The aim of this page is to organize the posts in the blog about my Ph.D. research developed from 2014 to 2018 in the School of Education, University of Nottingham.

My general interest is on lessons about fractions emphasizing visual representations for low achieving students. Thus most posts will be about these issues.

I do not have the intention of being systematic here, I only want to post particularly interesting papers and ideas that occurred to me during the research process and maybe some results and publications.

The lessons

In order to develop my research, I designed a series composed of 12 lesson plans about addition and subtraction of fractions for secondary students. The second version of them, refined after my data analysis, is available in the link below. If you are a teacher and want to teach fraction addition and subtraction using an approach heavily based on visual representations, you are more than welcome to try the lesson plans and send me comments, suggestions and opinions.

  • Download the lesson plans
  • My thesis

    My full thesis, containing a discussion of the whole process since my first contact with the British educational system, the development of my research question, the design principles behind the lesson plans, a discussion of the lesson taught to three different low achieving groups and my conclusions, can be downloaded from the University of Nottingham's repository in the link below.

  • Download my thesis
  • Papers and presentations about my research

    Possible parallels between visual representations and informal knowledge is the first paper I published (informal proceedings) related to my PhD research. It is based on a piece of data collected in a sort of pilot study in the same school I am carrying out my actual data collection. The paper shows a student reasoning about fraction addition and, I would argue, based on the visual representation that were being used in his instruction. This is paper is related to this post about the work of Nancy Mack.

    A teacher changing her practice: a tentative explanation for the reasons behind it is a paper by Rita Santos Guimaraes, whose research is focused on the same teachers that are taking part in my research. In fact, she is investigating the impact of all the process of discussing and enacting the lessons I am proposing in their practice in general.

    On March 2017 I presented some theoretical arguments that are emerging from my data collection at the BSRLM Conference. It is working-in-progress, but here is the presentation and here is the informal proceedings paper for those who may be interested.

    On August 2017, I read the paper "How to Make 'more’ Better? Principles for Effective Use of Multiple Representations to Enhance Students’ Learning about Fractions" by Rau and Matthews and it triggered some thoughts about the role of representations in the learning of mathematics. This may end up being a chapter on my thesis, but a preliminary version of these thoughts were summarized in this post.

    On April 2018, I presented my conclusions for the fist time at BCME. You can download the slide in here.

    Other relevant/interesting references

    Complementary to my research, Rita Santos Guimarães investigates how the teachers were affected by the intervention that served as basis for my research. Some of her conclusions are related to my conclusions. Her thesis can be downloaded here.

    I wrote a review of the book Mathematical Imagery by Dave Hewitt and others. This book came a bit late for my research, but it is a very interesting reference for teachers willing to use visual representations heavily. Also, I suggest the concept of de-numeration of mathematics, something that I want to develop further in the future.

    I wrote a review, in Portuguese, of the book Visual Thinking in Mathematics: an Epistemological Study by Marcus Giaquinto. This is a central reference for my theses and in this review I discuss connections between the book and ideas such as embodied cognition and mental number line proposed by other authors.

    The papers published by Nancy Mack about fraction operations and informal knowledge were quite inspiring for me.

    Mayer (1997) is a fundamental paper for my research because it justifies why use a visual approach to teach mathematics to low-achieving students.

    Ramani and Siegler (2008) analysed the relation between playing linear number board games, numerical knowledge and socio-economic status.

    I quite like the way Martyn Hammersley discusses methodological issues. Particularly, his view of what is a case study helped me a lot. His view on aspects of research design in social science is also very interesting.

    Post in Portuguese about ability setting as it is done in England inpired by the paper "Exploring the relative lack of impact of research on ‘ability grouping’ in England: a discourse analytic account" (Francis et al, 2016). This post is aimed at Brazilians that want to know a bit more about this practice.

    The paper Social class and the visual in mathematics (Gates, 2015) puts together arguments from different areas, such as sociology and cognitive sciences, in favour of more emphasis on visual representations in the teaching of mathematics, specially when it comes to socially disadvantaged students.

    Tool case

    If you have to transcribe audio (from interviews), I recommend the web app oTranscribe. Read more about it here.

    During my data collection, I collect almost 2000 worksheets. Or course I do not want to analyse all of them, but I knew from the beginning that I would need a good organizational system that included to possibility of applying tags (or codes) to the worksheets. Because I could not find a good open software for the job, I started to develop my own: OpenQDA. Although it is still not fully featured, it is available online for anyone to use.

    The scripts developed by Jonathan Schultz to export qualitative projects from Nvivo to a friendly and flexible format are very useful for those who used Nvivo but need something beyond they weird set of tools. More info about the scripts here.

    I use Zotero as my reference manager because it is open source and very effective. In this post, I discussed some of its features and explain how to start using it.


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    Leonardo Barichello