MathFoundations11: Laws of arithmetic for fractions

[steamednotfried]

By the way, much praise and respect for providing this generally excellent resource. I will be watching all of your videos. Thank you.

[njwildberger]

Hi steamednotfried,

Thanks for your comments. We must distinguish between the right way of setting up mathematics, and the right way(s) of teaching it.

I believe that the former is a prerequisite for the latter, but the latter need not follow the former. In other words, first we have to make sure we have a logically sound theory. Then we worry about how to teach beginners, perhaps in fact burying or at least delaying some of the logical structure. This series is aimed at the first goal.

[steamednotfried]

...And what are you not telling me? Why are we doing this in the first place? If a matrix is just a group of numbers, then why have you defined matrix addition to be this? And multiplication to be that? After all, those definitions do not arise naturally from any old groups of numbers.'

[steamednotfried]

I have just started a Computing Science degree at the University of Glasgow and maths is one of my three subjects for the first year. I had a very similar problem when my algebra lecturer introduced matrices. He introduced them simple as a group of numbers with no meaning and then started defining operations on them, proving commutativity, associativity and so on. I thought, 'I don't even know what a matrix is, why should I care if addition of matrices is commutative....

[njwildberger]

Hi steamednotfried,

I cannot let that slip pass by uncorrected---matrix multiplication is definitely NOT commutative. However that is unrelated to what we are doing here with fractions.

For more on matrix arithmetic, please see my series on Linear Algebra called

WildLinAlg.

[steamednotfried]

Otherwise, isn't it a bit like, say, to use a crappy example, teaching someone to read out loud (i.e.teaching them to make certain sounds based on what they see on a page) without them having any idea of what a word is?

[steamednotfried]

But, surely, operations on mathematical objects must arise from the meaning of the objects. For example, as you so elegantly conveyed, the operation of addition on natural numbers follows naturally from the meaning of a natural number. So, if you want to develop kids understanding in a natural and logical way, somewhat in parallel with the actual order in which humans understood these things, surely you must let the operations that may be performed on fractions come out of their meaning.

[dtv1966]

Very good video. You presented the material in a clear and understandable manner. Wish I had a teacher like back in public school. I wish you could have taught my undergrad calculus class. Maybe I would not have drop it.