Harel G Two dual assertions: the first on learning and the second on teaching or vice versa. Harel G The development of mathematical induction as a proof scheme: a model for DNR-based instruction.
In: Campbell S, Zaskis R eds Learning and teaching number theory: research in cognition and instruction. Harel G, Sowder L Toward comprehensive perspectives on the learning and teaching of proof. In: Lester FK ed Second handbook of research on mathematics teaching and learning. Information Age, Greenwich, pp — Google Scholar. Healy L, Hoyles C Proof conceptions in algebra.
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NCES Inglis M, Attridge N Does mathematical study develop logical thinking? Testing the theory of formal discipline. National Council of Teacher of Mathematics Principles and standards for school mathematics. Author, Reston Google Scholar. Stylianides G, Stylianides A, Weber K Research on the teaching and learning of proof: taking stock and moving forward.
In: Cai J ed Compendium for research in mathematics education. In: Hershkowitz R ed Proceedings of the seventh international conference for the psychology of mathematics education. To appear in Cognition and Instruction in press Google Scholar. If the example fits into the class of things previously mentioned, then deductive reasoning can be used. Deductive reasoning is the method by which conclusions are drawn in geometric proofs.
Deductive reasoning in geometry is much like the situation described above, except it relates to geometric terms. For example, given that a certain quadrilateral is a rectangle, and that all rectangles have equal diagonals, what can you deduce about the diagonals of this specific rectangle?
They are equal, of course. An example of deductive reasoning in action. Although deductive reasoning seems rather simple, it can go wrong in more than one way. When deductive reasoning leads to faulty conclusions, the reason is often that the premises were incorrect. In the example in the previous paragraph, it was logical that the diagonals of the given quadrilateral were equal. What if the quadrilateral wasn't a rectangle, though? Maybe it was actually a parallelogram, or a rhombus.
Yousef Essam Yousef Essam 73 1 1 silver badge 9 9 bronze badges. They are quite different things. Isn't it induction at all? I struggle to answer what the difference is, because there is little, if anything, in common. Please study at least one proof in maths that uses mathematical induction, then you may see for yourself.
Add a comment. Active Oldest Votes. Further help on induction - Daniel Velleman's quite terrific book How to Prove it. Peter Smith Peter Smith SK19 SK19 2, 6 6 silver badges 32 32 bronze badges.
When can induction be accepted as proofs for theorems? Again mathematical induction ISN'T inductive reasoning, but deductive. In general, induction in the broad sense "it seems to always hold, so it always holds" is never accepted in mathematics. Bary12 Bary12 2 2 silver badges 12 12 bronze badges. Or can they be proved like theorems?
And on what basis can we rely on them? For example, one of the Peano axioms is "The number 0 exists". This is not something one can prove from nothing. We are given an empty system, and start adding objects using axioms. We start with 0, then we add an axiom "each number in the system has a successor". That way we define 1 to be the successor of 0, which we know exists, because of the axiom we assumed.
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