This essay is a critical analysis of Realistic Mathematics Education (RM nontraditional way using real life context to make a proper concept in students' mind for our topic. Simultaneous linear equation is also a part of National Curriculum of England for mathematics at key stage 3 (KS3) but normally schools start this topic in year 8 and more formal manipulation of simultaneous equation in year 9. But I have decided to start it in year 7, so that students have enough time to develop algebraic thinking properly. Our aim is to familiarize this topic in such a way that students feel comfort during solving the system of linear equations .So I have decided to employ unit Comparing quantities examples from MiC textbook
Gender of the participant:
I have targeted my 15 students in the class of mixed gender and majority of them are boys and there are only 5 girls in the class. As Forgasz (2006) argues that girls are better academically over boys in mathematics and Arnold (1997) states that "there may be differences in the development of the function of brain hemispheres, and that language development follows a biological maturation 'timetable', where girls have a faster rate of progress than boys." (Library, 2001, p.1) The same case is here in our mathematics class, in all formative and summative assessments the achieved results of girls are always better than boys. Since the number of students is few in my class therefore I can easily focus on every student to achieve my targets.
Social background of the participant:
All students are belonging to entirely different cultural backgrounds, hailing from Somalia, India, Pakistan, China, Morocco and Great Britain. So I can say that my class represents multicultural group. Half of the Somalian students have just migrated from their country so they have severe language problem and mostly rest of the students also face the same problem because English is not their home language. Most of my students belong to the working class families and some of them are living in councils' houses. So their social background is not strong. Vast opportunities of learning are not available to them. They can't access the facilities like tutoring.
Algebra and its background
â€¢ Mason (1996) defines algebra and states that
"The word algebra is derived from the problems of al-jabr (literally, adding or multiplying both sides of an equation by the same thing in order to eliminate negative/fractional terms), which were paralleled by problems of al-muqabala (subtracting the same thing from or dividing the same thing into both sides)." (Wessels, 2009, p.16)
Teaching algebra seems to have been a coaching problem since Ancient Times (Radford, 2000). Many researches are in process to make the algebra easy to understand for the pupils but still algebra is considered a hard subject among the students
Traditional way of teaching and learning algebra has been a big factor in demise of its popularity. Lack of involvement of students during the class is a common picture. It has been observed that it has reached such heights that students dislike algebra. In worst scenario they even are scared of it. Bertrand Russell writes in his autobiography
"The beginning of Algebra I found far more difficult, perhaps as a result of bad teaching. I was made to learn by heart: The square of the sum of two numbers is equal to the sum of the squares increased by twice their product. I had not the vaguest idea what this meant, and when I could not remember the words, my tutor threw the book at my head, which did not stimulate my intellect in any way". (Kooij, 2001, p.1)
At present the situation is not different, still the students are passing out their GCSE without proper understanding of algebraic tools. In the result they have totally wrong perception of algebraic tools in the rest of their lives and they are become the part of formal traditional system. Algebra, which is a representative mathematical language, must be graspable for students as it is of vital importance. Being capable of solving algebra can lead the students to manage learning advanced. Thinking logically and analyzing accurately is a skill that a good mathematician can develop easily. Mathematics teaching should promote pupils' ability to solve problems in many spheres .The algebra comprises of following main divisions that are variables, relations, function, equations and in equations (inequalities), and graphs
The introduction of system of equations?
"Equations are mathematical statements that indicate equality between two expressions." (Wessels, 2009, p.18).System of linear equations means more than one linear equations having more than one variable. But here our target is not solving the equations; our target is only to develop system of equations through comparing quantities and relate them through real life context. So that students can feel comfort while manipulating system of equation. It has been observed by the researches that majority of the middle schools students face problems in understanding and solving the algebraic linear equations (Eric J. Knuth & Ana C. Stephens's et al., 2006).
Misconception of Equal sign:
Before introducing the simultaneous equations there are some issues and problems which need to be discussed. The first and very serious problem in year 7 students in misconception of equal sign '='.Since in National Curriculum of Mathematics algebra starts at year 7 and at this stage students have no proper concept of algebraic symbols and the common problem which are found in the investigation is the wrong perception of equal sign '=', As Stephens et al. (2006) also argues that "many of the difficulties that students have when working with symbolic expression and equations may be attributed to their misconceptions about the meaning of the equal sign." (p.299). Mexicia (2008) also states that a common mistake among student is misunderstanding the equal sign as a signal for doing something, instead of taking that sign as an operator. Generally for students, the equal sign flashes a button in their minds to start working on something rather than its own entity as a "symbol of equivalence" or "quantity sameness".
Alibali et al. (1999) argue that "many elementary and middle school students demonstrate inadequate understanding of the meaning of the equal sign, frequently viewing the symbols as an announcement of the result of an arithmetic operation rather than as a symbol of mathematical equivalence." (Stephens et al., 2006, p.298). He further explains by give an example from six grade students' i.e. 8 + 4 =? +5 (Falkner et al. 1999) and says that it was found by Falkner et al. that "many students provided answers of 12, 17, or 12 and 17 - answers that are consistent with an understanding of equal sign as announcing a result, students added all the numbers in the equation or added all the number before the equal sign, again indicating an operational view of the equal sign" (Stephens et al.,2006, p.298).
Utilization in daily life
The real purpose of algebra teaching is to facilitate the students to use algebraic symbols or tools to help in solving the real life problems (Wessels, 2009). So it is necessary for the students that they have pilot knowledge of all algebraic operations before manipulating the system of linear equations. Majority of the students at this level have problems to manipulate the system of linear equations. "Graeber & Tanenhaus (1993) says that "It has been found from the numerous studies that when students are working on making the algebraic equation they do not realize the situation modeled in the problem when determining which operation to apply, so they select the operation by guessing, by trying all operation and choosing one that gives what seems to them a reasonable answer, or by studying such properties as the size of the numbers involved." (Schifter, 1999, p.63)
Realistic Mathematics Education:
The development of the RME (Realistic mathematics education) evolved after thirty years of developmental research in teaching and learning mathematics in the Netherlands and is primarily based on Freudenthal's interpretation of mathematics as a human activity (Freudenthal, 1973)
Realistic Mathematics Education (RME) was the Dutch effort to change and reform the teaching of Mathematics all around the world in 1970s. It was developed by using extensive research and mainly applies the ideas of three mathematicians Freudenthal , Treffers and Gravemeijer (Autumn, 2003).This theory was developed by Wiskos project in Netherlands. The contemporary structure of RME is mostly determined by Freudenthals views on mathematics.
Two of its major points are
Mathematics must be connected to reality
Mathematics should be perceived as a human activity. (Zulkardi, 2002).
Freudenthal views on the method of mathematization were that pupils should be involved in guided reinvention of mathematics which in other words means that the pupils should develop their mathematical thinking themselves same as mathematicians did before them, beginning with informal strategies and then gradually moving to more formal strategies, so RME context is used to help students to understand Mathematics which is in keeping with Freudenthals theory about mathematics that it must be close to the real life, moreover the purpose of making it valuable for children, mathematics should be related to the children and ultimately be reflecting the society (Carol Marshall, 2003). Later on, Treffers (1978, 1987) gave the idea of two types of mathematization in an educational context and differentiated "horizontal" and "vertical" mathematization (Zulkardi, 2002) Freudenthal (1991) divides the horizontal and vertical mathematization and describes them as
"Horizontal mathematization leads from the world of life to the world of symbols. In the world of life one lives, acts (and suffers); in other one symbols are shaped, reshaped, and manipulated, mechanically, comprehendingly, reflectingly: this is vertical mathematization. The world of life is what is experienced as reality (in the sense I used the word before), as is symbol world with regard to abstraction". (Fauzan, 2002, p.39)
Principles and Characteristics of RME
The characteristics of RME are historically related to three Van Hiele's levels for of learning
Mathematics (de Lange, 1996). Here it is assumed that the process of learning proceeds
through three levels:
A pupil reaches the first level of thinking as soon as he can manipulate the known
characteristics of a pattern that is familiar to him/her;
As soon as he/she learns to manipulate the interrelatedness of the characteristics he/she will have reached the second level;
He/she will reach the third level of thinking when he/she starts manipulating the intrinsic characteristics of relations.
Traditional instruction is inclined to start at the second or third level, while realistic approach starts from first level. Furthermore, by the guided reinvention principle and progressive mathematization students are guided didactically and efficiently from one level to another level of thinking through mathematization. These two principles and the concept of self developed models (Gravemeijer, 1994) can be used as design principles in the domain specific instruction theory of mathematics education.
Characteristics or Tenets of RME
By summarizing three Van Hiele's levels and Treffer's modern mathematization and Freudenthal's didactical phenomenology, Zulkardi(2002) describes five fundamental features of Realistic Mathematics Education.
a) The use of contexts.
b)The use of models.
c) The use of students' own productions and constructions.
d) The interactive character of the teaching process
e) The intertwinement of various learning strands. (Zulkardi 1999)
RME'S KEY PRINCIPLES
According to Gravemeijer (1994) for instructional design there are three key principles of RME namely
â€¢ guided reinvention through progressive mathematization,
â€¢ Didactical phenomenology
â€¢ Self developed models or emergent models.
Guided reinvention through progressive mathematization:
According to the first principal of RME ,pupils should be given the chance to practice the same process by which mathematics was invented and a learning route has to be mapped out that help the pupils to find the intended mathematics by themselves and the reinvention of mathematical structure creates when the students start to use the daily routine language to make contextual problems into well organized and formal mathematics shape. (Armanto, 2002)
'Mathematizing' is a main movement in RME and this activity mainly involves generalizing and formalizing (Gravemeijer, 1994). Treffers(1987) defines the process of mathematization and states that formalizing includes modelling, symbolizing, schematizing and defining, and generalizing is to understand in a reflective sense and by solving the contextual problems in realistic approach students learn to mathematize contextual problems. (Fauzan, 2002)
De Lange (1996) described the process of conceptual mathematization in RME and states that the process of developing mathematical concepts and ideas starts from the real world, and at the end we need to reflect the solution back to the real world and in other words in mathematics education things are taken from real world , are methamitised and then brought back to real world. (Fauzan,2002)
Freudenthal (1983) present the thought of didactical phenomenology which means that learning mathematics should be start from the real life contexts which are meaningful for the pupils and it will motivate the students in learning processes. (Fauzan, 2002).But real life does not mean that only context through reality;
Self developed models:
The third key principle by Gravemeijer (1994) for instructional design in RME is self developed models.This principle plays an important role in making a connection between informal knowledge and formal knowledge. According to it we have to give the opportunity to the students to use and develop their own models when they are solving the problems. To begin with the students will develop a model which is familiar to them. After the process of generalizing and formalizing, the model gradually becomes an entity on its own(Armanto, 2002).
Gravemeijer (1994) describes "this process a transition from model-of to model-for. After the transition, the model may be used as a model for mathematical reasoning" (Fauzan, 2002, p.42)
Misconception of 'realistic'
There is a misconception about RME that its context are always from real life, it is not necessary always; it might be some imaginary stories and ideas to draw the concentration of the students towards the contextual problem. Hadi, (2002) defines that real world is the world which is outside the mathematics, it may be your school, your class or house or anything outside the mathematics. Gravemeijer (1999) explains the meanings of reality and states that
'The use of the label 'realistic' refers to a foundation of mathematical knowledge in
situations that are experientially real to the students. Context problems in RME do
not necessarily have to deal with authentic every-day life situations. What is central, is
that the context in which a problem is situated is experientially real to students in that
they can immediately act intelligently within this context. Of course the goal is that
eventually mathematics itself can constitute experientially real context for the students.'(Fauzan, 2002, p.42)
Realistic mathematics education is not only associate mathematics with real life , it has proper rules and principles under which we have to teach students.
According to Feudenthal (1973) students should not be passive receiver; they should be a part of learning through contextualizing process for their cognitive development. "In RME 'real world' is the starting point for the development of mathematics concept and ideas" (Hadi, 2002, p.32). Learning mathematics means doing mathematics, of which solving everyday life problems (contextual problems) is an essential part (Armanto, 2002). Freudenthal (1971) explain this activity which is performed in RME is:
An activity of solving problems, of looking for problems, and also an activity of organizing a subject matter. This can be a matter from reality, which has to be organized according to mathematical patterns if they have to be solved. It can also be a mathematical matter, new or old results, of your own or others, which have to be organized according to new ideas, to be better understood, in a broader context, or by an axiomatic approach. (Fauzan, 2002, p.34)
The concept of equal sign in RME:
As we have discussed that one of the issue related to algebraic system of linear equations is the misunderstanding of equal sign. Nicole M. McNeil (2006) also argues that clear concept of equal sign is necessary for understanding the algebraic linear equation moreover it has been proved by the researches that if students have proper understanding of equal sign then they have more possibility to solve system of linear equations correctly. The misunderstanding of the students can be resolved by using RME, because as Zulkardi (2002) states that the first characteristic of RME is the use of context so it can be possible that students can clear the concept of equal sign through context from real life, Also Alibali et al.(2005) emphasis on the educator to give more "attention to the contexts and formats in which they are presenting problems because small differences in how problems are presented can influence what students come to understand about the associated concepts."( NICOLE M. MCNEIL et al. , 2006, p.383). Furthermore Nicole M. (2006) put more stress on the teachers' professional development and change in mathematical curricula and improving the teaching methods to convey the proper concept of equality sign in algebraic equations from contexts.
To overcome this problem some curriculums "try to offer students some help to develop the algebraic ideas and to acquire and make sense of signs. For example, in the new Ontario Curriculum of Mathematics (Ministry of Education and Training 1997), students are introduced to a kind of 'transitional' language prior to the standard alphanumeric- based algebraic language and are asked to find the value of ' * ' in equations like: * + * + 2 = 8 or the value of ' â-¡ ' in equations like 32 +â-¡ + â-¡ = 54. This 'transitional language' approach, as any pedagogical approach for teaching algebra, relies on specific conceptions about what signs represent and the way in which the meaning of signs is elaborated by the students".(Radford, 2000, p 239)
The concept of basic arithmetic operations in variables and constants in RME:
During solving the system of algebraic linear equations students have some misconception about the basic arithmetic operations in constant and variable .it can remove by suing RME technique because the basic thing in RME is the context through real life. During RME teaching the teacher or instructor can use the context and easily solve this misconception.
Since in RME the first step is horizontal mathematization, and to achieve the first target the educator needs to make a realistic environment because according to Freudenthal's didactical phenomenology realistic situation is very important to start the development of mathematical concepts. Realistic situation can be develop by drawing pictures on the white board or making some stories or physically select some students to act. (Heeley, 2008) demonstrate the students by an excellent way to differentiate between variable and numbers and their addition. He puts some numbers and some bananas in a jar and invites a boy to find the variable and add them similarly the numbers, he can easily find out the bananas from the jar and separately add them, so this is the way to clear the concept of students in variable and numeric numbers' confusion. There are a lot of other methods to get the attention of students in the class and enhance the student's ability towards mathematics and according to Freudenthal (1973) students should be the active receivers not passives like other traditional classes .This is the best way to increase the cognitive level in the students learning . Kozulin (2003) also in the favor of using symbolic tools in learning and states that "symbolic tools have a rich educational potential, but they remain ineffective if there is no human mediator to facilitate their appropriation by the learner" (p.35) .So teacher's professional development is very necessary in teaching and learning through RME and it will be discuss later.
The role of RME for solving systems of equation:
As we have discussed above that real context enhance the student's ability and the learning aims of this essay is is how to create system of algebraic linear equation with informal and pre-formal methods using the scheme RME.
Corel (2003) present some examples of exchanging goods to develop the concept of equality and how to use system of equations in students' mind .Corel explain her personal experience of using RME and finds a positive result in students' progress. She fulfills the first principal of RME and create a context through real life by using an imaginary story in picture form and presents it in front of students. In the example it is given that a farmer has two sheep and one goat. He wants to sell them and buy some bags of wheat but when he goes to the market he finds that he can exchange his one goat by six chickens and same for one sheep and one bag of salt is equal to two chickens. Similarly three bags of salt can be exchanged by two bags of wheat.
According to RME rules after creating the realistic context, convert the problem in mathematical symbols which is called horizontal mathematization. At this step students should have some basic knowledge of the problem. For example here they should have some knowledge of equality sign and variables (as we have already discussed these issues). So first she introduce equal sign in algebra and named those things in algebraic notation say c for chicken , s for a bag of salt , h for sheep ,g for goat and w for wheat. So according to the question
1g = 6c & 1h = 6c & 2c = 1s & 3s = 2 w
These are four equations which are found by given data. Until this step students can understand how to make the simple equation.Now the next step solve the problem in mathematical symbols which is called verticle mathematization. At this step students need some supervision of any instractor or teacher .
So altogather the farmer can exchage his goat and sheep by 18 chicken.For 18 chicken we need to multiply both sides of third equation by 9 , because this is the rule of algebraic equation if you multiply any number on one side of equality you have to multiply the same number on the other side as well . so the third equation will become
9 * 2c = 9 * 1s
18 c = 9 s
As he has now 18 chicken so they are equal to 9 bags of salt .Now multiply fourth equation by 3 to get the answer i.e.
3s * 3 = 2w * 3
9 s = 6 w
It means the farmer can buy six bags of wheat when he exchanges his one goat and two sheep.
In this above example the students can learn three things in basic algebraic system of equation. The first thing is to understand the concept of equal sign and as second how to convert the real context in to algebraic term (like variables) and third how to solve the system of equation to find the required result.
The Difference between realistic approaches:
The realistic approach Vs the mechanistic approach
In the conventional (mechanistic) style of the learning process the teachers take control over each activity. In contrast, the RME approach suggests that the pupils are supposed to take responsibility for their own mathematical learning and they should be actively engaged in interactive discussion in the classroom. Guided by the teacher, the pupils reinvent informal and formal mathematics models in a process of mathematizing contextual problems (Armanto, 2002).
According to Heuvel- Panhuizen (1998) states that In the formal method of teaching mathematics, the learning subject matter is divided into insignificant small parts and the pupils are asked to solve these problems by fixed procedures and are trained by exercises, frequently to be done individually. This formal approach to mathematics education, which also rejects the mechanistic, is indeed a contrast to RME approach. RME has a more complex and meaningful conceptualization of learning. Active participation is expressed by students in teaching-learning process, in which they established mathematical tools and understanding rather than being the receivers of ready-made mathematics.
In this view mathematics education would be highly interactive in which the teachers would have to build upon the ideas of the students. It means they have to react based on what the students bring to the fore (Kooj, 1999).
There is only one step more in RME from other formal methods of teaching mathematics and this is context from real life. It is often observe that in formal teaching methods the teacher straight away start teaching mathematical problems without any introduction from context of real world as freudenthal (1973) declares mathematics as human activity and should be connected to real life ,so every problem in mathematics should be related to the real life and before starting the questions the educator should be create the environment from context of real life and then present it in front of students which in unfortunately not happen in our formal methods of teaching . "The jump to the formal level is made too quickly, and there is almost no time for students to develop their own schemes. The traditional algebra course is seen as sterile, disconnected from other mathematics and the real world" (Reeuwijk, 1995, p.1)
Difference between word problem and RME:
When we are talking about RME, the first questions comes in our mind about RME is this same like word problem? Hade (2002) describes that in solving word problem it is necessary for the students to follow the precise rules and regulation, moreover students apply the symbols without reflecting the specific context
The contextual problems in RME help in concept formation by providing the students an incentive and drive them with a natural approach towards understanding mathematics. These also provide a fundamental principle and foundation for learning the mathematical procedures, symbols, and rules linked with the other models which act as important aid for thinking.
Solving system of equations through Comparing Quantities:
Our target is to discuss the strategies for learning system of algebraic linear equation and to achieve our goal we are applying different schemes of ancient Greeks and Chinese's to solve system of linear equations presented by Reeuijk (1995),Autumn (2003) and Kooij (2001).
These are some basic strategies to develop algebraic thinking .They make a bridge between student's previous arithmetic's knowledge and algebra. By applying these strategies students can easily understand the tools of making system of algebraic linear equations. Reeuijk (1995) called theses strategies a tool which is used to make a link between the concrete and the abstract. All schemes are the basic informal strategies for manipulating system of linear equation from the chapter 'Comparing Quantities' of Mathematics in Context (MiC) textbook. An important aspect is that in unit Comparing quantities the ideas are usually presented as a story and thus the idea is to encourage the pupils to solve the problems in their own ways i.e. developing their own strategies. It has been noticed that by giving the problems in a natural context, students try to use the ideas that they may not have learned by their teachers at school. At this level more emphasis is on pre-formal methods and no formal algorithms are used because this unit is created for the11-12 years old students who start algebra for the first time. The more formal shape of simultaneous equations is used in next chapters of the textbook for the next classes. The scheme (Reuwijeck ,1995) mentioned four main strategies __ guess and check , reasoning , combination charts and notebook. .Reeuwijk (1995) presents an example is his paper, in which a diagram of two t-shirts and two drinks is given and their price is 44 $ .Similarly in another diagram the price of 30$ is given for one T-shirt and three soda water. Students are asked to find the price of one soda bottle and one T-shirt. This question is looks like a typical word problem but as we discuss above that the difference between simple word problem and RME is the realistic context and here the context is in the form of pictures. The 'guess and improve' strategy is commonly Figure 1. Structure of Comparing Quantities.
used by students who have just basic know-how of the subject. Mostly low cognitive level students attempted this question by using this strategy.
The second strategy is reasoning through exchanging, this scheme was used by ancient Chinese .It is same like barter system , Students took more interest in this strategy .The other two schemes are notebook notations and combination charts . These scheme mostly used by the students having high cognitive level. All above schemes are horizontal mathematization. According to Gravemeijer (1994) realistic environment is very important in the progression of mathematical concept. First, there are many problems in the world that are required to be solved ,later pupils apply their mathematical tools to resolve them realistically(Reeuwijk, 1995)
Reeuwijk(1995) states that once students endeavor to solve the problems by their own informal ways, it was summarized together (see fig.1) and students realize that all schemes are interrelated .Now the next step in vertical mathematization.
Our aim is to change the traditional concept of algebra and use context from real life to make the proper concept of mathematics in students' mind. Our dilemma is that algebra starts at KS3 in year 7, so at this stage if we provide the proper understanding to the students then they will never consider algebraic equations as a hard topic. Reeuwijk (1995) present the above methods to create the sense of simultaneous equations in the early stage of about 11 years old. He uses these strategies and got a very positive result.
According to Reeuwijk (1995) if we leave the students to allow them making their own ways to solve the problem with their previous knowledge of mathematics then students will use their mind more and becomes the active participants of the
class and can easily understand algebraic term. Moreover "the non-statutory guidance to the National Curriculum states that pupils at every stage should be encouraged to develop their own methods and pupils need to have opportunities to develop informal, personal methods of recording and an approach in which pupils are required to use and apply their developing knowledge and skills leads to more effective learning"( Antumn,2003,weblink) as Freudenthal (1973) declares that the students should be an active receiver in the class , so in this way students will become an active participant in the solving any problem .Moreover this thing will helpful in their real life problem solving. "By discussing and reflecting on the students' use of strategies, they start realizing that formal strategies of a more general character exist." (Reeuwijk, 1995, p.3)
In RME a context plays the important role, by using context students engage by using contextual problems the instruction is directed to the process of reinvention mathematical concepts through horizontal and vertical mathematization
Drawback of RME
There are some issues which are raised when we are working with RME.
RME is time consuming scheme:
The first is that when we are teaching by using RME scheme it will take a long time since the class timing is just an hour and when a teacher plan to teach this method it is certainly impossible to understand all the technicalities in a fixed period of time
Teaching this method once or twice a week will also not help the student along. The lessons should be precise and be given on daily basis. The teachers should make sure that the students are implying them practically as well.
â€¢Professionally trained teachers are required to teach RME
â€¢Are the students having language problem? Can they understand the stories or pictures? The students having language problem should let their teacher know about it so they can guide them at the right time. The English use in mathematics is simple, clear and concise
Differences in the learning and teaching activities made teachers reconsider their former personal thinking and beliefs about teaching and learning mathematics. At the one hand they believed that the RME approach gave more opportunities for students to learn different strategies, but on the other hand
It seemed that the teachers were afraid to lose their respects as teachers. It is true that the RME approach is not the only answer for teaching all subjects in the curriculum. However, the RME approach will guide teachers toward a new meaning of teaching in general terms. Teaching is not for the test only, but should build pupils' understanding and knowledge of the subjects (cognitive domains). It also gives pupils opportunities to develop their attitude toward learning activities (affective domains). These domains are remarkably essential to develop pupils' attitude toward gaining knowledge and learning new techniques. But the Netherlands experience shows us that changing from conventional to more realistic practice will be a long and difficult process (de Lange, 1996).