# Integrating concepts through hands-on mathematical exploration

## The isa approach:

Caveat: While it's a challenge to verbally demonstrate the process, this should convey the gist of it. The example is intentionally terse and the writing no paragon of exposition.

Problem-centric: We introduce topics based upon real business decision-making needs, not functional theoretic constructs. When students have a rough understanding of the real-world need that drives a new topic, the theoretic aspect quite generally takes care of itself. Having taught applied mathematics at a variety of levels, that goes as much for tensor calc as straight-line depreciation.

Before reviewing the zipped file containing both the .ppt and .xls files, here's a suggestion. Try reading the .ppt with the mindset of a student watching as the professor whiteboards the (PowerPoint) graphics. Each drawing, of course, would be accompanied by an oral description of the underlying concepts and how they interrelate — the .ppt text. The Excel model would then be computer-projected with oral explanation within the context of the .ppt "narrative" along with the still visible .ppt graphics.

In the example, the graphics, verbal explanation, and Excel model demonstrate how the most basic (from first principles) economic concepts of supply and demand lead, by simple steps, to (improving) a firm's stock market share price. Particularly, note that the specific concepts presented seamlessly span economics, accounting, finance, marketing, strategy, operations, and quant.

In practice, we've found that the integrated approach greatly facilitates learning while vastly compressing the time requirements. Further, as you're being exposed to the terminology (supply, demand, sales, costs, etc.) you're being immersed in both graphical and mathematical presentations. This bi-modal presentation engages both "sides" of your brain while dynamic Excel modeling brings to life the relatively "static" abstractions of the theoretic concepts as well as the normally deadly-dull financial statements.

In contrast, equivalent coverage in a typical function-based curriculum, occasions enormous overlap as each professor, in turn, haltingly attempts to put the concept du jour into the proper aggregate functional context. Sadly, it's been found that such repetition does nothing to improve either initial understanding or retention. Such repetition is nothing more than an utter waste of time.

The "perturbate" worksheet takes you into dynamic sensitivity analysis. Note that, once you've perturbated (altered) the input variables to your satisfaction, you re-run Solver to find the new optimal production mix. We find that it is the interactive use of the Excel model to explore the conceptual underpinnings which rapidly imprints the conceptual relationships. Further, the process bridges static financial statement and Gedankenversuch. You learn to rapidly construct/deconstruct models mentally. That's understanding; that's learning. So, now, how do we gain greater insight from our model?

You'll note that the model starts as a sparse framing of the concepts with a few lines of labels and very simple equations. With nothing more than estimated demand, quantities on hand, unit selling price, and unit cost, you're able to determine what quantity and combination (production mix) of products maximizes your operating profit and, (in theory), your company's share price. Is your model "right"? Of course not, because, for instance, we don't KNOW demand. It is, however, potentially extremely useful. Moreover, part of the point of the exercise is to recognize, for instance, that we'd undoubtedly do well to perform statistical market research to gauge demand sensitivity to price changes. Before performing that research, we would use the model to determine what, if any, reasonable pricing choices would result in financially significant overall changes. By doing so, we could more efficiently ask far more useful "questions."

Goal: Winning praise NOT arguments. Importantly, you've engaged in some fairly serious analysis. Yet, anyone should be able to follow your explanation of the findings using this Excel model format. Most importantly, the approach allows you to "defuse" the potentially disastrous, "my answer," peril. You are able to merely facilitate the joint exploration with your audience of THE situation. Should someone disagree with an input assumption, you merely change it on-the-fly and re-Solve.

"Gee, great point, Billy-Bob, I hadn't thought of that." Billy-Bob has to admit that you're an okay guy. The higher-ranking attendees have already started to see you as leadership material.

QED