```Numeracy Professional Development
Fractions, decimals and maybe a
bit on percentages
Michael Drake
Victoria University of Wellington
College of Education
Learning fractions
• If I say to you that I understand what
two thirds means, what am I telling you
that I know?
next to you
Representations
• A representation is one way of showing
a mathematical idea
• For fractions, a shape that is drawn
and shaded is one representation of
the idea two thirds
Issues of Representation
• Research has found that students have
difficulty translating between different
representations of a mathematical
object. For example the symbol 1/3 and
the fraction drawing showing 1/3, or the
fraction circle 1/3 and the number line
showing 1/3.
• If I say I understand what two thirds
means, I should be able to recognise the
key representations of two thirds and
translate between them
• If I say I understand what two thirds
means, I should be able to recognise
that the representations are equivalent
• Johnny says he understands what two
thirds means. It means two out of
three pieces.
• What do you want Johnny to be able
to do in relation to this concept of
fractions?
By the way
Did you know?
• In 2001 42% of year 7
& 8 students who sat
the initial NUMPA
could not name these
symbols
1
2
1
4
1
3
• More could not place them in size order…
What we want Johnny to be able to do is…
Recognise a range of drawings for two
thirds
Be able to read the symbol – and
relate it to the drawings
And be able to talk about it
Written words
Verbal/mental
picture
Materials
Symbol
Fraction sub-constructs
There are a number of ways we think of
fractions, all of which involve slightly
different understandings
• Part-whole
• Measure construct
• Quotient construct
• Operator construct
• Ratio construct
• Probability construct
• So what does Johnny really need to
understand if he is to be successful
with fractions?
Something to think about and discuss in
school
• How do you get these various
interpretations across so
students understand fractions
better?
Decimals
materials
imaging
Using number
properties
Generalising
from number
• Decimals can be introduced by
generalising from whole numbers
For example
• Addition by place value: 34 + 25
Does it work for decimals?
Investigate 3.4 and 2.5 [etc]
But several problems with student
understanding need to be checked
before hand
• 85% of students who haven’t been
formally taught decimals have a firm
idea of how they work. Its wrong, but
they have the idea
Bruce Moody
What’s the problem?
• Discuss what the student is doing, and
why it is a problem
• What teaching strategies do you
currently have that could help sort out
the problem?
1)
John reads the decimal 4.35 as four
point thirty five
2)
Carol says 1.50 is bigger than
1.5
3)
Tupu says 0.23 is the same as
0.023
4)
Martika and Karl are having an
argument. Martika says 1.5  10 =
1.50, while Karl says its 10.50
5)
On a trip you spend 98.5c on a
Litre of petrol, and \$0.95 on a
cheeseburger. How much have you
spent?
The moral of the story…
• Decimals need explicit teaching –
assuming students understand them is
• Don’t introduce them until students
understand how whole numbers work,
and have the concept of a fraction
• Spend time on decimal place
value
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