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mathematics in the brain - Christina - 03-04-2006
Triple Code Model
Dehaene proposes three different codes for representing number: a verbal code that is used to manipulate number words and perform mental numerical operations (e.g., multiplication); a visual code that is used to decode frequently used visual number forms (e.g., Arabic digits); and an abstract analog code that may be used to represent numerical quantities. Each of these codes is associated with a different neural substrate.
Do you see implications of this for teaching mathematics?
Thanks,
Christina
Dehaene, S. (1992). Varieties of numerical abilities. Cognition 44, 1-42.
Dehaene, S. (1996). J. Cogn. Neurosci. 8, 47-68.
Pinel, P., Dehaene, S., Riviere, D., and LeBihan, D. (2001). Neuroimage 14, 1013-1026.
mathematics in the brain - geodob - 05-04-2006
Thanks for that Christina,
These have great implications for the teaching and acquisition of maths skills.
Where each code as it is referred to here.
Needs to be understood individually in the developmental process, as well as the developmental process of integrating the 3 codes.
What is referred to as; an abstract analogue code that may be used to represent numerical quantities, no doubt refers to our inate Sense of Number and the ability to instinctively approximate quantities.
Though crucially, this inate ability which is present to some degree in 6 month old babies, needs further development and refinement.
A deficit in the development of a Sense of Number, is common amongst people with the maths disorder of Dyscalculia.
What this highlights, is that an effective Number Sense ability, needs to have been developed as a foundation. Prior to the introduction of the Verbal and Visual Codes.
If a child learns the Verbal/ Visual Codes, without a Sense of Number already established.
Then numbers, verbally, just represent an ordered sequence of words.
Without any sense of quantity being asociated with the words.
Which can be paralleled with learning to recite the alphabet.
Where the actual position of for example; 7 in 1,2,3,4,5,6,7.
Has as much meaning as G in A,B,C,D,E,F,G.
So that in practise, without a Number Sense, 5+2=7.
Has as much real meaning as, E+B=G.
In turn, basic arithmetic is not calculated. But instead memorised.
In terms of teaching mathematics, it is vital that a Sense of Number has been developed as a foundation for the introduction of Verbal/ Visual Codes.
As an analogy, it is parallel to teaching a child that was born blind, the names of Colours?
Where the order of colours on a colour wheel can be learnt verbally, but without any Colour Sense.
Mathematics needs to be recognised as a neural substrate/s developmental process, not simply something that is learned as declarative memory.
Geoff
mathematics in the brain - chrismom - 05-04-2006
I need to relearn algebra so that I can teach my son. I hope I can do it at my age. I am in my 40's. I have to confess that I don't really like math and I became calculator dependent at an early age.
My son is verbally gifted but he is also advanced in math. Around the time he turned five he asked me why he had to borrow or regroup to do a problem like 35-7. He wanted to know why he couldn't just do 5-7 which is negative 2and then negative 2 and 30 would be 28. I started to tell him he couldn't do it that way but then I realized that it does work. We even started using this method for doing mental math just for fun. For example in the problem 645-268 I just think 600-200 is 400. Then 40-60 is -20. 400 and -20 is 380. Then 5-8 is -3. 380 and -3 is 377. It is easier for me to do it this way than having to borrow or regroup. I did make my son practice subtracting the traditional way. I think it is good to know different ways of solving problems.
I never taught my son about negative numbers. An older boy in his musical theatre class told him about negative numbers at a rehearsal. The kids often brought their homework to do during breaks. My son was curious so the boy just told him about what he was doing and I guess my son thought it was interesting.
I remember that the kids in his acting class thought it was interesting that he could count handfuls of change including quarters, pennies, nickels and dimes correctly before he started kindergarten. I think learning to count by fives by counting nickels helped him with learning to tell time on an analog clock. He could also do that around the time he started kindergarten. We did tell the kindergarten teacher about the things he was doing before starting kindergarten but she was only concerned with his not coloring in the lines and not cutting very well and since he was also one of the youngest in the class she recommended T-1 (a year in between kindergarten and first grade with lots of coloring. She thought this was okay since he already knew how to read and do math and therefore didn't need to learn anything else). He would be in first grade this year if I had kept him in school and let him go to T-1 or 2nd if I had refused to let him be held back. Thankfully, a first grade teacher told me I needed to homeschool.
At the moment we are working on figuring percentages. He likes to calculate tax, discounts, rebates, etc. He does some of this mentally.
I worry that by next year I will be in over my head with trying to teach him math. So far I can tell him how to do something and get the correct answer but he already asks questions that I am not good at explaining. We are currently using Singapore math but I let him skip around and work on what he wants to since he seems to learn well that way. I welcome any suggestions for books or websites that might help us.
mathematics in the brain - Christina - 05-04-2006
Thanks so much for your responses.
Geoff: I love the analogy of teaching mathematical techniques without meaning as teaching a child who does not understand the meaning of “color†the names of colors. Delazer and colleagues have done some extremely interesting work on learning mathematics via drill (rote memory) vs. via strategies (application of arithmetic operations). Their work suggests that these different learning methods recruit different brain circuits: Delazer, M. et al. (2005) Learning by strategies and learning by drill-evidence from an fMRI study. NeuroImage, 25, 838-849.
Chrismom: The strategy for solving subtraction problems that you describe is quite creative! It sounds as through Chris enjoys philosophizing about mathematics, which is great. Have you heard of the mathematics curriculum called “Everyday Math?†It is very good and provides detailed and accessible explanations for the teacher (you), as well as creative extension games that Chris would enjoy.
All the best,
Christina
mathematics in the brain - John Nicholson - 06-04-2006
Hello Christina
I have been short of a decent computer and Dragon Speech for some time, as an old farmer with clumsy fingers I find it quite difficult to type, but through the modern systems of instant speech, I can write down immediately what I think. With this new computer I am able to write just as quickly as I can think, using this system of dragging speech previously I had learnt a great deal about how we speak and how quickly we can think using it to write letters and small papers with it has helped me a great deal to understand the workings of my own brain. On another thread you have placed a very informative website address giving information on the working brain, I have read most of it carefully.
However much we learn about the brain or what it is capable of, we have to consider what our own brain is capable of, and what the brains of those people around us and within our daily lives, is capable of. It is a most interesting exercise to break down the different aspects within our thinking, no doubt there is a great variation in the manner of how we learn anything, but in the end most of us learnt to speak and that is learnt purely by copying speech and then associating the word with the meaning of the word. My close friend Professor Winston Hagston is constantly fond of relating to me the words of Richard Feynman a mathematician and physicist of world renown who was a personal friend of his, Richard Feynman when asked by students what was the most Important thing he ever learnt, always replied the difference between words and the meaning of words.
I take this statement to mean that when we used words In normal speech, each of us may have a different understanding of the words we are using, but with mathematics the meaning of the words we use in relation to numbers has the same meaning for us all. Over 10 years ago I became enchanted by the study of how we learn anything, my interest was created purely from watching children who were conversant with using the Chinese abacus answering mathematical questions that would have been impossible for a European child who had no understanding of the abacus of a similar age.
I immediately understood that the children using the abacus were clearly building a mathematics map within the brain, from the research work that I have read and considered, it is obvious that the brain works at the speed of light, for my own explanation as to this I illustrate with the concept of speech, broadly we know what we wish to say, but the preparation of the words we use is left to the unconscious us mind, it is from the unconscious mind that we draw our ability within speech and preparation for conscious thinking, working In mathematics is a good example of conscious thinking, but it requires a great deal of unconscious ability in order to be able to consider the conscious questions we are dealing with in mathematics, or for that matter in any other conscious thinking.
over the last 2000 years the use of 10 symbols to represent numbers has become universal, I have devoted my own life, to the development of a system in teaching mathematics to all children in a similar and comprehensive manner.
In practical terms how we approach any subject we need to consider the working of the brain simply as a whole, we may be of course aware that different influences are assisting our senses in developing memory, it is my considered opinion that our unconscious abilities are far in excess of our conscious abilities. Obviously there are thousands of ways In which we teach anything , but the abacus presents us with a simply understood tool, it is a cheap resource and would be easily provided even to the most humble child, once it Is properly understood there is absolutely no need to have a professional teacher explaining how it works or what it can achieve. From my own observation I perceive that most of the important things we have to learn have to be learnt either on a one-to-one basis or within a small group, many generalised pieces of knowledge can be learnt immediately from the television screens from DVDs computers and newspapers.
Considering that any child virtually teaches Itself to speak by copying, the use of symbol’s in relation to the development of words and numbers is something completely different. It is my opinion that the understanding of symbols in the multitude of manners within which we use those symbols is the most vital part of any Childs education, most things can be easily understood by a child with clear verbal explanation, but the committing to memory of symbols is not something that can be done by explanation alone, my understanding of it is that probably only one or 2% of what we know has to be taught formally but when it comes to symbols and the meaning of those symbols virtually 100% of it has to be taught, some of this can be taught in classrooms and In groups, some of It must be assisted one-to-one between child and Tutor, schools are recognising that older children are good at listening to younger children read giving explanation and assistance in the pronunciation and understanding of words, it is exactly the same older children are excellent in explanation of mathematics and the order of numbers, especially where children are finding It difficult, mother a teacher of most consequence is usually more concerned with teaching reading then she is in teaching mathematics.
Every one of us adopts the posture of teacher and pupil thousands of times a day ,within our daily life. Formal education needs to be reorganised in order to take the best advantage of our clear human abilities, the abilities we have to both learn and teach, older children can reinforce the all educational concepts simply by teaching younger children on a one-to-one basis. Children being taught formally for half a day could spend the second half of the day teaching younger children on a one-to-one basis, in schools where teachers were not available, a handful of well-educated adults could organise schools with very little in available resources in order that they may teach many children effectively.
Where a child is taught mathematics directly from an abacus and the use of its fingers it is unlikely not to understand ongoing lessons in mathematics the unconscious abilities it developed from using the abacus and repetitive calculations combine with the wrote understanding of the times tables, which are also fully understood from a calculation angle providing the child with perfect mental understanding of numbers and how those numbers are achieved.
http://abacusone.net/
mathematics in the brain - geodob - 11-04-2006
Hi John
,I wonder whether you are raising, and trying to highlight perhaps the most important issue on this forum, given that it is an OECD forum?
I read your various posts, which I then consider in terms of developed countries?
Though very recently I listened to a lecture by a past UNESCO Director, speaking about the situation in Africa?
Where in terms of education, one simple statement explained what you are on about John?:
He stated:' I went to an African School, which had 2 teachers and 2 school rooms.
It also had about 4,000 [that's four thousand] students?
Then today I heard that the UK Govt has allocated 20 billion pounds, 'over so many years', towards education in developing countries?
But when we could be talking about a billion children, it suddenly becomes a very small amount for educating each child?
Where the cost of training enough teachers, would go beyond this? Before the cost of providing schools?
Though the idea that John often restates, is a teacher teaching children, where those children later in the day. Teach younger children.
Next morning, those younger children, teach even younger children.
Who teach younger ones again.
It seems that we have a flow-chart?
But really, to be honest, within the context of a school with 2 teachers and 4,000 students?
It'll be generations before enough money will be provided to make a difference?
Which raises the question about what to do in the meantime?
Without funding?
Where John's proposal for a Flow-chart model, presents an extremely low-cost approach to providing at least some basic education?
Where basic literacy and numeracy could be developed.
Which are the basic the Tools that anyone needs to enter the Modern World?
Also, whilst John's promotion of the Abacus, to develop numeracy skills, could be seen as self-promotion of his Abacus business?
On return from his recent trip to India, he designed a simple abacus which could be printed on recycled paper.
But the real issue that John has been attempting to highlight, is that in terms of countries where the expense for a Pen and Paper are beyond the parents and schools?
Let alone all of the Numeracy development and support materials?
A simple Abacus, can step in and fulfil this role comprehensively.
The durably of an Abacus is also a critical factor in terms of expense.
So that it can be passed on from child to child.
Forgive my carrying on, though if the only result from this forum, was to help provide an 'Affordable' basic education to the hundreds of millions of children who would never have had such an opportunity?
What a marvelous success!
Geoff
mathematics in the brain - Maulfry - 25-04-2006
John,
You say 'but the committing to memory of symbols is not something that can be done by explanation alone, my understanding of it is that probably only one or 2% of what we know has to be taught formally but when it comes to symbols and the meaning of those symbols virtually 100% of it has to be taught'. I'd agree that understanding of symbols does not come through formal teaching , but differ in that I don't believe that for the youngest children (birth to 8 years) it's about committing them to memory.
My research with a colleague (ongoing, since 1991) points to children as highly capable meaning makers who - when given the chance build on their earliest marks in socio-cultural contexts to develop their understanding.
From their earliest gestures, movement and speech infants begin to explore an increasing range od marks (Matthews, 1999). Somewhere between the ages of 3 and 4 years they begin to differentiate marks that they refer to as 'drawings', some which they identify as 'writing' and others occasionally to which they attach mathemaitcla meaning such as numbers / quantities. Over time they move through further explorations of numerals (their appearance and role) to the beginnings of early calculations (and we're not talking about formal written 'sums'). Children's personal explorations with symbols is highly revealing and they use a wide range of forms of visula representations thta are multi-modal (Kress, 1997). Our evidence is that by the time they are around eight years they have moved to standard written mathemaitcs. The most significant point about this is that they understand the symbols and calculations and their use at a very deep level. Of course they need adults who value, understand and support what they are doing. Our evidence is that when teachers give children pages of written maths to complete, this prevents children making these powerful connections.
Van Oers emphasises that ‘mathematics as a subject is really a matter of problem solving with symbolic tools' (Oers, 2001a, p. 63). Young children have amazing capability to make mathematical meanings and to understand symbols if only we can create contexts in which they can do this.
Kress, G. Before Writing: Re-thinking the Paths to Literacy. London: Routledge.
Matthews, J. (1999) The Art of Childhood and Adolescence: The Construction of Meaning. London: Falmer Press.
Van Oers, B. (2001) ‘Educational forms of initiation in mathematical culture’ in, Educational Studies in Mathematics, 46: 59-85,
mathematics in the brain - John Nicholson - 26-04-2006
High Maulfrey
I am hearing of twins who have developed a language that they understand and only them.
this is no suprise to me nor will it be to you.
a single child developing language has to establish meaning and sound in a joint manner every word we used is prossesed into meaning instantly, therefor i belive symbol recognition in maths 1 to 10 is vital, of course a reading child sees one and 1 as having the same meaning two ways of expressing it and the childs natural language sound, all three create the same meaning within a childs mind.
Quite naturaly a child unfocused will create its own vebal and visual signs and sounds for meaning but why let it.
We have hundreds of languages but only one common set of symbols for mathematic meaning, the sooner a child ties in the sounds of language to count and comprehend (establish meaning in a real form) its own fingers, and then goes on to tye in the the universal symbols, the better for me.
Thirty six symbols in our language must be easier to teach in our language then five to thirty thousand different pictures in chinese, but i would make a guess that those pictures are in fact words and not sounds.
I am intrested in what you are doing and saying and have responded as best i can from what you say. keep on teaching me please.
mathematics in the brain - Maulfry - 26-04-2006
Hi John
Please don't misunderstand - I was not for a minute suggesting that we do not teach children written numerals and mathematical symbols! However, what is clear from our research (and others including Ginsberg in the USA and Martin Hughes in England) that young children find these abstract symbols difficult to understand at a deep level unless they have a chance to connect them (and to build on) what they already understand from their earliest 'written' marks. Doing this build much deeper levels of understanding but just telling (direct teaching / showing) them what the numerals and symbols are results in only superficial understanding. An example of this is the '+' sign, which young children (outside of educational settings) will know as - for example - the sign on an ambulance or hospital; on the remote control for a video player or computer games - or rotated so that it appears as 'X' - as a kiss written on a birthday card. In mathematics it means something else and is used in standard addition calculations. If children have a chance to build these connections (understanding) in meaningful contexts (maths that makes sense to them, for real purposes) and explore through their own early marks, they can better make these connections. In themselves numerals and symbols have no meaning - it is humans who attach meaning to them and children need help to do this.
You say 'the sooner a child ties in the sounds of language to count and comprehend (establish meaning in a real form) its own fingers, and then goes on to tye in the the universal symbols, the better for me.'. Our evidence is that meaningful learning in mathematics needs to go deeper than copying or being able to write and with over 25 years experience teaching children 3 - 8 years, I'm not convinced that 'sooner' is best for young chidlren!
It's interesting to reflect that research on young children's early writing (non-mathematical) has explored this in great depth for over 30 years, but research on the beginnings of 'written' maths is relatively recent! Martin Hughes valuable study was published in 1986 and there have been a few papers in which researchers repeated his experiements (and came to the same conclusions). Our research is in natural contexts.
Hughes, M. (1986) Children and Number: Difficulties in Learning Mathematics Oxford: Blackwell.
Hope this is helpful!
mathematics in the brain - geodob - 27-04-2006
Hi Maulfry
,Good to see you here again!
Here's an excellent article that Christina posted, that you might find interesting:
http://72.14.203.104/search?q=cache:Gv4yZN4TmGsJ:www.gdn.edu/faculty/l_fowler/Fowler_article.rtf+%22asian+teachers+polish%22&hl=en&gl=us&ct=clnk&cd=1
Geoff.
mathematics in the brain - John Nicholson - 27-04-2006
Hi Geoff and Mal
i have posted your recomendation to my tame pysics proff i have asked him to sit around the abacus one map and develope a teaching stratergy fom it with his grandchildren they useually have more sense then we do.
i have just had almost a first for myself and it relates to SYMBOL AWARENESS
My first batch of low case cards as gone down well with a local school they are printed to maximise the letter in order to make up words with them
they have two light blue lines to give awareness, athick green line on the bottom to indicate which way up they are to be used and a small sun in the top left corner o reinforce that.
on the back of every card is the alphabet for chanting and recognition in a
3 card 4 card 6 card 4 card 4 card 5 card order
we need numbers for everything
above the alphabet lay out is the message to the parents
if i want to read
i must succeed
in knowing every
letter and sound
so for me
it has to be
the
a b c
d e f g
h i j k l m
n o p q
r s t u v
v w x y z
every parent of a reception class will be provided with 52 cards 2 of each letter and a printed simply layout
with my seven systemamatic steps in reading hanging onto the edges of the lay out, if any child does not manage to lay out the cards within a month properly. the mind police will be informed and BLAIR will arest them on an asbo order.
YOU guys need this stuff as soon as possible but use the abacus map on your kids right away together we shall perfect an international every family every child teach yourself kit based on abacus and alphabet.
my best regards john
mathematics in the brain - Maulfry - 28-04-2006
I'mj sorry John to disagree, but education in the past is full of sets of cards and kits of stuff that have been claimed to support children's understanding. Learning is much more complex that that - if understanding was down to drill and chanting then no adult or child would have difficulties with mathematics. But then I read on and see (hope) you were joking!
mathematics in the brain - Maulfry - 28-04-2006
Geoff - thanks for the link to the paper. I see that it refers to 'two powerful tools for representing mathematics: language and the manipulation of concrete objects' but omits an obvious one - children's direct engagement with symbols through exploring them iin their own ways with pen or pencil on paper.
Here's something on dyscalculia that may be of interest... http://www.qca.org.uk/downloads/mathematics_report.pdf
mathematics in the brain - geodob - 29-04-2006
Hi Maulfry,
I fully appreciate that the article omitted, or more likely was unaware of the "children's direct engagement with symbols through exploring them in their ways with pencil or pen on paper."
Yet hopefully your research into this will grow into greater global awareness?
Thanks for your link as well, which I found most interesting. Yet their appeared to be a primary focus on Dyscalculia as a brain disorder? Which I would question the extent of?
Where I would take your research into numerical symbol acquisition. Which I would suggest is a significant factor in Dyscalculia? Which is a result of an ineffective introduction to numerical symbols. Which is not a 'brain disorder', but a consequence of misdirected instruction?
Though on page 4 of the paper you gave a link to, I found the statement:
"Recent research with infants challenges previous theories that they can discriminate numbers of objects. Studies of young children suggest that mechanisms for tracking individual items and making approximate comparisons of amounts develop into visualising precise numerosities around the time that symbolising and number language develops, and linguistic representations are then built on visual ones. Understanding of counting and cardinality develops with experience and is also related to the understanding of numerals."
Where the key point is: "...making approximate comparisons of amounts develop into visualising precise numerosities..."
This is where I see the importance of your Emergent Maths, as it builds on the fundamental innate ability of 'approximation'.
So that the use of abstract symbols to represent approximations is allowed to develop.
I have only recently become aware of the importance of 'approximation' in the developmental process. Yet I would suggest that this is a critical factor for developing a 'Sense of Number'? Where the transition from sense of approximation to a sense of number, needs to be recognised as a developmental learning process of refinement.
Maulfry, I believe that this is precisely the developmental stage that your Emergent Maths addresses? Where it encourages the natural developmental transition from a sense of approximation to a sense of number.
Geoff.
mathematics in the brain - John Nicholson - 29-04-2006
I’m sorry John to disagree, but education in the past is full of sets of cards and kits of stuff that have been claimed to support children's understanding. Learning is much more complex that that - if understanding was down to drill and chanting then no adult or child would have difficulties with mathematics. But then I read on and see (hope) you were joking!
________________________________________
No healthy child or adult should have any difficulty whatsoever with learning mathematics, if they have been taught properly.
Nor should they have any difficulty in reading whatsoever, once again if they have been taught properly.
Students of accelerated learning are told very quickly for the mind to grove the body must move.
We all realize the past in education is littered with cards and devices, and I am equally as sure that the future will be just the same, until we can plug in the brain to a computer to transfer knowledge, we shall simply have to manage with what we have got.
Try a simple experiment just ask 20 people that are over 60 years of age, and you will find that they can all sing the alphabet, there were two separate ways of singing the alphabet, some people can sing it one way, some people can sing it another way, and some people can sing it both ways.
Most of them will not have been asked to sing it for 40 years, but they will all be able to remember how to sing it, in the manner in which they were taught.
My personal methods of teaching come entirely from observation and working with very young children my own grandchildren and their friends, careful use of rhythm will quickly establish the memory of sound.
Any child that can speak can easily be taught to count to ten and relate that count to its fingers, within fifteen minutes the basic exercise can be established, repetition for less than a week will establish that memory permanently.
Every child needs to prove that there are five fingers on each hand, once a child has established this fact clearly to itself, it is no problem for to it to understand that two hands together make the sum of 10, take one hand away they are left with five, every combination of fingers needs to be learned and committed to memory, purely by the use of fingers and toes any child can be taught basic mathematics, if the teacher is aware of the techniques and has the time available.
But there is a better way, I have spent 10 years of my life developing it, and it looks as though I shall have to spend the rest of my life explaining it to teachers, in practical terms, mathematics needs to be carried out in notation, the limited ability of short-term memory as regards the retention of a differing series of numbers, prevents anything other than notation (in any form) as being of practical use in the long-term application of practical mathematics.
Counting is the whole basis of mathematics, all we are trying to do in simple arithmetic is to understand the addition and subtraction of numbers, multiplication and division are simply faster ways of counting techniques.
The natural form of the decimal system, as assisted human beings for many thousands of years, after the birth and the creation of language development of a simple counting techniques were slowly established, pictorial definitions will have been used way beyond the 5000 years that we are aware of the beginnings of written language.
The development of the abacus and its simplistic counting techniques allowed an uneducated whole population to carry out simple mathematical calculations without having to resort to mental technique, other then the simply learned ability to read and manipulate it. The abacus is never wrong, the abacus tells no lies. Probably at least a quarter of the world's population still rely on its efficiency, in Western culture it has raised its head and been superseded many times by simple notation.
In practical mathematical terms notation is more efficient, eventually pen and paper always replace the abacus, and within high speed mathematics today, a simple calculator is more efficient.
Simply because there are more efficient ways of recording numbers and achieving a rapid calculation by electronic calculators the abacus has been ignored, only those populations where notation that was difficult and primary education restricted has the use of the abacus remained normal practical practice both within education, and day-to-day commerce, fortunately for my own grandchildren and the children of my friends I realized that without any effort and any formal teaching virtually every healthy child that uses an abacus develops within its own mind a practical working mathematic map, most of us are able to use a computer without too much training, the Internet is providing a thousands of families with a rapid information, but the development of the human brain is restricted when perfect mathematical techniques and abilities are failed to be established within the individual human brain, every healthy child can be taught to drive a car within a short time, every healthy child can be taught to use an abacus in a much shorter period of time, and every healthy child that uses an abacus for any length of time, developed the basis of perfect mental arithmetic which itself is the basis of all mathematics and all scientific explanation.
Children can handle simple resources perfectly ably long before they are able to write providing another child or a concerned adult can show them how, there is nothing difficult in teaching a healthy child how to read and calculate.
mathematics in the brain - Maulfry - 30-04-2006
Hi Geoff
My understanding from what I read (in the review of research on dyscalculia), was that the author had been asked to look at a range of evidence across the board - and that evidence from nueroscience was one aspect of this. I'm inclined to agree with you about the review of research's over-emphasis on this although there is still so much about the brain we don't know!
My own feelings are that there are likely to be a number of reasons for dyscalcula and that socio-cultural aspects are likely to be significant contributory causes, including aspects of teaching. Learning human constructs such as mathematics is complex and I don't think that there will ever be one simple solution to supporting children who experience difficulties - or to teaching mathematics to all children! I think we're in agreement here!
The quote to which you refer on p. 4 of the paper (which I find I had already marked) is, I think, referring to what is known as 'subitizing' - where young children of 3 and four years of age can recognise the total of a small number of objects without counting each item. And yes - I do think that approximation is significant - but I'm wondering if you are referring to children approximating ('estimating'?) a number of items? We would accept this - and also children's own marks as their current approximations of numerals and symbols. I do agree that approximation of quantity is an important aspect of developing a sense of number. I found it interesting that you comment that we enourage 'the natural developmental transition from a sense of approximation to a sense of number' - I like this description! However, we would say that very young children have their own sense of number. Presumably by referring to the 'sense of number' you mean a well developed sense of number? Linda Pound has described children's learning as 'partial' which we find helpful: that is to say that what they do need not be judged as 'wrong' or full of errors but rather that is shows their partial understanding at this point in time.
Hopfully our research will contribute to the educational debate on young children's learning (and learning mathematics in particular), and athough we are probably a long way from 'global awareness' interest is growing! I think we can all learn a lot from each other! Meanwhile in my own research I'm intending to focus on the ways in which young children represent their mathematical thinking multi-modally- and see that there is a good deal of interest growing in multi-modality in literature in Australia!
Maulfry
mathematics in the brain - Maulfry - 30-04-2006
John - it's good to hear of your involvement with your grand-children and others, in early mathematics. I am very aware that different forms of abacus have been used with great success in mathematics teaching around the world and wish you success in what you are doing - I'm sure that we all have a lot to learn from each other!
Maulfry
mathematics in the brain - John Nicholson - 01-05-2006
Thanks Mualfry
HI BUNNY
just a bit more
This is the first day of May, the year is 2006, and I am writing this with a speech recognition system. It is not a particularly brilliant first day of May in East Yorkshire. Unfortunately yesterday's news from America gave a report off the death of Kenneth Galbraith he was 97 years old; Kenneth Galbraith was a man who I admired greatly. He was born on a farm in Canada, hearing him speak about his family roots, drew me very much towards him, he described the hard work and the efficient economy on a Canadian farm, he explained to the world at large that after being born and brought up on a Canadian farm and having had to work extremely hard, the rest of his life was no more then an easy downhill slope.
Reading this if you are a farmer Reading this on a computer on an agricultural website in the United Kingdom spending a couple of minutes of your life looking at the life and influence of Kenneth Galbraith. After attending university in America Kenneth Galbraith was co-opted to economic research within the American government during the Second World War, Galbraith was much influenced by the British economist John Maynard Keynes, from the work and research of these two great economist, came success in the Second World War and the formulation of economic development which still holds true to this day. Every child and student needs to be made aware of the lives of at least 100 major figures, and when we are considering the history of the world , where ever we are in the world we should have to consider Kenneth Galbraith, and a major influence he played within the mentality and actions of John Kennedy, another of my great personal heroes.
I am nearing the age of 66, during the last decade of my life, I have devoted a major part of my thinking time entirely to education trying to understand my own mind, and the minds of others, the abacus and its efficiency as an educating tool, drew me into this line of thinking.
The efficiency of the human brain is far from being properly understood, our abilities in the first instance to teach ourselves to speak, gives a good illustration of the possibilities springing from the human mind. There are only three things of importance within education, we already have all the tools in the world available to us to develop universal education unfortunately, we have not realized the potential of the human brain to absorb knowledge instantly, nor have we realized what is necessary to make the best of the human brain. If we analyzed the most successful human beings on the planet, within every field of Endeavour, we would come to the same conclusion; very little practical information has been absorbed within formal education.
The human brain requires to perform at its best, communications systems, and the system was most important to it, is reading and understanding, understanding anything comes quite naturally to the human brain, reading does not however come naturally to the human brain we have to be taught to read, and therefore we need to be taught to read quickly, that means at the earliest point we are able to within our life, and we have to be taught to read thoroughly, only by going through logic or reading steps can we achieve efficient reading, and without efficient reading we shall never be able to master the depth of reading that is required to live within the modern world. We need to be able to read broadly, we need to understand many subjects; we need to understand the interrelation between everything that humanity as understood, has created, and has a need to create.
To encourage your child to read, is the most important part of parenthood, without the wish to read, for pure pleasure, no child will ever develop the efficiency in reading which is necessary to live and learn effectively. The most efficient method of teaching is the teaching of concepts within a story,
for thousands and thousands of years human beings have told each other stories, collected around campfires, huddle together in igloos, herded together in concentration camps, human beings rely on stories in developmental concepts,
Stories are far more effective in teaching, the then are our visual concepts for many things. Although we remember the most part of what we see in our daily lives, working with that memory is not something which we do naturally, we see and record many things, without developing conscious memory only the most important things we see become part of our conscious memory,. However much power from our visual memory that is available to us, only the most vital of that memory, is subconsciously stored, and very readily available, especially by the use repetitious consideration, during reading and verbal learning situations, especially when we're mentally and eagerly trying to absorb and understand the subject.
In a modern world we are losing out on reading ability, in a modern world where no one appears to have the time, to listen to a story .Within the modern world we appear to have very little time to consider the welfare of each other or the part we play in that welfare,
Very many new ideas are instantly available to a modern population; the Internet has created a far greater realization of global awareness, then anything else, John Kennedy and Keneth Galbraith were essential parts of the forefront within global thinking.
Returning again to the wonders of the human mind, there is nothing so special about teaching, we are all programmed to teach, just as we are programmed to learn.
There are only three things of importance,
SHOWING DOING REMEMBERING
There is nothing more then that, and nothing less than that, which is ever needed, by any human being to learn, anything.
Everything we do we need to do, we can be shown how to do,
In the manner of mathematics the decimal system is so efficient, it is so natural that the abacus being used for a couple of years between the ages of four years and six years establishes so effectively the basic grasp of all mental arithmetic, allowing any child that has speaking ability and normal health to absorb everything they need to know to be shown everything in mathematics to the point where being shown anything becomes irrelevant and personal consideration and thinking takes over.
Reading is far more complicated not the first steps in reading, symbol recognition is easily achieved by repetitious chanting and a visual repetition tying in the sound and the picture in ones memory., obviously with different children it takes differing lengths of time but it is absolutely pointless trying to teach a child to read that is not aware of the symbols necessary to read, my symbols are all in lowercase and it is impossible for a child to mix those letters. Every child will develop awareness of capital letters as it proceeds to learn reading they need not be taught it as a part of learning to read specifically. Capital letters are like many other things, we become aware of them slowly and when they are necessary to be established as part of our permanent thinking.
What we have to do is establish permanent memory of many things within our own thinking; we need to prove those things to ourselves, in just the same manner as a child proves to itself that it has five fingers on each hand.
Subconscious ability is where we are aiming in primary education, it is so central throughout our lives that primary education if ineffectively carried out will ruin the lives of any child if we take a simple look into the prison system 65 to 70% of people within the prison system have no ability to read and write, the pure frustration they endure is a large factor leading towards a life of crime. There is absolutely no problem in starting universal education tomorrow morning, but it cannot be done unless those that consider themselves to be at the forefront of educational thinking, accept the principle of essential basic skills teaching and are prepared to reconsider the direction of secondary education including the principles of university education which should run alongside reading a subject as well as practically carrying out of that subject whatever that subject may be.
Just as a have had to spend many hours considering education and developing my own ability to explain what is happening and why it happens and how it can efficiently be carried out simply by the utilization of one child showing another child.
Quite naturally this is a limited view of education, I have spent only ten years considering it, alongside 30 years farming and five years compressed into 15 years fighting for personal justice which at this moment is unobtainable. Only effective universal education will give us the possibility of universal democracy of universal housing and of universal peace I only asked one thing of this world, will you please utilize My research work immediately and adopt my abacus and the lessons I have learned from it.
At this point in my writing I am tired and need to refresh my batteries so I usually end with words here endeth the 157,000 lesson.
..
mathematics in the brain - Christina - 08-05-2006
Thanks so much for this great discussion on mathematics. If you haven’t read it already, I would highly recommend Stanislas Dehaene’s The Number Sense: How the Mind Creates Mathematics to anyone interested in mathematics in the brain. It is marvelous –very clear and extremely engaging.
Best wishes,
Christina
mathematics in the brain - Christina - 08-05-2006
Recent research suggests that the infant’s brain is equipped with a quantitative sense (Feigenson, Dehaene, and Spelke, 2004; Wynn, 1998). Infants seem to poses two core number systems that allow them to attend to quantity (Xu, 2003). One system appears to support the concepts of “one,†“two,†and “three.†Infants are able to precisely discriminate these quantities from one another and from larger quantities. Moreover, they may have an abstracted concept of these numerical quantities that is insensitive to modality as they seem to understand the “twoness†common to two sounds and two objects (Starkey, Spelke, and Gelman, 1990). The other core number system is approximate. It enables infants to discriminate among larger numbers with sufficiently high ratios. Therefore, infants can, for example, distinguish between eight and sixteen, but not eight and nine.
Moreover, there is evidence that infants can perform mathematical operations with these numbers. When one object is placed behind a screen followed by a second object, they expect to see two objects when the screen is removed, suggesting that they know that one plus one should equal two (Wynn, 1992). They can also perform approximate calculations, such as computing that five plus five equals about ten (McCrink and Wynn, 2004).
mathematics in the brain - Maulfry - 24-05-2006
Thanks for the references - I know Dehaene’s work but the others are new to me.
However, I do think that the fact that mathematics is a language - referred to example, by Ginsberg (1977), Cockcroft (1982) and Burton (1994), (and has also been referred to a a 'foreign' language), is at the heart of mathematics.
I've been doing some research on what is referred to in English schools as 'modelling' mathematics - the main outcome was that (in two separate studies) it was clear that showing children how to do something and then asking them to go and work on something similar immediatly afterwards, does not work. It appears that children (possibly adults too) need a time delay in order to (mentally) absorb new ways of working before they can integrate it and make it their own. I refer to this as 'adding to their mental toolbox' - rather like adding a new chisel or screwdriver and then (later) going to the toolbox to see if there is a tool that will do a particular task.
I came across some brain research that I felt supported this and will look it out.
mathematics in the brain - Christina - 26-05-2006
Hi Maulfry,
Thanks for you post. This seems to relate to the spacing effect, which states that distributive repetition (presentations spread out over time) work better than massed repetition (presentations closely together in time). I would be very interested to read more about the research you mention related specifically to mathematics. Please post the references if you come across some.
Many thanks,
Christina
mathematics in the brain - D.Woody - 20-07-2006
[QUOTE=Maulfry]"I do think that the fact that mathematics is a language - referred to example, by Ginsberg (1977), Cockcroft (1982) and Burton (1994), (and has also been referred to a a 'foreign' language), is at the heart of mathematics."
Hello All, I'd like to jump in on this conversation that is of great interest to me. It seems that most of the research that is coming out these days (esp. Dehaene and Butterworth amoung a slew of others) clearly demonstrates that mathematical thinking is not just "a motley set of language tricks" as Chomsky once put it. There seem to be specific neurological systems that are designated to quantity analysis, independent of language. In fact I would go so far as to say that many are out there right now working to demonstrate this.
One recent study done in Brazil among a group of people who have no words for larger numbers demonstrated that these people clearly had ways of working with larger numbers. (Dehaene et. al.) They also showed in a subsequent study that concepts of geometry for which they had no words were also intact.
Not that language doesn't play a very important role in our learning of mathematics and clearly aids in the development of mathematical concepts including the place value system, but to say that the heart and soul of mathematics lies in language I believe is inaccurate.
I'll add here one of my favorite quotes from Einstein.
“Words and language, whether spoken or written, do not seem to play any part in my thought processes. The psychological entities that serve as building blocks for my thought are certain signs or images, more or less clear that I can reproduce and recombine at will.”
David
mathematics in the brain - geodob - 20-07-2006
Hi David
, good to see some activity on this thread again.I think that you make some good points in relation to maths as a language.
Where in relation to:"There seem to be specific neurological systems that are designated to quantity analysis, independent of language."
I recently read of some research that identified a part of the brain near the ears that appeared to deal with quantity. Though it referred to 'approximation', where perhaps it is language that we use to define it further as quantity?
I would suggest that this 'approximation' is also the basis of what is termed; 'Sense of Number'. Where for people who have developed a 'Number Sense', they generally 'sense' quantities of up to around 5. If you look at a group of up to 5 random objects, you will probably be able to 'sense' 5. Though 7 objects becomes more than 5 and needs to be counted.
The old Roman Numeral system, was more suited to this 'sense of fiveness'.
Where V as a symbol, also has visual association with a hand?
Also X as 10 in turn, is two V's linked, 2 hands.
Though Maulfry is doing some very interesting research into how the association between our Number Sense and our visual and auditory symbols and words for numbers are established.
Which often doesn't effectively develop. As is evidenced in many people with Dyscalculia.
This can result from exactly what you are questioning, the idea of maths as a language. Where often very young children are taught basic numbers and arithmetic as a language. So that number symbols and words do not have an automatic association with a sense of number/ quantity.
Geoff.
mathematics in the brain - John Nicholson - 22-07-2006
HANDS ON TEACHING
THE PRINCIPALS OF THE ABACUS
S D K
SHOWING DOING KNOWING
THE PERFECT TEN
TEN INDIVIDUALS
TEN IN TWOS NOTHING TO LOSE
TEN IN THREES LEAVES ONE TO FREEZE
FOUR FOR ONE & ONE FOR FOUR
COUNT TO TEN 1 2 3 4 YOU WILL NEED NO MORE
TWO FIVES TO KEEP ALIVE
TWO LITLE KITTENS HAVE LOST THEIR MITTENS
PUT THEM TOGETHER AND REMEMBER FOR EVER
THE CENTER COLUMB OF THE ABUCAT
1 2 1 2 1 2 ONE
3 & 2 + 1&1&1&1&1
3 & 2 + 1 + 2 & 2
3 & 7 = heaven
SEVEN SEPARATE & THREE INTERGRATE
THIRTEEN PATERNS SO FAR AND SO GOOD
1 & 9
2 & 8
1 & 1 & 8
This is dedicated to Manisha and Louisa, it is almost a EURIKA moment in the life of abacus one, no I am wrong it is a Eureka moment, although I am not a religious believer I hold my hands together in prayer for the perfect 10. Ten together in both the Christian religion and the Hindu religion we have the hands together in prayer for the perfect 10.
Most teachers will develop there own method of hand demonstration, but simple repetition and recognition will enable us to cover the earth with the abacus, and the principles of peace which the Abacus represent, I have in my own mind a picture of the Russian rifle which is so famous in guerrilla warfare alongside it and I have in my own mind a picture of the abacus, underneath the picture of these two instruments we only need the question which one of these will lead to perfect peace.
Yes face your children one to one or one to a thousand and demonstrate with the teachers ten fingers the thirty three patterns that make ten.
children need to be confident about the numbers one to ten before they realy can progress,
making the differant patterns with their own hands realy establishes those patterns in the brain.
Children copy the numbers produced on the teachers hands insticntivly, they can do it even before they identify the numbers, then tye in the pattern to the number brain wise, always use rythm to establish easy permenent memory,
and then give explanation, times tables whatever!