View related sites
I was recently fortunate enough to be asked to travel to Oman to support some maths development work. I’ve little experience of this area of the world so it was a wonderful opportunity to expand my horizons. I enjoyed being immersed yet again in the sights, sounds, smells, tastes and experiences of another culture. I also enjoyed having my own perceptions, mathematical and cultural, challenged.
Discussing mathematics and mathematics education with academics and educationalists from another culture is always a fascinating experience and this week was no different. I’m not going to lie – at times it was stressful, heated and frustrating, but it was also jovial, comical and thoroughly enjoyable. By the end of the week, I had formed new friendships, developed collaborative relationships and widened my own window on the world.
There were obvious challenges in translation and notation, differences in beliefs and pedagogical approaches and often just complete miscommunications. But what an opportunity to explore another perception of mathematics.
A simple example is the term parabola. What does that word mean to you? When do your learners get introduced to it? How? Why? What does it represent? What does it imply knowledge of, or concern mathematically? Does its meaning take shape gradually over years, or in an instance?
For me, the term parabola has always been a name, much like the name of any other mathematical object such as octagon or denominator. It’s the name given to the graphical representation of a quadratic function. It’s the graph y=ax2+bx+c for some real values of a, b and c where a≠0. As such I begin using it with learners graphing quadratic functions, so generally around 12–13 years of age. In my use and conceptualisation of the word, when I hear the term parabola I think of the shape of a quadratic graph, generally ∪ or ∩ shaped. I’m comfortable that learners use the term in the same way, but what am I really expecting learners to mean by this? Digging into what I really mean, my intentions are that leaners recognise that the ∪ or ∩ shape has a maximum or minimum, it is symmetrical, smooth and its gradient gets steeper the further away you are from the turning point. Additionally, my intentions are that during the exploration of quadratics learners recognise that the algebraic representation is y=ax2+bx+c for some real values of a, b and c where a≠0; that the turning point and its value can be found; and that intercepts and, at a later time, the gradient function (or gradient at a point) identified. Some learners may go on to explore conic sections, exploring ideas of locus and directrix definitions of the parabola.
At some point the term parabola makes a shift from a name to something more (which may be abstracted), but I can’t quite put my finger on exactly what or exactly when. I’m realising it’s not even a step change – more a sliding scale. But interestingly, I’m accustomed and happy to use the term early on in that scale from when the graphical representation is first encountered.
For my Omani colleagues, this term was the cause of much concern. For the team I was working with parabolas sit firmly in the domain of conic sections and they were completely unprepared to use the term with learners who were not working in this area of mathematics. They agreed that the graphical representation of a quadratic function is a parabola but importantly were not prepared to include this name in learners’ experience when working with graphs and algebraic representations. For my Omani colleagues, as far as we were able to understand, the term parabola cannot just be used as a name – it’s not a term to be used without a thorough and complete experience of ideas within co-ordinate geometry and conic sections. The term parabola was the name given to parabolas in this context, not the one in which we were working. So, it seems we’ve identified a place where a parabola is not actually a “parabola” – or something like that.
So, what’s in a word? More than you could ever imagine, and what’s in it for you can be so completely different from what’s in it for someone else. That may be something you expect in more interpretive subjects, but not within mathematics, where often things are viewed as black and white, right or wrong, example or non-example.
The more I have these opportunities to work, discuss and argue about mathematics with colleagues around the world the more blurred – no, I’d say the more colourful – my interpretation of mathematics becomes. The effects of this on the way I consider curricula, resources, teacher and my own professional learning are that they are becoming more inclusive of others’ interpretations.
Join the conversation: You can tweet us @CambridgeMaths or comment below.