The affective dimension of learning and teaching mathematics and science

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.

Full text

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

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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

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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

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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

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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

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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:

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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).

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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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