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.
More about the video: ed.ted.com/lessons/how-does-caffeine-keep-us-awake-hanan-qasim
For those who does not know it, TED-Ed is the "educational branch of TED". Instead of lectures, they create and share "lessons that worth spreading". I have already contributed to TED-Ed with the lesson The last banana: A thought experiment in probability.
This weekend I discovered Boavista, a portuguese restaurant in Nottingham.
It is a pretty simple place, with a portuguese ambiance (much louder than a British restaurant, with TVs showing portuguese shows, and people talking in portuguese) and very good options. I tried the bacalhau a Gomes Sá, a traditional Portuguese dish with salted cod, potatoes, onion, eggs and lots of olive oil. My girlfriend tried spare ribs. Both dishes were very well done and following the Portuguese style of cooking (no sauce on the ribs, for instance).
But the real climax were the dessert: Portuguese custard tart (I cringe just by using custard to describe the filling of these tarts... it is much more than custard!). It was absolutely delicious.
And, to crown the visit, a very good espresso. I would not say it is the best coffee in town (as they state in the front show windows), but it is very very good.
This paper presents some theoretical arguments that I am developing for my thesis as a result of preliminary analysis of my data.
Abstract: In this paper, I will present some implications of Marcus Giaquinto’s ideas about visual thinking and its epistemology when combined with Toulmin’s layout of an argument. This is the result of an ongoing effort to discuss, from a theoretical perspective, some issues that emerged from my Ph.D. research about teaching and learning addition and subtraction of fractions to low achieving students. My claim is that visual representations can be effective for low achieving students when teaching is focused on a carefully chosen model and time is given to students to fully use it.
It was published in the Informal Proceedings of the British Society for Research into Learning Mathematics conference held in March 2017.