## Posts Tagged ‘Basic Algebra’

**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**:

- Mansell, W. (2006)
*Algebra at the age of four.*TES. [Online] [Accessed on 17th May 2011] http://www.tes.co.uk/article.aspx?storycode=2264060

**Original Article**:

- TES Online: Algebra at the age of four (Published in The TES on 21 July, 2006. Accessed on Tuesday, 17th May 2011)

**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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**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