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

FRACTIONS –Language confusions e.g.the third element in a set versus a third of a sharing item 1 st One third of the whole 2 nd 3 rd

FRACTIONS – Development of the concept ProcessOperation Concept Process:- Fractions begins life in the process of sharing e.g.

FRACTIONS 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) Error rate increases progressively

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 – what is it that makes each of these shaded areas worth one half of the whole shape?

FRACTIONS – Point of reference Approximation to zero, a half or one. Estimation (Note mix of decimal and fraction thinking)

FRACTIONS - Restricted vision 0 12 Typically work with fractions between 0 and 1 But what happens beyond 1?

FRACTIONS - Restricted numbers Typically work with “round” fractions May limit vision of fractions Limited understanding of more “difficult looking” numbers such as 3/13 (found in probability of cards)

FRACTIONS –Multiplication What do these mean visually? The processes used for whole numbers apply equally to fractions

Two thirds FRACTIONS –Multiplication

2/3 FRACTIONS –Multiplication ½ Two thirds x one half = two sixths

Represent these equations as diagrams 2½ ÷ ¼ = 3 ¼ ÷ ¾ =

FRACTIONS – Division “A NEW AND EXCITING WAY!! Look at Using equivalent fractions find fractions with common denominator Now introduce this new step!! 10 divided by 10 gives one …. so