[ColinWright]

People think of "drawing a graph of a function" as drawing a line, so if y=sin(x) they end up with a wriggly line going infinitely far left and right.

What they don't think of is that the line is actually the collection of points (x0,y0) such that the equation y0=sin(x0) is true. Kids in high school don't think of the graph of a function as being a subset of points of the plane, being:

  { (x,y) : y=sin(x) is true }

Realising that opens the doors to equivalences between different ways of thinking. We can think of a permutation of objects as both the act of permuting them, and as the result of applying that permutation to the default initial position. We can think of a vector (4,6,9) as a location in space, and as the movement to get from (x,y,z) to (x+4,y+6,z+9). We can think of "3" as a location on the number line, or as the action of adding 3 to something, or as the action of multiplying 3 my something, and so on.

We can think of the graph you draw as a line, or as a subset of the plane, and we shift effortlessly between them, deliberately blurring the distinction, and from that blurring can come power.

Does that help?

[sago]

This hurts when they deal with equations like

x^2 + y^2 = k

and can't even begin to understand how you could graph something like that.

It took me a while to grok it, so I second his experience of this not being well stressed (30 years ago, admittedly). I did further maths A-level (and got an A), but didn't grok it until uni.

[bitwize]

Having long been fascinated with computer graphics this bit was easy for me to get. I imagined a computer raster-scanning the plane and plotting a point when the condition in the equation was true. I just had to scale it up in my head from screen resolutions to infinite resolution and infinite extent and... oh boy, this is already starting to make my head hurt...

But the gist of it I got. And I admit, I'm unusual.

[xpinguin]

Thanks! Seems like I'd been getting it right since the beginning, so I've missed the point.

[kbenson]

In my first calculus class, they introduced integrals to us through Riemann sums, and ever increasing numbers of regions for those sums. It makes thinking in this manner somewhat easier to come by, as it was touched on in the introduction.

[thaumasiotes]

> Kids in high school don't think of the graph of a function as being a subset of points of the plane

This is literally the entire point of graphing inequalities.