3.3 MEANINGFUL LEARNING
3.3.5 Realistic Mathematics Education …
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need for formulation is the use of “the difference between eight and three” in spoken English which can be translated into the mathematical expression 8 – 3 and can be easily computed. Whilst by formalization Harel refers to the act of externalizing the exact intended meaning of an idea or concept. In this way the learner gains control over the idea so as to be able to talk like a mathematician (Pimm, 1987). Pimm calls this a Mathematics register. According to him a Mathematics register is a set of meaning that belongs to language of Mathematics. This set of meanings that constitutes a register does not refer only to words and structures but also to the styles of meaning and modes of argument.
For example the phrases/words ‘take away’, ‘top heavy’, ‘combine’, ‘divide into’, etc. have an everyday usage or meaning and an altered meaning or grammatical meaning in a mathematical discourse. When learners are learning Mathematics in school they are therefore attempting to acquire communicative competence in the Mathematics register.
Category 5: Need for structure
Harel mentions that the need for structure includes the need to reorganize learnt ideas or concepts into a logical structure. This need resonates with Piaget’s interrelated concepts of assimilation and accommodation as discussed earlier in the chapter. Harel’s notion of ‘reorganize’ is synonymous to accommodation which is the process by which a learner’s existing schema is modified to fit incoming ideas or concepts. The verb ‘organize’ in ‘reorganize’ means there are some ideas or concepts already existing as in Piaget’s pre-existing schema.
From the discussion thus far, what stimulates intellectual need depends on learners’ reinvention of mathematical ideas with the guidance of the teacher which is analogous to the notion of RME. Here, the teacher needs to facilitate and guide the learners as they attempt to reach conceptual understanding. And all Harel’s constructs of intellectual need would lead to meaningful learning.
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Mathematics’. The Netherlands was no exception as it started its reform movement in the early seventies when the first ideas for Realistic Mathematics Education (RME) were conceptualized. RME is a teaching and learning pedagogy in Mathematics education that was first introduced and developed by the Freudenthal Institute in the Netherlands. De Lange (1996) mentions that this pedagogy had been adopted by many countries across the world such as England, Germany, Denmark, Spain, Portugal, South Africa, Brazil, USA, Japan and Malaysia. According to Van den Heuvel-Panhuizen (2000), the present form of RME was mostly determined by Freudenthal’s (1977) view about Mathematics. Basically, the concept of RME is one of the theoretical aspects of learning which has a bearing to the conceptual understanding of meaningful learning.
Freudenthal (1977) felt that Mathematics must be connected to reality and Mathematics as human activity. His argument is that learners should be given the opportunity to reinvent Mathematics by mathematizing its content from learners’ everyday life experiences and by mathematizing its content from within the subject Mathematics. In both cases, the Mathematics content that is to be mathematized should be experientially real for learners (Gravemeijer, 2004). With regards to RME, Freudenthal (1977) asserts that it puts on offering the learners’ problem situations which they can envisage (context). According to him, the contexts should be sufficiently real for learners to be able to engage with the contexts. The contexts assist in solving problems which make sense to the learners, but also critical that they reflect the Mathematics structures the teacher wants learners to work out. Dickinson, Eade, Gough, and Hough (2010) had noted that rather than beginning with abstractions or definitions to be applied later, one must start with meaningful contexts that can be mathematized.
These contexts function as a basis for the learning process and for learners to make connections. Through staying connected with the context, learners are able to continue to make sense of what they are doing, and do not need to resort to memorizing rules and procedures which are meaningless to them (Dickinson et al., 2010). According to Dickinson et al., the contexts can be taken from the real world or from areas of Mathematics that learners have learnt, or from other subject disciplines as a starting point for learning the new content. This is what
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Ausubel et al. (1978) refers to as meaningful learning because new information is being linked to relevant, pre-existing aspects of what the learner already knows.
However, Bansilal (2009) had argued that during the use of learners’ everyday experiences, some learners may be disadvantaged. She had found that the learners may instead base their responses to their everyday experience. The question here is: Whose context is it? The intended context, therefore, may be out of learners' context.
RME also stresses to the idea of Mathematics as a human activity (Freudenthal, 1977) wherein the mathematics subject matter is sieved from a practical, real- life context. Furthermore, Mathematics must not be a subject matter that has to be transmitted rather Mathematics education should give learners ‘guided’
opportunity to‘re-invent’ Mathematics (Freudenthal, 1977). Freudenthal subscribes to the constructivist perspective of learning Mathematics. He believes that a teacher cannot transmit knowledge ready-made and intact to learners but rather learners should create their own conception of reality under the guidance and supervision of the teacher. Freudenthal (1968) further explains that in Mathematics education, the focal point should not be on Mathematics as a closed system but on the activity and process of mathematization. In short, teachers should help learners to make connections between new mathematical ideas to previous aspects of Mathematics ideas that they have learnt.
Treffers (1987) formulated two types of mathematization explicitly in an educational context. He called them horizontal and vertical mathematization.
Treffers argued that in horizontal mathematization, learners come up with mathematical tools which can help to organize and solve a problem located in a real-life situation, i.e. it involves a move from real world into the world of symbols in the context of Mathematics. On the other hand, Treffers described vertical mathematization as the process of reorganization within the mathematical system itself i.e. it involves moving within the world of symbols where learners find shortcuts and discovering connections between
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mathematical concepts and strategies. The table below illustrates the two types of mathematizations described above.
Table 3.3 Relationship between horizontal and vertical mathematization Horizontal mathematization Vertical mathematization Musa and Themba have similar
amounts of cake. Musa gave Thandi
3
7 of his cake and Themba gave her 2
7
of his cake. How much cake did Thandi get from the two boys?
Find 2
3+ 1
4 = Justify your answer.
In the mathematical problem on horizontal mathematization, the context gives meaning to the concept of adding fractions with same names. Learners may represent the problem by making paper cut-outs or any appropriate manipulative representation. Later, the presence of manipulative objects is no longer needed to answer the problem. On the other hand, in the problem on vertical mathematization, learners may use the concept of equivalent fractions and addition of fractions with same denominators to answer it. Here, learners use connections between Mathematics concepts to conceptualize the addition of fractions with different denominators.
In each of the mathematizations (vertical and horizontal), there is a link between pre-existing knowledge and new knowledge. With the context related-problem, the assumption is that context is meaningful to learners though this becomes a problem if learners are coming from different backgrounds or some learners may not be familiar with the chosen context. What is perceived as a context to one learner may not be a context to the other if they are coming from different backgrounds. Of note here is that both the constructs of horizontal and vertical mathematizations require the teacher to monitor the learning process. S/he, is in fact, supposed to be playing the role of a facilitator as learners attempt to learn meaningfully.
Gravemeijer (2004) elaborated on Freudenthal’s (1977) RME principle by using instructional design to reform Mathematics education. Gravemeijer argues that
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by using instructional design, learners develop a framework of relations or rather connections within Mathematics to do a problem easily by inventing the necessary tools for themselves. For example, in the problem 3
4 × 8
11 =, the procedure is multiplying the numerators and denominators and simplify to get6
11. However, learners can solve the problem by dividing the 8 by 4 to get 3 × 2
11
, and then multiply 3 by 2 and write 6
11 . What the learners have done here is to use flexible computation within the framework of number operations to arrive at the expected solution. Their operational procedure is tied to their pre-existing knowledge of simplifying fractions which is the fundamental basis of meaningful learning as perceived by (Ausubel et al., 1978). According to Gravemeijer’s (2004) elaboration of RME, learners should be given a problem and allowed an opportunity to think about and discuss possible solutions to the problem.
Also, of note about the Freudenthal’s (1977) RME reform to Mathematics education is that it resonates strongly with progressive approaches like problem- solving. Pólya (1945), who was the first scholar to discuss, analyze and promote problem-solving on a large scale, suggested the following stages for solving a problem for reforming Mathematics education.
understanding the problem
devising a plan
carrying out the plan
looking back
According to Polya, a learner begins with a problem. With the problem in front of him/her, s/he engages in minds-on activity to understand it. The learner then attempts to make a plan by finding the connection between given data or information and the unknown. Once the plan has been formulated, the learner may attempt to carry it out and finally s/he may examine the solution s/he obtained. It can be argued that Polya’s stages of problem-solving are embedded in Treffers’ (1987) notions of horizontal mathematization and vertical mathematization and are intrinsic in learners’ activity as they engage in solving mathematical problems.
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In education literature many researchers have attempted to clarify what is meant by problem-solving (Badger et al., 2012; Schoenfeld, 1992). Schoenfeld (1992) mentions that problem-solving has contradictory meanings and Badger et al.
(2012) concur with him and they state that problem-solving is widely recognized for its value in everyday life but what it means remain elusive. To Badger et al.
(2012), teaching problem-solving should focus on:
Letting learners ‘mathematize’ situations which they have not met previously.
Situations such that they are in accord with their pre-existing knowledge and situations that challenge them;
On the note of problem-solving, Ausubel (1962), argues that the most important single factor influencing learning as a learner is engaged into a problem situation is what s/he already knows. The learner’s previous conceptions save as the basis for what s/he is about to learn. S/he needs to make connections between prior knowledge and knew knowledge to enable understanding. Still within the notion of problem-solving, Cockcroft (1992) advocated for it as a means to develop mathematical thinking as a yardstick for everyday living. What this means is that problem-solving can provide a learner with a context for learning Mathematics and enhancing transfer of knowledge to new situations in everyday life. Once a learner has been empowered with problem-solving skills, s/he can apply them to a variety of novel situations. Therefore, RME and problem- solving are so much inter-connected such that they both emphasize on a learner pre-existing knowledge for meaningful learning to occur.
In sum, though RME has been part of the Mathematics education research filed for a long time but it can shed some ideas on how Mathematics teachers can enable meaningful learning within the framework of learner-centred practices.
According to Treffers (1987), RME is a theory in Mathematics education that stresses the idea of connecting Mathematics to learners’ meaningful contexts (horizontal mathematization) and connecting Mathematical matter to a higher level (vertical mathematization). Both RME’s horizontal and vertical mathematizations make strong emphasis on linking Mathematics to what the learner already knows (prior knowledge). Furthermore, the concept of RME
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resonates with the notion of Poyla’s (1945) problem-solving and both of which recognize the value of learners’ pre-existing knowledge during the process of learning.