Archive for May 4th, 2011
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. 395403
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 yearolds 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 prealgebraic 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. 1012