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Hoy en día, las Matemáticas se usan en todo el mundo como una herramienta esencial en muchos campos, entre los que se encuentran las [[ciencias naturales]], la [[ingeniería]], la [[medicina]] y las [[ciencias sociales]], e incluso disciplinas que, aparentemente, no están vinculadas con ella, como la [[música]] (por ejemplo, en cuestiones de resonancia armónica). Las [[matemáticas aplicadas]], rama de las matemáticas destinada a la aplicación de los conocimientos matemáticos a otros ámbitos, inspiran y hacen uso de los nuevos descubrimientos matemáticos y, en ocasiones, conducen al desarrollo de nuevas disciplinas. Los matemáticos también participan en las [[matemáticas puras]], sin tener en cuenta la aplicación de esta ciencia, aunque las aplicaciones prácticas de las matemáticas puras suelen ser descubiertas con el paso del tiempo.<ref>Peterson</ref>
== Etimología ==
ICOTS6, 2002: Batanero & Godino
La palabra '''"matemática"''' (del griego μαθηματικά, «lo que se aprende») viene del griego antiguo μάθημα (''máthēma''), que quiere decir «campo de estudio o instrucción». El significado se contrapone a μουσική (''musiké'') «lo que se puede entender sin haber sido instruido», que refiere a poesía, retórica y campos similares, mientras que μαθηματική se refiere a las áreas del conocimiento que sólo pueden entenderse tras haber sido instruido en las mismas (astronomía, aritmética).<ref name="Heath">
1
{{cita libro
TRAINING FUTURE RESEARCHERS IN STATISTICS EDUCATION: REFLECTIONS
| apellidos = Heath
FROM THE SPANISH EXPERIENCE
| nombre = Thomas
Carmen Batanero and Juan D. Godino
| título = A History of Greek Mathematics.
University of Granada
| año = 1921
Spain
| editorial = Oxford, Clarendon Press
A main point to assure the future of statistics education research is the training of researchers
| id = {{OCLC|2014918}}
through the Master's and Doctoral Programmes. Since in the majority of countries there are no
}}</ref> Aunque el término ya era usado por los pitagóricos en el siglo VI a. C., alcanzó su significado más técnico y reducido de "estudio matemático" en los tiempos de [[Aristóteles]] (siglo IV a. C.). Su adjetivo es μαθηματικός (''mathēmatikós''), "relacionado con el aprendizaje", lo cual, de manera similar, vino a significar "matemático". En particular, μαθηματική τέχνη (''mathēmatikḗ tékhnē''; en latín ''ars mathematica''), significa "el arte matemática".
specific departments of Statistics Education, this training is carried out from Mathematics
Education, Statistics, Education, Psychology and other related departments, and even there
La forma plural ''matemáticas'' viene de la forma latina ''[[:la:mathematica|mathematica]]'' ([[Marco Tulio Cicerón|Cicerón]]), basada en el plural en griego τα μαθηματικά (''ta mathēmatiká''), usada por [[Aristóteles]] y que significa, a grandes rasgos, "todas las cosas matemáticas".
starting a line of research in statistics education is not an easy task, due to the lack of trained
supervisors, specific bibliography and funds. In this presentation I will describe the experience of
starting the first Doctoral Programme in Mathematics Education at the University of Granada,
and developing there a research group in statistics education. The contents of the Doctoral
Programme will be analysed as a first step to establish what an ideal programme for training
future researchers in statistics education would be.
TRAINING OF RESEARCHERS IN STATISTICS EDUCATION
One main goal of the IASE is to promote research related to teaching and learning
statistics. As described in Jolliffe (1998), we still need to achieve academic recognition in the
different disciplines or programmes where we work. In Batanero, Garfield, Ottaviani, and Truran
(2000) and following discussion a group of researchers in Statistics Education reflected on their
views about what research is and about the main research questions that statistics education
research should address in the forthcoming years. In that paper we recognised that there is a
considerable amount of experience in the world about conducting research into statistical
education. We also suggested that we may now be at a stage where it would be possible to
develop some general principles about what background knowledge we need in order to conduct
quality research in statistical education and about how might we best train researchers to conduct
research in statistical education.
In the ICMI Study What is research in Mathematics Education and what are its results?
(Sierpisnka & Kilpatrick, 1998) it was recognised that scientific research should be guided by
principles, theories and conceptual frameworks. Furthermore, obtaining a relevant research result
(Nissen & Blomhoj, 1993) requires systematic and disciplined methodology to ensure the
research validity and reliability. We can translate these ideas to statistics education and then two
basic components of a Master’s or doctorate program in statistics education are the theoretical and
methodological courses.
Below we reflect on our experience in organizing a doctorate program in Mathematics
Education at the University of Granada, and starting there a research group in statistics education,
where 9 doctoral dissertation have been carried out in the period 1988-2001 and others are in
progress. We also take into account the analysis of 91 programs in Mathematics Education from
different countries (Batanero, Godino, et al., 1994) and our experience in teaching doctoral or
Master’s courses in several Spanish and South American universities.
THE DOCTORAL PROGRAMME AT THE UNIVERSITY OF GRANADA
The Doctoral Programme in Mathematics Education at the University of Granada was
started in 1988, with only three lecturers in charge of teaching the courses and supervising
research, one of them coming from the Department of Statistics and with no experience in
educational research. Fortunately we counted on the help of a very experienced French research
team in mathematics education: professors Michéle Artigue, Ives Chevallard, Regine Douady,
André Rouchier, coordinated by Guy Brousseau, who travelled to Granada to teach some of the
courses and helped in the orientation of the theses during the first four years of the programme.
Starting the doctoral programme would have been quite impossible without their help and no
doubt the influence of the French school in our research work is still visible.
ICOTS6, 2002: Batanero & Godino
2
There have been different regulations for doctoral programmes in Spain throughout this
period. From 1988 to 2000, students were asked to complete 320 hours of regular course work
and seminars, over two years part-time, with the possibility of spending up to 90 hours in the
second year to produce a preliminary written research monograph (similar to a Masters’ thesis).
In the new regulations, the amount of course work is reduced to 200 hours in the first year and the
second year is dedicated to produce a compulsory research monograph for a total of 120 hours.
When the student finishes these two years and his/her research work is approved, he must carry
out a Ph. D. dissertation (original research, supervised by a doctor expert in Mathematics
Education or a related field), which usually takes 2-4 more years to complete.
Students’ background
This Programme is offered within a Mathematics Education Department and most of the
students who throughout these years have carried out a thesis (24 at the moment) as well as other
students who are currently at different stages in their research are mathematicians. A few students
came from Education or Physics. In the last few years, we received South American students with
different background, although the majority still are related to mathematics. All the 9 students
who finished their doctoral thesis in statistics education took a speciality of statistics during the
University studies or have been teaching statistics for a number of years before entering the
programme. These students then had a solid basis in mathematics and theoretical statistics; the
majority of them were also acquainted with applied statistics, data analysis and statistical
software. As compared to their colleagues doing dissertations in other branches of mathematics
education, they had a better methodological background and were better prepared to analyse their
data and carry out an empirical research.
THEORETICAL CONTENTS IN THE PROGRAMME
Although some students have experience in educational research and in teaching
mathematics or statistics, the majority of them needed to complement their theoretical knowledge
about education and mathematics education. Two courses (60 hours) on Theory of Mathematics
Education include four main components. Although the discussion here is general, in the case of
students intending to produce a thesis in statistics education, the contents are contextualised to
this particular field.
Mathematics (Statistics) Education as a Scientific Discipline
We present a perspective of mathematics (statistics) education as a complex and
heterogeneous social system with three interrelated components:
1. Reflective practice about teaching and learning (teachers and lecturers).
2. Scientific research, which attempts to understand teaching in general, specific
didactic systems or its components (the teacher, the student, and the mathematical
and statistical knowledge).
3. Didactic technology, producing teaching materials to improve instruction.
The world of practice (teachers) is focused on a group of students. Scientific researchers
are engaged in building theoretical concepts. Didactic technology (or applied research) involved
the production of tools for action; this is the field for curriculum designers, text book writers, etc.
None of these three components operates independently from each other, nor is there a
hierarchical distinction between them, and moreover their frontiers are fuzzy. It is necessary to
make future researchers conscious of these distinctions, since most of them are University or
secondary school teachers with strong conceptions about practical problems.
Epistemological Foundations
The complexity of educational problems confronts mathematics and statistics education
with the dilemma of developing its own fundamental research to develop theories on which a
coherent and productive research agenda could be based. An essential question is whether we
should build an explicit conceptualisation about the nature of mathematical or statistical objects
(concepts, procedures, theories, etc.), and its development at an individual and institutional level.
As a part of our research work over the past 12 years and our theoretical reflections we started a
ICOTS6, 2002: Batanero & Godino
3
systematic inquiry into the nature, origin, meaning and understanding of mathematical objects
(Godino & Batanero, 1994, 1998). This conceptualisation is also applicable to statistical objects
and is based in the following assumptions:
Mathematics is a human activity involving the solution of socially shared problematic
situations, internal or external to Mathematics.
Mathematics is a symbolic language in which problems and solutions are expressed.
The systems of symbols, as culturally embodied, have a communicative function and
an instrumental role, which changes the very person who uses the symbols as
mediators.
Mathematics is a logically organised and socially shared conceptual system.
From this theory, we proposes a research agenda for Mathematics and Statistics education
(Batanero & Godino, 2001) and applied these ideas in many of the statistics education theses that
have been carried out in Granada.
Teaching and Learning Theories
Mathematics (Statistics) Education is a scientific discipline supported by Epistemology,
Mathematics, Psychology, Sociology, etc. This makes it difficult to select the teaching and
learning theories to include in a program for the preparation of researchers. Because of their
specificity for mathematics education and our relation with this research group we particularly
take into account the theoretical results of the 'French school' of Didactic of Mathematics. We
also take into account recent tendencies in the philosophy of mathematics and in Mathematics
Education, as well as constructivist theories of learning.
Curricular Theories
We conceive the curriculum as an operative plan that specifies what students need to
know, what teachers should do to make their students develop their knowledge, and what contexts
would be appropriate for teaching and learning. We assume the following pedagogical
assumptions:
The main goal of the teacher's action in the classroom is to help students develop
mathematical (statistical) reasoning, problem solving capacity, communication ability
and establishing relationships between mathematics (statistics) and other disciplines.
Special attention should be paid to the organization of teaching. Careful selection of
tasks should provide students opportunities to explore relevant problems, formulate
conjectures, use various representations, and communicate with classmates.
Students should recognize the level of development of mathematics (statistics) and its
applicability to other disciplines and human activities. The aim is the progressive
appropriation of knowledge, which is, students' construction of a network of concepts
and procedures, as well as the mastery of language.
Although the discussion is general, each student can choose a particular area (algebra,
statistics, probability to analyse). For the particular case of doctoral students trying to produce a
thesis in statistics education these ideas can be discussed using our books (Batanero, Godino, &
Navarro-Pelayo, 1994; Batanero, 2001; Godino, Batanero, & Cañizares, 1987), which contain
specific proposals for teaching Probability, Combinatorics, and Statistics according to the
assumptions described.
STATISTICS EDUCATION CONTENT IN THE PROGRAMME
The amount of specific courses in statistics education have increased from only a 30
hours course in statistics education in the period 1988-1994 to 3 different courses (didactics of
probability; didactics of data analysis; didactics of inference) with a total of 100 hours in the
period 2000-2002. These courses are intended to complement the general course of mathematics
education in providing a basic knowledge of statistics education as a specific research field,
present example of different types of research in this area and help student to identify possible
research problems to produce a dissertation in statistics education. Basic contents for these
courses are:
ICOTS6, 2002: Batanero & Godino
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Current situation of Statistics Education: the IASE and ISI works, other associations,
conferences, journals, discussion lists and other sources of specific bibliographic
information.
Epistemology of Stochastics: basic stochastical ideas, its historical genesis, different
conceptions and philosophical problems around them. Different approaches to
statistical inference; controversies around the use of inference and the problem of
validation for empirical knowledge; data analysis: main current approaches; statistical
modelling, experimenting, association and the search for causes.
Cognitive development: constructivism, social interaction and the role of language.
Piaget’s stages in cognitive development: the case of randomness, probability and
other stochastic ideas. Fischbein’s ideas about intuitions and the role of instruction.
Recent research on social interaction in the classroom and its influence on children’s
development of stochastic ideas.
The stochastic curriculum: reasons, aims and content of teaching stochastics at
different educational levels; didactic resources, technology, computers and the
Internet. Assessment; different assessment instruments. Methodologies for teaching
statistics and probability. Analysis of curricular materials and didactical units.
Research into students understanding and learning: main theoretical frameworks in
statistics education research. Research on students' conceptions before instruction and
on conceptual change induced by specific teaching experiences; research on students’
strategies and errors in problem solving or in data analysis.
Other research: attitudes, social factors, comparative studies, case studies, textbooks,
etc.
RESEARCH METHODOLOGY
During his /her research the student should produce measurement, survey, or observation
instruments, design educational interventions, collect and analyse data. Below we describe the
main methodological contents in the courses, which have ranged between 120 hours in the 1988-
1990 programme to 60 hours in the 2000-2002 programme.
Research paradigms: we discuss the notion of paradigm, and make a critical analysis of
the most relevant paradigms in educational research that can be situated between two extreme
positions. The positivist or process-product paradigm, which especially attempts to find laws and
test hypotheses about behaviours and procedures related to the students' academic achievement. It
is usually based on quantitative methods, systematic measurements, experimental designs, and
mathematical models. The interpretative or qualitative paradigm focused on searching for the
personal meaning of events, in studying the interaction between people and environment, as well
as the participants thoughts, attitudes and perceptions. It is associated with naturalistic
observations, case studies, narrative reports. In addition, the socio-critical paradigm tries to
connect research with practice.
The Research Process: When the student identifies a research area he/she needs to
systematically explore the bibliography and to focus progressively on one particular problem.
From the beginning he/ she has to decide on the particular approach or paradigm and define the
objectives and hypotheses, which will serve to select the variables and information needed. The
complexity of the problem might lead to reduce the number of independent and dependent
variables, and to control concomitant variables to guarantee the validity of conclusions.
Experimental and quasi-experimental techniques to assign subjects to different conditions, or to
develop questionnaires would be needed and sampling techniques should be applied to guarantee
representativeness and to increase generalisability. Time and effort dedicated to these themes are
very productive for doctoral students.
Data Collection: Subjects' mathematical knowledge is a complex system, not directly
observable, but should be inferred from their responses to assessment tasks. These empirical
indicators of the subject' s knowledge are qualitative and multidimensional. Consequently, we
need varied assessment tasks: questionnaires, problems, projects, individual interviews or
classroom observation. This diversity of data forces the researchers to choose different
observation, survey and measurement methods and techniques.
ICOTS6, 2002: Batanero & Godino
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Data Analysis: The study of the relations between the different variables in a research
requires a variety of data analysis techniques. Most researchers would require the collaboration of
expert statisticians to analyse their data. However, we try to provide every researcher with a basic
preparation that will allow him/her a certain degree of autonomy in the initial data analysis of
his/her own research; e.g., computing simple statistics or carrying out basic tests, to communicate
with and consult an expert statistician when they require more complex analyses and to
understand technical reports from the statistician. An important aim for future researchers is being
aware of the possibilities of new data analysis techniques. Although a narrative or interpretative
data analysis would be highly relevant in qualitative studies, this analysis could be complemented
by exploring the structure of interactions between variables, using correspondence analysis,
implicative analysis, cluster analysis, log-linear models, etc., which are applicable to qualitative
variables.
STARTING A RESEARCH GROUP IN STATISTICS EDUCATION
Starting in 1988 a doctoral programme with only three possible supervisors forced us to
concentrate the research topics on three basic research lines: Numerical Thinking, Theory of
Mathematics Education and Statistics Education. Two of the lecturers in charge of the programme
had carried out their dissertations in pure statistics and had extensive experience of statistics
consultancy work in different experimental research area, the field of statistics was familiar to
them in its different facets: teaching, applied and theoretical research. That experience had also
served to take consciousness of the problems in understanding and applying statistics and of the
interest to carry out didactical research in this area.
The fact that, after finishing their dissertations, our colleagues stayed in our department
or other cities close to Granada (Jaen, Melilla) served to increase the “critical mass” of lecturers
specialised in statistics education who could collaborate in supervising new students. From a few
starting problems: association, combinatorics, the test of hipotheses, we gradually moved to other
areas, such as probability or the normal distribution. Nine doctoral dissertation have been finished
in statistics education in this period and three more are currently in progress.
The doctoral programme at the University of Granada received a great support from
Spanish and other European academic authorities and this served to fund the visit from main
mathematics and statistics educators who gave us courses and discussed with us our research in
progress. Statistics education research received particular support through different funded
projects from the Spanish Ministry of Education. Another happy circumstance that served to
consolidate and impulse our work was the IASE decisions to hold the ICOTS-4 conference in
Marrakesh and the 1996 IASE Round Table Conference in Granada that helped to establish new
valuable contacts and collaborations that still continue. The list of theses, projects and
publications carried out in the group is available from our web page
http://www.ugr.es/local/batanero, which is also linked to the IASE and other statistics education
web servers.
FINAL REFLECTIONS
Having finished the training of all the members in our own department (where there are
now 18 lecturers with a Ph.D in mathematics or mathematics education, and also the training of
other colleagues in Andalusian universities (3 in Jaen, 2 in Cordoba, 2 in Malaga) we focus now
on the education of young students and South American colleagues, who, in the last years were
increasingly successful in getting a grant either from their own countries or from the AECI
(Spanish Agency for International Co-operation), OEI (Organisation of Iberoamerican States) or
other institutions. A number of doctoral dissertations by colleagues from Argentina, Colombia,
Chile, Mexico and Venezuela are being carried out in Granada. When returning to their countries
they usually take over responsibility for Master’s or doctoral programmes in education or in
mathematics education. It is then very likely that this collaboration can serve to spread
mathematics and statistics education research in South and Central America.
In this brief report we have reflected on our particular experience in the training of
researchers in statistics education from a mathematics education doctoral programme, which was
started with a lot of effort, but gradually developed to be the main research programme in
ICOTS6, 2002: Batanero & Godino
6
mathematics education in Spain and includes the larger statistics education research group in
Spain. International collaboration was very important at different stages to start up the doctoral
programme, and orientate the first theses. We hope this experience can encourage other
researchers to start new programmes even with modest initial resources and in this way can serve
to extend the interest towards statistics education research.
== Historia ==
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