Ofsted’s report ‘Mathematics: made to measure’ (2012) clearly showed that they believe that a stronger focus on problem solving is the key to improving mathematical ability. “Schools should increase the emphasis on problem solving across the mathematics curriculum.”
To what extent can a stronger focus on problem solving in Mathematics boost confidence and subsequent attainment in Year 11 students? Sinead McKeever, Teacher of Mathematics and KS4 Lead. Introduction I decided to examine the topic of problem solving in mathematics after I assumed responsibility for twenty-nine students in a Year 11 group that are designated as set 2 out of 4 sets for mathematics (Class A). From previous teaching experience, I had observed that Year 11 students seemed to lack the self-confidence required to tackle multistep contextual problems efficiently. The mathematics team identified a target of 80% at A*-A GCSE grades, and towards this my aim is to help ensure students develop higher order thinking skills and feel confident in speculation and creative processes. Research I carried out a review of learning theories and literature on problem solving in mathematics in order to decide what intervention to take. The constructivist learning view is that “we learn best when we actively construct our own understanding” and Piaget stated that intelligence was “knowing what to do when you don’t know what to do.” This was the important value that I wanted to teach the learners, who seemed reluctant to persevere when they struggled to see the starting point of a solution.
Teach Secondary magazine (Issue 4.1, page 54) stated “Mathematically high performing countries, such as New Zealand, Japan, Singapore and Finland, all have emphasis on problem solving in their maths curricula. Stigler and Hiebert in The Teaching Gap (1999) report that in these countries, problem solving is seen as an essential part of mathematics education; problem solving is used to actually drive learning, as opposed to merely testing it.” Polya (1945) provides a four-stage model of how to solve a mathematical problem, “understand the problem, create a plan, carry out the plan, and then look back at the problem.” This is expanded upon, with an inexhaustible list of problem solving strategies including ‘draw a diagram where possible, find a pattern, solve a simpler problem first, work backwards if you can, or act out the problem’. Again, these are all taken from Polya's ‘How to Solve It’ where he explains and gives many examples of various problem solving heuristics or general strategies. Intervention I concentrated on putting a strong focus on problem solving in Class A and I would later compare their GCSE results with their parallel set of equal ability (Class B). Before I began, I gave the class a questionnaire asking them questions about their confidence in mathematics and when solving problems; I would later compare this to their confidence levels at the end of year. My intervention strategy was split into two parts, implementing problem solving into my everyday mathematics lessons and planning lessons which solely focus on building problem solving skills. My day to day mathematics lessons would be planned so that I would embed the following practices: Minimise teacher talk
Effective open questioning Use an investigative approach to learning where possible Thinking time/discussion time Multistep problems with every topic Reinforce practices with algebra Promote independent learning including devising own problems Teach students to write up own findings and results
GCSE results comparison of class A and B
GCSE Results 25 20 15 10 5 0 B
My problem solving lessons would take place one lesson per fortnight. They would include open ended investigations which the students would do in differentiated groups where they are encouraged to discuss strategies openly. I would provide no help with the investigations, with the exception of open questioning. I picked six students from the class which covered the ability range and met with them once per half term in a focus group to discuss their progress and confidence with problem solving. Findings There was a huge improvement in the confidence of Class A. Data gathered from the questionnaire, focus group sessions and GCSE result analysis all confirmed that Class A were much more confident at attempting the problem solving questions on the GCSE paper.
My recommendations are: Continue both of the problem solving interventions with Year 11 students. Introduce problem solving lessons to Year 9 due to the changes of the content and questioning of the revised GCSE. References •
Pritchard, A. 2009, Ways of Learning. Learning Theories and Learning Styles in the Classroom. Routledge
Piaget, Jean (2001) The Psychology of Intelligence. London, Routledge
Ofsted (2012) ‘Mathematics: made to measure’
Teach Secondary magazine
Polya, George (1945) ‘How to Solve It’
20 15 10 5 0 B
A Class A
A* Class B
It is clear from GCSE results that the two intervention methods were very successful.
Target grade comparison of class A and B