About comparison

In this term , we use a new text book. In the book, teachers find a rule whih is not line with it used to be. The original sentence in textbook is as follows. "This pen is three times as long as that one. or we can say: This pen is three times longer than that one. but in the past, techer often explained this rule like this This pen is three times as long as that one. or we can say: This pen is twice longer than that one. We can find that in the text the sentence -This pen is three times as long as that one, is equal to this pen is three times longer than that one. Do they have the same meaning? Is the rule right in the past or right at prsent textbook?. I'm looking forward to your earliest reply.