Portraits of the Singapore Secondary School Mathematics Enactment: An Insider’s Perspective

Abstract

This chapter summarises the contributions of the preceding chapters of this volume and other publications related to the project. We bring together the various facets given by these reports by painting two portraits of the enactment of Singapore secondary mathematics curriculum. The first portrait draws from the well-known pentagonal model of the Singapore mathematics curriculum, and it seeks to address the question, “Does this model accurately depict what really goes on in the Singapore mathematics classrooms?” The second portrait borrows from the journey metaphor in describing the goal-driven nature and the deliberate planning involved behind the “examination-oriented” “look” of classroom instruction. This addresses another question: “How do we explain the seeming paradox of high performance of Singapore students in international tests while maintaining what looks like traditional modes of instruction?” We conclude the chapter by reflecting on these two portraits in relation to the wider cultural context in Singapore.

1 Bringing the Various Facets Together

In the earlier chapters of this volume, the various authors—who are the team members of the project as described in Chapter 2—report findings on different aspects of the overall study. In this chapter, we attempt the challenging task of pulling these rich and diverse perspectives together. Using the metaphor of painting a portrait (in this case, the portrait of the enactment of Singapore secondary mathematics teachers), we seek to draw together these various reports as facets that contribute to a coherent whole. We will also draw upon other publications arising from this same project that are not included in this volume.

At the same time, in the course of the project duration, we were frequently asked these two questions (both by educators within Singapore and international professionals looking into Singapore mathematics education):

• Is the (now internationally well-known) pentagon model, which is the framework of the Singapore school mathematics curriculum (see Fig. 1.2 in Chapter 1), really what goes on in the Singapore mathematics classrooms?

• How do you explain the “paradox” of Singapore’s high performance in PISA with the still very traditional modes of teaching in Singapore mathematics classrooms?

Thus, in developing the enactment portrait, we also attempt to address these questions. In fact, we present two portraits by structuring the rest of this chapter according to each of these questions—a portrait that is centred around discussions of the pentagon; and a portrait that unwinds the “paradox”.

2 Revisit of the Pentagon Model

Leong (2008) did a cursory comparison of the pentagon as intended and the pentagon as assessed. For our purposes, we find this pictorial comparison useful and it is included in Fig. 17.1.

Fig. 17.1

Pentagon as intended and pentagon as assessed

The pentagon on the right side of Fig. 17.1 shows a rough picture of the components that are directly assessed in typical high-stakes examination papers in Singapore. Attitudes and Metacognition are not directly tested while a high percentage of the items in the papers tests Skills, less on Concepts, and even less on Processes. It is arguable—depending on how one defines “problem solving” and whether students consider “problems” set by teachers as indeed problems or more like routine exercises—whether or how much mathematical problem solving items are in these papers. Interestingly, this picture of distribution appears to match some students’ assignment of value to what is important in mathematics lessons, as shown in the findings in Chapter 10. Leong (2008) asserted that, given the difference in emphases among the components in the pentagon between the official curriculum and the high-stakes examinations, it is not surprising that teachers feel “sandwiched” between these poles: do they teach more in accordance to the picture on the left—presumably, with roughly even distribution of emphasis on each of the components? Or, do they apportion their emphasis roughly according to that shown in the picture on the right? The Question Mark in the middle of Fig. 17.1 denotes this enactment dilemma of teachers. It is a question that this project sought to address too—how does enactment look like when juxtaposed between these two “models”?

The actual enactment picture is far more complex than what can be presented in a pentagon. Nevertheless, we begin to paint the picture by considering the respective components of the pentagon. Since Skills loom large (literally, in the diagram on the right), we shall begin the discussion on Skills.

It is clear that Singapore secondary mathematics teachers put a high priority on teaching Skills. They want students to be able to fluently carry out some standard methods as stipulated in the secondary mathematics syllabus—such as methods of solving quadratic equations (Chapters 13 and 14) and the use of the Pythagoras Theorem in solving right-angled triangles (Chapter 4). It is worth noting that Skills have been unjustly disparaged in literature that tends to equate Skills-talk with “merely drill-and-practise” mode of instruction. We think the importance of Skills has been given back its rightful position in mathematics teaching and learning when the influential report—commissioned by the American National Science Foundation—by Kilpatrick, Swafford, and Findell (2001) included “Procedural Fluency” as one of the core strands of their Mathematical Competence framework.

But the evidence in this project points towards this: Singapore secondary mathematics teachers do not think (nor enact the teaching) of Skills apart from the other relevant components of the pentagon. For example, in Chapter 4 of this volume, the description of the lesson on Pythagoras Theorem showed that the teacher devoted time not only in the application of the theorem, but also in its development. There was an emphasis in relating the theorem to other relevant mathematical concepts (such as its converse). In other words, the lesson trajectory is one where Skills and Concepts were developed together. Furthermore, neither is this a case of a one-off occurrence. Chapter 4 (on relating the formula of finding the distance of the line segment joining two points on the Cartesian Plane to Pythagoras Theorem) and Chapter 14 (on relating the quadratic formula to the method of Completing the Square) provide more illustrations of this Skills-Concepts co-development. Moreover, this intention that students learn Concepts alongside the development of Skills is not limited to the 30 experienced and competent teachers studied in the first phase of the project. In the second phase, where we collated the qualitative responses of 156 teachers from a broad spectrum of Singapore secondary schools, Leong et al. (2019b) reported, “[I]t is clear from the types of comments listed … that a substantial number of teachers were concerned not just with formula application but also with how it connected with related concepts in the topic” (p. 37).

In fact, this tight connection between Skills and Concepts is so ubiquitous in the teachers’ lesson enactment and instructional materials that we posit that Skills-Concepts form a central axis in the organisation of their lessons—as is supported by all the case studies reported in this volume (Chapters 4, 8, 9, 13, and 14); and they build other components on this organisational axis.

One of the pentagon components that were intentionally built into this axis is Processes. In this regard, the case of Teacher 13 as described in Chapter 13 is particularly illuminating. The authors of the report depicted the teacher’s move as one of using mathematical reasoning—one of the Processes in the pentagon—as a kind of “glue” to link the various solution methods and ideas together. Reinterpreted into our pentagon language, Teacher 13 wanted students to learn the various methods of solving different types of quadratic equations (Skills); he also wanted the students to learn the underlying conceptual underpinnings (and limitations) of each of the methods (Concepts). In addition, on top of this Skills-Concepts developmental axis, he sequenced his examples in such a way that mathematical reasoning was used as the main Process to link them together. While Teacher 13’s case provided an explicit foregrounding of how Processes—such as reasoning—shaped the enactment of his lessons, it is by no means an isolated case.

In Chapter 8, Teacher 5 was also clearly concerned that the students experience authentic mathematical Processes in learning the geometrical theorems. The students were given the opportunity, through inductive processes afforded by Dynamic Geometry software, to conjecture and test findings; towards the end, proof of the theorems was done so that students see the deductive process of explaining what was discovered earlier. Teacher 27 in Chapter 9 also included mathematical Processes such as justification in his math talk. His interest was not only that students were able to carry out the procedures; he wanted them to reason out the steps as well. In Chapter 14, Teacher 8 wanted students to “link everything together”; in other words, her goal was not limited to students’ ability to carry out each of the methods of solving quadratic equations; she also ostensibly aimed at students’ ability to connect (a mathematical Process) the various strands of Skills and Concepts together.

There is also evidence to suggest that this desire to teach mathematics in a way that mathematical Processes come to the fore was not restricted to the cases mentioned in the previous paragraphs. Table 11.6 of Chapter 11 indicated that out of 30 experienced and competent teachers we studied under Phase One of the project, there were evidence of “making connections” in the instructional materials of 21 of these teachers; the number is 19 for “support reasoning”. In Phase Two, where the survey was completed by more than 600 teachers, Table 9.2 of Chapter 9 shows that a significant proportion of the teachers indicated that they at least “frequently” require their students to use these Processes-related talks in their mathematics classrooms: “Explain” (70%), “Explore” (45%), “Analyse” (72%), “Evaluate” (55%), “Argue” (63%), “Justify” (52%).

The portrait with respect to the pentagon that is emerging up to this point of our discussion is this: the most explicit and visible parts of teachers’ planning and enactment are the Skills and Concepts. Indeed, they are so closely thought of and enacted that we posit a Skills-Concepts organisational axis in the way teachers develop their lessons. Less visible, less ubiquitous, but wherever there are opportunities, teachers also bring on board relevant mathematical Processes in their instructional materials and classroom talk.

If Skills, Concepts, and Processes are thought of as the disciplinary and cognitive aspects of the mathematics teaching enterprise, then Attitudes and Metacognition can be thought of as more generic and affective—hence, perhaps seen by secondary mathematics teachers as ‘further away’ from their core business as mathematics teachers. Admittedly, these latter aspects are not as visible or conspicuous as the former.

Nevertheless, there is substantial evidence through our project findings to indicate that the teachers were highly cognisant of student Attitudes when they thought of how they structured their lessons. When Attitudes are mentioned, often the first thing that comes to mind is students’ interest in the subject—that is, “How can we make mathematics more ‘cool’ for the students?” Indeed, some of the teachers attended to this aspect of Attitudes. Where relevant, teachers included real-life examples in class partly to pique students’ interest in the subject. This was mentioned in Chapter 7. Table 11.6 of Chapter 11 indicated that 20 of the 30 experienced and competent teachers included “Context” (that is, related to everyday experiences) items in their instructional materials.

But the evidence we obtained in the project pointed to students’ confidence (not interest—at least, not directly) that was the teachers’ main attitudinal focus. In particular, teachers were careful to adjust their materials and examples so as to build up students’ confidence in doing mathematics. This ostensible goal of confidence-building was highlighted in Chapter 7. This is colluded by the findings related to the instructional materials that teachers designed. Table 11.6 of Chapter 11 shows that 29 out of 30 teachers practise “deliberate sequencing of examples” in their instructional materials. The follow-up study of this observation in Leong et al. (2019b) indicated that one major consideration in this “deliberate sequencing” was the managing of cognitive load of students. Above other design considerations, the 156 teacher-respondents in the survey generally placed “start off with easier items to build confidence” as top priority when thinking about how to sequence practise examples. This sensitivity towards students’ affect was also evident when teachers thought of exposing them to challenging items (see Chapter 12). The teachers described strategies they adopted to alleviate students’ sense of intimidation when confronted with challenging items.

Here, we pause for a while (from the work of portrait-painting of the pentagon) to reflect on the still-common “teacher-centred versus student-centred” talk, which was also briefly referred to in Chapter 3. Up to this point, we wonder how readers would place the Singapore secondary teachers’ enactment across this “dichotomy”. [We think this is a false dichotomy]. If we grant that the disciplinary aspects of the enterprise (that is, Skills, Concepts, and Processes) reveal how “teacher-centred” Singapore teachers are—in that, the agenda in these aspects are largely teacher-determined and teacher-directed, what can be said of this premium placed on building students’ confidence—is it “teacher-centred” or “student-centred”? Should we then say that, cognition-wise, Singapore secondary teachers are teacher-centred; and affect-wise, they are student-centred? These questions reveal how over-simplistic this dichotomous talk is. In reality, we think teachers see themselves as intellectual authority with respect to the subject, and this renders them leaders in the class when it comes to learning disciplinary norms; but this does not translate into a careless ignoring of students’ learning needs—especially in the area of students’ affect; the teachers consciously factor in the building of their confidence in the design of their instructional materials and in the enactment of classroom instruction.

Finally, we come to the last of the five components—Metacognition. If attitudinal matters are less visible (than cognitive ones), then metacognitive moves by teachers are even more hidden from direct view. Nevertheless, Chapter 6 of this volume highlights specific metacognitive strategies that were used by some of the experienced and competent teachers. In particular, a common strategy used was to encourage students to compare different solution methods as a way to learn and reflect upon the affordances and constraints of each method. This strategy was also evident in Teacher 13 (Chapter 13) and Teacher 8 (Chapter 14), although it was not clear if these teachers had the goal of teaching metacognition explicitly in mind or if they saw multiple solution strategies as part of the norms of doing authentic mathematics. Table 6.3 of Chapter 6 shows that more than a majority of the 677 teacher-respondents in the survey indicated that they at least “frequently” provided opportunities for their students to learn different solution methods. These teachers also indicated that they regularly help their students to “reflect” on their learning and methods. Suffice to say, there is much more scope in the integration of Metacognition in the teaching of mathematics beyond specific metacognitive strategies that are reported in this project. Many areas in this domain (such as, the relation between “generic” metacognition and domain-specific metacognition, teachers’ metacognitive practices in their own learning of mathematics, the extent of teachers’ modelling of authentic metacognitive practices in the classrooms) are as yet under-explored by teachers. At the time of writing, the Ministry of Education of Singapore is commissioning a sizeable research project on metacognition. For now, we think Metacognition is the least understood by most Singapore mathematics teachers among all the components in the pentagon. Where teachers carry out metacognitive practices in their classroom work, they were more implicit (and hidden under other goals of instruction) rather than at the foreground of their agenda.

Based on the summary of the various components of the pentagon, we give our portrait of the pentagon as enacted (in place of the Question Mark in Fig. 16.1) in Fig. 17.2.

Fig. 17.2

Pentagon as enacted

The different font sizes of each of the components reflect roughly the weight we think teachers place in fulfilling them as goals of instruction. The bubbling up and the conjoining of them with a bold line segment show the tight Skills-Concept axis that is the central organisation frame in teachers’ planning and enactment. Where relevant, Processes are brought into the development axis of Skills and Concepts (as shown by the arrow). Likewise, students’ Attitudes are also considered when designing tasks. The different thickness of these arrows are meant to show the different levels of visibility—while Processes can appear at the foreground of teachers’ work, in that they would explicate it as a goal of instruction (such as, reasoning, making connections), dealing with students’ Attitudes are present but at the background of the teachers’ considerations, in that they mentioned its importance usually only when asked. Metacognition comes into play implicitly, often not even intentionally by the teachers, and hence represented by the perforated arrow.

The reader will notice that mathematical problem solving, which is supposed to be at the heart of the pentagon, is not included in Fig. 17.2. Why is it missing? The answer depends largely on how “problems” are defined. According to the Singapore Ministry of Education (2020), “Problems … include straightforward and routine tasks … as well as complex and non-routine tasks …. General problem solving strategies e.g. Polya’s 4 steps to problem solving and the use of heuristics, are important in helping one tackle non-routine tasks systematically and effectively” (p. 9). This quotation seems to keep the definition “open”: while the first sentence presents “problems” to include all kinds of tasks, the second sentence, by the very reference to Polya—and hence his commitment to problem solving (e.g. Polya, 1945) as an enterprise that presents actual problems to students—would necessarily exclude routine exercises. We can hence interpret this paragraph to mean that, while “problems” can be understood in its broadest sense as any mathematical tasks posed to students, there should be an emphasis on “problems” that are truly problematic to students—so that they will learn problem solving strategies. For our current purpose, we take “problems” to refer to the latter category. In this volume, only Chapter 12 provides some indication of the state of mathematical problem solving conducted in secondary mathematics classrooms. If we take “challenging items” to be a proxy for “problems”, then it appears that the picture is quite encouraging—a majority of the teacher-respondents indicated either “Sometimes” or “Frequently” when asked about the use of challenging items in their classes. The study does not go into the actual problem solving strategies—or the manner in which they were taught—to the students, and so we should interpret the frequency of use of challenging items with reservation. In fact, in other local projects that were specifically about helping teachers enact problem solving lessons (e.g. Ho et al., 2019, Leong et al., 2011, Toh et al., 2019), it was found that teachers require continual professional development to sustain problem solving as a regular activity in their classrooms. Nevertheless, we posit that in Singapore secondary mathematics classrooms, problem solving may not be as “elusive” as frequently claimed (e.g. Stacey, 2005).

3 The Singapore “Paradox”

Some authors write about an East Asian learner paradox (e.g. Biggs, 1996, Leung, 2001). Mok (2006) stated the paradox as “the apparent contradiction between the teaching methods and environment in East Asian schools (i.e. large classes, whole-class teaching, examination-driven teaching, focus on content rather than process, emphasis on memorisation, etc.) and the fact that East Asian students have regularly performed better than their Western counterparts in comparative studies” (pp. 131–132). We assume that, since Singapore is located in East Asia and fits roughly into the portrayal of traditional methods of teaching but with high mathematics performance, the paradox also applies to us.

First, we state the parts of Mok’s caricature that we think do not apply to the Singapore context. We reject the description of “focus on content rather than process”. As elaborated in the previous section and depicted in Fig. 17.2, Processes such as reasoning and making connections are explicitly interwoven into the Skills-Concepts development axis of the teachers’ lessons. And this is not restricted to the experienced and competent teachers; teachers in our survey in Phase Two of the project professed their commitment to teaching a number of relevant Processes in their mathematics lessons. Moreover, Mok’s use of “rather than” to juxtapose “content” and “process” presupposed that teachers have to choose one or the other—which is a false dichotomy. In reality, as we think is the case in most Singapore secondary mathematics teachers as reported in this volume, they intend to teach both content and processes. Chapter 8 and Chapter 14 provide compelling cases of how attending to both can be workable.

We also find “emphasis on memorisation” puzzling. What does this actually mean concretely? We take the case depicted in Chapter 8 as an example for the discussion here: Suppose Teacher 5, after she has developed the conceptual links among the various circle theorems, ask the students, “Would you like to learn an easy way to remember these theorems?” After which, she proceeded for 10 min to teach them quick ways to remember (or memorise) them. In this context, would we consider Teacher 5 to “emphasise on memorisation”? And even if we do consider it so, what is paradoxical about it—when the teacher follows up the content and process development with an effort to consolidate the learning by helping students to commit it to memory? Isn’t this sound pedagogy? Our point here is: based on the evidence of our study, Singapore secondary mathematics teachers focus on both Skills and Concepts (see Fig. 17.2); there is no evidence to suggest that they focus on memorisation of procedures in isolation from or excessively over conceptual understanding. Where “emphasis” is rightly done, there is no paradox.

Having clarified the aspects of the caricature that does not apply to the Singapore context, we think the remaining descriptions are indeed shared by us. But as “large class size” and “whole class teaching” are more structural givens than conscious pedagogical decisions, we will restrict our discussion on “examination-driven teaching”.

Indeed, all the 30 experienced and competent teachers who participated in the first phase of the project were “examination-driven”, in the sense that they set the target of their students being able to do well for subsequent (especially, high-stakes) examinations as one of their major instructional goals of teaching mathematics; also, because of this goal, the instructional contents and student tasks they planned for their lessons take their reference from typical examination items. Insofar as textbooks items are trusted as indications of standard examination items, there is also this associated adherence to textbook items as official proxies of what is expected in examinations. During the teacher interviews in the project, it is not uncommon for teachers to justify their inclusions of certain tasks in the lessons due to their being “included in examinations” or “included in textbooks”.

Before we proceed further into the heart of the paradox, we would like to make some related comments about being “examination-driven”. Often, this term has been used to lambast teachers for being narrowly focused in their work of teaching. But we should check such excesses against this reality: In a system where students’ examination results largely shape the career choices ahead of them, is it not socially responsible for teachers to take it as their primary role to help students attain the best they can in examinations? Conversely, would we think teachers are socially responsible to their charges if they ignore the importance of examinations for their students’ social-economic future just so that they can pursue their own educational ideas about teaching mathematics?

Also, from an education-system point of view, is it really so bad that teachers have a clear and concrete goal to prepare their students for? Perhaps this is clearer when contrasted against an alternative option: there are no high-stakes examinations; teachers can do what they think is professionally expedient as mathematics teachers; the individuality of the teacher and his/her conscience become the guide for what “drives” the mathematics instruction in class. To be sure, this is the utopian vision of many teachers—that they be ‘left alone’ to do what they like; but seen as a system, there remains no concrete goal to bring the many teachers in the system together with a common vision and accountability of what is to be taught and to what degree of rigour is to be expected. At its worst, teachers lose a sense of purpose and direction to “drive” their teaching and students are worse-off for it. We like to imagine that every teacher, left to himself/herself, knows what is best to teach for the students over the long term. This assumption has yet to be proven at scale. If the alternative to being “examination-driven” is being non-driven (a prospect we think very likely although few talk about it), then the choice is clear.

In addition, one who is “examination-driven” need not be solely driven by examination in his/her instructional work. Examination-orientedness may only be one of the goals of teaching. As seen in many of the case studies in this volume (Chapters 4, 8, 9, 13, and 14), the teachers are capable of keeping a keen eye on preparing their students to be examination-ready while concurrently pursuing other worthy goals of teaching mathematics.

Another point about being “examination-driven” is the actual assessment content of these examinations. If the examinations that we have in mind comprised merely items of low cognitive demand, then being examination-driven in such a context will indeed result in a great disservice to the students. It will indeed descend into a training of mechanistic “drill and practise” automatons—a portrayal common in the literature that presents being examination-driven as “bad”. But what if the “examination” to be “driven” towards typically consists of a significant number of items that are considered of high cognitive demand (see Chapter 12)? This seems to be the case in the Singapore context as depicted by the right pentagon in Fig. 17.1. For a teacher to be “examination-driven” in this context, he/she will have to regularly include items that are considered challenging to students so that what the students do in class approximates the kind of items they will encounter in subsequent examinations. From purely a content perspective, isn’t this kind of examination-drivenness “good” for the students—do they get to engage with items of high cognitive demand regularly? In fact, we think the content of Singapore mathematics examinations partially explain the paradox: If Singapore secondary students, regardless of ability bands (see Chapter 12), regularly engage in challenging items in preparations for examinations, is it surprising that they will perform well in similar “examinations”, such as the international tests presided by PISA?

4 The Heart of the Paradox

But, one may argue that simply giving challenging items to students does not necessarily result in good performance in these items. In other words, to unwind the paradox, we still cannot sidestep the heart of the issue: What actually takes place in these Singapore examination-driven classrooms that prepare the students so well for examinations such as the ones conducted by TIMSS and PISA?

The “examination-driven teaching” is a surface façade—and hence what usually catches the eye of a cursory observer. As a number of chapters in this volume has described, when we plunge beneath the surface, we find embellishments that may hold the keys to a reconsideration of the just-drilling-for-examinations first impressions.

In Chapter 3, the authors reported that Singapore secondary mathematics teachers commonly use or subscribe to the Development-Student work-Review (DSR) sequence in their instructional practice. What is particularly insightful to us is that these DSRs appear in cycles—and you can find a number of these cycles even within a short instructional episode that spans merely a few minutes. The teachers do not merely plan and execute at a broad-grained level; they also go meticulously into a mode of repeated small-step development, monitoring of students’ learning, and consolidations—all at fine-grained levels that approximate the incremental steps in students’ acquisition of knowledge. Translated to the examination-driven context, it means that the teachers do not merely give examination-type items (and some challenging ones) to the students and leave them to work on them; they execute detailed cycles of DSR moves to help students learn the requisite skills, concepts, and processes to be successful for these type of items. In the language of Chapter 12, teachers offered carefully planned “scaffolds” to help students gain access to and maintain engagement with these items (especially challenging items).

This careful fine-grained planning did not begin in the classroom. It was also conspicuous in the instructional materials developed by these teachers prior to their entering into the classroom. Chapters 11 to 14 of this volume report how the teachers thought about and carried out the design of their instructional materials. A common underlying thread among these teachers that stood out for us was how deliberate they were in weaving their agenda into the tasks they crafted for their students. In Chapter 13, Teacher 13 was deliberate through his instructional materials in building opportunities for students to use mathematical reasoning as they work on an item and move across items. In Chapter 14, Teacher 8 was deliberate through her instructional materials in helping students make connections among concepts and across solution methods. This deliberate weaving is also evident in the samples shown in Chapter 11—illustrations of how the teachers brought in “new” materials together with “modified” ones, and how these were “smoothened” so that they were presented as developmentally coherent to the students. This “deliberateness” challenges a narrow view of examination-orientedness—as if being “examination-oriented” necessarily results in a teaching mode where teachers just mindlessly hand out pages of examination questions for students to work on. In the case of the teachers in our study, there were deliberate and goal-driven efforts to transform textbook materials into carefully designed instructional materials that took into consideration the learning needs of their students.

Not only was the design deliberate, we were also fascinated at how much attention teachers put into very fine-grained levels of details which would not have caught the eye of most observers. Chapter 12 reports on the various strategies used by the teachers (particularly, teachers who specialise in the teaching of low-progress learners). The length to which teachers go—such as consideration for the placement of items, the actual scaffolds, the motivational prompts—shows the level of detail they thought about the items and how they are to be used in class. This attention to details applies to example sequencing as well. Leong et al. (in-press) reported how one of the experienced and competent teachers in this project (Teacher 10) carefully sequenced her practise examples in a way that affords variation (cf., Variation Theory) and took into consideration students’ cognitive load (cf., Cognitive Load Theory). But hers was not an isolated case; it was found that this careful attention to example sequencing was common across most of the teachers surveyed in Phase Two of the project (Leong et al., 2019b). This finding is colluded by the report in Chapter 11 where evidence of “deliberate sequencing of examples” was found in 29 out of the 30 teachers.

In summary, we think that “examination-driven teaching” can be viewed at two levels of zoom: one that is broad-grained and another that depicts the work of teaching as bringing the students to “drive towards” examination requirements (shown in the left side of Fig. 17.3). As mentioned earlier, this is not an untrue portrait of the work of Singapore secondary mathematics teachers—in that they view preparation of students for examination as a key part of their role as teachers, and they take reference from examination items in their selection of content for their students. However, there is a second and a more fine-grained level of zoom which reveals, as uncovered in many chapters in this volume and which we summarised in the preceding paragraphs, a portrait which is more nuanced, consisting of sub-destinations and carefully planned manoeuvres (as illustrated in the right diagram of Fig. 17.3).

Fig. 17.3

Examination-driven teaching viewed in two levels of zoom

This more nuanced portrait in Fig. 17.3 is largely motivated by the case of Teacher 2 as reported in Leong, Cheng, Toh, Kaur, and Toh (2019a). In his instructional practice, we find many of the features that are discussed in this volume and so we use his case as an illustration to pull together the various facets already discussed.

Before he began to teach a module on Vectors, he designed a full set of instructional materials for the whole topic. Through analysing the materials and his responses during interviews, we found that he had the whole development of the topic mapped out—represented as paths and sub-goals in Fig. 17.3. He “drove” towards the examination requirements by bringing the students to various sub-goals which served as milestones (following the journey metaphor). He did so through the careful and deliberate planning of tasks which was coordinated with how he would implement them in class (some of these strategies we reviewed in the earlier paragraphs). At these milestones, he checked through formative assessments if the students met the requirements of the sub-goals, and where necessary, he would zoom-into fill particular gaps in students’ knowledge. At suitable junctures, he would connect some of these various strands of knowledge (represented as different paths in the diagram) by helping students to work through tasks that required a coordination of these knowledge areas.

The diagram in Fig. 17.3 is a gross over-simplification of Teacher 2’s (and many other Singapore secondary mathematics teachers’) practice and as such does not do justice to the deliberateness and intensity of his (and their) work. But we constructed this portrait to uncover elements that are normally hidden in examination-driven talk. As soon as we see that being “examination-driven” is not incompatible with a pedagogy that attends to careful details of students’ learning development in conjunction with the content trajectory and which produces deliberately designed tasks that are student-friendly, the “paradox” dissipates. To us, there is no paradox between a pedagogy that is relentlessly “driven” towards examination content goals and towards students’ attainment of those goals, and their sub-goals and high performance in international comparison tests such as TIMSS and PISA. In fact, we think the former substantially explains the latter.

5 Conclusion

As editors of this volume, we used the task of writing this concluding chapter as an opportunity to reflect on Singapore mathematics education. It became clearer to us in the preparation of this chapter that Singapore mathematics education is a microcosm of Singapore herself. We think the ingredients which render Singapore “successful” are also the same ingredients that render Singapore (mathematics) education “successful” (if measured by the performance in international comparison tests). Interestingly, this perspective is shared by Seah (Chapter 16) as he viewed the findings in this volume from the broader lens of Singapore society’s strive for excellence.

We have heard numerous visitors to Singapore making this comment (or its equivalent), “I don’t know which ‘box’ to place Singapore in—it is neither East nor West, neither socialist nor capitalistic …” This might also be how readers of this volume feel, “Which pedagogical ‘box’ do I place Singapore mathematics education—it is neither teacher-centred nor student-centred, neither procedure-based nor concept-based …” If asked to use one word to describe Singapore (and concomitantly, Singapore mathematics education), most would choose “pragmatic”. However, we prefer the term “eclectic”. Our instinct as a people is not to merely follow the hollow theoretical models of others and assume that their avowed “success” would work for us. We have a habit of being open-minded: drawing upon the affordances of different models and mixing them—deliberately and experimentally—to see which configurations work best for us given our unique context. We think this deliberate eclecticism has been the underlying disposition for researchers and teachers to experiment with different pedagogical mixes all along, resulting in the portraits that we present today (and as illustrated in Fig. 17.2 and Fig. 17.3).

But the portraits will not stay “still”—it is a constantly “moving” portrait. Another feature embedded deep in the Singaporean psyche is this, “We cannot afford to stay still in this climate of tough global competition—we must keep working hard and constantly evolve in response to the changing challenges”. This “working hard” and “being ready for changes” also account for the practices of the Singapore secondary mathematics teachers that we have reported in this volume. The picture presented is one where the teachers are meticulous and hard working in attending to their planning and design work as well as to the learning needs of the students. They are also prepared to change—to improve—where the implementations do not work according to plan; that is, they are not easily content with low performance of students and the status quo; rather, they are “driven”—in the positive sense of the term—to help students attain their potential.

But constant evolvement also means that teachers do not easily rest on their laurels. There are still areas for Singapore mathematics education to develop further. As Fig. 17.2 shows, there are components of the pentagon that should fill the agenda of teachers and researchers in the near future.