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Putting 2 + 2 together

Developing confidence in mathematics

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WHAT IS CONFIDENCE AND WHERE DOES IT SPRING FROM?

 

Forget maths for a moment.  Think of an activity in which you have confidence in your ability to perform.  It may be playing the bassoon – public speaking – playing snooker – riding a bike - juggling – sketching cartoons –or whatever.  It is likely that your confidence springs from having performed previously and found yourself to be good at the activity in question.  You may have discovered a natural aptitude the first time you attempted a specific task – or, more likely, you had to put in a great deal of practice to perfect your skill.  The result, however, is that you believe in your capability to perform a particular task because you have done it before and achieved a successful outcome.  The greater the number of times you have achieved success the greater your confidence is likely to be.

 

Successful learners have often learned how to transfer this feeling of confidence across into other domains.  Initially, the transfer will be between similar domains.  “I have learned how to play chess to a reasonable standard – so there is no reason why I should not be capable of learning how to play bridge.”  Those who are confident in several domains may then transfer the feeling of confidence into dissimilar domains.  “I can play tennis well so I should be able to drive a car.”  Eventually, one’s confidence levels become so well developed that an individual will come to believe that there is nothing they cannot do if they set their mind to it.  It is all about mind-set.  Henry Ford is reputed to have said, “Whether you believe you can or whether you believe you can’t – you’re right,” and there is much truth in that aphorism.

 

 

 

Whether you believe you can or whether you believe you can't - you're right.

 

Henry Ford

 

Within days of him moving up into her class, Wayne’s primary school teacher noticed that he appeared to be having difficulty with addition.  Surprised by this discovery, since his previous teacher had rated him as “good at maths”, she observed Wayne closely as he performed the relevant operation.  She was intrigued to notice that Wayne started with the “hundreds column”, carrying across systematically to the right.  When she gently challenged Wayne on this, he was adamant that his previous teacher had taught him to do it that way.  “I remember what Mr. Smith told us,” said Wayne, then chanted confidently, “Start by the window then work your way across.”  Whereas this neat mnemonic may have been a useful reminder when Wayne was in Mr. Smith’s classroom (where the window was on the right), the incident shows how important it is that  children understand not only what to do but why they are doing it.

 

WHY DO SO MANY LACK CONFIDENCE IN MATHS?

 

For some, it may be that their confidence has been severely dented by someone who taught them maths in a forceful or unsympathetic manner, so that they came to believe that they were “no good at maths”.  A young woman with whom I spoke recently was able to track back her fear of mathematics to a specific incident (when she was 6 years old) during which she had her knuckles rapped for answering a question incorrectly.  Such an assumption, however, would suggest that large numbers of maths teachers – or, more probably, of primary school teachers – were to blame for our mathematical inadequacy.  But those same teachers managed to teach us how to write, spell and punctuate; even managed to instil in many of us a fascination for words and their meanings, and a love of literature and poetry.  Why were they not able to perform the same magic with maths?

 

In many respects, maths should be much easier to learn than English.  A maths problem has a “right answer” – whereas an idea can be constructed in many different ways by means of differently constructed English sentences.  Maths has rules which can be learned and applied – whereas English grammar (and spelling) is full of “exceptions to the rule”.  Performing well at maths is simply a matter of working through set processes – “carrying one across and paying one back” – “starting from the right and putting noughts under the numbers already used.”  And therein lies the problem.  Although we often view it as such, maths is not simply a matter of applying set processes.  There is often more than one way to solve a mathematical problem.

 

Unfortunately, if we are to learn some of the “mechanics” of maths we do need to practice.  Hence, pupils through the years have worked their way down columns of examples, applying rules rigorously in order to arrive at the right answer.  Whereas this may teach children the processes – and develop their confidence in performing those processes – it does not guarantee that they have learned the principles.  Nor does it ensure that they know which processes to apply to a set of data in order to arrive at a solution.

 

After all, if learning English comprised little more than learning how to apply a set of rules regarding spelling, punctuation and grammar, we would probably be just as “turned off” by English (as indeed, some are).  The difference with English (usually) is that we have to use it on a regular basis – for everyday communication in the home, classroom and playground (and later in the workplace).

 

Perhaps one of the reasons why so many lack confidence in maths is a failure to recognise that maths is about finding real solutions to real problems.  All too often, we can perceive something to be a mathematical problem and then seek to apply some half-remembered process.  Sometimes, we do this without proper consideration of the problem to be solved.  Whilst visiting a primary school recently, I noticed one child struggling to divide 29.5 by 4.  When I asked why she needed to divide the two numbers, she explained that she had to divide an A4 sheet into 4 equal horizontal bands.  Having dutifully measured the page, she was now grappling with the complexities of the associated arithmetic.  When I suggested that it might be simpler to fold the paper in half and then to fold each half again, she was adamant that it was “cheating”, oblivious to the fact that it was the simplest solution to her problem.

 

GET REAL

 

It is important – in all aspects of learning, but especially in maths – that we do not merely teach processes but that we enable learners to get to grips with underlying principles.

 

In order to develop confidence in maths, what we need to do is provide opportunity to use maths in a real context.  This is easier said than done, partly because mathematical tasks imply a certain level of responsibility that we do not normally expect to bestow upon children.  The kinds of everyday task that require mathematical solutions often relate to buying, selling and investing.  Or else they relate to practical matters such as household repairs and car maintenance; not something that the average seven-year-old is usually engaged in.  Yet, when I recall the formative experiences that helped shape my own love of maths, I realise that (quite inadvertently) I was supplied with opportunities to use maths in real contexts.

 

When I was a child, every Saturday my family would visit my grandparents.  The assembled cohort of uncles (and even a few of my aunts) spent those pleasant summer evenings playing darts in my grandfather’s cramped back garden, alongside the pigeon loft and the water butt.  Despite being the eldest child, I was deemed to be too small to throw darts – but I nevertheless had a role to play.  Clutching a piece of crumbling chalk, and no doubt straying perilously close to flying missiles, it was my responsibility to add up each player’s score (triple 17, double 11 and an 8, and so on); deduct the amount from each player’s diminishing total; scribe the result alongside the board; and remind players what they needed to aim for when they were “on a finish”.  Whereas the ostensible reward for my efforts was a “shilling” (or perhaps a Mars bar or a “Mivvi”) the real benefit was the status that such a task conferred – and the increase in my confidence at maths.

 

During the winter months, my uncles moved indoors to play cards.  Once again deemed too young to play, I was nevertheless valued as a “lucky mascot” by my favourite uncle.  Far from being merely tokenistic, my role was to keep count of which cards had been played so that Uncle Ernie could adjust his play accordingly, thereby winning more money and tipping me more liberally.  So adept did I become at keeping track of how many diamonds had been played, or who had played the Queen of clubs, that I was often called upon to arbitrate the occasional disputes that arose.  Whereas card-counting, of itself, may not be a valued ability outside the realms of professional gamblers, it nevertheless enabled me to develop a keen memory for process and, once again, contributed to the development of my confidence and self-esteem.

 

 

Check out the Maths resources elsewhere on this site.  You will find a variety of activities to encourage exploration of various aspects of maths.  Although aimed primarily at 11- & 12-year-olds, the open-ended nature of some of the activities means that they will also provide a challenge for adults.

 

You might also want to take a look at the "Make & do" section - which contains a selection of practical maths-orientated activities.

 

 

PLAY THE GAME!

 

Games provide an excellent opportunity to develop maths skills and thereby develop mathematical confidence.  The traditional trading game, “Monopoly” is a good example – but we need to recognise that games do not need to be overtly numerical in order to be mathematical.  Maths is not merely about numbers.  It is also about shape and pattern and sequence and order.  Maths comes into play (often without us recognising it) whenever we seek to apply logic to the solution of a problem.

 

There is now a plethora of computer games that challenge players’ reasoning skills as they seek to solve some cleverly contrived problem involving virtual landscapes, mazes, or game-boards with movable pieces.  Some of these are merely re-vamped versions of old favourites – but with the added benefit that you do not need to persuade other people to play with you: the computer plays the part of your opponent.  Many of them, however, involve such complex rules and processes that it is difficult to imagine how you would play them without the aid of the computer.  Whereas I am not advocating that we encourage young people to engage in the “shoot-‘em-up” mentality that pervades too many computer games, I believe that we could be overlooking a valuable resource for developing reasoning skills by dismissing all computer-games out of hand.

 

Similarly, practical activities can often require the application of simple mathematics.  Whether it be cooking, D-I-Y, sewing, decorating, or even gardening, maths often sneaks in unannounced.  Shopping is rife with calculation!  If children are encouraged to participate in these real-life activities, it is inevitable that they will become engaged in solving the myriad minor problems that require a mathematical approach.

 

Yet another avenue to explore as we seek to develop children’s (and adults’) facility with maths are those fascinating puzzles that engage, mystify and challenge.  Who can resist dipping into such diversions as magic number squares, pentominoes, Mobius strips, the tessellated designs of Escher, and the challenge of visual illusions – all of which offer countless opportunities to experiment and explore.  Suddenly, without realising it, you find yourself immersed in the world of maths and, strangely, not drowning but waving.

 

Teaching Maths to Pupils with Different Learning Styles

Tandi Clausen-May

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Checkout www.primarygames.co.uk for a good selection of maths games (including two darts games)