Welcome PGCE Core 2 Using and Applying Mathematics

Is it magic? Roll a die and make a note of the number Double the number Add three Multiply by 5 Roll again and make note of the number Add your new number Add 6 Multiply by 10 Roll again Add to the total Tell me the final answer

No it is mathematics! Roll the die and write down the numbern Double the number2n Add 32n + 3 Multiply by 510n + 15 Roll again and write down the numberm Add your new number10n m Add 610n + m + 21 Multiply by 10100n + 10m Roll againp Add to the total100n + 10m + p +210 The numbers rolled weren, m, and p

Classifying Problems - 1 As thinking skills (National Curriculum, 1999:p.22) Information processing skills Reasoning skills Enquiry skills Creative thinking skills Evaluation skills

Classifying Problems - 2 Key features Orton and Frobisher( 1996) Insights into Teaching Mathematics A mathematics problem can be said to be a situation in which an individual student a)Recognizes or believes that there exists a mathematical goal to be achieved b)Accepts the challenge to perform some mathematical task in order to reach the goal c)Has no readily recallable mathematical procedure available to enable the goal to be attained directly ‘It would appear that the distinction between a problem and an investigation is the existence of the clear goal specified in the statement of the activity’ (p.32)

Pupils need to be introduced to a wide range of problem solving situations Standard problems: stories requiring translation into mathematical operations Non-standard problems: situations for which the problem solver will not have a set routine or procedure for solving the problem Real world problems: situations where the pupil will have to select the relevant information and set up a mathematical model that simplifies the situation Puzzles: Problems depending upon,luck, insight or unusual strategies Askew, M and Wiliam, D. (1995) Recent research in mathematical education, London, HMSO

Standard problems Start unknown The initial quantity has to be found Mrs Green put some apples into bags. She ended up with 6 bags of apples, each containing five apples. How many apples did she start with? ? ÷ 5 = 6, possibly calculated as 5 x 6 = ? Change unknown The second quantity has to be found. Jo baked 20 currant buns. Someone sneaked down in the night and ate some. The next morning Jo found there were only 15 in the tine. How many were eaten? 20 - ? = 15, possibly calculated as 15 + ? = 20 Result unknown The final quantity has to be found Jo baked 12 lemon buns and 14 cherry buns. How many buns did she bake altogether? = ?

Problems and investigations Convergent Divergent A range of structures/models to support progress 82/11

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Problem solving models Scientific Process (Dewey)Classroom Model (Polya)Creative Process (Wallas)  Define the problem  Suggest possible solutions and identify alternatives  Reason about the solutions and implement  Test and prove  Understand the problem  Devise a plan  Carry out plan  Look back  Problem formulation and information gathering  Incubation – allowing the unconscious to work on it  Illumination – working to gain insight  Verification – testing for accuracy 12JMcC 2/10

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JMcC 2/1114 Read Understand Choose Solve Answer Check

Would you rather solve problems on your own or with others? Does it depend on anything?

Collaboration in mathematics promotes opportunities for;  discussion  justification of ideas  justification of the answer  communication of ideas  reflecting on learning  …..

Anthony, G. and Walshaw, M. (2009) ‘Effective Pedagogy in Mathematics.’. Educational Practices Series 19 On your Own  distraction  time to think  conflicting perspectives of others With Others  support  clarifies  forward thinking  engagement  exchange  argument

DCFS (2008) ‘Effective Teacher-Pupil Dialogue in Key Stages 2 and 3 Mathematics’. Full document available at: 08FR (E) 08FR Initiate (teacher) Response (pupil) Feedback (teacher).  Focusing attention on mathematics rather than ‘getting the answer right’  Working collaboratively with pupils  Transformative listening  Scaffolding

Today’s schedule 9.00 – – – Lead lecture HJB Gp 3 A 207 B 304 C HJB Gp 3 B 207 C 304 A Lunch HJB Gp 3 C 207 A 304 B Independent study

Independent Study Task Haylock, D and Thangata, F (2007) Key Concepts in Teaching Primary Mathematics London: Sage DCFS (2008) ‘Effective Teacher-Pupil Dialogue in Key Stages 2 and 3 Mathematics’. Full document available at: e6/DCSF-EPPI-02-08FR (E) e6/DCSF-EPPI-02-08FR DCFS (2006) Guidance Paper – Using and Applying Mathematics. Available at: paper-using-and-applying-mathematics-0 (E) paper-using-and-applying-mathematics-0