Prepared by

**The Curriculum Development Council**
Recommended for use in schools by
**The Education Bureau**

**HKSARG**
**2017**

**2017** **Mathematics Education ** **Key Lear** **ning Ar** **ea Curriculum Guide ** **(Primary ** **1 ** **– Secondary 6** **) **

**Mathematics ** **Education**

**Key Learning Area Curriculum Guide**

**(Primary 1 – Secondary 6) **

**(Blank Page) **

i

**Preambles **

The development of the Hong Kong school curriculum has advanced into a new phase of ongoing renewal and updating. It ushers in a new era for curriculum development to keep abreast of the macro and dynamic changes in various aspects in the local, regional and global landscapes in maintaining the competitiveness of Hong Kong. For the ultimate benefits of our students, schools are encouraged to sustain and deepen the accomplishments achieved since the Learning to Learn curriculum reform started in 2001, and to place new emphases on future needs in curriculum development for achieving the overall aims and learning goals of the school curriculum.

The eight Key Learning Area (KLA) Curriculum Guides (Primary 1 - Secondary 6) have
been updated and recommended by the Curriculum Development Council (CDC)^{1} to
support the ongoing renewal of the school curriculum at the primary and secondary levels.

In updating the KLA Curriculum Guides, the respective KLA committees under the CDC have taken into consideration the concerns, needs and suggestions of various key stakeholders including schools, principals, teachers, students and the public at large. A series of school briefing cum feedback collection sessions coupled with a territory-wide school survey were conducted in 2015 to gauge schools’ views on the major updates of the respective Curriculum Guides.

The eight KLA Curriculum Guides (2017) supersede the 2002 versions. Each KLA Curriculum Guide presents the updated curriculum framework which specifies the KLA’s curriculum aims, learning targets and objectives, delineates the direction of ongoing curriculum development at the KLA level, and provides suggestions on curriculum planning, learning and teaching strategies, assessment, as well as useful learning and teaching resources. In addition, updated examples of effective learning, teaching and assessment practices are provided for schools’ reference. Supplements to some KLA Curriculum Guides and subject curriculum guides are also available to provide further suggestions on their implementation at specific key stages. Schools are encouraged to adopt the recommendations in the KLA Curriculum Guides, taking into account the school contexts, teachers’ readiness and learning needs of their students.

1 The CDC is an advisory body offering recommendations to the Government on all matters relating to school curriculum development from kindergarten to secondary levels. Its membership includes heads of schools, teachers, parents, employers, academics from tertiary institutions, professionals from related fields or related bodies, representatives from the Hong Kong Examinations and Assessment Authority (HKEAA), and officers from the Education Bureau.

For a better understanding of the interface between various key stages and connections of different learning areas, and how effective learning, teaching and assessment can be achieved, schools should make reference to all related curriculum documents recommended by the CDC and the latest versions of the Curriculum and Assessment Guides jointly prepared by the CDC and the HKEAA for the senior secondary curriculum to ensure coherence in curriculum planning at the school, KLA and subject levels.

As curriculum development is a collaborative and ongoing process, the KLA Curriculum Guides will be under regular review and updating in light of schools’ implementation experiences as well as the changing needs of students and society.

Views and suggestions on the development of the Mathematics Education KLA Curriculum are always welcome. These may be sent to:

Chief Curriculum Development Officer (Mathematics) Curriculum Development Institute

Education Bureau

4/F, Kowloon Government Offices, 405 Nathan Road, Yau Ma Tei, Kowloon Fax: 3426 9265

E-mail: math@edb.gov.hk

iii

**Key Messages **

**Mathematics Education KLA **

Mathematics is essential in the school curriculum as it is a crucial mode of thinking that helps students acquire the ability to explore, conjecture and reason logically, a powerful means of communication, a foundation for the study of other disciplines, and an intellectual endeavor. Mathematics therefore plays an important role in helping students develop necessary skills for lifelong learning.

**The Direction of Curriculum Development in Mathematics **

In response to the changing needs of society, the rapid development of science and technology, the results of international studies on our education system, as well as views of stakeholders, the Mathematics Education KLA curriculum is developed in a direction to extend the existing strengths, to enhance students’ learning progression and to align with the focal points of ongoing renewal of school curriculum. The focal points that connect with the development of the Mathematics Education KLA include STEM education, information technology in education, Language across the Curriculum, etc.

**Aims of Mathematics Curriculum **

To develop students’ ability to conceptualise inquire, reason, communicate, formulate and solve problems mathematically; and their capability of appreciating the aesthetic nature and cultural aspects of mathematics.

**The Central Curriculum of Mathematics: An Open and Flexible Framework **

The central curriculum, in the form of an open and flexible framework, sets out what schools are encouraged to help students develop:

Subject knowledge and skills as embodied in the learning units under different strands or areas;

Generic Skills; and

Positive values and attitudes.

**Planning School-based Mathematics Curriculum **

Taking curriculum documents of the Mathematics Education KLA as major references

Taking into account the school contexts, the overall aims of the Mathematics curriculum and the focal points and major renewed emphases (MRE) of the ongoing curriculum renewal (such as STEM education and information technology in education)

Making use of the flexibility provided by the curriculum framework to cater for learner diversity, to enhance learning progression, and to plan learning and teaching sequences that facilitate cross-KLA learning activities

Adopting appropriate learning and teaching resources, such as textbooks, e-resources and community resources

**Learning and teaching of Mathematics **

Arranging diversified learning activities at different levels, such as hands-on exploratory activities, project work, mathematics reading activities, and activities that based on a topic in Mathematics to integrate relevant learning elements from other KLAs

Incorporating the use of information technology for interactive learning and self-directed learning

Adopting different strategies to cater for learner diversity, such as adapting the Mathematics curriculum and using the curriculum space created flexibly for consolidation and enrichment

Assigning quality homework to consolidate learning, and discouraging mechanical drilling

**Assessment **

Arranging assessments to collect ongoing information about the progress of student learning to provide timely and quality feedback for students to improve learning, and for teachers to adjust their teaching strategies

Providing diversified modes of assessment (such as classroom observation, questioning, open-ended questions, exploratory tasks and projects) for improving learning and teaching

Making use of suitable assessment tools, such as Learning Progression Framework (LPF) and Student Assessment Repository (STAR) to facilities assessment for learning and assessment as learning

*(For more information on various curriculum matters, please refer to Basic Education *
*Curriculum Guide – To Sustain, Deepen and Focus on Learning to Learn (Primary 1 – 6) *
*(2014) and Secondary Education Curriculum Guide (2017).)*

v

**Contents **

Page

**Preamble ** i

**Key Messages ** iii

**Chapter 1 ** **Introduction ** 1

1.1 What is a Key Learning Area?

1.2 Position of the Mathematics Education KLA in the School Curriculum

1.3 Rationale and Direction for Development 1.3.1 Rationale for the Development of the

Mathematics Education KLA

1.3.2 Direction for the Development of the Mathematics Education KLA

1.4 Strategies for Development

2 2 4 4 6 7

**Chapter 2 ** **Curriculum Framework ** 11

2.1 Aims of the Mathematics Education KLA Curriculum 2.2 Components of the Curriculum Framework

2.2.1 Strands, Learning Targets and Objectives 2.2.2 Generic Skills

2.2.3 Values and Attitudes

2.2.4 Developing Generic Skills, and Values and Attitudes

2.3 Curriculum Organisation 2.4 Core and Extension

11 11 12 28 29 31 33 35

**Chapter 3 ** **Curriculum Planning ** 39

3.1 A Balanced Curriculum

3.2 Central Curriculum and School Curriculum Development

3.3 Strengthening STEM Education

3.4 Strengthening Information Technology in Education 3.5 Strengthening Values Education

39 39 42 45 46

Page 3.6 Smooth Transition between Different Key Stages and

Multiple Pathways

3.6.1 Smooth Transition between Kindergarten and Primary

3.6.2 Smooth Transition between Primary and Secondary

3.6.3 Smooth Transition between Junior Secondary and Senior Secondary 3.6.4 Supporting Students in Educational and

Vocational Pathways 3.7 Cross-KLA Linkage

3.8 Time Allocation

48 49 50 51 52 53 55

**Chapter 4 ** **Learning and Teaching ** 57

4.1 Guiding Principles

4.2 Approaches to Learning and Teaching 4.2.1 The Four Key Tasks

4.2.2 Life-wide Learning 4.2.3 e-Learning

4.2.4 Meaningful Homework

4.3 Learning and Teaching for STEM Education 4.4 Embracing Learner Diversity

4.4.1 Central Curriculum Aspect 4.4.2 School Aspect

4.4.3 Classroom Aspect

57 58 60 66 67 69 70 75 76 76 79

**Chapter 5 ** **Assessment ** 87

5.1 Guiding Principles

5.2 Formative and Summative Assessment 5.2.1 Purposes of Assessment 5.2.2 Modes of Assessment

5.2.3 Promoting Formative Assessment 5.3 Internal and External Assessment

5.3.1 Internal Assessment 5.3.2 External Assessment

87 90 90 91 95 97 97 98

vii

Page

**Chapter 6 ** **Learning and Teaching Resources ** 101

6.1 Quality Textbooks and Other Learning and Teaching Resources

6.1.1 Textbooks

6.1.2 Learning and Teaching Resources Other than Textbooks

6.1.3 Resources in Support of Curriculum Development

6.2 Effective use of Learning and Teaching Resources 6.3 Resource Management in Schools

101 101 103 104 105 107

**Examples ** 109

**Appendices ** 207

**Bibliography ** 253

**Membership of the Curriculum Development Council Committee on **
**Mathematics Education **

257

**Examples **

Page 1. Hand in Hand

2. Sharing Game

3. Making Your Own Measuring Cup 4. Discovering Symmetry

5. Finding Area

6. Rubber Band Powered Car 7. Cross Sections of 3-D Shapes 8. Knowing Your Community 9. Extra-Curricular Activities 10. Four Seasons

11. Investigating Errors of Measurements by GPS Tracking Apps 12. Design a Healthy Diet Menu

13. Mathematics Magic

14. Open-ended Geometric Problem 15. Slopes of Perpendicular Lines 16. Volume of Frustums

17. Surface Areas of Prisms 18. Flipping Measure Spoons 19. Translations of Functions

20. Modelling the Spread of a Disease 21. Return and Risk

22. Mathematics Reading Scheme

112 117 122 126 130 133 138 141 145 148 153 156 158 162 164 167 172 178 182 189 195 202

ix

**Appendices **

Page 1. Seven Learning Goals of the Primary and Secondary Education

2. Development of Generic Skills in the Mathematics Curriculum 3. Integrative Application of Generic Skills

4. Examples of Websites and Application Software (Apps) for Learning and Teaching of Mathematics

5. Learning and Teaching Resources List for Mathematics

6. List of Collaborative Research and Development (“Seed”) Projects for Mathematics

207 209 232 239

242 249

**List of Tables **

Page 1. Overview of Learning Targets

2. Overview of Learning Units

3. Time Allocation for Mathematics Curriculum 4. Examples of Mathematics Competitions

5. Examples of STEM-related activities for selected topics in junior secondary Mathematics

15 22 55 67 72

**Chapter 1 **

**Introduction **

1

**Chapter 1 Introduction **

In response to the changing needs of society, the rapid development of science and
technology, the views of stakeholders collected through various surveys and engagement
activities as well as the need to align with the direction for the ongoing curriculum
*renewal of the school curriculum, the recommendations provided in the Mathematics *
*Education Key Learning Area Curriculum Guide (Primary 1 - Secondary 3) (2002) have *
been reviewed. Building on the strengths of Hong Kong students in mathematics, the
curriculum content of the Mathematics Education Key Learning Area (ME KLA) have
been updated to enhance students’ learning progression and to align with the focal points
of the ongoing renewal of the school curriculum, such as Science, Technology,
Engineering and Mathematics (STEM) education and IT in education, for further
enhancing the quality and effectiveness of learning, hence enabling students to become
more effective lifelong learners in the 21st century.

**The Mathematics Education Key Learning Area Curriculum Guide (Primary 1 - *** Secondary 6) (2017) (this Guide) is prepared by the Curriculum Development Council *
Committee (CDCC) on Mathematics Education. It is an updated version of the

*Mathematics Education Key Learning Area Curriculum Guide (Primary 1 - Secondary 3)*(2002) and has been extended to include the three-year senior secondary Mathematics Education curriculum to provide reference for schools in developing a coherent school Mathematics curriculum.

The direction of development in this Guide aligns with the Seven Learning Goals of
Primary and Secondary Education (see Appendix 1) and the major recommendations in
**the Basic Education Curriculum Guide – To Sustain, Deepen and Focus on Learning to ****Learn (Primary 1 – 6) (2014) and the Secondary Education Curriculum Guide (2017). **

This Guide provides the overall direction for the development of the Mathematics
Education curriculum in the five to ten years to come. It reinforces the curriculum
*emphases provided in the Mathematics Education Key Learning Area Curriculum Guide *
*(Primary 1 - Secondary 3) (2002) to further enhance learning and teaching and supports *
the focal points and major renewed emphases (MRE) of the ongoing renewal of school
curriculum which take into account the significant development in our society and around
the world in various fields, and for the ultimate benefits of student learning. This Guide
includes examples relevant to different key stages of learning to illustrate the concepts and
ideas introduced and to narrow the gap in curriculum implementation.

**1.1 ** **What is a Key Learning Area? **

A Key Learning Area (KLA) is an important part of a curriculum. It is founded on
**fundamental and connected concepts within major fields of knowledge which **
should be acquired by all students. A KLA provides a context for the development
and application of generic skills (e.g. communication, collaboration skills and
creativity) and subject-specific skills as well as positive values and attitudes
through appropriate use of learning and teaching activities and strategies. It serves
as a context for the construction of new knowledge and the development of
understanding. The studies offered in each KLA may have an academic, social or
practical orientation or a combination of these, depending on their purpose(s). They
can be organised into subjects, modules, units, tasks or other modes of learning.

The three interconnected components of the curriculum framework, i.e. Knowledge in Key Learning Areas, Generic Skills, and Values and Attitudes, can be represented in Figure 1.

**Figure 1 **

**1.2 ** **Position of the Mathematics Education KLA in the School Curriculum **

Students require knowledge and skills that will help them meet the dynamic challenges in the 21st century, which is a knowledge-based information era driven by technology and creativity. Knowledge of mathematics is a necessity for every individual if they are to contribute towards the development and prosperity of their society. Mathematics and its applications pervade all aspects of life in the modern world. Many of the developments and decisions made in industry and commerce,

Knowledge in Key Learning Areas

Generic Skills

Values &

Attitudes

3

the provision of social and community services as well as government policy and planning etc., rely on the use of mathematics.

The Mathematics Education KLA is essential in the Hong Kong school curriculum as it is:

*(a) a powerful means for developing various abilities in a technology-oriented *
*and information-rich society – It helps students acquire the ability to *
communicate, explore, conjecture, reason logically and solve problems using a
variety of methods.

*(b) a powerful means of communication – It can be used to present information in *
many ways (e.g. figures, tables, charts, graphs and symbols) which can be
processed to generate further information. The presentation skills help students
lay a strong foundation for lifelong learning and acquire new knowledge in
this rapidly changing world.

*(c) a tool for studying other disciplines – It helps students enhance their *
understanding of the world and provides a basis as well as a foundation for
studying other disciplines.

*(d) an intellectual endeavour and a mode of thinking – It is a creative activity in *
which students can be fully involved and through which students can
demonstrate their imagination, initiative and flexibility of mind.

*(e) a discipline, through which students can develop their ability to appreciate the *
*beauty of nature, manage uncertainty and make sound judgements – *
Mathematical experiences acquired in school enable students to become
mathematically literate citizens and contribute towards social prosperity.

Being one of the KLAs that play a more active role in promoting STEM education, the Mathematics Education curriculum provides students with a solid knowledge base in mathematics. It also strengthens students’ ability to integrate and apply the knowledge and skills of STEM-related subjects. As an integral part of general education, mathematics education supports the learning of other subjects. It contributes significantly to the whole-person development of students in primary and secondary schools, prepares them for multiple pathways to post-secondary education and future careers, and hence plays an important role in the Hong Kong school curriculum.

**1.3 ** **Rationale and Direction for Development **

**1.3.1 Rationale for the Development of the Mathematics Education KLA **

Before the Learning to Learn Curriculum Reform

In July 1997, an ad hoc committee was set up by the Curriculum Development Council (CDC) to carry out a review of the Hong Kong Mathematics curriculum.

Following two research studies^{2} conducted in 1998, the ad hoc committee
recommended in its final report (January 2000) that the Mathematics curriculum
should be designed according to a set of content-based strands, the learning of
abstract mathematical concepts should be backed up by adequate prior experience
of manipulating concrete objects and an abundance of examples, and thinking skills
should be developed through mathematical activities. These recommendations had
been incorporated into the Mathematics curriculum as stipulated in the
*Mathematics Curriculum Guide (P1 – P6) (2000) and Syllabuses for Secondary *
*School – Mathematics (Secondary 1 – 5) (1999). *

Learning to Learn Curriculum Reform and Implementation of the New Academic Structure

In 2001, the Learning to Learn curriculum reform was launched to promote a
curriculum and pedagogical change at the basic education level to help students
become lifelong learners capable of meeting the challenges of a knowledge-based
and changing society, globalisation and a competitive economy. In 2005, the report
*on The New Academic Structure for Senior Secondary Education and Higher *
*Education – Action Plan for Investing in the Future of Hong Kong proposed a *
3-year senior secondary and 4-year undergraduate academic system. A more
flexible, coherent and diversified senior secondary curriculum was implemented at
Secondary 4 in 2009. The curriculum and assessment reform at the senior
secondary level under the New Academic Structure (NAS) was regarded as an
*extension of the curriculum reform at the basic education level. The Mathematics *
*Curriculum and Assessment Guide (Secondary 4 – 6) (2007) provided details on *
the learning, teaching and assessment of the senior secondary Mathematics
curriculum (SSMC) under the NAS.

2* The two research studies were: (1) Comparative Studies of the Mathematics Curricula of Major Asian *
*and Western Countries conducted by The University of Hong Kong; and (2) An Analysis of the Views of *
*Various Sectors on the Mathematics Curriculum conducted by The Chinese University of Hong Kong. *

5

Review of the Senior Secondary Mathematics Curriculum under the New Academic Structure

There have been reviews of the SSMC in different aspects since its implementation in 2009. The first review was conducted in 2011 and in response to the views collected from the stakeholders, there was fine-tuning of the content and time allocation of the SSMC. Then a medium term review was conducted in the 2014/15 school year to solicit views of the Mathematics panel heads and teachers on the initial recommendations on the updating of the SSMC at the subject level. Teachers’

views on the curriculum framework of the SSMC were also collected through a questionnaire survey and focus group interviews.

Ongoing Renewal of the School Curriculum

Alongside the implementation of the Learning to Learn curriculum reform, there have been a lot of changes and challenges in society and around the world, including those observed in economic, scientific, technological and social developments. To maintain Hong Kong’s competitive edge and to prepare students well for the local and global changes taking place in various fields, it is necessary to enhance the Learning to Learn curriculum reform, to sustain and deepen the accomplishments achieved and to identify new focuses in the curriculum as we move to a new phase of curriculum renewal and updating.

In late 2015, a school survey was carried out to collect schools’ views on the promotion of STEM education and the updating of the Mathematics Education KLA curriculum (P1 - S6). The results of the survey indicated schools’ support for the promotion of STEM education as a key emphasis in curriculum development, the adoption of e-learning for effective learning and teaching, and the enhancement of the vertical continuity and lateral coherence of the Mathematics Education curriculum.

Stepping into the new phase of the ongoing curriculum renewal and following up the results of the NAS review in 2014/15, three ad hoc committees were set up under the CDCC on Mathematics Education in late 2015 to carry out a review of the Mathematics Education curriculum from P1 to S6 for updating purposes with due regard to the results of the previous school surveys. Following the completion of draft revised learning content of the Mathematics curriculum (P1 - S6) in late 2016, a multi-channel public consultation on the revised curriculum content and the curriculum framework of senior secondary Mathematics was conducted in 2016/17 school year. Views from different stakeholders, including school principals,

secondary career masters/mistresses, primary and secondary Mathematics panel heads and teachers, academics of universities and IVE, professional bodies and HKEAA were collected through focus group meetings, curriculum development visits, consultation forums and a school questionnaire survey. The revised Mathematics curriculum (P1 - S6) introduced in this Guide and its supplement was endorsed by CDC in 2017 after full consideration of the results of the public consultation.

The updated direction and strategies for the development of the Mathematics Education KLA are introduced in the following sections while the framework and content of the revised curriculum are introduced in the next chapter.

**1.3.2 Direction for the Development of the Mathematics Education KLA **

In face of the continual local and global changes in various fields, the rapid
development of technology, the views of stakeholders, the results of international
assessments (e.g. the Programme for International Student Assessment (PISA) and
the Trends in International Mathematics and Science Study (TIMSS)) which shed
light on mathematics education in Hong Kong, as well as the direction for ongoing
*curriculum renewal, the recommendations provided in the Mathematics Education *
*Key Learning Area Curriculum Guide (Primary 1 * Secondary 3) (2002) on
planning and implementing the school Mathematics curriculum are revisited. The
following focal points of curriculum renewal are put forth for primary and
secondary schools to incorporate into the school Mathematics curriculum to
accommodate students’ learning needs arising from the changing contexts and
education trends.

(a) Strengthening students’ ability to integrate and apply knowledge and skills through STEM education;

(b) Highlighting the importance of e-learning for enhancing learning and teaching effectiveness, facilitating self-directed learning and nurturing students’

competence in applying information technology (IT) in learning mathematics;

(c) Highlighting Language across the Curriculum (LaC) in the school Mathematics curriculum, such as promoting reading in Mathematics to develop students’ understanding of the connections between mathematics and

7

real life as well as other disciplines; and.

(d) Strengthening the development of generic skills and positive values and attitudes in an integrative manner through various Mathematics learning activities.

The above focal points are also some of the Major Renewed Emphases (MRE)
*introduced in Booklet 2 of the Secondary Education Curriculum Guide (2017). *

Besides, the development of the Mathematics curriculum is targeted at enhancing students’ learning progressions through the updating of curriculum content. It is also important for the development to be based on the existing strengths.

**1.4 ** **Strategies for Development **

In the ongoing renewal of the Mathematics curriculum, schools could build on their existing strengths, deepen and sustain the accomplishments achieved and identify new areas to focus on, to foster students’ capabilities for whole-person development and lifelong learning. Schools are encouraged to take into account the suggestions and focal points set out in Sections 1.3.2 when planning the school Mathematics curriculum. Schools may select those relevant to their needs, set priorities and integrate them into the school curriculum. The following table summarises schools’ existing strengths and suggested strategies for development to facilitate the ongoing curriculum renewal.

Existing Strengths Suggested Strategies for Development

Schools agree with the aims of the Mathematics Education KLA curriculum which cover the development of knowledge, generic skills, and positive values and attitudes.

Both students and parents show high regard for the Mathematics subject.

Developing the school curriculum continuously by identifying areas for focusing, deepening and sustaining

Participating in research and development activities (such as

“Seed” projects) to further develop the school Mathematics curriculum for enhancing students’ whole-person development

Existing Strengths Suggested Strategies for Development

Enhancing students’ interest and confidence in learning mathematics through various means, such as hands-on activities, STEM activities, mathematics reading and effective use of IT

As revealed in international assessments (e.g. PISA and TIMSS), Hong Kong students’ performance in mathematics has been ranked among the top four of the participating countries/ regions in the past decade.

The proportion of the top performing students is increasing in general, as revealed in TIMSS 1995, 1999, 2003, 2007 and 2011.

Sustaining the existing good practices in learning, teaching and assessment

Providing students with more opportunities and tools to apply mathematics in problem solving

Enhancing the design of learning and teaching materials and assessment tasks to cater for students’ diverse abilities, e.g. providing diversified e-learning resources for students to work at their own pace and receive timely feedback.

Most teachers support that STEM education is a focal point of the ongoing renewal of the school curriculum.

Providing STEM learning activities in the best interest of students and within their abilities, e.g. through activities based on topics in the Mathematics curriculum and project work

Most teachers support the incorporation of information technology for effective learning, teaching and assessment.

Using IT in a well-integrated, pedagogically sound and effective way for the learning and teaching of mathematics

Applying IT to facilitate students’

discussion and understanding of abstract concepts

9

Existing Strengths Suggested Strategies for Development

Supporting students’ self-directed learning by providing suitable e-resources and developing their e-learning strategies

Mathematics teachers are usually professionally trained.

Teachers welcome the in-service training provided by the EDB and other professional bodies.

Arranging teachers’ professional development on the focal points of the ongoing curriculum renewal, such as promoting STEM education and IT in education

Encouraging teachers to participate in collaborative research and development projects or community of practice for sharing of good practices among schools

Extending teachers’ understanding of the Mathematics curricula across different key stages for enhancing vertical continuity

Teachers show high regard for assessing students’ ability through formative and summative assessments.

Schools are making use of the internal and external assessment results to review and adjust their school curriculum and the learning and teaching strategies adopted.

Adopting diversified modes of assessment, such as hands-on tasks, open-ended questions and problem-based tasks to assess students’ diverse abilities

Deepening the use of more informative feedback to promote assessment as learning, in addition to assessment for learning, to help students set goals, and monitor, reflect on and evaluate their own learning

Existing Strengths Suggested Strategies for Development

Most schools have taken school-based measures to embrace learner diversity, e.g. organising remedial classes for students who are weak in mathematics.

There is flexibility in the curriculum design. Enrichment elements (e.g.

Enrichment Topics in the primary and junior secondary Mathematics curricula) and Further Learning Units are available in the Mathematics curriculum.

Offering enhancement/support measures to both ends of the student population

Planning a school curriculum which is adjusted and adapted to meet the needs of both less able students and more able students

Making use of the flexibility provided by the Mathematics Education curriculum to embrace learner diversity and adopt diversified learning and teaching strategies

Although the focal points of the ongoing curriculum renewal suggested in the previous section are not unfamiliar to schools, they are highlighted and supported through the following measures:

Professional development programmes on the focal points under different categories including curriculum planning, learning and teaching and knowledge enrichment organised for curriculum leaders and teachers;

Collaborative research and development projects and study groups organised for initialising, promoting, sustaining and improving the incorporation of the focal points in the learning, teaching and assessment of mathematics (e.g.

STEM education and IT in education); and

Others, such as provision of resource packages, newsletters and reading materials.

Suggestions on curriculum planning, learning and teaching, assessment and learning and teaching resources, and more elaboration on the focal points of ongoing curriculum renewal mentioned in this chapter are discussed in Chapters 3 to 6.

**Chapter 2 **

**Curriculum Framework **

11

**Chapter 2 Curriculum Framework **

**2.1 ** **Aims of the Mathematics Education KLA Curriculum **

Students need mathematics to meet the dynamic challenges of their future studies, careers or daily life in an information-rich society with rapid development in technology. The overall aims of the Mathematics Education KLA curriculum are to develop in students:

(a) the ability to think critically and creatively, to conceptualise, inquire and reason mathematically, and to use mathematics to formulate and solve problems in daily life as well as in mathematical contexts and other disciplines;

(b) the ability to communicate with others, express their views clearly and logically in mathematical language;

(c) the ability to manipulate numbers, symbols and other mathematical objects;

(d) number sense, symbol sense, spatial sense, measurement sense and the capacity to appreciate structures and patterns; and

(e) a positive attitude towards mathematics learning and an appreciation of the aesthetic nature and cultural aspect of mathematics.

The focal points of curriculum renewal mentioned in the previous chapter, including the promotion of STEM education, Information Technology in Education and Language across the Curriculum, echo the aims of developing students’

abilities to formulate and solve problems in daily life and other disciplines, ability to communicate with others clearly and logically. The development of Generic Skills and positive values and attitudes, being a continuous emphasis, is also a part of the curriculum aims.

**2.2 ** **Components of the Curriculum Framework **

The curriculum framework for Mathematics Education KLA is the overall structure for organising the learning, teaching and assessment of Mathematics together with curriculum management, leadership and planning to achieve the overall aims and

learning targets of the Mathematics Education KLA.

Figure 2 shows a diagrammatic representation of the Mathematics Education
curriculum framework. The central part of the framework comprises a set of
**interlocking components including subject knowledge organised under strands, **
**generic skills, and values and attitudes, which sets out what students should learn **
and develop in the Mathematics Education KLA.

Curriculum management, leadership and planning, as well as effective learning, teaching and assessment of Mathematics involve not only the central part, but also the learning needs of students in the contemporary contexts, including the development of students’ abilities in using language and information technology for learning. As the learning of Mathematics is also connected with other KLAs/subjects, one of the main concerns is integrating and applying knowledge and skills in different subjects, especially in promoting STEM education. Further, effective use of resources and partnerships between schools, the EDB and other organisations also lead to successful implementation of the Mathematics Education curriculum.

Further details on the knowledge, generic skills and values and attitudes of the Mathematics curriculum are given below.

**2.2.1 Strands**^{3}**, Learning Targets and Objectives **

**Strands are categories of mathematical knowledge and concepts for organising **
**the curriculum. Their main function is to organise mathematical contents for the **
purpose of developing knowledge, generic skills, and values and attitudes as a
holistic process. There are basically three strands in the Mathematics Education
curriculum, namely “Number and Algebra”, “Measures, Shape and Space” and

“Data Handling”. At the primary level, these three strands are subdivided into five strands (see Figure 2).

To ensure meaningful and effective learning, there must be a coherent plan for students’ learning at the primary and secondary levels. The learning targets and

3 The term “Strands” has been refered to as “Dimensions” in earlier curriculum documents such as the
*Mathematics Curriculum Guide (P1 – P6) (2000) and Syllabuses for Secondary Schools – Mathematics *
*(Secondary 1 – 5) (1999). *

13

*

**

**Figure 2 Diagrammatic Representation of **
**the Mathematics Education Curriculum Framework **

* Flexibility in curriculum content is provided by the setting of foundation topics, non-foundation topics and enrichment topics (see Section 2.3 and 2.4 for details). A Further Learning Unit is also included in each key stage.

# Module 1 of the Extended Part consists of “Foundation knowledge”, “Calculus” and “Statistics” and Module 2 consists of “Foundation knowledge”, “Algebra” and “Calculus”.

**Overall Aims and Learning Targets of the Mathematics **
**Education KLA Curriculum **

Information Technology in Education

Language across the Curriculum

**Resources & Partnership**

**Integration and Application** (STEM Education)

**Curriculum Management, **
**Leadership & Planning **

**+ **

**Effective Learning, Teaching and Assessment **

in response to the needs of students and the contemporary context

**Knowledge** *

**Generic Skills **

**Va** **lues** ** a** **nd A** **tt** **it** **u** **des**

Number &

Algebra

Measures, Shape & Space

Data Handling

Number &

Algebra

Measures, Shape & Space

Data
Handling
*Strands of KS1 and KS2 *

Algebra Measures Data

Handling

Number Shape &

Space
*Strands of KS3 *

*Strands of KS4 Compulsory Part *

*Areas of KS4 Extended Part ** ^{#}*
Algebra Calculus Statistics
Foundation

knowledge

objectives, which are geared towards the overall aims of the Mathematics Education curriculum, are organised progressively and systematically across Key Stage 1 (Primary 1 - 3), Key Stage 2 (Primary 4 - 6), Key Stage 3 (Secondary 1 - 3) and Key Stage 4 (Secondary 4 - 6).

As the content of the Extended Part of the senior secondary Mathematics curriculum are interwoven, they are not organised under strands but grouped under the areas of “Foundation knowledge”, “Algebra”, “Calculus” and “Statistics”.

Module 1 of the Extended Part consists of “Foundation knowledge”, “Calculus”

and “Statistics” while Module 2 consists of “Foundation knowledge”, “Algebra”

and “Calculus”. At each key stage, in addition to the curriculum content in the strands and areas, a Further Learning Unit is designed to enhance students’ ability to inquire, reason and conceptualise mathematical concepts and to allow students to integrate and apply knowledge and skills learned in different strands and areas.

The learning objectives of each strand or area are grouped and presented under different learning units which could, on one hand, reflect the relationship between learning content of similar nature, and on the other hand, enable teachers and students to relate the content in different units.

There are updates in the learning content of the Mathematics Education curriculum at different key stages in order to align with the direction of the development of mathematics education and the ongoing renewal of school curriculum as mentioned in Section 1.3.2 in Chapter 1 of this Guide. The main purposes of the updates include:

to enhance the interface of the curriculum across key stages;

to enhance the support to other subjects;

to enhance the organisation of curriculum content for betterment of learning and teaching; and

to provide more specific descriptions on the depth and breath of curriculum content.

The updated learning targets and learning units of the Mathematics curriculum for KS1 to KS4 are shown in the tables on the next pages. Readers may refer to the supplements of this guide for details of the learning content, including the learning objectives under each learning unit of the primary, junior secondary and senior secondary Mathematics curricula.

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**Overview of Learning Targets **

**Learning Targets of Primary Mathematics Curriculum (P1 – P3) **

**Number Strand ** **Measures Strand ** **Shape and Space Strand ** **Data Handling Strand **

Students are expected to:

recognise the concepts of whole numbers* and simple fractions;

recognise and use the commutative and associative properties of addition and multiplication;

perform four arithmetic operations of whole numbers and addition and subtraction of simple fractions, and check the reasonableness of results;

and

use numbers to formulate and solve simple problems.

recognise the concepts of length, distance, weight and capacity;

use different ways to compare the length, weight, capacity of objects and distance between objects, and record the results;

understand the need for using standard units of measurements;

choose and use appropriate measuring tools and standard units to compare the length, weight, capacity of objects and distance between objects, and record the results;

estimate the result of measurements;

recognise money, time and date, and their use in daily life;

and

integrate the knowledge in the strands of Number, Measures, Shape and Space to solve simple problems.

identify intuitively and describe 2-D and 3-D shapes;

recognise the properties of points and lines, and the concept of faces of 3-D shapes;

recognise the concepts of right angles, acute angles and obtuse angles;

recognise the concepts of perpendicular and parallel;

recognise the concepts and properties of squares, rectangles, parallelograms and trapeziums;

recognise the inclusion relations between parallelograms and squares, parallelograms and rectangles;

recognise the inclusion relations between different types of triangles;

make 2-D shapes and appreciate the beauty of geometric shapes; and

describe the relative position of different objects and recognise the four directions.

recognise the importance of the organisation and representation of statistical data;

collect and group statistical data according to given criteria;

use appropriate scales to construct simple statistical charts and interpret them;

and

formulate and solve simple problems arising from statistical data or statistical charts.

*** In the primary Mathematics curriculum, “whole numbers” refers to non-negative integers. **

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**Learning Targets of Primary Mathematics Curriculum (P4 – P6) **

**Number Strand ** **Algebra Strand ** **Measures Strand ** **Shape and Space Strand ** **Data Handling Strand **
Students are expected to:

recognise and use the distributive property of multiplication;

recognise the concepts of prime numbers and composite numbers;

understand the concepts of the highest common factors and the least common multiples;

understand the concepts of whole numbers, fractions, decimals, percentages and the relations among them;

perform four arithmetic operations of whole numbers, fractions and decimals, and check the reasonableness of results; and

use numbers to formulate and solve problems.

use symbols to represent numbers;

use algebraic expressions to represent the operations of and relations between quantities that are described in words and involve unknown quantities; and

use algebra to formulate and solve simple problems and recognise how to check the

reasonableness of results.

recognise the concepts of perimeter, area, volume and speed;

use different ways to compare the perimeter and area of 2-D shapes, volume and speed of objects, and record the results;

choose appropriate standard units to measure and compare the perimeter and area of 2-D shapes, volume and speed of objects, and record the results;

use the measuring tool and the standard unit to measure, compare and draw angles of different sizes;

recognise the degree of accuracy of measurements;

estimate the result of measurements;

recognise the concepts and properties of rhombuses and circles;

recognise the inclusion relations between different types of quadrilaterals;

recognise the concept of vertices and edges of 3-D shapes;

recognise the concept and property of sphere;

make 2-D shapes and 3-D shapes from given information and appreciate the beauty of geometric shapes;

and

recognise the eight compass points.

understand the criteria for organising and

representing statistical data;

use approximate values and appropriate scales to construct statistical charts and interpret them;

recognise relations of data and patterns on the changes of data from statistical charts;

recognise the concept of average and solve problems;

formulate and solve problems arising from statistical data or statistical charts;

choose appropriate statistical charts to represent given data; and

judge the appropriateness of the representation of statistical charts.

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**Learning Targets of Primary Mathematics Curriculum (P4 – P6) **

**Number Strand ** **Algebra Strand ** **Measures Strand ** **Shape and Space Strand ** **Data Handling Strand **
Students are expected to:

inquire and use

measurements formulae of 2-D shapes and 3-D shapes;

recognise the relation between volume and capacity and solve problems;

perform the

interconversion between units of time and solve problems related to time and speed；and

integrate the knowledge in the strands of Number, Measures, Shape and Space to formulate and solve problems.

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**Learning Targets of Junior Secondary Mathematics Curriculum **

**Number and Algebra Strand ** **Measures, Shape and Space Strand ** **Data Handling Strand **
Students are expected to:

recognise the concepts of negative integers, negative rational numbers and irrational numbers;

further use numbers to formulate and solve problems;

investigate and describe relationships between quantities using algebraic symbols, including patterns of sequences of numbers;

interpret simple algebraic relations from numerical, symbolic and graphical perspectives;

manipulate simple algebraic expressions and relations; and apply the knowledge and skills to formulate and solve simple

real-life problems and justify the validity of the results obtained; and

apply the knowledge and skills in the Number and Algebra strand to formulate and solve problems in other strands.

recognise errors in measurement and apply the knowledge to solve problems;

extend concepts and formulae of measurements of 2-dimensional figures and 3-dimensional figures and apply the knowledge to solve problems;

explore and visualise the geometric properties of 2-dimensional figures and 3-dimensional figures;

use inductive and deductive approaches to study the properties of 2-dimensional rectilinear figures;

perform geometric proofs involving 2-dimensional rectilinear figures with

appropriate symbols, terminology and reasons;

inquire and describe geometric knowledge in 2-dimensional space using algebraic relations and apply the knowledge to solve problems;

inquire and describe geometric knowledge in 2-dimensional space using trigonometric ratios and apply the knowledge to solve problems;

and

apply the knowledge and skills in the

Measures, Shape and Space strand to formulate and solve problems in other strands.

recognise the methods of organising discrete and continuous statistical data;

further choose appropriate statistical charts to represent given data and interpret them;

understand the measures of central tendency;

select and use the measures of central tendency to describe and compare data sets;

investigate and judge the validity of arguments derived from data sets;

recognise the concept of probability and apply the knowledge to solve simple probability problems; and

integrate the knowledge in statistics and probability to solve simple real-life problems.

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**Learning Targets of the Compulsory Part of Senior Secondary Mathematics Curriculum **

**Number and Algebra Strand ** **Measures, Shape and Space Strand ** **Data Handling Strand **
Students are expected to:

extend the concepts of numbers to complex numbers;

further investigate and describe relationships between quantities using algebraic symbols;

generalise and describe patterns in sequences of numbers using algebraic symbols, and apply the results to solve problems;

interpret more complex algebraic relations from numerical, symbolic and graphical perspectives;

manipulate more complex algebraic expressions and relations, and apply the knowledge and skills to formulate and solve more complex real-life problems and justify the validity of the results obtained; and

apply the knowledge and skills in the Number and Algebra strand to generalise, describe and communicate mathematical ideas and further solve problems in other strands.

use inductive and deductive approaches to study the properties of 2-dimensional figures;

perform geometric proofs involving 2-dimensional figures with appropriate symbols, terminology and reasons;

further inquire and describe geometric knowledge in 2-dimensional space using algebraic relations and apply the knowledge to solve problems;

inquire and describe geometric knowledge in 2-dimensional space and 3-dimensional space using trigonometric functions and apply the knowledge to solve problems; and

apply the knowledge and skills in the Measures, Shape and Space strand to generalise, describe and communicate mathematical ideas and further solve problems in other strands.

understand the measures of dispersion;

select and use the measures of central tendency and dispersion to describe and compare data sets;

further investigate and judge the validity of arguments derived from data sets;

acquire basic techniques in counting;

formulate and solve more complex probability problems by applying simple laws; and

integrate the knowledge in statistics and probability to solve more complex real-life problems.

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**Learning Targets of Module 1 (Calculus and Statistics) of Senior Secondary Mathematics Curriculum **

**Foundation Knowledge ** **Calculus ** **Statistics **

Students are expected to:

apply binomial expansion for the study of probability and statistics;

model, graph and apply exponential

functions and logarithmic functions to solve problems; and

understand the relationships between exponential and logarithmic functions and apply the two functions to solve real-life problems.

recognise the concept of limits as the basis of differential and integral calculus;

understand the idea of differentiation and integration through consideration of concrete phenomena;

find the derivatives, indefinite integrals and definite integrals of simple functions; and

apply the knowledge of calculus to solve real-life problems.

understand the concepts of probability, random variables, and discrete and continuous probability distributions;

understand the fundamental ideas of statistical reasoning based on the binomial, Poisson and normal distributions;

use statistical reasoning and thinking to know when and how to apply statistical methods to make inferences and justify conclusions; and

develop the ability to think mathematically about uncertainty and then apply such knowledge and skills to solve problems.

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**Learning Targets of Module 2 (Algebra and Calculus) of Senior Secondary Mathematics Curriculum **

**Foundation Knowledge ** **Algebra ** **Calculus **

Students are expected to:

recognise odd and even functions and their graphs;

understand the principle of mathematical induction;

expand binomials using the binomial theorem;

understand simple trigonometric functions, important trigonometric identities and formulae involving compound angles; and

*recognise e. *

understand the concepts, operations and properties of matrices and the inverses of square matrices up to order 3;

solve systems of linear equations;

understand the concept, operations and properties of vectors; and

apply the knowledge of vectors to solve problems in 2-dimensional space and 3-dimensional space.

understand the concept of limits as the basis of differential and integral calculus;

understand the concepts and properties of derivatives, indefinite integrals and definite integrals of functions;

find the derivatives, indefinite integrals and definite integrals of simple functions;

find the second derivatives of functions;

apply the knowledge of calculus to sketch curves; and

apply the knowledge of calculus to solve real-life problems.

22

**Overview of Learning Units **

**Learning Units of Primary Mathematics Curriculum (P1 – P3) **

**Number Strand ** **Measures Strand ** **Shape and Space Strand ** **Data Handling Strand **
1. Numbers to 20

2. Basic addition and subtraction 3. Numbers to 100

4. Addition and subtraction (I) 5. 3-digit numbers

6. Addition and subtraction (II) 7. Basic multiplication

8. 4-digit numbers

9. Addition and subtraction (III) 10. Basic division

11. 5-digit numbers 12. Multiplication (I) 13. Division (I)

14. Four arithmetic operations (I) 15. Fractions (I)

16. Length and distance (I) 17. Money (I)

18. Length and distance (II) 19. Time (I)

20. Length and distance (III) 21. Time (II)

22. Money (II)

23. Length and distance (IV) 24. Time (III)

25. Capacity 26. Time (IV) 27. Weight

28. 3-D shapes (I) 29. 2-D shapes

30. Directions and positions (I) 31. Angles

32. Directions and positions (II) 33. Quadrilaterals (I)

34. 3-D shapes (II) 35. Quadrilaterals (II) 36. Triangles

37. Pictograms 38. Bar charts (I)

**Further Learning Unit **
39. Inquiry and investigation

Note: Learning units in the overview are not arranged in the order of teaching.

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**Learning Units of Primary Mathematics Curriculum (P4 – P6) **

**Number Strand ** **Algebra Strand ** **Measures Strand ** **Shape and Space Strand ** **Data Handling Strand **
1. Multiplication (II)

2. Division (II)

3. Multiples and factors 4. Common multiples and

common factors

5. Four arithmetic operations (II) 6. Fractions (II)

7. Decimals (I)

8. Multi-digit numbers 9. Fractions (III) 10. Decimals (II) 11. Decimals (III) 12. Fractions (IV) 13. Fractions (V) 14. Decimals (IV) 15. Decimals (V) 16. Percentages (I) 17. Percentages (II)

18. Elementary algebra 19. Simple equations (I) 20. Simple equations (II)

21. Perimeter (I) 22. Area (I) 23. Area (II) 24. Volume (I) 25. Angle (degree) 26. Volume (II) 27. Perimeter (II) 28. Speed 29. Area (III)

30. Quadrilaterals (III) 31. Dissecting and forming

shapes

32. Directions and positions (III) 33. Circles

34. 3-D shapes (III) 35. Symmetry

36. Bar charts (II) 37. Bar charts (III) 38. Averages

39. Broken line graphs 40. Pie charts

41. Uses and abuses of statistics

**Further Learning Unit **
42. Inquiry and investigation

Note: Learning units in the overview are not arranged in the order of teaching.