ii c;, rt/u nr.iin 'i m I M U ' . N

I H I O W. /th <> V r . . V n M I i |

B . C . f l . D l . T . JOUfiilflL

V O L . 4, NO. 1 OCTOBER 1964

Published by B. C. Association of MathematlcH Teachers

of the Br i t i sh Columbia Teachers' Fori'-ration

1964-65 E X E C U T I V E C O M M I T T E E

President: J . F . Clark, Langley Vice-president: C. B , Kotak, North Surrey Past-president: N. K. Preston, Vic to r ia

Secretary: W. Abercrombie, North Vancouver Treasurer: J. W, Hal l , Burnaby 1

Membership Secretary: J. W. Hal l , Burnaby 1 Curr icu lum Representative: J, Hotell , Pr ince George Newsletter Editor: W, Abercrombie, North Vancouver

Journal Editor: R . Desprez, Nanaimo

Address a l l corjj^pj^denceJo_^he_editorj

M r , R . L . Desprez 848 Alber t Street Nanaimo, B . C.

Articles contained heroin refloat tho vlown of the authors and do not necessarily oxproBs tho official policy of tho D. C. AsBoclnllon of Mathematics Toacliors or tho 13. C. Teachers' Federation.

V.

C O N T E N T S

FOREWORD

A P S Y C H O L O G I C A L CONSIDERATION OF M A T H E M A T I C A L A B I L I T Y

- D i \ F , G. Robinson

F R O M T H E DIALOGS OF M Y O P Y N Y E N - J, A, McTaggart

M A T H E M A T I C S P R O G R A M FOR T H E N O N - C O L L E G E BOUND STUDENT

- O, F , Sehuaf

USING A L G E B R A TO T E A C H G E O M E T R Y A P A R A D O X ?

- W, Abercrombie

T H E R U L E R AND P R O T R A C T O R AXIOMS - J. F . Clarke

T H E P R O B L E M OF R E A D I N G AND T H E M A T H E M A T I C S STUDENT

- L , Johnstone

A GLOSSARY OF DEFINITIONS IN MOISE-DOWNS' G E O M E T R Y

. - W. Abercrombie

G E O M E T R Y PROOFS - HIDEBOUND B Y TRADITION?

- R. L . Desprez

'•Mjffi^

F O R E W O R D

1 n t h i s i s s u e o f o u r jo u r u a 1 we h a v e t r i e d to e m p h a s i z e two m a j o r a r o a B .

T h e f i r s t p a r t o f tho j o u r n a i i s d e v o t e d to a r t i c l e s w h i c h m i g h t be o f a c o n t r o v e r s i a l n a t u r e - m a t e r i a l to p o nd e r a nd p e r h a p s m a k e one p a u s e f o r a m o m e n t . If a n y of o u r r e a d e r s ' a g r e o o r d i s a g r e e s t r o n g l y e n o u g h w i t h s o m e of t h e s e c o m m e n t s , p e r h a p s t h e y w i l l se t d o w n t h e i r 1 de a & on p a p e r a n d s e n d t h e m to u s .

T h e s e c o n d p a r t o f the j o u r n a l Is k e y e d to the M o i s O - D o w n s G e o m e t r y to be i n t r o d u c e d i n the B . C , s c h o o l s y s t e r n t h i s S e p t e m b e r at the G r a d e 10 l e v e l . A c l o s e a p p r a i s a l of the t e x t h a s r a i s e d t h e q u e s t i o n t h a t p e r h a p s r e a d i n g w i l l be a p r o b l e m to s t u d e n t s . above a n d be,-y o n d a n y d i f f i c u l t i e s t h e y m a y h a v e w i t h t h e m a t h e m a t i c s . W i t h t h i s In. m i n d we h a v e a s k e d a r e a d i n g e x p e r t to g i v e us a f e w c o m m e n t s , F o r r e f e r e n c e , b y . ; t h e c l a s s r o o m t e a c h e r , W. A b e r c r o m b i e h a s c o m p i l e d a g l o s s a r y of d e f i n i t i o n s f r o m the t e x t . If any . of t h i s g e o m e t r y m a t e r i a l i s of u s e to t he G r a d e i 0 , t e a c h e r we s h a l l f e e l t h a t t h i s i s s u e h a s b e e n w o r t h ­w h i l e . •->

iij".":/^

•I.

I OF M A T H E M A T I C A L A B I L I T Y

? Dr. F loyd G. Robinson, • Director of the Canada Council

for Research in Education

This is Chapter 1 from A Psychological Consideration of Mathematical Abi l i ty by Dr. Rebinson. It is reprinted here with permission.

The argument pursued in this chapter may at first seem unrelated to the business of expounding a theory of mathematical ability, What we hope to convey here, however, is the idea that theorizing about the psychological processes underlying mathe­matics is necessary for the intelligent conduct of mathematical education.

A reasonable question which often is , or at least should be, put to mathematics teachors by stu­dents and parents is this: What does the high school student gain from his mathematics courses? If we examine the official courses of study, or i f we em-bar ras our colleagues with this question, wo w i l l invariably find that three sorts of answers emerge: The first deals with knowledge^ facts, or concepts; the second with social experiences; and the third with the alleged abili ty of mathematics to somehow ' t ra in the mind to think log ica l ly . ' '

In what follows, we shall examine these justifications in turn, in order to determine if they are supported by evidence, or whether they belong merely to the rea lm "of belief.

2 //

The Facta and Concepts of Mathematics - -Tho Few V B Tho Many

We w i l l probably al l agree that everyone ahould know the facta of arithmetic and should possess t'.ie ability to add and subtract, But If we ask how much one should know about geometry and algebra, then tho answer is not quite so obvious.

| If one examines the often-repeated c la im i that these subjects provide nocosaary facts, it bo­'s comes apparent that proponents of this argument moan | that tho facts are necessary for approximately five | percent of tho Grade 10 students who wi l l deal with I , ( mathematics beyond the high school level. This in-

' credible sacrifice of the interests of tho many to the needs of the few has boon perpotuatetl only because the high Bchool mathematics curr iculum has always boon superimposed from the university downward. In this connection, it is well to reca l l the words of Kinney and Pu rdy l in discussing the last riiajor r ev i ­sion of the high school mathematics curr iculum which took place about 1900:

Because the purposes of the Bequence were cultural and discipl inary, p r imar i ly designed

„ for college preparation, it is not surprising to find that no psychologist or sociologist was in ­cluded on the committee, and that none of the >t committee members had a record of close con­tact with youth, This may account for the fact

0 that no differentiation in purposes waa recognized between the secqndary school and college.

1. Kinney, L , B . and Purdy, G. R. , Teaching Mathe­matics in the Secondary School. New York,

o Rinehart and Co. , Iric. , 1952.

3

It now appears that the university mathema­ticians have decided to heed an argument advanced by Thorndiko as long ago as 1925, to tho effect that, the curr iculum contains a great deal of material that is unnecessary, even for university-bound students. Tho time saved by clearing away this dca'tlwood wi l l be given to work stressing the modern or axiomatic approach to mathematics. As laudable as these pro­posed alterations may be, there has really boon no basic change in outlook. Tho course revision taking place in Canada at the present time is s t i l l largely directed by university mathematicians, who cannot, by virtue of their training, or experience, have a vor real is t ic conception of the high school student except as a university-bound entity.

When prossod to justify the Btudy of mathe­matics by the average ( i . e . , non-university) student, the mathematician often resorts to the 'theory of functional competence. 1 The argument typically runB as follows:

Tho technological and scientific advances of our time have created a civi l isat ion in which mathematics plays an important role. It is essential that every citizen should have some knowledge of the forces which aro changing his life. Again, the demand for mathematical sk i l l s i s o n the increase. It follows that the level of mathematical competence of the future 'average' man should be raised,

This argument has received wide circulation, and sounds plausible enough. But what seems to be advocated here is a type of education which imparts a ' talking' knowledge without rea l comprehension, for we surely cannot believe that the everyday working man w i l l comprehend quantum mechanics, electromag netic theory, or relativity, to name three examples

4

of scientific advances which have tremendous imp l i ­cations for our present time, A H a matter of fact, one could equally well argue that not only aro science and mathematics advancing, but their areas, of necessity, aro being highly subdivided and specialized, There is, therefore, some justification for claiming that the average student should be taught loss mathe­matics, certainly less of tho highly Isolated bits of information which he now receives in the high school mathematics course, and which he shall l ikely never use. With regard to the demand for mathematical sk i l l s , even if wo contemplate a peiy,caplta quadrupling of our scientists and engineers, thot'p is s t i l l little reason to believe that more than a Bmal) proportion w i l l require post-elementary school mathematics. After a l l , the majority of out' school graduates w i l l becomo housewives or druggists, or lawyers, or laborers, and thes «,ople have made remarkably little use of mathematics in the past. And we roally have no reason to believe that they would make any more use of set theory than they have of,the rule for extracting the roots of a quadratic equation.

In summary, one 1B hard-pressed to believe that mathematics can be justified for everyone - or

• oven for the majority - on the grounds that it imparts useful knowledge,

Mathematics and Social Learning

We are now inclined to view the results of the educational process in broader terms than the mere acquisition of facts or concepts. We would l ist as legitimate functions of the school a wide array of per­sonal and social learnings such as: good work habits, the ability to work with a group, to e x e r c i B e leader­ship, and so on,

As to the efficacy of mathematics to develop

5

such behavior, wo must admit, that lack of evidence precludes the possibili ty of our reaching a decision ono way or another. About a l l one can say is that there is no evidence that the mathematics classroom offers opportunities for desirable development to a greater extent than any other school subject, or in­deed, than non-school activities. In fact, mathema­tics tends to be a solitary and competitive, rather than an interpersonal and co-operative activity, and tho total effect of its study - from a mental health point of view - may woll be negative.

Mathematics as a 'Mental Conditioner 1

Inevitably, a discussion of tho Justification for including mathematics In the high school cur r icu­lum reverts to a consideration of the relationship between participation in mathematics and tho enhance­ment of logical behavior. It is widely assumed by teachers - and almost universally believed by lay people - that as long as tho student struggles at mathematics, something Is happening to him which w i l l make him, in some sense, more ' l o g i c a l . 1 Many phrases of equivalent meaning are used, but the implication is alwayB the same,

How have we come by this great belief in the beneficial effectB of the study of mathematics? For one thing, the theory of mental discipline is not quite as extinct as many educators imagine it to be. Several generations ago it was believed that the mind consisted of a numMr of faculties, each of which could be en­larged through exercise, in much the same manner as the biceps grow through constant use. In this scheme, mathematics was part icular ly useful in' the develop­ment of the 'reasoning faculty. ' Although the faculty theory was long ago discredited by psychologists as an inadequate explanation of mental growth, the views of many teachers are, unfortunately) not far removed from this outdated idea.

6

It Is true that one oft on observes a correla­tion between participation in mathematics and logical behavior. Thus, mathematicians show a highly developed logic in their own field, And while indivi­duals who are highly irrat ional would l ikely be well advised to make a career of politics rather than of mathematics, one could not argue, however, that mathematics has made tho mathematician logical. It Is more l ikely that he possessed this characteristic before ho entered the mathematics field. In short, tho observation of a correlation between m ithomatlcs and logic need not Imply a causal relationship,

The reader should not conclude from what has gone before that it is generally believed that tho study of mathematics and logical behavior are unrelated. The situation is such that a person who has investiga­ted the meager evidence scarcely knows what to be­lieve. Our present confusion results from the fact that we have s t i l l to realize that the question of mathe­matical ability is not a mathematical problem, but rather a psychological one. If mathematics can have a beneficial effect on human thought, then it must follow that the structure of mathematics is related in some special way to the basic psychology of the brain. Unfortunately, we know so little about the psychology of mathematics that one recent author was able to state that 'we are s t i l l groping for the psychological principles which govern the learning of mathematics. '

In summary, we can say that one cannot seriously justify mathematics as a general course in the curr iculum of the secondary school on the grounds that it provides useful knowledge to a substantial pro­portion of those who study it, or that it leads to desirable social growth. When we dig down to the basic core of our feelings about mathematical educa­tion, we find a strong faith in the power of mathema­tics to influence or develop'logical growth,

Tho iixak of this report is now clear, if vvc are to justify mathematics, wc..must investigate the relationship which oxiBts between tho learning of mathematics and tho basic psychology of thinking. Along the way we .shall encounter and attempt to answer many qucstki is , of which the following arc typical:

1, How 1B axiomatic or modern mathematics related to thinking processes?

2. Is thero a special talent which we might cal l mathematical abil i ty? Is there more' than one kind of mathematical abili ty?

3. What factors influence or determine the l imit of the progress that a student can make in modern mathematics?

4, What Bpocific c lassroom and personal factors in­fluence the student1 s mathematical l imi t?

When we have obtained these answers, we ' can then (claim to have made a start toward,a science of mathematical education.* n

§ § § § § § § § § § § § § § § § § § § § § § § § § § § § §

D r . >F. G. Robinson comments on the teach­ing of geometry: :

. . .one, might be led to suspect that geometry affords the average student substantially more opportunity to practice the i l log ica l reduction of problems than to engage incorrec t analysis.

8

F R O M T H E DIALOGS OF M Y O P Y N Y E N

John A, McTaggart, Supervising Pr inc ipa l , Langley

" W e l l then, clear Glaucoma, having deter­mined that a l l things are to a purpose, we must examine the quality of worth to our present argument, 1

"I see, 1 1 he said,

"I perceive that there are those who are of the opposite persuasion. "

" T r u e , "

"But of these it is incumbent that they defend their position to the satisfaction of justice and truth, "

.,.•„. " C l e a r l y they must, 1 1

1̂ ^ "It appears that they have created a Divine

vimjige from which we, must.shrink but worship at the r/ar'.ae time. /These higher institutions presume to layout the path for the novice, neither consulting him regarding his needs nor assessing his aptitudes," .

| "Most aBsuredly they have demonstrated this.

i , . , " I f change is needed,, would it not be a safer course to follow, to be led by immediate require­ments than by, a dream? It is not difficult to under­stand their enthusiasm, but do we agreeihat the wisest-choices are not always made, at the height of enthusiasm?"

"We must agree. 1 1

9

"And while my own strivings are for tho preservation of tho Humanities, do I presume that a l l others aro thus persuaded?"

"Certainly not, "

"Yet at the cost of bewildering a nation of parents and children) not to mention a host ot Other­wise intelligent teachers, we must loam of sot theory for which there 1B but the remotest practical use."

"Only, it seems, as a manipulative exerc i se . "

"Further to add to our insecurity, they would have us learn of number systems to bases other than ten. We must concede that a study of the binary system has yielded Important progress in the f ield of automation, but is it not a doubt worthy of reflection that many of our students w i l l become in ­ventors of electronic digi tal computers or program­mers for these. 1 1

v' " B y the dog of Egypt, it is worthy of con­siderable deliberation 1. 1 1

l v " ? ' r ; ' ! . "Speaking in confidence, dear Glaucoma, for I should not like my words repeated to the mathema­ticians or the curr iculum builders , I admit that I am beneath a novitiate on a subject which charges their spir i t , but may I not offer a compromise to which none .could take exception?"

•• " Who is better qualified than a novitiate?"

• '" '•" "Then let us suppose that the wisdom of past cur r i cu lum builders is yet worthy of our study, They deemed it sound to make provis ion for tlk'se who had a bent for art or music by allowing them to accent their studies," v

10

"This was sensible. "

"Allowing that some aspects of the now mathematics aro essential to the study of higher mathematics, let us opon tho door of opportunity to those who would explore this fascinating world. Lot us allow them to accent their studies. Lot those who aro mathematically biased exploit their bias, 1 1

" A h , I see! Wil l you have your hemlock now, or la ter?"

, . . we must begin to make a real is t ic assess­ment of the amount of actual gain in analytic power which the vast majority of students make through a study of geometry. If this is as 'slight as we suspect, then we must re-examine the ch \ im that mathematics is a subject which is smfable for a majority of students.

- D r . F . G, Robinson

11

M A T H E M A T I C S P R O G R A M FOR T H E

N 2 N L C O L L E j ^ ^ ^

Oscar F . Schaaf

In recent yearB many different groups have made recommendations for mathematics programs for students with above average achievement in mathe­matics, A commensurate amount of attention has not been given to the courses for students with near average and below average achievement, While schools in Oregon may be guilty of having programs with an unbalanced emphasis, the Oregon State Depart­m e n t s Education, as ear ly as ten years ago, ex­pressed concern that every effort should be made to Improve the mathematics programs so that a l l ntudents would profit more fully than ever from their study of mathematics. Several years before Sputnik, a com­mittee was appointed to write a handbook, Mathematics in Oregon Secondary Schools, which was to give guidance to schools as they sought to revise their pro­grams so as to be more consistent with present day general education needs of 'a l l students, whether non-academic or non-mathematical in their goal or destined for specialization in the mathematical f i e ld , 1

In this publication, the committee suggests that the first step in planning a mathematics program is for a l l members of the mathematics staff and administra­tion to agree on certain assumptions upon which a comprehensive mathematics : program could be based. A s an aid,.the.committee offered some assumptions* for each school system to consider, These assump­tions which have a bearing on a. program for below average students are .listed below.

.........-.-fvThere, should be a mathematics course geared to the ability . level of any student who wishes

12

to study mathematics, Tho course should bo difficult enough to bo challenging, yet easy enough for him to experience some success,

Every school should have a carefully con­sidered student testing program which w i l l give data that can be used in placing students in mathematics courses which are most nearly consistent with their needs and abilit ies,

- - The sequence of courses should be built around a 'main track' of academic courses with 'side tracks ' introduced at various points to provide for varying intellectual, social, vocational, and remedial needs, including some with an acceleration and tele­scoping of contentfor the exceptionally capable student,

- - I n general, students who are deficient in mathematics should be given remedial instruction as soon as the deficiency becomes apparent,

— Nine years of mathematics should be re­quired (Grades 1-8 plus one in high school), but under certain conditions the student should be permitted to take his ninth year not necessari ly in the ninth grade but in the senior high school.

- - The last mathematics course a student studies should be in high school near the end of his high school career,

- - Every school of sufficient size should have a mathematics department chairman or head, or teacher assigned to leadership in the area of mathe­matics who in addition to teaching mathematics classes is given time to assist guidance personnel in the place­ment of students in mathematics classes, to plan for the in-service education of all l teachers in the depart­ment,1 and to administer the department,

13

P « « i f t W 8 » w « * * * ' » ~ - -

• It is obvious that tho handbook committee believes that mathematics coursoB for tho alow learner should not bo thought of as courses apart from the regular sequence. The 'main track' courses have value for a l l Btudents and they should be en­couraged, whom they are adequately prepared, to elect courses in this sequence. <

If the mathematics program in most Oregon school systomB were to be evaluated on the basis of the above principles, one would observe many inade­quacies. Probably such an evaluation would reveal that more attention should be given to courses in the

)i following categories.2 In some cases, it may mean that new courses should be added,

A. Seventh grade students whose mathema­tics achievement, is in the lowoBt quartile, For tho most part, the content should deal with topics usually taught in the elementary school, but tho problem situations should be taken from settings of interest to junior high school students, The goal of the course should be to provide instruction which helps studenta gain enough under atanding of elementary ma thematic e, part icular ly arithmetic, so .that he might succeed the following year in the f irst course of the regular se­quence for secondary Bchool student B ,

• B , Ninth grade students in the lowest quar­tile of mathematics achievement. In most ninth grade classes, students take either algebra or general mathematics. For the most part, those taking algebra are inthe upper 50% in mathematics achievement. A l l the other studentB„ usually take general mathema­tics, The committee feels that studenta in thiB lowest quartile should.not be in general mathematics but in a special course where considerable attention is given, to the, type of mathematics average students master in the fifth to the eighthjrrades.. A course such as this i s

14

especially needed in a four-year high school which serves students who have graduated from an eight-grade elementary school. Tho goal of this course should bo tho building of understandings and ski l ls which would enable students to succeed in a general mathematics course the following year,

C, Ninth grade students with mathematics achievement in tho 25 to 50 percentile range. Such a course should give students an overview qf algebra and geometry and, In addition, should contain content designed to improve student understanding of arithme­tic . It should help them understand how our algorithms are derived from the structural properties of the number system and the place-value properties of the Arab ic system of numerals. The teaching units should be designed to make explicit student deficiencies in understanding and to provide a remedial instructional approach based on a sound mathematical as well as pedagogical point of view.

D. Low achievers (not failures) in algebra and geometry who wish to continue their study of mathematics. At present, many of these students are in classes along with those who are superior achievers. The.results of such placement usually end in failure for the low achievers and an unchalleng-ing course for the superior achiever. It must be remembered that these 1 low-achievers, 1 for the most part, are s t i l l in the upper 50% in mathematics ability when compared with a l l students at the appropriate age level. At the present time, there is a great need for technicians trained in mathematics. If the heed is to be met, it w i l l have tobe met with these Viow achievers . 1 Mathematics teachers in general, have given up with these students too soon. The course should contain much new material - not merely a re­view of old topics. This content should give students an overview of different branches of mathematics and

••" ':. a . '

15

-^mjmatm.nimmit ' •» - ' - » » - - - - — - \ — i i — — . — ~ — '

should also prepare them for a mathematics course In the regular sequence (second year algebra or tho equivalent), Tho Introduction of students to new con­tent should be intuitive, but review of old material , insofar as possible, should be a formal point of view. At least the students should become aware of the mathematical structure of the courses they have pre­viously studied.

E , Eleventh and twelfth grade students who do not demonstrate a reasonable mastery of basic mathematics, Many high schools give mathematics

, achievement tests to their juniors, If their scoreB are high enough they have no other mathematics course requirements for graduation. If they score below a certain standard they are required to study mathema-ticaone more semester or year, This course should possess many of the characteristics of those for students in categories A and B , The problem material , however, should include situations of interest to juni-br.B' and seniors, The students who find themaelvea in Buch courses uaually have done poorly in the course in mathematics they have previously taken, Often by this time, students have had enough experience in the working world to be convinced that competence in mathematics is important, High schools have an obl i ­gation u to provide courses of this type for theae stu­dents,

' F . . Seniors who want'a terminal non-remedial course in consumer or pract ical mathematics. Many schoolshave courses of this type. Often such courses are not satisfactory from the point of view of both teacher and student. One reason may be that reme­dial students are also placed in the course. Much of the problem material involves ratio and proportion and percent concepts, and such concepts are probably too difficult for students who have serious deficiencies in arithmetic. The oourse content should contain

16; i

many problems taken from everyday situations. Also , some attention should be given to the explicit statement of mathematical principles invoked in the solving of tho problems.

In the publication, Mathematics for Oregon Secondary Schools, the courses listed below have boon described which w i l l fit into the categories men­tioned above.

Category A Category B Category C Category D Category E Category F

Arithmetic (Junior High) 3

Arithmetic (High School)4 General Mathematics5 Advanced General Mathematics 0

Basic Mathematics''' Consumer Mathematics^

In a 6-3-3 system, some possible sequences that below average students might follow are outlined below:^

Gr . 7 8 9

10

11 12

Ar i th . Math 1 Math 2 Gen,

Math Math 3 Math 4

A r i t h , Math 1

Consumer Math 2 Gen, Math

Math

Bas ic Math

Gen, Math Consumer Math

Math 1 Math 2 Gen. Math Math 3

Math 4 Adv. Gen,

Math

^The mathematics teachers in the Eugene Public Schools have developed course materials in a l l the categories. Other school distr icts have also developed some materials , It is unfortunate that commercia l texts are not available for a l l the courses in the above -categories, There is a strong possibi l i ty that'this deficiency w i l l b t f remedied in the hear future, The National Council of Teachers of Mathe­matics is actively pushing for such materials, At

present, they are sponsoring tho writing,of materials for Category C (General Mathematics), During the summer 1963 a writing team produced an"experimental text and at present it Is being tried in 75 classrooms at various schools throughout the United States, On the basis of data received from tho tryout, the text w i l l be revised during summer 1965, The National Council is presently seeking funds to finance tho writing of courses in the other categories, •«•

What has been mentioned thus far in'mapping out a program for the bel'ow average student is not something that can be put into effect at once. How­ever, 'there are two things which can be done imme­diately to improve courses for slow learners. Tho first concerns'the selection of the teachers for these students, and the second deals with numbers of stu­dents in the class. Generally, below average classes are assigned to teachers newest to the school or to those who need another course to f i l l out their teach­ing schedule, Such a procedure should not be followed, Teachers of slow learners require the best of teacherB who have a f i r m grasp of basic principles of-mathematics and who have the ability to apply generally accepted principles of learning, It is rather encouraging to see that many schools are assigning their bestteacher to the'most advanced classes and also to slow classes,

As a general practice, when schools have classes for below average students the class size is the largest in!;the;mathematics department. If a • ' school Is serious about improving the mathematics competence of'the below average student, .then the number-In,the class must be less than for other, classes. Students who have found mathematics difficult,gener- . a l ly need considerable individual attention.1 ' A l s o , but of'secondary importance, if a class size is kept between 20 and 25 students andthe;best teachers are

18

assigned to the c l a s B O B , the school administration should have little difficult/ in selling their mathoma tics program to parontB of below average students,

FOOTNOTES

1. Mathematics in Oregon Secondary Schools, State Department of Education, Salem, Oregon, 1961. pp. 22-23.

2. Based on an unpublished report of the National Council of Teachors of Mathematics Committee on Mathematics for the Non-college Bound Com­mittee, Chairman of the Committee - Oscar Schaaf, South Eugene High School.

3. Mathematics in Oregon Secondary F.lhools, State Department of Education, Salem, Oregon, 1961. p, 55.

4. Ibid, pp 64-65. ..

5. Ibid, p, 56.

6. Ibid, p. 59.

Ibid, p. 58.

8. Ibid, p. 58.

9. Other sequences are listed in Mathematics in .Oregon Secondary Schools, pp. 61 -69.

19

AJfAl^Aj30X?_

W, Abercrombie

One of tho major ways in which the new MathematicB 10 courBe in geometry differs from the traditional course ia the way it brings the use of algebra into its proofs, The algebra is not an, end in itself, but becomes one of the most Important tools the student has for solving equations.

It may'seem strange to see number featured so prominently in a geometry course, and it may appear that this is a new geometry. This is not the caae at a l l , for it is very important to remember that this is s t i l l a course in Euclidean geometry.

Why then use an approach that is not Euclidean? There is no one, I would hope, who would deny that there are mistakes in the work of Eucl id , Postulate 2, which states: "It is possible to produce a finite straight line continuously in a straight l ine , " points out the weakness in a system that attempts to be commensurable.

It is important to remember that in the time of Eucl id algebra had scarcely been developed. Since then algebra has developed to an unbelievably high degree, and yet the teaching of elementary geometry has involved almost none of it.

To quote the teacher's commentary to the SMSG Geometry course:

20

In thio book, algebra Is used in two Important ways, In the first place, it is used in tho postulates to make thorn easier to apply, If wo take for granted that the real numbers aro known, then it 1B possible to give a logically complete set of postulates, adequate for proving tho theorems, , , , Wo wi l l BOO also, as we go along, that a groat deal of the tradi­tional material was r ea l ly algebraic a l l along, and is much easier to handle when it is des­cribed algebraically. (This is especially true in the chapter on Proport ion,)

I would like to explore these two things in the new course, and part icularly show how traditional ma­ter ia l can be made easier through the use of algebra, Perhaps the best place to begin is with Chapter 2, 'Sets, Rea l Numbers, and L ines , '

Before any postulates are introduced, the real numbers are introduced and, more important, are ordered, On page 23, we are given these four laws: • '

(a) For .every x and y, one and only one of the following conditions holds:

x < y, x = y, y < x. (b) If x < y and- y < z, then x < z, (c) If a < b and x < y, then a + x < b + y, (d) l f x < y a n d a > 0 , then ax < ay,

We must have these before we can work with meaaure.

The real numbers are not introduced expl i ­c i t ly , but are essential for our f irst postulates, Let us examine Postulate 1:

To every pair of different points there . corresponds a unique positive number,

21

Our first-truly geometric statement is given almost completely in terms of algebra. We aro told that tho lengths of segments are expressible as real numbers and therefore have a l l the common properties asso­ciated with the real' numbers.

The words 'a unique positive number 1 intro­duces a problem not usually encountered at such an elementary level of geometry, The concept of absolute value Is a difficult one for Grade 10 Htudents to grasp, but here we can reverse our origlnalpolnt and say that this difficult algebraic idea can be shown simply by means of geometry. A student may have trouble seeing that !J8 - 5 | = |5 - 8 |, but should be able to see that a segment on the number line with endpoints 8 and 5 has the same measure whether we go from left to right or right to left,

One might argue that this postulate and the next two postulates are real ly just gimmicks to avoid the real substance of the theory of measure of seg­ments. But one has to decide somewhere just how, rigorous one wants a course like this to be, These early postulates can be proven as theorems, but at what an>expense to more important topics in an ele­mentary course I This I feel ia one of the major strengths in this courae, Difficult concepts such as distance and congruence are kept largely at an intui­tive level,

Our use of the real numbers allows us to correct one of the more important omissions in Eucl id - his failure to order points in space. The fact that ho does not define, 'betweennesa' ia apparent in The Theorem of the Exter ior Angle, So here is a place where SMSG and Moise and Downs go Euc l id one better and present a definition of 'betweenness':

B is between A and C if (1) A , B , and C are different points of the same line and (2) A B + B C = A C .

22

.""•'.i1.',"'.-;:."".1" '.".;."*: »—...-—.••••—-.-»-•••<• - —*~ ••*• •• «»>*'«(w.»iBMi>ti 1MiH'>wMrt»w*""'

You w i l l notice that tho second part of this definition is rea l ly an algebraic statement,

With a l l this in mind, let us examine two theorems presented in tho second chapter, These are not necessarily typical of tho typo of theorem the student w i l l latoi' bo expected to prove, but they do servo to illustrate the extent to which algebra can assist us. Theorem 2-2: Eve ry segment has exactly one mid-point,

Proof: We want a point satisfying the two conditions

AB + B C t A C and AB a B C ,

These two equations te l l us that

, A B = AC 2

We have previously proven that there is ex-' actly one point B of the ray AC that lies at a

distance A C / 2 from A , Therefore A C has exactly one mid-point,

l a m becoming so engrossed in'my topic that I am afraid L a m overemphasizing a point, We must not lose sight of the fact that this is a course in geometry, not algebra. Many topics, such as congru­ence/ are treated almost exactly as they are in a conventional course. But, even here, there are major differences. In a tradit ional course, congruence and equality are synonymous; in this text they are very different ideas.

Since I am in this deep, I might as wel l go a l l the way and examine Chapter 7, Geometry Inequali­t ies. At first sight, the mater ia l here is not radical ly different from that found in more conventional geometry courses, The major difference lies in the fact that Moise and Downs compare two segments or two angles

23

merely by comparing their lengths or measures, Therefore, although our theorems in this chapter describe geometric relations, thoy Involve only real numbers, More we have another advantage of our early introduction of real numbers,

Consider how the principles of inequalities can be applied to a problem like this!

Prove the following theorem; . Tho sum of the measures of any two angles of a triangle is less than 180,

Restatement, If tho angles of a triangle have measures a, b, and c, then

a + b < 180 b + c < 180 a + c < 180

I should point out that this chapter in the SMSG text was the most difficult in the course from the point of view of my students,, ,Yet many of them felt that this was one of the most rewarding in the text, However, we should not lose sight of our objec­tives and, should not let the geometry become merely a device for presenting algebraic ideas.

The use of algebra is carr ied to its most rewarding level in Chapter 12; Similar i ty . The • treatment of the mater ia l is for the most part conven­tional, but the student is constantly called upon to use his full knowledge of algebra,

, Perhaps ,the most amazing thing in this text iis,that;the most algebraic chapter in the text, Plane ••Co-ordinate Geometry, helps to show the student that there is an autonomous subject of geometry which is , logically equivalent to the autonomous subject of, algebra.,(•.••••. • • . , ! • , . - • . : . ••••••• ' . >

24

Through a l l my random thoughts, I have left tho most Important question unanswered: Is a l l this emphasis on algebra justified? I don't real ly know, I think it depends upon the teacher and how muchi emphasis he wants to place on it, I believe that this course can be a truly rewarding and enriching experi­ence both for the teacher and for the studont,

>;< >;< »;< >;< >\< >,< >;< >;< >;< ,;<•'>;<>!< >!<>!< >|< >j< >!< V >'< >;<>!< >;< >,'<

I T H E A R T I C L E , A Glossary of Definitions in • ' rt ,. Molse-Downs' Geometry, by W. Abercrombie

r ; l : • • : y-y^'Uwhichappears on page 36 of this journal, has ' 1

•)lf since been completed as a lesson aid. To y : ;. ) i { order copies of this lesson aid write to the'

•"*'••"'•'•'"•""•' •"" Lesson Aids Service, 1815 West 7th Avenue, •Vancouver 9. : i

'., ' ' ' , Please order by its lesson aid number.-3017.

' .ns*

.;>:<.

, >:<

#

'#

>:<

>:<;

*

>:<

25

J. F , Clarke, Alpha Junior Secondary School, Burnaby

Here they are! The Ruler Postulate!

The points of a line can be placed in corres­pondence with the rea l numbers in ouch a way that!

1, to every point of the line there corresponds ex-. actly one rea l number;

2, to every real number there corresponds exactly one point of the line; and

3, the distance between any two points is the absolute value of the difference of the corresponding num-

, bers.

. . \ \ The Angle Measurement Postulate:

To every angle there corresponds a rea l number be­tween .0. and, 180. , , ; •

; To these we may add: •-. , ,'

Postulate 19:

, > To every polygonal region there corresponds a unique positive rea l number,

, • . This marriage of the Rea l Number; System w i t h ^ the most significant differ in our new geometry course. The most immediate break with tradition is the com-

,plete submergence of the usual angle and line construc-

26

tion procedures. Our axioms provide us with the sensible tools - - a ruler and a protractor - - i'or the needed constructions.

A second peculiar result is tho exclusion of angles with a measure of zero or 180 degrees, There are no angle measures greater than 180 degrees either, This is an ar t i f ic ia l distinction which in no way l imits the developments of theorems, The inclu­sion of Postulate 19 in this survey carr ies the real number concept into area, Thi. . approach enables one to develop nimilar i ty theorems on a much more real is t ic foundation, The problems Eucl id encountered with incommensurable lengths are neatly avoided, The rather awkward and specific form of proof given on page 338 of the Morgan and Breckenridge text is replaced by a completely general development. F ina l ly , it should be obvious that the axioms relating points on a line with real numbers open the door to consideration of graphical representation of many geometric theorems. Chapter 13 of the Moise and Downs text gives an excellent introduction to analytic geometry,

Included in the topics which are either the basis or an extension of the postulates we are con­sidering are the F i e l d Axioms for the Real Number System, the laws governing Inequalities, the rules for dealing with Absolute Value, and ideas concerning Order on the number line, In addition, the solution of problems w i l l often Involve a knowledge of algebra. Definite efforts are made to bring to the pupil the realization that the study of geometry involves three dimensional figures as well as the usual two.

The l is t of variations from past courses may appear endless. The results A R E recognizable. Although there is a sizeable increase in the number of theorems (195 in all) nearly a l l of them are familiar

27

and appear in a familiar form, Readers interested in the background of reasons for the change are ad­vised to examine some of the publications listed at tho end of this report, In the SMSG publication w i l l be found a concise analysis of the recognized er rors in Euc l id ' s approach, It also pointB out very c lear ly that it would require a university level course based on Hilbert'.s axioms to lay down a completely valid iiystem, Despite the forbidding l is t of ideas which have been presented here, certain concessions have been made in order to make tho course palatable to school pupils, In particular, the concept of area is used in an intuitive manner. No mention is made of motion and itB consequence, superposition, This particular trap is avoided by assuming the SAS postu­late and thereby relating congruent angles and congru­ent segmentB, The remaining congruence theorems are easily derived once the SAS assumption is made, In fact considerable use is made of the basic congru­ence relations before the ASA and SSS theorems are shown to be provable,

The title of this report might have been 'Geometry With Rea l Numbers. ' These numbers are precisely those about which Eucl id knew little or nothing, Because the letters of the Greek alphabet were used as number symbols, the arithmetic of the Greeks was of dubious value, The wedding of ar i th­metic and geometry has taken a long time to consum­mate. W i l l the next development take 2,000 years? T r y this quotation: , " , , . Until recently it waB taken as a matter of course that a complete set of axioms

. for any branch of mathematics can be assembled, vIn part icular , mathematicians believed that the set proposed for arithmetic was complete, or, at worst, could be made-complete s implyby adding a finite

•. number of axioms to the original l is t . The discovery ' that th i s v wil l ; not work is one of Godel's major achieve ••; ments. " • • •. •

28

R E F E R E N C E S

1. Studio3 in Mathematics, Volume II, Euclidean Geometry, Based on Ruler and Protractor Axioms School Mathematics Study Group, Order from:

A, C, Vroman, Inc,„ 367 South Pasadena Avenue, Pasadena, California,

2. Godel'B Proof, Ernest Nagol and James R, New­man, New York University Press , I960,

3. P rog ram for College Preparatory Mathematics and Appendices, Report of the Commission on Mathematics, College Entrance Examination Board, New York, , 1959.

>!< >;< >!< >!< >!< >!< >!< >!< >!< >!< »!< >!< >!< >!< >!< >!< >!<

>!< >:< >i< >;< >;< >;< >|< >!< >!< >!< >!< >!<

>|< >!< >1< >]< >!< >!<

29

I S £ P£OJLLJ!M 0 F R E A D I N G AND

^ H E M A T H JDMATICS ST UDENT ***** *"** ""*** **"" **"" *~" » « • • « M M M,a (MD

Laura Johnstone, Nanaimo Secondary School

Because the suggestions I have to offer are varied, I have chosen to present them as a series of notes from the Reading Teacher to the Mathematics Teacher,

It has been estimated that over 75% of learn­ing at the secondary level Is acquired through reading, When it is suggested that the mathematics teacher is a teacher of reading, it is not to say he is concerned with the basic reading sk i l l s , but rather that he teaches the ski l ls necessary for understanding mathe­matics,

Bas ic reading ski l l s involve recognizing words and their meanings; fusing separate meanings into ideas; relating ideas one to another and to pre­vious knowledge and experience; responding thought­fully, c r i t i ca l ly or with appreciation to what is read; using ideas gained. Reading is a complex process of getting the meaning of words in combination and knowing what the author is trying to communicate. Experience, interest, motive or purpose, understand­ing of spoken words, a sense of language structure a l l enter the reading process.

Sk i l l s are part ial ly acquired during elemen­tary school years and expanded in high school. What part does the mathematics teacher have in this pro­cess?

30

. . . . . ™ " ' ' « » > ^ m m m m * * * t M M M * > m * i ^

F o r every new text the teacher discusses tho author, chapter headings, student helps, indexes, and glossary, He makes sure the studont roads the preface and introduq'ion and learns tho stated purpose of tho author, or editor, and the overal l pattern of presentation, I have found tho major portion of some pupils' difficulty lies in their lack of ability to handle their texts,

Then the teachor recognizes difficulties inherent in the mathematics problem, Problems are stated concisely, They contain technical vocabu­lary, Details are important, It is essential to grasp pertinent Information, What is told? What is asked?

P rob lem solution requires concentration and clear thinking, Ask the student having trouble where his difficulty l ies, "I don't know where to begin - - I panic, 1 1 These are reading difficulties, Sometimes it is wise to warn that reading should be slow - - even the eye progression might be necessary to gain details,

To test ability to read arithmetic problems, a series might be presented with questions: What is told? What is asked for? What computations are necessary, and in what order ? How can correctness be tested?

Solution need not be required,

To offset panic with the algebra problem, it might be wise to point out that few problems can be solved by one quick reading.

F i r s t reading - gain a general idea Second reading - find the factors in the

problem

31

Thi rd reading - supply numerical and symbolic detail

Fourth reading - see problem as a whole and use inflight related to it,

Fifth reading - cautious chock 1

OR. Read the problem phrase by phrase, noting pertinent details,

Making Better Readers, by Ruth Strang and Dorothy Kendall Bracken, outlines exorcises!

Pages 255 - 256

Two automobiles start at the same time from towns 330 miles apart and travel toward each other, If one averages 30 miles an hour and the other aver­ages 25 miles an hour, in how many hours w i l l thoy meet?

if One teacher, Douglas .'J, Erath, when a

graduate student at Teachers' College, described his method of teaching the reading of the above problem,

The f irst impulse of many students upon reading a verbal problem is to get panicky, Where does one start? One way to allay this fear is to point out that very few mathematical problems can be solved by one quick reading. A l l we can hope to gain the first time we read the problem is.the general idea. The subsequent readings must be more deliber-ate; .During the second reading we can select a l l the

.act ive forces which appear in.the problem and list them on the right side of our paper. If the nature of the problem alters any of these forces we must l is t the altered form also.

F r o m the f i rs t reading we learn!

32

- r a t e of speed for f irs t auto - rate of speed for second auto » distance travelled

Ori our third reading we must supply the numerical and symbolic description for each of those factors. Wo must also determine what our unknown is to bo. If the unknown is not immediately apparent, it w i l l unfold as tho result of the process of el imina­tion as we f i l l in the data for a l l our factor a. This leads us to;

30 m.p .h , - rate of speed of first auto 25 m,p, h. - rate of speed of second auto x - number of hours travelled 330 miles - distance separating automobiles

We now have the elements but lack perception of the entire problem, Let us read again (fourth read­ing) with the aim of gaining this understanding and' setting up an equality, This is the crux of the prob­lem, The teacher may help by aBking such questions as, "Have you ever seen a s imi la r problem?" or "What are a l l the plausible relationships you can think of?", Insight must ariBe at this point, In this problem it is,the r eca l l of the formula relating diutance, rate and, t ime in order to confirm our understanding and to serve as a safeguard against a mechanically defective

...solution,,... ,

Solution Check.',

30x plus 25x equals 330 30(6) plus 25(6) equals 330 • • , . ; ;''»'; !.55x equals 330 v" ;-: 180 plus 150 equals 330

'r''':Y!K'yf»^ 330 equals 330

• : > , ; One more cautious reading must be made in. ••• . which each'detail of the preceding steps is carefully ^hdckedi^.It ' i is 'h 'ere'^hat the 'care less ' 'mistake is-

'found, The mathematical 'check' only ver if ies the

I . . . . . . . . . . 3 3

mechanical manipulation of. the preceding informa­tion, Tho verbal chock verifies the correct interpre­tation of tho information,

A high Bchool teachor, Matilda S, Cohen, taught her students to read an algebra problem in phrases, F i r s t they road the problem through once silently, Then one member of the class read it.aloud, one phrase at a time, while others volunteered to translate the phrase into algebraic terms, The following problem il lustrates the method:

Twice a certain number increased by 5 equals 18 diminished by the unknown number, F ind the number,

The last sentence tells the class what the unknown is, so they begin by writing:

Let X equal the number,

Reader:

twice a certain number increased by 5 equals 18 diminished by the number

Secretary writes on _ board: 2x 2x plus 5 2x plus 5 equals 18 2x plus 5 equals 18

minus x

In this way the students get the whole meaning of the problem, and realize how it must be read to form an equation,

, „ , Instruction given deliberately as reading "lessons can help the student inclined to panic,

' ',, ,, Sometimes, after clearing vocabulary d i f f i ­cul t ies , to gain understanding the reader must resort to'grammatical .analysisi verb, subject, object and .connectives, , ,

"•'\V.i.:'.i';.V''.i \ - " 34

Recently Tho B , C, Teacher carr ied a human intoroQt story, An eighty-five year woman was searching for lessons to help a boy in scholastic diff i­culties, Sho remarked, 'To help a boy, you find what he lacks and begin, 1 This basic principle of remedial work can be translated into the field of mathematics, If that which is lacking is reading s k i l l , the mathematics teacher becomes the reading instruc­tor,

I I I I i I B I B L I O G R A P H Y

Fay, Leo C , , Reading in the High School, Depot of Class room Teachers, Amer ican Education Research Associat ion of the NEA, 1956,

Simpson, Elizabeth A, , Helping High School Students Read Better, Science Research Associates Inc., Chicago, 1954,,

Strang, Ruth and Bracken, Dorothy Kendall, Making Better Readers, D, C, Heath and Company, Boston, "1957,

Witty, Pau l , How to Improve Your Reading, Science Research Associates, Chicago, 1956,

What We Know About High School Reading - prepared by National Conference on Research in English, Champagne, Illinois, 1957-1958,

i,.

35

MOISErDOWNS' G E O M E T R Y

W, Abercrombie

Acute angle; An angle whose measure Is less than 90.

Alternate Interior Anglos; Given two linos L and L ' , cut by a t ransversal T at points P and Q, Lot A be a point of L and let B be a point of L 1 , Buch that A and B lie on opposite sides of T. Then M P Q a i u U P Q B aro alternate interior angles,

Altitude of a Triangle; .' ' A perpendicular segment from a vertex of tho triangle to the line containing the opposite side,

Angle'. If two rays have the same end point, but do not lie on the same line, then their union is an angle, The two rays are called its sides, and their common end point is cal led Its vertex.'

Angle Bisector of a Triangle; A 'aegment'is an anglebisector Of,a triangle if (1) it l ies in the ray which bisects an angle of the triangle and (2) its end points are the vertex of this angle

.and a point of the opposite side,

Area of C i r c l e ; The l imit of the areas of the inscribed regular polygons,

36. ,

A r e a of a Polygonal Region! Tho number assigned to it by postulate 19. Tho area of tho region R is denoted by aR, This is pronounced area of R,

Base of Isosceles Triangle! See isosceles triangle.

Base of a Trapezoid! Either of tho two paral le l sides.

Between;' B is between A and C if (1) A , B , and C are dif­ferent pointB of the same line and (2) AB •!• BC a A C .

Bisector of an Angle! If D is ij\ tho interior of Z B A C . and ^ B A D B ^ D A C , then AD bisects ^ B A C , and AD 1B called the bisec­tor of L B A G .

Bisector of a Segment! The mid-point of a segment is said to bisect the segment.

Chord! A chord of a c i rc le is a segment whose end points lie on the c i rc le .

Circles ' Let P be a point in a given plane, and let r be a positive number. The c i rc le with center P and radius r Is the set of a l l points of the plane whose distance from P is equal to r,

Circumference! The circumference of a c i rc le is the l imit of the perimeters of the inscribed regular polygons.

Col l inear : A set of points is collinear if there is a line which contains a l l the points of the set.

37

Complement; If the sum of the measures of two angles Is 90, then tho angles aro called complementary, and each of them Is called a complement.

Concentric; Two or more spheres or c i rc les with tho same center are called concentric.

Concurrent; Two or more lines are concurrent if there is a single point which lloa on a l l of them, Tho common point o is called tho point of concurrency,

Congruence! Angles are congruent if they have the same measure, Segments' are congruent if thoy have tho same length,

C i r c l e s with congruent radi i are called congruent,

Given a correspondence - ABC<—) D E F - between the vert ices of two triangles, If every pair of corresponding sides are congruent, and every pair of corresponding angles are congruent, then the correspondence ABC<—> D E F is called a congru­ence between the two triangleB,

Consecutive! Two sideB of a quadrilateral are consecutive if

' they have a common end-point, Two angles are consecutive if they have a side of the quadrilateral in common,'

Convex; A set A is called convex if for every two points P and Q of. the set, the entire segment P Q lies in A .

A polygon is convex if no two points of it lie on opposite sides of a line containing a side of the

" polygon, • •'

. ' 38' ' ' • u • ,' •.•v'.' ' • • •••'.' • \ i '

Co* ordinate; A correspondence of tho sort described in tho Ruler Postulate is called a co-ordinate system, The number corresponding to a given point is called the co-ordinate of tho point,

Coplanari A sot of points is coplanar if there is a plane which contains a l l points of the Bet,

Corresponding Angles: If two lines aro cut by a transversal , If ,6c and <£y are alternate Interior angles, and i f Ly and Iz are ver t ica l angles, then ,6c and are corresponding angles,

Diagonal! A diagonal of a quadrilateral Is a segment Joining two nonconsecutive vertices,

Diameter! A diameter of a c i rc le or sphere Is a chord con­taining the center,

Distance! The distance between two points is the number given by the Distance Postulate. If the points are P and Q, then the distance is denoted by PQ.

The distance between a line and an external point is the length of the perpendicular segment from the point to the line. The distance between a line and a point on the line is defined to be zero.

The distance between two para l le l lines is the dis­tance from any point of one to the other.

End point: See segment and ray.

Equi la teral : A triangle whose three sides are congruent is called equilateral.

- . : 39, T

Exter ior ; L e t ^ B A C bo an angle in a plane E , The exterior of <£BAC ia the aet of points of E that Ho neither on the angle nor in its interior,

Tho exterior of a c i rc le is t in set of a l l points of the plane whoso distance from the center is greater than the radius,

A point lies in the exterior of a triangle if it lies in the plane of the triangle but does not lie on the triangle or in the interior,

Exter ior Angle; If C is between A and D, then^BCD is an exterior angle of triangle A B C ,

Face; The two Bets described in the Space-Separation Postulate are called half-spaces, and tho given plane is called the face of each of them,

Geometric Mean; If a, b, c are positive numbers, and a b

b 8 c ' then b is called the geometric mean between a andc,

Great C i r c l e ; The intersection of a sphere with a plane through its center is called a great c i rc le of the sphere,

Half-plane; • Given a line L and a plane E containing it, the two sets described in the Plane Separation Postulate are called Half-Planes or Bides of L , and L is called the edge of each of them. If P lies in one of the half-planes and Q lies in the other, then we,say that P and Q lie on opposite sides of L ,

40

Half-Bpacoi See face.

Identity Congruence; Every segment or angle is congruent to itself.

Included; A side of a' triangle Is said to be Included by the angles whose vertices aro the end points of the segment.

An angle of a triangle is Baid to be included by the sides of the triangle which lie In the sides of the angle.

Inscribed! An angle Is inscribed in an arc if (1) the sides of the angle contain the end points of

the arc, and (2) the vertex of the angle Is a point, but not an

end point, of the arc.

A quadrilateral is inscribed in a c i rc le if the ver­tices of the quadrilateral lie on the c i rc le . If each side,of the quadrilateral is tangent to the c i r c l e , then the quadrilateral is c i rcumscr ibed about the

i , / ' .ApA;, Ih^er"c"ept!;'v;'</;/:v:i. 7. • If/a t ransversal intersects two lines L , L ' in points

) ; 'A" and B,'then,we say that L a n d L 1 intercept the;' • ' • segment A B oh the transversal . ; ' .// //'

'-,//• A n ahgle Intercepts an arc if • .//y/v ^ i ' ; ^ ^ lie on the angle, , yA:':yh':}::.y;y:)(2) ali*other points of the arc are in the Interior

of the angle, and (3) each' side of the atigle contains/an •end'point'.of

? t l 1 if 41

LlftWtf<WL*ijHlti>*>'l"***'r'* 'ft^"«ii>'i» ir r IT i i i - i ) - - - i in - i l l - . 1 — — „ , I W J M W W , u^,^^ t - w m n M n i w i <w«. m . — -

Interior*. Let ZBAC be an angle in a plane E , A point P lies i n the interior of ^ B A C if (1) P and B are on the same aide of the line A C , and (2) P and C are on the aamo aide of the line A B ,

A point Ilea in the interior of a triangle if it Ilea in the interior of each of tho angles of the triangle,

Tho interior of a c i rc le IB tho act of a l l pointa of the plane whose distance from the center is less than the radius,

Isosceles: A n isosceles trapezoid ia one having at least one peiir of opposite sides congruent,

A triangle,;,with two congruent aides is called i sos­celes, Theremaining Bide is the base, The two angles that Include the base are base angles, The angle opposite the base is the vertex angle,

Leg: .A"

See right triangle,

Length of a Segment:

/R-JS-KGD If A B <CD,

Linear. Pa i r :

If AB* and AD* are opposite rays, and AC* is any other ray, then ^ B A C and ACAD for a linear pair .

^Measure: ;; ;•.••••> . , ;L: - The number given by the Angle Measurement Postu-; 1 ; late i s v ca l l ed the measure of 2BAC, and is written

rn^BAC,

Median: The median of a trapezoid i s the segment joining

-••^••••'•r:the,'mid^points'-:of'the';two non-parallel sides,

^i^vAjtme^ian'iOf. a'triangle i s a segment whose end • ^^pbints are a vertex of the triangle and the mid-point

of the opposite side, Eve ry triangle has three medians,

42

Mid-point: , A point B is called tho mid-point of u segment AC if B 1B botwoon A and C and A B « B C ,

Obtuse Angle: An angle with measure greater than 90,

Opposite: Two sides of a quadrilateral are opposite if they do not intersect,

Two angles of a quadrilateral are opposite if thoy do not have a side of the quadrilateral in common,

If A is between B and C, then A B and AC are called opposite rays,

Ordered Pa i r : The x-co-ordinate of a point P is the co-ordinate of the foot of the perpendicular from P to the x-axis, The y«co-ordinate of P is the co-ordinate 1 ' of the foot of the perpendicular from P to the y-axis, If P has co-ordinates x and y, then we write P(x ,y) ,

Origin: . , . The point whore the x-axis intersects the y-axis ,

P a r a l l e l Lines: Two lines are paral le l if (1) they are coplanar and (2) they do' not intersect.

P a r a l l e l Planes: ' Two planes, or a plane and a line, are parallel1, if

•'they do' not intersect,

Pa ra l l e log ram: A paral lelogram is a quadrilateral in' which both pairs of opposite sides are para l le l .

43

Perimeter ;

Tho sum of tho longtlia of tho sides of a polygon.

Perpendicular; A*B* and AC* form a right angle, then they aro

called perpendicular, and wo write -

A B ^ A C 4

A line and a plane are perpendicular if they inter­sect and if every line lying in the plane and pass­ing through the point of intersection 1B perpendicu­lar to the given line. If the line L and tho plane E are perpendicular,, then',we write L J _ E or E ^ L , If P is their point of intersection, then we .jay that L J__ E at P ,

Perpendicular Bisector; In a given plane, the line which 1B perpendicular to a segment at its mid-point,

Point of Tangenqy; See tangent,

Polygon; Let P . , P . , . , , P be a sequence of n distinct l i n points in a plane, with n__ 3, Suppose that the n

segments P . P . , P . P - , . . , P„ ,P . P P . have 1 c co n-1 n n l

the following properties: > i <>

• ' (1) No two of the segments intersect except at their end points,

(2) No two segments with a common end point are ... r collinear, , .. .

Then the union of the n segments is called a polygon.

• Polygonal Region: , A/,plane figure formed by fitting together a finite

•;• 'number of triangular, regions.

11 's V 1 <' ' 4 4

Proport ional; Given two sequences a, b, c, . . . and p, q, r, . . . of positive numbers, If ,

* a b c < p q r

then the sequences a, b, c, . , , and p, q, r, , , , are called proportional,

Pyramid! Given a polygonal region R in a plane E , and a point V not in E . The pyramid with base R and vertex V is the union of segments VQ for which Q belongs to R, The altitude of the pyramid is the (perpendicular) distance from V to E ,

Quadrilateral,' Let A , B , C, and D be four points of the same plane, If no three of these points are colllnear, and the segments A B , B C , C D , and DA intersect only at their end points, then the union of these four segments is culled a quadrilateral. The four segments are called itB sides, and the points A , B , C, and D are called its ver t ices . The angles ZDAB Z A B C , ^ B C D , and^CDA are cal led its angles,

Radius; A radius of a c i rc le is a segment from the center to a point of the c i rc le . (S imi la r ly for spheres,)

Ray! Let A and B be points of a line L . The ray A B is the set which is the union of (1) the segment A B and (2) the set of a l l points C for which it is true that B is between A and C. The point A is called the end point of A B .

Rectangle; A paral le logram a l l of whose angles are right angles.

45'

Rogular Polygon! A polygon Is regu'.'.ar if (1) It is convex, (2) a l l of its sides aro congruent, and (3) al l of Its angles are congruent.

Remote Interior Angle1. Lk and £B of M B C f a r o called the remote Interior of the exterior angles Z B C D and / A C E ,

RhombuB!

A parallelogram a l l of whose sides are congruent,

Right Angle! If the angles in a l inear pair have tho same meas­ure, then each of them is called a right angle. A right angle is an angle whose measure is 90,

Right Triangle! A triangle one of whose angles is a right angle, The side opposite the right angle is called the hypotenuse, and the sides adjacent to the right angles are called the legs,

Scalene!

A triangle no two of whose sides are congruent.

Sector; Let AB be an arc of a c i rc le with center Q and radius r , The union of a l l segments QP^where P is and point of A B , i s called a sector, A B 1B called the arc of the sector, and r is called its radius.

Segment! Fo r any two points A and B , the segment A B is the set whose points are A and B , together with a l l points that are between A and B . The points A and B are called the end points of A B . -

46

l.H,<,

1

,-i/>'

Semic i rc le ; Lot C bo a c i rc le , and lot A and B be tho end points, of a diameter, A semicirc le A B is the union of A , B , the points of C that lie in a given half-plane with AB as edge, The points A and B aro the end points of tho semicircle,

S imi la r i ty ; Given a correspondence between two triangles, If corresponding angles are congruent, and corres­ponding sides are proportional, then the correspon­dence is called a s imi lar i ty , and the triangles are

' said tb be similar ,

Skew Lines; Two lines which are not coplanar and do not Inter­sect,

Slope; If'P.j = (Xj. , yj) and ? 2 = (x 2 , y 2 ) , and P j P g Is

nonvert ica l , then the Blope of P , P 9 is

vz - vi m a X 2 " X 1

The slope of a nonvertical line is the number which is the slope of every segment of the line,

Sphere! Let P be a point, and let r be a positive number, The sphere with center P and radius r is the set of a l l points'of. space whoso distance from P is equal to r,

Square: A rectangle a l l of whose sides arr, congruent.

Supplementary: If the sum of the measures of two angles is 180,

47

Supplementary (aontlnuod) then tho angle a are called Bupplomentaryi and each 1B called a supplement of tho other.

Tangent! """ A tangent to a c i rc le Is a line (in the same plane)

which Intersects tho c i r c l e In one and only one point. This point 1B called the point of tangency. , or point of contact. We say that the lino and c i rc le are tangent at this point.

Transversal! A transversal of two coplanar lines is a line which intersects them in two different points.

Trapezoid!

A quadrilateral which has two paral le l sides.

Trlanglei If A , B , and C are any three noncoll lnear points, then the union of the segments A B , A C , and B C is called a triangle, and Is denoted by A A B C . The points A , B , and C are called its vert ices , and

> the segments A B , A C , and B C are cal led Its sideB, Every triangle A B C determines three angles, namely^BAC, ^ A B C , a n d M C B . These are called the angles of triangle A B C .

Triangular Region! The union of a triangle and its interior.

Ver t i ca l Angles; . Two angles are v e r t i c a l angles if their sides form two pairs of opposite rays,

48

Hm^BjDUND BYJ^J^Jl^EL

Roger L . Desprez, Nanaimo Secondary School

'Intelligible expression postulates a language, and geometry has a symbolic language which allows us considerable scope for presenting proofs, which possess both originali ty and s tyle . ' So states Duncan MacAdam in his ,ar t icle ' A P lea for Style in the Writ ing of Geometry' ( B C A M T Journal, May 1964). He then proceeds to present a series of useful thoughts on the subject, Here are some further ideas on this thorny problem of 'proof and the methods by which assessments and evaluations might be made by the teacher,

In high school, geometry has traditionally been the course in which formal proofs have been used extensively, With the introduction of the axio­matic approach to algebra in our schools it becomes necessary to develop a deeper understanding of what the student is expected to produce as a 'proof for a problem, It is also necessary that we recognize the fact that the use of province-wide governmental exam­inations imposes a regimentation upon,the teacher whether he desires it or not, since a marking com­mittee may be expected to allow only certain degrees of .latitude, in what, they wi l l accep t . from a student.

• Any teacher who stubbornly ignores this fact, whether his method be superior or not to the accepted techni­ques, • is-running the r i sk that hie: students w i l l be penalized for his stubborness rather than for their ...

: lack of mathematical sk i l l s .

49

Thtu'e are some objections to the usual two-column traditional types of proofs, They require too much timo and space, but oven worso, they roquire that the student laboriously restate definitions, theorems and postulates committed to memory, Thus it becomes convenient In many cases for tho Btudent to memorize the proofs of theorems, making him grossly conscious of tha 'number of steps' re­quired for a proof, and in many casoB giving h im no understanding of the true nature of tho logical sequences involved.

We must recognize from the beginning that a ' formal ' proof'consists essentially of two distinct streams of thought - and the studont should be made fully aware of this. The first of these streams of thought,1 and to my mind the more important, is the logical sequence of steps starting with the hypotheses, running through additional concepts garnered from previously-gained knowledge, and ending with the desired conclusion - preferably with as direct and as economical a line of ideas as possible,

• This is the' essence, the flavor of mathema­t ica l proof. C a n the student think sequentially? Can he recognize that certain items may be grouped so that'they can give r ise to a following'thought? Can he recognize that'multiple t r a i n s o f concepts may ul t i ­mately converge to give h i m h i s desired conclusion? ThiB is the type of constructive thinking that we, as •teachers, are supposed to develop in the minds of future cit izens. ' '

'The second s t reamof thought is an essential oi'.e. from the mathematician's point of view - the 'justification'- of each of the step's inthe p r imary stream. 1 ' This justification-is !supposedto show that the student recognizes the fact that no mathematical concept should be used in a-proof unless it comes -

50

f rom tho existing framework that has boon rigorously developed from fundamental postulates, definitions, and previous proofs. The student also is made aware that his present proof is going to add to and expand this framework oven further. While no one denies the mathematical correctness of this second stream, nevertheless one must recognize that tho very act of transcribing it to paper involves difficulties, In the vast majority of instances in which a student uses a concept in his pr imary, or ' l o g i c , ' stream, he does so because he is already conscious of its existence, When he util izes tho idea that opposite sides of a paral lelogram are equal, it is most frequently because he has been exposed to this concept previously. It is quite true, of courBe, that sometimes he w i l l use a thought that he has accepted intuitively, or even a wild bolt-from-the-blue guess, but most ideas in his proof come from the deep, murky wel l of his memory, f i l led with true statements, half-truths, and d im recollections of text diagrams, past mental struggles, and teacher pronouncements.

It is with the thought of these two para l le l streams in 'mind that we should approach the problem of having students present proofs, and I would like to offer the following suggestion, During a. two-year session with Mathematics 91 Experimental classes, in which the students were required to give both formal and informal proofs of algebraic material , I t r ied a 'flow-chart ' method of presenting proofs which seemed to have some merit , This idea is certainly not new to some, and I cannot c la im to be its origina­tor - it has been outlined in several mathematics periodicals - but I present it here for those who may not have seen it before. Offered is an example of a problem as it might be done by a student. (See page 52.) Naturally this is not as easy an example as many,the beginning student w i l l encounter, since the intent here is to show the use of the technique for i l lustrat ing the thought-stream of the student.

51

One disadvantage of this technique is that aB tho problems become more lengthy and complex, there develops an increasing difficulty In arranging brackets and arrows with a minimum of crossing and confusion. Nevertheless, studenta become quite adept at presenting proofs this way so that the eye can follow them easily,

Another disadvantage is that the technique lends itself best to presenting tho pr imary of ' logic ' stream of tho proof rather than to showing the secon­dary or ' jus t i f ica t ion ' stream, I have circumvented this in my classes by having the students add the justification for each statement in parentheses be­neath each one, I emphasized to tho students that this was'best done after the logical sequence had been developed to their satisfaction, They may even use a different color for their justification statements if they wish,

This method of proof presentation does have several considerable advantages which outweigh the disadvantages, i ts 'chief merit being that it outlines the thought processes involved much more c lear ly than does the tradit ional method, This leads to class dis­cussion of blackboard work so that students can recognize and point out e r rors with greater ease, to clearing of misunderstandings with greater dispatch,

.. and to facilitation of marking student assignments by fjj the teacher. The proofs are more economical of

students' t ime, thus allowing for greater depth in the course.

Once the proof of a problem has been pre­sented by a 3tudent, the second major area of diffi­culty is encountered - that of an adequate system of marking and testing this type of exercise or examina­tion question. This is another aspect of geometry proofs that has needed serious attention for some

53

time. -I am constantly appalled when I BOO tho manner In which loca l and departmental examinations f l a ­grantly break one of the most fundamental proceptB , of examination procedure - the.t a test item should be designed as much aB possible to measure only one thing at a time, whether it be simple reca l l of factual knowledge, ability to organize thoughts and material , ability to express oneself, or any other, Naturally it is not always possible to follow this rule - this is the chief problem In the sotting and marking of eBBay or subjective questions - but it does not mean that the precept should be so completely ignored as it is , This sin is found running with abandon through the nether world of geometry examinations,

Since the two major partB of a geometry proof demand two different modes of thought and attack by the student, it is only logioal that they be examined differently, by the use of two separate test itemB, To teBt the abili ty of a student to sequential-ize or organize mathematical concepts, two examina­tion methods offer themselves, One method is to give the student such basic materials as the hypothe­sis, diagram and To Prove parts of a problem, followed by the necessary steps for the proof, but with the proof steps in a jumbled order - perhaps even including afew unnecessary steps for him to eliminate, This avoids the problem of reca l l , which can much better be measured by reca l l questions, and throws , di rect ly at the student one problem only - organization i In a logical manner, If this method is shunned by \ some because of the bui l t - in guessing factor, then j perhaps a modification could be used, Following the ] presentation of the necessary materials for the prob- f lem (hypothesis, diagram and conclusion), the student 1 should be asked to supply from his memory the steps j in the proof, in the correct order, but should not be I asked for justifications. This avoids guessing to a | greater extent but actually measures more than one j

thing - both reca l l and logical organization.

'But what of the "justification" part of a proof? 1 you w i l l probably ask. 'Is it not important? Doesn't it meri t examination?' Of course It is i m ­portant, but since it Involves a different type of thinking process on the part of tho studont, it do servos a different examination technique, Two possibil i t ies offer themselves, One method is to recognize that this is largely a r eca l l problem and measure it by means of well-designed completion or multiple-choice questions. Another method might appeal to those to whom the terms 'completion' and 'multiple-choice ' are abhorrent. Supply the student with the complete proof to'a problem, and perhaps certain obvious justif ica­tions, but leave blank those justifications which are to be measured,

The above suggestions on examination techni­ques do involve the difficulty that more time w i l l have to bo spent on the construction of test items, it is true, but they w i l l result in a more val id assessment of the student, They also offer the advantages that students w i l l have presented to them a more concise, less confusing idea of what is expected of them on an examination, and that the greater thought given to the item w i l l probably include the problem of how many marks to allot for questions, thus making this problem less thorny from the marker ' s point of view,

In short, in our examination technique let us measure the mathematics we set out to measure and let us avoid the problem, as stated by Duncan MacAdam, that enunciations of proofs and theorems too frequently are exercises in Engl ish syntax, clumsy redundancies detracting from the deductive nature of proof, present­ing Btumbling blocks to many of our students and con­tributing nothing to the solution.

55'

.... W M W D> twrtwrl. fcPWMH* JfllW.

B I B L I O G R A P H Y

MacAclam, Duncan, 1 , 1 A P i c a for Style in tho Writing of Geometry, 1 1 B C A M T Journal, Vo l , 3, No, 1, May 1964,

Nous, Harald M , , "A Method of Proof for High School Geometry',"1 The Mathematics Teacher, V o l , 55, No, 7, November 1962,

# # i # # # # # # # # # # # # # # # # # # tf # #

# # # # # # #: r # # # # # # # # # # # # # # #

, , .Perhaps the Freudian view is correct in assuming that man i s basical ly an i r ra t ional entity, and that logical behavior is possible only; under certain circumstances, of which the classroom may not be one.

- Dr, F . G, Robinson

56