Chapter 3: Methodology
3.3 Study Instruments
3.3.1 The Proposed Mathematics Enrichment Program (MEP)
The proposed MEP was an intervention program that was designed to help the students improve their mathematical literacy to fulfill the purpose of this study.
3.3.1.1 The Proposed MEP Development Principles
Based on the previous literature and the related studies, the following principles were identified to underline the development of the proposed enrichment program to improve the students’ mathematical literacy.
• The proposed MEP was designed to build on students’ prior knowledge, so it should demonstrate the most important prior knowledge related to the studied topic.
• The proposed MEP was based on contextual problems that address the modeling problems that would improve the high order thinking skills (problem-solving and reasoning).
• The proposed problems consisted of PISA problems of different levels. However, there are few problems with low levels like one and two, as such were the upper levels of 5 and 6 based on the PISA problem classification. Most of the problems lied in levels 3 and 4 to be fulfilled by students. Adopting PISA problems was to ensure that levels of problems are at the appropriate cognitive level for the students.
• The teacher’s role was to facilitate students' construction of mathematical knowledge; support and expand student thinking by fostering discussions and encouraging students to develop their own problem-solving strategies and use informal or prior knowledge to help develop their conceptual understanding and use of alternative solution methods. The problem-based learning was a perfect fit for this role of the teacher in implementing the enrichment program that was based on constructivism and social constructivism where the students could interact and help each other.
3.3.1.2 The Content of the Proposed Enrichment Program
This proposed MEP aimed to improve the students’ mathematical literacy.
Nevertheless, mathematical literacy is a very broad and cumulative area. So, the scope of content for this enrichment program was identified to be restricted to the comprehensive framework of mathematical literacy in PISA. The MEP can be found
in Appendix A. Mathematical literacy was the major topic in 2012 for PISA assessment that focuses mainly on the processes of problem-solving (modelling cycle) and it will return to be in 2021 with extra focus on reasoning as it is the core of the problem-solving processes (OECD, 2018a). Considering the framework of mathematical literacy by PISA, the main components of mathematical literacy involve mathematical thinking such as reasoning, modelling, and making connections between ideas (Stacey, 2007). According to Piaget's theory of constructivism, students of this age are cognitively capable of reasoning and solving problems that support the relevance of the MEP program because students at the age of 15 are in the operational stage (Piaget, 1957).
The development of the Enrichment Program design relied mainly on two components, which were a review of the basics of prior knowledge required for each lesson as well as relevant PISA elements released. Using released items from PISA was appropriate to the students’ cognitive level as it was designed to test the 15 years age students. This proposed enrichment program consisted of the four PISA mathematical literacy content areas which were change and relationship, space and shape, quantity, and uncertainty. In addition, addressing reasoning was embedded in these four content areas as the processes of solving these problems might be through formulating, employing, and interpreting in which reasoning was essential to all these processes.
Two lessons were developed for each of the four “overarching ideas”: quantity, space and shape, change and relationships, and uncertainty. The developed eight lessons and allocated time range for each lesson of the program were determined as shown in Table 5.
Table 5: Enrichment program content and time-range for lessons
Content Area Lessons No. of
sessions Change and relationship 1- Functions and variations
2- Numerical trends and patterns
2 2 Space and shape 1- Geometric approximation
2- The visual and physical world
2 2
Quantity 1- Percentages
2- Quantification
2 2 Uncertainty 1- Probability
2- Statistics
2 2
Total 8 lessons 16 sessions
This enrichment program, as shown in Table 5, consisted of eight lessons where two lessons were assigned to each of the content areas, each lesson was designed based on the most addressed topics from the released PISA items of different levels in addition, these released items were collected to build on students’ prior knowledge.
The time allotted for each lesson was two periods of 45 minutes. Only one lesson was discussed each week. As a result, eight weeks of time was required to implement the enrichment programs.
3.3.1.3 Appropriateness of the Proposed Enrichment Program
In light of what has been reached in the theoretical framework and previous studies, the development of the enrichment program took several steps to reach its final form. After the initial development of the enrichment program, it was presented to a group of experts, who are experienced in teaching and learning mathematics. The group of experts consisted of one professor in mathematics education, one professor in mathematics, and five expert mathematics teachers.
For judging the appropriateness of the proposed MEP, these experts were asked to decide to what extent it was appropriate to the level of students in the advanced tenth grade. Experts indicated that the proposed enrichment program was suitable for advanced tenth grade students. They remarked that released items for PISA were suitable to the group age of the students as they were specially developed for this age group and reviewed by several international experts all around the world. They also mentioned that this enrichment program included problems that might challenge most levels of students. In addition to the most important prior knowledge was necessary for every lesson. However, they suggested that more than two periods might be required to discuss each lesson. Unfortunately, due to the intensive curriculum of mathematics and time constraints, it was difficult to provide more than two periods for each lesson. Therefore, after discussing this point with the experts, we agreed that not all problems should be discussed with students in the classroom as the teacher's role as a facilitator of student learning. Therefore, students were encouraged to self-learn the rest of the problems independently.
Another point that was mentioned by the experts about the language level used in the problems. They stated that it involved difficult words that the students might not understand and might affect their ability to solve problems. After discussing this point, we agreed not to modify the language of the problems as the students needed to be exposed to the same level of problems that PISA provided. Additionally, this could be considered another challenge for students to use their skills to just anticipate the meaning of the difficult words from the context of the problem as using words was an essential part of the mathematical contextual problems.
The last point was about the order of the lessons, the experts suggested teaching the lessons about “quantity” just after “change and relationship” not after “space and shape”. This was because “quantity” might include concepts that are considered basics to other content areas. The experts’ suggestion was met, and the order of lessons was changed as they said.