Math and math education : A vision of its evolution

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Math and math education A vision of its
evolution

• Ricardo Cantoral and Rosa María Farfán
• TA-C, ICME 10
• Cinvestav IPN - Mexico

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Aims and focus, TA-C

• How do new developments in Mathematics influence
  the teaching of mathematics?
• How are teachers in mathematics trained in
  Mathematics?
• How can mathematicians and educators collaborate
  to construct better curricula and improve
  teaching methods?
• (Final programme of ICME 10, p. 103)

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Looking towards the future

• Testimonies Some examples of attempts.
• Plans Constructing web sites, cooperative
  events, getting users of mathematics to testify
  about their etc.
• New ideas Understand the complex relationship
  between Mathematics and mathematics education and
  construct a vision of its evolution.

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How math influence to teaching math
Mathematics
Teaching of mathematics
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We are using some references

• Cantoral, R. and Farfán, R. (2003). Mathematics
  education a vision of its evolution. Educational
  Studies in Mathematics 53 (3).
• Cantoral, R. et Farfán, R. (2004). Sur la
  sensibilité a le contradiction en mathématiques
  et lorigine de lanalyse complexe. Recherches en
  Didactique des Mathématiques 24 (3).

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We are using some reference

• Cantoral, R. e Ferrari, M. (2004). Uno studio
  socioepistemologico della previsione. La
  matematica e la sua didattica 2.
• de Guzmán, M. El papel del matemático frente a
  los problemas de la educación matemática. Spain
  Complutense de Madrid, 1993.
• Holton, D (ed.). The teaching and learning of
  mathematics at university level. The Netherlands
  Kluwer Academic Publishers, 2001.

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the fourth International Congress of
Mathematicians Rome, 1908

• The Congress, recognizing the importance of a
  comparative study on the methods and plans of
  teaching mathematics at secondary schools,
  charges Professors F. Klein, G. Greenhill, and
  Henri Fehr to constitute and International
  Commission to study these questions and to
  present a report to the next Congress (Lehto,
  1998, p. 13)

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A Call for ChangeMathematical Association of
America

• On the mathematical preparation of teachers
• The mathematical experiences recommended for
  teachers at the K-4 level require that
  mathematics departments offer courses
  specifically designed for these audience.
  (Leitzel, 1991, p. 11)

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So, we prefer talk about

• The dialogue of mathematics education research
  with other scientific communities in particular
  the mathematics research communityas Sierpinska
  and Kilpatrick said in 1998 it was one of the
  issues raised at the outset of recent ICMI Study
  on research in mathematics education. (Hodgson,
  B. 2001, p. 516).

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are connected
Research in Mathematics Education
Mathematics instruction
Research in Mathematics
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Aims and focus, TA-C

• How do new developments in Mathematics Education
  and Mathematics influence the teaching of
  mathematics?
• How are teachers in mathematics trained in
  Mathematics Education and Mathematics?
• How can one create community between
  mathematicians, mathematics educators and
  mathematics teachers, to construct curricula,
  textbooks and improve mathematics teaching?

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From teaching to research

• Collatz conjecture
• ? n ? IN , ? k ? IN ? ? k (n) 1
• Infinitesimal models for teaching calculus
• Let be i a formal expression, i is an positive
  infinitesimal if ? x ? IR it follows 0 lt i lt x

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Our problematic

• we will deal with shall be those relating to the
  evolution of the study of educational phenomena
  that take place when mathematical knowledge,
  socially produced outside of school environments,
  is introduced into the educational system,
  forcing it to undergo a series of modifications
  that directly affects both its structure and
  functionality.

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• This process of incorporating highly specialized
  knowledge into the educational system creates a
  series of non-trivial theoretical and practical
  problems, which require methodological approaches
  and suitable theorists will allow us to
  understand the mechanisms for the adaptation of
  mathematical and scientific knowledge into
  practice both for teachers as well as students.

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• We shall present a serie of examples that
  demonstrate the evolution at different times,
  which we have called
• didactics without students
• didactics without school
• didactics without sociocultural settings and
• didactics without
• didactics as pedagogical approaches

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Didactics without students

• The classic problematic in mathematics education
  was the designing of presentations with
  mathematical content for schools considered more
  accessible to students and teachers than those
  so-called traditional presentations. It was
  assumed that a presentation better adapted to
  schools and their employees could only be created
  by means of reflection by mathematics
  professionals.

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Didactics without students

• textbooks and educational materials were
  produced, which systematically failed to take
  into consideration other factors such as those of
  a cognitive or emotional nature or those relating
  to the sociocultural issues of knowledge.
  Instead, they sought to produce that which the
  school ought to use, without carrying out an
  in-depth study of school culture.

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Traditional methods
? r 2
b ? h
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Didactics without students

• Example Students were offered various learning
  activities in order to estimate the value of a
  given area, such as the area covered by the
  following

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Didactics without students

• The introduction of a cover made up of elements,
  of which the area is known, was proposed. For
  example, a rectangle with sides of 3 by 6 cm.
• Then if the area sought is denoted as A cm2, it
  thus fulfills the relationship 0 ? A ? 18.

A
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Didactics without students
A
4 ? A ? 18
A more refined approximation
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Didactics without students
A
4 ? A ? 18
A more refined approximation
a1 ? A ? b 1 a 1 ? a 2 ? A ? b 2 ? b 1 a 1 ? a 2
? a 3 ? A ? b 3 ? b 2 ? b 1 a 1 ? a 2 ? a 3 ? a 4
? A ? b4 ? b 3 ? b 2 ? b 1
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Didactics without students
During this procedure, the student is not in
charge of the learning process, but only of its
execution. The new question was how the people
learn mathematics?
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Didactics without students

• it can be seen that, mathematically, the limit
  of the sequences an and bn is, in both cases, A,
  so the approximation process would lead, by a
  kind of educational sensualism, to students being
  convinced that such limit exists and that their
  conceptions of the area and what its
  representation through approximations

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Didactics without school

• In the 1980s, an action program was presented at
  the ICME 4 around which our discipline
  gradually developed. It was based on approaches
  such as that of Professor Freudenthal who
  presented questions such
• How do people learn?
• How can we learn to observe learning processes?

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Didactics without school

• this led to a new paradigm of research that
  modified its purpose and method of study. This
  has led to a cognitive approach to investigation
  with the systematic observation and description
  of the achievements of students and various
  learning experiences.

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Didactics without school
One of the classic examples in research on
teaching and learning of Calculus, consisted of
explore the answers for two questions on a single
sheet given to students finishing their high
school diploma or starting university, which
would lead to contradictory mathematical answers
without this contradiction being noticed by the
students
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Didactics without school
a) Compare the numbers 1 and 0.999?
Regular answer 0.999... ? 1
b) Calculate the sum of the series
Regular answer 1
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where is ?(x) gt 0?
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where is ??(x) gt 0?
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where is ??(x) gt 0?
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where is ???(x) gt 0?
?
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Didactics without sociocultural settings
In a research project, we reported that sought to
express the concept of convergence of infinite
series, making use of new educational approaches
among university professors, in order to find the
association of the notion of convergence with the
scientific study of conduction of heat.
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Didactics in school, but without the school
sociocultural settings
The phenomenon of heat conduction was an issue
dealt with both by Rational Mechanics and
Mathematical Analysis during the eighteenth
century and for which, at that time, no
definitive answer was found.
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Didactics in school, but without the school
sociocultural settings

• Definition (Marsden, 1974, p. 47)

A serie infinite ? Xk where Xk ? Rn, converges
to x ? Rn
If the sequence of partial sums
Converge to x and write
Convergence of sequences (opcit, p. 44)
Teo. A sequence Xk en Rn converges to x ? Rn if
for every ?gt 0 exists N ? k ? N implies ?? Xk -
x ??lt ?
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Didactics in school, but without the school
sociocultural settings
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Didactics in sociocultural settings
The prediction idea as a fundamental tool for
understanding variation. Newtons binomial
expression is first written as

and not as
This notion of prediction is socially constructed
from the daily experiences of individuals, since
in certain situations we need to know the value
that a magnitude will acquire with time.
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Didactics in sociocultural settings
In our opinion, these findings favor the
discussion and preparation of proposals for
teaching that deal with what should be taught and
not only, as has been customary, with how it
should be taught. In summary, the purpose of our
research is to study that which is
socioepistemological in mathematical knowledge
and includes the primary intuitions of the
student in order to redesign the scholar
mathematical discourse.
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Bibliography
Artigue, M. (1992). Didactic Engineering. RDM.
Selected Papers, 41- 66. Artigue, M. (1999).
Lévolution des problématiques en Didactique de
lAnalyse. RDM 18(1), 31 - 63. Biehler, R., et
al. (Eds.) (1994). Didactics of Mathematics as a
Scientific Discipline. Dortrecht Kluwer Academic
Publishers. Brousseau, G. (1986). Fondements et
méthodes de la didactique des mathématiques. RDM
7(2) 33-112.
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Cantoral, R. (2000). El futuro del cálculo
infinitesimal. ICME 8 Sevilla. Mexico
GEI. Cantoral, R. (1997). An example of the
Sociological Point of View in Math Education The
Case of Analytical Functions at the University
Level. Principal speaker, Conference on Research
in Mathematics Education. MSU, USA. Cantoral, R.
and Farfán, R. (1998). Pensamiento y lenguaje
variacional en la introducción al análisis.
Épsilon 42, 353 369.
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DAmore, B. (1999). Elementi di Didattica della
Matematica. Italy Pitagora Editrice. Douady, R.
(1995). La ingeniería didáctica y la evolución de
su relación con el conocimiento. In P. Gómez
(Ed.). Ingeniería didáctica en educación
matemática, (pp. 61-96). Colombia Editorial
Iberoamérica. Dubinsky, E. and Harel, G. (Eds.)
(1992). The concept of function Aspects of
Epistemology and Pedagogy. The Mathematical
Association of America, Notes Vol. 25.
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Farfán, R. (1995). Ingeniería Didáctica,
Pedagogía 10 (5), 14-23. Farfán, R. (1997). La
investigación en matemática educativa en
Latinoamérica. Alme 1(0), 6-26. Farfán, R.
(1997a). Problemática de la enseñanza de las
matemáticas en América Latina. In D. Calderón and
O. León (Eds.) La didáctica de las disciplinas en
la educación básica 123-146, Bogotá Universidad
Externado de Colombia.