Making Your Own Sense

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

Tag: visualisation

  • Zooming in to see the slope

    A lot of people introduce the derivative at a point as the slope of the tangent at that point, which to me is quite confusing. It seems to me that the reason we want the derivative is that it is a measure of the slope of our actual function at that point, not the slope of a completely different thing. To me, the thing is that the function itself is pretty much straight if we are close enough to it, so when we’re looking really close, saying it has a slope at this point is a meaningful thing to say.

    For years I got at this idea by drawing a little magnifying glass on my function, and a big version nearby to show what you could see through the magnifying glass. But last year I decided to get technology to help me do this dynamically and I made a Desmos graph with a zooming view window that will show you the view on a tiny circle around a point on a curve. (Some of the bells and whistles here were provided with a little help from Andrew Knauft .)

    Here’s what the graph looks like, and a link to the graph so you can play with it yourself: https://www.desmos.com/calculator/pa1cudpc07 

    A screenshot of a Desmos graph. There is a red curve which is the graph of a function, and a small circle centred at a point on the function. A dotted line joins this to a bigger circle showing a zoomed in version of what's in the small circle. There is a zoom factor slider.

    I’ve used this to help students see that the curve itself really does look straight when you’re very close, and so treating it like it is straight there and saying it has a slope is a reasonable thing to do. Students are usually most impressed and love to play with it. It’s also interesting to put in a function with a kink in it like  |x2-1| to show that the kink never goes away no matter how closely you view the function, so having a single slope there isn’t really reasonable.

    Every time I’ve had to search my own twitter account to find the tweet where I shared it , and I couldn’t keep doing that, so here it is in a quick blog post for posterity.

  • Flipping absolute values

    Every semester I talk to students about what the absolute value does to the graph of a function.

    You can read the rest of this blog post in PDF form here. 

  • Obscuring the GST by making it simple

    I was helping out at Roseworthy Campus yesterday as the Vet Medicine students were learning about budgeting for a Vet Clinic as a business. One aspect of this was calculating the amount of the cost of goods and services that was GST (stands for “Goods and Services Tax” – in other countries it’s known as VAT or Sales Tax). The Excel sheet they were working in already had the formula worked in and it was this: GST = (Total Price)/11.

    You can read the rest of this blog post in PDF form here. 

  • 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.