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Innovative
Teaching Materials for AS Mathematics |
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School: Wilberforce College
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Fellow: Susan Wall |
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Email: susan@wal.karoo.co.uk |
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Final report |
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In April 2000, an article in the TES was
headlined “ ‘Too easy’ maths
A-level to be made more difficult”.
The specifications for curriculum 2000 had
just been published and there were changes
to the maths specifications that caused concern
to many teachers. More algebra and greater
mathematical rigour were to be included. There
was to be a restriction on the use of graphic
calculators (at a time when the use of ICT
was supposed to be encouraged). There was
now a section of ‘assumed knowledge’
that had previously been included in the A
level syllabus and the formula booklet allowed
in exams was to have many useful formulas
removed. The exams were to be “somewhat
more demanding”. At a time when there
was already concern about the numbers studying
A level maths, the numbers applying for maths
degrees and the shortage of maths teachers,
this was rather worrying. |
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In August 2001 the TES reported “Nearly
a third fail ‘too hard’ AS-level
maths.” This was followed in the autumn
by “Maths in crisis as students drop
out” commenting on the very high drop
out rate from AS to A2 maths and “AS
chaos hits maths degrees” which showed
that the numbers of sixth formers applying
to study maths at university has plummeted
following the AS results. The shortfall in
the numbers applying for teacher training
was worse in maths than in any other subject. |
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QCA have had an enquiry and acknowledged
that “There is some evidence that the
content of the AS mathematics specification
is too great to be taught and to be mastered
by students in the time available before the
May/June of their first year of post-16 study.
For some students the gap between GCSE and
AS is such that time was taken up acquiring
important background knowledge that was not
itself part of the AS specification.”
A rewrite of the specification has been ordered
but it will not be ready until September 2004
– assuming no delays! |
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Thinking Behind The Project |
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I love maths. I find it a fascinating subject
to study and to teach and I have seen many
students enjoying the challenges that it offers.
I desperately did not want to see our numbers
studying maths drop and maths only being a
subject for the very able. Therefore I decided
that the only way forward was to confront
the difficulties. I decided to develop a new
teaching and learning approach. As a member
of the Mathematical Association and Association
of Teachers of Mathematics, I had read about
many of the exciting developments in teaching
and learning for younger age groups with a
strong emphasis on an interactive approach.
Why not for post 16 mathematics? |
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Initially I decided that a connectionist
approach would be an important feature. This
approach would have the advantage that topics
would be continually revisited and developed
rather than been done once in detail and then
left – an approach that many textbooks
use. Exams require students to put ideas together
and really that is part of the fascination
of the subject. It is a set of interrelated
ideas not a set of separate elements, which
can be taught in isolation from one another.
Therefore no textbooks! |
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Instead of using textbooks I wanted to develop
a more interactive and activity based approach
to teaching and learning whereby students
would be much more involved in their own learning.
This would involve group work, problem solving,
identifying misconceptions at an early stage,
encouraging an ethos of ‘have a go’
and developing both oral and written communication
skills. I felt that communication was one
of the keys to success. If students could
explain their thoughts they would understand
better or alternatively misconceptions could
be identified. |
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The Project Itself |
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I produced a weekly pack containing outlines
of the activities for teachers and materials
for students. The worksheets are not just
a list of questions but have activities such
as graphs and equations to match up (and justifying
choices!) or missing bits to fill in or finding
errors. Other activities range from cards
to be matched up by pairs or groups of students
to problem solving in groups using big pieces
of sugar paper and large felt tips. Individual
whiteboards are used to encourage student
to ‘have a go’ and graphic calculators
are used for mini-investigations. We have
even done 'cutting and sticking' to sort out
the steps of equation solving. With everything
there is an emphasis on “why do you
think that?” or “how did you reach
that solution?” and lots of whole class
discussion. Using open-ended questioning is
another technique that is used in a wide range
of situations to encourage communication and
understanding. E.g. It is easy to give the
answer to “What is the gradient of the
line y = 4x + 1?” but more challenging
is the question “Give me the equations
of 2 lines that are perpendicular –
and you are not allowed the same two as the
person sitting next to you!”. The question
“What have the graphs of y = ÷2x
- 1÷ and y = 2÷x÷ - 1
got in common and what are the differences
between them?” requires an understanding
of the structure of the graphs and the need
to express it in suitable language. It has
been really interesting listening to students
working together in pairs or groups. They
really talk 'maths' and get much less distracted
than when they are working through an exercise. |
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The Future |
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The teaching of this has been great fun
and the students seem to have enjoyed it as
well! Retention has been good and the numbers
applying to do AS maths for September 2002
are up again for the second successive year,
which is against national trends. Inevitably
some of the activities worked better than
others and new ideas were always being generated
throughout the year. Therefore I am currently
reviewing and rewriting the weekly packs in
the light of experience. |
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However it has became apparent that writing
all the activities down on paper does not
always result in a successful transfer of
what I had in mind to other teachers. In a
sense that did not matter too much this year
as we were all working in the same college
and therefore communicated regularly, but
it made me think again about dissemination. |
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I have decided that rather than disseminating
a long list of activities it was more important
to transmit the ethos or culture of what I
was trying to do illustrated with a range
of activities. |
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Therefore I am also currently working on
putting together a package that will include
a discussion of what my approach is trying
to achieve, how I am aiming to achieve this
with a wide range of example to illustrate
each point. I believe that once these ideas
have been conveyed then anyone can think up
further activities to fit into what they are
trying to teach. |
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This year at college has been very exciting
and stimulating. However we have encountered
a familiar stumbling block at the end. We
finished the course with time for revision
but the language of the exam papers has prevented
students from being able to demonstrate the
mathematics that they have learned. Unfortunately
for both teachers and students success is
measured only in examination success but phrases
like “Write down an expression for …”
and “ Express your answer in the form
of …..” ”Hence or otherwise
…” or “deduce that …”
and the difference between “necessary”
and “sufficient” is causing students
who have relatively poor linguistic skills
to leave questions undone. Part of the future
of this project must be to consider whether
on top of everything else the problem of language
can be overcome. I would also like to develop
these ideas into A2 maths and GCSE maths. |
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