a vicennial walk through ‘a’ level mathematics in...
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A Vicennial Walk through A Level Mathematics in Singapore
Asst/Prof. Ho Weng Kin Mathematics and Mathematics Education
National Institute of Education
Keynote Lecture VII, MTC2017

Outline
Introduction Curriculum ideologies Evolution of ALevel Math Syllabus Findings and implications Conclusion

Introduction

Introduction

Introduction

Introduction

Intrinsic tension

Evolution of ALevel Math Syllabi

Evolutionary history Period (Exam Years)
Syllabus Developed By
Subject code
Earlier years 1979
University of Cambridge Local Examinations Syndicate (UCLES)
840 Mathematics (Syllabus A) 842 Pure Mathematics 844 Applied Mathematics 846 Statistics 848 Mathematics (Syllabus B)
1980 1985 UCLES 9202 Mathematics (Syllabus B) 1986 1998 UCLES 9205 Mathematics (Syllabus C)
9225 Further Mathematics 19992001 UCLES 9205 Mathematics (Syllabus C)
Interim syllabus; for Singapore candidates only
UCLES 9225 Further Mathematics

Evolutionary history Period (Exam Years)
Syllabus Developed By
Subject code
20022006 MOE, Singapore, in collaboration with UCLES
9233 Mathematics
MOE+UCLES 9234 Further Mathematics 20072016 MOE+UCLES 9740 H2 Mathematics
FM withdrawn
2007  Present MOE+UCLES 9758 H2 Mathematics MOE+UCLES 9649 H2 Further
Mathematics

848/Jun 84/P2/Q15 Numerical calculation
The three points A, B and C, on a spherical surface of radius R are at the vertices of an equilateral triangle of side a. A line perpendicular to the plane of the triangle, and through the centre G of the triangle, meets the surface of the sphere at D, where GD = d (d < R). Show that
=12 +
2
6.
If the value of d is subject to a small error , obtain an expression for the corresponding approximate error in R. When a = 60, d = 18, and = 0.5 (all lengths being in mm), calculate R and , and hence estimate the limits for R.

848/Dec 74/P1/Q5 Pure Mathematics
Write down 1, where
= 5 44 3 .
Verify that if
= 14 2012 17 and =2 00 1 ,
then = 1.

848/Dec 74/P1/Q5 Pure Mathematics
Show (without using the numerical forms of the matrices) that 3 = 31, and explain the advantage of using similar results to find high powers of M. Show that if M and D are given as above, and if
= 5 44 3 ,
where 0, 0, then = 1.

848/Jun 75/P1/Q17 Algebraic Structure
For the sets P, Q, A, B, give the formal definitions of
and , and formal proofs of the relations
= , = .
[No credit will be given for the use of Venn diagrams.]

848/Nov 79/P1/Q14 Numerical Analysis and Computation
The flow diagram given below indicates, for suitable values of the input variables x and n, a procedure for calculating and summing the successive terms of a power series in x. The summation is terminated when the magnitude of the next term to be added is less than a given small positive number .

848/Nov 79/P1/Q14 Numerical Analysis and Computation

848/Nov 79/P1/Q14 Numerical Analysis and Computation
Use the flow diagram to write down, in algebraic form in terms of n and x only, the first four expressions for S given by box (i) on the assumption that remains greater than . Write down, in terms of n and x, the expression whose series expansion is being considered.

9202/Nov 81/Q4 Pure Mathematics
Find the general solution of (i) the equation 3 sin 2 cos = 2.5, giving your answer in degrees to the nearest 0.1, (ii) the equation cos 3 = sin , giving your answer in radians.

9202/Jun 83/Q10 Particle Mechanics

9202/Jun 83/Q10 Particle Mechanics
A small bead B of mass m is free to slide on a fixed smooth vertical wire, as indicated in the diagram. One end of a light elastic string passes through a smooth fixed ring R and the other end of the string is attached to the fixed point A, AR being horizontal. The point O on the wire is at the same horizontal level as R, and AR = RO = a. Prove that, in the equilibrium position, OB = 1
2.

9202/Jun 85/P1/Q13(b) Probability and Statistics
The petrol consumption of a new model of car is being tested. In one trial, 50 cars chosen at random were driven under identical conditions, and the distances, x miles, that were covered on precisely 1 gallon of petrol were recorded. The results gave the following totals: = 2685, 2 = 144 346. Calculate a 99% confidence interval for the mean petrol consumption, in miles per gallon, of cars of this type.

9205/Nov 86/P1/13(i) Pure Mathematics

9205/Nov 86/P1/13(i) Pure Mathematics
The sides of the square ABCD are each of length a. The rectangle BKLC lies in a plane perpendicular to the plane of ABCD and BK = CL = 2a (see diagram). Find each of the following angles, giving your answers to the nearest tenth of a degree: (i) the angle between the line AL and the plane AKB,

9205/Nov 87/S/Q5 Pure Mathematics
Evaluate
(i) (ln )32
,
(ii) (+1)2
(2+3+3)21 ;
(iii) 14 cos +3 sin 3 cos +4 sin 20 .

9205/Jun 88/P1/Q17 Pure Mathematics
Prove by induction that the following results are true for all positive integers n.
(i) 2( 1)=1 = 112 2 1 3 + 2 .
(ii) Given that = , then
= + .

9205/Nov 90/P2/Q11(b) Pure Mathematics
The equation, in polar coordinates, of a curve is = asin cos2 ,
where a is a positive constant, and 0 . Show that the greatest value of r is 2
33 , and
sketch the curve.

9205/Jun 91/P1/Q16(b) Pure Mathematics

9205/Jun 91/P1/Q16(b) Pure Mathematics
The point P, with xcoordinates 1, lies on the curve C with equation = 2. The tangent to the curve at P cuts the xaxis at the point T, as shown in the diagram. Show that the coordinates of T are (1
12
, 0). Deduce that any straight line which passes through a given point (, 0) on the xaxis and which cuts C twice has gradient greater than 22+1.

9205/Jun 91/P2/Q15(i) Pure Mathematics

9205/Jun 91/P2/Q15(i) Pure Mathematics
The diagram shows a garden shed with horizontal rectangular base ABCD, where AB = 3 m, BC = 2 m, and vertical walls. There are two rectangular walls, ABBA and DCCC, where AA = BB = CC = DD = 2 m. The roof consists of the planes ABSR and CDRS, where RS is horizontal. Each section of the roof is inclined at an angle to the horizontal, where tan = 3
4.

9205/Jun 91/P2/Q15(i) Pure Mathematics
The point A is taken as origin and vectors i, j, and k, each of length 1 m, are taken along AB, AD, AA respectively. (i) Verify that the plane with equation . 22 + 33 12 = 65 passes through B, D and S.

9233/Nov 00/P1/Q4
By considering the derivative as a limit, show that the derivative of 3 is 32.

9233/Nov 01/Q11 (Section I)
Given that = tan (12
tan1 ), show that
1 + 2
=12
1 + 2 .
By differentiating this result twice, show that up to and including the term in 3, Maclaurins series for tan(1
2tan1 ) is
12 1
83.

9233/Nov 01/P2/Q7(i) (Option b) Statistics
The continuous random variable X has a uniform distribution on 0 . Find the cumulative distribution function of X. Two independent observations of X1 and X2 are made of X and the larger of the two values is denoted by L. (i) Use the fact that L < x if and only if both X1 < x and X1 < x to find the cumulative distribution function of L.

9233/Nov 06/P1/Q14 EITHER(i) Pure Mathematics

9233/Nov 06/P1/Q14 EITHER(i) Pure Mathematics
A curve has parametric equation = , =
, where c is a positive constant. Three
points ,
, ,
and ,
on the curve are shown in the diagram.
(i) Prove that the gradient of QR is 1
.

9233/Nov 06/P2/Q24 (Section D) Probability and Statistics
A group of 10 pupils consists of 6 girls and 4 boys. The random variable X is the number of girls minus the number of boys in a random sample of 3 pupils from the group. For example, if there are 2 boys and 1 girl in the sample then X = 1. Find the probability distribution of X, giving each probability as a fraction in the lowest terms. Find Var(X).

9233/Nov 06/P2/Q13 EITHER (Section B) Applied Mathematics
A researcher is investigating the distribution of the amount of time per week that teenagers spend playing computer games. Using the data from a large random sample, the researcher obtains the result that (3.52,4.14) is a 95% confidence interval for , the population mean number of hours in a week that teenagers spend playing computer games. Explain what is meant by (3.52,4.14) is a 95% confidence interval for , and explain why, in obtaining the confidence interval, it is not necessary to make any assumptions about the distribution.

9740/Nov 07/P1/Q9

9740/Nov 07/P1/Q9
The diagram shows the graph of = 3.
The two roots of the equation 3 = 0 are denoted by and , where < . (i) Find the values of and , each correct to 3 decimal places.

9740/Nov 07/P1/Q9 A sequence of real numbers 1, 2, 3, satisfies the recurrence relation
+1 =13
for n > 1. (ii) Prove algebraically that, if the sequence converges, then it converges to either or . (iii) Use a calculator to determine the behaviour of the sequence for each of the cases 1 = 0, 1 = 1, 1 = 2.

9740/Nov 08/P2/Q3(a)
The complex number w has modulus r and argument , where 0 < < 1
2 , and
denotes the conjugate of w. State the modulus and argument of p, where =
.
Given that 5 is real and positive, find the possible values of .

9740/Nov 10/P2/Q2(ii)
(a) Prove by the method of difference that
1
( + 2)=
34
12 + 1
1
2 + 2.
=1
(b) Explain why 1(+2)
=1 is a convergent
series, and state the value of the sum to infinity.

9740/Nov 14/P2/Q11(ii) An art dealer sells both original paintings and prints. (Prints are copies of paintings.) It is to be assumed that his sales of originals per week can be modelled by the distribution Po(2) and his sales of prints can be modelled by an independent distribution Po(11). (ii) The probability that the art dealer sells fewer than 3 originals in a period of n weeks is less than 0.01. Express the information as an inequality in n, and hence find the smallest possible integer value of n.

9740/Nov 16/P1/Q4(i) Pure Mathematics
An arithmetic series has first term a and common difference d, where a and d are nonzero. A geometric series has first term b and common ratio r, where b and r are nonzero. It is given that the 4th, 9th and 12th terms of the arithmetic series are equal to the 5th, 8th and 15th terms of the geometric series respectively. (i) Show that r satisfies the equation 510 83 +3 = 0. Given that < 1, solve this equation, giving your answer correct to 2 decimal places.

9740/Nov 16/P1/Q10(b) Pure Mathematics
The function g, with domain the set of nonnegative integers, is given by
=
1 for = 0
2 + 12 for
1 + 1 for
(i) Find g(4), g(7) and g(12). (ii) Does g have an inverse? Justify your
answer.

9740/Nov 16/P2/Q6(i) Probability and Statistics
The number of employees of a company, classified by department and gender, is shown below. (i) The directors wish to survey a sample of 100 of the employees. This sample is to be a stratified sample, based on department and gender.
(a) How many males should be in the sample? (b) How many females from the Development
department should be in the sample?
Production Development Administration Finance Male 2345 1015 237 344 Female 867 679 591 523

Curriculum ideologies

Curriculum ideologies
Scholar Academic Social Efficiency Learner Centered Social Reconstruction
M.S. Schiro (2013)

Scholar academic
Over centuries our culture has accumulated important knowledge that has been organized into the academic disciplines found in universities.

Scholar academic
The aim of education is to get children to acquire this body of accumulated knowledge, i.e., the various academic disciplines.

Scholar academic
Education involves making youth members of a discipline by first moving them into it as students and then moving them from the bottom of hierarchy towards its top.

Social efficiency
The purpose of schooling is to efficiently meet the needs of society by training youth to function as future mature contributing members of society.

Social efficiency
Train youth in the skills and procedures they will need in the workplace and at home to live productive lives and perpetuate the functioning of society.

Social efficiency
Youth achieve an education by learning to perform the functions necessary for social productivity.

Learner centered
Focuses on the needs and the concerns of the individuals.

Learner centered
The goal of education is the growth of individuals, each in harmony with his or her own unique intellectual, social, emotional and physical attributes.

Social reconstruction
Come from a social perspective. Awareness of societal problems, e.g.,
racial, gender, social and economic inequalities.

Social reconstruction
The purpose of education is to facilitate the construction of a new and more just society that offers maximum satisfaction of its members.

Orientation Scholar Academic
Social Efficiency
Learner Centered
Social Reconstruction

Eisners model
Helps us understand the various contexts of curriculum decisionmaking by considering two sets of dilemmas: the scale and scope of our attention and the time at which and for which curriculum decisions are made. (Eisner, 2002)

Eisners model
Twodimensional representation of the scope and timeliness of curriculum decisions, organized into four quadrants.

Eisners model
Quadrant 3
Time at which and for which
Quadrant 1
Decisions are made
Quadrant 4
Quadrant 2
FUTU
RE PR
ESEN
T
GENERAL
PARTICULAR
SCAL
E AN
D S
CO
PE O
F
CU
RR
ICU
LUM
DEC
ISIO
NS

Eisners model
Quadrant 3
Time at which and for which
Quadrant 1
Decisions are made
Quadrant 4
Quadrant 2
FUTU
RE PR
ESEN
T
GENERAL
PARTICULAR
SCAL
E AN
D S
CO
PE O
F
CU
RR
ICU
LUM
DEC
ISIO
NS

Quadrant 1
Impact on the masses and for long term National or international policy concerns
for education Teach Less, Learn More Thinking Schools, Learning Nation Every School is a Good School

Eisners model
Quadrant 3
Time at which and for which
Quadrant 1
Decisions are made
Quadrant 4
Quadrant 2
FUTU
RE PR
ESEN
T
GENERAL
PARTICULAR
SCAL
E AN
D S
CO
PE O
F
CU
RR
ICU
LUM
DEC
ISIO
NS

Quadrant 2
Focuses on the particular but longterm decisions
Longterm implications in a school context Ascertain implications of nationwide policy on
the scope of content, pedagogy and assessment
Consider how implications impact on schools particular setup and needs

Eisners model
Quadrant 3
Time at which and for which
Quadrant 1
Decisions are made
Quadrant 4
Quadrant 2
FUTU
RE PR
ESEN
T
GENERAL
PARTICULAR
SCAL
E AN
D S
CO
PE O
F
CU
RR
ICU
LUM
DEC
ISIO
NS

Quadrant 3
Focuses on general scope but with immediate implications
Subject syllabi that MOE publishes Provides overview of the curriculum, common
to all schools and classrooms Organized according to level addressed by
the syllabus

Sing
apor
e Ed
ucat
ion
Syst
em A
t A G
lanc
e Junior College / Centralized Institute
Ages 1619 (23 Years)

Evolutionary history Period (Exam Year) Exam Syndicate Subject code 19741979 Cambridge 848 (M) 19801985 Cambridge 9202 (M) 19862001 Cambridge 9205 (M)
9225 (FM) 2002 CambridgeSingapore 9233 (M: interim)
9225 (FM) 20032006 SingaporeCambridge 9233 (M: New)
9234 (FM) 20072017 SingaporeCambridge 9740 (M)
FM removed 2017Present SingaporeCambridge 9758 (M)
9649 (FM: reintroduce)

Year of policy
Policy Year of syllabus change
Syllabus change / Focus
1997 TSLN Thinking Schools, Learning Nation
1999 9205 to 9233 (Interim) GCs were introduced in the Alevel math curriculum in the 2001 FM (9234) syllabus.
2001 9205 (Interim) to 9233 (Revised) 9234 FM (Revised)  GC in Math  GCneutral exam

Year of policy
Policy Year of syllabus change
Syllabus change / Focus
2002
Review of JC/Upper Secondary Education Landscape It was actually recommendation from this review that led to changes in the Alevel education landscape and the 2006 Alevel Curriculum. Recommendations from this review were reported in 2004.
2006 9233 (New) to H2 Math 9740  a broader and more flexible JC
curriculum and a more diverse JC/Upper Sec landscape
 Contrasting subjects  H1/H2/H3 There are some features of the 2006 Alevel Curriculum. Removal of Further Math (which meant removal of a double math option for students)
2004
TLLM Teach Less, Learn More TLLM came later on and one of the underlying ideas was to encourage teachers to rethink how they teach.

Year of policy
Policy Year of syllabus change
Syllabus change / Focus
2015 ESGS Every School is a Good School Studentcentric, values driven education. The Every School is a Good School is part of this phase of education in Singapore and is one piece of the 4 Everys. This is at a more macro level of the whole education system.
2016 9740 to 9758  21st Century Competencies  STEM Education
Specific to JC education, there was a JC Curriculum Review Committee recommended an emphasis on deep disciplinary understanding and the use of constructivist pedagogies that promote deep thinking and active learning and support the development of 21st Century Competencies.
Reintroduction of Further Math

Significant features
Revised 2001 Math Syllabus (9233) 2006 Alevel Math Syllabus (H2 Math 9740) 2016 Alevel Math Syllabus (H2 Math 9758)

Revised 2001 Alevel Math Syllabus (9233)
Reduction in content to provide schools with more time to incorporate more thinking activities and infuse IT into lesson
GCs allowed in FM, although questions set are GCneutral. The use of GCs is in line with the IT initiative as well as to expose students to the use of a powerful tool
Assessment format changed; EitherOr option for the last question of Paper 1.

2006 Alevel Math Syllabus (H2 Math 9740)
Emphasis on solving realworld problems, including communication about the mathematics involved in solving a problem and interpretation of the solution in the context of the problem
GCs will be used for teaching and learning as well as in the examination
No question choice in assessment format

2016 Alevel Math Syllabus (H2 Math 9758)
An expanded suite of syllabuses, with H2 FM, to give more options to students and better cater to their different interests and needs
Emphasis on mathematical processes such as mathematical reasoning, mathematical modelling and communication
Emphasis on learning experiences, which are stated in the syllabus to influence the ways teachers teach and how students learn so that curriculum objectives can be achieved
Teachers are also encouraged to use pedagogies that are constructivist in nature

Porters framework

Intended curriculum
The content target for the enacted and assessed curriculum, statements of what every students is to know and be able to do.
Often captured in content standards, i.e., official syllabus, that define specific points in time when students are to learn the knowledge and skills.

Framework for the Singapore Mathematics Curriculum

Syllabus aims
Comparison of the syllabus aims across the following Math Syllabi: 9205 9233 9740 9758
Use the lens of curriculum orientation to guide us in this comparison

Scholar academic
Subject disciplinarity How does a mathematician think and work? What characterizes the practices of a working
mathematician?

Scholar academic
9205 (c) encourages clear thinking and accurate working; 9233 6. develop their ability to think clearly, work carefully and communicate mathematical ideas successfully;

Scholar academic
9205 (g) develops a logical and coherent view of mathematics; 9233 4. appreciate mathematics as a logical and coherent subject with rich interconnections

Scholar academic
Equipping youth members with the disciplinarity of mathematics by first moving them into it as students and then moving them from the bottom of hierarchy towards its top
Focus on training to attain subject skills and techniques, problem solving

Scholar academic
9205 (d) provides as much as possible of the mathematics necessary for the students concurrent study at Alevel; (e) provides a suitable foundation for beginning a degree level course in mathematics or a related discipline;

Scholar academic 9233 9. acquire a suitable foundation for further study of mathematics and related disciplines. 9758 (a) acquire mathematical concepts and skills to prepare for their tertiary studies in mathematics, sciences, engineering and other related disciplines

Scholar academic 9205 (b) enables students to acquire and become familiar with appropriate mathematical skills and techniques; (h) develops the students ability to formulate a problem in mathematical terms, to solve the resulting mathematical problem without error, to present the solution clearly and to check and interpret the results;

Scholar academic
(i) develops an understanding of mathematical concepts and of mathematical argument.

Scholar academic
9233 3. acquire and become familiar with appropriate mathematical skills and techniques; 7. develop their ability to formulate problems mathematically, interpret a mathematical solution in the context of the original problem and understand the limitations of mathematical models

Scholar academic 9740 develop the mathematical thinking and
problem solving skills and apply these skills to formulate and solve problems
9758 (b) develop thinking, reasoning, communication and modelling skills through a mathematical approach to problemsolving

Social efficiency
Learn mathematics that is functional and useful at workplace Financial mathematics Engineering mathematics Econometrics Mathematical and theoretic biology Interdisciplinary stance Applications

Social efficiency
9205 (h) presents at least one major area of application of mathematics either particle mechanics or probability and statistics so that students can see examples of the usefulness of mathematics in the real world;

Social efficiency
9233 8. appreciate how mathematical ideas can be applied in everyday world;

Social efficiency
Learn mathematics that is functional and useful at workplace Collaboration Communication Invention

Social efficiency
9740 Produce imaginative and creative work
arising from mathematical ideas
develop the abilities to reason logically, to communicate mathematically, and to learn cooperatively and independently

Social efficiency
9758 develop thinking, reasoning,
communication and modelling skills through a mathematical approach to problemsolving

Social efficiency
9740 make effective use of a variety of
mathematical tools (including information and communication technology tools) in the learning and application of mathematics

Learner centered 9205 (a) develops further the mathematical knowledge of students in a way that encourages confidence and provides understanding and enjoyment; 9233 1. develop further their understanding of mathematics and mathematical processes in a away that encourages confidence and enjoyment;

Learner centered
9233 2. develop a positive attitude to learning and applying mathematics; 9740 develop positive attitudes towards
mathematics

Learner centered
9758 (d) experience and appreciate the nature and beauty of mathematics and its value in life and other disciplines.

Learner centered
4 Everys:
Every School a Good School Every Student an Engaged Learner Every Teacher a Caring Educator Every Parent a Supportive Partner

Learner centered
These 4 Everys form part of the efforts towards developing a learnercentric, valuesdriven education.
Since 2011, theres been a specific focus on schools, students, teachers and parents.

Eisners model
Quadrant 3
Time at which and for which
Quadrant 1
Decisions are made
Quadrant 4
Quadrant 2
FUTU
RE PR
ESEN
T
GENERAL
PARTICULAR
SCAL
E AN
D S
CO
PE O
F
CU
RR
ICU
LUM
DEC
ISIO
NS

Quadrant 4
Focuses on curriculum decision with a particular scope and immediate implications
Daily activities in the classroom and for each child

Enacted curriculum
Perceived part (by the teachers)

Qualitative feedback by teachers
Is there a real need for changes in syllabus? If so, why?

Qualitative feedback by teachers
I think the syllabus should change according to the changing needs. Beauty, suitability, relevance and applicability are some parameters for consideration when deciding on the contents of the syllabus.

Qualitative feedback by teachers
At the level you are operating, what are some of the benefits you reap, and what are the challenges you face whenever there is a change of syllabus?

Qualitative feedback by teachers
Teaching the new H2 Further Mathematics syllabus has given me the opportunity to pick up topics like Conics and Numerical Methods.
Whenever the syllabus changes there is often a necessity to learn new topics, let alone teach them. I have to pick up Conics from scratch as I have never learned it with such depth myself as a student.

Qualitative feedback by teachers
Describe the changes in the learning outcomes of the students over these years of changes in the syllabus.

Qualitative feedback by teachers
The changes made to the syllabus over the last few years have been trying to make the learning of mathematics more relevant to the student by relating it to the realworld contexts. The implementation of Learning Experiences and the introduction of Application Questions in the examination aim to achieve this goal.

Qualitative feedback by teachers
I do agree that mathematics should be seen as an effective tool in solving realworld problems but I also do not discount the fact that studying mathematics should be an end itself. That is, mathematics should be pursued regardless of whether it has potential for applications. But it almost always turns out that some obscure piece of pure mathematics holds the key to the answer of some deep questions in science.

Qualitative feedback by teachers
Sum up your experience/opinions concerning the change in Alevel mathematics syllabus over these 20 years.

Qualitative feedback by teachers
I think the H2 Math syllabus should provide opportunities for students to think logically and articulate mathematically. One area where students can develop these good qualities is Proofs. I am actually saddened that Mathematical Induction (MI) has been removed from the H2 Math syllabus. MI is an important tool in proving mathematical statements.

Qualitative feedback by teachers
Students can also appreciate the beauty of this technique and its logical foundation. Its a beautiful piece of mathematics. I feel that that some of the proofs in the current H2 Math syllabus are not rigorous enough. For example, to proof that a function is 11, one uses the horizontal line test which is incorrect.

Qualitative feedback from teachers
Mechanics is another subject which in my opinion lends itself perfectly to mathematical modelling and applications in realworld contexts. It also has crossdisciplinary interaction with Physics.

Qualitative feedback from teachers
I strongly maintain that traditional pure mathematics topics like Group Theory should be brought back to the A level FM syllabus to let students have a taste of handling mathematical proofs and understanding what a mathematical structure is. This idea of structure in mathematics is an important one and permeates almost all branches of mathematics Group structure, Measure Spaces, Normed Spaces, and Topological Spaces etc. are all mathematical structures.

Qualitative feedback from teachers
I remember the Math B and Further Math B syllabus I did as a student have given me a strong foundation to study mathematics in the university. I cant really say the same about our current syllabus.

Enacted curriculum
Operational part
The content actually delivered during instruction (i.e., instructional content), as well as how it is taught (i.e., instructional practices).
Typically, the content targets are based on the intended curriculum.

Attained curriculum
Perceived curriculum part of the Attained Curriculum
Learning experiences

Attained Curriculum
Learned curriculum part of the Attained Curriculum.
The knowledge and skills acquired by students during the schooling process.

21 century competencies

Learning experiences
It matters how and not just what students learn.
Mathematics Teaching and Learning Guide (Pre
University: H1, H2 and H3).

Learning experiences
Learning mathematics Learning concepts and skills Equipping with cognitive and metacognitive
process skills

Learning experiences
Engage in mathematical discussion where students actively reason and communicate their understanding to their peers and solve problems collaboratively;

Learning experiences
Construct mathematical concepts (e.g., to develop their own measure of linear relationship before being taught the formal concept) and form their own understanding of the concepts;

Learning experiences
Model and apply mathematics to a range of realworld problems (e.g., using exponential growth model to model population growth) afforded by the concepts and models in the syllabus;

Learning experiences
Make connections between ideas in different topics and between the abstract mathematics and the realworld applications and examples; and

Learning experiences
Use ICT tools to investigate, form conjecture and explore mathematical concepts (e.g., properties of graphs and their relationship with the algebraic expressions that describe the graph).

Constructivist classroom
A blend of pedagogies
Greater student participation, collaboration and discussion
Greater dialogue between teachers and peers

Constructivist classroom Students take on a more active role in
learning, and construct new understandings and knowledge.
Teachers role is to facilitate the learning process through more indepth dialogue and questioning, and guide students to build on their prior knowledge, and provide them with opportunities for more ownership and active engagement during learning

Constructivist classroom
Activity based learning, e.g., individual or group work, problem solving

Constructivist classroom
Teacherdirected inquiry, e.g., demonstration, posing questions

Constructivist class
Flipped classroom, e.g., independent study, followed by class discussion

Constructivist class
Seminar, e.g., mathematical discussion and discourse

Constructivist class
Project, e.g., mathematical modelling, statistical investigation

Constructivist class
Lab work, e.g., simulation, investigation using software and application

Assessed Curriculum
The knowledge and skills (i.e., the content) that are measured to determine student achievement.

Assessment
The role of assessment is to improve teaching and learning.

Assessment
For students, assessment provides them with information about how well they have learned and how they can improve.
For teachers, assessment provides them with information about their students learning and how they can adjust their instruction.

Assessment in math
Understanding of mathematics concepts Ability to draw connections and integrate
ideas across topics Capacity for logical thought, particularly,
the ability to reason, communicate and interpret, and
Ability to formulate, represent and solve problems within mathematics and other contexts.

Types of assessments
Summative Formative Diagnostic

GCE ALevel National Examination
Assessment objectives

9205 AO
The objectives are to test (a)understanding of relevant mathematical
concepts, techniques, terminology and notation;
(b)familiarity and facility with techniques; (c)ability to translate a verbally posed
problem into an equivalent mathematical problem and to solve this problem;

9205 AO
(d) ability to present mathematical work in a clear and logical way; (e) ability to present results clearly and check and interpret these results; (f) ability to work accurately in algebra, in calculus and in numerical work.

9233 (AO) The assessment will test candidates abilities to: 1. recall, select and use their knowledge of
appropriate use of precise techniques in a variety of contexts;
2. construct rigorous mathematical arguments through appropriate use of precise statements, logical deduction and inference and by the manipulation of mathematical expresssions;

9233 (AO)
The assessment will test candidates abilities to: 3. evaluate mathematical models, including
an appreciation of the assumptions made, and interpret, justify and present the results from a mathematical analysis in a form relevant to the original problem.

9740 (AO)
There are three levels of assessment objectives for the examination.

9740 (AO)
The assessment will test candidates abilities to: AO1 understand and apply mathematical
concepts and skills in a variety of contexts, including the manipulation of mathematical expressions and use of graphing calculators

9740 (AO)
The assessment will test candidates abilities to: AO2 reason and communicate
mathematically through writing mathematical explanation, arguments and proofs, and inferences

9740 (AO)
The assessment will test candidates abilities to: AO3 solve unfamiliar problems, translate
common realistic contexts into mathematics; interpret and evaluate mathematical results, and use the results to make predictions, or comment on the context.

9758 (AO)
There are three levels of assessment objectives for the examination.

9758 (AO)
The assessment will test candidates abilities to: AO1 Understand and apply mathematical
concepts and skills in a variety of contexts, including those that may be set in unfamiliar contexts, or require integration of concepts and skills from more than one topic.

9758 (AO)
The assessment will test candidates abilities to: AO2 Formulate realworld problems
mathematically, solve the mathematical problems, interpret and evaluate the mathematical solutions in the context of the problems.

9758 (AO)
The assessment will test candidates abilities to: AO3 Reason and communicate
mathematically through making deductions and writing mathematical explanations, arguments and proofs.

Use of a graphing calculator (GC)
Use of approved GC (w/o CAS)
Exam questions assume students accessibility to GC
Unsupported answers from GC allowed unless otherwise stated
No calculator commands Evidence of using GC counted
are credited for methods marks

Scheme of papers

9205 (SOP)
Paper 1 (3 hours) 50% Consisting of a first section of short questions
with no choice together with a second section of seven longer questions from which candidates may answer not more than 4.
Scoped from Section A, and additional topics from Section B

9205 (SOP)
Paper 2 (3 hours) 50% Containing 5 questions on each of the 3 options:
Particle Mechanics Statistics Pure Mathematics
Section C of the syllabus Answer no more than 7 questions, chosen from
any three options; no more than 4 questions can be chosen from one option

9233 (SOP)
Paper 1 (3 hours) About 14 questions of different marks and
lengths, based on Pure Math syllabus (Sections 1 to 16).
Answer all questions, except the last question which has a choice of two alternatives (12 marks each alternative).

9233 (SOP) Paper 2 (3 hours)
4 sections: A, B, C and D. Section A (Pure Math: 34 marks), about 5
questions of different marks based on Sections 1 to 16 of Pure Math Syllabus
Section B (Applied Math: 66 marks) About 4 Mechanics questions for 27 marks About 4 Statistics questions for 27 marks A final Either/Or questions with two alternatives (12
marks each: one Mechanics, one Probability & Statistics

9233 (SOP)
Paper 2 (3 hours) Section C (Particle Mechanics: 66 marks)
About 6 to 7 questions for 54 marks A final Either/Or question worth 12 marks each
alternative Section D (Probability & Statistics: 66 marks)
About 6 to 7 questions for 54 marks A final Either/Or question worth 12 marks each
alternative

9233 (SOP)
For Paper 2, candidates will Answer all questions in Section A. Choose one of Sections B, C or D, and
answer all questions in that section, except the last question, where candidates choose one out of the two alternatives.

9740 (SOP)
Paper 1 (3 hours) Consisting of 10 to 12 questions of different
lengths and marks based on the Pure Math section of the syllabus
Answer all questions

9740 (SOP)
Paper 2 (3 hours) Consisting of 2 sections, Sections A and B Section A (Pure Math 40 marks)
About 3 to 4 questions of different lengths and marks based on Pure Math section
Section B (Statistics 60 marks) About 6 to 8 questions of different lengths and
marks based on Statistics section Answer all questions

9758 (SOP) Same as 9740 Scheme of Papers except that, in addition, we have: There will be at least two questions on [Paper 1 & Section B of Paper 2] application of Mathematics in realworld contexts, including those from sciences and engineering. Each question will carry at least 12 marks and may require concepts and skills from more than one topic.

Item comparison

A very old FM question A student has a calculator that will no longer find natural logarithms. The student finds ln (3) by solving the equation
3 = 0 using the NewtonRaphson method with 0 = 1. Use this method to obtain a value for ln (3) to 5 significant figures.

9233/Nov 06/P2/Q5 Using a graphical argument, or otherwise, show that the equation
3 + = 100 has exactly one real root, . The iteration
+1 = (100 )3 ,
with initial approximation 1, converges. Explain why the iteration converges to , and use the iteration to find correct to 3 decimal places.

9740/Nov 07/P1/Q9

9740/Nov 07/P1/Q9
The diagram shows the graph of = 3.
The two roots of the equation 3 = 0 are denoted by and , where < . (i) Find the values of and , each correct to 3 decimal places.

9740/Nov 07/P1/Q9 A sequence of real numbers 1, 2, 3, satisfies the recurrence relation
+1 =13
for n > 1. (ii) Prove algebraically that, if the sequence converges, then it converges to either or . (iii) Use a calculator to determine the behaviour of the sequence for each of the cases 1 = 0, 1 = 1, 1 = 2.

9740/Nov 07/P1/Q9
(iv)By considering +1 , prove that +1 < if < < ,
+1 > if < or > .
(v) State briefly how the results in part (iv) relate to the behaviours determined in part (iii).

Findings and implications

Eisners model
MOE
Quadrant 3
Time at which and for which
SCHOOL
Quadrant
1
Decisions are made
SCHOOL
Quadrant 4
SCHOOL
Quadrant
2
FUTU
RE
PR
ES
EN
T
GENERAL
PARTICULAR
SC
ALE
AN
D S
CO
PE
OF
C
UR
RIC
ULU
M D
EC
ISIO
NS

Findings and implications
The school takes a larger share of responsibility in terms of curriculum decisionmaking!

Findings and implications
Changes in Alevel math syllabus can be caused by or associated to different ideological orientation
Changes in Alevel math syllabus arise from decisionmaking in curriculum; short vs longterm, and general vs specific

Findings and implications
Changes in Alevel math syllabus manifest themselves in different domains: Intended Enacted Perceived & Learned Assessed

Some research questions At what levels are the changes in the A level
mathematics syllabus initiated and being operated?
Do the usual orientations, i.e., curriculum ideologies (academic scholar, social efficiency, studentcentered, social reformation) account for these changes? Any one which is dominant, or should it be a hybrid?

Some research questions
Is the generic framework offered proposed by Eisner (2007) rigorous enough to account for the changes specific to the context of A level mathematics in Singapore?
Are there anything peculiar to A level mathematics teaching in the schools that are different from those at Primary and Secondary Schools?

Some research questions
How best can the schools top management, its middle leadership and its teachers (and perhaps its students) best make out of these changes, cope with them, and even thrive and benefit from them?

Some research questions
To what extent and how well all the policy elements in the educational system working together to guide instruction and, ultimately, facilitate and enhance student learning?
In other words, how well do the different forms of curriculum for Alevel Mathematics align?

Some research questions
How do the theoretical considerations presented here (e.g., ideological orientations, and Eisners model) help the (a) policy maker, (b) school leader, (b) middle leader, and (c) classroom teacher achieve a better alignment between the different forms of the Alevel math curricula?

Some research questions
Will there be a new worldorder in which one day we shall be applying social reconstruction to our education system? Or are we already doing that now?

Conclusion
Blind men and the elephant

References Eisner, E. (2002). The educational
imagination: On the design and evaluation of school programs (3rd ed.). Upper Saddles river, NJ: Pearson Education.
Porter, A. C. (2006). Curriculum assessment. In J. L. Green, G. Camilli, & P. B. Elmore (Eds.), Complementary methods for research in education (3rd edition). Washington, DC: American Educational Research Association.

References
RatnamLim, C. (2017). Decisionmaking in curriculum leadership. In Kelvin Tan H. K., Mary Anne Heng and Christina RatnamLim (Eds), Curriculum Leadership by Middle Leaders (pp. 4257): Routledge.
Schiro, M. (2013). Curriculum theory: Conflicting visions and enduring concerns (2nd Ed.). Los Angeles: Sage Publications.

Thank you
Special acknowledgements: June Tan, Dennis Yeo (MOE) Wang Juat Yong (DHS) Leong Chong Ming (NYJC) Chan Puay San (IJC) Dr. Christina RatnamLim (NIE/CTL) MProSE/MInD Team (all the bosses) Prof. Berinderjeet Kaur (NIE/MME) A/P Ang Keng Cheng (NIE/MME)
A Vicennial Walk through A Level Mathematics in SingaporeOutlineIntroductionIntroductionIntroductionIntroductionIntrinsic tensionEvolution of ALevel Math SyllabiEvolutionary historyEvolutionary history848/Jun 84/P2/Q15 Numerical calculation848/Dec 74/P1/Q5Pure Mathematics848/Dec 74/P1/Q5Pure Mathematics848/Jun 75/P1/Q17Algebraic Structure848/Nov 79/P1/Q14Numerical Analysis and Computation848/Nov 79/P1/Q14Numerical Analysis and Computation848/Nov 79/P1/Q14Numerical Analysis and Computation9202/Nov 81/Q4Pure Mathematics9202/Jun 83/Q10Particle Mechanics9202/Jun 83/Q10Particle Mechanics9202/Jun 85/P1/Q13(b)Probability and Statistics9205/Nov 86/P1/13(i)Pure Mathematics9205/Nov 86/P1/13(i)Pure Mathematics9205/Nov 87/S/Q5Pure Mathematics9205/Jun 88/P1/Q17Pure Mathematics9205/Nov 90/P2/Q11(b)Pure Mathematics9205/Jun 91/P1/Q16(b)Pure Mathematics9205/Jun 91/P1/Q16(b)Pure Mathematics9205/Jun 91/P2/Q15(i)Pure Mathematics9205/Jun 91/P2/Q15(i)Pure Mathematics9205/Jun 91/P2/Q15(i)Pure Mathematics9233/Nov 00/P1/Q49233/Nov 01/Q11 (Section I)9233/Nov 01/P2/Q7(i)(Option b) Statistics9233/Nov 06/P1/Q14 EITHER(i)Pure Mathematics9233/Nov 06/P1/Q14 EITHER(i)Pure Mathematics9233/Nov 06/P2/Q24(Section D) Probability and Statistics9233/Nov 06/P2/Q13 EITHER(Section B) Applied Mathematics9740/Nov 07/P1/Q99740/Nov 07/P1/Q99740/Nov 07/P1/Q99740/Nov 08/P2/Q3(a)9740/Nov 10/P2/Q2(ii)9740/Nov 14/P2/Q11(ii)9740/Nov 16/P1/Q4(i)Pure Mathematics9740/Nov 16/P1/Q10(b)Pure Mathematics9740/Nov 16/P2/Q6(i)Probability and StatisticsCurriculum ideologiesCurriculum ideologiesScholar academicScholar academicScholar academicSocial efficiencySocial efficiencySocial efficiencyLearner centeredLearner centeredSocial reconstructionSocial reconstructionOrientationEisners modelEisners modelEisners modelEisners modelQuadrant 1Eisners modelQuadrant 2Eisners modelQuadrant 3Slide Number 70Evolutionary historySlide Number 72Slide Number 73Slide Number 74Significant featuresRevised 2001 Alevel Math Syllabus (9233)2006 Alevel Math Syllabus (H2 Math 9740)2016 Alevel Math Syllabus (H2 Math 9758)Porters frameworkIntended curriculumFramework for theSingapore Mathematics CurriculumSyllabus aimsScholar academicScholar academicScholar academicScholar academicScholar academicScholar academicScholar academicScholar academicScholar academicScholar academicSocial efficiencySocial efficiencySocial efficiencySocial efficiencySocial efficiencySocial efficiencySocial efficiencyLearner centeredLearner centeredLearner centeredLearner centeredLearner centeredEisners modelQuadrant 4Enacted curriculumQualitative feedback by teachersQualitative feedback by teachersQualitative feedback by teachersQualitative feedback by teachersQualitative feedback by teachersQualitative feedback by teachersQualitative feedback by teachersQualitative feedback by teachersQualitative feedback by teachersQualitative feedback by teachersQualitative feedback from teachersQualitative feedback from teachersQualitative feedback from teachersEnacted curriculumAttained curriculumAttained Curriculum21 century competenciesLearning experiencesLearning experiencesLearning experiencesLearning experiencesLearning experiencesLearning experiencesLearning experiencesConstructivist classroomConstructivist classroomConstructivist classroomConstructivist classroomConstructivist classConstructivist classConstructivist classConstructivist classAssessed CurriculumAssessmentAssessmentAssessment in mathTypes of assessmentsGCE ALevel National Examination9205 AO9205 AO9233 (AO)9233 (AO)9740 (AO)9740 (AO)9740 (AO)9740 (AO)9758 (AO)9758 (AO)9758 (AO)9758 (AO)Use of a graphing calculator (GC)Scheme of papers9205 (SOP)9205 (SOP)9233 (SOP)9233 (SOP)9233 (SOP)9233 (SOP)9740 (SOP)9740 (SOP)9758 (SOP)Item comparisonA very old FM question9233/Nov 06/P2/Q59740/Nov 07/P1/Q99740/Nov 07/P1/Q99740/Nov 07/P1/Q99740/Nov 07/P1/Q9Findings and implicationsEisners modelFindings and implicationsFindings and implicationsFindings and implicationsSome research questionsSome research questionsSome research questionsSome research questionsSome research questionsSome research questionsConclusionReferencesReferencesThank you