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S.88-86

SIMON FRASER UNIVERSITY

MEMORANDUM

TO:

Senate ?

FROM:

J.W.G. Ivany

Chair, SCAP

SUBJECT: Graduate Curriculum Revision

DATE: ?

Nov. 17, 1988

- Dept. of Math & Statistics

Action undertaken by the Senate Committee on Academic Planning/Senate

Graduate Studies Committee gives rise to the following motion:

Motion:

that Senate approve and recommend approval to the

Board of Governors as set

forth

in S.88-86 the following

curriculum revisions:

New courses: ?

MATH 603-4 Foundations of

Mathematics

MATH 604-4 Geometry

MATH

605-4

Mathematical Modeling

Deletion of: ?

MATH

601-5

Trends and Developments

in Mathematics I

MATH

602-5

Trends and Developments

Sc. in SecondarU School Mathematics Education

This is a proposal for introducing an M. Sc. degree in Mathematics

Education into the Faculty of Education. Discussions have taken place with

Harvey Gerber of the Mathematics Department and Bernice Kastner of the

Faculty of Education. Sandy Dawson has also commented on the structure

and content of the program. The Chair of the Mathematics and Statistics

department is very supportive.

Rationale

In 1975, the SFU Faculty of Education offered a graduate program in

education with'emphasis on secondary school mathematics." It was

intended to be an on-campus program with students meeting twice a week

for 5 hours a night. Students would take two courses per semester I or a

total of 7 courses, and would also undertake a special project. The details

are attached as Appendix A. A number of candidates applied, but there were

too few qualified applicants, and the program was not initiated.

As part of the 1975 program, the Mathematics department created

two special courses, Mathematics 601 and Mathematics 602. Each was to

deal with recent developments in mathematics. They are still listed in the

Calendar under Mathematics Graduate courses, although

all

references to

the Mathematics graduate program have been deleted from the Faculty of

Education section of the Calendar.

At the present time, there appears to be renewed interest among

secondary mathematics teachers In such a program. Secondary mathematics

teacflers are racing a reviseci curricuium that Will oe rung in piece in 1990.

This curriculum will contain a renewed emphasis on geometry, a new strand

on probability and statistics, and a new unit on calculus. The graduate

program will be designed to help teachers develop insights into the nature

of mathematics, its place in the school curriculum, research on how

secondary students learn mathematics, and current ideas on how best to

teach the subject. This is an opportune time for teachers to examine

curriculum changes and upgrade their qualifications.

The theme of the graduate program stresses the human aspects of

mathematics." Emphasis will be placed on the role of mathematics In

society and the natural development of mathematics as a growing, changing,

entity. Developments in the school mathematics curriculum, and in.

pedagogy, will be related to historical, cultural, and psychological forces

operating within society. The goal is to produce teachers who have a broad

understanding of mathematics and mathematics education, and who will be

qualified to deal with rapid curriculum change in the next several decades.

This is different from a traditional program in which teachers progress

through a intensive, but narrow, specialization In mathematical topics,

followed by a brief exposure to a collection of recommended teaching

procedures:

Course Structure of the program

Students will require at least 23 credits and a thesis for a M. Sc.

(Education). Equal credits will be taken in the mathematics department and

in the faculty of education. Three new courses will be developed by the

mathematics department, and two new courses will be developed in the

faculty of education. All courses are four credits. Outlines for the five

courses are contained in Appendix B.

Mathematics Department

Foundations of Mathematics (new course)

"Crises in mathematics, their historical and philosophical background

and their resolution." This is- a non-technical course in which all necessary

mathematics would be taught as part of the course. The intent is to show

mathematics in the making rather than as a finished product.

Geometry (new course)

Euclidean and non-Euclidean geometries. Klein's Erlanger programme."

A look at the development of geometry to the present time. Emphasis on

how geometry was interpreted at various times in history, including the

influence of Euclidean geometry on philosophy, and the crisis precipitated

by the discovery of non-Euclidean geometry. Modern geometrical treatment

including transformations of the plane.

Mathematical Modeling (new course)

Introduction to mathematical modeling using algebraic and geometric

techniques, along with techniques using Calculus." Designed to give

students experience in creating and fitting mathematical models to real

world problems. Based on recommendations of the MAAs committee on the

Undergraduate Program In Mathematics. Includes modeling using the

computer program Minitab.

All three mathematics courses should have appeal to other

students in the Mathematics department, hence the offerings would not be

entirely dependent on enrolment in the proposed graduate program.

Education

Foundations of Mathematics Education (new course)

An examination of historical, cultural, and psychological forces

shaping the secondary school mathematics curriculum. Current

developments in mathematics curriculum and in mathematics education

research." The emphasis will be on the historical underpinning of the

curriculum and the cyclical

nature of reform in mathematics education.

The course will have a structure similar to the one on the foundations of

mathematics and will focus on critical periods in the development of the

school mathematics curriculum.

Teaching and Learning Mathematics (new course)

"The theory and practice of mathematics teaching at the secondary

level. Emphasis on the nature of the learner and the function of the

teacher." Implications for instruction of the ideas of various mathematics

educators and schools of thought, for example, Dienes, Gattegno, Skemp, and

the constructivist school. Emphasis on teaching geometry reflecting the

content of the Geometry course in mathematics. Emphasis on applications

and problem solving reflecting the content of the Mathematical Modeling

course.

One elective

- possibly Education 864-3: Research Methods in Education

Education 816-5: Developing Educational Programs

Educatlofl 623-5: &I in an InOlvicluffi Teaching Specialty

Education 651-5: Computer-Based Learning

Timetable for Implementation

The program is designed for the students as a cohort. The entire

program is a cooperative venture between Mathematics and Education, and

each course Is designed to complement the other courses. The initial intake

will be In the Fall 011989, with a second intake planned for two years after

that time. This schedule may change, subject to demand.

fdIl/89 ?

- Mathematics: Foundations of Mathematics

Spring/90 - Education:

Foundations of Mathematics Education

Summer/90 - Mathematics: Geometry

Fall/90 ?

- Education: ?

Teaching and Learning Mathematics

Spring/91 - Mathematics: Mathematical Modeling

Summer/91 - Education: ?

Elective

One year to complete a thesis. Members of the Faculty of Education

and the Mathematics Department will shore responsibility for supervision of

students theses. Students will be encouraged to undertake work in whicli'

they can apply the ideas from the courses to curriculum

development.

FTE and other implications

Two new courses for Education.

One

FTE per year (O'Shea or Dawson)

No additional Library books or resources will be required for the

Mathematics courses, as each will have its own textbook. All references

listed for the Mathmatics Education courses are available through Faculty

members.

Conclusion

The program outlined would be unique in Canada, if not in North

America. Most graduate programs in Mathematics Education are housed

solely in the Faculty of Education where the emphasis is on pedagogical

problems in teaching mathematics. The

proposed program is designed to

expand the teachers mathematical knowledge, to encourage teachers to look

at historical and philosophical influences on the school mathematics

curriculum, and to provide alternative, research-based approaches to

teaching. This program follows SFUs tradition of innovation and creativity.

SItION FRASER 1141VERMY 

New Graduate Course Pronairorin 

c,i.iimi 

Department: 

?

Mathematics and Statistics 

?

Cniirse Number:_____________ 

Title: ? 

Foundations of Mathematics 

Description ? 

Crises in mathematics, their historical andphilosophical 

background and their resolution. 

Credit hours: 

4 ?

_

Vector:_40'0    

Prerequisite(s) If anv 

Acceptance into the Masters Programme in Mathematics Education, 2L_Rt.rmission 

the department. 

ENR0LLIrNT 

AND

SCHEDULING: 

Estimated Enrollment: 

?

When will 

the

course first he offered: 

?

Fall 1989 

Nov often will the course be offered: 

?

Every other year 

JUSTIFICATION: 

This coursewilI be one of three mathematics courses that will comprise the 

. mathematical content of the Masters Degree in Mathematics Education. Note Math 601 

will Oe

dropped. 

RESOURCES!_ 

Which Faculty member will normally teach the course: 

?

•

erber, Berggren 

What 

are the budgetary implications of mountin 

g

the

course: 

At

least a 4 credit 

sessional stipend to cover the teaching. 

Are there sufficient Library resources (a 

pp

end details): ? 

Yes

Appended:, a) Outline of the Course 

b)

An indication of the com 

p

etence of the Faculty member to give the course. 

c)

Library resources 

of

**

Approved ? 

Dc'pnrtmCntl*l Graduate Studies Committee: 

Faculty Graduate Studies Committec 

c2CfZl*e 

V

flntr/. gg 

Dnte: ? 

1

41a.rc4

Date: ? 

ijP

fUlJL7• ?

Senate Graduate StudiesCommitteew 

..

_Date: I 

Senatel 

** Graduate students in the Department of Mathematics and Statistics cannot 

take this course to satisfy their degree requirements. 

Math 603-4

Foundations of Mathematics

Proposed text:

TO BE DETERMINED

Topics ?

Incommensurability and Eudoxus

The Greek theory of proportions, Pythagoreans,

incommensurability, Eudoxus.

Zeno's Paradoxes and the Calculus

The beginnings of Greek speculation on

infinitesimals, Zeno's paradoxes, the exhaustion

method of the Greeks, Newton's infinitesimals, the

arithmetization of analysis.

Naive and Formal Set Theory

Naive set theory, the Russell contradiction, basic.

relations and operations, finite and infinite sets,

the Choice axiom.

Intuitionism vs. Formalism

Basic philosophy of intuitionism, spreads, species,

intuitionistic logic. Hubert's proof theory, Godel's

incompleteness theorem, consistency, formal

systems.

Transfinite Arithmetic

Countable sets, uncountable sets, diagonal

procedures and their applications, cardinal

numbers and their ordering, the continuum

hypothesis, order types, well-ordered sets,

ordinals.

Note: The instructor may have to develop notes for this course.

SItION FRASER UNiVE'IT?

New Graduate Course Prw,oqAlmO.E!!! 

CM.ENDAR I NFORMATION:

Department 

?

Mathematics and Statistics 

?

Course Number: 

?

604-4 

Title: ? 

Geometry 

Description: 

?

Euclidean and non-Euclidean Geometries. Klein's Erlangen Programme 

Credit Hours: 

?

4 ?

Vector: ? 

4-0-0 

?

_Prerequisite(n) If any:________ 

Entrance into the Masters in Math. Education Programme or permission. ** 

ENROLL IrNT AND SCHF.rnILlNr.:

Eat1nLcd Enrollment: 

?

10 ? 

When will the course first 

be

offered: 

Summer 

1990

How often will the course be offered: 

?

Every other year. 

JUSTIFICATION:

This course:'1'be one of three mathematics courses that will comprise the 

mathematical content of the Masters degree 

_in Mathematics Education'. 

Note: Math 602 will be dropped.' 

RESOURCES:

Which Faculty member will normally teach the course: 

?

Gerber,Berggren 

What are 

the

budgetary implications of mountin 

g

the course: 

At least a 4 credit sessional stipend to cover the teaching.- 

Are there sufficient Library resources (a 

pp

end details): ? 

yes

Appended. 

n)

Outline of the Course 

b)

An Indication of the com 

p

etence of the Faculty member to give the course. 

c)

Library resources 

Approved: DcpnrtmCfltfll Graduate Studio; Committee: 

?

cd2.(/I),,te:2 

,c4

Faculty Graduate Studies Committee: 

?

______Th e : 

Date: 

VntiI v:

Son;;; 

Senate:,

Graduate Studio; Committeet q 

2 ?

,_I)nte/ 

** 0

Graduate students in the Department of Mathematics and 

'

Statistics cannot take 

• ?

enticfv

their de 

g

ree requirements. 

Math

604-4

Geometry

Proposed text:

Euclidean and Non-Euclidean Geometries -

Development and History

by Marvin Jay

Greenberg, W. H. Freeman and Company, Second

Edition, 1974

Topics ?

. ?

Euclid's Geometry

The origins of geometry, the axiomatic method,

Euclid's five postulates, attempts to prove the

* ?

parallel postulate.

Logic

Theorems and proofs, techniques of proof, incidence

geometry, models, isomorphisms of models.

Hubert's axioms

Flaws in Euclid, axioms of betweenness, congruence,

continuity, and parallelism.

Neutral Geometry

Geometry without the parallel axiom, alternate

interior angle theorem, exterior angle theorem,

measure of angles and segments, Saccheri-Legendre

theorem, equivalence of parallel postulates, angle

sum of a triangle.

History of the Parallel Postulate

Non-Euclidean Geometry

Hyperbolic geometry, angle sums, similar triangles,

parallels that admit a common perpendicular,

limiting parallel rays, classification of

parallels.

7.Independence of the Parallel Postulate

Consistency of hyperbolic geometry, the Beltraini-

Klein model, the Poincare 'models, perpendicularity

in the Bel.trami-Klein model, inversion in circles, the

projective nature of the Beltrami-Klein model.

8. Philosophical Implications

9. Geometric Transformations

Klein's Erlanger Programme, groups, applications to

geometric problems, motions and similarity,

reflections, rotations, translations, half-turns.

• ?

SIHON FRASER UNIVE'TT? 

Necrndunte_Course Prc,nosnl rorin 

CALENDAR 1NFOR9ATlON:

flpnartrteflt 

?

Mathematics and Statistics ? 

Cnurae Number: 

?

605-4 

Title: 

?

Mat6ematical Modeling: 

Description: 

?

Introduction to Mathematical 

Modeling using algebraic, geometric 

techniques along with techniques using ca1cults. 

Credit hours: 

?

4 ?

Vector: ? 

4-0-0 ? 

Prerequisite(s) if any:________ 

?

Acceptance into the Masters Programme in Mathematic Education, one year of Univ. 

r.NRoWIr.NL

SclIEDhILiNc.

Estimated Enrollment: 

?

10 ? 

When will the course first be offered: 

?

Spring 1991 

How often will the course be offered: 

Everyotheryear 

JusTTrlcATlo:4:

This course-will be one of three mathematicLco u 

5es that will 

rnmpr se

the mathematical content of the Masters Degree in Mathematics Education. 

• ? 

RESOURCES: 

Which Faculty member will normally teach the course: Faculty 

What are the budgetary implications of mountin 

the course: 

At least a4 credit sessional stipend will berequiredto cover the teaching. 

Are there sufficient Library resources (a 

pp

end details): ? 

_Yes 

Appended:. n) Outline of the Course 

b)

An indication of the com 

etence of the Faculty 

member to give the course. 

c)

Library resources 

Approved: flcpnrtmefltfll Graduate Studies Commtttne: 

?

/5_

'q-,

Date: 

3,1A8'

Faculty Crndunte Studies Committee: 

________________________ ?

/?QFtA' ''

?

Unto: 

?

AQ Q '?'(

Faculty: 

Senate Graduate 

?

Studies committee 

C)rh,

___________________________I)nte:' 

Senate! ? 

• ? 

_flsstr: 

** This course may not be used for the satisfaction of degree requirements in the 

Department of Mathematics and Statistics. 

level 

calculus*.* 

Math 605-4

Mathematical Modeling

Proposed text

First Course in Mathematical Modeling

Frank R. Giordano and Maurice Weir,

Brooks Cole Publishing Company, 1985

Topics:

.Graphs

of Functions as Models

The Modeling Process

Mathematical modeling, examples.

Modeling using PrOl)Orl ionality

Proportionaltiy and geometrc similarity.

Model Fitting

Fitting models to data graphically, analytic

methods of model fitting, applying the least-

squares criterion, examples.

Models

requiring Optimization

Classifying optimization problems,

formulation of optimization problems,

examples.

Experimental Modeling

One-term models, interpolation using higher-

order polynomials, smoothing using

polynomials, cubic spline interpolation.

Dimensional Analysis and Similitude'

Dimension as products, the process of

dimensional analysis, examples.

X. Simulation Modeling

Modeling deterministic behavior, area tinder a

curve, developing submodels for probabilistic

behavior.

• ?

9. Modeling using the Derivative

Examples using the derivative, numerical

approximation methods.