Abstract
The popular teaching pattern of multiplication of division following textbooks in Japan is making an expression at first and solving its answer. For example in division by decimal a problem are shown as follows:
"A ribbon is 72 yen for 2.4 m. How much is it per 1 m?"
The teaching stages are based on principle of the Permanence of Equivalent Forms and like these:
(1) making the expression 72 2.4 by "the word expression"
(2) seeking the solution of 72÷2.4 with thinking the price of 24 m ribbon using a line segment figure.
(3) calculating 72÷2.4; 72÷2.4=(72×10)÷(2.4×10)=720÷24=30
But this way isn't necessarily natural and understandable to students, as they mistake like this: 72÷2.4=72÷24÷10=0.3
I propose a hypothetical learning trajectory for division by decimal based on the chain of signification as follows:
(1) understanding the problem using the picture of a ribbon or a ribbon itself.
(2) estimating the price, if it is 72 yen for 2 m or for 3m, using a belt figure.
(3) trying how much for □m, based on 72 yen for 2.4 m, using a line segment figure
(4) understanding 30 yen per m, on 720 yen for 24 m.
(5) expressing the formula of division by decimal based on a expression to figure out the price per m at 2m, 3m.
The chain of signification and hypothetical learning trajectory suggest us a mathematics teaching based on constructivism. I would like to investigate a teaching experiment under the learning trajectory.
