Pedagogy

Part I - On Social Justice

1.1 Introduction
Between the means and the end, a common view on social justice is that it establishes a combined goal and objective this is to say that social justice is posed as the end point, the state in which it encapsulates an effective ideology as represented in practice.  Although such lofty perception is only fair, especially as social justice is perceived to address a multitude of social issues, social justice can be deemed ineffective as an end, but rather it should be taken as the force that determines the means.  The end, in this case, has to be defined according to the social realities today which include the continuous pull of capitalism and the call for individualism, the supposed threat of socialist ideologies, globalization and diversity.  Social justice can therefore function effectively more if designed as a framework and not necessarily as a platform.

What is interesting is that social justice can be considered as a paradox because it can be simplified as the mechanism that is just across social sectors.  However, the definition of just and the dynamics of social sectors have further highlighted the complexity of social justice.  For instance, in Starrs (1991) discussion on the three views of social justice --- conservative, liberal and socialist --- it seems that the premises of each lens is only fair however, in close inspection, the propositions of each of these perspectives can contradict one another.  An interesting proposal by Fraser (2003) suggests an integration of paradigms --- the redistribution of resources and recognition of cultural differences --- as a working means to establish social justice grounds.  Other authors also cited integrative approaches according to current social relevance such as globalisation (Martines, 2000) and the interweaving of global and local policies (Taylor  Henry, 2002).

In the aspect of education, social justice has served as a force that influence policy-making, especially when it comes to addressing accessibility, effectiveness and efficiency.  Evidently, education has been subject to a debate which evaluates as to how education can be function as both ends and means that are representative of social justice.  This generally boils down to the discussion whether education is privilege or a right, and in the context of social justice, the latter is continuously challenged by the greater pull of the former.

Hence, social justice may serve as the grounds for policy-making in education but the issue is that given the social realities today, what is just may need to be redefined.  Although this does not mean that the ideologies get thrown out of the window, what is important is that social justice is modified according to the changing social conditions and the factors that make it relevant. 1.2 The Different Lenses Coming up with a Definition

Many literature admittedly pointed out that social justice does not have a universal definition (Starr, 1991 Gewitz, 2001 McIrney, 2004 Rizvi, 1998).  However, common threads can be cited which defines social justice based on the grounds of fairness and equality.  Basically, social justice can be simplified as an ideology that is meant to bring about good to all (Gewitz, 2001), and how it represents equity trends.

Although social justice can be simplified in these terms, what can be defined as good, the same was as just, is subjective.  For instance, Starr (1991) presented a view on social justice according to the conservative, liberal and social perspectives.  Evidently, the differences among these three set social justice against different backgrounds.  The conservative view, which is founded on the libertarian principles which include capitalism, individual freedom, and free will, suggests that social justice is a means to remedy any injustices that may have emerged from these conditions.  This is to say that the conservative view upholds the influence of capitalism, which, in close inspection, can further polarise the rich and the poor.  There is the idea of capitalism, free will and individual freedom being the priority of the conservative, and social justice can be mostly seen in programs that are supposed to bridge the holes and the gaps as to the discrepancies of the supposed equities of the modern capitalist societies.  The discrepancies and inequalities are due to a persons birthright, but given the opportunities brought by capitalism and the free economy, this somehow gives everyone equal footing.

Although this landscape may seem fair to all, there are still the factors that make society an unequal system, and interestingly, this is also caused by the forces of the privileged.  Economic successes, despite the conservative idea, do not trickle down, and this is especially prevalent in poorer societies (Starr, 1991).  Which is why social justice does not necessarily exist as an essential element of the society, but rather it is an ideology that is pulled out in order to remedy the inequalities.

In a liberal perspective, social justice is expected to at least balance the individualist values of liberalism.  However, the lack of political stance of liberalism would prove to be problematic according to Starr (1991, p. 22), Underlying neutrality is an implicit acceptance of the current social order and a view of the way society should be, which is underpinned before, by a politically fertile ideology.  One cannot be ideologically neutral if one exists within the social order and absorbs and accommodate its values and expectations.  Apparently, to establish social justice, it needs to have a strong political pull because it establishes the relationship between ideology and society.

Last but not the least, Starr (1991) discussed social justice in the context of socialism.  The authors description can be regarded to have given a more workable lens for social justice especially as socialism refers puts social justice in the platform of socialist emancipation which strives for collective emancipation as a social reality, not for competitive individual accomplishments (p. 23) which is emphasised both in the conservative and liberal contexts.  Hence, social justice emphasises more the collective in terms of impact and influence.  However, the problem with this concept is that it overturns the ability of the individual to determine what is best or what is a functionally just society especially as the socialists are the likely ones to have the upper hand as to the definition of what is socially just.

It can be gathered that social justice based on these lenses are caught in the middle of a paradox.  The common ground among these three views can be traced at how social justice is still supposed to work for the people, but achieving it may also mean jeopardising the welfare of other groups, the individuals and the entire collective.  In putting these thoughts in the context of education, it can be easily traced as to how education policies are also caught in the same paradox especially in terms of serving the best interests of the majority.  On one hand, although the majority gets to benefit from the policies of education, what about the rest  Dont they deserve the same amount of benefits as the rest  Is social justice about pushing the welfare of the majority and leaving the rest behind, or having the majority stay put at a certain level in order for the rest to catch up

In putting this topic in a different context, what comes to mind is Frasers (2003) discussion on redistribution and recognition.  The author mentioned that social justice has been divided into two claims the redistribution of wealth and the recognition of the diverse social components that does not require assimilation to majority or cultural norms as a means to achieve respect.  In this case, these two claims can be looked at based on the context of identity politics.  What is interesting is that in close inspection, redistribution and recognition can work side by side mainly because by means of redistribution, there is a sense of assimilation, and by means of recognition, the act of recognition is substantiated.  For instance, immigrants are collectively identified according to their ability to assimilate to the dominant culture, hence, one of the initiatives of redistribution, out of this recognition, is through the design of assimilation programmes such as ESL studies.  In any case, there is always the push and pull as to the degree of what should be recognised (i.e. ethnic and cultural diversity), but at the same time, the aspect of redistribution is still driven by the fact that its main purpose is to establish a sense of equality not only in terms of value and capacity, but also in the context of acculturation.

It can be seen that in education, the redistribution and recognition is all about the intention to achieve a specific balance of acknowledged diversity and acknowledged need for integration.  This is to say that education policies, in the framework of social justice, has been all about recognising the different abilities of the students and at the same time, its curriculum and strategy design aims for a general integration.  What more can an educator want but to have his or her students above the passing grade

Other lenses that tackle social justice can be summarised based on individual rights, the extent of these rights, and how these rights affect others (Rizvi, 1998).  For Luke (1999), social justice can be simplified
through equality.

1.3 Challenges
It can be observed that the problem with social justice is that it seems it does not have a place in the current realities.  Equity in social justice is still limited which basically shows that social justice cannot exist in absolute terms.  For instance, even though Luke (1999) pointed out the importance of equality, the definition of equality needs to be specific in its context.

Social justice in education may be deemed simple at localised levels, but in setting it against the larger scope such as national standards, problems can possibly arise.  For instance, a districts above-average performance may rate poorly on a national scale.  This can be further challenged should performance is set against the international level.  Which is why, as Martinez (2000) suggested, a more global perspective may help in the formulation of more effective strategies should social justice is addressed.

1.4 Reconciling the Differences a Personal Stance
What is confusing about social justice is that as an ideology, it is an important goal that should be achieved, especially with the current state of the world.  Social justice seems unrealistic, but then it can be achievable this accomplishment, however, needs to address the hurdles of social realities especially in terms of the social, political, economic and even global aspect.

My stance about social justice is that at this point, it should be emphasised more as a process in which its design is based on larger goals (i.e. global perspective) with strategies enforced at grassroots level and pace.  In a sense, this reflects Martinezs (2000) point that social justice should be a goal and a process.  However, it should be noted that with this set up, social justice elicits a redefinition, hence, it should not be confined to the ideals of equality and equity.

From this, my stance is that a working definition of social justice, especially in the context of education, is to create policies that are effective, efficient and accessible.  Education should be effective as can be seen in its well-designed strategies following the established mission, vision, objectives and goals.  This effectiveness highlights the need for quality and at the same time, relevance this is where the integration of a more global perspective comes in.  Effectiveness, in this case, should therefore identify the current needs based on the current realities and at the same time, the projected future.

From effectiveness, the other step is to have education be efficient.  This time, it emphasises the processes involved which work hand-and-hand with effectiveness.  Efficiency also takes into consideration the processes needed in order to make education accessible as well.  For instance, this can be seen in programs which enable children that are challenged in certain subjects to have more support and access to remedial courses and assistance.  Efficiency thereby manifests the effectiveness of education.

Last but not the least, accessibility becomes the equity this basically makes education a working product of social justice, but it only works if accessibility is substantiated by both content and process.

Hence, I believe that social justice is redefined not by concept and principle but by framework.  This is because it is through a working framework that translates the concept into more concrete terms that make this ideology both relevant and realistic which will eventually pave the way to equal education outcomes.

Part II - Mathematics and Gender

2.1 Introduction

It is a common perception that the likely person who excels best in mathematics is a male.  Hilton and Berglund (1974) mentioned a study by Macoby that based on an assessment of academic performances in the context of gender, females were found to perform better in verbal and linguistic studies while males had the advantage in arithmetic and spatial aptitudes.  What is interesting is that in this study, the degree of differences did not diverge until late in the grade school years, which would become more evident in high school and then more common in college.  In Hilton and Berglund (1974) study, the authors found that the performance in mathematics is based on the perception of the student, whether male or female, as to the usefulness of math in their professions.  Hence, at that time, more males found mathematics more useful, but this does not discount the fact that although there were fewer females who were as good in math, they can compete as well with the opposite sex.

This notion can be deemed inescapable in a society that has been mostly gendered, skill sets have been also associated whether it is female or male.  Unfortunately, this has also affected the attitudes of the students towards certain subjects for example, engineering and certain sciences are more male whereas liberal arts subjects are supposed to cater better to females.  In the realm of mathematics, such perception has been somehow ingrained as well.

In this specific context, social justice becomes affective of student behaviour.  This educational issue spans the perceptions and the behaviours of both instructors and students, which is why as the executors of the curriculum, instructors need to understand how social justice is applicable even in this particular subject.

2.2 Social Justice and Mathematics
Social justice in the context of education establishes venues in which the students are guaranteed effective, efficient and accessible education which will lead to equal outcomes.  As previously mentioned, even though this sets up an ideal scenario, conditions have made this concept far from reality.  Basically, gender has been cited as a main social justice issue in mathematics especially as there have been the automatic presumptions as to which gender is likely to perform in mathematics better.  Jungwirth (2003) pointed out that as early as the 1970s, feminist movements have been aware of social inequalities due to the prevalence of male-dominated societies.  Even though this can be assumed in the professional field, the education system also became subject to scrutiny whether there was indeed some gender bias in the field.

However, in addition to gender, more studies have also cited additional factors which would further magnify the need for social justice in mathematics classrooms.  Eventually, other factors have been explored such as ethnicity and social class.  Basically, social justice in the context of mathematics education may be due to the fact that student performance in mathematics have been varied, in which case the percentage of above-average and average students are still somehow threatened with the substantial number of students who struggle in the subject.  This is why in looking at mathematics in the context of social justice, it is important to look at the different relevant factors in addition to gender.

What is interesting is that even the notion of social justice as an integral consideration in the pedagogy framework has been put into question.  This is mainly because social justice in education can be inevitably viewed because of its political intent (Atweh, 2010).  Gutstein (2003) also mentioned two points in the exploration of social justice in mathematics instruction how to concretise and factor in social justice in the teaching and learning of mathematics, and how this process is related to the standards enumerated in the general and specific curriculum.

So far, Atweh (2010) mentioned that even though there have been a considerate amount of studies examining social justice and mathematics, the engagement in social justice itself was observed to be lacking, although there have been initiatives in the aspect of policy-making.  The author raised an important point in observation, Perhaps the debate between equal opportunity and the different ways of knowing is a reflection of the need to engage with an understanding of social justice that will allow us to engage in evaluation of the merits and limitations of measures to counteract injustice (p. 47).  From this, it is evident that the main issue between the concept of social justice and mathematics is that their relationship is a challenge to establish mainly because essentially, teaching and learning are not necessarily equal.  As Atweh mentioned, there are different ways of learning, and even though a teacher applies similar strategies, the students as variables can already pose a challenge especially in terms of how their reception towards the subject can also greatly affect their performance.

One way of examining this is one of the common approaches in which students are divided into classes based on performance for instance, classroom A contains the advanced students, classroom B houses the average students and classroom C is for the below average students.  In a sense, this system already determines what kind of instruction these classrooms will receive from their respective instructors.  This system can be perceived to create a more democratic means in instruction because of the degree of education each classroom will receive according to the needs of the students.  However, the question is that, in order for the entire classes to be equalise does that mean the development and progress of classroom A will be jeopardised in order to make room to the progress of classrooms B and C  Or will classrooms B and C go through rigourous processes in order to meet the established standard similar to classroom A

However, it should be noted that social justice is not basically about student performance, but rather it becomes a derivative based on student performance.  This is to say that in the context of social justice, performance is evaluated according to prevailing social factors such as gender, ethnicity and social class.  What thereby makes social justice a growing issue is that apparently, there have been patterns that point out that there are indeed differences in performance and interest based on these components.

These differences are therefore supposed to be resolved by integrating social justice in the mathematics pedagogy this is based on the influence and impact of social justice as explained by Povey (2003, as cited in Christiansen, 2008, p. 71)

To harness mathematical learning for social justice involves rethinking and reframing mathematics classrooms so that both the relationship between participants and the relationship of the participants to mathematics) as well as the mathematics itself is changed.

It can be gathered that mathematics classrooms and instructors are tasked to develop and design a pedagogy that steps outside the conventional notion when it comes to student performance.  What can be highlighted in this concept is how mathematics is not just going to be a subject to measure intelligence and capability, but rather, it is a subject that is meant to arm all the students when it comes to their future roles in the society.  A concept that can enforce this is the idea of ethnomathematics as discussed by Dambrosio (2008).  The author pointed out that mathematics is not just a school subject and a component of some professions, but rather, it is most universal mode of thought (p. 38).  Dambrosio further elaborated how education should be also considered especially in terms of its purpose he enumerated (p. 39)

To promote creativity, helping people to fulfil their potentials and raise to the highest of their capability, but being careful not to promote docile citizens.  We do not want our students to become citizens who obey and accept rules and codes which violate human dignity.

To promote citizenship transmitting values and showing rights and responsibilities in society, but being careful not to promote irresponsible creativity.  We do not want our student to become bright scientists creating new weaponry and instruments of oppression and inequity.

As can be seen from these, dAmbrosios thesis points at how mathematics and education in general have a greater set of goals which are not just directed towards the development of the traditional view on intelligence, but rather education is enforced for the purpose of citizenship.  Social justice in mathematics, based on Dambrosios view, is both the means and the end in the attainment of the greater social goal of peace.  Because it is universal both in process and result, mathematics as education should be therefore directed towards its role in the achievement of lofty social goals.

An example is explained in a study by Goos, et al. (2008) in which the authors looked at a case in Australia.  Goos, et al. examined the presence of community partnerships based on numeracy education these partnerships are participated by schools, families, and communities in order to address and respond to the learning and development of the students for their intellectual foundation.  This program is actually based on a national policy which highlights the importance of mathematics education the Department of Employment, Education, Training and Youth Affairs emphasises, to be numerate is to use mathematics effectively to meet the general demands of life at home, in paid work, and for participation in community and civic life (1997, as cited in Goos, et al., 2008, p. 13).

In this study, Goos, et al. (2008) mentioned how the effectiveness of these partnerships is mostly based on how each program is designed according to the specific needs and conditions of the community.  These differences can be seen in the cultural and ethnic diversity of the community, the differences in the perception on mathematics and community, and the variations of the instructional approaches.  This thereby highlights the diversity of the society which Atweh (2009) enumerated the big 8 factors race, gender, ethnicitynationality, organisational role, age, sexual orientation, mentalphysical ability and religion (p. 113).

Gutstein (2003) pointed out that in order for social justice to work, it is important that policies also include students as part of the solution.  Interestingly, one may look at social justice in the classroom as an instrument for the student, and not necessarily a process which also requires involvement from the students.  Hence, the author suggests that in order to be more effective as an integral part of social justice initiatives, the teacher must also integrate in the instruction how the students view their respective surroundings, especially their communities.  For instance, whether there are gangs in the community or how women are generally treated in their socio-cultural group and family.  Through this exercise, the student is able to have a greater understanding and awareness about these social conditions this underlining substantial meanings of their lives.

How is this therefore relevant in effective classroom pedagogy  An important foundation in performance is from these reflections, the student should then eventually realise a sense of agency.  This is to say that with the awareness about their social conditions, the student is able to accept and understand how he or she can be instrumental to changing and improving the communities.  In addition to these, through the context of social justice, the student is able to establish a better relationship with his or her respective cultural identities.

Such concept, according to Gutstein (2003) can be further translated into effective instructional strategies especially when it comes to mathematics.  Normally, numbers may be perceived to be on a different plane as compared to the social sciences.  However, the author pointed out that in order to have the students feel more involved in the learning process, the sense of social awareness can make them feel more relevant especially if the mathematics instruction is designed according to the context they are familiar with.  For instance, Gutstein mentioned the design of real-world mathematics projects that connected to students lives and experiences (p. 42).

Social justice in mathematics thereby aims to redefine the mathematics beyond the intellectual sphere hence, as a subject, mathematics education should thereby address social factors in order to create a more effective impact among the students.  Although social justice is also about equality, equity and access, applying this in the context of the mathematics curriculum also requires relevance.  This thereby explains the need to explore the social factors involved that may affect the reception and performance of the students, thus issues in gender, ethnicitysocio-cultural background and even social class have elicit inquiries in order to solve the current prevalence of inequalities in term of curriculum, instruction, learning and outcomes.

2.3 Mathematics and Gender Bias Motivational Factors
Catsambis (1994) identified three factors that may have caused a gender gap in mathematics education opportunities, achievement and choice.  It can be observed that these factors are more directed towards a persons end goal which dictates the process.  This is to say that there can be a psychological influence when it comes to the interest of the student towards the subject, and apparently, prevalence is present among genders (Appelbaum  Davila, 2009).

Catsambis (1994) generally points out that the issue is that the applicability of mathematics in the context of opportunity most likely point at activities and tasks that may interest the males more.  However, this is just an assumption but basically, this assumption affect how female students generally perceive mathematics.  Hence, if the opportunity structure appeals more to the males, and this opportunity has strong applications of mathematics, females find underrepresented, thus perceiving that the opportunity is irrelevant to them.  Thus, this highlights the relationship between interests and achievement.

It is important to point out, however, that such perceptions are also relevant depending on the age of the students.  In Catsambiss (1994) study in which the author surveyed 10th graders, it was found that in terms of performance, the female students are as good as the male students and that the perception that females lag behind males in maths can be challenged.  However, interestingly, psychological factors highlight how, despite age and performance, there is a greater amount of lower esteem and confidence when it comes to their perception towards mathematics this may be also driven by the obervation that females are generally less interested in the subject.

However, if it is observed that the general female perception on mathematics is less as compared to the males, does that mean that the traditional approaches to the subject should be maintained  Although this may be true at some point, it is important to note that the trends have been changing over the years.  This is to say that the growing interest in mathematics may be due to its growing relevance not only in the math-intensive subjects as normally perceived such as engineering.  Evidently, as dAmbrosio (2008) previously pointed out, mathematics is a universal mode of thought this can be observed to be especially true today especially with the emergence of the information and digital age.

Such development can be seen in a study by Forbes, et al. (1998) in which it was found that there has been an increase of female participation in mathematics both at secondary and university levels.  However, despite the improvements of the girls performance in the subject (i.e. studies by Couppland  Wood, 1998 Zevenbergen, 1998 and Leder  Forgasz, 1998), the image that associates mathematics with males remain to be strong.  In a study conducted by Brew, et al. (1998) which involved the participation of more than 370 students, males and females rate themselves differently when it comes to their performance in mathematics these ratings are then compared to the ones given to them by their teachers.  Apparently, the males over-rate themselves whereas the females under-rate themselves even though their teachers ratings illustrate otherwise.  What can be gathered from this study is that many students still have a gendered perception about mathematics in which it is still identified as a male domain.  In a similar study by Forgasz (1998), the author explained the presence of gender-stereotypped behaviour and expectations ith respect to mathematics (p. 42).

It can then be seen that when it comes to the issue of gender in the mathematics classroom, the main issue is perception and psychology.  Because this has been in place, this thereby highlights the dynamics of gender, equity and power in the context of mathematics.  According to a study by Johnston (1998), the kind of mathematics education that has been taught has been mostly for the purpose of applying the knowledge for quantification and reductionism, ideas which are mostly associated with the male dominated-world.  Because of this, there is also the aspect of identity in play when it comes to this subject especially as this positions both men and women in a certain place per function in the society.

With these issues at hand, social justice remains to be a pressing issue because this calls for a change as to what mathematics should be.  It is important to encourage a sense of change especially in terms of how the subject is not necessarily gender-oriented or gender-influenced.  Hence, as Fennema pointed out (as cited in Hart, 2003, p. 29)

At the end of schooling, there should be no differences in what females and males have learned, nor should there be any gender differences in how students feel about themselves as learners of mathematics.  Males and females should be equally willing to pursue mathematics-related careers and should be equally able to learn new mathematics as it is required.  The definition of equity as the achievement of equal outcomes offers the greatest promise for achieving social justice.

Interestingly, as Jungwirth (2003) cited a few studies addressing this basic issue, one of the initial solutions was co-education, but this did not bring about solutions in bringing forth balance in the classroom.  In this case, a common means to approach equity in mathematics pedagogy is through gender-sensitive approach in which both males and females receive the same degree of education, attention and interaction with the teacher.  The author pointed out that in this approach, the teacher applies a gender sensitive measure because of the uniformity.  Hence, the teacher  designs the same curriculum for both boys and girls in terms of topics, problems and tasks.  In addition, the teacher also gives equal amounts of social interaction with the students in which the genders are assumed to be equal in principle with respect to their internal conditions to learn mathematics so that it is assumed that interacting with them equally leads to an equal outcome of their achievements, interests and so on (p. 6).

As can be observed, social justice in the mathematics classroom is faced with the challenge in terms of how a teacher approaches and applies effective teaching according to certain gender equality frameworks.  This is to say that give there has been a prevailing norm as the gender bias in mathematics, an instructor needs to address the possibility that in a co-ed classroom, the male students may have the advantage over the girls, and in turn, girls are aware that their male classmates are better in mathematics than them.  Again, this shows that the existence of the general belief affects behaviour and performance.  Although it can be said that there may be more boys who are better in math, this does not discount the fact that some girls can have the same or better grades than them.

This thereby shows how the normal approach to mathematics teaching, which is described above, is probably the safest way to apply gender sensitivity in the classroom.  This time, gender sensitivity is about giving all students the same amount of instruction, material and interaction.  According to Jungwirth (2003), should the teacher notice of some students lagging behind, such case is no longer a gender issue but rather an issue that is specific to the student.  Although this brings up an important point, it cannot be helped that there are some aspects of this set of students that a teacher may eventually take note of, and that is whether these students that need additional attention and assistance are mostly boys or girls.

What is interesting is that in a study by Leder and Forgasz (1998), the authors found that there is a certain effect when classes are split into single-sex groups.  In their evaluation across a number of mathematics classrooms of this context, the authors mentioned that the girls performed better or as good as the male students.  However, this still does not change the perception on the subject as a male domain.

2.4 An Assessment of Studies on Social Justice and Mathematics
As can be seen in the previous section, the examination and exploration of social justice and mathematics step beyond the issue of gender as these also embrace other social factors.  What is interesting is that thse studies seem to emphasise similar points a) there is a variation on the student performance of students b) one of the variables of performance is gender c) there have been notable changes and developments as to the performance levels between males and females in the mathematics classroom d) despite the improvement of female students, there is still the prevailing notion that mathematics are mostly suited more for males.

Very few of these studies actually offered solutions and recommendations, but there is a general consensus that a means to tackle the problem is to acknowledge the real problem the challenge is not because mathematics is a hard subject but rather there are the social forces that can affect a students participation and performance.  These social forces, outside the big 8, also include the greater socio-cultural, -economic and -political factors that can affect the student.  This is why, as Gutstein (2003) suggested, it is important to make mathematics relevant to the students and not have the subject be placed as a mere subject that focuses on quantification and reductionism.

Part III - Action Research Approach

3.1 About Action Research
Research serves as the means to gather data in order to process information that will result to knowledge.  However, knowledge is useless if not applied and translated into a set of actions.  Action research thereby enables the researcher to look beyond the data and the information, and eventually come up with wisdom that will be applied as a means to respond to the posed problem.

In definition, Carr and Kemmis (1986, p. 162) define action research as,
... a form of self-reflective enquiry undertaken by participants in social situations in order to improve the rationality and justice of their own practices, their understanding of these practices, and the situations in which the practices are carried out.

Action research is popular in the field of education especially in terms of policy- and decision-making.  When it comes the design of the curriculum, action research is considered critical especially as there is inherently the presence of social conduct in the educational sphere.  In addition, education is also highly involved when it comes to social issues especially as this requires the development at a greater context.  Given that education is dynamic, especially as this aims to ensure continuous progress and improvement of the students, education is constantly subject to changes as enforced by social forces.

Another important factor of action research is how it is found to be also impactful to the one conducting research.  This is especially important especially among teachers who need to improve and refine his or her actions in order to be more effective.  Furthermore, according to Sagor (2000), action research can be also empowering to the teacher especially as he or she is the primary consumer of the findings (p. 3).  Sagor also enumerated the following steps involved in action research (pp. 3-4)
Selecting a focus
Clarifying theories
Identifying research questions
Collecting data
Analyzing data
Reporting results
Taking informed action

The last step in action research can be considered as an important step especially as the results are translated into a set of strategies.  In education, teachers engage in action research in order for them to find solutions to the identified problems in the classroom. Hence, this shows how this initiative is mostly used in specific settings and conditions that are only particular in that class.  In this case, a teacher may use the results for a specific class or subject depending on the identified needs.

Moreover, according to (Selener, 1997), what makes action research important is the degree of participation that the teacher actually has with this overall inquiry.  As the author pointed, this activity catalyzes practitioners creative potetial by encouraging them to create and apply their own knowledge rathen than merely implement results derived from academic research studies (p. 96).  Because the research is based on their classrooms needs, and that the inquiry can be done through various channels such as reflection which will result to a deliberative action, this approach to research can be deemed more effctive in coming up with classroom-specific solutions.

3.2 Action Research for Motivational Gender Split in Mathematics Academic Success and Social Justice
In the previous section, it was found that the gendered perceptions in mathematics has been caused by perceptions that the subject is male-dominated.  Although in some cases this has not refected in the performance of some female students, a potential issue with this is that this may affect the motivation of the female students to pursue mathematics.  Teachers can be therefore tasked to examine how the female students in their classroom view mathematics, and from there, compare how perception is translated to motivation from the motivational element, the teacher can examine the performance of the student.

This sphere of inquiry can be related to social justice as a component of action research based on the fact that action research itself can provide answers as to how to make the subject in a specific classroom or school effective, efficient and accessible.  Basically, according to Sagor (2000), action research has the following purposes 1) building the reflective practitioner 2) establishing progress in the priorities at school and 3) to build professional cultures.

Through action research, the teacher is able to reflect not only in the performance of the class but also explore the different dynamics at work.  Normally, classes are rated according to grades and standards, but basically, an effective classroom  shows substantive learning and application of the students.  Teachers can therefore reflect on the social justice factor in the class by looking at the effectiveness, efficiency and accessibility of the education.  These elements can be seen in how the teacher is able to come up with measures that are substantial to learning as determined by the set standards, learning as determined by the defined social goals, and learning as determined by the specific goals and objectives of the teacher as influenced by the conditions of the classroom.

In reflective practice, action research begins without a definite focus.  According to Sagor (2000), the inquiry can begin with the teachers general refection about the class.  For instance, if the teacher wants to know why there is a critical variation in participation and performance in mathematics, the researcher cannot automatically assume why rather, through the use of a reflective journal, the researcher is able to determine potential directions of more specific inquiry.

The teacher can apply the reflective activity by taking note of the occurrences in class.  Should the teacher notice that there are certain patterns such as, for example, more male students are actively participating in recitation, this can open the venue for reflection and inquiry that can become the subject of the research.  The action, eventually, is to make sure that the female students will be also as active as their male classmates.

Action research also includes more involvement with the subject.  The teacher can therefore conduct interviews and surveys which can further help the daily reflections as based on the happenings in class.  These data can then help the teacher understand and explore other issues which may not manifest in the classroom but instead another issue seems to more evident in test scores and overall performance.  Hence, there is a strong participatory aspect among the students in this research.  In fact, the teacher can also ask certain students to get involved in the research in order for the students to be also aware of their own social and educational practices (Kemmis, 2007).

Hence, in the mathematics classroom where it requires an inquiry as to the gender gap in terms of performance and motivation, it is important for the teacher to also explore the motivational factors of the students not only in terms of their participation but also their perception towards the subject.  As previously mentioned, although there have been female students who perform well in mathematics, the perception that it is a male domain remains to exist (Leder  Forgasz, 1998 Forgasz, 1998 Catsambis, 1994).  Although these studies establish a reality about mathematics classrooms, it is important to test this in a specific classroom.  As noted, trends may change over the years, thus many female students may have better motivation to participate and perform better in math.  In this specific research, the teacher should note of potential motivational elements in addition to perception such as, for example, the increasing interest opportunities among the female students towards opportunities that have a strong mathematics utilisation.  At the same time, it is also important to compare these information with the motivational forces among the male students.

In the reflective practice, the data can be gathered through direct inquiry or interaction with the students, and then by means of observation.  A reflective teacher thereby looks across the many sources of competencies among the students, and with effective observation, the teacher may even point out certain patterns and trends that can contribute significantly to the ongoing inquiry in this field.

From the gathered data, the teacher is now able to evaluate and come up with deliberate action.  With action research, it is integral that after the knowledge, a set of strategies and tasks are formulated.  When the teacher comes up with a set of results, solutions are then designed and applied.  In action research, the teacher can thereby deliberately impose changes in order to encourage a certain set of results andor behaviour (Kemmis, 2007 Selener, 1997),

In applying deliberate action in this particular action research, especially in the context of social justice, the teacher is then able to consider social factors as influences to the overall motivational and performance dynamics in the mathematics classroom.  In this case, the teacher may consider how female students motivation and perception may be affected by how females are generally viewed in the community.  Another possible example is that females are less motivated to study mathematics because their environment influences them to study subjects that are supposed to be irrelevant to mathematics such as liberal arts and domestic sciences.  From these results, the teacher is now able to come up with actions that may be also influential to the overall school and community dynamics. In this case, solutions and actions can possibly step outside the classroom.

Social justice in action research thereby help point out issues that are not only exclusive inside the classrooms.  Given that mathematics has been so far a male-dominated subject and practice, it is important to determine how female students respond to this.  Although at the eighth grade level many female students may be more determined to do better in mathematics, it is important to establish among the students, inside the school and inside the community that mathematics is universal and is not gendered.  Hence, encouraging students to be more open towards mathematics is a good motivational step especially should the teacher open all the students minds when it comes to the relevant opportunities, achievement and choice.

3.3 Important Notes on Action Research Ethics and Education
Since action research can open venues for more open and direct inquiry between the teacher and the student, it is important to take note of the ethical requirements that should be applied during the entirety of the activity.  The first step is for the teacher to inform the participating students the purpose of the research and what will happen after the research.  This is especially important especially if this will impact the students in the future. The second step is to define the boundaries of the research such as the teacher making it clear that participation does not lead to better grades.  Another boundary is to establish the degree of participation of the students such as the amount of time required in the participation and what kind of information that they can divulge.  Third, it is important to inform the school and even the parents of the students about this research.  This is especially important if the research requires direct involvement, and for example this may require the students to spend some time on the research or the students are required to certain access such as a computer with Internet access.  Hence, this protocol is an important practice that every teacher should not forget.

Last but not the least, the teacher should take note of the implications of the research especially should the data eventually lead to community-wide action.  This is possible if there is a strong social justice factor involved that may be outside the sphere of gender.  In this case, should the teacher decide to take the research and the action to the next level, it is important to have the school involved in order to gain support before bringing the issue to the community.

4.0 Implementation of Two Cycles of Action Research

4.1 Overview
Physical activity can be observed to be integral in any school curriculum.  In addition to giving the students the opportunity to step outside the classroom and get some exercise, physical activity has been actually found to be beneficial to the overall learning development of the students.

Physical movement and brain development have been associated together as critical components to the overall learning of a student.  Biological and physiological evidence show how the physical activity contributes to brain development by helping it prepare for the learning process.  As Feinstein (2006, p. 373) further explain

The brain and body connect to navigate the environment by predicting, planning and executing physical movements.  The brain uses this framework to sequence, order, practice, and rehearse cognition.  Being physically active prepares the brain for learning by increasing oxygen and glucose to feed the brain nutrients and by integrating, organizing, and energizing key components of brain function through physical activity.

Many studies have therefore supported the importance of physical activity to maintain excellent brain conditions how this affects learning is, however, based on the physiological benefits of physical movements to the brain.  This is to say that physical activity can help a person learn better because of the improved conditions of the brain, however, whether this affects the motivation, participation and engagement to learn still needs to be explored.

4.2 Cycle One
4.2.1 Overview
The action research tests the connection between physical activity and motivation, participation and engagement as precursors to effective and positive learning.

The connection between physical activity and motivation, participation and engagement has been found in the tasks themselves.  This is to say that getting involved in physical activities such as exercise and sports are connected to motivation and determination.  These two factors are mostly for the purpose of getting into and committing physical activities (Markland  Ingledew, 2007).  Moreover, physical activity is found to bring about a sense of achievement especially as these tasks set certain goals.

Based on this, it can be gathered that regular physical activity can help an individual develop his or her own practice of motivation, participation and engagement.  It can then be hypothesised that this concept can be applied in learning.

4.2.2 Set-Up and Metric
This concept is tested in an eighth grade class consisting of 12 boys and 12 girls.  The hypothesis is that greater involvement in physical activity can result to better motivation and participation in learning.  What is interesting in this set-p is that it can be easily assumed that girls tend to shy away from physical activity, thus, based on the hypothesis to be tested the girls may perform less as compared to the boys.

To test this, the following metric is designed as follows
In a class of 12 boys and 12 girls, each student is given an identifier B1, B2, and so on for the boys, and G1, G2, and so on for the girls.

Boys and girls are observed according to their participation in a physical education class.  Through this observation, notes are taken down as to their degree of participation in the PE class such as enthusiasm, participation, sportsmanship, etc.

The students are then observed based on their performance in other classes based on participation and performance.

The distinguished students are then identified six will be chosen from each.  From there the students are compared.  Based on the hypothesis, students who perform well in PE can also demonstrate a similar degree of performance in other classes.

The results then look at the gender differences.
This measure is a simple to way to look at whether performance in PE is comparable to performance in the regular classroom.  In a sense, this determines how participation in both areas work hand-in-hand with each other.

4.3 Cycle Two

4.3.1 Results
Based on the testing, it was found that there were certain inconsistencies based on PE performance and classroom performance.  This is to say that there are some students who show excellence both in PE and in the classroom whereas there are those who perform better in PE but are more on the tail-end when it comes to academic classroom performance.

The breakdown of the results is as follows
Classroom Performance (Boys) B3, B4, B6, G1, G3, G10, G12
PE Performance B2, B3, B4, B7, B10, G3, G7
Classroom and PE B3, B4, G3

The results show that the difference is not substantial enough to demonstrate that boys perform better than girls, and that they represent the physical activity and learning performance relationship.  What is interesting, rather, that classroom performance is more dominated by the girls rather than the boys.  In this case, this shows that although physical activity is integral to positive learning, the assumption that girls getting lesser physical movement does not follow that they are not as competitive as the males when it comes to learning and classroom performance.

4.3.2 Analysis
Although the results showed that greater physical activity does not necessarily result to better learning performance and demonstration, this does not discount the fact that physical movement is integral to better learning conditions.  When it comes to the aspect of gender, it cannot be easily assumed that just because the girls perform less than the boys, they are not armed with the same degree of motivation and determination.  This goes to show that there are other factors to consider.  Hence, when it comes to establishing the relationship between physical activity and learning, the connection is there but it cannot be gendered in measuring motivation, participation and engagement in learning as compared with physical movement.