Archive for May 15th, 2011
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 & JohnstonWilder, 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: “Picturepattern 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. 5794
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 JohnstonWilder, 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 learnergenerated examples. Mahwah: Lawrence Erlbaum Associates.
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Key points:
 “Recent research on the status of student knowledge based in the traditional arithmeticthenalgebra 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. 518
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. 4966. 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. 1532. 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. 390419. New York: Macmillan.
 MacGregor, M., and Stacey, K. (1997) ‘Students’ understanding of algebraic notation: 1115′ Educational Studies in Mathematics, Vol. 33, No. 1, pp. 119.
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“This paper describes a lesson taught to Judy’s Y2 class in a rural Hertfordshire primary school. It shows how meaningful mathematics can be made accessible to young children and, importantly, how fundamental issues of variant and invariant properties might be developed. An important component of this understanding is the position to term relationship – the relationship (or rule) between the position, n, of a number in a sequence and the number itself. In short, it is our view that young children, when given appropriate opportunities, can operate at levels substantially higher than the curriculum would indicate was likely.”
The teacher in this case made an animal, a dog, using linking cubes. The general shape was expanded upon to make two larger animals in a ‘family’ of three dogs.
“We discussed the dogs we had made. My questions and prompts were intended to alert them to an awareness of those elements of the dogs that remained constant (invariant) and those that changed. In respect of those elements that changed the intention was to encourage their understanding as to the systematic, and therefore predictable, nature of that change.”
The key to the children’s understanding seems to be labelling each of the parts of the dog, head, legs, shoulders, body and bottom. Once they could apply these labels to each of the three dogs, they could see how the animal was made up. They then used this knowledge to predict the make up of the fourth dog.
After this, they applied their knowledge to their own animals – working out the 4th and 10th members of the family. Some did this in tables, some did this in sentences, but they all could do it regardless of their mathematical abilities. Some were making generalised statements linking the term in the sequence to the body parts. All good stuff and remarkable when you consider that these children are in Year 2…
Reference:

Andrews, P., Sayers, J. (2003) ‘Algebraic Infants’ Mathematics Teaching, Vol. 182, pp. 1822