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CAPTURING THE
LEARNING
Janine Blinko explores building evidence of mathematical
understanding from assessments of responses to ‘open tasks’.
How did we get here?
There are many advantages to having a national
approach to teaching and learning mathematics.
This country adopted this later than many. In
England the approach brought with it a number of
elements; a national curriculum, national tests and
later a framework for teaching mathematics. There
have been many great things emerge from these
developments... an enhanced understanding of
progression; clarification of many aspects of mathematics; a wealth of written resources; a clearer
agreement about levels, standards and entitlement;
clear guidelines about what children should be
learning and when; and many more.
However, there is always a price to pay for such
development. In the case of the teaching and
learning of mathematics, the price has been that
many children are receiving the ‘delivery of mathematics’ in packages, rather than finding themselves
in an environment where they can grow into
excited, confident, and competent mathematicians.
On the other side of the table, there are many
teachers who have only ever known the structure
for planning that emerged from the national
strategy, and have become accustomed to teaching
mathematics in blocks, and making the assumption
that, in so doing, they are ensuring coverage, and
catering for children’s mathematical needs. This
interpretation of the primary strategy, together with
a similarly prescriptive interpretation of the
suggested model for the three-part lesson has led to
piecemeal teaching of mathematics, and has, in
some cases, been unsuccessful in growing happy
and successful mathematicians. Additionally, the
excessively broad use of national tests to make
judgements on children, teachers, schools and local
authorities has reduced teacher confidence and
freedom to decide what is the best way to support
8
the budding mathematicians for whom they are
responsible. These interpretations of the situation
many schools, and teachers, find themselves in
were not, in any way, the intention of the national
strategy. The revised Ofsted schedule, which came
into effect in September 2009, and the subjectspecific inspection proposals – in draft form at the
time of writing – rightly focus on learning rather
than teaching, and although this has been a shock
for many schools, it is serving to refocus the profession, back on to what is received by the learner,
rather than what is delivered by the teacher. This
shift is a challenge, and for many, it is particularly
challenging in mathematics.
There is also the matter of evidence. The
teaching profession, as with many others, has
become increasingly data driven. Unlike many
other professions, measurable data is not always
easy, or indeed appropriate, as so many aspects of
teaching are subjective and instinctive. Thus we
find ourselves in a position where we have to
quantify, that which is not easily quantifiable ...
and to cite evidence to support any judgements that
we make about children’s development and mathematical understanding.
What do we want to capture?
It is not necessary here, to discuss the range of
purposes of assessment, just to acknowledge that
teachers are under enormous pressure to submit
data, in many cases six times a year as part of
school and local authority tracking procedures, and
yet to maintain up-to-date assessments at all times,
to ensure that teaching is based on what children
need to learn next. The introduction of APP –
Assessing Pupil Progress – has opened up a door
that is intended to enable schools, and more
importantly, individual teachers, to generate both
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accurate data, and keep track of more detailed
information about children’s mathematical understanding. This brings with it challenges for teachers
of mathematics, about how to ‘capture’ mathematical understanding. Unlike writing, secure mathematical understanding, number sense, and the
ability to apply these in a range of contexts is not
easily judged from recorded work. The need for
evidence of children’s mathematical prowess can
tempt teachers down the dangerous path of relying
on written mathematics and tests as the key source
of data for APP. This information can be misleading;
a list of two correct subtractions does not necessarily reflect mathematical understanding ... and is
often just a procedural success! The same would
apply to adding ten to a list of 2-digit numbers ...
they can all be correct procedurally, but when put
in the context of 10 centimetres more ... or 10
hours more ... in a real life or mathematical
problem, for example, my holiday starts on the 14th,
and I’m away for ten days, the success diminishes.
Not for a moment should written recording of
mathematics be dismissed as a source of valuable
information about children’s understanding, but it
is the more free child-centred activities, which give
us the most information.
There is another key thing that tells us what
children really know ... in the deep learning sense
of knowing. As we ‘helicopter’ around the class, as
children are engaged in mathematics, there are
moments when children say wonderful things that
tell us that they ‘get it’, that they have the mathematical understanding and structures securely
enough in place to access them, and use them,
when they need them, or they have just figured out
something for themselves, that they would never
have remembered if they had been told.
There are some challenges related to both
interpreting children’s written mathematics, and
capturing the understanding they demonstrate
when they talk about mathematics. As a profession,
we are building some brilliant strategies for this,
some of which are shared and discussed below.
Interpreting children’s written
mathematics
If we want loads of information and evidence from
children’s recording of mathematics, we need to
give them the kind of activities that enable them to
show us. In the same way as teachers can ask open
and closed questions, we can give children open or
closed tasks. Open tasks take no more planning
than closed tasks, but as a profession, we need to
step away from expected outcomes, and move into
a more open approach which asks ‘what can this
tell me?’
Take this example from a year 3 class. The
children had been grappling with the mathematical
interpretation of finding the difference, and were
giving all the familiar signs of not really understanding the terminology, or recognising the
connection with subtraction. They were introduced
to the following task orally.
Think of 2 numbers with a difference of 2...
...(that means one number is 2 more, or 2 less
than, the other’.
They were asked to agree on a pair of numbers
with a partner – and if they couldn’t figure it out,
they could ‘pinch’ a pair of numbers from another
pair. The teacher was then at liberty to ask anyone
in the class to contribute their numbers. These
pairs of numbers were collected, and the teacher
modelled a way of recording the pairs of numbers on
the board – without directing or even mentioning
it. After a few pairs had been collected, the teacher
asked if anyone had used numbers bigger than 22 –
which is the biggest number used so far, or if they
could think of a new pair that used larger numbers
... a few of these were collected ... they then
noticed that most of the numbers used were even,
and thought of some pairs that were odd. The
children were then set the challenge of working in
pairs and thinking of their own difference. The
teacher suggested they start with an easy difference,
and easy number ... then they should try some
middle ones, and then they should challenge
themselves with some hard differences, and hard
numbers. There was one pair of children who used
sticks of cubes to find pairs of sticks with the same
difference. for example, with a difference of 3.
The rest of the group worked mentally, and
recorded their own pairs of numbers as they chose.
Here are some things worth noting from what
the children did ... and the kind of information
that would be great APP fodder.
General observations:
The
• children were engaged with the task, and
very happy to discuss the numbers they had
chosen and why
• Many children thought that bigger meant
harder
• All children recorded their pairs of numbers in
a list as the teacher had modelled, even though
they had not been directed to do so.
There were other interesting snippets from
other children’s work.
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This showed a fantastic understanding of place
value, and mental calculation.
What if one of the numbers is 1000 000, can you
find more than one number to be its partner if the
difference is 24?
Evidence of particular understanding: writing in italics indicates next
steps and possible follow-up questions.
1 Very secure use of the five times table, extending into numbers far
beyond 50 (5 10). But only using multiples of 5 until the
teacher interjected with the challenge of trying to use numbers
that were not a multiple of 5.
2 Discovering a pattern and testing it with other numbers. One of
the pair was also able to explain that it would always be true ...
so was able to make a general statement about adding 5 to
numbers that end in 3.
Here the children are very bravely challenging
themselves to extend the pattern into numbers
beyond 1000. A little insecure in recording these
numbers, and could not maintain the pattern.
What if numbers had only 3-digits, could the
pattern work?
What if neither of the numbers ended in 4 or 0?
What would happen if one of the numbers ended with 4 instead of 3?
3 Arguably finding a pair of numbers with difference of 10 should
easier than a difference of 5 if a child has secure understanding of
the number system, so they are not very secure in 10 more/less
other than using multiples of 10. The teacher intervened part way
through and suggested children try the challenge of using at least
one number that ended in 2 rather than zero. She was successful
with a couple of pairs of numbers, but thought they were wrong
and reverted to finding a difference of 2.
Use a number line or a hundred square to find pairs of numbers with
a difference of 10 ... don’t use any numbers that end in zero.
Record them and look for patterns.
Cut lengths of string, there must be a difference of 10 centimeters
between the two lengths.
4 Successfully finding pairs of numbers with a difference of 100, but
not secure enough to use numbers that do not end in zero.
After the activities suggested in 3 above, try cutting lengths of string
that have a difference of 100 centimeters in their length.
Find a way to check that the difference is exactly 100 centimeters.
Measure and record the lengths.
10
Impressive understanding of difference with
some pairs of numbers start with the lowest, and
some with the highest, demonstrating that difference is not just more than, or less than. Also a
good understanding of negative number that had
not been taught in school!
Think of two numbers with a difference of 17, one
of the numbers must be 58. Is there only one
answer?
The most effective mathematical recording is
when it stems from activities that can give teachers
information about if, and how, children can use the
skills and knowledge they have. This informs the
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teacher about which activities can support the
children in building the next steps in their learning.
This can all be recorded as suggested by the primary
strategy on the APP sheets.
Capturing what the children say
If children are to become confident mathematicians,
much of mathematics teaching has to be about
supporting children in developing, and articulating,
ideas and understanding. If we ask the right
questions – the ones that don’t just lead to a right
or wrong answer – ‘How do you know...?’ ‘Does it
always work...?’ etc, and have a culture of building
understanding in the classroom, rather than passing
on information, the children learn to explain their
thinking with confidence which in turn gives us
insights into their development. The challenge is to
capture the magical moments when children give us
those insights, in such a form that it counts as
robust evidence of understanding. We can learn
many lessons from Foundation stage models of
recording, and modify those strategies to suit older
children. Post-it notes seem to be working in a
number of classrooms. For example, in a year 4
class, pairs of children were working with a set of
1–100 cards. Their challenge was to:
• pick a random card, for example 38
• find either the next number card or the
previous one
• find the total of the two numbers
• mark the number on a 100 square
• Will all the numbers on the square eventually
be marked? If so, why? If not, why not?
• Would the answer be the same if you used both
the number before and the number after?
This is a consecutive number problem. For
pairs of consecutive numbers, only odd numbers
are marked on the hundred square. For trios of
consecutive numbers, only multiples of 3 are
marked. Of course, this begs the question, ‘What
happens for 4, 5, 6 consecutive numbers?
As the children worked, the teacher asked them
about what they had found out, but instead of
asking them to record it, she used a pad of speechbubble shaped sticky notes and wrote down what
they said and gave
it to them.
For example: indicating an understanding of
properties of number; an ability to organise
work; explaining thinking; recognising, and
explaining patterns.
Recognising and explaining patterns,
and properties of number and making a
general statement. Justifying and
explaining results.
Making and explaining predictions. Using knowledge of the
properties of numbers.
At the end of the lesson, the
children dated these speech
bubbles and stuck them in their
books, which the teacher used
as a record of the learning,
and as evidence for the APP record. The most
exciting thing was that children were very clear
about what the sticky notes were for, which was to
record their mathematical thinking. So as the
teacher moved amongst them they deliberately
tried to explain ‘their thing’, so that they could have
a speech bubble. Later in the term, children used
the speech bubbles themselves. The stage before,
where they actually said things to the teacher and
this was recorded, had clarified for them what they
were looking for as ‘mathematicians’, so now they
were able to identify key ideas for themselves. The
notes had provided motivation, focus, and evidence
for the teacher – a bargain at twice the price!
In conclusion
Change is always a challenge, and the change of the
Ofsted spotlight from teaching to learning is a
huge, but an exciting one. It means that teachers
can really focus on the development of mathematical understanding. However, this raises questions
about how we capture and ‘evidence’ the learning.
In a Year 5 class, the children had cut a strip of
newspaper that was exactly the length of their hand
span. They measured this, and recorded the length
exactly. Then they cut/made a strip of paper that
matched their height exactly. The challenge was to
calculate their height, without using a ruler, but
using their hand span as a measure. This led to
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•
•
12
much debate and calculation. The
children had been asked to justify/prove
their answers. One boy produced this
record of his calculation. He could
explain why this was accurate and
correct, pointing enthusiastically at the
numbers, and each of the steps he had
taken, and convincing anyone who would
listen that his answer was spot-on.
This record looks different to the
traditional view of what mathematics
should look like, and is a long way from
recording mathematics in books with
small squares. However it is a true
record of some thoroughly understood
mathematics, and is the outcome from a
confident, competent and engaged Y5
mathematician, who has demonstrated
understanding in:
calculating using decimal numbers
using the relationship between units of measure
•
•
•
•
reasoning about number
recording mathematics
choosing and using appropriate calculation
strategies
explaining and justifying answers.
Maybe these children could go on to write a
report about their findings in their literacy lesson!
Recording mathematics is often a different skill
from ‘doing it’, and is most useful when it is
emergent, rather than prescriptive.
So maybe we need to expect, and encourage,
assessment evidence in mathematics, to look
different to evidence in other subjects, and
different from the ‘traditional’ view of recorded
mathematics, in order to accommodate the development of adventurous and confident mathematicians.
Janine Blinko works as a consultant in
mathematics education with The Maths Hub.
info@themathshub.co.uk 07909 984751
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