The international trends in mathematics education call for mathematics teaching and learning to enable learners in seeing, connecting, and applying mathematics in real-life. This call is advocated in a movement to teach mathematics as a subject matter closely related to other subject matters such as science, commerce and daily life experiences.
Freudenthal (1973; 1983, 1991) promotes the philosophical idea of mathematics teaching and learning as a human activity. Freudenthal‘s notion of ‗mathematics as a human activity’ was developed further in collaboration with his colleagues in Freudenthal Institute. It is now accepted as a well known theory in mathematics education called Realistic Mathematics Education‘ (RME). The basic notion of RME is that mathematics should be undertaken as an activity for students to experience mathematics as a meaningful subject under the guidance of teachers. One of the basic principles in RME is guided reinvention principle, which stresses the importance of learners to experience learning process as a process where in they get to ‗reinvent‘ mathematical properties and notion under the guidance of others. In 1993, Freudenthal noted the link of guided reinvention principle with the aim of developing common sense in learning process:
…I have pointed out that, in invented and reinvented on a manifold of places on earth independently, mathematics, unlike any other science, has been and still is a matter of common sense: in the course of individual histories and that of mankind, gradually refined common sense. So didactically, it seems to be no exaggerated requirement to have this knowledge reinvented by the learner, albeit under guidance. (Freudenthal, 1993, p.72)
Gravemeijer and Doorman (1999, 116) contend that in guided reinvention process, ―the learners come to regard the knowledge they acquire as their own private knowledge, knowledge for which they are themselves responsible‖. In this case, contexts from real-world or a story serve as a starting point where students explore and reinvent mathematical notions in a situation that is ‗experientially real‘ for them (see Gravemeijer and Doorman, 1999).
In a similar vein, Schoenfeld (1994) argues that thinking mathematically is far more important than just building an inventory of mathematical topics. He discriminates between ‗knowing‘ a list of mathematical contents and ‗knowing‘ to do and think mathematically. In his view, knowing mathematics is characterized by his ability to use mathematics in dealing with both novel and familiar situations. Hence, he advocates the use of problem solving as a strategy to enable students in developing mathematical thinking.
In line with RME approaches, a more general approach to teaching and learning, known as Contextual Teaching and Learning (CTL) (see e.g., The cornerstone of tech prep, 1999) also underscores the use of contexts for teaching and learning approaches. This approach advocates the use of context as a tool to help learners in making sense of the content as reflected in the following quotation:
According to the contextual teaching and learning theory, learning occurs only when students (learners) process new information or knowledge in such a way that it makes sense to them in their own frames of reference (their own inner worlds of memory, experience and response). This approach to teaching and learning assumes the mind naturally seeks the meaning in contexts – that is in relation to the person‘s current environment- and it does so by searching for relationships that make sense and appear useful. (The cornerstone of tech prep, 1999, p.1)
Similarly, NCTM (2000) promotes the use of daily-life contexts that allow students to experience mathematics from an informal setting to a more formal and abstract mathematics. The emphasis on assisting students to see and make connections among various mathematical ideas is written in Chapter 3 of the NCTM Standards (2000). Furthermore, it underscores the role of teaching and learning process in making students aware of the mathematical connections through the use of probing questions:
By emphasizing mathematical connections, teachers can help students build a disposition to use connections in solving mathematical problems, rather than see mathematics as a set of disconnected, isolated concepts and skills. This disposition can be fostered through the guiding questions that teachers ask, for instance, "How our work today with similar triangles is related to the discussion we had last week about scale drawings?" Students need to be made explicitly aware of the mathematical connections.
The notion of mathematical literacy promoted by OECD (2004) also emphasizes on the function of mathematics in daily life. OECD‘s (2004) definition of mathematical literacy depicts a broader spectrum of what constitutes mathematics. This definition goes beyond school mathematics curriculum:
Mathematical literacy is an individual‘s capacity to identify and understand the role that mathematics plays in the world, to make a well-founded judgment, and to engage in mathematics in ways that meet the needs of that individual‘s current and future life as a constructive, concerned and reflective citizen. (p.72)
Mathematical modelling and the process involved in it are considered as one of the most central notions in mathematical literacy (Kaiser & Willander, 2005; Stacey, 2009). The modelling cycle involves a process of translating real-life problem into a mathematical model (mathematisation) and a process of reinterpretation the mathematical solutions back to the real world problems. Going through this cyclic process, it is expected that mathematics could be explored as something that is closely related to our daily life situations. A didactical modelling
process proposed by Kaiser and Blum (in Kaiser & Schwartz, 2006, 197) illustrates this in Figure The use of contextualised tasks in interdisciplinary settings that are meaningful for students is often used in mathematical modelling to promote mathematical literacy (see e.g. Stillman, 2000; Ng & Stillman, 2009).
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