In addition to looking at the role of proofs in mathematics education, this article also talks about the importance of dynamic geometry software in teaching mathematical proofs. According to Hanna, as educators in mathematics, a key task for us is to first understand the role of proofs in teaching mathematical concepts and ideas so that we are in a better position to promote conceptual understanding of mathematics through proofs.
For instance, the use of dynamic geometry software can provide more effective ways of challenging students and also teaching sound and logical mathematical reasoning to produce valid proofs for geometrical propositions and objects. The author goes on to suggest that dynamic software tools can not only provide an avenue towards heuristics, exploration, and visualization of mathematical concepts and ideas, but these tools can also pose challenges to teaching proofs.
The author notes that the term 'mathematical understanding' is somewhat elusive. By this, I think that the author is wondering what it really means to gain an understanding of mathematics, here, through proofs. For instance, expert mathematicians view proofs to be most valuable when it leads to understanding. That is, understanding proofs assists them to clearly verify, explain, systematize, discover, construct, communicate, and explore mathematical concepts and ideas. However, students and novices view proofs as alien, unnatural, not useful, or simply too boring as a human activity. That is, proofs are not appealing to the learners. One reason that novices feel this way about proofs is because, according to Hanna, teaching of proofs in traditional classrooms amounts to rote learning of mathematical proofs.
In order to move away from learning mathematical proofs through lower order thinking skills, dynamic software tools are available in order to explore, discover, and construct mathematical proofs, properties, and conjectures through dynamic investigations, explorations, demonstrations, constructions, problems, arts, or puzzles. For example, these tools can provide students to "check a large number of cases or even ... an infinite number of cases" through dynamic manipulation of mathematical objects.
It will be interesting to see if students' views of proofs as intuitive collection of properties change to that of viewing proofs as a system of theoretical ones through dynamic geometry software.
For instance, the use of dynamic geometry software can provide more effective ways of challenging students and also teaching sound and logical mathematical reasoning to produce valid proofs for geometrical propositions and objects. The author goes on to suggest that dynamic software tools can not only provide an avenue towards heuristics, exploration, and visualization of mathematical concepts and ideas, but these tools can also pose challenges to teaching proofs.
The author notes that the term 'mathematical understanding' is somewhat elusive. By this, I think that the author is wondering what it really means to gain an understanding of mathematics, here, through proofs. For instance, expert mathematicians view proofs to be most valuable when it leads to understanding. That is, understanding proofs assists them to clearly verify, explain, systematize, discover, construct, communicate, and explore mathematical concepts and ideas. However, students and novices view proofs as alien, unnatural, not useful, or simply too boring as a human activity. That is, proofs are not appealing to the learners. One reason that novices feel this way about proofs is because, according to Hanna, teaching of proofs in traditional classrooms amounts to rote learning of mathematical proofs.
In order to move away from learning mathematical proofs through lower order thinking skills, dynamic software tools are available in order to explore, discover, and construct mathematical proofs, properties, and conjectures through dynamic investigations, explorations, demonstrations, constructions, problems, arts, or puzzles. For example, these tools can provide students to "check a large number of cases or even ... an infinite number of cases" through dynamic manipulation of mathematical objects.
It will be interesting to see if students' views of proofs as intuitive collection of properties change to that of viewing proofs as a system of theoretical ones through dynamic geometry software.
Hanna, G. (2000). Proof, explanation and exploration: An overview. Educational studies in mathematics, 44(1-2), 5-23.
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