sxperimen tal s of a cyclic

; (relate to

h a y ratios

geometrical

6. Specific Proposals for Stage C (Senior)

6. 1. INTERIM RECOMMENDATIONS FOR GRADES 11-13 One of the first undertakings of the Geometry Committee was its study

of proposals for the Grade 10 curriculum committee which at that time (January, 1966) was drawing up the revised mathematics course for the five-year Grade 10 program. Since the proposals made by our committee were submitted directly to the curriculum committee, we shall not repro- duce them here. However, since in the next three years revised programs for Grades 1 1, 12, and 13 will be promulgated, we shall include in this section some suggestions about the geometry content that might be con- sidered for these courses.

The following suggestions are given in outline only. Many of the topics are now included in the courses for Grades 11, 12, and 13.

6.1.1. Grade 11 Analytic geometry in 3 dimensions (3 weeks)

coordinates, equations of planes, simple inequalities, equations of lines, length of a segment of a line

Vectors in 3 dimensions (3 weeks) * components of vectors relative to a Cartesian system, dot pr&ct of

two vectors, angle between two vectors

Trigonometry (3 weeks) * circle definition of the sine and cosine functions, relation to right-

angled triangles, graphs of sine and cosine functions, and of sin 2x, cos (%)x , etc.

Transformations in 2D and 3D (1 or 2 weeks) translations, rotations, reflections in 3 dimensions

* group of motions in 2D and 3D, simple examples of symmetry groups

Quadratic functions and their graphs ( 3 weeks) * properties of parabolas and their equations

differentiation of y = ax2 -+ bx + c with applications to tangents and minimum values of the parabola

Geometry of circles ( 3 weeks) chord and angle properties of circles (see 5.3.2.) (This topic is intended to provide examples, in considerable scope and detail, of deduction in geometry. )

94 / GEOMETRY: KINDERGARTEN TO GRADE THIRTEEN

SUPPLEMENTARY

Polyhedrons and lattices polyhedrons, lattices, crystallography in 2 dimensions (see Coxeter, Introduction to Geometry (43 ) , Chapter 4. )

Synthetic geometry of circles * incircle, circumcircle, nine-point circle, Euler line, problems of Apol-

lQMus, and related topics

6.1.2 . Grade 12 Analytic geometry in 3 dimensions (4 weeks)

planes and lines, intersections, solutions of equations using matrices

Vectors (3 weeks) linear dependence, parallelism, coplanar vectors, orthogonal projec- tions, basis vectors

nometry (3 weeks) sine and cosine laws, addition theorems, amplitude and phase

* periodicity, polar coordinates

Transformations in 3 dimensions ( 2 weeks) rotation group about origin in 3D

* motions, linear transformations and matrices, affine properties

Geometry of conics (4 weeks) parabola, ellipse, hyperbola (synthetic definition and mainly analytic treatment) ; applications including space orbits

SUPPLEMENTARY TOPICS

Topology surfaces, deformations, equivalence of surfaces (intuitively) sphere, torus, and pretzel maps, and colouring problems loops, links, and knots and their classification

Convexity and linear programming * linear inequalities, vertices, feasible points, extreme points

6. 1. 3. Grade 13 (Analysis) Quadric surfaces (1 week)

description of equations, and graphical work for ellipsoid, hyperboloids, paraboloids, and cones

Reduction of conics and quadrics ( 2 weeks) simple cases on1 y of transformation to principal axes

6. 1 .4 .

geoi solu am

Vector mtr

Axiom. * Eu(

pro * exa * Par * SUB * net

elli (TI var Prc cot VIs

SUPPL

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matrices

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yperboloids,

PROPOSALS FOR STAGE C (SENIOR) / 95

6.1.4. Grade 13 (Algebra) Linear algebra (4 weeks)

geometrical language applied to 4 or more dimensions solution of linear systems

* applications to programming

Vector spaces (3 weeks) introduction to vector space of 4 or n dimensions, with applications

Axioms and geometries ( 3 weeks) * Euclid's axioms and postulates

projective axioms example of a finite geometry parallel axioms in various cases sum of angles of a triangle

* neutral or absolute geometry elliptic and hyperbolic geometry and their models (This unit is suggested because of the importance of the possibility of various geometries that are equally consistent, and because this topic provides a natural climax to the work on deduction in various earlier courses. A small amount of deductive work and of exercises is en- visaged. )

SUPPLEMENTARY TOPICS

Topology graphs, circuits, networks, combinatorial problems, boundary, dimen- sion, deformation of curves on surfaces &e ff

'6 Polyhedrons

semi-regular polyhedrons and vtallography in 3 dimensions

6 . 2. LONG-TERM RECOMMENDATIONS FOR SENIOR-LEVEL STUDENTS

The suggestions we shall make here will be based upon the interim recommendations for Grades 11-13 given above. If the system of grade levels remains in effect, we would suggest the gradual addition of a number of selected supplementary topics to these courses, as the teachers gain experience with the course material. We have provided some suggestions for supplementary topics along with the interim, recommendations; these sug- gestions are by no means complete, nor are they intended to be exclusive. - A further study, perhaps when more experience of the interim recornen-, dations themselves has become available, is needed in this connection.

c - , Following upon our main recommendation of an ungraded course of study, we shall give some suggestions as to the possible adaptation of-the interim recommendations. Generally speaking, the material recommended

I

for Grades 11, 12, and 13 in the interim will be suitable for the three stages (Cl, C2, and C3) of the Senior or C division. However, some modifications will be in order in view of the Stage B recommendations given in the preceding section. For example, in trigonometry at the C l level we would plan to have mainly graphical work connected with the phase and amplitude of functions related to the sine and cosine, and perhaps also the sine and cosine laws for triangle's. At the C2 level more emphasis would be expected on polar coordinates, and perhaps an intro- duction given to complex numbers in polar form.

6.2.1. S t a g e d Analytic geometry in 3 dimensions

coordinates, equations of planes, simple linear inequalities, equations of lines, length of a segment of a line

:

Vectors in 3 dimensions . ponents of vectors relative to a Cartesian system, dot product of vectors, angle between two vectors

Trigonometry the sine and cosine functions and their graphs, relation to right-angled triangles, direction cosines.

Transformations in 2D and 3D translations and dilatations of graphs of sine and cosine, translations, rotations, reflections in 3D, groups of motions in 2D and 3D, examples of symmetry groups

Quadratic functions and their graphs parabola, tangents to a parabola as an application of the differentiation ,of a quadratic function

SUPPLEMENTARY TOPICS

Polyhedrons polyhedrons, lattices, and crystallography in 2 dimensions (see Coxeter, Introduction to Geometry (43), Chapter 4 )

Topology surfaces, deformations, equivalence of surfaces (intuitively), sphere, torus, and pretzel maps and colouring problems, loops, links, braids, and knots and their classification

Synthetic geometry of circles * incircle, circumcircle, nine-point circle, Euler line, problems of Apol-

lonius, and related tonics

6.2. 2 . Analytic

plant

Vectors * linea

basis

Trigon0 * addii

phas with

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SUPPLE

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PROPOSALS FOR STAGE. C (SENIOR) / 97

6.2.2. Stage C 2 Analytic geometry in 3 dimensions

* planes and lines, intersections, solutions of equations using matrices

Vectors linear dependence, parallelism, coplanar vectors, orthogonal projection, basis vectors

Trigonometry * addition theorems, sine and cosine laws, periodicity, amplitude and

phase, polar coordinates-graphical and analytical exercises, relation with complex numbers

Transformations rotation group about origin in 3D; motions, linear transformations and matrices in 2D and 3D, further transformations in 2D: inversion and reciprocation

Geometry o f conics parabola, ellipse, hyperbola (synthetic definition and mainly analytic treatment) with applications

SUPPLEMENTARY TOPICS

Topology graphs, circuits, networks, combinatorial problems

* boundary dimension deformation of curves on surfaces # ,

Lr

Convexity convexity and linear programming linear inequalities, vertices, feasible points, extreme points

Inversive geometry inversion with respect to a circle, and related theorems

6.2.3. Stage C 3 Axioms and geometries

Euclid's axioms and postulates * example of a finite geometry

projective axioms parallel axioms in various cases

* sum of angles of a triangle neutral or absolute geometry

* elliptic and hyperbolic geometry and their models (This work on non-Euclidean geometry provides a natural climax to

the work on deduction in various earlier courses. A modest amount of deductive work and exercises is intended to be incorporated with descriptive material. )

Linear algebra geometrical language applied to 4 or more dimensions

* solution of linear systems with applications to programming Vector spaces

introduction, to vector space of 4 or n dimensions, with applications linear dependence

Quadric surfaces description and graphical work for ellipsoid, hyperboloids, paraboloids, and cones

Reduction of conies and quadrics simple cases only of transformation to principal axes, with applications such as to statistics ?k

SUPPLEMENTARY TOPICS

Topology * 'building block" description of simplexes and complexes

calculation of boundaries, boundary of a boundary, continuity, vector fields, fixed points

Polyhedrons * semi-regular polyhedrons and crystallography in 3 dimensions

Synthetic affine and projective geometry * theorems of Ceva, Menelaus, Pappus, and Desargues

The chief need in connection with an ungraded program, we believe, is for advanced topics that will be reached by only a few selected students. The following partial list has purposely been made short, to focus attention upon the most significant advanced topics. For enrichment purposes the advanced student will need to read more widely and should no~t be restricted to these topics.

1. Calculus. An early acquaintance with intuitive differential calculus is highly advantageous to the advanced student. The Cl stage is by no means too early for this purpose. We include this topic among those connected with geometry because we wish to emphasize the graphical, qualitative, and intuitive aspects of calculus, with the aim of fostering the student's ability to visualize.

For students of moderate ability, at least a contact with the basic notions

in calculi tion with

Becaui reference widely a'

2. Qu, Here we prescribe and the2 this unit. the cone gives an and of tl

3. Gel tions wit mined b] formula, elements

4. Pro axe: axi cross-rati tic, and 1

It she into thes students,

6. 4. C 6.4.1.

From features offered t( as curre] that give the O r g ~ reprinted "~uclid deficienc Dieudoni a way a; the scha of "Eucl. list some hold a 1 example,

amount of rated with

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pplications

ity, vector

as

believe, is I students. 3 attention rposes the restricted

salculus is no means connected ualitative, students

* ' . ic notions

PROPOSALS FOR STAGE C (SENIOR) / 99

in calculus is suggested, for the Cl or interim Grade 11 course, in connec- tion with quadratic functions and parabolas.

Because calculus is now included in the Grade 13 analysis course, reference and source materials for the initial portion of the subject are widely available.

2. Quadric surfaces and transformations (in 3 or more dimensions'). Here we include material for the advanced student who has taken all the prescribed stages through C3. The classification of conics and quadrics and their transformation to principal axis (standard) form is the basis of this unit. Matrix notation for quadratic forms should be employed, and the concepts of eigenvalue and eigenvector introduced and applied. This gives an opportunity for the study of matrices as linear transformations, and of the invariant subspaces associated with each eigenvalue.

3. Geometry of complex numbers. This comprises study of the opera- tions with complex numbers in the complex plane, of simple loci deter- mined by equations in complex numbers, de Moivre's theorem and Euler's formula, linear fractional transformations, matrices, introduction to elements of complex variables.

4. Projective and non-Euclidean geometry. The main topics suggested are: axioms of projective geometry, perspectivities and projectivities, cross-ratio and invariance, together with further deductions in affine, ellip- tic, and hyperbolic geometry.

It should be noted that exceptionally able students will be able to go into these and the other topics to a much greater depth than average students, and they should be given every opportunity to do so.

6.4. 1 . Synthetic geometry and deduction From the long-term historical viewpoint, one of the most significant

features of the recent changes in school mathematics has been the challenge offered to the teaching of "Euclid," that is, to synthetic Euclidean geometry as currently taught in schools. The most dramatic of these challenges is that given by Professor Jean Dieudonn6 at the Royaumont Conference of the Organization for Economic Cooperation and Development in 1959, reprinted in New Thinking in School Mathematics ( 8 5 ) under the title "Euclid must go." While this committee sees some grounds for certain deficiencies pointed out by Dieudonne, we feel that his case was overstated. Dieudonn6 adopted the point of view of a research mathematician in such a way as to defy the necessary pedagogical principles that must govern the school program. A.kbough the limitations of. this and similar criticisms of "Euclid" are now more generally understood, we think it worthwhile to list some reasons why synthetic geometry as blown to Euclid should still hold a prominent place in the secondary school curriculum. (See, for example, Budden and Wormell (691, p. 23.)

100 / GEOMETRY; KINDERGARTEN TO GRADE THIRTEEN

1. Basic nature of spatial relations. The primitive relationships of spatial situations are the foundation of physical and mathematical thought. Spatial intuition and perception are antecedent to numerical estimation or calcula- tion in the experience of every individual. Synthetic geometry develops and refines spatial intuition.

2. Ease and scope of deduction. No field of mathematics surpasses synthetic geometry as a field of application of elementary deduction. The concepts used in geometry are basic and of frequent application in every- day life; therefore a great variety of theorems, deductions, and applications are open to the student. The very richness of Euclid was one reason why, for so many centuries, no great need was felt for extensions of mathematics into new domains.

3. Applications. We need scarcely point out that in every branch of science, and in nearly every branch of knowledge, references in the language of Euclidean geometry are frequently necessary. Every scientifi- cally educated person uses the Euclidean concepts without being conscious 9f)the long process required to learn them. Such training increases our intuitive ability to manipulate and control our environment.

6 . 4 . 2 . Relationship of work on vectors and transformations to synthetic and analytic geometry

The introduction of new topics such as vectors and matrices raises the question as to how closely they should be related to the existing material on coordinate or synthetic geometry. We might at the same time consider the more general question of the relations between different methods or outlooks in mathematics, and of the degree of flexibility or interchange- ability with which they should be used.

To give a relevant example, we may recall that Descartes is famous today for his method of analytic geometry, which was based on setting up a relationship between two branches of mathematics, geometry and algebra. Generally speaking, mathematics flourishes most often through just such an interrelating process. The learning of mathematics, and the learning of the spirit and nature of mathematics, also rely heavily upon the employ- ment of previously developed ideas in conjunction with one another. We would therefore recommend that sharp separations or distinctions between two topics should be avoided wherever possible. If an analytic solution to a given problem is neater or more concise, we should not for that reason exclude from consideration a synthetic solution supplied by a student. Only by experience and judgment does one learn which of several methods works best in given circumstances. We would regard various methods, such as the synthetic, analytic, or vectorial, as tools in the carpenter's chest or weapons in the soldier's armoury.

In the light of later discovery it is often possible to see that two methods are inseparable, neither being able to succeed without the other. For instance, the notion of transformation, which is new to the school program,

underlies transform However, related o pence , i

We th cross-refe of solutio the compi

6.4. 3. The ui

intended sets of ge previous 12 or 13 by a give many stu some extt try serve suggest c( should nc

In con discovery velopmen d eveloprn contempc

We w< history o parisonn o should bi individual Part 2, C 86; 94)

6.4. 4. While

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hips of spatial ought. Spatial 3n or calcula- develops and

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PROPOSALS FOR STAGE C (SENIOR) / 3.01

underlies the basic geometrical notion of congruence, while the special transformations known as translations can be represented by vectors. However, the explicit concept of a transformation did not precede these related concepts, but was derived from concepts of superposition, con- gruence, and motion based on synthetic geometry.

We therefore urge that the various topics be taught with continual cross-references and comparisons. Under ordinary circumstances, methods of solution should not be restricted. Although sometimes the-consuming, the comparison of solutions by two or more different techniques is valuable.

1 6.4. 3, Non-Euclidean geometry The unit dealing with geometry on a sphere, included in Stage B, is

intended partly as a preparation for the later introduction of alternative sets of geometric axioms. Having studied geometry "in the small" at several previous stages, the advanced student reading at stage C2 or C3 (Grades 12 or 13) should be able to handle the notion of a geometry as defined by a given list of axioms. I t is difficult to motivate this type of work for many students, because the axioms appear self-evident and therefore to some extent pointless. Thus the comparison with other schemes of geome- try serves a dual purpose: a) to motivate the Euclidean axioms and suggest comparison of the different geometries, and b ) to show that axioms should not be regarded as absolute truths.

In connection with the latter purpose, we should emphasize that the discovery of the non-Euclidean geometries was the most significant de- velopment in the whole field of geometry since Euclid. Neglect of this development would be incompatible with the position of geometey in contemporary liberal education.

We would like to see attention concentrated on the foundations and history of the developments in non-Euclidean geometry, and upon com- parison of different systems of geometry. A certain amount of deduction should be included but extended work in this direction should be for individuals to undertake on their own enterprise. (32, Chapter 10; 78, Part 2, Chapters 6, 7, 8, and Part 3; 46, Chapters 3, 4; 43, Part 111; 86; 94)

6 . 4 . 4 . Topology While the great contribution of the early nineteenth century to geometry

was the discovery of non-Euclidean geometry, the contribution of the twentieth century has been the development of topology. While the depths of topology are sophisticated and subtle, the subject contains elementary topics well suited for inclusion as secondary school material. Several such topics have been proposed as supplementary topics for the interim Grade 11-13 course, and these topics are intended to be part of the ungraded curriculum proposed for the long term. (108, Chapter 8; 43, Chapter 21; 130, Chapter 5; 103, Book 2, Chapter 1)

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The work of the present K-13 Geometry Committee in planning for long-term curricular development and revision is not an isolated pheno- menon. The Development Division of The Ontario Institute for Studies in Education has some sixteen or more committees all concerned with curri- culum development. The ferment is not limited to Ontario, as witness the International Curriculum Conferences held (in Toronto) in November 1964 and February 1966.

background for our recommendations a brief survey of current about curriculum development and educational change is worth

First, curriculum can he seen to involve both content and process. On the one hand, we think of curriculum as a program of studies designed to "provide a learning person with a coherent sequence of impressions, exercises and cognitive subjects . . . "I1 On the other, it can be viewed as

place. The current emphasis in curriculum development seems to be on the idea of process. Recommendations on curriculum development, then, must necessarily relate either to content or ways of experiencing.

Second, any curriculum will reflect the aims of education held in a particular society. It is generally agreed that in democratic societies, one of the most important purposes of education is to help the individual to develop his talents and personality to the full, in a responsible way, within the developing framework of society.

In the culminating chapter of the Report of the (then) Ontario Curricu- lum Institute Committee on the Scope and Organization of the Curriculum, (1 1; p. 42), the following views are stated as part of "A Cumculum Credo" : The process of intellectual growth is essentially the development of ways of thinking. (1 1, p. 42) Intellectual growth might be said to occur as the learner develops resourceful- ness, the power to observe carefully, to ask critical and creative questions, and to formulate reasonable conclusions without regarding them as ultimate truth. (11, P. 43) Growth in independence is fostered when the learning situation brings experi- Thus, 3 enca in gathering, recording, presenting, and applying information, as a library curricul 11. R. Ulich, "Social and Individual Aspects of the Curriculum" (82, p. 1 ) . feedback

banning for ated pheno1- ir Studies in 1 with curri- witness the November

of current ige is worth

process. On designed to

impressions, ie viewed as vidual takes is to be on anent, then, % z held in a eties, one of idividual to way, within

do Currim- Curriculum, Curriculum

of ways of

resourceful- lestions, and imate truth.

ings experi- as a library

RECOMMENDATIONS AND THEIR IMPLEMENTATION / 103

research project related to a real problem identified by a class and researched by one of its members. Growth in independence and interdependence are fostered when the learning situation brings experience in interacting with people, including experience in expressing, receiving, sharing, questioning, and testing ideas, as in a classroom symposium where divergent thinking is encour- aged but thoughts are carefully examined. ( 1 1, p. 44) Young learners are naturally inquisitive and it is essential to allow the process of inquiry to generate discovery. The discovery of basic relationships lies at the heart of learning. Teaching as "telling" may fail to develop the potential of the inquiring mind. . . . (1 1, p. 44)

Essentially, these quotations argue that the curriculum must take into significant account the individuality of each student. This means that the curriculum must be fitted to the learner and not vice versa.

Third, in any curricu!or development, or reform, of more than a minor nature, the whole educational system is involved. Co-operative action of society, its institutional and instructional agencies, and teachers is essential. This is urgently required because of the economic and organizational com- plexity of society today.

Thus Derek Morrell of the Schools Council, England, says:

What is emerging, and finding practical expression through the establishment of the Schools Council, is the view that the instructional freedom of the teachers needs the positive support of societal action, with the institution-the school~occupying an intermediate position. (82, p. 27) and all must share in creating the necessary basis of knowledge, and in developing new applications of knowledge. (82, p. 27)

In Ontario, the same trend is evident. In discussing "A New M a e l for Educational Research and Development", Dr. R, W. B. Jackson gives his view that the whole educative process should be continuously evaluated and improved. (81, P. 59) and that The major problem in effecting change is undoubtedly that of communication. (81, P. 59)

Goodlad, one of the most outstanding workers in the field of curriculum development in the United States, .underlines Dr. Jackson's first remark when he says The time is come to rise above parochial considerations in the creation of cooperative approaches to curriculum study and improvement which bring together research, facilities and techniques for field testing, and machinery for implementation across the whole length and breadth of the curriculum. (17, P. 81, Thus, if we want to minimize radical change, and achieve more smooth curriculum development than in the past, procedures for commmlcation, feedback, and self-modification must b e built into contemplated changes.

7.2. IMPLEMENTING CHANGE

The previous discussion of change was concerned with assumptions, definitions, and principles related to the achievement of educational change. Concurrently with this thinking, studies have been conducted which seek to identify what constitutes improvement as well as ways and means of achieving desired changes.

Halpin has done considerable research on organizational climate in schools. He views the task of changing this climate as "one tiny example of the fundamental issue: permanence and change." He and Crofts devised the Organizational Climate Description Questionnaire (OCDQ) and, on the basis of further analysis, invented a typology of "climates," with the "open" climate at one end of a "continuum" and the "closed" climate at the other.

The "open" and "closed" climates are characterized by Halpin as follows :

Open Climate describes an energetic, lively organization which is moving its goals, and which provides satisfaction for the group members' social

needs. , . . The Closed Climate is characterized by a high degree of apathy on the part

of all members of the organi~ation.~~ On the basis of relevant OCDQ studies, and separate work on "change-

induction'? by Miles, Brown, and Maslow, he distinguishes between reme- dial and preventative courses of action, and makes these statements: It is time for us to recognize that successful efforts at planned change must take as a primary target the improvement of organization health-the school system's ability not only to function effectively, but to develop and grow into a more fully-functioning system.l3 I am trying to puncture a myth-the myth that every man does, indeed, want to lead and innovate. Most administrators will proclaim this as their purpose. Yet research on the behavior of administrators repeatedly gives the lie to this proclamation. Consequently, I suggest that we differentiate between those men who are more disposed toward being administrators-in Lipham's sense-and those who are more disposed toward making innovations. The one group will emphasize the stability of the organization; the other will emphasize change.l4

Halpin is also highly critical of the quality, personal attitudes, and personal value-patterns of many prospective teachers and teacher educators, and sees these as sources from which "closed" climates tend to be gen- erated. He suggests as one remedy the exaction of tighter control on the selection and promotion of teachers.

12. Andrew W. Halpin, "Escape from Leadership" (48, p. 57). 13. Halpin, "Change and Organizational Climate," Ontario Journal of Educational

Research, V I I I (Spring, 19661, p. 234, quoting Matthew 3. Miles, "Planned Change and Organizational Health: Figure and Ground" in Change Processes in the Public Schools (Eugene, Oregon: The Centre for the Advanced Study of Educational Administration, University of Oregon, 19651, p. 11.

14. Halpin, "Escape from Leadership" (48, p, 63).

Halpin t (a) the or, between ad. o f teachers

Another of The On tion of cha literature o begin to te; good exan factors ma

a) Staff unpi

b ) Free

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Educational ;, "Planned e Processes id Study of

RECOMMENDATIONS AND THEIR IMPLEMENTATION / 105

Halpin thus identifies as critical in the process of educational change (a) the organizational climate of the school, (b) the need to differentiate between administrators and innovators, (c) the quality, attitudes, and values of teachers and teacher educators.

Another committee of the curriculum and development studies section of The Ontario Institute for Studies in Education is studying implementa- tion of change in the classroom. Its main work so far has been to study the literature on change, develop a theoretical model based on this study, and begin to test the validity of the model by visiting schools judged to provide good examples of educational change. Out of these visits, two critical factors making for readiness for change were tentatively identified as

a) Staff dissatisfaction, combined with optimism that things can be improved

b ) Freedom for the teacher to make innovations in the school itself.

7 . 3. SPECIFIC RECOMMENDATIONS FOR CHANGES AND THEIR IMPLEMENTATION

7.3. 1. Coordination o f the work of different curriculum- development groups

As indicated earlier, the increasing uGty of higher mathematics in the twentieth century underlines the need to treat school mathematics as a unity. At the same time, there is great need to unify all the curricular experiences which together comprise the education of the student. Of par- ticular importance are correlations among mathematics, science, and social studies. /'<* We therefore recommend that R l measures be taken centrally by The Ontario Institute for Studies in

Education to ensure continuing and effective communication and coordination among the various curriculum-development committees.

We are in general agreement with the recommendations of the two Mathematics Committees which met in the summer of 1965. In particular, we would endorse recommendation 8 of the K-6 Committee on p. 29 of its report and recommend that

7.3.2. Basic general recommendations in geometry . . . .

I n line with the general discussion and theses advanced earlier in this report we recommend that ' , . . . .

R3 planned &@ences of experiences with geometric ideas andrelations, . . . . based on the outlines given in this report, be developed as an integral , .

. ..

as close a liaison as possible be established immediately between the work on the geometry curriculum in Ontario and similar develop- ments elsewhere, particularly in Britain.

part o f the mathematics curriculum throughout the entire schooling of the child

and that R4 the totality o f these geometric experiences be made up of a common

care to be undertaken by all students, together with supplementary experiences provided for students with more mathematical aptitude.

The precise nature of these experiences will, of course, be continually subject to modification, but, in our view, the general content and develop- ment should approximate the topical and sequential outlines presented in Sections 4.2.1.-4.2.3., 5.1.1.-5.1.4., and 6.1.-6.3.

We further recommend that

R5 flexible and varied approaches to- learning and teaching geometry be employed with emphasis on active individual learning experiences leading to increasingly independent study.

Tafacilitate the successful working of such approaches to learning and, teaching geometry we recommend that R 6 a movement towards an ungraded system of instruction be made by

means of homogeneous grouping in mathematics and more flexibility in timetabling

and that R7 provision be made for the practically and non-verbally oriented

student as well as for the academically and verbally oriented. It is hoped that such provision has been incorporated in our suggested

curricular outlines, but the problem is significant and should be re- examined continually.

We further recommend that

RS promotion in mathematics be determined by the students record in mathematics

and that R9 a minimum level of mathematical competence with respect to ideas

and techniques corresponding to the completion of Stage B be re- quired of all students for graduation from second9 school.

The reasons for these recommendations are outlined in Section 3.1.

7 . 3 . 3 . Experimentation and related curricular development The changes proposed in this report are far-reaching, both for the con-

tent and approach to school geometry, and for the relation of geometry to the rest of the curriculum.

In order that the changes be implemented in an orderly manner, we strongly recommend that

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R 10 a study group be formed to continue the development of the geometry curriculum begun by this committee

and that R 11 this group begin its work by selecting some of the specific proposals

and testing them at a pilot level to determine their suitability and optimum place in the sequences of experiences comprising the geo- metry curriculum.

We see this process of experimentation and evaluation as a continuing one, and therefore recommend that, as soon as possible,

R 12 the group above become a full-time working group, by means of full- time appointments and temporary appointments for periods with a normal minimum of one year.

We propose the scheme outlined below as a plan for initial experimen- tation at Stage B and Stage C. The rationale behind the suggestions is that since material which can be used for experimental purposes at Stage I3 (Grades 7-10) is readily available in suitable form (e.g. School Mathe- matics Project Material), teachers can prepare immediately for experimen- tation. On the other hand, suitable material for Stage C (Grades 11-13) may have to be written. In addition, it is logical to phase experimentation so that knowledge of Stage B learning is available in experimenting at Stage C. Initially, therefore, it is envisaged that experimentation at Stage C will be carried out by members of the geometry development group men- tioned above, without the involvement of classroom teachers.

Spring-Summer 1967

Stage B (Gr. 7-10) Stage C (Gr, 11-13)

1967-68 Stage B (Gr. 7-10) Stage C (Gr. 11-13)

Summer 1968 Stage B (Gr. 7-10)

Stage C (Gr. 1 1-1 3 )

Thereafter

Preparation and First Testing of elected Materia Is Some teachers as well as students involved Testing with students only

Limited Experimentation in Schools Testing and further revisions Initial pilot testing in schools

Further Revisions and Testing Evaluation and possible beginning of a new cycle of experimentation with different content Revisions. Involvement of teachers to prepare for extension of testing

Continued Experimentations and Revision, with involvement of larger numbers o f teachers.

We further recommend that

R13 close liaison and communication be maintained between this group,

The Ontariolmfttute fm Studies in Education, the Ontario Depart- ment of Education (curriculum, supervision, teacher education), the Ontario Teachers' Federation, and other relevant bodies so that appropriate basic research, field testing, evaluation, and implementa- tion may be adequately coordinated.

7.3.4. Communication with educators and the public Lines of communication must be set up so that the dissemination of the

proposed changes, the reasons for them, and the results of experimentation and modification reach all those concerned with change in the educational system (primarily this means university professors, teacher educators, administrators, school trustees, supervisors and consultants, inspectors, principals, teachers, parents, and educational publishers) at the optimum time.

Since the changes proposed are major, we recommend that

Rl+,varied measures be employed to ensure adequate continuing com- ^~munication of current and proposed curricular developments to all

concerned with educational change. In particular, we recommend that

R 15 demonstrations of samples of developments in school geometry be provided in various forms as an on-going part of the process of communication

and that R16 a newsletter giving up-to-date news of current developments and

curriculum materials be established, and circulated regularly to all concerned.

A newsletter may be relatively brief and to the point, but should give sources from which fuller details might readily be obtained. Demonstra- tions should include successful teaching, and the use of TV and film to save duplication of effort. The need for personal contact, with its accom- panying advantages of discussion and questioning on the spot, means that the pattern should include local as well as central demonstrations.

7.3. 5. Teacher Education Apart from keeping educators informed of curricular developments in

geometry, there remains the crucial specific need to ensure that mathematics teachers become sufficiently involved in and identified with these develop- ments to understand them and contribute significantly to their improvement and implementation.

Experience shows that teachers cannot easily make or test innovations without the co-operative support of principals, administrators, inspectors, and school trustees.

We therefore recommend that

R 1 7 .

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R 18

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R 19

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R 17 special attention be given to acquainting principals, school adminis- trators, inspectors, and trustees with the reasons for the proposed changes.

and that

R18 seminars aimed specifically at informing these groups be arranged in 1967, and annually thereafter.

In order that mathematics teachers may play the crucial role which is necessarily theirs in further curricular reform in geometry, we strongly recommend that

R 19 every possible positive measure be taken to ensure continuing oppor- tunities for practising mathematics teachers at all levels to learn about, observe, discuss, and experiment with new geometry materials and approaches.

We envisage re-orientation and re-education of teachers as occurring in two major phases. In the first, the main emphasis will be on in-service education, initially of key personnel such as teacher educators, inspectors, consultants, and principals, who have responsibilities for teaching and advising others. In the second phase, the emphasis will fall more on pre- service education in the teachereducation institutions. The first phase will merge into the second, however, and in-service education will continue in the second phase.

In particular, we recommend that

R20 a seminar of at least one week in duration be arranged in o f 1967 for personnel responsible for the further edu&tion o f teachers

R21 further seminars o f this kind become an accepted practice to ensure continuing dissemination and critical evaluation of curricular and methodological innovations

R22 regional seminars under the guidance o f key personnel, and those already involved in the proposed experimentation, be set up as required after the summer of 1967 so that gradual implementation of the major proposals of this report can be facilitated

R23 "travelling kits", comprising suitable materials, be made up for use with such seminars

R24 teacher educators incorporate into their pre-service offefings materials and activities relating to the proposed changes o f content and ap- proach

R25 every encouragement be given to first-rate teachers to form voluntary '

mathematics clubs for both elementary and secondary school students -