## Posts Tagged ‘Kaput’

**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**:

- 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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