Procept

A procept, in the context of mathematics education, refers to a mental object that encapsulates both a process and the concept that is produced by that process. The term was coined by Eddie Gray and David Tall to describe a key characteristic of advanced mathematical thinking, particularly in algebra and calculus.

Essentially, a procept is a "chunk" of mathematical knowledge that can be unpacked either to reveal the process by which it was obtained or treated as a single entity for further manipulation. For instance, the expression "3 + 4" can be seen as a process (the addition of 3 and 4) or as the concept "7". The ability to flexibly move between the process and the concept perspectives is considered crucial for success in more abstract mathematics.

The development of proceptual thinking is often contrasted with procedural thinking, which focuses primarily on the steps involved in performing a calculation or applying a formula. While procedural knowledge is important, it can become a barrier to understanding more advanced concepts if students are unable to see the underlying meaning or represent the process as a manipulable object.

Proceptual thinking is considered a cornerstone of advanced mathematical fluency, allowing learners to handle complex ideas efficiently and effectively. It highlights the dynamic interplay between actions and mental objects in the understanding of mathematical concepts. A weakness in proceptual understanding can manifest as an inability to solve problems requiring flexible manipulation of mathematical expressions and a dependence on rote memorization of procedures.