



 
We have two responses for you Uno, I have three suggestions for you.
Harley
I feel that algebra and proofs are very important in our world, because they deal with higher levels of abstraction and train us to be better at thinking abstractly. It does require training, just like baseball and music, but we do get better at it with training. For instance, you understand 3 + 4 = 7, that is "concrete enough" for you. But it is already abstract; it says that if you add a collection of three objects to a collection of four objects, you get a collection of seven objects, no matter what the objects are, cats, balls or newspaper. A dog can understand what is a cat, what is a ball or what is a newspaper (as in "Rex, fetch the newspaper"), but not 3 + 4 = 7; it is too abstract. Now an algebraic equation like x + y = y + x is one level of abstraction higher still. It talks about abstractions of numbers, that is, abstractions of abstractions. This one says that the order in which you add up two numbers doesn't matter, no matter what the two (abstract) numbers are. In word problems, you are supposed to set up your own abstract variable, translate the textual information into equations that this variable must satisfy, and manipulate the equations to find the "concrete" number that this variable must equal. In algebraic proofs like the irrationality of the square root of 2, you are supposed to dream up an imaginary world where the square root of two is a fraction a/b, then use algebraic manipulations to derive consequences. Talking and thinking rationally about imaginary worlds, that is, hypotheses, comes later in life. Babies first learn to talk about the present ("want cookie!") which is most concrete. Then, they will learn to talk about yesterday and tomorrow, which are more abstract. There are greater levels of abstraction yet: Thinking and talking about hypotheses, things that are not past, present but that could be. It is very practical. For instance, imagine a world where on September 12, 2001, the newspapers delivered a terrible news: "The US air force shot down commercial planes, killing hundreds. They claim that these planes were hijacked by terrorists who planned to crash them in the world trade center and make it collapse". What then? It is terribly abstract; we are to imagine the inhabitants of this hypothetical world trying to imagine a hypothetical world of their own, that is, our own reality where the terrorists did make the world trade center collapse. Algebra seems simple in comparison. And every news brings with it the possibility of other actions, other worlds and their consequences. For instance, our government that do little about climate change say it is an hypothesis that is not proven. An hypothesis is not something that will happen or will not happen, but something that might happen. We need to understand this abstraction adequately to decide whether we are taking appropriate action. As citizens in a democracy, we need not only to understand and suffer the laws imposed on us by our "rulers", but we have the power discuss laws, explore possibilities and vote wisely for the best representatives. So, thinking abstractly is essential in our world, and algebra and mathematical proofs are a good start at training for it. Claude  


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