# Realistic Mathematics Education; a solution to prevent the linearity illusion (Extended)

## Introduction

Linearity illusion, a tendency to see any relation between two quantities as a proportional or linear one, is a phenomenon in mathematics education, especially among students in lower education level. Students tend to improperly utilize the linear model in solving word problems related to lengths, areas, and volumes of similar planes, figures, and solids (De Bock et al, 2002). The best known example of this misconception (Van Dooren et al, 2004), originated from what is called “synthetic model of linearity”, is that if a geometrical shape is enlarged k times, its area or volume will also become k times larger too.

In some extend, the linearity illusion can be considered as an obstacle in promoting the mathematical literacy since it provides some restriction to the students in solving ‘real’ problems related to mathematics. Referring to de Lange (2006), mathematical literacy is about dealing with ‘real’ problems. That means that these problems are not ‘purely’ mathematical but are placed in some kind of a ‘situation’. In short, the students have to ‘solve’ a real world problem requiring the use of the skills and competencies they have acquired trough schooling and life experiences. A fundamental role in that process is referred as ‘mathematization’.

Realistic mathematics education (RME) is based on the idea that education should give students the “guided” opportunity to “re-invent” mathematics by doing it. It means that in mathematics education, the focal point should not be on mathematics as a closed system but on the activity, on the process of mathematization (Freudhental, 2002). Given such an opportunity to experience mathematics, students will experience what is so called meaningful learning and according to the conceptual change theory (Vosniadou, 2003), it is a source of motivation for the students. In the light of it, RME seems to be a promising approach to prevent student falling to the linearity illusion. Furthermore, RME should be started in primary school since it might be more appropriate to intervene much earlier in the students’ school career in order to prevent, rather than remedy, the illusion of linearity (Van Dooren et al, 2004).

## Main Question

In this essay, I would like to answer the following question: How can the elaboration of RME prevent the students from the linearity illusion?

## Addressing the linearity illusion by RME

The linearity illusion, as a misconception, mostly occurs because students use availability heuristic, based on how easily the case comes to mind, and representative heuristic, assuming each case is homogenous, (Gleitmen et al, 2007), as their strategy when facing problems related to relation between two quantities. It is supported by the fact that linear functions immediately appear in the students’ mind due to their simplicity (Rouche in De Bock et al, 2007). In addition, most students are not immediately given the idea of non linearity after acquiring the concept of linear relation. Hence, they do not have sufficient understanding in the proportion which is caused by not given opportunity reinventing the concept and learning from the contextual problem.

One of RME characteristics is the use of contexts in phenomenological exploration (Freudhental, 2002). It provides a theory that the starting point of mathematics instruction should be experientially real to the student which allows them to become immediately engaged in the contextual situation. In this fashion, the students are given the chance to reinvent the idea about linearity, and also non linearity later, when using contextual problems that they can imagine. This certainly will help them to cope with this concept. While reinventing the concept, students will be guided by the teacher to construct the understanding by themselves, so the students will experience a meaningful learning activity in which they struggle and choose their own way to solve the problem and acquire deeper insight.

The advantage of using contextual problem to overcome the linearity illusion was also proved by De Bock et al (2004). The students who were introduced in the real problem context with the concrete materials have the tendency to successfully avoid the linear error. Given problem to get the exact number of tiles to cover the floor of the doll house, five pairs out of seven pairs involved in his study correctly solved the problem. In addition, this problem is used since it has the tendency to lead students into the linearity illusion. Students tend to think that the number of tiles used to cover the floor will become k times larger when the floor’s width and height are enlarged k times.

In this essay, the goal of the use RME approach in teaching is to prevent students from deep-rooted tendency to give linear responses in non-linear situations, more specifically in the context of the relationship between the linear measures of a figure and its perimeter, area, and volume. For this goal, a series of lessons, including all teacher and learner materials, need to be developed for 5th graders since in this grade students, for the first time, deal with relation problem.

**RME Principles**

There are several instructional design principles for developing a learning environment with the above-mentioned goal, which is derived from the RME approach. These principles are guided reinvention, didactical phenomenology, and self-developed model (Gravemeijer, 1994).

Guided Reinvention

In the guided reinvention principle, the students should be given the opportunity to experience a process similar to that by which mathematics was invented (Gravemeijer, 1994). Regarding to this principle, a learning route has to be figured out such that allow the students to find the intended mathematics by themselves. The learning route should be emphasized on the nature of the learning process, rather than on inventing mathematics concepts or results. It means that students must be given the chance to gain the knowledge so that it becomes their own private knowledge, knowledge in which they are responsible. Hence, it implies that in the teaching learning process, students should be given the opportunity to construct their own mathematical knowledge on the basis of such a learning process.

Didactical phenomenology

Freudenthal (2002) advocates the didactical phenomenology. It asserts that learning mathematics activity must be started from phenomena or contexts that are meaningful for the students and stimulate the learning processes. In didactical phenomenology, situations where a given mathematical concept is applied are to be investigated for two reasons (Gravemeijer, 1994). The first reason is to reveal the kind of applications that have to be anticipated in the instruction. The second one is to consider their suitability as points of impact for a process of mathematization.

The use of imaginable or real context also has another advantage. It can serve as a source of motivation for the students in learning some concepts since it potentially stimulate students’ interest and curiosity (Vosniadou, 2003). Motivation itself can be understood as an internal state that arouses, directs and maintains behavior (Woolfolk, 2008).

Self-developed Model

The third principle is self-developed model. It plays a prominent role in connecting students’ informal knowledge and formal knowledge. Students need to be given the opportunity to use and develop their own model when solving mathematical problems. At the beginning, students will develop model that familiar to them. Their models, in this stage, are mostly the representation of the action or the situation (Dolk & Fosnot, 2001). After generalizing and formalizing the model of situation, students will get into model for mathematical reasoning. This process is referred as a transition from *model-of *to* model-for *(Gravemeijer, 1994)*.*

Using these three principles, it will be given an example of a series of lessons utilizing RME approach that can be used to prevent the linearity illusion in 5^{th} grade students.

**An example of RME lesson**

Here is given an example of a lessons series with RME approach that can be used to prevent the linearity illusion. Teacher can choose doll house problem as the context was used by De Bock et al (2004). At the beginning of the lesson, students are introduced to the linear proportion problem before finally given juxtaposing problem including linear and quadratic proportion. The idea of juxtaposing problem gives students the opportunity to make comparison and draw a conclusion between two different concepts (Dolk & Fosnot, 2001), in this case linear and non linear proportions.

Given sketches of doll house floor with its squares tiles, students are told to determine the perimeter of the floor. This problem is a good context since it is experienced real by the students and promotes mathematizing. In order to get the perimeter, students are asked to consider the number of tiles representing the floor’s width and height. For example, the sketch shows the floor has 2 tiles as its height and 3 tiles as its width. Using this information and their self-developed models, students are expected to find 10 as the perimeter of the floor. Next, students are told to complete a table including width, height, and perimeter of the doll house. The table guides students to find perimeter of floors with width and height 2 and 3, 4 and 6, 6 and 9, and 8 and 12 respectively.

In doing such an activity, students, working in group of 3, are encouraged to use their own models of the given situation to solve the problem, finding the perimeter of the floor. They can use playing square tiles provided by the teacher to form the floor asked. Another possibility is they make drawings of the floor or only the edges of the floor. The more advance students only need to calculate numbers representing floor’s width and height to get the perimeter. Giving students chance to do modeling is based on the fact that modeling activity involves a complex, cyclical process consisting of a number of subsequent steps: understanding the situation described, selecting the elements and relations in this situation that are relevant, building a mathematical model and working through it, interpreting the outcome of the computational work in terms of the practical situation, and evaluating the results and the applied model itself (Verschaffel et al., 2000).

After completing the table, students are asked to find the relation between enlargement in the width and height of the floor and enlargement in the floor’s perimeter. Based on the information they already write in the table, they are expected to grasp the idea that whenever floor’s width and height are enlarged k times, its perimeter will also become k times larger. Students, who have not got this kind of relation, are supposed to get it on math congress session where some groups of students present their results and ideas and question one another’s solutions. The best way for teacher to establish an inquiry approach during whole-group discussion is giving the students opportunity to talk one another about their works that leads them into productive discussions around mathematical ideas (Kline, 2008).

When the students have already succeeded in perceiving the relation between the enlargement of floor’s width and height and the enlargement of floor’s perimeter, teacher will introduce the new term, linear, as the type of the relation or proportion to the students. Teacher will emphasize that the relation they just established is a linear relation since one quantity become k times larger when another is enlarged k times.

Juxtaposing problems is what teacher should do next. Still considering the result of the first activity, students are given the next problem that is intended to promote the idea of non linearity with juxtaposing the linear growth of perimeter and quadratic growth of area (Van Dooren et al, 2004). In this activity, students, again, work with doll house problem, but with a modification. Students are not asked about the perimeter of the floor anymore. Instead, they are told to determine the number of tiles needed to completely cover the floor if its width and height are given. This problem has the same idea with finding the area of the floor. As an example, students are asked to find the number of tiles needed if the width and height of the floor are 4 and 2, respectively.

Similar with the previous problem, in this occasion, students in groups are also told to complete the table and encouraged to develop their own models of the situation. They are freely choosing the way to solve the problem which represents their cognitive development. After completing the table they will discuss about the relation between the enlargement of floor’s width and height and the enlargement of the number tiles needed. Students are also asked to compare the relation they find in the first problem and the last one.

After discussing in small group, presenting idea or paying attention to the idea presented, and questioning one another’s solutions, students are expected to grasp the idea that the relation in the second problem is different with the previous one. In the second problem, the number of tiles needed to cover the floor will become k^{2 }times larger if the floor’s width and height are enlarged k times. It is different with the perimeter that only will become k times larger if the same thing happens. At the end, teacher will introduce the non linier term to refer the second relation. Teacher will also tell that there are a lot of non linear relations and what they just dealt with only one example of non linear relation, namely quadratic relation.

Experiencing such activities students will grasp the idea that there are more than one relation and they need to observe carefully to get the type of the relation. If it is necessary they should make the model of the situation in order to decide whether it is a linear problem or not. Teacher is suggested to give students other contexts that will provide them opportunity to more clearly understand the relations between quantities.

As the next activity, it is appropriate to promote a “harder” context that challenges students to use the knowledge they just constructed. It also can be used to asses whether they already perceive all the ideas involved in the previous lesson, so that they do not fall into the linearity illusion. One that can be tried is “Gnome problem” (Van Dooren et al, 2004). Students are told to assume that a gnome is similar to a human being, but 12 times smaller. Then, using knowledge about their own world, students are asked to answer the following questions about the world of gnomes: (1) how long is the belt of a gnome? (2) What is the area of the sole of gnome shoe? (3) How much coffee is there in a cup for gnomes? (4) How much fabric do you need to make a skirt for a woman gnome? and (5) How much water does a gnome need to take a bath? Students are also suggested to collect additional information when necessary.

Even though it seems hard at the beginning, students will easily get engaged to the situation. It is based on the fact that the problem is experienced real for them, so they will have intrinsic motivation, that comes from their interest and curiosity (Woolfolk, 2008), to work with this intriguing problem. In doing this problem, students are working in the small groups of 3 or 4 students. They will discuss and make a model of the situation given in the problem in their groups before finally presenting their works in front of the class. Of course, their answers would not be the same at all and this fact can lead them to the rich mathematical discussion. With the teacher serving as a facilitator, making sure students keep on the track, they will question one another’s solutions, give reason and justify their answers. The discussion encourages children to think more deeply (Kline, 2008) so they will construct deeper understanding related to the concept of relation and go further from the tendency to fall into the linearity illusion.

The lesson with RME approach like what exemplified above will make students meaningfully aware about linearity and non linearity. In addition, by the use of the intriguing context, students will be motivated and easily get engaged into the learning activity. Furthermore, after perceiving idea about non linear relation and working with other contexts, they will be able to make reasoning and clear distinction so that they will not fall to the linearity illusion. Since students have been promoted about non linear relation, also in a meaningful way, just after grasping the linear proportion, it will reduce the possibility of misconception that comes from improper reasoning heuristic availability and representatives. In this stage, both relation, linear and non linear, are easily come to students’ mind and of course student will not generalize that every relation is linear anymore.

## Conclusion

Applying RME as an approach in learning mathematics with appropriate context and sequence of activities in the classroom seems to be a very reasonable way to prevent young students from falling to the linearity illusion. Besides providing students with a chance to experience mathematical big ideas, in this case linearity and non linearity, in such a meaningful way, it enhances students’ reasoning. Hence they will not improperly use availability and representativeness heuristic to judge every relation between two quantities is linier. In addition it also motivates students since the use of appropriate context is also source of motivation in the classroom.

Overall, this essay has given insight how to prevent the linearity illusion in early grades students. Since the linearity illusion also happens in adults (De Bock et al, 2002), further research need to be conducted to find the solution for this problem that also has relation with mathematical literacy.

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