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mathematics in the brain - John Nicholson - 30-07-2006
WHY IS LEARNING MATHS EASY WITH AN
ABACUS
All mathematic concepts are based on numbers.
All mathematic processes have a relationship with counting.
The abacus is the world's earliest counter, the first human technical tool, over eight thousand years old,
Simply by counting numbers and then adding or subtracting the process of calculation takes place.
Those calculations are achieved by physical movement, apart from creating a number and then either adding it to an existing number or taking it away from an existing number the abacus user achieves an answer without thought, in just the same manner as we utilise a modern calculator.
But with one vital difference, the process of calculation is visual on the abacus and unseen on the calculator.
Visualisation is the secret of the abacus. My development of
ABACUS
ABACUS ONE is the perfect teaching tool.
Association with this abacus is possible as the child learns to speak.
The parent or the child simply moves up the first counter, counting and movement are combined, one two three four five six seven eight nine ten.
Perfect visualisation, we then exchange the symbol of ten, with the reality of ten counters, with our left-hand we move the symbol of ten to the top of the abacus where it represents ten, and at the same time moving ten counters in exchange to the bottom of the abacus.
Transferring Ten quickly becomes a subconscious action carried out by both teacher and pupil.
With only the written word ten in place on the abacus we continue our count, starting again with one as we move it upwards we say ten and one are eleven, continue ten and two are twelve, ten and three are thirteen, ten and four are fourteen, ten and five are fifteen, ten and six are sixteen, ten and seven are seventeen, ten and eight are eighteen, ten and nine are nineteen, ten and ten are twenty.
With two hands we exchange the symbol of twenty, with the reality of ten.
It is only when we reach 110 that we need to repeat the addition
110 + 1 is one hundred and eleven.
In ten years I have tried to improve on this layout, I have never produced any better lay out, only in this edition of explanation
Have I managed to utilise the concept of the teens.
This abacus can be produced in any language or overwritten
In stickers in any language or symbol of numbers.
This written explanation is as near perfect as any I have ever made. If you are able to improve on it please feel free.
Starting to use this abacus at the same time as the child is learning to read is a vital part of reading instruction every healthy person can drive a car, every healthy child can drive an abacus.
Comprehension of numbers takes place within the human brain in exactly the same manner as the comprehension of every other word, every number is no more or less then an idea, but an idea clearly understood by a child that is taught arithmetic on an abacus.
EVERY CHILD SHOULD HAVE ONE
mathematics in the brain - geodob - 30-07-2006
Hi John
, Good to see that the heat you're having in the northern hemisphere hasn't been slowing you down! It's been freezing down here.
Though I must disagree with your statement:"Comprehension of numbers takes place within the human brain in exactly the same manner as the comprehension of every other word, every number is no more or less then an idea.."
Also you wrote:"Visualisation is the secret of the abacus."
But this is only part of the secret of the abacus?
Visualisation can simply be recalling an image, as in a 2D photograph.
Yet in terms of Maths, it actually involves Visual-Spatial thinking.
Where the Spatial element turns the 2D Visual image into a 3 Dimensional image/s.
It is the Spatial element that our brain comprehends as quantity, rather than the purely visual. So that we can distinguish between a photo of a shoe and the real shoe. Yet in early childhood, tests have shown that children are unable to distinguish the difference and will try to put their foot into the photo of a shoe.
Which reflects a yet to be developed Spatial thinking ability.
Recent research has identified that Numbers are represented Spatially in the mind. Which does not occur with words when they are Visualised.
So that for example, to mentally picture the word; go. It is a single image.
But to mentally picture the number; 47. The 4 and the 7 are spatially located in the mental picture.
This is where the abacus is most helpful, as it develops not only a Visual concept of numbers, but more importantly a Spatial concept of numbers.
Where the Spatial directly connects to the primate sense of quantity.
The physical manipulation involved with using an abacus, also links our sense of touch with numbers. Where Touch is highly tuned to detect Quantity.
Touch is also essentially Spatial.
So an abacus creates a numerosity Loop.
So John, it seems that my disagreement, is that we need to make a stronger case for the abacus. Or maybe that's an agreement?
Geoff.
mathematics in the brain - John Nicholson - 30-07-2006
starting with the abacus was what i was about in that piece,my abacus trained daughter has had two fifteen year old friends with her here over night,
obviously the fifteen year daughter, old is mightily browned of with abacus talk, her mother worse, i am reminded of mrs marks stop talking about capital
and make some,in my case stop talking about abacus and sell some, i need something to eat.
Even with world war three going on here i managed to get my daugters two friends to understand those words of explanation.
So Blake was maybe at work someware within my brain, i would think it would be extremly unlikely that iether of us could prove our individual opoinion on what part of the brain is activated we must both keep working on this.
Visualisation we both agree that the abacus is far more then visualisation alone, the kinesthetic action of using an abacus is i belive at the heart of its abity to build effective maths memory,
I am working with my maths teaching graduate on developing a simple routine in hand demonsration routines for different age groups, she is dealing with children from four to twelve years old they all apear to have benifitted from doing these exercises, i have been evolving these exercises over a number of years and i know how usefull they are.
My latest exercise is for four year olds, after tapping the finger tips of the two hands together illustrating the equal properties of both hands, we can encourage the child to do the same, then get the child to put the little fingures together starting the two times table it is a real visualisation of the two times table up to 5 times two.
The two abaci i have got from China are perfect, have you had any good responce from the abaci i sent you, i am thrilled that you are still intrested.
a bit of cold in western aus must mean you are down to 75 degrees.
i am leaving here for China on the first of september or so they wiil have shed loads of abaci by then, do you want some. i will provide some for any public trial.
mathematics in the brain - John Nicholson - 01-08-2006
you can use a paper abacus one map to try out these instructions
WHY IS LEARNING MATHS EASY WITH AN
ABACUS
All mathematic concepts are based on numbers.
All mathematic processes have a relationship with counting.
The abacus is the world's earliest counter, the first human technical tool, over eight thousand years old,
Simply by counting numbers and then adding or subtracting the process of calculation takes place.
Those calculations are achieved by physical movement, apart from creating a number and then either adding it to an existing number or taking it away from an existing number the abacus user achieves an answer without thought, in just the same manner as we utilise a modern calculator.
But with one vital difference, the process of calculation is visual and kinesthetic on the abacus and unseen on the calculator.
Visualisation is the secret of the abacus. I believe my development of the Abacus and years of research work on developing permanent mathematic memory make
ABACUS ONE
The perfect teaching tool.
Association with this abacus is possible as the child learns to speak.
The parent or the child simply moves up the first counter, counting and movement are combined, one two three four five six seven eight nine ten.
Perfect visualisation, we then exchange the symbol of ten, with the reality of ten counters, with our left-hand we move the symbol of ten to the top of the abacus where it represents ten, and at the same time moving ten counters in exchange to the bottom of the abacus.
Transferring ten quickly becomes a subconscious action carried out by both teacher and pupil.
With only the written word ten in place on the abacus we continue our count, starting again with one as we move it upwards we say ten and one are eleven, continue ten and two are twelve, ten and three are thirteen, ten and four are fourteen, ten and five are fifteen, ten and six are sixteen, ten and seven are seventeen, ten and eight are eighteen, ten and nine are nineteen, ten and ten are twenty.
With two hands we exchange the symbol of twenty, with the reality of ten.
It is only when we reach 110 that we need to repeat the addition
110 + 1 is one hundred and eleven.
JUST HOW DO WE ESTABLISH PERMENANT MEMORY
Simply by doing something.
My conception of Abacus One education, is that it leads humanity
towards universal primary education on a step by step process.
Nothing we will ever do is easier then mental arithmetic provided we know the process, Abacus One demonstrates that process.
Simply by learning to count on the abacus we have grasped the concept and manner of addition.
Addition is at the centre of all arithmetic process, when we multiply we are simply adding one number a set number of times, in division we simply reverse that process.
Subtraction is also only addition in reverse process.
When we ask a child to takeaway five from ten, a child with its finger tips touching separates the two hands, it is by the use of the hands that we show the patterns of basic arithmetic.
Everyone learns to count on their fingers, so as we introduce a child to the abacus we illustrate counting and the concept of ten.
Ten symbols are now universal in identifying numbers, the column arrangement used on the abacus for eight thousand years uses a perfect circle to represent zero in any written representation of the ABACUS.
As we introduce any child to the abacus we naturally introduce it to written numerals, and the concept of columns, international use of numerals simplifies information exchange on a world basis, with the use of only thirty words and ten numerals we can identify over a million individual numbers. No other language is so exact or so simple, and with only three columns on Abacus One, we can ensure that every child understands the significance of the word million and the seven columns of writing or abacus needed to express it.
There are three columns on Abacus One, reminding us that there are three separate teaching/showing processes to combine.
To teach effectively we only have to remember three things.
SHOWING DOING KNOWING
Permanent memory is only established as permanent memory by
DOING
SHOWING a child how to write numbers is straightforward
The child simply copies, an example of numbers written illustrating the perception of the column, this also demonstrates the efficiency of the abacus as the abacus latter demonstrates the efficiency of written numbers.
What do I mean? I will show you, in one column on one page we write the numbers 1 2 3 4 5 6 7 8 9 0 ZERO is meaningless in this order but when we add a second column alongside and write one we have meaning.
Imagine that before the use of the abacus, to record a number we had two choices for one hundred we made that number of marks or collected one hundred stones, early Egyptians evolved marks for the decimal system which evolved into a simple working tool also developed in China and South America the Abacus, it is still in use.
Indian mathematicians evolved notation around two thousand years ago from where we now have our universal notation system.
Our number system is perfect, extremely efficient both for recording numbers and carrying out calculations, modern electronic calculators provide us all with instant answers but they do not teach the basic maths concepts as efficiently as an abacus, children brought up in countries where national versions of the abacus are still in use ( Russia China and Japan) have natural maths ability conferred on them by regular use of the abacus both in the home and in the school.
With Abacus One and efficient training every child will grasp mathematics easily. Simple explanation and demonstration will secure universal maths ability.
KINESTHETIC LEARNING
This approach is researched-based. Studies in neurophysiology have shown that physical experience creates especially strong neural pathways in the brain. When students participate in tactile/kinesthetic activity, the two hemispheres of the brain are simultaneously engaged. This type of learning experience helps assure that new information will be retained in long-term memory.
THREE KINESTHETIC DEMONSRTATIONS AT THE SAME TIME.
The first demonstration is using the abacus from the most simple counting to the most complicated sums possible.
The second demonstration is writing numbers to illustrate the sums and answers that the abacus performs.
The third demonstration uses both of the two previous demonstrations to reinforce the lessons but it is the most important
Demonstration of all. MATHEMATIC PRINCIPALS
The simple patterns of numbers, 33 ways to produce 10
Every which way to add 9 to 9 and any digit to any other digit.
Instantaneous ability in times tables.
Those three written lines are all we need to know, every other mathematic concept springs from this permanent memory we have to establish within every human brain.
Step by step number awareness is best conceived by the Childs own fingers first, then with the abacus then with pen and paper.
Naturally mothers teach their children to count, the sound of the Childs natural language naming each number, but mothers think that is sufficient they concentrate on life skills and reading.
Teaching any child perfect arithmetic awareness between four and six years of age (or at any age) is an essential stage in reading.
Every child we have taught perfect arithmetic to on the abacus has developed quickly in reading ability, the brain of a child automatically working out mental arithmetic problems obviously develops neurological pathways that are useful in reading.
ROTE LEARNING BY CHANTING IS PERFECTY NATURAL.
Parents and teachers can combine wrote learning with physical demonstration of proof in mathematics, it is the most powerful teaching format for any child learning basic skills.
Starting to count early, just tap the fingers and count in rhythm.
1 2 345 67 89(10)
This usefull starting point transfers easily to the right hand column of the abacus (the finger column) the only physical reality on the abacus. The child needs to understand this proof.
The child needs to create the thirty three patterns of ten on the hands and the abacus and with pen and paper. In this manner the child will create the mental patterns of numbers as permanent memory.
Singing the times tables and adding them on the abacus at the same time commits the times tables to perfect permanent memory.
mathematics in the brain - Maulfry - 01-08-2006
OECD expert Wrote: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
Christina - so sorry that I am only just replying to your May response (above)! Firstly, I'm not sure about the 'spacing effect' since what I have looked at on the internet appears to emphasise the significance of repetition and practice - (I wasn't thinking of children repeating or practicing tasks, but rather that children appear to understand and use new models and different ways of representing of written mathematics that a peer or adult has modelled, if they are not expected to use it immediatly (sometimes they do so after an interval of weeks or months). In contrast, my evidence suggests that when young chidlren are expected to to use what they have just been shown, they experience considerable difficulty.
However, if you could let me have any references to research papers on the spacing effect, I'd be intersted to read more!
Meanwhile, here's my references (I'm interested in research on visual representation so the following reference relate to this):
Stefan Gais, Werner Pilahl, Ulrich Wagner and Jan Born, (2000). 'Early Sleep triggers memory for early visual discrimination tasks', Nature Neureoscience 3, pp.1335 - 1339 (01 Dec 2000).
Bower, B. 'Certain memories may rest on a good sleep' http://www.sciencenews.org/articles/200000722/fob8.asp - but I have just checked this link and it is no longer there so don't have a date for it.
Stickgold, R., James, L. and Hobson, J.A. 'Visula discrimination leaning requires sleep after training'. Mature Neuroscience. Volume 3, Number 12 (December 2000), pp. 1237 - 1238.
Dr. Jan Born. Nature 427, pp. 352 - 355 (22 Jan, 2004).
Dr Carl Hunt of the National Centre on Sleep Disorders Research, believes that these recent studies on slpeep and the brain are going to have 'potentially important results for children for school performance and for adults in terms of work performance (BBC News, 23 Jan, 2004).
Interesting to see that there is now a discussion thread on this website on sleep!
mathematics in the brain - Maulfry - 02-08-2006
... and having 'slept on it' I feel that part of the problem is that studies on spacing appear to focus on learning that is aurally transmitted and I can find very little on the brain and visual representations (there are plenty on perception and visual/sight but this is not what I'm searching for).
Perhaps other possibilities might be 'consolidation theory' or 'dual store theory'? I think so far what I have found that best supports my focus is the research (above) is the need for periods of time that include sleep.
While considering this issue (how the brain processes new visual material that can later be retrieved and actively used by the learner), I would like to raise my concerns about many of the current 'fashions' in education. These include 'accelerated learning', 'brain gym', and 'learning styles' and with respect to 'learning styles' has been promoted by the DfES. My concern is that teachers and educators are devoting a great deal of time and money on these at the expense of drawing on quality research related to learning theories (e.g. socio-culturalism). The evidence to support these fashions appears to be slim - see for example the major review:
Coffield, F., Mosely, D., Hall, E. and Ecclestone, K. (2004) Should we be using learning styles? What research has to say in practice. London: Learning and Skills Research Centre, LSDA.
mathematics in the brain - Christina - 02-08-2006
Hi Maulfry,
Thanks very much for these.
This article, which applies the spacing effect to mathematics education, provides some good references: http://72.14.207.104/search?q=cache:nBoDI_ciE_4J:everydaymath.uchicago.edu/educators/Practice.pdf+spacing+effect&hl=en&gl=us&ct=clnk&cd=6
Your criticism of ‘fashions’ in education that are not research-based is an important one as this is a real problem in education. In fact, OECD/CERI’s forthcoming publication on Brain and Learning will include an entire chapter devoted to dispelling these types of neuromyths.
All the best,
Christina
mathematics in the brain - Maulfry - 02-08-2006
Hi Christina!
Thanks for reference - I recognise some of the references. Well - maybe spacing could be a part of what I'm looking at, but I'm still not entirely convinced. It seems to me this research (and the theory) related more to memorising and recalling facts.
I feel that perhaps these three things may be related:
1. 'dual store theory'
2. the research on sleep needed to process new learning
3. Also when many of us are working on something that we get stuck on (e.g. a piece of writing) stopping to do something totally different such as going for a walk or digging the garden (thereby often totally 'switching off' from the problem we had) can often help unblock things (mentally) when we return to them.
I am very encouraged to hear of your stance on dispelling neuromyths for educators and do hope you can also direct this to the DfES too. I have just been going through the draft of the Early Years Foundation Curriculum and am very disappointed to see that they refer to the need to 'meet all children's needs, learning styles and interests' (SureStart, 2006, p. 16). It's very sad that this pseudo-science is promoted by the government and also by colleges of education - equally worrying is that so many teachers and educators accept the latest fads unquestionably!
mathematics in the brain - 4th grade teacher - 02-08-2006
Interesting discussion going on about teaching math. In an effort to get to the point of some of these messages, I have skimmed their surface and perhaps haven't given justice to the arguments. So, instead of continuing a dialog that is ongoing, I will just add some brief ideas going on my head.
I recently started to read The World is Flat by Thomas Friedman. As a teacher, I am quite interested in the premise of the book- that we are taking the steps of producing commodities apart and distributing them around the world. In terms of math, businesses are giving responsibilities of memory tasks (computing easy tax returns) to people outside of the U.S. and Europe, and giving responsibilities requiring "people skills" (problem solving) to those inside the us and Europe. Perhaps this has something to do with the three ways math is categoized in the brain that Christina brought up in the beginning of this forum.
Outsourcing is becoming the way of the times, and if we are going to prepare our students to become successful in competing globally, we should educate ourselves as teachers not only in how the brain functions, but in how the division of math tasks plays out in the real world.
mathematics in the brain - geodob - 03-08-2006
Hi Maulfry and welcome back
,Though I was interested to read your dismissal of different Learning Styles. Given its relevance to your maths research. Yet perhaps it is the interpretation that DfES have on it?
But basically these different learning styles refer to Working Memory and the spectrum of different techniques that it employs for processing information from each of our senses.
Where a deficit in any of these sensory techniques, is partially compensated by utilising the other W/M techniques. Though for this to be successful, instruction needs to be offered in the appropriate learning style.
You might recall my involvement with Dyscalculia, where typically these people have a strong Verbal thinking/ learning style.
Which is commonly a consequence of a deficit in either or both Visual and Spatial W/M abilities.
Therefore if they are to successfully acquire maths skills, they need to be taught maths as a verbal rather visual-spatial process.
But on the other hand, I do share your concern about DfES recognising different learning styles. Where I suspect that they are thinking of these different learning styles, as something one is born with?
Whilst a physical deficit in the brain, may be the causative factor?
My position that I'm researching, is that more often this is a result of some yet to be developed sensory W/M skills.
The crucial issue, is that we are not born with sensory W/M skills, but that we have to learn and practise them. Where a deficit may just be a result of never having learnt the skill.
In terms of maths, I have been researching the Visualisation W/M skill, which is essential for mentally processing maths. With some successful trials so far.
But what this has highlighted, is the crucial role of Spatial thinking as both a separate and related W/M skill. Where Visual/Spatial has been bundled together. Yet they work in parallel with each other but involve different W/M skills. I'm currently developing some trials with adults to identify whether Spatial W/M skills can still be aquired as an adult.
Though Maulfry, having looked at your research examples, it is a perfect example of using an integrated Visual/Spatial approach to introducing maths.
Which links to Dehaene's research into the way that numbers as opposed to words are presented in the W/M. Where the mental 'Number Line' is not just Visually presented, but also Spatially.
Where the Spatial factor gives a number; Magnitude.
Which suggests to me that our Spatial W/M is also critical to our; 'Sense of Number'.
Though to come back to the issue of Learning Styles, what I am pushing for is a recognition that the spectrum of Sensory Working Memory Skills need to be taught. Be it directly or indirectly. Which I hope that this recognition of different Learning Styles will lead to?
Yet I noticed that you expressed a concern about BrainGym, though have you looked at the research that it adopted in its earlier days. Which some other Corporations have also adopted.
Where basically they just use simply physical exercises to develop Lateralisation [left-right brain coordination].
As well as the Vestibular balance mechanism in our ears.
Also our sense of Proprioception [awareness of our body parts in space.]
Which are all directly related our Visual and Spatial Working Memory.
So it does have some value.
Though in east Asia they have been using such techniques for centuries, where they have the martial arts of Tai Kwon Dho, Tai Chi, Jujitsui and others.
A considerable amount of Learning Difficulties that exist in Western Schools, could be largely resolved by a greater recognition of Physical Education and the Arts as critical to children's development.
Geoff.
mathematics in the brain - Christina - 03-08-2006
Thanks for this important discussion. Yes, one of the difficulties with neuromyths is that they usually develop from some truth. For example, as Geoff notes, research does support the notion that exercise supports learning functions of the brain. However, does this translate to, tapping your toes will make you better at long division? Probably not. It means that incorporating physical education into the curriculum is critical. The problem is that it can be difficult for teachers to distinguish between a fact and a commercial spin-off a fact.
Cheers,
Christina
mathematics in the brain - geodob - 04-08-2006
Hi Christina
,Whilst 'tapping your toes' might not make you better at long division.
It can be helpful in terms of literacy?
More particularly in relation to developing a 'sense of Rhythm'.
Where rhythm is an essential element of speech and words.
Perhaps as an example, you might re-read the last sentence, and pay attention to rhythm of each word, as well as the overarching rhythm of the sentence.
This is particularly relevant to the sub-types of Dyslexics with Phonetic difficulties. Where the phonetic segmentation within words, is rhythmically defined. Where a lack of a sense of rhythm, makes an awareness of phonetic segmentation within words difficult. I have read of studies where developing dyslexic high school students 'sense of rhythm' caused some reduction in their dyslexia. Drums were used for this study.
But 'tapping toes' or fingers, would no doubt be equally as effective.
Though it does also raise a question in relation to Working Memory and how it processes text in blocks/ chunks, rather than word by word. Where rhythm could be implicated in this chunking process as well?
So we better restore Music Education to the curriculum.
Geoff
mathematics in the brain - Christina - 04-08-2006
Hi Geoff,
Thanks for bringing up this interesting body of research, which I do not believe has been discussed yet on this forum. A team of researchers led by Ush Goswami at University College London are investigating this hypothesis in 10 languages. It would be interesting to assess the relative effects across languages.
For those who are not yet familiar with this topic:
This BBC article summaries the hypothesis: http://news.bbc.co.uk/2/hi/uk_news/education/2144790.stm
This PNAS article provides a more technical explanation: http://www.pnas.org/cgi/content/abstract/99/16/10911
All best wishes,
Christina
mathematics in the brain - John Nicholson - 05-08-2006
Here you are discussing rhythm and dyslexia on a website headed mathematics in the brain, I well understand that human beings are slow learners, the problem here is only of a technical nature, it is the tendency to learn one small piece of information and add it to what we already know which cannot be anything other then very little and assume we have the answer.
Individually we have the answer to very little, that is why this website is here, and some one some where is looking for knowledge without research.
We can make little mental progress unless we first of all accept our own mental deficiencies; I was over sixty years of age before I knew this obvious fact that the first four numbers we use to count with add up to ten.
Silly old sod I can hear you now,
But some of you still believe that mathematics has little to do with reading
And here you are discussing rhythm and dyslexia, on a thread headed mathematics in the brain.
Howard Gardner identified seven intelligences but within every human these seven intelligences work together, and maybe Howard missed a few hundred more intelligences that humans are capable of.
There are some things we cannot learn naturally symbol recognition is the earliest stumbling block we have, the language of mathematics is simple, we have perfected the written description of numbers it mirrors mans earliest calculator the abacus which makes the abacus itself the perfect tool to actually understand the process of calculation on.
is there any wonder then that perfecting a Childs concepts of numbers assists and corrects a Childs concepts of symbols collected together in words, but the child has to know the formulation of the alternative sound symbol alternatives just as exactly as it knows numbers.
We are under using Howard Gardner’s clearly defined rhythmic intelligence in teaching the vital steps needed to build basic skills, leave any step out of a ladder and maybe you will never reach the top.
BASIC SKILLS ARE ESSENTIAL TOOLS WITHIN EVERY HUMAN MENTAL TOOL BOX. What use is a Plummer without his blow torch or mechanic without his spanners a surgeon without his knife or a child without perfect reading and counting ability, or a teacher without minute step by step explanation?
mathematics in the brain - Christina - 14-08-2006
As we have discussed before, infants are equipped with a mathematical sense. A recent BBC article on this topic: http://news.bbc.co.uk/1/hi/health/5253040.stm
Cheers,
Christina
mathematics in the brain - Maulfry - 30-09-2006
Hi Geoff
I'm all for serious research but my concern remains the same. In England (at least) teachers and colleges of education have rushed to promote learning styles in schools. Several serious studies are now questionning their wholesale adoption and it was this to which I referred.
I recognise that there may be some valid reasons for using brain gym with some students: my concern is that every few months there is yet another educational 'fashion' that teachers rush to adopt.
I believe that teachers (and the children they teach) would benefit more if they developed their knowledge of child development, read to understand therories of learning and were up-to-date research evidence on pedagogy.
I believe that the research on rythm and music is of a different order and I know there have been a number of studies that have shown strong links between music and mathematics. It's intersting to hear that it can also be of value for those with dyslexia.
mathematics in the brain - Maulfry - 30-09-2006
John - the Russian psychologist Vygotsky described symbols as 'symbolic (or cultural) tools'. I believe that symbols can be learnt in a 'natural' way, provided that the child is able to build on her own marks and meanings and make personal sense of symbols in meaningful contexts (often in play) and for real purposes. It is the meaning that is missing from rote learning.