An USA, Introduction Ability grouping is historically

An exploration of differentiation
within the mathematics classroom

KEY WORDS:
differentiation, ability grouping, mathematics, fixed mind set, fixed/fluent, UK,
USA,

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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