Overview of Site
New neuroscience findings illuminate the functions of specific brain
centers in learning math -- in both typical learners and those with different
math learning difficulties (see Introduction this
page).
Math learning difficulties, affecting more than 3 in 10 students in
school today, are due to:
• Poor preparation
for current studies due to inadequate learning of the subject at an earlier age.
Category includes poor school or home environment, etc., but not factors below.
• Dyscalculia,
a specific math learning disability of neurological origin, affecting 6% of the
population.
• Math anxiety,
including any negative emotional reaction to learning math. Can be a primary cause
if significant enough by itself to limit educational or career opportunities.
• Dyslexia, also
affecting 6% of the population. Not only affects reading but causes significant
problems learning math, with own cluster of symptoms.
• Low IQ (<80), affecting 9% of the population.
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New:
Welcome to the new forum for math learning difficulties, for
students, parents and educational professionals.
We value your input:
If you would like to post information or report experiences helpful to others,
or ask a question, please open the comment
box. Some previous comments are posted at lower right.
How children naturally develop math skills and where difficulties arise: Introduction.
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INTRODUCTION
Natural Development of Arithmetic Skills
The most exciting discoveries to come from the new field of cognitive mathematics
include how human children naturally develop mathematical skills, which different
brain centers are involved, and how.1 It is very important in understanding
math learning difficulties to first learn how it goes when it goes well!
The first arthmetic skill naturally present in human children and
most animals from birth is the ability to "subitize." Subitizing
is knowing from a brief glance, accurately but without counting, how many objects
of a kind are present in a small group. Obviously this ability evolved due to
the need to make quick survival decisions when faced with several predators, etc.
Typical human children can subitize accurately 3-4 objects from infancy. By adulthood,
with the addition of the skill of counting from visual memory, the number increases
to an average of 7.
Subitizing has been shown to rely on a area of the brain called
the intraparietal sulcus (IPS), part of the brain's innate "Approximate Number
System" (ANS). This system gives us the ability to understand number magnitudes and
to "estimate" -- to approximately compare relative numbers of
even large groups, instantly and without counting. Researchers have concluded
that this area is vital to the function of an innate "sense of number"
or numerosity.
The next skill to be learned for human children is counting. This
skill is partly cultural, as children effortlessly seem to learn the words for
numbers from their society as they learn to talk, and partly natural, as children
seem naturally to learn to count using their fingers. At least, they are observed
to do so, and the brain center involved in exact counting is located near the
center associated with finger control and sensation (and relatively distant from
the "number module"). The sketch of the brain gives an idea of the relative
locations of areas involved with acquiring a sense of number.
A sense of the relative relationship between number magnitudes --
a sense of the "number line"-- is the next ability to develop. This requires developing
connections between the number module (IPS) and the counting area. Because these
two brain regions are relatively distant, this may be a potentially difficult
ability for the brain to develop fully.1
Children next seem to acquire, naturally and without
specific training, the ability to subtract numbers by counting between
them. For both adding and subtracting, they usually learn on their
own to choose the most efficient counting method for a particular
case -- a choice which depends on the relative magnitude of the
two numbers. This second skill obviously requires quickly comparing
number magnitudes using the number line sense.
After this point, acquiring subsequent arithmetic skills becomes
largely cultural, as children begin to learn these skills from the adults in their
society-- skills such as multiplication, division, manipulating fractions, etc.
And here is where the trouble most often begins, as many factors can interfere
with this learning, including complete lack of educational opportunity, poor school
environment, poor teaching techniques, poor attitudes toward math by teachers
and/or parents, inability of parents to help explain how to do these skills, and
so on -- in addition to significant neurological dysfunctions affecting about
1 in every 5 people.
Main Causes of Math Learning Difficulties
More than 30% of students2 in school today have significant
difficulties learning math, in spite of normal or above-normal intelligence. There
has been a wide range of problems or "symptoms" observed within this group, leading
educators initially to propose that there were a number of different types of
math learning difficulties.3 However, a careful review of the recent
literature4-7 suggests that most symptoms can be ascribed to one of
four main, well-established causes, each having its characteristic cluster of
symptoms.
• The largest fraction of the students
having math learning difficulties suffer from
indequate
preparation or ineffective early education in the underlying basic
math operations that are required for math studies at their current level (or
for their desired education or occupation, activities of daily living, etc.).
• A specific math learning disability of neurological,
genetic/developmental origin (called dyscalculia)
affects 5-7% of the population. 8
• "Math anxiety," including any negative emotional
reaction or attitude toward learning math, can be considered a primary cause if
significant enough by itself to limit educational or career opportunities.
• Dyslexia not only affects
reading but contributes to significant problems learning math. It affects the
same number of people as dyscalculia 9 but causes a different cluster
of math learning difficulties.5
Relationships Among Causes
The category of poor preparation includes causes like poor teaching
techniques, poor school or home environment, and so on. A student would not be
placed in this category when one of the other causes listed above is the primary
cause for the existence of poor preparation. Similarily, the diagnosis of
math anxiety only applies when it is the primary cause of the poor performance
in math.10 Such distinctions are important because choosing the most effective
remedial action in an individual case depends very much on establishing the primary
cause.
Two of the specific conditions listed above do occur together: over
half (60%) of people with either dyslexia or dyscalulia have both.
Learning math is doubly difficult for this group11, because each dysfunction
results in a different cluster of symptoms, which therefore are additive.5
One may well ask how these two syndromes, each affecting different
brain centers, can both occur separately but do occur most often
together. One way that this could occur would be for both dysfunctions to require
two genetic or developmental factors, with one (and only one) factor ("A")
being common to both dysfunctions, the other two factors ("B" and "C")
varying independently.
The resulting proportions of each syndrome are llustrated in the
figure on the left. The area of each colored box is proportional to the number
affected out of 100 (top box).
In sum: out of every 100 people, 7 have dyscalculia and 7 have dyslexia,
in agreement with current estimates. The total number affected is not 14, however,
but rather 10, because 4 people by chance end up in both groups, and so are doubly
affected.
Distribution of Cause with Age
Good assessment takes into account both the age of onset of the
learning difficulty and the fraction of students expected in different categories
at a given age. For example, dyscalculia affects people from infancy or before12
and, while its effects can be moderated with remedial education, the underlying
condition appears to remain throughout the affected person's lifetime.
In constrast, math anxiety is unknown in normal young children,
who are as eager sponges for number information as they are about every other
aspect of life. Students who test as math anxious -- who "hate math"
or consider themselves poor at math -- begin to appear about fourth grade. The
number affected grows steadily through high school and then levels off.
A sketch at the left shows the estimated distribution of the different
causes of math learning difficulties in the population as a function of age. At
the bottom of the sketch is represented the 9% of the population who have an IQ
less than 80, usually sufficent to interfere with normal math learning. The illustration
as drawn suggests that there is no change with age in the fraction of population
with this condition.
Next come the cases of dyslexia and dyscalculia as discussed in
the previous section. About 3% of the population have dyslexia alone, 4% have
both dyslexia and dyscalculia, and 3% have dyscalculia alone; these fractions
are also constant with age.
A small fraction of preschool children are put into the category
of poor preparation when extreme environmental deprivation (and only that) prevents
them from the natural learning of numbers and counting that normally occurs before
starting school. Once school begins, poor classroom environment and poor teaching
can and, unfortunately, do add to the fraction of students affected. The rate
of loss of children to this factor seems to peak at the times when formal manipulations
are introduced such as learning muliplication tables, "long" subtraction, multiplication
and division; and, to a large degree, manipulating fractions.
The high school student or adult affected by poor preparation usually suffers
from not ever having really understood or become efficient at the most basic operations
supposed to have been learned in grade school. The fraction of people affected
by poor preparation is therefore generally set by the end of grade school and
remains approximately constant through high school and beyond.
The introduction of formal math operations, and the way they are often
taught, are also responsible for the appearance of "math anxiety" beginning
about fourth grade. As will be seen, students very often become adverse to learning
math or afraid of taking math tests when they are faced with a difficult subject
combined with a poorly trained and/or overcritical teacher. As defined here, "math
anxiety" includes all negative emotions and attitudes about math but it is to
be considered a primary cause only when it by itself interferes with learning
math, taking math tests, taking math classes, or when it restricts educational
or career choices otherwise desirable to the person.
The category in the middle of the figure includes all cases of
acquired dyscalculia, properly termed acalculia. These
are syndromes with similar effects to early-onset developmental dyscalculia but
caused by head injury, stroke or other illnesses. One must also include in this
category any other causes of math learning difficulties presently unknown but
which do not develop early, e.g., a "late-onset" type of developmental dyscalculia
At the very top of the figure is placed, for completeness, other
possible neurological conditions which may affect math learning ability. These
syndromes include attention deficit disorder (ADD), attention deficit hyperactivity
disorder (ADHD), and developmental dyspraxia. This last is a disorder affecting
motor coordination and body awareness. This particularly can affect math learning
if it includes finger agnosia, a lack of awareness of the fingers which may interfere
with the ability to learn to count. At present, the overall impact of these different
conditions on math learning is unknown, as is the degree of overlap with the better-known
neurological conditions, dyslexia and dyscalculia.
An Apparent Contradiction
The figure above illustrates how two often
quoted (but apparently contradictory) statistics can both be approximately
correct: These statistics are:
1) More than 30% of students in school today
suffer from significant math learning difficulties not attributable to low IQ;
2) More than half (60%) of American adults are deficient in math ability.
As the figure is drawn, averaged over ages 5-18, about one-third
(33%) of students with IQ above 80 suffer from dyslexia, dyscalculia or poor preparation,
with math anxiety adding relatively little. By adulthood, now including significant
contributions from low IQ, math anxiety, acalculia and other causes, the figure
is in agreement with the estimate that about 60% of the adult population suffer
from significant math learning difficulty.13
From Here:
Further discussion of each of the four main categories of math learning
difficulties for students with normal IQ are found in sections listed in the Directory
at the top right of this page. For each category, the latest research on causes
and possible remedial actions are explored.
The last section discusses the overall process of diagnosis and
summarizes the choice of interventions for individuals, parents and professionals.
How do you determine exactly what you, your child or your student has, and
what can you do about it?
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Comments
Special Needs Educator / Adult Dyscalculic re: Subitizing Practice
I have been doing the sub program. Not sure if I'm getting
any better at math but I am better at recognizing the number of
objects up to 5 or 6. Four has become a piece of a cake whereas
it wasn't before.
I tried using the program with one of my students... Unfortunately
the exercises didn't hold his attention or provide the kind of
immediate feedback kids get with video games. I'm not sure most
kids would work through the program. That's a shame... I recommend
providing more positive feedback between levels and dividing the
levels .. so that there is a chance of accomplishment sooner.
-- Dawn Romano at All-Ways
Learning, New Orleans.
Number Line Should be Vertical
...My experience developing math approaches over the last
27 years had led me to the conclusion that the horizontal number
line should be banned. Apart from the fact West being negative
and East being positive makes little sense in a modular sense,
most if not all children perceive higher and lower numbers as
they relate to the real world in a vertical plane.
Has anybody else researched whether or not horizontal number lines
are associated with directional confusion? Thank you,
Jonathan Crabtree, Project Director, Australian
Numerals
----Many neurological conditions -- dyscalculia and dyslexia
among them -- can cause right-left confusion. My students with
dyscalculia applaud your idea, and it seems to help them, but,
unfortunately, they still have to be able to contend with the
prevailing approach at school.-- Ed.
(The above comments were originally posted to the
Dyscalculia blog.)
Many Students Learn Math by Memorizing Procedures,
Don't Understand Concepts
... In my
experience many (students) have an acquired difficulty with math;
they have been surviving (by) memorizing procedures. Many of them
don't comprehend what numbers mean, and things break down a bit
with multiplication and completely with division. Others have
things collapse sooner or later along the way. Pre-algebra is
sometimes survivable if they can memorize enough rules... Just
before the holiday break two staff members approached me in confusion
because they didn't understand why a question about the area of
something was expressed as in^2 . For the one, a reminder of the
difference between one dimension and two sufficed; for the other,
it took some explanation and visual demonstration. These folks
passed their math courses. I've been entertaining the hypothesis
that an awful lot of students have fallen off the concept train,
and an article about why students struggle with concepts will
be up on my blog (by someone doing research on the topic; not
just my blatherings) before the year is out...
Sioux Geonz
(Please send us your blog address!)
Subtraction
What are some of the causes of pupils inability to do simple
subtraction?
Shei Osman
---Young children spontaneously learn to add using their
fingers, and learn to subtract the same way. They usually also discover on their own
the most efficient methods to do each, which
depends on the relative size of the two numbers. (If they
do not, this may be an early sign of a possible problem with basic understanding
of numbers, including dyscalculia.) When in school, they are usually
encouraged to remember (memorize) the addition "facts" and once they do they
can do addition "in their head" and subtraction by the reverse
process.--- Ed.
Dyscalculia Test
Here's quite a comprehensive test for Dyscalculia.
Hope this helps. http://dyscalculianomore.com
Dyscalculia test
---This is not a test, but a questionaire designed to lure
parents into buying software (The
Number Race) which is not only available for free from the
authors but is NOT specifically designed to treat dyscalculia,
which in any case probably cannot be "cured" but only compensated
for.
We want your input: If you would
like to pass on information or experiences helpful to others,
add a comment or ask a question, please open the comment
box.
INTRODUCTION:
Notes and References
1. Sousa, "How the Brain Learns Mathematics," (2008).
Excellent introduction to recent research findings with concrete applications
to teaching students. This book, as does (7), begins with a review of
why achieving math competency is important for individuals in modern society
and for society.
2. Statistics are interpreted in human terms in these notes when appropriate.
For example, this statistic suggests that math learning difficulties seriously
affect more than ten children in a typical classroom containing 33 children.
The estimate is from the Introduction to (4) below.
3. To hypothesize that the variety of symptoms observed in different individuals
represents different causes is a common pitfall during early stages of development
in any scientific field. In reducing the bewildering amount of data and theories
to this short list of primary causes, I have primarily relied on the most recent
results by research scientists in cognitive mathematics, particularly the reports
immediately below (4-7).
The situation in dyscalculia today is similiar to that in the
history of research in dyslexia. In the early days it was suggested, because of the wide variety
of symptoms expressed, that there were several distinct subtypes of dyslexia. However, now it is
generally accepted that all dyslexia is caused by a defect in a particular brain center called the
phonological module. The wide variety in the way that different symptoms are expressed can be thought of as
simply a consequence of the complexity of the brain, that different people have different cognitive
strengths and weaknesses due to multiple genetic factors, that all these factors act to modify the effect of
the single phonological defect.
4. Berch and Mazzocco, Eds, "Why is math so hard for some children:
The nature and origins of mathematical learning difficulties and disabilities"
(2007). A collection of excellent review articles by active researchers.
5.Landerl, Fussenegger, Moll, Willburger, "Dyslexia and dyscalculia:
Two learning disorders with different cognitive profiles," J. Exp. Child
Psychology, 103 309-324 (2009).
6. Fischer, Gebhardt, Hartnegg, "Subitizing and visual counting in children
with problems in acquiring basic arithmetic skills," Optom. & Vision Dev.,
39 24-29 (2008).
7. Butterworth, Varma, Laurillard, "Dyscalculia: From brain to eduction,"
Science, 232 1049-1053 (2011). Excellent comprehensive review of
recent research on this specific math learning disability followed by practical
suggestions for remedial education, including reviews of available training software.
8. Two children in every classroom!
9. Despite there being virtually the same number of people affected by
dyscalculia as dyslexia, and the obvious importance of mathematical ability to
modern life and commerce, NIH over the last decade has spent fifty times
more money on dyslexia than on dyscalculia research (7).
10. For example, dyscalculics are usually anxious about math for good reason,
it is very difficult for them -- but their math difficulties are the reason
for their emotional reactions and not the other way around. See Letter to My Math Teacher.
11. On average, every classroom has one child with this serious
"co-morbid" condition.
12. At least for the established "early-onset" type of dyscalculia (7). Research
has not yet ruled out the possibility that there exists neurological dyscalculia
which produces effects which are only evident at later ages.
13. 60 of every 100 American adults say they "hate math" or are "not good at math."
This is many times the number who say the same for reading.
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