Explorer The integration of a problem-solving framework for Brunei high school mathematics curriculum in increasing student’s affective competency

A mathematics framework was developed to integrate problem-solving that incorporated simulation of real-life problems in the classrooms. The framework coined as the RECCE-MODEL emphasised understanding and thinking with a view on mathematics embedded in real-life. The RECCE which stands for Realistic, Educational, Contextual, Cognitive, and Evaluation encompass the underlying principles of teaching problem solving and guide teachers in planning, designing, developing, and facilitating real-life activity tasks in developing students’ problem-solving competencies in mathematics lessons. It also explores students’ cognitive competency in their application of abstract mathematical knowledge into real-life problems based on students’ developmental status of their thinking and reasoning skills correlating to Meanings, Organise, Develop, Execute and Link ( MODEL ). This study investigated the affective development of the students through activity tasks developed by the sampled teachers using the principles within the framework. In total, 94 students from two high schools in Brunei Darussalam responded to a students’ questionnaire constructed to address the MODEL aspect of the framework. In particular, the analyses involved the students’ affective competencies that corresponded to a 19-item instrument within the questionnaire. The findings showed that Brunei high school students have stimulated beliefs and positive attitudes towards non-routine problem-solving in the learning of mathematics. Meanwhile, meaningful activities developed by the teachers encouraged the development of cognitive-metacognitive and affective competencies of the students. The RECCE-MODEL framework paved the way towards understanding the relationships between effective pedagogical approaches and students’ learning, and between attitudes and cognitive abilities, and also for teachers to make better-informed decisions in the delivery of the curriculum.

Kata kunci: Kerangka Kerja Matematika, Pemecahan Masalah, Kurikulum, Kompetensi Afektif How to Cite: Chong, M.S.F., Shahrill, M., & Li, H-C. (2019).The integration of a problem solving framework for Brunei high school mathematics curriculum in increasing student's affective competency.Journal on Mathematics Education,10(2), 215-228.Mathematical modelling is one of the applied mathematical tools that support real-life problem solving in mathematics education that has emerged from several perspectives.Blomhøj (2008) identified five main perspectives of research on the teaching and learning of mathematical modelling: 1) The realistic perspectiveauthenticity of real life modelling in designing problems where students learning is supported by relevant technology, and assess the model and its results against the reality; 2) The epistemological perspective -the development of more general theories and practices in the teaching and learning of mathematics; 3) The contextual perspectiveto include research on problem solving and deepening the philosophical role of word problems in its connection to learning theories; 4) The cognitive perspective -students' modelling processes are analysed with the purpose of understanding the cognitive functions and cognitive barriers of the individual going through the modelling process; and 5) The educational perspective-integrating mathematical modelling in the teaching of mathematics, and discuss problems related to assessing students' learning processes using mathematical modelling activities from different types of mathematics curricula.Barbosa (2012) adopted mainly the education perspective in Brazil where the focus of learning mathematical concepts and the development of 'modelling competencies' are viewed as a way to teach mathematical concepts, in relation to the idea that mathematics education must take part in efforts to educate students be critical, engaged citizens.In the 21 st century, it is not sufficient for students to be only competent in applying mathematical knowledge in the context of the framework of the curriculum, which describe the cognitive and educational perspectives.Instead wider perspectives that include embedding real world contexts into the curriculum are needed to support students' cognitive development in engaging new ideas, supporting earlier understandings, and mathematical reasoning from abstraction to solutions.Consequently, it would be appropriate to adapt all five perspectives proposed by Blomhøj (2008) in developing the mathematics framework for Brunei mathematics education.Our teachers need not only teach the curriculum, but continuous support and guidance from relevant stakeholders in educating the future generation is crucial, especially the kind of support and guidance that may elicit confidence and relevance in raising the quality of teaching and learning.Thus, one of the way forward for our mathematics education will be to have our own relevant framework, which guides teachers in preparing their lessons that is realistic, educational, contextually relevant, cognitively challenging for their students.Anthony and Walshaw (2009) identified ten principles of effective mathematics pedagogy, namely an ethic of care, arranging for learning, building on students' thinking, worthwhile mathematical tasks, making connections, assessment for learning, mathematical communication, mathematical language, tools and representations and teacher knowledge, that were found to develop mathematical capability and disposition within an effective learning community.They believed that holistic development of productive students depends highly on effective mathematics pedagogy, which acknowledges the mathematical potentials in all students in optimising a range of desirable academic outcomes, and also enhancing a range of social outcomes in classroom.Thus, the ten principles encompass the complex dynamic of a classroom environment within the western education system, where the nature of classroom mathematics teaching focus mainly on students' learning in a safe and supportive environment.This corresponds highly to Brunei's current education system model entitled the National Education System for the 21st Century or Sistem Pendidikan Negara Abad Ke-21 or termed as SPN21 (Ministry of Education, 2013).Accordingly, the primary goal of the SPN21 curriculum is based on the principle that each learner is the centre of all teaching and learning through the process of knowledge and understanding, essential skills, and attitudes and values in a well-balanced education system.

The Mathematics Framework: RECCE-MODEL
In conducting a lesson on problem solving, Lester, Garofalo and Kroll (1989) also proposed that teachers focus on creating a classroom culture of mathematical inquiry through connection and relevant discourse.A design by Lester, Garofalo and Kroll (1989) was also explored to study the effect of instruction on students' cognitive self-regulation of the problem solving processes.This also helps to build the foundation of the current framework.In addition to the ten principles of effective teaching by (Anthony & Walshaw, 2009), the five perspectives proposed by Blomhøj (2008), and the additional five fundamental elements of education by Novak (2013aNovak ( , 2013b)), which are the learner, the teacher, the curriculum, the context, and evaluation, had been incorporated in developing the present mathematics framework.Novak (2013aNovak ( , 2013b) ) recognised that in enhancing any successful educational event, each of these five elements must be optimised.Underpinning these principles and perspectives; Pólya's Model (1945), Garofalo and Lester (1985) cognitive and metacognitive framework, Carlson and Bloom (2005) Mathematical Problem Solving (MPS) framework and modelling cycle by Blum and Leiβ (2007), an emerging mathematics framework representing Realistic, Educational, Contextual, Cognitive, and Evaluation -RECCE and Meanings, Organise, Develop, Execute, Link -MODEL (see Figure 1) was developed for this present study applicable to the mathematics curriculum of Brunei.
The RECCE-MODEL is a framework developed to encompass the underlying principles of teaching problem solving by incorporating simulation of real-life problems in classrooms, which emphasized contextually relevance, understanding and expressing thinking with a view on mathematics embedded in real-life.Furthermore, the framework sets direction in learning and assessment of mathematical knowledge and skills in developing students' cognitive, metacognitive and affective competencies.RECCE aims to guide teachers in planning and designing their mathematics lessons, developing non-routine activity tasks and evaluate the implementation process of the lesson plans to subsequently make improvement.It is important that the assessment of the learning process in providing information about the progress of students in achieving learning goals are conducted through learning activities between the teacher and the student (Kenedi, et al. 2019;Shahrill & Prahmana, 2018;Khoo, et al. 2016).The RECCE-MODEL framework also echoed similar importance between teaching problem solving and developing competencies through the use of real-life activities and eventually achieving the learning goals.Therefore, this aspect of the framework is focusing on the structuring and development of meaningful lessons to maximize learning in the classroom.
Two theoretical perspectives were drawn in developing the conceptual design of the RECCE-MODEL framework.Both constructivism and Ausubel's (2000) assimilation of cognitive learning provided the theoretical perspectives in guiding this present study.Students are encouraged to communicate and share their ideas and methods of workings to others as a way of developing their communicating skills.The Evaluation principle refers to teachers reflecting their teaching approaches and lessons conducted to effectively improve students' competencies in learning and applying mathematics.In addition, the earlier four principles (Realistic, Educational, Contextual and Cognitive) must be reviewed to subsequently make improvement in designing lessons that contribute to the success of teaching and learning mathematics.
The RECCE-MODEL framework proposes that teachers create a mathematics classroom based on the five guiding principles of RECCE, to engage students in mathematical thinking and problem solving through constructivist approach.This is the approach where knowledge is constructed by learners in new experiences from previous learning and propositions of the learning environment, which leads to deeper understanding and flexibility in their mathematical thinking.The key elements of the teacher's role involved planning an overall course of lesson plans; selecting appropriate resources and mathematical problems following the three fundamental requirements for meaningful learning by Novak (2013aNovak ( , 2013b)); monitoring process and progress; and evaluating results.
Therefore, the RECCE-MODEL framework aims to create a strong link between teachers' approaches to specifying the mathematical problem solving processes from mathematical content of the curriculum to the mathematical reasoning required in problem solving.Teachers are also expected to foster classroom climate that includes non-routine tasks which, enhances students' beliefs and affects in further contributing to their metacognitive competency towards successful problem solving.
Meanwhile, the MODEL framework is used to examine and evaluate students' cognitivemetacognitive competencies in completing a mathematical task.While, students also used MODEL in assessing their Level of competencies in completing a task through creating meaning from the real-life problem posed (Level 1); identifying the dependent and independent variables in the problem posed (Level 2); deciding which variables and appropriate mathematical formulae are feasible and possible to use in solving the problem (Level 3); obtaining mathematical solution(s) and contextualise the solution(s) in order to justify for interpretations (Level 4); and finally linking to validate the solution(s) to the problem and reflecting on any error(s) encountered (Level 5).Furthermore, the MODEL framework explores students' cognitive competency in six levels, in their application of abstract mathematical knowledge into real-life problems based on students' developmental status of their thinking and reasoning skills correlating to Meanings, Organise, Develop, Execute and Link (MODEL) (shown in Table 1).Knowing why it is applied.
In L1 -Meanings (M), students must present some fragments of their abstract knowledge into diagrammatic representation of the problem using concept map, mind map, flowchart, diagrams of all sorts and also any relevant figures.At this Level, students will demonstrate memory recall and reinforced prior knowledge or learning into the real-life problem posed.In L2 -Organise (O), students must identify the dependent and independent variables in the real-life problems posed.They will explore and generate ideas, parameters and break down the problem into simpler task by asking questions and linking ideas.In L3 -Develop (D), students make relevant assumptions based on their ideas and decide which variables are feasible and possible to solve this problem.Students will learn creative decision-making at this Level by choosing the appropriate mathematical formulae to use in solving the problem.In L4 -Execute (E), students will obtain mathematical solution(s) at this Level, and will need to contextualize the solution(s) in order to justify for interpretations at the final Level.The learning outcome at this Level is that students will demonstrate their metacognitive competency in reflecting back into the problem.And the fifth Level, L5 -Link (L), the metacognitive Level, and students must be able to link and validate their solution(s) to the problem and finally reflecting on any error(s) encountered.
The MODEL framework proposes that students to self-scaffolding by following the five levels of problem solving in helping them to become self-aware and self-regulate in their thinking, thus supporting their use of knowledge to help solve a problem.Therefore, with the development of the RECCE-MODEL framework, this study aims to investigate the affective development of the students through activity tasks (Chong, et al. 2018) developed by the sampled teachers using the principles within the framework.A pilot study was conducted in identifying the affective competencies of Brunei pre-university students (or high school equivalent of Year 12 in the United Kingdom or the 11 th Grade in the United States), prior to the development of the RECCE-MODEL framework.The pilot study concluded that the affective competencies of Brunei students are stimulated and can be further developed through structured activities in a learning environment (Chong & Shahrill, 2015).Thus, the development of this framework will provide the structure in designing realistic, educational, contextual and cognitive challenging tasks to develop students' affective competencies.

METHOD
A mixed (qualitative and quantitative) research methodology was employed in this study, to engage teachers and students in working with RECCE-MODEL in integrating perspectives on problem solving of realworld examples through activity tasks (Chong, et al. 2018).The quantitative data were collected using a students' questionnaire, and the qualitative data gathered from semi-structured interviews involving all the participants using open-ended questions and were conducted in groups of four to six students, following the recommendation from Creswell (2013) in relation to focus group interviews.The questionnaire was designed in three sections: the first section consists of questions regarding students' demographic and academic characteristics; the second section consider students' perceptions of the five aspects of the MODEL framework; and the last section consider students' affective domain of learning mathematics (beliefs and attitudes).
The students' questionnaire was developed addressing the MODEL aspect of the framework and how it interconnects between students' cognitive and metacognitive competencies as they go through the process of problem solving.All the items developed also provided opportunities to critically reflect on individual's attitudes and beliefs of learning mathematics.The development of the questionnaire followed the requirement and criteria set out by Cohen, et al. (2011) to obtain as much personal information and academic background of the students as possible and also to assess students' affective competency in learning mathematics.The questions that are designed to capture students' affective competency are in rating scales following Likert scale ranging from never = 1 to always = 5.The design of the questionnaire was concise such that five items that describe the experience of doing and learning mathematics within the context represented each category of the MODEL framework.The questionnaire only required students to read the questions, read the possible responses and mark their responses accordingly.At the start of administering the questionnaire, for ethical considerations, students were informed and assured of the confidentiality, anonymity and non-traceability as all information and data were aggregated into categories.Piloting of the questionnaire was conducted in one of the pre-university institutions prior to implementing the main study.
Meanwhile, the use of activity tasks in this study was to enhance students' cognitive, metacognitive and affective capabilities through communication, self-regulation, and facilitating discovery in enhancing understanding of the problem, and thus supporting students' cognitive, metacognitive and affective development towards non-routine problem solving being part of their learning experiences in mathematics.
The subsequent results of the pilot study was also reported in Chong and Shahrill (2016), and the findings showed that Brunei high school students have stimulated beliefs in learning of mathematics and positive attitudes towards non-routine problem solving being part of learning in mathematics.
In reporting the findings in this paper, the students' affective competencies were explored from their responses to a set of 19 questionnaire items that described their beliefs and attitudes towards mathematics and problem solving in general.The 19 items appeared at the last section of the students' questionnaire.In total, the sample size comprised of 94 students from which 42 students were from the first participating high school and the remaining 52 students were from the second high school.There were 33 male students (35.1%) and a total of 61 female students (64.9%).The participating students' ages ranged from 15 to 20 years old.

RESULTS AND DISCUSSION
The reliability score of the 19-items instrument was in the acceptable range of Cronbach's alpha value of 0.76.The results were confirmatory with all 19 items as they fit all the six dimensions in Table 2 below.The questionnaire of this present study was administered after the intervention has been completed.Therefore, the participating students' views of learning mathematics and problem solving in this study was reflective of their attitudes and beliefs after the intervention has been carried out.This was to measure the extent of how Earlier work by Ernest (1988) has distinguished three conceptions of beliefs about mathematics teaching and learning into the instrumentalist view, Platonist view and the problem-solving view.The significance of these views is that a learner with instrumentalist view will view mathematics as collection of facts, skills and rules with no connection, Platonist will view mathematics as a static body of knowledge, and problem-solving learner will view mathematics as dynamic with content continually growing (Allen, 2010;Shahrill, et al. 2018).In her study, Allen discussed that teachers need to shift their views to one of the problem-solving view in order to be effective teachers of mathematics.Similarly, in the context for a student to be effective learner, one must view mathematics as a process of enquiry and exploration, not just mastery of facts and procedures.Further analysis of the interview excerpts showed that the participating high school students have very strong perception of the purpose and importance of learning mathematics.They believed that mathematics will be able to support their future career paths and is essential for life.During the intervention, the mathematics pedagogical approaches developed by the sampled teachers using RECCE-MODEL framework have shaped the students' beliefs and their behaviour in learning problem solving.In particular, the teachers' actions in scaffolding students' learning during the interactions using the activity tasks (Chong, et al. 2018), the technology, the resources and their peers, were crucial to the success of solving the tasks.This seemingly simple findings have important implications on how students learn and apply the metacognitive processes and strategies during the activity tasks.For example, a task on designing a school car park was viewed as the most challenging task for majority of the students, but it enriched their metacognitive experience as the students continually check the appropriateness of their solutions and justifying the final solution.The task was designed to give students the opportunities to reflect on their strategies following their engagement in the problemsolving task with their group members.Furthermore, they had to test, redesign if necessary and review their solutions repeatedly during the problem solving process guided by the MODEL framework.Consequently, all groups persevered and managed to complete this task through good discussion and strategic collaboration.This was attributed by the teachers' influences on changing the culture of the classroom by bringing the realistic experiences of learning mathematics through non-routine problem solving in the classroom.

CONCLUSION
The sampled teachers in this study provided meaningful tasks that encouraged the development of cognitive-metacognitive and affective competencies of the students.The progress of the RECCE-MODEL framework has paved the way towards understanding the relationships between effective pedagogical approaches and students' learning, and between attitudes and cognitive abilities, and also for teachers to make better informed decisions in the delivery of the curriculum.Goos, et al. (2017) identified mathematical knowledge base, heuristics, self-awareness, self-regulation, beliefs, affects and classroom environment are the factors that contribute to successful problem solving.And these factors are inter-connected to one another.Evidently, a teacher plays a critical role in shaping students' beliefs and attitudes towards a learning environment.Therefore, a simple change in teachers' classroom practice in this study appeared to influence and articulate students' beliefs and dispositions in deepening their mathematical engagement.Through synthesis of researches, Lesh and Zawojewski (2007) pointed out that developing a productive problem-solving persona involves complex, flexible, and manipulatable profile of affect.Therefore, co-developing affective and metacognitive competency can contribute to how cognition develops in learning mathematics.Sari and Mutmainah (2018) also highlighted similar significance of teacher's role in delivering the subject matter to motivate learning of mathematics for students through creative, open and joyful learning.It can be suggested that with high cognitive demand tasks, students may be more engaged and become active in the exploration stage, and may be able to use strategies that were meaningfully connected to concepts.To conclude, this present study marked the beginning of integrating a mathematics framework called the RECCE-MODEL into the Brunei school curriculum in developing students' affective competencies in the learning of mathematics.

Figure 1 .
Figure 1.The emerging RECCE-MODEL mathematical problem-solving framework RECCE-MODEL helps to develop students' affective competency in relation to their cognitive and metacognitive development in solving non-routine problems.McLeod (1989) viewed emotion as one of the critical factor influencing the process of solving nonroutine mathematical problem.The emotion described by McLeod was the feeling of frustration with each unsuccessful attempt; the feeling of anger when a solution cannot be reached; and the feeling of satisfaction and joy when solution is obtained.Therefore, this domain of feelings described by McLeod plays a critical role in influencing the cognitive processes of solving problem, in particular non-routine problems.This is because the extent of the willingness of an individual to solve a problem is greatly dependent on the individual understanding of the problem posed, the kinds of decision-making made during the process and also the working conditions.Schoenfeld (1983) also presented similar views, where he discussed that students manage their cognitive resources through students' belief systems which, included attitudes towards mathematics and confidence about mathematics.Consequently, McLeod (1992) has re-conceptualized beliefs and attitudes towards mathematics as the affective domain in mathematics education and instruction.He categorized beliefs into beliefs about mathematics (importance, difficulty, and based on rules), beliefs about self (self-concept, confidence and metacognition), beliefs about mathematics teaching or mathematics classroom instruction, and beliefs about the social context (home environment, parental and peer influences).

Table 1 .
MODEL cognitive-metacognitive framework categorised in 6 levels in performing a mathematical task

Table 2 .
The six dimensions of the students' affective competency in learning mathematics and problem solving Presented in Table3are the descriptive statistics of the six dimensions of the students' affective competency in learning mathematics and problem solving.Entries from Table3were evident that students have strong beliefs about mathematics and also positive beliefs.These two dimensions recorded the highest mean values in comparison to its total maximum score.

Table 3 .
Numerical variables between the six dimensions of the students' affective competency

Six dimensions of students' perceptions Total Minimum score Total Maximum score Mean (SD)
These findings were further supported by students' comments from the interviews when asked these questions: why study mathematics and what use of mathematics is important for you to learn?The following are excerpts from the interviews that were relevant to support the findings: T1 For me, it is my best subject.I like it and it also gives a

lot of help in my other subjects.
Physics, there's all these Maths, also in Computer Science, there's all these calculations where we're converting numbers in a system to another system.It's very (cradles his head in his hands).Maths is definitely helping in all my other subjects and also one of my goals.H1 Because one, it's easy and-second, it's most job requirements.

useful for my Economics because
I plan to take Economics degree C1 I take Maths because it is important.Because it is related to Physics.F1 Hmmm.I find that it is interesting and sometimes I can release my stress by just doing the past year questions.B1 I love maths, and I think I'm good in maths and that's why I'm doing Maths.F1 When I ask my friend, they say that maths is really important when you want to get a job.Nowadays, I think it is the most important subject.D1 Because I like mathematics and doing calculations G1 Because my father said... Like, maths is important for all.Like, any course you want to take.Maths is important.V1 Because it might be helpful in the future.O1 I have the interest to study Maths in A-Level.It's actually because of my career.I have two career basically either become engineering or the doctor.So to be engineering, engineer, so I need to take Maths.K1 Basically we use mathematics everyday either we do realize or not so if we don