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DIRECTED NUMBERS
Emma Armstrong reflects on different teaching approaches
for directed numbers.
As a PGCE student, it quickly became apparent
that students, as old as year eleven, were having
trouble with addition and subtraction involving
negative numbers. The greatest issue that arose
involved questions with a double negative such as 4
–5. A common response from teachers was the
phrase; ‘two negatives make a positive,’ and
although this seemed to appease students for that
moment, it was later forgotten. When attempting
to put the double negative problem into context,
the notion of ‘taking away debt’ seems to be
suggested frequently. However, many teachers can
find this analogy confusing, rendering it both a
poor explanation, and learning tool for students.
Near the end of my main placement, I had the
opportunity to teach the topic of directed numbers
to a Y7 low ability class, and instead of hunting out
the old number line, I tried to find something a
little different. It proved difficult to find an
approach that was appropriate for all types of
question – negative plus a negative, positive plus a
negative, etc. however, a PowerPoint presentation
involving building sandcastles and digging holes was
found that seemed to fit the bill1.
The key to this model is that one sandcastle is
equivalent to 1, and one hole is equivalent to 1.
I adapted this resource to ensure it was suitable for
all students, and for each question, and then
students worked in pairs to discuss how the answer
could be found using the sandcastle model. Students
could draw on mini-whiteboards to give a visual
form for their explanations, and then these whiteboards were used as an assessment tool to enable
me to see the whole class’ numerical answers.
The different scenarios were approached as
follows, with animations from the PowerPoint
presentation to demonstrate (see Figure 1).
Lesson evaluation
Students seemed to find adding a negative number
easy to comprehend. For the question 4 2,
pupils were happy to draw 4 sandcastles and 2
holes, and decide what would happen.
2 3 was also a simple task, as pupils
drew 2 holes and another 3 holes. Here, it was
Calculation Analogy
1 1
A hole plus a sandcastle gives a level surface
– since the sand from the sandcastle fills the hole, so answer 0
3 3
3 sandcastles plus 3 holes gives a level surface, so answer 0
35
3 sandcastles take away 5 sandcastles, which is the same as flattening 3
sandcastles, then taking away 2 more sandcastles, thus making 2 holes,
so answer 2
4 2
4 sandcastles plus 2 holes. 2 of the sandcastles fit in 2 of the holes,
leaving 2 sandcastles remaining, so answer 2
3 5
3 sandcastles plus 5 holes. The 3 sandcastles fill 3 of the holes, leaving
2 holes remaining, so answer 2
2 3
2 holes plus another 3 holes. This gives 5 holes in total, so answer 5
7 4
7 holes take away 4 holes. Taking away a hole means filling in the hole
with the sand of one sandcastle. Filling in 4 holes leaves 3 holes, so
answer 3
2 3
2 holes take away 3 holes. Having taken away (or filled in) 2 holes, we
have enough sand to make one sandcastle, so answer 1
Figure 1
pleasing to observe that pupils were seeing the
number 3 as 3 holes, and not ‘taking away 3
sandcastles’.
Certain double negative questions such as 7
4 were tackled well, as it was understood that
taking away 4 holes was the same as filling 4 holes
in, leaving 3 holes. The main issues to arise
involved crossing the zero boundary between
positive and negative numbers for instance, 2 3; as once the sand was level there were no more
holes to fill.
Through formative assessment during the
lesson, it became apparent that lower ability
students found the model easy to understand,
however higher ability students became confused.
This reversal of aptitude was highly motivating for
the low attainers, while causing frustration to those
that usually excelled in the class – an interesting
situation to witness. The weakest student in the
class who is working at level 2/3, found the lesson
particularly accessible, and was highly enthused at
his pictures and explanations being used as
examples, when he typically struggles in maths
MATHEMATICS TEACHING 217 / MARCH 2010
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© ATM 2010 • No reproduction (including Internet) except for
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lessons. His satisfaction was confirmed by the
comments written on a subsequent questionnaire
that;
“I think it was fun and exciting. I was good at it
but the other lessons I just get it wrong but this
I got right so I enjoyed it. I want to do it
again.”
In contrast, a high ability student asked at the
end of the lesson whether this approach would be
used throughout later years in school, as she was
concerned that she found the ideas confusing.
Iding (1997) summarises research that shows
novices and low ability students being helped most
by analogies. The reason given for this is that
higher ability students already possess representations for certain ideas, and being subjected to
different frameworks is therefore superfluous. It is
low ability students that particularly relish the
explicit scaffolding that an analogy provides. It was
clear that a student with EAL found the visual
representations beneficial, and although he found it
difficult to put his ideas into words, could use his
hands to show the effect of sandcastles falling into
holes and sandcastles being flattened. He responded
afterwards that the moving sandcastles and holes
on the board really helped him to understand.
Different approaches
When members of the maths department at my
placement school were asked how they would
approach teaching the topic of adding and
subtracting with negative numbers, particularly with
a low ability Y7 set, many different approaches
were suggested and are summarised below:
Number line
adding moves to the right, subtracting moves to
the left. A double negative changes the direction to move right
Following patterns
if we know that
3 2 5
3 1 4
3 0 3,
we can extend to give
3 1 2
3 2 1
Hot air balloon
one sandbag is 1, and taking away sandbags
causes the balloon to rise. One blast of hot air
is 1
4
Temperature on a thermometer
a vertical number line. The difference is how
far apart the two temperatures are
Temperature in a bath
hot and cold water represent positive and
negative numbers respectively. Taking away hot
water ( ) or adding cold water ( ) gives
a lower temperature; taking away cold water
( ) results in a hotter bath
Mood
beginning with an initial ‘mood’, positive
number being happy, negative number being
sad, adding 1 is represented by adding one
bad event, resulting in a sadder, more negative
mood. Subtracting 1 is represented by
‘getting rid’ of a bad event, giving a more
positive result
Friends going out
if neither friend wants to go to the cinema (
) or both do want to go ( ), the signs –
friends – agree, giving a positive effect; if one
friend wants to go but the other doesn’t ( or ), the signs disagree, giving a negative
effect
By considering these approaches, some characteristics of what makes a good analogy were revealed.
Analogies, by their novelty, may compel attention
and enhance memorability. However their effectiveness can be reduced if they are not meaningful, or
if students are not familiar with the secondary
subject, for example the idea of making sandcastles
and holes. In relation to directed numbers, a
distinction between the words ‘subtract’ and
‘negative’ is of great importance. Here, ‘adding’ or
‘subtracting’ is the imperative verb or action, and
‘positive’ or ‘negative’ is the adjective describing
what type of number is being dealt with. Problems
also arise if the analogy is restrictive and does not
apply to advanced contexts such as the case of a
double negative.
Table 1 gives each approach evaluated against
these characteristics. Table 2 evaluates each
approach against how well it represents different
calculations that involve crossing the zero boundary,
and result in a positive or negative answer:
Looking at both tables, the most effective seem
to be the sandcastles, hot air balloon, and mood
analogies. A further disadvantage of the mood and
hot air balloon analogies is that the nouns within
the calculations are not consistent. For example,
when using the mood analogy with a calculation
MATHEMATICS TEACHING 217 / MARCH 2010
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Table 1
Approach
Meaningful
Novelty
Applies to
double
negative forms
Students
familiar with
subject
Distinction of
minus as a
verb/noun
Sandcastles
✓
✓
✓
✓
✓
Number line
✗
✗
?
✓
✗
Following patterns
✗
✗
✓
✓
✓
Hot air balloon
✓
✓
✓
?
✓
Temperature:
thermometer
✓
?
?
✓
✓
Temperature in bath
✓
✓
✓
✓
✓
Mood
✓
?
✓
✓
✓
Friends going out
✓
✓
✓
✓
✗
Table 2
Approach
2 5
2 5
2 5
2 5
5 2
2 5
Sandcastles
✓
✓
✓
✓
?
Number line
?
?
?
?
?
✗
✗
Following patterns
✓
✓
✓
✓
✓
✓
Hot air balloon
✓
✓
✓
✓
✓
✓
Temperature:
thermometer
✓
✓
✓
✓
?
?
Temperature in bath
✓
✓
?
?
?
?
Mood
✓
✓
✓
✓
✓
✓
Friends going out
?
?
?
?
?
?
such as 2 5 -3, the first number refers to
mood, the second number 5 refers to a number
of events happening, and the result 3 is the
resultant mood. With the hot air balloon strategy,
the first number refers to the position of the
balloon, the second number relates to five
sandbags, and the result is the final position of the
balloon. In contrast, with the sandcastles analogy,
the objects within them are consistent – either
holes or sandcastles.
I believe the sandcastle analogy could be one of
the most effective ways to approach the subject of
adding and subtracting with negative numbers.
Further experimentation by conducting this lesson
with classes of different ages and abilities would
give additional evidence to contradict or support
this claim.
It would also be interesting to determine
whether other topics in the mathematics
curriculum could benefit from the use of analogies
or pedagogical metaphors, thus supporting students
with many parts of a subject they may struggle to
access.
At the time of writing Emma Armstrong was a
Mathematics NQT at Selby High School; she now
teaches at St Mary’s Catholic High School.
Note
1 ‘Sandcastles,’ available from the number resources section
of www.mrbartonmaths.com
Reference
Iding, M. K. (1997). How analogies foster learning from
science texts. Instructional Science, 25, 233-253.
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