Archive for May, 2011

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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“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. 18-22

The article gives a history of the mathematics landscape in the UK since 1837. Algebra has been part of this throughout.

We see algebra as a key tool to help solve problems now, but historically “problem solving was seen as a specialized skill, only for mathematicians in opening up new fields.”

Key points:

  • “A problem solver needs a rich, connected understanding of mathematics and the abilty to see patterns of similarity and association, as well as the skills to carry out the planned attack, and to check that the results make sense in the context of the problem.”
  • “A Royal Commision, reporting on the state of mathematics teaching in nine leading Public (i.e., private) Schools in 1837, noted that the typical two weekly hours of mathematics consisted of Arithmetic, a little manipulative Algebra, and “Euclid“, learned by rote.”

Reference:

  • Burkhardt, H., Bell, A. (2007) ‘Problem solving in the United Kingdom’ ZDM, Vol. 39, no. 5, pp. 395-403

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 3a + 3b stands for 3 apples and 3 bananas instead of a and b standing for any number. 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.”

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

The majority of the next few posts are going to be based on my findings from a variety of reading.

Part of the MaST course is to produce a couple of assignments, based on personal research, designed to meet the aims and objectives of the programme, reflecting school based activities, readings, synthesis of theory and practice, analysis of teaching and learning in mathematics and personal and collegiate professional development.

Assignment 1:

Select an area of mathematics or an issue in mathematics teaching that they have identified as an area for development and write a report of a mathematical enquiry (equivalent to 4500 words) that you have undertaken that critically explores content, concepts and relationships in the chosen area.

My chosen area is based around algebra and how children use it to solve problems.

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