Analysis of the Learning Disabilities in Math

2. Which actions are to be carried out in teaching practice with reference to the Mathematical Learning Disabilities?

It is widely believed, among science and math teachers in particular, that many of the students' comprehension and learning difficulties depend on linguistic factors.
Knowing formulas by heart, however, is not enough in the face of the text of a problem, although the students remember them, sometimes they not seem to be able to recognize the question, interpret the instructions, identify the necessary elements to reach the solution, etc.
Sometimes, teachers also complain about children's difficulty in expressing: correcting math homework often requires interpretation and integration of disconnected and linguistically incorrect texts. The linguistic or textual organizational error, in a mathematics task, has to be considered serious, as if the knowledge of the contents could be independent of the ability to express them. The tools needed by teachers of scientific disciplines must be efficient to address language difficulties as a source of difficulty in mathematics.
In learning mathematics, communicative component is central, to express and transfer knowledge, skills, attitudes, experiences, which are continually reworked and intertwined; the learning path is therefore the result of a work in which language links different components to interact with each other.
Martha Fandiño analyses the aspects of learning process from another point of view, focusing on the strategies. She distinguishes among "conceptual learning", "procedural or algorithmic learning", "semiotic learning or management of representations", "strategic learning" and finally "communicative learning", which have decisive impact in the final phase of the learning process, when they switch to effective learning and final understanding. Communicative learning of math is an aspect of education that regards the ability to express logical ideas, telling, validating, justifying, arguing, demonstrating mathematical concepts (both orally and in writing) and visually representing them with figures.
The most important international survey in the framework of mathematical skills, the OECD-PISA survey, gives a definition of ‘Mathematics performance’ which, for PISA purposes, measures the mathematical literacy of a 15 year-old: “Mathematical literacy is an individual’s capacity to reason mathematically and to formulate, employ and interpret mathematics to solve problems in a variety of real-world contexts. It includes concepts, procedures, facts and tools to describe, explain and predict phenomena. It helps individuals know the role that mathematics plays in the world and make the well-founded judgments and decisions needed by constructive, engaged and reflective 21st Century citizens”.
For instance, students should be able to manage three mathematical processes:

• Formulating situations mathematically;
• Employing mathematical concepts, facts, procedures and reasoning; and
• Interpreting, applying and evaluating mathematical outcomes;

Linguistic competence plays a fundamental role. The frame of reference of the OECD-PISA survey explains this role and can be a very useful tool, also for teachers, to better define the phenomena and interpret the students' behaviours.
In the framework of the OECD-PISA survey, the importance of communicative competence is underlined in the three different aspects of mathematical processes – formulating, employing, interpreting. In the process of formulating language is fundamental considering reading, decoding and interpreting statements, questions, tasks in order to create a mental model of the situation; in the process of employing it is clearly stated that linguistic skills are necessary to articulate a solution, illustrate the work necessary to arrive at the solution and summarizing and presenting the intermediate results; finally, in the process of Interpreting, we need language to elaborate and communicate explanations and arguments in the context of the problem. Argumentative competence also manifests itself in the three phases of the problem-solving (modelling) cycle – formulate, employ, interpret and evaluate – in particular through providing, explaining and defending justifications for mathematical modelling, important for the acquisition of linguistic competence.
In light of all this, it is evident that it is necessary to build an interdisciplinarity link between the work of the language teacher and that of the scientific disciplines, and of mathematics in particular, in all school grades.
Since linguistic difficulties and disabilities actually interfere with the understanding of the text of a problem, the ability to identify the instructions, the possibility of finding an effective solution strategy, the ability to control the correctness and the sensibility of the result, the ability to justify the chosen strategy and to discuss and justify the final solution, the strategies to contrast language difficulties and disabilities should be taken into consideration to contrast as well the problems related to the role of language in learning mathematics. First of all the linguistical aspect of manuals: they should be balanced to reflect the age and the social environment of the students, furthermore it should be taken into consideration the use of other communication techniques, such as images, and other teaching strategies, on the one hand tackling learning disabilities, on the other hand strengthening of awareness of the potential of work on mathematical texts, for the improvement of language skills.
Among the motivational aspects that should be taken into consideration during the learning process, in all disciplines, and in particular in math, there is that regularity and harmony, for example in formulas and geometrical figures, are aspect related to the concept of beauty. Statements as "nice theorem", "nice proof" or "nice theory” are common among mathematicians: for a theorem, "beautiful" means short and clear, and in the case of a demonstration "beautiful" means not too short, because it refers to a well-expressed result. Harmonic and mathematical proceedings are strictly connected in art and music, as is evident for example in the golden ratio and musical harmonic scale, thus making students aware of how the characteristic features of "emotional" mathematical elements find themselves amplified and systematized in music and, more generally, in arts, should be implemented in school learning process. In the words of Leibniz: "Music is a hidden arithmetic exercise of the mind, which does not know that it is counting". Since learning styles may differ among students, real life examples stimulating different sensorial activities are considered extremely useful to tackle students’ difficulties. Different anecdotes are related to the mathematical elements of the music theory, since Pythagoras and his disciples noticed that by vibrating two strings subjected to the same tension but of different length (1/2, 2/3 and 3/4 of the first one respectively), they obtained sounds that were particularly pleasant to the ear (consonants, in fact). It is the physiological structure of our hearing that makes us perceive the frequencies of sounds in a multiplicative rather than additive way: in short, with the ear we "count" in geometric progression, while with the fingers, adding units to units, we count according to an arithmetic progression. The scale is constructed from the fundamental frequency of a string taken as a unit and multiplied or divided by 3/2. Proceeding in this way, by ascending or descending fifths, multiplying by 3/2 or 2/3, we obtain the ratios of what is called the Pythagorean scale (although it actually dates back to Eratosthenes, in the 3rd century BC). For example, the note emitted by a string stretched by a quadruple weight has a double frequency: it will be said that it is one octave from the previous one and will be perceived as "equal", but more acute. The same observation can be repeated in terms of length: by shortening a string and in particular by pressing it to half its length and then pinching one of its halves, you will obtain a note at a higher octave. In today's piano keyboard, between two adjacent keys, black or white, there is an interval called "tempered semitone". The strings of any chosen semitone are in the same ratio. The scale produced according to the equable temperament is therefore obtained by dividing the octave into twelve equal parts. Since the octave is represented by the ratio 2:1, with a chain of simple proportions we obtain the value of the smallest interval, called temperate semitone, equal to image near the value of the diatonic semitone E-F=256/243=1.053 and the value of the temperate tone image near the value 9/8=1.125 of the tone of the diatonic scale. As we know today, the fundamental frequency (note) of the sound emitted by a tight string in vibration is directly proportional to the square root of the tension to which the string is subjected; it is inversely proportional to its length, the square root of its density and its section. This solution somehow saved the consonance of the intervals of the Pythagorean system and made the steps of the scale uniform, allowing composers and instrumentalists much more freedom and with more ability to play and compose, but it had to make use of the irrational, a concept rejected by Pythagoras because it denied the possibility of expressing any relationship by means of natural numbers. In the course of the 18th century, mathematicians better understood the nature of sound and were able to describe its propagation analytically. The French mathematician J.-B. J. Fourier came to the conclusion that – after Daniel Bernoulli (1700-1782), who believed he had described, by means of a trigonometric series, a particular type of sound – every periodic function can be expressed through a trigonometric series. Mathematical elements are present in the music of Arnold Schönberg (1874-1951), and his disciples, following atonality and dodecaphony, a compositional method that uses the twelve sounds of the chromatic scale free from reciprocal and hierarchical harmonic relations and reorganized, even with the use of combinatory techniques, according to the principle of the series. During the 19th century once undermined the principle of consonance/dissonance of musical chords, we have several examples of mathematical principles introduced into music with stochastic music based on Markov's chain theory, in 1955 Iannis Xenakis introduced probability into music: musical composition is processed by formal processes defined in probabilistic terms.
Imagination is, as well, an absolutely necessary element in mathematical thinking. It requires to be educated by a correct and fine interpretation of the language and rules though which mathematical objects are structured. It is possible to help a student in the right imaginative construction of an abstract mathematical question by choosing, in the history of mathematics, the paradigm closest to his/her cultural models and the most suitable to educate to think with a structure (build a model) that will remain consistent and functional even in further continuation of the study, when different branches, such as the geometric and algebraic ones, will mix together, for example in Zeno's paradoxes of Plato’s Socratic dialogues: In order to reach the turtle, Achilles should be able to travel a sum of infinite segments (and this is true); but the sum of infinite segments is not necessarily an infinite segment if they have zero length. Here it lies the paradox, indeed the false paradox: in order to execute sums of almost infinities of a distance of almost zero, the answer is not infinite. Zero and Infinity are two numbers like all the others, however, unlike the common ones, have some exceptional requirements: Zero, for example, multiplied by any number, always gives zero as a result, and Infinity, also multiplied by any number, can only give rise to infinity. What happens then when zero and infinity multiply? The result that emerges remains undefined. To understand that such sums can be finite, one had to wait until the 18th century. Only then it was started to put the basis for a rebellion against the diktat of the Peripatetic, which finally led Georg Cantor (1845-1918) to the creation of a satisfactory and coherent theory of mathematical infinity.
It is important to clearly express the meaning of the words through which a mathematical concept will be presented and the corresponding images that will be chosen to illustrate it: the construction of an abstract concept cannot be separated from examples. In mathematics an image can never be representative of the concept to which it refers, but simply serves to evoke it, nevertheless a wrong or misused image could lend itself more easily to misunderstandings than a poorly written text.
The word "imagination" is linked to the image, and provides the ability to create mental images, which are something certainly different from the figure seen on a book, or on a PC screen and is actually more abstract, for example, when it comes to those images that are the two-dimensional representation of a three-dimensional structure. The sense of sight is often connected with understanding, "I see" is, in many languages, under may circumstances synonymous of "I understand it": it does not necessarily refer to the sense of sight or to a real image, sometimes it can be a mental image or an abstract concept. Nevertheless, imagination could deal with all the five senses: this statement has to be taken into consideration when structuring different teaching strategies to reach the different students’ predispositions and abilities.