ability in results with different levels of effort can lead to discussions about how learning mathematics depends on effort and practice and that everyone can get better at it if they practice and try hard. Effort creates competencies that are the building blocks for the next steps in the learning path for numbers, relations, and operations.
The teaching-learning path described in this chapter shows how young children learn, integrate, and extend their knowledge about cardinality, the number word list, 1-to-1 counting correspondences, and written number symbols in successive steps from age 2 to 7. Much of this knowledge requires specific cultural knowledge—for example, the number word list in English, counting, matching, vocabulary about relations and operations. Children require extensive, repeated experiences with small numbers and then similar experiences with larger and larger numbers. Counting must become very fluent, so that it can become a mental representational tool for problem solving. As we have shown, even young children can have experiences in the teaching-learning path that support later algebraic learning. To move through the steps in the teaching-learning path, children require teaching and interaction in the context of explicit, real-world problems with feedback and opportunities for reflection provided. They also require accessible situations in which they can practice (consolidate), deepen, and extend their learning and their own.
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