Blog

Mathematics Teaching has just published my review of the book Mathematical Imagery. For editorial reasons, the review was shortened but they authorized the publication of the full version online and here it is.

Review of the book Mathematical Imagery, written by Dave Hewitt and others

Initially, the book focuses on 5 processes that the authors identify as related to imagery, keeping in mind the almost self-evident but still very oblivious perception that one does not learn just by looking at an image but has to do some work on it. The processes are: Reconstruction/Construction (Deconstruction?), Moving from imagery to abstraction/symbolism, Awareness of awareness, Stressing and ignoring, and Images that provoke a need for…. The authors do not claim the list is exhaustive, but claim it illustrates, with examples of tasks and experiences, the types of process that may lay hidden behind imagery when it is used in mathematics classroom as more than just mere illustration.

The most important aspect of the book, in my opinion, is the fact that the authors do not evoke the well-known instructional approach of Concrete-Pictorial-Abstract (or any of its variations). Instead, the authors place imagery by itself as a central issue in teaching and learning mathematics and use it as a springboard to promote other skills, such as communication (what do you see in the image?), pattern seeking (what would come next?) and generalization (can you extend the image?).

This approach to imagery is coherent with the common view among mathematicians that there is something visual behind mathematical thinking and discovery; something that is often related to imagination, that has a more holistic character than symbolic representations and formal reasoning. But it is also consistent with new findings suggesting that our brains seem to rely on essentially visual models to represent basic mathematics concepts, such as a number line to represent quantities.

Beyond that, while reading the book something else got my attention. Something subliminal that resonated with the feeling I got after John Mason’s seminar “Teaching More by Teaching Less - Getting Learners to Make Use of Their Natural Powers” in November 2015. I am going to call it the de-numeration of mathematics. Despite the fact that several activities suggested in the second part of the book involve actual numbers, they are never central neither in the question nor in the solution methods I imagined while trying to solve the activities. It feels that the authors are saying, and I deeply agree with them, that there is a lot of mathematics that can be done without numbers or, more precisely, without arithmetic skills and knowledge.

This view seems to be especially relevant when we think about students struggling with prior mathematical knowledge. I would say that the approach suggested in the book offers a way out of the typical situation where a teacher avoids a topic because he has the impression that the students do not master the pre-requisites well enough. This seems to be the reason why students placed in low sets are often stuck revisiting the same topics over and over again. I do understand the dilemma; in a subject so hierarchically structured as (academic) mathematics, how can I move to a new topic if students are not fluent enough with the previous? I believe this book hints at an intriguing solution: why not do some de-numerated mathematics and, progressively, build new topics on the top of such knowledge? An interesting example is given by Louise Orr and her proposal of an algebraic model using Cuisenaire Rods.

This process of de-numerate mathematics could be a path to reduce the frustration caused by difficulties with times-tables and other arithmetic skills and to promote imagery as well as other visual skills, that are knowingly correlated to mathematical achievement and choice of STEM carriers.

The second part of the book consists of more than forty well illustrated images accompanied by prompts and questions to promote discussions that starts in what can be seen in the images and drifts to topics ranging from geometry, arithmetic, algebra, number theory and so on. All low threshold and high ceiling ideas. In conclusion, the authors say that:

We suspect that many successful mathematicians develop their own imagery for mathematical processes and, in the absence of such imagery, others are left mystified by where mathematical insight comes from. (p. 35)

If you agree with that proposal, this book offers a rich and intriguing set of images, tasks, experiences and thoughts to inspire you to explore a more de-numerated mathematics.

Comic strip straight from xkcd: https://xkcd.com/2025/

PS: if you do not know xkcd, there is a hidden joke when you leave your mouse over the image for a couple of seconds.

Acaba de ser publicada a minha revisão (em português) do livro Visual Thinking in Mathematics: an Epistemological Study de Marcus Giaquinto.

A revisão trata rapidamente do conteúdo do livro e depois traça paralelos entre as ideias desse autor sobre visual thinking e as ideias de autores como Lakoff sobre embodied cognition e Dehaenne sobre mental number line.

O texto pode ser lido na íntegra no site do períodico Educação Matemática em Revista.

Atendendo a pedidos, eis a receita (em português) de cookie que eu adaptei ligeiramente do livro The Violet Bakery Cookbook, de Claire Ptak.

A receita é bastante simples e o resultado é realmente muito bom! Rende 6 cookies de tamanho médio

85 gramas de manteiga sem sal derretida;
100 gramas de açucar (eu uso 2/3 mascavo e 1/3 refinado, mas tenho certeza de que outras proporções não vão comprometer a qualidade);
1/4 de colher de chá de baunilha (eu uso em pasta; essência não funciona pois o cookie vai pro forno);
1 gema de ovo;
110 gramas de farinha de trigo;
Uma pitadinha de sal;
1/4 de colher de chá de bicarbonato de sódio;
75 gramas de chocolate meio-amargo quebrado em pedaços pequenos.

Bata a manteiga e o açucar com um mixer até que a mistura fique homogênea e comece a ficar cremosa. Adicione a baunilha e a gema. Bate até ficar homogêneo. Misture os ingredientes secos restantes, exceto o chocolate, em um recipiente a parte e acrescente pouco a pouco à mistura cremosa mexendo com um garfo. Quando todos os ingredientes secos estiverem incorporados, acrescente pouco a pouco os pedaços de chocolate usando o garfo para misturá-los na massa.

Divida a mistura em 6 partes e faça bolinhas. Coloque em uma forma forrada com papel manteiga e leve ao congelador por pelo menos uma hora. Quando quiser assá-los, tire os cookies do congelador por 10 minutos e leve ao forno pré-aquecido a 160 graus (essa temperatra funciona para fornos com ventilador interno; o livro sugere 180 graus para fornos convencionais, mas eu não testei esse caso). Asse por 20 minutos (os cookies ainda estarão um pouco gordinhos) e deixe esfriar por pelo menos 15 minutos antes de servir.

Os cookies já assados ficam muito bons até o dia seguinte. Os congelados duram bastante tempo antes de serem assados.

Aos que vierem pra Londres, fica a sugestão da Violet Bakery. Além da sede oficial (um pouco fora do centro), eles também fazem aparições em feiras em regiões mais centrais (cheque no site).