s m a l l High S c h o o l s , however, and i n the l a r g e r i n s t i t u t i o n s many students o f mediocre a b i l i t y to attempt  the f o r m a l c o u r s e s .  continue  S i n c e heterogeneous groups a r e  common, there i s need f o r some means of a p p l y i n g the m u l t i p l e t r a c k p r i n c i p l e w i t h i n these s i n g l e c l a s s e s ; of p r o v i d i n g b o t h t h e o r e t i c a l and p r a c t i c a l mathematics w i t h the amount o f each v a r i e d a c c o r d i n g t o student  ability.  D a r e l l t s p r o p o s a l , t h a t a l l p u p i l s s t a r t a t the same p l a c e and some proceed f u r t h e r than o t h e r s , o f f e r s towards o r g a n i z a t i o n .  guidance  Modern psychology and the t r e n d of  common p r a c t i c e I n the lower grades  suggests p r o g r e s s i o n f r o m  the concrete o r the p r a c t i c a l a p p l i c a t i o n t o the a b s t r a c t o r the t h e o r e t i c a l , b u t the amount of s c i e n t i f i c the advantages o r disadvantages is  decidedly l i m i t e d .  data concerning  of such an i n d u c t i v e approach  16  Statement of the Problem I n the Present I n v e s t i g a t i o n Search f o r a means of p r o v i d i n g f o r heterogeneous groups has l e d t o the q u e s t i o n of t h e r e l a t i v e e f f i c i e n c y of two t e a c h i n g methods, and the problem f o r i n v e s t i g a t i o n i s now s t a t e d as f o l l o w s : General Problem; mathematics  I n the t e a c h i n g o f f o r m a l o r s e q u e n t i a l  i n S e n i o r High S c h o o l , does an i n d u c t i v e method  i n which p r o g r e s s i o n i s f r o m the c o n c r e t e or p r a c t i c a l  appli-  c a t i o n t o the u n d e r l y i n g t h e o r y ( h e r e i n a f t e r r e f e r r e d t o as "the i n d u c t i v e method" o r Method A) o f f e r advantages over a deductive method i n which p r o g r e s s i o n i s from theory t o app l i c a t i o n ( h e r e i n a f t e r r e f e r r e d t o as "the deductive method" o r Method B) when a p p l i e d t o heterogeneous c l a s s e s ? S p e c i f i c Problems: 1. W i l l t h e r e be s t a t i s t i c a l l y s i g n i f i c a n t d i f f e r e n c e s between t h e mean gains i n g e n e r a l l e a r n i n g r e s u l t i n g under Method A as oompared w i t h Method B?„ 2. W i l l t h e r e be s t a t i s t i c a l l y s i g n i f i c a n t d i f f e r e n c e s between the mean gains r e s u l t i n g under Method A and Method B: (a) i n t h e . t h e o r e t i c a l (b) i n the p r a c t i c a l  aspects?  aspects?  3» W i l l there be a h i g h e r c o r r e l a t i o n between a b i l i t y and l e a r n i n g g a i n under Method A t h a n under Method B?  CHAPTER I I PLAN, SETTING, AND LIMITATIONS OF THE STUDY General Plan In order to Investigate the problem, i t was planned to conduct a c o n t r o l l e d experiment using f o r the sample two equated groups of students, one taught under Method A, the other under Method B.  The subject matter, time, and working  conditions would be the same f o r both groups, while the method of presentation would form the v a r i a b l e .  Evidence as to the  advantages or disadvantages of the Inductive method would be sought i n comparison both of the learning gains i n general and of the t h e o r e t i c a l and the p r a c t i c a l aspects considered separately.  Since the problem concerned a s p e c i f i c type of  s i t u a t i o n , i t was planned to draw the sample from a common heterogeneous population and the subject matter from the normal material i n a sequential type course. Setting f o r the Experiment Two classes i n the same mathematics course were being taught by the writer at Kamloops, B r i t i s h Columbia. This course, known as Mathematics 30 i n the B r i t i s h Columbia Programme of Studies, formed the second year»s work on the sequential l i n e of a double track program and was compulsory f o r students seeking entrance to u n i v e r s i t y i n the province. It contained selected topics i n geometry and algebra, graphs, elementary trigonometry, and logarithms.  Although  18  c o n s i d e r a b l e s t r e s s was p l a c e d upon a p p l i c a t i o n s and upon c r i t i c a l t h i n k i n g , i n l i n e w i t h modern t r e n d s , o r g a n i z a t i o n of the s u b j e c t matter tended t o remain t r a d i t i o n a l i n s t y l e w i t h the deductive approach g e n e r a l l y dominant,  1  One o f these c l a s s e s c o n t a i n e d 17 g i r l s and 10 boys of approximately 16 t o 18 years of age whose I.Q.'s ranged from 90 t o 1 2 5 , the other had 16 g i r l s and 10 boys s i m i l a r l y aged from 16 t o 18 years w i t h i . f t , a from 93 t o 1 2 5 . f  were i n a J u n i o r - S e n i o r High School of composite a t o t a l enrolment  of about 900 p u p i l s .  type h a v i n g  In the s e n i o r grades  t h i s s c h o o l p r o v i d e d a v a r i e t y of academic courses some s p e c i a l i z a t i o n i n languages, social studies.  These  including  s c i e n c e , mathematics, and  I t a l s o o f f e r e d a f a i r program I n commercial  s u b j e c t s , home economics, i n d u s t r i a l a r t s , music, and a r t . The m a j o r i t y of the students i n the two mathematics c l a s s e s were attempting u n i v e r s i t y entrance b u t , as there was a c o n s i d e r a b l e d i v e r s i t y of courses w i t h i n t h e entrance program, no common p a t t e r n predominated  among the members of e i t h e r  group. The community s e r v e d by the s c h o o l I s a r a p i d l y growing  c i t y and suburban v i l l a g e of c l o s e t o 1 0 , 0 0 0 people,  and the surrounding country w i t h i n a r a d i u s of approximately 30 m i l e s .  O n e - t h i r d o r more of the p u p i l s are t r a n s p o r t e d by  s c h o o l bus.  1.  The c i t y i s a r a i l r o a d d i v i s i o n a l p o i n t and a  B r i t i s h Columbia Department of E d u c a t i o n , D i v i s i o n of C u r r i c u l u m , Mathematics 1950»  19  commercial d i s t r i b u t i o n c e n t r e f o r a wide area; i t a l s o has r e s i d e n t a l a r g e number of government s e r v i c e employees.  In  the surrounding country s e r v e d d i r e c t l y by the s c h o o l , i n t e n s i v e f r u i t and vegetable growing, c a t t l e r a n c h i n g , and lumbering  a r e i n d u s t r i e s of c o n s i d e r a b l e importance.  one-quarter  Perhaps  o f the a d u l t s i n t h e s c h o o l d i s t r i c t a r e o f  foreign birth.  Thus, the occupations of t h e parents and the  s o c i a l and economic backgrounds o f the p u p i l s v a r i e d c o n s i d e r a b l y and t h i s  d i v e r s i t y was common through b o t h mathematics  classes. In t h i s s i t u a t i o n , t h e two c l a s s e s i n a s e q u e n t i a l course, s i m i l a r l y heterogeneous as t o background and a b i l i t y , appeared  t o s a t i s f y r e a s o n a b l y w e l l t h e requirements  of the  p l a n f o r a sample, and they were s e l e c t e d f o r the experiment. Design of t h e Experimental Study Examination  o f the two c l a s s e s r e v e a l e d t h a t matched  p a i r s were n o t o b t a i n a b l e i n any q u a n t i t y , and i t was d e c i d e d to equate t h e groups by mean and s t a n d a r d d e v i a t i o n on t h e bases o f b o t h  and p r e v i o u s achievement i n mathematics.  To compensate f o r the smallness of the sample, i t was planned t o attempt  a s e r i e s o f s h o r t experiments  e q u i v a l e n t o f s e v e r a l p a i r s o f groups.  as the  Here an o p p o r t u n i t y  arose t o s t r e n g t h e n c o n d i t i o n s f o r e q u a l i t y o f f a c t o r s  other  than the v a r i a b l e , and the experimental design now was s t r u c t u r e d so as t o a l t e r n a t e t h e two methods w i t h each c l a s s .  20  The idea of conducting the experiment i n several schools was  considered*  While t h i s would provide a l a r g e r  sample, the conditions of the experiment would be much more d i f f i c u l t to control*  The plan was  not discarded, however,  but l e f t i n abeyance as a possible addition l a t e r * The short unit experiments allowed f o r more r i g i d control of working conditions*  O r i g i n a l l y four of these were  considered but, because the length of time i n operation would reintroduce problems of control and because preparation of four complete sets of exercises and tests would be necessary, the f i n a l plan employed only two units* Subject Matter From the material regularly prescribed f o r the Mathematics 30 course two sections were chosen: one  on  elementary trigonometry dealing with the theory of simple trigonometric r a t i o s and t h e i r a p p l i c a t i o n to i n d i r e c t measurement, the other on plane geometry dealing with both t h e o r e t i c a l and p r a c t i c a l c a l c u l a t i o n aspects of chords i n a circle*  While t h i s material was  selected a r b i t r a r i l y , an  attempt was made to choose portions of the course that would lend themselves to inductive treatment neither more nor l e s s r e a d i l y than others*  21  Measurement and  Comparison  S i n c e the experiment was  designed as a s e r i e s of  s h o r t u n i t s w i t h a l t e r n a t i o n of method between the two groups, i t was planned t o measure the l e a r n i n g gains by u n i t The mean and s t a n d a r d d e v i a t i o n would be employed,  tests.  and  comparisons of the u n i t means under each method viewed over the  whole s e r i e s .  L i m i t a t i o n s of the Study This study seeks evidence as t o whether the i n d u c t i v e method o f f e r s advantages over the deductive method. The g e n e r a l problem r e s t r i c t s i t s scope t o the t e a c h i n g o f s e q u e n t i a l mathematics High S c h o o l .  i n heterogeneous c l a s s e s of the S e n i o r  The sample groups appear heterogeneous, and  s t a t i s t i c a l procedures w i l l determine the extent t o which t h e i r r e s u l t s may be a p p l i c a b l e t o a l a r g e p o p u l a t i o n , y e t these procedures can not e s t a b l i s h t h a t such p o p u l a t i o n i s the  g r e a t mass o f S e n i o r High S c h o o l students i n s e q u e n t i a l  mathematics  classes*  The s u b j e c t matter f o r the experiment c o n s i s t s of two u n i t s from a s p e c i f i c course and, as b o t h course and u n i t s were chosen s u b j e c t i v e l y , i t may  o r may not be  truly  r e p r e s e n t a t i v e of a l l f o r m a l o r s e q u e n t i a l mathematics. Any c o n c l u s i o n s drawn from t h i s experiment, t h e r e f o r e , must be a p p l i e d c a u t i o u s l y t o the t e a c h i n g of mathematics  g e n e r a l l y or even t o heterogeneous c l a s s e s  s e q u e n t i a l mathematics  In general.  Yet r e s u l t s may  and  be viewed  22  t o g e t h e r w i t h those from other o b j e c t i v e  s t u d i e s as  bits  evidence c o n t r i b u t i n g t o knowledge of a g e n e r a l p i c t u r e .  CHAPTER I I I PROCEDURE S i n c e method was t o c o n s t i t u t e the s i n g l e v a r i a b l e * the p l a n r e q u i r e d t h a t s u b j e c t matter, time, and working c o n d i t i o n s s h o u l d be e q u a l i z e d under r i g i d c o n t r o l . Preparation of Material To ensure t h a t s u b j e c t matter would be a c o n s t a n t , a d e f i n i t e s e l e c t i o n was made a t the beginning content p r e s c r i b e d f o r each u n i t .  from the  T h i s m a t e r i a l was then  d i v i d e d i n t o l e s s o n s e c t i o n s , each c o n t a i n i n g b a s i c a l l y e i t h e r t h e o r e t i c a l o r p r a c t i c a l work.  G e n e r a l l y one s e c t i o n f i t t e d  a s i n g l e c l a s s p e r i o d o f f o r t y minutes, b u t some r e q u i r e d two periods.  The aim of t h i s  d i v i s i o n was t o p r o v i d e  t h a t the  time devoted t o each type o f m a t e r i a l as w e l l as the t o t a l time would be the same f o r each group; i t a l s o paved the way for  the next s t e p . Order of p r e s e n t a t i o n formed a main f e a t u r e of the  d i f f e r e n c e i n method.  A c c o r d i n g l y , the l e s s o n s e c t i o n s were  arranged i n two sequences, one f o r Method A h a d p r a c t i c a l e x e r c i s e s f i r s t and theory l a s t , t h e o t h e r f o r Method B h a d the same m a t e r i a l i n r e v e r s e o r d e r .  F o r example, i n the f i r s t  u n i t Sequence A began w i t h i n d i r e c t measurement o f r e a l o b j e c t s while the f i r s t l e s s o n of Sequence B d e a l t w i t h the theory of t r i g o n o m e t r i c r a t i o s . > The l a t t e r m a t e r i a l i n Sequence A, however, i n Lesson  occurs  O u t l i n e s of b o t h  2k sequences showing the g e n e r a l s u b j e c t matter content are g i v e n i n Tables I and I I . TABLE I O u t l i n e of Lesson Sequences - Elementary  Sequence A Lesson No.  Trigonometry U n i t  Subjeot Matter  Sequence B Lesson No.  1  I n d i r e c t measurement of r e a l o b j e c t s u s i n g tangent  6  2  I n d i r e c t measurement u s i n g s i n & cos  7  3 &k  C a l c u l a t i o n problems as above but from g i v e n data  k & 5  5  Theory - r a t i o s c o n s t a n t f o r same angle  1  6  Theory - r a t i o s v a r y as angle changes  2  7  C o n s t r u c t i o n of angle from g i v e n function  3  25 TABLE II Outline of Lesson Sequences - Chords i n a C i r c l e Unit Sequence A Lesson No, 1 & 2  5  Subject Matter Calculation exercises - chord, distance from centre, and radius  Sequence B Lesson No,  5 & 6  3  Construction exercises - c i r c l e through three points, etc.  k  k  Two  1  theorems on chord perpendicular  and  Third theorem and t h e o r e t i c a l exercises  &-6  2 & 3  The i n d i v i d u a l lessons then were organized f o r presentation.  Those f o r each sequence were prepared separate-  l y since method, the v a r i a b l e , frequently required differences i n explanation of the same subject matter to accord with the order of presentation.  As a means of control over the  teacher  f a c t o r , explanations and work f o r each lesson were set down In s p e c i f i c d e t a i l , and to guard against departure from the pattern during class operation, the material as f i n a l l y arranged was mimeographed f o r d i s t r i b u t i o n to students; these lesson sheets formed a combination text and work-book. Three lessons from the f i r s t u n i t are given on the next pages f o r i l l u s t r a t i o n .  Comparison of Lesson 1 of  Sequence A with Lesson 1 of Sequence B shows the difference i n approach between the two methods.  Comparison of the l a t t e r  Elementary Trigonometry Unit.  Group  Period 1. Introduction; Certain dimensions which are d i f f i c u l t or impossible to measure directly, such as the height of a tree, a building, or a room, can often be calculated i f we can measure one related distance find one related angle. Demonstration Example; (The working of this w i l l be shown by the teacher, one step at a time, with students following and carrying out the operations step by step). Fhat i s the height of this classroom? Using a- sighting protractor an^ level placed on a desk, sight the intersection of wall and ceiling and read the angle of elevation. Mea-sure the distance along the floor from point under the observer's eye tb the vertical wall. Record these t^o measurements; Ahgle of elevation j horizontal distance Make a dia-gram in the space at the right; mark the angle, base distance, anr' unknown to be found on i t . In a? right-angled triangle, the ratio of vertical side to base i s called the tangent of the lower angle; we can find the value of this from a table on page 512 of the text-book.  t" ite: tangent of r  We then write an equation  ° is  =  and solve i t  Table Practice; Find tangent of 7°;  16°;  30°;  53°;  72°;  80°.  Practice Exercises; Students work i n pairs; sight angle and measure distance together but each work out calculations and check result with erech other. 1. Find height of a teee immediately outside school. 2. Find height of s pole » » " 3. Find height of school building. 4. Find height of any -joint on classroom wall (in case weather does not allow 5. Find height of electric light in classr~>om. outside work)  (GROUP B*) SEQUe/t/cF  Elementary Trigonometry Unit.  B  Period 1. Introduction; Recently we proved the theorem: " t f two triangles are equiangular their corresponding sides are proportional. The ratios of sides of equiangular right-angled triangles are of great importance i n mathematics and are widely used. n  Exercises; The following are to be read and worked or completed by each student. As'these are demonstration and study examples , explanations w i l l be given and results checked as the work proceeds". 1  1. BAC and EDF, shown immediately below^ are equiangular right-angled triangles  Complete:  ( i i i ) BC  ( i i ) AC ^ AB ~  (i) BC _  AXT  ~  2. Construct a right-angled triangle lettered like the sample BAC ajpove but having sides: a> » 3 cm; b » 4 cm. (Use the l e f t side space below)  With protractor, measure angle A and write its^ sixer here degrees. Calculate the length of side^"c (Pythagoras theorem) j check by measuring. Write i n figures, f i r s t as common fractions, then as decimals, the ratios: n  (i) a? _ . (ii) b _. (iii) a . c c b 3. In the right hand space-above} construct another right-angled triangle, lettered the same, but having side- "bj 6 cm; maker angle A" the same size as i n No. 2 by using your protractor. Measure^ the other two sides after the triangle i s drawn and write the ratios", f i r s t as common fractions, then as decimals: =  =  m  1  :  (i) a _ a * (ii) b _ _ . (iii) a _ _ . c dr b Since angle A remained constant, would you expect these ratios to be the same for both triangles? 3  4.  If side b i s 10 f t . and angle A the same, calculate side a  5.  If side c i s thirty miles and angle A the same, calculate sides a -  and b  Definitions: Because, triangles can be lettered i n many ways, a- standard means of naming sides has been adopted to avoid confusion. One of the acute angles i s taken as a reference point and the sides are spoken of as : the hypotenuse the side opposite to the angle the side adjacent to the angle The ratios have been given the following names: -  Slde  "SStSuS  side adjacent to angle hypotenuse  i S  C a l l e  *  i 8 ; o m l l e d  *** <*»>  of the angle.  COSINE (Cosln; Cos) of the angle,  side opposite-to angle TANGWT (Tan) of the angle, side adjacent to angle Exercise: Identify the ratios Sin A, Cos A% and Tan A' of No* 2 above . ' Cheek their values with those given for angle A i n tables at back of textbook. i j r o a l l e d  1  Elementary Trigonometry Unit.  (Group  £j] Sequence  A  Period 5. ^ttiat are'-these ratios: Sine, Cosine, and Tangent? A standard method of naming the sides of right angler) triangles has been adopter!. One of the acute angles i s taken as a reference point and the sides are called: the hypotenuse the side opposite to the angle the side adjacent to the angle The SINE (Sin) of the angle i s alrays  side opposite to angle hypotenuse side adjacent to angle The COSINE (Cosin, Cos ) of the angle i s always' hypotenuse -  7  The TANGENT (Tan) of the angle i s always-  side opposite to angle side adjacent to angle  What happens to these ratios when triangles differ i n length of sides but angles remain constant? In the figures immediately below, BA(T and E7?D are equiangular right-angled t r i angles? side BC'is 5 cm., side ACT i s 4 cm., side- ED iff 4-| cm., side AD i s 6 cm.  t  p ct O »  M H H  o  et ca  o  S3 VA % CO ®  g  A  C  A  (In the exercises below, the length of the hypotenuse may be found by calculation of measurement 1. Write: tangent of angle A i s ?4 " ~ s  e  side For triangle BAC, tangent R 2. Write:  =  triangle EAD, tangent A =  F o r  triangle EAD, sine A =  sine of angle A i s side  For triangle BAC, sine A =* 5. Write:  F o r  cosine of angle A i s  For triangle BA'C, cosin R -  s i d e  " p r triangle EAD, cos A = ^ 0  4. Reduce- each of the above-ratios to i t s lowest terms and complete this statement: If an angle remains-' constant then the sd.no> cosine", and tangent each no matter how large the-triangle. 5. Reduce" each of the above ratios tb a decimalj 1  Sin A" Cbs A Tan A 6. Measure angle A with protractor, find i t s sin, cos, and tan from tables and check your values of No. 5. 7. Measure angle B (note that A + B must total 90°) and find from table the values of sin B , cos B and tan B 8. From the triangle BACT above write the values of sin B, cos B, and tan B from the lengths of the sides ; Reduce each to decimal and compare with No. 7. -  7  to  00  29  w i t h Lesson 5 of Sequence A I n d i c a t e s the d i f f e r e n c e i n treatment of the same s u b j e c t matter*  A copy of the mimeo-  graphed d e t a i l f o r a l l l e s s o n s i s a t t a c h e d as Appendix A. Classroom  Procedure As a f u r t h e r c o n t r o l of the time f a c t o r , a l l work  during the experiment was  c o n f i n e d t o the r e g u l a r c l a s s  p e r i o d s , no homework b e i n g assigned* f o r the l e s s o n was  The mimeographed sheet  d i s t r i b u t e d a t the b e g i n n i n g of each p e r i o d  t o g e t h e r w i t h a l l p r e v i o u s pages but none i n advance* Students  1  work was w r i t t e n on these sheets or on f o o l s c a p  and  a l l papers were taken from them a t the end of the p e r i o d t o be r e t u r n e d on the f o l l o w i n g day* S i n c e these f e a t u r e s of the procedure were f o r e i g n t o normal r o u t i n e and the students would r e a l i z e t h a t some unusual type of t e s t was  o c c u r r i n g , i t was f e l t t h a t a more  s t a b l e s i t u a t i o n would p r e v a i l i f they were taken i n t o confidence*  A c c o r d i n g l y , b o t h groups were informed t h a t a  s p e c i a l p i e c e of work was  t o be conducted which r e q u i r e d no  homework, t h a t the course was  b e i n g t e s t e d r a t h e r than them-  s e l v e s , and t h a t they c o u l d c o n t r i b u t e t o the success of the p r o j e c t by working as n o r m a l l y as p o s s i b l e * The l e s s o n s of the f i r s t u n i t o c c u p i e d seven o r d i n a r y c l a s s p e r i o d s and those of the second u n i t s i x * each case one a d d i t i o n a l p e r i o d was f i n a l t e s t was  In  taken f o r review, and the  a d m i n i s t e r e d on the f o l l o w i n g day*  30  Measurement Teacher-made t e s t s were used t o measure r e s u l t s . For  the u n i t on elementary trigonometry, t h e s u b j e c t matter  was c o n s i d e r e d t o be e n t i r e l y new m a t e r i a l hence o n l y a f i n a l t e s t was g i v e n and the marks on t h i s were t r e a t e d  as g a i n .  Items of t h i s t e s t were b a l a n c e d between p r a c t i c a l and t h e o r e t i c a l types of work so as t o p r o v i d e a b a s i s f o r  considering  the q u e s t i o n of whether Method A would r e s u l t i n h i g h e r p r a c t i c a l achievement a t t h e expense of t h e t h e o r e t i c a l . The  u n i t on chords i n a c i r c l e  p r e v i o u s l y covered so b o t h a p r e t e s t employed here.  included material  and a f i n a l t e s t were  In c o n s t r u c t i n g these, p a i r s  of s i m i l a r Items  were made ready and one of each p a i r a l l o t t e d t o the p r e t e s t or t o the f i n a l t e s t by t o s s i n g measured as the d i f f e r e n c e of each student.  a coin.  L e a r n i n g g a i n was  between the f i n a l and p r e t e s t  An attempt was made t o balance these  marks tests  between t h e o r e t i c a l and p r a c t i c a l b u t was abandoned as unsatisfactory. final  However, a t h e o r e t i c a l p r o o f was added t o t h e  test. A copy of each of the t h r e e t e s t s i s a t t a c h e d as  Appendix B. Administration The  f i r s t p a r t of the experiment was conducted a t  Kamloops e a r l y i n 195>2 w i t h Group A taught under Method A and Group B under Method B.  F o r the second u n i t , c a r r i e d on about  s i x weeks l a t e r , the p l a n n e d r e v e r s a l was made; Method A was  31  used f o r Group B and Method B f o r Group A. i n the mornings, one immediately  a f t e r the o t h e r , and a l l  p e r i o d s were f o r t y minutes i n l e n g t h . was  r e c o r d e d throughout  Both c l a s s e s met  Individual  attendance  so t h a t absentees might be e l i m i n a t e d  as s u b j e c t s or, as an a l t e r n a t i v e , the groups e q u a l i z e d i n this respect. Before c l a s s work was  begun an i n t e l l i g e n c e  the O t i s Q u i c k - S c o r i n g Gamma, was  test,  given to a l l students.  Groups were equated by mean and s t a n d a r d d e v i a t i o n u s i n g b o t h I.Q. who  and f i r s t term achievement In mathematics. were r e p e a t i n g the course as w e l l as two  Pour students  chronically  i r r e g u l a r attendants were not c o n s i d e r e d , and two were made t o secure a b e t t e r b a l a n c e .  transfers  T h i s gave p r o s p e c t i v e  groups f o r the sample as shown i n Table I I I .  1  TABLE I I I  P r e l i m i n a r y Equating of Kamloops Groups •  1 s t Term Marks  Number in Group  Mean  S.D.  Mean  S.D.  Group A  23  110.0  8.k  66.6  11}-. 5  Group B  23  109.0  8.5  66.9  12.8  I.'Q.  32  F i n a l s e l e c t i o n of p e r s o n n e l was postponed, however, u n t i l the r e c o r d of attendance h a d become a v a i l a b l e *  I t was  then found t h a t e x c l u s i o n o f a l l absentees, w h i l e i d e a l , would g r e a t l y reduce the s i z e of the sample and s e r i o u s l y the balance between the groups*  disturb  Y e t students having any  a p p r e c i a b l e number o f absences c o u l d h a r d l y be c o n s i d e r e d as participants*  In a r a t h e r a r b i t r a r i l y determined compromise,  a l l those absent f o r any t e s t o r during more than two p e r i o d s of e i t h e r u n i t were e l i m i n a t e d , and two others dropped i n equating*  The r e s u l t of t h i s procedure  i n the next  i s shown i n Table IV  chapter*  In November of the same y e a r , the work of the f i r s t u n i t was r e p e a t e d w i t h c l a s s e s a t C h i l l i w a c k and a t Langley i n the lower F r a s e r V a l l e y of B r i t i s h Columbia*  This area i s  a r i c h d e l t a where d a i r y i n g and the growing and p r o c e s s i n g of s m a l l f r u i t s and vegetables are b a s i c i n d u s t r i e s * amount o f lumbering  a l s o Is c a r r i e d on*  A fair  Both towns a r e  commercial c e n t r e s , and each has a l a r g e composite  High  School  w i t h approximately h a l f the p u p i l s urban r e s i d e n t and the others conveyed by bus from r u r a l  territory*  In each o f these schools the experimental work was conducted by the r e g u l a r t e a c h e r of two groups who used the two sequences of mimeographed l e s s o n sheets and the a c h i e v e ment t e s t under the d i r e c t i o n of the w r i t e r .  As a v a i l a b l e  mathematics marks f o r the p r e v i o u s y e a r were i n l e t t e r grade form, equating was p o s s i b l e o n l y on the b a s i s o f I.Q.'s as  33  r e g u l a r l y o b t a i n e d and used I n each s c h o o l * A t C h i l l i w a c k , Group A o r i g i n a l l y c o n t a i n e d 3k p u p i l s and Group B 38*  each i n one c l a s s *  In s e v e r a l cases  I.Q.'s were u n a v a i l a b l e ; these p u p i l s were e l i m i n a t e d along w i t h absentees, and two were dropped i n equating* r e s u l t i n g groups are shown i n Table V of the next  The chapter*  In t h i s s c h o o l , because 55 minute p e r i o d s were customary, the f i n a l review p e r i o d was  omitted.  A t Langley, two s m a l l c l a s s e s of Ik and 15  pupils  . r e s p e c t i v e l y were c o n s i d e r e d Group A, w h i l e Group B had p u p i l s i n one c l a s s *  E l i m i n a t i o n s as b e f o r e gave  groups as shown i n Table VI of the next chapter*  33  equated Here the  seven kO minute p e r i o d s p l u s one f o r review were employed*  CHAPTER IV ANALYSIS OP RESULTS E q u a l i t y of Croups Each p a i r of groups was equated by the mean and s t a n d a r d d e v i a t i o n of I.Q.'s.  Those a t Kamloops were equated  a l s o on the b a s i s of mathematics marks f o r the p r e v i o u s term, the mean and s t a n d a r d d e v i a t i o n a g a i n being employed. Table IV shows a comparison of the two Kamloops groups as f i n a l l y equated.  TABLE IV P i n a l E q u a t i n g of Kamloops Groups Number i n Group  1st  I.  Boys  Girls  Total  Group A  7  11  18  Group B  6  12  18  Term Marks  S.D.  Mean  109.3  8.8  68.5  13.9  109.2  8.7  69.1  11.1  Mean  S.D.  The d i f f e r e n c e between the means f o r I.Q. i s 0.1, and f o r f i r s t term marks 0.6.  In the l a t t e r case the s t a n d a r d  e r r o r of the d i f f e r e n c e has been computed as lj.»3» u s i n g the formula t b  SEt, M' " ^ 5 k p  statistically.  ^**  t  Z  , as O.llj..  J  /V  ' ^  $ and the c r i t i c a l r a t i o ,  The d i f f e r e n c e i s not  significant  C a l c u l a t i o n s are shown i n Appendix C.  35  A comparison o f the C h i l l i w a c k groups Is shown I n Table V, and o f the Langley groups i n Table V I .  TABLE V Equating of C h i l l i w a c k Number i n Group  Groups  Mean I.Q.  S.D. I.Q.  Group A  25  110.2  10.7  Group B  32  108.k  10.3  The d i f f e r e n c e between these means Is 1.8.  The  s t a n d a r d e r r o r of the d i f f e r e n c e and the c r i t i c a l r a t i o have been computed as b e f o r e a t 2.88 and O.63 r e s p e c t i v e l y . d i f f e r e n c e i s not s i g n i f i c a n t  The  statistically.  TABLE V I Equating o f Langley Groups  Number i n Group  Mean I.Q.  S.D. I.Q.  Group A  22  108.0  9.1+  Group B  28  108.0  10.9  There i s no d i f f e r e n c e between the c a l c u l a t e d means.  Group A, however, was composed o f two s m a l l c l a s s e s  whereas Group B was a s i n g l e c l a s s .  36  General Achievement o f Groups The l e a r n i n g g a i n s o f s t u d e n t s taught under b o t h methods were measured by the teacher-made t e s t s p r e v i o u s l y d e s c r i b e d i n Chapter I I I ,  Comparison i s made b y the mean and  s t a n d a r d d e v i a t i o n of the raw t e s t soores f o r each group* F o r U n i t I these a r e the f i n a l  t e s t marks w h i l e f o r U n i t I I the  d i f f e r e n c e between the f i n a l  and the p r e t e s t marks has been  taken f o r each student* For  each p a i r o f groups the s t a n d a r d e r r o r o f the  d i f f e r e n c e between means and the c r i t i c a l r a t i o " t " have been c a l c u l a t e d u s i n g the formulas p r e v i o u s l y g i v e n on page 3k»  in  o r d e r t o determine the s i g n i f i c a n c e o f any d i f f e r e n c e , (N - 1) degrees of freedom f o r the combined sample have been c o n s i d e r e d i n each case* The g e n e r a l achievement o f the Kamloops groups i s shown i n Tables V I I and V I I I .  TABLE V I I General Achievement of Kamloops Groups i n U n i t I  Number i n Group  Mean Test Score  Group A  18  22.9  Group B  18  22.3  The d i f f e r e n c e between the  "S.B.  k.k  two means i s 0*6*  s t a n d a r d e r r o r o f the d i f f e r e n c e i s 1.5  and the c r i t i c a l  The ratio  37  i s G-.ij.C-.  For N - 1  not s i g n i f i c a n t  B  35 degrees o f freedom the d i f f e r e n c e i s  statistically.  TABLE V I I I General Achievement o f Kamloops Groups i n U n i t I I F i n a l Test  Number in Group  Mean  S.D.  Pretest Mean  Gain  S.D.  Mean  S.D.  Group A  18  27 3  5.6  6.8  M  20.5  5.3  Group B  18  27.9  5.0  8.2  5.9  19.7  5.2  ?  The d i f f e r e n c e I n mean gains and a l s o those between the means o f f i n a l t e s t and p r e t e s t scores a r e shown, t o g e t h e r w i t h the standard  e r r o r and c r i t i c a l r a t i o f o r e a c h , i n  Table IX. TABLE IX T e s t and Gain D i f f e r e n c e s - U n i t I I F i n a l Test  Pretest  Gain  D i f f e r e n c e i n Means  0.6  l.lj.  0.8  S.E. D i f f e r e n c e  1.8  1.8  1.8  C r i t i c a l Ratio  0.33  0.78  O.I44  No d i f f e r e n c e above i s s i g n i f i c a n t  statistically.  38  Tables X and XI show the general achievement of the Chllliwack and Langley groups i n Unit I.  TABLE X General Achievement of Chilliwack Groups i n Unit I Number i n Group  Mean Test Score 26.1*  Group A Group B  S.D.  32  Z.k  26.6.  • 2.0  The difference In means i s 0.2, the standard error of the difference 0.59, and the c r i t i c a l r a t i o 0.31+. N - 1 f o r the combined sample i s £6. significant  The difference Is not  statistically.  TABLE XI General Achievement of Langley Groups In Unit I Number i n Group  Mean Test Score  Group A  22  21*.6  Group B  28  23.6  S.D. 3.1  -  3.9  The difference i n means i s 1.0, the standard e r r o r l.Ol*, and the c r i t i c a l r a t i o 0.96. difference i s not s i g n i f i c a n t  N - 1 i s lj.9. The  statistically.  39  The f o r e g o i n g d a t a on mean g a i n s i n g e n e r a l l e a r n i n g are summarized i n T a b l e X I I .  TABLE X I I Comparison of Mean Gains i n G e n e r a l  Learning  Kamloops Unit I  Kamloops Unit I I  Chilliwack Unit I  Langley unit I  Method A, Mean  22.9  19.7  26.1+  21+.6  Method B, Mean  22.3  20.5  26.6  23.6  / 0.6  - 0.8  - 0.2  / 1.0  D i f f e r e n c e (A - B) S.E. D i f f e r e n c e  1.1+6  1.79  0.59  1.01+  C r i t i c a l Ratio  0.1+0  o.kk  0.31+ '  0.96  Degrees o f Freedom Significance  35 nil  35 nil  56  k9  nil  nil  Achievement i n P r a c t i c a l a n d i n T h e o r e t i c a l Work The attempt t o measure l e a r n i n g g a i n s i n p r a c t i c a l and i n t h e o r e t i c a l work s e p a r a t e l y was c o n f i n e d t o U n i t I as the U n i t I I t e s t s were c o n s i d e r e d u n s a t i s f a c t o r y f o r t h i s purpose.  Because of f a u l t y communication w i t h t h e o t h e r two  s c h o o l s , r e s u l t s became a v a i l a b l e f o r t h e Kamloops groups only.  These r e s u l t s a r e shown i n Table X I I I .  TABLE X I I I R e s u l t s i n T h e o r e t i c a l and P r a c t i c a l Work Shown S e p a r a t e l y  Theoretical  Practical  Mean  S. D.  Mean  S.D.  Method A  6.3  2.1*  8.3  1,8  Method B  6.I4.  2.7  8.1  1.1*  Difference  (A - B)  / 0.2  - 0.1  As the d i f f e r e n c e s between means f o r the two methods appear n e g l i g i b l e i n each case, no c a l c u l a t i o n s o f s t a n d a r d e r r o r and c r i t i c a l r a t i o have been made.  F o r e i t h e r method,  however, the mean score on p r a c t i c a l Items was l a r g e r than t h a t on t h e o r e t i c a l items.  A summary of these d i f f e r e n c e s i s  shown i n Table XIV.  TABLE XIV Mean Score D i f f e r e n c e s  Between P r a c t i c a l and T h e o r e t i c a l  Method A  Method B  Mean Score - P r a c t i c a l  8.3  8.1  Mean Score - T h e o r e t i c a l  6.3  6.1*  2.0  1.7  0.714-  0.71+  2.70  2.30  •ei  .05  Difference S.E.  (Prae, - Theor.) Difference  C r i t i c a l Ratio L e v e l of S i g n i f i c a n c e  kl C o r r e l a t i o n Between A b i l i t y and Achievement Coefficients  of c o r r e l a t i o n between I.(£. and l e a r n -  i n g g a i n under each method as measured by t e s t scores have  £ been computed u s i n g the f o r m u l a  r s  1  . .  .  Those which  i n c l u d e g e n e r a l o r t o t a l l e a r n i n g g a i n are shown i n Table XV, TABLE XV C o r r e l a t i o n Between I.Q. and Test Score Gain Number i n Group  Corr. C o e f f i c i e n t Method A  Method B  Method A  Method B  18  18  .63  .55  32  .19  .17  22  28  •13  .13  18  18  .06  .Ik  Unit I Kamloops Chilliwack Langley Unit I I Kamloops  Use of the c r i t i c a l r a t i o , t Z •{< j' ._— , shows t h a t v  for  /  N - 2 degrees of freedom the c o e f f i c i e n t s of .63 and  .55  are s i g n i f i c a n t a t the .02 l e v e l w h i l e the others have no s t a t i s t i c a l significance.  1.  I t s h o u l d be noted, however, t h a t  This formula was used because the components were a l r e a d y a v a i l a b l e from p r e v i o u s c a l c u l a t i o n s .  1*2  Kamloops I.Q. «s were from one r e c e n t t e s t but t h a t the o r i g i n of the others i s not d e f i n i t e l y known.  Also that Unit I  scores were from one f i n a l t e s t w h i l e U n i t I I s c o r e s were the d i f f e r e n c e s between marks on two Coefficients  tests.  have been computed a l s o of  between I,Q, and achievement I n the t h e o r e t i c a l  correlation  a s p e c t s , and  between I.Q. and achievement i n the p r a c t i c a l aspects of the work of U h i t I as evidenced by t e s t marks f o r the Kamloops groups.  These are shown i n Table XVI,  TABLE XVI C o r r e l a t i o n Between I.Q, and Test Marks P r a c t i c a l and T h e o r e t i c a l - U n i t I - Kamloops Groups Corr,  Number i n Group  Coefficient  Method A  Method B  Method A  Method B  I.Q. - Theor,  18  18  .54  *k$  i.a. - Practical  18  18  .37  .35  Use of the c r i t i c a l r a t i o as b e f o r e shows t h a t f o r N - 2 degrees of freedom the c o e f f i c i e n t of .54 a t the .02 l e v e l and t h a t of .1+5  a t the .10  i s significant  level.  CHAPTER V SUMMARY AND CONCLUSIONS Summary of the Problem and I t s Background T h i s study c o n s i d e r s  an i n d u c t i v e approach t o t h e  s e q u e n t i a l mathematics of the S e n i o r High School as a method whereby b e t t e r p r o v i s i o n may be made f o r the d i f f e r e n c e s i n a b i l i t y and I n t e r e s t s of students I n heterogeneous c l a s s e s . The  secondary s c h o o l of to-day attempts t o serve a  g r e a t l y i n c r e a s e d and v a r i e d p o p u l a t i o n and t o t h i s end o f f e r s as d i v e r s i f i e d a program as f a c i l i t i e s w i l l p e r m i t . mathematics; of the h i g h e r system p r o v i d e s  In the  grades the double o r m u l t i p l e  track  f o r m a l mathematics f o r some p u p i l s and  vocational o r s o c i a l u t i l i t y arithmetic f o r others. s i n c e these a r e o r g a n i z e d  i n separate courses,  But,  the system  appears l i m i t e d t o the l a r g e r High Schools and i t s success I n these dependent on adequate guidance. Schools and t o a c o n s i d e r a b l e  extent  In the many s m a l l High i n the l a r g e r ones  heterogeneous c l a s s e s a r e common, and the means of p r o v i d i n g f o r i n d i v i d u a l d i f f e r e n c e s w i t h i n these groups remains a problem. Modern psychology and the t r e n d of common p r a c t i c e i n lower grades suggests t h a t a l l p u p i l s i n such c l a s s e s might s t a r t together w i t h p r a c t i c a l a p p l i c a t i o n s and each p r o g r e s s as f a r i n t o t h e o r e t i c a l work as he i s a b l e . view concerning  But a l o n g  t h e l o g i c a l sequence of f o r m a l  held  mathematics  seems t o c o n f l i c t w i t h t h i s i d e a , hence i t i s necessary t o  kk c o n s i d e r whether o r n o t such an i n d u c t i v e approach would r e s u l t e i t h e r i n l o s s o f l e a r n i n g g e n e r a l l y o r i n l o s s of achievement i n the t h e o r e t i c a l a s p e c t s of mathematics. S t u d i e s made thus f a r comparing the i n d u c t i v e and deductive methods of t e a c h i n g mathematics i n d i c a t e  little  p r e f e r e n c e f o r e i t h e r , but the amount o f such r e s e a r c h i s limited. In an attempt t o add t o e x i s t i n g knowledge a c o n t r o l l e d experiment was undertaken,  based on the problem  s t a t e d as f o l l o w s : General Problem:  I n the t e a c h i n g of f o r m a l o r s e q u e n t i a l  mathematics i n S e n i o r High S c h o o l , does an i n d u c t i v e method i n which p r o g r e s s i o n i s from the concrete o r p r a c t i c a l  appli-  c a t i o n t o t h e u n d e r l y i n g theory ( h e r e i n a f t e r r e f e r r e d t o as "the i n d u c t i v e method" or Method A) o f f e r advantages over a deductive method i n which p r o g r e s s i o n i s from theory t o app l i c a t i o n ( h e r e i n a f t e r r e f e r r e d t o as "the deductive method" or Method B) when a p p l i e d t o heterogeneous c l a s s e s ? S p e c i f i c Problems: 1, W i l l there be s t a t i s t i c a l l y  significant differences  between the mean gains i n g e n e r a l l e a r n i n g r e s u l t i n g under Method A as compared w i t h Method B? 2. W i l l t h e r e be s t a t i s t i c a l l y  significant differences  between the mean gains r e s u l t i n g under Method A and Method B: (a) i n the t h e o r e t i c a l (b) i n the p r a c t i c a l  aspects?  aspects?  3.  W i l l there be a h i g h e r c o r r e l a t i o n between a b i l i t y  and l e a r n i n g g a i n under Method A than under Method B? The  Experiment The  p r o j e c t was designed i n the form of a s e r i e s of  s h o r t u n i t experiments i n each of which two equated groups would c a r r y on a s e c t i o n of t h e i r r e g u l a r mathematics course, one  of these c l a s s e s being taught by Method A and the other by  Method B. for  The method was t o be a l t e r n a t e d between the groups  succeeding u n i t s . A s e r i e s of two u n i t s , each occupying e i g h t lj.0 min-  ute p e r i o d s , was c a r r i e d out w i t h sample groups a t Kamloops, B. C.  L a t e r the f i r s t u n i t was r e p e a t e d w i t h two c l a s s e s a t  C h i l l i w a c k and w i t h two others a t Langley i n the same The  province.  s u b j e c t matter c o n s i s t e d of two s e c t i o n s from a  course i n the B. C. c u r r i c u l u m  designated Mathematics 3 0 , i n  which a l l students p a r t i c i p a t i n g had e n r o l l e d . s e l e c t e d f o r each u n i t was; f i r s t  The m a t e r i a l  divided into a definite  number of l e s s o n s which then were o r g a n i z e d i n two sequences according  t o method.  S u b j e c t matter, time, and t o some e x t e n t  emphasis upon each type of work were thus c o n t r o l l e d as constants. time.  E l i m i n a t i o n of homework a l s o a i d e d i n c o n t r o l of  A d d i t i o n a l r e s t r a i n t of emphasis upon e i t h e r type of  work and r e g u l a t i o n of the teacher f a c t o r were p r o v i d e d by mimeographing the complete d e t a i l and p r e s e n t i n g c l a s s e s l e s s o n by l e s s o n .  i t t o the  1*6  E q u a l i t y of Groups Kamloops groups were equated by the mean and ard  stand-  d e v i a t i o n on the bases of p r e v i o u s marks I n mathematics  and of I.Q.'s o b t a i n e d f r o m a s t a n d a r d i z e d t e s t g i v e n s h o r t l y b e f o r e the experimental work began*  The  only on the b a s i s of I.Q.. 's as on f i l e schools.  others were equated  i n t h e i r respective  In a l l oases the d i f f e r e n c e between the means  s l i g h t and not s i g n i f i c a n t s t a t i s t i c a l l y IV, v, and V I .  In each group t h e r e was  as shown i n Tables a c o n s i d e r a b l e range  and a s i z a b l e d e v i a t i o n i n d i c a t i n g t h a t i t was as to a b i l i t y .  was  heterogeneous  Economic and s o c i a l background of students  appeared to be s i m i l a r l y heterogeneous. In g e n e r a l i t may  be f a i r l y c l a i m e d t h a t i n each of  the three samples the groups were w e l l equated b u t , because of v a r i a t i o n i n the means of o b t a i n i n g I . Q . ' s t h e r e ?  i s some  q u e s t i o n about combining the t h r e e p a i r s i n t o a s i n g l e sample of two  groups.  Measurement A l l t e s t s employed t o measure l e a r n i n g gains were c o n s t r u c t e d by the author. used, as trigonometry was  F o r U n i t I only a f i n a l t e s t e n t i r e l y new  t o the s t u d e n t s ,  was and  t h i s t e s t c o n t a i n e d s e c t i o n s on theory and on p r a c t i c a l work of equal score v a l u e .  F o r the plane geometry of U n i t I I ,  however, the d i f f e r e n c e between a f i n a l t e s t and a p r e t e s t formed the score counted as g a i n .  An attempt i n these t o  measure theory and p r a c t i c a l work s e p a r a t e l y was  abandoned.  Summary of Results Results of the experiment  stated as direct answers  to the s p e c i f i c questions of the problem are as follows: 1, There were 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 between the mean gains i n general learning  differences resulting  under Method A as compared with Method B. 2, There were 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  differences  between the mean gains r e s u l t i n g under Method A and Method B i n either the t h e o r e t i c a l or the p r a c t i c a l aspects,  (Information here came from one sample only),  3, In one case the c o r r e l a t i o n between  and test  score gain under Method A was s l i g h t l y higher than that under Method B,  In the other three eases a l l  c o e f f i c i e n t s were n e g l i g i b l e . Two additional results were obtained from a single p a i r of groups, information on the same features not being available from the others.  These r e s u l t s , numbered with  reference to those above, are as follows: 2A, Under both methods the scores f o r p r a c t i c a l work were s i g n i f i c a n t l y higher than those f o r t h e o r e t i c a l work, 3A, The c o e f f i c i e n t s of c o r r e l a t i o n between I.Q, and scores i n t h e o r e t i c a l work were higher than those between I.Q, and scores i n p r a c t i c a l work, with some significance, and of the former, that f o r Method A was s l i g h t l y higher than that f o r Method B,  1*8  I n t e r p r e t a t i o n of R e s u l t s On the q u e s t i o n of advantage i n g e n e r a l l e a r n i n g , the r e s u l t s of t h i s experiment do not f a v o u r e i t h e r method. This f i n d i n g i s In a c c o r d w i t h t h a t of M i c h a e l .  1  It differs p  from views f a v o u r i n g the I n d u c t i v e method by Luchins McCreery^ but these were only s u b j e c t i v e o p i n i o n s .  and  by  In t h i s  c o n n e c t i o n i t i s worthy of note t h a t a t l e a s t two t e a c h e r s p a r t i c i p a t e d i n the experiment l i k e w i s e expressed  who  preference  f o r the i n d u c t i v e approach, y e t the mean t e s t scores of t h e i r c l a s s e s d i d not show s i g n i f i c a n t d i f f e r e n c e s . To r e f e r back to Bodes' summary,^" there i s s t i l l no s t r o n g evidence  in  f a v o u r of e i t h e r method. But the f a c t t h a t s i g n i f i c a n t d i f f e r e n c e s d i d not appear i s not to be accepted as c o n c l u s i v e evidence were none.  t h a t there  I t Is p o s s i b l e t h a t d i f f e r e n c e s i n l e a r n i n g g a i n  e x i s t e d which the t e s t s f a i l e d t o d e t e c t , o r t h a t d i f f e r e n c e s due to method were c o u n t e r a c t e d by other f a c t o r s .  1.  M i c h a e l , op. c i t . (See page  13).  2.  L u c h i n s , op. c i t . (See page  13)*  3. McCreery, Gene S., Mathematics f o r A l l the Students i n High S c h o o l , Math. Tch., Nov. 191*8: 302-308. 1*.  Dodes, op. c i t . (See page l l * ) .  1+9  In t h i s experiment the r e s u l t s were from three s m a l l but w e l l equated samples. limited.  The amount of s u b j e c t matter  Time and working c o n d i t i o n s were r e a s o n a b l y  controlled.  weir  But the v a l i d i t y of the t e s t s from which r e s u l t s  were o b t a i n e d i s open to some q u e s t i o n . of one  was  However, i n the  case  sample where I.Q. s were taken from a s t a n d a r d i z e d t e s t , f  r e c e n t l y administered, the r e l a t i v e l y h i g h c o r r e l a t i o n between I.Q,  and t e s t scores g i v e s some support t o the v a l i d i t y of the  latter.  T h i s may  appear to be c o n t r a d i c t e d by the low  f i c i e n t s i n the o t h e r cases, but due t o circumstances noted these can not be g i v e n as much w e i g h t ,  coefalready  1  Taking a l l f a c t o r s i n t o c o n s i d e r a t i o n , the w r i t e r concludes  t h a t , w i t h r e s p e c t to advantage i n g e n e r a l l e a r n i n g  between the two methods, the r e s u l t s of t h i s experiment g i v e a d e f i n i t e i n d i c a t i o n i n f a v o u r of the n u l l  hypothesis.  The q u e s t i o n of d i f f e r e n c e s I n l e a r n i n g g a i n i n the t h e o r e t i c a l and the p r a c t i c a l aspects c o n s i d e r e d s e p a r a t e l y was  o r i g i n a l l y r a i s e d because of a seeming p o s s i b i l i t y t h a t  the i n d u c t i v e method might emphasize the l a t t e r a t the expense of the former.  No such disadvantage  f o r the I n d u c t i v e method  i s i n d i c a t e d from t h i s experiment, although once a g a i n the f a c t t h a t d i f f e r e n c e s were not s i g n i f i c a n t s t a t i s t i c a l l y not c e r t i f y t h a t there were none. I.Q.  1,  does  The h i g h e r c o r r e l a t i o n of  t o t h e o r e t i c a l than to p r a c t i c a l , shown I n Table  XVI,  See Table XV and f o l l o w i n g comment; a l s o pages 3 1 - 3 3 .  50 coupled w i t h the s i g n i f i c a n t and r e l a t i v e l y h i g h c o e f f i c i e n t s between I.Q. and t o t a l score by the same sample, lends some reinforcement  t o the d i s t i n c t i o n made s u b j e c t i v e l y by the  w r i t e r i n composing the t e s t , s i n c e I.Q. Is a c c e p t e d as our best p r e s e n t measure of a b i l i t y i n h a n d l i n g a b s t r a c t i o n s . But the t e s t items c o v e r i n g each aspect of the work were few i n number.  Moreover, r e s u l t s were a v a i l a b l e o n l y f o r one of  t h e . t h r e e samples. The w r i t e r concludes ment t h e r e i s no evidence  t h a t from incomplete measure-  of l e a r n i n g l o s s i n the t h e o r e t i c a l  aspects under the i n d u c t i v e method.  T h i s c o n c l u s i o n Is a t  v a r i a n c e w i t h t h a t of. Michael' ' who f o u n d the deductive group 1  significantly better i n generalizations. The q u e s t i o n a b l e v a l i d i t y of the t e s t and the p e c u l i a r i t y of the s i n g l e source of i n f o r m a t i o n a f f e c t a l s o any I n t e r p r e t a t i o n t o be p l a c e d on the r e s u l t t h a t under both methods the mean scores f o r p r a c t i c a l work were s i g n i f i c a n t l y 2 h i g h e r than those f o r t h e o r e t i c a l work. q u a l i f i c a t i o n , the w r i t e r would conclude  Subject to t h i s t h a t the p r a c t i c a l o r  more concrete aspects form an a r e a of^ b e t t e r achievement f o r heterogeneous c l a s s e s . psychology to proceed,  Prom the viewpoint  of modern  t h a t success causes s a t i s f a c t i o n and encouragement t h i s suggests  i n d u c t i v e method. 1. M i c h a e l , op. c i t . 2. See Table XIV.  a p o s s i b l e advantage f o r the  51  The t h i r d s p e c i f i c q u e s t i o n of the problem d e a l t w i t h c o r r e l a t i o n s between I.Q. method.  and t e s t score g a i n under each  By comparison of these i t was  sought t o d i s c o v e r  whether g r e a t e r e f f i c i e n c y i n matching achievement t o a b i l i t y would occur under the i n d u c t i v e method than under the deductive. Table XV,  The r e s u l t i n g c o r r e l a t i o n c o e f f i c i e n t s , shown i n have a l r e a d y been d i s c u s s e d .  r e s u l t s no c o n c l u s i o n can be  Summary of  Prom such  conflicting  drawn.  Conclusions 1. The r e s u l t s of t h i s experiment show n e i t h e r  advantage nor disadvantage  i n g e n e r a l l e a r n i n g f o r the  I n d u c t i v e method as compared w i t h the d e d u c t i v e .  There i s  a d e f i n i t e i n d i c a t i o n i n f a v o u r of the n u l l h y p o t h e s i s . 2. Prom incomplete measurement there i s no  evidence  of advantage i n the t h e o r e t i c a l aspects under e i t h e r method. 3. Prom incomplete measurement there i s an i n d i c a t i o n t h a t the p r a c t i c a l aspects form an a r e a of b e t t e r achievement f o r heterogeneous c l a s s e s . !+• Because of c o n f l i c t i n g evidence no c o n c l u s i o n can be drawn as to whether e i t h e r method r e s u l t s i n a h i g h e r c o r r e l a t i o n between a b i l i t y and achievement.  52 C r i t i c a l Note and Suggestions  T o r F u r t h e r Study  The experiment d e s c r i b e d h e r e i n was  undertaken t o  secure, i n f o r m a t i o n on the r e l a t i v e m e r i t s of two methods of t e a c h i n g mathematics, e s p e c i a l l y as a p p l i e d to heterogeneous c l a s s e s i n S e n i o r High S c h o o l s ,  The q u e s t i o n seems of  c o n s i d e r a b l e Importance y e t , a l t h o u g h s u b j e c t i v e o p i n i o n s are common, the number of o b j e c t i v e s t u d i e s Is s m a l l and the  total  of t h e i r f i n d i n g s i n c o n c l u s i v e . F o r an a c c e p t a b l y complete study of such a problem under s t a t i s t i c a l procedure,  random sampling  and s u b j e c t matter would be e s s e n t i a l ,  m  of both  students  the w r i t e r ' s  s i t u a t i o n the forraer p a r t i c u l a r l y had to be s e l e c t e d from those r e a d i l y a v a i l a b l e , hence the p o t e n t i a l accomplishment was  a v e r y s l i g h t a d d i t i o n t o e x i s t i n g knowledge.  As  the  r e s o u r c e s of the average i n v e s t i g a t o r s i m i l a r l y l i m i t  the  c o n t r i b u t i o n from any s i n g l e experimental  study of t h i s  problem, there seems need f o r a c o n s i d e r a b l e number of t o be  these  undertaken. In the s e l e c t i o n of students f o r the sample i n t h i s  experiment, the q u e s t i o n arose of whether to r e s t r i c t i t to c l a s s e s taught by the w r i t e r or to employ a l s o groups under other teachers i n schools a t some d i s t a n c e .  P a r t i c i p a t i o n of  c l a s s e s under a number of teachers i s normal t o mathematics i n s t r u c t i o n g e n e r a l l y and p r o v i d e s a l a r g e r sample, but u n l e s s these are a c c e s s i b l e f o r c l o s e s u p e r v i s i o n i t i s d i f f i c u l t even i m p o s s i b l e f o r the i n v e s t i g a t o r to observe  adequate  or  53  c o n t r o l of the experimental procedure.  I t may be t h a t the  d i f f e r e n c e i n c o r r e l a t i o n r e s u l t s between the Kamloops and the other.groups  was due t o such l a c k of c o n t r o l .  Incomplete  measuring of the p r a c t i c a l and the t h e o r e t i c a l gains s e p a r a t e l y must be charged t o n e g l e c t on the p a r t of t h e w r i t e r , b u t d i f f i c u l t y of access t o t h e o t h e r groups and t h e i r teachers p r o v i d e d the s e t t i n g f o r t h a t n e g l e c t . Inadequacy of measurement g e n e r a l l y i s p r o b a b l y the most s e r i o u s adverse c r i t i c i s m which can be made of t h i s experiment.  Employment of normally o b t a i n a b l e . s t a n d a r d i z e d  t e s t s was not f e a s i b l e as these cover a much broader range of s u b j e c t matter than was d e a l t w i t h h e r e .  The v a l i d i t y of  teacher c o n s t r u c t e d t e s t s as a measure of achievement i n any s i n g l e f e a t u r e might have been e s t a b l i s h e d , but t o do t h i s f o r t h e o r e t i c a l a s p e c t s , p r a c t i c a l a s p e c t s , and g e n e r a l l e a r n i n g t o g e t h e r would have been a tremendous i f not i m p o s s i b l e undertaking.  An obvious  c o n c l u s i o n i s t h a t t o o many questions  were attempted i n one p r o j e c t . The b a s i c q u e s t i o n from which t h i s experiment was c o n c e i v e d was whether o r n o t use of the i n d u c t i v e approach would be advantageous I n heterogeneous c l a s s e s .  I t was  suggested t h a t a l l p u p i l s i n such groups might s t a r t  together  w i t h p r a c t i c a l a p p l i c a t i o n s and each proceed i n t o theory according t o a b i l i t y .  The experiment attempted t o t e s t o n l y  p a r t of t h e q u e s t i o n , i . e . whether l e a r n i n g l o s s would r e s u l t from the i n d u c t i v e order and s t y l e of p r e s e n t a t i o n .  54 I n d i v i d u a l d i f f e r e n c e s i n b o t h r a t e and extent of p r o g r e s s i o n i n t o t h e o r y were a v o i d e d .  F u r t h e r study i n v o l v i n g t h i s  vital  f e a t u r e of the o r i g i n a l l y suggested i n d u c t i v e approach s h o u l d be  worthwhile.  BIBLIOGRAPHY A.  A r t i c l e s , Books, and Reports t o Which D i r e c t Reference  Is Made i n T h i s T h e s i s ; 1. A l b e r s , M. E . and Seagoe, M. V., Enrichment f o r S u p e r i o r Students i n A l g e b r a C l a s s e s , J n l . Ed. Rsch., Mch. 191+7: 1+81-1+95. 2. B i l l e t t , Roy 0 . , Fundamentals of Secondary S c h o o l Teaching, Cambridge, The R i v e r s i d e P r e s s , 191+0. 3. B r i t i s h Columbia Department of E d u c a t i o n , D i v i s i o n of C u r r i c u l u m , Mathematics 1950. 1+. B r e s l i c h , E . R., How Movements of Improvement Have A f f e c t e d Present Day Teaching of Mathematics, Sch, S c i . & Math.,  Feb. 1951: 131-11+1.  5.  Brownman, David E., Measurable Outcomes of Two Methods of Teaching Experimental Geometry, J n l . Exp. Ed., Sept. 1938: 31-31U  6. Commission on Post War P l a n s , N a t i o n a l C o u n c i l of Teachers of Mathematics, Second Report, Math. Teh., May 191+5:  195-221.  7. Dodes, I r v i n g A l l e n , The S c i e n c e of Teaching Mathematics, Math. Tch., Mch. 1953: 157-166. (A summary of r e c e n t developments) • 8. D u r e l l , F l e t c h e r , Mathematical Adventures, Boston, Bruce Humphries Inc., 1938. 9. Fawcett, H. P., The Nature of P r o o f , T h i r t e e n t h Yearbook, - N a t i o n a l C o u n c i l of Teachers of Mathematics. 10. F e h r , Howard F., A Proposal f o r a Modern Program i n Mathematical I n s t r u c t i o n i n the Secondary S c h o o l s , S c h . S c i . & Math., Dec. 191+9: 723-730. 11. Fowler, Wynette, An Experiment i n the Teaching of Geometry, Math. Tch., Feb. 191+7: 81+-88. 12. Gager, Wm A., F u n c t i o n a l Mathematics, Math. Tch., May  1951:  297-301.  56  13•  I r v i n , Lee, The o r g a n i z a t i o n of I n s t r u c t i o n i n A r i t h m e t i c and B a s i c Mathematics i n S e l e c t e d Secondary S c h o o l s , Math. Tch., Apr. 1953: 235-21*0.  11*. Jones, P. S» & o t h e r s , Report on Progress i n Mathematics E d u c a t i o n , Sch. S c i . & Math., June 191*9: lj.65-ll.7il-* 15.  Lane, Ruth, The Use o f Graded O r i g i n a l s i n Plane Geometry, Math. Tch., Nov. 191*0: 291-300.  16. Lee, Wm, P r o v i s i o n f o r I n d i v i d u a l D i f f e r e n c e s i n High S c h o o l Mathematics Courses, Math. Tch., O c t . 19il-7: 291*297. 17. L u c h i n s , A. S. & E . H., A. S t r u c t u r a l Approach t o t h e Concept o f Areas i n I n t u i t i v e Geometry, J n l . Ed. Rsch., Mch. 191*7: 528-533. 18. McCreery, Gene S., Mathematics f o r A l l the Students I n High S c h o o l , Math. Tch., Nov. I9l*8: 302-308. 19. M i c h a e l , R. E», The R e l a t i v e E f f e c t i v e n e s s o f Two Methods of Teaching C e r t a i n Topics i n N i n t h Grade A l g e b r a , Math. Tch., Feb. 191*9: 83-87. 20. Reed, Homer B., Psychology and the Teaching o f Secondary S c h o o l S u b j e c t s , New York, P r e n t i c e H a l l Inc., 1939. 21.  Reeve, W. D», S i g n i f i c a n t Trends i n Secondary Mathematics, Sch. S c i , & Math., Mch. 191*9: 229-236.  22. S c h o r l i n g , R., What's Going On i n Your School? Apr. 191*8: 11*7-153. 23.  Math. Tch.,  U n i v e r s i t y of Chicago (Laboratory S c h o o l s ) , Mathematics I n s t r u c t i o n i n the U n i v e r s i t y High S c h o o l , P u b l i c a t i o n No. 8, Nov. 191*0.  21*. Wren, F . Lynwood, What About the S t r u c t u r e of the Mathematics Curriculum? Math. Tch., Mch. 1951: 161-169.  57  B. 1.  A d d i t i o n a l Works C o n s u l t e d f o r General Background; Adkins, Dorothy C., C o n s t r u c t i o n and A n a l y s i s of A c h i e v e ment T e s t s , Washington, U. S. Government P r i n t i n g O f f i c e , 19l*.7. .  2. Betz, Wm, F i v e Decades of Mathematical Reform - E v a l u a t i o n and Challenge, Math. Tch., Dec. 1950: 377-387. 3. Betz, Wm, F u n c t i o n a l Competence i n Mathematics - I t s Meani n g and I t s Attainment, Math. Tch., May 191*8: 195-206. I*. B r e s l i c h , E. R., Problems i n Teaching Secondary School „ Mathematics, U n i v e r s i t y of Chicago Press, 191*0. 5. B r e s l i c h , E i R., C u r r i c u l u m Trends i n High School Mathematics, Math. Tch., Feb. 191*8: 60-69. 6. Brown, K. E., What Is General Mathematics? Nov. 191*6: 329-331.  Math. Tch.,  7. Brown, K. E., The Content of a Course I n General Mathemati c s - Teachers' Opinions, Math. Tch., Jan. 1950: 25-30. 8. .Brown, K. E., Why Teach Geometry? 103-106.  Math. Tch., Mch.  1950: .  9. Brueckner, L . , J . , The N e c e s s i t y of C o n s i d e r i n g the S o c i a l . Phase of I n s t r u c t i o n i n Mathematics, Math. Tch., Dac. 191*7: 370-371*. 10. Cook, Inez M., Developing R e f l e c t i v e T h i n k i n g Through Geometry, Math. Tch., Feb. 19l*3: 79-82. 11.  Carpenter, Dale, Planning a Secondary Mathematics Curriculum, to Meet the Needs of A l l Students, Math. Tch., Jan. 191*9: 1*1-1*8.  12.  Commission on Secondary School C u r r i c u l u m of P r o g r e s s i v e E d u c a t i o n A s s o c i a t i o n , Mathematics i n General E d u c a t i o n , New York, D. Apple ton - Century Co. Inc., 191*0.  13. Fawcett, H. P., A U n i f i e d Program i n Mathematics, Sch. S c i . & Math., May 1950: 31*2-31*8. llj.. Gager, Wm A., Concepts f o r C e r t a i n F u n c t i o n a l Mathematics Courses, Sch. S c i . & Math., Oct. 1950: 533-539. 15.  Gager, Wm A., Mathematics f o r the Other E i g h t y - f i v e Per Cent, Sch. S c i . & Math., Apr. 191*8: 296-301.  58  16.  G a r r e t t , Henry E., S t a t i s t i c s i n Psychology and E d u c a t i o n , Toronto, Longmans Green & Co., 191+7.  17. G a r r i s o n , S. C. & K. C , Fundamentals of Psychology i n Secondary E d u c a t i o n (Chapter X I I I ) , New York, P r e n t i c e H a l l Inc., 1937. 18.  H a s s l e r , J . 0. & Smith, R. R., The Teaching o f Secondary Mathematics, New York, The M a c M i l l a n Co., 1930.  19. Jones, D. M., An Experiment i n A d a p t a t i o n t o I n d i v i d u a l D i f f e r e n c e s , J n l . Ed. Psy., V o l . 38, 191+9: 257-272. 20. Kinney, L . B., C r i t e r i a f o r Aims i n Mathematics, Math.  Tch., 21.  Mch. 191+8: 99-103.  L a u g h l i n , B u t l e r , F r o n t i e r s i n Teaching Mathematics and S c i e n c e , Sch. S c i . & Math., Mch. 1951: 211-215.  22. N a t i o n a l C o u n c i l of Teachers of Mathematics, The L e a r n i n g of Mathematics, T w e n t y - f i r s t Yearbook, 1953. 23. N a t i o n a l C o u n c i l of Teachers of Mathematics, The P l a c e o f Mathematics i n Secondary E d u c a t i o n , F i f t e e n t h Yearbook,  191+0.  2l+. Norberg, C a r l G., Mathematics i n the Secondary School Curriculum, Math. Tch., Nov. 191+6: 320-321+. 25. Reeve, W. D„, General Mathematics f o r Grades 9 t o 12, Sch. S c i . & Math., Feb. 191+9: 99-110. 26. Schaaf, Wm L,, New Emphases i n Mathematical E d u c a t i o n w i t h Reference t o Recent L i t e r a t u r e , Sch. S c i . & Math.,  Nov. 191+9: 639-61+9.  27. S c h o r l i n g , R., L e t ' s Come t o G r i p s w i t h the Guidance Problem, i n Mathematics, Math. Tch., Jan. 191+9: 25-28. 28. Schmid, John J r . , A Mathematics Course f o r Any Student, Math. Tch., May 191+9: 227-229. 29. Walker, Helen M., Elementary S t a t i s t i c a l Methods, New York, Henry H o l t and Company, X91+-6. 30. Wrightstone, J . W., Comparison of V a r i e d C u r r i c u l a r P r a c t i c e s i n Mathematics, Sch. S c i . & Math., Apr. 1935:  377-381,  59  APPENDIX A MIMEOGRAPHED DETAIL OB LESSONS 1  Elementary Trigonometry Unit.  •  {Group  S£(pU/SA/ceA  i  Period 1. Introductions Certain dimensions which are d i f f i c u l t or impossible to measure directly, such as the height of a tree, a building, or a room, can often be calculated i f we can measure one related distance and one related angle. Demonstration Example; (The working of this w i l l be shown by the teacher, one step at a time, with students following and carrying out the operations step by step). Fhat i s the height of this classroom? Using a-' sighting protractor and level placed on a desk, sight the intersection of wall and ceiling and read the angle of elevation. Measure the distance along the floor from point under the observer's eye tb the vertical wall. Record these t^o measurements; Angle of elevation j horizontal distance Make a diagram in the space" at the right; mark the angle, base distance, and unknown to be found on i t . In ai right-angled triangle, the ratio of vertical side to base i s called the tangent of the lower angle; we can find the value of this from a table on page 512 of the text-book,  Writes tangent of  is  We then write an equation and solve i t  Table Practices  Find tangent of 7°;  16°;  50°;  53°;  72°;  80°.  Practice Exercisess Students work in pairs; sight angle and measure distance together, but each work out calculations and check result with erech other. 1. Find height of a teee immediately outside school. 2. Find height of as pole " » " 3. Find height of school building. 4. Find height of any noint on classroom wall (in case weather does not allow 5. Find height of electric light in classnom. outside work)  Elementary Trigonometry Unit.  Group£.  Period 2. Review Exercise; Find height of a tree, viewed through the classroom window. Three students w i l l measure- outside distance (horizontal) while three- others measure" angle of elevation, ^ o l e class' then works" problem from their dajrep. Demonstration Exercise: "orked by teacher anc students together as in that of period 1, 1  (a ) The distance up the slope of a h i l l can be measured directly; the angle of e l evation of i t s top can be determined by sighting from the foot. Find the vertical height of the h i l l . (A cross-section diagram of the h i l l , drawn on blackboard, w i l l be used) 1  r  Measure and record slope distance Measure and record angle of elevation Make right-angled triangle diagram in speech at right, marking data. In this case we use the ratio of vertical side to hypotenuse; i t i s called the sine of the angle. ^ r i t e equation as before (obtaining value of sine from table) Solve equation (b) Calculate the horizontal distance from foot of h i l l to a noint directly under its top. This i s worked in a similar manner, but the ratio of the base of the triangle to the hypotenuse i s called the cosine of the angle.  Practice Exercises:  (forking in pairs as in period 1).  Measure the angle of ascent of a flight of steps; measure the sloping distance along the steps from bottom to top. Calculate the vertical rise. Measure thelength and angle of elevation of a sloning board. Calculate the height of i t s upper end and also the horizontal distance i t covers. Review of differences between tangent, sine", and cosine.  '  Elementary Trigonometry Unit.  Group A.  Periods 3*& 4. Brief review of sine, cosine, andtangent from blackboard diagram. Brief review of system of writing equation from diagram with data, and solution. .Practice Exercises; Each student i s to work at his or her own rate. Individual h e l | w i l l be given by teacher as;requested. Papers w i l l be collected at end of period 3 ,and reissued for period 4. The last half of period 4 w i l l be devoted to class *checking of answers?and corrections^ Work on foolscap. '  m  a  a  u  »  and BC  2, In the above diagram, i f angle A i s exactly 27 degrees and AB is 200 yds, find the length of AC using trigonometric function value from table. H  is  of angle A; that i s ,  ± 200 Then AC = .  AB Practice Exercises^; Each student i s to work help w i l l be given by teacher as requested. period 4 and reissued at beginning of period w i l l be given at the beginning of period 5. the end of period 5. Work on foolscap. 1.  Find "h»  2.  Find "w"  3.  Find "1"  l0  *  3  27° = . x 200  at' his or her own rate. Papers will be collected 5. Answers to the- f i r s t A l l work i s to be handed 5  =  Individual at end of * five exercises in again at  jh  4. Find "a* 5.  Find "1"  6. From a point on level ground 250 feet from the foot of a tree, the angle of elevation of i t s top i s 12 degrees. How t a l l i s the tree? 7. A highway slope's upward on a steady climb at an angle of 7 degrees for a distance of one-half mile. How many feet i s the top higher than the foot of the h i l l ? 8. A fire-truck ladder i s raised u n t i l i t i s at an angle of 52 degrees from horizontal, and i s extended u n t i l i t s length is*85 feet. How high i s the upper end of the ladder above i t s base? How far out horizontally does i t extend? 9. If the fire>-truck ladder i s extended to a length of 100 feet' and raised u n t i l its upper end i s 75 feet above- the base , what i s the angle- of elevation? 1  -  1  10. An observer on a-bridge;160 feet above the water sees a boat downstream at an angle of depression of 15 degrees. How far i s the boat downstream from the bridge? 11. An ordinary ladder i s considered safest when i t i s placed at an angle of 75 degrees from horizontal." How far from the foot of a vertical wall should a 60 foot ladder be placed i f this ruleis followed? 12. Two towns A and B are 350 miles apart in a straight l i n e . west of due north from town A (a) By how many miles i s B further North than A? (b) By how many miles i s B further West than |A? 13. 14. 15.  Problem 1, page 319 of text-book Problem 3, « » » Problem 4, " " «  Town B i s 37 degrees  Elementary Trigonometry Unit.  Group £.  Periods. •> what are--these ratios: Sine, Cosine, and Tangent? A standard method of naming the sides of right angled triangles has been adopted. One of the acute angles i s taken as a reference point and the sides are called: the hypotenuse the side opposite to the angle the side adjacent to the angle 5  The SINE (Sin) of the angle i s alrays.  opposite to angle  s i d e  hypotenuse The COSINE (Cosin, Cos ) of the angle i s always 7  g l d g  gjl^ggg j°  ^  The TANGENT (Tan) of the angle iff always: side opposite to angle Side adjacent to angle What happens to these ratios when triangles differ i n length of sides but angles remain constant? In the figures immediately below, BAC and EAD are equiangular right-angled t r i angles? side BC'is 5 cm., side ACT i s 4' cm., side ED i s 4§ cm., side AD i s 6 cm.  t  s  A  A  C  (In the exercises below, the length of the hypotenuse may be found by calculation of measurement 1. Write: tangent of angle A i s ^ ~ s=ide For triangle BAC, tangent A = triangle EAD, tangent A = s  2. Write:  sine of angle A i s  cosine of angle A i s  For triangle BAC, cosin A =  F  o  r  F  o  r  8 i d e  For triangle BAC, sine A . 5. Write:  e  . s i d e  "  ' .  triangle EAD, sine A *= •  For triangle EAD, cos A =  4. Reduce each of the above-ratios to itis lowest terms and complete this statement: If an angle remains: constant then the sine-, cosihGP, and tangent each _ no matter how large the triangle. 5. Reduce^ each of the above* ratios tb a decimal: Sin A" Cbs A Tan A 6. Measure angle A with protractor, find i t s sin, cos, and tan from tables and check your values of No. 5. 7. Measure angle B (note that A + B must total 90°) and find from table the values of sin B , cos B ' • and tan B 8. From the triangle BAC above write the values of sin P, cos' B, and tan B from the lengths of the sides . Reduce each to decimal and compare with No. 7. 7  Elementary Trigonometry Unit.  Gram)A.  Period^ . »  *  Review; Study from previous page followed by oral d r i l l on; (1) Definitions of Sine, Cbsine, anri Tangent. (2) If angle & remains constant, what happens" to these ratios for different sized right-angled triangles? Class Exercises; The following are to be worked by each student with explanations and checking at frequent intervals as* for those of period 1: A.  In each of the three right-angled triangles below, the hypotenuse AP i s 5 cm. Measure the other two sides of each triangle and write, f i r s t as common fractions, then as decimals, the values of Sin Aj Cos A; Tan A.  Sin A Cos A Tan PI Complete the statement; Cosin A becomes  As angle A increases, Sin A becomes , Tangent £ becomes  , .  Check your statement from the values* given of these functions for various sized angles i n the tables at the back of the text-book. 0  From these same tables and the ratios worked out above-, find, to the nearest degree the size of each angle A in the above figures. B. 1, Construct an angle whose tangent i s 5/8; that i s , draw a right-angled triangle which has a base of 8 units and an altitude of 5 units, Mark the angle.  2. Reduce 5/8 to a decimal and find from table what size the angle should be. Check your construction by measuring the angle with protractor. 3. Calculates (Pythagoras) the length of the hypotenuse of this triangle to the nearest tenth. 4. Write the value of the sine of the angle", using hypotenuse just calculated, reduce i t to a decimal and compare with the value given i n table. 5. Write the value of the cosine of the angle, using hypotenuse calculated, reduce i t to a? decimal and corn-Dare with table. C. Write from memory the definitions of sine, cosine, and tangent.  ,..  Element aery Trigonometry Unit.  Group /[.  Period 7. Practice Exercises: Work on this sheet or use additional paper as necessary. Each student i s to proceed at h i s or her own rate. Help may be gained by restudy of previous page; individual assistance"will be given also by teacher as' requested. A l l work i s to be handed i n a-t the close of the period. 1. Construct an angle whose sine i s 3/7. Measure with protractor and check with table 2. Construct an angle whose cosine i s 4/9. Measure and check. 3. Construct an angle whose tangent i s 2/3. Measure and check. 4. I f the tangent of an angle i s 5/12, calculate : (a) i t s sine (b) i t s cosine 5. I f the cosine'of am angle is? 4/9, cra.lcru.late": (a) i t s sine (b) its--tangent. 1  1  6. Construct an angle whose sine" is- .5 7. Construct an angle- whose- tangent" i s 2§. 8. Exerciser 5, 6, and 12 on page 316 61F ttext-book.  Elementary Trigonometry Unit.  bJ  (GROUP  j^poe/i^ces  Period 1. Introduction; Recently we proved the theorem: "If two triangles are equiangular their corresponding sides are proportional." The ratios of sides of equiangular right-angled triangles are of great importance i n mathematics and are widely used. Exercises; The following are to be read and worked or completed by each student. As" these are demonstration and study examples, explanations w i l l be given and results checked as the work proceeds'. 1. BAC and EDF, shown immediately below, are equiangular right-angled triangles  2. Construct a right-angled triangle lettered like the sample BAC above but having sides: a» » 3 cm; b => 4 cm. (Use the l e f t side space below)  With protractor, measure angle A and write i t s sixer here degrees. Calculate the length of side~"c" (Pythagoras theorem) ; check by measuring. Write i n figures, f i r s t as common fractions, then as decimals, the ratios: 1  (i) a- _ . (ii) b . (iii) a . c c b 3. In the right hand snace-abovej construct another right-angled triangle, lettered the same, but having side" "bJ' 6 cm; make- angle A" the same size- as i n No. 2 by using your protractor. Measurer the other two sides after the triangle i s drawn and write the ratios-, f i r s t as common fractions, then as decimals: m  (1) a c  _  =  _ i ~  s  (ii) b _ c ~  =  .  =  (iii) a _ b  ~  .  Since angle A remained constant, would you expect these ratios to be the same for both triangles? 4.  If side b i s 10 f t . and angle A the same, calculate side a  5.  If side c i s thirty miles and angle A the same, calculate- sides a  and b  Definitions? Because triangles can be lettered i n many ways, a standard means of naming sides has been adopted to avoid confusion. One of the acute angles i s taken as a reference point and the sides are spoken of as: the hypotenuse the side opposite to the angle the side adjacent to the angle The ratios have been given the following names: side °PP^lte^to angle side adjacent to angle hypotenuse  l g  i g ;  c  a  U  e  d  ^  ( g S a ) o  f  ^  a n g l e #  COSINE (Cosin; Cos) of the angle,  side, opposite- to angle ( ) side adjacent to angle Exercise: Identify the ratios Sin A, Cos A, and Tan A' of No. 2 above-. Check their values with those given for angle A i n tables at back of text ±  T A N G H N T  1  T a n  o  f  t  h  e  1  Elementary Trigonometry Unit.  Ground.  Period £ Review; Study from previous page followed by oral d r i l l on: (1) Definitions of Sine, Cosine, and Tangent. (2) I f angle i? remains constant, what happens to thesre ratios for d i f f e r ent sized right-angled triangles? 7  Class Exercises; The - following are to be worked by each student with explanations and checking at frequent intervals as, for those of period 1: 7  A.  In each of the three right-angled triangles below, the hypotenuse AP i s 5 cm. Measure the other two sides of each triangle and write, f i r s t as common fractions, then as decimals, the values of Sin Aj Cos A; Tan £.  P  Sin A Cos A Tan A: Complete the statement: Cosin A becomes  As angle A' increases, Sin A becomes , Tangent A becomes .  , .  Check your statement from the values' given o f these functions for various sized angles in the tabbies at the back of the text-book. Prom these same tables and the ratios worked out above, find, to the nearest degree the size of each angle A in the above figures. B. 1, Construct an angle whose tangent i s 5/8; that i s , draw a right-angled triangle which has a base of 8 units and an altitude of 5 units* Mark the angle.  2. Reduce 5/8 to a' decimal and find from table what size the angle should be. Check your construction by measuring the angle with protractor. 3. Calculate* (Pythagoras) the length of the hypotenuse of this triangle to the nearest tenth. 4. Write the value of the sine of the angle , using hypotenuse just calculated, reduce i t to a decimal and compare with the value given i n table. 5. Write the value of the cosine of the angle, using hypotenuse calculated, reduce i t to as decimal and compare with table. -  C. Write from memory the definitions of sine, cosine, and tangent.  Elementary Trigonometry Unit.  Group3_.  Period3 . t  Practice Exercises; Work on this sheet or use additional paper as necessary. Each student i s to proceed at his or her own rate. Help may be gained by restudy of previous page; individual assistance w i l l be given also by teacher a3S requested. A l l work i s to be handed i n at the close of the period. 1. Construct an angle whose sine i s 3/7. Measure with protractor and check with tffble 2. Construct an angle whose cosine i s 4/9. Measure and check. 3. Construct an angle whose tangent i s 2/3. Measure and check. 4. If the tangent of an angle i s 5/12, calculate* (a) i t s sine (b) i t s cosine 5. I f the cosine of an angle is? 4/9, calculate: (a?) its' siner (b) its'tangent. 6. Construct an angle whose sine i s .3 7. Construct an angler whose- tangent*" i s 2§. 8. Exercises'5, 6, and 12 on page 316 6fF text -book. 1  Elementary Trigonometry Unit.  Group B.  Periods 4 & 5 Demonstration Exercises: These are to be worked by each student with teacher showing steps and method on blackboard as necessary. 1. In the accompanying right-angled triangle BPC, " of angle A  or  3  — = .45  If AB actually i s 100 yds, then: BC 9 100 " 20 and BC »  or, using decimal,  BC' 100 " ' ° and BC  2. In the above diagram, i f angle A! i s exactly 27 degrees and AB i s 200 yds, find the length of AC using trigonometric function value from table. H AB  is  of angle A; that i s ,  Practice Exercise?* Each student i s to work help w i l l be given by teacher as requested. neriod 4 and reissued at beginning of period w i l l be given at the beginning of period 5. the end of period 5. Work on foolscap. 1.  Find »h"  2.  Find "w"  is 200 Then ACT = .  27° = . x 200  at his or her o^m rate. Papers w i l l be c ollected 5. Answers to thff f i r s t A l l work i s to be handed  =  Individual at end of five exercises i n again at  S. Find "1" 4. Find "ar» 5.  Find "1"  6. From a point on level ground 250 feet from the foot of a tree, the angle of elevation of i t s top i s 12 degrees. How t a l l i s the tree? 7. A highway slopes upward on a steady climb at an angle of 7 degrees for a distance of one-half mile. How many feet i s the top higher than the foot of the h i l l ? 8. A fire-truck ladder i s raised u n t i l i t i s at an angle of 52 degrees from horizontal, and i s extended u n t i l i t s length is 85 feet. How high i s the upper end of the ladder above i t s base? How far out horizontally does i t extend? 5  9. If the fire>-truck ladder i s extended to a length of 100ffeet'and raised u n t i l i t s upper end i s 75 feet abover the baser, what is the angle of elevation? -  10. An observer on a'bridge:160 feet above the water sees a boat downstream at an angle of depression of 15 degrees. How far i s the boat downstream from the bridge? 11. An ordinary ladder i s considered safest when i t i s placed, at an angle of 75 degrees from horizontal. How far from the foot of a vertical wall should a 60 foot ladder be placed i f this ruleis followed? 12. Two towns A and B are 350 miles apart in a straight l i n e . west of due north from town A (a-) By how many miles i s B further North than A? (b) By how many miles i s B further West than 'A? 13. 14. 15.  Problem 1, page 319 of text-book Problem 3, " " » Problem 4, " » "  Town B i s 37 degrees  Elementary Trigonometry Unit.  Grous _ ,°  Period 6. Check answers and make corrections to exercises of last two neriods (15 minutes) Demonstration Example; (This w i l l be shown by the teacher with students using his observed data to comnlete the solution). "tiat i s the height of this classroom? Using a sighting protractor and level placed on a desk, sight the intersection of wall and ceiling and read the angle of elevation. Measure the distance along the floor from point under the observer's eye to the vertical wall. Record these two measurements; Angle of elevation ; horizontal distance ^orking on this -oaper, make a diagram, mark dita, and work out the height.  Practice Exercises: Students work in pairs; sight angle and measure distance together, but each work out calculations and check result with each other. 1. Find height of a tree immediately outside school 2. Find height of a pole " " » 5. Find height of school building. 4. Find height of any ooint on classroom wall (in case of unsuitable weather for 5. Find height of electric light in classroom. outside rork)  Period 7; Completion of unfinished work of period 6 (5 to 10 minutes) Demonstration Exercise; Teacher demonstrate? and students work from his data as before. In this case one or two students may make the actual measurements for the class. , The distance up the slope of a h i l l cnn be measured directly; the angle of elevation of i t s top can be determined by sighting from the foot. ImmmmibmBrTgifflBitomramm (A cross-section of the h i l l , drawn on the blackboard, w i l l be used) (a) Find the vertical height of the h i l l .  (b) Find the horizontal distance from foot of h i l l to a "-joint directly unr'er i t s to i. T  Practice Exercises;  parking in nairs as- in period 6).  Measure the angle of ascent of a. flight of stairs; measure the sloping distance along the steps from bottom to top. Calculate the vertical ripe. Measure length and angle of elevation of a sloping board. of i t s upper end and also the horizontal distance i t covers.  Calculate the height  60  Chords i n a Circle Unit.  (Group As.)  5^M^- B  Period 1^ Introduction; Any straight line joining two points on the circumference of a circle is called a- chord. In studying these, the line joining the mid-point of any chord to the centre of the circle i s a key line, ^e can prove certain features important enough to be classed as theorems. Theorem: Ther straight line joining the centre of a c i r c l e to the mid-r>oint of a chord i s at right angles to the chord. Data;  Let ""B be~ any chord with C as i t s nid-point, and 0 be the centre^ of the circles  Aim:  To prove that OCT i s perpendicular to .'J?.  Construction: Proof: (to be completed with help)  Join OA and  In the triangles OCT and  Brief review questioning on main features of theorem and proof: What two things are given about the l i n e OCT? What third feature i s to be Droved? "faat i s the general method of oroof? What f i n a l sten i s necessary 'after proving triangles congruent? Write again on foolscap the comlete"'oroof of t h i s theorem. Theorem: The~line drawn from the centre of a circle perpendicular to a chord bisects the chord. What two things are given i n this theorem? What i s to be proved? On foolscap, write the general enunciation, draw the figure, write down i n proper form the data and aim. (Upon completion these w i l l be checked with blackboard sample) Can this theorem probably be proved by congruent triangles? What construction i s necessary? Students w i l l attempt to coanlete this proof, fflta Sanrole for checking w i l l then be provided on blackboard.  ,  Group As  Chords in a Girale Unit.  Perioda*?^ 3 Theorem; The perpendicular bisector of a chord passes through the centre of the c i r c l e . On foolscap, draw a suitable figure, and write down in good form the data and sim. (To be checked with sample on blackboard before proceeding further. Demonstration of proof w i l l be given. Students then write out proof for themselves. Exercises;  1. Prove that the .distance of any chord from the centre of BJ circle i a equal to the square root of (the square of the radius minus the square of half the chord). -  2. If two equal chords are drawn in a circle proven that they are equidistant from the centre. 3. State and prove?the converse of exercise 2. 4. AB and CD are two chords•> of ee circle. AB is-longer than CD. Prove-that AB i s closer to the centre? than CD.  5. Two circles whose centres are 0 and Q respectively intersect at two points, A and B. The straight line AB i s called the common chord of the two circles. Draw two such circles with their common chord and mark i t s mid-point C. Join OC and QC, and prove that OGQ forms one atraight line. (TMs l i n e i s called the line of centres. 6. In the accompanying diagram, 0 and Q are centres of the two^cixfiles" and OQ the line of centres. AB i s the common chord. . XBY i s a straight line perpendicular to AB. Prove that XY equals twice the line of centres. 1  7.  s  D  In the diagram at the right, AB i s any straight line and CD i s the perpendicular bisector of AB. Quote a proven reference to show that Mm i f the circumference of any circle' passes through points A and B, i t s centre must l i e on the line CD. -  8. In the diagram a^Uifec-jssgfat, 0 is? the centre of the circle of which only an arc is shown. AB i a a chord r„nd OCD the perpendicular bisector. Using Pythagoras rule, work out a formula for the length of CD i n terms of radius and chord. 1  o  Chords i n a Circle Unit.  Group  /\  Period ^ / Construction Exercises: (Geometrical construction using ruler and comoasses only) 1. A circle i s to be drawn through two given points, A, and B. (1) Join AB ( i i ) Construct the perpendicular bisector of AB ( i i i ) Taking any ooint, 0, on this perpendicular bisector SES- centre, with radius OA, draw the c i r c l e . How many such circles aan be drawn?  2. A circle i s to be drawn through three" given points, A, B, and C, which are not in one straight line. (i) Join AB and construct the perpendicular bisector, ( i i ) Join BC and construct the perpendicrular bisector. ( i i i ) With 0, the point where these lines intersect, as centre and radius OA, draw the c i r c l e .  /\ •  ' £•  /  ^  /\ • fi>  -  V  3. Given an arc of a circle, locate the centre and complete the c i r c l e .  4.  A circle i s to be drawn through the three vertices of a given triangle: (a) Acute angled triangle. (b) light angled triangle. (c) Obtuse angled triangle.  Chords i n a Circle Unit. Periods  Group _/\  'b  Review Exercise of Right-angled Triangle Calculations; This i s to be worked by a l l students immediately. Fork w i l l be checked and explanations given before next section i s begun. _ OCB i s a right-angled triangle having the right angle at C. (a) If OC i s 4 cm and CB i s 7 cm, calculate the length of OB.  (b) If OCT is 5 cm and OB i s 8 cm, calculate thelength of CB.  Calculation Exercises Involving Chords:  ( ork on foolscap) w  1.  In the figure below, 0 i s the centre of the circle and OC the peroendicular bisector of the chord AB. Join OB and name the rightangled triangle. 0 ! If OCT i s 4 cm and chord A'B i s 14 cm: (a.) How long i s CB? A\ 7* 7*B (b) Calculate the length of the radius OB.  2.  In the figure below, 0 is the centre of the circle and OC i s the perpendicular bisector of the chord AB. Complete a right-angled triangle and name i t . 0 ^ If OC i s 5 cm and the radius i s 8 cm: (a) Calculate the length of CB. C / (b) How long is the chord AB? y  3. In the figure_below, 0 i s the centre of the circle and OCT) the perpendicular bisector. If' CD is 5 inches and the radius 9 inches: (a) Calculate the length of OC. ^ j (b) Calculate the length of CB and then of AB. 4. A certain c i r c l e has a 16 foot chord placed so that its-greatest distance from the circumference i s 3 feet. Calculates the'radius of the c i r c l e . 5.  An arch type' bridge i s to be built over a canyon. The diagram shows AB, the span C*""" of the bridge which i s 240 feet, and the arch ACB, an arc of a circle whose radius i s 135 feet. Calculate the height of the middle point of the arch above AB.  6. The cross-section of a tunnel, circular except for a f l a t bottom, i s shown i n the * ^ accompanying diagram. If the chord AB i s 10 feet, and the diametor of the circle i s 20 feet, calculate the height CD.  7. A chord of a circle i s 24 inches long. The radius of the circle i s 15 inches. How far i s the chord from the centre of the circle?  Q  8. Two chords of a circle, AB and CD ers shown i n the diagram, are 4 inches apart. Chord AB i s 24 inches and CD is 16 inches long. Cal13 culate the-radius of the c i r c l e . 0  Additional problems w i l l be found on pages 456-7 of text-book: Education Through Mathematics.  f^Group  Chords i n a Circle Unit. Periods / v l  Review Exercise of Right-angled Triangle Calculations: This i s to be worked by a l l students immediately. Fork w i l l be checked and explanations given before next section i s begun. OCB i s a right-angled triangle having the right angle at C. (a) If OC i s 4 cm and CB i s 7 cm, calculate the length of OB.  (b) If OCT i s 5 cm and OB i s 8 cm, calculate thelength of CB.  Calculation Exercises Involving Chords:  (Work on foolscap)  1.  In the figure below, 0 i s the centre of the circle and OC the perpendicular bisector of the chord AB. Join OB and name the rightangled triangle. 0 1 If OCT i s 4 cm and chord AB i s 14 cm: (a) How long i s CB? A\—7*. ~f& (b) Calculate the length of the radius OB.  2.  In the figure below, 0 i s the centre of the circle and OC i s the perpendicular bisector of the chord AB. Complete a right-angled triangle and name i t . 0 \ If OC i s 5 cm and the radius i s 8 cm: (a) Calculate the length of CB. C / ^ (b) How long i s the chord AB?  3i  In the figure_below, 0 i s the centre of the circle and OCD the perpendicular bisector. If" CD i s 5 inches and the radius 9 inches: (a) Calculate the length of OCT. (b) Calculate the length of CB and then of AB.  4. A certain c i r c l e has a 16 foot chord placed so that its-greatest distance from the circumference i s 3 feet. Calculates thar radius of the c i r c l e . 5.  A  c  An arch type bridge is' to be built over a canyon. The diagram shows AB, the span C~"~---\ of the bridge which i s 240 feet, and the arch ACB, an a r c of a circle whose radius i s 135 feet. Calculate the height of the middle point of the arch above AB.  6. The cross-section of a tunnel, circular except for a f l a t bottom, i s shown i n the f^-^ accompanying diagram. If the chord AB i s 10 feet, and the diametor of the circle i s 20 feet, calculate the height CD.  7. A chord of a circle i s 24 inches long. The radius of the circle i s 15 inches. How far i s the chord from the centre of the circle?  Q  8. Two chords of a circle, AB and CD as shown i n the diagram, are 4 inches apart.Chord AB i s 24 inches and CD i s 16 inches long. Cal3 culate the-radius of the circle. 0  Additional problems w i l l be found on pages 456-7 of text-book: Education Through Mathematics.  Chords i n a Circle Unit. Period  Group  &  3  Construction Exercises: (Geometrical construction using ruler and comoasses only) 1. A circle i s to be drawn through two given points, A, and B. (1) Join AB ( i i ) Construct the perpendicular bisector of AB ( i i i ) Taking any Doint, 0, on this perpendicular bisector as centre, with radius OA, draw the c i r c l e . How many such circles aan be drawn? 1  2. A circle is to be drawn through threegiven points, A, B, and CT, which are not in one straight l i n e . (i) Join AB and construct the perpendicular bisector, ( i i ) Join BC and construct the perpendioular bisector, ( i i i ) With 0, the point where these lines intersect, as centre and radius OA, draw the c i r c l e .  A '  ' &  A  6  • c 3. Given an arc of a circle, locate the centre and complete the c i r c l e .  4,  A circle i s to be drawn through the three vertices of a given triangle: (a) Acute angled triangle. (b) Eight angled triangle. (c) Obtuse angled triangle  Chords i n a Circle Unit.  Group B.  Period 4. Introductions In the calculation and. construction exercises we have just completed, a line similar to OC in the accompanying figure apparently! (i) passes through the centre- of the circle, ( i i ) passes through the mid-point of the chord, ( i i i ) is" perpendicular to the chord. have assumed that i t does a l l these three at once, and while high probability assumptions have to be used as the basis of action in many features of l i f e , yet i f proof i s possible then we have a mmga stronger basis. For example, a construction comnany spending many thousands of dollars' on an arch type bridge such as i n exercise 5 of page 1 would appreciate such a proof before investing their money. Theorem; The straight line joining the centre of a c i r d e to theraid-pointof a chord i s at right angles to the chord. 1  Data; Let ^13 be any chord with C as i t s mid-point, and 0 be the centre^ of the circle". -  Aim;  To prove-that OC i s perpendicular to AB.  Construction: Proof: (to be completed with help)  Join OA and  In the triangles OCA and  Brief review questioning on main features of theorem and proof: What two things are given about the line OC? What third feature i s to be proved? What i s the general method of oroof? What f i n a l step i s necessary after proving triangles congruent? Write again on foolscap the completer proof of this theorem. Theorem: The l i n e drawn from the centre of a circle perpendicular to a chord bisects the chord. What two things are given i n this theorem? What i s to be proved? On foolscap, write the general enunciation, draw the figure, write down i n proper form the data and aim. (Upon completion these w i l l be checked with blackboard sample) Can this theorem probably be proved by congruent triangles? What construction i s necessary? Students w i l l attempt to complete this proof, fflh Sample for checking w i l l then be provided on blackboard.  Group 3  Chords in a G i r d e Unit.  PeriodsJf v L , Theorem; The perpendicular bisector of a chord passes through the centre of the c i r c l e . On foolscap, draw a suitable figure, and write down in good form the data and sim. (To be checked with sample on blackboard before proceeding further. Demonstration of proof w i l l be given.  Students then write out proof for themselves.  Exercises:  1. Prove that the distance of any chord from the centre of a circle- i a equal to the square- root of (the square of the radius minus the square of half the chord). 2. If two equal chords are drawn i n a circle prover that they are equidistant from the centre. 3. State and proves the converse- of exercise 2. 4. AB and CD are two chords-of & c i r c l e . .'B i s longer than CD. Prove that AB i s closer to the centre? than CD.  5. Two circles whose centres are 0 and Q respectively intersect at two points, A and B. The straight line AB i s called the common chord of the two circles. Draw two such circles with their common chord and mark i t s mid-point C. Join OC and QC, and prove that OCQ forms one straight line. ( H i s line i s called the line of centres. 6. In the accompanying diagram, 0 and (J are centres of the two^^cirfiles and OQ the line of centres. AB i s the common chord. _ , XBY i s a straight line perpendicular to AB. ^ Prove that XY equals twice the line of centres.  O  7.  D  s  In the diagram at the right, AB i s any straight line and CD i s the perpendicular bisector of AB. Quote a proven reference to show that Mm i f the circumference of any circle passea through points A and B, i t s centre must l i e on the l i n e CD.  8. In the diagram a.£uthoTasagnt, 0 is* the centre of the c i r c l e of which only an arc is shown. AB i a a chord and OCD the perpendicular bisector. Using Pythagoras' rule, work out a formula for the length of CD i n terms of radius and chord.  O  61  APPENDIX B TESTS  Elementary Trigonometry Unit.  TEST.  Student's name  X  Part A. (Values, l_each) The accompanying triangle BAC i s right-angled at C. XB and ACT are parallel horizontal lines. Underline the correct answer to each of the f o l lowing: 1.  The angle of elevation of point B i s :  ABX;  ABC:  AC;  BC;  AB;  i i s : AC;  BC;  AB;  is:  4.  c  The tangent of angle A i s : BC AC  BC AB  AC AB  AC BC  BC AB  AB AC  AC AE  AC BC  6.  The jjpsine of angle B i s :  AB AC  FB BC  AC BC  AC AF  7.  The^s-ine of angle A i s :  AF PC  AB BC  AC EC  AC AF  8.  AC i s : If AB i s 7 and BC i s 3, theni AC  4;  40;  9.  If | | = | and FA is'IS units,, then then AC AC is-. is-.  10;  If H - .3652 and AF i s 200 units, then BC i s :  10.  Part B.  BAC;  XBA.  «V40~j  V5B.  12|;  14;  18.26;  182.6;  18. 70;  73.04  Work ae directed i n spaces provided. (Value of each i s given in margin at right).  Values  1.  Using the given framework of straight lines in which the angle C i s a right angle, construct accurately an angle Tfhose sine i s 3/5.  £.  -c 2.  Wishing to determine the height of a t a l l building, an observer, from a position 500 feet away on level ground, sights the angle of elevation of i t s top as 16 degrees. On the diagram given immediately below: A (a) Which i s the angle of elevation?  (b) T%ich side represents the building?  /  I  if 3.  ~&  C  (<r) What function of the angle would you use to solve' the problem?.......  (Do not work any further on this question)  If sine A i s 2/7, cnlculate cosine A (leaving answer in surd form)  2  4.  Using the given framework of straight lines which has angle C a right angle, construct ac*curately an angle whose tangent i s .4 1  Elementary Trigonometry Unit 5.  TEST Page 2.  Student's name.  An observer in an aeroplane, known to be 5000 feet above level ground, sights a town ahead at an angle of depression of 4*0 degrees. Complete the given diageam and: (®) Mark the angle of depression.  o  (b) To solve this problem you should use the ( t e l l which function) °Z degrees.  i j ;  (No other work i s required on this problem) 6.  If tangent A is* 5/6, cffllculst.-, sin A (Lsr.ving answer in surd form)  z  7. Sighted from the bottom of a h i l l , tho an^le off' .Vlevfticv. of i t s top i s 24 degrees. The measured distance up the- ever* slope la-1800 tZ*t. Wh'.t i s the-vortical height of the h i l l ? (Work t h i a problem usin^ on-3 of t w <*ivon function value::;: Sin 24 ia. .4067; aos- 24 ie t-r. ;»4. i s .4452)  8.  z  Complete* the following: and i t s  #a an angler bscomeRr larger, its- tangent deoreaistras.  PRETEST.  Exp. Unit II - Chords i n a Circle-.  Namer  (Items 1 to 6 are valued at 1 mark each; items' 7 to 10 at 6 marks? each). 1. Complete this statement: bisect at chord is^ also  The straight line drawn from the centre? of a circle* to *  2. Complete the statement: the centre i s  Of two unequal chords in a* aircler, the one farther from •  3. Complete the statement: centre, they  If two chords of a circle are? the same distance from the .  4. In the accompanying figure, the line AB i s called the  .  5. The figure at the right shows an arc of » circle and a chord. If the chord i s 9 inches from the centre of the circle, and the radius i s 15 inches, what i s the greatest height of the arc at CD? 6. In the accompanying right-angled triangle, AB is? 9 units and AC i s 5 units. Calculate BC, leaving answer in surd form. BC = 7. Construct a circle (geometrical construction) to pass through the-vertices of the given triangle ABC.  B 8. On the reverse side of this page, prove: that: The-perpendicular drawn to a chord from the centre of a circle bisects the chord. In the accompanying diagram, 0 i s centre of the circle whose- radius i s 12 feet. If chord AB i s 9 feet from 0, calculate* the length of AB.  10. The tunnel ACB is m. circle"! whose? lower part has been cut off by the-chord AB. If CD, the-greatest'-height of the tunnel, i s 20 f t . , and AS i'ss 16 f t . , osilculateF ther radius of the circle* -1  Exp. Unit II.- Chords in a Circle.  Final TEST.  Name  (Items 1 to 6 are valued at 1 mark each; items 7 to 10 at 6 markd each). 1. Complete the statement; dicular to ai chord also  The straight line drawn from the' centre of a. c i r c l e perpen-  2. Complete the statements centre i s  Of two unequal chords of a c i r c l e , the one nearer the •  3. Complete the statement:  If two chords of a circle are equal, they are the centre.  4. In the accompanying figure, the line CD is called the  .  5. The figure at the right shows an arc of a- circle and a chord. If the greatest height of the arc at CD is 5 inches, and the radius i s 13 inches, how far i s the chord from the centre of the circle? 6. In the accompanying right-angled triangle, AB i s 7 units and BC i s 4 units-. Calculate ACT, leaving answer in surd form.  AC =  a 7. Using ruler and compasses only, construct a c i r c l e that w i l l pass through the three given points A, B, and C.  •0  8• 8. On the reverse side of this page, prove that: The straight line joining the centre of a circle to the mid-point of a chord is perpendicular to the chord. 9.  A  In the accompanying diagram, 0 i s centre of the circle whose radius is 12 feet. If chord AB i s 16 feet long, calculate the distance from 0 to AB.  8  10. ACB represents a curved mirror whose chord AB i s 20 inches long, while CD, the perpendicular bisector, i s 3 inches. Calculate the-radius of the arc. D  s^Zv-d a-J-^-^-i -A  * e^^>4  ^*>^i-t  &  62  APPENDIX C SAMPLE OF CALCULATIONS E q u a t i n g of Kamloops Groups - I.Q. Group B  Group A  P u p i l No.  2  X 1 "2 3 1+ 5 6 7 8 9 10 11 12 13 11+ 15 16  /16 /13 /12 /10 / 7 / 7  125 122 121 119  116 116 115 110 108 108 101+ 103 103 102 101 99 98 97 1967  17  18 X a Assumed X\  a  t  6  - 1 - 1 - 5 -6 - 6 - 7 - 8 -10  -11 -12  X  256 169 11+1+ 100 1+9 1+9 36 1 1 1 25 36 36 1+9  125 122 118 117 116 115 112 111  73  •  S /3ZI  81 61+  k?  7  6  / 3 / 2 0 - 1 - 3 - 3  - k - 5 - 5 - 7 -16 -17  J"?  109.2  109.3  109  A3  256 169  8  108 106 106 105 101+ 101+ 102 93 92 1965  61+  (*>)  A I A t A  109  100 121  X'  109  N J /3V7  S.7  _  /JL  36 9  h 0 1 9 9  16 25 25  1+9  256 289 T3H7  63  Equating o£ Kamloops Groups - 1 s t Term Marks P u p i l No.  Group A  1 2 3  k  5 6 7 8 9 10 11 12 13  tk  15 16 17 18  X  x'  (x»)  81 61+ 81 77 82 75 63 78 80 k2 87 56 51 68 86  /12 - 5 A2  25  kk 60 ?8  1*93 X z  Assumed X s  Group B  6ff 169 36 36 81 121 729 324 169 32if 1 289 625 81 121  < A3 8  - 6 ^  9  Ai  -27 /18 -13 -18 - 1 A7 -25 - 9 -11  m  2  /3.?  x«  (x')  93 ?k 68 79 82 72 56 62 73 76 72 66 71 58 53 51 70  frk  576 225 1 100 169 9 169  A5 -1  /10  A3 / 3 -13 - 7  / 3 - 3 / 2 -11 -16 -1.8 t 1 -11  3m  68.5  69  X  69.1  -  - -  69  =  //•/  It? 16  k-9 9 9  121 256 32% 1 121  __15  2  6k  E q u a t i n g of Kamloops Groups - 1 s t Term Marks  (continued)  E s t i m a t e d S t a n d a r d D e v i a t i o n of a p o p u l a t i o n from the combined I n f o r m a t i o n of two samples: (N.B» S i n c e the d i f f e r e n c e between the assumed mean and the a c t u a l mean^Is v e r y s l i g h t i n each case, the ^ x ' and the ^ ( x > ) , immediately a v a i l a b l e from the p r e v i o u s page, have been used f o r ^ x and ^ j t i n the f o r m u l a below).  Standard E r r o r of the D i f f e r e n c e between the two samples when assumed t o be drawn from the same p o p u l a t i o n :  C r i t i c a l Ratio:  t =  A/1, - Niz  4.3  =  0.\H  65  General Achievement of Kamloops Groups i n U n i t I Pupil  1 2 3  27 21  28  29  \  29  25 27 21+ 23 16 25 17 19 23 24 15 23  6 7 8 9 10 11 12 13  15 16 17 18  Gi*oup B  Group A  No,  11  4 2 5 6 6 2  16  1 0 7 2 6 4 0 1 8 0 6  1 0 k9  25 27 24 29 26 22 12 28 25 21  k  25 36 36  Jt  3^ 16 0 1 64 0  2k 21 20 ^  2  15 17 18  16  m  X s 22.9 Assumed X g 23  22.3 22  -  5 =  5=  St.  3 yy + 35L_  2 3 5 2 7 ll0 -10  A  6  3 1 7 2 1 2 2 7 5  4  9 25 4 49 16 0 100 36 9 1  4 l  X % 16  66  C o r r e l a t i o n Between I.Q. a n d U n i t I T e s t S c o r e s (Kamloops Group A) (I.Q. dev. f r o m page 62) x  P u p i l No* 1 2 3  k  5 6 7 , 8 9 10) 11 12 13 1*  / k  M  A  A * A  /60 /60  ^  A  - 2 6  7  A  7  6  ^  / 1 - 1 -1 - 5 - 6 - 6 - 7 - 8  -  Prom page 62,  x  2  6  k  0 - 7 - 6  - k 0  - 8 0 - 6  -12  = 1381  -26  2  /1  -mo>i i  16 17 18  xy  A6 A3 A2 /10 ^  1$  ( T e s t S c o r e dev. f r o m page 6£) y  Prom page 65, ^ _ x  / 1 0 / 7 -10 /2k  0 - 8 /80 0  2 s  3kk  (N.B. As n o t e d on page 6k, here a g a i n t h e d i f f e r e n c e between assumed mean a n d a c t u a l mean i s so s l i g h t i n each case t h a t x has been u s e d f o r x ) . 1  r  =-.  2*  +  

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