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INTRODUCTION AND PURPOSE OF THE PROJECT

In mathematics education, mathematical problem solving has been established as the central theme of the Singapore primary and secondary mathematics curriculum since the 1990s. The primary aim of the curriculum is to develop students’ ability to solve mathematics problems, commonly seen as a measure of spatial-logical cognitive ability. This goal remains elusive. Yet, indeed, if students successfully learn mathematical problem solving, which encapsulates within it the processes of exploring a problem, devising and modifying plans of attack, and adapting and generalising the solution, they certainly will have learnt how to think well.

Researchers such as Schoenfeld (1992),Lester (1994) and Stacey (2005) point out that the findings have been less conclusive than desired. M-ProSE aims to revisit research in this area for several reasons. First there was limited research done in Singapore in this area. Second, although international comparative studies such as the recent TIMSS studies have revealed that Singapore has achieved a high level of competence in mathematics in schools, these studies have also noted a relatively weaker performance of Singapore students on the problem solving items.

Singapore teachers were provided with preservice preparation or professional development in teaching problem solving, largely drawing on overseas resources as well as resources developed locally. However, these resources tended to emphasise the learning of heuristics but do not focus on mathematics content at a deep level or the kind of mathematical thinking used by mathematicians such as conjecturing and proving. Towards upper secondary and junior college levels, on the other hand, the students concentrated on national exam-type mathematics problems, and in the project team’s experience, the heuristics learned at lower levels tended to be ignored by both students and teachers instead of being applied in their mathematical engagements. Therein lays the lack of success of our attempt to teach problem solving, indeed of any attempt anywhere else on this globe to implement problem solving successfully into the taught curriculum.

PARADIGM SHIFT

We think that the way out of this perennial quandary is by making a paradigm shift. In a pilot project at an Integrated Programme school, we decided to construct a worksheet like that used in science practical lessons and told the students to treat the problem solving class as a mathematics ‘practical’ lesson. In this way, we hoped to achieve a paradigm shift in the way students looked at these ‘difficult, unrelated’ problems which had to be done in this ‘special’ class. The science practical lesson is very much accepted by students as part of science education and many have an understanding that it is to teach them how to ‘do’ science.

Despite much debate of how exactly it is to be carried out (Woolnough and Allsop, 1985), practical work is accepted as a mainstay in science education. It is certainly conceivable that similar specialised lessons and materials for mathematics may be necessary to teach the mathematical processes, including and via problem solving.

AIM OF RESEARCH PROJECT

In view of the above, this research project aims:

to design and implement a Problem Solving Curriculum (PSC) in Integrated Programme (IP) schools;
to train teachers for implementing the PSC in their classrooms; and
to develop an appropriate assessment system for gauging the effectiveness of the PSC.

This project will benefit both the students and the teachers.

Students

As the problem solving involves higher order thinking skills, the students will benefit from developing skills, understandings, dispositions and values that young people are likely to need to effectively negotiate 21st century institutional environments (NIE Research Framework, 2008). There will be a focus on

the development of higher order thinking skills through exposure to a special curriculum;
the development of desired other traits, such as engaged learning and deep understanding which Hogan (2008) has highlighted as important student outcomes.

Teachers

The teachers will benefit from the training and implementation of the curriculum as well as their interaction with the researchers. It is to be noted that Hedberg et al. (2007) had found that teachers valued and benefited from their collaboration with researchers. This is in line with OER’s research priority on capacity building of teachers that is necessary for improving the quality of teaching and learning in Singaporean schools. Of the six MOE’s research priorities, the project will address “Curriculum. Assessment, Pedagogy and Instruction” and to some extent “Teacher Learning and Development”. The focus on problem solving skills will address the issue of “a growing mismatch between the kinds of skills and understandings learnt in school and those arguably required by 21st century institutional settings”. This will certainly be beneficial to the whole country.

Prominent conception of the role of problem solving	Stage of the project	Stage of the project
Teaching for problem solving	Pre-project years	Pre-project years
Teaching about problem solving	Students’ early familiarization with problem solving	In students’ initial experience with problem solving, the emphasis is on generic processes and strategies to attacking problems
Teaching through problem solving	Students later apply problem solving approach to learning content	After students are more familiar with the problem solving paradigm, they infuse it as a habit-of-mind in the learning of other mathematical content

Table 1: Adopting different conceptions of the role of problem solving in the project

In M-ProSE, the team advocates the apportionment of significant resources for teachers’ professional development. Prior to teachers’ implementation of problem solving in the classroom, we plan to have a substantial component (lasting about a year) on teacher education which draws on the innovations of the earlier cited projects, such as providing teachers opportunities to solve problems, discuss plans, and participating in Lesson Study cycles.

RESEARCH DESIGN

RESEARCH METHODOLOGY

M-ProSE, being a design research, attempts to change or modify the educational environment and, at the same time, conduct empirical studies of curricular or classroom innovations. The methodology assumes that many forms of learning, which are important objects of study, cannot, in fact, be investigated unless the conditions for their successful learning are supported first. This is appropriate to the target of inquiry in the M-ProSE project - problem solving is multi-faceted learning. A key ingredient of design research is the cyclical interaction between two complementary aspects of design and research. Using previous research and theory, the M-ProSE project team of researchers will design, develop and implement a learning environment which may vary from teacher to a teacher or classroom to classroom within a school. The M-ProSE design research will entail conducting a systematic program of research on the learning that results from the classroom (or teacher, or school) interventions which are aimed at deep understanding of how student outcomes are related to contextual factors, and not just at uncovering and examining the relationships among the factors without deep explanation.

For M-ProSE, we rely on our experience from our previous attempts to introduce problem solving into the mathematics curriculum in Singapore schools. To us, it appears necessary that the teachers must make the proposed instructional approach a routine sufficiently familiar to them so that the approach becomes classroom practice. To reach this stage, it seems essential for the teachers to adapt the researchers’ ideas to make them their own, in the sense that their beliefs of mathematics and of problem solving in mathematics are transformed. Such a process apparently passes through a stage where the teachers negotiate and change the problem solving lesson (for which we are proposing to investigate with a lesson study approach). Finally, a community of practice (or learners) develops to support the change process by providing opportunities to learn to engage the proposed ways – thinking, talking and reflecting on the new teaching experiences and ways of doing mathematics. The entire process of transforming an externally proposed instructional approach and curricular change into classroom and school practice appear to be cyclical, incremental and emergent in nature.

Procedure

Curriculum Design and Development
Teacher Capacity Building Workshops
Student and Teacher Participants
Data Collection

Competitive Advantage of the Project in Proposed Area of Research

The Polya processes to be taught are sound because these are the same processes professional mathematicians use. The other innovations are generic in nature, a one mould-fits-all concept for thinking. In fact, when Hammerness, Jaramillo, Unger and Wilson (1997) reported their analysis of 38 students’ understanding in four different subjects (History, Physics, English and Mathematics), the mathematics class performed the worst of the four classes.
The methods of teaching are generally not complicated. A course on problem solving is compact in terms of time and does not involve immense unsettling school changes.
The methods of teaching are generally not complicated. A course on problem solving is compact in terms of time and does not involve immense unsettling school changes.
The science practical, though with its problems, is an accepted part of science education. The paradigm shift to a mathematics practical is psychologically less disturbing when viewed in comparison with the science classes.