Visible Language

We know that the confusion between meaning and sign (in French: signifié/ significant) is the root of a great number of mistakes in mathematics. Particularly, instead of making easier the approach to the mathematical concept represented, the sight of the design often produces a disturbance to understanding; it leads to mistaking the drawing for the presented idea, as idolatrous people do. I will demonstrate — by presenting many genuine examples which I have met in mathematical classrooms at every level — the mathematical roles of synonymy and homonymy.

Special difficulties often arise in reading mathematics because of the symbols and notation that are used. This is caused not only by the range of symbols and their density of meaning (interiority) but also by strong emotional responses raised by certain symbols or combinations. These feelings may reflect unpleasant memories of when the symbols were first encountered, but may even derive from an unease with the shape of some of them.

Problem solving in mathematics may require different kinds of language: the verbal or mathematical language in which the problem representation is available to the solver, and the planning language for heuristic reasoning and formulation of strategies. This paper explores some relationships among these languages, with examples of ways they can influence problem-solving processes.

Extracts are taken from the biographies of Hobbes, Rousseau, Darwin, and Russell which refer to their mathematical education. The common feature of an attraction toward geometry and an aversion to elementary algebra is noted. These experiences are analyzed using theoretical positions promulgated by Davis, Hersh, Skemp, and Bruner. The central thesis is that these men probably have had difficulty learning elementary algebra because they had failed to develop a strong image or iconic representation of the concepts involved. This thesis is developed in relation to “squaring a binomial,” the concept which troubled both Rousseau and Russell.

Behind the formal symbols of mathematics their lies a wealth of experience which provides meaning for those symbols. Attempts to rush students into symbols impoverishes the background experience and leads to trouble later. In conjunction with manipulating objects it is essential to provide time for talking about their activities and developing their own informal records before meeting the formal symbols of adult mathematicians. We present three examples of children’s work which demonstrate these steps in the struggle to move toward recording perceived patterns.

Experiments by psychologists have led to the conclusion that images play an indispensable, if subordinate, role in thought as symbols. An analysis is begun of the mental images that are judged to be properly evoked by certain number symbols in arithmetic. A variety of graphical models are suggested for use in linking these symbols to the desired mental construct. Some of these models have been found to be advantageous and may prove to be critically essential in certain mathematical contexts. Their assets and liabilities are discussed, and suggestions are made for modifications of conventional curriculum practice. A rich field of investigation exists in the visual imagery that can be associated with elementary mathematics. Progress here holds promise of extending mathematical competence to a larger portion of society.

We noticed that the use of a non-verbal formalism can favor cognitive development (in the frame of the elementary school) in problem children as well as in normal children. An example is given to show how a formalism inspired by mathematics can be used to aid the development of the verbal language of 8- to 9-year-olds. We will then analyze the results and try to discover the cause of success we observed.

A distinction is made between the surface structures (syntax) of mathematical symbol-systems and the deep structures (semantics) of mathematical schemas. The meaning of a mathematical communication lies in the deep structures — the mathematical ideas themselves, and their relationships. But this meaning can only be transmitted and received indirectly, via the structures; correspondence between deep and surface structures is only partial. Some resulting problems of communicating mathematics are discussed, and some remedies suggested.

One of the essential distinguishing features of mathematics is its eventual dependence upon symbols and symbolic expression. Few attempts to determine those processes, activities, or contents which uniquely identify mathematics have succeeded. It is indeed questionable whether human knowledge can be classified into such self-contained categories. The many diverse activities of mathematicians do, however, have symbolic expression as their common feature, and the extent to which modern disciplines depend upon mathematics could be measured by their growing reliance on symbols. It is reasonable to surmise that much of the difficulty experienced by children in mathematics, and the lack of popularity of physical as opposed to biological sciences in higher education, could be traced to the problem of symbolization. It will be interesting to watch the effect on, say, geography as the school syllabuses move toward mathematical as opposed to descriptive aspects. There is surprisingly little apparent research into the use and learning of symbols, except for the many investigations into both the problem of how children learn to read and adult perceptual experiences with words (e.g., Coltheart 1972). There is, however, a real distinction between the use of symbols as a verbal language (spoken or written) and the use of symbols in the mathematical sense. It will indeed be suggested below that one activity interferes with the other.