Critique of an approved textbook for the Singapore Mathematics Curriculum
(5,000 words essay)
1. An Introduction
The mathematics textbook is an important teaching and learning resource in a Mathematics classrooms. This practice is not only true in Singapore, but also is also prominent worldwide. Mathematics classrooms in countries like Germany (Rezat, 2009), the United States (Shih, 2008) and Sweden (Skolverket, 2003) reported similar practices, too.
In Singapore, all schools with students siting for the national examinations would deliver the respective subject curriculum that are developed by the Ministry of Education (MOE). To ensure that no part of the syllabus was left out, textbook publishers must seek the approval from the Curriculum Planning and Development Division in order to have their titles ‘rubber-stamped’ to be deemed suitable for use in classrooms.
Singapore is recognised worldwide for her success in producing students with excellent achievements in mathematics. In the Trends in International Mathematics and Science Study (TIMSS), Singapore students secured the top 3 places in both Grade 4 and Grade 8 categories since 1999. Since 2000, some schools in the US and Israel started adopting the Singapore mathematics textbook in their mathematics classrooms, too.
Over the past 10 years, the profile of the learners has changed. Studies were carried out on the ‘new’ set of skills that are going to be relevant to the 21st century. Technology use has become ubiquitous. Being responsive to the global changes, the MOE has also drawn up a set of 21st century skills, known as “Competencies for the 21st Century” whereby all learners are expected to acquire through the curriculum before they leave the school. As a result, it also demands changes in the way we teach. More importantly, how we frame our minds to teacher in a whole new way that opens up ‘spaces’ for the learners to learn, not just adhering to the requirements of the mathematics syllabus, but also the 21st century skills. These changes demand teachers to revisit their beliefs in the way mathematics is learnt, as well as the change in the teachers’ pedagogical practices.
Since mathematics textbook is one of the essential items in the classrooom, to what extent has it cater to the changing needs of the learner as well as the key areas spelt out in the curriculum? This key focus of this paper is therefore to critique an approved textbook used in a typical mathematics classroom in Singapore. In my critique of the textbook, I shall make reference to the framework for publishing textbooks (Fan, 2010):
§ Curriculum Principle
§ Discipline Principle
§ Pedagogy Principle
§ Technology Principle
§ Context Principle
§ Presentation Principle
In fact, the key preposition is to examine, “To what extent, has the design of the textbook been relevant and adequate to support the desired outcomes in the Singapore Mathematics curriculum, in today’s content?”
To do this, we would have to first understand and identify the needs of today’s learners in order to understand of how they learn. At the same time, we would examine the curriculum documents closely to understand the desireable outcomes of the mathematics curriculum before we could take into consideration when designing the learning resources. The curriculum outcomes emcompasses the syllabus (which are knowledge and matematical skills) as well as the soft skills and dispositions that we hope to cultivate through the subject discipline.
These 2 key components, learners and curriculum, would shape how a mathematics classroom would look like, and hence influencing the approach and learning theories that teachers would adopt deliver the lessons, hence the students’ experiences in the mathematics classroom. Being the primary resource in the most of the mathematics classrooms, the textbook used should therefore be designed to cater to the above needs.
In this critique, recommendations would be also made, when possible, to better areas that that have been identified as areas for improvement.
2. Knowing Our 21st Century Learners
“Today’s students are no longer the people our educational system has designed to teach” (Prensky, 2001). They are “native speakers” of the digital – they grow up in an environment evolving around computers, video games and the Internet. They possess many educational traits that many older teachers may not be familiar or comfortable with. Their value system and characteristics are largely shaped by the environment that they are brought up, could also be quite different from learners in the past. Rodgers (2006) described, “today’s learners are digitally literate, mobile, always on, experiential and social. To them, computers aren’t technology – they are just part of their life and experience background”. Today’s learners are good at learning with visuals and mutlimedia and they love interactivity and social activities; and they have the tendency to multitask.
Both Prensky (2001) and Rodgers (2006) pointed out the need of teachers, in this case, mostly the “Digital Immigrants” to be responsive to the digital natives, which includes the communication and learning experiences crafted out for the learners. This implies that lesson materials, like textbooks, ought to be designed such that they take into consideration the ‘changed’ learning style (that is, those of the digital natives). It should also factor in pedagogy approaches that teachers are most likely to employ in today’s classrooms.
The 21st century curriculum and instruction place great emphasis on the skills, knowledge and expertise that students should master to succeed in work and life in the 21st century. According to the framework for the 21st century learning, the outcomes of 21st century learners is a blend of specific skills, content knowledge, expertiise and literacies.
There is a paradigm shift from the traditional “knowledge and content” towards more holistic approach of equipping learners with “life skills”.
3. Understanding the Singapore Mathematics Curriculum
Mathematics is believed to be the vehicle for development and improvement of one’s intellectual ability in logical reasoning, spatial visualisation, analysis and abstract thought. Hence, it is clearly spelt out in the Singapore Mathematics curriculum document that “students develop numeracy, reasoning, thinking skills and problem solving skills through the learning and application of mathematics”. This whole approach was driven by the idea that to remain competitive to meet the challenges of the 21st century, learners must be euipped with skills valued in everyday living, further learning and at the work place.
Mathematical problem solving is central to mathematics learning. It involves the acquisition and application of mathematical concepts and skills in a wide range of situations, including non-routine, open-ended and real world problems. Accoridng to the framework, the development of mathematical problem solving ability is dependent on five inter-related components, namely, Concepts, Skills, Processes, Attitudtes and Metacognition.
Being a hierarchical subject, Mathematics learning involves having higher concepts and skills built upon the more foundational ones and have to be learnt in a sequence. The design of the syllabus obviously echoed Piaget’s Stages of Cognitive Development. From the primary to secondary mathematics syllabus, the concepts and skills covered progress from concrete to formal operational. For example, concrete model drawing is introduced at primary level for simple problem solving before algebra was progressively introduced and widely adopted at secondary levels to deal with more complex and abstract problem solving.
Below is an example of the Strand, Probability and Statistics, in secondary mathematics, that illustrates how the spiral approach is adopted in building up the content across the levels:
At secondary one: At primary level, students are introduced to the three basic graphical representations (bar graphs, pictograms and pie charts). They are able to recognise the graphs and read information from the graphs.
Building on this knowledge, In secondary one, students are now introduced to the data collection methods so that they learn and appreciate the processes behind the data sets that they have been dealing with since primary levels. They also moved into more complex tasks like constructing the graphs and interpreting the graphs, apart from just ‘reading’ information. More graphical representations are also introduced at secondary one, where students learnt that more ‘complex’ graphs such as histogams are used to handle real world situations that could not be addressed by simple graphs that handle discrete information only.
At secondary two: More complex graphical representations are introduced, and students now learn a ‘variation’ of averages and gain an understanding the appropriate use of the different kinds of averages, depending on the context and situation arises. The expanded repetoire of graphs also brings students into the next level – discussion of the spread of data, where they would make use of this knowledge and understanding in the upper secondary mathematics syllabus.
At upper secondary (secondary 3 and 4): Students learnt to interpret grouped data and gain deeper understanding on the characteristics of each set of data before making comparisons of the means to draw out meaningful inferences from the data, substantiate and supported by statistics. Through this, students are able to exercise disconcernment on the reliability of data.
The knowledge and skills developed over the four years of secondary mathematics on this strand would contribute to the development of students’ statistical thinking, in preparation of their mathematics learning at higher levels.
On top of this, the introduction of new knowledge takes into consideration of past learning, which aligns itself very well with Vygotsky’s (1978) “Zone of Proximal Devleopment” (ZPD) – to faciltiate connections of knowledge. Vygotsky’s often quoted definition of ZPD, “the distance between the actual development level as determined by independent problem solving and the level of potential development as determined through problem solving under adult guidance, or in collaboration with more capable peers”.
In the mathematics framework developed by the Mathematics Unit in the Curriculum Development and Development Division (CPDD), Ministry of Singapore, two components, Mathematical concepts and Processes also placed emphasis in being able to make connections across topics and disciplines, as well as real world applications:
Mathematical concepts: “Students should develop and explore mathematics in depth, and see that mathematics is an integrated whole, not merely isolated pieces fo knowledge.”
Mathematical processes – Connections: “refer to the ability to see and make linkages among mathematical ideas, between mathematics and other subjects, and between mathematics and everyday life. This helps students make sense of what they learn in Mathematics”.
While it is not explicitly articulated in the curriculum document, elements of site-based curriculum design have been weaved into the curriculum. This could imply the possible incorporation of applied learning approach in the design of the curriculum, where creating learning opportunities where students could make connections to their prior knowledge, learn through hands-on activities that are linked to real world context and subsequently able to apply knowledge and skills to new situations and contexts.
4. Initiatives to be infused into the Subject Discipline (Mathematics)
Apart from the syllabus that addresses to the discipline content and skills, there are two other major educational initiatives that are to be infused into curriculum (across all subjects in Singapore), Information and Communication Technology (ICT) and National Education.
Information and Communication Technology (ICT)
Since 1997, the MOE rolled out its ICT MasterPlans to drive the use of ICT in education. The underlying philosophy of the MasterPlans is that education should continually address to the needs of the future and prepare learners to meet those needs. At present, the emphasis of the third MasterPlan for ICT in education (2009-2014) is “to enrich and transform the learning environments of our students and equip them with the critical competencies and dispositions to success in a knowledge economy”. Students should develop competencies for self-directed and collaborative learning through the use of ICT.
As a methodology for instruction, self-directed learning refers to a learning situation in which an individual student or trainee works with instructional materials on his or her own time, without direct supervision or guidance from either instructor or fellow students. (Keirns, 1999)
Collaborative learning refers to the social process whereby students learn through interacting with others. It entails social construction of knowledge (Nussbaum, 2008)
§ Perform inquiry and solve problems together
§ Propose ideas/ theories
§ Support and evaluate claims with evidences
§ Offer multiple perspectives, rise above, etc
§ Build on each others’ ideas
In 2008, the Educational Technology Division in the MOE rolled out the “Baseline ICT Standards for Pupils” initiative to all primary and secondary schools. Students are expected to acquire and be able to apply a defined set of ICT skills, defined by 4 key stages. A mapping of skills to subjects was provided in the implementation guide as we recognise that some skills present a more natural fit in certain subjects, topics and activities. At secondary level mathematics, the following are proposed:
Basic operation skills where students would maniputlate in an interactive media environment such as Geogebra to explore and investigate geometrical properties.
§ Learning with searches where they could search for online information for analysis.
§ Learning with spreadsheet where students process data and interpret results to draw inferences.
§ Learning with Data Collection Tools where students select appropriate ICT tools to measure, collect and process data to support an investigation.
If we make referece to the mathematics framework closely, there are plenty more ICT applications or platforms, in particular Web 2.0 tools, that we could leverage on to develop students’ attitudes and processes (apart from the generic tools suggested by ETD). For example, the use of blogs for students to communicate their ideas using the mathematical language and vocabulary; the use of google form and google spreadsheet where for formative assessment, to check for understanding and clarify misconceptions. Other tools and platforms such as Google Map and Wallwisher could also be creatively used to support activities that promote critical thinking skills in the subject. The integration of ICT into the teaching and learning of mathematics can help develop and motivate students’ interest in the subject, enrich their learning experience and spur them to learn independently.
National Education (NE)
National Education is one of the cornerstone in Singapore’s holistic education. It aims to develop national cohesion, cultivate the instinct for survival as a nation and instill in students, confidence in the nation’s future. It also emphasizes cultivating a sense of belonging and emotional rootedness to Singapore. Teachers are encouraged to integrate NE messages into instruction by drawing examples from the prevailing national and current issues during the mathematics lessons, and these examples can serve as the context for problems or activities to motivate and generate interest. Teachers have been reminded that the infusion of NE messages should be guided by appropriateness of the topics and not one that is contrived.
5. Approach to undertake to Design Learning Materials?
Having discussed and understood the two key areas, namely “Knowing Our 21st Century Learners” and “Understanding the Singapore Mathematics Curriculum”, it gives us a clearer direction to identify the approach and strategies to employ when designing the learning experiences.
Learning in the 21st Century is no longer a passive process. The learner actively participates in the search and construction of knowledge. Immersed in the technologically-rich enviroment, he interacts with information, media and technology at all times. He likes to be touch with peers and be engaged in discussions and idea generation.
According to Audrey Gray (1997), the characteristics of a constructivist classroom are as follows:
§ learners are actively involved
§ environment is democratic
§ activities are interactive and student-centered
§ teacher facilitates a process of learning in which students are encouraged to be responsible and autonomous
Having described how learning would look like in a classroom, it is obvious that “constructivism” would be an appropriate approach to adopt in the design of the learning materials and learning experiences.
As discussed in the earlier segment (page 7), Vygotsky’s (1978) “Zone of Proximal Devleopment” (ZPD) should be another learning theory to be adopted in the design to facilitate the building of knowledge, skills and undestanding.
As technology became more common in the education landscape, Papert (1980) introduced two theories, instructionism and constructionism, that were then considered as approaches to promote educational innovation. He highlighted that instructionalism was a means where teachers improve their instructions by making computers to do the instruction; while constructionism means “Giving children good things to do so that they can learn by doing much better than they could before.” and he anticipated technologies would make mathematics learning more realistic, where learners could connect to real world application. Today, as technology become more ubiquitous and accessible in classrooms, the world could be ‘brought’ into the classroom, making mathematics learning more realistic. Technology, when well harnessed and leverage on, would certainly able to level up the learners’ level understanding and leading them to learn more deeply.
6. Significance of textbook in Mathematics classroom
Having described the approach and understood the potential and benefits to today’s learners, I am proposing to examine if the desired approaches and strategies have been carefully crafted and weaved into the design of the materials in the textbook. The approach and strategies should be explicitly articulated so that both teachers and learners could use to meet their needs.
Why bother to embed the approach and strategies in the textbooks when teachers are supposed to be equipped with the skills when they deliver the lesson? Based on findings from a study that involve 44 mathematics teachers, McNaught (2009) reported that multiple data sources revealed that teachers were generally more faithful to the content contained within the textbook than with presentation of material. Indeed, to many beginning mathematics teachers, the textbook is the primarily resource they rely on, where they seek guidance in not just the content, but also ‘how’ to deliver the lessons.
James (2008) pointed out that in the United States, schools believe that “one avenue for strengthening school mathematics programs is through the selection and implementation of high quality textbooks.” While we agree that there are other important factors that influence students’ learning outcomes, textbooks still have significant influence in the way students learn.
In US, textbooks must comply to the local standards before they made it to the approved textbook list. Similarly, in Singapore, textbooks must be reviewed by the CPDD in MOE in order to be ‘rubber stamped’ and make its way to the “approved” textbook list. The emphasis on the quality by the education authorities is an indication that “textbooks are viewed by many as an important lever for change – a tangible tool for impacting what teachers do and therefore what students learnt.” (James, 2008)
While researchers face challenges in studying the relationship among teachers, curriculum, and student learning, a growing body of studies provides evidence that NSF-funded curricular materials influence teacher decisions and actions (e.g., Remillard, 1999; Remillard & Bryans, 2004; Sawada et al., 2002) and positively affect student learning (e.g., Briars, 2001; Griffin, Evans, Timms, & Trowell, 2000; Huntley, Rasmussen, Villarubi, Sangtong, & Fey, 2000; Mullis et al., 2000; Reys, Reys, Lapan, Holliday, & Wasman, 2003; Riordan & Noyce, 2001). (James, 2008)
In Singapore, locally developed mathematics textbooks are not only used in Singapore schools, they received much attention from schools overseas, too. As pointed out by the local textbook publisher, Marshall Cavendish, schools over 35 countries in the world have adopted the Singapore Maths textbooks, with several adaptations to local curriculum requirements and lanagues. This has reaffirmed the belief of high quality textbooks have an impact on the quality of learning in mathematics classrooms worldwide.
While the mathematics textbooks are popular both locally and overseas, are the materials in the textbooks designed such that the intended outcomes of the curriculum are addressed? In a website that promotes the use of Singapore textbooks, “Try Singapore Math Textbooks. Your students will learn Math.” Jerome (2000) listed several reasons why many US educators looked to Singapore mathematics textbooks:
§ Do an especially good job in training students in Basic Skills.
§ Do an especially good job in providing students with Conceptual Understanding.
§ Provide an especially good background in Arithmetic and Arithmetic word problems, for the learning of Algebraic calculations and for learning how to solve Algebraic word problems.
§ Do an especially good job in training students in non-trivial Arithmetic word problems; while American texts largely avoid non-trivial Arithmetic word problems.
Does the list of ‘reasons’ show if the textbooks developed really cater to the curriculum needs as spelt out by the Singapore MOE (since these textbooks were primarily produced for Singapore students)? According to Fan (2010), researchers have found there is still much foom for improvement.
7. Putting things together: Key features of the mathematics textbook
Having learnt that textbook plays an important role, not just the learner, but also the teachers, the next important question is, have the textbooks (we used) been adequate to addrss to the curriculum requirements and provided sufficient support (to both learners and teachers)?
In the previous segment (page 9), we discussed the various approaches that are desireable to develop our learners. Are today’s textbooks designed with these approaches in mind? Do they come with activities designed that caters to the learning styles of today’s learners? Do they make use of the learner-familiar technology-enriched environment
More often than not, we associate “Constructivism” as a process play out in the classroom and it has to be facilitated by the teacher. With technology, the approach can now be possibly weaved into the design of the materials.
Here is an analogy: In the absence of technology, collaboration is possible only when learners meet up face to face to work together to complete a task. Today, with technology, learners could now collaborate with each other to complete a task without having the need to meet up physically, but via online platforms and tools such as online chat (that could be text, audio or a combination of both).
The same applies to strategies that are previously employed to conventional classrooms without technology. With a wide variety of technology platforms and tools, such “classroom strategies” could be facilitated with technology. Of course, the smooth execution implies implementation plans must be well thought through and learners have to be prepared and are ready to learn in the digital environment.
8. Critique: Mathematics Textbook “Discovery Mathematics 1B”, by making reference to Chapter 16: Data Handling
As a classroom practitioner, I use the textbook adopted by the school. The textbook comes with a digital version, too. Nonetheless, it does not leverage on the digital features to ride on the materials or resources available in the internet.
I will draw on examples from one chapter of this textbook to illustrate my discussions. The chapter focuses on the topic, “Data Handling”, which is the chapter that launches the Probability & Statistics strand in secondary mathematics. Screen shots of selected pages of the chapter are included in Annex 1 for reference.
I will make reference the six inter-related principles proposed by Fan (2010) for developing and publishing mathematics textbooks:
(1) Curriculum Principle
This principle requires that textbooks must be developed for the implementation and realisation of the intended curriculum.
Apart from schools that do not offer the GCE “O” level programme, the mathematics programme in all schools are guided by the syllabus outlined by CPDD, MOE. All textbooks therefore must align themselves to the syllabus.
According to the mathematics framework, “Concepts” and “Skills” are two areas that are covered in detail in the textbook. This is evident in the Chapter identified for critique. Higher order thinking questions that challenge the metacognition seemed to be rare.
(2) Discipline Principle
This principle requires that school mathematics textbooks must provide solid foundation for the students to understand, apply, and study mathematics in their daily life, further learning and workplace.
It is clear that textbook writers must have sound knowledge base in mathematics as a discipline. Indeed, it is not difficult to find at least one mathematician from the National Institute of Education (Singapore’s only teacher training institution) being part of the team of editors or writers.
(3) Pedagogy Principle
This principle requires that textbooks must be developed to facilitate the teaching, learning and assessment in mathematics.
Textbooks, as a learning resource could communicate the different pedagogical messages to the users. The approach and strategies adopted could provide users with either an encouraging or discouraging learning experience.
At the same time, the pedagogical orientation provided could be implicit or explicit. For example, in this chapter, the practices are clearly marked “Try It”, “Basic Practice”, “Further Practice”, “Maths@work” and “Brainworks” to give learners an indication on the level of difficulty of the questions.
(4) Technology Principle
This principle requires that the textbook developers be familiar with the development of technology.
Nonetheless, in this chapter, many potential opportunities were not surfaced to teachers to engage students to learn deeper with technology use. Recommendations will be made on how ICT could be infused into learning.
(5) Concept Principle
This principle requires the textbook developers have reasonable knowledge of local contexts as contextualised information is incorporated to motivate and engage learners in their learning of the subject as they are familiar and can make connection with the contexts.
An attempt was made in the extension activity “Extend Your Learning Curve” where students could do a seemingly authentic task with the context that they are familiar with. The task also requires students to apply the knowledge and skills acquired in this chapter to accomplish the task. While the task looked authentic, a more realistic task whereby data collected would lead to recommendations that are possible to implement in real world.
In the activity “Write in your Journal”, students were presented with two questions that require them to exercise higher order thinking skills (as the ywould need to make some inference). On the other hand, it lacks the context to show how the understanding could be applied in a situation that the learner is familiar with.
(6) Presentation Principle
This principle requires the presentation of the contents in textbooks must suti the level and needs of teaching and learning.
Within Chapter 16, the various examples and activities are clearly marked out using colour codes and symbols. This is consistent across all chapters.
9. Recommendations
The following changes were suggested focused largely on leveraging technology to learn:
As the first chapter to introduce “Data Handling” at the secondary level, a connection to their prior knowledge acquired at primary school would help students to see the topics they are learning do not exist in isolation.
Tapping on the internet, students could be tasked to search for graphs used in real life related to the Singapore context. They would post the images (of the graphs) as well as the website addresses on digital post-its on an online collaborative platform known as “Wallwisher”.
(http://sstclass101maths.blogspot.com/2010/04/chap-16-data-handling-when-to-use-what.html)
(http://sstclass101maths.blogspot.com/2010/04/chap-16-data-handling-when-to-use-what.html)
With the visuals on the ‘wall’, learners could reorganise the post-its so that similar kinds of the charts would be grouped together where they will identify the common attributes for each kind fo graphs. With this, students draw, from their observations and by making comparisons, attributes and unqiue characteristics of each type of graph. On top of that, they could click at the hyperlink posted up previously to read more about how the data are used in the real world context.
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References
- Fan, L. (2010). Principles and Processes for Publishing Textbooks and Alignment with Standards: A case in Singapore. Replicating Exemplary Practices in Mathematics Education among APEC Economies.
- James E. T., Robert, E. R.,Barbara, J. Reys, Osar, C. (2008). The Impact of Middle-Grade Mathematics Curricula and the Classroom Learning Environment on Student Achievement. Journal for Research in Mathematics Education (2008, Vol 39, No. 3)
- Jerome Dancis (2000). Try Singapore Math Textbooks. Your students will learn Maths. Retrieved from http://www-users.math.umd.edu/~jnd/Singapore.Math.htm
- McNaught, Melissa D. (2009). Implementation of integrated mathematics textbooks in secondary school classrooms. Retrieved from https://mospace.umsystem.edu/xmlui/handle/10355/6146
- Ministry of Educaton, Singapore (MOE). (Last updated 2010). ICT Connections. Retrieved from http://ictconnection.edumall.sg/
- Ministry of Education, Singapore (MOE). (Last updated 2010, Oct 6). National Education. Retrieved from http://www.ne.edu.sg/
- National Center for Education Statistics. U.S. Departement of Education Institute of Education Sciences. Trends in International Mathematics and Science Study (TIMSS). Retrieved from http://nces.edu.sov/timss
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- Partnership for 21st Century Skills. (2004). Retrieved from www.p21.org
- Prensky, M. (2001). Digital Natives, Digital Immigrants. On the Horizon (MCB University Press. Vol 9 No. 5, October 2001).
- Rezat, S. (2009). The Utilization of Mathematics Textbooks as Instruments for Learning. Proceedings of CERME 6. Retrieved from www.inrp.fr/editions/cerme6
- Rodgers, M., Runyon, D., Starrett, D., Holzen, R. V. (2006). Teaching the 21st Century Learner. The Annual Conference on Distance Teaching and Learning. The Board of Regents of the University of Wisconsin System.
- Skolverket (2003). Nationella kvalitetsgranskningar 2001-2002: Lusten att lara – med focus pa matematik (Report No. 221). Stockholm: Skolverket.
- Soh, C. K. (2008). Mathematics Curriculum in Pacific Rim Countries – China, Japan, Korea and Singapore: Proceedings of a conferenece. An overview of Mathematics Curriculum in Singapore. Edited by Zalman Usiskiin and Edwin Willmore. US: IAP – Information Age Publishing, Inc. Retrieved from Google Books Database (http://books.google.com/sg)
- Vygotsky, L. S. (1978). Mind in Society: The Development of Higher Psychological Processes. Chapter 6: Interaction between Learning and Development. Harvard University Press. England: London.
