**Abstract**

In this chapter, I will argue that the proposition of "math is a language" has beneficial implications on the way we conduct scientific inquiry and math education. I posit that model-based discovery and learning are the acts of "replacing a theory-less domain of facts by another for which a theory is known." Data collected over 30 years of mathematical modeling research demonstrate that the creative process of model *formulation* is much more taxing than the solution of mathematical artifacts with scientific computing methods. Approaching mathematics as an acquired language also has profound benefits for engineering math education. Evidence from 20 years of teaching engineering students further reveals a shockingly *short-lived* retention of math literacy skills. A novel course pedagogy that treats 'math as a foreign language' was able to improve long-term learning outcomes at the undergraduate and graduate level. The chapter closes with an outlook on existing scientific frontiers in neuroscience that may be overcome by more math-eloquent scientists and engineers.

**Keywords:** mathematical modeling, ontologies, mathematical education, math literacy, scientific computing

## **1. The art of mathematical modeling**

Mathematical models appear everywhere in science. Models explain our understanding of the natural world even beyond the frontiers of science. For example, our worldview of the space/time continuum of Einstein's theory of relativity is not merely a study subject in theoretical physics, but is present in popular knowledge. What exactly constitutes a mathematical model? Can the art of generating a mathematical model about the physical world be learned or taught? Wikipedia offers the following on mathematical modeling: "A mathematical model is a description of a system using mathematical concepts and language." The Wiki page proceeds to characterize mathematical terms such as types of equations, but has no information on the role of language in modeling, or how this language would be employed. Despite its significance, mathematical modeling is not a term readily defined. A rare exception is Aris' book entitled mathematical modeling techniques [1]. This book entirely devoted to the art of mathematical modeling offers a definition as a set of equations corresponding to a physical biological or economical prototype. Aris also cites the logician Tarski who sees it as realizations in which all valid sentences of a theory are satisfied. I prefer a more general definition: mathematical modeling is the replacement of a theory-less domain by another for which theory is known.

The replacement of theory-less domain by another for which the theory is known has hardly ever been more successfully demonstrated than in the revolutionary discoveries of Alan Turing whose model of mathematical operations establishes the logical basis of modern computing. Turing realized that algebraic operations in our decimal system, additions and multiplications in our common natural number system, can be replaced with AND or OR logic operators within a binary number system. Accordingly, natural and real numbers could be digitized. More importantly, expensive tasks such as evaluating long lists of summation or multiplication operations could be executed by clocking thousands of AND and OR operations on a simple electronic machine, which became later known as a *computer*.

His contribution fits the definition of modeling as the replacement of decimal operations in the common number systems with the extremely simple and fast binary AND/OR logic operations; his breakthrough achievement rang in the digital age. His seminal work rooted in an ingenious mathematical model that combined well-known facts of mathematical logic with electronic principles, heralded a new era of computing.

Discoveries like Turing's spring from inspiration and creativity; thus, mathematical modeling is an art. But is there a formal way to generate modeling art? The sciences offer a useful template for model-based discovery and learning. I propose an analogy with a feedback circuit depicted in **Figure 1**. The learning cycle begins with a problem formulation conceived in the mind of the modeler that described properties and possible transformations about a novel physical prototype such as a chemical process, whose extent and critical parameters may not yet be fully known or understood. Problem formulation implies the transformation of domain-specific states and transition into suitable mathematical relations. When the physical domain, its state descriptions, and transitions are adequately mapped into a suitable mathematical surrogate, the algorithmic machinery is invoked to make mathematical predictions about the system states, X, typically using digital computers. Mathematical analysis involves the solution of mathematical equations for which we typically use computers. This involves the numerical solution of algebraic equations, nonlinear models, and optimization or system dynamics using well-known rules in the mathematical domain.

But model-based learning does not stop here: properties of the mathematical surrogate, X, are not the object of study. Instead, it is critical to interpret (=or back-transform) predictions in terms of the physical states in the problem domain. In engineering, this means inferences about the states of the matter such as pressure, temperatures, and concentrations (P, T, and C). Since the range and deliberate manipulation in the original prototype space are not fully known, or may be just vaguely known, it is often necessary to sharpen the original mathematical problem formulation or assumptions. The incorporation of feedback about the original

**25**

**Figure 2.**

*techniques [1].*

*Mathematical Modeling: The Art of Translating between Minds and Machines…*

cycles. Learning requires frequent model adjustments and reformulations.

**2. The cost of problem formulation with mathematical models**

Unfortunately, model formulation is a rather strenuous task performed in the creative mind of a modeler. The graphs from Aris' book in **Figure 2** illustrate profound modeling insights generated about reactive systems. The wonderful illustrations were not created spontaneously, but were conceived by intensive contemplation. From a practical point of view, it is legitimate to ask how taxing is the art of model generation, and whether industry can afford to pay for the labor of modeling artists? Let me offer an empirical estimate for the cost of mathematical model generation using examples of my past 20 years in chemical engineering research. My PhD thesis on coal gasification mathematical problem formulation took 2 years: its resulting nonlinear equations were encoded in Fortran. Today, this task may be accomplished a few months faster using modern modeling languages such as gPROMS or GAMS [2, 3]. My second example concerns chemical process flowsheets typically analyzed in

*Collage of Aris' figures about reactive systems. Assembled from Aris' book entitled mathematical modeling* 

domain realized in the domain mathematical analysis actually constitutes the essence of model-based learning and discovery. Let us also emphasize that the model-based learning paradigm proposed here requires frequent translations between the prototype domain and the mathematics. The need to frequently translate between different reference systems characterizes mathematical modeling as a linguistic activity. Mathematical modeling strongly relies on math literacy, which has implications on how we should teach mathematics to engineering students. This point will be discussed more at the end of this section. When feedback is omitted and mathematical predictions are taken at face value about the actual system, a gap between reality and its mathematical surrogate may open. This undesirable phenomenon is called model mismatch. Model mismatch is the most severe problem affecting mathematical models and is often unavoidable in the early stages of a study, but needs to be gradually mitigated by repeated reformulation, testing, and interpretation feedback

*DOI: http://dx.doi.org/10.5772/intechopen.83691*

**Figure 1.** *Schematic of the model-based learning paradigm.*

*Mathematical Modeling: The Art of Translating between Minds and Machines… DOI: http://dx.doi.org/10.5772/intechopen.83691*

domain realized in the domain mathematical analysis actually constitutes the essence of model-based learning and discovery. Let us also emphasize that the model-based learning paradigm proposed here requires frequent translations between the prototype domain and the mathematics. The need to frequently translate between different reference systems characterizes mathematical modeling as a linguistic activity. Mathematical modeling strongly relies on math literacy, which has implications on how we should teach mathematics to engineering students. This point will be discussed more at the end of this section. When feedback is omitted and mathematical predictions are taken at face value about the actual system, a gap between reality and its mathematical surrogate may open. This undesirable phenomenon is called model mismatch. Model mismatch is the most severe problem affecting mathematical models and is often unavoidable in the early stages of a study, but needs to be gradually mitigated by repeated reformulation, testing, and interpretation feedback cycles. Learning requires frequent model adjustments and reformulations.
