Good point about the clarification about "less than 1" versus "between 0 and 1." Of course, if you are talking about multiplying by a positive number, it's still true that multiplying by anything less than 1 makes it smaller. Math is tricky because just when you've developed an intuition about one domain, say positive integers, some other situation comes and breaks it.
A picture is worth 1000 words, or 1000 tutoring hours. I tried what you mentioned. She thought taking 1/2 of something was division, and didn't have enough grasp of fractions to see that multiplying by 1/2 was the same as dividing by 2. Needless to say, her pre-algebra was weak. Amazingly, she had passed all her math classes up to an including algebra 2--the school district had an easy grading policy. Her SAT score was very low, however. Can't fake the SAT. That's why her parents sought me for help initially, when they realized her math was actually very weak.
She had basically developed this intuition that multiplication is always bigger and division is always smaller. That was a simple way of looking at it. I wanted to replace that with an equally simple was of looking at it, but one that was more correct.