Research of Educational Approach Which Lets Learners to Understand Algebraic Ideas and Methods Deeply

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Although black box approach seems to have popularity in high-school mathematics learning in Japan, it certainly devalues the learning of not only mathematics but also engineering subjects that use mathematics. If teachers encourage their students to use a formula as a convenient tool to solve mathematical problem without teaching the hidden mechanism enough, the students tend to memorize it blindly and use it as a black box. They use it in the examination effectively and forget it in several months. After that, there remains only vague memory that they have learnt something. In such a learning style, the simple use of technology might cause disaster because it only increases the effectiveness of black boxes.After surveying the low performance of our students in linear algebra, the authors have confirmed the cause as black box approach in their learning of abstract ideas. In the lessons of linear algebra, too many formulas were learnt without understanding the background mechanism and possible applications in the real world. There were few modeling, numerical confirmations, and linguistic explanations of the relations between symbolic & numerical representation and graphic objects. As the result, large number of students applied black box approach especially in linear algebra.This research tries to show the importance of linguistic explanation in avoiding black box approach by describing the authors' recent attempt to connect symbolic expressions and graphical representations in linear algebra through linguistic explanation with the help of technology. The further focus on linguistic explanation we have, the lesser black box approach the students apply in their learning, we believe.