**Progression and Differentiation in the Topic of Fractions**

What is Progression?

"The process of developing gradually towards a more advanced state." (oxforddictionaries.com, 2014)

"The term learning progression refers to the purposeful sequencing of teaching and learning expectations across multiple developmental stages, ages, or grade levels." (The Glossary of Education Reform, 2014).

**What is Differentiation?**

"...The process by which differences between pupils are accommodated so that all students have the best possible chance of learning."

(TES Magazine, 2013).

"Differentiation is an approach to teaching that attempts to ensure that all students learn well, despite their many differences."

(Suffolk Learning, 2013.)

"The process by which curriculum objectives, teaching methods assessment methods, resources and learning activities are planned to cater for the needs of individual pupils." (National Council of Teachers of Mathematics, 2013.)

Why Fractions?

"It has been said that fractions have been responsible for putting more people off mathematics than any other single topic." (Nuffield Maths, 2007, p.11)

"Many children consider [fractions] and their concepts difficult, and too often children have had difficulty understanding why they are carrying out a particular procedure to solve a calculation involving fractions...which can contribute to later anxieties surrounding the topic." (ncetm.org.uk, 2009, p.6)

"It is vitally important to shift from a narrow emphasis on disparate skills towards a focus on pupils' mathematical understanding using number. The fundamental issue for teachers is how to better develop that foundation."

(Department for Education, 2010, p.27)

What is a Misconception; and Where and Why do They Occur?

"A view or opinion that is incorrect because based on faulty thinking or understanding." (oxforddictionaries.com, 2014)

"Errors in Mathematics may arise for a vairety of reasons. They may be due to the pace of the work, the slip of a pen, slight lapse of attention, lack of knowledge or a misunderstanding." (Swan, 2001, p.32)

"One of the most important findings of mathematics education research carried out in Britain over the last twenty years has been that all pupils constantly ‘invent’ rules to explain the patterns they see around them." (Askew and Wiliam, 1995, p.17)

"[Misconceptions] may be simply due to lapses in concentration, hasty reasoning, memory overload or a failure to notice salient features of a situation. Others, however may be symptoms of deeper misunderstandings." (Gates, 2012, p.76)

Progression Map

"Progress maps can be used to gain a sense of current student achievement of outcome and where improvement is required. Reference to progress maps can ensure evidence of student performance is valid for provinding feedback and making judgement about student progress." (Ofsted, 2008)

Step 6

Objective 1:

Recognise the equivalence of percentages, fractions and decimals;

calculate simple percentages and use percentages to compare simple proportions.

Objective 2:

Multiply and divide a fraction by an integer.

Step 7

Objective 1: Use the equivalence of fractions, decimals and percentages to compare proportions; calculate percentages and find the outcome of a given percentage increase or decrease.

Objective 2:

Order fractions by writing them with a common denominator

or by converting them into decimals.

Step 8

Objective 1:

Add and subtract fractions by writing them with a common denominator. Begin to multiply and divide fractions which do not share a common denominator.

Objective 2: Use proportional reasoning to solve a problem, choosing the correct numbers to take as 100%, or as a whole.

(webarchive.nationalarchives.gov.uk, 2010)

Prior Knowledge - What Should our Students Already Know?

Step 1

Objective 1:

Recognise unit fractions such as One half, One third, One quarter, One fifth, One tenth … and use them to find fractions of shapes and numbers.

Step 2

Objective 2:

Recognise simple fractions that are several parts of a whole, such as Two thirds or Five eighths, and mixed numbers, such as Fifty three over four; recognise the equivalence of simple fractions (e.g. fractions equivalent to One half, One quarter or Three quarters).

Step 3

Objective 1:

Relate fractions to division

and to their decimal representations.

Objective 2:

Order a set of fractions such as: three quarters, one half, one quarter and position them on a number line.

Step 4

Objective 1:

Reduce a fraction to its simplest form by canceling common factors.

Objective 2:

Use a fraction as an 'operator' to find fractions of numbers or quantities (e.g. Five eighths of 32, Seven tenths of 40, Nine hundredth of 400 cm).

Step 5

Objective 1: Solve simple problems involving ratio and proportion.

Objective 2:

Begin to add and subtract simple fractions and those with common denominators.

Objective 3:

Simplify fractions by canceling all common factors and identify equivalent fractions.

(webarchive.nationalarchives.gov.uk, 2010)

Steps 6 and 7: Fractions Cluedo

Progress Level 8

eChalk: Rocket Rooster

Improve your ability to add fractions as you choose the right set of rockets to fire Ernie into the nest. In this hilarious game there are fifty levels to complete and each one will help you get to grips with addition and multiplication of fractions.

(eChalk, 2014)

In Conclusion

Addressing misconceptions during teaching does actually improve achievement and long-term retention of mathematical skills and concepts, hence contributing to progression. Drawing attention to a misconception before giving the examples was less effective than letting the pupils fall into the ‘trap’ and then having the discussion.

(Askew and Wiliam 1995)

"Excellence in Mathematics education requires equity, high expectations and strong support for all students." (NCTM, 2013, p.12)

After Completing Final School Placement and Being Reflective...

• Are the objectives designed to ensure that the work provides an appropriate

challenge

for all learners? (Ofsted, 2008)

• Does the work offered build on

prior learning

? (Ofsted, 2008)

• Does the work offered allow for learners to

succeed at their own level

? (Vygotsky, 1978)

• Are a range and variety of quality resources available?

• Can all learners access and use the resources they need? (Swan, 2001)

• Are

all learners

able to

participate fully

in the activity? (Gates, 2012)

• Is/are the activity(s) the most

effective way

of

achieving

the outcome?

• Can the learner work without continual reference to the teacher? (Stremmel & Fu, 1993)

• Is the process of assessment an integral part of the learning? (Piaget, 1947)

• Are learning outcomes used to plan future work?

•

Is the learner involved in an assessment of their learning and progress?

• Does the process result in the

learner gaining a greater understanding

of their future needs?

University of

Southampton (2012)

citing Capel et al (2001, p.136)

Trouble Spots and Misconceptions That Arise Within Fractions

Student writes fraction as part/part instead of part/whole.

Student does not understand that when finding fractions of amounts, lengths, or areas, the parts need to be equal in size.

Student does not understand that fractions are numbers as well as portions of a whole.

Has a lack of understanding with terminology, subsequently leading to misinterpretation of questions and ideas

Misapplies additive ideas when finding equivalent fractions

When adding fractions, generalizes the procedure for multiplication of fractions by adding the numerators and adding the denominators

Using the numerator and ignoring the denominator

When multiplying fractions, multiplying the numerator of the first fraction by the denominator of the second, and adding the product of the denominator of the first and the numerator of the second

When dividing fractions, dividing the numerators and dividing the denominators

**Zoe Wilson 1105005**

A very common error in the addition of fractions was to use a rule ‘add tops add bottoms’.

This [method] was more prevalent in examples where the two denominators were different.

It was also interesting to note that this particular error occurred more when the question was posed in computation form than in [word] problem form.

(Hart, 1981)

“The ability to solve addition and subtraction computations [with fractions] declines as the child gets older” (Hart, 1981)

(Department for Children, Schools and Families, 2013).

(Real Life, 2013)

(The University of British Columbia, 2014)

“In Vygotskian perspective, the ideal role of the teacher is that of providing scaffolding to assist students on tasks within their zones of proximal development”

Dahms et al (2007)

(Department of Education, 2013; Mathematics Navigator, 2011)

(Youtube, 2011)

(Strauss, 2013)

A fraction may mean:

A piece of undeveloped land

In church, the breaking of Eucharistic bread

"All but a fraction of people voted in the general election" the word fraction means a small part

"The stock rose fractionally" means less than one pound.

(Lamon, 2012, p.55)

"...to understand is to discover, or to reconstruct through rediscovery, and such conditions must be complied with if in the future individuals are to be formed who are capable of production and creavtivity and not simply repitition." (Piaget, 1975)

"Thought and understanding are not merely expressed in words, it comes into existence through them." (Vygotsky, 1997)

“[Discussing Zone of Proximal Development] The distance between the actual developmental level as determined by independent problem solving and the level of potential development as determined through problem solving under adult guidance or in collaboration with more capable peers.” (Vygotsky, 1978)

What Does The Curriculum State?

Key Stage 3: Working Mathematically

Develop fluency

Consolidate

their numerical and mathematical capability

from key stage 2

and

extend their understanding of the number system

and place value to include decimals,

fractions

, powers and roots

Select and use

appropriate calculation strategies to solve

increasingly complex problems

Move freely between different numerical

, algebraic, graphical and diagrammatic

representations

[for example,

equivalent fractions, fractions and decimals

, and equations and graphs]

On a broad scale...

(Department for Education, 2013, p. 89)

What Does The Curriculum State?

On a refined, mathematics orientated, scale...

Number

Pupils should be taught to:

use the

four operations

, including formal written methods, applied to integers, decimals,

proper and improper fractions, and mixed numbers

, all both positive and negative

work interchangeably

with terminating decimals and their

corresponding fractions

interpret fractions

and percentages

as operators

Fractions

c) use

fraction notation

; understand

equivalent fractions

,

simplifying a fraction by cancelling

all common factors;

order fractions

by rewriting them with a common denominator

Decimals

d) use decimal notation and

recognise that each terminating decimal is a fraction

Ratio and proportion

f) use ratio notation, including reduction to its simplest form and its various

links to fraction notation

g)

recognise where fractions or percentages are needed to compare proportions

; identify problems that call for proportional reasoning, and choose the correct numbers to take as 100%, or as a whole.

Number operations

c)

calculate a given fraction of a given quantity

, expressing the answer as a fraction;

express a given number as a fraction of another; add and subtract fractions by writing them with a common denominator; perform short division to convert a simple fraction

to a decimal

d)

understand and use unit fractions as multiplicative inverses multiply and divide a given fraction by an integer, by a unit fraction and by a general fraction

e)

convert simple fractions of a whole to percentages of the whole

and vice versa, then understand the multiplicative nature of percentages as operators

...Verses Old

New...

(Department for Education, 2007, p.145)

(Department for Education, 2013, p. 132)

"Pupils who work in groups learn how to compromise and resolve petty arguments as well as making rapid progress in maths. The Institute of Education at London University suggests that teachers should act as "guides on the side" of the groups, rather than directly teaching children in the traditional whole-of-class way."

(Smith, 2012)

Benefits of ICT in Mathematics

The interactive nature of multimedia

software motivates pupils and leads

to improved performance.

(Moseley et al., 1999)

ICT-based tools provide pupils with an

advanced communication capability, allowing them to use graphics, images and text together, to demonstrate their understanding of mathematical concepts.

(Jarrett, 1998)

Receiving instant feedback from

computer programs when trying out

ideas, encourages pupils to use

conjecture and to keep exploring

(Clements, 2000)

Used in conjunction with an interactive

whiteboard, software can be used in

whole-class teaching to overcome

pupils’ apprehensions, to reward them,

and let them demonstrate their ability

(Richardson, 2002)

Maths curriculum software has been

shown to motivate both teachers and

pupils, leading to a deeper

understanding of the subject matter

and enhanced learning opportunities

(RM, 2001)

In the best practice, teachers know when and when not to use ICT to enhance teaching and learning in mathematics lessons. These more confident teachers provide evidence of use of ICT to support more effective learning in a number of

areas of mathematics. (Ofsted, 2002)

Ofsted - ICT in Mathematics

[ICT] enables the teacher to provide different levels of challenge for pupils according to their prior attainment. (Ofsted, 2002)

[ICT] is available beyond the lesson so pupils can decide if they wish to have more practice to consolidate their understanding or pursue the

challenge further. (Ofsted, 2004)

(Department for Children, Schools and Families, 2013).

"...Whatever affects one directly, affects all indirectly.

Differentiation supports

the classroom as a

community to which

peers belong and be

nourished as

individuals."

(Lawrence-Brown, 2005, p.5).

(EduGuide, 2014)

(Youtube, 2014)

'Deprived Children':

Level 5 KS2 - 43% chance of achieving A/A*

Level 4 KS2 - 5% chance of achieving A/A*

Level 3 KS2 - 0% chance of achieving A/A*

(Department for Education, 2013)

62%

44%

22%

11%

For FSM Pupils:

Level 5 KS2 39% DID achieve A/A* at GCSE in 2013.

For 'Non-Deprived' Pupils:

63% DID achieve A/A* at GCSE in 2013.

(Department for Education, 2013)

"Pupil participation means opening up opportunities for decision-making with young people as partners engaging in dialogue, conflict resolution, negotiation and compromise – all important life skills."

"Children and young people’s personal development and our democracy will benefit from their learning about sharing power, as well as taking and sharing responsibility"

Department for Education (2004)