Problem solving – the ultimate goal

Blog: Dr Helen Drury, Executive Director

Problem solving – the ultimate goal

Posted: 3/01/19

To teach mathematics for mastery is to teach with the highest expectations for every learner, so that their understanding is deepened, with the aim that they will be able to solve non-standard problems in unfamiliar contexts.

In my first book, Mastering Mathematics: Teaching to transform achievement, I described mastery – the ultimate aim of teaching and learning mathematics – like this:

“A mathematical concept or skill has been mastered when, through exploration, clarification, practice and application over time, a person can represent it in multiple ways, has the mathematical language to be able to communicate related ideas, and can think mathematically with the concept so that they can independently apply it to a totally new problem in an unfamiliar situation.”

Problem solving is the purpose of teaching for mastery. 

Problem solving is at the heart of mastering mathematics. Teaching for mastery involves holding problem solving as the ultimate aim of learning mathematics for every student, whatever their home background or prior attainment. Every student can learn to solve complex problems in unfamiliar contexts.

In addition to ensuring that students become fluent in the fundamentals of mathematics and reason mathematically, the National Curriculum for mathematics in England aims to ensure that all students: “can solve problems by applying their mathematics to a variety of routine and non-routine problems with increasing sophistication, including breaking down problems into a series of simpler steps and persevering in seeking solutions.” [1]

According to these aims, then, problem solving means all students:

  • applying the concepts and skills they have learnt to problems;
  • learning to tackle both routine and non-routine problems;
  • independently breaking down problems into a series of simpler steps;
  • persevering in seeking solutions.

Setting aside the strong case to be made that once a problem becomes ‘routine’ it ceases to be a problem at all, this is a list that many could agree with. 

Problem solving in Singapore

In Singapore, problem solving is the focus of the mathematics curriculum. Their national curriculum documents clearly state that:

“The learning of mathematics should focus on understanding, not just recall of facts or reproduction of procedures. Understanding is necessary for deep learning and mastery. Only with understanding can students be able to reason mathematically and apply mathematics to solve a range of problems. After all, problem solving is the focus of the mathematics curriculum.”[2] 

They neatly summarise this in their first principle of mathematics teaching:

“Teaching is for learning;
learning is for understanding;
understanding is for reasoning and applying and,
ultimately problem solving.”[3]

It is interesting to note that holding problem solving up as the ultimate aim of mathematics education does not inevitably equate to teaching by discovery, or inquiry-based learning. Naturally, experience of solving problems is vital for success in problem solving, but Singapore’s commitment to problem solving does not result in problem solving taking centre stage every moment of every lesson. We can distinguish, then, between the ultimate goal – problem solving – and the classroom pedagogy that will best achieve that goal.

This blog is based on an extract from Chapter 1 of ‘How to teach mathematics for mastery.

[1] Department for Education (2013) Mathematics Programme of Study. Quote is on page 2.

[2] Ministry of Education (2012) Mathematics Syllabus Secondary One to Four, Ministry of Education, Singapore. The quote is taken from page 21.

[3] Ministry of Education (2012) Mathematics Syllabus Secondary One to Four, Ministry of Education, Singapore. The quote is taken from page 21.

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