1.Purpose of this study

In order to excite children's motivation for learning and increase their intellectual curiosity from mathematics, cognitive conflicts of children are need to be manifested on the way to acquisition of mathematical knowlege. The purpose of this study is to clarify how such cognitive conflicts of children generate and dissolve. Concretely, this study will also clarify "the problem of situations that generate cognitive conflicts" concerning "the existence of digonal lines in polygons".

2.Problem definition and cognitive conflicts

Learning contents for the fifth grade contain the subject"the number of diagonal lines in polygons". Diagonal lines are usually taught in the forth grade using quadrangles. Based on this, the fifth grade childeren learn how the number of diagonal lines changes for different polygons, such as pentagon, hexagon, and heptagon. The total number of diagonal lines can be summarized as follows(Table1), and its generalization is also possible by marking use of expression of variables.

However, such contents should be taught in the final stage. Before manifesting cognitive conflicts associated with diagonal lines that appear inside of children, we have other important problem that must be considered further. What is it? It is the existence of diagonal lines. Certainly, children know fine that a triangle does not have diagonal lines and a quadrangle has two diagonal lines. However, this does not absolutely guarantee that children can draw diagonal lines in a pentagon. Surprisingly, some students say, "a pentagon does not have any diagonal lines", and other students say, "a pentagon has two diagonal lines" or "a pentagon has three diagonal lines".
When the author first used this subject in teaching, he met a child who claimed that a pentagon should have three diagonal lines(Fig.1).

When the author asked him the reason, he gave the following answer.
Aihara: The angle faces the angle A is the angle C. The angle that faces the angle B is the angle E. But, since the angle that faces the angle D is on the intercection point of the line segments AC and BE, I have connected the points D and F.(Fig.2).
This answer from the child caused unexpected development of the class. Immediately, many adverse opinions were given one after another by children who had answered with the Fig.3. However, these opinions were not accepted without objections. Based on the answer shown in Fig.3, a student who had answered with Fig.2 asked an other question.
Sawaki: Figure3 is wrong. The angle that faces the angle A is the angle C or D. It breaks the rule to draw two lines from one angle to both the two angles.
In this way, the class was ended without reaching a conclision. On the following day, Y amamoto, a boy, showed me the following description of his impression:
"I cannot guess the conclusion without asking you a question. I want to confirm if I can draw two diagonal lines from one angle. If it is OK, I will have a result shown in Fig.3.
If not, I will have a conclusion that pentagons cannot have diagonal lines. However, I cannot agree with Aihara and Sawaki. They said that the rest angles can be connected to the center point shown in Fig.2, but the Chinese character "taikaku-sen"(diagonal lines)should literally indicate 'sen'(lines) that connects 'taikaku'(facing angles). If lines are connected as with Fig.2, they should be called "tai-kaku-kaku-sen"(diagonal-angle lines). So, I cannot agreee on Fig.2."
Cognitive conflicts concerning the existence of diagonal lines are discribed vividly in this Yamamoto's impression. He could'nt agree on Fig.2, but he also didn't accept Fig.3 in the existing circumstances. This hesitation is the said mental conflicts.
From this viewpoint, the author interprets cognitive conflicts as follows:
"Cognitive conflicts are mathematical activities to stabilize lost mental equibrium due to recognition of differences between considered target and related already learned contents when confronting new things."
That is, cognitive conflicts need to be considered from two viewpoints of generation of cognitive conflicts and decision making on selection and action.

3. Situation that generate cognitive conflicts

A situation that causes cognitive conflicts also affects how to generate them.
Although the author had student's response concerning diagonal lines of a pentagon as shown in Fig.2, I did not have expectaion that he would probably have such student's response. If anything, I learned unknown existing problems on diagonal lines from children when coming in contact with impressions by Aihara, Sawaki, and Yamamoto. After this, in order to highlight this point, I performed empirical study to pursue how the best ordering of presentation of teaching materials and setting of situation should be. As a result, the following ordering was obtained.

Quadrangle ���@hexagon���@pentagon���@heptagon�@

�@In this teaching material, the order of pentagon and hexagon was changed from the usual one.

Quadrangle ���@pentagon���@hexagon���@heptagon

Furthermore, arbitary shapes cannot be chosen for hexagon next to quadrangle, and I restrict myself to the shape of hexagon shown in the following figure, where diagonal lines that connect facing vertices cross on a point(Fif.4). Two phases proposed by Bruner, "the best ordering of presentation of teaching materials" and " the best experienceand situation for exciting children's motivation for learning", lurk in this inverted presentaion and specified hexagon.
Why are the ordering of pentagons and hexagons inverted? If we present pentagons after quadrangles and instruct students to draw all posiible diagonal lines in a pentagon, this instruction itself suggests that multiple diagonal lines can be drawn from one point in a pentagon, resulting in obstruction of children's spontaneous acqirement of knowledge. However, if we present children the above-mentioned hexagon that has diagonal lines crossing on a point after quadrangle, they always show the following responses.

Among them, the answer A is an overwhelming majority. When the author had classroom teaching opening to general teachers at Chubu Elementary School (Sagae City,Yamagata Prefecture) in October 1991, I had students' responses where twenty-six among forty children belonged to A, five to B, three to C, and six to D. Certainly, even if a hexagon is presented after a quadrangle, answers such as in the following figure are sometimes given.(Fig.6)
However, such student's response is quite rare and often treated as a stumble. We cannot expect conflicts among children's viewpoints as long as children are much familiar with such discussions. However, if the above-mentioned hexagon is presented after a quadrangle, we can expect the students' responses (A)-(D), especially, (A), and promote conflicts among children's viewpoints in a natural way.

4. Conflicts among children's viewpoints

How was conflicts among children's viewpoints promoted by the teaching that had been conducted by the author in Chubu Elementary School? I will consider it based on protocol of the teaching.
The author explained a term "polygon", wrote "the numbers of diagonal lines of various polygons" on the blackboard, and asked children the following question.
T: As a matter of fact, you have already learned diagonal lines in the forth grade. We will here consider the number of diagonal lines by drawing these figures. It is of course important to draw figures with a ruler, but it is also important to draw figures contained in your head without a ruler. This is called freehand drawing. Today,Iwill let you do freehand drawing without a ruler. Well then, draw a quadrangle and all possible diagonal lines on it with freehand drawing but not a ruler.
The purpose of stressing freehand drawing is to make children express their imagined figures outside and to save time.
In this way, the following figures were then presented.(Fig.7)

Subsequently, I asked them, "What kind of polygons do you imagine for drawing diagonal lines?" and then children answered, "Pentagons and hexagon". Based on this, I presented a problem, "First of all, draw all possible diagonal lines in a hexagon", and wrote the following figure on the blackboard.(Fig.8)
As a result, the above-described student's response (A)-(D) was obtained.
Then, conflicts among children's viewpoints were observed originating in the three types of responses (A)-(C) but not in (D). I expected that the response (D) would appear through discussions. Actually, (D) appeared after doing some discussions.
We can classify such conflicts among viewpoints into the following three situations.
-Situation 1: Conflicts between the viewpoints (A) and (B).
-Situation 2: Conflicts between the viewpoints (A) and (C).
-Situation 3: Conflicts between the viewpoints (A) and (D).
We will next show you aspects of these conflicts between the viewpoints.
(i) Situation 1: Conflicts between the viewpoints (A) and (B)
C_1-1: My answer is (A). I think that (B) is wrong, because a quadrangle has just one intersection point. But this figure has two intersection points.(Fig.9)

This difference is also difference between interpretations of diagonal lines of quadrangles. Child C_1-2 simply inpterpreted diagonal lines as "lines that connect facing vertices", and considered that "diagonal lines must intersect on a single point". This difference between their viewpoints caused conflicts between answers (A) and (B).
(ii) Situation 2: Conflicts between the viewpoints (A) and (C)
C_2-1: I would like to ask people who support the answer (C). (Fig.11) I think that you should be able to draw more diagonal lines because you have drawn so much lines. Is this right?

C_2-2: I think that (C) might be OK because diagonal lines are lines that connect facing vertices.
C_2-3: But, there are other lines that have not been drawn, aren't they?
C_2-4: Say it again. What does it mean?
C_2-5: (Going to the blackboard) I can also draw lines like this.(He draws dotted lines).(Fig.12)
C_2-6: I don not agree such thing.
Child C_2-1 who supported the answer (A) did not agree that two diagonal lines had been drawn from a single vertex. Children who supported (B) had also the same opinion. Accordingly, child C_2-1 asked the question "I think that you should be able to draw more diagonal lines because youm have drawn so much lines. Is this right?" by assuming that the former opinon is right.
However, child C_2-6 replied "I do not agree such thing" to this question. This reply shows that he did not have a viewpoint "multiple number of diagonal lines can be drawn from a single vertex". Certainly, the answer (C) has multiple number of diagonal lines drawn from a single vertex.
After seeing it, children C_2-1 and C_2-3 asked questions, "I think that you should be able to draw more diaglnal lines because you have drawn so much lines. Is this right?" and "there are other lines that have not been drawn, aren't they?", whereas child C_2-3 couldn't understand the intention. Basically, in a similar manner to C_2-1 and C_2-3, child C_2-6 had understood that "diagonal lines are line that connect facing vertices" by guessing diagonal lines of a hexagons based on already learned quadrangles. In this sense, this situation applies to conflicts among children's viewpoints due to differences in understanding of the statement "diagonal lines are lines that connect facing vertices". Conflicts among children's viewpoints can be caused by differences in understanding of a statement even if children have the same viewpoint.
(iii) Situation 3: Conflicts between the viewpoint (A) and (B)
C_3-1: I would like to ask people who support (A). I think that you should be able to draw more diagonal lines because they are lines that connect angles. (Fig.13) Is this right?

T::Which answer do you support?
C_3-2:Both are not my answer.
T: Tell me what your answer is.
C_3-3:(He went to the front of the blackboard ans drew the following figure on it.) (Fig.14)

C_3-4: Those lines are too much because the number of angles is only six.
C_3-5: I support (D). Any number of lines can be drawn because diagonal lines are lines that connect angles.
C_3-6: I support (D), too. The answer (D) is OK because teacher said,"draw all possible lines".
C_3-7: I would like to ask people who support (D). Is the sentence "all possible lines" same as "all nonsense lines"?
C_3-8: Those lines are not nonsense because they connect angles.
C_3-9: Because diagonal lines should connect facing angles that have mutual correspondence, the answer (D) is wrong.
C_3-10: I think that (D) is wrong. A quadrangle has four angles and draws two diagonal lines connecting tow pairs of angles. Since a hexagon has six angles and diagonal lines have to connect pairs of angles, the answer (A) with three lines is correct.
C_3-11: If three lines are OK, the answer (B) with three lines also seemes to be OK. The answer (A) has three, but (B) has also three. Is this right?
C_3-12: Diagonal lines should connect corresponding angles. Diagonal lines of the answer (B) do not connect corresponding angles.
The viewpoint conflict in the situation 1 and 2 were triggered off by children who supported the answer (A). The answer (A) seemed to be superior because many children supported it. Among them, child C_3-1 asked a question about the answer (A). This child supported the answer (D) that had a new viewpoint. Whereas some children immediately supported the answer given by child C_3-1, many children were surprised at this new state of thing. Among them, child C_3-6 who supported this child explained its basis as, "The answer (D) is OK because teacher said, "draw all possible lines". However, this remark also had repercussions, since children who supported the answer (A) and (B) had different understanding concerning the statement "Draw all possible lines". As a result, one side claimed "nonsense", and the other side "not nonsense". In this deadlock, child C_3-10, wh supported the answer (A), presented as argument that claimed the relationship between the numbers of angles and diagonal lines. However, this opinion also was not supported. Child C_3-11 who supported the answer (D) persisted giving a counter example, "If three lines are Ok. The answer (A) has three, but (B) has also three. Is this right?"
Conflicts among viewpoints concerning the existence of diagonal lines are quite complicated.

5. Conclusion

I have considered generation of cognitive conflicts that lurk inside of children. It is exactly intended evocation of children's bewilderment. We have clarified the details of coginitive conflicts concerning the existence of diagonal lines, by considering that "ordering of presentaion of teaching materials" and "setting of situation" play important roles in their generation and evocation.

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