The concept of epistemological obstacle emerges in philosophy of science in the works of Bachelard (1938) who is the first to interpret the genesis of scientific knowledge with the support of this concept: “It is in terms of obstacles that one must pose the problem of scientific knowledge […] it is in the very act of knowing that we will show causes of stagnation and even of regression, this is where we will distinguish causes of inertia that we will call epistemological obstacles.”

The examples given by Bachelard are typical of the prescientific thinking and connect to what he calls the obstacle of primary experience. In this, the substantialist obstacle consists in referring to a substance equipped with quasi magic properties in order to explain the observed phenomena: as an example, the attraction of dust by an electrically charged surface will be explained by the existence of an electric fluid. Bachelard rightly explains that the obstacle arises from the fact that this is not a metaphor but indeed an explanation of the situation created by what our senses tell us: “We think as we see, we think what we see: dust sticks to the electrically charged surface, so electricity is an adhesive, is a glue. One is then taking a wrong way where false problems will generate worthless experiments, the negative result of which will fail in their role of warning, so blinding is the first image […].”

Brousseau (1976, 1983) is the first to transpose the concept of epistemological obstacle to the didactics of mathematics by highlighting the change in status for the error, that this notion generates: it is not a “result of ignorance […] or chance” but rather an “effect of prior knowledge that was relevant and had its success, but which now proves to be false, or simply inadequate” (Brousseau 1983). Among the obstacles to learning, Brousseau distinguishes indeed the ontogenic obstacles, related to the genetic development of intelligence, the didactical obstacles, that seem to only depend on the choice of a didactic system, and the epistemological obstacles from which there is no escape due to the fact that they play a constitutive role in the construction of knowledge. At one and the same time, the concept of epistemological obstacle extends to the didactics of experimental science (Giordan et al. 1983).

The pioneering works in didactics deal with, among others, obstacles related to extensions to sets of numbers – relative numbers in Glaeser (1981), rational and decimal numbers in Brousseau (1983) – with obstacles related to the absolute value in the research from Duroux (1983), with those that tend to hide the concept of limit, as studied by Cornu (1983) and Sierpinska (1985), with obstacles related to learning the laws of classical mechanics according to Viennot (1979) and with those arising from a sequential reasoning in solving electrical circuits, of which Closset (1983) shows the excessive strength. From these works and others, Artigue (1991) conducts an analysis in which several questions arise, that are subject to debate when trying to characterize the concept of epistemological obstacle: can we talk about epistemological obstacles when there is no identification of errors and but simply of difficulties? Should we look for their appearance and their resistance in the history of mathematics? Look for their unavoidable character in the students’ learning process? What does their status of knowledge consist of, having its domain of validity? Can we talk, in certain cases, about a reinforcement of epistemological obstacles due to didactical obstacles?

Other studies also ask the question of the scale at which it is appropriate to look at the epistemological obstacles, as well as that of their cultural character. The works of Schneider (1988) raise these two questions in an articulated manner by showing that the same epistemological position, namely, empirical positivism, can account for multiple difficulties in the learning of calculus: errors when calculating areas and volumes in relation with misleading subdivisions of surfaces into lines and of solid surfaces into surface slices, a “geometric” conception of limits leading students to think of segments as being “limits” of rectangles, and of the tangent line as being “limit” of secants without reference to any topology whatsoever, and their reluctance to accept that the concept of derivative will provide the exact value for an instantaneous velocity. This empirical positivism which, mutatis mutandis, converges with the primary experience from Bachelard in the sense of “experience placed before and above criticism” goes well beyond learning calculus (Schneider 2011). This example illustrates indeed, on the one hand, an obstacle considered at a large scale, with its interpretive scope covering errors or multiple difficulties and, on the other hand, its cultural aspect which can be considered as a pure product of Western modernity. It also shows that, despite the opinion of Bachelard, the notion of epistemological obstacle applies to mathematical thinking, at least on a first level.

The debate on the scope and cultural character of epistemological obstacles, of which the examples above illustrate the probable dependence, is animated and most probably not closed. Regarding the first aspect, Artigue insists on the interest in considering what she calls “obstacle-generating processes,” including “undue formal regularization” that, as an example, leads students to the misapplication of linearization processes such as “distributing” an exponent on the terms of a sum, or “fixing on a familiar contextualization or modeling,” such as the excessive attachment to the additive model of losses and gains when considering relative numbers. About the second aspect, Sierpinska (1989) puts back in a theory of culture some sayings of Bachelard who thinks that, if empirical knowledge of reality is an obstacle to scientific knowledge, it is because the first acts as an unquestioned “preconception” or as an “opinion” based on the authority of the person who professes it. Johsua (1996) continues to believe that some spontaneous reasonings, like those transgressing the laws of classical mechanics, have a cross-cultural character, while Radford (1997) argues that the so-called epistemological obstacle refers more to local and cultural conceptions that one develops on mathematics and science in general. And presumably, we cannot settle this debate without specifying it, example after example, as cautiously proposed by Brousseau 20 years earlier: “The notion of obstacle itself is beginning to diversify: it is not that easy to propose relevant generalizations on this topic, it is better to perform studies on a case by case basis.” All this without yielding to the temptation of qualifying as epistemological obstacle whatever is related to recurring errors for which we did not analyze the origins (Schneider 2011).

The identification of epistemological obstacles brings forward the question of their didactical treatment: should we have students to bypass them or, on the contrary, should we let them clear the obstacle and what does that mean? Let us first turn to “educator” Bachelard (as described by Fabre 1995). It is the intellectual distancing that Bachelard emphasizes as major learning issue, when he writes that “an educator will always think of detaching the observer from his object, to defend the student against the mass of affectivity which focuses on certain phenomena being too quickly symbolized […]” (1949). An echo hereof is the psychological shift of perspective (“décentration”) of Piaget that, among children, the interpretation of an experience assumes: as such, it “does not obviously make sense” that sugar dissolved in water has disappeared on the account that one cannot see it anymore! One of the primary goals of education would thus be to promote, among students, the detachment from “false empirical objects” born from the illusion that the facts and observations are given things, and not constructed, that is to say to get them to pass from world 1 of physical realities, in the sense of Popper (1973), to world 2 of states of consciousness and to world 3 of concepts that contain “more than what we did put in them.” It is presumably those connections that lead Astolfi and Develay (1989) to place Piaget, Bachelard, and Vygotski at the origin of the constructivist movement in didactics of science, the first explaining “how it works,”, the second “why it resists,” and the third pointing out “how far one can go.” Brousseau (1983), as for him, provides clear-cut answers to the questions above: “an epistemological obstacle is constitutive of achieved knowledge in the sense that its rejection must ultimately be mandatorily justified.” There resides, according to him, the interest of “adidactical situations” whose fundamental nature with respect to the target knowledge will allow invalidating an old knowledge that proves to be an obstacle to new knowledge, by highlighting the limits of the scope of operation of the former. Martinand (1986) goes further by making obstacles – be these from the works of Bachelard, Piaget, or Wallon – a selection mode for objectives: the concept of “objective-obstacle” appears then in opposition to the usual idea of blocking point. One can think today, together with Sierpinska (1997), that an equivalent coupling may have been too systematic or even normative at a given time in didactics of mathematics, but it is probably advisable that the teacher should manage, at least by a vigorous heuristic discourse, the epistemological obstacles identified on a large scale (Schneider 2011).

The notion of epistemological obstacle has some kinship with that of conception or more precisely that of misconception, but also with that of cognitive or socio-cognitive conflict as illustrated in the acts of an international symposium on knowledge construction (Bednarz and Garnier 1989). The concept of misconception itself may be related to the mental object from Freudenthal (1973) or to the image-concept in Tall and Vinner (1981) who, despite some differences, indicate that the mind of students being taught is not in a virgin state but is a host of intuitions keen to facilitate learning but also to hinder it. In some examples, misconceptions converge with epistemological obstacles in an obvious manner. As such, some of the probabilistic misconceptions identified by Lecoutre and Fischbein (1988) are explained by causal and chronologist conceptions of the notion of conditional probability which, according to Gras and Totohasina (1995), are obstacles of epistemological nature. As for the concepts of cognitive or socio-cognitive conflicts that underpin the Piagetian and Vygotskian theories, they also rely on the assumption that learning is motivated, on the one hand, by an imbalance between the reality and the image that an individual makes up of it and, on the other hand, by confronting his opinion with that of others or with a contradictory social representation. The transfer of the concept of epistemological obstacle to the didactics of mathematics is then bringing a new contribution to the theories mentioned above, in terms of close dependency between the evolution of conceptions among students and the didactical situations they are confronted with: “[…] the crossing of an obstacle barrier requires work of the same nature as the setting up of knowledge, that is to say, repeated interactions, dialectics of the student with the object of his knowledge. This remark is fundamental to distinguish what a real problem is; it is a situation that allows this dialectic and that motivates it” (Brousseau 1983). And this is indeed what makes the link between didactics and epistemology to be so tight.