24

Start by talking about functions in general, not only about functions that can be expressed by a simple formula in x and y. Examples: The function that maps every non-empty list to its first element. The function that maps every finite set to its size. The function that maps color names to RGB triples. The function that maps days to sunrise times at a ...


23

You might remind them that $y$ is just a name for a number. When they draw a plot, they draw a bunch of points: maybe $y=3$ here, $y=5$ there, and $y=-2$ over there. But at some point (no pun intended) we want to talk about the entire shape: we want to say that $f$ is symmetric, that $f$ is concave, that $f$ has an asymptote. We can't do that with $y$; ...


20

You should tell them these two main benefits: (1) Function notation is concise! For example, instead of writing "Find $y$ when $x=5$" one can simply write "Find $f(5)$" This becomes very appreciable when dealing with long or complicated problems asking for a lot of information. We also shorten things like this all the time. For instance, instead of writing $...


16

The crucial thing the students need to realise is that the (e.g.) $x$ that turns up in the function definition is a bound variable. That's what allows it to be freely renamed or indeed omitted without changing the semantics. Unfortunately, education tends to completely obscure this facet by a) always using the same dumb variable names as if there were a ...


15

Because x and y are just variable names It happens that sometimes y=f(x), but other times z=f(x,y), w=f(x,y,z), or x=f(y) for that matter. All of these variable names are syntactically equivalent, and the mere existence of "x" and "y" in an equation does not necessarily connote that "x" is the independent variable and "y" the dependent. Thinking of the ...


11

TL;DR: A function is a verb. It's an action. Variables are nouns, objects. Verbs (functions) connect nouns semantically, i.e. how A (or x) relates to B (or y), how to get from here to there. Long version Some context: I learned maths from my father who was a physics / engineering guy at heart, so everything had to be 'tangible' or 'observable' to him. ...


8

(First, I should mention that I've never taught this, so my approach does not come from experience.) So you have students who think of something like $y = f(x) = x^2 + 3$ as a relationship between two “specific” quantities $x$ and $y$. As intuitions go, that's not so bad: it serves physicists quite well. But it's incomplete, and you're looking for ways to ...


6

I have never worked with students of that skill-level, so take this with a grain of salt. I like to thinks of functions as values, just a different kind of value from numbers. This can help demystify stuff like $\circ$ as it just like $+$, except it works on a different type of value. Once you get to vectors you also have a very nice parallel, since they ...


5

Prior to my final year of high school, I was sent to a maths tutor for a couple of sessions, to give me a headstart on calculus. It helped a lot. He introduced me to the concept of functions. He described it as a monster, living inside a box, that accepted a thing through one (!) tube, and pushed out a thing through another tube. The monster was consistent -...


5

I've noticed a few issues when students solve problems of the form, "Find the inverse of this function", and not all of the issues are necessarily because of the students' misunderstanding of what an inverse function means! Misunderstanding/forgetting the "one input $\to$ one output" defining feature of a function. This issue arises because implicitly-...


4

From a comment by the OP: I'm trying to come up with "plausible" wrong answers for a multiple choice question about finding inverses. Per an answer given to this question, you might be able to collect data on your students' possible answers by giving them a fill-in-the-blank quiz on inverse functions. Then, keep track of the most-common wrong answers by ...


3

Different notation for different things The key thing that the students seem to be missing is the conceptual distinction between f and y (in this example), so this apparently needs to be explicitly explained to them. IMHO the way to go at this is to tell them that there are two "things" that we may want to talk about - the transformation process (the ...


2

Function notation is a next step in mathematical maturation. In the language of Dubinsky et al., your students are in the process of encapsulating functions as primary objects. 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 ...


2

One example that comes to mind is modelling the position of a diver (or of a diver's head). Let $t$ be the time elapsed since jump. Let $d(t)$ be the diver's distance from the water and define $d(t)>0$ to be "above water" and $d(t)<0$ to be below water. (It should be obvious why $d(t)=0$ represents being "at the surface"). If we use a quadratic model, ...


2

Looking at the analogy in your question, suppose someone was confused about whether mymoney = yourmoney + 1 made me richer, or you richer, compared to mymoney = yourmoney. How would you help that person understand? I think this is pretty clear: you would tell them to try some values. Ask them: If mymoney = 5, what is yourmoney in the two cases? Now looking ...


2

From a computer science perspective understanding that functions are first class objects is also pretty difficult. There are cases where functions can be parameters to other functions, the classic example being sort accepting a compare function. This case would be impossible to explain using just the y output. In the case of sort you don't even need to know ...


2

To answer the question of why you need f... ... have them consider the region between two graphs. Unless you have a way to distinguish between the different y values, you're going to be hopelessly lost. Now you don't need to use f... you could use subscripts: $y_{1}, y_{2}$ (and in fact, that's how graphing calculators handle it). But it's nice to ...


2

Well... $ \def\zz{\mathbb{Z}} $ Let $f : \zz → \zz$ defined by $f(n) = n+1$ for every $n∈\zz$. Then $f(0) = 1$ and $f(f(0)) = 2$ and $f(f(f(0))) = 3$ and so on. It is now obvious why having functions as first-class objects is useful, since we can repeatedly apply them. Similarly, the Mandelbrot fractal is defined in terms of iterating an elegant function.


1

Many answers already, so I'll keep this one short: it has been realized by researchers in didactic that one difficulty in the concept of function is that it changes status: at first each function is considered as a process (a verb in @ΦDev's answer); they meet several of them, each being akin to a (unitary) operation, not very different from addition or ...


1

Some rudimentary programming exercises might make it obvious why it's useful to encapsulate functionality. When you write y = f(x) in Python, for example, it's clear that y is just a static result, while f is the thing that does the work. You can't reuse y to change another variable z in the same way - you have to refer to f to do that.


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