Learning mathematics at
school....and later on

Elisabetta
Monari Martinez

Elisabetta
Monari Martinez, Dipartimento
di Matematica Pura ed
Applicata, Universita
degli Studi di Padova,
via Belzoni, 7, 35131
Padova, Italy. E-mail:
martinez@math.unipd.it

This article
presents evidence of teenagers
with Down syndrome learning
algebra, and of one 51-year-old
learning how to count
and tell the time for
the first time, suggesting
that we need to be careful
in our assumptions about
what numerical skills
people with Down syndrome
can attain. The author
outlines the main approaches
she has drawn from working
with these individuals.

Introduction

Do students
with Down syndrome learn
mathematics?

In 1974 Cornwell[1]
said they only learn to
count by rote, with no
conceptual understanding.
These difficulties in
counting were used to
account for their general
difficulty in concept
formation and abstraction.
In 1978 Gelman and Gallistel[2]
defined 5 principles to
divide the counting skill.
In 1987 Sue Buckley and
Ben Sacks[3] published
a study on 90 teenagers
with Down syndrome who
had received little early
intervention and did not
attend mainstream schools:

* only 18%
of the sample were able
to recite numbers or count
objects beyond 20,

* about 50% could only
do some simple addition,

* few could do simple
multiplication or division,

* only 6% were able to
use money and to manage
independently in a shop.

In 1988 Janet
Carr[4] presented data
taken from a longitudinal
study of 41 young adults
with Down syndrome: at
an average age of 21,
the maths skills compared
to those of a typical
5 year old child, but
the reading skills compared
to those of an 8 year
old. In 1988 Gelman and
Cohen[5] found that children
with Down syndrome learn
the 5 principles late,
but they could be trained
by exercises (see Macquarie
programme of Thorley and
Woods, 1979[6], and research
by Joanna Nye and Gillian
Bird, University of Portsmouth,
1995[7]). In 1991 Caycho,
Gunn and Siegal[8] found
counting skill had a correlation
with the development of
receptive language and
then it could depend on
the educative program
and on the adult-child
interactions.

In 1994 Billie
Shepperdson[9] studied
the reading and number
abilities of two groups
of English people with
Down syndrome, the people
of the first group were
born in the sixties and
the other were born in
the seventies: as teenagers
the seventies group had
better performances in
mathematics than the sixties
group at the same age,
maybe due to the better
learning opportunities
and, if teaching continues,
they keep on improving
into the adult years.
In more detail, in 1995,
Nye, Clibbens and Bird[10]
found a correlation between
numerical ability and
receptive grammar. Some
researchers found a significant
correlation between mental
age and numerical ability,
others (as Baroody, 1986[11]
) did not.

For the above
reasons, special education
teachers considered teaching
academic skills in mathematics
not useful but frustrating,
and preferred to focus
on community living skills
such as the practical
use of money and the ability
to tell time.[12, 13]
This was a notable idea,
which helped many disabled
people to reach autonomy!
My questions are: Are
we sure they cannot learn
more? Are we sure about
what is basic in mathematics
and what is the best path
to follow in order to
teach to each student?

School inclusion in Italy

In Italy,
the inclusion of students
with disabilities in mainstream
classrooms has been warranted
by law in elementary and
middle schools since 1977
and in secondary schools
and universities since
1992.[14] Students with
Down syndrome may go on
to secondary school, but,
until now, no one has
been able to get a secondary
school diploma: they usually
get a certificate of their
achievements. So far no
one with Down syndrome
has attended University.
In any case, in an inclusive
environment, the teachers
and children are motivated
to try more academic skills,
following the achievements
of the typically developing
students.

Inclusion
in mainstream schools
gives the child the opportunity
to study different topics
and try various activities.
[15, 16, 17, 18, 19] The
desire to work in the
same way as their peers,
gives them the will to
do their best!

To have successful
inclusion we need two
conditions:

* to have
the freedom and the imagination
to close the handbook
of special education sometimes
and to try a real daily
adaptation of the program
of the class to the child;
to prepare individualized
tests, which have to be
taken at the time of the
class test;

* collaboration between
the teachers of the class
and the support teacher
of the child with special
needs, so that the child
is a pupil of all the
class teachers and not
only of his support teacher.
Even in severe cases,
when a support teacher
is present all the school
time, students with special
needs are very sensitive
to the class teacher’s
attitude towards them:
the class teacher has
the authority of the ‘true’
teacher and, on the contrary,
the support teacher is
considered a ‘friend’.

When these
two conditions take place
together, the child becomes
a ‘genius’
in mathematics, as nobody
would have believed before!

Despite good
intentions, successful
inclusion is not easy.
In 1985 Anna Contardi[20]
studied how the inclusion
of all the students with
Down syndrome in the middle
schools of Rome works.
In the logical-mathematical
achievements the students
with Down syndrome retained
what they had learned
in elementary school,
but did not improve. Sometimes
they did worse. The author
wondered whether these
results were due to the
characteristics of students
with Down syndrome or
to the manner of inclusion.
We believe we have to
adapt inclusion to the
character of the students,
and the teacher’s
expectations of the student’s
abilities play a basic
role in improving learning.
Hence better formation
of the teachers is needed.
On the other hand, Contardi
tested the logical-mathematical
achievements mostly with
indicators of arithmetical
skills, where students
with Down syndrome have
more difficulties.

Our experience

Since 1994
I have been counselling
teachers in schools to
adapt the mathematical
program for students with
Down syndrome. Usually
the parents contact me
at the Associazione Down
Padova, to which I belong
(also as a parent) and
ask me to meet the teachers.
Often my intervention
is not limited to mathematics,
but involves the entire
curriculum and other aspects
of inclusion. This activity
has allowed me to follow
up 20 cases of students
with Down syndrome, included
in regular classrooms,
as summarized in the box
below.

This evaluation
was done taking interviews
with teachers and parents,
school records and homework
into consideration.

20 students with Down
syndrome attending regular
schools

* Chronological
age: from 7 to 18 years

* Gender: 14 males and
6 females

* School: 12 in elementary
school, 6 in middle school,
2 in secondary school.

* Good social integration
in the school: 18 students

‘Good social integration
in the school’ means
the student remains in
his or her classroom most
of the time, is happy
to stay there (often does
not want to leave), is
welcomed and loved by
his or her peers and participates
in the social activities
of the class.

* Follows most of the
class programme with some
changes: 11 students (5
females)

‘Follows most of
the class programme’
means the student follows
the class programme, adapted
at a lower level, with
simplifications and changes
and takes class tests
(with simplified tests).
In some topics the programme
can be shifted backwards
usually one or two years
to fill some gaps. Even
with these limits, they
improve their mathematical
academic skills, as nobody
would have believed before.

* Follows partially the
class programme: 6 students

‘Follows partially
the class programme’
means the student follows
the class programme less
than the above group,
this is often the choice
of the teachers or for
temporary problems and
not due to the student’s
difficulties.

* Follows little of the
class programme: 3 students

‘Follows little
of the class programme’
means the student has
particular difficulties
and needs special education
programmes, different
from the class programme.
In spite of these limits,
he or she shares many
activities with the class
(social, musical, sports,
and recreation).

* Severe speech problems:
6 students (1 deaf)

* Other problems in addition
to Down syndrome: 6 students
(2 with hyperactivity,
1 with depression, 2 with
relational problems, 1
autistic).

The adapted
mathematical programme
is usually followed conforming
to the global programme,
but differs for two students
who are weaker in mathematics
that the others and two
students who are stronger
in mathematics.

Hence we have
students with Down syndrome
who learn to solve problems,
to use fractions, to solve
algebraic expressions
(see Figure 1), to measure,
to solve geometric problems,
to draw geometric figures
and diagrams, to use the
computer. They are given
slightly different tests
when tested with their
peers and if there are
difficulties in mental
calculation we suggest
visual prompts, such as
some simple memory device
to carry, or a pocket
calculator.

Example algebra worksheet
for 15 year old

Example algebra worksheet
for 15 year old

Figure 1: Examples of
work by Italian teenagers
with Down syndrome learning
algebra [24]

In difficult
cases, the desire to use
money or the ability to
tell time gives them the
motivation to begin to
study mathematics and
enjoy it.

Since 1996
I have organized and managed
continuing education courses
at the Associazione Down
Padova for adults with
learning difficulties.
In a course for illiterate
adults, a 51-year old
man with Down syndrome
is learning to read, write,
count and tell the time
(see Figure 2).[21] He
is so enthusiastic to
learn and that gives us
a joyful feeling! Other
adults maintain and improve
their mathematical knowledge.
Adults who have attended
mainstream schools have
much more mathematical
ability than those who
have not.

Example worksheet for
the number 7

Example worksheet for
counting up to 5

Figure 2: Learning to
count at 51 years of age

What difficulties (and
strengths) do we observe
in mathematics?

* Difficulty
in reciting fluently the
sequence of numbers beyond
20 for errors in the change
of the tens. The use of
rulers can help.

* Difficulties in understanding
the decimal system and
the positional value of
the digits.

* Difficulties in counting
backwards.

* There are no difficulties
in learning the procedure
of counting objects.

* Difficulties and slowness
in remembering multiplication
tables and in doing mental
calculations. When the
aim is not the operation,
we suggest the use of
a pocket calculator.

* There are no difficulties
in working on sets, which
means the classification
of objects according to
one or two or more characteristics,
and in logic, i.e. negation,
relations, tables of true
and false, short chronological
sequences, cause-effect
relations.

* In problem solving,
there are no difficulties
if we first teach the
students to visualize
the problem, by making
a sketch or by objects,
and write the arithmetical
solution afterwards.

* Difficulties in measuring
lengths because they do
not properly fix the zero
point of the ruler.

* The students learn the
procedures slowly, but
when they have learned
them, they perform the
procedures carefully and
in the right order. In
the beginning, a visual
representation of the
procedure helps.

* There are no limits
of age to learn mathematics:
students need the opportunity
and an aim, for instance
autonomy.

These difficulties
were observed in many
students, though not in
everyone, and sometimes
they overcame them. For
instance there are students
who do mental calculations.

What are the possible
reasons for these difficulties?

* Problems
in short term memory (span
and organization)

* Problems in long term
memory or in explicit
memory

* Possible problems in
receptive grammar, i.e.
in understanding sentences
[10] and in language

* Instability of learning,
i.e. sometimes students
forget what they have
learned. Hence special
attention should be paid
to ensuring that learning
is strongly consolidated

* Delayed maturation:
progress is often observed
into the second decade
of life, in areas where
learning is usually completed
into the first. This suggests
that educational efforts
should not be abandoned
after the early years,
but have to be continued
through the teens and
after

* There are problems in
remembering all the items
of a sequence, but, after
they are learned, there
are no problems in remembering
the right order of a sequence

* There are no problems
in implicit memory, hence
they learn well by ‘doing’

* Problems of self-esteem

Discussion

Sometimes
the difficulties in arithmetic,
the slowness in learning
and the difficulty in
retrieving what they have
learned, discourages teachers
from continuing in the
study of mathematics,
and they prefer to go
back and repeat what the
child is not able to do.
I do not agree with this
attitude, because they
need to do exercises,
but they also need to
have a positive image
of themselves: hence it
is better to follow their
interest and go on with
the program, helping them
in what they are not able
to do or allowing them
to use some device to
fill the gaps.

What is basic in mathematics?

Usually people
believe that arithmetical
skills are the basis of
the entire mathematical
knowledge, because they
are the first step in
learning mathematics and
because all people know
them. This is illustrated
in the ‘old’
maths tree in Figure 3.
Arithmetic is important,
but there is much more
that can be learned satisfactorily,
and supports such as calculators,
visual and other aids
can be used to fill the
gaps for the arithmetical
difficulties. Persons
with Down syndrome have
notable logical abilities,
abilities in space organization,
in performing procedures,
in understanding and using
symbols: these are basic
abilities, more important
in daily life than arithmetical
skills, and taking these
as a basis, mathematical
knowledge will be built
up. This suggests a different
conception of maths, illustrated
in the ‘new’
maths tree in Figure 3.
Here arithmetic is seen
as one branch of maths
but not a prerequisite
for learning other aspects
of mathematics. I have
reached this conviction
by observing persons with
Down syndrome and by listening
to their teachers and
parents, and I am doing
research in this area.

Example numeracy tree
- old

Example numeracy tree
- new

Figure 3: The 'old' maths
tree and 'new' maths tree

How can we stimulate mathematical
learning?

The more important
motivation to study mathematics
is to learn what everybody
else is learning, which
means to be included in
the mainstream school.
Other stimulation can
be given by social or
individual games[22] and
by aims of autonomy, also
for children in school
(for instance, to learn
the schedule of the day
and of the week, calendars,
telling time, use of money.[23])
It is important also to
give an active role to
the child and believe
in his/her abilities.

Conclusion

Most of the
students have a successful
inclusion in school and
follow the class program
in mathematics, adapted
at a lower level, with
simplifications and changes
and take class tests (with
differentiated tests).
Usually they are not weaker
in mathematics than the
other subjects: it may
happen, but the converse
may also occur. We have
to change our attitude
about what is basic in
mathematics and give more
value to logical and mathematical
abilities which do not
involve computations.
There are children who
are good in mathematics
and who like mathematics.
Courses of mathematics
for adults are also useful,
because they can learn
what they have not learned
before. There are no limits
of age in learning mathematics.

Acknowledgements

Many thanks
to the persons with Down
syndrome, to the parents,
to the teachers of the
schools, to Anna La Rosa
and Francesca Cella, teachers
in the continuing education
courses of the Associazione
Down Padova.

Resources

For information
about available resources,
see the Number resources
list.

References

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