mathematics: Ways of knowing
Reasoning
Language (notation and syntax)
Faith
Imagination
Intutiton
Memory
Senses./ everyday experience
Emotion
The ability to logically reason is at the heart of mathematics.
Deduction fro the 5 axioms to where we are today.
Sometimes this lets you down though. Counting is intuitive, but our use of negative numbers is less so. Why is negative x negative a positive? It's really hard to convince anyone who hasn't thought too deeply about it.
The representation of mathematical ideas can help or hinder development of knowledge. Representing a problem in a different way can lead to new insights.
Part of deduction - prior knowledge must be retained (in written form?) to be referenced in future work.
This depends on how you view creation - are our representations of our models really a model of some divine creation?
Is there some book of mathematics that we are discovering?
To misquote Einstein - God does not play dice with the universe.
Without imagination there is no discovery or invention - complex numbers for example
The ability to work on even when it appears hopeless when solving a problem
Mathematics: teaching thinking skills
Reflection: Students must reflect on the statement and think about what knowledge means in maths.
Difficulty is a subjective term in maths, some students will find understanding pythagoras theorem very challenging compared to others - does this make it less valued? It still solves problems,
Reflection is an important skill in maths - you must practice the skill regularly - checking whether answers are sensible, and evavluating the approaches taken to solve problems are good ways to encourage it.
Analysis: You have to analyse the knowledge that you have in maths - how are you putting a value on it? Academic difficulty? Usefulness in a range of contexts? To do this you need to build the links between areas of maths,
evaluation: evaluating methods and their advantages and disadvantages
Needs regular practice and a classroom environment in which students are happy to share methods - a safe space, where noone is seen as the keeper of knowledge.