É como se o sax de Coltrane nos forçasse a lidar com o fato de que nem tudo é calma, ritmo, melodia e estabilidade; ao mesmo tempo que nos lembra sutilmente de que ainda há calma, ritmo, melodia e estabilidade por baixo de toda superfície agitada.

The text below is a section of the book "Where mathematics comes from: how the embodied mind brings mathematics to being" by George Lakoff and Rafael Núñez.

As we have just noted, a significant part of mathematics itself is a product of historical moments, peculiarities of history, culture, and economics. This is simply a fact. In recognizing the facts for what they are, we are not adopting a postmodernist philosophy that says that mathematics is merely a cultural artifact. We have gone to great lengths to argue against such a view.

The theory of embodied mathematics recognizes alternative forms of mathematics (like well-founded and non-well-founded set theories) as equally valid but about different subject matters. Although it recognizes the profound effects of history and culture upon the content of mathematics, it strongly rejects radical cultural relativism on empirical grounds.

In recognizing all the ways that mathematics makes use of cognitive universal and universal aspects of experience, the theory of embodied mathematics explicitly rejects any possible claim that mathematics is arbitrarily shaped by history and culture alone.

Indeed, the embodiment of mathematics accounts for real properties of mathematics that a radical cultural relativism would deny or ignore: conceptual stability stability of inference, precision, consistency, generalizability, discoverability, calculability, and real utility in describing the world.

This distinguishes an embodied view of mathematics from a radical relativist perspective. The broad forms of postmodernism recognize the effects of culture and history. But they do not recognize those effects of embodiment that are not arbitrary. It is the nonarbitrariness arising from embodiment that takes mathematics out of the purview of postmodernism.

Moreover, the embodiment of mind in general has been scientifically established by means of convergent evidence within cognitive science. Here, too, em- bodied mathematics diverges from a radically relativistic view of science as just historically and culturally contingent. We believe that a science based on convergent evidence can make real progress in understanding the world.

I think that just the fact that the authors felt the need to clarify why their ideas do not align to the postmodern view is quite interesting. But also, I enjoyed their explanation.

I agree with the following statement by Duval (2006):

From an epistemological point of view there is a basic difference between mathematics and the other domains of scientific knowledge. Mathematical objects,2 in contrast to phenomena of astronomy, physics, chemistry, biology, etc., are never accessible by perception or by instruments (microscopes, telescopes, measurement apparatus). The only way to have access to them and deal with them is using signs and semiotic representations. (pp. 107)

If one accepts that statement, it seems reasonable to conclude that representations are particularly important in the teaching and learning of mathematics. However, it does not imply that **multiple** representations should be at the core of teaching, as several scientific papers and recent official documents place them.

One of the justifications often presented to support the use multiple representations is the following:

Because no single visual representation perfectly depicts the complexity of mathematical concepts, instructors often use multiple visual representations, where the different representations emphasize complementary conceptual aspects. (Rau and Matthews, 2017, pp. 531)

I fundamentally disagree with this view because I understand that some representations are very powerful and may be able to communicate a wide enough (for educational purposes, for instance) range of conceptual aspects of a given concept. Two examples: Hindu–Arabic numeral system to represent quantities and flat drawings made with pen, paper, ruler and compass for euclidean plane geometry.

A second common argument is the idea that multiple representations promote conceptual understanding. The problem with this argument is that since there is no instrumental definition of conceptual understanding, been able to use multiple representations to present a given concept became the definition of conceptual understanding. So, it is not a matter of multiple representations promoting conceptual understanding, but conceptual understanding being multiple representations.

From my perspective, multiple representation is a matter of curriculum: we, teachers, teach multiple representations because they are included in the curriculum directly, as a topic on its own, or indirectly, as a pre-requisite for another topic. That is my stand point in the paper Implications of Giaquinto’s epistemology of visual thinking for teaching and learning of fractions, where I defend the adoption of a carefully chosen visual representation (instead of multiple representation) especially when it comes to low achieving students.

Curiously, my position finds support in the paper published by Rau and Matthews (2017), where the authors draw some recommendations to promote learning through multiple representations. When discussing the limitations of their recommendations, they state that "some visual representations may be intuitively more accessible than others because they align with the structure of human cognitive architecture" (pp. 540). The authors call these representations privileged and point out that "deploying them as anchor representations might help optimize the web of meaning that emerges from use multiple representations" (pp. 540).

That is my point! For some representations there are reasons to use them that go beyond curriculum. Therefore, these representations should stand out of the pool as tools that can actually support learning and, then, the other representations may come to complement specific aspects (if some) or to cover curricular goals.

Duval, Raymond. ‘A Cognitive Analysis of Problems of Comprehension in a Learning of Mathematics’. Educational Studies in Mathematics 61, no. 1–2 (2006): 103–31. doi:10.1007/s10649-006-0400-z.

Rau, Martina A., and Percival G. Matthews. ‘How to Make `more’ Better? Principles for Effective Use of Multiple Representations to Enhance Students’ Learning about Fractions’. ZDM 49, no. 4 (August 2017): 531–544. doi:10.1007/s11858-017-0846-8.

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