"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

Pat Thompson has done research on students' and teachers' understandings of functions and variation. (He's a co-author on the article mentioned in this answer, though he's not listed for some reason in the link you gave in your comment.)

In a short 2013 article, Thompson calls the statement $C=f(x)=3x+2$ a misuse of notation that can lead to "[confusing] the idea of function and variable." In the same article, he relates this to when students see a function definition like $V(u)=u(13.76-2u)(16.42-2u)$ and think that the name of the function is $V(u)$, i.e. $V(u)$ is simply a label for the right-hand side. He also makes the following claim about function notation that agrees with the sentiment of your answer to another question.

A primary source of students’ difficulty with function notation is that they only see it where “𝑦” could be used just as well. The textbook says they must use 𝑓(𝑥) when there really is no need for it. They rarely see function notation used in settings where using it actually enables them to do things that they otherwise could not.

Yoon and Thompson (2020) offers several more references on students' and teachers' difficulties with function notation.

Sajka’s (2003) case study involved a 45-minute interview with one high school student, Kasia, who had seen function notation for three years. [The study] showed Kasia moving from thinking "𝑓" means “the beginning of a function formula” to thinking of “𝑓(𝑥)” as serving the same role as “𝑦”... Sajka’s report that Kasia understood “𝑓(𝑥)” as the start of a function formula is in line with Thompson’s (1994) report of students thinking a function definition consists of the “𝑓(𝑥)” on the left side, the symbol “=”, and an algebraic expression on the right side. Thompson (1994) also pointed to this way of thinking as behind the common mistake of mismatched letter on the left and letter on the right, such as 𝑓(𝑥) = 𝑛(𝑛 + 1)(𝑛 + 2).

Musgrave and Thompson (2014) and Thompson and Milner (2019) shifted focus from students’ to teachers’ understanding of function notation. Musgrave and Thompson (2014) and Thompson and Milner (2019) found that, for many teachers, function notation served as a label or a name for the defining formula rather than a representation of one quantity’s values in relation to another quantity’s values. While students meaning of function notation as a label might be a root of so many reports of students’ difficulties with it (e.g., Carlson, 1998; Dreyfus & Eisenberg, 1982; Vinner & Dreyfus, 1989), it is important to investigate the possibility that teachers hold similar meanings.

Yoon and Thompson's study asked teachers in the US and in South Korea to respond to tasks involving function notation, and they give many examples of what I think aligns with your idea of "blurring between functions and variables."

A large majority of US teachers seemed to think that “𝑣” in 𝑐(𝑣) is a part of the function name and they were therefore free to use other letters in the function’s defining rule. Thompson (1994, 2013b) suggests a reason for the teachers who used 𝑠 or 𝑡 at least one blank in Level 2 and Level 1. Teachers thought of function notation as a four-character symbol that is used in place of the letter “𝑦” (Thompson, 2013b). Teachers who filled the blanks with 𝑠 or 𝑡 might consider “𝑤(𝑡)” as one symbol because they thought they could replace “𝑤(𝑡)” with “𝑦”.

Teacher 1’s response... tells us that she thought 𝑞(𝑠) was one inseparable symbol instead of thinking 𝑞 is a function’s name and 𝑠 is an independent variable... Although she said 𝑞 is a function of 𝑠, she thought 𝑞 always accompanies with 𝑠, and viewed 𝑞(𝑠) as one entity.

Teacher 1 first wrote 𝐴 = π𝑟² , and then added (𝑡) after 𝐴. Her final answer was 𝐴(𝑡) = π𝑟² . She used function notation only to represent the area on the left hand side and used 𝑡 in the function notation and r in the defining rule. Her statement “when I am using function notation I am thinking 𝐴(𝑡)” and “I was just noting the area of the circle which was what I was starting” is consistent with our hypothesis that teachers who used function notation only to represent the area think of 𝐴(𝑡) as one symbol, and 𝐴(𝑡) is a label for the formula on the defining rule.

One of the study's interesting conclusions is that

SK data shows that US teachers’ difficulties when reasoning with function notation are not due to epistemological obstacles to understanding meanings of function notation. The SK data suggests that US teachers’ problematic responses are a systemic aspect of mathematics education in the U.S. Thus, it is plausible that US teachers convey problematic meanings to students unintentionally.

In Thompson and Carlson (2017), the authors discuss the concepts that you call "function" and "function of" and argue that many student difficulties stem from an overemphasis in school mathematics on the modern, set-theoretic definition of function and on a static conception of variables. Instead, they argue that there should be more focus on the notion of what you call in this MO question "variable quantities."

We argue that ideas of continuous variation and continuous covariation are epistemologically necessary for students and teachers to develop useful and robust conceptions of functions. Put another way, we argue that variational and covariational reasoning are fundamental to students’ mathematical development. We ground this claim in research that highlights difficulties students experience regarding function relationships by not having the ability to reason variationally or covariationally and in research that shows productive shifts in teachers’ and students’ conceptions and uses of function that result from reasoning covariationally.

They review U.S. and Japanese school mathematics textbooks and find that

Our cursory review of 17 U.S. secondary precalculus level textbooks ranging from algebra 1 to precalculus, revealed, consistent with Cooney and Wilson’s (1993) textbook review, that all the textbooks used a correspondence definition of function. The research we have cited in this chapter further supports the fact that U.S. curriculum and instruction are failing to develop students’ quantitative and covariational reasoning abilities, contributing to many weaknesses in students’ conceptions of fundamental mathematical ideas, such as variable, function, and rate of change, that are essential for understanding calculus and modeling dynamically changing phenomena in the sciences and engineering.

The Japanese primary mathematics texts have a clear, coherent focus on having students think about quantities whose values vary and about ways that quantities’ values vary together. By high school, Japanese texts’ authors presume that students think with images of variation and covariation and rely on this assumption as a matter of practice. In contrast to Japanese textbooks, many popular U.S. textbooks do not emphasize or support students in conceptualizing quantities and viewing function formulas and graphs as representing how two varying quantities change together. The idea of variable often is presented as representing a single unknown value.

Based on this, I think there is substantial support for the claim that the "function" vs. "function of" issue and the "function" vs. "variable" issue do indeed affect student understanding, and that teachers' actions contribute to the confusion.

Answer 2

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 3

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 4

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 5

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.