Reasoning & Problem-solving – The New Black? The New Brain Gym? Or Just The New Curriculum?

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.

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.

The above are excerpts from:
http://www.australiancurriculum.edu.au/mathematics/content-structure

What’s Philosophy Got To Do With It? Or… We’re Mates With Maths

When I told people that my work with philosophy for children had moved me into mathematics for children, and how we teach it, a lot of them were surprised. They are still surprised when I insist that philosophy and maths are closely related. For many, those two subjects would seem opposite ends of the spectrum: at one end is cold hard mathematics with its truths set in stone, and the other is philosophy, as vague and elusive as a puff of smoke. But this is to misunderstand them both.

Historically, mathematics and philosophy went hand-in-hand for centuries. The mathematician whose name is probably known to the most people, Pythagoras, was a Greek philosopher. And one of the most famous of all philosophers, Descartes, was also a mathematician.

These days, both disciplines have come so far that it would be asking a lot for even a genius to master both. So there probably won’t be another Descartes or Pythagoras, with a foot in both camps. But the two subjects are still linked. One reason why is that they both combine the mental abilities of logic and imagination.

As most of us know from school, we need logic to solve the mathematical problems we are set. When we say to ourselves, ‘The answer to this multiplication must be an even number, because we are multiplying two even numbers together’, that is pure, simple logic. But where does the imagination come in? Well, if you speak to mathematicians who work on the research side, trying to map out parts of the universe of numbers that have not been fully explored, they all maintain that imagination is essential. They mean the imagination to ask ‘What if…?’ and follow a train of thought onto new ground. Or the imagination to conceive of shapes and forms that lesser brains simply boggle at.

The same is true of philosophy. In philosophy, the illogical is not allowed. There are, of course, schools of philosophy which claim that the illogical is a necessary part of thinking. But even this discussion over whether logic is the be-all-and-end-all still puts logic at the heart of things. And as for imagination, yes, the great philosophers have all been hypothetical thinkers, able to picture the world in a multitude of ways.

This is important for the study of maths. Every single school child studies it. To get the maximum from each one, we need to make sure that the imaginative, curious children have the chance to explore maths in that way. Yes, they too need to be drilled and practised on calculation methods. But they need more if they are going to take maths to their hearts.

In the UK, and the US too, taking maths to your heart is quite a comical notion for many. But that’s not the case elsewhere. Many children in East Asia, for example (China, Japan, Korea) claim to love maths. There is no social stigma to liking it, or being good at it. And that is a fact about the whole society, much more than its teaching methods.

Until we in the West take curiosity, imagination, wonder and mystery as essential parts of mathematics, we will lag behind.

We can do this by seeking out questions in maths that are difficult and starting to think about them. And the difficult questions are not only in the difficult areas of maths. Just asking yourself if zero is a number, and trying to prove your answer is enough.

Here are some others to get you started:

What happens if you divide a number by zero?
Does Pi go on for ever? How do we know?
Are there more fractions or more integers (whole numbers)?
How many shapes are there?
How many lines of symmetry does a circle have?

Some of these big questions we can answer. Some we can’t. Some no-one ever will but we don’t know which ones those are.

Philosophy is the missing link that makes maths meaningful, which is why philosophers are very good at sniffing out questions like this. But do you know what? They are not as good as children. Given the chance, children will bamboozle you with queries that strike right at the heart of what numbers actually are. And you won’t be able to answer them sometimes. And that’s great, because it keeps the flame of curiosity alive – for them and you.

**Check out my book The Numberverse: How Numbers Are Bursting Out Of Everything And Just Want To Have Fun.
You can ‘Look Inside’ on Amazon here:
http://www.amazon.co.uk/The-Philosophy-Foundation-Numberverse-everything/dp/1845908899

You can also buy it here:
http://www.philosophy-foundation.org/resources/philosophy-foundation-publications/the-numberverse

When Numbers Won’t Behave

When numbers won’t behave

Some people like Maths because it feels like a land of certainty. Where other subjects – indeed life in general – teem with doubts and contradictions, numbers are cool, hard and permanent. You know where you are with them.

But is this how numbers really are, or just how we want them to be?

Once we know that two plus two equals four it can be quite comforting. We feel confident that there will never be a time that teachers, the government, our parents or anyone else turn out to have misled us about this simple fact. George Orwell, in 1984, floated the idea that ‘freedom is the freedom to say that two plus two make four’. So for him, or the protagonist of his novel at least, it was the perfect example of objective truth.

That may well be so, and certainly I myself can’t think of a better example. But despite that, students of maths can find what they are being told vague, or illogical. And some of the most confusing moments come with the most basic concepts.

I once got into a discussion with a class about whether 7 was a number or a digit. One child said ‘both’. We then started to discuss whether a digit and a number are the same thing. Some children said ‘yes’, some ‘no’. Their teacher sighed and crinkled her brow, having gone to some pains to teach them recently that a digit and a number are not the same thing.

Except that sometimes they sort of are…

Think of it like this:
51 is a number made of two digits.
5 is a number made of one digit

In the first example, 51 is a number and I have used two digits to write it. A computer working in binary, on the other hand, might communicate it as 110011, using six digits to express the same number. So it is clear to me that one number can be expressed with different digits.

I could, of course, communicate it another way altogether: ‘fifty-one’. That’s two words. Is it two digits? Maybe. What if I make up a single word to express the number… let’s say, ‘flimp’. If my new word caught on then people would be referring to the number 51 without using any digits. So this proves that numbers and digits are not the same thing. Doesn’t it?

But what about a single-digit number? The 5 in 51 was a digit, we said, but what about the 5 in the number 5? It’s a digit because it’s in a column where it means ‘5 units’. But isn’t it a number as well? And if it isn’t… what is a number?

Let’s widen this out a bit. Because ‘5’ (and I am talking about that bendy line on the page here) is also a numeral. A numeral is a mark, and can vary between languages. In Arabic, for example, 5 is written with a circle much like our sign for zero.

So 5 (European etc) and 0 (Arabic) are two different numerals, but they stand for the same digit. It helps to compare numbers with words here. The Arabic for ‘book’ is ‘kitab’, apparently. So ‘kitab’ and ‘book’ are two different words for one thing – a book.

The problem is that numbers aren’t quite like books. I have no doubt what both ‘book’ and ‘kitab’ stand for because there are some books on the shelf two metres away from me, and I could pick one up and drop it down on the table to illustrate. However, if I look at the numerals ‘5’ and ‘0’, there is nothing I can drop on the table to show what they refer to. So what do they refer to? If what they refer to is a number, how can I demonstrate what that number is without going back to the numerals all over again?

This may sound as if I am looking for a problem where there isn’t one – or at least not if you don’t go looking. But in my experience, people learning new concepts or procedures in maths come up against precisely these weird questions as they try to bend their brains around the new idea. This can happen when they are introduced to double-digit figures. Or decimal fractions.

Wittgenstein, a philosopher who was fascinated by the fundamentals of mathematics, wrote that as a philosopher he was trying to ‘show the fly the way out of the fly-bottle’. I always wondered what a fly bottle was but apparently it’s something they used to use to trap flies. Anyway, you can imagine the predicament of a fly inside a bottle. It buzzes around, ever more frantically, blocked by an invisible wall and unable to locate the exit.

That is a fitting metaphor for intellectual confusion. And you don’t have to be an intellectual to suffer from it. An ordinary 8-year-old can be unable to progress because a new concept is, basically, doing her head in. At that moment, numbers are doing things she thought they weren’t meant to do. They are behaving irrationally.

What can a teacher do? What a philosopher does. That is to:

• listen to the confusion
• try to understand the confusion rationally (i.e. how it stems naturally from the student’s current knowledge and beliefs)
• sympathise with the confusion (e.g. by saying ‘I see why you’re confused – that must be annoying’)
• consider why you the teacher believe what you believe instead, and try to explain it

Remember, we all have an upper limit of understanding in any subject, a point where the next stage doesn’t make sense to us. Reaching that point is anxious and frustrating. A guide who takes those feelings of ours seriously rather than brushing them aside is more likely to boost us up to that next level of understanding.

**Check out my book The Numberverse: How Numbers Are Bursting Out Of Everything And Just Want To Have Fun.
You can ‘Look Inside’ on Amazon here:
http://www.amazon.co.uk/The-Philosophy-Foundation-Numberverse-everything/dp/1845908899

You can also buy it here:
http://www.philosophy-foundation.org/resources/philosophy-foundation-publications/the-numberverse