Imagining the puzzle

In maths the ability to imagine the problem is key. A couple of years ago I was studying in the company of a group of postgraduate students the eldest of whom was 14 years my junior. As a result all were too young to remember when the UK currency consisted of pounds, shillings and pence. These were all highly intelligent people. I proposed a pub challenge, in which I would ask the questions and they would compete against the clever clogs [this guy was a Physics PhD, who got 780 (out of 800) on his GMAT exam score] to do sums in pounds shillings and pence, the winner being the one who could furnish the correct answers first. My proposed quiz never took place because the PhD was unwilling to try his brain at calculating in bases 12 and 20 (there being 12 pennies in a shilling and 20 shillings in a pound) in such a public arena, where he might lose face. What he lacked, because he had never been previously required to possess it, was the ability to imagine counting in these bases. If he had tried, he would quickly have been able to imagine piles of copper pennies growing in height until they were 12 high when they turned into a silver shilling, and piles of shillings growing in height until they were 20 coins high, when they turned into a pound. Had he developed this picture in his mind he would have been able to solve any questions that I would ask involving the old UK currency, but until he did, until he could imagine the problem, he was not in a position to attempt the questions I might ask. So often, maths is not related to a real situation. As a result children are unable to imagine the problem. Like the Physics PhD, they just cannot get their heads it. At Puppet Maths we couch the puzzles we give our pupils in terms of familiar situations, so that they feel at home with the puzzle and can imagine what’s being asked of them and apply their minds to finding a solution. We mean to help them develop their ability to visualise in their mind’s eye what needs to be done to arrive at a solution.

Problems and puzzles

What’s the difference between a problem and a puzzle? The answer is… the consequences. If there are no dire consequences to getting the answer wrong, then you’re dealing with a puzzle, and it is an enjoyable mental challenge. If there are unfortunate consequences then you’ve got a problem, which is an unpleasant experience. But what constitutes unfortunate consequences? Well that rather depends on your position in life, however, for children, things that an adult would find trivial can take on great significance. Not being able to do an exercise of sums at school because the child has not understood what is required of them, can be a disaster. They may not want to ask for help, as that would mean admitting their inability to do the work, and they expect to be told off for it. They may fear being teased by other children if word gets out that they can’t do it. So they hide their inability, and what should be a maths puzzle and fun to do becomes a maths problem and a headache. This can turn them off maths for life. At Puppet Maths we don’t give our pupils maths problems, we don’t want them to experience maths adversely, Puppet Maths is the home of maths puzzles, puzzles that will develop our pupils and teach them to enjoy maths.

School as a parallel world

Children used to learn in the real world. My own grandfathers left school at 12 and went to work. One grandfather said that he learnt nothing of importance until he left school. I think that that is not quite true. He would have learnt to read, write and do arithmetic at school, which was all that was expected of schools in those days, but he was taking those skills for granted when he made that comment. However, there is some truth in what he said. As a practical man he valued skills that he could apply to making a living, to bettering his family, and these were not taught in the world of the schoolroom. Indeed today, 100 years later, schools do not teach children how to survive and prosper in the real world. Schools have become a parallel existence and when young people leave education the realities of life in the real world often come as quite a shock. This parallel existence is true in maths education. We ask pupils to calculate 3 + 5. This is an abstract calculation. It has not been put into any context. They are not being asked to calculate the sum of £3 and £5, or 3 boxes and 5 boxes. Such examples belong to the real world. Without context maths loses its meaning. If maths loses its meaning then it becomes simply an exercise that children feel they have to do to please their teacher and stay out of trouble. If we expect children to calculate in an abstract manner, then they will do so, but they will not always appreciate the uses of what they are doing, it will not catch their imagination, and they will not enjoy doing it because they fail to understand the purpose for doing it. This is especially true as children become teenagers. At Puppet Maths we put the maths we teach in the context of solving real world problems. We home in on the relevance of the maths to situations that the pupils can imagine either at home or in an everyday situation. This way they will engage with the subject, and achieve good results.

Programmed to succeed at maths

Children are programmed to do whatever the adults around them do. When she was small, my eldest daughter was placed in the care of a lady who came to our home and looked after her while we went to work. This lady, when not caring for her and her baby sister, used to clean the house compulsively. That year, when we went to the Ideal Home Exhibition, we placed her in the children play room, which was being sponsored by the local toyshop. The last thing I saw on leaving her was her taking hold of a toy vacuum cleaner and starting to hoover the carpet. On my return some 90 minutes later, she was just arriving at the opposite corner of the carpet, having spent the whole time playing at vacuuming. Shame it didn’t survive throughout her teenage years. Why do they stop doing these useful things? Because adults do not expect them to continue. Children will learn to do the things that they are expected to do. If they live in a society where they are expected to stand up in a boat and spear fish, then they will learn the balance needed to stand in a boat at a young age. If they are expected to count in bases 12 and 20 (as was required for using UK currency before decimalization in 1971 – there being 12 pennies in a shilling and 20 shillings in a pound) then they will learn to count in bases 12 and 20 when still young. It is common for adults to think of maths as being hard. As a result it is common for people to expect children to find maths hard… so children find maths hard. At Puppet Maths we believe that maths is easy. It is not only easy it is also fun. We make it easy, we make it fun. We expect children to do well in maths, and as a consequence we expect that they will indeed do so.

Children are cleverer than adults used to think 2

After writing yesterday’s post on what researchers have observed about babies ability to do maths, I was reminded of the behaviour of my oldest daughter when she was a baby. When she was 5 weeks old, her mother went into hospital to have her tonsils removed, so the poor child was left in my tender care. To make life as easy as possible for myself, I quickly developed a routine. I brought her home from the child minder’s, and fed her her tea. He clothes then went straight into the washing machine, and she went straight to the bath. I washed her while singing the bathing song. This determined what part of her body I washed… something along the lines of “now we go and wash the feet, and now we go and wash the knees” set to the old Irish folk tune “Let no one steal away your thyme”. I remember that it finished with the line “It’s time to ring the baby out” when I would lift her out of the water and swing her from side to side while saying “Ding, dong, ding, dong, ding dong”. But I digress from the purpose of my script today. The point of this story comes with the drying song. I would lay her on the bed and dry her systematically, while singing the drying song. This process required me to lift her right arm and dry her armpit, then to lift her left arm to dry her left armpit. On the third day of doing this no sooner had I dried her right armpit than she raised her left arm herself. At the age of 5 weeks my daughter knew what was happening to her, and able to anticipate what was going to happen next. To what extent the use of her song important in this? Would she have recognised the routine without it? There is work for the child psychologists to do there, but I believe that it played a crucial part. I think that babies and children respond to the use of music and song, that it gives them patterns to recognise, it helps them learn. Surely this is obvious, don’t we all remember poetry that scans better than prose that is just a connection of words? This is why at Puppet Maths we have adopted the use of the Sands Daniels musical times tables. The times tables comprise 80 facts that children have to learn (or 40 facts if you consider that half of them are the same as the other half - 2x6 being the same as 6 x 2). These are boring to learn by rote. But if they’re set to music, if they scan, then they are memorable and children can pick them up without difficulty. At Puppet Maths our aim is to help children use their natural ability to learn easily.

Children are cleverer than adults used to think

Children are more capable than people used to think. Recent research has shown that babies only weeks old can count. Interestingly this experiment involved the use of puppets. Babies, will stare at a situation whenever something unexpected occurs. This enables researchers to know whenever a baby encounters something they find surprising. The experiment involved the researchers showing two puppets to a baby. They then dropped a screen and very obviously removed one of the puppets from behind the screen, so that the babies could see it being taken away. On some occasions when the screen was raised again there was only one puppet there, but on others the researchers surreptitiously introduced another puppet so that when the screen was raised there were two. The baby was found to be expecting to see only one puppet, because on those occasions when there were two, the babies stared at the occurrence for a significantly longer time than they did when only one puppet was there, indicating that they were surprised. This indicates that even very young children can count. So why do so many children find so much difficulty with maths? It must be a function of the way maths is taught. Maths teachers simply are not connecting and communicating with young children well. This is why we have developed Puppet Maths, to communicate the ideas of maths with children and develop their natural ability to succeed in the subject.

Sudoku and overcoming the fear of numbers

Numbers are just shapes drawn on paper. What they symbolise is relatively straight forward too. Unless you are 5 or 6 years old that is. Children who are at school but who have not grasped how numbers work, find that they are not getting approval for their work, but they don’t really know why, they don’t know what they’re doing wrong… but there are all these other children around them who seem to know what going on, who seem to be able to get their sums right. This is intimidating. The child is not one of the group, and even worse is not getting approval from the adult in the room. They feel isolated. They hide their inability, because they need to be part of the group, they need approval. The teacher should be identifying these children and helping them to understand, but with a classroom full of children, perhaps the child’s problem get missed for a short while. Unfortunately, that short while is long enough for some children to develop a fear of numbers. Whenever they see numbers they cease to be able to think logically because fear sets in, and their minds close down. These children have learnt that these written shapes can separate them from the social group. This can lead to many years of underachievement. One remedial method to help a child overcome a fear of numbers is to teach the child to play sudoku. No maths is involved in sudoku, the numbers are simply used as different shapes, any other 9 ideograms could be used (take your pick from the Chinese dictionary). However, getting a child to play sudoku teaches them that they can manipulate numbers and get them to do what they want them to do within the game, and so helps them overcome their fear of numbers.