1.Introduction
In the lower grades(1-2graders), to solve addition and subtraction word problems
is one of the difficult things for students. Especially, compare-problems trouble
students more or less. What obstacles troubles students in solving word problems?
They first may read a word problem when they will solve it. And then somewhat
representation about the problem will appear in their brains. And they will transfer to solving actions on the basis of the representation. As observing their actions, we find that they do unconsciously the process of starting the solution after reading a word problem. But the unconsciusness consists of many elements, which are learnings,e.g, knowledges and strategies, and so on, gotten in the past. Although they get to solve working these elements, I think that it is at the word-understanding when they feel the difficulty of solving them. In other words, it can be said that the fact, what representation appears, will fix the direction to solution. However, teachers can not grasp what representaions appeared in students' brains. Therefore, we set the first part as a blackbox and will observe the next actions, solving ones. Then, I thought I would have students use 'Tape-figure' as a strategy and support their solution. In orther words, I hope that students would form the schema to be able to solve addition and subtraction word problems by themselves through using tape-figure.

2.Contents of the study
(1)Tape-figure is appropriate or not for representing the construction of word problems. Tape-figure is used in the text-book, but I research whether it can be an appropriate model for students. And also I research what meanings tape-figure as model have to students from the view point of metaphor.
(2)I used the tape-figure with two tapes(fig.1), which was appropriate or not?
Tape-figure is used as one tape or two tapes accoding to the content of problems. In this study, I used it with two tapes. The intention is that I hope students would solve the problems using the same tape-figure. That is, I expected that students would activate the problem structure of fig.1(Whole-Part Schema) as a model and solve through operating the model. Nevertheless, I do not know whether this model work well as a schema to solve every addition and subtraction word problems.
(3)Does using tape-figure promote students' problem-solving ability? We make sure of the effectiveness to use tape-figure.
(4)Researching the points in instruction from tape-figure to number straight line. Students use tape-figure while lower grades, but from the third grade they use number straight line. Then, what points we should note? I research the points and what we should consider in teaching tape-figure.

3.Practice records

In starting mathematics lessons, I first thought that it is necessary for students to use tape-figure well. So, I thought, they should learn word problems after they have understood the meaning and function of tape-figure to some extent. And I used the first time to practice how to write and use tape-figure.

(1)Practice to use tape-figure
At the first time, I had students write a tape-figure with easy formula, 3+2=5.
[3] and [2] are wrote as following right with [3] longer in the upper.
Next, I had students write down [5] whithin the lower tape. This is a addition-model. And moreover, I had students write as fig.3 of subtraction 6-2=4. As the sample was onle one, it may be not sufficient for students. But we could not have the time more than a lesson and so we advanced next learning by considering that students had understood tape-figure operating.
(2)Explanation of how to learn
I explained to them that they should learn using tape-figure, stepping these 7 orders.

1)Read the problem more than twice.
2)Draw a underline at the sentence to ask an answer.
3)Surround some elements with a box, which you can understand well after reading.
4)Make a tape-figure on the basis of the former elements and write them down.
5)Judge from your tape-figure whether you should add or subtract.
* Addition in looking for C, subtracttion in looking for A or B
6)Make a formula and calculate it.
7)After finding an answer, ascertain whether the answer is fit for thing asked(as 2))
As the above, I explained how to learn using tape-figure in (1),(2), and prepared for students the basis to start text-book problems.

(3)Record of lessons

Problem1

Takashi(boy) have read 18 pages of a story book 

                                   today.  Adding 18 pages to the pages read 

                                                    yesterday, all are 46 pages.

                                    How many pages did Takashi read yesterday?

Problem2

Threre are 16 goldfish in the pond. Then someone
                                   put some goldfish into the pond, so there became
                                   to be 24.  
                                    How many goldfish were put into the pond later?

To this problem, most students could write the tape-figure like fig.5. But 4 students were wrong in the place to be written 24 down(fig.6). This causes from the fact that I did not instruct them well about how to write the tape-figure. Rather, observing their actions, they seemed not to understand the meaning of tape-figure. That is, they do not understand what tape-figure represents and how to use tape-figure. Even they do understand what they should search after reading the problem. Nevertheless, they can not translate what they can understand into the tape-figure. Therefore, even if they could make the formula 24-16, I wonder if they did so by understanding well. And a student could not draw tape-figure by himself.

Problem3

                         Children made some decoration rings.  The length of 
                         the ring that Akira(boy) made was 1m20cm ,and which 
                         was 30cm as long as Michiko's(girl).
                          Then how long Michiko's ring was?

This is the subtraction problem that students need to think reversely to addition. And also they need to devise how to write tape-figure. They troubled about where they should write 1m20cm and 30cm in the tape-figure.

Problem4

                          Tadashi(boy) folded 13 pieces of paper into the figure 
                         of a crane.  The numer 13 was 8 less than the number of 
                         cranes that Toshiko(girl) folded.
                           How many cranes did Toshiko fold?  

This is an addition problem to need students' reverse-thinking. As my hope I thought that students could write the tape-figure like fig.7, but I found it was rather difficult.
Because the relation among the numbers 13, 8, and Toshiko are not integrated within students' understandings. As a cause of it, I guess that it is rather affected by dealing with subtraction word-problems which were learned the last lesson. That is, there remains the newest learning in students' memories, their representaions rose up.

We can find out from the right two figures that Ayumi and Yuichi do not understand which is more than, the number of cranes folded by Tadashi and the number of cranes folded by Toshiko.
And, Tadako do not understand the relation among 21, 8, 13, however, she made a formula, 13+8=21. By the way, I wonder what role tape-figure really held on her case. Did she write a tape-figure because the teacher(I) asked her to do so? The fact that a formula and a figure are not related with each other, does show that tape-figure does not work as a tool of thinkings to her. And also it is not fixed that tape-figure represents the volume of quantity by length.

Problem5

We can consider that students have mastered learning to use tape-figure if they can solve the problem which is a exercise problem for Problem4. Nevertherless, also with this problem, a student could not write a correct tape-figure. That figure was correct in the relation among the sizes of 28,140,168. But they did calculate such as 168-28=140 and showed that they did not understand which cost more expensive, Noriko and Takeshi.
But I felt that as the lowest boy "Ryo" wrote a correct tape-figure and calculated correctly, learning to use tape-figubecause ofre may have penetrated into students.

4. Concluding Remarks

Although I researched how to use tape-figure effectively through practicing lessons, I found some problems. Perhaps some students may solve the problem even if they would not use tape-figure. Or a few students may not have understood the problem because of using tape-figure. In this sense, I can not judge the effectiveness of tape-figure at present. (1) The meaning of tape-figure as a model
The thing that I considered most on teaching is whether tape-figure was really a model mapping the problem structure for students. Certainly, tape-figure is more concrete than number line. But when we think whether it is clear for students, we can not say so. The degree of abstraction as model, I think, is an important element. Rather it may be more difficult than nember line. On this case, we cannot consider that the closeness to concrete will connect with students' understanding easily.
I wonder how the problem structure and tape-figure are connected each other in students' minds. If it does not work as a model well, it will defeat its own end.

(2) How to write tape-figure
I had students write tape -figure of "Part- Whole schema" about all problems, but some problems remained. For example, as for problem1 we may write tape- figure as fig.9. I had them write tape-figure as 3 elements intentionally, on the case of problem1 students cannot find "set C" through the problem. Accordingly, the problem is whether writing set C on tape-figure was appropriate fro students. And furthur it is the problem whether it was integrated in students' understanding. It could be used on the case of " the compare problem", but it may have rather problem about using 2 tapes on the case of "the change problems".

(3) The effectiveness of using tape-figure
I can not judge correctly about whether using tape-figure was effective for learning. But as children said to me,"Using tape-figure is interesting.", I guess that it was useful for them to increase their learning-will.
(4) The relation between tape-figure and number line
Students learn tape-figure in the 2nd grade and in the 3rd they learn number line. I think that these 2 models are not always the same and there exists other meaning besides changing from surface to line. Both models represents the size of volume by length but students' receiving is not the same. But I can not reply to the question soon what different these are but I think number line is more clear than the other. Anyway I can say that there are both the time to understand easily and the time to feel more difficult in using tape-figure for students. And I could not have students master using tape-figure as learning tool because of less time.

5. End

It is not easy that we go side by side with educational practices and theoretical study. If we follow any theory, we will lose sight of students. If we practice only from actual state of students, it will be technique-principle without obvious foundation. We need balancing sense what is more important, but it is difficult to work in harmony.
In this lessons," Learning to use tape-figure", theoretical study may have preceded and parts of learning with students were rather lack. Therefore, it cannot be said that I taught them hearing from their thinkg well. We should have learned creating the strategy of solution keeping much time. I have to practice mathematics lessons grasping students' thinkings and breathing. �@