To allow Elesis to use a spear, she must complete the Spearman quest first. Go to the shop and buy the Spearman quest scroll in the Acadamy tab. There are 2 scroll, one can be bought by GP and another by Cash. I don't know about the Cash scrolls but the GP scroll took me lots of hours to complete. Also you need to have only 4 quests on Elesis questlist or less in order to use the quest scroll.