Reasoning and Problem Solving

When the new Australian curriculum goes live in 2015, teachers across the country are being asked to – among other things – get students thinking mathematically. Two of the four proficiency strands are Problem Solving and Reasoning, (the others being Understanding and Fluency).

There is a great deal of overlap between Problem Solving and Reasoning. After all, it would be hard to solve problems without reasoning about them and it would be hard to demonstrate reasoning ability without some sort of problem to get our teeth into.

At the foot of this post is a link to the helpful ACARA curriculum website and I have also pasted in the clear definitions they give there.

In England the new KS3 curriculum says students should:
• become fluent in the fundamentals of mathematics, including through varied and frequent practice with increasingly complex problems over time, so that pupils develop conceptual understanding and the ability to recall and apply knowledge rapidly and accurately
• reason mathematically by following a line of enquiry, conjecturing relationships and generalisations, and developing an argument, justification or proof using mathematical language
• 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

To help any teachers trying to incorporate these aims into their practice, here are some thoughts about Problem Solving and Reasoning, their difference and similarities, and what to do…

We can begin with problem-solving. People use certain skills, or develop certain habits, to solve problems. Take a jigsaw puzzle, for example. There are certain things that a skilled puzzler might do, such as: search for corners and edges, group pieces of the same colour, stand the box lid up to compare its picture to the one being formed, use both piece-shape and picture detail to match pieces, and so on.

However, it is possible to use some of these skills without reasoning all that much. So someone may ‘do the edges first’ because that’s the way they were taught to do it, but they would be unable to explain why it is better to start with the edges. So when pupils do exam questions, they may be solving problems quite effectively because they have encountered problems of that type before (like the puzzler who’s done lots of jigsaws) and ‘know the rules’.

Problem-solving skills are very important. An electrician uses them to identify a fault, logically testing until the cause of a problem emerges. A doctor uses them to diagnose a single condition from a set of symptoms and test results. And although I have given a very narrow picture of problem-solving above, with someone knowing the rules and being familiar with the problem, it is fair to say that really good problem-solvers attack unfamiliar problems effectively too. According to ACARA ‘Students formulate and solve problems when they use mathematics to represent unfamiliar or meaningful situations’.

So how does reasoning differ? Basically, it is more abstract. Whereas we can judge problem-solving according to outcome, to a large extent, (so electricians or doctors who fix things – or have good ways of fixing things – are good problem-solvers), someone could possibly demonstrate good reasoning without managing to find a solution. A scientist might observe a phenomenon and show that it needs explaining, but have no answer. He/she may observe ‘Sunrises look different to sunsets: sunsets have a warmer, redder light and are more likely to cast spectacular colours across the sky. This must be because of a difference in the air the light is passing through or a difference in the objects it strikes, as the sun’s rays themselves would not be affected by the movement of the Earth.’ There is no problem to solve here, as such. And the scientist may be wrong about the reasons too, or even the observation itself, but he or she is reasoning: noticing, asking why, and attempting to work out where explanations might be found. Think of that great reasoner Isaac Newton: when he saw the apple fall out of the tree and wondered why, it wasn’t because gravity was a problem!

In my opinion, though, there is no need to ponder at length over the difference between problem-solving and reasoning, as so much of it is the same – especially in practice. The one useful distinction for me is that reasoning ability is a far more open thing, and problem-solving might not be enough. Reasoning leans more towards independence and creativity.

Can teachers teach these kind of skills? Those that say No have a point when they say that a lesson on ‘problem-solving’ is unlikely to be effective. I agree with that, but you could say the same about things like discipline, imagination, initiative and conscience. None of these things are taught as topics in themselves but… and here is the key thing… children learn them from adults or peers and they learn them (if indeed they do!) the whole time. The adult can teach by being a role model or by guiding children through experiences in such a way that these skills, or virtues, are acquired almost imperceptibly.

And so I believe that I have, on occasion, taught both problem-solving and reasoning through enquiry. An enquiry is an attempt by a group of learners to explore a topic. The exact question or problem to be explored is not automatically provided by the teacher. The teacher is more likely to present a stimulus that is rich in possibility and controversy. The question the pupils will be trying to answer will emerge from their own reactions and reflections.

Here are three possible stimuli, or starting points, for enquiries:
1. The children make a number line on the floor with each number on a separate piece of paper. Questions that might arise are:
Where will the line stop? How do we know?
Is there anything before zero? How do we know?
Is there anything in between the numbers? What?
2. Give the children a 10×10 number square. Challenge the children to find patterns in it. The class vote for which patterns they find most interesting. Questions or observations arising from the class or teacher might be:
If you draw a straight line through any group of numbers can you always say what the pattern is? (e.g. if you draw a vertical line, you always add 10 to get the number on the next line; if you draw a horizontal line, you add 1 – a diagonal line… then what?)
3. Show the children a set of different shaped drinking glasses. See what emerges. If nothing, then prompt them by asking which glass is ‘the biggest’. Then ask if the biggest one will take the most water, and how we can find out.
The process of facilitating an enquiry mainly involves reducing the length and frequency of teacher interventions to the minimum, and – more importantly – withholding judgment of the content of the children’s discussion.

Teaching Maths – or anything else – through enquiry gives a teacher a chance to nurture reasoning in the children. It treats mathematics as a field of discovery – as it once was for earlier mathematicians in history and still is for those high-fliers at the cutting edge of maths today. It allows students to pursue their own thoughts rather than rely on the teacher always to present them with a problem, still less a ready-made solution.

This is not to say that they wander aimlessly; the mere fact that children have freedom would not guarantee useful learning. Nor is it to say that teaching styles that are not enquiry-driven are ineffective or wrong. It is more that if children never have freedom to follow their own curiosity, then learning – if it is taking place – is more limited in scope, and less likely to allow young minds to fully flourish.

For more on how to introduce Enquiry into your set of teaching skills, try my book The Numberverse: How Numbers Are Bursting Out Of Everything And Just Want To Have Fun.
http://www.amazon.co.uk/The-Philosophy-Foundation-Numberverse-everything/dp/1845908899

The ACARA definitions:
Problem Solving
Students develop the ability to make choices, interpret, formulate, model and investigate problem situations, and communicate solutions effectively. Students formulate and solve problems when they use mathematics to represent unfamiliar or meaningful situations, when they design investigations and plan their approaches, when they apply their existing strategies to seek solutions, and when they verify that their answers are reasonable.
Reasoning
Students develop an increasingly sophisticated capacity for logical thought and actions, such as analysing, proving, evaluating, explaining, inferring, justifying and generalising. Students are reasoning mathematically when they explain their thinking, when they deduce and justify strategies used and conclusions reached, when they adapt the known to the unknown, when they transfer learning from one context to another, when they prove that something is true or false and when they compare and contrast related ideas and explain their choices.