This project ended a while ago. I am aware that this has been a very neglected blog.

I am now a qualified MaST Specialist and use my skills with colleagues on a regular basis. I guide my work mates through many aspects of maths.

At some point, I will upload some highlights of my more recent maths work. With the changes in curriculum coming up, I have a lot to organise in my current school.

The taught session of Study Block 9 discussed a lot of the history of counting and of the different number systems used in various times throughout history. We looked at Babylonian, Egyptian, Greek, Roman and Mayan counting systems, all of which relate to our Hindu-Arabic system in some way.

 

English: babylonian numbers Español: Números b...

The Babylonian number system - Image via Wikipedia

Early european variants of the arabian digits

Early european variants of the arabian digits - Image via Wikipedia

The hardest to get my head around was Babylonian which uses a base 60 system, which we still find in our time units today. Once I began thinking like that, I began to be able to see the beauty of the system and its inherent usefulness, however clumsy it is to work out at first!

 

The main thing I took from this was the urge to share some of this maths from other cultures with my class – we are currently studying Ancient Greece as our main topic, and were about to start a Literacy unit called stories from other cultures in which I focus on tales from the middle East (Mesopotamia, now Iran and Iraq – precisely where the Babylonian system originated). This seems like an opportunity not to be missed to create a link between Literacy and Numeracy as well as discover a little about the history of the numbers we use today.

 

I planned two lessons based on the work we covered – one looking at Ancient Egyptian numbers and the other based on the Babylonian system.

Egyptian Numeric System

My class were able to understand and use the Egyptian system well – by the end of the hour they could add, subtract and complete multiplication sums using their symbols. I deliberately didn’t teach them the Egyptian way of multiplying as I was unsure whether their knowledge of the systems we use today was strong enough and wanted to avoid confusing them! The children were able to see links between the Egyptian system and ours easily, with them both using base 10, and could therefore think in English numbers, as it were, and convert into Egpytian…

The Babylonian lesson went less well. The process of converting from our numerals into Babylonian ones a simple process up to 59. After that, it becomes a little tricky as my class made clear with exasperated cries of frustration!

I still think there is merit in discussing these systems with the children – if only to look at the practicality of different ways through history!

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Other tasks we have been asked to complete include looking at how numerate my school is and to consider what children know about numbers by asking them to consider what the biggest number they know is.

Numerate school

It turns out that the classrooms at my work place are generally well set up children to encounter numbers. There were many examples of number lines, multiplication squares, and so on. It was also clear that these opportunities occur far more often the lower down school children are – with the upper Key Stage 2 classes having far fewer examples of numbers around the rooms. The other areas particularly lacking in number were the additional parts of the building – things such as corridors, the hall, the reception area and the whole outdoor area. Plans are now in place to make these areas a little more interactive number-wise. There is now a dedicated numeracy display board in the hall, there will be a series of number challenges around the corridors and there will be an outdoor number trail created around school when the weather is a little nicer.

“What’s the biggest number you can think of?”

I asked this question to children in all classes from Reception upwards with the results showing a wide range of ideas. The concept of billions appeared as early as Year 1, but even in Year 2 there was a little confusion over the order of numbers (2000 and a billion being one answer there). In Year 3, they struggled to think of anything larger than one million. Infinity made its first appearance in Year 4, although I also got the answer “20 thousand, 5 million and 2 million” from a child in this class. Year 5 produced some spectacular answers, each trying to outdo the previous child. After “a trillion 9s”, we had “infinity times a trillion”. I couldn’t really argue with that…

This exercise shows me that there are some issues with place value in school – many children gave garbled answers which I had to reorder to make into a number – but also that children struggle to read larger numbers when written down. This was the extension task I gave them and the children were OK when reading four digit numbers, but problems crept in with the majority of children once more digits were used.

 

The title here implies I’m about to write something about pendulums, however, it’s more to explain why there has been such an absence of posts from recent MaST work.

The main reason is that I have just moved house, and other factors are an increased work load lately as well as a distinct lack of time to work with children in school on MaST type things.

So, things are just about settled down now, so I feel able to post here a little more. I also have things to write about based on my work in school and from the most recent study block.

But, thinking about pendulums – and chaos theory (I have been reading a lot lately about oscillations and magnitudes of vibrations creating wonderful changes as discussed by David Acheson) – here is a fabulous video.

Wonderful!

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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Over the course of the next couple of weeks in school, I intend to begin my work with the children. I haven’t really finished my reading yet, but I have a solid foundation to begin from – also I have the time available to me in school over these two weeks and after that it will be increasingly difficult to be able to do these sorts of things.

Session 1: Show a pattern made from multilink cubes. 1st, 2nd and 3rd stage of the pattern. Following this concrete modelling, introduce the use of an input-output table to organise data about the number of blocks used for each stage of the pattern. The table helps quantify the pattern so that children see both the growing pictures and the growing numbers in the table. They can note the change from stage to stage and work to explain how the change in the table matches the change in the picture of the growing pattern. Finally, children can try to write a general rule that will work for any stage of the pattern without having to build it or know how many blocks were used in the stage before it. This is an important abstraction of the pattern and the rule must make sense to children and be in their own words or in their own mathematical notation that reflects the level of their current understanding. These may not be accurate at this stage – but that’s ok (and the whole point of the sessions…)

Fir Tree investigation: uses pattern block triangles to create growing fir trees. Children must extend the pattern, complete the table of values and describe the 10th tree.

Session 2: use ideas from algebraic infants. Modelling how a family of dogs can be constructed. Legs, shoulders, bum, body, head – with only legs and body growing, Children need to show each stage in a table for the dog. They then create their own animal, growing it in the same way. Multilink cubes needed.

Session 3: Tables & Chairs investigation challenges students to find a rule to describe the relationship between the number of small square restaurant tables placed together in a line and the number of diners that can be seated at the larger table if only one person sits on each side. Can model with shapes if needed (squares and circles). As before, children make a table to shapes used and differences to aid thinking.

Session 4: Part 1: Ships Ahoy. Children look at a simple pattern of horn blows and predict future patterns.

Part 2: Rockets. Children look at a pattern for building rockets. They are challenged to see how many parts will be needed for the 50th stage. Whiteboard/paper drawings probably best – stickers might be helpful though?!

Session 5: Task Cards. Children choose a task card then construct it using squares or triangles. Encourage them to look at the patterns and decide how it is growing. Create and complete a table to show the growth pattern. Use the tiles to make the next two shapes in the pattern. Be prepared to explain how the pattern is growing. If there is time, choose another task card. Cards attached at the end.

Session 5 – Challenge Cards (PDF)

Session 6: Same as session 1. Can the children identify the growing pattern and express the rules algebraically?

Show a pattern made from multilink cubes. 1st, 2nd and 3rd stage of the pattern. This pattern must be similar but not the same as session 1. Following this concrete modelling, introduce the use of an input-output table to organise data about the number of blocks used for each stage of the pattern. The table helps quantify the pattern so that children see both the growing pictures and the growing numbers in the table. They can note the change from stage to stage and work to explain how the change in the table matches the change in the picture of the growing pattern. Finally, children can try to write a general rule that will work for any stage of the pattern without having to build it or know how many blocks were used in the stage before it. This is an important abstraction of the pattern and the rule must make sense to children and be in their own words or in their own mathematical notation that reflects the level of their current understanding.

Hexagon dragon investigation: dragons made from equal numbers of hexagons and triangles, each new term adding on more. Requires children to extend the pattern, create an input/output table to describe the growing pattern, then draw and/or describe the 10th dragon in words. An extra challenge asks students to generate a rule for this pattern so that Miguel can figure out how many blocks he will need to build a dragon of any size.

Each of these sessions will be completed during the children’s Numeracy time. I am likely to choose 4 or 5 children to work with from across ability groups and a mix of genders. I will take copies of their written work and record conversations to help me analyse their progress and thinking over the course of the sessions – I feel that this will give me the data I need to go further. Session 6 is a repeat of session 1 in order to set a baseline and see the progress, if any, the children have made in between.

Clearly, this is such a small scale research project that results can’t be read into too much. However, it is a beginning for my school to look at how we can use algebra in wider contexts. I have been careful to choose similar tasks throughout the sessions as I will only have at most half an hour to complete the tasks with them.

Key points:

  • Algebra is a third of the Malaysian Secondary Mathematics curriculum.
  • The article lists a large range of sources relating to misconceptions in the understanding of algebra. It states that, because algebra is an extension of previously acquired mathematical learning, it is linked to the use of symbolic representations and that “Warren (2003) felt that understandings of basic arithmetic operations could assist sucessful transition from arithmetic to algebra.”
  • “Students use the equals sign in both contexts: arithmetic and algebra. The concept of the equals sign in the framework of arithmetic is that of a ‘do something’ signal (Bodin & Capponi, 1996). Many students tried to add “= 0″ to algebraic expressions when they were asked to simplify (Kieran, 1997).”
  • The article cites Booth, 1981, as a refernce for children not understanding the idea of a letter as a variable, and they rather see each letter as representing a digit.
  • The article looks in detail at a range of errors (12 types) made by students. The examples given are from the secondary curriculum and so may not apply to any errors made by children in a primary school – but a worthwhile source to help identify the sorts of errors that could be made.

Main Reference:

  • Lim, K. (2009) ‘An Error Analysis of Form 2 (Grade 7) Students in Simplifying Algebraic Expressions: A Descriptive Study’ Electronic Journal of Research in Education Psychology, Vol. 8, No. 1, pp 139-162

Key points:

  • The article discusses how algebraic equations can be shown as visual sentences. The example of x+y=4, where x>y is used. The Reception aged children are given two rules to colour them in: “they have to colour in four snails, and the number of brown-coloured snails must be more than the number of yellow-coloured ones.”
  • The author states how remarkable it is that the children of this age can complete this algebraic idea and that staff argue it should make it easier for the children to manipulate equations later in life.
  • The approach to teaching also requires children to discuss their ideas in groups, challenging each other’s answers, explaining their reasoning and arguing with the teacher who deliberately makes mistakes to generate such discussion.
  • It seems that both children and teachers are capable of exceeding perceived expectations through innovative thinking. Clearly, this is just one example and it’s hard to draw conclusions but it would be interesting to see where those children of 2006 are now in Year 5.

Main Reference:

Original Article:

Key points:

  • That  children struggle to make the move from being in a situation where “not knowing answers (to arithmetic calculations) is treated negatively, and then suddenly introduced to algebra in which not knowing is treated positively as an opportunity to use symbols, as a way of working with not knowing.”
  • “Whenever a learner solves a problem, there is available the question ‘What is the method that was used?’, which in intimately tied up with the question ‘What can be changed about the problem and still the same technique or method will work?’ or ‘What is the class of problems which can be solved similarly?’ S. Brown and M. Walter (1981) suggest asking ‘What if … something changed?’ or ‘What if not…?’ Watson and Mason advocate explicitly asking learners to consider what dimensions of possible variation and corresponding ranges of permissible change they are aware of (Mason & Johnston-Wilder, 2004; Watson & Mason, 2004) as stimulus to becoming aware of, and even expressing features of, the general class of problem of which the ones considered are representative.”
  • Awareness of and Expressing Generality: “Picture-pattern sequences (Mason, 1988b; Mason, Graham, Pimm & Gowat, 1983; South Notts, n.d.) provide just one context for generalizing.” Learners are shown a sequence of pictures, and then specify a method describing how the pattern is growing through each term in the sequence, extrapolating these ideas to fit further terms in the sequence
  • “Getting learners to make use of their powers is not simply an approach to algebra or even approach to mathematics. It is mathematics.”

Main Reference:

  • Mason, J. (2008) ‘Making Use Of Children’s Powers To Produce Algebraic Thinking’ in Kaput, J., Carraher, D. and Blanton, M. (eds.) Algebra In The Early Grades. New York: Lawrence Erlbaum Associates, pp. 57-94

Citations:

  • Brown, S. and Walter, M. (1982) The art of problem posing. Philadelphia: Franklin Institute Press.
  • Mason, J. (1988b) Expressing generality [project update]. Milton Keynes: Open University Press.
  • Mason, J. and Johnston-Wilder, S. (2004). Designing and using mathematical tasks. Milton Keynes: Open University Press.
  • Mason, J., Graham, A., Pimm, D. and Gowar, N. (1985). Routes to, roots of algebra. Milton Keynes: Open University Press.
  • South Notts Project. (n.d.) Material for secondary mathematics. Nottingham: Shell Centre, University Of Nottingham.
  • Watson, A. and Mason, J. (2004) Mathematics as a constructive activity: The role of learner-generated examples. Mahwah: Lawrence Erlbaum Associates.

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Key points:

  • “Recent research on the status of student knowledge based in the traditional arithmetic-then-algebra regime has pointed to specific obstacles to algebra learning that computational arithmetic creates for the learning of algebra. For example, limited approaches to equality and the “=” sign in arithmetic as separator of procedure from result (Kieran, 1992) and now known to interfere with later learning in algebra (Fujii, 2003; MacGregor & Stacey, 1997).”
  • The majority of the chapter discusses the uses of algebra and tries to define it – “algebra needs to be described both through a snapshot of its structure and function in mathematics today and in mathematically mature individuals, and through a dynamic picture of its evolution historically and developmentally.” “Most attempts to describe algebra historically…tend to be oriented toward progress in solving equations, where the origin of the equations might be the problem situations or simply assertions about numbers or measurement quantities, often surprisingly similar across millennia (e.g., Katz, 1995).”
  • The article continues to describe two core aspects of algebra – generalisation and “syntactically guided action on symbols within organized systems of symbols” (which I take to mean reasoning). When these two core  aspects are introduced to children is another area of discussion – with practitioners giving reasonable arguments for each aspect to be given favour.
  • Movement from arithmetic to algebra depend on the understanding of the “=” sign. Children must realise that sign shows equivalence: 18 plus 3 is the same as 3 plus 18 just as a plus b is the same as b plus a.

Main Reference:

  • Kaput, J. (2008) ‘What Is Algebra? What Is Algebraic Reasoning?’ in Kaput, J., Carraher, D. and Blanton, M. (eds.) Algebra In The Early Grades. New York: Lawrence Erlbaum Associates, pp. 5-18

Citations:

  • Fujii, T., (2003) ‘Probing students’ understanding of variables through cognitive conflict problems: Is he concept of a variable so difficult for students to understand?’ in Pateman, N., Dougherty, B. and Zilliox, J. (eds.) Proceedings of the 27th Conference of the International Group for the Psychology of Mathematics Education, Vol. 1, pp. 49-66. Honolulu: University of Hawaii.
  • Katz, V. (1995) ‘The development of algebra and algebra education’ in Lacampagne, C., Blair, W. and Kaput, J. (eds) The algebra initiative colloquium, Vol. 1, pp. 15-32. Washington, DC: U.S. Department of Education, Office of Educational Research and Improvement.
  • Kieran, C. (1992) ‘The Learning and teaching of school algebra’ in Grouws, D. (ed) Handbook of research on mathematics teaching and learning, pp. 390-419. New York: Macmillan.
  • MacGregor, M., and Stacey, K. (1997) ‘Students’ understanding of algebraic notation: 11-15′ Educational Studies in Mathematics, Vol. 33, No. 1, pp. 1-19.

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