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Keywords: Abstraction, Arithmetic, Symbols, Representing, Generalizing, Synthesizing, Learning Mathematics, Advanced Mathematics
Thinking abstractly is a prerequisite of advanced mathematical learning
A very common definition of abstraction views abstraction epistemically, that is, as part of the process by which we come to know things about the world. For example, when we encounter several instances of chairs, we learn to abstract away the peculiar differences between them and start to focus only on their common physical properties. Over time we form the concept of a chair over and above all these instances and their differences, and this concept is an abstraction of chairs. Hence abstraction is a form of generalizing about something, a way of coming to see the essence of a thing.
While this definition of abstraction is one possible definition, it is heavily weighted towards an empirical viewpoint and is not as useful when applied to advanced mathematics. The abstraction we are primarily concerned about is the one involving symbols and mathematical notation, because it is here that the real difficulties start for the student of arithmetic and not in the concept of a counting number, for example. Symbols, unlike chairs, tables, or dogs, cannot be probed empirically to determine their meaning or 'essence'. Symbols have an element of arbitrariness about them, and their meaning is often in relation to other arbitrary symbols. Hence symbols by nature are both arbitrary (in the sense that they can take on any form) and autonomous (in the sense that they get their meaning only from their usage as symbols and not from anything we encounter in experience). To understand these two properities of symbols is to appreciate why abstraction is an especially difficult thing for students of arithmetic.
Definitions of abstraction in mathematics often make reference to these properties of symbols, as is the case in Michael Mitchelmore's paper "Abstraction in Mathematics and Mathematics Learning":
"We claim that the essence of abstraction in mathematics is that mathematics is self-contained: An abstract mathematical object takes its meaning only from the system within which it is defined. Certainly abstraction in mathematics at all levels includes ignoring certain features and highlighting others, as Sierpinska emphasises. But it is crucial that the new objects be related to each other in a consistent system which can be operated on without reference to their previous meaning. Thus, self-containment is paramount." (Mitchelmore 2004)
This definition helps explain why Mitchelmore in his paper calls mathematical objects "abstract-apart", whereas notions of abstraction relating to empirically-known objects he calls "abstract-general". When students are first taught arithmetic in school, the abstraction that is taught is similar to the abstract-general sort, because symbols that are employed to represent numbers and number operations first need some concrete connection to the world at large in order to give them meaning. However, every advancement made in mathematics learning from this point on will be at higher and higher levels of abstraction, and at some point the symbols will take on a meaning that can only be properly understood within an entire system of such symbols. For this reason it is fundamental to doing advanced mathematics that the symbols employed are used consistently. It is in such a system of representations that the meaning of these symbols exists — and nowhere else.
"[T]his capability to abstract may well be the single most important goal of advanced mathematical education."
Of all the skills that students need in order to attain adequate competence in advanced mathematics, the ability to think abstractly is probably the most important. Dreyfus, in his paper "Advanced Mathematical Thinking Processes", places the ability to think abstractly above even the ability to represent or generalize when it comes to advanced mathematical reasoning.
"If a student develops the ability to consciously make abstractions from mathematical situations, he has achieved an advanced level of mathematical thinking. Achieving this capability to abstract may well be the single most important goal of advanced mathematical education." (Dreyfus 2002)
According to Dreyfus, distinct cognitive abilities such as representing, generalizing, and synthesizing are all prerequisites to thinking abstractly. Representing is said to be the employment of either symbolic representations or mental representations while visualizing a particular mathematical problem. Generalizing, as mentioned above, is the process by which we come to know the essence of a thing and is therefore connected to deriving or inducing from instances towards common properties or generalities. Synthesizing then combines these representations or generalizations about subjects from various developed areas of mathematics in order to create a unified picture of a larger mathematical subject. Dreyfus gives the example of linear algebra to show how synthesizing works in practice:
"[I]n linear algebra, students usually learn quite a number of isolated facts about orthogonalization of vectors, diagonalization of matrices, transformation of bases, solution of systems of linear equations, etc. Later in the learning process, all these previously unrelated facts hopefully merge into a single picture, within which they are all comprised and interrelated. This process of merging into a single picture is a synthesis." (Dreyfus 2002)
All these different thinking processes (representing, generalizing, synthesizing) are interwoven together with abstracting in order to advance mathematical thinking. As we learned earlier, even abstraction begins early for the student of arithmetic in the form of generalizing and learning basic mathematical notation. But abstraction is not simply the end result of a long chain of thinking processes. At each stage of mathematical learning, abstract thinking helps reconstruct what we have already learned into something we have not seen before. In a sense it is a leap from one stage of understanding to another, more "systematic" level. In other words, we begin to see one and the same thing in different ways, or diverse things we have learned separately as facets of the same underlying object. We begin to see seemingly disparate things as parts of a larger system, all interconnected in a consistent way. Abstraction thus becomes the means by which the student advances in mathematical learning from one level of complexity to that next, higher level, which somehow simplifies that complexity under a new, self-contained and self-consistent system.
The patent-pending Tetractys Number Puzzle incorporates some aspect of all of these processes in order to help students learn how to think abstractly. Using these three features of the puzzle — the Rule, the Template, and the System — the student is able to encounter all the elements of thinking that are necessary to make the leap to abstract thinking, particularly as it relates to the digital representation of numbers. By understanding in a deeper way the multifunctional nature of these digits under an arbitrary rule, students are able to learn to see these digits in a new way — even as a new type of mathematical object. If students can accomplish that much, then they have also learned in the process how to think abstractly about these same digits.
Dreyfus T. (2002). Advanced Mathematical Thinking Processes. In: Tall D. (eds) Advanced Mathematical Thinking. Mathematics Education Library, vol 11. Springer, Dordrecht.
Ferrari, Pier Luigi (2003). Abstraction in Mathematics. Philosophical Transactions: Biological Sciences Vol. 358, No. 1435, The Abstraction Paths: From Experience to Concept, pp. 1225-1230.
Mitchelmore, M., & White, P. (2004). Abstraction in mathematics and mathematics learning. In M. J. Hoines, & A. B. Fuglestad (Eds.), Proceedings of the 28th conference (Vol. 3, pp. 329-336). Bergen, Norway: Bergen University College.
If a student develops the ability to consciously make abstractions from mathematical situations, he has achieved an advanced level of mathematical thinking. Achieving this capability to abstract may well be the single most important goal of advanced mathematical education.