Transcript Math and math education - Cimate

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
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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
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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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Mathematics
How math
influence to
teaching
math
Teaching of
mathematics
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Mathematics education Mathematics as a
as a scientific domain scientific domain
Mathematics teaching
as a practical domain
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We are using some references:
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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 l’origine de l’analyse
complexe. Recherches en Didactique des
Mathématiques 24 (3).
6
We are using some reference:
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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
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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 Change
Mathematical Association of America
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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 …
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The dialogue of mathematics education
research “with other scientific communities in
particular the mathematics research
community”…as 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
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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 …
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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 < i < x
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Our problematic…
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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
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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 socalled “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
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… 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…
r2
bh
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Didactics without students
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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
A
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Didactics without students
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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.
A
3 cm.
6 cm.
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Then if the area sought is denoted as A cm2, it
thus fulfills the relationship 0  A  18.
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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
a1a2Ab2b1
a1a2a3Ab3b2b1
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
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… 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
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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
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… 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
9
9
9


 ... 
10 100 1000
Regular answer 1
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where is (x) > 0?
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where is (x) > 0?
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where is (x) > 0?
+
+
+
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where is (x) > 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
k
If the sequence of partial sums

Converge to x and write
Sk   X l
l 0
 X k  x"...
k 0
Convergence of sequences (opcit, p. 44)
Teo. A sequence Xk en Rn converges to x  Rn if for
every > 0 exists N  k  N implies  Xk - x < 
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Didactics in school, but without
the school sociocultural settings
SALE AGUA HIRVIENDO
CARA A
(tapa sup.)
ANILLO
CARA C
CARA LATERAL
CARA B
(tapa inf.)
C
CONDUCTO POR EL QUE
CIRCULA AGUA HIRVIENDO
.
.
B
ENTRA AGUA HIRVIENDO
. . . .
A
D
B
C
A
.
D
A
B
A´
B´
A
B
A´
B´
CORTE RADIAL
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Temperatures on AB line, at
time t0
T
t = t0
Temp
A
B
x
Radial Position
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Didactics in sociocultural settings
The prediction idea as a fundamental tool for
understanding variation. Newton’s binomial
expression is first written as :
( P  PQ )
m
n
and not as
(a  b)
n
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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42
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