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2 Formal logic
3 Word problem
There is no royal road to geometry; a famous quote by Euclid to Ptolemy or by Menaechmus to Alexander the Great. Why and how to teach Euclidean geometry is always questioned by educators.
Some educators take the "either the whole truth or none" approach in the teaching of Euclidean geometry in high school. However, some feel that the complete axiomization of Euclidean geometry is too complicated for high school students, and hence Euclidean geometry should not be taught in high school.
There were movements of putting formal logic in high school curriculums. But some feel that the language of formal logic is too abstract and too boring for young teenagers. They found that a college student having bad formal logic training performs worse than a classmate with no previous logical training!
Use of the word problem to foster mathematical reasoning in less abstract terms has been argued to also bring with it a commonplace misunderstanding that mathematics and the language describing it are two different and separate things when in fact they are not. However, some feel that word problems are useful in motivating students to learn mathematics, making the problem feel more practical instead of mechanical.
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