a vicennial walk through ‘a’ level mathematics in...

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, MTC’2017

Outline

• Introduction • Curriculum ideologies • Evolution of A-Level Math Syllabus • Findings and implications • Conclusion

Introduction

Intrinsic tension

Evolution of A-Level 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 1999-2001 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

2002-2006 MOE, Singapore, in collaboration with UCLES

9233 Mathematics

MOE+UCLES 9234 Further Mathematics 2007-2016 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FM withdrawn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

𝑅 =12𝑑 +

𝑎2

6𝑑.

If the value of d is subject to a small error ±δ, obtain an expression for the corresponding approximate error ρ in R. When a = 60, d = 18, and δ = 0.5 (all lengths being in mm), calculate R and ρ, and hence estimate the limits for R.

848/Dec 74/P1/Q5 Pure Mathematics

Write down 𝐓−1, where

𝐓 = 5 44 3 .

Verify that if

𝐌 = −14 20−12 17 and 𝐁 = 2 0

0 1 ,

then 𝐌 = 𝐓𝐓𝐓−1.

848/Dec 74/P1/Q5 Pure Mathematics

Show (without using the numerical forms of the matrices) that 𝐌3 = 𝐓𝐓3𝐓−1, 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

𝐓 = 5𝜆 4𝜇4𝜆 3𝜇 ,

where 𝜆 ≠ 0, 𝜇 ≠ 0, then 𝐌 = 𝐓𝐓𝐓−1.

848/Jun 75/P1/Q17 Algebraic Structure

For the sets P, Q, A, B, give the formal definitions of

𝑃 ∩ 𝑄 and 𝑃 ∪ 𝑄, and formal proofs of the relations

𝐴 ∩ 𝐴 ∪ 𝐵 = 𝐴, 𝐴 ∪ 𝐴 ∩ 𝐵 = 𝐴.

[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 δ.


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 𝑡 remains greater than δ. 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 sin𝜃 − 2 cos𝜃 = 2.5, giving your answer in degrees to the nearest 0.1°, (ii) the equation cos 3𝜃 = sin𝜃 , 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

2𝑎.

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: Σ𝑥 = 2685, Σ𝑥2 = 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 𝑥)3𝑑𝑥𝑒2

𝑒 ,

(ii) ∫ (𝑥+1)2

𝑥(𝑥2+3𝑥+3)21 𝑑𝑥;

(iii) ∫ 14 cos 𝜃+3 sin 𝜃3 cos 𝜃+4 sin 𝜃

𝑑𝜃𝜋20 .

9205/Jun 88/P1/Q17 Pure Mathematics

Prove by induction that the following results are true for all positive integers n.

(i) ∑ 𝑟2(𝑟 − 1)𝑛𝑟=1 = 1

12𝑛 𝑛2 − 1 3𝑛 + 2 .

(ii) Given that 𝑦 = 𝑥𝑒𝑥, then 𝑑𝑛𝑦

𝑑𝑥𝑛= 𝑥 + 𝑛 𝑒𝑥.

9205/Nov 90/P2/Q11(b) Pure Mathematics

The equation, in polar coordinates, of a curve is 𝑟 = asin𝜃 cos2 𝜃,

where a is a positive constant, and 0 ≤ 𝜃 ≤ 𝜋. Show that the greatest value of r is 2𝑎

3√3 , and

sketch the curve.

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

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

The point P, with x-coordinates 𝑥1, lies on the curve C with equation 𝑦 = 𝑒2𝑥. The tangent to the curve at P cuts the x-axis at the point T, as shown in the diagram. Show that the coordinates of T are (𝑥1 −

12

, 0). Deduce that any straight line which passes through a given point (ℎ, 0) on the x-axis and which cuts C twice has gradient greater than 2𝑒2ℎ+1.

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, ABB’A’ and DCC’C’, where AA’ = BB’ = CC’ = DD’ = 2 m. The roof consists of the planes A’B’SR and C’D’RS, where RS is horizontal. Each section of the roof is inclined at an angle θ to the horizontal, where tan𝜃 = 3

4.


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 𝐫. 22𝐢 + 33𝐣 − 12𝐤 = 65 passes through B, D and S.

9233/Nov 00/P1/Q4

By considering the derivative as a limit, show that the derivative of 𝑥3 is 3𝑥2.

9233/Nov 01/Q11 (Section I)

Given that 𝑦 = tan (12

tan−1 𝑥), show that

1 + 𝑥2𝑑𝑦𝑑𝑥

=12

1 + 𝑦2 .

By differentiating this result twice, show that up to and including the term in 𝑥3, Maclaurin’s series for tan(1

2tan−1 𝑥) is

12𝑥 − 1

8𝑥3.

9233/Nov 01/P2/Q7(i) (Option b) Statistics

The continuous random variable X has a uniform distribution on 0 ≤ 𝑥 ≤ 𝑎. 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 𝑥 = 𝑐𝑡, 𝑦 = 𝑐

𝑡, where c is a positive constant. Three

points 𝑃 𝑐𝑐, 𝑐𝑝

,𝑄 𝑐𝑞, 𝑐𝑞

and 𝑅 𝑐𝑟, 𝑐𝑟

on the curve are shown in the diagram.

(i) Prove that the gradient of QR is 1𝑞𝑟

.

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 µ, 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 µ’, 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 𝑦 = 𝑒𝑥 − 3𝑥.

The two roots of the equation 𝑒𝑥 − 3𝑥 = 0 are denoted by 𝛼 and 𝛽, where 𝛼 < 𝛽. (i) Find the values of α and β, each correct to 3 decimal places.

9740/Nov 07/P1/Q9 A sequence of real numbers 𝑥1, 𝑥2, 𝑥3, … satisfies the recurrence relation

𝑥𝑛+1 =13𝑒𝑥𝑛

for n > 1. (ii) Prove algebraically that, if the sequence converges, then it converges to either α or β. (iii) Use a calculator to determine the behaviour of the sequence for each of the cases 𝑥1 = 0, 𝑥1 = 1, 𝑥1 = 2.

9740/Nov 08/P2/Q3(a)

The complex number w has modulus r and argument θ, where 0 < θ < 1

2𝜋 , and 𝑤∗

denotes the conjugate of w. State the modulus and argument of p, where 𝑐 = 𝑤

𝑤∗ .

Given that 𝑐5 is real and positive, find the possible values of θ.

9740/Nov 10/P2/Q2(ii)

(a) Prove by the method of difference that

�1

𝑟(𝑟 + 2)=

34−

12 𝑛 + 1

−1

2 𝑛 + 2.

𝑛

𝑟=1

(b) Explain why ∑ 1𝑟(𝑟+2)

∞𝑟=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 non-zero. A geometric series has first term b and common ratio r, where b and r are non-zero. 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 5𝑟10 − 8𝑟3 +3 = 0. Given that 𝑟 < 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 non-negative integers, is given by

𝑔 𝑛 =

1 for 𝑛 = 0

2 + 𝑔12𝑛 for 𝑛 𝑒𝑒𝑒𝑛

1 + 𝑔 𝑛 − 1 for 𝑛 𝑜𝑑𝑑

(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

Eisner’s model

• Helps us understand the various contexts of curriculum decision-making 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)

Eisner’s model

• Two-dimensional representation of the scope and timeliness of curriculum decisions, organized into four quadrants.

Eisner’s model

Quadrant 3

Time at which and for which

Quadrant 1

Decisions are made

Quadrant 4

Quadrant 2

FUTU

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PARTICULAR

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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

Eisner’s model

Quadrant 3


Quadrant 1

Decisions are made

Quadrant 4

Quadrant 2

FUTU

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GENERAL

PARTICULAR

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Quadrant 2

• Focuses on the particular but long-term decisions

• Long-term implications in a school context – Ascertain implications of nation-wide policy on

the scope of content, pedagogy and assessment

– Consider how implications impact on school’s particular setup and needs

Eisner’s model

Quadrant 3


Quadrant 1

Decisions are made

Quadrant 4

Quadrant 2

FUTU

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GENERAL

PARTICULAR

SCAL

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CU

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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 16-19 (2-3 Years)

Evolutionary history Period (Exam Year) Exam Syndicate Subject code 1974-1979 Cambridge 848 (M) 1980-1985 Cambridge 9202 (M) 1986-2001 Cambridge 9205 (M)

9225 (FM) 2002 Cambridge-Singapore 9233 (M: interim)

9225 (FM) 2003-2006 Singapore-Cambridge 9233 (M: New)

9234 (FM) 2007-2017 Singapore-Cambridge 9740 (M)

FM removed 2017-Present Singapore-Cambridge 9758 (M)

9649 (FM: re-introduce)

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 A-level math curriculum in the 2001 FM (9234) syllabus.

2001 9205 (Interim) to 9233 (Revised) 9234 FM (Revised) - GC in Math - GC-neutral exam

Year of policy



2002

Review of JC/Upper Secondary Education Landscape It was actually recommendation from this review that led to changes in the A-level education landscape and the 2006 A-level 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 A-level 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 re-think how they teach.

Year of policy



2015 ESGS Every School is a Good School Student-centric, 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 ‘Every’s. 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.

Re-introduction of Further Math

Significant features

• Revised 2001 Math Syllabus (9233) • 2006 A-level Math Syllabus (H2 Math 9740) • 2016 A-level Math Syllabus (H2 Math 9758)

Revised 2001 A-level 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 ‘GC-neutral’. 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; Either-Or option for the last question of Paper 1.

2006 A-level Math Syllabus (H2 Math 9740)

• Emphasis on solving real-world 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 A-level 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

Porter’s 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 student’s concurrent study at A-level; (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 problem-solving

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 problem-solving

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 learner-centric, values-driven education.

• Since 2011, there’s been a specific focus on schools, students, teachers and parents.

Eisner’s model

Quadrant 3


Quadrant 1

Decisions are made

Quadrant 4

Quadrant 2

FUTU

RE PR

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T

GENERAL

PARTICULAR

SCAL

E AN

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PE O

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CU

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DEC

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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?


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.


• 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?


• 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.


• Describe the changes in the learning outcomes of the students over these years of changes in the syllabus.


• 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 real-world contexts. The implementation of ‘Learning Experiences’ and the introduction of ‘Application Questions’ in the examination aim to achieve this goal.


• I do agree that mathematics should be seen as an effective tool in solving real-world 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.


• Sum up your experience/opinions concerning the change in A-level mathematics syllabus over these 20 years.


• 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.


• Students can also appreciate the beauty of this technique and its logical foundation. It’s 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 1-1, 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 real-world contexts. It also has cross-disciplinary interaction with Physics.


• 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.


• 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 can’t 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 mathematics – Learning concepts and skills – Equipping with cognitive and metacognitive

process skills


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


• 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;


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


• Make connections between ideas in different topics and between the abstract mathematics and the real-world applications and examples; and


• 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.

• Teacher’s role is to facilitate the learning process through more in-depth 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


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


• Teacher-directed inquiry, e.g., demonstration, posing questions

Constructivist class

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


• Seminar, e.g., mathematical discussion and discourse


• Project, e.g., mathematical modelling, statistical investigation


• 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 A-Level 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 real-world 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 real-world 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

𝑒𝑥 − 3 = 0 using the Newton-Raphson method with 𝑥0 = 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

𝑥3 + 𝑥 = 100 has exactly one real root, 𝛼. The iteration

𝑥𝑛+1 = (100 − 𝑥𝑛)3 , with initial approximation 𝑥1, converges. Explain why the iteration converges to 𝛼, and use the iteration to find 𝛼 correct to 3 decimal places.

9740/Nov 07/P1/Q9

9740/Nov 07/P1/Q9

The diagram shows the graph of 𝑦 = 𝑒𝑥 − 3𝑥.

The two roots of the equation 𝑒𝑥 − 3𝑥 = 0 are denoted by 𝛼 and 𝛽, where 𝛼 < 𝛽. (i) Find the values of α and β, each correct to 3 decimal places.

9740/Nov 07/P1/Q9 A sequence of real numbers 𝑥1, 𝑥2, 𝑥3, … satisfies the recurrence relation

𝑥𝑛+1 =13𝑒𝑥𝑛

for n > 1. (ii) Prove algebraically that, if the sequence converges, then it converges to either α or β. (iii) Use a calculator to determine the behaviour of the sequence for each of the cases 𝑥1 = 0, 𝑥1 = 1, 𝑥1 = 2.

9740/Nov 07/P1/Q9

(iv)By considering 𝑥𝑛+1 − 𝑥𝑛, prove that 𝑥𝑛+1 < 𝑥𝑛 if 𝛼 < 𝑥𝑛 < 𝛽 ,

𝑥𝑛+1 > 𝑥𝑛 if 𝑥𝑛 < 𝛼 or 𝑥𝑛 > 𝛽.

(v) State briefly how the results in part (iv) relate to the behaviours determined in part (iii).

Findings and implications

Eisner’s model

MOE

Quadrant 3


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

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The school takes a larger share of responsibility in terms of curriculum decision-making!


• Changes in A-level math syllabus can be caused by or associated to different ideological orientation

• Changes in A-level math syllabus arise from decision-making in curriculum; short- vs long-term, and general vs specific


• Changes in A-level 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, student-centered, 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?


• How best can the school’s 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?


• 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 A-level Mathematics align?


• How do the theoretical considerations presented here (e.g., ideological orientations, and Eisner’s 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 A-level math curricula?


• Will there be a new world-order 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

• Ratnam-Lim, C. (2017). Decision-making in curriculum leadership. In Kelvin Tan H. K., Mary Anne Heng and Christina Ratnam-Lim (Eds), Curriculum Leadership by Middle Leaders (pp. 42-57): 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 Ratnam-Lim (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...

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