C
ONTENTS1 Introduction … … … . … … … . … ... 1
2 Background … … … . … … . 2
3 Rationale for Development … … … . … … … . … … … .. … … 3
4 Phases of Development … … … . … … … . … … 5
4.1 Short-term (2000-2005) … … … . 5
4.2 Medium-term (2005-2010) … … … 6
4.3 Long-term (2010+) … … … . 6
5 The Framework … … … ... 7
5.1 Overall Aims … … … … . … … … . 7
5.2 Learning Targets … … … ... 7
5.3 Components of the Framework … … … . . 7
5.3.1 Strands … … … .. 8
5.3.2 Generic Skills, Values and Attitudes … … … ... 9
5.4 Modes of Curriculum Planning … … … . . 12
5.5 Teaching, Learning and Assessment … … … . . 12
5.6 School-based Curriculum Development … … … ... 15
5.7 Life-wide Learning … … … ... 15
5.8 Connections with Other Key Learning Areas … … ... … … … 17
6 Conclusion … … … .... 19
Appendices 1 Findings of the Research Studies Conducted in 1998 … … … 21
2 List of Tryout Topics/Teaching Strategies Conducted since 1997 … . . 25
3 An Overview of Learning Targets of the Mathematics Curriculum... 29
4 Developing Generic Skills in the Mathematics Education Key Learning Area … … … . . . 35
5 Related Values and Attitudes … … … . . . 57
6 Exemplars … … … . 59
1 INTRODUCTION
This document on the key learning area of Mathematics Education is written in support of the consultation document Learning to Learn prepared by the Curriculum Development Council (Nov 2000) and should be read together with it. The Learning to Learn document is the outcome of the Holistic Review of the School Curriculum conducted by CDC beginning in 1999, which is done in parallel with the Education Commission’s Education System Review.
2 BACKGROUND
2.1 Students require knowledge and skills that will help them compete in the society of the 21st century, which is an information age. To reflect the rapid growth of knowledge in the information age, mathematics has unquestionably become a necessity for every individual to contribute towards the prosperity of the society. Mathematics pervades all aspects of life and it would be very difficult to live a normal life without making use of mathematics of some kind. Many of the developments and decisions made in industry and commerce, the provision of social and community services as well as government policy and planning etc., rely to some extent on the use of mathematics.
2.2 Mathematics is essential in the school curriculum of Hong Kong as it is:
• a powerful means of communication – It can be used to present information in many ways like figures, tables, charts, graphs and symbols, which can be further manipulated to deduce further information.
• an analyzing tool for studying other disciplines – It helps students enhance their understanding of the world.
• an intellectual endeavour and a mode of thinking – It is a creative activity in which students can be fully involved and display their imagination, initiative and flexibility of mind.
• a discipline, which can develop students’ abilities to appreciate the beauty of nature, think logically and make sound judgement – These mathematical experiences acquired in schools enable students to become mathematically literate citizens and contribute towards the prosperity of the society.
Mathematics is valuable to help students develop the core competencies for lifelong learning. It is an integral part of the general education and hence a key learning area (KLA) of the school curriculum of Hong Kong.
3 RATIONALE FOR DEVELOPMENT
3.1 The CDC Committee on Mathematics Education holds the following views:
• High technology like computers and calculators has profoundly changed the world of mathematics education.
Students should master information technology to become more adaptive to the dynamically changing environment.
Mechanical drilling and impractical topics are no longer essential and relevant in mathematics learning.
• It is important for our students to gain experience and acquire foundation knowledge and skills, develop capabilities to learn how to learn, think logically and creatively, develop and use knowledge, analyze and solve problems, access and process information, make sound judgements and communicate with others effectively.
• Students should be able to build up confidence and positive attitudes towards mathematics learning, to value mathematics and to appreciate the beauty of mathematics.
3.2 According to two research studies1, which were conducted in 1998 for supporting a holistic review2 of the Hong Kong mathematics curriculum, although the current mathematics curriculum is well supported by various stakeholders, there are problems which should be resolved for improvement: The existing school mathematics curriculum is generally content- oriented3, rather packed and difficult. The mathematics curriculum at S.1 repeats some of the material at P.5 and 6.
Some of the content areas in Additional Mathematics overlap
1 The two research studies were Comparative Studies of the Mathematics Curricula of Major Asian and Western Countries and An Analysis of the Views of Various Sectors on the Mathematics Curriculum. The former was conducted by The University of Hong Kong while the latter The Chinese University of Hong Kong. The two research studies
with those of Secondary Mathematics (1985), AS-level Mathematics & Statistics and A-level Pure Mathematics. Such redundancy has brought difficulties to teachers teaching these subjects and generated an unnecessary inclination among students to take Additional Mathematics even if they do not have the competence. Moreover, the similarity between A-level and AS-level Applied Mathematics, the lengthiness of A-level Pure Mathematics and the unclear target students of AS-level Mathematics & Statistics have drawn grievances from the school teachers.
3.3 Teaching at the senior primary, senior secondary and sixth form levels is examination-driven. Most teachers and students only focus on the teaching and learning of those materials which are included in the examination syllabuses. Mathematical knowledge which can arouse students’ interest is seldom introduced if it is beyond the examination syllabuses. The scope of learning is therefore confined.
4 PHASES OF DEVELOPMENT
4.1 Short-term (2000 – 2005)
4.1.1 Both the primary and secondary mathematics curricula have been revised with the aim of shifting the emphasis of students from rote procedures and meaningless drilling to the development of their thinking abilities, catering for the different needs and abilities of students and strengthening their learning. In terms of content, the revised primary and secondary mathematics curricula have been trimmed down by about 15% and 11% respectively. If necessary, schools/teachers could further adapt the curricula to create more space for
• using IT in mathematics teaching/learning;
• conducting project learning;
• doing exploratory work;
• organizing consolidation/enrichment activities, etc.
4.1.2 Tryouts of some selected topics (see Appendix 2 for the list of tryout topics/teaching strategies) in some schools have been conducted since 1997. The results of the tryouts and the comments and suggestions from the schools concerned, teachers’ opinions from related surveys and seminars, and the recommendations of the Ad hoc Committee were adequately considered before the finalization of the revised curricula. The revised secondary mathematics curriculum4 will be implemented at S.1 in September 2001 and the primary one5 at P.1 in September 2002.
4.1.3 The Education Department will organize relevant in- service training programs to familarize teachers with the focuses of the revised curricula. Reference materials including exemplars, useful web sites and CD Roms will
4.2 Medium-term (2005 – 2010)
To reduce the effect of overlapping between Additional Mathematics and the related mathematics subjects at the sixth form levels, the curriculum of Additional Mathematics will be revised in the near future.
4.3 Long-term (2010+)
In the long term, pending the review of the new senior secondary structure, the sixth form mathematics curriculum will be re- structured to cope with the revised primary and secondary mathematics curricula, to ensure better continuity among various mathematics subjects and to suit the diversified needs of students.
5 THE FRAMEWORK
5.1 Overall Aims
The aims of mathematics education are to develop:
• our youngsters’ knowledge, skills and concepts of mathematics and to enhance their confidence and interest in mathematics, so that they can master mathematics effectively and are able to formulate and solve problems from a mathematical perspective; and
• their thinking abilities and positive attitudes towards learning mathematics and build related generic skills6 throughout their life time.
5.2 Learning Targets
Both learning processes and content of mathematics are important in mathematics learning and hence reflected in the learning targets of the mathematics curriculum. Details of the learning targets in the four key stages can be found in Appendix 3.
5.3 Components of the Framework
Diagrammatic Representation of the Framework of the Mathematics Curriculum
In general, the framework of the mathematics curriculum can be represented diagrammatically as shown in Figure 1 on page 9.
5.3.1 Strands7
It can be seen from Figure 1 that the essential learning experiences for achieving the aims of mathematics education are organized into 5 learning dimensions at the primary level and 3 at the secondary.
Primary Secondary
• Number • Number & Algebra
• Algebra • Measures, Shape & Space
• Measures • Data Handling
• Shape & Space
• Data Handling
The learning dimensions are merged from 5 dimensions to 3 because less emphasis will be put on the Number and Measures dimensions at the secondary level. Moreover, it is not easy to include certain learning areas in one single dimension in Key Stages 3 and 4. For example, trigonometry involves measures of angles and lengths as well as spatial concepts. Similarly, problem-solving strategies usually interweave at the higher level and it is not easy to solve problems by using only numerical or algebraic approach.
On the other hand, since students use mathematics to a different extent as they move along the grade levels, the use of dimensions will not be extended to the sixth form mathematics curriculum.
7 The term “learning dimension” has been used in the revised primary and secondary school mathematics syllabuses.
Diagrammatic Representation of the Framework of the Mathematics Curriculum
Mathematics Curriculum
provides content knowledge which can serve as a means to develop students’ thinking abilities
and foster students’ generic skills and positive attitudes towards mathematics learning
Learning Dimensions provide a structured framework of
learning objectives
which enable students’ progress across various levels to be represented systematically
9 Generic Pri Number Algebra Measures Shape &
Space
Data
Handling Pri Skills
Sec Number & Algebra Measures, Shape &
Space
Data
Handling Sec
Values and Attitudes
effective linkage of
teaching, learning and assessment
Overall Aims and Learning Targets of Mathematics Figure 1
manipulated data, the mastering of generic skills and the fostering of positive values and attitudes should also be stressed. They are expected to develop/be fostered through the learning of mathematical knowledge in the content areas and it is desirable for teachers to help students cultivate these learning elements through planned learning activities. Figure 2 illustrates how these learning elements intertwine to form a reference grid.
Figure 2
The list of generic skills, which are regarded as paramount for lifelong learning in a world where knowledge is ever changing, is attached in Appendix 4. Examples showing how the skills can be experienced through the course of mathematics learning are provided. Similarly, related values and attitudes which are expected to be developed throughout the mathematics course at various stages of the Mathematics Education are shown in Appendix 5.
Appendix 6 gives six detailed exemplars which illustrate how the learning targets and generic skills can be linked.
Exemplars 1 to 3 are at the primary level and exemplars 4 to 6 at the secondary level. The linkage between the exemplars and the main learning targets and generic skills is summarized in the following table.
HOTs/
Generic Skills
Values and Attitude s
Content-based Learning Dimensions
Exemplar Main learning targets linked Main generic skills linked 1
“To understand … whole numbers” of the Number Dimension in Key Stage 1
• critical thinking
• communication
• problem solving 2 “To group and make … 3-
dimensional shapes” of the Shape
& Space Dimension in Key Stage 2
• communication
• critical thinking
• problem solving
3
“To formulate and solve problems arising from collected data and constructed graphs” of the Data Handling Dimension in Key Stage 2
• collaboration
• critical thinking
• problem solving
• creativity
• numeracy
4
“To interpret simple algebraic relations from numerical, symbolic and graphical perspectives” of the Number and Algebra Dimension in Key Stage 3
• information technology
• numeracy
• critical thinking
• problem solving
5
“To investigate and describe relationships between quantities using algebraic symbols and relations” of the Number and Algebra Dimension in Key Stage 4
• information technology
• numeracy
• critical thinking
6
“To select and use the measures of central tendency and dispersion to compare data sets” of the Data Handling Dimension in Key Stage 4
• communication
• numeracy
• critical thinking
Schools could develop their own school-based mathematics curriculum along the same line as the framework with reference to the detailed set of learning objectives provided in the two recently revised Syllabuses.
5.4 Modes of Curriculum Planning
5.4.1 The principle of proposing different modes of curriculum planning is to build on strength of schools and teachers, and provide them with curriculum flexibility and diversity to meet different purposes of teaching/learning. At the primary and secondary levels, the curricula are planned by objectives and a “dimension approach” is adopted.
Although the dimensions are content-based, the high-order thinking skills, generic skills, values and attitudes are expected to be incorporated into the content (see Figure 2) during teaching.
5.4.2 Enrichment activities/topics are provided in the revised primary mathematics curriculum, while the foundation part, non-foundation part and enrichment activities/topics are provided in the revised secondary mathematics curriculum to suit the different abilities of students.
5.4.3 Spare periods are also reserved at each key stage to provide teachers with curriculum space to rearrange or to adapt the content and depth of the teaching materials, using IT in mathematics teaching/learning and organize exploratory activities. Teachers have the flexibility to design their school-based mathematics curriculum to suit the needs of their students. However, it is not desirable to utilize the spare periods to do unnecessary drilling.
5.5 Teaching, Learning and Assessment
5.5.1 To take better account of the changing needs of the community, the main focus of the revised primary and secondary mathematics curricula is not on what mathematical topics should be learnt but on how mathematics is learnt. Therefore, the acquisition of high- order thinking skills and generic skills and the fostering of positive attitude towards mathematics learning are strongly advocated and should be allied with the learning of mathematical content. The teachers’ role should be to facilitate students to learn how to learn through using the said skills in mathematics.
5.5.2 Teaching strategies should be progressively changed through different levels of schooling, say from concrete to abstract, so as to cope with students’ development. A thematic approach is encouraged at junior primary levels.
Similarly, investigational work is encouraged at all the primary levels and should be continued in secondary schools. The teaching of abstract mathematical ideas should be supported by students’ concrete experiences at earlier stages as far as possible. This is because students need time to play around with concrete objects before proceeding to more abstract notions at the senior level.
On the whole, exposure to concrete objects and gaining personal experiences are important and should therefore be planned and built into the teaching programme as much as possible to support discussion in the mathematics curriculum.
5.5.3 Diversified teaching/learning activities including projects are encouraged. The application of mathematical concepts receive greater attention as it provides students with motivation for learning mathematics and experiences in using mathematical models. Therefore, daily-life situations are suggested as the means to help students realize the need for mathematics and its applications. In the revised secondary mathematics curriculum, further applications of mathematical knowledge in more complex real-life situations, which require students to integrate their knowledge and skills from various disciplines to solve problems, are provided through the new module
“Further Applications”.
5.5.4 Since each student has his/her own strengths and weaknesses, the mathematics curriculum can be duly adjusted across different levels of learning to cater for the different abilities of students. The structures of the
teaching too many topics in a single year and making learning fragmented should be avoided. In addition, measures like organizing bridging programmes can be taken to ensure that students of different abilities can follow. In the process of developing the school-based curriculum, a flowchart indicating the inter-relation of topics at different levels is highly desirable to ensure continuity. The flowcharts attached in Syllabuses for Primary Schools: Mathematics (Primary 1-6) (2000) and Syllabuses for Secondary Schools: Mathematics (Secondary 1-5) (1999) can be referred to.
5.5.5 Apart from the formal mathematics curriculum, mathematics-related activities also play an important role in mathematics learning. It is generally agreed that well chosen and organized mathematics-related activities help to promote students’ interest in learning the subject.
Examples include mathematical games/puzzles, mathematics competitions/quizzes, mathematics workshops, projects, talks, plays, film shows, mathematics bulletins, newspaper cutting and board displays, etc.
5.5.6 Assessment is an integral part of the teaching-learning cycle. Valid and reliable assessment should reflect the objectives and goals of the curriculum. It should be used as a tool both to collect data and to influence instruction.
Since student performance cannot be described by a single set of scores or by a single type of assessment activity because of its complexity, evidence of learning should be collected through various modes of assessment activities, to reflect students’ achievement in mathematics.
Formative assessments, which have come to be considered increasingly important in the teaching and learning processes, are also recommended, to provide a complete picture of student performance and help improve the learning of students. Students should be given a chance to demonstrate what they learn and how they apply their knowledge and skills of mathematics. Project work, class discussions, oral presentations and observations of students’ performance during lessons are useful assessment activities and should be integrated with other classroom activities. A description of minimal competence, which is the pre-requisite to the learning of
mathematics at the next advanced stage, is helpful to teachers for reporting student performance in terms of the basic knowledge, concepts and skills acquired.
5.5.7 Communication is also one of the skills which students need to learn in mathematics. It is essential to assess students’ ability in communicating findings, presenting an argument and explaining an intuitive approach to a problem either in an oral or a written form. From the written and oral presentation of students, teachers are able to identify their strengths and weaknesses. Over-drilling for examinations, on the other hand, will hamper effective learning and should thus be discouraged.
5.5.8 Schools need to formulate their assessment policy according to their culture, teachers’ experiences, students’
needs and interests. The design of learning objectives, learning activities and assessment tasks should be aligned to ensure that what is intended will be properly taught and successfully learned.
5.6 School-based Curriculum Development
Although both the revised primary and secondary mathematics curricula are recommended by CDC, schools are encouraged to carry out curriculum adaptation and integration, if deemed necessary, to meet the different needs, abilities and interests of students. The flexible elements introduced in the said curricula (i.e. foundation and non-foundation parts, and enrichment topics/activities) aim to help schools to design their school-based mathematics curriculum. For details, the two documents Syllabuses for Primary Schools: Mathematics (Primary 1-6) (2000) (Second Draft) and Syllabuses for Secondary Schools: Mathematics (Secondary 1-5) (1999) can be referred to.
organized/developed by tertiary institutions/professional bodies/the government. Some popular activities are:
Activities Specific Purposes Organizing Bodies Mathematics
Competition for Hong Kong Primary
Schools – a mathematics competition for primary students
The Competition aims to promote students’
interest in studying mathematics and hence improve
mathematics learning in primary schools.
Organized by the Professional Teachers’
Union
Mathematics Trails – a mathematics
competition for secondary students
Mathematics Trail aims to enhance students’
mathematical problem solving skills through applications in real life situations and physical environment.
Organized by the HK Association of Science and Mathematics Education.
HK Mathematics Olympiad (HKMO) – a mathematics
competition for secondary students
The Olympiad aims to promote and sustain students’ interest in the study of
mathematics.
Jointly organized by HKIEd and the Mathematics Section of ED.
Statistical Project Competition for Secondary School Students – a mathematics competition for secondary students
The Competition aims to promote students’
interest in studying statistics and to encourage them to understand the community in a scientific manner through the use of statistics.
Jointly organized by the HK Statistical Society and the
Statistics Department.
Po Leung Kuk
Primary Mathematics World Contest
– an international event for primary students
The Contest aims to discover
mathematically gifted primary students. It also aims to create an opportunity for the exchange of learning experiences among gifted students.
Organized by the Po Leung Kuk.
Training is provided to potential
participants by Po Leung Kuk.
International Mathematical Olympiad – an
international event for secondary students
The Olympiad aims to discover
mathematically gifted young people who have not formally enrolled at any university. It also aims to foster friendly relations between gifted students.
Organized by the International Mathematical Olympiad (HK) Committee which is affiliated to the HK Mathematics Society.
Training is provided to potential
participants by the Committee.
More details can be found from the booklet “全方位學習活動簡 介” (pp.8-9) published by “教育統籌委員會”. The booklet can also be downloaded from the Internet at the web site http://www.life-wide-learning.org.
5.8 Connections with Other Key Learning Areas
5.8.1 Mathematics is the foundation and supporting knowledge to many other disciplines. It is linked to the other 7 KLAs by providing a basis for making investigations as well as a tool for analyzing data, representing findings and models with symbols, graphs and charts, and theorizing
mathematical ways, skills and instruments. For some selected topics (like percentage and statistics), integration with other KLAs (like Science or Personal, Social and Humanities) is one of the ways of organizing students’
learning experiences mathematically. Integrated learning removes the boundaries of subjects and reflects the interdependent nature of reality and the complexities of life. It provides students with a holistic context for learning and enables students to make connection with the real world in solving problems. An example showing how this is done can be found in exemplar 3 in Appendix 6.
Some of the links between the Mathematics KLA and the other KLAs are exemplified in the following paragraphs.
5.8.3 In the Chinese Language and English Language KLAs, mathematical concepts are essential to understand essays with mathematical and statistical ideas. For the Arts Education KLA, lines and shapes are important elements to create pictures and models, and patterns and symmetry are often explored in creative dance. In the Physical Education KLA, mathematics can help to analyze sport data and design an appropriate strategy for striving for sporting excellence.
5.8.4 In the Personal, Social and Humanities KLA, a variety of mathematical tools and procedures are used in making rational and responsible social decisions, such as identifying patterns and trends in statistical data and assessing validity in personal and social issues.
Mathematical models are also used in theorizing knowledge in Social Sciences, in particular, Economics.
5.8.5 In the Science KLA, laws and formulae are represented in mathematical language, mathematical methods are employed to solve problems and generalize experimental findings and mathematical models are used to represent physical phenomena. In the Technology Education KLA, mathematical models are used in computer simulations to explore the feasibility of applying design ideas to investment decisions, and tables and charts are usually important tools to represent technical information.
6 CONCLUSION
6.1 To meet the challenges of an ever advancing knowledge-based society and a dynamically changing environment, mathematics has unquestionably become one of the necessities for enabling every individual to contribute towards the prosperity of society.
The curriculum framework of the Mathematics Education KLA not only aims to provide students with mathematical knowledge, but also aims to equip them with a repertoire of skills, to help them develop thinking abilities and to foster positive attitudes, so that they can develop capabilities to learn how to learn and the confidence to face the knowledge-based society.
6.2 The revised secondary and primary mathematics curricula have been developed along these lines and will be implemented in September 2001 and September 2002 respectively. The sixth form mathematics curriculum will be revised once the review of the new senior secondary structure is clear. Additional Mathematics will be revised in the near future to enhance its continuity with the mathematics subjects concerned.
6.3 In this framework, diversified teaching/learning activities and adaptation of the curriculum to suit the different needs, abilities and interests of students are recommended. Multifarious assessment instruments can also be used to get complete profiles of students’ performance. These profiles should be used to improve teaching/learning.
6.4 Apart from formal mathematics education in schools, mathematics-related activities organized by schools and local/international bodies also provide opportunities for students to acquire learning experiences in mathematics.
Students should be encouraged to participate actively in these activities.
You are welcome to send your views to the Curriculum Development Council Secretariat by post, by fax or by e- mail on or before 15 February 2001.
Address: Curriculum Development Council Secretariat Room 1329, Wu Chung House
213 Queen’s Road East Wan Chai
Hong Kong
Fax Number: 2573 5299 / 2575 4318 E-mail Address: cdchk@ed.gov.hk
Appendix 1
Findings of the Research Studies
Conducted in 1998
Appendix 1 Findings of the Research Studies
Conducted in 1998
The summaries8 of the two research studies are as follows:
Research Study 1 – Comparative Studies of the Mathematics Curricula of Major Asian and Western Countries
The study consists of three components: a literature review, an analysis of curriculum documents, and a summary of the HK results in the TIMSS. The main findings are:
(a) The revised Secondary Mathematics Syllabus (1999) in HK is generally in line with worldwide trends.
(b) The HK mathematics curriculum attempts to strike a balance between process abilities (which are very much emphasized in the West) and basic skills and content (which are stressed in Asian countries).
(c) In HK, the introduction of topics into the curriculum is on average 2 years earlier than the international average.
(d) The textbooks in HK focus much of their attention on students’ performance of
“knowing” and “using routine procedures”.
(e) A “canonical” curriculum is usually stipulated by the governments in Asian countries and is followed closely in schools.
(f) East Asian countries put a lot of emphasis on textbooks; by contrast, Western countries are more flexible in their use of textbooks.
(g) Tracking for mathematics teaching is common, and there are various ways of implementing tracking in different countries.
(h) HK is probably the place with the least flexibility and choice in its mathematics curriculum.
The results of TIMSS, which are relevant to the theme of the study, are also summarized:
(a) HK students came fourth both in the 26 countries in grade four and the 41 countries in grade eight. They performed very well in routine problem solving, not so well in solving exploratory problems, and significantly worse in the TIMSS Performance Assessment, where students were required to conduct hands-on activities.
8 The summaries are abridged from Chapter 3 of the final report of the Ad hoc Committee, namely Report on Holistic Review of the Mathematics Curriculum.
Appendix 1 (b) Students in HK, like their counterparts in the rest of the TIMSS countries, found
mathematics important, but they did not particularly like mathematics.
(c) Contrary to the common belief that students in East Asian countries attribute success more to hard work than to natural talent or ability, and that they attach a lot of importance to memorization, the TIMSS results indicate that students do not totally support these stereotypes. Teachers in HK however did not tend to believe in natural talent.
(d) Students in HK did not think that they did well in mathematics and in general, girls had a lower perception of their ability than boys.
(e) Compared to their counterparts elsewhere, HK students spent more out of school time doing mathematics homework, studying mathematics or attending extra mathematics lessons, especially at the primary school level.
The results show that HK students did extremely well in the TIMSS mathematics tests, but some students did not display the corresponding level of positive attitudes towards mathematics and some lacked confidence in doing mathematics.
Research Study 2 – An Analysis of the Views of Various Sectors on the Mathematics Curriculum
The main findings are:
(a) Both students and parents showed high regard for mathematics.
(b) Different stakeholders held a positive view of the mathematics curriculum.
(c) Mathematics education should address a wider objective. HOTs should be addressed and teaching should provoke student thinking.
(d) The interest of students has to be maintained.
(e) The curriculum should be re-designed with epistemological and pedagogical considerations, so as to strengthen thinking and conceptual understanding.
(f) The problem of learner differences has to be addressed, including curriculum differentiation at senior secondary level.
(g) The idea of core and extended curriculum is worth further exploration.
Appendix 1 (l) Collegiate exchange in the field should be promoted.
(m) Different stakeholders should be well informed of future curriculum changes, so that they provide support.
(n) The workload of teachers should be carefully considered.
Appendix 2
List of Tryout Topics/Teaching Strategies
Conducted since 1997
Appendix 2 List of Tryout Topics/Teaching Strategies
Conducted since 1997
Key Stages Topics/Teaching Strategies
1 An Investigation of the Composition of Numbers up to 18 Developing an Understanding of Fractions
2 Squares and Square Roots Equivalent Fractions
Simple Problems on Fractions
Simple Problems on the Multiplication of Fractions
Fractions, Decimals, Percentages and Recurring Decimals (Using Calculators)
Recurring Decimals
Multiplication of Decimals
Multiplying Decimals by Decimals Chance : Its Meaning and Applications Chance
Developing an Understanding of Chance Speed
3 Geometry – Transformation and Symmetry 3-Dimensional Solids
Estimation
Data Handling – Construction and Interpretation of Graphs Using IT in the Teaching and Learning of Geometry
Using IT in Teaching & Learning Mathematics High-Order Thinking Skills
Appendix 2
4 Data Handling – Simple Statistical Surveys Further Applications
Using IT in the Teaching and Learning of Algebra Using IT in Teaching & Learning Mathematics
Appendix 2
Appendix 3
An Overview of Learning Targets of the
Mathematics Curriculum
Appendix 3 An Overview of Learning Targets of the Mathematics Curriculum
General Aims
To enable students to cope confidently with the mathematics needed in their future studies, workplaces or daily life in a technological and information-rich society, so that each student is ready for lifelong learning, the curriculum aims at developing in students:
(a) the ability to conceptualize, 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 and 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 and a sense of measurement as well as the capability of appreciating structures and patterns;
(e) a positive attitude towards mathematics learning and the capability of appreciating the aesthetic nature and cultural aspect of mathematics.
Appendix 3
Knowledge and Skills
The Learning Targets for Key Stage 1 (P1-P3) Number and Algebra
Dimensions
Measures, Shape and Space Dimensions
Data Handling Dimension
Number Algebra Shape & Space Measures Data Handling
l To understand and manipulate whole numbers
l To understand simple fractions
l To examine the reasonableness of results
l To formulate and solve simple problems involving numbers
The ALGEBRA Dimension is not included at this key stage.
l To identify, describe and group lines, angles, 2- dimensional & 3- dimensional shapes
l To recognize intuitively the elementary properties of 3- dimensional shapes
l To recognize the properties of 2-dimensional shapes
l To make 2-dimensional and 3-dimensional shapes from given information
l To recognize and appreciate shapes
To identify the four
l To choose and use a variety of non-standard units to record results in basic measuring activities
l To understand the need to use standard units of measurement
l To select appropriate measuring tools and standard units of measurement
l To integrate knowledge of Number, Measures, Shape & Space to solve simple problems in
l To collect, compare and group discrete statistical data according to given criteria
l To construct and interpret simple statistical graphs showing relations among data
l To formulate and solve simple problems arising from collected data and constructed
Appendix 3
The Learning Targets for Key Stage 2 (P4-P6) Number and Algebra
Dimensions
Measures, Shape and Space Dimensions
Data Handling Dimension
Number Algebra Shape & Space Measures Data Handling
l To understand whole numbers, fractions, decimals, percentages and the relations among them
l To manipulate
numbers and examine the reasonableness of results
l To formulate and solve problems involving numbers
l To use symbols to represent
unknown numbers
l To communicate simple
mathematical facts and relations using symbols
l To formulate and solve simple problems and examine the results
l To understand the properties of 2- dimensional and 3- dimensional shapes
l To group and make 2- dimensional and 3- dimensional shapes
l To identify the eight compass points
l To choose and use a variety of non-standard and standard units to record results in various measuring activities
l To select and justify appropriate measuring tools and standard units of measurement
l To recognize the degree of accuracy and the approximate nature of measurement
l To inquire and use simple measurement formulae
l To integrate knowledge of Number, Measures, Shape & Space to formulate and solve simple problems in measurement
l To understand the criteria for organizing and grouping discrete statistical data
l To apply simple arithmetic and
appropriate scales in constructing and interpreting more complex statistical graphs
l To show relationships among data using a variety of statistical and graphical representations
l To recognize relations and patterns from graphs
l To formulate and solve problems arising from collected data and constructed graphs
Appendix 3
The Learning Targets for Key Stage 3 (S1-S3) Number and Algebra
Dimension
Measures, Shape and Space Dimension
Data Handling Dimension
l To experience rational and irrational numbers
l To develop various strategies in using numbers to formulate and solve problems, and to examine results;
l To develop and refine strategies for estimating
l To extend the use of algebraic symbols in communicating mathematical ideas
l To explore and describe patterns of sequences of numbers using algebraic symbols
l To interpret simple algebraic relations from numerical, symbolic and graphical perspectives
l To manipulate algebraic expressions and relations; and apply the knowledge and
l To understand the nature of measurement and be aware of the issues about precision and accuracy
l To apply a variety of techniques, tools and formulas for measurements and solving mensuration problems
l To explore and visualize geometric properties of 2-dimensional and 3-dimensional objects intuitively
l To use inductive reasoning, deductive
reasoning and an analytic approach to study the properties of 2-dimensional rectilinear shapes
l To formulate and write simple geometric proofs involving 2-dimensional rectilinear shapes with appropriate symbols,
terminology and reasons
l To understand the criteria for organizing discrete and continuous statistical data
l To choose and construct
appropriate statistical diagrams and graphs to represent given data and interpret them
l To compute, interpret and select the appropriate measure to describe the central tendency of a set of data
l To judge the appropriateness of the methods used in handling statistical data
l To understand the notion of probability and handle simple probability problems by listing
Appendix 3
The Learning Targets for Key Stage 4 (S4-S5) Number and Algebra
Dimension
Measures, Shape and Space Dimension
Data Handling Dimension
l To recognize different types of numbers
l To investigate and describe
relationships between quantities using algebraic symbols and relations
l To generalize and describe patterns of sequences of numbers using algebraic symbols; and apply the results to solve problems
l To interpret more complex algebraic relations from numerical, symbolic and graphical perspectives
l To manipulate more complex algebraic expressions and relations, and apply the knowledge and skills to formulate and solve a variety of practical
problems and justify the validity of results
l To interconnect the knowledge and skills in various Learning Dimensions to solve problems
l To use and select inductive reasoning,
deductive reasoning or an analytic approach to study the properties of 2-dimensional shapes
l To formulate and write geometric proofs involving 2-dimensional shapes with appropriate symbols, terminology and reasons
l To inquire, describe and represent geometric knowledge in 2-dimensional space using algebraic relations
l To inquire, describe and represent geometric knowledge in 2-dimensional and 3-
dimensional space using trigonometric functions
l To interconnect the knowledge and skills in various Learning Dimensions to solve problems
l To understand and compute the measures of dispersion
l To select and use measures of central tendency and dispersion to compare data sets;
l To investigate and judge the
validity of arguments derived from a data set
l To formulate and solve further probability problems by applying simple laws
l To integrate knowledge in statistics and probability to solve real life problems
Appendix 4
Developing Generic Skills in the
Mathematics Education Key Learning Area
Appendix 4 Developing Generic Skills in the
Mathematics Education Key Learning Areas Collaboration Skills
(The expected achievements of the learners in this type of generic skills cannot be suitably classified according to key learning stages)
Descriptors of Expected Achievements
across the School Curriculum Exemplars of Implementation in Mathematics Education Understanding working
relationships
Learners will learn to
s clarify and accept various roles and responsibilities of
individual members in a team and be willing to follow team rules
s recognize that individuals as well as the team have to take the consequences for their own actions
Learners
1. share responsibilities and understand the roles of individual members in doing mathematical group work like collecting data, measuring objects and presenting projects
2. understand and accept that members with different cultural backgrounds may have different interpretations of a
mathematical problem (e.g. analyzing statistical data) 3. accept and follow the group decision in doing mathematical
group work
Developing attitudes which contribute to good working relationships
Learners will learn to
s be open and responsive to others’ ideas; appreciate, encourage and support the ideas and efforts of others s be active in discussing and
posing questions to others, as well as in exchanging, asserting, defending and rethinking ideas
s recognize and avoid stereotyping; withhold
premature judgement until the facts are known
s be willing to adjust their own behaviour to fit the dynamics of various groups and situations
Learners
1. discuss and exchange ideas openly with others in completing tasks and solving mathematical problems 2. be patient and listen to others in the discussion of
mathematical problems like experience-sharing in the processes of investigating number patterns or formulating proofs of geometric problems
3. value the contributions of others in accomplishing mathematical tasks or solving mathematical problems together
4. appreciate different solutions to mathematical problems presented by others, for example, in using different approaches to proving mathematical theorems
5. participate actively and pose questions in clarifying one’s arguments in solving mathematical problems, for example, in the discussion of strategies to be adopted in investigating practical statistical problems
Problem solving, planning and making decisions in a small group require the necessary collaboration skills, namely the skills of listening, appreciation, communication, negotiation, making compromises, asserting leadership, making judgement, as well as influencing and motivating others. Learners with these skills will be able to effectively engage in tasks and teamwork as well as working with others. Ultimately, learners will be able to form relationships that are mutually beneficial.
Appendix 4
Descriptors of Expected Achievements
across the School Curriculum Exemplars of Implementation in Mathematics Education Achieving effective working
relationships
Learners will learn to
s select a strategy and plan cooperatively to complete a task in a team
s understand the strengths and weaknesses of members and build on the strengths to maximize the potential of the team
s liaise, negotiate and compromise with others s reflect on and evaluate the
group work strategy and make necessary adjustments
Learners
1. share experience in solving mathematical problems and select cooperatively a suitable strategy to solve a mathematical problem
2. clarify their arguments objectively and rationally in solving mathematical problems, for example, in examining the appropriateness of the strategy adopted in solving mathematical problems
3. liaise, negotiate and compromise with others in selecting a suitable strategy for solving a mathematical problem (e.g.
use a synthetic or analytic approach in solving a geometrical problem)
4. make adjustment, if necessary, to the strategy adopted in solving mathematical problems (e.g. is it necessary to solve a given quadratic equation to prove that it has no real roots?)
Appendix 4 Communication Skills
Descriptors of Expected Achievements
across the School Curriculum Exemplars of Implementation in Mathematics Education Key Stage One (Junior Primary)
Learners will learn to Ÿ comprehend and act
appropriately on spoken instructions
Ÿ use clear and appropriate means of communication, both verbal and non-verbal, to express meaning and feelings Ÿ read and write simple texts
Learners
1. describe objects such as cubes and prisms orally with simple and appropriate mathematical terms (e.g. a cube has six faces)
2. interpret drawings, tables, graphs (e.g. pictograms) and symbols (e.g. ＋, －, ×)
3. present findings with drawings and symbols
4. present data with tables and graphs (e.g. block graphs) 5. describe drawings and symbols in plain language (e.g. 2＋3
as 2 plus 3)
6. express simple daily-life problems in mathematical language (e.g. use symbols like $2×3 and graphs like bar graphs)
Key Stage Two (Senior Primary) Learners will learn to
Ÿ comprehend and respond to different types of texts Ÿ use spoken, written, graphic
and other non-verbal means of expression to convey
information and opinions, and to explain ideas
Ÿ work and negotiate with others to develop ideas and achieve goals
Learners
1. interpret drawings, symbols (e.g. %), tables and graphs (e.g.
broken line graphs)
2. describe and explain findings/results/data of mathematical tasks in both oral and written forms (e.g. the average score of a student’s performance in a test, the favorite fruit)
3. present results of tasks with appropriate drawings and symbols
4. present data with tables, charts and graphs (e.g. broken line graph, straight line graph)
5. describe and analyze data
6. present solutions of problems logically (e.g. use of “=”
properly)
7. express simple problems in mathematical language (e.g. the percentage of discount is 10%)
8. discuss with others in accomplishing tasks (such as projects)
Communication is a dynamic and ongoing process in which two or more people interact in order to achieve a desired outcome or goal. In learning to communicate effectively, learners should learn to speak, listen, read and write effectively. They should learn to select the most appropriate means to convey a message in accordance with the purpose and context of the communication. They should use accurate and relevant information and organize it systematically and coherently for their audience. They should also evaluate the effectiveness of their communication and identify areas of improvement for action.
Appendix 4
Descriptors of Expected Achievements
across the School Curriculum Exemplars of Implementation in Mathematics Education Key Stage Three (Junior
Secondary)
Learners will learn to
Ÿ understand, analyze, evaluate and respond to a range of different types of texts Ÿ use appropriate language
and/or other forms of communication to present information and different points of view, and to express feelings
Ÿ reflect and improve on the effectiveness of their own communication
Ÿ work and negotiate with others to solve problems and accomplish tasks
Learners
1. interpret numeric, symbolic and graphical presentations 2. describe findings or explain conjectures in both oral and written forms using mathematical language (e.g. the two triangles are congruent)
3. choose appropriate statistical diagrams/graphs to present given data and use appropriate mathematical terminology or symbols in explaining ideas
4. formulate and write simple geometric proofs involving 2-D rectilinear shapes with appropriate symbols, terminology and reasons
5. interpret and respond appropriately to others’ mathematical arguments in both oral and written forms
6. distinguish the difference between the language used in a mathematical context and daily life (e.g. rate, similar) 7. use mathematical language including graphs, figures and
symbols to analyze and present possible solutions to a problem and discuss with others
Key Stage Four (Senior Secondary)
Learners will learn to
Ÿ listen and read critically, and speak and write fluently for a range of purposes and audiences
Ÿ use appropriate means of communication to inform, persuade, argue and entertain and achieve expected
outcomes
Ÿ critically evaluate the effectiveness of their communication
Ÿ resolve conflicts and solve problems with others to accomplish tasks
In addition to points 1-6 in KS3, point 4 is modified as follows:
Learners
4. formulate and write geometric proofs involving more complex 2-D shapes with appropriate symbols, terminology and reasons
Further points include:
Learners
8. investigate and judge the validity of arguments presented in statistical reports from various sources, including those from the media (e.g. select appropriate samplings for statistical surveys)
9. resolve problems with others in accomplishing tasks such as projects
Appendix 4 Creativity
(The expected achievements of the learners in this type of generic skills cannot be suitably classified according to key learning stages)
Descriptors of Expected Achievements
across the School Curriculum Exemplars of Implementation in Mathematics Education Learners will learn to
Ÿ strengthen creative abilities:
fluency2, flexibility3, originality4, elaboration5, sensitivity to problems6, problem defining7, visualization8, imagination, analogical thinking9, analysis, synthesis, evaluation,
transformation10, intuition, logical thinking, etc.
Ÿ develop creative attitudes and attributes: imagination, curiosity, self-confidence, independent judgement, persistence and commitment, tolerance for ambiguity, openness to new and unusual
ideas/methods/approaches, deferment of judgement, adaptability, willingness to take sensible risks, etc.
Ÿ use and apply the Creative Problem Solving (CPS) Model and creative thinking
techniques: brainstorming, 6W thinking technique, 6 hats method, attribute listing11, idea checklists, synectics12, mind mapping, etc.
Learners
1. create geometric patterns with different shapes and tell stories with given mathematical sentences
2. devise their own way/strategy in solving problems such as different solutions to a plane geometry problem
3. adopt different approaches to a task or problem, such as proving a geometrical theorem using a synthetic or an analytical approach
4. pose related problems such as “Can triangles other than equilateral triangles be used in tessellation?” and “Will the same relationshipa2 + b2 = c2in Pythagoras’ Theorem still hold if the triangle is not right-angled?”
5. formulate hypotheses such as that the value of a fraction decreases as the denominator increases if the numerator is kept constant
6. be persistent when solving problems 7. be imaginative in visualizing 3-D shapes
8. be open-minded to new methods and approaches in accomplishing tasks and solving problems, such as using a synthetic or an analytic approach in solving geometrical problems
9. use and apply the technique of synectics to relate different given information, and utilize analogies to help analyze problems, such as deducing the formula of the volume of a cylinder from that for a prism
A brief description: Creativity is an important but elusive concept. It has been defined in a variety of ways. Some people define it as an ability to produce original ideas and solve problems, others see it as a process, and yet others take it as certain personal qualities. In fact, creativity is a complex and multifaceted construct. Within the individual, creative behaviour is the result of a complex of cognitive skills/abilities, personality factors, motivation, strategies, and metacognitive skills.
Person’s creative performance may not correspond to his/her developmental stages.
General Principles: Although the demanding process of teaching for creativity is hard to make routine, some principles apply in general. To develop students’ creativity, we ask them to go beyond the given information, allow them time to think, strengthen their creative abilities, reward their creative efforts, value their creative attributes, teach them creative thinking techniques and the Creative Problem Solving model, and create a climate conducive to creativity1. These principles can be employed in all key learning areas (KLAs).
Appendix 4
Notes:
1. Climate conducive to creativity: Respecting the novel and unusual, providing challenges, appreciating individuality and openness, encouraging open discussion, absence of conflicts, allowing time for thinking, encouraging confidence and a willingness to take risks, appreciating and supporting new ideas, etc.
2. Fluency: The ability to produce many ideas in response to an open-ended problem, question or task.
3. Flexibility: The ability to take different approaches to a task or problem, to think of ideas in different categories, or to view a situation from several perspectives.
4. Originality: Uniqueness, nonconformity in thought and action.
5. Elaboration: The ability to add details to a given idea, such as to develop, embellish, and implement the idea.
6. Sensitivity to problems: The ability to identify problems, list out difficulties, detect missing information, and ask good questions.
7. Problem defining: The capability to 1) identify the “real” problem, 2) isolate the important aspects of a problem, 3) clarify and simplify a problem, 4) identify subproblems, 5) propose alternative problem definitions, and 6) define a problem broadly.
8. Visualization: The ability to fantasize and imagine, “see” things in the “mind’s eye” and mentally manipulate images and ideas.
9. Analogical thinking: The ability to borrow ideas from one context and use them in another; or the ability to borrow the solution to a problem and transfer it to another.
10. Transformation: The ability to adapt something to a new use, to “see” new meanings, implications, and applications, or to change an object or idea into another creatively.
11. Attribute listing: A creative thinking technique that involves listing out all the important characteristics of an item and suggesting possible changes or improvements in the various attributes.
12. Synectics: The joining together of apparently unrelated elements. This technique utilizes analogies and metaphors to help the thinker analyze problems and form different viewpoints.
Appendix 4 Critical Thinking Skills
Descriptors of Expected Achievements
across the School Curriculum Exemplars of Implementation in Mathematics Education Key Stage One (Junior Primary)
Learners will learn to
Ÿ extract, classify and organize information from a source Ÿ identify and express main ideas, problems or central issues
Ÿ understand straightforward cause-and-effect relationships Ÿ distinguish between obvious
fact and opinion
Ÿ recognize obvious stereotypes, assumptions, inconsistencies and contradictions
Ÿ formulate questions, make predictions/estimations and hypotheses
Ÿ draw simple but logical conclusions not contradictory to given evidence and data
Learners
1. sort objects using various criteria such as shapes and sizes 2. choose the right tools to measure objects such as using
measuring tapes to measure the “circumference” of a head 3. reason inductively such as when exploring the
commutative property of addition
4. check the reasonableness of the answer to a problem (e.g.
the number of apples eaten by a boy per day found in one problem is too large to be realistic)
Critical Thinking is drawing out meaning from given data or statements. It is concerned with the accuracy of given statements. It aims at generating and evaluating arguments. Critical thinking is the questioning and inquiry we engage in to judge what to and what not to believe.
