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How can we understand what it is like to be dyscalculic?
Understanding how a person feels when she or he cannot grasp a matter that everyone else finds so easy to understand that it is “obvious” is vital. Because if the whole subject matter is outside our comprehension, it is hard to the point of impossible to teach.
For example, if we were faced with a child who believed that the sun went around the earth, then we could at once see that, because the child sees the sun travelling across the sky each day, from the child’s perspective the sun obviously goes around the earth.
That would give us a starting point. We would know how the child sees the issue, and from that see where the problem lies, and as teachers we would be able to start at that point.
But supposing we meet a child who simply doesn’t understand numbers – how can we relate to that? How can some people not understand that six take away four leaves two? It is, after all, obvious.
When we are in this position, it is clear that something has to be done to help us understand the child’s perception (or lack of it) of numbers. But this is difficult since quite obviously we as teachers cannot unlearn the maths we know and which we use as a fundamental part of our everyday life. And indeed, even if we could unlearn the maths we know, we could quickly relearn maths, because for most of us, maths is something we immediately get. It is out there, it is obvious, it is central to our everyday existence.
That gives us a problem when we come to working with dyscalculic pupils. But fortunately there are two ways of trying to understand what it is like to be dyscalculic.
First we can take the issue of number itself. Imagine you are asked to define “six”, you might first do this by reference to external objects such as “six cars” or “six people.”
But supposing I was being rather difficult, and said to you, “I don’t want you to tell me about six cars or six people, I want to know what six is,” then you might find it harder. You could try and say that six is one more than five – but that is no good if I don’t know what five is. And if you find that comment frustrating, that’s good, because this is what it is like to work with a dyscalculic person. It is very frustrating simply because most of us simply “get” numbers. They are out there. They are obvious.
So, if our first attempt to unravel dyscalculia by relating numbers to other things fails to get to the heart of the matter, we might try a different route. And it is this second approach that I will describe below.
We use, in the west, a numbering system that is generally known as Hindu-Arabic numerals, and which we might call today the “decimal number system”. The system originated in the 6th century in India, was then adopted in the Middle East, and got to Europe around the 12th century.
And you will probably realise that this raises a really interesting question: how did people count before that?
In fact that turns out to be a very important question, because before the 12th century the west had already had two great civilisations of its own: the Roman Republic, and then from the time of Julius Caesar onwards, the Roman Empire.
These civilisations installed order and governance across vast parts of Europe and north Africa, and they did it without the Indo-Arabic system of counting. But, of course, they had their own system: Roman numerals.
Now I am labouring this point, because there is a fundamental difference between Roman numerals and the Indo-Arabic system of numbers in that the Indo-Arabic system has a number that does not appear in the Roman system. You might care to close your eyes and think what that number is, before reading on.
And that number is…
In Roman numerals there was no zero. So if you had five oranges and gave them to your friends, the calculation of five minus five could not and would not be undertaken. It was not a part of reality. The answer was nothing, it was nowhere, it was like asking today, “what lies outside the universe?” The question is meaningless.
Now this lack of zero might strike you as a bit surprising given that the Roman Republic on its own lasted almost 500 years (or “D” if you want to go native), and the Roman Empire lasted for 1000 years beyond that (“M”). 1500 (MD) years, and not a zero in sight.
We would find it impossible today to have any conversations about anything involving numbers without zero, and yet for well over 1000 years the accountants of the Roman Republic and Empire (and believe me there were a lot of accountants in the Roman Republic and Empire) did exactly that. They ran European civilisation without the zero.
I go through all this because most people brought up with our numbering system find this rather odd. And then they often find it odder if asked to explain what zero is without simply using another word for zero (such as nothing).
And again this gets us a bit closer to understanding how a dyscalculic person feels. It is not that they are not very good about numbers but rather that they don’t get numbers at all, in the way that most of us today can’t get the idea of a system that simply fails to acknowledge zero.
But to help our understanding of this oddity, we can take this matter further.
Everywhere in our universe there is something. Even in the depths of space there is something; as any physicist will tell you, there is no such thing as a total vacuum. There is always something. Nothing does not and cannot exist.
So in that sense “zero” or “nothing” is a construct – something we have invented to signify an impossible state, but it is also a notion that is quite handy when we have five oranges, take five away and then want to say what is left.
Not surprisingly, this totally artificial construct of “zero” turns out to be not a proper number at all. If you try to multiply by it, you get zero. If you try to divide by it you get zero. If you add or subtract it, nothing happens. Three minus two equals one. Zero minus zero equals zero. How can that be?
Now as I have said, we all accept this, because it is a way of doing maths. And with a lot of help the dyscalculic person can come to accept it too. But initially for that person zero, and all the other numbers that can exist, are all odd, because the gene that allows most of us to understand numbers and accept the oddity of zero without thinking about it, is faulty. It is not their fault, that is just how it is.
It does us no harm to puzzle over the oddity of expressing nothingness as “0” because it does remind us how odd all maths seems to a dyscalculic person, and helps us understand why conventional ways of teaching maths, which just assume all this oddness is, well, normal, make all sorts of assumptions.
When we recall that the Romans had no concept of “zero” and yet could raise massive armies, count the men, and take over much of the civilised world, imposing law, order and civilisation as they went, it reminds us that maybe maths is not quite as natural as we might want to assume in order to make our everyday lives easier.
If we want to teach dyscalculic pupils and students we need to take ourselves back to the very basics and realise that if we walked into a class of students who were learning maths in the days of the Roman Republic and started talking about zero, the whole class – indeed the whole civilisation – would appear to be dyscalculic to us.
Dyscalculic pupils and students need a different way of teaching. One that moves slowly through the physical world and relates that physical world to numbers in a way that makes no assumptions. It is a completely different approach from that which we can use with the 97% of the population that is not dyscalculic. Such teaching can be done, but it needs to be done slowly, with patience, and with a different approach from the norm.
Tony Attwood C.Ed., B.A., M.Phil (Lond) F.Inst.A.M
Head of the Dyscalculic Centre
Understanding dyscalculia: why the question “is maths real, or just a man-made invention?” is so important.
By Tony Attwood, chair of The Dyscalculia Centre.
There is a fundamental debate in maths over the question of the origin of maths, and it comes down to two basic points.
One side of the argument says that maths is something fundamental and is part of the universe as we know it. It is natural and “out there”, just as, for example, the elements like hydrogen, oxygen, carbon, iron, etc are “out there” as the fundamental building blocks of the universe itself..
The alternative argument is that maths does not exist of itself, but that it is created when people try to explain what is going on in the world. It is a way of seeing the world that human beings have invented to help along their understanding.
Of course, it is true that whichever way we look at it, maths is an amazingly successful way of describing the universe. It is logical (most of the time) and can be used to explain the world and the universe we see around us.
According to some (and this is an argument that goes back to the Greek philosopher Plato) maths suggests to us that there is an order in the universe, in every way we look at it – and that suggests that if humankind disappeared the maths, as well as the universe, would still be out there.
But others argue that we have constructed maths to describe the universe, so of course maths is uniquely set up to reflect the universe. In other words we make up maths as we go along, and as we discover more and more about the universe around us.
But there is a problem because maths is not very successful at describing all the universe. Take, for example, that whole business of the area of a circle, which most of us learn in school as the area equals pi multiplied by the radius of the circle squared.
Which raises the question, why “pi” - a number that we cannot calculate exactly but which starts as 3.14159265359 and keeps on going on and on and on and... And we might be tempted to ask, if maths is all that natural, why is the relationship between the radius of a circle and the area of a circle dependent on such an odd, never ending number? Wouldn’t it be more reasonable for it to be an obvious number like “5”?
After all, a circle is a perfect construction, and the diameter and radius are perfectly straight lines which touch the centre of the circle. So why is all this “pi” business involved?
This sort of thinking leads to the notion that maths is just a way of reflecting reality in numbers, to make it easier to understand and easier to make predictions about things.
And indeed it can be argued that all mathematical models of reality do ultimately break down. I have written elsewhere about how zero is not a number like other numbers – it doesn't work in the same way. We have to accept it in our way of calculating things, but it doesn't actually work as a number.
To take this further, through the investigation by physicists into the way the universe works at a very small level, we find all sorts of strange things going on, with particles affecting each other instantly over enormous distances in a way that cannot be explained by any form of conventional maths. There is also that problem of objects behaving in totally different ways until we measure the object, which seems irrational. How can measuring an object change it?
Now this argument about maths being a natural part of the universe, or a description invented by humankind to reflect what we find, may seem a bit remote and necessary, especially for a site devoted to dyscalculia. But there is a point here.
If one sees maths as part of the natural order of things, and one sees human beings also as part of the natural order of things, it would seem logical that human beings can understand maths.
But if we see maths as a human invention then it is much more likely that there will be some people who simply find that human creation difficult to understand. After all, why should everyone be able to understand everything that people have invented over time? Certainly we don’t expect everyone to be able to understand all art, or all music, or all foreign languages.
I think that this idea, which many people still hold, that maths is natural and obvious, is the reason why so many people find it difficult to understand why dyscalculics can’t grasp maths as readily as most people.
If, however, the thinking were reversed and maths was seen as much a human creation as works of art are, or the ability to play a musical instrument is, then there would be far less argument about whether dyscalculia were real or not.
To start from the premise that maths is a human-invented construction set out to describe the universe is a perfectly reasonable and valid position, and it has the benefit of taking us into a view of the world where we are as understanding of people who find maths difficult as we are of working with people who can’t sing in tune, find modern art incomprehensible, or find learning a foreign language intolerably difficult.
6. Dyscalculia: the everyday activities
Dyscalculia: the everyday activities that can be undertaken to help pupils and students overcome their difficulties and enhance other areas of learning.
While it is possible to make many specific points about dyscalculic pupils and students and how they can be helped, it must also be admitted that not every dyscalculic child faces exactly the same problems or can be helped in the same way.
In other articles we have mentioned the fact that for many dyscalculic pupils and students helping them work with sequences of events can help them begin to focus on the ordering of events and issues that they encounter in everyday life.
Thus asking a child to write down the order in which clothes are put on in the morning can be a very good way of focussing on the fact that there is a natural order in many things - an order that they will obey but which they have probably never thought about.
Elsewhere we have mentioned the benefit of encouraging pupils and students to spell out the exact order of events that is undertaken when taking a bath - including all the details that can be easily forgotten in the telling of the tale, such as putting the plug in, taking off one’s clothes, and so forth.
This can be extended into asking the pupil or student to describe the journey to school in ever increasing detail, describing the left and right turns if the individual walks part of the way, or the arrangements for getting in the car and how matters proceed once in the car.
Of course the child may well not know about unlocking the door, putting the key in the ignition (if that is required on this make of car), reversing out of the garage, etc, but this can be explored step by step, ideally with some input from the parents who can also help by discussing the sequence each day with their son or daughter.
A separate approach comes with the problem of solving mazes, for many dyscalculic people find solving mazes very hard and often end up just guessing randomly about how to find a way out rather than trying to work on a solution.
To help with this one can provide simple mazes at first which can just be drawn on a piece of paper showing the child, if it is helpful, the technique of working backwards from the end, which can often be easier than trying to find a route from the start.
Then the mazes can become more complicated - but never so complicated that the child is not going to be able to solve the maze fairly quickly. It may seem like a game with little educational benefit, but it does show that through planning and logical thought something that looks impossible can be solved. It encourages the child to slow down and work step by step.
Working with sequences also seems to enhance the use of memory which for some dyscalculic people is not naturally activated. Where one starts is dependent upon the child and what can be achieved - presenting sequences that the child cannot remember is simply going to cause frustration, rather than enhance the child’s sequencing ability.
If the child is particularly challenged over number sequences one might start with a sequence of just three numbers. In each case one should, in the early stages, have a sequence that comes from real life. It might be the numbers of the roads on the signpost passed on the way to school. Or the number of your house and the house opposite. Or the birthdays of three members of the family.
Where little progress is being made one might work with a very limited number of times table cards - which on the front have part of a sum or an element from a multiplication table.
Thus you might have
3 x 1 =
3 x 2 =
3 x 3 =
The child sees the question, reads it aloud, says the answer and turns the card over to reveal the answer.
Everything in this process is dependent on the ability of the child and the willingness of the child to undertake the task. When each task can be completed without fault, it can be expanded.
Ultimately one might work on the child’s mobile phone number or the home’s landline number. Somehow the numbers need to be given a meaning - and often working in pairs is a good way to proceed.
Thus a phone number that reads
Could be broken down into
79 25 97 33 71
There are many ways of giving a “meaning” to this sequence, but if we take it that the child is nine years old (or even that the child was nine last year and is now 10) we could ask...
What number appears most in your phone number? The answer is 7, so we start with that.
What age were you last year? The answer is 9, and so we are off 79.
How old is your baby sister? The answer is 2.
Tell me your five favourite friends at school. 5 becomes a remembered number.
So we have 79 25. 7 turns up most, 9 is last year’s age, 2 is your sister and your 5 favourite friends. Alternatively you might pause at this point and do some maths with each question only involving those numbers.
- 9 minus 7 is 2
- 5 add 2 is 7
And the reverse of these sums, so 5 plus 2 is 7 and 7 plus 2 is 9.
Moving on, 97 becomes easy now because it is 79 backwards.
And so on. The meanings don’t have to make too much sense, they simply have to relate to something. Indeed one can play the game of working out a meaning for each number. It doesn’t matter if one takes the ten allocated minutes just to work out a way to remember 79 25, for one can come back to it tomorrow and start applying the method.
So what we have here is a way of playing games with number sequences and spatial sequences as a way of training the brain’s “working memory” and getting it to be more active and to take information forward into the long term memory.
Incorporated within this is the notion of “overlearning” by going over the sequences and mazes over and over again even when the child clearly knows them. If difficulties arise try to find something visual that links numbers that turn up in sequences. For example, the number four as written by most children (with the flat line at the bottom) is the number seven upside down. Six can be seen as nine upside down. Simply playing with the shapes can help the child remember.
All of this work will expand the effectiveness of the short term memory (the working memory) into which information goes for processing and moving onto the long term memory.
Children who undertaken just 10 minutes a day short term memory using any of the methods above find they can expand the capacity of their working memory considerably within a matter of weeks. The key requirement is that it is done for a short intensive spell every day without interruptions or distractions.
Eventually as the sequences become more complicated one can ask, in relation to a long sequence (such as the mobile phone number), where did the number six come, and the child will be able to say “it is the third number in the sequence” without seeing the sequence written down.
An extra benefit in all this is that research has generally shown that children who do undertake this sort of training for ten minutes a day also then show a rise in their general intelligence scores. That’s not what the activities are there for - but it is a handy bonus along the way.
We might also note that even where these activities aimed at stretching short term memory are undertaken regularly by adults they once again are shown to enhance the score the adult gets in IQ tests.
Certainly the more these activities can be seen as fun by those taking part the faster there will be progress. And we should not be too surprised by the rise in IQ test scores from regular activities undertaken in this way. Research among children who learn to play chess, and who enjoy playing chess shows a similar rise in IQ.
In short using the memory by solving puzzles that involve shape, position and their own sets of rules, enhances the ability to handle the subject matter (in our primary example, maths), enhances the activity of the short term memory and then enhances the ability to score well in IQ scores.
Of course, what then also happens is that the pupils and students who do appreciate that they are improving their ability to do something also tend to feel better about themselves, and have in fact started the journey towards being lifelong learners, which is no bad thing in itself.
We might also note in conclusion that these types of activities are the complete opposite of playing games on a mobile device where nothing is learned save the ability to do the game. There, the game is devised to become hypnotic - to make the youngster keep playing the game. Here the activities enhance the workings of the brain, and thus enhance intelligence and the response to later activities.
5. Can brain training help overcome dyscalculia?
Because it is widely established that dyscalculia originates from a genetic issue within the individual and we cannot change our genes, it is sometimes felt that nothing can be done to help dyscalculics improve their maths.
This is not the case at all, and over time two different approaches to helping to improve maths in dyscalculic children have been developed. One involves teaching maths in a very different way from that used generally in schools, and the other involves helping dyscalculic children develop and enhance the way they use their brains.
In this article I deal with this second point: helping children utilise their brains in different ways.
This approach can appear to be somewhat surprising, in that the brain is considered to be the brain - it is what it is and what goes on in the brain is fixed. But in reality the brain is a highly adaptive organ which can adjust to changing circumstance and changing situations.
Different people react to the world in many different ways. Some, if told that dinner tonight will be at 8pm and not 7pm as usual, will automatically remember that and adjust to the change of events. Others will instantly forget. Some of these will subsequently deny they were ever told of the change, and indeed in doing so they are not lying because as far as the accessible part of their memory is concerned they were not told.
Of course over time some will realise that they are poor at remembering specific pieces of information that they are given out of context, as it were, and so whenever this happens they write the information in a notebook kept just for this purpose.
To take a different tack, some people if asked to describe the scene outside their window will give only a generalised report. Others will be able to describe the situation in great detail. And some of those who are poor at giving descriptions will work hard on the issue and train themselves to take much more note of what they see around them.
Yet another example: some find it incredibly hard to remember other people’s faces, while most retain a clear image of what a person recently met looks like. Some, when involved in a road traffic accident, can immediately describe the other car. Others find it hard to conjure up any image at all.
And here is one final example: many people when asked to describe a sequence of events can run their memory back and go through the events. Others find this very difficult, even if the events in question are fairly commonplace such as the act of preparing to have a bath or preparing to go out on a winter’s day.
The contention that those of us who work at the Dyscalculia Centre make, is that it is possible to train the brain of almost everyone to be more proficient than it is naturally in virtually all of these areas. Further we take the view that this is not hard, but it does require regular practice, so that ultimately the practice becomes a habit.
Now at this point I must deviate from the main thrust of this piece to say a word about habits.
All of us have habits - many more habits than we like to admit. Indeed it is a very common human trait to believe that we don’t have habits and that every action and decision is under our direct control, but this is not true. For without habits we could not function. Habits mean that we don’t have to think about most of what we do most of the time, and thus can focus our attention on the more important and more unexpected issues in life.
Indeed one only has only to think about something complex that we do every day to realise how habitual many of our actions can be. Take driving, for example. As one can see with a learner driver, driving is a phenomenally complex task, and yet for those of us used to driving, we can do it without thought while listening to the radio or talking to a passenger.
But habits come at a cost - they are easy to pick up and hard to get of. Anyone who has got into the habit of biting his fingernails or repeatedly checking the time or adjusting one’s glasses or saying “err” in each sentence or … well anything in fact - will know that habits are incredibly hard to stop. It can be done but the whole point about habits is that they are so deeply embedded that they cannot easily be removed.
Thus quite often when we want to change a dyscalculic person’s behaviour we need to break a habit. And if we don’t have that habit, that is going to be very hard to do.
I am going to describe approaches to helping dyscalculic people which flow from the view of the world set out above.
First I am going to consider sequences and second keeping track of time.
If you ask some dyscalculic children to describe to you the exact sequence of events involved in having a bath they may well get it wrong while those of a similar age and intellect who are not dyscalculic can get it right. Even if you stress that you want all the details of everything that has to be done, some can still miss vital elements out.
They might for example start off by saying, “I’d turn the taps on, and check that the water is not too hot nor too cold. But actually one really ought to put the plug in first. They might say that as the bath got filled they would test the temperature of the water carefully, and then if it was ok, they’d get in the bath - without suggesting that first they would need to take their clothes off.
And so on. Going through scenarios such as this in great detail, and then repeating the exercise later can help all children focus - and it can help dyscalculic children particularly, because they often find sequencing difficult. And that’s important, because sequencing is a central part of maths.
Of course the child might well get frustrated if you ask her or him to run through the same sequence too often, so you need to change the scenario. For example the exact sequence of getting up and having breakfast. The sequence involved in going to school...
All such activities push the child’s brain into handling sequences - something that some find incredibly natural and normal and others find takes a lot of thought. But, and this is the key point, the more sequencing is practised the easier it gets, and the easier it gets to learn mathematical sequences such as the times tables.
Second, the issue of time. Not all dyscalculics have difficulty with time, but some do. It is an important skill to have, and getting to grips with time can be very helpful in all walks of life. Here again we may start with sequencing and talk about what happens through the day at certain times including for example typical times for meals, homework, getting up, favourite TV programme, etc. Then one can ask the individual to put together a sequence of events for a certain day, with their times.
All such activities can do two things. First they will help identify the individual’s areas of difficulties, and secondly they strengthen and stretch the memory processes by forcing the memory into areas of work that the individual might shy away from because it is the area of difficulty.
Even if the memory strengthening activity has no benefit in itself (such as asking the child to remember where everyone sits in class) it helps strengthen the memory. Indeed even encouraging the child to remember the lyrics to favourite songs can be a helpful strengthening if you feel that the child is having greater problems with parts of the memory than others in the class.
There is one final benefit from this type of work. It both encourages the child to work on getting the short term memory (the working memory) more and more active, and it allows you to see where the individual is having particular difficulties with processes that you might consider to be normal. Together these two developments can be very useful, for it allows you to focus on areas of memory where the child has a problem, while at the same time improving memory without endlessly asking the child to work on her or his maths times tables.
4. The link between working memory problems and dyscalculia
Making sense of maths problems and undertaking maths calculations in one’s head can put an enormous strain on the “working memory” - the short term memory that also allows us to remember the start of a sentence while we are talking through the end of the sentence.
Most, if not all, people who suffer from dyscalculia have problems with their working memory, although working memory problems are not necessarily an indicator of dyscalculia.
Many researchers have observed that the failure of children to keep abreast of maths lessons and learn at the level that might be expected for their age and intelligence is associated with difficulties in processing data in this “short term” or “working” memory. This is because in our everyday lives we tend to undertake quite a range of mental arithmetic activities related to everything from remembering our phone number to deciding if one has enough money left to buy something.
Thus children who find maths difficult often do so because they struggle to get past the mental arithmetic stage. They may learn part of their “times tables” and be able to recite the table one day, but then forget it completely the next day.
For the dyscalculic child the cause of this problem is the dyscalculia itself which always brings with it a difficulty in handling numbers. For the non-dyscalculic child the problem can be an overload of the short term memory and lack of practice at using this memory other than in a classroom situation where other children may be more adept at handling numbers.
Obviously since maths knowledge is always cumulative (that is to say, for example, one can’t really learn division without understanding subtraction) children with short term memory problems can fall behind very early on in their studies.
So it is helpful if children who suffer from short term memory problems which result in poor performance in mental maths can undertake one of more of these four activities which encourage the development of this memory.
1: Take it slowly
For times tables never progress to the next table until the first one is learned, as this only adds to the confusion. One way to help the children who are having difficulty is to give them a few cards with the multiplication written on one side (eg “2 x 4”) and the answer on the other side.
The child says the multiplication and then says the answer while turning over the card. If the answer is not known the child can see it, read it and say it, and so does not spend time guessing at the wrong answer.
If the child is struggling with a table do not progress to the introduction of further cards and don’t move onto another table. The point is for the individual to be answering correctly and have the knowledge secure in his/her brain. By all means stop after a few steps within the table. Thus one might only get to 5 x 4 before the knowledge breaks down, and thus the pupil or student only has five cards. When those five can be said correctly then a sixth card can be added, but again there is no further progress until the six cards can be read and the answer said without pause.
Eventually the repetition of the table can move the table into the long term memory so that in future the individual with working memory problems no longer has to recite the whole table to find the answer to a multiplication question.
This has a double benefit since it means that with a multiplication question there is less dependence on the working memory, leaving it free for other work, while in some situations the child will not have to say the whole table but simply find the answer in the long term memory as an established fact (along with the child’s name, the child’s age, address and other key facts).
2: An alternative to cards
Working with cards is helpful because cards are easy to transport, but if this causes difficulty for the pupil or student, then moving counters of the type used in games like snakes and ladders and ludo can be used. The individual says “once three is three” and moves three counters from the central pool to one side, putting them in a line.
Then the individual quickly selects another three and says “two three’s are six” and adds these three counters o the initial group of three.
3: Giving numbers meaning.
It is obviously helpful if the pupil or student can remember the phone number of a parent, but phone numbers can be difficult to remember because they are long and generally meaningless.
Therefore if the phone number to be remembered can be broken into meaningful sections the number itself is easier to remember.
For example if the individual’s age, or the number of the dyscalculic person’s house, or a number on the parent’s car or any other number that the pupil or student has an association with turns up in the number of the phone, that can be easier to read.
Certainly the number has to broken down into sections to make this work, and it can take a while to learn all ten numbers after the zero, but it can be worth the effort, since the whole activity can strengthen the meaning of the number to the individual. The year of the person’s birth, the number of a nearby main road travelled each day on the way to school, the age of a relative, a number that turns up in a song - we are looking in all cases for anything that has a meaning to the child. This can even work by always saying a particular two digit number in a singularly odd voice!
Remember the individual numbers should be meaningful and the sequence to be remembered should have some meaning as well.
4. Practise focussing on numbers.
Here one might show the pupil or student four cards each with a number on. The cards are turned over and immediately the individual is asked, “what is the number on the third card?” Most people start by reading the numbers left to right and then reciting the answer. Practising doing this again and also gradually reducing the time the pupil or student is given to see the numbers helps strengthen the ability of the memory to focus on numbers .
3. Spatial reasoning and dyscalculia
There are a number of experiments that have been reported which suggest that people with dyscalculia have particular difficulty with spatial reasoning.
However it should be noted at once that the link is not absolute. A lack of spatial reasoning difficulties does not mean that the person does not have dyscalculia, and the appearance of spatial reasoning difficulties does not by itself mean the person has dyscalculia. It is an indicator.
Spatial reasoning relates to the way we can think about objects in three dimensions, and the ability of an individual to deal with spatial reasoning is often measured by asking individuals what an object would look like if seen from a different perspective.
This is obviously a helpful ability in everyday life as it allows us to imagine, almost instantly, what the hidden part of an object probably looks like, and it also helpfully allows us to perceive illustrations drawn on paper as having a 3D perspective.
The fact that this difficulty link is often spotted with dyscalculic individuals shows the extent to which a problem with maths can affect many aspects of a person’s life. In this case it restricts the ability to draw conclusions from limited information.
Thus in a test in which pupils or students are asked what an object would look like if rotated, dyscalculic children generally do far worse than others in their class. Similarly if a child is told that one person has five coins and is then given ten more, while another person has 25 coins, the dyscalculic child may well be unable to appreciate who has the most while the non-dyscalculic child of an appropriate age can do so.
In another test relating to this issue the child or teenager may be shown a number line. for example between 0 and 10 or 0 and 100 but with no grid marks, and be asked where certain numbers should go on the line. After doing such a test a child without dyscalculia can then normally explain why she or he put a number in a certain place. Dyslexic children often find it difficult if not impossible to explain why a number has been allocated a specific spot.
However it should not be thought that such tests are definitive methods of saying that an individual is dyscalculic. Again these are indicators.
However such indicators can be helpful in suggesting the help an individual needs with spatial awareness. For example, in the case of a person with a real problem with a number line one might ask the individual to draw a straight line and write in the numbers from 1 to 10 spaced equally along the line. If it is helpful you might mark the 0 at the start of the line and 1 a short distance along with 10 at the far end.
Some pupils and students do find it difficult to place the numbers equally, and bunch them at one end or the other. If that is the case, regular practice at this can help the individual see that it is helpful to space out all the numbers equally, which in itself gives a sense of the equality of “distance” in numerical terms between for example 3 and 4 on the one hand nad 9 and 10 on the other.
If this proves very difficult one can start with a page marked with 0 at the start of the line, 1, 2, and 3 at appropriate points and then 5 at the end, and ask where “4” will go. The aim of course is to get the individual to “see” that when the numbers are spaced equally the resultant graph has a particular look. The child or student then begins to see, sometimes after a lot of practice, the “look” of a series of numbers which are both spatially and in terms of mathematics equally spread out.
Once this is mastered one can undertake the same task with tens and 100s.
Spatial reasoning is itself related to an ability to read maps and plans and appreciate them as a representation of the real world. Again the dyscalculic individual might well find the notion of drawing a plan of a room hard, let alone a map of the streets between or around the school and their home.
Asking an individual who has such a problem to jump straight into map reading can be very daunting for the individual, and therefore it can be advisable to work with very simple mazes, asking the pupil or student to plot the route from the beginning to the end of a very simple maze, and later ask the individual to draw his/her own mazes and then to show that the exit can be found only by following one unique route. In this regard the pupil or student has to try to solve the maze by taking the wrong routes, in order to show there is only one solution.
All such activities can help the pupil or student evolve a sense of space and distance while playing a game. As long as the tasks are kept at first within the boundaries of what the child can achieve, and then those boundaries are only slowly challenged, the individual is normally able both to enjoy the process and make progress.
For many pupils and students this approach to learning about spaces and locations can represent a significant step on the journey to being able to appreciate the relationship between numbers - something that non-dyscalculic people grasp as they learn about the numbers themselves.