• School Mathematics
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• v.8 no.3
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• pp.327-339
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• 2006

In this study, a teaching units for teaching and learning of secondary preservice teachers' mathematising is designed, focusing on reinvention of Bretschneider's formula. preservice teachers can obtain the following through this teaching units. First, preservice teachers can experience mathematising which invent a noumenon which organize a phenomenon, They can make an experience to invent Bretscheider's formula as if they invent mathematics really. Second, preservice teachers can understand one of the mechanisms of mathematics knowledge invention. As they reinvent Brahmagupta's formula and Bretschneider's formula, they understand a mechanism that new knowledge is invented Iron already known knowledge by analogy. Third, preservice teachers can understand connection between school mathematics and academic mathematics. They can understand how the school mathematics can connect academic mathematics, through the relation between the school mathematics like formulas for calculating areas of rectangle, square, rhombus, parallelogram, trapezoid and Heron's formula, and academic mathematics like Brahmagupta's formula and Bretschneider's formula.

• Journal for History of Mathematics
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• v.16 no.2
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• pp.43-54
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• 2003

In this article we study various proofs of Heron's formula, extract some didactical ideas from these proofs, and didactically enlarge Heron's formula. In this paper we in detail introduce five different proofs from various articles and textbooks, and suggest our proof of Heron's formula. Enlarging this proof we are able to prove Brahmagupta's formula and generalized convex quadrangle's area formula.