## Posts Tagged ‘Warren’

**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

**Key points**:

- “Early algebraic thinking in a primary context is not about introducing formal algebraic concepts into the classroom but involves reconsidering how we think about arithmetic.”
- “arithmetic thinking focuses on product (a focus on arithmetic as a computational tool) and algebraic thinking focuses on process (a focus on the structure of arithmetic) (Malara & Navarra, 2003)”
- “The aim was to assist 5 year-olds to come to an understanding of the structure of equations, and in particular the use of the equal sign.”
- “When asked to find the unknown for 7 + 8 = ? + 9, many students express this as 7 + 8 = 15 + 9 = 24.”
- “Three add 2 is different from 4 add 2. There is a different number on each side. Three add 2 is not equal to 4 add 2. It is different from 4 add 2″. Arithmetic thinking is required when computing the value of the two expressions. Algebraic thinking is required when placing the appropriate language between the expressions. Equals is only applicable if the two expressions are equivalent, that is, have the same value.”
- “It is interesting that both Brianna and Ethan spontaneously introduced symbols as short hand for the object:
*p*for pears and*b*for bananas or an iconic picture of each. In the secondary context this is commonly referred to ‘fruit salad’ algebra where the letter stands for an object instead of variable, and is thinking that we want to discourage.**For example, a common misconception in the secondary context is that 3**In the early years it is important to make the distinction between how we verbally describe number problems and how we represent these problems with symbols. While we say, “Three cars and five trucks,” the convention is to represent this as 3 and 5. Number sentences are made up of numbers.”*a*+ 3*b*stands for 3 apples and 3 bananas instead of*a*and*b*standing for any number.

**References**:

- Malara, N., Navarra, G. (2003) ‘ArAl Project: Arithmetic pathways towards favouring pre-algebraic thinking.’ Bologna, Italy: Pitagora Editrice.
- Warren, E., Mollinson, A., Oestrich, K. (2009) ‘Equivalence and Equations in Early Years Classrooms.’
*Australian Primary Mathematics Classroom,*Vol. 14, no. 1, pp. 10-12