Analysis of the Learning Disabilities in Math

3. Analysis of the language used in Math text books

Studies highlight how the difficulties of linguistic communication can make any type of direct intervention on mathematical contents vain. They stress also the fact that, for the teacher, they need to continually switch, during classroom work, from the use of languages to represent mathematics to languages to interact with the class that requires considerable metalinguistic awareness.

Beliefs about math in many cases affects motivational engagement of young students, with regard to textual problems, in particular those related to the formulation of the text of a problem, stereotypes (also linguistic) and misconceptions in the formulation of school problems induce erroneous beliefs and generate deviant attitudes towards problems and math itself. Critics about the use of the term 'misconceptions' have a theoretical foundation, and are the result of a progressive refinement of research in mathematical education. In particular, the idea of misconception and the approach to error is the starting point to a radical change that towards this definition which has put the student and their learning processes at the centre of attention. It is this shift of point of view where the learner is now considered as an active subject who builds his or her own knowledge. More precisely, this model undermines the traditional interpretation of errors. In fact, the student interprets the experience with mathematics, in particular the messages that the teacher continuously sends: the student gives meaning to these messages, a sense that naturally depends on the knowledge he has but also on many other less obvious elements. That algorithm, that term, that symbol, that property, that concept, will be internalized according to the sense attributed by the student, and it may happen that this meaning does not coincide with what the teacher intended to communicate. The learner, and more generally the individual, continually interprets the world, relating the observed facts with previous experiences: the beliefs are precisely the result of this continuous attempt to make sense of reality and, in the same time, determine the patterns with which the individual approaches the world and therefore interprets the future experience. In mathematical education, therefore, students' beliefs are seen as the result of their continuous process of interpretation of experiences with mathematics; on the other hand, in turn determining the patterns according to which future experience is interpreted, they act as a guide in selecting the resources to be activated; but, in particular, they can prevent from using adequate knowledge and resources. Beliefs and misconceptions act as a filter or as a simplification of a theory (as well as reality).

In the words of Lev Vygotsky:
“The scientific concepts evolve under the conditions of systematic cooperation between the child and the teacher. Development and maturation of the child’s higher mental functions are products of this cooperation. Our study shows that the developmental progress reveals itself in the growing relativity of causal thinking, and in the achievement of a certain freedom of thinking in scientific concepts. Scientific concepts develop earlier than spontaneous concepts because they benefit from the systematicity of instruction and cooperation. This early maturity of scientific concepts gives them the role of a propaedeutic guide in the development of spontaneous concepts. The weak aspect of the child’s use of spontaneous concepts lies in the child’s inability to use these concepts freely and voluntarily and to form abstractions. The difficulty with scientific concepts lies in their verbalism, i.e., in their excessive abstractness and detachment from reality. At the same time, the very nature of scientific concepts prompts their deliberate use, the latter being their advantage over the spontaneous concepts. At about the fourth grade, verbalism gives way to concretization, which in turn favourably influences the development of spontaneous concepts. Both forms of reasoning reach, at that moment, approximately the same level of development”.
The example of the INVALSI Math tests administered in Italian schools in recent years have provided a great mass of results and highlighted many macro phenomena attributable to textual or linguistic difficulties of Italian students. As stated by Branchetti and Viale, the syntactic dimension of the mathematical text, does not seem to be particularly analysed in didactic studies, neither in mathematical nor in linguistic-educational ones. Only in recent years the opinion that the linguistic dimension represents a fundamental component of the study of mathematics has become increasingly solid, in contrast to a certain idea inherited from the school tradition, which wants language and mathematics separate and incommunicable areas.

Traditional math schoolbooks in Italy use problems which are often characterized by long periods with a complex syntax, the use of an implicit subordinate introduced from a past participle (e.g. given the trapezoid ...) or a gerund, verbal mode are very frequent in mathematical texts. Typical of the traditional style of mathematics is also the use of the passive impersonal form (in Italian: ‘si passivante’) and the high frequency of embedded parenthetical clauses, which increases the information density of the sentence; the use of the form so that subjunctive falls within the typical style of the traditional mathematical text. It is interesting to note that some textbooks reproduce syntactic modules typical of the Italian tradition of mathematical text of this genre also in the English version of some exercises, such as the following example from the English section "Test your skill" of a reference textbook used in a technical school during the first two-year of study: “Each day a company can produce a maximum of 300 tons of a certain product. For each ton produced the cost of manufacturing and raw materials is € 1,6 and the standing daily expenses are € 36,00. Find the maximum profit and the minimum amount so as not to be in deficit knowing that each ton is sold at € 4,00”.

A good example of reference textbook is given in Italy by the work of Massimo Bergamini, Graziella Barozzi & Anna Trifone, Manuale di Matematica blu, rosso, azzurro and verde, edited by Zanichelli, which is aimed to a course that highlights the connections between mathematics and reality; the theory is expressed with particular attention to the use of a clear language, expressed with rigorous and precise criteria, and presents many exercises set in everyday life, a balanced use of images and references to activities linked to an online site. On the sides the formulas are represented with different system, to address students’ different abilities and learning strategies and customs, and a part dedicated to those students with Special Educational Needs (SEN, in it. BES).

The Polish poet Wisława Szymborska dedicated several poems to mathematics, in her book Wszystkie lektury nadobowiązkowe (Nonrequired reading) on the prince of geometry theorems, she writes:
I can easily imagine an anthology of the most beautiful pieces of world poetry making room for Pythagoras’s theorem. And why not? It sets off the sparks that are the mark of great poetry, its form is pared beautifully to only the most necessary words, and it has a grace with which not even every poet has been blessed…