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Mathematics Education
Org: C. Kieran (UQAM) and M. Santillana (UPN)
[PDF]
- TENOCH CEDILLO, Universidad Pedagógica Nacional
The potential of CAS in promoting changes in teachers'
mathematical knowledge, practices and conceptions
[PDF] -
This presentation will discuss a study aimed at providing plausible
answers to the following questions:
- In which ways do middle school mathematics teachers take
advantage of the algebraic transformation facilities offered by a
CAS?
- In which ways does the use of CAS influence teachers'
mathematical knowledge?
- In which ways do mathematics teachers change their practices
as a result of using CAS in their teaching?
From the beginning of the study the teachers seemed to be convinced of
the potential of CAS facilities to stimulate their students to
explore, put forward and test mathematical conjectures. At this point
it seemed interesting for the aims of this study to investigate why
they also seemed to be reluctant to use those CAS facilities to carry
out algebraic transformations. During the interviews it was found
that teachers' reluctance was due to the value they gave to students'
learning of algebraic transformation rules by paper-and-pencil
techniques. However, they acknowledged that most of their students
learn the algebraic algorithms meaninglessly.
I will discuss how teachers changed their view during the study and
found how to take advantage of CAS in order to help their students
learn in a more meaningful way such notions as how to simplify similar
terms within an algebraic expression, the laws of exponents, and some
non-conventional strategies to factor polynomial expressions "without
making their students become key pushers".
- ALEJANDRO DIAZ-BARRIGA, Universidad Nacional Autónoma de México, Instituto de
Matemáticas, Circuito Exterior de Ciudad Universitaria,
México, DF, CP 04510
A mathematician's perspective on teacher training in Mexico
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Teacher training is done in Mexico at schools that are exclusively
devoted to this aim, the "Escuelas Normales" and "Normales
Superiores", depending on the level: the former for primary school
(ages 6-12), the latter for secondary school (ages 12-15).
Mathematics teachers of the baccalaureate level (ages 15-18) do not
usually receive a pedagogical training but have degrees in subjects
related to mathematics. Generally speaking, primary school
pre-service teachers do at most one course in mathematics, and those
who will become mathematics teachers in secondary school do two or
three. There is an agreement among mathematicians that teacher
training (especially future maths teachers') lacks mathematics.
In this paper we will expand on this subject, stressing the importance
of the manner in which mathematics should be taught. We will give
some examples of this, particularly for the secondary and
baccalaureate levels.
The session will be shared with Silvia Alatorre (Universidad
Pedagógica Nacional).
- OLIMPIA FIGUERAS, CINVESTAV
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- FLORENCE GLANFIELD, College of Education, University of Saskatchewan, 28 Campus
Drive, Saskatoon, SK S7N 0X1, Canada
A Reflection about Mathematics Teacher Education Programs in
Canada
[PDF] -
In this session I will first start by providing an overview of
mathematics teacher preservice programs in Canada. These programs
differ for those who will be elementary mathematics teachers and for
those who will be secondary mathematics teachers. In almost all
teacher education programs in Canada, preservice teachers are
required to take at least one mathematics content course as well as a
mathematics `methods' course (a course that teaches you how to teach
mathematics). Mathematics educators in Canada ask the following
questions about the preservice programs: Are these requirements
sufficient to become a mathematics teacher? Should preservice
teachers take more mathematics courses? What mathematics should
preservice teachers know? In my talk I will propose the question,
"Are these the questions that mathematics educators should be
asking?" In my experience and research many elementary (and some
secondary) preservice teachers do not see `themselves' in mathematics;
they come to preservice education courses with certain understandings
and perspectives about mathematics and what it means to teach
mathematics. These understandings and perspectives are framed by the
experiences that preservice teachers have had prior to entering and
while enrolled in their teacher education programs. Perhaps the
questions that mathematics educators should be asking are not related
to the content... but related to what we do in our programs to
`shift' the understandings and perspectives of preservice teachers.
- FERNANDO HITT, Université du Québec à Montréal, Département de
Mathématiques, Case Postale 8888, Succursale Centre Ville,
Montréal, PQ, H3C 3P8, Canada
Reflexión sobre diferentes acercamientos en la enseñanza
de las matemáticas en ambientes tecnológicos
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Los profesores de matemáticas frente al uso de la tecnología en
los procesos de enseñanza tienen, en general, posiciones extremas;
una que va en contra del uso de la tecnología porque el profesor
considera que su uso inhibe el aprendizaje y, la otra, con un
entusiasmo desbordado, que la tecnología juega un papel
fundamental en el aprendizaje de las matemáticas. Nos podemos
preguntar cuál es la influencia que tienen los investigadores en la
promoción implícita o explícita de estas posiciones.
Quisiera, en esta exposición, analizar algunas investigaciones
realizadas en ambientes tecnológicos en donde podríamos
encontrar indicadores que nos permitan tener una posición
crítica y reflexiva sobre el uso de tecnología, tanto en el
terreno de la investigación sobre el estudio de fenómenos ligados
al aprendizaje de las matemáticas como en su uso en los procesos de
enseñanza.
- BERNARD R. HODGSON, Département de mathématiques et de statistique,
Université Laval, Québec G1K 7P4, Canada
The mathematical education of primary and secondary school
teachers: the experience of Université Laval
[PDF] -
In most Canadian universities, the education of pre-university
teachers represents an important component of the task of the
mathematics department, at least in terms of the number of students
involved. Such is the case at Université Laval, where the
Department of Mathematics and Statistics is responsible for several
courses offered in the context of the pre-service education of
teachers. These include two compulsory math courses given to
prospective primary school teachers, and seven courses specific to
prospective secondary school teachers.
The aim of this talk is to describe briefly the general framework of
the mathematical education of school teachers at Université Laval
and to discuss the main themes around which the content of the
mathematics courses is articulated. Examples will be given of the
mathematical topics presented in these courses; some of the
pedagogical approaches will also be discussed.
Comments about both the departmental support of these activities, and
the collaboration with colleagues from the Faculty of Education and
from other departments of the Faculty of Sciences and Engineering,
will be presented. A brief comparison will be made with other
Canadian universities.
- CAROLYN KIERAN, Département de mathématiques, Université du Québec
à Montréal, CP 8888, succ. Centre-Ville, Montréal, QC
H3C 3P8
Using Computer Algebra Systems (CAS) in Teaching High School
Mathematics: Two Research-Based Examples from Classroom
Practice
[PDF] -
The integration of new technologies in mathematics education has been
an ongoing issue for the last two decades. Teachers and teacher
educators are struggling with questions regarding the use of
technological tools and their relation to required paper-and-pencil
skills. The original optimism regarding the benefits of technology,
which would allow a focus on conceptual understanding at the expense
of calculation techniques, has become quite nuanced. This
presentation will address the dialectical relation between theoretical
thinking and technique, as they co-emerge in a combined computer
algebra and paper-and-pencil environment. More particularly, it will
focus on two Grade 10 teaching experiments involving CAS technology:
the first one on equivalence, equality, and equation; the second one
on generalizing and proving within factoring. Attention will be given
to the nature of the tasks in which the students engaged and to
students' ways of thinking within these tasks. Even though the topics
are quite different, findings indicate the importance of the
co-emergence of theory and technique in both cases.
- LUIS RADFORD, Ecole des sciences de l'éducation, Université
Laurentienne, Sudbury, Ontario, Canada, P3E 2C6
The Investigation of Motion and its Symbolic Mathematical
Expression
[PDF] -
Generally speaking perceptual activity, gestures, concrete actions,
and natural language provide one with the basic resources to achieve a
certain understanding of motion. However, as the studies conducted in
the late Middle Ages suggest, the mathematical investigation of motion
rests on a process of idealization achieved through the use of
signs. This is why a mathematical investigation of motion requires not
only the overcoming of the concrete experience and its intuitive,
phenomenological key concepts (e.g. space, time, velocity),
but also the understanding of new subtle concomitant forms of
mathematical symbolization. In this presentation we pay attention to
this idealization considered as a dialectic process between concepts
and signs. We analyze some classroom excerpts that point to some of
the students' difficulties in their attempt to understand and make
sense of motion and its symbolic mathematical expression. It is
suggested that rather than merely "representing" motion, algebraic
symbolism (in its graphical or formulaic form) is an artifact.
Algebraic symbolism mediates new ontogenetic ways of reflecting about
the world that emphasize certain qualitative and quantitative
relationships and leads to specific cultural conceptions of space,
time and motion.
- ARTURO RAMIREZ, Centro de Investigación en Matemáticas, A.P. 402,
Guanajuato, Gto., CP 3600, México
Una propuesta didáctica usando geometría dinámica
[PDF] -
En esta presentación mostraremos cómo motivar el teorema de la
recta de Simson utilizando para ello el GeoLab, un laboratorio de
geometría dinámica. Mostraremos primero cómo, utilizando un
método de Monte Carlo podemos conjeturar el teorema de Simson:
"Dado un triángulo arbitrario y un punto en su circuncírculo,
las proyecciones de dicho punto a los lados del triángulo están
alineadas", y recíprocamente "Si las proyecciones de un punto a
los lados de un triángulo están alineadas, entonces el punto
está en el círcuncírculo del triángulo".
Tenemos entonces que dado un triángulo, a cada punto de su
circuncírculo se le puede asociar una recta, su recta de Simson.
Mostraremos luego que la envolvente de la recta de Simson, conforme el
punto se mueve en el circuncírculo es una deltoide que es
tangente al triángulo dado.
Finalmente, construiremos el triángulo de Morley del triángulo y
veremos cómo está relacionado con la deltoide mencionada en el
párrafo anterior.
Si el tiempo lo permite veremos algunos otros teoremas geométricos
difíciles de visualizar sin la ayuda de GeoLab.
- MARCELA SANTILLAN, Universidad Pedagogica Nacional
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- THOMAS STEINKE, Ottawa-Carleton Catholic School Board, Ottawa, Ontario
From Toys to Tools: Reflections of a Math Consultant on
Implementing Technology in K-12 Math in Ottawa and Ontario
[PDF] -
Tom will share his reflections on his school board and Ontario,
province-wide experiences related to the implementation of technology
in K-12 mathematics in policy, teacher professional learning, and
resource development. The experiences relate predominantly to the
challenges and opportunities related to the insertion of CAS (TI 89),
dynamic statistics software (Fathom and TinkerPlots), dynamic geometry
software (GSP4) in our provincial mathematics policy documents. Some
recent hopeful and innovative models for professional learning that
focuses the effective use of technology to support improved student
learning will be highlighted.
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