"Function" vs "Function of ...": how much does it contribute to students difficulties?

Question

Most textbooks I've seen (and teachers I've met, myself included) are rather careless about the distinction between variables and functions.

For example, when we write $y=f(x)$ we all know that $f$ is the function, while $x$ and $y$ are variables. We also know that $y\neq f$. Still, we go on to call $y$ a function of $x$ in front of students, even though $y$ is not a function in the mathematical sense.

The terminology "$y$ is a function of $x$" has a long tradition that seems to predate the moment when people decided to also call $f$ a function. So there is no hope of changing this. In principle there is also no need of changing terminology, if we all agree never to call $f$ "a function of $x$" nor $y$ "a function".

Unfortunately, people soon drop the "...of $x$" and call $f(x)$ a function, they call $x^2$ a function and they call the temperature a function. Or they write $y(1)$ and $df/dx$, even though the first one is meaningless when $y$ is a variable, while the second would probably be zero in most cases, since $f$ does not depend on $x$. In many applied areas it is common to go as far as writing $y=y(x)$, completely blurring the difference between the function and the dependent variable. Here is a nice example from Dray and Manogue that illustrates the effect this can have (click "Next").

When I'm teaching introductory calculus to engineers and see their difficulties with function application notation, composition of functions and the chain rule, I sometimes wonder how much of that is caused by this constant blurring of notions by the teachers.

Question: Is there any research in mathematics education that tries to measure the effect the common blurring between functions and variables has on students, while they are learning the concept of a function?

Answer 1

I really appreciate your question. It gets to something I think about often when teaching calculus. You should read about Ed Dubinsky's notion of reflective abstraction, rooted in Piaget.

At one point in mathematical development, after learning to count, positive integers are "encapsulated" by children as primary objects. Later, while learning algebra, variables such as $x$ and $y$ become encapsulated as primary objects, after a struggle which some never surpass. Then when learning calculus, functions like $f$ and $g$ might become encapsulated (abstracted) as primary objects. But it's a messy process, complicated by the convenient, historical "abuse of notation" blurring $f$ and $f(x)$.

Teaching calculus with computers can help college-aged students encapsulate functions as primary objects. In Dubinsky you can read about the language ISETL. I think teaching calculus using Maple also helped students reach this level of abstraction, but that was when the only input was "classic." There you set up functions by notation such as $f:=x\rightarrow x^2$. In Church's lambda calculus, this is equivalent to distinguishing between $x^2$ and $\lambda x.x^2$. Once 2-d input was put into Maple with document mode, this pedagogical tool was lost to "disambiguation dialogue boxes."

The question you bring up is very important. Keep thinking about it.

Answer 2

This MAA Notes volume has a couple of good articles about related topics, including the article "Foundational Reasoning Abilities that Promote Coherence in Students' Function Understanding" by Oehrtmann et al. which I think has some valuable connections. I think the references for the articles would lead you further - unfortunately I don't have a print copy in front of me, just a few pages I photocopied long ago that reminded me of it.

I don't think it directly addresses this issue but definitely deals with the issue of function versus variable, which I think is at the root of f versus y versus f(x) or even y(x) (though the latter is so darn useful in differential equations).

Answer 3

The issue you point out is an example of Abuse of notation. Quoting Wikipedia:

In mathematics, abuse of notation occurs when an author uses a mathematical notation in a way that is not formally correct but that seems likely to simplify the exposition or suggest the correct intuition (while being unlikely to introduce errors or cause confusion). Abuse of notation should be contrasted with misuse of notation, which should be avoided. [...]

One encounters, in many textbooks, sentences such as "Let f(x) be a function ...". This is an abuse of notation, as the name of the function is f, and f(x) denotes normally the value of the function f for the element x of its domain. The correct phrase would be "Let f be a function of the variable x ..." or "Let x ↦ f(x) be a function ..." This abuse of notation is widely used, as it simplifies the formulation, and the systematic use of a correct notation becomes quickly pedantic.

A similar abuse of notation occurs in sentences such as "Let us consider the function $x^2 + x + 1$.". In fact $x^2 + x + 1$ is not a function. The function is the operation that associates $x^2 + x + 1$ to $x$, that is $x \rightarrow x^2 + x + 1$. Nevertheless, this abuse of notation is widely used as, generally, it is not confusing.

Of course one should tell the students that f is the function and f(x) is the function value at x. But if one can prevent misunderstandings, then abuse of notation is a very useful thing. Without it, we would be like machines who understand everything literally. To quote Bourbaki:

We have made a particular effort always to use rigorously correct language, without sacrificing simplicity. As far as possible we have drawn attention in the text to abuses of language, without which any mathematical text runs the risk of pedantry, not to say unreadability.

Answer 4

I'm not sure whether this is useful or not, but if the students are used to the notation $f(x)$ and you wish to allow them mental continuity, you could always try the technique of saying:

"Let $x$ be a variable whose domain is the set of real numbers. Then we define the function $f(x)$ to be [whatever formula]."

In that way you kill two birds with one stone: you keep the familiar notation without which a student can often get (at least temporarily) lost, while at the same time adding some small aspect to the rigor which is required when you get deeper into the subject.

At that stage you can then start talking about "domain" and "codomain" and "image" (but steer clear of "range" as the term is dismayingly ambiguous), and at that point you can then introduce the idea that a function is actually a triple consisting of domain, codomain and the subset of the cartesian product of the latter -- by which time they are also now probably ready to consider the concept of the general relation.

Sooner or later, many students may well find themselves on a course which explores the abstract concepts themselves: injections, surjections, bijections, etc. and will need to understand the subtleties like the image of a union equals the union of the images, while intersections are more complicated than that ... for physics students it is probably not appropriate to explore this, at least at this stage, but having the proper language for the various "parts" of a function is always useful to have in one's toolkit.