Posts Tagged ‘2007’

Key points:

  • This article is full of ideas to support my action research with children when the time comes. I intend to use some of the ideas within to help structure my input and form the base and review tasks.
  • This article also grabs me as it contains citations of other authors I have read so far – not necessarily the same articles/sources as the Kaput writing I have read is a later date than this article, however the 2008 Kaput source seems to be a development of the 2001 piece used here. These connections confirm to me that my thinking and research is along the right lines and hasn’t been as unfocussed as I first feared.
  • “Difficulties occur with adolescent students stem from a lack of early experiences in the elementary school” – relates to functional thinking and how students find it difficult to spot generalisations easily. They lack apporpriate language to describe what’s happening, generally focus on a single data set rather than comparing information and have “an inabilty to visualise spatially or complete patterns.” (Warren, 2000). The researchers found that children had limited experience with visual growth patterns and had rarely used arithmetic for anything other than finding answers.
  • It continues to state that recording data in a table inhibited the children’s thinking, encouraging “single variational thinking, finding relationships along the sequence of numbers instead of find the relationship between the pairs.” … “The patterns chosen here were those  where links between the pattern and its position were visually explicit…to focus in particular on the relationship between the position number and the pattern.” The article gives examples of the patterns used (shown below) and describes the aims of the questions in detail.

Main Reference:

  • Warren, E., Cooper, T. (2007) ‘Generalising the pattern rule for visual growth patterns: Actions that support 8 year olds’ thinking’ Educational Studies in Mathematics, Vol. 67, No. 2, pp 171-185


  • Kaput, J., Blanton, M. (2001) ‘Algebrafying the elementary mathematics experience’ in Chick, H., Stacey, K., Vincent, J. and Vincent, J. (eds.) The future of the teaching and learning of algebra. Proceedings of the 12th ICMI study conference. Melbourne: ICMI, Vol. 1, pp. 57-94
  • Warren, E. (2000) ‘Visualisation and the development of early understanding in algebra’ in Nakahara, T. Koyama, M. (eds.) Proceedings of the 24th conference of the International Group for the Psychology of Mathematics Education. Hiroshima. Vol. 4, pp.273-280
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The article gives a history of the mathematics landscape in the UK since 1837. Algebra has been part of this throughout.

We see algebra as a key tool to help solve problems now, but historically “problem solving was seen as a specialized skill, only for mathematicians in opening up new fields.”

Key points:

  • “A problem solver needs a rich, connected understanding of mathematics and the abilty to see patterns of similarity and association, as well as the skills to carry out the planned attack, and to check that the results make sense in the context of the problem.”
  • “A Royal Commision, reporting on the state of mathematics teaching in nine leading Public (i.e., private) Schools in 1837, noted that the typical two weekly hours of mathematics consisted of Arithmetic, a little manipulative Algebra, and “Euclid“, learned by rote.”


  • Burkhardt, H., Bell, A. (2007) ‘Problem solving in the United Kingdom’ ZDM, Vol. 39, no. 5, pp. 395-403

After Study Block 1, taught session 2, we were given a reading to look through and digest.

We were given Chapter 5 (pp. 79-104) from “Children’s mathematics 4-15: learning from errors and misconceptions” by Julie Ryan, Julian Williams. (McGraw-Hill International, 2007)

The mistakes children make in mathematics are usually not just ‘mistakes’ – they are often intelligent generalizations from previous learning. Following several decades of academic study of such mistakes, the phrase ‘errors and misconceptions’ has recently entered the vocabulary of mathematics teacher education and has become prominent in the curriculum for initial teacher education.

The popular view of children’s errors and misconceptions is that they should be corrected as soon as possible. The authors contest this, perceiving them as potential windows into children’s mathematics. Errors may diagnose significant ways of thinking and stages in learning that highlight important opportunities for new learning.

This book uses extensive, original data from the authors’ own research on children’s performance, errors and misconceptions across the mathematics curriculum. It progressively develops concepts for teachers to use in organizing their understanding and knowledge of children’s mathematics, offers practical guidance for classroom teaching and concludes with theoretical accounts of learning and teaching.

Children’s Mathematics 4-15 is a groundbreaking book, which transforms research on diagnostic errors into knowledge for teaching, teacher education and research on teaching. It is essential reading for teachers, students on undergraduate teacher training courses and graduate and PGCE mathematics teacher trainees, as well as teacher educators and researchers.

Anyway, I found an online version of the text via Google Books, which is embedded below. However, pages 82, 83, 87, 88, 94, 95, 101 and 102 are not included in the preview due to copyright reasons. Although the content that is there gives a reasonable account of the subject and includes many examples of good practice. Clearly, the lack of the full chapter really doesn’t help!

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