- Mathematics arises from our bodies and brains, our everyday experiences, and the concerns of human societies and cultures
- Metaphor is both what enabled mathematics to grow out of everyday activities, and what enables mathematics to grow by a continual process of analogy and abstraction
- Mathematics is not transcendent or out in the world waiting to be discovered (Platonic view)
- Math is a human creation, and thus was created uses the cognitive domains of linguistic experience
- It the result of human evolution and culture
- During experiencing of the world a connection to mathematical ideas is going on within the human mind
- Mathematics develops by means of metaphor
Mathematics is connected to how we view the world. It is a result of our biological and evolutionary history. We are human beings. We stand upright on two feet. We have eyes set into the front of our head giving us 3D vision. We have four fingers and a thumb that enable to use multiple grips. We can see a small segment on the color spectrum. We hear at certain wave length. And on and on.
Lakoff and Núñez maintain that mathematics is a product of human beings and is shaped by our brains and conceptual systems, as well as the concerns of human societies and culture. We have evolved so that our cognition fits the world as we know it.
For them, mathematics starts in the world around us and is embodied in our physical being. They suggest four main grounding metaphors. They claim that all mathematics, no matter how abstract, can be traced back to one of these four metaphors.
For simple arithmetic like addition, it is easy to see how these metaphors apply to our experiences in the world:
Object Construction - items are piled on top of each other and there are more of them (addition)
Object Collection - items are grouped into categories that are similar (patterns)
The Measuring Stick - when physical segments are put in a line, they measure more (addition) when taken away, then measure less (subtraction)
Moving along a path - moving along a path changes my place on the overall path (number lines)
Once we get into more complex mathematics, these grounding metaphors link to more conceptual metaphors. As mathematical understanding grows, a web of connections and networks form.
When students make errors in their mathematical understanding, often these errors can be traced back to the use of these four grounding metaphors. Students may be looking at the concept of fractions using the motion along a path metaphor, when they should be using the object construction metaphor.
In conclusion, this is an exciting piece of work. There is something here that I cannot put my finger on, but just screams to me and says, 'there is something incredibly important here.' Again, their writing is not populist, and it is not easy to understand. It is dense. It is complex.
It feels like I will need a lifetime to untangle this knot. I better get started.