Some problems associated with learning fractions
• Whole number confusion
• Language confusions
• Concrete models/abstract concepts
• Materials used for teaching
• Point of reference
• Restricted vision
• Restricted numbers
FRACTIONS – Whole number confusion
1. Transfer of whole number arithmetic processes to fractions
2+3 = 5
754332
75
43
32
FRACTIONS –Language confusions
e.g. the third element in a set versus a third of a sharing item
1st One third of the whole2nd 3rd
FRACTIONS – Development of the concept
Process Operation Concept
5
3
Process:- Fractions begins life in the process of sharing e.g.
FRACTIONS
5
3
•Begins as a process of counting or sharing
•Eventually welding together to a concept
•Adds, not replaces, a layer to the process of understanding.
FRACTIONS-Discrete vs continuous material
Circle two-sevenths of the faces
Mark off two -sevenths of the plank
FRACTIONS - Problems associated with materials used for teaching
Perceptual distractors
½ + ½ = 1
Why might this be a distractor?
Inconsistent cue (need to ignore all lines and reconstruct the diagram)
FRACTIONS – Perceptual distractors
Shade three quarters of this shape…
Complete cue
Incomplete cue (need to add lines)
Irrelevant cue (need to ignore some lines)
Err
or r
ate
inc r
eas e
s pr
o gre
ssiv
ely
Problems with materials used for teaching Structured vs Unstructured Materials
Many commonly used fraction ‘kits’
•are continuous quantity materials
•approximate the idea of equal units that students have so much trouble with
Context - what to focus on?
Draw lines to cut the cake into 21 equal pieces
Focus on the lines drawn to explore what misconceptions this student may have
Context - what is the big picture?
Colour 2/5 of these hearts
When interviewed, this student saw 5 hearts as the ‘whole unit’. She found 1/5 of each row and combined them to obtain 2/5
Fractions - open ended tasks:
Name a fraction•between ½ and 5/8•between ½ and ¾with a denominator of 12•between 0 and 1/3 with a numerator › 1
•The answer is 3½, what is the question?•The answer is 4x + 1 what is the
(x+1)(x-2) question?
FRACTIONS – Point of reference
Approximation to zero, a half or one.
one thangreater bemust sum so
5
2.553
and 3
1.532
?53
32
Estimation(Note mix of decimal and fraction thinking)
FRACTIONS - Restricted vision
0 1 2
Typically work with fractions between 0 and 1
But what happens beyond 1?
FRACTIONS - Restricted numbers
Typically work with “round” fractions
43
,41
,32
,31
,21
May limit vision of fractions
Limited understanding of more “difficult looking” numbers such as 3/13 (found in probability of cards)
FRACTIONS –Multiplication
341
41
3
34
43
What do these mean visually?
The processes used for whole numbers apply equally to fractions