My thesis, **An investigation into how low achieving secondary students learn fractions through visual representations**, developed at the University of Nottingham and funded by the university and CAPES, is finally available online for download:

The abstract is:

The gap between low and high achievers is a worldwide concern in Education, especially when it comes to mathematics. One way of facing this issue is by investigating the learning processes of those disadvantaged students at a classroom level. Bearing this in mind, I started my research by observing lessons for low achieving students in an underperforming school in England. After getting acquainted with the context, I designed lesson plans to teach fraction addition and subtraction following three design principles: lessons should enable students to build their knowledge about fractions on visual representations, students should have opportunities to solve tasks without being told how to do it beforehand and lesson plans should maintain some coherence with participant teachers’ current practices. The first principle is the most relevant for my findings, and its choice was based on the growing evidence pointing out the relevance of visual representations for mathematical learning and as a potential pathway to overcome some difficulties faced by low achieving students. Three teachers enacted the lesson plans with a different low achieving group each. Data was collected of the pupils’ working out, as registered in the worksheets, and also in the form of audio recordings, taken during the lessons, of my interactions with students about their thinking while solving the tasks. The data analysis revealed aspects of students’ learning through visual representations that were grouped into two major findings. Firstly, the lessons were successful in promoting reasoning anchored in visual representations, and enabled students to extend their knowledge beyond what was explicitly taught to them. Secondly, an apparent lack of visual skills and prior knowledge on multiplication restricted their engagement with some tasks. The final discussion focuses on the role of visual representations in the learning of mathematics in general, but mainly for low achieving students, and how this can be implemented in classrooms.

Sometime ago I found Ian Gordon on Flickr. I did not manage to find much information about him online, but I really enjoyed his drawing, which he published regularly on Flickr, especially those portraying Nottingham scenes.

Some of his work is available at Dukki, a great Nottingham-themed gift store.

Currently, his drawings are the background of my PC and mobile. Thanks, matte!

I finally found a pizza that deserves to be publicly recommended: Corner Stone Pizza, at Sherwood

The place is quite peculiar. Small and simple, but the pizza is amazingly well executed! Thin crusty dough with a delicious fluffy border and great combinations for toppings. They only have about 10 options, but all sounded delicious and with a good degree of variation. The price is also very good.

I only went there once and they were not serving drinks, even though they suggested I could buy something near and bring it to drink at the place. It is clearly a independent business with a strong local identity: no need of signs saying "Neapolitan pizza", just a very well crafted product.

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.

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.