Matching not patching: primary
maths and children’s thinking
Anne Watson
June 2009
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In this talk:
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Children’s spatial understanding
Children’s understanding of quantity
Measure
Relations between quantities
Roots of algebra
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Spatial understanding
• Pre-school knowledge of space is
relational, not just descriptive:
size and transitivity
distance between
corners and edges
fitting in and together
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Talk about relations between shapes: size,
corners, edges, fitting
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Number as quantity
• Pre-school knowledge of quantities
and counting develop separately:
interacting with objects
stretching/scaling
fitting
sharing out
pouring
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Elastic: stretching and scaling
• Comparing lengths
• Same shape different size
• What makes it the same?
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Success in mathematics is related
to understanding:
• Addition/subtraction as inverses
• ‘Undoing addition’ feels different to
‘adding on’
• Relations as well as quantities,
e.g. difference
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Additive relationship (fitting)
a+b=c
b+a=c
c–a=b
c–b=a
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c=a+b
c=b+a
b=c-a
a=c-b
Difference
• Write down two numbers with a
difference of 3
• … and two more numbers with a
difference of 3
• … and another very different pair
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Sharing
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Sharing by counting out
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Relations involved in
multiplicative reasoning
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One to one
Many to one
One to many
Stretching and scaling
How many …? (fitting and
measuring)
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Actual measurement
• Iteration of standard units has to be
understood – and is difficult
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Exact measurement:
multiplicative relationship
a = bc
a = cb
b=a
c
c=a
b
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bc = a
cb = a
a=b
c
a=c
b
Fractions
• 5 is the multiplicative relation between 5 and 3
3
• measurement (inexact units) and division (as
when sharing one to many)
• transferring understanding between division to
measurement is really hard
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Inexact measurement: what do
children know?
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Sharing by chopping up
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One to many
Many to one
Fairness
Iterative process of dividing and
distributing
‘Continuous’ quantities: pouring
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Pouring questions are about
multiplicative relations
• How many …. in ….?
• How many times ….?
• How much is left over?
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Relational reasoning
53 + 49 – 49 = ?
2x2+2x4=2(2+4)
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Implications for teaching?
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Implications about shape and
space
• Use their knowledge of comparisons
and relations between 3D shapes and
spaces
• Use their experience of 3D to develop
spatial reasoning and ideas about size,
and scaling, and multiplication
• Measuring is about comparing one unit
to another – and is hard
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Implications for teaching number
• Additive understanding does not
precede multiplicative
• Very young children can reason
multiplicatively from everyday
experiences of sharing one between
many, distributing many to one,
comparing quantities, and measuring
• Multiplication is not only repeated
addition; this meaning can get in the
way of understanding it fully
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Implications about relations
• Understanding relations between quantities,
shapes and measures is a strong foundation
for later learning
• ‘=‘ expresses a relation
• With many quantities it makes more sense to
talk about <, > and = at the same time
• Young students can use letters to express
relations between quantities
• Understanding addition and multiplication as
two kinds of relation, rather than knowing four
operations, draws on ‘outside’ knowledge and
also helps in understanding scaling, ratio,
proportion …
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