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Question
(a) Explain the following two initiatives in detail:
i ‘investigation’ in the British mathematics curriculum
ii ‘problem solving’ in the US mathematics curriculum.
(b) Discuss the extent to which recent curriculum development in Hong Kong secondary school mathematics aligns with the two curriculum initiatives above. Discuss also any difficulties in introducing them into local mathematics classroom.
(a)
(i)
INVESTIGATION
After the National Curriculum was introduced in the British, the mathematics skills were outlined by this curriculum. This mathematics curriculum stress mathematics processes such as investigation and apply mathematics. Mathematical investigations were involved in the much work of students. (OUHK, 2001, 3:21) Since investigation work is believed that will foster the various mathematics skills of students, it is viewed as central to the British mathematics curriculum. Moreover, the Cockcroft Committee approved the teaching practices, one of these mathematics teaching practices at all levels should include opportunities for investigational work. (Roper and Carter, 1994, 59) Cockcroft (1982) emphasized that the idea of investigation is basic both to the study of mathematics itself and also to an understanding of the ways in which mathematics can be used to solve problems and to extend knowledge in many fields.
A mathematical investigation is an extensive task of work, which usually last for long time to finish. Investigation can also be undertaken by individual student or by small group formed by students. The problem under investigation should be opened and need not have a known goal. The essence of an investigation work is the application of communication, reasoning, operational and recording processes to a study of the core topics which make up the content of a mathematics curriculum. (Frobisher, 1994, 169) In the processes of the investigation work, the students should be encouraged to use the skills of talking, questioning, collecting, analyzing, sorting, ordering, writing, and graphing and so on to explore the problem.
As investigation is a very important teaching process in mathematics teaching, the role of the teachers are changed as an instigator, enabler, facilitator, listener, questioner, positive evaluator and observer. (Pirie, 1987) Investigation is basically activity based, and not simply as lecturing and listening in the teaching and learning experiences of students. Investigation is also a teaching practice of student-centered.
(ii)
PROBLEM
SOLVING
Since the late 1970s, problem solving has been
advocated as foundation of the mathematics curriculum in the United States. (OUHK, 2001,
3:29) From the result of the students in
state and district mathematics examinations, the ability of problem solving was poor than
that of arithmetic. (Lui, 2000, 133) Thus,
problem solving began to be emphasized in mathematics curriculum. The ability to solve problems is very important in
mathematics. In the NCTM (1977) position
paper, learning to solve problems was regarded as the main reason for studying
mathematics.
Problem solving can be defined as the ability to
solve unknown and non-routine problems. Problem
solving must go through the thinking of the steps and methods of solutions. It also has to try the new and different methods
to solve the problems. The students may meet
the several frustrations, and they will begin to solve the problem again. (Lui, 2000, 133)
Polya (1945) thought the problem solving was an
heuristic approach to solving the problems of mathematics.
Problem solving can be summarized as a four-stage process that are
understand the problem, devise a plan, carry out the plan, and look back. Also, there are some examples of heuristic
strategies of problem solving. These examples
include guessing, checking and improving, looking for patterns, making a model or drawing
a picture, making an organized list or table, restating the problem, separation out
irrelevant information, identifying, and attempting subtasks, solving a simpler version of
the problems and eliminating possibilities. (OUHK, 2001, 3:30) If the students can be solving a wide range of
problems and understanding that a particular problem may be solved using a number of
different strategies, they can develop their problem solving abilities. Moreover, students should be encouraged to solve
the problem using different methods and strategies, and also to pose a new or related
problem. (OUHK, 2001, 3:30)
The concept of problem solving has great impact to
the mathematics curriculum since the knowledge component of mathematics is no longer
viewed as the only element in the mathematics curriculum.
The mathematics curriculum is no longer a fixed unchanged body of knowledge
because the mathematical process is viewed to be an important part of the curriculum.
(OUHK, 2001, 3:30)
(b)
The recent mathematics curriculum
development in Hong Kong secondary school can be viewed in the two documents that are the
Syllabuses for Secondary Schools: Mathematics (Secondary 1 to 5) and the Learning to
Learn: Key Learning Area Mathematics Education (Consultation Document).
The new aims and objectives of Syllabuses adopt a perspective on mathematics as both process and content oriented, and treat mathematics as dynamic, reacting to social changes and technological changes. For example, it emphasize the principles of curriculum design that are target-oriented, catering for learner differences, relevance of study to students, impact of information technology, and fostering general abilities and skills. (CDC, 1999, 1-3)
In
the document of Learning to learn, 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 skills
throughout their life time. (CDC, 2000, 7)
The past emphasis on content knowledge
and skill proficiency have been replaced by emphasis on experience in mathematical
processes and developing higher cognitive skills, together with affective development. This greatest change in the mathematics curriculum
has been found in the philosophy of mathematics held by the mathematics teachers. (OUHK,
2001, 4:31) Since these two documents have great change in future
development of mathematic curriculum, we can discuss the extent to which that aligns with
two curriculum initiatives discuss above.
In syllabuses and document, the principle of curriculum design indicate that “It is important that students need to develop their capabilities to learn how to learn, to think logically and creatively, to develop and use knowledge, to analyze and solve problems, to access information and process it effectively, and to communicate with others so that they can meet the challenges that confront them now and in the future”. (CDC, 1999, 3) Also, “fundamental and intertwining ways of learning and using knowledge such as inquiring, communicating, reasoning, conceptualizing and problem-solving are considered important in mathematics education.” (CDC, 1999, 3) Student are not only expected to learn mathematics to enhance the development of these skills, but also expected to use these learning strategies to construct their mathematics knowledge. Teachers should plan and implement the variety of learning activities to develop these general abilities and skills.
In the chapter 5 of syllabuses, there are some teaching suggestions. In the process of learning, the important of problem solving in mathematics education was stressed. It involves understanding the problems, considering possible strategies and choosing an appropriate one to solve the problem, carrying out the plan, and justifying or evaluating the solution. (CDC, 1999, 37)
In the document of Learning to Learn, the teachers’ role should be to facilitate students to learn how to learn through using the high-order thinking and generic skills in mathematics. Moreover, diversified teaching and learning activities should be encouraged for 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. (CDC, 2000, 13) This point is similar with the role of teacher in investigation as facilitator discussed above. Also, the rationale of problem solving is application of different skills to solve unknown and not-routine problem. Such abilities can apply in real-life situation.
In the syllabuses, the application of communication, reasoning, operational and recording processes are also emphasized and the students are also encouraged to think, analyze, solve problem, access information, and communicate with others but ‘investigation’ is not shown directly. In the syllabuses, the stages of problem solving are stressed in its teaching suggestions.
Therefore, above content of the syllabuses and document quite align with the initiatives of investigation and problem solving. Of course, Hong Kong is influenced by Britain and United State in many aspects. In mathematics education, Hong Kong is inevitable influenced by those countries, especially Hong Kong is a colony of Britain before 1997. Although there is no firm stating the initiatives of investigation and problem solving in government policy documents, the contents of these document usually mention the rationale and principle of investigation and problem solving. These rationale and principle may be help to resolve problems that mathematics teachers in Hong Kong may have in their teaching. Thus, the recent curriculum development in Hong Kong secondary school mathematics quite aligns with the two curriculum initiatives mention above.
In Hong Kong, many teachers are preferring a more traditional ‘chalk and talk’ approach, so few teachers would use investigation and problem solving for the teaching of mathematics. It is believed that the activities of investigation and problem solving can foster the mathematical learning of students. But, the teachers of Hong Kong are too less experienced and clear about the thinking processes of students through the investigation and problem solving for making the students useful. (OUHK, 2001, 3:29)
Moreover, there are about 40 students in the classroom in Hong Kong, so it is difficult for teachers to allow students to proceed to individual investigation and problem solving activities. For example, the teachers are difficult to observe the processes of about 40 students to investigate and solve their problems in the same time. (OUHK, 2001, 3:29)
Also, the individual difference of the students is very great in low banding secondary school. Investigation and problem solving are too difficult for those students who even do not understand the foundation of mathematics.
In Hong Kong, the students of senior
secondary school only have one and half years to study at least six subjects and then take
the examinations of certificate. The
investigation and problem solving need more time to let students to experience and think
in these processes. Indeed, the teachers are
not sufficient time to implement the learning activities of investigation and problem
solving for completing the whole syllabuses of mathematics in this short time.
The students of Hong Kong have
habituated as a receiver in classroom. They
seldom ask and answer the question. They are
very passive from the stages of primary school. However,
many teachers like this kind of students because it can make the lesson run smoothly. The student has said, “teacher, please don’t
play the game and activities, we are not enough time to learn the whole syllabuses of
mathematics”. Thus, introducing
investigation and problem solving into local mathematics classroom need to match with
change of teacher training, syllabuses, classroom size, and students’ learning culture. Otherwise, the best initiatives are also difficult
to implement.
Cockcroft, W H (1982) Mathematics Counts, Her Majesty Stationery Office.
Curriculum Development Council (1999) Syllabuses for Secondary Schools: Mathematics (Secondary 1-5), Hong Kong: Printing Department.
Curriculum Development Council (2000) Learning to Learn: Key Learning Area Mathematics Education (Consultation Document), Hong Kong: Printing Department.
Frobisher, L (1994) ‘Problems, Investigations and an Investigative Approach’ in Orton, A and Wain, G (Eds) Issues in Teaching Mathematics, British: Cassell.
National Council of Teachers of Mathematics (1977) ‘NCTM position statement on basic skills’, Arithmetic Teacher, Vol. 25, 19-22.
Pirie, S (1987) Mathematical Investigation in Your Classroom, Basingstoke: Macmillan.
Polya, G (1945) How to Solve It, Princeton: University Press.
Roper, T and Carter, D (1994) ‘The National Curriculum in Mathematics’ in Orton, A and Wain, G (Eds) Issues in Teaching Mathematics, British: Cassell.
The Open University of Hong Kong (2001) ‘Mathematics Curriculum’ in ES361 Teaching Mathematics in Secondary Schools, Hong Kong: OUHK.
呂秋文
(2000) 《新數學科教材教法》，台北：五南圖書公司
