- 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.
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.
Thus the lesson would always begin with an "ELI5" overview of the essential concept (and indeed, I was between 6-18...), involving drawings and a lot of gestures, but more importantly was always based on a real-world tangible use-case/example.
Typically, simple physics. That's how I learned most math: as a tool to solve some problem I could 'feel' (often even solve mentally for simple/extreme cases, which is great to build initial understanding). To this day I think¹ of maths mostly in visual terms.
It helped me "internalize" or "own" mathematical objects exactly the way you seek for your students to understand. So that's my recommendation: make them 'feel' it through real stuff.
Below (in ““quotes””) is just one extremely simple example of how I'd do it (it's about going from a tone to the same tone 1 octave higher in music). [my remarks in sq. brackets]
““A musical tone is defined by its frequency in Hertz: for instance, 440 Hz is an
When you double the frequency, you obtain the same tone, just 1 octave higher: so 880 Hz is an
A too, higher in pitch. We usually number the tones to know which octave on a piano, 440 Hz is
A4 and 880 Hz is
So when we go from
A5, we apply a function which doubles the frequency. That's what going 1 octave higher "does".
Let's write this in math:
we define the function
Oct of "going one octave higher" as:
Oct(x) = 2x with
x the frequency of a music tone.
[notice, "going" is a verb, as it should: it expresses a relation between nouns, between 'things' aka variables; also we do away with
F and go more literal to open the mind to meaningful abstractions — thereby laying some groundwork for an other important metaconcept, how to select/create/cookie-cut variables to solve problems]
If we input
x=A4 we get twice that in result:
That's the equivalent of moving your finger 12 keys to the right on a piano, or some length down on a cello: that's what this function does in those contexts.
[bonus points if you can show them the movement of the arm, the three physical elements:
A5; in other common words
Notice that, we can take any tone, we get the same tone 1 octave higher: this particular function works for all pairs between/from the origin space (say x is bound to octaves 0-7) and/to the destination space (
Oct(x), the 'y's (all of them), would thus be bound to octaves 1-8).
Forget about an "equality" between some terms of x and y, only consider the movement from x to y: a relation that takes x and transforms it into y. This relation, this transformation, this change is called a function.””
The important part here, imho, is to de-associate the function itself from variables. Have them see:
- a "definition space" (terms like 'origin' or 'from' work well because they prepare concepts like bijection, surjection etc).
- a "result space" (again, 'destination' or 'to' work well)
- a "connection" or "relation" between the two, a way to get from 1 to 2, from the origin to the destination.
A function is a movement, a process, a change, a variation, a computation, a derivation, an integration, a representation, an abstraction. A fucntion is a verb, an action, something we do. It's not a thing, it's what changes things. It's a very-transparent-box that takes stuff in and chews some other stuff out.
Have them think like:
- don't ask "how much of x is equal to y?" because that's entirely static, there is no function (verb, change) even if y=2x
- rather wonder "what do I need to do to x to obtain y?" and that's an action, a verb, you take x and change it into something else and call that thing y.
- Well, this change is a function.
Later on you can expand on the idea of "for any X" (domain/space of definition).
You could elaborate further on the musical (or whatever) example you used in introduction:
““Now you can also see how there are tons of possible functions in music to describe going from some tone to some other, and these variations more than the tones themselves are what makes music: for proof we can totally change the tonality of a song (play it in "D" instead of playing it in "C") and it's still the same song to everyone listening: because we're in fact listening to the functions, the variations).””
I'd caution against using non-static variables (involving time, progressions), it may be confusing in this particular context (you want 'x' and 'y' as static as can be, like music tones don't ever change by themselves: the only 'moving part' in the picture should be the function).
I don't know where you're going with this (I'm not American so I don't really know in which order you learn math topics), but there are interesting functions to visualize. I personally like:
i as a new object defined by a function (
sqrt(-1) ). How cool is that.
- Complex functions, rotations in the complex plane etc.
Plug with trig or not (I'd say as little as necessary in this context) or compare x,i with traditional x,y linear, show them how functions can appear under different forms yet may remain identical (or close enough for the "feel" like multiplication by i; and how it's also just a rotation of coordinates themselves if you'd rather fix variables in the plane).
[note: also great electricity math to be done with these, and current works much like water for the most part (e.g. "tension" is a difference of potential between two points, much like gravity/altitude on Earth) so it's very easy to picture the effects of some well-crafted circuits/functions, mentally.]
- "Divide by two" is more complex but more fun that our
2x example above.
Like halving 1 infinitely approximates to 0 (or 1/2+1/4+1/8... goes to 1).
- Likewise the
α=1+1/α equation (https://math.stackexchange.com/a/315376). Generally these infinite expressions can be represented as infinite compositions of the same function, infinite recursions, and these kinda 'fractal' movements or evolutions can not only be mentally pictured but via computers as well (or even simple pieces of paper, add one half of the remaining half until you can't and everyone gets the point). Think animation, step by step, whatever the medium.
Tell the story of "this specific function" in "this specific context", then shift one (function or context), and again, and again, to further isolate the set of variables from the set of functions, in the student's mind.
I'm sorry I took so long but I wanted to arm you with all the terms, views and 'tricks' I could think of, so you can choose. I feel that's easily in the top 5 most important concepts to really grasp in math and any applied field.
¹ Although this is slightly off-topic, I need to plug 3Blue1Brown's video on the Euler identity, it's a visual graal if there ever was one on a math relation.
Edits: clarifications, typos