Problems with standard algorithms
As older children are introduced to standard algorithms as in long division, multipli-
cation and decomposition methods, more confusion can arise since they sometimes
forget the procedures and have no strategy to fall back on. Children often abandon
effective mental methods to do a calculation in the standard written way. The standard
algorithm has been under attack and is accused of making students dependent and
cognitively passive (Zarzycki, 2001). Zarzycki recommends that students create their
own algorithms and compare them to standard methods. He found that children do
not understand why the algorithms work and believes that the logic behind them
should be taught. Others would disagree. For example, Merttens and Brown (1997)
advocate that children should practise algorithms so that they become automatic: chil-
dren then do not have to think about what they are doing, or why. Askew (2001) still
has questions about the standard algorithm and invites other responses. The debate
continues but the important point is that the standard algorithm is in question: it is
not an absolute mathematical teaching method for solving equations. This then puts
in doubt other standard procedures and the rightness of imposing them on children.
We know that standard algorithms are a recognised written mathematical language
that children can work towards, but there is no need to hurry children into these pro-
cedures to the detriment of their own mathematics and understanding.When do you teach sums?
Gifford (1997) raised the question of when to start teaching formal ‘sums’. Formal
approaches have from the early 1960s given way to a Piagetian perspective of delay-
ing any forms of teaching abstract mathematical concepts, until children are ‘ready’.
Practical mathematics became the main focus of many Early Years classes. There has
been and remains an abundance of commercially produced mathematical materials.
Sorting and sets and matching – in Piaget’s terms ‘logico-mathematics’ – created
classrooms full of plastic bits and structured apparatus. The use of what are known
in the USA as ‘manipulatives’ was an attempt to bridge the gap between children’s
informal knowledge and the formalism of standard calculation (see Barratta-Lorton,
1976). This practical approach is now questioned (see Chapter 7) and Hall’s review
emphasises that: ‘the cumulative evidence suggests that the value of manipulative
materials in mediating understanding is at best unclear and may indeed be adding
to the difficulties which children experience in making the transition from total
dependence on informal knowledge to the use of the formal notational system’
(Hall, 1998, cited in Maclellan, 2001, p. 75).
With the advent of the National Numeracy Strategy, teachers in England have now
moved away from concrete to mental mathematics in the Early Years, with less
emphasis on writing numerals or practical mathematics. The modelling of calcula-
tions by the teacher is encouraged. The teaching of mathematics in other countries
such as Hungary and Holland has heavily influenced this (see Chapter 1). The role
of ‘the sum’ and the question of when to introduce are still unclear, but encourag-
ing mental mathematics is a shift in the right direction, since we are helping chil-Bridging the gap between home and school mathematics 758657part 1b.qxd 04/07/2006 18:11 Page 75