Ed.D. EDM9102 CURRICULUM
ORGANISATION - MATHEMATICS
Sino Building 239 5:30-8:00 p.m. 11, 25 October 1999
(1999-2000)
Ngai-Ying Wong 黃毅英
Room 421, HTB, Voice: 2609-6914, Fax: 2603-6724, nywong@cuhk.edu.hk
INTRODUCTION
 What's difference with those offered in PGDE/MEd courses ? Difficult to
understand ? Something to understand ? Wants a discourse.
 Why we have arithmetic, algebra, geometry ... in the mathematics curriculum, and
are they really indispensable, should they be taught separately or in an integrated
style ?
 Why mathematics is a "major subject" in the school curriculum worldwide ?
- Why we have to "force" Arts students to take mathematics ?
 Why mathematics is often taken as an indicator whether someone is smart or not ?
- We always have the impression that those who are good in mathematics can pick
up other subjects very quickly but not vice versa
- And yet adults are often proud of forgotten everything in mathematics [(i) 蕭文強
(1988)。
「數學冷漠症」文匯報 8/29；(ii) 蕭文強(1990)。
「數學普及工作面臨的困難」
。
《無
盡之旅》38-41；(iii) 蕭文強(1990)。
「從數學奧林匹克談起」《遠見雜誌》74 期 143。]
-
"…it is generally perceived that mathematics is a subject for all so that its role in
mass education becomes all the more prominent. On the other hand, it is a
common belief that the acquisition of mathematical concepts requires special
talent, a belief which creates a seemingly contradictory image of a 'subject for
all'. … if they (the students) do not see the relevance of the subject and cannot
cope with the level of sophistication, will fast become indifferent to, or
apprehensive of, the subject and very likely leave school with an unpleasant
imprint of this nightmare called mathematics" [Siu, F.K., Siu, M.K., & Wong, N.Y.
(1993). Changing times in mathematics education: The need of a scholar-teacher. In C.C.
Lam, H.W. Wong, & Y.W. Fung (Ed.s). Proceedings of the International Symposium on
Curriculum Changes for Chinese Communities in Southeast Asia: Challenges of the 21st
Century, 223-226. Hong Kong: Department of Curriculum and Instruction, The Chinese
University of Hong Kong.]
 To answer these questions, we must look at
- Why study mathematics ? (the goal of mathematics education)
- What is mathematics ? (the nature of mathematics)
If we say that we enable students to see thing in a "mathematical way", what is
meant by that ?
The latter can only be understood through the historical development of
mathematics
 What shape the current mathematics curriculum ?
- Knowledge on mathematics learning
- societal/individual needs
- curriculum trend
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PROLOGUE: WHAT IS MATHEMATICS - GENERAL PERCEPTIONS
 Students' concepts [(i) 黃毅英、林智中、黃家鳴(待刊)。數學是甚麼─學生的角度；(ii) Wong,
Marton, Wong & Lam (under preparation). The lived space of mathematics learning]

Set of rules
Image of a mathematician [(i) Spangler, D.A. (1992). Assessing students' beliefs about
mathematics. Arithmetic Teacher, November, 148-152; (ii) Snapper, E. (1988). What do we
do when we do mathematics ? The Mathematical Intelligencer, 10, 53-58]
Mathematics is a science object ? Is an art ? [(i) Priestley, W.M. (1990). Mathematics
and poetry: How wide the gap ? The Mathematical Intelligencer, 12, 14-19; (ii) 蕭文強(1999)。數
學與詩(等)(pp. 7-18, 25-26)。載黃毅英(編)。《數學內外》
。香港：天地圖書。]
-
B.A. Mathematics
"One of the elements of beauty of mathematics is the feeling of surprise" (H.
Poincaré, 1845-1912)
- "Beauty is the first test: there is no permanent place in the world for ugly
mathematics" (G.H. Hardy, 1877-1947)
- 「數學與邏輯沒有甚麼關係」(小平邦彥)
- "The moving power of mathematical invention is not reasoning but imagination"
(A. DeMorgan, 1806-1871)
 Mathematics is a plural [Freudenthal, H. (1991). Revisiting Mathematics Education, Chapter 1:
Mathematics phenomenolgically (pp. 1-44).
Dordrecht: Kluwer Academic Publishers.] (1905-1990)
 Mathematics is an exact science (I. Kant, 1724-1804)
 Applicable in the real world and work place "I am not thinking of the 'practical'
consequences of mathematics. I have to return to that point later: at present I will
say only that if a chess problem is, in the crude sense, 'useless', then that is equally
true of most of the best mathematics; that very little of mathematics is useful
practically, and that that little is comparatively dull" (Hardy, G.H., 1940). - will
elaborate more on real life mathematics later.
 Rigorous (AL Pure Math 79 II 6(a) simplified: Let q( x)  3x 2  5x  4 . Show that
every quadratic polynomial p(x) can be expressed as kq(x)+s(x), where k is a real
number and s(x) is polynomial of degree < 1), formal, exact (2 instead of 1.414,
1 1 1
1
1    ... ( 1) n 1 ( ) ... or log 2 ?) and highly symolised (  B  Z - find
2 3 4
n
B( Z )
an example of a uncountable union of thin sets which is still thin, ei = -1 - mystic
combination of all symbols)
 "Doing mathematics": involves discovery and invention (the case of F.Y. Sing)
DEVELOPMENT OF MATHEMATICAL IDEAS
 What is mathematics ?
- "Mathematics is the science of space and numbers" Oxford dictionary
 What areas it encompasses ? - axiomatics, fuzzy mathematics, catastrophe, topology,
non-standard analysis, chaos, fractals …
Some landmarks:
 "Everything is number"
Mathematics (Pythagaros, 580 B.C. - 500 B.C.)
- Discrete
= absolute (arithmetic)
= relative (music)
- continuous
= stable (geometry)
= moving (astronomy)
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 Euclid and the axiom system
- (5 common notions, 5 postulates, 465 propositions)
- Newton
(mechanics),
Langrange
(analytic
mechanics),
Clausius
(thermo-dynamics), Spinoza (ethics)
 I. Kant (1724-1804) and the non-Euclidean geometries
- Mathematics is "synthetic a priori", "the only science that could give exact
knowledge".
- "God creates the universe according to Euclid's Elements"
- Physical interpretation [Gamow, G. (1961). 1,2,3,…  . N.Y.: Viking Press.]
 The second crisis of mathematics
- Call for rigor
- The arithmetisation of mathematics (elimination of geometrical intuition)
= B. Bolzano (1781-1848): father of arithmetisation
= J. Fourier (1768-1830): try to reduce all of analysis to arithmetic
= Epsilon-delta definition of limits: A-L. Cauchy (1789-1857), K. Weierstrass
(1815-1897)
 Rise of symbolic logic (G. Boole: 1815-1864)
 Finitism: L. Kronecker's (1823-1891)
- "Plato said, 'God always geometrises', Jacobi (1804-1851) changed this to 'God
always arithmetises. Then came Kronecker and created the memorable
expression 'Die Ganzen Zahlen had Gott gemacht, alles andere ist
Menschenwerk' God made integers, all else is the work of man" (F. Klein,
1849-1925: Jaresbericht der Deutschen Mathematiker Vereinigung)
- Man's creation, e.g. artificial numbers: negative numbers, n-dimensional
geometry, imaginary numbers
 The third crisis of mathematics
- Intuitionism: original intuitiveness (natural numbers), constructibility, doesn't
admit law of excluded middle, undecidability. (L.E.J. Brouwer, 1881-1966)
- Forerunner of axiomatics: G. Peano (1858-1932)
1. 0  N.
2. If m  N, m+  N.
3. If m, n  N and m  n, then m+  n+.
4. There does not exist m  N such that m+ = 0.
5. If S  N such that 0  S and (m  S  m+  S), then S = N.
- "In the first place, Peano's three primitive ideas - namely, '0', 'number', and
'successor' - are capable of an infinite number of different interpretations… We
wil give some examples. (1) Let '0' be taken to mean 100 and let 'number' be
taken to mean the numbers from 100 onwards in the series of natural numbers.
Then all our primitive propositions are satisfied …" (Russell, 1919) ("arbitrary
structures" - lack of algebraic structure).
- Logicism: Principia Mathematica (B. Russell, 1872-1970, & A. Whitehead,
1861-1947)
- Formalism: Grundlagen der Arithmetik (1884, G. Frege, 1848-1925),
Grundlagen der Geometrie, Grundlagen der Mathematik (D. Hilbert, 1862-1943),
Grundlagen der Analysis (E. Landau, 1877-1936)
- "Pure mathematics consists entirely of such assertions as that, if such and such a
proposition is true of anything, then such and such another proposition is true of
that thing. It is essential not to discuss whether the first proposition is really
true, and not to mention what the anything is of which it is supposed to be
true … If our hypothesis is about anything and not about some one or more
particular things, then our deductions constitute mathematics.
Thus
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-
mathematics is defined as the subject in which we never know what we are
talking about, nor whether we are saying is true"
"Let us take, for instance, the natural number two. We all have an intuitive idea
of twoness. For example, we say that each of the unordered pairs {a,b}, and
{c,d}, where a b and cd, has exactly two elements. Thus a common property
of these sets is their towness; and their having this common property may be
expressed by a bijective mapping between them. Therefore we can define, in
particular, the natural number two as a certain standard set that conveys the idea
of towness and, in general, the natural number n as a certain standard set that
conveys the idea of n-ness." [Leung, K.T., Chen, D.L.C. (1967). Elementary Set Theory.
Hong Kong: Hong Kong University Press.]
-
"The number of a class is the class of all those classes that are similar to it, a
number is anything which is the number of some class" (Russell, 1919).
x
x ( x  1)
 2
The mathematical equal (ABC  BCA ?,
, 3/4=6/8: Q = Z2/~)
x 1
x 1
[Wong, N.Y. (1996).
ABC  BCA ?
EduMath, 2, 22-24.]
- Proof theory and K. Godel's (1906-1978) incompleteness theorem
- The essentials of axiomatics: postulates and undefined terms
- Metamathematics
 Mathematical structures
- F. Klein and the Erlanger programme (invariance under geometric
transformations): Metric (measurements), equiaffine (area), affine (congruence),
projective (projection), topology (topological)
- N. Bourbaki (1935-1968, 3 among the 12 Field's medallists between 1950-1960):
Element de Matheqmatiques
- Mathematical structures: set, algebraic, order, topological, composite, multiple,
mixed [胡作玄(1984)。《布爾巴基學派的興衰》第九章：Bourbaki 的選擇。上海：知識出
版社。]
- the extension of the number system
 Beyond absolutism
- Mathematics as a cultural activity [Wilder, R.L. (1952).
An Introduction to the
Foundations of Mathematics, Chapter XII: The cultural setting of mathematics (281-299). New
York: John Wiley & Sons.]
-
-
= mathematics changes across time
= Kronecker's remark on Lindemann's (1852-1939) work on  "What is the use
of your beautiful theorem concerning  ? After all, irrational numbers (hence
) does not exist." - what is mathematical existence ?
Empiricism
= concepts of mathematics have empirical origins
= the truths of mathematics have empirical justification from the physical world
Quasi-empiricism (I. Lakatos)
= mathematical knowledge is fallible
= mathematics is hypothetico-deductive
= history is central
= the primacy of informal mathematics is asserted
= a theory of knowledge creation is included
= weaknesses: no account of mathematical certainty, no account of the nature of
the objects of mathematics, does not account for application of mathematics,
does not establish the legitimacy of bringing history into the heart of the
matter
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-
-
Social constructivism: link between objective and subjective knowledge
= the basis of mathematical knowledge is linguistic knowledge, conventions and
rules, and language is a social construction
= interpersonal social processes are required to turn an individual's subjective
mathematical knowledge, after publication, into accepted objective
mathematical knowledge
= objectivity itself will be understood to be social
Mathematics education in different ideologies (Ernest)
WHY STUDY MATHEMATICS ?
 Hirst & Peters: Seven areas of knowledge (logic and mathematics, physical science,
mind, ethics and moral, aesthetic experience, religion, philosophy) [Hirst, P.H., &
Peters, R.S. (1970).
& Kegan Paul]
The Logic of Education, Chapter 4: Curriculum (pp. 60-).
London: Routledge
 Griffiths and Howson's discussion [Griffiths, H.B., & Howson, A.G. (1974). Mathematics:
Society and curricula, Chapter 2: Why teach mathematics ? (pp. 7-26).
University Press]
Cambridge: Cambridge
 Axiomatic approach and "Why Johnny can't add" (1973, M. Kline, 1908-)
 New math and back to basis: problem solving [(i) NCTM (1980). An Agenda for Action.


Reston, VA: Author; (ii) NCSM (1977). Position paper on basic skills. ***: Author; (iii) NCSM
(1989). Position paper on essential mathematics for the twenty-first century, Reston, VA: Author;
Hilbert's address]
Cockcroft's report [Cockcroft, W.H. (1982). Mathematics Counts, Chapter 1: Why teach
mathematics (pp. 1-4). London: H.M.S.O.]
Product and process [Howson, G., & Wilson, B. (Ed.) (1986). School Mathematics in the 1990s,
Chapter 2: Mathematics and general educational goals, Chapter 3: The place and aims of
mathematics in schools (pp. 7-35). Cambridge: Cambridge University Press.]
 Not a matter of content ?
- 「怎樣在教授數學知識之同時，以之作為培養深層能力的基礎」 [黃毅英
-
(1995)。普及教育期與後普及教育期的香港數學教育。於蕭文強(編)《香港數學教育的回顧
與前瞻》內，頁 69-87。香港：香港大學出版社。]
H. Wu, impact of hi-tech and Stigler on blackboards [(i) Stigler, J., & Hiebert, J.
(1999). The Teaching Gap: Best ideas from the world’s teachers for improving education in the
classroom. New York: Free Press; (ii) Curriculum Development Council (1992). Learning
Targets for Mathematics. Hong Kong: Education Department; (iii) 黃毅英(1996)。香港數學
教育改革另類報告。
《香港數學教育會議-96》研討會專題演講。香港：香港大學，1996 年
12 月 23 日。後載馮振業(1997)(編)。
《香港數學課程改革之路》
，141-160。香港：香港數學
教育學會；(iv) 黃顯華(譯)(1993)。劍齒虎的課程。載蓮華(編)《教無止境》。香港：廣角
鏡出版社。]
- 朱門狗肉臭
 蕭文強《為甚麼要學習數學》
- From concrete to abstract, abstraction. What is "twoness" ?
- From induction to deduction
- Generalisation and specification: application.
- Mathematical modeling
- 數學意識 [蕭文強(1999)。//(pp. //)。載黃毅英(編)。
《數學內外》
。香港：天地圖書。]. E.g.
restaurant mathematics, 曾榮光問題 lion rock tunnel: contour, topology: rubber
sheet geometry [黃毅英(1997)(編)。《邁向大眾數學的數學教育》，185-216。台北：九章
出版社。]
-
Real life mathematics, artefects, expanding definitions, local axiomatics [黃家鳴
(1997)。生活情境中的數學與學校的數學學習。《基礎教育學報》7 卷 12 期，161-167。]
 Doing mathematics vs receive in an organised way ( ab 
-5-
a b
)
2
 Math-in-the-making and Math-as-an-end-product (M.K. Siu)
 Methematisation:
- problem solving, connection, communication, number sense, etc.
- abstraction/theorising (IT affects capacity of abstraction ?)
- sense making (J. Cai's study) [黃毅英(編)。《數學內外》(pp. 19-24)。香港：天地圖書。;
鄭毓信(1995)。
《數學教育哲學》(pp. 338-341)。四川：四川教育出版社。]
HOW DID THE CURRENT MATH CURRICULUM COME ABOUT
 New Math reform (basically elite)
 Basic mathematics
 Math for all
 The unofficial curriculum [Ad-hoc committee on holistic review of the mathematics curriculum
(1999).
-
Position Statement No. 4: Learning dimensions in the mathematics curriculum]
Developmental stages
Task analysis (累積性) and learning hierarchy
Epistemological consideration
Theoretical basis for the identification of the core
SUGGESTED READINGS
Nil. Just follow my "guided tour" to the mathematics world in these two lectures. If
you want to do some further readings, please start with those with "*" below if you have
some foundations in mathematics and "#" if otherwise.
REFERENCES
Cockcroft, W.H. (1982). Mathematics Counts. London: H.M.S.O. QA14.G7A3 (CCED)
Ernest, P. (Ed.) (1991). The Philosophy of Mathematics Education. Hamsphire: The Falmer
Press. QA11.E74 1991 (CC)
(*) Freudenthal, H. (1991). Revisiting Mathematics Education: China lectures. Dordrecht:
Kluwer Academic Publishers. QA12.F74 (CC)
Fung, C.I., & Wong, N.Y. (1997). (Unofficial) Mathematics Curriculum for Hong Kong, P.1 to
S.5. Hong Kong: Hong Kong Association for Mathematics Education.
(*) Griffiths, H.B., & Howson, A.G. (1974). Mathematics: Society and curricula. Cambridge:
Cambridge University Press. QA11.G82 (UL)
Hardy, G.H. (1948). A Mathematician's Apology. Cambridge: Cambridge University Press.
QA7.H3.1967 (UL)
Hirst, P.H., & Peters, R.S. (1970). The Logic of Education. London: Routledge & Kegan
Paul. LB880.H59 (CC)
Howson, G., & Wilson, B. (Ed.) (1986). School Mathematics in the 1990s. Cambridge:
Cambridge University Press.
National Research Council (1989).
Everybody Counts. Washington, D.C.: National
Academy Press. QA13.394 1989 (CCED)
National Research Council (1990). Reshaping School Mathematics. Washington, D.C.:
National Academy Press. QA13.R467 1990 (CCED)
Russell, B. (1919). Introduction to Mathematical Philosophy. London: G. Allen & Unwin.
QA9.R8 (UL)
(*) Wilder, R.L. (1952). An Introduction to the Foundations of Mathematics. New York:
John Wiley & Sons. QA9.W58 (UL) (Especially the last chapter)
《為甚麼要學習數學》。香港：學生時代出版社。第二版 (1992) 香港
(#) 蕭文強 (1978)。
新一代文化協會。增訂本 (1995)，台灣：九章出版社。QA7H83 (UL)
蕭文強(1995)(編)《香港數學教育的回顧與前瞻》
。香港：香港大學出版社。QA14.H6 H75
1995 (UL/UCH)
鄭毓信(1995)。
《數學教育哲學》
。四川：四川教育出版社。QA11.C4825 1995 (CC)
鄭毓信、李國偉(1999)。
《數學哲學中的革命》。台灣：九章出版社。
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