PGCE (Primary) - Assignment 1

Student Name: Thomas GroveI.D. Number: 22661727Programme: PGCE (Primary)

With Reference To Social Constructivism: How Does

Grouping by Gender Influence Attainment within the

Mathematics Classroom?

‘Mathematics equips pupils with a uniquely powerful set of tools to understand and change

the world. These tools include logical reasoning, problem-solving skills, and the ability to

think in abstract ways. Mathematics is important in everyday life, many forms of

employment, science and technology, medicine, the economy, the environment and

development, and in public decision-making. Different cultures have contributed to the

development and application of mathematics. Today, the subject transcends cultural

boundaries and its importance is universally recognised. Mathematics is a creative

discipline. It can stimulate moments of pleasure and wonder when a pupil solves a problem

for the first time, discovers a more elegant solution to that problem, or suddenly sees

hidden connections.’

(DfEE, 1999: 60)

Student Name: Thomas Grove

Programme: 6270 PGCE (Primary)

Professional Tutor: Hilary Bowen

Assignment 1.


Introduction

Vygotsky (1978) believed that during the construction of knowledge two complementary frameworks

are in operation, a dual stage process of internalisation:

‘Every function in the child’s cultural development appears twice: first on the social level,

and later, on the individual level; first, between people (interpsychological), and then inside

the child (intrapsychological). This applies equally to voluntary attention, to logical memory

and to the formation of concepts. All the higher functions originate as actual relations

between human individuals.’

(Vygotsky, 1978: 57)

This assignment will review the assessment data, literature, and research on grouping by gender

and how this influences attainment within the mathematics classroom, with specific reference to

social constructivism based on a Vygotskian theory of mind, however, I personally believe it would

be appropriate to initially review the extensive research on gender differences in mathematics

learning and achievement, and investigate the supposed existence of a ‘gender gap’. This critical

review will challenge the assumption that there is a difference between boys and girls that makes

boys predisposed to do better in mathematics; that there is an intrapsychological difference

between males and females that influences the construction of knowledge and cognitive

development in mathematics, as defined by Vygotsky (1978). This initial review will be constructed

in a clear and concise manner with relevant and appropriate research to provide a firm foundation

for, and consequently support, the critical review on grouping by gender and how this influences

attainment within the mathematics classroom.

At this point, the assignment will progress and review the assessment data, literature, and research

on grouping by gender and how this influences attainment within the mathematics classroom, with

specific reference to social constructivism from a Vygotskian theory of mind, and investigate the

supposed negative relationship between group size and mathematics learning and achievement.

This critical review will investigate the hypothesis that the difference in learning styles of boys and

girls influences mathematics learning and achievement in groups of different gender composition;

that the intrapsychological difference between males and females influences the interpsychological

construction of knowledge and cognitive development in mathematics, as defined by Vygotsky

(1978).

Assignment 1. 1.


Theoretical Framework

Constructivism

For the purpose of this assignment constructivism shall be regarded as a theory about knowledge

and learning (Brooks & Brooks, 1993) in which learning is an active process wherein learners

construct new concepts based upon their current and prior knowledge (Haylock with Thangata,

2007), however, Jaworski (1994) argues that constructivism should be regarded as a ‘philosophical

perspective on knowledge and learning’ (Jaworski, 1994: 14) since ‘neither its key terms, nor the

relationships between them are sufficiently well or uniformly defined for the term “theory” to be

strictly applicable’ (Ernest, 1991 quoted from Jaworski, 1994: 35).

Scott (1987), and Ollerenshaw and Ritchie (1995) provide an outline of the principles which underpin

the various aspects of constructivism as defined by Driver and Bell (1986). These principles are:

what is already in the learner’s mind matters;

individuals construct their own meaning;

the construction of meaning is a continuous and active process;

learning may involve conceptual change;

the construction of meaning does not always lead to belief;

learners have the final responsibility for their learning;

some constructed meanings are shared.

Ollerenshaw and Ritchie (1995) argue that every child, independent of ability, background, and

gender, will have a diverse repertoire of ideas and concepts that to some degree will affect

subsequent learning. Constructivism identifies that as educators, with significant professional

expertise, it is important to bring child and personal, meaningful experience together and to ensure

that interaction between them ensues. This interaction results in the abandoning of individual ideas

and concepts, or the modification of them in the light of the new experience. In this sense the

teacher is of central importance and is seen to facilitate numerous roles within the mathematics

classroom environment such as the enabler, catalyst and challenger.

‘The teacher’s role is not merely to convey to students information about mathematics. One

of the teacher’s primary responsibilities is to facilitate profound cognitive restructuring and

conceptual reorganizations.’

(Cobb, 1988: 89)

Constructivism, as a theory about knowledge and learning, has been recognised as having much to

offer to mathematics education (Jaworski, 1994) and as such has relevance to what constitutes an

effectual mathematical classroom environment (Haylock with Thangata, 2007). Malone and Taylor

(1993) are amongst numerous educators that believe constructivism has significant implications for

mathematics teaching and learning. Ollerenshaw and Ritchie (1995) provide evidence in a clear and

Assignment 1. 2.


concise manner to support the argument that children with evident behavioural and/or social

problems have, through implementation of constructivism within a practical situation, frequently

shown significant improvement in their individual behaviour.

Modern ideas and discussion about constructivism identify two separate components which need to

be considered; these being radical constructivism and social constructivism (Harries & Spooner,

2000). Jaworski (1994) indicates this movement from a radical to a social view of knowledge

construction ‘might be seen to parallel the move from a Piagetian to a Vygotskian view of learning’

(Jaworski, 1994: 25).

Radical Constructivism

‘Constructivism is a theory of knowledge with roots in philosophy, psychology, and

cybernetics. It asserts two main principles whose application has far reaching consequences

for the study of cognitive development and learning, as well as for the practice of teaching,

psychotherapy and interpersonal management in general. The two principles are:

knowledge is not passively received but actively built up by the cognising subject;

the function of cognition is adaptive and serves the organisation of the experimental

world, not the discovery of ontological reality.’

(von Glasersfeld, 1989 quoted from Jaworski, 1994: 15-16)

This constructivist theory about knowledge and learning, termed radical constructivism,

encapsulated by the principles stated above, has been developed from the Piagetian view of

learning that knowledge is purely subjective. von Glaserfeld (1989) argues that knowledge is

constructed, that learning takes place, during cognitive disturbance and the resultant adaptation

through processes, as defined by Piaget, of assimilation, accommodation, and equilibration to fit the

learner’s experiences.

Not all of Piaget’s influences are regarded favourably however (Bruner, 1985), as much of his work

prioritises the individual aspect of learning. Radical constructivism thus ‘regards other aspects, such

as the social, to be merely a part of, or reducible to, the individual’ (Ernest, 1994: 62). Numerous

educators (Ernest, 1991, 1994; Merttens, 1996; Harries & Spooner, 2000) provide an excellent and

compelling argument that further empathise needs to be placed on the social influences of learning,

and the role of discussion and negotiation.

Social Constructivism

Ernest (1993) provides an account of social constructivism as a theory about knowledge and

learning, identifying two central features:

Assignment 1. 3.


‘First of all there is the active construction of knowledge, typically concepts and hypotheses,

on the basis of experiences and previous knowledge. These provide the basis for

understanding and serve the purpose of guiding future actions. Secondly there is the

essential role played by experience and interaction with the physical and social worlds, in

both physical action and speech modes.’

(Ernest, 1993 quoted from Jaworski, 1994: 24)

To summarise, Ernest argues that knowledge cannot purely be subjective, as developed within

radical constructivism, but that objective knowledge needs to be acknowledged, as determined by

its social acceptability.

Social constructivism continues to be underpinned by the principles that encapsulated radical

constructivism, however, emphasises is placed on the social influences of learning, and the role of

discussion and negotiation. Jaworski (1994), and Harries and Spooner (2000) acknowledge that,

while individuals will develop their own meaning to knowledge constructed, in a social environment

they will be engaged by other individuals. Interaction within this social environment results in the

abandoning of individual ideas and concepts, or the modification of them in the light of the new

experience. This social interaction and exchange develops an apparent common meaning shared by

the individual learners often referred to as ‘common’ or ‘intersubjective’ knowledge (Jaworski, 1994).

Ernest (1994), and Harries and Spooner (2000) suggest that while numerous educators attribute

differing characteristics to the term ‘social constructivism’, with many others developing theoretical

perspectives on knowledge and learning under alternative names which might be usefully

characterised as social constructivist, there are only two possible forms of social constructivism in

principle; these being a form of extension to radical constructivism that ‘adds’ a classroom-based

social dimension and a form which emanates from the work of Vygotsky which ‘views individual

subjects and the realm of the social as indissolubly interconnected, with human subjects formed

through their interactions with each other (as well as by their internal processes) in social contexts’

(Ernest, 1994: 69).

‘The construction of knowledge in the classroom goes beyond interaction between teacher

and students, to the wider interaction between students themselves in the social and

cultural environment of the classroom and beyond. It seems crucial for mathematics

teachers to be aware of how mathematical learning might be linked to language, social

interaction and cultural context.’

(Jaworski, 1994: 28)

For the purpose of this assignment social constructivism shall be regarded as a theory about

knowledge and learning based on a Vygotskian theory of mind.

Although chronologically Vygotsky’s work precedes that of Bruner, Vygotsky extended Bruner’s

concepts and ideas to develop an additional dimension. Vygotsky placed great emphasise on social

Assignment 1. 4.


interaction and communication being perquisites for learning, and in particular on the role of the

teacher in the process:

‘From the very first days of the child’s development his activities acquire a meaning of their

own in a system of social behaviour and, being directed towards a definitive purpose, are

refracted through the prism of the child’s environment. The path from object to child and

from child to object passes through another person. This complex human structure is the

product of a developmental process deeply rooted in the links between individual and social

history.’


Vygotsky (1978) believed that during the construction of knowledge two complementary frameworks

are in operation, a dual stage process of internalisation:

‘Every function in the child’s cultural development appears twice: first on the social level,

and later, on the individual level; first, between people (interpsychological), and then inside

the child (intrapsychological). This applies equally to voluntary attention, to logical memory

and to the formation of concepts. All the higher functions originate as actual relations

between human individuals.’


For the purpose of this assignment, although an extremely interesting concept within the

mathematical classroom environment, the zone of proximal development, as a measure of a

learner’s cognitive development related to ‘problem solving under adult guidance or collaboration

with more capable peers’ (Vygotsky, 1978: 86), shall be principally overlooked as the focus remains

that of grouping by gender and how that influences attainment within the mathematics classroom,

not that of grouping by ability. Consequently, as an extension of Vygotsky’s notion of a zone of

proximal development the concept of scaffolding shall also be largely overlooked, although

reference will be made within the concluding remarks of this assignment.

Assignment 1. 5.


Gender and Mathematics

The ‘gender gap’ in mathematics has been an issue of national and international concern since the

mid-1970s with numerous evaluations of the educational system (e.g. Cockcroft, 1982)

acknowledging the disadvantages encountered by girls in areas such as mathematics and science

(Leder, 1992). Considerable concern has been raised over ‘gender issues in the mathematics

education research community’ (Leder, 1992: 599). Reyes (1983) reported that during the two-year

period 1981-82 ‘the most common topics for manuscripts (submitted to the Journal for Research in

Mathematics Education) were problem solving, attitudes, student achievement, and sex differences

– with each accounting for 8%-10% of all manuscripts’ (Reyes, 1983: 146). Leder (1992) reveals that

the majority of articles submitted to the Journal for Research in Mathematics Education between

1978 and 1990 focused on the nature and extent of differences in the mathematics achievement of

males and females, with varying areas and contributing factors examined. Articles of particular

interest and relevance include those submitted by Fennema and Sherman (1978), Armstrong (1981),

Marshall (1983), and Hanna (1986), as these focus on a variety of measures to explore a possible

connection between gender and mathematics learning and achievement. Not all studies submitted

follow the general consensus of a statistically significant difference in mathematics achievement

between males and females (e.g. Callahan and Clements, 1984; Ethington, 1990), however many of

these studies have been discredited within recent studies (Leder, 1992).

Thom (1987) argues that although initiatives had been introduced and developed, aimed at

achieving a more equitable education system, numerous political and educational pressure groups

had been principally concerned with class inequalities rather than gender inequalities however,

although Thom provides an excellent and compelling argument, the ideas and concepts presented

are now considered dated and largely irrelevant, and for the most part have been addressed by the

introduction, and consequent revisions, of the National Curriculum, which provides provision for both

class and gender equality.

Friedman (1989), and American Association of University Women (1998) argued that the gender gap

in mathematics achievement had been declining steadily since the mid-1970s and could be

considered negligible, or even non-existent, however numerous studies have reported that this is

simply not the case.

Trends in International Mathematics and Science Study (TIMSS)

Neuschmidt et al. (2008) investigated changes in gender differences evident in the performance of

pupils participating in the Trends in International Mathematics and Science Study (TIMSS)1 between

1995 and 2003, focusing on 16 countries that ‘had non-annotated, fully approved data for all three

1 The TIMSS surveys are conducted by the International Association for the Evaluation of Educational Achievement (IEA) to measure trends in mathematics and science achievement (Neuschmidt et al., 2008);

Assignment 1. 6.


cycles’ (1995, 1999, and 2003) (Neuschmidt et al., 2008: 57). Neuschmidt et al. (2008) compares

gender specific results and patterns found in TIMSS 1995 with later assessment cycles in order to

‘address the question of how far the mathematics and science gender gap has narrowed over time’

(Neuschmidt et al., 2008: 56).

Neuschmidt et al. (2008) reveals that, as published in the TIMSS international reports (e.g. Beaton et

al., 1997a, 1997b; Mullis et al., 2000, 2004), ‘several countries showed significant gender differences

in mathematics and science achievement in each testing cycle’ (Neuschmidt et al., 2008: 56).

Neuschmidt et al. (2008) provides interesting and compelling evidence that, using the regression

approach as defined by Gonzalez and Miles (2001), gender differences in achievement in

mathematics and science and the content areas2 within each are complex. Neuschmidt et al. (2008)

reports:

‘At the subject level, the results showed a relatively stable pattern in terms of the number of

gender differences across the three TIMSS cycles. A look at the content areas, however,

revealed marked changes in the patterns, especially for the current areas of chemistry and

physics in science and of measurement in mathematics. In all three areas, the dominance of

boys distinctly decreased over time.’

(Neuschmidt et al., 2008: 63)

Therefore, Neuschmidt et al. (2008) concludes that the significant gender differences in

mathematics achievement, as reported by Beaton et al. (1997a, 1997b), and Mullis et al. (2000,

2004), remain in all content areas other than that of measurement, where a noticeable reduction in

the advantage that boys held over girls can be seen. This article would seem to support the

assumption that there is a difference between boys and girls that makes boys predisposed to do

better in mathematics; that there is an intrapsychological difference between males and females

that influences the construction of knowledge and cognitive development in mathematics.

Hastedt (2004), and Hastedt and Sibberns (2005), referring to results in TIMSS 1995, reported that,

‘depending on the subject, boys outperformed girls in several countries, especially on multiple-

choice items’ (Neuschmidt et al., 2008: 1), however, Neuschmidt et al. (2008) reports that although

some difference in mathematics and science achievement was detected in relation to multiple-

choice items these differences were within the margins of standard errors. Therefore, Neuschmidt et

al. (2008) concludes that there is no noticeable support for the assumption that ‘gender differences

would have been affected by the different composition of the TIMSS assessment in terms of the

number of multiple-choice and of constructed-response questions’ (Neuschmidt et al., 2008: 72).

Neuschmidt et al. (2008) also found ‘no effect in relation to changes in the distributions of the

respective populations of students who participated in the three TIMSS cycles’ (Neuschmidt et al.,

2008: 72).

2 Although between the 1995, 1999, and 2003 assessments the definition of the content areas for each subject changed, all three testing cycles had the following content areas in common for mathematics: algebra, geometry, and measurement.

Assignment 1. 7.


National Curriculum Assessments at Key Stage 2 in England, 2007 (Provisional)

This Statistical First Release (SFR) (DCSF, 2007a) provides assessment data on ‘the achievement of

eligible pupils (typically 11 year olds) in the 2007 National Curriculum assessments at Key Stage 2

(KS2)’ (DCSF, 2007a: 1).

The National Curriculum assessment tests measure pupils’ attainment in relation to the National

Curriculum programmes of study with pupils awarded levels in the National Curriculum scale to

reflect attainment (DCSF, 2007a). The National Curriculum standards have been carefully

constructed and developed so that most eligible pupils will progress by approximately one level

every two academic years, therefore, eligible pupils would be expected to achieve Level 4 (or

higher) by the end of Key Stage 2.

The statistics in this Statistical First Release (DCSF, 2007a) are based on the outcomes of the

National Curriculum assessments administered during May 2007, however, participation by

independent schools was voluntary, therefore data analysis only includes results from independent

schools which chose to contribute, consequently ‘data underpinning the figures for teacher

assessments3 are based on approximately 77% of 11 year olds nationally’ (DCSF, 2007a: 3).

The figures presented (Table 1) show that in 2005 76% of boys achieved Level 4 or above in

mathematics compared to 75% of girls. The figures identify progress for boys in 2006 with the level

of attainment in mathematics improving by 1% to 77%, however, girls failed to show any

improvement. The provisional figures identify progress in 2007 with the level of attainment in

mathematics improving by 1% for both boys and girls to 78% and 76% respectively.

The National Curriculum assessment test figures of pupils achieving Level 4 or above seem to

support the supposed existence of a gender gap in mathematics, however, the teacher assessment

figures seem to indicate that the difference between boys and girls could be considered negligible,

or even non-existent. This could indicate that the learning style of boys in relation to mathematics

and the method of assessment offered afford them a more advantageous position.

In addition, the figures presented (Table 2) show that in 2005 33% of boys achieved Level 5 or

above in mathematics compared to 28% of girls. The figures identify progress in 2006 with the level

of attainment in mathematics improving by 3% for both boys and girls to 36% and 31% respectively.

However, in 2007, the provisional figures identify retreat with the level of attainment in mathematics

falling by 1% for both boys and girls to 35% and 30% respectively.

The National Curriculum assessment test figures of pupils achieving Level 5 or above again seem to

support the supposed existence of a gender gap in higher level mathematics, however, the

difference between boys and girls is significantly greater. The teacher assessment figures would

seem to support this argument, with a significant difference between the percentage of boys and

3 Teacher assessments (TA) provide complementary information about eligible pupils’ attainment in relation to the National Curriculum programme of study and are ‘the teachers’ judgement of pupils’ performance in the whole subject over the whole academic year’ (DCSF, 2007b: 4).

Assignment 1. 8.


girls achieving Level 5 or above in mathematics noticeable. This again could indicate that the

learning style of boys in relation to higher order mathematics and the method of assessment offered

afford them a more advantageous position.

‘The Government has set itself the following Public Service Agreement (PSA) targets for the

achievement of 11 year olds:

To raise standards in KS2 English and mathematics tests so that, by 2006, 85%

achieve Level 4 or above, with this level of attainment sustained until 2008.’

(DCSF, 2007a: 2)

The figures presented (Table 1) show that this target was not met in 2006, with 79% of pupils

achieving Level 4 or above in English and only 76% in mathematics (77% for boys and 75% for girls).

The provisional figures identify progress in 2007 with the level of attainment in both English and

mathematics improving by 1% to 80% in English and 77% in mathematics (78% for boys and 76%

for girls); however, this still falls below the PSA target.

Assignment 1. 9.


Table 1:

Percentages of pupils achieving Level 4 or above in Key Stage 2 tests and teacher assessments by

gender.

Percentage of pupils at Level 4 or above

Boys Girls

2005 2006 2007 2005 2006 2007

English Test 74 75 76 84 85 85

Reading Test 82 79 81 87 87 87

Writing Test 55 59 60 72 75 75

Mathematics Test 76 77 78 75 75 76

Science Test 86 86 87 87 87 88

English TA 70 72 73 81 82 83

Mathematics TA 76 78 78 76 78 78

Science TA 82 83 84 84 85 85

(DCSF, 2007a)

Table 2:

Percentages of pupils achieving Level 5* or above in Key Stage 2 tests and teacher assessments by

gender.

Percentage of pupils at Level 5 or above

Boys Girls

2005 2006 2007 2005 2006 2007

English Test 21 26 28 33 39 39

Reading Test 39 41 44 47 53 52

Writing Test 10 13 15 21 23 24

Mathematics Test 33 36 35 28 31 30

Science Test 48 45 47 46 46 46

English TA 21 23 24 32 34 35

Mathematics TA 32 34 35 28 30 30

Science TA 37 38 38 35 37 38

* Level 5 or above means Level 5 in Key Stage 2 tests, and Level 5 or Level 6 in Key Stage 2 teacher

assessments. Level 6 cannot be attained in Key Stage 2 test.

(DCSF, 2007a)

Assignment 1. 10.


The figures provided within this Statistical First Release (DCSF, 2007a) are based on provisional

National Curriculum assessment data provided to the Department for Children, Schools and Families

(DCSF) by the National Assessment Agency’s (NAA) contracted external data collection agency.

Amendments on the national results, that result from the checking exercise for the 2007 Primary

Achievement and Attainment Tables, is ‘typically of the order of plus or minus one percentage point,

although slightly bigger revisions may be seen for the TA figures’ (DCSF, 2007a: 5), however it is

recognised that some teacher assessment data has not been included due to technical issues during

file submission.

The Qualifications and Curriculum Authority (QCA) have responsibility for maintaining the high

professional standards set out in the National Statistics Code of Practice.

‘The rigour of QCA’s standards maintenance procedures has been endorsed by external

observers, including the independent Rose panel, which found that they bear comparison

with best practice in the world, and have not been subject to any political interference. The

processes rest on a range of evidence about test standard, brought to bear at the level

setting meeting. This evidence includes pre-test evidence from experienced markers on how

pupils performed in the live test and statistical evidence about that pupil performance.’

(DCSF, 2007a: 5)

Haylock with Thangata (2007) argue that it is important to understand that, ‘whereas statistical

trends may show some overall differences in achievement, individual pupils of both sexes are

positioned right across the range’ (Haylock with Thangata, 2007: 74).

This Statistical First Release would again seem to support the assumption that there is a difference

between boys and girls that makes boys predisposed to do better in mathematics; that there is an

intrapsychological difference between males and females that influences the construction of

knowledge and cognitive development in mathematics. However, this could also indicate that the

learning style of boys in relation to mathematics and the method of assessment offered afford them

a more advantageous position.

Boys, Girls, and Learning Styles

Geist and King (2008) present the argument that, in many mathematics classrooms, ‘the classroom

climate, learning style, instructional style, and experience offered to boys and girls may not address

the needs of either gender’ (Geist & King, 2008: 1), and that traditional methods of teaching the

mathematics curriculum based on the traditional skills model of memorisation and rote recitation

rather than open, problem solving environments is having a detrimental effect on both boys and

girls (Kindlon, 2000; Gurian, 2005). However, Boaler (1997) argues that although both boys and girls

are negatively affected by this, ‘the greatest disadvantages were experienced by the girls, mainly

because of their preferred learning styles and ways of working’ (Boaler, 1997: 110).

Assignment 1. 11.


Belencky et al. (1986), and Becker (1995) suggest that boys tend to value a ‘separate’ concept of

knowledge construction and learning involving logic, rigour, and abstraction, whilst girls tend to

value a ‘connected’ concept of knowledge construction and learning involving intuition, creativity,

experience, and consequently understanding (Boaler, 1997). Becker (1995) claims that the

traditional methods of teaching the mathematics curriculum place emphasise on a ‘separate’

concept of knowledge construction and learning, and thus girls are ‘denied access to success in

mathematics because they tend to be ‘connected’ thinkers’ (Boaler, 1997: 111). Head (1995)

extended Belencky et al. and Becker’s concepts and ideas to suggest boys prefer competitive,

pressurised environments, whilst girls prefer co-operative, supportive working environments. Boaler

(1997), however, argues that although boys actually prefer an open, problem solving environment,

when confronted with a traditional, closed approach of teaching the mathematics curriculum, they

seem able to ‘adapt’ with greater success than girls.

Geist and King (2008) acknowledge and support research which has shown that ‘girls tend to feel

less confident about their answers on tests and often express doubt about their performance. Boys,

however, tend to show more confidence and sometimes overconfidence’ (Geist & King, 2008: 1).

Bevan (2001), and Leedy et al. (2003) reveal that these differences between boys and girls extend

beyond the individual problem solving within the mathematics classroom into their general view of

mathematics, with girls enjoyment of mathematics deteriorating much more drastically than that of

boys. Geist and King (2008) report that although boys and girls are competing at a similar level,

there are significant differences in their experiences of learning mathematics (Bevan, 2001).

Haylock with Thangata (2007) reported that teachers, particularly male teachers, are:

‘sometimes observed to be more protective towards girls in the way they deal with pupils’

problems and errors in mathematics. Such behaviour by the teacher could serve to reinforce

a non-risk-taking approach to problem solving in girls, while giving the boys the advantages

of more opportunities to sort things out for themselves and thereby to construct their own

meaning more securely.’

(Haylock with Thangata, 2007: 78)

As a reflective practitioner implementation of the Professional Standards for Qualified Teacher

Status such as Q1, Q10, Q27, Q29, and Q31, for example, should address and resolve issues such

as those stated above.

Boaler (1997) presents compelling evidence that it is a ‘desire’ to understand and construct

meaning, rather than a difference in their ability or potential to understand mathematics, that

differentiates girls from boys, an argument supported by numerous educators, including D’Arcangelo

(2001), and Geist and King (2008), with the latter concluding that ‘while their (boys and girls) ability

and potential to understand higher level mathematics is equal, their brains are different and more

importantly, their approach to learning may be different’ (Geist & King, 2008: 2).

Assignment 1. 12.


Linn and Peterson (1986), however, examined biological explanations for gender differences in

mathematics attainment and concluded:

‘Some have suggested that spatial ability difference might be biologically determined and

provide the mechanism for a biological influence on mathematics and science. No evidence

for that view can be found.’

(Linn & Peterson, 1986: 94)

The assessment data, literature, and research on gender differences in mathematics learning and

achievement supports the supposed existence of a ‘gender gap’. However, rather than a biological

difference in ability or potential to understand mathematics between boys and girls, that makes

boys predisposed to do better in mathematics, the extensive research indicates that the learning

style of boys in relation to mathematics and the method of assessment offered afford them a more

advantageous position, as boys seem able to ‘adapt’ to the traditional, closed approach of teaching

the mathematics curriculum with greater success than girls.

Therefore, rather than a difference in ability or potential to understand mathematics, their approach

to learning mathematics may be different; there is an intrapsychological difference between males

and females that influences the construction of knowledge and cognitive development in

mathematics. This intrapsychological difference, however, does not make boys predisposed to do

better in mathematics, nevertheless, boys seem able to adapt to the traditional curriculum and

pedagogy of schools with greater success than girls.

Assignment 1. 13.


Gender, Group Composition, and Mathematics

‘The benefits to learning of working in groups have been known for some time’ (Edwards & Jones,

2003: 135), with numerous educators (e.g. Koehler, 1990; Lou et al., 1996) acknowledging that

attainment within mathematics may be influenced by the classroom organisation. Lou et al. (1996)

reveals that the variety of students within the classroom ‘means that teachers face difficult

pedagogical decisions if students are to learn effectively and enjoyably’ (Lou et al., 1996: 423).

Peterson (1988) conducted a study into the development of higher-order mathematics, and

recommended ‘teaching approaches which analyse children’s thinking and place greater emphasis

on problem-solving and more active learning, including work in small cooperative peer groups’

(Jaworski, 1994: 9).

In addition, Peterson (1988) emphasised classroom processes such as:

a) a focus on meaning and understanding mathematics and on the learning task;

b) encouragement of student autonomy, independence, self-direction and persistence in

learning; and

c) teaching of higher-order cognitive processes and strategies.

(Peterson, 1988 quoted from Jaworski, 1994: 9)

Wilkinson and Fung (2002) examined ‘the extent to which the grouping of students within classes

affects their learning processes and outcomes’ (Wilkinson & Fung, 2002: 426), and presents the

argument that ‘learning is socially constructed during interaction and activity with others, so there is

interdependence of social and individual processes in the co-construction of knowledge’ (Wilkinson &

Fung, 2002: 426).

Group Size

Wilkinson and Fung (2002) reveal that few studies have ‘examined systematically the relationship

between group size and learning outcomes’ (Wilkinson & Fung, 2002: 436), however, although

individual study findings appear varied, there appears to be a negative correlation between group

size and learning outcomes. Levine and Moreland (1990) concluded that:

‘as a group grows larger, it also changes in other ways, generally for the worse. People who

belong to larger groups are less satisfied, participate less often, and are less likely to

cooperate with one another.’

(Levine & Moreland, 1990: 593)

Vygotsky placed great emphasise on social interaction and communication being perquisites for

learning, with the ‘developmental process deeply rooted in the links between individual and social

Assignment 1. 14.


history’ (Vygotsky, 1978: 30). Therefore, this relationship between group size and learning outcomes

seems to be of central importance in relation to the implementation of social constructivism as a

theory about knowledge and learning within the mathematics classroom.

Numerous educators (Barnes & Todd, 1977; Kagan, 1988; Cohen, 1994; Lou et al., 1996; Wilkinson &

Fung, 2002) report the optimal size group for learning and achievement consists of four members,

however, Wilkinson et al. (2002) acknowledge that ‘evidence on students’ interaction and learning in

groups of different sizes is equivocal’ (Wilkinson & Fung, 2002: 437). Lou et al. (1996) reports that

‘pairs achieved significantly more than students in ungrouped classes’ (Lou et al., 1996: 448), as

interaction was a perquisite for learning (Wilkinson & Fung, 2002), however, the limitation of pairs is

that ‘if the task is a challenging one requiring academic and other creative abilities, there is a strong

chance that some pairs will not have adequate resources to complete the task’ (Cohen, 1994: 73).

Cohen (1994), and Wilkinson et al. (2002) reveal that groups consisting of three members often

results in two members forming a ‘coalition’ and subsequently ignoring questions, concepts, and

ideas proposed by the third member. Barnes and Todd (1977), and Cohen (1994) report that groups

consisting of five members results in members remaining silent, rather than participating. Finally,

Lou et al. (1996) argues that groups of six to ten members ‘did not learn significantly more than

students from ungrouped classes’ (Lou et al., 1996: 448), thus negating the theoretically purposed

benefits of social constructivism.

Lou et al. (1996) meta-analysis concludes that:

‘there are small but positive effects of placing students in groups within the classroom for

learning. On average students placed in small groups achieved more, held more positive

attitudes, and reported higher general self-concept than students in nongrouped classes.’

(Lou et al., 1996: 446)

At this point, the assignment will progress and investigate the hypothesis that the difference in

learning styles of boys and girls influences mathematics learning and achievement in groups of

different gender composition; that the intrapsychological difference between males and females

influences the interpsychological construction of knowledge and cognitive development in

mathematics, as defined by Vygotsky (1978).

Grouping by Gender

‘Clearly, one issue underlying group composition is whether or not groups should be

heterogeneously composed according to ability, interests, liking, gender, ethnicity, and so

on. Unfortunately, a scant number of studies exist to integrate findings on group

composition criteria other than relative ability or prior achievement.’

(Lou et al., 1996: 426)

Assignment 1. 15.


However, I personally believe the issue of group composition on the basis of gender provides an

extremely interesting field of research, particularly in relation to mathematics. Unfortunately,

research which has been conducted in this area has ‘partly yielded contradictory results’ (Busch,

1996: 125), however, ‘there are some insights to be gained that relate to gender differences’

(Koehler, 1990: 143).

Barnes and Todd (1977) suggest that homogenous gender groups are ‘slightly more comfortable,

and less challenging, because children at this age seem to define as ‘friends’ members of their own

sex only’ (Barnes & Todd, 1977: 86). In addition, Barnes and Todd (1977) report that where the

teacher defines heterogeneous gender groups the task ‘inevitably becomes that of coping with

working in this unaccustomed situation’ (Barnes & Todd, 1977: 86), with the possibility that the

gender groups form separate coalitions, thus encouraging a competitive rather than co-operative

environment. As previously reported, boys prefer competitive, pressurised environments, whilst girls

prefer co-operative, supportive working environment, therefore, formations of separate coalitions

within a teacher defined heterogeneous gender group would seem to afford boys a more

advantageous position in relation to mathematics learning and achievement.

Barnes and Todd (1977) conclude that:

‘where groups are making an initial exploration of difficult material, it would be wiser to

allow children to explore their half-intuited knowledge tentatively, and to reformulate as

they go along. But where the material is not so difficult, or where the children have got

beyond the initial, exploratory stage and the teacher wishes them to tighten up their

argument, and justify their thinking, then the mixed group will be more appropriate, as it is

here that arguments are more likely to be challenged, and justification required by group

members holding opposite views.’

(Barnes & Todd, 1977: 86)

Cohen (1994) extended Barnes and Todd’s concepts and ideas suggesting:

‘Mechanically insuring that each group has equal numbers of males and females or one or

two students of color has the disadvantage of making the basis of your decision clear to the

students. They will tend to focus on their fellow members as representatives of their race or

gender and are much less likely to respond to them as individual persons.’

(Cohen, 1994: 74)

Dalton (1990), and Underwood et al. (1990) found that homogenous gender groups achieved

significantly more than heterogeneous gender groups, with the latter concluding that ‘only single

gender pairs improved their performance in comparison with individuals working alone’ (Busch,

1996: 125). This concluding statement by Busch, however, contradicts previously reported literature

and research that report the optimal size group for learning and achievement consists of four

members. Although Dalton, and Underwood et al. provide an excellent and compelling argument,

Assignment 1. 16.


the ideas and concepts have been developed within a computer-based environment, and can

therefore be considered largely irrelevant within the mathematics classroom environment.

Webb (1984) found that ‘effects of gender might depend on the precise composition of mixed-

gender groups’ (Wilkinson & Fung, 2002: 435). Conducting the study within the mathematics

classroom environment, Webb compared the interaction and achievement of 77 students across

‘two above-average eighth-grade mathematics classes and two below-average ninth-grade

mathematics classes’ (Koehler, 1990: 143). Students worked in co-operative, small mixed-gender

groups for two weeks that either had a male majority (three males and one female), a female

majority (one male and three females), or were gender balanced (two males and two females)

(Koehler, 1990; Wilkinson & Fung, 2002). Webb (1984) reports that within the above-average

mathematics classes males achieved significantly more than females. In addition, ‘girls and boys in

the balanced-sex group showed similar patterns of interaction and similar amounts of learning’

(Wilkinson & Fung, 2002: 435), whilst:

‘girls suffered in both the majority-girls and the majority-boys groups. In the majority-girls

groups, the girls directed most of their requests for help to the boy, but he tended not to

respond appropriately to their requests. In the majority-boys groups, the boys simply

ignored the girls. In both cases, the breakdown in interaction impeded the girls’ learning.’

(Wilkinson & Fung, 2002: 435)

In contrast, Webb (1984) reports that within the below-average mathematics classes ‘there was no

significant difference in achievement, nor was there a difference in the interaction patterns of males

and females’ (Koehler, 1990: 143). Webb and Kenderski (1985) argue that within the above-average

mathematics classes the males may have been exhibiting autonomous learning behaviours.

Busch (1996) concludes:

‘A general weakness of these studies is that only a few variables at the individual level are

considered in the analyses. Thus, it is difficult to ascertain whether there exist hidden

variables which would explain the documented differences between males and females and

between groups of various gender compositions.’

(Busch, 1996: 126)

The assessment data, literature, and research on grouping by gender, and how this influences

attainment within the mathematics classroom, supports the supposed negative relationship between

group size and mathematics learning and achievement. The existing research indicates that the

difference in learning styles of boys and girls influences mathematics learning and achievement in

groups of different gender composition; that the intrapsychological difference between males and

females influences the interpsychological construction of knowledge and cognitive development in

mathematics.

Assignment 1. 17.


Therefore, the intrapsychological construction of knowledge and cognitive development is

‘indissolubly interconnected’ to, and therefore influences, the interpsychological construction of

knowledge and cognitive development.

Assignment 1. 18.


Conclusion

The assessment data, literature, and research on gender differences in mathematics learning and

achievement supports the supposed existence of a ‘gender gap’. However, rather than a biological

difference in ability or potential to understand mathematics between boys and girls, that makes

boys predisposed to do better in mathematics, the extensive research indicates that the learning

style of boys in relation to mathematics and the method of assessment offered afford them a more

advantageous position, as boys seem able to ‘adapt’ to the traditional, closed approach of teaching

the mathematics curriculum with greater success than girls.

Therefore, rather than a difference in ability or potential to understand mathematics, their approach

to learning mathematics may be different; there is an intrapsychological difference between males

and females that influences the construction of knowledge and cognitive development in

mathematics. This intrapsychological difference, however, does not make boys predisposed to do

better in mathematics, nevertheless, boys seem able to adapt to the traditional curriculum and

pedagogy of schools with greater success than girls.

The assessment data, literature, and research on grouping by gender, and how this influences

attainment within the mathematics classroom, supports the supposed negative relationship between

group size and mathematics learning and achievement. The existing research indicates that the

difference in learning styles of boys and girls influences mathematics learning and achievement in

groups of different gender composition; that the intrapsychological difference between males and

females influences the interpsychological construction of knowledge and cognitive development in

mathematics.

Therefore, the intrapsychological construction of knowledge and cognitive development is

‘indissolubly interconnected’ to, and therefore influences, the interpsychological construction of

knowledge and cognitive development.

The existing research conducted within the mathematics classroom environment seems to support

the use of co-operative, small gender balanced groups consisting of four members. However, as

previously reported, a scant number of studies exist to integrate findings on group composition

criteria other than relative ability or prior achievement, and research which has been conducted in

this area has partly yielded contradictory results. I personally believe there is a need for more

research that examines group composition in relation to gender within the mathematics classroom.

Therefore, this review of the assessment data, literature, and research on grouping by gender and

how this influences attainment within the mathematics classroom, constructed in a clear and

concise manner with relevant and appropriate research, will provide a firm foundation for, and

consequently support, a small-scale classroom-based research project. This small-scale classroom-

based research project will investigate how groups of different gender composition influence

mathematics learning and achievement.

Assignment 1. 19.


Q3 a); Q6; Q7 a); Q8; Q9; Q10; Q13; Q15; Q18; Q21 a); Q324

Word Count

6,515

4 See ‘Professional Standards for Qualified Teacher Status’.

Assignment 1. 20.


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Assignment 1. 24.


Professional Standards for Qualified Teacher Status

Professional Attributes

Frameworks

Q3 a) Made aware of the professional duties of teachers and the statutory framework within

which they work;

Communicating and Working With Others

Q6 Have a commitment to collaboration and co-operative working;

Personal Professional Development

Q7 a) Reflected on and improved personal practice, and took responsibility for identifying and

meeting personal developing professional needs;

Q8 Have a creative and constructively critical approach towards innovation, being prepared

to adopt practice where benefits and improvements were identified;

Q9 Acted upon advice and feedback, both positive and negative, and was open to coaching

and mentoring;

Professional Knowledge and Understanding

Teaching and Learning

Q10 Gained a knowledge and understanding of a range of teaching, learning and behaviour

management strategies and how to use and adapt them, including how to personalise

learning and provide opportunities for all learners to achieve their potential;

Assessment and Monitoring

Q13 Gained knowledge on how to use local and national statistical information to evaluate

the effectiveness of teaching, to monitor the progress of those taught and to raise

levels of attainment;

Subjects and Curriculum

Assignment 1. 25.


Q15 Understood the relevant statuary and non-statuary curricula, frameworks, including

those provided through the National Strategies, for subjects/curriculum areas, and other

relevant initiatives applicable to the age and ability range;

Achievement and Diversity

Q18 Understood how children and young people develop and that the progress and well-

being of learners is affected by a range of developmental, social, religious, ethnic,

cultural and linguistic influences;

Health and Well-being

Q21 a) Made aware of current legal requirements, national policies and guidance on the

safeguarding and promotion on the well-being of children and young people;

Professional Skills

Team Working and Collaboration

Q32 Worked as a team member and identified opportunities for working with colleagues,

sharing the development of effective practice with them.

Assignment 1. 26.

Documents

PGCE (Primary) - Assignment 1