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Selecting Teaching Strategies (at key stage 4)
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Selecting Teaching Strategies(at key stage 4)
Teaching strategies that lead to successful mathematical learning.
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.
Main Section
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, this nearly 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. At key stage 4 there are various strategies to use in the approach to the exam season. Assessment for Learning techniques based on exam questions can give learners the opportunity to take charge of their own revision, build confidence and explore mathematics further. Teachers need to prepare learners as thoroughly as possible for their exams and a variety of strategies are needed to avoid the pitfalls of boredom and repeating work that can already be done (see Speed teaching revision and Easy-Hard-General for ideas).
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.
Probes and promptsWhat 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?
When did you last try something new?
Taking actionChoose a question from the list above. Arrange a meeting with colleagues in which you will ‘brainstorm’ your responses to the question. Arrange to work with a colleague in the classroom, then discuss again your responses to the question.
Watch a colleague teach. Keep a record of strategies used in the lesson. Which could you try in your classroom?
Ask a colleague to observe you and keep a tally of questions asked. Look through afterwards and sort into open, closed, probing. What have you learned? Case studies
A teacher is looking for ways to make algebra more interesting with a year 10 class. The main objective is to ensure the learners have sufficient strategies in place to be able to solve quadratic equations of any type.
His first activity was a “pair share” type where each learner was given the same problem (x2 + 8x + 15 = 0 type) to solve, worked on it individually for 3 minutes and then shared results with a partner, then with the couple behind. As the learners worked the teacher was able to walk around the room listening and noting who was acting as mentor (so understood the problem and how to solve it) and who was in need of more support.
After some feedback and questioning, targeting different approaches to “getting an answer” the class was regrouped into fours, each group containing two stronger and two weaker students. Each group was given a different problem from a GCSE paper and asked to find a solution. The solution had to be presented clearly so that anyone else could follow and they were warned that one “random” person from the group would be asked to explain the problem and the solution to the rest of the class.
Ten minutes later the class was called to order and given five minutes to put together a final solution. During the reporting back time (with original questions on the board), the teacher ensured that a weaker learner from each group was asked to explain the solution and actively encouraged groups to question each other to probe for understanding.
The final activity was to put question and solution together on a sheet of A4 for display on the classroom walls so that at any point that a similar problem was encountered a learner could simply get up to look at the example. Research SourcesBell A., 'Some experiments in diagnostic teaching', Educational Studies in Mathematics, vol. 24, no. 1, 1993, pp. 115-137.
Swan M, Improving learning in mathematics: challenges and strategies,
University of Nottingham, Standards Unit
Swan M. and Green M., Learning mathematics through discussion and reflection, 2002, Learning and Skills Development Agency.
Muijs D. and Reynolds D., Effective teaching: evidence and practice, 2001, Paul Chapman Publications
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
Categories
Curriculum, Didactics, Pedagogy, Professional Development
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