Problem Solving as Tool Formation Processes: A Preliminary Case Analysis.

Hideaki Takahashi

The University of Tsukuba, Tsukuba, Ibaragi, 305 JAPAN

In this paper I describe mathematical problem solving processes. The experimental task is a figure-drawing problem as following (see Figure 1). I named this task Ozawa-Anno (Illusory) Figure-Drawing Task according to its originality (Takahashi, 1989).


Insert Figure 1


There is a 3-D space where the lattice points of coordinates are dotted. Define one graduation of the lattice point as the distance 1. Line AB has been drawn and the coordinates of point A and point B are given. Draw a rectangular parallelepiped that has the line AB as its diagonal. Let each surface of the rectangular parallelepiped be parallel with each coordinate plane.

In solving this task, subject spontaneously takes some additional actions other than 'drawing the rectangular parallelepiped on graph paper' action as main task action. I regard these actions as psychological tools (Vygotsky, 1979) and I describe the tool formation processes of solving the task 4 times repeatedly. Because of the limit of paper length, I report only results of one subject.

The subject is an undergraduate student(Subject T). She is not expert in mathematics. The subject receives an instruction paper, one piece of A3-sized paper, and a pen. The A3-sized paper has two areas; left half is graph paper area, and right half is blank paper area. The experimenter instructed the subject to be able to use blank paper as she likes. The subject is right-handed.

Four materials are ready from same rectangular parallelepiped, as a rectangular parallelepiped has four diagonals. The subject receives repeatedly four trials that have same answer. The experimental method is personal observation. I record subject's actions on the desk by a video camera.

Results and Discussions
Subject T's performances are perfect. Her solution time decreases as a function of number of trials. She states that all materials may have same answer after fourth trial.

I identify her actions and interpret their corresponding psychological functions. She takes various actions, particularly on first trial. Subject T seems to try to solve the task by trial and error during this trial. As she repeats trials, she takes only unique actions.

Subject T takes two kinds of additional actions on different media; one is verbalizing actions on speech medium and another is writing actions on pen and hand media.

Verbalizing actions have many psychological functions; monitoring comprehension, checking problem conditions, imagining rectangular parallelepiped, checking figure image, regulating counting, regulating marking x, regulating drawing rectangular parallelepiped. Regulating functions are that verbalizing actions regulate some writing actions and main task action.

There are two kinds of writing actions according to whether the subject really writes something; one is 'kusho'(writing in the space, by Sasaki & Watanabe (1983)) and another is writing on paper. Kusho has two forms, finger-kusho and pen-kusho. Functions of kusho are imagining rectangular parallelepiped, checking figure image, and counting. Other writing actions (for example, drawing rough sketch on blank paper, note-taking on blank paper, marking x on graph paper) have each functions.

Functions of verbalizing actions become limited as the subject repeats trials. On first and second trial verbalizing has 6 and 4 functions, but on third trial it has 2 functions, both are regulating functions. On fourth trial the subject takes no verbalizing action. This means that verbalizing actions are internalized gradually; firstly monitoring and checking functions are internalized and secondly regulating functions are internalized.

Subject T takes 'drawing rough sketch on blank paper' actions on first and second trial, but not after third trial. These actions help her making clear the direction of rectangular parallelepiped and the position of diagonal in rectangular parallelepiped. After third trial she can do this internally, using other tools, say inner speech.

For Subject T, 'marking x on graph paper' action is most useful tool. She predetermines the position of vertex on graph paper by this action. On third and fourth trial, she mainly repeats alternatively 'counting the length of edge by pen-kusho on graph paper' action and 'marking x on the point of vertex' action.

Sasaki, M. and Watanabe, A. (1983). An experimental study of spontaneous writinglike behaviour ("kusho") in Japanese. Japanese Journal of Educational Psychology, 31, 273-282. (In Japanese with English Summary.)

Takahashi, H. (1989). Solving processes of Ozawa-Anno (Illusory) Figure-Drawing Task. Proceedings of the 6th Annual Meeting of Japanese Cognitive Science Society, 6, 74-75. (In Japanese.)

Vygotsky, L. S. (1979). The instrumental method in psychology. In J. V. Wertsch (Trans. & Ed.), The concept of activity in Soviet psychology (Pp. 134-143). New York: M. E. Sharpe.