A plan for a series of lessons, however detailed, cannot specify exactly what teachers and learners will do in the lessons. First, the teacher needs to know how learners have been introduced to the topic in earlier years, and to assess individuals’ prior learning. Teaching rigidly towards particular (possibly age-related) objectives may result in some learners being unable to grasp essential ideas because they build on understanding that the learners do not have, and other learners may be taken through steps that they have already understood. Most importantly, the teacher must be prepared to support learners in the lessons, through individual discussion and observation, identifying any difficulties and providing appropriate challenges. Therefore an effective, flexible, teacher cannot know in advance exactly what mathematics a lesson, or series of lessons, will contain.
Also, research shows that, in a lesson, learners often make progress in topics and techniques that are not part of the current topic. This is because a complex network of mathematical concepts and ideas feed into any topic, and any topic feeds into a further network of ideas. So a teacher cannot know in advance exactly what learning will result from a lesson or series of lessons. The teacher can only offer opportunities for people to learn from their experience, and cannot do the learning for them. For example, during a lesson focussing on the concept of area, a learner might make progress in using and understanding decimals and place value, while hardly altering their previously established ideas about area.
It is also a fact that a task as intended by the designer may not be the same as the task as imagined by the teacher, which again may not be the same as the task as presented to learners or as construed by them. Therefore the learning outcomes of a task may not be what was expected.
Further, it is unwise to assume that a pupil has learned some particular thing just because they have completed a task. Usually learners need, subsequently, to make sense of what was observed while they were doing the task. They need time to reflect on what they observed. Doing is not the same as construing.
Have a think about how you currently plan your mathematics lessons and what you might do to change your approach after reading the example in this section.
You might like to make some notes and possible changes to your current practice in your Personal Learning Space
You might be interested in reading the Mathemapedia article entitled The role of planning in teaching mathematics
. This discusses the idea that planning is a key part of a teacher’s role and that it takes place at different levels and in different forms. It suggests that effective planning is a skill that can be learnt and developed over time.