Modeling the cost of a sandwich. The food operations controller of a catering company that supplies sandwiches and lunches both through mobile vans and as special orders for external customers has developed a spreadsheet that
enables the cost of sandwiches and similar items to be calculated.

Please notice the word “spreadsheet”. Anyone who ever did a spreadsheet of complexity of a sandwich should know that the key mathematical skill needed is awareness of the role of brackets in arithmetic expressions and an intuitive feel of how brackets are manipulated, something that is sometimes called “structural arithmetic” or “pre-algebra”. At a slightly more advanced level working with spreadsheets requires understanding of the concept of functional dependency in its algebraic aspects (frequently ignored in pre-calculus), but very prominent in abstract algebra (and in computer programming, say, in the form of polymorphism in C++ and other programming language).

To illustrate this point, I prepared a very simple spreadsheet in Apache OpenOffice Calc (it uses essentially the same interface as Microsoft Excel).

Look at the picture above: if the content of cell C14 is SUM(C8:C13) and you copy cell C14 into cell D14 (look at the next picture),

the content of cell D14 becomes SUM(D8:D13) and thus involves change of variables. What is copied is a structure of the algebraic expression, not even an algebraic expression itself. And of course this is no copying of the value of this expression: please notice that the value 85 becomes 130 when moved from cell C14 to cell D14!

At a very elementary level, abstract algebra provides intuition about such things as a structure of an algebraic expression.

Intuitive understanding that SUM(C8:C13) is in a sense the same as SUM(D8:D13) is best achieved by exposing a student to a variety of algebraic problems which convince him/her that a polynomial of kind ${\mathrm{x^}}^2+2x+1$ is, from an algebraic point of view, the same as ${\mathrm{z^}}^2+2z+1$, and that in a similar vein, the sum

C8 + C9 + C10 + C11 + C12 + C13

is in some sense the same as

D8 + D9 + D10 + D11 + D12 + D13 .

In the terminology of abstract algebra, it is called isomorphism. Abstract algebra studies structure of algebraic entities up to isomorphisms. I do not claim that everyone who uses a spreadsheet should know this terminology, but it is desirable to have some intuition about what is going on. Even if the number-crunching is passed to the computer, use of spreadsheets still requires mastering, at an intuitive or semi-intuitive level, some mathematical concepts, like understanding that arithmetic expressions have certain structure, or developing some basic intuition of functional dependency. And the most acute problem of mathematical education in our time is that this intuitive component of mathematics is being lost.

My former student who now works as a project manager in a serious engineering company once told me that many his colleagues cannot handle macroses in Excel spreadsheets with time-dependent entries (a basic tool of project management, an IT version of the proverbial clipboard) because they suffer from “brackets overload” — I love this formulation!