One of the most delightful aspects of working at Cambridge Mathematics is the diverse range of queries about mathematics we receive in our inbox. Recently, we received a lovely example of this, where we were asked to weigh in on what the definition of an ‘edge’ is and therefore how many edges a cone ‘should’ have. As with any such question, our first port of call was a coffee and a chat – and here are the issues we discussed.
What is mathematics and what is it for?
The answer to this seemingly innocuous question actually goes quite deep, in that it depends on your view of what mathematics is and what it is for, as shown by my colleague Lynn Fortin in her blog ‘What is mathematics?’. Broadly, Cambridge Mathematics sees mathematics as connected, creative and joyful, meaning we resist oversimplification and learning in a procedural way. This includes the rote learning of definitions in a way that does not question or challenge them, in particular what they are for. We also resist the idea that mathematics is about right and wrong, black and white; rather, we see it as containing much more interesting uncertainty than perhaps has been communicated in the past. That means we also challenge the use of questions such as ‘How many edges does a cone have?’, especially for assessment, because it is a closed question with an implied right or wrong answer which does not necessarily further the student’s understanding of mathematics.
Ambiguity
The very ambiguity of what could be construed as an edge on a 3D shape is the point for us, and depends on what you would be using it for. Whether or not mathematicians think it is ‘right’ or ‘wrong’ to count the edges of a cone in a certain way is something we are trying to move away from; instead, we are asking questions such as how and when might we use different definitions for different purposes, and what might be the consequences. In this example we might need to name an edge if we were constructing a cone (such as an ice cream cone or a funnel) but equally we might never need to use that word at all. Similarly, whether or not a face ‘can’ be curved or not depends on what you are using it for – this is a point of some disagreement among mathematicians. There is rarely a single, agreed-upon definition for any mathematical object or process, although some have less flexibility than others. This is also partly because we tend to add nuance and detail to the definitions we use as we develop our own mathematical understandings and study mathematics further – if you asked a primary school student, a biology undergraduate or a mathematics professor what ‘the definition of average’ meant you are likely to get very different answers; even if you asked several members of the same group we would expect some variation! It is also true to say that definitions are not used consistently across different geographic areas of world – our former colleague Rachael Horsman wrote about this in her blog ‘Where in the world is a trapezoid a trapezium?'.
Definitions in service of arguments
It is important to say here we are not advocating that anything can be an edge (for example) but that a clear and logical definition should be in service of a particular purpose, meaning that multiple definitions can and do co-exist within the system of mathematics, and across countries and levels of mathematics. In terms of geometry, we have found in our work that finding the properties of a shape is one way to define it, but another way is to think about how to generate the shape itself. In the case of a cone, that might be beginning with a flat shape or net, which likely does have an edge, but then the process of constructing the cone might change this in an interesting way. We like to consider that definitions are not particularly important per se, but become so when they are in service of arguments, exploration, chains of reasoning, or proofs. So, for example, you might want to define ‘a circle’ if you are trying to make an argument that all circles are similar to one another, or explain to someone else how to draw one, or ask them whether they consider a ‘perfect’ circle can ever exist in the real world. Thus, the definition you choose serves as a kind of bundle of assumptions, a starting point from which you can explore and reason.
Definitions and gatekeeping
In short, what we can and can't call things in mathematics is an important site for us to challenge gatekeeping and people's views of mathematics as right and wrong. It is perfectly reasonable to disagree on definitions, so long as we apply our mathematical reasoning to what the consequences of this might be. So, we would like to challenge perspectives that suggest students learning definitions is something to focus on, and instead shift the focus to constructing them: in other words, thinking about the use and utility of definitions in mathematical communication. A better question for us might be, ‘Could you make the case that a cone has zero, one, or two edges?’ Another one might be simply, ‘How might you describe a cone?’ The key here is not just how the question is posed, but also how it might be answered.
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