As in so many things Jewish, it is a machloket between the poskim. Both Rav Moshe Feinstein and Rav Ovadya Yosef ruled that a bracha should be made. However, others (Rav Dov Lior, the sefer Birkat Hashem, et al) differ and say that no bracha is required; indeed, it may even be a "bracha l'vatala," either because the gum's taste is absorbed into the saliva & then swallowed (and saliva has no bracha) or because gum is not considered to be food in the normal sense. Still others differentiate between regular and sugar-free gum, the former needing a bracha because sugar is considered a food that we enjoy, while sugar-free gum needs no bracha. Related to this topic is the widespread opinion that smoking has no bracha (even flavored tobacco that one "enjoys" when inhaled); clearly, it is rather a blessing NOT to smoke & to live longer!