Having made some arguments for quantifier packaging in earlier posts, I think I should also point out an argument against overdoing it.
I do have some evidence that suggests that, although my second-year students appear to understand many of the basic concepts of mathematical analysis better as a result of my quantifier packaging approach, many of them are then not ready to work with multi-quantifier statements for themselves in third year. In fact, by making the second-year material “easier to understand” I may have increased the gap between the second-year and third-year material.
Now I could introduce quantifier packaging in the third-year too, but this does not seem appropriate. Students should surely learn to work with multi-quantifier statements confidently at some stage, inventing their own packaging if necessary.
I had hoped that my suggested quantifier packaging would help the students learn how to work with multi-quantifier statements. But there is a danger that the students become dependent on the lecturer to do their quantifier packaging for them.
I still think that quantifier packaging is an excellent way to help students to gain a solid understanding of any particular topic. But I do think that it needs to be backed up by plenty of practice in working with multi-quantifier statements. Maybe it would be an interesting exercise to get the students to invent their own temporary names for various intermediate concepts, so that they can learn to package quantifiers for themselves? Or maybe quantifier packaging is more like a crutch that students need to learn to do without.
Joel Feinstein 23/1/09