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In: Handbook of Lifelong Learning Developments ISBN: 978-1-60876-177-7

Editor: Margaret P. Caltone, pp. 265-287 © 2010 Nova Science Publishers, Inc.

Chapter 10

THE AFFECTIVE DIMENSION OF LEARNING AND

TEACHING MATHEMATICS AND SCIENCE

Lorenzo J. Blanco1, Eloisa Guerrero2, Ana Caballero1, María

Brígido1 and Vicente Mellado1

1Dept. Science and Mathematics Education, Faculty of Education, University of

Extremadura, Spain.

2Dept. Psychology and Anthropology, Faculty of Education, University of Extremadura,

Spain.

ABSTRACT

Learning scientific and mathematics concepts is more than a cognitive process.

Learning and teaching is highly charged with feeling. Nevertheless, in schools and

universities, science and mathematics is for the most part portrayed as a rational,

analytical, and non-emotive area of the curriculum, and teachers, texts and curricular

documents commonly present images of science and mathematics that embody a sense of

emotional aloofness.

In the chapter, we review the most significant research on the affective domain in

teaching and learning Science and Mathematics. We describe the research program

carried out in the University of Extremadura (Spain) on the influence of emotions in

primary pre-service teachers' process of learning to teach science and mathematics. We

present the diagnostic studies that we conducted to determine the emotions that these

primary pre-service teachers felt when they were learning science and mathematics in

school, relating them to different variables (sex, education, topic, problem solving, etc.),

and the emotions they feel when they are teaching science and mathematics during their

practice teaching. Finally, we show the results of an intervention program for primary

pre-service teachers which focuses on emotional control in solving mathematics problems

in order to foster change in attitudes, beliefs, and emotions towards mathematics and its

learning and teaching.

Lorenzo J. Blanco, Eloisa Guerrero, Ana Caballero et al.

266

1. COGNITION AND AFFECT

Learning and teaching science is more than a purely cognitive process and is highly

charged with feelings. Nevertheless, in schools and universities, for the most part science and

mathematics is portrayed as a rational, analytical, and non-emotive area of the curriculum.

Referring to research in mathematics education, De Bellis & Goldin (2006) and Furinghetti &

Morselli (2009) note that studies of students' performance and problem solving have

traditionally concentrated primarily on cognition, less on affect, and still less on cognitive–

affective interactions.

In our work, we consider it to be a significant achievement in education that the affective

domain is accepted as a key aspect in teaching science and mathematics. Indeed, we believe

that recognizing the role of affect in the processes of knowing, thinking, acting, and

interacting is essential in the teaching–learning process. We concur with the definition of

McLeod (1989) that the affective domain in education covers a wide range of feelings

different from pure cognition, including attitudes, beliefs, and emotions as basic components.

Since the nineties, the study of affective processes has found a new conceptual

framework in the fields of neurobiology and the psychology of emotion. There were findings

that the intelligence quotient and psychometric tests are not predictors of professional and

personal success. An important contribution was the theory of multiple intelligences (Gardner,

1995) according to which cognitive skills are defined as a set of abilities. From this

perspective, intelligence is conceived of as an ability needed to solve problems. Gardner

identified seven different types of intelligence, including interpersonal and intrapersonal

intelligences.

In the classical models of school learning, the personal variables encompassed capacity

(intelligence and aptitude), motivation, personality, and the skills and strategies of learning.

This view has changed substantially with the recognition that the pupil's competence also

depends on such other factors as prior knowledge, styles of learning, attitudes, beliefs,

attributional styles, and emotional and affective factors. A study of the characteristics of

learners necessarily involves considering their individual differences in learning behaviour,

and this can be found very useful in adapting teaching to these differences.

School success depends on many social and emotional factors that have little to do with

the early development of intellectual capacities. Thus, emotional security, interest, self-

assurance, knowing what kind of conduct is expected of you, control of your impulses, and

the expression of your needs will be the catalyst of your establishment of relationships as a

learner with others, and of the effectiveness and quality of your academic, personal, affective,

and social development.

It is therefore advisable to distinguish between intelligence in its classical sense as an

aptitude, and the concept of emotional intelligence as a skill (Mayer & Salovey, 1995;

Shapiro, 1997; Goleman, 1995).

"Emotional intelligence" is defined as "that which involves the ability to monitor and

understand one's own and others' emotions, to discriminate among them, and to use this

information to guide one's thinking and actions" (Salovey & Mayer, 1990, 57). Goleman

(1998) elaborated on this idea, understanding emotional intelligence to be the capacity for

recognizing one's own and others' feelings, but also for self-motivation and the appropriate

handling of emotions. The origin of this new construct lay in the line of work begun by

The Affective Dimension of Learning and Teaching Mathematics and Science

267

psychologists in the seventies on the interaction between emotion and thought, with variables

that had not at first been suspected of being related (Anadón, 2006).

In recent years, cognitive neuroscience with the support of new brain imaging techniques

has contributed decisively to the development and modification of theories arguing that good

adaptation to the environment requires both declarative and affective information (Damasio

2005, 2006). Likewise, Bermejo (1996) and Pérez (2008) find a significant relationship

between variables relating to emotional intelligence and academic performance, and Páez &

Rigo (2008) relate the construct to the self-control of learning.

All this points to the need to study and to include emotional intelligence in the academic

and school contexts (Galindo, 2005; Salmurri 2004). In general, there is widespread

agreement about connecting the cognitive and the affective, since "emotions influence

knowledge, but [also] knowledge influences emotions" (Marina, 2004, 53). Indeed, recent

results have called into question the independence of the rational and the emotional since,

according to the theory of "affective cognitive moulds" of Hernández (2002), the cognitive

configures the affective and vice versa.

It is only a decade ago when the cognitive aspect still overshadowed the affective, and

LeDoux (1999) argued for the idea that emotion and cognition are best understood when

considered as separate but complementary mental functions. In recent years, the explanation

of the relationship between cognitive and affective variables has become oriented to

considering the cognitive and affective as mutually conditioning each other, as was put

forward in the aforementioned theory of affective cognitive moulds of Hernández (2002).

2. THE AFFECTIVE DOMAIN IN SCIENCE AND

MATHEMATICS LEARNING

Research in science and mathematics education also recognizes the importance of

emotions in teaching and learning, and advocates the need to consider the cognitive and

affective dimensions (Gómez & Chacón 2001; Zan, Bronw, Evans & Hannula, 2006; Koballa

& Glynn, 2007; Furinghetti & Morselli, 2009).

Much of current research on the influence of the affective dimension on mathematics

teaching and learning has grown out of the work of McLeod (1986, 1989, 1992). This showed

how important it was to consider the basic descriptors of the affective domain – beliefs,

attitudes, and emotions (McLeod, 1989). De Bellis & Goldin (2006) extended these

descriptors to a fourth subdomain of values / morals / ethics.

Studies have shown that pupils' affects are key factors to understanding their behaviour in

relation to mathematics and science. Pupil's experiences in learning provoke in them feelings

and emotional reactions that influence the formation of their beliefs. Moreover, the beliefs

that they already hold have a direct bearing on their behaviour in situations of learning and on

their ability to learn (Gil, Blanco & Guerrero, 2006).

During their learning, pupils receive continuous stimuli associated with science and

mathematics – problems, the teacher's actions, social messages, etc. These cause them a

certain tension to which they react emotionally either positively or negatively. If these

situations recur under similar conditions producing the same affective reactions, then the

activation of the emotional reaction (satisfaction, frustration, etc.) may become automated

Lorenzo J. Blanco, Eloisa Guerrero, Ana Caballero et al.

268

and "solidified" in a set of attitudes that influence the conformation of their beliefs. Thus,

their beliefs, being associated with their experiences, will condition their ideas about their

abilities.

Following the classification that McLeod (1992) gave for mathematics education, one

could say that the learner generates beliefs about the discipline and its teaching and learning,

beliefs about him or herself as a learner of mathematics or science, and beliefs reflecting the

social context.

Beliefs are cognitive structures that allow individuals to organize and filter the

information they receive, and to progressively construct an understanding of reality, and a

form of organizing and viewing the world and of thinking (Gilbert, 1991). On the one hand,

beliefs are indispensable because they structure the meaning we give to things. But on the

other, they act as a filter with respect to new realities or certain problems, limiting the

possibilities for action and understanding (Blanco & Barrantes, 2006). For Schoenfeld (1992),

beliefs form a particular view of the world of mathematics, setting the perspective from which

each person approaches that world, and they can determine how a problem will be tackled,

the procedures that will be used or avoided, and the time and intensity of the effort that will

be put into the task.

It is therefore important to describe pupils' conceptions and beliefs about the discipline.

Many authors have pointed out that pupils conceive of mathematics as a difficult,

authoritarian, abstract, and rule-based subject, in which memorization and routine, and

algorithmic, algebraic, and analytical procedures predominate, with exercises having to be

solved that usually have little application in the real world (Mtewa & Garófalo, 1989; Flores,

1999; Schoenfeld, 1992). In this regard, according to Szydlik, Szydlik & Benson (2003)

research has shown that prospective teachers tend to "see mathematics as an authoritarian

discipline, and that they believe that doing mathematics means applying memorized formulas

and procedures to do textbook exercises" (Szydlik, Szydlik & Benson, 2003, 254).

All this leads pupils to regard mathematics as dispensable, and although they do not

doubt the true value of mathematical knowledge, they consider it to be external to their world

(Flores, 1999). These beliefs have a negative influence on mathematics activity, being a cause

of attitudes of wariness and mistrust.

Beliefs about oneself as a mathematics or science learner carry a strong affective load

with respect to confidence, self-image, and the causal attribution of success and failure in

class. Pupils who feel competent, who trust in their capabilities, and have expectations of self-

efficacy, involve themselves in the learning process. Moreover, learning is more satisfactory

if failures as well as successes are attributed to internal, variable, and controllable causes

(e.g., personal effort, perseverance, planning, …). It will be less satisfactory, however, if the

successes are attributed to external, uncontrollable causes (e.g., luck, easiness of the task,…)

and the failures to internal, stable, and uncontrollable causes (lack of ability) (Miras, 2001).

Confidence in the willingness and ability to want to learn mathematics plays an essential

role in pupils' achievements in mathematics (McLeod, 1992; Reyes, 1984).

For González-Pienda & Núñez (1997), subjects' active involvement in the learning

process is enhanced when they feel competent, i.e., when they trust in their own abilities and

have high expectations of self-efficacy, value the work they are set, and feel responsible.

Moreover, the beliefs of self-efficacy influence the activities in which they involve

themselves, the amount of effort they put in, their perseverance in the face of obstacles, their

capacity to overcome or adapt to adverse situations, the level of stress and anxiety they

The Affective Dimension of Learning and Teaching Mathematics and Science

269

experience when they are set some task to do, their expectations about the results, and the

process of self-regulation. They generally attribute success in mathematics to the teachers'

attitudes towards the pupils, and to greater commitment and effort in studying the subject, but

they reject the influence of luck. The conclusion is therefore that they attribute both success

and failure largely to internal, unstable, and controllable causes, an attribution that is

favourable for learning. These findings are similar to the conclusions drawn by Gil, Blanco &

Guerrero (2006).

Vanayan et al. (1997) and Kloosterman (2002) note that the beliefs that most influence

motivation and achievement in mathematics are pupils' perceptions about themselves in

relation to mathematics. Thus, self-confidence in mathematics is an important indicator of

learners' positive views of studying the subject, and hence of their active participation and

regulation in the learning process. Pupils who believe that mathematics is only for those with

mathematical talent, and is based on infallible and mechanical procedures of solution, have

less confidence in themselves in learning situations than those who do not think in this way.

We understand attitude to be an evaluative (positive or negative) predisposition which

determines personal intentions and influences behaviour (Hart, 1989), and which has four

components – cognitive (knowledge), affective (feeling), intentional (intentions), and

behavioural (behaviour).

Pupils on the whole show rejection, denial, frustration in the learning process. Sarabia

(2006) observed that secondary education pupils exteriorize discontent, displeasure, and lack

of enjoyment, as well as little motivation or interest in learning mathematics.

In science education, affective aspects have been addressed only infrequently, and even

then generally relating them more to attitudes than specifically to emotions. In the first and

third handbooks on science education by Gable (1994) and Abell & Lederman (2007), there

are two extensive reviews of attitudes in science learning (in which the emotions are

included). These are the respective chapters of Simpson et al. (1994) and Koballa & Glynn

(2007).

For Sanmartí & Tarín (1999), attitudes and science learning are not only strongly related,

but attitude is the starting point for any meaningful learning. However, pupils' attitudes can

not be conceived of as an isolated fact, but are correlated with a wide range of variables, both

internal and external to the classroom (Espinosa & Román, 1995). Several studies indicate

that interest and positive attitudes towards science and mathematics decrease with age,

especially during secondary education (Beauchamp & Parkinson, 2008; Murphy & Beggs,

2003; Osborne et al. 2003; Ramsden, 1998; Vázquez & Manassero, 2008; Yager & Penick,

1986). Boys usually have a more positive attitude to science than girls (Caleon &

Subramaniam, 2008; Koballa & Glynn, 2007). These last authors also note that the interest of

girls is far more focused on biology than physics.

This worrying attitudinal depression towards science and mathematics is attributed to

school science creating a negative image in pupils' minds over time, it being described as

authoritarian, boring, hard, or irrelevant for everyday life (Vázquez and Manassero, 2008).

The study of emotions is complex because we are all different, with different

personalities whose interactions between the cognitive and the affective-emotional form a

mosaic of individual factors and peculiarities. Nonetheless, it is a vital aspect of learning.

Studies of emotion have focused on the role of anxiety and frustration and their impact on

achievement in mathematics (Marshall, 1989; McLeod, 1989; De Bellis & Goldin, 1997,

2006). According to Ojeda et al. (2003), emotions take part in our learning because they block

Lorenzo J. Blanco, Eloisa Guerrero, Ana Caballero et al.

270

our intelligence and hinder success in life. Frustration is a very common emotion at the

secondary education level. This may be related to the belief systems that guide pupils'

behaviour and conditions the goals they set themselves to reach. Fear, emotional block, and

anxiety have often been confirmed as being able to influence the mental processes involved in

the development of mathematics skills (De Bellis & Goldin, 1997). For Richardson &

Woolfolk (1980), anxiety in mathematics consists of the feelings of tension, mental

disorganization, and helplessness that a pupil experiences when set mathematics problems to

solve. Anxiety leads to abandonment, avoidance of the task, and the pupils' protecting

themselves in some way (Guerrero, Blanco & Vicente, 2002).

3. THE AFFECTIVE DOMAIN AND LEARNING TO TEACH

Zevenbergen (2004) cites various studies that indicate that prospective primary teachers

show "low levels of mathematics knowledge as well as considerable anxiety towards the

subject" (Zevenbergen, 2004, 5), even with respect to important aspects of the content that

they will have to teach in their future career (Brown & Borko, 1992). This lack of

mathematical knowledge is important since the ability to manage the class and the choice of

curriculum depend directly on the mastery of the content. Teachers with a low level of

knowledge of the subject they are to teach find it difficult to implement educational changes,

avoid teaching the topics they are unsure of, lack self-confidence, and reinforce the pupils'

conceptual mistakes (Mellado, Blanco & Ruiz, 1999). Conversely, when they acquire this

knowledge, they are more confident in their ability to teach (Manoucheri, 1998).

The same pattern of results is found in research on specific aspects of the curriculum –

geometry and measurement (Baturo & Nason, 1996; Blanco 2001; Blanco & Barrantes,

2006), the use of new technologies in mathematics (Walen, Williams & Garner, 2003;

Wachira, Keengwe, & Onchwari, 2008), arithmetical aspects (Thipkong & Davis, 1991;

Tirosh & Graeber, 1989; Putt, 1995), etc.

In another sense, prospective primary teachers, as a consequence of their own learning

experiences in school, carry a baggage of conceptions and attitudes that are inconsistent with

today's curricular proposals and recommendations (Johnson, 2008). These conceptions and

attitudes almost always appear with strongly negative influences on the process of learning to

teach (Ernest, 2000). Student teachers use their conceptions, consciously or unconsciously, as

a kind of lens or screen to filter and occasionally block the mathematics teaching content of

their teacher education courses and interpolate their own educational process (Barrantes &

Blanco, 2006). Also, during the teaching they receive in their initial courses, they will feel no

need to express or reflect on their conceptions about the teaching of mathematics if they have

no practical references to compare with. And they typically show signs of "wishful thinking"

that teaching is easy and they will have no great difficulty in doing it (Flores, 1999).

To learn how to teach science and mathematics, prospective teachers need to understand

that the emotions they feel about learning are also related to those they feel about teaching. In

a study conducted at the University of Extremadura we identified the emotions aroused in a

sample of pre-service primary teachers during their period as secondary school pupils and

when doing their teaching practice in the Education Faculty regarding the subjects of Physics

/ Chemistry and Nature Sciences. The exploratory, descriptive study was conducted by means

The Affective Dimension of Learning and Teaching Mathematics and Science

271

of a survey presented to 63 students of primary education in the Faculty of Education at the

University of Extremadura during the academic year 2007/8. The instrument used was a

questionnaire in which the subjects noted from among those offered the emotions aroused in

them by the different subjects of science, both in their time as secondary school pupils and in

their teaching practice. The resulting data were subjected to the necessary processes of

checking, coding, and digital storage in order to proceed with their descriptive analysis using

SPSS (Statistical Product and Service Solutions) 13.0.

The results showed a great difference between the emotions related to the subjects of

Physics/Chemistry and Nature Sciences.

Their memory of the subjects of Physics or Chemistry at secondary school suggested

fundamentally negative emotions: nervousness, anxiety, tension, worry, or despair, and only

rarely positive emotions such as confidence or enthusiasm. During their practice teaching,

teaching topics related to physics or chemistry also suggested to them more negative than

positive emotions, but to a lesser extent than when they were at school.

Their memory of the subjects of Nature Sciences during their time in secondary school

suggested to them fundamentally positive emotions: fun, tranquillity, joy, satisfaction,

congeniality, capacity, etc. On teaching topics related to Nature Sciences during their teaching

practice, they also experienced positive feelings, even to a greater extent than when they were

at school. Figure 1 presents the percentage of each emotion chosen, both at school and in the

science teaching practice at the University. With regard to Nature Sciences, there was a high

correlation between their emotions when learning at school and as teachers during their

practice teaching.

Since conceptions influence attitudes, and both influence the teacher's behaviour (Ernest,

2000) and the pupils' learning (Georgiadou & Potari, 1999), in order to foster change in our

prospective teachers' views of teaching we will have to incorporate conceptions and attitudes

as part of a process of discussion and reflection in our initial teacher education programs

(Stacey, Brownlee, Reeves & Thorpe, 2005; Johnson, 2008).

Figure 1. Emotions aroused in topics related to Nature Sciences as school pupils and as teachers.

The above points suggest there is a need to consider in greater depth activities that

promote the critical analysis of prospective primary teachers' knowledge, conceptions, and

attitudes about mathematics and its learning and teaching. These activities should allow them

Lorenzo J. Blanco, Eloisa Guerrero, Ana Caballero et al.

272

to share, discuss, and negotiate the meanings that they generate, so that they will be able to

reinterpret their previous experiences and knowledge about the learning and teaching of

mathematics. This will enable them to develop metacognitive skills with which to analyze the

processes of their own learning as student teachers and of teaching–learning in primary

education. Their theorizing process will also give them a sound basis for their professional

knowledge and decisions. In this regard, it has to be noted that our students apply, consciously

or unconsciously, the models of teaching and learning that they themselves experienced at

school. For this reason, the reflection process has to be implemented explicitly, following

models that lead them to think about their own and the group's learning process, and about the

context in which this learning took place. They also need to be helped to verbalize and reflect

on the main variables in this process. And it has always to be borne in mind that the activities

must establish links between the students' cognitive and affective dimensions (Zevenbergen,

2004).

4. AN INTEGRATORY MODEL FOR LEARNING TO SOLVE

MATHEMATICS PROBLEMS

Mathematics Problem Solving and the Affective Domain

Since the 1980s (NCTM, 1980), the level of the presence and importance of problem

solving (PS) in curriculum proposals has been maintained and has even increased both

nationally and internationally (Castro, 2008). In these proposals, it is regarded as specific

content, as application of knowledge, and as a methodological approach (Schoenfeld, 1985;

Schroeder & Lester, 1989). Its importance is that it promotes analytical skills, comprehension,

reasoning, and application.

Recent international assessment reports (PISA, 2003; MEC, 2006) find poor levels of

performance in mathematics, and have again highlighted the importance of PS in school

mathematics.

Various studies have found that pupils see mathematics PS solving as a rote and

mechanical procedure, that they have few resources to represent and analyze problems, and

that they neither use different strategies or methods to find a solution nor do they make use of

the suggestions they are given to help them (Garofalo, 1989; Blanco, 2004; Córcoles and

Valls, 2006; Harskamp & Suhre, 2007; Santos 2008). Probably, "the prospective teachers'

relative ignorance of problem solving and the difficulties they manifest as solvers is also one

of the causes of their resistance to considering problem solving as a suitable context for

learning sciences" (Blanco & Otano, 1999, 295).

Also, it seems important to emphasize the lack of attention in textbooks to learning

heuristic problem solving strategies (Schoenfeld, 2007, Pino & Blanco, 2008).

The literature references given above that relate the affective domain to the teaching–

learning of mathematics are also relevant to PS. Thus, McLeod (1986, 1992) shows that the

cognitive processes involved in PS are susceptible to the influence of the affective domain in

the three areas identified previously: beliefs, attitudes, and emotions. Thompson & Thompson

(1989) add that certain emotional states experienced by pupils during the PS process tend to

be regarded as undesirable affective states. Pupils make negative comments about

mathematics before starting to solve problems, which is construed as a signal of distress and a

The Affective Dimension of Learning and Teaching Mathematics and Science

273

revealing indicator of their negative attitude towards mathematics (Marshall, 1989).

Nicolaidou and Philippou (2003) establish relationships between PS performance and beliefs

and attitudes. For Richardson & Woolfolk (1980), anxiety about mathematics comprises

feelings of tension, mental disorganization, and helplessness that a pupil experiences when set

PS tasks in mathematics, and that these arise in everyday and other school situations as well

as in PS.

Several recent studies have looked in greater depth into the problem (Gómez-Chacón,

2001; Gil, Blanco & Guerrero, 2006; Sarabia, 2006; Harskamp & Suhre, 2007). They reveal

the influence of the pupil's self-efficacy on performance (González-Pienda, Núñez & García,

1998; Hoffman & Spatariu, 2008). Hernández, Palarea & Socas (2001) and Caballero (2007)

note prospective primary teachers' lack of confidence in solving mathematics problems, and

that they do not consider themselves capable and skilled in this area. The great majority of

them experience insecurity, despair, and nervousness, which seriously hinders or even blocks

their performance of the task.

These literature references abound with observations on the need to relate cognition and

affect in PS. Specifically, there is seen to be a need for the affective and cognitive factors to

be developed simultaneously in teacher education programs (Furinghetti & Morselli, 2009).

"The role of teacher education is to develop beginning teachers into confident and competent

consumers and users of mathematics in order that they are better able to teach mathematics"

(Zevenbergen, 2004, 4).

Program of Intervention on PS and Emotional Control

The research discussed above has not yet led to the development of an integrating process

of teaching–learning that includes cognitive, emotional, and affective aspects. Knowledge and

learning are the products of the mental activity of the learner who perceives, evaluates, and

interprets the facts, the reality, the object, or the situation concerned. Similarly, we understand

that the learners themselves are key and active agents in managing their own knowledge,

since it is they who will generate new knowledge on the foundation of their previous

knowledge (Guerrero, 2006). The basis of school learning lies not in the amount of content

learnt, but in the degree of autonomy, how meaningful it is to the pupils, and the sense they

attribute to it.

Our current line of work is an integration of teaching–learning about PS and emotional

education. The latter is understood as a continuous and permanent educational process, aimed

at enhancing emotional development as an indispensable complement to cognitive

development, the two of which constitute the essential elements of the development of the

whole personality (Bisquerra, 2000).

We also believe that the development of problem-solving skills should be an attainable

goal given a suitable educational environment. Moreover, the approach to PS is very personal,

so that we shall have to help each student to discover his or her own style, capabilities, and

limitations. We must not only convey to them some given method or set of heuristic rules, but

the attitudes and emotions towards mathematics PS based on their own experiences. At the

same time, however, we recognize that attempts to teach PS strategies have failed

(Castro, 2008; Santos, 2008). We therefore considered it important to design a program of

Lorenzo J. Blanco, Eloisa Guerrero, Ana Caballero et al.

274

intervention on PS that integrates the above aspects into a process of action and

reflection.

Oliveira & Hannula (2008) discuss three ideas for consideration in teacher education,

which we also consider in our model. The first is to challenge the prospective teachers'

beliefs, many of which are implicit. They therefore have to be made explicit and reflected

on, creating the opportunity for change. The second is to involve the stude nts actively as

learners of mathematics, usually in a constructivist setting. And the third, also aimed at

producing changes in belief structures, is to provide them with experiences of

mathematical discovery, which seems to have a profound and immediate transformative

effect on their beliefs regarding the nature of mathematics, as well as its teaching and

learning (Oliveira & Hannula, 2008).

González-Pienda, Núñez, Álvarez & Soler (2002) present a model that is a

combination of cognitive and constructivist paradigms. It is based on the following

assumptions: (a) learners bring frameworks of reference to the learning process as a result

of their previous experiences, their social context, interests, beliefs, and ways of thinking;

(b) they present major individual differences (abilities, learning styles, cognitive styles,

expectations, etc.); (c) learning is a constructive process that is facilitated when the

material to learn is meaningful, and when the learners are actively involved in creating

their own knowledge and understanding, connecting what they want to learn with their

prior knowledge and experiences; (d) learning is fostered by positive interpersonal

relationships and when the learner feels appreciated, valued, and recognized; and (e) the

teaching methods have to take the learners' goals, interests, and prior knowledge into

account.

In our review of the literature, although though there were studies relating PS to the

affective domain, we found no research applied to the design and development of

programs of intervention that consider aspects of cognition and emotional control

conjointly, and that evaluate their effectiveness in the initial teacher education classroom.

Objectives

Our aim is to describe the beliefs, attitudes, and emotions of prospe ctive primary

teachers, and to analyze how they confront them when they come to reflect on the

emotional states that accompany mathematical activity, given that their emotions will

affect their participation in the activities (Thompson & Thompson, 1989). Affects

towards mathematics exert a decisive influence on students' learning, on their perception

of the subject, and on their view of themselves as learners (which is a key element in

determining their behaviour). In this sense, affects play four roles: as a regulatory system

of learning in the classroom, as an effective indicator of the learning situation, as inertial

forces of resistance to or impulse in favour of activities and educational changes, and,

given their diagnostic nature, as a vehicle of knowledge (Gómez-Chacón, 2000).

We therefore wish to: "Provide prospective teachers with an educational tool that will

enable them to learn to solve mathematics problems, taking into account aspects of

cognition and emotional education."

The following specific objectives were considered within the overall program:

The Affective Dimension of Learning and Teaching Mathematics and Science

275

To assess attitudes, beliefs, affects, emotions, and attributional styles of the research

participants.

Training in emotional and cognitive skills related to the different steps in the process

of PS.

To provide resources for the management of the emotions, stress, and anxiety that

arise in the PS process.

To encourage the prospective teachers' self-esteem and professional self-efficacy in

relation to teaching about PS.

Methods and Population

The nature of the research suggested the use of qualitative and quantitative methods with

a focus on action-research, since the ultimate goal is to help the participants develop their

thoughts, modify attitudes, and find solutions to the "problem" that solving mathematical

problems represents for them.

Guerrero & Blanco (2004) proposed a theoretical model based on general models of PS

(Polya, 1957; Schoenfeld, 1985), on the cognitive-behavioural models of Zurilla & Goldfried

(1971) and Meichembaum (1974), and on the systemic model of De Shazer and the

Milwaukee group (De Shazer, 1985). Beginning with the 2006–07 academic year, we began a

research project that has enabled us to design an integrated model which forms the basis of

the study we are currently carrying out.

This study consists of a fifteen-session workshop divided into two distinct parts, which

was implemented in the 2007–08 and 2008–09 courses of prospective primary teachers in the

Faculty of Education, University of Extremadura. The first workshop had 55 participating

students, and the second 60.

In the first part, we work on knowledge, conceptions, attitudes, attributional styles,

expectations, and emotions on the basis of questionnaires and activities related to specific

problems.

The second consists of a process of experimentation and reflection based on the general

model, and structured in five steps: (i) accommodation / analysis / understanding / familiarity

with the situation; (ii) search for and design of one or more problem solving strategies; (iii)

execution of the strategy or strategies; (iv) analysis of the process and the solution; (v) How

do I feel? What have I learnt? In the first three steps, we consider two phases: control of the

situation (relaxation and instructing oneself), and the use of mathematical concepts and

processes based on heuristics specific to each case. In the fourth step, we evaluate the process

and its outcome in order to learn and to transfer knowledge to new situations. And finally in

the fifth step, we lay emphasis on the solver's situation to modify, in so far as possible, his or

her affects (conceptions, beliefs, attitudes, self-concept, etc.) regarding mathematics PS.

At all times, we take into account the need to experiment and to reflect on the experience

as the basis for acquiring new knowledge and to provide specific activities to put into practice

in the primary classroom.

Lorenzo J. Blanco, Eloisa Guerrero, Ana Caballero et al.

276

Research Instruments

To verify the reliability of the research, we carried out a comprehensive and detailed data

collection process. The validity of the study was monitored using different data collection

instruments in order to relate, compare, and contrast different types of evidence. We used

various research tools to access the informants in depth. These were:

Questionnaires, both open and closed, which are analyzed qualitatively or

quantitatively depending on their structure:

– Adaptation to mathematics PS of the BEEGC-20 Questionnaire (Battery of Scales of

Generalized Expectations of Control) of Palenzuela et al. (1997). This instrument

will allow us to determine the students' causal attributions in relation to mathematics

PS.

– Adaptation to mathematics PS of the STAI (State-Trait Anxiety Inventory) of

Spielberger (1982).

– Adaptation to mathematics PS of the questionnaires of Gil, Blanco & Guerrero

(2006), Sarabia (2006), and Caballero (2007) on the affective domain in

mathematics.

– Re-elaboration of open questionnaires designed to extract from the participants the

sensations, feelings, attitudes, motives, reactions, etc., which they experience in the

different phases of solving a problem and at different stages during the workshop.

Observation of the behaviour in the classroom of both teacher and students,

videorecorded with two cameras, with subsequent transcription and analysis. These

recordings have a dual purpose: to be a source of data to analyze the evolution of the

participants' teaching strategies, and to constitute the fundamental material for

reflection with the students and teachers.

The Moodle Platform is a useful tool for the presentation of information and

communication. It allows information to be stored for later analysis (both qualitative

and quantitative), with the date and the subject contributing the information being

reliably logged. It allows one to evaluate the participation, and to see whether the

students have attained specific learning objectives, providing feedback as well as

motivation to the students (Rodríguez, 2005).

Diaries (Nichols, Tippins, and Wieseman, 1997; Volkmann & Anderson, 1998) kept

on the Moodle virtual platform. These allow the collection of observations,

sensations, reactions, interpretations, anecdotes, introspective remarks about feelings,

attitudes, motives, conclusions, etc.

Discussion Groups to facilitate debate (Watts & Ebbutt , 1987), since people who

share a common problem will be more willing to talk to others with the same

problem (Lederman, 1990). The prospective teachers require a group context and a

researcher for this information to emerge, be expressed, and deciphered in words

(Lederman, 1990). The discussion groups yield data of a type that would be hard to

obtain by other means since it corresponds to natural situations in which spontaneity

is possible, and in which, thanks to the tolerant atmosphere, there come to light

opinions, feelings, and personal desires that would not be expressed in rigidly

structured experimental situations (Gil, 1992-93).

The Affective Dimension of Learning and Teaching Mathematics and Science

277

For the data analysis, we used the program packages SPSS 15.00 program and Aguat for

the quantitative and qualitative methods, respectively, following the recommendations and

suggestions set out in various works, including those of Miles & Huberman (1984) Goezt &

Lecompte (1984), and Wittrock (1986).

Some Results

The data showed there to be a contradiction between the expectations, actions, and

reflections the participants make when solving problems. Thus, in the pre-workshop open

questionnaires they state that: "Maths is never learnt by memory, it must all be reasoned out",

"It is not enough to know all the formulas to apply". The questionnaire responses during and

after solving problems, however, implicitly considered it to be mechanical learning, as they

indicated that knowing how to do some school problems you can solve others by just

changing the data.

We also found contradictions between the attitudes they said they felt and those that they

manifested during the PS, and which we observed in the complementary videorecordings. For

example, they claim to look for different ways and methods to work on the problems, but in

reality there was evident abandonment in the face of difficulties in finding a solution. Their

statements show the relationship between mathematics PS and the emotions and beliefs

generated. "When I got it (solved the problem) I felt very satisfied", and "Solving problems

correctly also gives you more security and confidence". In the contrary sense: "When you do a

problem and it does not come out, you leave it, and you think that mathematics is very hard".

The findings derived from the questionnaires indicated that these prospective primary

teachers consider the results that occur in their lives will depend on their actions, i.e., they

have a great expectation of contingency or internality, pointing to effort, perseverance, and

patience as key aspects in mathematics PS. Hence they express such statements as: "With a lot

of effort and dedication I managed to get it out", and "Also, it is due to my own attitude".

However, despite their responses to the questionnaires declaring effort, perseverance, and

patience to be necessary factors in mathematics PS and saying that they persevere in that task,

the videorecordings at the beginning of the workshop showed that many of the participants

gave up easily in mathematics PS tasks. This attitude had improved, however, after the

workshop.

As against this high locus of internal control, we observed a low locus of external control.

I.e., they attach little importance to the influence of external factors on the succession of

events or the attainment of their goals. There was a low score on helplessness (capacity for

control), which means again that they do not expect that the events or outcomes that may

happen to them will be independent of their actions. This was also the case, although to a

lesser extent, with the expectation or belief in luck. Thus, the degree to which they believe the

things that can happen to them in life will depend on chance and coincidence was practically

insignificant. They do not consider that the results achieved primarily derive from other

external sources excepting the teachers, to whom they assign a key role in the teaching and

learning of mathematics PS: "It depends on how they explained it to you", or "The attitude of

the teachers is decisive".

The prospective primary teachers do not feel very confident about their personal abilities

(expectations of self-efficacy). This confidence was favoured by tasks relating to everyday

Lorenzo J. Blanco, Eloisa Guerrero, Ana Caballero et al.

278

life, and disfavoured in situations of significant difficulty. Thus, they note that they lack

security and confidence in mathematics PS ("I have always been null with mathematics

problems"), but these factors increased with their working on the problems in groups. The

security and self-confidence improved after the workshop on solving mathematics problems,

as illustrated by the results of the STAI pre-test and post-test shown in Figures 2 and 3.

Figure 2. Security. (“I feel secured”)

The Affective Dimension of Learning and Teaching Mathematics and Science

279

Figure 3. Confidence. (“I have self-confidence”)

They have favourable expectations of success for goals of a general type, i.e., they expect

to get the desired results given the objectives we set them. However, when they are actually

faced with a mathematics problem, they show no such expectations of success, and their

confidence is lower. This again shows the discordance between what the subjects see as

Lorenzo J. Blanco, Eloisa Guerrero, Ana Caballero et al.

280

desirable and the reality, i.e., their responses to certain questionnaires not reflecting what they

really think or do, but what they believe to be the most positive, what is expected of them, or

what they would like their attitude to be like.

These results do show that the students are well predisposed to the learning situations.

The students state that they are calm when faced with mathematics problems. However,

when they get stuck or blocked with the solution, their insecurity and nervousness (anxiety)

increase. This could mean that it is the blocks in solving the problem rather than the problem

itself which provokes their anxiety, which would mean that there is a need to learn to

intervene when such blockages occur.

The workshop's evaluation conducted by means of questionnaires, discussions, and

specific PS activities showed that, in general terms, there was an increase in feelings of

security, satisfaction, and self-confidence.

Both at the beginning and at the end of the workshop, the subjects were asked to locate

themselves on a scale of 1 to 10 as problem solvers. The results are shown in Figures 4 and 5.

A Student's t-test for related samples showed there to exist statistically significant

differences between pre-test and post-test scores (p = 0.0000). This means that the prospective

primary teachers believed that they had significantly improved in their PS performance after

carrying out the workshop.

Figure 4. Problem solver.

N

Mean

St. Dev.

Pretest. Solver

problem (before the

whorkshop)

33

4,02

1,661

Postest. Solver

problem (after the

whorkshop)

33

6,64

1,377

Figure 5. The mean scores as problem solvers.

The Affective Dimension of Learning and Teaching Mathematics and Science

281

As well as these results, we would note that the students valued the workshop positively,

emphasizing the importance of combining psychology and mathematics. The following are

some of their statements that together summarize the evaluations the students made of the

workshop: "It helps to see mathematics differently, not as a threat but as a challenge", "Now

we know how to look for different ways of solving problems", "I stop longer on the wording of

the problem to understand it better and I know how to analyze it better, before I faced it with

more nervousness, but now with these steps and with the strategies for relaxation I face it in

another way", and "The most important was the affect that you teachers had brought to the

workshop, by being understanding with the students, because you have given us more

keenness about learning".

5. CONCLUSIONS

The work that was carried out with prospective teachers, including their evaluation of the

workshop, reaffirms our conviction that there is a need for PS to be studied in greater depth,

considering cognitive and affective aspects as complementary. It is not easy to design and

implement a workshop on PS that includes in all of its sessions and activities specific aspects

of cognition and emotional education. Nonetheless, although difficult, we consider it

necessary because teachers in their actions in the classroom can not dissociate the two aspects

when they are dealing with some specific activity for pupils of a specific level.

It is true that some of the prospective teachers stated that they still lacked confidence as

problem solvers. But it is no less true that these same pupils showed more willingness to

tackle a problem that they were set than at the beginning of the workshop. This opens the way

to changes in their values concerning PS, and they will be better disposed to initiate changes

in this activity along the lines set out in the current curricular proposals.

6. ACKNOWLEDGEMENT

The present study was funded by Research Projects SEJ2006-04175 of the Ministry of

Education and Science (Spain), and PRI08B034 of the Junta de Extremadura (Spain), and

European Regional Development Fund (ERDF).

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