An exploration of differentiation

within the mathematics classroom

KEY WORDS:

differentiation, ability grouping, mathematics, fixed mind set, fixed/fluent, UK,

USA,

Introduction

Ability grouping is historically how maths classrooms are

differentiated (*) and is still commonplace at present throughout primary and

secondary schools. Initially, grouping by ability seemed the logical way to

give students the best environment to learn in, surrounded by comparable

students at a similar level; judged through pre conceived notions of intellect

assessed through formal assessment, regardless of individual back ground. Some have

found grouping by ability in maths can be beneficial to students to help

individualise their learning and allow all to work at a challenging level to

maximise understanding and progress. However research and theory dictates that

grouping by ability has adverse effects on students regarding attainment and

anxiety of the subject (*). This research suggests that grouping by ability is

looking at maths as a ‘fixed concept’ with the idea that all children have ‘immutable

ability’ (Marks 2014). Over recent years, research and case studies have found

numerous problems with grouping by ability in the maths classroom. Some of the

problems explored are types of pedagogy, pace of lesson, mind set of pupils and

attainment gap. The two main concerns that are dominant throughout reading are:

the attainment gap of those from with low attainment to those with the highest

and; mind set of pupils and their perceptions of mathematics through varied

experiences. These, I believe, can be changed through teaching mathematics in a

challenging and engaging way, in which students understand the mathematical concepts

behind their learning in a fun environment. (*) argues that majority of teachers and pupils

approach maths with a ‘fixed ability’ mind

s even when actively trying to employ a ‘mixed ability’ mentality towards

collaborative learning. The notion of fixed ability thinking versus mixed

ability practices will be explored and how this could be implemented within a

school setting. Finally I will look at the Concrete – Pictorial – Abstract Heuristic

and how this method comes away from fixed or mixed abilities and teaches maths

in stages with the notion at all students learn at the same time.

Methodology

This essay started at looking at differentiation within the mathematics

classroom and how

The main source of information was through the use of

academic journals. Using keys words as mentioned above I was able to narrow

down the search to articles only relevant to differentiation in maths.

Considering in all five primary schools I have worked at all use ability

grouping to differentiate in their maths lessons, there was surprising very

little research or theory justifying why schools group their pupils by ability

for maths. Initially I tried focusing on English primary education and was able

to find case studies surrounding the negative effects of ability grouping. However,

this was all very similar research so I widened the search to include

elementary education in the United States of America. This allowed for a varied

of opinions of how differentiation can work in the maths classroom. I chose the

USA as they have a very similar educational system to the U.K. Most research is

restricted to the past five years however to allow a fuller picture of the

progression of theory and practice, articles relate back to 1970’s.

Mathematics in U. K.

Curriculum

Maths in British culture has changed over recent years to

compete with countries around the world in the ranking tables (PISA). The

national curriculum focusing on 3 main areas of maths : Fluency, problem

solving and reasoning skills. These 3

areas together will lead to ‘mastery’ of the subject allowing pupils to use

maths with confidence in real life situations. The national curriculum (DfE, 2013)

states that pupils should be ‘secure’ in their ‘understanding’ and ‘readiness

to move on’. In order for pupils to become secure then they need an environment

that motivates and engages them with quality teaching allowing misconceptions

and mistake s to be addressed and allowing pupils to find the answers on their

own and the teacher only leading them. Mathematics is deemed to be a hierarchical

subject (*) meaning that learning and understanding is built upon previous

knowledge. Misconceptions and gaps will not allow pupils to progress to deeper

learning and therefore children will fall behind or not be able to access

learning further down the line.

Ability Grouping in

Mathematics

Ability grouping is referred to as splitting a group of

people into similar attainment levels usually based on formal assessment. This

could be groups within a classroom (Highers, middles, lowers) or within a year

group (set 1, set 2, set 3, etc…). Groups are usually between 2-4 sets within a

primary school however can increase substantially once in high school. Historically

grouping students by ability was thought to be the best environment to give all students the greatest opportunities

to learn with the assumption being that some people exceed better at maths than

others.

The problem with setting is that in many cases pupils don’t get

given a chance to move up or down in those sets. (*) found that 92% of students

who were setted at the beginning of their school career were still in the same

groups by the end of high school.

(*) suggests that in

a ‘traditional’ classroom few students learn at the same pace: some students

will be rushed whilst others will get bored waiting. Grouping by ability allows

all students to work at a more synchronized speed. Grouping by ability also

allows for all students to learn at a challenging level designed for their

capabilities as a class allowing for all pupils to work together in a collaborative

group usually sharing similar strengths and weaknesses. This arguably allows

for students to work in an environment similar to a real life work situation

where those at a similar level work together.

Although ability grouping in math is very prominent

throughout English education, there seems to be an overwhelming consensus

within the educational community that this type of differentiation has a

negative impact on all pupils. Firstly,

research by (*) found that ability grouping in a maths classroom didn’t benefit

any one. (*) suggested that ability grouping was used to help those deemed to

be low ability to have specialist individualized learning aimed to allow these

pupils to access the curriculum. However several case studies and research (*)

describe ability grouping in math to only help those ‘on the cusp’ of passing

the exam(*). Those in lower ‘ability groups’ made average progress and (*)

states that all that took the exam in the lower set failed. Moreover the

ability grouping widened the attainment gap quite dramatically from assessments

taken at the beginning of the year, 1.45 years, to assessment taken at the end

of the year, 2.22 years. Furthermore Marks (2014) criticizes that the intention

of ability grouping is to ‘push as many pupils as possible to achieve government

targets’ leaving the lowest attaining pupils with ‘reduced mathematical

learning experiences’. Moreover, (*) states that ability grouping restricts

learning and progress to only the gifted and talented (G) pupils. S/he says that

Overall ability grouping seems to be more for the convenience

of the teacher and less beneficial to the pupils

Heterogeneous Grouping

in Mathematics

Heterogeneous grouping in basic terms is a ‘traditional’

classroom filled with pupils with different abilities and different strengths

and weaknesses. The teacher looks at the differences between the pupils as an

advantage, allowing all students to learn from each other in a respectful and

equal environment. Pupils are encouraged to have an active role in the class

and are responsible for their own learning. Math problems are discussed and the

teacher leads the class to the correct answer without revealing the answer. Quality

dialogical talk is the main tool used to help consolidate learning and quality

questions are asked to help reveal any misconceptions the pupils may have.

Mistakes are looked at learning opportunities and are welcomed to enable

students to figure out why they are wrong giving them a good conceptual

knowledge for future learning.

Heterogeneous grouping as opposed to homogeneous grouping

looks at maths as more of a fluid concept (*). That pupils may have gaps or

misconceptions in a certain area for example, mental maths, however may be

secure in their knowledge and understanding of area and shapes (*). This allows

students to share strengths with their peers in addition to working on their own

weaknesses simultaneously.

(*) compares maths to

any other subject and gives the example that within an English lesson, there

will be children who can articulate really well in comparison to pupils who are

quite introvert and shy. There will be children who are great readers but poor

writers or great writers but struggle with spelling. The point that (*) is

trying to get across is that these other core subjects use heterogeneous groups

when teaching so the same can be done for when teaching mathematics.

This ‘progressive’ method of teaching is born out of Vygotsky’s

Theory of Social Construction using peer scaffolding to help consolidate their

own knowledge whilst being the more ‘knowledgeable other’ for peers (*). (*) shows

how Vygotsky’s Zone of Proximal Development (ZPD) can be used in the maths

classroom to help students work together with a shared goal.

Kagans collaborative learning theories can also be used to

help children work together in a heterogeneous environment. Kagan grouping is

using co-operative learning and strategies and collaboration to enable those

with different abilities to work together and support each other. An example of

this could be ‘Think – Pair – Share’. For this stratagy to work, the educator

has straegicaly positioned students in specific places to allow maximum quality

diologue. This strategy is introducing shoulder partners (the person sitting

next to the pupil) and face partners (the person facing opposite the pupil). The

shoulder partners are deemed to be at similar abilities and the face partners

are of different ability. A problem is then given to the class. First the

individual thinks alone about what

the solution could be. Next, pair

with their shoulder partners and discuss what they thought the solution could

be and their process. Finally they share

their collective thoughts with their face partners. Armed with their collective

thoughts, irrespective of their ability, the

pupil will be confident to share their thoughts with their face partners

with a shared solution. If correct then this can reinforce their knowledge and

if wrong then they can have a discussion with their partner to the process.

Using ZPD they are able to peer scaffold in a mutually respected environment. This

stratagy can also be used to introduce a new topic and to assess any

misconceptions early on eg. 3d shapes,

(*) suggests that this approach is working towards a more

fluid mind set but needs more than an open mind to work. For this approach to

work, there needs to be a whole school ethos towards collaborative learning and

to move away from the notions that children have immutable ability when it

comes to math.

Concrete – Pictorial –

Abstract (CPA) Heuristic

Singapore maths is highly rated as an approach to teaching

math with Singapore and Hong Kong – who also employ the Singapore method – both

ranking 1st and 2nd in the Programme for International

Student Assessment (PISA) tables in mathematics. Professor Lianghup Fan, the

author of the textbook used for Singapore Maths uses the CPA heuristic to

demonstrate how, when used in conjunction with the other aspects of the

approach, allow all students to exceed in math, regardless of ‘ability’.

The CPA method is a partition of the ‘Singapore Maths’

approach. Underpinned by Piagetian, Bruner and Vygotskian theories of using manipulative

resources to help develop and consolidate learning of mathematical operations,

this approach takes a step away from looking at pupils as having varying

abilities.

Concrete: The concrete stage states that for a pupil to

understand a concept then they must first be able to have concrete or real life

objects to see and to hold. This allows the pupil to start making sense of the

concept and allows them to attach meaning. Piaget’s theory states that the ‘concrete’

object must be tangible otherwise the pupil is unable to attach meaning. His

theory states that to move from’ sensori-motor to operational thinking depends

on the internalisation of actions’ In contrary to that, Bruner’s ‘enactive’

stage of representation suggests that a picture of the object is sufficient enough

for the child to attach meaning. Professor Fan sides with bruner’s representational

stages theory which is evident throughout his textbook.

Pictorial: The pictorial stage links to visual

representation. Once a pupil has been able to relate meaning to the object in

their task, the next step is for them to assert meaning to a picture. I.E. 3 +

8 is represented as 3 red dots and 8 blue dots. Students can now use their previous

knowledge of manipulable objects to aid them into visual representations.

Piagetian theory dictates that these picture should be the same pictures of

what the tangible object was, whereas Bruner theoty suggests that this isn’t required

and that dots is sufficient.

Abstract: The abstract stage allows pupils to come away from

objects and pictures and understand that 3 + 8 = 11.

There is varied opinion on how the CPA should be taught

Instead it suggests that all pupils are able to access the

learning regardless of perceived ability.

The Singapore Maths Approach is more complex than just using

the CPA method. Other aspects of the approach are