I believe we develop our understanding of mathematical concepts in two main ways: making connections, and building familiarity. New ideas are never wholly new – they build on previous ideas, or take one aspect of something we know and develop it somehow. But until we spend time messing around with them, our understanding will only remain skin-deep.
Making connections – linking the new idea to prior understanding – will involve well-chosen language, examples and analogies on the part of the teacher. However incomplete or flawed these are, if chosen carefully so as not to actively hinder future development of ideas, they can form a vital (if temporary) bridge to otherwise inaccessible concepts. I have found the ADEPT method outlined on www.betterexplained.com to be incredibly helpful in developing and adapting my own explanations of tricky concepts to students.
Building familiarity – playing around with mathematical ideas – should involve both problem solving and open investigation . Because of the time-consuming nature of play, and the fact that we can’t know in advance what the ‘learning outcomes’ may be, this method of teaching often takes a back seat in the maths classroom. Fortunately this is not the case everywhere. http://www.inquirymaths.co.uk/ champions the practice of providing students with a well-chosen prompt and giving them the freedom to explore the ideas that arise.
Bringing together both of these aspects – making connections and building familiarity – the rich tasks provided on www.undergroundmathematics.org are designed to emphasize the links between A-level topics and also encourage more open investigation than is found in the typical textbook or exam question. The hard part is providing sufficient direction for students to access the problem, but leave enough room for the creative investigative problem solving that maths is really all about.
I recently used a couple of tasks from Underground Maths in a plenary lesson on functions for a year 12 class. By this point in the year, students have worked with functions and graphs in the context of straight lines, circles, trigonometry, polynomials and calculus, and even statistical distributions or kinematic graphs, and a solid grasp of functions will underpin much of the upcoming year’s content, from parametric curves to circular motion, so I wanted to make sure students weren’t just memorizing the rules for translations and stretches, but developing a sense for functions that goes a bit deeper.
This activity encourages students to consider how transformations affect gradient. It provides sufficient direction to be accessible while keeping the questions broad enough to promote discussion – there’s no one correct approach, and working in a group gave them the freedom to discuss possible methods and gain alternative perspectives.
One student used differentiation to investigate the effect of stretches and translations on cubic functions, and was then able to make the next step, differentiating to , proving the same results for a much more general case. We also had discussion about how the same sort of process governs transformations of data (specifically why standard deviation is invariant under translation).
The second part of this activity turned out to be a great way for students to grapple with what changes and what stays the same when functions are transformed. In particular, the fact that functions were only described in terms of a general function meant students didn’t get bogged down in specifics, which can make it hard to keep a clear view of the problem, and could focus instead on how the input and output were affected by each transformation.
Can you find…
Many students of physics already had a good grasp of frequency and amplitude, and were quick to apply their understanding in this area to the mathematical descriptions of horizontal and vertical stretches of trig graphs. Again, students worked in small groups, but were encouraged to use GeoGebra up at the front to test predictions. Some students dived straight in, finding valid solutions straight away for the one graph version, while others weren’t sure where to begin. As they compared their approaches within groups, those who struggled saw methods they could attempt, and those who already had one solution were challenged to find alternatives, and then to generalise further. Some students ruled out horizontal stretches for the tan graph problem because they wanted to maintain the same vertical asymptote, while others tried anyway, and spotted that the right scale factor would simply move a new asymptote into its place. Some solutions were considered more elegant than others, but this was often a matter of personal opinion, and it was so nice as a teacher to replace “No, that’s not the right answer” with “Yes, that also works – and can anyone spot any more? Can we describe them all?”
One student decided that moving a sine wave up by 2 was too boring a solution to the first challenge, so spent a happy 10 minutes looking for ever more obscure ways to solve the same problem. I love lessons where students ask me questions I can’t answer, and it took me some considerable messing around afterwards for me to find an equation that would create the graph one student had sketched:
Unless you get the constants just right, the graph unravels like a broken slinky, and if you try to get too fancy you might just end up with this weird scribble:
This graph prompted a discussion about what it means to be a function, and provided a nice precursor to the whole area of parametric curves. Whether or not it fit the requirements of the question was no longer the issue – the original activity provided a vital prompt, but when it was no longer needed to direct students’ curiosity, it was simply put to one side.
Keeping with the making connections theme, one student drew a periodic graph that wasn’t quite a trig function, and, with a bit of messing around with modulus function and vertical translations, we were able to reverse-engineer a partial equation:
An equation for the endless version was beyond us during the lesson, but the floor function came in very handily when I tackled it again after class:
What’s the difference between an abacus, a calculator and a smartphone?
What’s the difference between , and ?
Some tools do one job, some many, and some have potential beyond what their designers ever imagined. Functions are to algebra what algebra is to number – they provide a wide angle lens that allows us to ‘see’ the algebraic structures and patterns which would be almost impossible otherwise. Now every student with a phone can use GeoGebra to investigate functions, I can’t wait to see the impact on understanding.