The strategies that teachers use make all the difference. They may enable learners to do mathematics enthusiastically, enjoying their achievements, or they may inhibit learning, giving the impression that mathematics is about ‘getting right answers’ following rules set up by ‘authorities’. There are many effective strategies for facilitating the many varied aspects of rich learning environments.
As explained in paragraph 242 of the Cockcroft report 'Mathematics Counts' (HMSO 1982), it is not possible or desirable to indicate a definitive style of teaching mathematics.' See Effective Mathematics Teaching
But a good teacher will select appropriately from the range of teaching strategies that have been found to be most likely to facilitate learning, and will avoid strategies that are less effective or that may even discourage learning.
Mathematical activity may be initiated by demonstrating, explaining, or presenting a model, such as an image or diagram, or by a combination of these. While doing this, the teacher can use strategies such as starting in silence, or presenting an example and inviting learners to tackle their own variations. See PresentingTasks. It is important to use a variety of ways of initiating activity rather than usually adopting the same strategy, however effective.
Teaching strategies should encourage learners to understand that the source of the authority for truth and correctness in mathematics is not the authors of text books or the teacher, but is reasoning and logic; the authority lies in the structure of mathematics itself. Therefore teachers' strategies should encourage learners to justify conclusions by, for example checking calculations by working backwards; teaching strategies should not encourage learners to rely on the teacher or a book as a check for 'correctness'. See Authority in mathematics
There are strategies for initiating and sustaining whole class discussion. Whole class discussion is not easy. It is usually most successful when the teacher builds up to it by, for example first inviting learners to direct actions in an open situation presented on an interactive whiteboard. Whole class discussion will be more successful in an atmosphere in which it is understood that everything anyone says is an idea that may be adjusted or changed, than in a classroom in which learners knock down ideas or try to overcome hurdles. How the teacher responds when learners disagree is vital. For example 'that's interesting, how else could we look at it?' is more likely to encourage discussion than 'No, that¹s wrong!'. See Conjecturing Atmosphere
Whole class discussion is also difficult, if not impossible, when asking individual learners to explain their answers always leads to a 'ping-pong' closed interchange between the teacher and an individual learner. To encourage discussion the teacher needs to use the 'volleyball' approach in which a response from one learner is 'opened up' for all learners to think about. See Volleyball NOT Ping Pong. An effective strategy is to respond to a learner's idea by inviting conjectures, examples, counter-examples, then pulling back and letting the learners' momentum carry them along, possibly occasionally making suggestions. See Being Alongside.
Good teachers will frequently use open questions to which there is more than one possible answer, so that learners have opportunities to formulate their own ideas rather than trying to 'guess what's in the teacher's mind'. But, in classrooms in which learners are encouraged to generalise whenever possible, 'closed' questions that have a right answer can also be opportunities for exploration. In these classrooms closed questions will be treated, for example, as invitations to consider other similar questions, or to look for other questions with the same answer . See Open & Closed Questions and Question and Answer Interactions.
In effective teaching, the teacher's questions are NOT mostly questions just involving recalling and applying facts. The teacher frequently asks higher order or more demanding questions that encourage pupils to explain, analyse and synthesise. They also sometimes ask questions that surprise learners, while enquiring about how they are thinking, such as 'what question am I going to ask you now?'. See Metacognitive Questioning.
Whatever kind of question it is, research has shown that in mathematics classrooms the wait time for a response is usually less than 3 seconds. Increasing the wait time to between 3 and 5 seconds can greatly improve results. See Increasing teacher wait time.
Effective teachers intervene in the independent work of an individual or group to prompt their thinking, rather than to tell them what to do. Therefore an important strategy is to free up time to spend listening to learners as they discuss and write. See Teaching by Listening. An effective teacher, roaming the classroom while learners work, knows when to 'step in' and when to 'step out'. She observes, sometimes responds and sometimes intervenes, perhaps asking a prompting question to help the learners get going, keep going or extend the scope of what they are doing. For example, 'What information do you have?', 'Would it help to draw a diagram?', 'What could you change?'. See Roaming While They¹re Working. To encourage self-reliance teachers use increasingly indirect prompts so that learners begin to prompt themselves, spontaneously. See Directed-Prompted-Spontaneous, and ScaffoldingFading.
Good teachers use a variety of strategies to enable learners to review the learning points in a lesson or sequence of lessons. For example learners might pair up and discuss what they think the lesson was about, while the teacher 'roams' and picks up particular points for general discussion.
It takes time to develop effective teaching strategies. But it can be inspirational to read and consider detailed reports of lessons, study filmed teacher-learner interactions, or observe or share the teaching in other teachers' lessons.
What do you do to encourage a conjecturing atmosphere in which:
- what learners say is always taken to be a conjecture;
- those who are sure wait so as to allow those who are unsure to try to express themselves;
- if what is said is in need of modification, it is not 'wrong'.
Do your learners have opportunities to try to express themselves and to listen to others expressing themselves?
Do learners have time to reflect upon their activity and the things others say and do?
How do your learners justify mathematical facts and conventions?
Do you have a standard way of presenting tasks? What are its strengths and weaknesses?
How often do you manage whole class discussions? What happens when you do?
Are you a 'volleyball' or a 'ping pong' questioner?
Do the questions you ask provoke mathematical thinking?
Do you ask learners questions that have many answers?
When might it be appropriate to ask 'closed' questions¹?
Do you ask questions that help learners to become aware of general mathematical strategies?
Do your learners sometimes respond to your questions as if it were a game of 'guess what's in my mind'? If so, what do you do?
How much time do you spend listening to learners in a lesson?
A teacher had been struggling for weeks to get whole class discussions going in her Year 7 group. Usually the 'discussions' were unsuccessful because they consisted of a series of dialogues between the teacher and individual pupils, during which other pupils became bored and restless.
This teacher¹s first good whole class discussion was generated when she set her pupils the 'Bag of flour' task described by Alan Bell in 'Diagnostic teaching: 3 Provoking Discussion' in Mathematics Teaching 118. Her aim was to generate conflict discussion and resolution, as pupils tried in small groups to reach conclusions with which all four members of the group agreed.
Each pupil was given a copy of the same sheet on which there was a picture of a 1.5 kg bag of flour with a price tag showing 42p. There were also given the following information. 'With this bag of flour Grandma can make fourteen large cakes or one hundred and twenty-six small cakes. The large cakes were sold for £1.25 and the small cakes for 15 pence each. Here are some of the calculations that Grandma did.' Seven different calculations were then listed, each consisting of two numbers combined either by multiplication or division, such as 42 x 5, 42 ÷ 1.5, 1.5 ÷ 42. The challenge for the pupils was to decide, for each calculation, what Grandma was trying to find out.
The pupils first worked individually for a short while, before pairing up with the aim of reaching agreement in their pairs. After a period of earnest pair discussion, each pair of pupils joined another pair to make a group of four pupils. They were challenged to reach a decision about each calculation with which everyone in the group agreed. The teacher explained that, where they did not at first all agree, they were to try to convince each other of the correctness of their views by explaining them to each other very carefully.
When the groups had got as far as they could in their efforts to reach genuine agreement, a representative of each group was invited to give the group's conclusions, describing and explaining exactly how the group had arrived at them. The teacher made it clear that, while a group was reporting, anyone in the class could ask a question, add to what was said or put an alternative view. In this way a very rich class discussion developed, with some pupils going to the board, writing and drawing things to help explain their reasoning and their own examples. As in the lesson that Alan Bell refers to in his writing, the calculation 42 ÷ 1.5 generated much discussion. Everyone was involved and keen to explain the thinking that they had been doing.
Bell A., 'Some experiments in diagnostic teaching', Educational Studies in Mathematics, vol. 24, no. 1, 1993, pp. 115-137.
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University of Nottingham, Standards Unit
Swan M. and Green M., Learning mathematics through discussion and reflection, 2002, Learning and Skills Development Agency.
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Wigley A., 'Models for mathematics teaching', in Bloomfield A. and Harries T. (eds), Teaching and learning mathematics, 1994, Association of Teachers of Mathematics, pp. 2225.
Anne Watson and John Mason (1998), Questions and Prompts for Mathematical Thinking, Association of Teachers of Mathematics
Tahta D, et al Geometric Images Association of Teachers of Mathematics