In this section we present five features which characterize realistic mathematics. At first we are dealing with learning in a context and second with the use of models. The third point (the mathematical subjects are not atomized but interwoven) is not of so much relevance for this book, while the three characteristics of the process of mathematization (construction, reflection, and interaction) are analysed in the following sections.
The new realistic approach to learning and thought process in children has far-reaching consequences. Mathematization is viewed as a constructive, interactive and reflective activity. To begin, the point of departure for education is not learning rules and formulas, but rather working with contexts. A context is a situation which appeals to children and which they can recognize in theory. This situation might be either fictional or real, and forces children to call upon the knowledge they have gained by experience − for example in the form of their own informal working methods − thereby making learning a meaningful activity for them, A well-chosen context can induce an active thought process in children, as the following example shows.
Let us start to give children of, say, 11 years the following formal and bare problem, not presented in a context: 6 ÷ 43. Many of them will have a great deal of trouble finding a solution (Streefland, 1991). Some will answer, for example: 42, 243 or 421. They manipulate at random with the given numbers, for instance 6 ÷ 3 = 2, so 6 ÷ 43 must be 42. This child views fractions as whole numbers and so do other students (Lesh et al., 1987). But some students will calculate that 6 × 4 = 24 and that 24 divided by 3 equals 8. It is true that the latter answer is correct, but when these children are questioned more closely, it turns out that they understand almost nothing about the operation which they themselves have just performed. They just remembered a rule they learned by heart, they know that the given solution is correct however they don’t know why.
Now, the same children are next given the following context problem which is accompanied by a picture: a patio is 6 metres long; you want to put down new bricks and the bricks you are going to use measure 75 centimetres in length (43 of a metre). How many bricks will you need for the length? This problem is the same as the previous one, but it has now been presented within a context, a picture of a patio and the bricks to put down. This presentation elicits a child’s own, informal approach: measuring out. This approach provides insight into the problem, something which the symbolic form (6 ÷ 43) did not do. Some students even manipulated and took the measure in reality, this means they measured out step by step 75 centimetres and after 8 steps they counted 6 metres. So the answer must be ‘eight’, they concluded. This example demonstrates that working with contexts − which, if carefully constructed, can be considered paradigmatic examples − form the basis for subsequent abstractions and for conceptualization. That is because thinking must achieve a higher, abstract level and at that level these particular contexts no longer serve a purpose. That is not to say that a process of decontextualisation occurs, but rather recontextualisation. The children continue to work with contexts, but these contexts become increasingly formal in nature; they become mathematical contexts. Their connection with the original context, however, remains clear. The process by which mathematical thinking becomes increasingly formal is called the process of progressive mathematization. Contexts, thus, have various functions. They may refer to all kind of situations and to fantasy situations (Van den Heuvel-Panhuizen, 1996). It is important that the context offer support for motivation as well as reflection. A context should indicate certain relevant actions (to take measures in the example above), provide information which can be used to find a solution-strategy and/or a thinking-model.
Of course, leaving the construction to the students does not guarantee the development of successful strategies. However it guarantees that students get the opportunity to practice mathematician’s thinking and problem solving processes. Strategies are tried, tested and elaborated in various situations.
In the previous discussion we have not argued that a student presented with ‘bare’ numerical tasks (like 6 ÷ 43) will necessarily fail to solve the problem. Hence we were not suggesting either that students who are given context problems will necessarily produce the right solution. In recent research there is found a strong tendency of children to react to context problems (‘word problems’) with disregard for the reality of the situations of these problems. Let us give two examples of items used in research (Greer, 1997; Verschaffel et al., 1997):
− ‘An athlete’s best time to run a mile is 4 minutes and 7 seconds. About how long would it take him to run 3 miles?’
− ‘Steve has bought 4 planks of 2.5 metre each. How many planks of 1 metre can he get out of these planks?’
In four studies, discussed by Greer (1997), the percentage of the number of students demonstrating any indication of taking account of realistic constraints is: 6%, 2%, 0% and 3%. The student’s predominating tendency to apply rules clearly formed an impediment to thoroughly understanding the situation.
Verschaffel et al. (1997) confronted a group of 332 students (teachers in training) with word problems and found they produced ‘realistic’ responses in only 48% of cases. Moreover the pre-service teachers considered these ‘complex and tricky word problems’ as inappropriate for (fifth grade) children. The goal of teaching word problem solving in elementary school, after their opinion, was “...learning to find the correct numerical answer to such a problem by perforn1ing the formal-arithmetic operation(s) ‘hidden’ in the problem” (Verschaffel et al., 1997, p. 357).
When solving word problems students should go beyond rote learning and mechanical exercises to apply their knowledge (Wyndhamn & Säljö, 1997). Their research showed that students (10-12 years of age) gave in most cases logically inconsistent answers. The authors interprete these findings by claiming that the students focus on the syntax of the problem rather than on the meaning. That means that the well-known rule-based relationship between symbols results in less of attention being paid to the meaning. The students follow another ‘rationality’, that is, they consider word problems as mathematical exercises “… in which a algorithm is hidden and is supposed to be identified.” (Wyndhamm & Säljö, p. 366). Hence they do not know or realize that they are expected to solve a real life problem.
Reusser and Stebler (1997) discuss another interesting research finding namely the fact that pupils ‘solved’ unsolvable problems without ‘realistic reactions’. For example:
− ‘There are 125 sheep and 5 dogs in a flock. How old is the shephard?’ (Greer, 1997).
A pupil questioned by the investigators gave as his opinion: ‘It would never have crossed my mind to ask whether this task can be solved at all’. And another pupil said: ‘Mathematical tasks can always be solved’. One of the author’s conclusions is that a change is needed from stereotyped and semantically poor, disguised equations to the design of intellectually more challenging ‘thinking stories’. What we need are better problems and better contexts. Finally, Reusser and Stebler (1997) − following Gravemeijer (1997) − give as their interpretation of the research findings that the children are acting in accordance with a typical school mathematics classroom culture.
Second, the process of mathematization is characterized by the use of models. Some examples are schemata, tables, diagrams, and visualizations. Searching for models − initially simple ones − and working with them produces the first abstractions. Children furthermore learn to apply reduction and schematization, leading to a higher level of formalization. We will demonstrate, once again this using the previous example. To begin, children are able to solve the brick problem by manipulating concrete materials. For instance, they might attempt to see how often a strip of paper measuring 43 of a metre fits in a 6-metre-long space. At the schematic level, they visualize the 6-metre-long patio and draw lines which mark out each 43 of a metre or 75 centimetres. The child adds 75 + 75 + 75... until the 6 metres have been filled The visualization looks as follows:
An example of reasoning on a formal-symbolic level is as follows: 75 centimetres fits into 3 metres 4 times. We have 6 metres, so we need 2 × 4 = 8 bricks. The formula initially tested can also be applied, but this time with insight: 41 metre fits 4 times into 1 metre, so it fits 24 times into 6 metres. But I only have 43 of a metre, so I have to divide 24 by 3, and that makes 8. At this formal level, moreover, the teacher can also explore the advantages and disadvantages of the two methods with the children.
Third, an important element of realistic mathematics instruction is that subjects and curricula (such as fractions, measurement and proportion) are interwoven and connected, whereas in the past, the subject matter was divided − and so atomized. Fourth, two other important characteristics of the process of mathematization are that it is brought about both by a child’s own constructive action and by the child’s reflections upon this action. Finally, learning mathematics is not an individual, solitary activity, but rather an interactive one.
One of the most enduring ideas concerning mathematics instruction is the following: mathematics consists of a set of indisputable rules and knowledge; this knowledge has a fixed structure and can be acquired by frequent repetition and memorization. In the past twenty-five years, far-reaching changes have taken place in mathematics instruction. More than in any other field, such changes were influenced by mathematicians who had come to view their discipline in a different light. Their observations went a long way towards stimulating a process of renewal in mathematics instruction. New consideration was given to such fundamental questions as: how might mathematics best be taught, how might children be encouraged to show more interest for mathematics, how do children actually learn mathematics, and what is the value of mathematics?
According to Goffree, Freudenthal, and Schoemaker (1981), the subject of mathematics is itself an essential element in ‘thinking’ through didactical considerations in mathematics instruction. Moreover, the notion is emphasized that knowledge is the result of a learner’s activity and efforts, rather than of the more or less passive reception of information. Mathematics is learned, so to say, on one’s own authority. From a teacher’s point of view there is a sharp distinction made between teaching and training. To know mathematics is to know why one operates in specific ways and not in others. This view on mathematics education is the basic philosophy in this chapter (Von Glazersfeld, 1991) In order to understand current trends in mathematics education, we must consider briefly the changing views on this subject.
The philosophy of science distinguishes three theories of knowledge. Confrey (1981) calls these absolutism, progressive absolutism and conceptual change. In absolutism, the growth of knowledge is seen as an accumulation, a cumulation of objective and empirically determined factual material. According to progressive absolutism a new theory may correct, absorb, and even surpass an older one. Proponents of the idea of conceptual change have defended the point of view that the growth of knowledge is characterized by fundamental (paradigmatical) changes and not by the attempt to discover absolute truths. One theory may have greater force and present a more powerful argument than another, but there are no objective, ultimate criteria for deciding that one theory is incontrovertibly more valid than another (Lakatos, 1976). Mathematics has long been considered an absolutist science. According to Confrey (1981), it is seen as the epitome of certainty, immutable truths and irrefutable methods. Once gained, mathematical knowledge lasts unto eternity; it is discovered by bright scholars who never seem to disagree, and once discovered, becomes part of the existing knowledge base.
Leading mathematicians however have now abandoned the static and absolutist theory of mathematics (Whitney, 1985). Russell (in Bishop, 1988) once explained that mathematics is the subject in which we never know what we are talking about, nor whether what we are saying is true. Today mathematics is more likely to be seen as a fluctuating product of human activity and not as a type of finished structure (Freudenthal, 1983). Mathematics instruction should reveal how historical discoveries were made. It was not (and indeed is still not) the case that the practice of mathematics consists of detecting an existing system, but rather of creating and discovering new ones. This evolving theory of mathematics also led to new ideas concerning mathematics instruction. If the essence of mathematics were irrefutable knowledge and ready-made procedures, then the primary goal of education would naturally be that children mastered this knowledge and these procedures as thoroughly as possible. In this view, the practice of mathematics consists merely of carefully and correctly applying the acquired knowledge If, however, mathematicians are seen as investigators and detectives, who analyse their own and others’ work critically, who formulate hypotheses, and who are human and therefore fallible, then mathematics instruction is placed in an entirely different light. Mathematics instruction means more than acquainting children with mathematical content, but also teaching them how mathematicians work, which methods they use and how they think. For this reason, children are allowed to think for themselves and perform their own detective work, are allowed to make errors because they can learn by their mistakes, are allowed to develop their own approach, and learn how to defend it but also to improve it whenever necessary. This all means that students learn to think about their own mathematical thinking, their strategies, their mental operations and their solutions.
Mathematics is often seen as a school subject concerned exclusively with abstract and formal knowledge. According to this view, mathematical abstractions must be taught by making them more concrete. This view has been opposed by Freudenthal (1983) among others. In his opinion, we discover mathematics by observing the concrete phenomena all around us. That is why we should base teaching on the concrete phenomena in a world familiar to children. These phenomena require the use of certain classification techniques, such as diagrams and models (for example, the number line or the abacus). We should therefore avoid confronting children with formal mathematical formulas which will only serve to discourage them, but rather base instruction on rich mathematical structures, as Freudenthal calls them, which the child will be able to recognize from its own environment. In this way mathematics becomes meaningful for children and also makes clear that children learn mathematics not by training formulas but by reflecting on their own experiences.
In the 1970s, the new view of mathematics, often referred to as mathematics as human activity, led to the rise of a new theory of mathematics instruction, usually given the designation: realistic. As it now appears, this theory is promising, but it is not the only theoretical approach in mathematics instruction; three others can be distinguished: the mechanistic, the structuralist and the empirical (Treffers, 1991).
− The mechanistic approach reflects many of the principles of the behaviouristic theory of learning; the use of repetition, exercises, mnemonics, and association comes to mind. The teacher plays a strong, central role and interaction is not seen as an essential element of the learning process. On the contrary, mathematics class focuses on conclusive standard procedure.
− According to the second approach − the structuralist − thinking is not based on the children’s experiences or on contexts, but rather on given mathematical structures. The structuralist tends to emphasize strongly the teacher’s role in the process of learning.
− The outstanding feature of the third trend − the empirical − is the idea that instruction should relate to a child’s experiences and interests. Instruction must be child-oriented. Empiricists believe that environmental factors form the most important impetus for cognitive development (Papert, 1980). Empiricists emphasize spontaneous actions.
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