It was a joke all right.

But let's take another angle on it. Imagine, if you will, that you lived in a world where anywhere you went there were cupboards, full of chemicals. Actually, such a world is not too imaginary. If there were a healthy demand for such cupboards, there would be a healthy supply of those cupboards. Now, the joke about preparing hydrochloric acid does not seem like a joke anymore. On the contrary, wasting time concocting or learning to concoct a chemical that you can get off-the-cupboard seems foolish.

"But wait," you interject, "someone has to make those chemicals in the cupboards and that person needs to know how to concoct chemicals."

True, but lets say you are not a chemist in charge of stocking cupboards, but, say, an engineer who occasionally needs certain chemicals. In the past, you concocted those chemicals yourself, but now, thanks to enterprising chemists, you don't have to. With the time you save, you can learn more engineering skills, or listen to music, say, Mahler's ninth symphony, or just put on your running shoes and see how fast you can run.

"I have at least three objections," you exclaim. "One, cupboards cannot contain every chemical that an engineer will ever need. What if a chemical he desperately needs is not in the cupboards?

"Two, even if an engineer never has to concoct chemicals, learning to concoct them can teach him many useful laws of nature. The learning will only make him wiser. Who knows, he might mix his engineering and chemistry ideas and concoct a chemical that even chemists could not concoct.

"Three, as a chemist, I learned to synthesize complex chemicals and the experience has been so enlightening. The creative aspects of the experience have made me see the art in that science and I have jumped with joy at some of my art. It would be a shame to deny those opportunities to engineers."

Well, OK. All three points are valid, notably the third, and you have put me on the spot. Like any professor put on the spot, I am going to tell you another joke. I read this one in a joke book long ago.

Two engineers, close friends, are taking a walk in the woods. Suddenly, out of nowhere, a lion appears and charges at them. One of them starts to put on his running shoes. The other says, "You're wasting your time putting on your running shoes. You cannot outrun the lion."

"I don't have to outrun the lion," says the first. "I only have to outrun you!"

This joke teaches at least three moral lessons. First, it shows that we can think of circumstances in which almost any skill could become vital. What is relevant therefore is how likely is it that a particular skill can become vital. If the land of cupboards is also infested with lions, then the engineers would be wise to learn how to escape charging lions.

Second, it shows that there will always be competition. Which skill becomes vital is determined, in the most part, by competition. If competition hinges on advanced engineering skills, then the engineers should learn advanced engineering skills. Our own history confirms how gun powder, fighter aircraft and nuclear weapons have influenced wars. Sometimes the competitive edge is more subtle. In 1814 Sadi Carnot, a French soldier and engineer, fought a war for Napoleon in the outskirts of Paris. When the French lost the war, he attributed the defeat to the low efficiency of France's steam engines. Although steam engines were not directly used in the war, he argued that the steam engines determined a country's prosperity and thereby its ability to win a war. We cannot discard him as uninformed because he went on to discover the fundamental principle about the efficiency of heat engines: the more reversible a heat engine the more efficient it will be. Today, we teach that to every engineering student.

Third, it reminds us that life is full of sudden and deadly events that are so unpredictable. To a business, for instance, an economic shock wave is like a charging lion. Confronting the lion with what the competition might do is unpredictable. The unpredictability of competition is so intricate that in the land of cupboards what the engineers should learn in the time saved by cupboards is theoretically undecidable. Practically, this means that no solution is permanent. A solution hatched from a hunch may be good for a few months, one hatched from a consensus, good for a few years, and one hatched from a vision, good for a few decades. The education administrators in a competitive land would do well if they find a leader with a true vision. Even then, in a few decades, there will come fancier cupboards and deadlier lions that confuse them all. And they will look for yet another leader with yet another vision.

"Well, maybe," you shrug your shoulders. "What was that about Mahler's ninth symphony?"

What else? Because listening to it can be so thought provoking [8].

The Spreadsheet Cupboards

Thanks to the popularity of spreadsheets, the spreadsheet cupboards are ubiquitous. As a result, it is possible to argue convincingly that it is not necessary to teach business students any of the algebraic approaches listed on the left side of the Cupboards 1 through 3. Instead of teaching the students the simplex method we may give them an overview of the Solver; instead of teaching regression through the solution of normal equations we may teach them how to set up a spreadsheet to calculate SSE and how to invoke the Solver to minimize it. When we teach exponential smoothing later, the students automatically understand how to invoke the Solver to find the best smoothing constant(s). Some common objections to the spreadsheet approach, such as it being a black box, can be taken care of through proper treatment of spreadsheet based materials in textbooks, lectures, assignments and tests.

Enthused by how well spreadsheet methods substitute for algebraic methods, some scholars have claimed that algebra itself is unnecessary. My first reaction to the claim was, not knowing algebra, how will a student be able to understand the formulas in spreadsheets? But it seems quite possible that, say, in junior high school, we teach spreadsheet formulas from first principles and later teach algebra as an extension to spreadsheets. Seen in this light, the claim does have validity.

Let me now clearly state the question I wish to address. Most academics, including me, are convinced that spreadsheet approaches listed in Cupboards 1 through 3 can be substituted for the corresponding algebraic approaches. The substitution saves time. Some academics, not including me, have claimed that if spreadsheets are fully used then algebra itself is not necessary. For them, there will be even more savings in time. Depending upon where one stands in the use of spreadsheets and in the elimination of other topics, one is going to have some amount of time saved. In that time, what additional topics, if any, should be taught to business students?

We do see some "lions" around such as the denunciation that we do not teach enough cross functional integration, leadership skills, teamwork skills, international business and communication skills. We also see much competition among business schools. If the lions are a real threat, we might take care of them first. If not, we might go for teaching advanced skills or even art appreciation to business students. It is impossible to prove that a certain course of action is the right thing to do, but we can look for "visions."

In the next section, I'll describe my own vision (if you can call it one) that with the time saved by spreadsheets we should teach additional, rigorous algebra.

My Vision

After the standard algebra lessons, we ask Sara to prove

if a + b + c = 0

then {{a sup 3 + b sup 3 + c sup 3} over 3} = abc.

Unlike Sal, Sara has no problem in proving it. We then tease her with, is it also true that

if a + b + c = 0

then {{a sup 5 + b sup 5 + c sup 5} over 5}

= ({a sup 3 + b sup 3 + c sup 3} over 3)

= ({a sup 2 + b sup 2 + c sup 2} over 2).

When she proves it to be true, the beauty of the relationship is too engaging for her to sit idly. She wants to explore higher powers. She knows that trying even powers on the LHS is pointless. So she checks whether

then {{a sup 7 + b sup 7 + c sup 7} over 7}

= ({a sup 5 + b sup 5 + c sup 5} over 5)

= ({a sup 2 + b sup 2 + c sup 2} over 2).

is true. And golly, it is true! She gets ready to prove a theorem, but before doing so she wants to test just one more case, the ninth power. Alas, the relationship does not hold for the ninth or any higher power. She is disappointed but not resigned. She will try different forms on the RHS, or move on to four or more variables. There is no telling what she might discover, but one thing is sure. She is at the artistic level.

Now let us put Sal and Sara together in some scenario, after their graduation. Suppose they are both asked to solve a plant re-layout problem. Sal uses her spreadsheet skills to find the best layout, uses her leadership skills to convince the concerned people and produces a neat report. The report depicts the proposed layout, details all the relevant costs, and shows how much money and effort the new layout would save. Attached to her report is an invoice for a handsome fee for all her services.

Sara knows how to find the best layout on the spreadsheet, but she is not doing it. She ponders the problem for days. Most of the time she is just gazing; other times she is working with paper and pencil. She rarely uses spreadsheets. Suddenly there is a spark and she has invented a solution. She too writes a neat but shorter report. The report proposes a different sequence of operations rather than a different layout. Following a description of the modified sequence, it contains a mathematical proof that the new sequence will have a higher efficiency regardless of the layout used. Sara has not attached an invoice because she is busy jumping with joy at her invention. After she stops jumping, she will worry about where to apply for a grant to buy time to generalize and publish her invention.

Even if Sara's case is one in 10,000, it is still wise to impose college-level algebra on our students and to keep grant programs alive. If we do not, we might lose out to the competition or worse yet, lose a war to an enemy. And that could affect millions of us.

Conclusion

Well, maybe. I shrug my shoulders. Let me conclude with some supporting arguments in favor of my vision.

Although the spreadsheet is powerful in many ways, it cannot teach methods of proof. Rigorous algebra can. Many significant discoveries and inventions to be made in business fields need theorem proving skills. Algebra is thus a complement to spreadsheets, and not a substitute. Of course, other subjects such as rigorous geometry can also teach methods of proof, but algebra has a clear edge. Advanced algebra will not only reinforce the basic algebra that students learn, but it will also enable us to teach some advanced concepts not presently taught due to their mathematical complexity. (For instance, we teach hypothesis testing but not Lindley's paradox [1, 3]. We teach CAPM but not the embarrassments of mean-variance preferences [e.g., 2].) Considering all possible complements to spreadsheets that are not already in the curriculum, I deem that algebra has the most appeal. I therefore dare say that among two societies that are otherwise identical, the one that successfully incorporates rigorous algebra into its business curriculum would be the one that prospers better.

References

[2] Borch, K. 1969. A Note on Uncertainty and Indifference Curves, Review of Economic Studies, 36(1), pp. 1-4.

[3] Lindley, D.V. 1957. A Statistical Paradox, Biometrika, 44, pp. 187-192.

[4] Sounderpandian, J. 1996. StatSheets: An Excel Supplement to Accompany Complete Business Statistics by Aczel, Irwin, Chicago.

[5] Sounderpandian, J. 1997. Business Statsheets: An Excel Supplement to accompany Applied Statistics by Bowerman and OConnell, Irwin, Chicago.

[6] Sounderpandian, J. 1997. OM Sheets: Excel Problems and Cases to accompany Operations Management by Markland et al., West Publishing Co., Minneapolis.

[7] ftp://cs.uwp.edu/pub/jay/cupboard.html

[8] Thomas, L. 1983. Late Night Thoughts on Listening to Mahlers Ninth Symphony, Viking Press, New York.

Download julyfigs.zip for TIF version of Figures 1-3.

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