The aim of this study was to examine if there is a difference in successfulness of non-linear problems solving between younger and older, male and female students and between a group of students who had an offered linear solution for non-linear problems and a group that was not offered a linear solution.

For the requirements of this study three lists of mathematical problems were constructed. Form A contained five non-linear problems, and for every problem five answers were offered. Among these five answers, one was a correct solution; one was incorrect linear solution, while the remaining solutions served to reduce the probability of guessing. Form B was identical to Form A, but a linear solution was not offered in it. Problems in the Form C were classical proportionality problems with five offered solutions.

One half of participants were asked to solve forms A and C, and the other half to solve forms B and C. A convenience sample of high school students was examined. The sample consisted of 112 first grade (N=52) and fourth grade students (N=60), 53 girls and 59 boys.

There were no differences between participants in the solving of linear problems. The older students were more successful in the solving of non-linear problems than the younger ones. The students in the group without the linear solution were more successful than those who had an offered linear solution. The interaction effect of age and solving situation showed that older students were somewhat better than younger students when a linear solution was offered, but that difference was even larger when a linear solution was not offered. The results suggest that methods of education should be re-examined if we want the students to learn different models that can be applied successfully in the school and real-life situations.

The illusion of linearity is an error that occurs in solving mathematical problems when people (wrongly) believe that when a certain length is enlarged by factor k, area and volume should also be enlarged by factor k. A correct solution would be that when a certain length is enlarged by factor k, areas are enlarged by factor k², and volumes by factor k³. Many studies point out the power of this illusion (De Bock et al, 1998, 2002; Van Dooren, 2004).

In previous studies the illusion of linearity was examined mainly with open ended problems, i.e. without any offered solutions. After the subjects answered wrongly, researchers tried to encourage them to reevaluate their answers (e.g. with diagrams or information on the other students’ answers). In addition to these procedures, it is possible to encourage the answer and verification of solution by offering the answers to the participants in advance, i.e. by using multiple choice problems. Such answers serve then as a reference frame for the verification of the solutions. If a solution that corresponds to the illusion of linearity is removed from the answers offered, a dissonance will be created, due to which the participants will have to reexamine the linear procedure of solving and then the linear image of the problem as well.

The aim of this study was to examine if there is a difference in the successfulness of non-linear problems solving between younger and older students, male and female students and between a group of students who had an offered linear solution for non-linear problems and the students who didn't have an offered linear solution.

With regard to the results of the previous studies (De Bock et al, 1998, 2002), we assumed that the older participants will be more successful in solving non-linear problems than the younger participants, that there will not be gender differences in the problems solving and that the group without the offered linear solution, due to the created impossibility of linear model application, will be more successful than the group with an offered linear solution.

For the requirements of this study three lists of mathematical problems were constructed. Form A contained five non-linear problems, and for every problem five answers were offered. Among these five answers, one was a correct solution; one was incorrect linear solution, while the remaining solutions served to reduce the probability of guessing. Form B was identical to Form A, but a linear solution was not offered in it. Problems in the Form C were classical proportionality problems with five offered solutions. One half of participants were asked to solve forms A and C, and the other half to solve forms B and C.

A convenience sample of high school students was examined. Participants were 112 students from two classes of the first grade (N=52, age 15-16) and two classes of the fourth grade (N=60, age 18-19), 53 girls and 59 boys. One class of the first and the fourth grade became at random the group with the offered linear solution and the other two classes became a group without the offered linear solution. In order to reduce the possibility of guessing, we also asked the participants to write down the procedure or an explanation of how they solved the problem. An answer was accepted as correct only if the subject besides it wrote an acceptable mathematical, graphical or verbal explanation.

Since 111 from 112 participants solved correctly all five linear problems, we concluded that there are no differences in solving linear problems with respect to gender, age and solving situation.

Results in non-linear problems were analyzed using 2x2x2 between-subjects analysis of variance (age x gender x solving situation).

The main effect of age was significant (F_(1/104)=34.34; p<.01). The older students were more successful than the younger students. Similar results were also obtained by De Bock et al. (1998) on somewhat younger samples. We assume that this effect occurs partly because the older participants are more cognitively mature; they have more experience in solving different mathematical problems, more knowledge in different areas of mathematics and they probably create correct problem schemas more successfully.

The main effect of solving situation was also significant (F_(1/104)=114.89; p<.01). Students who didn't have an offered linear solution were more successful. These results suggest that the removal of the possibility to apply the linear model really encourages the participants to use the non-linear model. In other words, the absence of the linear solution prevents students to solve problems routinely, without thinking and verifying the image of the problem.

The interaction effect of age and situation was also significant (F_(1/104)=9.37; p<.01). The older students were somewhat better than the younger students when a linear solution was offered, but that difference was even larger when a linear solution was not offered. If solving according to the linearity illusion is prevented in non-linear problems, it seems that the older students create more adequate problem schemas than the younger ones. An inspection of the procedures used by students showed us that the older students use drawing in non-linear problems in a somewhat greater extent than the younger students. However, since that strategy was not generally used very often, it is probably more a consequence of an individual discovery than of a systematic teaching of this strategy in school.

The illusion of linearity is the example that demonstrates that methods of education in the area of solving word problems should be reexamined. The results suggest that students are accustomed to a limited and stereotypical set of problems and without tendency to deal with verification of the problem image and solution. On the other hand, this study shows that the process of verification can be encouraged and that students can then solve problems significantly better. Mathematical education should teach students to analyze problems and go through every phase of problem solving; thus they will be able to gain a better quality of knowledge and to apply it in real-life situations more easily and more successfully.