This document discusses teaching mathematical problem solving. It begins by defining what constitutes a problem versus a routine exercise, noting that problems are unfamiliar, unstructured, and complex. It then discusses how teaching problem solving requires going beyond memorization and standard techniques to focus on conceptual understanding. Several examples of complex, unfamiliar problems are provided. The document emphasizes the importance of supporting student engagement through scaffolding, modeling, and valuing explanation over just obtaining answers. Finally, it discusses the critical role of metacognition in problem solving, providing examples of metacognitive questioning techniques and heuristics students can use to monitor and regulate their problem solving activity.

1. Teaching for Problem Solving:
Teaching for (Mathematical) Problem
Solving: The Challenges and Some
Solutions
Stirling Secondary Mathematics Conference
Dr Jennie Golding, UCL IoE j.golding@ioe.ac.uk

2.  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

3. 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.

4.  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)

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

6. Maths teachers and spiders:
10 heads, 56 legs. Insects and spiders:
How many of each? 11 heads, 80 legs.
How many of each?

7. 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?

8. 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)

9. 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?

10. 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)

11. Activity by expert problem solvers = monitoring activity
(Schoenfeld 1987)
Read
Analyse
Explore
Plan
Implement
Verify
Elapsed time (minutes)

12. Activity by novice problem solvers = monitoring activity
Read
Analyse
Explore
Plan
Implement
Verify
Elapsed time (minutes)

13. 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

14. 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)

15. 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

16. 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

17. 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

18. 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