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How Engineers Learn to Develop and Apply Math Modeling as Part of CPD&M
Earlier in my career, I taught undergraduate and graduate mechanical engineering and product development courses at Rochester Institute of Technology (RIT). During my eleven years as a professor there, I developed an approach to teach students how to generate models of their designs prior to building, metrifying and experimenting with physical prototypes. The following steps illustrate that approach. They outline the flow of work that underwrites the integrity of analytical models. A balanced iteration of analytical and empirical models should integrate and culminate in a synergistic explanation of how the various factors and parameters govern and control how the design works under both nominal and stressful conditions.
Special note: I cannot over-emphasize the usefulness of Jack Lewis's text, Modeling Engineering Systems, High Text Publications, ISBN: 1-878707-08-6; a superb, highly readable text on generating math models and NOT excessively theoretical! This book is a high water mark in the communication of how to do this in as clear a way as I have ever seen!
Teaching How to Generate Math Models
1. Explain the principles of physics and engineering that apply to the particular domain on which you are working; e.g. fluid flow, heat transfer, structural load bearing, vibration control, etc.
2. Identify clear, generic math models (Y=f(Xs)) that exist for the domain.
a. Fundamentals and mechanisms are translated from words to analogs to equations. See Modelling Engineering Systems (note above).
b. Write equations using differential calculus, if possible (differential equations)
c. Explain how to solve the equation(s) in closed form or open form using iteration methods.
3. Show student a stair-step flow of easy, moderate and complicated examples of how to do this with detailed step-by-step procedures.
- MOST IMPORTANTLY! - Provide correct intermediate and final answers so they know exactly what "right" looks like across this easy-to-difficult spectrum. If there is more than one way to arrive at the right solution, share them with the students.
4. Repeat solution paths with students on a variety of types of scenarios to dwell on the proper paths - use repetition to set-the-hook in their minds.
a. Homework with ANSWERS provided; don't let them go down the false paths yet! Don't let them get stumped or flounder. If a student repeats a series of wrong solution paths you have to go back and erase all that mis-practice!
b. The gifted will get it quickly, the average at a slower learning rate. A good teacher teaches to the average and searches for the struggling students so they can be saved.
Keys to sustained learning:
- 1st principles are grasped and used as a base-line of building blocks.
- Foundational math structures are transformed into functional math; convert theory to applications by specific domains.
- Transition from words/requirements to block diagrams and schematics to basic underlying equations to differential equations to solutions forms. See Modelling Engineering Systems (note above).
- Provide lots of clear, detailed solution paths from start to finish for easy, moderate to difficult scenarios. My best professors showed us many solved problems before letting us off the leash for more challenging attempts. And when we got stuck, he was there to quickly show us where we had gone wrong so we avoided learning bad habits.
- Don't try to stump, frustrate or stymie a student. It is a bad practice in most of our universities and I have nothing but criticism for those who do that to their students! Constructing good models is hard enough without trying to intentionally be tricky just to weed out weaker students!
- Repeat successful solution structures and paths-over and over! In a sense you want to "brainwash" the student into developing neural pathways that harden into good habits and best practices that are embedded for a lifetime.
These are the steps to transform a student into a competent professional as they enter their domain-specific engineering career. It's also how an experienced engineer can re-train themselves when it comes time to refresh the skills needed to generate math models. Happy model building!
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