Making Your Own Sense

Reflections on maths, learning and maths learning support, by David K Butler

Tag: concepts

  • A function is not a graph

    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.

  • Vector philosophy

    What is a vector?

    When students first meet vectors they are pretty much told that vectors are arrows. They move arrange the arrows head-to-tail to add them, and they lengthen the arrows when they multiply them by numbers. Sometimes the vector is represented using coordinates, but they are told that this is shorthand for the arrow that goes from the origin to that point. Lovely.

    And then, in first year maths at uni, they are told that vectors are a list of coordinates. To add them, you add the coordinates, and to multiply them by a number, you multiply each coordinate. You create sets of vectors using equations, which are represented as coloured regions in space – a point is coloured in if its coordinates satisfy the equation. The only way to make sense of this is to think of your set of coordinates as a point, not an arrow. Oh dear.

    So which is it then? Are vectors points or arrows?

    There are at least four answers to this:

    1. Vectors are points, and the arrow picture is simply a way to helps us visualise what the result will be when we add them and multiply them by numbers. So when I see a set of coordinates, I think of the point in space they represent, and when I draw a set of them, I just colour in the points. And when I add them I am just using the rules for how they add to find the answer, but I can visually picture where in space the answer will be by imagining them as arrows.

    2. Some vectors are points, and other vectors are arrows, depending on context. So when you add two vectors, the first one is a location – a place to start, if you will – and the other one is an arrow – a direction to go. The result is a the place you get to by starting at the point and moving the arrow, and is of course a point again.

    3. Vectors live in two separate worlds, one where they are points, and another where they are arrows. There is a world where vectors represent locations in space and they happily find themselves inside our outside sets. But there is another world, where vectors are arrows and they happily arrange themselves head-to-tail when they want to add. When you have two points, and you want to add them, they quickly move over into arrow-world, do their adding, and then the answer comes back as a point again.

    4. Vectors are neither points nor arrows. They are mathematical objects with no innate physical reality at all. The concept of drawing a vector as a point or an arrow is simply a way to represent them on paper and it is not what they actually ARE.

    And which answer do I think is the right one? All of them, none of them – this sort of discussion is philosphy, not maths, and in philosophy there’s no right or wrong. You pick what works for you. Indeed, being able to see it from different viewpoints only enriches your understanding.

    Still, I do quite like the “separate worlds” idea.

    This comment was left on the original blog post: 

    David Roberts 23 July 2013:

    I think the main difference is that a vector as an arrow is an inherently geometric idea: you draw them, move them around the page and so on. The list of numbers is more algebraic, and more static: you can’t pick up (2,5,1) and stick it on the end of (22,7,0) like you can with those vectors represented as arrows.

    There are really different things going on when you move vectors as arrows around the page that don’t happen in the list of numbers approach. The trick is that those concepts don’t get formalised for several years after R^n is formally introduced, and are usually couched in a much more complicated and abstract setting.