Dept. of Educational Studies
University of Oxford
Dept. de Didáctica de la Matemática
Universidad de Granada
Joaquin Giménez Rodriguez
Dept. d'Educació Lingüística i Literària, i Didàctica de les Ciències Experimentals i la Matemàtica
Universitat de Barcelona
João Filipe Lacerda de Matos
Instituto de Educação da Universidade de Lisboa
School of Teacher Education, Charles Sturt University
Learning in an increasingly complex world
How can we re-conceptualise learning with understanding in a complex world?
How can we promote learning with understanding in an increasingly complex world?
What features should a task have in order to promote learning with
understanding? How to research the complex dynamic of learning with
understanding promoted by such tasks? What can we learn from this
research to use within the classroom and in designing lessons/tasks?
How can we establish connections in mathematics learning: Between
different areas of mathematics? Between mathematics and other subjects?
Between mathematics and everyday life?
What implications does the increasingly complex world have in terms of
numeracy or mathematics literacy? How does this inform our practices
within the classroom and in designing lessons/tasks?
Mathematics Teacher Education
What kind of mathematics training should teachers have in order to be able to promote learning with understanding?
How can teacher training contribute to establishing connections between the various areas of Mathematics?
How can teacher training contribute to establishing connections between Mathematics and other subjects?
How to promote connections between school mathematics and academic mathematics, in teacher training?
What type of competences do we need to include in professional training
programs for mathematics teachers to cope with the increasingly complex
Teaching for connections and understanding
In relation to connections and understanding, what kind of teaching methods are more appropriate?
How do we evaluate and/or research the resources from the perspective of
the connections and the understanding they try to promote?
How can we promote mathematics education as a means to explore environmental issues?
How can we promote mathematics as a means to reflect on the sustainability of the world?
How can mathematics promote "living together"?
Mathematics Education with Technology
How can ICTs contribute to learning rich in connections, in an increasingly complex world?
How can ICT be used in teacher training to promote understanding in mathematics?
How can we use ICT as teaching-learning tools, rather than instruments that replace students’ cognitive efforts?
Connections with culture
Is it possible to understand peoples’ lives from an ethnomathematics perspective?
How can school mathematics take into account the culture developed by young people in their everyday lives?
How to take advantage of cultural aspects to enrich the teaching and learning of mathematics?
How can we create hybrid spaces linking school-mathematics to mathematics situated in cultural, everyday contexts?
What dos it mean to develop a critical approach to mathematics and culture in an increasingly complex world?
THEME OF THE CONFERENCE
Phrases like “mathematics is the language in which God has written
the universe” (Galileo Galilei) or “all things in nature occur
mathematically” (René Descartes) express the idea that if we want to
understand the world, then we need to use mathematics. But can we use
mathematics without understanding? John von Neumann once said “Young
man, in mathematics you don't understand things. You just get used to
them.” One way to interpret this statement would be to say you could use
mathematics (with success) without understanding it. Or, perhaps we can
speak of a kind of understanding that is merely instrumental instead of
relational (Skemp, 1976) or intuitive, or formal (Byers &
Herscovics, 1977). Another different way to read von Neumann’s statement
is to take it as a clarification that understanding is not a black and
white issue. There may be degrees of understanding. And there may also
be a form of understanding that impedes better understanding. In the
words of Richard Skemp, “to understand something means to assimilate it
into an appropriate schema. This explains the subjective nature of
understanding, and also makes clear that this is not usually an
all-or-nothing state” (Skemp, 1971, p. 46). Pragmatically, the power of
adaptability of a schema results from its connection to a greater number
of concepts, but it may happen that what is an appropriate schema at
one particular time may be obsolete and turn into an obstacle later on
Let’s get back to René Descartes: “All things in nature occur
mathematically”. A different idea implied by this saying would be that
to understand mathematics we need to connect our mathematical
understandings with our understandings of the world we live in (natural,
psychological and socio-cultural; see also Skemp, 1979). This idea is
at the base of the concept of mathematization, or, more precisely,
horizontal mathematization (Freudenthal, 1991). Concurring with this
idea is the belief many have that Mathematics is a cultural product
based on human experiences, such as counting, measuring, locating,
designing, explaining, and playing (Bishop, 1988). Nevertheless,
mathematical understanding has to do with both the learning of
invariants and the acquisition of cultural tools in which children can
represent mathematical ideas, in a dynamic and interconnected process
(Nunes & Bryant, 1997). This idea is in line with a recent
formulation of understanding in epistemology, in which understanding of a
given phenomenon has to be maximally well-connected and it may have
degrees of approximation (Kelp, 2015).
Concerning the learning and teaching of mathematics in the complexity
of our world, we can revalue the ideas of Galileo, Descartes and Von
Neumann on the central role of mathematics in the context of the genetic
approach of epistemology proposed by Piaget to the logical-mathematical
dimension of the construction of scientific knowledge. Piaget proposed
replacing the positivist hierarchization of science with an
interdisciplinary cyclic epistemology. This approach to epistemological
interrelationships in the context of learning, conceived in the digital
environment of education, calls into question not only the connections
of mathematics as a scientific discipline, but also the connections of
mathematics as an academic subject. How is it possible to make the
presence of mathematics visible in the understanding of other school
subjects? How to collaborate with other teachers of mathematics and of
other courses? This question of interdisciplinarity is in close
interaction with the learning and teaching of the complexity and variety
of the natural and social phenomena of our era.