Developing students' mathematical thinking: how far have we came?

Malaysian students' poor performance in the newly implemented Form Three Assessment (Pentaksiran Tingkatan 3, PT3), TIMSS and PISA has spurred many debates and criticism on the quality of our students' learning of mathematics and science. Since the inception of the Integrated Curriculum for Secondary Schools in 1989, the aim of secondary mathematics has been steadfastly on "developing individuals who are able to think mathematically and who can apply mathematical knowledge effectively and responsibly in solving problems and making decisions". So how far have we come? Are the classroom activities, assessment tasks and assessment questions geared towards cultivating mathematical thinking and decision making? Evidence show that there has been vast misalignment between the intended curriculum (the one prescribed by policy makers), the implemented curriculum (the one that is actually carried out by teachers in their classrooms) and the attained curriculum (the one learnt by students or on what was examined). Thus can the large number of students getting As in mathematics for Lower Secondary School Evaluation (PMR) and Malaysian School Certificate (SPM) be used as indicators of the success in developing students' thinking? This paper discusses what is meant by mathematical thinking. There has not been a conclusive definition on mathematical thinking because some view it as a process and others as an outcome. Mathematical thinking is the foundation to do reasoning and problem solving and to develop conceptual knowledge, as opposed to procedural knowledge. Several findings from studies that focused on students' ability to provide reasoning and give meanings to concepts and algorithms are highlighted. Students' development in geometric thinking based van Hiele's levels of geometric thinking in learning, shapes and spaces is also discussed. van Hiele's levels of geometric thinking includes higher order thinking and decision making skills and acquisition of mathematical concepts to enable learners to operate at higher levels in van Hiele's theory. The role of metacognition in facilitating mathematical thinking is also deliberated. Metacognition refers to higher order thinking which involves active control over the cognitive processes engaged in learning. The paper also touches on the role of technology in facilitating mathematics thinking. This paper highlights that generally, Malaysian students are not meeting the international benchmark for mathematics performance. This may be partially attributed to students' inability to think mathematically and thus their inability to translate the contexts given and to use mathematics to provide solutions. Factors that may contribute to this phenomenon are further examined from the perspectives of the curriculum, the instruction and the assessment. The present curriculum may not be able to support the initiative to focus on higher order thinking skills (HOTS), as can be concluded from the expected outcomes stated in the curriculum guides. In terms of instruction, although it has been suggested since 20 years back, constructivist teaching has not been a preference of Malaysian teachers. Nevertheless, the shift to school based assessment, the HOTS initiative and the change in examination format to include more challenging questions, such those given in PT3, TIMSS and PISA, may well elicit a classroom atmosphere that cultivates HOTS as well as critical and creative thinking. To provide support for learning school textbooks need to be greatly improved as the current contents, contexts and examples do not stimulate students' thinking. As for assessment, as long as the 'teaching for examination' culture is strong and the right concept of school-based assessment is not implemented, not much change will happen in classroom instruction and the aspiration to get our students to think mathematically will remain far from reality.

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