• Journal of Educational Research in Mathematics
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• v.20 no.2
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• pp.145-161
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• 2010

Concepts in mathematics have been formulated for unifying and abstractizing materials in mathematics. In this procedure, usually some developments happen by necessity as well as for their own rights, so that various interesting materials can be produced as byproducts. These byproducts can also be established by themselves mathematically, which is called byproduct mathematization (sub-mathematization). As result, mathematization and its byproduct mathematization interrelated to be developed to obtain interesting results and concepts in mathematics. In this paper, we provide explicit examples：the mathematization is the continuity of trigonometric functions, while its byproduct mathematization is various trigonometric identities. This suggestion for explaining and showing mathematization as well as its byproduct mathematization enhance students to understand trigonometric functions and their related interesting materials.

• The Mathematical Education
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• v.49 no.2
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• pp.149-160
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• 2010

Composition of functions are important tool for producing associativity in mathematical model. However it is not properly treated in dealing together with the other operation, the addition +, of functions defined on real numbers. In this note, we will study mathematization of the construction of nearring axiom from relationships between the addition + and the composition $\circ$ of functions, comparing with those between the addition + and the multiplication of functions. Furthermore, we will suggest some helpful teaching methods of these mathematization in the secondary school mathematics.