see a child make marks on the sandy ground outside, even in play.
Teaching in primary schools was of a very similar style, although funding permitted
children in some classes to use slates and older children to use exercise books. They
copied standard algorithms from the blackboard and filled in the answers. Alexander
includes a transcription of one spelling lesson (in a Hindi-speaking area) with children
aged 5–6 years: this is typical of what I saw in every nursery and primary school I visited:
four children come to the blackboard at the teacher’s invitation, to write ‘A’. Teacher
then writes ‘A’ herself and asks the class to recite the sound, over and over again.
Teacher writes, ana, Anamika, aachi(pomegranate; a girl’s name; good) on the board.
Three pupils come forward to circle the ‘A’ in these words. Class applauds. Teacher asks
questions to recapitulate and children chant in response. (Alexander, 2000, p. 282)Alexander reported that lessons he had observed in England in 1998 were as tied to
textbooks and published schemes as those in Russian schools. Teachers in France and
the USA were moving away from the domain of textbooks. Alexander emphasised
the tension in the USA between those schools wanting to move away from the dom-
inance of standard textbooks, and the concern of school boards and government to
raise standards. The change noted was that instead of using workbooks exclusively,
teachers created their own worksheets for children to record their mathematics set
by the teacher. This finding is mirrored by our study (see Chapter 5).Mathematics education in the Netherlands
In the Netherlands the main influence in mathematics education has been ‘Realistic
Mathematics Education’ (REM). This was initiated by Freudenthal who professed that
mathematics must be connected to society and children should learn mathematics by
a process of ‘progressive mathematization’ (Freudenthal, 1968). Treffers built on this
idea and describes two types of processes, horizontal and vertical (Treffers, 1978). ‘Hor-
izontal mathematization’ is Freudenthal’s term to explain the way in which the gap
between informal mathematics and formal mathematics is bridged. Horizontal math-
ematization helps children move from the world of real life into the world of symbols.
In teaching terms, for example, a picture of a real-life problem is given to the children,
perhaps people getting on and off a bus. This would later be shown with symbols and
then again, after a period of time, shown without any picture cues (Heuvel-Panhuizen,
2001). The term ‘vertical mathematization’ refers to the children working within the
world of symbols. Children move on to models such as the empty number line which
the National Numeracy Strategy in England has adopted (QCA, 1999).
Mental arithmetic is at the ‘heart of the curriculum’ in schools in the Netherlands
that use the REM curriculum (now the majority), (Buys, 2001). Children try to do
every calculation mentally and are also encouraged to write their thinking down on
scrap paper so that they remember the steps for more difficult calculations where it
is impossible sometimes, to keep track. Children’s own ways of thinking are encour-
aged through their mental work, and informal recordings through the empty
number line have a high priority.
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