نوع مقاله : مقاله پژوهشی
1 دانشجوی دکتری آموزش ریاضی، گروه ریاضی، واحد علوم و تحقیقات، دانشگاه آزاد اسلامی ، تهران، ایران
2 استادیار آموزش ریاضی، گروه ریاضی و علوم کامپیوتر، واحد تهران مرکزی، دانشگاه آزاد اسلامی ، تهران، ایران.
3 دانشیار آمار، گروه آمار، واحد علوم و تحقیقات، دانشگاه آزاد اسلامی، تهران، ایران.
عنوان مقاله [English]
Objective: Mathematical modeling is an interconnecting process between mathematics and real-world problems that, as a tool, can be applied to increase motivation, develop cognitive competencies, and enhance the ability to transfer mathematical knowledge to other areas such as engineering disciplines. This study aimed to investigate the effect of applying the modeling approach on the problem-solving performance of engineering students during a differential equation course and compare the results with those of traditional teaching.
Method: A quasi-experimental method with a pretest-posttest design comprising a control group was applied in this study. The statistical population was 1225 engineering students of the Islamic Azad University Central Tehran Branch. The available statistical sample contained two experimental and control groups, each comprising 33 engineering students who randomly attended the course. The modeling approach was applied to teach the first-order separable, linear and homogeneous differential equations in the experimental group, whereas, in the control group, the traditional approach was used. Since the antecedent knowledge of calculus directly affects the ability of students to learn differential equations, a placement test was first applied to both groups. A combination of the results derived from the placement test and the modeling pre-test was used as the covariate in the Analysis of Covariance (ANCOVA) test, and the modeling performance of both groups and their problem-solving skills were eventually examined.
Findings: The data extracted from pre-test and post-test results were analyzed using the SPSS version 26. The results of this study indicate that apart from the fact that teaching differential equations based on the modeling approach can increase the ability to apply mathematical knowledge in the field of engineering, it also can significantly improve their problem-solving performance. In contrast, in the traditional approach, students cannot transfer the knowledge attained from the course to applied and real-world problems. The research findings can be used in reviewing the differential equation syllabus.
Abstract (Extended): Due to the lack of connection between real-world applications that students encounter in their future careers and the knowledge attained from traditional mathematics courses at the university, students are virtually incapable of interpreting complex structural systems as the future workforce. Therefore, the primary purpose of introducing major mathematics courses in the engineering curriculum should be to prepare students for mathematical reasoning on the issues they will face in their related engineering courses and/or real-world applications. Studies over the past few decades have shown that mathematical modeling as an interconnecting tool between mathematics and real-world problems can help students fulfill this goal effectively. Due to this fact, many international associations around the world, such as GAIMME (2016), have suggested guidelines and instructions for the inclusion of mathematical modeling as a new subject in the syllabus of typical mathematics courses such as calculus, linear algebra, and differential equations that are generally taught in non-mathematical STEM (Science-Technology-Engineering-Mathematics) disciplines. It has also been suggested, in many cases, to include mathematical modeling in the curriculum as a separate mathematics course. In many international reports such as PISA OECD and CCSSM, the mathematics literacy of students and their skills and general perceptions are assessed by their ability in modeling, which confirms the importance and necessity of applying mathematical modeling at different educational levels. Differential equations (DEs), to a great extent, may be considered as a modeling procedure that interrelates applied and real-world phenomena with abstract mathematical concepts. Therefore, as the results of this study reveal, using the modeling approach in instructing DEs may improve the students’ problem-solving performance.
Like other mathematics courses, which are generally incorporated into the engineering curriculum, the syllabus of differential equations is designed, coordinated, and taught by mathematicians. As a result, the main contents are often presented in the form of abstract and general examples, and the instruction is through introducing a set of well-known equations solved by specific algebraic techniques. Therefore, many researchers now acknowledge that the relationship between differential equations and their solutions for most students is more procedural than conceptual. This study aimed to investigate the effect of applying the modeling approach on teaching differential equations to students and its preeminence compared to the conventional method considering their problem-solving performance and modeling abilities. The research was conducted on two groups of students who had randomly attended the course from different engineering disciplines. Initially, a modeling pre-test and a calculus placement test were performed to assess their background knowledge in mathematics, which as a covariate, it is anticipated to have a statistically significant impact on the analysis of covariance. The results showed that the two groups had almost equal performances, both in the calculus (sig = 0.803) and the modeling (sig = 0.708) tests.
In the experimental group, students were introduced first to different stages of the mathematical modeling cycle, including understanding the task, simplifying/structuring, mathematizing, working mathematically, interpreting and validating through an example, and achieving competence to form and solve a differential equation during the stages of mathematizing and working mathematically. The students in the experimental group learned to solve first-order separable, linear and homogeneous differential equations based on the modeling cycle procedure, whereas, in the control group, the conventional algebraic approach was instructed; and the performance of both groups in modeling was then assessed by performing a test comprising real-world problems. Since not being introduced to modeling skills, students are virtually unable to translate existing real-world mathematical concepts into the form of differential equations; it is anticipated that the null hypothesis of equality of the means of the two groups should be rejected by selecting any significance level. This assertion was confirmed by the research findings, which reveal that the experimental group significantly outperformed the control group in the modeling post-test (sig<0.001).
Therefore, in addition to evaluating the students’ modeling ability, it was decided to assess the effect of applying the modeling approach on their overall problem-solving performance. Hence, other course topics were taught to both groups by the conventional method. Similarly, the use of both ANOVA and ANCOVA tests shows that the hypothesis of equality of the means of the two groups in the final exam was also rejected (sig = 0.009 and sig <0.001).
The results of this study clearly show that students in the control group lacked a conceptual perception of applied problems and were incapable of connecting real-world conditions to abstract mathematical concepts. In contrast, students in the experimental group attained the ability to interpret and analyze existing physical and abstract mathematical concepts in real-world problems and model them through integrating conceptual and procedural understanding.