Concept-Based Modeling as a Method Combining Digital and Analogue Means for Problem-Solving

Abstract

In this paper we present Concept-Based Modeling (CBM), an innovative pedagogical method for problem-solving in engineering education, which combines analogue and digital tools. We outline the scientific rationale for CBM and discuss how it compares to traditional teaching with respect to optimizing the pedagogical value of both analogue and digital means. CBM is based on conceptual modeling of quantities derived directly from first principles and streamlined for the use of computer algebra systems (CAS). The method was evaluated in a pilot survey in a statics course for engineering students in their freshman year at Halmstad University. We conclude that CBM improves students’ problem-solving skills by the reciprocal action between conceptual understanding and modeling of a problem. Student evaluations suggest that CBM enables students to handle more realistic problems and that CAS as a professional tool prepares them for their future working life. Future studies will address CBM for more advanced courses, as the students’ knowledge develops over time.

Appendix

Sample Statics Problem Resolved Using CBM

Here, we work through a typical statics problem using CBM (see Fig. A1).

Fig. A1.

A typical problem in a university level course in statics.

• a) Modeling

  Given the question, we embark on the first task of the solution process, the conceptual modeling of the problem. We need to decide the first principles that governs the problem. This is a static problem for which it is know that the stool needs to be in equilibrium with its surroundings, that is, the resulting forces and force-couples are zero ($\sum {\mathbb{F}}=0$ and $\sum {\mathbb{M}}_{C}=0$). This means that a free-body diagram must be drawn, and all the forces and force-couples must be modeled.

  A free-body diagram is a careful drawing of the specific part (the body) that is to be studied (see Fig. A2). Here, we introduce the known lengths and angles along with an appropriate coordinate system (x, y, z). This is the part of the solution process where we really see the problem. Remember that we cannot draw what we have not seen.

  Once the stool is drawn, we start to model the forces conceptually by adding vectors (arrows) in the direction of their application onto the stool. Generally, assuming that each of the three legs are in contact with the floor, their reaction forces must be a combination of normal and friction forces. In finding the minimum static coefficient of friction we must assume that the friction forces are fully developed and equal to the coefficient of friction times the normal force. The load P is already given in the problem.

  Fig. A2.

  

  The conceptual modeling of the loaded stool. The shaded equations represent the first principles valid for equilibrium. Each of the physical concepts are here modelled in steps all the way down to the most basic, the base vectors of the coordinate system.

• b) Computation

  The next step of the solution process is to transform the conceptual models into a code that can be processed by a computer. In this example we make use of the CAS Mathematica ®. The coding always starts with defining the most basic of all concepts, the base vectors. Once we have these base vectors, we can define our rotation matrix and so on.

  

  After the mathematical concepts are defined, we are ready to go on and model all our physical concepts. Note that the concepts always appear on the left-hand side of the equal signs and their definition and models on the right:

  

  Now we are ready to state the 1^st principles, the equations for equilibrium:

  

  And finally, we state the unknowns and apply the solver:

  

  From the solver, we get the answer that the minimum static friction coefficient, needed for the stool to remain at rest, is $\mu =0.27$. This concludes the CBM study.