The notion of ‘procept’ began its life as an idea generated by looking at a symbol such as 3+2 both as a process (of addition) and a concept (of sum). It was extended (**Gray & Tall, 1994**) to include different symbols and different processes that give rise to the same mental object in the mind of a particular individual. Thus 3+2, 4+1, 5, 7-2 can all represent the same procept, involved in composing and decomposing arithmetic processes that give 5.

The definition of procept is given in Gray and Tall (1994) is as follows:

An

elementaryprocept is the amalgam of three components: a process which produces a mathematical object, and a symbol which is used to represent either process or object.

Aproceptconsists of a collection of elementary procepts which have the same object.

The definition caused us a great deal of heart-searching because we wanted it to reflect the observed cognitive reality. In particular we wanted to encompass the growing compressibility of knowledge characteristic of successful mathematicians. Here, not only is a single symbol viewed in a flexible way, but when the same object can be represented symbolically in different ways, these different ways are often seen as different names for the same object. ... In this sense we can talk about the procept 6. It includes the process of counting 6, and a collection of other representations such as 3+3, 4+2, 2+4, 2x3, 8/2, etc. All of these symbols are considered by the child to represent the same object, though obtained through different processes. But it can be decomposed and recomposed in a flexible manner.

Gray & Tall distinguished between the specific *procedure* as an explicit sequence of steps and the input-output *process* where different procedures can have the same input-output effect. Thus 'count-all' and 'count-on' as procedures for evaluating 3+2 are different procedures to give the same arithmetic process. In various papers we have studied the procedure-process-procept spectrum of performance and linked it to a growing bifurcation between success and failure that occurs when some individuals cling to familiar procedures whilst others develop the flexibility that comes through the use of procept.

In my more recent papers (2003c, 2004a, 2004b, 2004d below) I have framed the theory of procepts within a wider theory of ‘three worlds of mathematics’, and in particular, I have worked with Anna Watson (subsequently Anna Poynter) on a very important paralled between the focus on the 'effect' of an embodied action and the notion of compression of a procedure into a process and on to a generic embodiment of a procept. This idea arises first in 2002j.

More recently, the notion of procept has been enhanced by introducing a notation that distinguishes between an expression as a process or concept by placing an object in a box.

as the operation of addition of two mental objects (numbers)

or

as the single mental object (the sum of the two numbers).

This introduces a technique to interpret an operational expression sucessively as operations on objects (see, for example, 2019a, 2019b, 2019c, .2020a.

**Papers on procepts and related ideas:**

1991h (with Eddie Gray), Duality, Ambiguity & Flexibility in Successful Mathematical Thinking, *PME 15*, Assisi, 2 72-79.

1992d Success and failure in arithmetic and algebra, *Mathematics Teaching 1991*, Edinburgh University, September 1991, 2-7.

1992h Mathematical Processes and Symbols in the Mind, in Z. A. Karian (ed.) *Symbolic Computation in Undergraduate Mathematics Education*, MAA Notes 24, Mathematical Association of America, 57-68.

1992k (with Eddie Gray): Success and Failure in Mathematics: Procept and Procedure - A Primary Perspective, *Workshop on Mathematics Education and Computers*, Taipei National University, April 1992, 209-215.

1992l (with Eddie Gray): Success and Failure in Mathematics: Procept and Procedure - Secondary Mathematics, *Workshop on Mathematics Education and Computers*, Taipei National University, April 1992, 216-221.

1993b (with Eddie Gray), Success and Failure in Mathematics: The Flexible Meaning of Symbols as Process and Concept, *Mathematics Teaching*, 142, 6-10.

**1994a (with Eddie Gray). Duality, Ambiguity and Flexibility: A Proceptual View of Simple Arithmetic, The Journal for Research in Mathematics Education, 26 (2), 115-141.
**1993g Success & Failure in Arithmetic and Algebra,

1995d The Psychology of Symbols and Symbol Manipulators,

1997c (with Michael Thomas & Garry Davis), What is the object of the encapsulation of a process?,

1997f Metaphorical objects in Advanced Mathematical Thinking,

2001m Chae, S. D. and Tall, D. (2001). ‘Students’ concept images for period doublings as embodied objects in chaos theory’.

2002m Ehr-Tsung Chin, David Tall (2002), Proof as a Formal Procept in Advanced Mathematical Thinking,

2003c Watson, A., Spyrou, P., Tall, D. O. (2003). The Relationship between Physical Embodiment and Mathematical Symbolism: The Concept of Vector.

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