The Notion of Epistemological Obstacle
in My Works [1995a] & [1995b] -reprintingAt first, it is a main purpose that [1995a] develops categories to describe
the process of overcoming the epistemological obstacle of the student. The
reason to make categories of description for is as follows. How can we recognizethat a student overcomes an epistemological obstacle? I think that a theoretical(or “theory-laden” by words of Hanson) viewpoint is needed for us so that we
can understand that a student overcomes an obstacle. The categories ofdescription are prepared for as such our viewpoint.
As a result, we need to distinguish three categories for description of
students’ overcoming process; “notion”, “event”, and “conviction”; The “notion”
category includes students’ ambiguous ideas, images, and mental models. The“event” category means students’ concrete experience which “notion” is laden.
What is described by the above two categories means the overt behavior of a
student. The conviction means the covert and comprehensive value judgment of
the student which explains why a student shows such a behavior. In otherwords, the “conviction” category means a student’s attitudes towards
mathematical knowledge.
A set of “notion” and “conviction” forms a student’s way of knowing. The“notion” also corresponds with some “event”. A student doesn’t evaluate the“notion” itself directly, but through the “event” as indeed. Hence the“conviction” is also a criteria to evaluate the event. These constitute thefollowing C(C, N, E) model.(see fig.A)

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The process of the student's overcoming epistemological obstacle isexplained as follows by this model. If a model is consistent, the student's
cognitive activity based on it lasts. Because the consistency of the model is not
kept, the learning activity is caused. This means such situation that though astudent was able to lade a “notion” (N1) to a “event” (E1) in the former activity,
which was based on “conviction” (C1) he/she is going to lade the same way ofknowing to the “event” (E2) that he/she confronts newly, but ends in failure. In
other words an existing way of knowing functions as an epistemologicalobstacle(EO).

To overcome the epistemological obstacle is not achieved by the revision of
the error or the supplement of a necessary matter, if, at least, a way of knowingthat a student achieved was satisfied enough in past learning. That is, only to
prepare a new “notion”(N2) is not enough. On this account it calls for total
reform (or “refonte totale” by words of Bachelard) of the knowing for overcoming
epistemological obstacle. That is, it is considered that the overcoming isaccomplished by the “notion”(N2) which is applied to the “event”(E3) confronted
newly and the “conviction”(C2) which evaluates (N2 & E3) occurring newly. E2 E3 may be the same object superficially, but they are distinguished asastudent looks at from a different viewpoint because his/her way of knowingchanges.
When I observe the student's activity with C(C, N, E) model, he/she is notgoing to overcome an epistemological obstacle anytime even if a teacher2expected it, rather many cases which he/she does not so is seen. Wheninvestigating how a student confronts an obstacle, we can identify four different
kinds of state of knowing as follows. [1995b]a) Persistence to the subjective ease
It leads clear decision making for a student that he/she maintains anexisting way of knowing of his/her own which functioned effectively till now atleast. Because such a way of knowing is suitable for the past experience of thestudent. We can call a state of knowing as such a decision making of thestudent "persistence to the subjective ease".
The state of knowing is characterized such that the student's existing wayof knowing does not change and therefore he/she is not going to accept “event”
which it does not suit well. (see fig.C1)b) Justification as the social adaptation of “event”
When a student persists in subjective ease, the relations with anotherperson do not occur. On the other hand, it is seen in the lesson that thestudent accepts “event” to meet newly with by relations with other(s). We cancall such a state of knowing “justification as the social adaptation of “event” ”.
The social adaptation to say here is the action of the student who takes it to
avoid that it becomes difficult to continue communal living because of the ill-3adaptative action. When we run a social life, we perform some kind of social
adaptation. This is similar about students in the classroom, and it isnecessary. However, students are enough by increasing as information by
new knowledge additionally if they merely aim at the social adaptation. (seefig.C2) Hence, the state of knowing is characterized by dependence toother(s) of the justification surfaced as words of the students that “I wastaught it by my teacher like that”.

c)Becoming aware of an epistemological obstacle
It is required that a student becomes aware of an epistemological obstacle
so that he/she overcomes it. We can call such a state of knowing of thestudent “Becoming aware of an epistemological obstacle”. What is important
here is not that a student accepts a new “notion”, but that student oneself
becomes aware of an epistemological obstacle. Actually, as a result of
becoming aware of an obstacle, the student may dismiss a new “notion” orotherwise reserve it. However, we cannot expect the attainment of theovercoming an epistemological obstacle in this state of knowing immediately.
It is shown in fig.C3, even if “notion”(N3) which is consistent in “event”(E2)that a student confront with newly was temporarily accepted by him/her, hecannot justify N3 and therefore he/she may dismiss it since it is evaluated byformer “conviction”(C1). Therefore, this state of knowing is characterized as
the activity/thinking of the student is delayed, as a result that he/she
becomes aware of his/her own obstacle.

4d)Overcoming an epistemological obstacle
When a student is going to overcome an epistemological obstacle, though it
is required to become aware of it at first, however why he can overcome an
obstacle is that it is by shifting “conviction”(C1 to C2) that he can justify“notion”(N3) which was generated or shown newly and “event”(E3) which“notion”(N3) is laden. Hence, the state of knowing that we call “overcoming an
epistemological obstacle” is characterized by shifting “conviction”.

Above-mentioned four kinds of state of knowing was all gained from theactual activities of the students by qualitative studies.

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The state of knowing that we aim at is “overcoming an epistemologicalobstacle” obviously. However, the student shows various relations with the
obstacle as mentioned above. Four kinds of state of knowing doesn't always
have order except that becoming aware of the obstacle is essential forovercoming it, rather they have advantage to each. Nevertheless, if we aim at
overcoming the obstacle for the student, it is necessary to explain how it issuperior to other relations with the obstacle by laying some kinds of point ofview.
1) Social context: social nature of the mathematical knowledge and of
the construction of the mathematical knowledge
The epistemological obstacle should be overcome by an individual student.
However, we cannot have criteria to judge as a student accomplishes theovercoming of the obstacle if it means quite relative change. Even if a student
changed as a result, it may be judged that he/she overcame an obstaclesuperficially at least, if the obstacle comes not to function as itself. If a
student accomplishes such a change, he/she would construct “anothermathematics” which is different from the mathematical knowledge that weapprove now. (cf. Bloor, 1976) That is, it is “mathematics” considered as todeviate from the direction of the development of the knowledge that we aim
at in normal education. However, we do not expect the overcoming in such ameaning. Therefore we usually lade certain rationality for “overcoming”
tacitly. In other words the overcoming should be done in an individual, but
the knowing accomplished by overcoming is not to personal. Because thenature of “another mathematics” that Bloor points out is neither the personalconsistency nor the social agreement. That is why we hope that a student
learns mathematics as social knowledge as Balacheff(1990) insists. In otherwords, it is necessary for a student that the solution of the problem isguaranteed in social context. Therefore, the social context is the first point ofview to describe relation with the obstacle of the student when an observer
judges as a student faces an epistemological obstacle, whether or not a
student is aware of it. Then is it desirable for us to expect any kind of social
context for the mathematics learning of the student?
A student participating in no social context means that he/she does nothave any relations with another person and then he/she persists in his/her
subjective ease. As mentioned above, the persistence to the subjective ease6

has a superior point to arrive at clear decision making for a student.
However, when a student meets with another person insisting on a different
thought, the student must persuade his thought to another person or do somekind of reaction for the insistence of another person.
Therefore, it is
impossible that the student ignores the relations with another person, and we
also do not expect so. Therefore, there will be seen some kind of social context
unless a student persists in his/her subjective ease.
Justification as the social adaptation of “event” means the social context tobe concluded among a student with a teacher in particular. This is going to be
overt by a student evading a risk with not accepting the taught by a teacher.
However, there is not the necessity of the social construction of themathematical knowledge which is another insistence of Balacheff. As
mentioned above, the social adaptation is needed in itself, and if authority ismore excellent, the social adaptation based on it is more certain andcontinues for a long time. (cf. Peirce, 1877) However, we do not wish to carryout such authority in the mathematics learning of the student. Rather we
expect that a student evolves his/her knowing through judging validity of his/her mathematical knowledge by a confrontation with another person. The
phase of such a confrontation occurs in a state of becoming aware of an
epistemological obstacle and in a state of overcoming it by assuming theformer. Because, in these states, a new “notion” has some kind of context of
purpose and meaning, so that the student confronts his/her “conviction” forsuch a purpose and a meaning. The “notion” is socially demanded based on
examining a certain reasoning and its conclusion(s), and the student who is ina state of becoming aware of the obstacle cannot justify such a “notion”. In
other words, even if the student in such a state accepts the need of the social
construction of the mathematical knowledge, he/she cannot accomplish thisenough successfully. On the other hand, the state of overcoming
epistemological obstacle means that social construction of the mathematicalknowledge is achieved.
In summary, we need social nature of the mathematical knowledge and of
the construction of the mathematical knowledge as social context in themathematics learning of the student. The former is not seen in persistence tothe subjective ease. It is seen by a way to accept social knowledge withauthority in justification as the social adaptation. However, in becoming
aware of or overcoming the epistemological obstacle, the student will confront

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another person not only for accepting social established knowledge, but also
for his/her knowledge to become socially.
2)Reflection on one’s knowing
It is necessary for a student to be able to become aware of difference
between of his/her knowing and of another person so that a student confronts
another person. In other words, it is necessary for the student to be able toreflect to his “notion” and “conviction”. When a student learns mathematics,
it is indispensable for him/her to reflect on his/her knowing so that he/she
considers it as not to be fix but to be changeable. Therefore, reflection on his
knowing is the second point of view to describe relation with the obstacle ofthe student. Obviously because the phase of the confrontation with anotherperson does not occur in a state of persistence to the subjective ease or ofjustification as the social adaptation, a student does not reflect on his/her
knowing.
What is emphasized in particular by this point of view is the distinction
between a state of persistence to the subjective ease and of becoming aware of
the obstacle. Let's suppose that a student dismisses a new “event” as a resultthat he/she becomes aware of an epistemological obstacle. At least, therejection by a student is not different as a result that he/she persists insubjective ease. However, we distinguish two kinds of state of knowing in aprocess, not considering both as the same with the result. Then, we candistinguish both definitely by adopting a point of view of reflection.
3) The “conviction”-shift
We saw that overcoming epistemological obstacle is accomplished by astudent shifting his/her “conviction”. It means that overcoming the obstacle is
not accomplished only by a student merely correcting his/her old “notion”
through reflection of his/her knowing and changing it into a new one. Shifting“conviction” means that conceptual change should be total reform of theknowing not the change of the “notion” only - C(C, N, E) model totally
becoming new [1995a] -, and there we see evolution of the knowing of thestudent. Therefore, the “conviction”-shift is the third point of view to describe
relation with the obstacle of the student.

8What is emphasized in particular by this point of view is to distinguish astate of overcoming epistemological obstacle from a state of becoming awareof the obstacle. In this case also, both advance in comparison with the past sothat a student accepts a new “notion” if we see only a result. However, in astate of justification as the social adaptation, his/her “conviction” ismaintained because there is not reflection on his/her knowing. On the other
hand, overcoming the obstacle means to be able to shift old “conviction” to thenew one which justifies a new “notion” so that a student understands social
nature of the construction of the mathematical knowledge and then reflects
on his/her knowing.
The above-mentioned argument is arranged as the following table. “­”means that the right side is not considered when the left is not seen, because
the left becomes the premise of the right. And in a point of view of social
context, “△” means that social nature of the mathematical knowledge is seen
but social nature of the construction of the mathematical knowledge is not. In
the table, it is shown that state of overcoming epistemological obstacle satisfies
all points of view. This means that we may not sense that a student follows aprocess of overcoming epistemological obstacle if his/her learning ofmathematics is not interpreted at each point of view. Therefore, three above
points of view are suggested as the significance which a student follows aprocess of overcoming epistemological obstacle.

‣ Mizoguchi, T. (1995a). Featuring the process of overcoming epistemologicalobstacle by categories of description: A case of the notion of limit. Journal ofJapan Society of Mathematical Education: Reports of Mathematical
Education, 63/64, 27-48. (in Japanese)9‣ Mizoguchi, T. (1995b). A study of the significance of overcoming
epistemological obstacle to the learning of mathematics: Focus on thelearner's way of concerning with epistemological obstacle. Bulletin of Instituteof Education, University of Tsukuba, 20(1), 37-52. (in Japanese)10