Assignment kokkaliari

Uninterested, hard-to-reach pupils in secondary school mathematics:

Teacher’s puzzle

ABSTRACT: The aim of this assignment is to claim uninterested student’s attitude towards

mathematics as a central concern within mathematics classroom in secondary education, not

in the traditional sense usually teacher’s use it as an unchangeable, stable one, that causes

their students failure in mathematics, but as a powerful and vital aspect of both teacher and

researcher for further collaboration. Drawing on research evidence, a case is made that it is

never too late to change students’ attitude towards mathematics and this implies that it is

never late to change a teacher’s attitude towards his own teaching practices. The article also

raises concerns about the ability of the education system to positively address collaboration

between researchers and teachers in the classroom in a permanent basis for development

learning environments for students in mathematics classrooms with an identity of self-

efficacy and values, happy citizen of the future world.

Keywords: identity of uninterested learner, students’ attitudes towards mathematics,

engagement, motivation, teacher, researcher

In the fourth age BC Aristotle states: ‘those who educate children well are more to be

honoured than parents, for these only gave life, those the art of living well.’, and his grateful

student Alexander the Great some years later declares: ‘I am indebted to my father for living,

but to my teacher for living well’. Of course what is not mentioned so frequently is that,

Philippe, Alexander’s father searched a lot for his son’s teachers among the most famous

academics.

Since then a lot of things changed but families have still the same concern for their children’s

education, looking for the best schools and teachers to guarantee their well-living. Education

has become a human right for almost every child all over the world and Mathematics is

honoured to be - after pupils’ native tongue - the second compulsory subject for most years

of schooling. If we consider a Mathematics classroom as ‘a social space in which gender,

ability and other social identities are played out’ his role as a teacher is, nevertheless, crucial.

Educators, researchers and teachers work continuously to make sense of school, classroom,

mathematics, curriculum, children, knowledge, learning, teaching, tasks, activities, etc. For

any one of these notions a mass of information has been gathered, a lot of theories have been

developed, most of the time in contradiction to each other, in the best case complementary.

But how can theories, derived from psychology, sociology, philosophy and mathematics, help

the teacher solve the problems he faces in a mathematics classroom? It is well known that

every theory includes some and excludes others. Hence no matter which of them, consciously

or not, he adopts, a not belonging-marginalised student identity is formed, who is the black

sheep of every educational system, including the teachers who attribute the responsibility to

the student’s attitude towards mathematics, the family background and the previous teachers’

attitudes. Uninterested, in mathematics, students and sometimes whole classes are one of the

most popular topics discussed among teachers in professional meetings. Every teacher feels

uncomfortable with these students who most of them cause problems, that make teacher’s life

difficult, behaving in ways that teaching learning processes do not include. They ask their

colleagues how they handle these difficult cases, especially the less experienced ask for

advice. They express emotions of disappointment, when they try to apply strategies of

classroom management unsuccessfully. They ask the head teachers’ help and they, in turn,

try to make them be aligned with school’s rules, they punish them and they ask their parents’

help. In this unpleasant teaching condition, the teacher expects urgently material assistance in

the form of innovations, as Brousseau states: ‘there is a controversy involving innovators and

the defenders of action research about what didactique is, what it can do and what it should

do’ (Theory of Didactical Situations in Mathematics Didactique des Mathématiques, 1970–

1990). Is there any way in which mathematics education. Why during their pedagogical

meetings at school they talk about the same difficulties as their colleagues did many years

ago, in spite of the large development of Maths Education? Are they right when they say:

‘Yes, this is a fantastic idea, but how can I implement it in my class?’ winking at the teacher

next to them? Or when they complain that the research should provide them with ready-to-

use tools for their problems in the classroom?

Even though mathematics classrooms have similarities, they also have differences.

As Anna Sfard and Anna Prusak claim ‘different individuals act differently in the same

situations and differences notwithstanding, do different individuals’ actions often reveal a

distinct family resemblance’. Hence the teacher drawing on evidence of a previous research

should make his own research for his own unique classroom which ‘like every human

community, is an individual at its own scale of organization. It has a unique historical

trajectory, a unique development through time. But like every such individual on every scale,

it is also in some respects typical of its kind’ (Lemke 2000).

The teacher should take into account the complexity of the interactions between knowledge,

students and himself within the context of a particular class. His particular pedagogy which

positions makes available for his student? ‘Ideas, emotions and actions of participants are

shaped by the dynamic of interactional practices, and how positions available in discourse

can be realised as positioning in practice. It provides evidence of excitement and anxieties

felt by these pupils, showing how they are associated with their positioning in different

discursive practices. By analysing the positions occupied by each pupil in interaction, we

understand how hierarchical positions are (re)produced, as well as the role that emotions

play in adopting, modifying, ‘submitting to’, or claiming, a position’ (Jeff Evans, Candia

Morgan, Anna Tsatsaroni, 2006). And how these positions interact with his? To which extent

it is responsible for ‘the uninteresting, hard-to-reach student position’ which some students

occupy’ and what are the emotions he experiences adopting the position of not participating,

of the follower who expects his classmate’s help, who is not consistent with his homework

and his achievement is not as he expected?

Does he feel he wants to change his students’ attitude towards mathematics? This is the

turning point. What does he want to do? What about should he be aware? The first thing he is

aware of is that behind all these questions is hidden a whole mass of information that

Mathematics Education has gathered. Usually disappointed he stops trying under the shelter

of ideas putting the blame on family, previous teachers, not existing prerequisite knowledge,

lack of time, difficulty of Mathematics.

As Marshall and Drummond (2006) argue, teachers’ conceptions of learning are central to

understanding and enacting these practices. Or he starts questioning everything he does,

starting from the tasks; whether they are meaningful, the teaching methods; whether they are

engaging, in a few words he tries to find ways to change student’s attitude, to engage him in

working, to motivate him. But what are finally attitudes, engagement, motivation; these

controversially discussed multifaceted and mutually interacting notions?

We can say that in everyday life by the term attitude we mean someone’s liking or disliking

of a familiar target. The term ‘attitude’ is often used by teachers as a negative one to attribute

their students’ difficulties or failure in mathematics. The last forty years researchers make

experimentally studies on the construct ‘attitude’ giving it a multidimensional definition.

Everyone has his own point of you but it seems that they agree on the fact that there is a

relationship between students’ attitude towards mathematics and achievement in that.

For example, Frost et al., 1994; Leder, 1995 state that girls tend to have more negative

attitudes towards mathematics than boys,

McLeod, 1994, states that attitudes tend to become more negative as pupils move from

elementary to secondary school. Tony says: ‘In the primary school maths was my favourite

subject, but later I could not get it, I should work hard without success…... so disappointing’

and Haladyna et al., 1983, that the general attitude of the class towards mathematics is

related to

1) the quality of the teaching ‘since the day this new teacher came, mathematics has

proved to be the most boring subject, because he speaks continuously and we do nothing else

except coping what he is writing on the blackboard’

and to

2) The social-psychological climate of the class ‘although the teacher tries hard, all the

boys try to make noise and they don’t attend the lesson. They are afraid of being teacher’s pet

and when a student says he likes the lesson they laugh at him during the break’.

In this way theory justifies practice: As Ruffel, Mason and Allen (1998) state ‘Teachers'

attitude to mathematics is increasingly put forward as a dominant factor in children’s

attitudes to mathematics’. When the teacher feels that his students’ attitudes to his lesson are

negative and decides to intervene, the first thing he does is to be problematized for his own

attitude and practice. He tries to find out what is wrong, starting from his own positioning in

the class: his emotions and methods. ‘I always like teaching older students. I cannot stand

very young pupils.’

Markku Hannula has developed a framework for analysing attitude and can be used by the

teacher as an analytical tool for exploring his students’ attitudes. She builds a foundation

from the background of psychology of emotions and separates the observable category

‘student’s attitude towards mathematics’ into four different evaluative processes:

1. the emotions the student experiences during mathematics related activities;

It is remarkable that emotions in the mathematics class are not stable, but may

include both pleasant ‘well multiplications, additions, subtractions it was fun…..the best

time’ and unpleasant ones ‘it started being a little bit confusing, difficult and could not get it,

so disappointing,’, as John comments.

2. the emotions that the student automatically associates with the concept 'mathematics';

(‘all these letters instead of numbers….and geometry, with all these proofs…..was so

confusing and embarrassing’ )

3. evaluations of situations that the student expects to follow as a consequence of doing

mathematics;

4. The value of mathematics-related goals in the student's global goal structure. ‘Me,

now, I am going on with History I don’t need any maths.’

Goldin and DeBellis 1997 suggest four facets of affective states: emotional states, attitudes,

beliefs, and values/morals/ethics, which provide insight into the development of attitude and

Hannula uses to reconceptualise attitude with emotion and cognition as two central concepts

so intensely mutually interacting that can be seen as two sides of the same coin.

Although there is not simple recipe for the teacher, this framework of emotions,

associations, expectations, and values seem to be useful in describing attitudes and their

changes. The most important think is the way it is constructed as a theoretical view point for

an accurate interpretation of students’ behaviour, capable of steering future action. As Di

Martino and Zan say (2009) ‘The relationship is rarely told as stable, even by older students’

and, in contrary to what mathematics teachers in higher secondary classes think, it is never

too late to change students’ attitude towards mathematics.

Researchers have provided teachers with a powerful instrument for analysing, their students

attitude, but mainly theirs. They can take the responsibility of their teaching methods,

question their classroom culture. They first have to be aware of every student’s needs, in

order to help him reach his own potential. Mary is a competent mathematics teacher. She

never forgets the day her new mathematics teacher came to school. ‘She was a fantastic

teacher’ she says. ‘I used to hate mathematics before . I thought it was so boring and

nobody used to care about it in my class. It was a really terrible class, with many problems

due to the multi ethnicity of the students. She never discriminated anyone and in a few weeks

she gained the class’s respect, which had an influence on student’s attitude, they were

motivated and soon we turned to be the most creative mathematics class in the school.

Almost everybody was engaged and nobody could understand how she managed it.’, as

Berstein (1990) states: ‘If the culture of the teacher is to become part of the consciousness of

the child, then the culture of the child must first be in the consciousness of the teacher.’

But now how he can go along with that? What can be the trigger in students’ development?

What is engagement and how can the teacher engage his students with mathematics in a way

that mathematical learning takes place?

Again he is not alone in this attempt. There is a mass of research about engagement the

teacher can use to find his ‘unique’ way. A mass of information and interpretations is

available. Finn (1989, 1993) proposed the “participation-identification model” that describes

students' identification with school. In addition he suggested that students’ academic

engagement comprises three constructs: cognitive, affective and behavioural engagements.

1. Affective engagement implies a sense of belonging and an acceptance of the goals of

2. Cognitive should be:

Flexible vs. Rigid Problem Solving

Active vs. Passive Coping with Failure

Independent vs. Dependent Work Styles

Independent vs. Dependent Judgement

Preference for Hard Work vs. Preference for Easy work

3. Emotional

In the form of Anger Interest Nervousness

Happiness Sadness Curiosity

Boredom Discouragement Excitement

Behavioural

Class Participation vs. Uninvolvement

On-task vs. Off-task Behaviour

Extra-curricular Academically Oriented vs. Extra-curricular Non-academically Oriented

Career Plans

Classes Skipped

Tardiness

A framework for conceptualising and measuring engagement in mathematics was developed

by Kong, Wong, and Lam (2003) through research and validation, identify significant

markers of engagement. In this study they adopt these markers as a framework for

investigating, categorising and interpreting student engagement, and are as follows:

Affective engagement (Interest and achievement orientated by Anxiety and frustration)

Behavioural engagement (Attentiveness, Diligence, Time spent on task, Non-assigned time

spent on task

Cognitive engagement and strategies that could be drawn to succeed in effective learning, in

the form of:

• Surface Strategies (memorisation, practising, and test taking strategies)

• Deep Strategies (understanding, summarising, making connections, justifying)

• Reliance (on teachers/parents)

One would state that it is not realistic to say that a teacher can use all these constructs. But

every experienced teacher finds in all this research ways he acts, consciously and/or, most of

the time, unconsciously in his everyday practices. Such constructs are results from teachers’

practices and are powerful up to the day that a more viable idea can emerge: Because

Mathematics education is ‘a theory in my work, or, better, a set of thinking tools visible

through the results they yield, but it is not built as such … It is a temporary construct which

takes shape for and by empirical work.’ as Bourdieu claims in Wacquant, 1989. And this is

the only way teachers can use it in their attempt to interpret their students’ negative attitudes,

consequently to trigger them in mathematics learning. The means they use is the pedagogic

task which they should plan taking into account the fact that it should be useful and

purposeful. But how can this be feasible? In Connecting engagement and focus in pedagogic

task design Janet Ainley, Dave Pratt and Alice Hansenb (2004) warn the teachers of the

problems they can face in planning tasks. The central idea is defined as ‘the planning

paradox’ in their one word: ‘If teachers plan from objectives, the tasks they set are likely to

be unrewarding for the pupils and mathematically impoverished. Planning from tasks may

increase pupils’ engagement but their activity is likely to be unfocused and learning difficult

to assess.’ The point they make is how the teachers can produce tasks, which give their

students the chance to be engaged in essential content set out by the curriculum in focused

and motivational ways. These two latter can be afforded by carefully selected tools which

connect the knowledge that students must gain with their everyday experiences.

Motivational ways raise an issue of what motivation is. For one more time we can find a huge

amount of research on this new construct, which is created to help us explain, predict and

influence behaviour. Within psychology one important approach to motivation has been to

distinguish between intrinsic and extrinsic motivation (Deci & Ryan, 1985).

Intrinsic motivation has emerged as an important phenomenon for educators— a natural

wellspring of learning and achievement that can be systematically catalysed or undermined

by parent and teacher practices. In students’ narratives as story tellers we can find witness of

this kind. ‘I like solving mathematical problems. It reminds me of my holidays spending

hours with my father playing cards, answering puzzles and solving strange problems, which

derived from mathematics, as I understood later. Our family’s habits….’ Intrinsic motivation

must not be undermined because it results in high-quality learning and creativity. However,

equally important can be the different types of motivation that fall into the category of

extrinsic motivation and is present in students’ narratives about their experiences. ‘I am not

sure that I really like maths, but I study hard because a good grade is important for studying

medicine which is my deepest desire’. In the classic literature, extrinsic motivation has

typically been characterized as a pale and impoverished (even though powerful) form of

motivation that contrasts with intrinsic motivation. In mathematics education not many

researchers have focused on motivation (See Evans & Wedege, 2004; Hannula, 2004b), and

only a few researchers have distinguished between intrinsic and extrinsic motivation. Holden

(2003) makes a distinction between intrinsic, extrinsic and contextual motivation. She

suggests that the students’ motivation always is governed by some kind of “rewards”.

According to her, students who are extrinsically motivated engage in tasks to obtain extrinsic

rewards, such as praise and positive feedback from the teacher. The students’ intrinsic

motivation is governed by intrinsic rewards, which concern developing understanding,

feeling powerful and enjoying the task. Students who are contextually motivated are doing

something to obtain contextual rewards, such as acknowledgement from peer students,

working with challenging tasks and seeing the usefulness of the task. Goodchild (2001)

relates extrinsic and intrinsic motivation with ego and task orientation and with performance

and learning goals. According to him a student is extrinsically motivated when he is doing

something because it leads to an outcome external to the task, such as gaining approval or

proving self-worth. A student is intrinsically motivated when he considers the task to have a

value for its own sake; he is engaging in the task in order to understand. Evans and Wedege

(2004) consider people’s motivation and resistance to learn mathematics as interrelated

phenomena. They present and discuss a number of meanings of these two terms as used in

mathematics education and adult education. In Hannula’s dissertation his approach to

motivation involves needs and goals, rather than intrinsic and extrinsic motivation (Hannula,

2004a).

Kjersti Wæge in Intrinsic and Extrinsic Motivation Versus Social and Instrumental Rationale

for Learning Mathematics (2007) discusses the relation between two different concepts of

motivation for learning mathematics: intrinsic and extrinsic motivation as defined in Self

Determination Theory and Mellin-Olsen’s concept of rationale for learning mathematics in

activity theory’s point of view. Within Self Determination Theory, as she claims, one

suggests that extrinsic motivation varies considerably in its relative autonomy and thus can

either reflect external control or true self-regulation, comparing to Mellin-Olsen’s two

rationales for learning mathematics in school; an S-rationale (Social rationale) and an I-

rationale (Instrumental rationale). Both points of views are very interesting for the teacher

and can help him decide in which way he could motivate his student by analysing his attitude

and at what extend he could use them depending on his teaching practices and his own

ideology.

What is undoubtedly clear is the fact that mathematics teachers cannot always rely on

intrinsic motivation to foster learning. Many of the tasks, their students should perform,

despite their efforts, are not inherently interesting or enjoyable. That’s why knowing how to

promote more active and volitional forms of extrinsic motivation becomes an essential

strategy for successful teaching.

When the teacher first meets all these information about these meaningful, for his work,

constructs usually has the same feeling with his uninterested in mathematics, unwilling to

cooperate student. How can he cope with all this information? Is it worthy to go on or the

time available is so short that the results are doubtful? The most interesting thing that is

revealed by Alexander’s and Aristotle’s quotes in the beginning of this assignment is the hint

that students and their teachers talk about the same stories. Their attitudes towards the

meaning of teaching and learning are shaped in a strange interaction. The teacher who gives

up in front of this difficulty putting the blame on other factors beyond him (‘I can do nothing

but wait’) is honoured with the most difficult and failing students. Others find some of these

ideas charming and decide to use it. ‘Collaborative learning is very interesting’ or ‘Tasks

with computers are very motivating for students’ ‘The curriculum is very demanding, I

should expect less from them’ are some ideas they have, especially those who decide to

interview their students and take for granted everything they say about what they think that is

discouraging for them in being more successful.

Every attempt is important and gives the teacher insight for parts of the work. A teacher

whose teaching method is traditionally oriented in teacher centred methods finds interesting

the change in some students’ positioning when he decides to ask them work collaboratively.

He likes it, although he faces problems by others who prefer working individually, or by

ways students’ new positionings interact in a disharmonic way: ‘He tells me what to do and

does not listen to me’, ‘she behaves as if she were my teacher, I cannot stand it!’. Working

with digital technology is very popular among students (‘maths seems to have a different

dimension, it is a fantastic experience to see how functional and easy to understand are

graphs when using computers’) but new problems appear for the teacher who is not familiar

with it like the ‘planning paradox’, which we have mentioned before when talking about

Ainley and Pratt’s work. Some in front of these difficulties give up and others begin to realise

that teaching is a complicated process, in which sometimes a teacher needs to be a student in

new learning processes. The latter goes on trying new methods, enjoying insight and skills he

had not before. This is the moment that becomes aware of the difficulties that all, without

exception –hopefully- students face in different fields of the teaching learning process. Every

student can be uninterested in different ways and under some special conditions can shape a

hard-to-reach identity.

This is the moment we can state that in turn, it is never late for a change in a teacher’s

attitude towards teaching mathematics. In Brousseau’s sense, the didactic contract is broken

this time for the teacher. The situation reminds him of The Nine Dot Puzzle, when he should

draw no more than four straight lines (without lifting the pencil from the paper) which will

cross through all nine dots, that shape a square. A solution was difficult to find until he was

willing to ‘think outside the box’. Again Mathematics education research is going to give him

the ‘tool’.

How his student from a curiosity machine, as every child is, turned to a mathematical idiot?

Brouseau gives an interesting theoretical framework for what he calls situation didactic: it

consists of the learners, the teachers, the mathematical content and the classroom ethos, as

well as the social and institutional forces acting upon that situation, including government

directives such as a National Curriculum statement, inspection and testing regimes , parental

and community pressures and so on (milieu).

He states that learning takes place when the didactic contract is broken. For every student it

can happen in different ways, for which is a teacher’s duty to look individually. When he

neglects, an uninterested, hard-to-reach identity is shaped that sounds loudly, even in silence.

Every experienced teacher can identify it watching, for example, video clips from classrooms

practices all over the world in the most different didactic contacts, even conducted by the best

teachers. The ways depend on his pedagogy, ideology, and teaching practice. Ways derive

from psychology and can be applied even in short time.

What is in question is how every student can reach his potential and become the more

mathematically literate he can. He must be the researcher of his own practice, of his students’

special needs, gathering information from other colleagues, the parents/carers, the students. A

new learning contract among teacher and student must be signed. He must take into account

everything he is said, starting from questioning his own expectations (individualised), tasks

(adapted to everyone’s potential- giving the chance everybody to be motivated), guarantee

that the contract is kept by both parts so that new habits could be formed with the hope new

engaged –not marginalised- identities are possibly shaped. Even if it does not happen the

commitment will give emotions worth experiencing and insight for facing new undesired

situations.

It sounds utopic and perhaps it is, but doing something is always more than doing nothing.

All the students have something to learn in Bernstein’s ‘totally pedagogised society which is

shaped through pedagogy rather than productive processes’. Mathematics is a very important

subject which supplies skills of competence, especially for those who finish compulsory

education mathematically illiterate (‘I have learned some maths (generally) which is good

enough, but then in Year 10-13 you have to learn some advanced maths, which I don’t think

that a lot of people are going to use in their everyday life. I am going on with IT, Computers

etc. The only thing I need to know from maths is how many wires you need 1,2,3,4,5,6,7,8

nothing else, you only need practical skills’, says George who after 10 years of schooling

cannot recall anything else). Of course it is a difficult process for the teacher and sometimes

unsuccessful but he is not alone in this attempt. In the era of globalisation new conditions are

formed that need to be interpreted. Researchers from all over the world in collaboration with

the teachers exchange aspects of every facet of the educational systems, using qualitative,

quantitative and mixed methods they measure students’ performance in mathematics. The

role of parents, teachers, students and curriculum in countries where students have a high

attainment is researched. On the other hand, digital technology provides new teaching tools

and environments that create new mathematics. Constructivism gave birth to constructionism

and other learning theories in community of practice. A very interesting aspect of

participants’ engagement , imagination and alignment in the activity and practice of this

specified community with its own purposes and goals, is stated by Wegner 1998 and Lave

1991 in a community that sounds perfect. In universities the creation of inquiry communities

between didacticians and teachers in a co learning inquiry, a mode of developmental research

in which knowledge and practice develop through the inquiry activity of the people engaged

(Jaworski, 2004a, 2006), to explore ways of improving learning environments for students in

mathematics classrooms. Research both charts the developmental process and is a tool for

development learning environments for students in mathematics classrooms.

This seems to be the greatest hope for the teacher, who expects research of his own unique

environment by experts to provide him and his students with powerful and vital aspects of

understanding the ways in which this ‘complex site of political and social influences, socio-

cultural interactions, and multiple positioning involving class, gender, ethnicity, teacher–

student relations, and other discursive practices in which power and knowledge are situated’’

(Lerman, 2001a, p.44), for helping with shaping an identity of a citizen of the world with

values and competence.

Until then professional mathematics teachers will share with researchers their inquiries and

fears in MA Mathematics Education classes in the same strange, charming and unique

relationship of the kind of teacher and student.

ACKNOWLEDGEMENTS

I should like to thank most sincerely Candia Morgan, Cathy Smith, Melissa Rodd, Eirini

Geraniou, Dave Pratt for their interesting stories I heard during the sessions of ‘Issues in

Mathematics Education’ and last, but not least I want to thank all my students of so many

years who trusted me for their mathematics educations.

MAGDALINI KOKKALIARI

KOK11094464

MMAMAT_04

MA STUDENT ()

MATHEMATICS EDUCATION

INSTITUTE OF RDUCATION

UNIVERSITY OF LONDON

FEBRUARY 2012

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