The author's claim is that students should have more recess time. That is the point he/she is trying to prove. "Without more recess time" means "if the students don't have more recess time." The answer is not 'will,' because the author is trying to prove that students are already struggling with their problems and won't get better unless they have more recess time. It is not 'might,' because that is still negative for their point that students should have more recess time to not struggle with their problems. It is 'shall continue to' because the author is proving that students are already struggling with problems and will continue to struggle without more recess time.
A logical fallacy is an error in reasoning: a conclusion is drawn (they quickly... get to work, because... another recess is coming) when there is no real proof of its argumentation. The use of the defective auxiliary Might removes the logical fallacy, because it expresses possibility
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