Key Questions for Focusing on Mathematical Communication

Mónica Miyagui
Ministry of Education - PerúWhy do we focus on mathematical communication?
Currently, the publications of massive character - as newspapers, magazines or advertisingleaflets - include in the presentation of the information: geometric graphs, statistics graphics,
numerical tables and other schemes of mathematical type that allow people to accede to theinformation in a compact, synthetic and precise way. Due to this increase in the use of symbols andmathematical concepts in the processes of social communication, the basic education inmathematics has been forced to incorporate the codification and decodificación of these elements,
the interpretation of the same ones and the aptitude to argue using foundations of mathematicalbase. This is one of the reasons as which, in the curriculum for the Basic Regular Education in Peru,
we have identified three fundamental capacities, one of which is the Mathematical Communication,
another one is the capacity of Reasoning and demonstration(proof) and the Resolution of Problems.
We know that, in the frame of the process of mathematical thinking, these are not capacities
developed independently, but rather complementary.
We depart from the premise that the knowledge must allow to take decisions about aproblem that it is necessary to resolve, also it must allow to communicate the chosen procedures,
defend and validate what has been done, to confront it and to compare it to what others did. It is aquestion then, of generating a space in which the child has the possibility of learning to argue infavour of his products, to thinking if he agrees or not with the results of his classmates, to taking theinteresting ideas of others and to checking his own ideas.
The communication, that is not exclusive of mathematics, forces to returning on the own
thinking to specify it, to justify it, to clarify it. The communication in mathematics includes the own
expression, in a variety of ways, of a problem with a mathematical content, so much in oral as
written form.
To communicate a resolution allows to make explicit what was implicit and there makes
possible the recognition of this knowledge by the individual. To report on the produced necessarily
implies the reconstruction of the realized action (Ressia, 2003).
Defined in our curriculum, the mathematical communication is one of the capacities of thearea that acquires a special meaning in the mathematical education because it allows to express, toshare and to clarify the ideas, which get to be an object of reflection, development, discussion,
analysis and readjustment. The process of communication helps to, also, give meaning andpermanency to the ideas, to express them, as also to listen to the explanations of the others, giving
opportunities to develop the comprehension, that is to say, to establish the mathematicalconnections between the ideas. We also consider that this capacity contributes to the developmentof the language itself, which allows to express the mathematical ideas accurately.
Due to the fact that the mathematics is expressed through symbols, the oral and written
communication of the mathematical ideas is an important part of the mathematical education.
As the degrees of education go on, the communication increases its levels of complexity.
It is necessary to consider the autonomy of the mathematical language in relation with thedaily language. For example the "equal" term in mathematical language means that two different
expressions designate the same mathematical object; this way in the equality " 3+4 = 9-2 ", both

- Mónica Miyagui 1 -1

"3+4" and "9-2" represent the number "7", and for that we say that " 3+4 equal 9-2 "; whereas in theCastilian language we use daily, "equal" means "similarly", "familiar".
To understand and to use the mathematical ideas is fundamental the way in which they are
represented. Many of the representations that currently seem natural to us, such as the numbers
expressed in the decimal system or in the binary one, the fractions, the algebraic expressions andthe equations, the graphs and the spreadsheets, are the result of a cultural process developed
throughout many years.
The term representation refers both to the process as to the product (result), this is, to theact of catching a mathematical concept or a relation in a certain way and to the way itself, forexample, the student who writes his age using his own symbols, uses a representation. On the other
hand, the term is applied to the processes and to the externally observable products and also, tothose that take place "internally", in the mind of those who are doing mathematics. Nevertheless, it
is important to consider that the students who speak an original language and they do not have the
Spanish as a mother language, need additional help to understand and to communicate theirmathematical ideas.
The forms of representation, as the graphs and the symbolic expressions, must not be
considered as goals of the learning itself, because of being ways of mathematical communicationand not either capacities or contents.
27 By fault, they must be treated as essential elements to sustain the comprehension of the concepts
and mathematical relations, to communicate approaches, arguments and knowledge, to recognize
connections between mathematical concepts and to apply the mathematics to real problems.
The reading of the mathematical language helps the students to develop their skills toformulate convincing arguments and to represent mathematical ideas in a verbal, graphical or
symbolic way. It also refers to the aptitude to obtain and to cross information from different sources(texts, maps, graphs, etc.) to be able to:
PENSAMITOCRBLEMAS To organize and to consolidate their mathematical thinking to communicate.
 To express mathematical ideas in a coherent and clear way to their classmates, teachers andothers.
 To extend their mathematical knowledge together with the thinking and strategies to otherareas.
To use the mathematical language as an economic and precise way of expression.

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2. What are your components of mathematical communication to develop?
The Mathematics has its symbolic and formal language, possesses linguistic forms
that express operations or transformations and refers to certain reasoning that must be
motivated by specific concepts.
In its condition of matter of study, the Mathematics establishes itself since the firstyears of education, with a series of codes that invade all the spaces of the language; thechild acceds to the meeting of laws and procedures that indicate him mathematicalbehaviors very defined to find solutions that get to be simple goals of the commonness
and that includes numbering, counting, ordering, classifying and even inferring, and there
is where the verbal communication represents the most effective way to explain themathematical ideas orientated to the comprehension of the concepts. In this respect, it isimportant to accentuate that because of the complexity of the Formal Language,
constituted by the incorporation of strange symbols more than words, is what makes thechildren do efforts to understand the Mathematics, because they do not manage toestablish relations between the daily language and the formal one. On the other hand, thewidespread utilization of the formal language in the classroom by the teachers has serious
consequences, since instead of molding the mathematical uses attending to its informal
language, emphasizes this special language of the Mathematics in a confusing way (Pimm,1999). Also, there are the multiple meanings, typical of many mathematical terms, this isdue to the fact that in occasions words of daily use are taken to interpret any symbol, butthey do not always they adjust to them accurately.
In this order of ideas, Fennell (mentioned by Ruiz, 2003) indicates that “in themathematical communication the standardized symbols and the definitions of theterminology are necessary, but the education of the mathematics in very formalized
language, sometimes, causes a kind of blockade in the comprehension ". This situationmust be handled carefully by the teacher who thinks that the pupil is understanding themathematical concepts, nevertheless the results obtained in the evaluations applied byhim demonstrate the weaknesses in the acquisition and comprehension of Mathematicalknowledge. On the other hand, Gonzalez (1998), expresses the negative consequences of
the traditional approach of the education of the mathematics where there is no emphasisin the transmission and comprehension of the formalized language, where thecommunication processes are unilateral and the transmission of the information that is
acquired by the pupil mechanically without understanding prevails.
With this preamble, I might identify that, among the componentes of theMathematical Communication we would have to consider: the acquisition of the oral
language, which develops across cognitive processes, but depending on the social andcultural environment; the representation and symbolic interpretation.
It is so the quantity of experiences that the child realizes - consciously of his
sensory perception - with himself, in relation with the others and with the objects of thesurrounding world, transfer to his mind some facts on which he elaborates a series ofideas that are useful to relate himself with the outside world.

- Mónica Miyagui 3 -3The interpretation of the mathematical knowledge is obtained across experiences inwhich the intellectual act is constructed through a dynamics of relations, about thequantity and the position of the objects in space and time. The logic - mathematicalthinking must be understood since the development of the cognitive (internal) processes
that imply:•••

Aptitude to generate ideas which expression and interpretation on what it isconcluded is: truth for everyone or false for everyone.
Utilization of the representation or set of representations with which the mathematicallanguage refers to these ideas.
To understand the environment that surrounds us, with major depth, through theapplication of the learned concepts.

A). Cognitive development
Attending to the different statements presented by Piaget, the stage of the concrete
operations coincides with the principle of the education and is decisive in the intellectual
development of the child, that means; in this stage are generated the more sophisticatedconducts in relation to the quantity and to the reasoning, especially in the area of themathematics.
Because of that, it is necessary to have supremely care when starting to teacharithmetical, algebraic and geometric knowledge before the appearance of the operative
thinking, otherwise the children would have a very limited comprehension to generalize
and to reason correctly.
To order, to classify, to bring together or to separate is fundamental so then raise
the thinking. With that said, a long exercise of action is necessary to construct thesubstructures of the later thinking.
For Vigotsky (1978), the cognitive development does not take place in an isolated
form, but side by side with the development of the language, with the social development
and even with the physical development. Thus, the comprehension and the acquisition of
the language and the concepts by the child are made through the meeting with thephysical world and by the interaction between the persons that are around him. The
acquisition of the culture with sense and meaning supposes a form of socialization where
the teachers, the family and friends, with their mediating of the learning function, facilitate
the acquisition of the social culture and uses, both linguistic and cognitive.

B.) Cognitive development in the Mathematical Language- Mónica Miyagui 4 -4Piaget (1967) supports: " The language can constitute a necessary condition for thedevelopment of the logic - mathematics operations without being a sufficient condition of
education. " (P. 59).
In this order of ideas, we can conclude that the derivation of principles of
constructive learning, of concrete representations, of comprehension of the language, ofsocial response and of interaction between teacher and pupil is what would contribute to
the construction of the mathematical thinking. This way, the mathematical language isconsolidated and acquires great strenght in the measure that is revealed as an effective
representation of certain deep structures; because of that, one way or another, themathematics appear in each of the manifestations of the culture.
To explain the acquisition of the oral language, the social lingüists consider not
only the acquisition of structural rules of the language but also the characteristics of thesocial group, because both are relevant factors for the process of acquisition anddevelopment of the language, besides it is also affected by the individual and groupdifferences (Halliday, 1986).
C.) Social and cultural component of the Mathematical Language
The language and the numerical system are communicated from the early infancy: In what
concerns the mathematical field several researches have been done, such as the Saxe (1991) who
conceived the number as a system constructed culturally, in the same way as Vygotsky understoodthe construction of any system of signs. He extended his research to relate the social class and thechildren's numerical competition in pre-school age, concluding that the social class was adeterminer of the numerical environment.

His work considers that some linguistic phenomena that can be found in thecontext of the mathematic lessons, are about 3 general points of meaning; symbols,
symbolized things and syntax. As for the meaning; the pupil uses an important plot ofmathematical knowledge to provide consistency to the meaning; as for the symbols andsymbolized things; he shows that confusions can arise when the pupil focus on thesymbols themselves, instead of what they mean, and as for the syntax in mathematics; it ispossible to formulate some transformations in an analogous way, in which case thealgebra can be considered as a manipulation of symbols according to certain rules. For
such a reason in algebra many mistakes take place, because it could be focused in an
abstract way and manipulating symbols without paying attention to the possible
meanings.
D). Capacities of the Symbolic language
The school mathematics materializes in a language constructed thanks to theincreasing human capacities for the symbolization, not verbally in the first stage andverbally later, wich can be observed in the first infancy. The language of the schoolmathematics therefore, uses a kind of signs constituted by verbal suitable expressions, forexample: four, equal, plus, minus, over, under, etc. And notational signs such as: 4, =, +, -,>, <, etc. The school mathematics is learned, according to our understanding, following the- Mónica Miyagui 5 -5model of symbolically through construction, of wich the major or minor control of thesemiotic instruments of mediation, for example: the linguistic and notational signs, will
favor or not the advance of this learning. Nevertheless, before the formal education of themathematics and even in absence of this education, the children through all the cultures
have experiences of informal mathematics that are easily articulated by their natural
aptitudes to observe and understand the phenomena of the control of the mathematics, forexample, the estimation of quantities and the not verbal calculation. To this respect, weunderstand that the mathematics constitutes a natural control of the human thinking,inherited from million years of evolution of our brain in a cultural context in which thenumbers (as the words) are an essential parameter.
E.) The mental representation of the quantity
Recent Experiences allow us to suppose that the two-year and a half-old children
possess some aptitudes for the counting. It is therefore possible to think that the aptitude
to count or solve simple problems of arithmetic is as natural as the language.
Actually, until the 50’s, specialists of the cognitive development of the childthought that the numerical capacities were appearing late in the child, nevertheless, nonverbal tests have demonstrated that the one-year-old babies can identify small quantities,
add them and reduce them (Gelman. 1983). We will begin wondering if the sense of thequantity is specifically human. There has been demonstrated in research with animals thatthe monkeys, the dolphins and the birds have an elementary sense of the quantity, similar
to the babies one. The aptitude for the numerical perception in the animals and the humanbeings is observed in simple tasks of discrimination (1) as the comparison of two
quantities, on having changed the height (size effect) and on having changed the distancethat separates them (distance effect). For example, the human beings of 6 months-old andthe animals can differentiate 8 from 16 or 16 from 32 but they cannot differentiate 9 from
10. These numerical capacities shared between human beings and animals support the
hypothesis that they are the result of a long evolutionary history. The researches insurmise, on the other hand, the evolutionary advantage that contributes to the perceptionof the quantities when it is a question of estimating a quantity of food or the size of agroup of compatriots.
F. The capacity for the non verbal calculation
On the other hand, the neuropsychology based on the research of the functional
nets of the brain cortex allows to advance the hypothesis that there would be at least two
cerebral systems involved in the mental calculation (Dehaene, S., Molko, N. and Wilson, A.
2004): a non verbal system, based on the sense of the numbers and the manipulation of thequantities, and another one, verbal, based on the memorization of calculations
independent from the perception of the numbers, for example, simple addings: 2+2, 20+20
and tables of multiplication. The first one of these systems (intraparietal) is activated inboth cerebral hemispheres (according to observations in the cerebral imagery) in all thetasks that require manipulation of quantities, for example, in the visual presentation ofquantities and the numerical estimation of groups of objects. In effect, the calculation does
not always need the verbal memory; many operations as the adding, the subtraction or thecomparison need to manipulate only quantities without resorting to the memorization oftables. Nevertheless, though we share the visual estimation of quantities with other- Mónica Miyagui 6 -6animals, the learning of the language and the writing allow our human intraparietal
region to be activated equally by effect of the symbolic notation of quantities or numbers( For example the Arabic numbers, the marks) in sticks or ropes for the count). Thischanges radically the process of symbolization and mathematical abstraction that takes asa base the human capacities constructed in the course of the evolution.
So that, the language of the mathematics has a non verbal original stage that wemust not scorn because in normal conditions the children from a very early age canestimate quantities, manipulate groups, compare, add up, etc. This is confirmed by the
researches that demonstrate that up to the age of five, the simple problems of adding andsubtraction are solved better when they appear in a non verbal form. For example, anexperience carried out by Starkey, P. (1983) allowed to study the adding and thesubtraction with 4-year-old children. The child had to place from 2 to 4 identical objects ina box, one after another one. Later, the experimenter in front of the child extracts an object,adds other one or does not do anything. Later, he asks the children to extract the objects ofthe box one at a time. It is a question of knowing how many times the child puts the handin the box. He observed that 80 % of the children introduces the hand the times
corresponding to the number of objects that remain in the box. Nevertheless, some of thesame children failed when the operation was explained to them verbally in the shape ofterms of reference: in the shape of history or numbers of objects to add up. Only at the ageof 5 years and more, independently of the type of presentation (visual or verbal) of theoperation of adding, the results are positive.
G) The use of symbolic materials
With the entry in the school education the mathematical symbolism reaches thefirst stage of the learning of the mathematics. Before this happens, the apprentice has usedenvironmental materials to count (stones, small sticks), has learned to use his fingers as
symbolic material. Nevertheless, in the school, to construct informal experiences ofcalculation and to favor his learning, we resort to the direct education guided by the adult,with objects or symbolic resources as the representations (abstractions) of the quantities orof the numbers. These representations can be analogical, that is to say, those which aresimilar to the represented (constellations of points, dices, blocks) and are used as
representations of quantities or representations of the numbers themselves. The otherrepresentations are conventional, that is to say, they are not similar to the represented
quantity, they are the notations written as the notational symbolism of figures and related
signs. In the school, therefore, the formal education is favoured to get to operate with, tounderstand, to handle other abstractions: the numbers. And the most important thing isthat the action itself is mediatized by the symbolic used instruments: the verbal language,
the symbolic material (environmental or constructed), the graphical representations ofquantities. To operate with symbolic instruments, to graphically represent quantities is anecessary experience of mathematical abstraction in the way towards the exclusively
notational.
H) The notational symbolism
The use of the language and symbolic materials are two instruments, two ways ofrepresentation of the number that lead to another one: the notations and related signs. In
effect, the figures refer to other previous symbolizations and acquire meant as a result ofprevious learnings: verbal knowledge of figures, count and manipulation of materials,- Mónica Miyagui 7 -7graphical representations. Now then, the learning of the notational expression of thenumber is a new way of expressing the already possessed knowledge on quantities. In
effect, the evolution of this learning is ending in the possibility of formalizing, ofexpressing with arithmetical writing, increases and decreases of quantities: thedecomposition and numerical alteration and other operations of increasing complexity.
A constant process of symbolization is necessary for the full construction of themathematical language, process that begins, as we have seen, with the perceptive
estimation of quantities and it continues with a meaningful learning of the first numerals ),of the notations (figures) and of the related signs. In effect, the school learning of themathematics is done properly when the children are capable of assimilating themathematical language, that is to say, when is possible for them the assimilation ofsymbols: the use of symbols and of symbolic structures that is done most of the time withthe words (for example, the ones that refer to quantities: two, three, etc., or the relations
between quantities, for example, over, under, equal, etc.) because the words are theexpressive support of the mathematical meanings that are constructed.

- Mónica Miyagui 8 -8What kinds of approach will you prefer to develop the communication in classroom?
The use of oral and written communication as a tool that allows the pupils to reflect
their comprehension of the mathematics, will help them to personalize and realize
connections between mathematical concepts. When the pupils communicate mathematicalinformation, they remember it, understand it and use it to discover and to find more
information (Perkins 1992).
The teachers need to know how to help the pupils to turn into competent mathematicalcommunicators who can describe their process of thinking in a clear way. The teachers
must help them to make their thinking visible to others, stimulating them to speak andwrite about the process that they follow when they solve a problem.

It could be an activity in theclassroom in which each of thechildren possesses a complete
game of Tangram:.

A child, whom we will call A, after
constructing a mathematical figure
with all or some of the pieces of thetangram (for example, a triangle or aparallelogram),
explains
tohiscompanions how he has done thatwithout showing them the constructedfigure at any moment.

The others will have to reproduce the figure that there has been formed by the child A
through the indications.
The children will be able to do the explanatory questions in case of being necessary.
Once achieved the goal, another child will be able to expose his work.
The quantity of questions that the children ask will be registered. Finally, we will be able
to identify the degree of mathematical communication of the children, analizing if they
have received more or less questions to solve the figure.
1. Approach of a situation.
•
The children listen to the explanation of the activity.

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9

•
•
•

They listen to the indications of the participant who describes the construction of the
figure.
The children, mentally or in writting, re-formulate the indications with their own
words and propositions.
They form their ensambles of figures.

1. Formulation of inducting questions for the analysis of the situation.
−

−
−

−

He/Sheidentifies
thecompositions
fromtheindications.
They do explanatory questions:
A: " two triangles (a big one, amedium one) are put together by
their bases, remaining the biggest
below and the smallest above ".
B Asks: how can we know which
are the bases that are joined?

− He relates graphs, drawings and verbalizes the relations found.
3. Comparison and relation of the registered information.
− He relates written records.
− Construction and use of graphical schemes to read and interpret the symbolic
propositions.
4. Construction and use of oral and written records to elaborate propositions and tocommunicate results, decisions, opinions. They show the formed figure and explain theprocess that they followed to achieve it.

- Mónica Miyagui 10 -10Approach of a situation.
 Recognition of the meaning of words
and symbols.
 Ordained formulation of the steps to
obtain information

Formulation of inducting questions forthe analysis of the situation.
Use of oral records to transmit
their ideas.
Uses written records to transmit
their ideas.
Graphical or ideographical torepresent their ideas.
Use of graphical and symbolic
records to transmit their ideas.

Comparison and relation of the registeredinformation.
•
He relates written records.
•
Construction and use of graphical schemes
to highlight the existing relations (tables,graphs, conceptual maps).
•
Construction and use of symbolic
propositions to establish relations.

Construction and use of oral and written
records to elaborate propositions and tocommunicate results, decisions, opinions.

- Mónica Miyagui 11 -To verbalize, to annotate
known
terms,
previous
knowledge,
relations,characteristics.

To explain with propositionsor their own words, to show
the object, to verbalize, toexplain the situation and goal.
To represent graphically

Selection of the relevant
information: He identifies
propositions, variables,
information.
Relates words and symbols.

He relates graphs, drawings. He
verbalizes the relations found.

He completes pictures ofsimilarities
anddifferences,using
symbols and graphs.
He formulates symbolic
propositions: equations,operations.
He reads and interprets
the symbolic propositions.
He elaborates oral
and written records:
tables,numbers,symbols.
Graph, uses numerical orgraphical
schemestocommunicate ideas.
They explain the process stepby step to sustain an answer(they argue their answer)11