Epistemological Obstacles
What are Epistemological Obstacles (EOs)?
Epistemological obstacles (EOs) are necessary challenges in a student's mathematical development that persist even in the face of research-based instruction. They are learning challenges that cannot be resolved within a single lesson, or even a unit, or an entire course. Resolving EOs requires a structural change in the way one logically reasons.
Bachelard (1938) was the first to coin the phrase "obstacle epistemologique," and he asserted that "it's in terms of obstacles that we must pose the problem of scientific knowledge" (p. 16). In 1990, Balachef reposed EOs from the perspective of students' individual development: "old knowledge can turn into an obstacle to the constituion of new conceptions, even though it is a necessary foundation" (p. 264). Brusseau (2002) further clarified EOs by distinguishing three kinds of obstacles that students might experience when learning new concepts:
- Ontogentic obstacles - related to neurological growth and maturation.
- Didactical obstacles - products of instruction
- Epistemological obstacles - challenges that remain even when students are developmentally ready and even in response to the best research-based instruction.
The Proofs Project further characterizes EOs as a persistent tension during instructional interactions that is initially recognized by teachers and then, ideally, experienced by students. We believe that teachers play an essential role in both eliciting students' experience of EOs and facilitating classroom interactions that support students in addressing these EOs.
Documented EOs Experienced in Introductory Proofs Courses
Here we highlight some of the EOs that have been identified in prior research on students' challenges in introductory proofs courses.
Video Example of EOs
Here we present a video example of a student, Shivani, reasoning about the following task in her clinical interview with Andy.
As you watch the video, try to identify what EOs Shivani might be experiencing. Her written work is shown below the video.
Shivani's written work produced in the video.
- NO: Treating the Negation as its Opposite. When asked for the negation, Shivani responds with the opposite implication: P(x) → ~Q(x), instead.
- Qh: Hidden Quantification. Shivani does not explicitly attend to the quantification of her negation. While she states her negation symbolically, the quantification of x in her statement remains hidden.
Note that Shivani does not experience Qneg because quantification remains hidden. For a student to experience Qneg, they must be explicitly attempting to quantify their negation.