**Author details**

*Technology, Science and Culture - A Global Vision*

to be taught.

this language.

proposed model-based learning model delineated a continuing feedback cycle of sharpening the problem formulation, solution, and interpretation of results. Accordingly, the rigorous solution of mathematical properties is only a subtask, but not the essence of mathematical modeling which requires translation between physical prototype and mathematical relations and between computational predictions and actual process system states. It is a key that the interpretation of mathematical results (predictions) informs knowledge of the behavior of the original study system. The repeated translations pose a linguistic, more than a mere logical challenge. We therefore suggest that problem formulation of process models is similar to a communication and composition task. The realization about the linguistic nature of mathematical modeling has implications on how it ought

Mathematical modeling involved frequent translation between the physical and mathematical languages. The view that mathematical modeling is a form of translation and composition between languages gives indications on how modeling can effectively be learned and taught. First, let us appreciate that languages requires a grammar and syntax. In the world of mathematics, these are the mathematical properties that need to be studied before any serious composition can commence. In this aspect, students are often at a loss, not because they fail to comprehend the logic of mathematics, but because they fail to parse its terminology. Even if the logic is clear, we do not comprehend wisdom written in a foreign tongue. It requires good reading practice before students are able to compose in

I have implemented the "math-as-a-foreign language" pedagogy in several course offerings in the past 10 years, for instance, a course in biological system analysis. Accordingly, we have reading exercises to make sure that students are familiar with the words of the mathematical syntax. Grammatical rules are introduced as the properties of linear and nonlinear systems. All assignments are given as a natural language memo, which forces students to translate instruction in natural language into mathematical expressions. There are translation exercises in which students practice physical prototypes conversion into quantitative expressions. This involves the choice of mathematical entities (scalars, vectors, and matrices) and a suitable mapping into biological properties. Temperatures are an example of scalar fields, velocities form vector fields. For the description of the state of stress, a tensor field is needed. The characterization of an anisotropic porous medium requires a diffusion tensor field. We also have composition exercises in which multi-physical phenomena are transcribed from the physical world of reactors with connecting processing streams into networks of mathematical relationships using property vectors and connectivity matrices. Finally, students are tasked in the validation steps in which mathematical predictions are interpreted in terms of physical model

This course has been running the last 10 years with a high success in terms of student retention rates. Typically, our students are more confident about mathematical modeling than they were before the course or the sequence of math courses before. The biggest obstacle to mathematic learning was removed by recognizing

An earlier version of this text was delivered as a presentation of the 2040 Visions of Process Systems Engineering—A Symposium on the Occasion of George

Stephanopoulos's 70th Birthday and Retirement from MIT, June 2, 2017.

**36**

behavior.

that mathematics is a language.

**Acknowledgements**

Andreas A. Linninger University of Illinois, Chicago, United States

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