Problem Solving

Problem-solving has been a major topic in mathematics education since the publication of the seminal booklet “How to solve it” by George Pòlya (1945).

He proposed the following steps when solving a mathematical problem:

• First, you have to understand the problem.
• After understanding, make a plan.
• Carry out the plan.
• Look back on your work. How could it be better?

If this technique fails, Pólya advises: “If you cannot solve the proposed problem, try to solve first some related problem. Could you imagine a more accessible related problem?”

The question now is how applicable this is for a numeracy class in adult education. The problems to solve are not the textbook school mathematics problems, but problems from real life.

• How to use the ideas behind (Polya’s) problem-solving to improve the quality of numerate behaviour of the learners?
• Problem-solving requires a disposition of resilience. Learners have got the idea for earlier educational experiences that you have yto work towards one answer, which is right or wrong. And that there is one way to reach such an answer. However, nor in mathematical problem solving, nor in daily problem solving this is true. In most cases you have to try different approaches, reflect on the process and change your approach when necessary.
• Problems brought into the lesson by learners are more effective than problems from a textbook constructed by educational authors. It helps when learners can relate to the problem.

PD – Activity 1

Collect from the participants examples from their own life in which they use numeracy to solve a problem. Describe the situation and the (numerate) action taken.

Try to extract which specific skills are being used in these situations.

Refer this to the CENF and see if you can find match the skills to those mentioned in the CENF.

PD – Activity 2

Find the pieces in the  Assessment Frameworks for Cycle 2 of the Programme for the International Assessment of Adult Competencies, that refers to problem-solving. Try to illustrate those pieces with everyday situations.

Source:  OECD. (2021). The Assessment Frameworks for Cycle 2 of the Programme for the International Assessment of Adult Competencies. OECD. https://doi.org/10.1787/4bc2342d-en

I

We elaborate a bit on the first two principles of Polya’s approach

First principle: Understand the problem

Pólya taught teachers how to prompt each student with appropriate questions, depending on the situation, such as:

• What are you asked to find or show?
• Can you restate the problem in your own words?
• Can you think of a picture or a diagram that might help you understand the problem?
• Is there enough information to enable you to find a solution?
• Do you understand all the words used in stating the problem?
• Do you need to ask a question to get the answer?

Second principle: Devise a plan

Pólya mentions that there are many reasonable ways to solve problems. The skill at choosing an appropriate strategy is best learned by solving many problems. You will find choosing a strategy increasingly easy. A partial list of strategies is:

Pólya lays a big emphasis on the teachers’ behaviour. A teacher should support students with devising their own plan with a question method that goes from the most general questions to more particular questions, with the goal that the last step to having a plan is made by the student. He maintains that just showing students a plan, no matter how good it is, does not help them.

II

The problem-solving in real-life situations has been mentioned explicitly in the frameworks used for  the large scale assessments on literacy and  numeracy, like IALS, ALL, and PIAAC

This scheme is used to unravel the aspects of numerate behaviour.

In this definition the concept of managing a situation or solving a problem is broken down in meaningful and concrete details.

The following definition of numerate behaviour, was adopted for PIAAC Cycle 1:

“Numerate behaviour involves managing a situation or solving a problem in a real context, by responding to mathematical content/information/ideas represented in multiple ways.”

Gal, I., van Groenestijn, M., Manly, M., Schmitt, M. J., & Tout, D. (1999). Numeracy Framework for the International Adult Literacy and Lifeskills Survey (ALL).

Groenestijn, M. van. (2002). A Gateway to Numeracy: A Study of Numeracy in Adult Basic Education. University of Utrecht.

Hoogland , K. (2010). Realistic Numeracy problems: in Maths At Work – Mathematics in a Changing World; Proceedings of the 17th International Conference of Adults Learning Mathematics (ALM); Oslo, 28th – 30th June 2010, p 58

Hoogland, K., Diez-Palomar, J., & Maguire, T. (2019). Towards a second cycle of PIAAC. In B. Kelly, D. Kaye, G. Griffiths  Dalby, Diane, & J. Stacey (Eds.), Boundaries and Bridges: Adults learning mathematics in a fractured world. Proceedings of the 25th International Conference of Adults Learning Mathematics: A Research Forum (ALM) (pp. 67–68). UCL Institute of Education.

Madison, B. L., & Steen, L. A. (2003). Quantitative Literacy: Why Numeracy Matters for Schools and Colleges. National Council on Education and the Disciplines.

https://research.acer.edu.au/cgi/viewcontent.cgi?article=1033&context=transitions_misc

OECD. (2016). Skills Matter: Further Results from the Survey of Adult Skills, OECD Skills Studies, OECD. In OECD (Organisation for Economic Co-operation and Development). https://www.oecd.org/skills/skills-matter-9789264258051-en.htm

OECD. (2021). The Assessment Frameworks for Cycle 2 of the Programme for the International Assessment of Adult Competencies. OECD. https://doi.org/10.1787/4bc2342d-en

PIAAC Numeracy Expert Group. (2009). PIAAC Numeracy: A Conceptual Framework. In OECD Education Working Papers, No.35 (Issue 35). OECD. https://doi.org/10.1787/220337421165

http://www.maa.org/sites/default/files/pdf/QL/WhyNumeracyMatters.pdf

Pólya, G. (1945). How to solve it. Princeton Univerity Press.

Schoenfeld, A. H. (1992). Learning to think mathematically: Problem solving, metacognition, and sense making in mathematics (D. Grouws, Ed.; pp. 334–370). McMillan.

Tout, D., Coben, D., Geiger, V., Ginsburg, L., Hoogland, K., Maguire, T., Thomson, S., & Turner, R. (2017). Review of the PIAAC Numeracy Assessment Framework: Final Report. Australian Council for Educational Research (ACER).