Mathematics educators have been busy studying mathematical discourse in the past two decades.

It is commonly agreed within this community of researchers that intuition and operational experimentation serve successfully the development of higher level understanding, however, reaching this level necessitate reflective thinking which is anchored in verbal discourse. (E.g., Skemp 1973, Hiebert, and Carpenter 1992, Sfard 1994 and others)

There is a tendency to communicate about visible entities by visual representations, and about formal mathematics notions, by symbols and verbal arguments.

Although 3-d geometrical entities, are real and visible, the examples given above illustrate that in order to develop a sense of meaning for their structures, their properties and the inter-relationships among them, it is necessary to combine language and pictorial representation.

As Vygotsky (1986) said:

"...thought unexpressed remains immature and eventually dies out. Language therefore, is not just an expression of otherwise independent and fully formed thought, but rather is a necessary form of the thought's realization."

While the visual representation provides an intuitive infrastructure, it is the language that guarantees the transition from the intuitive perception to an analytic-synthetic understanding, and even to innovative findings.

For example, while describing in words the process of truncating (cutting off) the vertices of a tetrahedron through mid-edges, students observed that truncating the “top” vertex first, yielded a triangular face parallel to the “basis” of the original tetrahedron. This observation led to the insight that by truncating the other three vertices, a triangular antiprism is obtained.

A second thought brought them to the conclusion that the resulted 8 triangular face polyhedron is actually an octahedron, and from here the way was short to the insightful finding that the triangular antiprism and the octahedron are nothing but two sides of the same coin. 

The property of the octahedron as an antiprism emerged as a result of the discourse among the group of students who had a tetrahedron in front of them and were challenged to study the results of various ways of truncating it.

The students arrived at their discovery, totally on their own based upon the combination of the mental image they were able to operate on, and their linguistic ability to follow their imaginary action of truncating and connect it to a recently acquired concept of the antiprism. 

Visual representations and mental (pictorial) images, being wholesome and compact supported students’ overall structural conception, that can be grasped at one glance, while verbal encoding enabled a process of decomposition of the whole into its parts, analyzing the inter-relationship among the parts and the complete new structure.

Language played an important role in coming to grips with spatial relationships and structural properties.

Using 'literary' forms in making connections, helped students in making sense of mathematical constructs, and in remembering it.

Language is the medium within which the creation of new concepts takes place. We are not questioning the power of visualization. As Sfard (2000) noted: "Availability of visually manipulable means, either actual or only imagined, underlies our ability to communicate on the objects and operate them discoursively".

On the other hand, we suggest that verbal representations for visual entities are crucial for the understanding of those entities.

The concepts we are dealing with belong to the domain of concrete objects or processes, usually, accessible to us through perceptual experience.

As these concepts are more complicated, the role of language and verbalization becomes more crucial.

Students come to recognize the properties of  a solid, by describing them verbally. While verbalizing, students organized their visual impressions by comparisons: distinguishing which aspects of the shape are to be noticed and which ones are to be ignored, which properties are similar to already known objects, and which are different.

The immediate implication is that whoever strives to become a professional, must develop the ability to express verbally, visual observations and to visualize verbal ideas mentally .

Describing an object verbally is a process that involves the decomposition of the whole, followed by sequential reassembling of the parts.

As early as 1969, Paivio observed that visual imagery has many of the properties of a spatially parallel system, whereas verbal processes are better suited for handling sequential, serial information (Paivio 1969).

It takes cognitive processing to make sense of visual information, be it a 2d representation or a concrete 3-d model.

The interplay between visualization and verbalization is the key to cognitive processing of that sort. The enhancement of this combination is therefore at the heart of a literate approach to any profession that has to do with spatial relations - - Architecture, Engineering, Mathematics and Arts.

Finally, to balance the claim made so far, we leave you with a dilemma we started this paper with:

Is verbal description always appropriate? Is it necessarily needed with no exception? Or is it sometimes simply redundant? Possibly disturbing???

References

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