In mathematics we often present work as simple to hard problems with the goal being for learners to arrive at specific solutions. However, perseverance can be a better indicator of problem-solving success than a solution.

Some problems, like the famous Four colour problem, took several hundred years to solve. It took perseverance to solve it and there are still many more problems just like it that are going to take cooperation and perseverance to solve. Learning to grapple effectively with problems has always been a vital skill for learners, and that’s not looking like it will change any time soon.

When George Poyla wrote his foundational text “How to solve it” in 1945, he said that problem-solving has four basic steps:

  1. Define the problem. Diagnose the situation so that your focus is on the problem, not just its symptoms.
  2. Generate alternative solutions.
  3. Evaluate and select an alternative.
  4. Implement and follow up on the solution.

It has been over 80 years since he wrote that book, and it’s still in print because we’re still finding new applications for his basic problem-solving process, one of which is problem-based-learning.

When I’m working with schools on raising math and science outcomes, problem-based learning is one of my go-to techniques because it creates such a strong bedrock for learning and because it scaffolds with other techniques so well.

What is problem-based learning?

On a fundamental level, we encounter problems as the endless stream of challenges that make up daily life. Problems present themselves as simple “Will I be on time for my train?”-style challenges to more demanding ones like “How do we tackle plastic pollution in 21st century” or “What will the impact of global warming be and how can we combat it?”. We do not even know some of the problems the Mars Exploration Team will face one day, but the fact that they will is definite and their survival will depend on their ability to do so!

Developing a positive mindset towards solving problems helps in all aspects of life. I would much rather have a student respond “I have not solved this problem… yet,” to all other options.

Problem-based learning involves tasking students with applying a problem-solving approach to challenges they may not have encountered or solved before. The problem is not always “new” (since the teacher may well have encountered it before), but there are many useful old problems.

When solving a problem, a student, or group of students collaborating, critically use the knowledge they have collectively acquired to develop a solution to the problem, seeking and developing new knowledge as needed.

The process of collaborative problem-solving develops skills that are critical for both their academic success and their success in the wider world, such as creativity, critical thinking, collaboration and various communication skills.

An example would be that I like to tease my older students who think they know what “place-value” is all about by asking them to write down the number “eleven thousand, eleven hundred and eleven”. Try it!

Everyone attempts this willingly and answers vary wildly including 11,000, 1100, 11. The extra learning comes when all eventually agree that the answer is about the way we read and say numbers.

When solving a problem, it is quite normal for students to encounter a place where there is a mismatch or lack of knowledge and understanding. Extra learning then happens as ideas are shared – group knowledge is always more than what is possessed by individuals.

Getting started in your own classroom

Professor Jo Boaler of Stanford University and others have created a wonderful resource around problem-based learning, which you can access at The resources contain advanced learning and teaching aspects with courses for both teachers and students. The website is collecting a variety of interesting problems such as “The Painted Cube”, which are all nicely exemplified in video.

All achievement objectives in mathematics in the New Zealand Curriculum are set in the frame of “solving problems”, which sets up mathematics as a very special learning area. All problems have more than one solution and I will end this article with a statement George Polya famously stated:

“It is better to solve one problem five ways that five problems one way.”

Jim Hogan is an accredited facilitator specialising in mathematics and statistics with Team Solutions and has extensive experience both as a teacher and an educational researcher.


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