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 about the complex dynamic of learning with understanding promoted by such a tasks? What can we learn from this research to use within the classroom and in designing lessons or 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 has the increasingly complex world in terms of numeracy or mathematics literacy? How does this inform our practices within the classroom and in designing lessons/tasks?

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 world challenges?

In relation to connections and understanding, what kind of teaching methods are more appropriate?

How can we promote mathematics education as a means to explore environmental issues?

How do we evaluate and/or research about the resources from the perspective of the connections and understanding they try to promote?

How can mathematics promote "living together"?

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?

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 life?

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?

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What does it mean to develop a critical approach to mathematics and culture in an increasingly complex world?

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 white or black issue. There may be degrees of understanding. And there may also be 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 (Brousseau, 1997).

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 use the world. This idea is at the base of the concept of mathematization, or, more precisely, horizontal mathematization (Freudenthal, 1991). Also 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 logico-mathematical dimension of the construction of scientific knowledge. Piaget proposed to replace 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.

Dept. of Educational Studies

University of Oxford

Dept. de Didáctica de la Matemática

Universidad de Granada

Dept. d'Educació Lingüística i Literària, i Didàctica de les Ciències Experimentals i la Matemàtica

Universitat de Barcelona

Instituto de Educação da Universidade de Lisboa

School of Teacher Education, Charles Sturt University

Alexandra Gomes

Ema Mamede

José António Fernandes

Helena Martinho

Pedro Palhares

Alexandra Gomes - Portugal

Ana Serradó – Spain

Andreas Moutsios-Rentzos - Greece

Ema Paula Mamede - Portugal

Fragkiskos Kalavasis - Greece

Gail FitzSimons - Australia

Gilles Aldon - France

Javier Díez-Palomar - Spain

Lisa Boistrup - Sweden

Marcelo Bairral - Brasil

Monica Panero - Switzerland

Pedro Palhares - Portugal

René Screve - Belgium

Phasellus interdum

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