Originally Posted by Anarion
Deadly, we're actually talking about two different things here. You're saying that people should be comfortable with fractions at the point when they're getting into circles. I agree with you that they should, and that Tau has several notational advantages.
However, I am making a statement about my belief as to the state of the world. I believe it is a fact that even though people should know fractions, many do not. And I believe it is a fact that many people become uncomfortable when they see an expression with the appearance of a fraction.
Because I believe those facts to be true, I think that the introduction of tau would be counterproductive and would make it harder for young people to catch up due to their discomfort. Thefore, I think Tau should only be introduced after or in concurrence with other efforts to improve mathematics teaching.
And I say I think it's wrong not to address the actual problem itself. Keeping pi for the sake of a single formula, so that you can avoid something so trivial as taking half of a value, is ... it just can't be justified, in my head.
Say you're a teacher, and you're teaching your students circles when you realize that some of them are scared off by that little fraction because of bad experiences or something. I don't dispute that this might be a common problem, but what do you do? Do you hide the fraction with a magic trick which is only going to be helpful for that one
case and will cause lots of confusion later on, or do you do your best to help those students not fear that little fraction?
You could keep pi for that one instance and use tau everywhere else, but that's going to confuse people a lot too, I suspect.
And what about pi (and tau) itself? Pi is defined as a fraction, are you going to hide that and just give them a magic number without any kind of explanation? You could do that, either option might cause confusion, but isn't it better to present the definition and try to help them not fear it but to see the elegance of it all?
Trying to hide things that the students might fear will only hurt them and stop them from overcoming that fear. It will also make it harder to grasp future material, which will lead to further fears.