Stillman (2000) elaborates on the different classes of, and the importance of, prior knowledge in the way students approach a task. Prior knowledge of the academic

variety consists of knowledge gained through outside academic experiences, such as content learned in another class or study. Encyclopaedic prior knowledge

includes general facts and trivia of the world, and episodic prior knowledge

is that which is gained by a learner from personal experiences outside of an academic setting.

Sloyer (2004) offers a strategy he calls the extension-reduction strategy

. This is a pedagogical strategy in which a teacher will present a problem that requires his or her students to use their prior knowledge to construct new knowledge. The goal of the strategy is to get the students to reduce the new problem down to a simpler and better understood problem (e.g., finding the area of a polygon by subdividing it into triangles). The teacher's role is to help and guide the students in activating their prior knowledge. Even though the students may have this prior knowledge, they may not always know how to use it productively. The example that Sloyer gives is a problem in which students try to find the volume of a segment of a cone. The prior knowledge here was that of finding a part of a whole. The value of the desired part is found by taking the value of the whole (whether that be a region's area, a finite series, a solid's volume, etc.) and subtracting the *extra* amount.

Hare's (2004) example is a bit more complex. In this study, students learned implicit differentiation by reinforcing and expanding their concept of a function. Students had varying misconceptions of the definition of *function* and thus could not take the implicit derivative of an equation properly. Through guided questions and activities, the students were not only able to use their prior knowledge but also improve on it, while at the same time gain new knowledge.

References

- Hare, A. & Phillippy, D. (). Building mathematical maturity in calculus: Teaching implicit differentiation through a review of functions.
*Mathematics Teacher, 98*(1), 6–12. - Sloyer, C. W. (). The extension-reduction strategy: Activating prior knowledge.
*Mathematics Teacher, 98*(1), 48–50. - Stillman, G. (). Impact of prior knowledge of task content on approaches to applications tasks.
*Journal of Mathematical Behavior, 19*, 333–361.

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