In 1994 Eddie Gray and David Tall of the University of Warwick wrote a research paper extolling the virtue of ambiguity in mathematics. They identified that those who can flexibly handle different perceptions of the one mathematical object are those who find mathematics much easier and who seem to achieve at a higher level. Their central ideas revolve around the duality between process and concept. They distinguish “process”, as in the “process of addition”, and a “procedure”, which refers to a specific algorithm for carrying out a process (such as, for addition, “count-all” or “count-on”). And while procedures (routine manipulation of mathematical objects) and concepts (knowledge rich in relationships and connected ideas) are often considered disparate areas of learning, Gray and Tall argue that they are organically connected: procedures are interiorised by the learner and become objects in themselves, in turn manipulated within other procedures.
The authors draw attention to mathematical symbolism, highlighting that the same notation is often used “to represent both a process and the product of that process”. Examples they give include:
The symbol 34 stands for both the process of division and the concept of fraction.
The trigonometric ratio sine A = length of opposite ÷ length of hypotenuse represents both the process for calculating the sine of an angle and its value (see From the Library in Issue 125).
The algebraic symbol stands both for the process “add three times and two” and for the product of that process, the expression “”.
The function notation both gives the instructions as to how to calculate the value of the function for a specific value of as well as encapsulating the complete concept of the function for a general value of .
The authors argue that “the ambiguity in interpreting symbolism in this flexible way is at the root of successful mathematical thinking”; conversely, the absence of such ambiguity leads to meaningless procedures (“two negatives make a plus”, “change sides, change signs”). The reification of the process into a concept, and the ability (which is rooted in the meaning provided by the process) to flexibly shift between the two complementary perspectives, supports the learner to make the crucial step from thinking about the concrete to thinking in the abstract. Gray and Tall name this idea procept, “the amalgam of three components: a process which produces a mathematical object, and a symbol which is used to represent either process or object.” Critically, they argue, it is the use of procept that distinguishes performance at mathematics.
Procedural thinking, they explain, focusses “on the procedure and the physical or quasi-physical aids which support it”, while proceptual thinking “is characterised by the ability to compress stages in symbol manipulation”. Procedural thinking provides guaranteed success in a limited range of situations but is unlikely to lead to success in more complex problems. Gray and Tall identify the proceptual divide, separating those who think procedurally and those who think proceptually. Procedural thinking limits the formation of concepts at the next level up; the proceptual divide creates a chasm between those for whom mathematics provides great power and those for whom it becomes a subject of “spiralling complexity”.
Fig 1 from Gray and Tall (click to enlarge)
For example, the authors argue that the procedure of multiplication, e.g. 3 × 4, will be almost impossible for a pupil to grasp while still considering addition as a (separate) procedure, 4 + 4 + 4. In contrast, proceptual thinking collapses the hierarchy of thought into a single level. The symbol 4 + 4 + 4 simultaneously represents the process of adding 4 repeatedly and the sum of 12, enabling progression to the concept of product.
Fig 2 from Gray and Tall (click to enlarge)
Compression of mathematical ideas makes them simpler to handle. The “whole of mathematics may therefore be thought of in terms of the construction of structures … mathematical entities move from one level to another, an operation on such entities becomes in its turn an object of the theory” (Piaget, 1972, p.70). The compression of a process into a compact idea allows the short-term memory to manipulate a mathematical object that is rich in conceptual meaning.
So what does their theory imply in the classroom for teaching and learning? It suggests that new concepts need to be grounded in sufficient examples of the appropriate type to enable the learner to interiorise the procedures, implying, as Weber points out (Weber 2005, p.95), that an opportunity and structure for reflection is required. Fields Medallist William Thurston wrote about the importance of “work[ing] through some process or idea from several approaches” in order to be able to see the mathematical object as a whole. Applying this to the procept of “34”, this suggests that pupils need to experience different procedures such as representing as different equivalent fractions and ratios, calculating the fraction as a decimal, as a percentage, as a fraction of something, visualising it as cutting an object (an area, a length, a volume) into 4 equal parts and then choosing 3 of them, experiencing dividing 3 objects between four people. For the function , pupils should calculate a table of “input-output” values, sketch the , sketching the graph of using pencil and paper, solving the equation and using a dynamic geometry package to slowly trace out the curve. These are just a few suggestions; let us know (firstname.lastname@example.org or Twitter @NCETMsecondary) what you try with your pupils, and what impact and outcomes you observe.
Beth, E. W. and Piaget, J. (1966). Mathematical Epistemology and Psychology (W. Mays, trans.), Dordrecht: Reidel.
Boaler, J. (2009) The Elephant in the Classroom: Helping Children Learn and Love Maths.
Gray, E. M. and Tall, D. O. (1994). Duality, Ambiguity and Flexibility: A Proceptual View of Simple Arithmetic. Journal for Research in Mathematics Education, 26 2, 115 – 141.
Piaget, J. (1972). The Principles of Genetic Epistemology (W Mays, trans.) London: Routledge and Kegan Paul.
Sfard, A. (1991). On the Dual Nature of Mathematical Conceptions: Reflections on Processes and Objects as Different Sides of the Same Coin. Educational Studies in Mathematics, Vol. 22, No. 1 (Feb., 1991), pp. 1-36.
Thurston, W. P. (1990) Mathematical Education, Notices of the American Mathematical Society, 37 (7), 844 – 840.
Weber, K. (2005) Students’ Understanding of Trigonometric Functions, Mathematics Education Research Journal 2005, Vol. 17, No. 3, 91–112