Transcript COMMON MISCONCEPTIONS - morelandnumeracyaiznetwork

COMMON MISCONCEPTIONS
in Space, Measurement,
and Chance & Data
Adrian Berenger
14 July 2010
Teaching & Learning Coach
Moreland Network
SPACE
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Language
Orientation & Modelling
Limited Definitions
Mathematical Tools
Language
• Imprecise – words in mathematics take on
different meanings
– What does volume mean?
– What’s the difference between volume and capacity?
– What is a solid in mathematics?
• Van Hiele Levels (recognition, analysis, ordering, deduction, rigour)
– Language is developmental and so therefore it is
vastly different at levels 1 & 2 than at levels 3 & 4
Recognition, Analysis, Ordering, Deduction, Rigour
Students at level 1&2
Students at level 3&4
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corners, pointy
like a square
diamond
a square has four
sides
angles
rectangular
rhombus or kite
a square has four
equal sides and
at least one R.A.
Mathematical language is and should be different at different developmental levels.
Orientation & Modelling
• Coordinates in mapping are
commonly reversed, so the order
of ‘horizontal then vertical’
needs to be consistently
represented and emphasised.
• Map reading depends on the
orientation of the viewer with
respect to the map itself.
• Difficulties with perspective
representations
• Hidden surfaces in 3D
representations
Shapes & Solids
• Common shapes are not recognised unless
they are upright or in their usual orientation.
• Younger children tend to describe aesthetic
qualities rather than mathematical attributes.
Limited Definitions
• What is a hexagon?
But most would not agree that the
following shapes are also hexagons
– The first figure is only a regular hexagon.
• What is a polygon?
Common Misrepresentations
• What is a solid?
Mathematically, stability
or rigidity does not
define a solid. A solid is a
region of space enclosed
by a 3-D figure. It may be
a rigid structure but need
not be. It may be open or
closed. It may be regular
or irregular.
• What is a shape?
 A shape is the appearance of
something especially its
outline that is not dependent
on size, position or
orientation. It need not be 2dimensional.
 A regular shape is not simply
one that is common. A regular
shape is one where all its
sides and angles are equal.
REGULAR
REGULAR
IRREGULAR
Symmetry
• Students find more lines of
symmetry than actually exist.
– Simple or convenient definitions
that lines of symmetry ‘chop’
shapes into half do not necessarily
imply that these lines must also
create one half the exact mirror
image of the other
• How can ICT be used to overcome
this misconception?
• Angles should be defined as the ‘amount of turn’.
– Modeling with two stick joined together of different
lengths is important to overcome this common
misconceptions with angles.
• What other terms can be used to develop completeness of
definition in relation to angles? eg. pivoting, rotation
• What other modeling can be used? eg. clock hands
• The diagonal of a square is the same as its side
Mathematical Tools
• Reading and using the scale of a protractor
Misconceptions
• ‘…a rectangle is a long shape…’
• ‘…a square is not a rectangle…’
• This is a square, this is a diamond.
a
b
• angle  a is smaller than  b
• a right angle and left angle
Children associate
the word right with
directional
language.
Misconceptions
• A triangle is not a polygon.
• Using a protractor.
• The diagonal of a square is the
same length as its side.
• 3D shapes have diagonal lines.
• All lines which divide a shape
into two equal 'halves' are lines
of symmetry.
MEASUREMENT
• Students perceive volume as a solid
measurement and capacity as a liquid
measurement.
• Mathematical rules for calculating perimeter,
area and volume and their units get confused.
They often believe that rulers
• Squared units for shapes that are not square.
Language
• Volume and Capacity