What plans do you actually use? What is their form?

What plans to you devise, or help to devise?

What aspects of your plans are useful, and what aspects are not so useful?

Think about a lesson that you will be teaching soon.
Read all the plans that you have that are relevant to that lesson.
'Play the lesson through' in your mind.
If you think other reminders or prompts would help you immediately before or during the lesson, write them in the plan.
After the lesson, look again at your plans and any reminders and prompts that you wrote.  What was useful, and what was not?

The teachers in a department developed their long and medium term plans together.  At no time were the plans regarded as complete or finished. They were regarded as 'living' documents that evolved as teachers added to and modified them, sharing, reflecting on, and discussing their experiences and ideas.   

Teachers individually devised their own short term plans for lessons.  These plans were based on the medium term plans. Because short term plans were also shared and discussed, experienced teachers supported less experienced teachers, although more experienced teachers often changed their plans when new teachers suggested fresh ideas.

A teacher in this department was planning the first of a series of lessons for year 9 learners working towards level 7.  During these lessons the teacher wanted learners to build on their understanding of theoretical probability by estimating probabilities and comparing experimental and theoretical probabilities.

The teacher decided that during the first lesson she would find out as much as possible about the students' prior learning.  She would devise an activity in which learners had to calculate probabilities using equally likely outcomes.  She made a note to remind herself to set up the activity so that some probabilities would be easier to calculate by first finding the probability of an event not happening, then subtracting that probability from 1, than by trying to find the probability in one step.  She would look out for evidence that learners were doing this, and encourage them to explain and discuss different methods.

For the next task she would set up a puzzle that could be solved by carrying out an experiment, estimating probabilities, and comparing the estimates with theoretical probabilities. She thought that the learners were unlikely not to know that different outcomes may result from repeating an experiment, but she made a note to check that this knowledge was evident.

At the end of the lesson she would invite learners to talk about their findings, and describe their reasoning.  This would prepare learners for the next lesson. 

Having established, and made notes about, the structure of the lesson, the teacher set about planning each activity in detail.  The teacher remembered that, when they were developing the medium term plans, someone had mentioned dominoes.  The teacher found a set of dominoes. She set them out on a table, arranged and rearranged them.  She began to devise tasks, and eventually she felt ready to create her lesson plan.

She decided on this occasion not to begin the lesson with any kind of discussion about learning objectives.  She would try to generate interest by focussing straight away on dominoes. The teacher would explain that each domino in a complete double-6 set shows a pair of numbers taken from the set of integers from 0 to 6, all possible pairs are shown, and each pair is shown only once.  She would challenge the learners to work out mentally the number of dominoes in a complete set. Because some learners might just know the answer, she would stress that she was looking for explanations.  She would encourage learners to discuss with each other ways of working it out, before inviting them to justify their answers and explain their thinking.  This introduction might be extended by generating discussion about the total number of dots on subsets of a complete set of dominoes.  She wrote brief notes summarising this first part of the lesson. 

The teacher wrote a reminder that, for the rest of the lesson, learners would need sets of dominoes, and that she would display a picture of a complete set on the whiteboard.  The picture was the first item on her list of resources that she would need to prepare.

A learner would be invited to pick out one domino at random and state a fact about the numbers on it.  The teacher would challenge learners to work out the probability of that fact being a true fact about the numbers on any domino picked at random.  She would ask learners to explain their thinking, and encourage discussion.  She would then invite learners to state other facts about the numbers on the domino and to work out the probability of those facts being true.  This could be repeated by replacing the domino and inviting another learner to pick a different domino, for which different facts about the numbers would be true.  By repeating the procedure several times the probability of many different events would be discussed.  The teacher wrote a summary of this part of the lesson.  She also wrote a note to remind herself that, when generating discussion of the learners' methods, she would, if necessary, prompt discussion about first finding the probability of an event not happening, then subtracting that probability from 1.  She might, for example, ask learners how they would work out the probability that the total number of dots is greater than 2.

She made another note to remind herself to observe whether, and how, learners use the words 'outcome' and 'event' and the phrases 'possible outcomes', 'mutually exclusive' and 'theoretical probability'.  If necessary she would prompt discussion about their meanings. 

Learners, perhaps working in pairs, would now have the opportunity to choose their own domino, write true facts about the numbers on their domino, and work out the probability of each fact being true for any domino picked at random.  She wrote a line to remind herself that learners would do this.

After this the class would be divided up into small groups to attempt a task in which each group uses one set of dominoes.  In order to be able to introduce it clearly, the teacher wrote for herself a description of the task: "Each group of learners chooses a whole number between  0 and 6, and removes from their set all the dominoes with that number on it. They place the remaining dominoes in a container, and groups exchange containers.  The challenge for each group is to reach a conclusion, without taking all the dominoes out of the container and looking at them, about which dominoes have been removed from the set that they receive. They may take one domino at a time out of the container, recording the numbers on the dominoes before replacing them."

The teacher added containers to her list of necessary resources.

The teacher, anticipating that groups might need to be prompted, spent some time thinking about, and writing, prompts that might be helpful.  The teacher's list of prompts included 'If an event  is impossible, how often will it happen?', 'If, when you do an experiment, A happens much more often than B happens, what would you conclude about the probabilities of A and B?', 'Could you make use of the theoretical probabilities of some simple events?', 'Will the probability of an event that we discussed earlier still be the same now that some dominoes have been removed?'.

It was while thinking about prompts to get learners started, or to keep them going, that the teacher decided to conclude this lesson, when the groups had time to discuss their first thoughts, by generating a class discussion of possible ways of tackling the task.  This would be an opportunity for learners to reflect on what they had been doing earlier in the lesson, and on how they could use it in the present task.  They would also have time to think about ways forward before the next lesson.  The teacher wrote a line to remind herself to try to end the lesson in this way. 

Jones, K. and Smith, K. (1997), Student Teachers Learning to Plan Mathematics Lessons. Paper presented at the 1997 Annual Conference of the Association of Mathematics Education Teachers (AMET1997). Leicester. 15-17 May 1997.

Perks, P. and Prestage, S. (1994), Planning for Teaching. In B Jaworski and A Watson (eds), Mentoring in Mathematics Teaching. London: Falmer.

Ref: Mason, J. (1993) with Hewitt, D. & Brown, L. Ways of Seeing, in M. Selinger (Ed) Teaching Mathematics, Routledge & Kegan Paul, London.

Oliver Caviglioli & Ian Harris (2001), Mapwise: Accelerated learning through visible thinking, Network Educational Press

Robert Powell (1997),  Active Whole-Class Teaching, Robert Powell Publications

Griffin, P. & Gates, P. (1989). Project Mathematics UPDATE: PM753A,B,C,D, Preparing To Teach Angle, Equations, Ratio and Probability, Open University, Milton Keynes.

Mason, J. & Johnston-Wilder, S. (2006 2nd edition). Designing and Using Mathematical Tasks, Open University, Milton Keynes.

Mason, J. (2002). Researching Your own Practice: The Discipline of Noticing. London: RoutledgeFalmer

Perks, P. (2002). Progression in mathematics. In Haggarty, L. (Ed) Aspects of teaching secondary mathematics: perspectives on practice. London: RoutledgeFalmer

Mason, J. & Johnston-Wilder, S. (2004). Fundamental Constructs in Mathematics Education, RoutledgeFalmer, London.

Gardiner, A. 1992, Recurring Themes in School Mathematics: Part 1 direct and inverse operations Mathematics in School, 21(5) p5-7.

Gardiner, A. 1993a, Recurring Themes in School Mathematics: Part 2 Reasons and Reasoning, Mathematics in School, 23(1) p20-21.

Gardiner, A. 1993b, Recurring Themes in School Mathematics: Part 3 generalised arithmetic Mathematics in School, 22(2) p20-21.

Gardiner, A. 1993c, Recurring Themes in School Mathematics, Part 4 infinity, Mathematics in School, 22(4) p19-21.