Communication and the development of mathematical thinking
Francisco Cerda Bonomo
CHILE
The need of communication led humans to create languages
It is well established that all human groups communicate and that all cultures have a language.
(Bishop, A. 1999) Communication with others requires a language; this is oral, written or
representational.

Geoglifos1, Representational communication
Creating and developing mathematical language. Processes trigged off by certain kinds ofactivities.
Following Bishop, (1999, pg. 42 and 78.) We can say that it has been found that in all cultures
human groups develop certain activities that lead to a cultural product we know as mathematics.
These activities rise from relationships with the physical environment (counting, measuring,
locate and design) or with the social environment (playing and explain). Likewise, all culturesgenerate and structure language at the service of communication, all classify objects andphenomena, all have explanatory links, they all have ways of connecting ideas through speech
and
all
have
a
fundamental
referenceforvalidating
explanations.
Accordingly, to explain is as universal as language and certainly has a crucial for mathematicaldevelopment.

Geoglifo
Across northern Chile there exist gigantic and stylized manifestations of prehistoric art that adorn thehigh slopes of the hills, known as geoglifos According to the archaeologists these geoglifos represent a mysterious
and enigmatic past, those relating to artistic expressions, signs or sanctuaries.

1- Francisco Cerda Bonomo 1 -Accionesde activities
las personas
HumangroupsExplainexplicar
contarcounting

conceptualizations
conceptualización

locatelocalizar

matemática
MATHEMÁTICS

Problema
intramatemático

Syimbolic Language

diseñardesignLenguaje matemático

arpleayingjugmeasuring
medir

Solving problems

In a characterization of Mathematics, proposed by Godino (2000) we can recognize the elements
listed above and even to complement them:(A) The mathematics are a human endeavour, produced in response to some kind of problem
situations on the real world, social or mathematical itself. In response or solution to these external
and internal problems, mathematical objects (concepts, theories, etc.), raise and evolve gradually.
Actions of individuals should be considered, therefore, as the source of genetic mathematicalconcepts, according to piagetan constructivist theories.
(B) The mathematical problems and their solutions are shared within specific groups or
institutions involved in the study of certain classes problems. Accordingly, the mathematicalobjects are socially shared cultural entities.
(C) The Mathematics creates a symbolic language in which they express the situations problems
and their solutions. The systems of symbols, given by the culture, not only have a communicative
function, but an instrumental role, which change at the same subject that uses them as mediators.
(D) The mathematical activity pursues, among other aims, the construction of a conceptual
system logically organized. Therefore, when adding new knowledge to the existing structure, notonly increases the structure, but all relationships are altered.

- Francisco Cerda Bonomo 2 -The communication in the process of construction of mathematical knowledge in the classroom
One of the central assumptions of the current constructivist said that the sociocognitive conflicts
between members of the same social group may facilitate the acquisition of knowledge as they allow
students aware of other different responses to yours, which force they to consider another point of sightof their initial response. The different answer from others carries information and draws attention on
the subject about mathematical aspects of the task that had not considered until then.
Thus, sociocognitive conflicts provoke a double imbalance: interindividual imbalance due to thedifferent responses of subjects; intraindividual imbalance due to the awareness of different responses,
which invites the subject to doubt his own response. (Chamorro, M.2005)
In a scenario in which the teacher has proposed solving certain types of problems, an importantmoment of the "pooling" of the developed mathematical work in the classroom. At least for that group,
pooling is to make public, the solutions, procedures and the reasoning. In this process the language as ameans of social communication, is crucial. The language will allow students to structure what has been
done, appropriating the new meanings, identifying concepts and procedures, and prepares the way forthe demonstration. The demonstration, from the point of view of mathematical communication, it is asocial act, as it is directed to other individuals whom it is necessary to convince and demands a verbal,
written or representative record.
Other authors as Nunez, M. (2007) speak of the co-construction of knowledge. This concept is
described as a combination of individual and social processes that occur during the process of teaching
and learning science. In particular, the authors refer to the processes of interaction (dialogue) that
occur between the teacher and students or between students. Through the study of the conversation inclasses or in small groups they have attempted to characterize the reasoning of the students in terms ofprocesses or the flow of argument propositions that an individual performs with it which allows theidentification of specific forms of argument.
It is alleged that the conversation between children, which seeks to explain their methods contributes totheir own understanding. Obviously, talk about their own actions leads to children having to think
about it, which is not always attained otherwise, and therefore allows become more aware of them.
Other authors as Nesher (2000) distinguish between talking about mathematics in the classroom andtalk mathematically. the author refers to "Mathematically speaking," to using mathematical ideas freely,
as function, equal, proportions, to manipulate them according the syntax of mathematical language
and to be able to apply them in various contexts. In contrast, to "talk about math" carry out otheractions. We use natural language as a metalanguage to express all kinds of thoughts about math. It isdifferent, because the vocabulary, grammar, syntax and semantics, belong to natural language.
Under the constructivist approach, it is very important that teachers listen to the explanations his
students gave about his mathematical work. This requires to open spaces and to give opportunities totalk about mathematics in the classroom.
Recall that on the Cockroft Report (1985) it was stated that the students successful in mathematics
came from those classrooms in which it was usual to participate in discussions of mathematical type,
between teacher and students, as well as between students. This endorses Nesher (2000) noting thatstudents develop learning math when discussing among themselves about valid arguments in- Francisco Cerda Bonomo 3 -mathematics or explain their strategies. For this purpose they use it in mixed-mode: Natural Language
ideas partially expressed in mathematical terms.
Example of a communication experience focused on mathematical thinking2.
Students know the schematic representation of the binomial square:(a + b) 2 = a2 + 2ab + b2
a
a

b

b

a2

ab

ab

b2

They are asked to look at this other representation based on an equilateral triangle, and comparing itwith the previous, setting differences and similarities between the two. They must justify in writing
their conclusions.

2

Set of activities created and tested by Francisco Cerda B, on 8º y 9º grade.

- Francisco Cerda Bonomo 4 -Once the work in pairs is ended, a pooling of some achieved answers was made, generating aninteresting discussion among students.

On the equilateral triangle we can recognize amultiplication of the binomial square by a factor 3 ,

3 ab
4

4

namely:3 (a+b)2 = 3 a2 + 2• 3 ab + 3 b2
4
4
4
4

3 ab
4
3 b2
4

3 a2
4

Then the students were challenged to work with other regular polygons as rhombus, hexagons toconclude for the circle:

• a

•

b

π(a+b)2 = πa2 + 2πab πb2This activity continued searching for similarities in the different representations with regular polygons,
such as:(a2 - b2) = (a + b )(a - b)

- Francisco Cerda Bonomo 5 -Communication an mathematical communication in the Chilean curricular framework
In the Chilean educational curriculum are described the general purpose of education, i.e. theknowledge, skills, attitudes, values and behaviors that students are expected to develop at a personallevel, intellectual, moral and social development. In this curriculum framework, it is defined as acentral objective, the development of communication skills, which are linked to the ability to present
ideas, opinions, beliefs, feelings and experiences in a coherent and informed manner, making use ofmany and varied ways of expression.
Spanish Language and Communication is a subject proposed to develop the most of the communicative
abilities of students, so that they can function properly and effectively in the various situations they
have to face. This subject highlights oral and written communication:
Oral Communication: It refers to promote student participation in oral communicative interaction
situations in which they have the opportunity to analyze communicative situations interaction oral ofargumentative type (debates, controversial, taped discussions radio or television) in order todistinguish: the structure of speeches (assumptions, arguments, conclusions), procedures (types ofarguments, validity of them, and so on.), and the results and effects.
Written Communication: Reading and production of written argumentative texts produced in public
situations, to perceive the overall structure of the text, the internal organization of their parts andcomponents, with special emphasis on types of arguments used and validity of them, incorporation in
the text of illustrations and statistical tables.

Mathematical communication
In the subject of Mathematics, among other things, it is stated that his learning contributes to thedevelopment of communication skills, which makes more precise and rigorous the expression of ideas
and arguments, incorporating on the language and usual arguments forms of mathematical expression(numerically, graphically, symbolic logic, probability and statistics) and understanding quantitative andqualitative mathematical elements (data, statistics, graphics, drawings, etc.) present at the news, views,
or advertising, and analyzed them independently. It relieves teamwork, communication and theexchange of ideas, the foundation of views and arguments, examining its logical connections and withthe support technological elements. Moreover, education must contribute to a better performance ofpeople in everyday life, through the use of mathematical concepts and skills that will enable them toreinterpret reality and solving everyday problems of family, social and professional circle, contributing
to establish a language for understanding the phenomena of science and technology.
In an explicit way, there are some fundamental objectives which are related to mathematicalcommunication. For instance, students will be able to:• Identify and interpret the information provided by numbers present in the environment and used
numbers to communicate information in both oral and written form.
• Communicate and interpret information relating to the place where objects or persons are placed(positions) and giving and following instructions to get from one place to another (trajectory).

- Francisco Cerda Bonomo 6 -• Manage basics features of problem solving, such as formulating the problem in their own words, takeinitiatives to resolve it and communicating the solution.
• Collect and analyze data in local, regional and national levels situations, and communicate results;
selecting ways to present information and results according to the situation.
• Use systematically arranged and communicable reasoning in problem solving
Communication in the National Assessment System (SIMCE)
Since this year (2007) the national tests SIMCE (levels 4th, 8 th and 10 th), include open-ended
questions, whose focus is to assess aspects that are more difficult to measure with alternatives items
In mathematics, students are requested to explain by words, the reasoning they make to obtain thesolution of a problem.
Sample question (SIMCE 1st cycle)
Loreto dice:“Cuando se suman dos números, la respuesta es siempre un
número impar”¿Es correcto lo que dice Loreto?
SI

NO

Explica tu respuesta, usando uno o más ejemplos.

Loreto says:“When two numbers are added, the answer isalways an odd number”
Is it correct what Loreto says?
YES
NO
Explain your answer, giving one or more
examples.

Below are two answers given by children to that question:

YES-NO X
1- I say that Loreto’s answer is wrong,
because, recently I added 24 + 16 and I
obtained 40, and then it is not ok.

YES-NO X
Because if you add two odd numbers,
you obtain an even one, for instance:
103 + 103 = 206- Francisco Cerda Bonomo 7 -These answers, give a first approach about mathematical reasoning level that those children employed.
Maps of progress
In the framework of contents standards, the maps of learning progress (MPA) is a new curricular
instrument that is in validation and development process. They described the typical sequence inwhich learning is progressing in certain areas or domains considered essential in the training ofstudents in various subjects. The maps described 7 learning levels, ranging from primary school tosecondary school.,
As an example we can noted that the learning map described in Numbers and Operations axis progressconsidering three dimensions that are developed in interrelated way:
A. Understanding and using the numbers. .
B. Understanding and using operations.
C. Mathematical Thinking (this dimension involves skills related to the selection, implementation andevaluation of strategies of problem solving; argumentation and communication strategies and results)
From Numbers and Operations Axis we extracted the progression of communication and argumentationin their 7 levels. (From low to high complexity)
grade
Description of the communication
Level ofachievement
1 ° and 2º
1
Describes and explains the strategy used in problem solving
3
y4
2
Justify the strategy, explaining his reasoning or verifying conjecture throughexamples.
5y6
3
Argues the validity of a process, strategy or conjecture raised.

4

7y8

5

9 y 106

11 y 127

exc.

Justify the strategy, the guesses made and the results obtained using
concepts, procedures and mathematical relationships.
Argues its strategies or procedures and uses examples and counterexamples
to verify the validity or falsity of conjecture.
Make conjecture that assume generalizations or predictions and argues thevalidity of the procedures or conjecture
Use mathematical language to present arguments demonstrating
mathematical situations.

- Francisco Cerda Bonomo 8 -•Enhancingmathematicalcommunication

in

the

lessonstudy

It has become apparent close relationship between mathematical thinking and mathematicalcommunication. We had talked about lesson study focused on the development of mathematicalthinking, now we are going to add a special regard to the mathematical communication of thisprocess. The question is how teachers can help their students to become competent mathcommunicators in order they can describe his thought process in a clear way. Teachers can help
them make their thinking visible encouraging others to speak and write about the process being
carried out to solve a problem.
On the below diagram (adapted from Wang-Iverson and Makoto, 2005), we have the full process oflesson study. The proposal is to enhance the mathematical communication on each and every one ofits stages:

Lesson Study Group

Research Lesson
Participants
Knowledgeable other

Develop a lesson plan
Research lesson
LessonStudycycle.

ReviseOther teachers from theschoolpost-lesson discussion

Thought

Pedagogicalcoordinator

Teachers from otherschools

New ideas for teaching andlearning
At each stage, the approach to the topic will be specific and may require some instruments. The
following questions can guide the work at every stage:- Francisco Cerda Bonomo 9 -Stage I when the lesson study group meets to design the research lessono What is the language and reasoning level of students that they are going to live in the lesson?
o How or when to give space to students for sharing procedures and reasoning they use to solve
the proposals mathematical tasks?
o When we can foster discussions among students and between students and teacher?
o On which specific points, we which to focus mathematical thinking and therefore mathematicalcommunication?
o What we understand by “mathematics competition”?
o Can we draw up a list of mathematical communicative skills that includes sub-competencies?
•

Stage II. When lesson is applied and it is observed by other teacherso How the teacher promotes the dialogue in the classroom?
o How are the discussions and the flow of propositions, when students argue or justify
the techniques they employ?
o How is the teacher's role in promoting math communication?
o How teacher balance the students communication, (oral, written and representational)?

•

Stage III When they examine and evaluate the implementation of the lesson.
o Did thinking and mathematical communication have an outstanding role in thedevelopment of the lesson observed?
o What milestones can be stressed on the mathematical thinking and communication
developed in that lesson?
o What kinds of changes to the lesson plan are suggested to improve the mathematicalthought and communication?

- Francisco Cerda Bonomo 10 -

REFERENCES
Bishop, A. (1999) Enculturación Matemática. Piados. Barcelona
Buschman, L. (1995) El lenguaje matemático en el aula. En Teaching children mathematics, vol. Nº6
NCTM, Page. 324-329.
Cockroft, W.H. (1985). Las matemáticas sí cuentan. Informe Cockroft.: M.E.C. Madrid.
Chamorro, M. (2005) Didáctica de las matemáticas para la Educación Preescolar. Pearson. Madrid
Cerda, F. (1999) Representación de expresiones algebraicas mediante polígonos regulares. En edición.
Stgo.
Godino, L. (2000) Significado y comprensión de los conceptos matemáticos. Revista Uno. Nº5.
Barcelona.
Nesher, P.(2000)) Posibles relaciones entre el lenguaje natural y el lenguaje matemático. En“Matemáticas y educación”. Grao. Barcelona.
Núñez, M. (2007) Validación de metodología co-constructiva basada en la construcción de modelosmentales para la enseñanza de las ciencias. ProyectoFondecyt Santiago.
Sol, M. Y Jiménez, J. (2007) Competencias y proyectos matemáticos realistas en la ESO. En revista
UNO, Nº 46. Grao. Barcelona.
Wang-Iverson, P. and Makoto, Y. (2005) Building our understanding of Lesson Study. RBS.
Philadelphia.

Francisco Cerda Bonomo,e-mail: cerda.fran@gmail.com- Francisco Cerda Bonomo 11 -