• Journal of the Korean School Mathematics Society
• /
• v.18 no.4
• /
• pp.371-385
• /
• 2015

In this paper we investigate the axioms defining area and volume so that revisit area formula for triangle, volume formula for triangular pyramid, and related contents in school mathematics from the view point of axiomatic method and Hilbert's third problem.

• Education of Primary School Mathematics
• /
• v.17 no.3
• /
• pp.253-263
• /
• 2014

In this paper we review an axiomatic definition of the area of plane figures with area axioms, discuss what the area axioms mean, and analyze the contents about the area of plane figures in elementary school mathematics from the view point of area axioms. So we evaluate which aspects of the concept of area are emphasized or deemphasized in the current elementary school mathematics textbook.

• Journal for History of Mathematics
• /
• v.31 no.6
• /
• pp.315-337
• /
• 2018

Pythagorean thoerem exists in several equivalent forms in the Euclidean plane, that is, the Hilbert plane which in addition satisfies the parallel axiom. In this article, we investigate the truthness and mutual relationships of those propositions in various non-Hilbert planes which satisfy the parallel axiom and all the Hilbert axioms except the SAS axiom.

• School Mathematics
• /
• v.7 no.4
• /
• pp.391-402
• /
• 2005

The purpose of this study is to resolve misunderstanding for centroid of a triangle and to clarify several logical problems in finding the centroid of a Polygon. The conclusions are the followings. For a triangle, the misunderstanding that the centroid of a figure is the intersection of two lines that divide the area of the figure into two equal part is more easily accepted caused by the misinterpretation of a median. Concerning the equilibrium of a triangle, the median of it has the meaning that it makes the torques of both regions it divides to be equal, not the areas. The errors in students' strategies aiming for finding the centroid of a polygon fundamentally lie in the lack of their understanding of the mathematical investigation of physical phenomena. To investigate physical phenomena mathematically, we should abstract some mathematical principals from the phenomena which can provide the appropriate explanations for then. This abstraction is crucial because the development of mathematical theories for physical phenomena begins with those principals. However, the students weren't conscious of this process. Generally, we use the law of lever, the reciprocal proportionality of mass and distance, to explain the equilibrium of an object. But some self-evident principles in symmetry may also be logically sufficient to fix the centroid of a polygon. One of the studies by Archimedes, the famous ancient Greek mathematician, gives a solution to this rather awkward situation. He had developed the general theory of a centroid from a few axioms which concerns symmetry. But it should be noticed that these axioms are achieved from the abstraction of physical phenomena as well.