Teaching Mathematics Ch 5: Learning mathematics

  • Piaget argues that learning consists primarily in being faced with experiences that do not fit into our current understanding, meaning that such understanding needs to be adjusted (constructivism).
    • By implication, the teacher’s role is to discover pupils’ current levels of understanding, and offer learning experiences that challenge that understanding.
    • This will encourage pupils to develop new theories, and the teacher’s role is then to prompt pupils to test and refine those theories to reduce errors.
  • Social constructivism (e.g. Vygotsky) takes this theory further, arguing that effective learning can only take place in a social context.
    • By implication, mathematical discussion is much more important than transmission of knowledge from the teacher to the pupils.
    • The teacher’s role is to provide the scaffolding on which pupils construct their learning, by providing an appropriate level of challenge to each pupil.
  • Drawing this together, the teacher should: make the purpose of the lesson clear; build non-threatening relationships and environments; select challenging activities; intervene to help pupils develop their understanding of key ideas.
  • Possible lesson format based on constructivist approach:

Presentation – Exploration – Reflection – Consolidation

  • Another implication of the social constructivist approach is the value of group work, and particularly posing problems that generate disagreement and thus discussion amongst pupils.
  • Most misconceptions arise from over-generalisation of earlier mathematical learning.
    • Simply explaining that a method is wrong and showing the correct method often isn’t enough to address the misconception.
    • Instead the teacher needs to create uncertainty in the pupils through cognitive conflict, and be brave in doing so!
    • This can be achieved through designing activities that will encourage misconceptions to come to light by pupils getting two contradictory answers.
  • Swan discusses this model in five steps:
  1. Assess pupils’ initial understanding

  2. Pupils complete a task that exposes misconceptions and causes cognitive conflict.

  3. Pupils share their methods and solutions through discussion.

  4. The teacher organises whole-class discussion to resolve conflict.

  5. Pupils consolidate learning through applying it to new problems.

  • Activities that suit this type of teaching include sorting activities, and always, sometimes, never, classifying into two-way tables, pupils making up questions, pupils marking answers to questions.
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