The important one in learning mathematics is student should be able to use mathematical idea to further their understanding of other mathematical idea, and they should be able to apply mathematical thinking and modeling to solve problems that arise in other disciplines. Furthermore, they should understand the role of mathematics in our multicultural society and the contributions of various cultures to the advancement of mathematics.
Following we will review some works of mathematics educationist from different context of culture in relation to the aspects of mathematical thinking
1. Australian context: the works of stacey kaye
Being able to use mathematical thinking is solving problems (Stacey, K. 2006), is one of the most the fundamental goals of teaching mathematics. It is an ultimate goal of teaching that student will be able to conduct mathematical investigation by themselves and they will be able to identify where the mathematics they have learned is applicable in real word situation. She indicated that mathematical thinking is important in three ways: as a goal of schooling, as a way of learning mathematics and for teaching mathematics
In Australian context, Stacey K (2005) have found it helpful for teachers to consider that solving problems with mathematics requires a wide range of skills and abilities, including:
(1) deep mathematical knowledge;
(2) general reasoning abilities;
(3) knowledge of heuristic strategies;
(4) helpful beliefs and attitudes;
(5) personal attributes such as confidence, persistence and organization;
(6) skills for communicating a solution.
She then identified four fundamental processes, in two pairs, and showed how thinking mathematically very often proceeds by alternating between them:
• specialising – trying special cases, looking at examples
• generalizing - looking for patterns and relationships
• conjecturing – predicting relationships and results
• convincing – finding and communicating reasons why something is true.
Stacey, K. (2005) also draws on important general mathematical principles such as :
· Working systematically;
· Specializing – generalizing : learning from example by looking for the general in the particular;
· Convincing: the need for justification, explanation, and connection;
· The role of definitions in mathematics.
2. British context: the works of David Tall
David Tall (ibid.), in the case of long-term learning of mathematical concepts, strived to
explain how do students learn about mathematical concepts and how do they grow over the years to learn to think mathematically in sophisticated ways? He referred to Piaget that there are distinguished two fundamental modes of abstraction of properties from physical objects: empirical abstraction through teasing out the properties of the object itself, and pseudoempirical abstraction through focusing on the actions on the objects, for instance, counting the number of objects in a collection as well as reflective abstraction focusing on operations on mental objects where the operation themselves become a focus of attention to form new concepts. Accordingly, he distinguishes two ways of building mathematical concept:
1) the first is from the exploration of a particular object whose properties he focus on and use first as a description – ‘a triangle has three sides’ – and then as a definition – ‘a triangle is a figure consisting of three straight line segments joined end to end’.
2) the second arises from a focus on a sequence of actions and on organizing the sequence of actions as a mathematical procedure such as counting, addition, subtraction, multiplication, evaluation of an algebraic expression, computation of a function, differentiation, integration, and so on, with the compression into corresponding thinkable concepts such as number, sum, difference, product, expression, function, derivative, integral.
3. Taiwaness Context: the works of Fou Lai Lin
Fou Lai Lin (2006) has developed a framework for designing conjecturing activity in mathematics thinking. The ultimate results of his work suggest that conjecturing approach can drive innovation in mathematics teaching. He concluded that conjecturing activity encourages the students:
1. To construct extreme and paradigmatic examples,
2. To construct and test with different kind of examples,
3. To organize and classify all kinds of examples,
4. To realize structural features of supporting examples,
5. To find counter-examples when realizing a falsehood,
6. To experiment,
7. To self-regulate conceptually,
8. To evaluate one’s own doing-thinking,
9. To formalize a mathematical statement,
10. To image /extrapolate/ explore a statement,
11. To grasp fundamental principles of mathematics involves learners in thinking and constructing actively.
References
www.staff.uny.ac.id/dosen/marsigit-dr-ma. Mathematical Thinking Across Multilateral Culture_Makalah Marsigit Semnas di Jurdik Matematika UNY 2007.
www.staff.uny.ac.id/dosen/marsigit-dr-ma. asumsi-dasar-karakteristik-matematikasubyek-didikdanbelajar-mat-sbg-dasar-pengemb-kur-mat-berbasis-komptensi.
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