5:00AM Tuesday
May 20, 2008
By Vaimoana Tapaleao
Children at the lower end
of a streaming system at school do worse, but it has no affect on brighter
students, a British study has found. Less-able
children achieve poorly when they are placed with
children of the same ability but achieve better if they study with
the rest of the class. More-capable students perform well regardless of whether
they are in an exclusive class of high achievers or taught in a mixed-ability
class, according to the latest reports of the Primary Review - led by Cambridge
University.
Auckland Primary Principals Association president Owen Alexander says
streaming children can be
"devastating". "You end up labelling them. Even though
schools don't tell students which class is the lowest and the highest, kids
know. For those at the bottom, their self-esteem can be affected," he
said. Mr Alexander said he did
not know of any primary schools with a streaming system, and said there has been a move away from it in secondary schools. "They [kids] come out of primary school, where they are
encouraged. But when they get to high school, streaming definitely has limited them." Children are
usually placed in ability groups according to their performance in national
tests or other assessments. However, those with behavioural problems are often
placed in the bottom groups no matter how highly they achieve.
Secondary Principals Association president Peter Gall acknowledged
that streaming makes no difference to a motivated
child. Mr Gall, principal of
Papatoetoe High School in Manukau City, says the new curriculum has had an impact
on the way schools function.
"Streaming is probably something that has happened as a
result of NCEA, which virtually organises students into streaming subjects like
maths, English and science." The study found that while in theory children
should be able to move between the groups, in practice children rarely move,
damning pupils in the bottom groups to low achievement.
Professor John Hattie, of Auckland University, who has researched
classroom composition and peer effects, says schools would do better to re-group students
once in a while, to prevent "locking" them in. He said there was also "a
difference in the distribution of teachers".
"There certainly was a tendency for better-quality teachers
to be assigned to the top classes," Professor Hattie said. The reports, by academics from the
University of London's Institute of Education and King's College, also called
for smaller classes within schools, arguing that pupils can fall behind when they
are forced to move to larger classes as they progress through the school. They called for classes of fewer than 25 pupils for the youngest
children and for first-year secondary school students, to stop pupils falling behind when they
transferred from primary to college.
The University of Auckland's dean of education, Dr John Langley,
said although most people thought a smaller class would be better for a child, class size
was not the issue, rather the teacher was. "You
can have a class of 25 kids or fewer and a bad teacher, where on the other hand
you can have a class of around 30 kids and a good teacher, who can still handle
and teach those kids well. It's really around the
quality of the teacher," Dr
Langley said.
- Would a good teacher give 25 children a better educational outcome, than 30 children? In fact would a bad teacher provide better for a class of 25 children, than a class of 30 children? Would any teacher, good or bad, do better with less children?
Together literacy and
numeracy takes up the greatest proportion of the school day – one UK commentator has
said that ‘literacy and numeracy are the evil twins that have gobbled up the
entire curriculum’. Sir
Ken Robinson writes that schools only ‘mine’ children’s heads for literacy and
numeracy ignoring other important learning areas – he believes creativity is as
important as literacy and numeracy. Guy Claxton
echoes Sir Ken’s thoughts saying that ‘learnacy’ is as important as literacy
and numeracy.
For all this emphasis too many students leave
school unable to do basic numeracy and see themselves as maths cripples.
I have always believed two things have caused this
situation: an emphasis on
‘school’ rather than ‘real’ maths and the use of ability grouping.
When I taught I refused to use ability grouping and did my best to introduce
real or interesting maths topics to my class– integrating maths, where
possible, with the current inquiry study. To cover myself we also did basic
computation daily but I taught the students that this was to be seen as
‘practice’ not ‘real’ maths. The majority of the learning in my room was centred around the inquiry
based study topics we chose. Literacy and numeracy were seen as ‘foundation skills’
necessary for students to complete quality learning in selected study areas -
as well as for their own intrinsic value
I wish I had access at the time to Jo Boaler’s book
(and also the research of Australian maths educationalist Charles Lovett
) but I gained support from computer educationalist Seymour Papert
who had written that ‘all science and maths ought to be applied not pure'. Dr
Z P Dienes (of Dienes blocks) had also written that it is time to 'shift
from teaching to learning, from our experience to the children's, in fact, from
our world to theirs'. I had also come to the conclusion that the secret was to
do fewer maths topics well – a lesson we now are now learning from Japanese
maths teaching.
Jo Boaler writes, ‘far too many students hate maths. As a result adults all over the
world fear maths and avoid it at all costs…. It’s the subject that can make
them feel both helpless and stupid….Maths more than any subject has the power
to crush children’s confidence.’
It’s time to remedy this situation and her book
provides the knowledge and the details of the real maths children should be
learning. The ‘elephant in the room’ is
the belief held by teachers that some people can do maths and some can’t and
so, to help ‘slow’ learners, children are grouped by ability. As a
result the maths that is taught, she writes, has little in common with real
maths. When ‘real’ maths is taught in context many more children are
successful.
But changing maths teaching has proven to be very
difficult and as a result ‘many children are still subjected to an outdated and
narrow form of teaching. With
the imposition of national testing in the UK English students’ results (along
with students in the US) are falling behind other countries and, as well, more
student end up disliking maths. And to compound the problem ability grouping is increasingly being seen as
the ‘way’ to teach maths. Ironically these ‘failing’countries are the
ones our government is emulating with its standards agenda!
The purpose of Boaler’s book is to make students and teachers
excited about maths and to ensure more students are competent, equipped for a
future that will require maths ability.
A key question to ask is what is maths? Real mathematicians see it as a
study of patterns and relationships –a way of thinking and making meaning.
One clear
difference between the work of mathematicians and school maths is that mathematicians work on tasks
that take a long time to complete usually collaborating with other mathematicians to devise
solutions while at school students work alone on short ‘unreal’ questions with known answers. Teachers of‘school’ maths might be
surprised, Boaler writes, to know that mathematicians involve themselves in
guessing at first and then setting about problem solving. This provides a clue
for teachers who want to have their children working like junior mathematician;
an approach encouraged by American educator David Perkins.
‘Bringing
maths back to life for school children involves giving them a sense of living
mathematics….posing and extending problems of interest
to students means they enjoy mathematics more, they feel ownership of their
work and they ultimately learn more’.
There seems like the need to make school maths a
perpetual ‘maths fair’. ‘If
students were to work in the ways mathematicians do, for at least some of the
time- posing problems, making guesses and conjectures, exploring with and
refining ideas, and discussing ideas with others they would be given a sense
of true mathematical work….to enjoy mathematics…and learn in the most
productive ways.’
Check out this video clip of good maths teaching
Check out this video clip of good maths teaching
The biggest problem in
school work in the UK, Boaler believes, is the desire to label children and assign them to a level
(standard) and teach them in ability groups. Recent UK
government policies, she writes, has ‘pushed the situation out of control’.
Schools are now deciding who can and can’t do maths when they are only four
years old and research show that most children stay in the low groups until they
leave school. This
grouping ignores all that is known about the variety in the individual
development of children. Since target setting has taken over in schools
children in the UK have dropped from eight to twenty fourth in international
tests of mathematical problem solving. A lesson for New Zealand! Boaler
believes it is important to replace this target driven approach, which labels
and prejudges children producing ‘can’t do’ maths students, with real maths
experiences.
What is
required, she
writes, ‘is to provide
stimulating environments for all children; environments in which children’s
interests can be peaked and nurtured, with teachers who are ready to recognize,
cultivate and develop the potential that children show at different times and
different areas.’ This is difficult if children are placed in low
groups – such students come to believe in their lack of ability. In NZ - once a
weka always a weka!
Such destructive ability grouping is not used in
high achieving countries like Finland and Japan where
schools communicate that everyone can be good at maths and teachers work hard
to see it happens – working collaboratively to plan exciting maths and doing
fewer things well.
It appears to me that in New Zealand schools the
use of ability grouping is unquestioned and that many schools, following the UK ‘lead, are’
moving into greater setting and streaming for maths. Boaler points out that the
International Mathematics and Science Study shows countries ‘who leave
grouping to the latest possible moment or who use the least amount of grouping
by ability are those with the highest achievement.’ Japanese educators are
‘bemused by the Western goal of sorting students into high and low abilities’.
In Japan what is important is balance. ’Everyone can do everything…..so we
can’t divide by ability…Japanese education emphasizes group education, not
individual education…we
want students to help each other, to learn from each other…to get along and
grow together.’Add to this teachers who plan maths collaboratively and do
fewer topics well and you have recipe for success.
‘Research’, Boaler writes,’ tells us that
approaches that keep students as equal as possible and that do not group by
ability not only helps those who would be placed in low sets, which seems
obviously, but those who would be placed in high sets too.’
‘Researchers consistently find that the most
important factor is what they call opportunity to learn. If students are
not given challenging and high level work then they do not achieve at high
levels.’ ‘Students who
struggle in mathematics are helped by engaging in discussions about maths with
students who are working at higher levels.’ ‘In Japan students are
not all expected to learn the same thing, which the unrealistic
expectations in many countries; instead they are given challenging problems and each student get the most that
they can.’
This reflects my own teaching approach.
Boaler writes ‘the impact that ability grouping has
upon students’ lives – in and beyond school – is profound. Researchers in England found that 88%
of children placed in ability groups at age four remain in the same groupings
until they leave school. This is the most chilling statistics I have ever read.’
For schools to make these sort of grouping decisions, she writes, ‘is
nothing short of criminal.’
Only by
offering opportunities for rich challenges, by flexible system of grouping that
does not pre-judge a child’s achievement, and by using multi-level mathematical
that each student takes to the highest level will we ‘encourage smart and
competent maths learners in our classrooms.’
"the opportunity to learn" - if students are not given opportunities to learn challenging and high level work then they do not achive at high levels.
when teachers have low expectations for students and they teach them low level work, the children's achievement is suppressed
students who struggle in maths are helped by engaging in discussions with students who are working at higher levels
in a mixed ability group the teacher has to open the work, making it suitable for students working atdifferent levels and different speeds - work should be multi-leveled so students can work at the highest levels they can reach
Maybe it is time
for New Zealand schools to re-think their use of ability grouping in
mathematics- and reading?
Porritt??
Why Do Problem Solving?
Using a problem solving approach to teaching and
learning maths is of value to all students and especially to those who are high
achieving. The reasons for using problem solving are summarised below.
- Problem solving places the focus on the student making sense of mathematical
ideas. When solving problems students are exploring the mathematics
within a problem context rather than as an abstract.
- Problem solving encourages students to believe in their ability to think mathematically.
They will see that they can apply the maths that they are learning to find
the solution to a problem.
- Problem solving provides
ongoing assessment information that can help teachers make instructional
decisions. The discussions and recording involved in problem solving
provide a rich source of information about students' mathematical
knowledge and understanding.
- Good problem solving activities provide an entry point that allows all students to be
working on the same problem. The open-ended nature of problem
solving allows high achieving students to extend the ideas involved to
challenge their greater knowledge and understanding.
- Problem solving develops
mathematical power. It gives students the tools to apply their
mathematical knowledge to solve hypothetical and real world problems.
- Problem solving is enjoyable. It allows students to work at their own pace and make decisions about
the way they explore the problem. Because the focus is not limited to a
specific answer students
at different ability levels can experience both challenges and sucesses on
the same problem.
- Problem solving
better represents the nature of mathematics.
Research mathematicians apply this exact approach in their work on a daily
basis.
There are several reasons why it is important to do
problem solving or to take a problem solving approach to mathematics,
especially with more able students. We discuss these under three headings
below: to benefit the student, to better represent the subject and to benefit
the teacher.
To benefit the
student
Probably the first way that a
student’s exceptional ability comes to the fore is because they solve sums
accurately, efficiently, and speedily. Certainly in primary school, students
stand out because they learn new concepts quickly, manipulate numbers
intuitively and easily remember number facts.
Learning and applying algorithms in standard
situations can quickly become not very interesting for them. Introducing a
problem solving approach to mathematics can present bright students with a challenge. It gives
them an opportunity to think outside the square and develop their confidence in
themselves.
Not all bright students’ ability thrusts them to
the fore. Some gifted children may, for whatever reason, quietly sit in a class
and be content with the regular programme. Problem solving activities can give
these students more
motivation and challenge and provide them with an interest in the subject and
renewed confidence in themselves.
To better represent
the subject
Some observers see mathematics as a subject where
success depends on learning rules that are to be followed without thought. We
have tried in the Bright Sparks section to show the research aspect of
mathematics. This aspect is the
creative side of mathematics and is what research mathematicians do on a day to day
basis. The highs and lows of their creative activity are felt by mathematicians
in the same way that they are by artists, musicians and sports people.
In the Bright Sparks section we have tried to show
that solving problems parallels very closely the creation of mathematics. By
encouraging problem solving in mathematics classrooms we are helping show that
there is another aspect to mathematics that has a much more human face and is more interesting than
simplying following rules.
To benefit the
teacher
There are also some advantages to implementing a
problem solving approach for teachers. Being the teacher of a really
exceptional student can be very challenging. It may be in the best interests of
all if a mentor can be found to extend the students and relieve the teacher of
some pressure. However, the Bright Sparks and problem solving sections of this
website may be of some help to a teacher ‘in need’. Once an able student is familiar
with using a problem solving approach they should be able to work independently
on a mathematical problem for an extended period of time, choosing and
exploring generalisations and extensions to the problem that interest them. We
hope that it works for you and that you can see how to use it to develop the
next bright spark that comes into your class.
THE ESSENCE OF MATHS.
‘Mathematical
notation no more is mathematics
than musical notation is music. A page of
sheet music
represents a piece of music, but the notation and the music are not the same;
the music itself
happens when the notes on the page are sung or performed on a musical
instrument. It is in
its performance that the music comes alive; it exists not on the page but
in our minds. The
same is true for mathematics.’
Mathematics is a
performance, a living act, a way of interpreting the world. Imagine music
lessons in which
students worked through hundreds of hours of sheet music, adjusting the notes on
the page, receiving ticks and crosses from the teachers, but never playing the
music. Students would not continue with the subject because they would never
experience what music was.
Yet this is the situation that continues in mathematics classes, seemingly
unabated.
Those who use
mathematics engage in mathematical performances,
they use language in all its forms, in the subtle and precise ways that have
been described, in order to do something with mathematics. Students should not
just be memorizing past methods; they need to engage, do, act, perform, problem
solve, for if they don’t use mathematics
as they learn it they will find it very difficult to do so in other situations,
including examinations.
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