The "Rule of Four" in Calculus: A Short Critique

[Ackbach]

The "Rule of Four" in calculus is the teaching philosophy that all concepts should be taught algebraically, graphically, numerically, and verbally. In particular, when dealing with functions, the teacher should emphasize the algebraic, graphical, numerical, and verbal representations. I could be wrong on this, but it seems to me that this emphasis is meant to imply that all four of these representations are equally valid and useful.

Baloney.

I want to present a little table. This table is meant to illustrate which representations can be accurately inferred from other representations. The vertical column on the left is the origin representation, and the horizontal row on the top is the target representation. In other words, this table is attempting to answer the question, "Which representations can accurately generate which other representations?"

$\implies$	Algebraic	Graphical	Numerical	Verbal
Algebraic	Of course.	Yes.	Yes.	Yes.
Graphical	With difficulty. Often impossible without additional information.	Of course.	Yes, but requires much effort.	Yes, if enough data is present.
Numerical	With difficulty. Often impossible without additional information.	Yes, if enough data is present.	Of course.	Yes, if enough data is present.
Verbal	Yes.	Yes, but often must be done through another representation.	Yes, but often must be done through another representation.	Of course.

So one of these origin representations can generate the other three with very little difficulty: the algebraic/analytical representation. Therefore, when it is available, the algebraic/analytical representation is the most superior.

Am I saying that the other representations are useless? By no means. Of course they're not. Sometimes, all you get is numerical, like when you're doing data acquisition in a control setting. Other times, all you have is graphical, like one time when I needed the VI curve for a diode and was given a graph of it in the diode's data sheet.

What I am saying is that the algebraic/analytical is not equal in value to the other three - it is superior. Therefore, in teaching calculus, I emphasize the algebraic/analytical approach whenever it is available.

 

[I like Serena]

When an engineer needs to know the area of a specific piece of land (real life problem), what he does is:
- draw it,
- cut it out of paper,
- weigh it.

Very simple, very straight forward, and sufficiently accurate.
Who needs any of the other representations. ;)
(I'm not really joking.)

Note that it is near impossible to describe a piece of land algebraically, a numerical approximation is a lot of unnecessary work, and any verbal method is not reliable enough (although a good salesman can probably do a good job).

 

[Ackbach]

When an engineer needs to know the area of a specific piece of land (real life problem), what he does is:
- draw it,
- cut it out of paper,
- weigh it.

Very simple, very straight forward, and sufficiently accurate.
Who needs any of the other representations. ;)
(I'm not really joking.)

Note that it is near impossible to describe a piece of land algebraically, a numerical approximation is a lot of unnecessary work, and any verbal method is not reliable enough (although a good salesman can probably do a good job).

Sure: in this situation, you could say that the algebraic representation is not available. But I've already covered that possibility in the OP.

 

[Evgeny.Makarov]

By the verbal form of concepts, do you mean something like this?

"To the absolute number multiplied by four times the square, add the square of the middle term; the square root of the same, less the middle term, being divided by twice the square is the value". (A version of the quadratic formula from the 628 AD Indian book, see Wikipedia.)

Then clearly it does not have the same value as the algebraic representation.

I often need to see graphs of functions to understand their behavior, so drawing graph sketches from formulas is important. Also, I wish we had more numerical exercises and examples. It is one thing to know that logarithm grows much slower than the identity function, but another to realize that the common logarithm of one million is a mere 6 (which is obvious, of course). In particular, I am wondering if it would make a good first part of an exercise on proving continuity by definition to find numerical deltas for several epsilons, before coming up with a general formula.

 

[HallsofIvy]

By the verbal form of concepts, do you mean something like this?

"To the absolute number

What does 'absolute number' mean here?

multiplied by four times the square, add the square of the middle term; the square root of the same, less the middle term, being divided by twice the square

Twice the square of what?

is the value". (A version of the quadratic formula from the 628 AD Indian book, see Wikipedia.)

Then clearly it does not have the same value as the algebraic representation.

I often need to see graphs of functions to understand their behavior, so drawing graph sketches from formulas is important. Also, I wish we had more numerical exercises and examples. It is one thing to know that logarithm grows much slower than the identity function, but another to realize that the common logarithm of one million is a mere 6 (which is obvious, of course). In particular, I am wondering if it would make a good first part of an exercise on proving continuity by definition to find numerical deltas for several epsilons, before coming up with a general formula.

 

[Evgeny.Makarov]

The "absolute number" apparently is the constant term $c$ in $ax^2+bx=c$, the form of the quadratic equation from the link. "The square" means the coefficient of $x^2$, i.e., $a$. This description should not be judged too strictly: it dates from the first millennium AD.

 

[I like Serena]

This description should not be judged too strictly: it dates from the first millennium AD.

Hmm, isn't Euclid's Elements from the first millennium BC? ;)

 

[Evgeny.Makarov]

Who said anything about Euclid?

 

[Jameson]

Could you give an example of how you use the "numerical method" when teaching calculus? I think you make a great point that these are not equally useful in practice, but maybe for conceptualizing what is going on they are more equal. Having a verbal description of a function or of what a derivative is can't do very much for concrete problems but it is worth covering for sure. You wrote that you use the conclusion that all four methods shouldn't be equally weighted in your teaching. Can you give an example of how you do this? As a teacher and tutor of various subjects, this is very interesting for me.