When students learn about functions at school, we spend a lot of time forging the connection between functions and graphs. We plot individual points, and we find x-intercepts and y-intercepts. We use graphing software to investigate what the coefficients do to the graph, and discuss shifting along the x-axis and y-axis. We make reference to the graph to define derivatives and integrals. Some teachers help students to recognise from the formula of a function what general shape its graph ought to have, such as recognising that a quadratic function must have a parabola-shaped graph. (I wish this last point was much more strongly pushed, actually.)
However, there is a problem with all of this that students come to think that functions are graphs you can draw. Their idea of a function is a curve drawn on a piece of paper, or at least something that can be drawn as a curve. And this is can cause some serious issues later on.
The first problem comes when they investigate certain pathological functions where the graph is not drawable, but they are still perfectly good functions. For example, consider the Dirchlet function which is 1 when the input is rational and 0 when it’s irrational. It’s a perfectly good function but good luck trying to draw it!
This one’s not insurmountable — students can usually imagine a graph of two ghostly lines with the property that a vertical line which meets one of them in an actual point misses the other. They’re just extending their definition of what it is to “draw” when they draw a graph.
The real problem comes when we move on to functions where the inputs and/or the outputs aren’t ordinary numbers. The simplest case is a function like f(x,y) = xy. This takes a point in R2 and produces a number. Many students struggle to understand these functions because they don’t have a way to draw them. “A function is a curve”, says their experience, but where is the curve here? We get around this by extending their picture of what it means to “draw” a graph: We locate the point (x,y) on a plane and then the output we draw as a vertical height or depth. What this produces is not a curve but a surface.
This is a good start, but unfortunately at this point we also often tell them about level curves (or indifference curves if they’re in Economics). Many students at this time simply come to see a specific one of the level curves as the function itself, instead of all the level curves together as a description of the function, because their experience says that a function is a curve.
And now the real trouble starts: what about a function which takes a vector of 3 or 4 or 7 variables and outputs a number, like they meet in microeconomics or statistics? We don’t have enough dimensions to “draw” the graph then. And what about a function that takes a vector in 3D and produces a vector in 2D, like they meet when doing linear transformations in Maths 1B? And what about a function that takes a real number and produces a point in 3D, like they meet in geophysics? And what about a function that takes a complex number and produces a complex number, like they meet in Engineering Maths? What hope do those students have of understanding functions like that when their only understanding of function has an x-axis, a y-axis and a curve?
These functions are most emphatically not graphs, at least not in a way that you can draw. (I can hear pure mathematicians saying something about the definition of function being a subset of the cartesian product and hence essentially a graph, but you can’t draw it can you?) At the very least they are certainly not the curves students are familiar with!
I believe we need ways to represent functions that don’t involve drawing a curve on two axes, even for functions that can be drawn this way. When we introduce non-curve functions we place a huge burden on the students’ imagination, which can prevent them from understanding what’s going on. My idea is that if they can be familiar with multiple ways of imagining an ordinary number-to-number function, then new types of function will be a little less alien to them, because they will have ready-made ways to imagine them.