Teaching for Problem Solving:

Teaching for (Mathematical) Problem Solving: The Challenges and Some SolutionsStirling Secondary Mathematics Conference

Dr Jennie Golding, UCL IoE j.golding@ioe.ac.uk

My background Content A conference such as this might be to inform, to

suggest tools or approaches for the classroom, and/or to recharge

We shall meet a variety of problems suitable for the secondary classroom

Ppt will be available from www.m-a.org.uk

What is a problem? Mathematical routines (skills and knowledge) are important. ‘word problems’ may be used to introduce concepts

by harnessing informal knowledge, or as worded contexts in which standard techniques should be selected and applied, or as a worded context in which there is no standard relationship or algorithm to apply

For me, an exercise becomes a problem when it is Unfamiliar and/or Unstructured and/or Complex, whether within or beyond mathematics

The solution of problems depends, among other things, on conceptual understanding as well as routines.

Typically exactly enough information is given, and the ‘problem’ has a unique solution.

Many students admit that memorisation, and practice of given standard techniques, are the most important skills they need to succeed in mathematics classrooms (Schoenfeld 1992).

PISA (OECD, 2004, 2013, 2014): in a genuine mathematical problem the content brings up the mathematical big ideas, the context might relate to authentic real-life situations ranging from personal to public and scientific situations, and the constructs are more complex than in traditional problems.

Example of a CUN (complex, unfamiliar and non-routine) task:

Several supermarkets currently advertise that they are the cheapest supermarket in town. Please collect information and find out which of the advertisements is correct. (Mevarech and Kramarski 2014)

Share this chocolate cake fairly between a) 4, b) 5, c) 29 maths teachers

Maths teachers and spiders:

10 heads, 56 legs. Insects and spiders:

How many of each? 11 heads, 80 legs.

How many of each?

Unstructured

Which odd numbers can be written as the sum of two primes?

What is the first even number > 2 which cannot be written as the sum of two primes?

The problem of meaning

An army bus holds 36 soldiers. If 1128 soldiers are being bused to their training site, how many buses are needed? (to stratified sample of 45,000 USA 15yo nationwide)

31 remainder 12 (29%) 31 (18%) 32 (23%) Incorrect computation (30%)(Schoenfeld 1987)

Unfamiliar and unstructured

Which has the greater perimeter, a square or the circle to which it is a tangent at a mid-point of a side and which

passes through two of its vertices?

Supporting engagement with mathematics at a high level:

Build on students’ prior knowledge Scaffolding Appropriate amount of time Teacher modelling of mathematical habits* Sustained valuing of explanation and meaning Teacher draws conceptual conclusion (Henningsen and

Stein 1997)

So teachers must know students well

* including organisation and heuristics for PS, regulation of execution, and evaluation (Stillman and Galbraith, 1998)

Activity by expert problem solvers = monitoring activity (Schoenfeld 1987)

Read                                         

                          

Analyse                                         

                                

Explore                                         

                                    

Plan                                         

                                          

Implemen

t

                                         

                                        

Verify                                         

Elapsed time (minutes)                                  

Activity by novice problem solvers = monitoring activity

Read                                         

                          

Analyse                                         

                                

Explore                                         

                                    

Plan                                         

                                          

Implement                                          

                                        

Verify                                         

Elapsed time (minutes)                                  

Ideally… Students using inappropriate strategies that lead to

incorrect or unreasonable answers should check their calculations for errors and, if none are found, consider a change of strategy

Students using inefficient strategies that do not lead to an answer at all should review their progress and choose an alternative approach

Students using appropriate strategies that, nevertheless, produce an incorrect answer should find and correct their errors

Students review correct solutions for completeness, efficiency (and elegance!)

But Worthy intentions will be foiled if students are unable to

recognise when they are stuck, have no alternative strategy available, cannot find their error (or cannot fix it if they do find it), or fail to recognise nonsensical answers

So students need to employ metacognition

What is metacognition, and what does it look like in maths education?

• A) Knowledge of person, task or strategy in relation to cognition (Flavell, 1979) or of one’s own cognition (Brown, 1987)

• B) Regulation of cognition: planning, monitoring, control and reflection. (cf. Piaget’s ‘reflexive abstraction’, Skemp’s ‘reflective intelligence’)

• We’ll concentrate on the latter.• Predicts school achievement in various academic areas and at a

variety of ages, even when ’intellectual ability’ is accounted for. • To some extent domain-specific, and in mathematics particularly

important in the solution of complex, unfamiliar and non-routine problems: successful mathematics learners are metacognitively active (Schoenfeld, 1992)

Metacognition in mathematics education: some evidence and heuristics

- Polya 1949/1957 (understand problem/devise plan/carry out plan/look back)

- Schoenfeld 1985/1989: elaborates Polya, and uses self-directed questions: what exactly are you doing?/why are you doing it?/how does it help you? (builds up flexibility and persistence)

- IMPROVE (Mevarech/Kramarski 1997 on, in a variety of domains): develops m/c self-directed questioning in parallel with (higher and lower level) cognition, with lots of teacher modelling and m/c questioning: Comprehension (what’s the qu about?)/Connection (what’s the same and what’s different from other problems and why?)/Strategic/Reflection (does the solution make sense, can it be solved differently, am I stuck?) Low on monitoring? Evidences improved socio-emotional outcomes as well as enhancing mathematics achievement. With ordinary teachers, age 4+ and range of prior attainment, and evidences lasting significant gains.

- Verschaffel 1999 small group + whole class, develops scaffolding- Singapore’s pentagonal framework for mathematical PS

What can teachers do?• Build genuine PS into their classrooms on a regular

and frequent basis: there’s a wealth of evidence it will not diminish learning of routine knowledge and skills, and will enhance deep conceptual understanding , flexibility, reasoning and affect and is also desirable in its own right.

• Teach domain-specific metacognitive protocols; teacher modelling is powerful.

• Cooperative learning methods can enhance metacognitive learning since they require students to articulate thinking, use mathematical language, work within ZPD, provide elaborated explanations, and be involved in conflict resolutions

• *Hattie: es 0.69

Sources of problems

http://allaboutmaths.aqa.org.uk/attachments/2050.pdfFMSP KS4 resources

www.suffolkmaths.co.uk/pages/Problem solving http://www.furthermaths.org.uk/gcse_problem_solving www.ukmt.org.uk www.primarymathschallenge.org.uk www.nrich.maths.org Problem Pages 11-14, 14-19 Challenge your pupils 1,2 (More) Creative Uses of Odd Moments

To distract your department from marking….. *Fold a rectangle a) along a diagonal or b) corner to diagonally opposite corner. Which of the two resultant pentagons (one concave and one convex) has a larger area?

*What is the area of the biggest triangle that can be inscribed in a circle of radius r? *What is the shaded area in this diagram?

3u2

2u2