The Singapore Mathematics Curriculum and Mathematical Communication
Yeap Ban Har
National Institute of Education
Nanyang Technological University
Singapore
Introduction
In this paper, we consider various aspects of the curriculum to determine the extent to which
the Singapore mathematics curriculum includes mathematical communication. The termcurriculum can be interpreted and investigated at different levels. First, we can consider theintended curriculum by examining the curriculum document. Second, we can consider thehidden curriculum by examining the nature of tasks in the national examination. Third, wecan consider the interpreted curriculum by investigating the textbooks. Finally, we canconsider the implemented curriculum by studying actual classroom lessons. In this paper, we
do the first three. The proposed study, to be reported in the next meeting, focuses on the last.
Based on an analysis of the curriculum, and a review of the literature, a framework that
describes various aspects of mathematical communication is described along withsuggestions on how mathematical communication can be infused into the mathematics
classroom. The paper concludes with a proposal to use lesson study as a platform for teachers
to gain a deeper understanding of mathematical communication, its relevance inmathematical thinking and problem solving, and how to organize classroom activities topromote it.
The Singapore Mathematics Curriculum
The Singapore mathematics curriculum (Ministry of Education of Singapore, 1990, 2000,2005) has mathematical problem solving as its primary goal. According to the curriculum, thedevelopment of mathematical problem solving is dependent on the development of five interrelated components, namely (1) concepts, (2) skills, (3) processes, (4) attitudes, and (5)metacognition.
The processes listed in the revised curriculum framework (Ministry of Education ofSingapore, 2005) include reasoning, communication and connections. For primary schools,
this revised framework was implemented in 2007 for grades one to four. Subsequently, therevised framework will be implemented in 2008 for grade five and in 2009 for grade six.
Among the stated aims of the curriculum is to “develop the abilities to reason logically, tocommunicate mathematically, and to learn cooperatively and independently” (Ministry ofEducation of Singapore, 2005, p.5). As the curriculum is designed to “ensure that we have anincreasingly competitive workforce to meet the challenges of the 21st century (Ministry ofEducation of Singapore, 2005, p. 5)”, communication has to be an important part of thecurriculum.
In the curriculum document, communication is used to “refer to the ability to use
mathematical language to express mathematical ideas and arguments precisely, concisely andlogically” (Ministry of Education of Singapore, 2005, p.8). Communication is described as away to help students develop their own understanding of mathematics and to sharpen theirmathematical thinking.

- Yeap Ban Har 1 -The curriculum also made a call for changes in the way teachers teach and students learn
mathematics due to advances in technology. Students should have opportunities to discover,reason and communicate mathematics by engaging in “stimulating discussions and activities(Ministry of Education of Singapore, 2005, p. 2)” to explore possibilities and makeconnections.
These qualitative changes require a change in the teaching and learning approaches. Lesson
study holds promise as a professional development tool to effect such a change.
It is evident that the intended mathematics curriculum in Singapore places a great deal ofimportance on mathematical communication.
The Role of the National Examination
At the end of six years of primary schooling, students in Singapore sit for Primary School
Leaving Examination (PSLE). Only 20% of total score comes from selected-response(multiple-choice) items. Among the constructed-response items, the majority require students
to communicate their methods.
The following are selected-response items that assess ability to solve problems.
Alice bought some guppies. She gave away 11 of them. Benny gave her the same number ofguppies as the number of guppies she had left. She put all the guppies equally into 8 bowls.
Each bowl contained 13 guppies. How many guppies did Alice buy?
(1) 52(2) 63(3) 104(4) 115(SEAB, 2007, p. 9)
The chairs in a hall were arranged in rows. Each row had the same number of chairs.
Weiming sat on one of the chairs. There were 5 chairs to his right and 5 chairs to his left.
There were 7 rows of chairs in front of him and 7 rows of chairs behind him. How manychairs were there in the hall?
(1) 140(2) 150(3) 154(4) 165(SEAB, 2007, p. 9)

While knowledge-level items require minimal level of reading and comprehension ability,problem-solving items in the selected-response format expect students to be able readreasonably complex paragraphs.
The following are constructed-response items that assess ability to solve problems.
Siti started saving some money on Monday. On each day from Tuesday to Friday, she saved
20 cents more than the amount she saved the day before. She saved a total of $6 from Mondayto Friday. How much money did she save on Monday?
(SEAB, 2007, p. 40)- Yeap Ban Har 2 -At 09 00, a lorry started from Town X and travelled towards Town Y at a speed of 55 km/h
for the whole journey. At 11 00, a car started from Town Y and travelled towards Town X.
The speed of the car remained the same until it passed the lorry at 13 00. At this point, thelorry had travelled of the journey. After passing the lorry, the car decreased its speed by 8km/h and travelled at the new speed for the remaining journey. At what time did the car reach
Town X?
(SEAB, 2007, p. 41)

Similarly, these constructed-response problem-solving items require students to be able toread reasonably complex paragraphs. In addition, students are expected to be able tocommunicate their thinking and methods by showing how they reason and arrive at theanswer. Typically, up to 80% of the total score for an item is awarded for the ability tocommunicate one’s method.
In Singapore, students learn mathematics (and all other subjects except mother languages) in
the English language from the first grade. This is despite the fact that only 21% of thesampled students reported that they always use English at home in PIRLS1 2006 (Ministry ofEducation of Singapore, 2007). The prevalent use of English in schools probably contributed
the increased proportion of students who use English at home. In PIRLS 2001, 37% of thesampled students reported that they used English at home. In PIRLS 2006, 42% did.
The nature of the national examination has made it necessary for students to be able tofunction at both ends of the communication process, i.e. to be able to understand and to beable to be understood.
The Role of the School Textbooks
Textbooks used in Singapore schools have to be approved by the Ministry of Education
before schools can elect to use them. A complete analysis of textbooks used in Singapore is
not within the scope of this paper. However, several textbook studies had shown that one
common method used and taught in the textbooks for problem solving is the so-called ‘model
method’. Figure 1 shows an example from a textbook for the fourth grade.
Ravi and Samuel collected 1775 stamps altogether. Samuel bought another 85 stamps. Ravithen had thrice as many stamps as Samuel. How many stamps did Samuel have at first?

Figure 1 Word problem and the ‘model method’The use of ‘model method’ provides students with a means to handle information, deal withcomplexity and, at the same time, communicate their thinking through the use of visuals
which they can manipulate.

1 Progress in Reading Literacy Study (PIRLS) was conducted in 2001 and 2006.

- Yeap Ban Har 3 -The extensive use of this problem-solving heuristic in the textbooks is probably the main
reason why most teachers include it in their lessons and why many students are comfortableusing it as a problem-solving and communication tool.
From an investigation of the curriculum document, national test items and textbooks, it
becomes clear that consistency among the three elements of the curriculum is probably thereason why you would expect mathematics teachers in Singapore to place emphasis on being
able to handle the language aspects of word problems and communicating their solutions inwriting and using diagrams. In this case, the curriculum document does demand mathematicalcommunication. The national test expects students to be able to engage in mathematicalcommunication. The textbooks provide tools for students to do so, even in fairly complex
tasks.
Aspects of Mathematical Communication
In this section, we further clarify the different aspects of mathematical communication. Thereis the ability to understand and ability to be understood. The ability to read and comprehend
word problems is an example of the former. The ability to present one’s thought in solving
problems is an example of the latter.
In either aspects, there are the skill aspects, the metacognitive aspects and the attitude aspects.
In understanding, examples of skill aspects of communication include ability to read. In
making oneself understood, examples of skill aspects include ability to draw ‘models’ torepresent one’s thoughts during problem solving. In understanding, examples ofmetacognitive aspects include the ability to monitor one’s reading so as to achievecomprehension. In making oneself understood, examples of metacognitive aspects include themonitoring of one’s solution as one is solving it. In understanding, examples of attitudeaspects include not possessing negative feelings towards lengthy word problems. In making
oneself understood, examples of attitude aspects include the willingness to explain oneself inwriting although one may be able to see the entire solution mentally.
The above focuses on aspects of mathematical communication in the process of solving
problems. There are also aspects of communication in mathematical activities other thanproblem solving.
Research Plan
The general research proposal is to use lesson study to help teachers learn more aboutmathematical communication, its development and its role in mathematical problem solving(and mathematics learning).
During the Tokyo-Kanazawa session, we will have opportunities to benefit from several
keynote lectures. We will also have opportunities to see how different economies andresearchers interpret what mathematical communication is, what types of instructional
strategies can promote communication and the roles of communication in mathematics
learning including problem solving.

- Yeap Ban Har 4 -