Archive for October, 2011

Animated construction of Sierpinski Triangle

Image via Wikipedia

A tricky one this.

Have barely written much in quite a while. In fact, since I completed my first assignment – something I will post here once the course is over.

We’re going back to geometry with study block 7. For me, shape and geometry is fascinating. I haven’t particularly studied this aspect of maths in a long while, but my degree’s dissertation was based around the Fibonacci sequence, the golden ratio and its appearance in art and nature.

Absolutely wonderful stuff, with some complex maths.

My recent fascination, a clear one given the theme of this site, are fractals.

Given 4 shots of a fractal, can you order them from least zoomed in to most zoomed it?  The point is that this is hard to do. Fractals are objects which are equally complex and look similar on all scales – therefore it is inherently difficult to tell how zoomed in you are. If you looked over someone’s shoulder, and saw them looking at a shot of the Mandelbrot set, it is entirely possible that at their zoom-level the entire set would span the size of the observable universe!

The above quote and image are from Matt Henderson’s maths and science blog.

Again, the maths is complex, literally, and not something I totally understand (that knowledge has somewhat left my mind). However, I feel that something like this would be a good thing to explore in the primary school setting. Not only are they incredibly beautiful, they can provide a stimulus for ordering exercises, they provide a new and exciting set of shapes to explore, they can make great displays.

Let’s expand:

  • Sierpinski Triangles can be made using equilateral triangles – which in turn can be created through paper folding.
  • Investigating the Sierpinski Triangles can lead to such questions as: what fraction of the triangle is left after one step, two steps…? Is there a pattern?
  • Linking Sierpinski’s Triangle to Pascal’s Triangle, see below. This involves investigating the pattern in Pascal’s Triangle and shading all even numbers.
To me, this would make a good series of a couple of lessons at the upper end of Key Stage 2.
Also, part of the Eiffel Tower is similar to a fractal!

 

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

Citations:

  • 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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