# Realistic Mathematics Education; A Solution to Prevent Linearity Illusion

## Introduction

Linearity illusion, 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. They 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 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.

Realistic mathematics education (RME) has an 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). 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 RME prevent the students from linearity illusion?

## Analysis

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 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 linear after acquiring the concept of linear relation. Furthermore, 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 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, giving the students chance to reinvent the idea about linearity, and also non linear later, using contextual problems that can be imagined will certainly 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). In his research aimed at breaking the illusion of linearity by a new manipulation of the experimental context, the students who were introduced in the real problem context with the concrete materials have tendency to successfully avoid the linear error. Giving problem to get the exact number of tiles to cover the floor of the doll house, five pairs of seven pairs involved in his study correctly solve the problem that mostly leads students to the linearity illusion.

The RME approach will make students aware about the linearity. Furthermore, after promoted idea about non linear relation, they will be able to make distinction and 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

RME as an approach in learning mathematics in the classroom seems to be a very reasonable way to prevent early students from falling to the linearity illusion. Besides giving student chance to experience mathematical big ideas, in this case linearity, in such a meaningful way, it also enhances students’ reasoning. Hence they will not improperly use availability and representativeness heuristic to judge every relation between two quantities is linier.

## Sources

De Bock, D., Van Dooren, W., Janssens, D. & Verschaffel, L. (2002) Improper Use of Linear Reasoning: An In-depth Study of The Nature and The Irresistibility of Secondary School Students’ Errors. *Educational Studies in Mathematics*, 50(2002), 311-334.

De Bock, D., Van Dooren, W., Van Parijs, K., & Verschaffel, L. (2004). Overcoming Students’ Illusion of Linearity: The Effect of Performance Tasks. *Proceeding of the 28th International Conference of the International Group for the Psychology of Mathematics Education*. Bergen: PME.

Freudenthal, H. (2004). *Revisiting Mathematics Education, China Lectures*. New York: Kluwer.

Gleitman, H., Reisberg, D., & Gross, J. (2007) Chapter 8: Thinking. In: Gleitman, H., Reisberg, D., & Gross, J. *Psychology* (7^{th} ed., pp. 270-311). New York: Norton.

Van Dooren, W., De Bock, D., Hessels, A., Janssens, D., & Verschaffel, L. (2004) Remedying Secondary School Students’ Illusion Of Linearity: A Teaching Experiment Aiming At Conceptual Change. *Learning and Instruction*, 14(2004), 485-501.

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