According to Blum (2011), mathematical modelling is the translation between the real world and mathematics and from mathematics back to the real world. Blum and other studies Nourallah and Farzad (2012) for example, have indicated that this process of alternating between reality and mathematics during mathematical activities has impacts on students’ mathematical knowledge.

This study investigated the effects of mathematical modeling instruction on precalculus students’ performance in a Rational Function Exam (RFE) and their attitudes toward rational functions. It was an exploratory embedded single case study design using both quantitative and qualitative methods. A sample of 54 precalculus students enrolled in two sections of precalculus at a local college in one major southern city of the United States was used for this study. The two precalculus sections were purposefully selected from five sections, with 24 students in the treatment group and 30 students in the comparison group.

Quantitatively, participants completed a pre-post Rational Function Exam (RFE) and an Attitude Toward Mathematic Inventory (ATMI) survey (Tapia & Marsh, 2004) before and after the study. Qualitative techniques were employed to determine the type and cognitive complexity of representations. These qualitative methods included interviews, a questionnaire, artifacts of students’ work and the researcher’s memos. The interviews and questionnaire responses were used to gather demographic and in-depth information about students’ experiences with the method of instruction. ANCOVA and reliability analysis were used to analyze quantitative data while coding (Saldaña, 2013) was used to analyze qualitative data.

Quantitative analysis results using ANCOVA showed a statistically significant difference (p < 0.001) between the posttest mean score on the RFE of the treatment group and the mean posttest score of the comparison group. The ANCOVA results also showed a statistically significant difference (p = 0.004) between the ATMI mean posttest score of the treatment group and that of the comparison group.

Qualitative data analysis of the artifacts, interviews, researcher’s memos and the questionnaire by coding revealed three important themes describing the effects of modeling instruction on students’ types and cognitive complexity of representations of rational functions: 1) Students tend to have positive views of rational functions and display engaging and immersed attitudes towards learning mathematics in a modeling instructional setting, 2) teacher’s guidance during modeling instruction tend to help students’ mathematical representations of functions and real-world scenarios & 3) mathematical modeling instruction tend to foster critical thinking and conceptual understanding of rational functions, increasing students’ representations capabilities and cognitive complexities.

These results suggest that mathematical modeling instruction had positive effects on students' learning and understanding of rational function concepts, their attitudes towards learning rational functions and the cognitive complexity of their representations of functions.