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Iceberg
When you see an iceberg, you only see the top. Most of the iceberg is hidden beneath the surface. However, this part is very important, in order to let the iceberg float.
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Teachers have a tendency to invest a lot in the teaching and training of the top of the iceberg (formal mathematics, sums) whereas most of the important insights and skills are developed before that (floating capacity). Especially for the lower achievers, one should take time to create a solid basis. Some children with special needs are for example not able to automatise formal sums upto 20, but they are able to work on a more basic level on fractions.

This idea arises from the Speciaal Rekenen (Special Arithmetic) project in the Netherlands which supports teachers' thinking about learning processes and the strategies used by children. The iceberg has proved to be a very powerful metaphor to suggest how children need a broad range of mathematical representations and experiences to make sense of the formal mathematical representations.
Pathways
In order to offer the teacher an oversight of important activities in the textbook series, in the Speciaal Rekenen project (Mathematics for children with special needs) we scanned student activities out of the textbook (recognizable) and placed them into the iceberg: you see the top and you see the arithmetic that is below the top (grade 2).
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Some of the activities are more important than others. They are marked by the red boxes. We call them crucial learning moments: at that moment certain skills should be evolved. If not, the student can get serious problems in the remaining part of the process.
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Conferences
(c) Freudenthal Institute US & Freudenthal Institute for Science and Mathematics Education 2013
FIUS: mathematics education, development and research