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The context

In my country, when the concept of function needs to be introduced in math classes, most teachers will simply talk about $f(x)=c$, $f(x)=ax+b$ and $f(x)=1/x$ (constant, linear and inverse functions). (This is enough for the first year; the following $2$ years deal with second degree polynomials, exponential and logarithmic functions, trigonometric functions, etc.)

The basic properties of functions (domain, range, maximum, etc.) as well as the concept of reciprocal functions ($f^{-1}(x)$) are also briefly discussed.

However, it is rarely explained what functions truly are. Most students don't understand the usefulness of the "$f(x)$" notation versus "$y$" (for example, $f(x)=ax+b$ versus $y=ax+b$). When asked: "explain what a function is", most will say "it's a line", "it's a curve", "it's a graph" or "it's a formula" (these are not too far from the truth, but not quite right), or worse, will not be able to say anything about what functions are.

There is even a scary amount of students who have studied functions for more than two years and still struggle to explain the concept clearly.

Source of the problem

I believe part of the problem stems from the fact that it is never quite explained clearly what functions are. For example, it is never mentioned that functions are a rule that associates elements from one set to another. Incidentally, students never develop a real intuition of what functions do; they think "formula" and not "input" $\Rightarrow$ "output".

Note: I might be making generalizations here, but it is my perception as the same problem was there years ago when I was in school and is still present today.

Example of one way to introduce functions

One example that I've thought of but I'm sure is not optimal is to present a set:

$$\mathbb{L}=\{a,b,c,d,...,x,y,z\}$$

and another set:

$$\mathbb{E}=\{\text{vowel},\text{consonant}\}$$

and invent a function $f$ whose job, whose "function", is to assign each letter to the right category (vowel and consonant).

Then I can explain many things like why $f(x)=y$ works (because for example, $f(a)=\text{vowel}$ but $f(b)=\text{consonant}$). I can also explain that $\mathbb{L}$ represents the domain and $\mathbb{E}$ the range of the function. However, the example would be bad to explain what $f^{-1}(x)$ is and there is ambiguity if I try to say that $f(x)=y$ because really $f(x)=\text{consonant}$; so it is definitely not a perfect example.

The question

Do you know of any simple (think "short"), elegant, unforgettable and ideally "all-encompassing" ways to explain the concept of functions before actually starting to teach about real functions, formulas and graphs?

Thank you for your time and propositions.

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    $\begingroup$ Recommended: The presentation in OpenStax Intermediate Algebra: cnx.org/contents/AndhM9Sd@4.13:XlSGJo8P@2/… $\endgroup$ – Daniel R. Collins Jun 30 '18 at 5:27
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    $\begingroup$ "There is even a scary amount of students who have studied functions for more than two years and still struggle to explain the concept clearly." The function concept is much more modern than most of the mathematics students are taught. My own impression is that most university engineering students don't understand it - this is why things like change of variables and the chain rule is so difficult for so many of them. One has to start with graphs, formulas, and so forth, and slowly build to the notion of mapping. A top down approach will fail. Elegance is not a useful goal at this level. $\endgroup$ – Dan Fox Jun 30 '18 at 7:34
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    $\begingroup$ I'd be very careful about using the word reciprocal for $f^{-1}(x)$ . This is the notation for 'inverse function'. In my opinion, it's very important to not confuse the two, even when students know what they are doing, they should be encourages to use the right words. $\endgroup$ – JTP - Apologise to Monica Jun 30 '18 at 11:04
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    $\begingroup$ In my country, when the concept of function needs to be introduced in math classes, most teachers --- FYI, this is extremely dependent on the country and curriculum. For example, in my experience both as a student and as a teacher, pretty much every high school and early college level text made a BIG DEAL about functions being arbitrary rules, input and output issues, and probably more than half of the texts even went so far as to define functions as subsets of the Cartesian product (of the domain and range sets) that have "the function property". $\endgroup$ – Dave L Renfro Jun 30 '18 at 13:56
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    $\begingroup$ Also, is it REALLY that important at this level to worry so much about arbitrary functions? Besides, why stop there ... why not start with relations in general (inverses of them, compositions of them, etc.), then specialize to functions? And finally, what does "arbitrary rule" mean anyway? I don't think you want to open this can of worms. $\endgroup$ – Dave L Renfro Jun 30 '18 at 14:05
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I think that the phrase "input-output machine" strongly suggests that the function must correspond to a formula. If there weren't a formula, how would you describe how the machine works? It is a machine after all, so when students hear that word they think of the internals of the machine as having to be mechanical and formulaic. So for the sake on not further ingraining the impression that a function has to involve a formula, and so must involve numerical inputs and outputs, I would avoid this concept. Instead, I draw this picture to show students what a function is, and say

A set theoretic illustration of a function, with arrows pointing from the inputs in one set to the outputs in another set

A function is something that takes any object from your set of inputs (point to the oval labelled with X) and will tell you the output (follow the arrows with your finger to the oval labelled Y) that corresponds to it.

And then I would draw this picture for many different examples! The above example inputs a (colored) shape and returns its color. You can draw this picture with your example of the function $\mathbb{L} \to \mathbb{E}$ that classifies a letter as a vowel or consonant. Of course you have to use the example where your function is going from $\mathbb{R} \to \mathbb{R}$; pick a specific formula for the function, and pick some sample inputs and draw the arrow going from that input to its output. I think that drawing a picture like this, and relating this picture to the more specific case of functions $\mathbb{R}\to\mathbb{R}$ that they typically see is the key to getting students to understand the distinction between a function and the graph of a function. There are plenty of other wacky examples of functions you can invent just to hammer home the fact they don't have to be based on numbers and formulas.

While giving the accompanying explanations to any of this, I would avoid all mathematical jargon. Don't say ordered pair, or domain, or codomain, or range. Just get the students comfortable with this idea of a function. Saying set is fine because that word will have non-mathematical meaning to the students.

So when teaching functions for this first time, I would personally spend maybe only day on this, since usually the curriculum demands you talk about real-valued functions rather quickly. But then at each new concept I teach about functions, like inverse functions, and the composition of functions, and one-to-one and onto functions if you get to those ideas, I would pull this picture out again and draw what these concepts mean in terms of this picture. There are very nice illustrations of these ideas in terms of this picture.

A few key points/pitfalls regarding showing students this picture:

  • That picture has only finitely many things in each oval (set), and when you draw this picture you can only draw finitely many things in your domain and codomain. But since many functions have an infinite domain and codomain, like functions $\mathbb{R} \to \mathbb{R}$, it's probably a good idea to put some ellipses in the domain and codomain ovals and to say explicitly in these examples that although you're drawing only finitely many sample inputs and outputs, there are many many more there.

  • The bold word "any" in my explanation above is important. A function has to tell you the output of any possible input, otherwise it's not technically a function. The rigorous way to avoid this is to define a function to be a domain equipped with a rule for assigning outputs (the domain is included in the definition). But most elementary math textbooks ignore this, like when they present classic exercises such as, "What is the domain of the function $\log(x)$?".

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    $\begingroup$ I like this very much. What I was aiming for specifically was to hear about these wacky examples (is there one that stands out more than the others?). I think it's pertinent to at least explain that a function doesn't need to be $\mathbb{R}\rightarrow\mathbb{R}$ before saying "okay, now we just deal with real functions". Your approach is basically the one I think would help. Don't use the jargon right away, show the idea. And like you said, we don't spend hours on this; it's just a preamble. $\endgroup$ – orion2112 Jun 30 '18 at 19:28
  • $\begingroup$ I have yet to establish wacky examples of functions that I come back to each time I teach functions this way; I usually just invent new ones each time. You examples is good because it shows that functions can be used to classify things (is it a vowel or consonant). Related but similar, my example picks out a trait of an input (it color), which if you know about object-oriented programming can be thought of as a function that returns one of an objects member variables. (I've thought about much of this while teaching Discrete Math to computer science majors) $\endgroup$ – Mike Pierce Jun 30 '18 at 19:45
  • $\begingroup$ An example of an invertible function I suppose would be the function that takes a book in a library and returns it's call number. A head-scratcher you could give students would be this: Consider the function from the set of all people on earth to the set all people on earth that for each person, the function will output the person nearest to their location. Is this function invertible? $\endgroup$ – Mike Pierce Jun 30 '18 at 19:45
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An expectation that there is a simple and elegant way to teach the concept of function seems unrealistic. There is much literature about this. The APOS (Action-Process-Object-Schema) theory of Dubinsky et al. holds that the student’s concept image of function must evolve and undergo revision before “functions” can be “objects.” There are many references, one of which is APOS Theory: A Constructivist Theory of Learning in Undergraduate Mathematics Education Research, by Ed Dubinsky, Michael A. Mcdonald.

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  • $\begingroup$ I certainly agree that functions (and probably many other subjects in math) must evolve and can't completely be taught in one year. I'm not expecting perfection, just a better basic understanding. As I mentioned in my post, I'm noticing many students have, at best, an unclear understanding of functions. It seems to me it shouldn't take much to give them a stronger basis on which to build upon. A few good and strong examples should do the trick (hence this post). There is always room for improvement and we shouldn't stop here just because someone wrote that it's unrealistic. $\endgroup$ – orion2112 Jul 1 '18 at 23:22
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I'll be provocative and claim that the modern notion of function is not inherently more difficult to learn than other concepts in mathematics. What makes it really difficult is an additional man-made problem: we don't practice what we preach in front of students! This has historical reasons, and is hard to fight against.

I've written about this elsewhere, but I'll summarise: for more than 200 years (since Bernoulli until ~1930), the word function was not defined as a rule that transforms inputs to outputs. Instead, the official definition involved a relation between two things $a$ and $b$ and went roughly like this:

$a$ is a function of $b$ iff the value of $a$ is uniquely determined as soon as the value of $b$ is given.

This is still the predominant use of the word function among physicists, engineers and other scientists (including some mathematicians). Think of expressions like "the area of a circle is a function of the radius", "the temperature of the gas is a function of the pressure" etc. Moreover there is a long tradition (at least since Euler) of dropping the "of ..." when it's clear from the context, and just calling the temperature a function etc. But notice that the area or the temperature are definitely not rules that transform inputs to outputs. This follows from the fact that the area of a circle is not only a function of the radius but also of the diameter or of the circumference etc. So whatever the $a$ in the original definition is, it's not a function in the modern sense.

Add to this the very popular notation $y=y(x)$ which goes back to at least 1830 (and was meant to express that $y$ is a function of $x$, not that $y$ is a rule that maps input to outputs) and I think the confusion for students must be quite big. How for example should they interpret something like $v=v(t)=v(h)$? Such confusions are are further supported by notations like $\frac{\partial f}{\partial x_1}$ for the partial derivative of a "modern function" $f:\mathbb{R}^n\to \mathbb{R}$.

I've been trying to address these points when I teach calculus to engineers, but I've found that students already arrive with all these contradicting uses of the word function and the notation f(x) in their head. They also constantly see these contradicting uses in their other courses. So it would take me significant more time to get things straight in their heads. Even if I'd manage, I suspect they still would not fully understand what they see in other courses. So I'm rather pessimistic about the effectiveness of the well meant suggestions you have received here.

Finally, let me mention that Euler wrote his whole Institutionum calculi differentialis without using the $f(x)$ notation (as far as I can tell). So I'll make another provocative claim: the $f(x)$ notation is not more useful than the $y$ notation. At least not for those parts of mathematics which most engineers or natural scientist need.

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    $\begingroup$ Interesting read. I guess the $y$ vs $f(x)$ notation really is a matter of personal preference. I prefer dealing with $x(t)=x_0+v_0t+\frac{a}{2}t^2$ than with $\Delta x=x_0+v_0\Delta t+\frac{a}{2}(\Delta t)^2$ for example, at it's clearer with respect to what I can calculate derivatives. I agree with you that functions shouldn't be harder to learn than, say elementary geometry, provided they are taught "the correct way". I was deploring the fact, as you say, that teachers don't "practice what they preach" and, to my opinion, don't actually explain what functions are, thus all the confusion. $\endgroup$ – orion2112 Jul 1 '18 at 23:10
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I would be careful not to generalize from your own situation (good or bad). Many courses teach the concept of the input/output machine and teach kids multiple ways of thinking of functions:

An input/output machine An equation of y and x An equation of f(x) and x A graph A graph that has no more than one y for a given x, unlike relationships like circle graphs (more precise) Ordered pairs Probably some more

My recommendation is to introduce the concept with the input/output machine. It is the most intuitive and most related to the word function. Ordered pairs is probably the least intuitive. All that said, eventually students should be able to think of a function in several ways. But pick your battles and the timing of them.

There is a lot of math to teach (and sometimes definitions and concepts are clearer in retrospect) so don't think that killing yourself over the definition of the function is the most important thing. Get the concept across and then start teaching manipulation, graphing, calculus, etc.

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  • $\begingroup$ I agree with you that the input/output machine is pretty straightforward. I would definitely not talk about ordered pairs; this seems like a huge detour to the subject at hand. $\endgroup$ – orion2112 Jun 30 '18 at 17:35
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    $\begingroup$ @orion2112: And yet, that essentially is the subject. $\endgroup$ – Daniel R. Collins Jun 30 '18 at 18:30
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I use the example of a vending machine to describe the concept of a function, v(b), where "b" is the value of the button that is pressed and v(b) outputs the type of food you receive.

A vending machine has clear inputs: Buttons that can be pressed. It has clear outputs: The food that it dispenses. The rule can be summarized by a table: Press this button, get this soda.

You can talk about domain and range: Is there a button labeled "X" on this machine? No? So "X" is not in the domain at all, and v("X") is undefined. Perhaps the vending machine itself tells you that there is no valid output for button "C". You can then say that v("C") is undefined, even though you could think about trying to press "C". Can you get coca-cola from this vending machine? Perhaps it's a pepsi machine, so coca-cola is not in the image or range of this function. Is it an invertible function? Perhaps there's actually three different buttons that will give you Pepsi, so it isn't.

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Suggestion: Introduce calculator functions. Very concrete. Use function notation to record the results.

Examples

sin(1) = 0.017 (rounded)

sin(2) = 0.035

log(1) = 0

log(2) = 0.301

Then introduce simple functions like f(x) = 2x.

f(1)=2

f(2)=4

Try substitutions:

f(f(1)) = f(2) = 4

Generalize: For any real number x, f(x) is a unique real number. (Logically, "unique" is redundant here. Added for emphasis.)

If x=y, then f(x) = f(y), by substituting in f(x)=f(x).

Converse not necessary true, e.g. g(x) = x^2. g(-1) = g(1)

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  • $\begingroup$ The upside is that it's a "hands-on" activity in which everyone can participate. The downside is it's not concrete enough. Also, chances are some kids will think that functions absolutely have to do with sine and log (but its the other way around) and this could lead to even more confusion. $\endgroup$ – orion2112 Jul 1 '18 at 23:25
  • $\begingroup$ @orion2112 What could be more concrete? You have a physical machine in your hand, not just a drawing or a table of values. You physically key in the input. You physically press a button on the machine to get it to display the result. Then you can "invent" your own function, e.g. f(x)=2x as in my suggestion. Then maybe f(x)=x^2 or 1/x, for which there should be buttons on the calculator. $\endgroup$ – Dan Christensen Jul 2 '18 at 14:11
  • $\begingroup$ I was referring to sine and log. They'll understand that there is an input and an output, but not what the output is concretely. I prefer your example perhaps with the $x^{-1}$ and $x^2$ buttons. $\endgroup$ – orion2112 Jul 2 '18 at 16:01
  • $\begingroup$ @orion2112 The problem seems to be associating an abstract symbol like sin, log or f with different functions. Build on their familiarity with calculators.I understand they don't have trouble with 1/x; they have trouble with f(x). Using a calculator, they will have inexhaustible supply of examples like sin(0)=0, log(10)=1, etc. And they can make up examples like f(x)=2x, g(x)=x^2, etc. $\endgroup$ – Dan Christensen Jul 2 '18 at 16:54
  • $\begingroup$ I don't remember students in high schools having so much trouble with the notion of a function. On calculators, they even have "function buttons." You enter a number, press a certain function button, and you will get a result. If, later, you enter the same number and use the same function button, you will always get the same result. $\endgroup$ – Dan Christensen Jul 2 '18 at 17:13

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