• The Mathematical Education
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• v.52 no.4
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• pp.549-565
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• 2013

To help students understand the deduction system in the proof, we analyzed the textbook on mathematics at first. As results, we could find that the textbook' system of deduction is similar with the Euclid' system of deduction. The starting point of deduction is different with each other. But the flow of deduction match with each other. Next, we searched for the example of circular argument and analyzed. As results, we classified the circular argument into two groups. The first is an internal circular argument which is a circular argument occurred in a theorem. The second is an external circular argument which is a circular argument occurred between many theorems. We could know that the flow of deduction system is consistent in internal-external dimension. Lastly, we proposed the desirable teaching direction to help students understand the deduction system in the proof.