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Application of Number
How do I teach Application of Number?
Teaching Key Skills Application of Number - a few suggestions
These have been separated into four different sections. This is not intended
to be an exhaustive list, but maybe the ideas will spark off other thoughts
that will generate a wealth of activities.
See also
Student starting points
Although Application of Number students are usually taught in groups,
it is likely that each student will have a different mathematical background,
a different social background and a different set of learning preferences.
As a consequence, each will have a different set of needs, which we as
teachers have to recognise so that we can develop teaching and learning
strategies that take account of these differences. Much of Application
of Number teaching time is taken up with filling gaps in students' understanding
and skills and this can create difficulties when teaching the whole group.
Whatever strategies are adopted, these individual needs should be at the
forefront.
Initial diagnostic assessments
Initial assessment
Initial assessment is an assessment that helps you and the student to
identify the level of study that is appropriate for them. This process
is particularly important in a college setting where the student is often
new to the course and to the institution. It usually takes place at interview
or at the beginning of the course during induction. In a school, the student
and their prior achievements will be known to the teaching staff but it
is still helpful to review this in the context of the Application of Number
standard.
The best initial assessments:
- identify the level of the student at the time of the test
- give an indication of what might be achieved by the end of the year.
Often, initial assessments include an element of self-assessment and
can take into account the student's background, such as GCSE results and
school reports, although most also include a written or computer-based
element. The results allow you to group students into levels or clusters
of levels. Although not impossible, running a group with students from
Level 1 and Level 3 can be a challenge. Many institutions try to 'set'
the Application of Number groups so that all the students in the group
are working at roughly the same level. Where this is not possible, it
is sometimes helpful to organise the students into Level 1/2 groups and
Level 2/3 groups.
Diagnostic assessment
Diagnostic assessment is an assessment that identifies which topic areas
the student can deal with well and which are the areas that may need to
be worked on further. These tests are usually quite detailed and often
take rather longer to work through than the initial assessment. Some diagnostic
assessments also include a section on learning styles.
Diagnostic assessment usually takes place at the beginning of the course,
although some institutions spread the diagnostic process over the first
few weeks. Inevitably diagnosis, both formal and informal, will continue
throughout the course. Don't neglect the views of the students themselves;
they will often be able to tell you which skills they feel comfortable
with, and which need some attention.
These tests give both the teacher and the student detailed information
and provide a starting point for the individual programme of learning.
Collectively, the results of these tests for the group, will give the
teacher an idea of topic areas that most of the students can cope with
well and, at the other end of the spectrum, topic areas that most students
need help with. This will allow the teacher to plan work for the group
making the best use of time and resources. This process is important for
both colleges and schools. A GCSE pass in mathematics doesn't tell us
anything about the particular skills the student has got, so it is important
to carry out a more detailed assessment.
Which initial assessment or diagnostic test to use?
There are many initial assessment tests available, including from commercial
sources. Similarly, diagnostic assessments are also available commercially.
Many are computer based, and the best ones will automatically produce
an action plan for each student, identifying areas of strength and areas
to be worked on. Whether considering initial or diagnostic assessments,
you should look at a range of tests, ask colleagues in other institutions
which ones they use and always ask for inspection copies to pilot the
tests before choosing which to purchase.
Learning, practising, reinforcing and applying skills
The underpinning skills (Part A of the standards) need to be practised
so that they are learnt and become 'second nature' to the students. As
is the case when learning any other skill, this usually involves some
repetition. Just as in learning any skill, this usually involves some
repetition. This practice is best carried out using graduated worksheets,
usually out of context. This allows the students to practise the skills
without a context getting in the way. The external assessment (the test)
is itself written to be accessible to everyone and so is not set in any
particular vocational or subject context. However, many of the questions
(especially at Level 3) focus on everyday life situations such as kitchen
design, travel surveys, holiday planning and shopping.
Remember that each student will have different needs, so giving the whole
class the same practice sheet will almost certainly not be appropriate.
Throughout this section you will find ideas that address this issue. Coping
with mixed ability, achievement, background, interests, motivation and
attitude will be a challenge.
Variety in your teaching
The best mathematics teachers use a variety of approaches in their teaching.
Each student has a different mathematical background and the group will
include people with a variety of learning preferences. There are lots
of topics to cover and a range of skills that needs to be practised and
learned.
Different skills may be best taught using different methods. This allows
you to vary your activities during a lesson so that interest is maintained
and more of the differing needs of each individual student are likely
to be met.
Look out for computer programs that could be used for individual topic
areas. A good source of information is the TES and in particular
the maths curriculum supplement, which appears periodically. The Mathematics
Association and NANAMIC
will also be very helpful.
See also
Talking
of Number (section 4)
Teaching
and Learning: Application of Number (section 4)
Teaching Tip
Variety is the spice of life
Students at Levels 1 and 2 need lots of practice working without calculators,
so activities that encourage mental arithmetic skills and quick pencil-and-paper
methods are required. Try setting short, quick-fire quizzes. They are
fun and students really do enjoy them! They can also be used to review
what the group knows at the beginning of a topic, or as a way of assessing
what has been learnt at the end of a lesson or topic.
Traditional pencil-and-paper methods can be the best way of getting
some messages across. At Level 3, for instance, when covering rearranging
equations, there really is a need to practise with pencil and paper. It
is a bit like music; all the great musicians spend hours practising their
scales. There is a place for this sort of practice in number work - though
we shouldn't overuse it!
Try using a groupwork exercise to cover volume and area. Set small groups
the task of designing packaging for a product (perhaps related to a vocational
area). It is surprising just how much of the standards are covered by
this exercise. If you get time, go through this task and identify all
the skills it covers in Part A of the standard at Level 2.
Active learning
"I hear and I forget
I see and I remember
I do and I understand"
Confucius (551-479 BC)
Most people learn best by doing. As you can see from the quotation, this
has been known for a very long time. What is surprising, therefore, is
that this principle is so often disregarded by teachers and learners.
Always be on the lookout for ways in which your lessons can become more
active.
- Where students need to read something, get them to paraphrase what
is said, make a poster or explain what they have learnt to a peer.
- Instead of telling students something, get them to find it out.
- Instead of using a written problem to calculate an area or volume,
get learners to do the problem for real.
- Ask a student to 'teach' a point or even a short topic for revision.
- Encourage discussion in the classroom and use groupwork as well as
individual work to promote this.
All these ideas will lead to real activities (with the emphasis on 'active')
and this is more likely to lead to a deep understanding of number and
its concepts.
Ah - but it will take up too much time!
Time is always an issue. There is never enough. However, rushing through
the standards with fewer and fewer students keeping up, and never giving
time for concepts to be assimilated, is a really poor use of time. There
should always be time to teach something well. Good methods encourage
self-reliance and autonomy in learning and, as the students become better
learners, their pace of learning will increase and you may well find that
these methods are rather more time-efficient than they appear at first
sight.
Differentiation of levels
There is a whole raft of strategies that can be used to help you to accommodate
differences in your teaching group.
Graduated tasks
Set activities that start with short, easy, closed tasks that get progressively
harder, longer and more open-ended. Students will then work through the
activity and, if it is well designed, there will be a challenge for everyone.
For example in a Level 2 Construction GNVQ group, if the topic is ' fractions
conversions', the first questions will be dealing with easy fractions:
1/2, 1/4 and so on. Later questions should deal with more challenging
fractions (3/7 for example). The final questions might include some written
problems set in context, or an open-ended task, producing a number line
poster with different fractions clearly marked, and related to the construction
industry.
Choice
Offer a selection of tasks to the group so that students can select
one that interests them and is at a level that will challenge them but
is not out of their reach. Students will sometimes need some guidance
in making the choice. Many topics in number can be studied at a range
of levels. It is therefore possible to teach a short introduction to the
whole class followed by different worksheets set at different levels,
or even with different contexts to suit the variety of interests amongst
the group. This would allow you to deal effectively with a mixed group
made up of students from two vocational areas, or a number of different
A-levels.
Differentiated resources
A well-resourced number classroom will have a whole variety of resources.
See Appendix 5 in Teaching and Learning: Application of Number.
If you have a good indexing system, students can seek out resources and
help that is appropriate for them. If you have a computer in the classroom,
there are various learning packages that could be used in this way.
Extension tasks
Always try to have available a task that is demanding - one that would
challenge the most able in the group. Some of the best extension tasks
are open-ended. Don't be afraid if you are not sure of the answer yourself.
There will be people in your institution that can help and in some cases
there may not be a definitive answer anyway. Mathematics teaching has
sometimes been criticised for not paying enough attention to the most
able students. This strategy is one way of addressing the problem.
Question and answer
Try directing questions to particular students. General questions to
the group usually get answered by the quickest, most able, or (sometimes)
the most vociferous students. By selecting individuals and asking questions
that are at an appropriate level for that individual, you will challenge
more of the group, more of the time.
Another good Q and A strategy is to ask the question and ask everyone
to write down the answer. This also allows more of the individuals in
the group to think through their response rather than have the question
answered immediately by someone else.
Groupwork
Using groupwork is one strategy that can help to overcome lack of confidence.
Working individually can be rather daunting for some students. A group
can be a safer environment for many. It is usually wise for you to keep
some control of the mix in each group, although you may feel that, for
some classes, self-selecting groups may work equally well. Groups can
be good for students needing support - other group members will take on
some responsibility for explaining tasks and topics. It is also good for
the more able members of the group. There is a lot of truth in an adage
that 'you don't know that you understand something properly until you
have taught it'.
Groupwork is particularly useful when the students have learnt the underpinning
skills and are starting to develop the skills in context.
Relevance
As the name suggests, the emphasis in Application of Number is on applications
- how number can be used in real contexts.
Whist there is some evidence that certain underpinning skills are best
introduced in an un-contextualised form. Once the skill has been learned,
however, students need to learn how to apply it in everyday life, in other
subjects or in the vocational context of the main programme of study,
or future career.
Always be on the lookout for possible scenarios on which to base questions,
tasks or assignments. Possible sources may include the teachers who deliver
the main programme, exam papers, textbooks, the internet. Most importantly
of all, ask. Vocational or subject tutors will usually be pleased to discuss
how Application of Number can be used to support the work that they are
covering.
View
assignments that incorporate Application of Number in vocational settings.
Teaching number
Fractions
This activity is designed to help teachers to explore the concept of
a fraction and how fractions can be used and taught in a more concrete
and fun way. However, the activity can easily be adapted for students.
Other links
see also
Teaching algebra
Lots of ideas for teaching algebra in an active and exciting way making
good use of discussion and groupwork are in Learning mathematics through
discussion and reflection (sections 3, 10)
View
more information.
View other links
Teaching shape and space
Teaching handling data
The National
Statistics web site is a really good source of data that can be used
in assignment work. It includes data from the censuses as well as a range
of financial and social data on the UK. In particular look at the learning
zone area of the site.
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Summary of this section