¥ mastery of specific content areas (for example number, algebra, measures,
shape and space, data) is assessed through the application of these
areas in practical, everyday contexts.
¥ the ability to communicate and express mathematical ideas and processes and
the correct use of mathematical language in oral and written form can
be assessed by observation while the children are engaged in a
mathematical task. Discussion of their own work can reveal gaps in their
knowledge and skills. Incomplete understanding of mathematical
terminology or processes can also be identified. When recording, children
can communicate pictorially, orally or in written form using words and/or
¥ attitudes towards mathematics, including confidence, interest, willingness
to take risks, and perception of the usefulness of mathematics, are
assessed by observing the enthusiasm with which the child approaches a
task. Attitudes also encompass the interest the child shows in completing
tasks and in using mathematics confidently in other curricular areas and
in real-life situations. TeachersÕ observations of such attitudes contribute
to an overall picture of the childÕs mathematical development and are
continuing and informal.
Assessment tools: how to assess
Although proficiency in computation is essential, assessment should
encompass examination of the childÕs understanding of mathematical
concepts and skills and his/her ability to verbalise that understanding.
Assessment tools must also consider the childÕs use of mathematical language
A broad range of assessment tools is available in mathematics. It is suggested
that teachers use a variety of tools in assessing mathematics, for example
a portfolio that includes samples of a childÕs work, observation records,
mastery check-list results, and the results of both teacher-designed tests and
Teachers assess children every day as they observe them at work, correct
homework or class work, and engage them in discussion. Many of these
observations are done informally but indicate to the teacher how the child
is responding to a particular topic as it is being taught.
This type of continuing assessment includes observation of the childÕs
activity, written work, discussion and questioning during class or group work.
It is useful to have a notebook to hand in which to note the strengths or
difficulties a particular child may have during an activity, for example a
child who frequently chooses an inappropriate measuring tool or a child who
constantly approaches addition tasks by adding the tens first. These short
observations help teachers in planning the next step of a lesson or in
assessing the childÕs readiness for a new topic and in the building up of a
Discussing a childÕs work with him/her can be very revealing, particularly
when he/she is asked to explain how an assignment was completed, either
individually or in groups. The responses will often indicate gaps in
knowledge and skills, and appropriate action can then be taken.
This type of observational assessment also includes analysis of the childÕs
written work to identify types and patterns of error and is a useful way of
establishing how he/she is performing in relation to his/her peers.
Teacher-designed tasks and tests
Teacher-designed tasks and tests, used regularly, provide information useful
in planning for children of differing ability and in matching the programme
and methodology to the needs of those children. They also enable teachers
to determine the level of progress of each child and provide information for
reappraisal and modification of the mathematics programme. They are
directly linked to the instructional objectives of a particular class and can be
used to provide formative, diagnostic and summative data on childrenÕs
progress. By providing a variety of formats in the presentation of teacher-
designed tasks and tests the teacher can help the child become comfortable
with assessment. A broad range of presentations helps children who have
different learning styles.
Some examples of such presentations would be:
¥ oral tests of recall skills (tables, continuation of number patterns)
¥ written tests of numerical competence
¥ problem-solving exercises that use a variety of mathematical skills
¥ projects that require compilation of data, construction of a model or
drawing a diagram.
In examining and recording the results of these tests and tasks the teacher
can also note the processes used by the child in performing the task, for
example using a separate sheet for rough estimates or choosing the correct
tool for the task (long ruler, protractor, number line).
Work samples, portfolios and projects
These are systematic collections of childrenÕs work kept in a folder or file,
and they provide a tangible record of development over a term or a year.
They provide a basis for discussion with both the child and the parent and
can be passed on to the next teacher. Models of portfolio assessment include
representative sampling of progress through written work or subject-based
portfolios that contain all work done in that area. Manageability is an issue
in the compilation of a portfolio, and consideration must be given to the
quantity and value of the work that is kept. The child can take an active part
in the compilation of his/her own portfolio by sometimes choosing a piece
of work for inclusion.
Curriculum profiles allow the teacher to make an overall judgement about
the achievement of an individual child. They allow for the interpretation of
a wide span of learning outcomes. This requires the teacher to look at the
childÕs ability to select materials and processes for particular mathematical
tasks, to select and use appropriate strategies for completing a task, or to
identify the solution to a simple problem. The teacher then decides whether
the child in question has developed these skills or whether they are still in
the developmental stage.
Diagnostic tests identify learning difficulties in particular areas of mathe-
matics, and the results can then be used in the remediation of a problem.
Commercial diagnostic testing kits often provide schemes of work that are
specifically aimed at the skill or skills that the child needs to improve. This
type of assessment is often undertaken by a remedial teacher. However,
analysis of a childÕs work can also fulfil a diagnostic function, and tests can
be designed by the teacher. Persistent errors in a childÕs work can be
analysed to identify areas of difficulty. The use of early screening tests at
infant level means that children who are experiencing problems in
mathematics can be identified at an early stage and appropriate remediation
provided at this point. This type of analysis also indicates the childÕs
strengths, and the results can be used by the teacher in providing extension
Standardised tests comprise norm-referenced tests and criterion-referenced
tests. Norm-referenced tests compare pupils with other pupils or with national
standards. They consist of highly structured tasks that have associated with
them a set of scoring rules. Standardisation refers to the uniformity of
procedures in administering a test. All children take the same test under the
same time limits and instructions. These rules must be adhered to rigidly in
order to produce a standard score and maintain the validity of the test.
Administering the same test to all children under the same conditions means
that achievement can be judged independently of external factors.
Criterion-referenced tests provide information on the childÕs functional
performance level, but, unlike norm-referenced tests, this is not made in
relation to the performance of others. They allow a teacher to estimate the
amount of specified content an individual pupil has learned and are based
on sets of instructional objectives or on course content.
Mastery records and check-lists are one type of criterion-referenced test and are
used to keep track of mastery in certain elements of the curriculum in a
structured manner. This form of assessment can be based on teacher-made
tests or may be part of a mathematics textbook or scheme. Unlike more
formal tests, these are not administered in a strictly standardised manner,
and the childÕs scores cannot be interpreted with reference to class or age-
level norms. They are, however, extremely useful in providing diagnostic
information on a pupilÕs achievement.
Standardised tests should be used judiciously. They can be diagnostic if
errors are analysed and are used as a means of identifying childrenÕs
strengths and their readiness for further learning.
A balanced approach to assessment
Tests must be evaluated with regard to their aims and suitability for the
children for whom they are intended. Teacher-made tests, purchased tests
and check-ups in textbooks all have different purposes and applications. It is
important to consider variety in the types of test given to children, for
example a dictated test that requires short written answers, tests where the
child has to show how they worked out the answer, and multiple-choice tests.
The language used in a test must also be considered, as it can militate
against the performance of a child with a reading difficulty.
Manageability of tests
The manageability of tests is an important issue. Tests that can be
administered to a whole class are useful for screening but are not usually
diagnostic. Where an area of weakness has been identified, a more detailed
test will need to be given to a smaller group or an individual child. Tests must
be easy to administer, as many teachers operate in a shared or multi-class
Recording and communicating
Reporting the results of assessment
The results of assessment must be meaningful. At school level it can be
decided to have a common format for reporting to ensure that accurate
information is carried from class to class. Assessment results for parents
should also cover more than just numerical proficiency. The use of a
portfolio-type system that includes areas such as perseverance, presentation
of work and ability to work in a group gives an informative and rounded view
of the childÕs mathematical ability. This provides an opportunity for parental
feedback. The analysis of results on a school or class level can show areas of
weakness or strength, which can then be developed.
Pupil profile cards
Pupil profile cards allow the teacher to systematically record the progress of
the children and include some examples of observations that the teacher has
noted throughout the year. These profiles provide an overall description of
the childÕs progress in mathematics and are completed over the course of the
school year. They contain information derived from various forms of
assessment, for example standardised tests, teacher-designed tests and tasks,
and teacher observation. They are then used to provide accurate information
for parents and other relevant parties. The recording system should
complement sound instructional practice and reflect the breadth of learning
outcomes implicit in the curriculum. Each school should develop a
co-ordinated policy on record-keeping, which sets out the types of
information to be gathered, the frequency of the data-gathering, and the
uses to which it will be put.
T h ese descriptions are intended to be a help to primary te a c h e rs and are not neces s a r i ly
the full mathematical definition of the te r m .
a logical, arithmetical or computational pro c e d u re
that, if corre c t ly applied, ensures the solution of a
p ro b l e m
a clock on which hours, minu tes and sometimes
seconds are indicated by hands on a dial
a unit of area equal to 100 square metres
an operation such as multiplication or addition is
as s o c i a t i ve if the same answer is produced re g a rd l es s
of the order in which the elements are grouped, e.g.
(2 +3) + 5 = 10, 2 + (3 + 5) = 10
a number denoting quantity but not order in a set
giving the same result irres p e c t i ve of the order of the
e l e m e n ts in addition and mu l t i p l i c a t i o n
6 + 2 = 8, 2 + 6 = 8; 5 3 7 = 35, 7 3 5 = 35
a number with more than two facto rs that is not a
prime nu m b e r, e.g. 6, 10
the divisor in a fra c t i o n
a straight line connecting the centre of a circle with
two points on the perimete r
the same result is produced when multiplication is
performed on a set of nu m b e rs as when performed
on the members of the set individually, e.g.
5 3 4 = (3 + 2) 3 4 = (3 3 4) + (2 3 4 )
a number or quantity to be divided by a n o t h e r
number or quantity
a number or quantity to be divided i n to a n o t h e r
number or quantity
a mathematical sentence with an equals sign
a unit of area equal to 100 ares
a shape has line symmetry if one half of the shape
can be folded ex a c t ly onto the other half
Documents you may be interested
Documents you may be interested