If it provides a boost, it's short-lived.
It's pretty well established that a burst of glucose provides a brief cognitive boost (see review here), so the question is whether chewing gum in particular provides any edge over and above that, or whether a benefit would be observed when chewing sugar-free gum.
Researchers reported some positive effect of gum-chewing for four of the tests. It's a little hard to tell from the brief write-up, but it appears that the investigators didn't correct their statistics for the multiple tests.
This throw-everything-at-the-wall-and-see-what-sticks may be a characteristic of this research. Another study (Smith, 2010) took that same approach and concluded that there were some positive effects of gum chewing for some of the tasks, especially for feelings of alertness. (This study did not use sugar-free gum so it's hard to tell whether the effect is due to the gum or the glucose.)
A more recent study (Kozlov, Hughes & Jones, 2012) using a more standard short-term memory paradigm, found no benefit for gum chewing.
What are we to make of this grab-bag of results? (And please note this blog does not offer an exhaustive review.)
A recent paper (Onyper, Carr, Farrar & Floyd, 2011) offers a plausible resolution. They suggest that the act of mastication offers a brief--perhaps ten or twenty minute--boost to cognitive function due to increased arousal. So we might see benefit (or not) to gum chewing depending on the timing of the chewing relative to the timing of cognitive tasks.
The upshot: teachers might allow or disallow gum chewing in their classrooms for a variety of reasons. There is not much evidence to allow it for a significant cognitive advantage.
EDIT: Someone emailed to ask if kids with ADHD benefit. The one study I know of reported a cost to vigilance with gum-chewing for kids with ADHD
Kozlov, M. D., Hughs, R. W. & Jones, D. M. (2012). Gummed-up memory: chewing gum impairs short-term recall. Quarterly Journal of Experimental Psychology, 65, 501-513.
Onyper, S. V., Carr, T. L, Farrar, J. S. & Floyd, B. R. (2011). Cognitive advantages of chewing gum. Now you see them now you don't. Appetite, 57, 321-328.
Smith, A. (2010). Effects of chewing gum on cognitive function, mood and physiology in stressed an unstressed volunteers. Nutritional Neuroscience, 13, 7-16.
Wilkinson, L., Scholey, A., & Wesnes, K. (2002). Chewing gum selectively improves aspects of memory in healthy volunteers. Appetite, 38, 235-236.