This study accepts that beliefs ‘are largely cognitive in nature and are developed over a relatively long period of time’ (McLeod, 1992, p. 579), and since the ‘cognitive and affective domains intersect in the area of beliefs’ (Schoenfeld, as cited in McLeod, 1992, p. 590), this is a significant realm of research for understanding and improving mathematical learning. This statement assists in understanding that beliefs influence what is learnt by learners in mathematics classrooms as well as how it is learnt.

According to Walberg’s (1984) theory of educational productivity, the following nine factors are considered to be ‘potent, consistent and widely generalisable’ (p. 20), and if optimised could develop learners’ affective, behavioural and cognitive domains. The nine factors, divided into three groups, are as follows:

In the main study (Biccard, 2010) learners were involved in collaborative work on modelling tasks. This meant that one factor from each of Walberg’s groups was directly affected by the programme. Learners were highly motivated during the programme; they spent more time on quality tasks, whilst the grouping focused on shared ideas and opinions. This was a change from the traditional approach to mathematics tasks that they had experienced.

Traditional teaching methods in mathematics classrooms have led to learners holding common negative beliefs about mathematics (Schoenfeld, 1992). This is because a traditional teaching approach focuses on drilling specific procedures – that are not necessarily meaningful to learners and which they do not alwaysunderstand or relate to anything else in their lives. Traditional teaching is entrenched in what Ernest (1988) called an instrumentalist view of mathematics, which sees the subject as well defined and as a correct set of procedures that need to be practiced and memorised. Traditional teaching methods also have learners working individually, which according to Cobb (1986, p. 8) means they lose valuable feedback mechanisms that they would experience working with peers, such as dealingwith inconsistencies, expressing concepts and comparing procedures.

Interaction assists learners to build a socially rich view of mathematics. Cobb (1986) therefore suitably defines learner beliefs about mathematics as their ‘attemptedsolutions to problems that arise as they interact with the teacher and their peers’ (p. 8). Cobb further clarifies that learner beliefs are not simply communicated from teacher to learner, but that students construct beliefs that fit into the classroom norms that they have experienced. In this sense, Cobb explained learner beliefs as both social and cognitive, developing in response to the learners’ experience of classroom mathematics. Changes in classroom experiences may hinge aroundtasks set for the classroom. The hypothesis of this study is that interaction around good-quality tasks develops more positive beliefs about the value and use of mathematics in learners.

In the main study, the development of modelling competencies in primary school learners was documented over a period of 12 weeks. Three main areas of competence – cognitive, meta-cognitive and affective – were considered. Within the affective domain, it was decided to determine if a modelling perspective on learning mathematics could result in more positive beliefs being held by learners. Traditional classrooms tend to encourage rote learning and drilling of abstractconcepts; very often problems are routine and removed from anything that resembles real-life problems. To change pupil perceptions and beliefs, problems set in real life are necessary. Modelling problems present real situations that need learner engagement and structuring. Very often modelling tasks result in a product that is different to traditional exercises (Lesh & Doerr, 2003a, p. 16), often a letter or report to someone (a client) who has a real need for the problem to be modelled. Thismeans that learners may experience mathematics as useful and connected to real-life problems and to real people who make decisions based on models.

In terms of the main study, the focus was on changing the quality of not only the teaching experience but what was considered to be the ‘learning experience’.Modelling tasks require that learners work in groups, and much more time is needed to complete them because of their complex and real nature.

This study considered whether well-formulated modelling tasks in teaching could positively transform learner beliefs about mathematics. During the 12-weekperiod that learners were involved in the modelling programme, positive results to support this hypothesis were found, which are presented in this article.

Goldin’s (2008, p. 188) view of affect is that it should be ‘regarded as an internal representation system’, and ‘to see affect as representational is to hypothesize that emotions encode information’. He adds that it is ‘detailed, context-dependent, rapidly changing information essential to the doing of mathematics’. To uncover learner-affective internal representations is not an easy task, but whilst modeling, learners do leave ‘auditable trails of documentation’ (Lesh & Doerr, 2003a, p. 31).These are the many written and verbal products of learner interactions when modelling, which allow glimpses into what learners are thinking even on an affective level. This trail of documentation that learners produce whilst modelling allows a certain rendering of learner-affective factors not always possible with more traditional teaching approaches. As explained by Lesh and Doerr (2003a), problem-solving leads to significant forms of learning. This learning can also includeaffective learning or an awareness of the use and value of mathematics through tasks that would encourage learners to think mathematically. Lesh and Doerr further clarify that thinking mathematically involves constructing, describing, explaining, and making patterns in complex systems (2003a, p. 16). This suggests that the content of learners’ mathematics classroom experiences may have an impact on the beliefs that learners would hold as a result of these experiences.

The relationship between experiences and beliefs is a premise for this study. If emotion encodes information, then all learning would have an affective element and all learning experiences would allow learners to formulate certain beliefs about mathematics. This means that learning through tasks such as modelling may enablelearners to develop more positive beliefs about their abilities to interact successfully with mathematics, and more positive beliefs about the role of mathematics in real-life situations.

When learners become personally involved in their learning and reflect on this learning, then a formation of beliefs takes place. Learners develop beliefs about the role-players in the learning situation. They develop beliefs about themselves and their abilities and beliefs about the relevance of the subject matter to their lives and real problems in general. Learners also develop beliefs through the role and nature of content in the mathematics classroom experience.

Ernest (1998) reminds us that since the time of Plato, mathematical knowledge has been thought of as secure and infallible. This means that in a traditional classroom, mathematical knowledge is handed over from teacher to learner as absolute truths that learners have to memorise and learn to apply. From a modelling perspective, learners produce, sort out and develop conceptual tools that describe or explain a mathematically significant system (Lesh & Doerr, 2003a, p. 9, 2003b, p. 532). When learners experience the former in their classrooms, their beliefs about mathematics may be very different to when they experience modelling in mathematics classroom activities. The learners involved in this study often used phrases such as ‘your method’, ‘my method’ or ‘what did you do?’, which showed more ownership of the problem than just memorising of rules.

Österholm (2009, p. 158) asked how knowledge of the relationship between the diameter and circumference of a circle is ‘less’ affective than beliefs about mathematics teaching. It may be that the latter is personally reflected knowledge and this adds the affective component, whilst the former is not personally reflected knowledge but knowledge that is accepted as true. It may be that knowledge is unlikely to be disputed, whilst beliefs are not consistent for all people: for any belief that may exist, a diametrically opposing view may also exist. Wilson and Cooney (in Speer, 2005, p. 365) propose a different perspective than insisting on definitions for beliefs. In a study on teacher beliefs they stated that concerns over precise definitions of beliefs are not as important as issues of understanding teacher knowledge. Schoenfeld (2011, p. 52) establishes that beliefs are ‘very powerful explanatory constructs’.

The aim of this article is to understand and provide an explanation for the development of learner beliefs relating to mathematics during a modelling programme, and not so much to provide new definitions for beliefs. Answers to the questions posed in a pre-modelling and post-modelling sessions indicate learner beliefs, and their responses are reported on. If learner beliefs do change as a result of modelling tasks, then it can be inferred that modelling provides learners with the evidence, experience and repetition they need to modify some of their beliefs. Furthermore, it can be inferred that if beliefs do change, then modelling provides learners with a means for personal reflection that leads to improved beliefs about mathematics.

Schoenfeld (1992, p. 359) lists typical learner beliefs about the nature of mathematics:

These six categories relating to learner beliefs were used to code the data for this study. Although Schoenfeld generated these from work on problem-solving, whilst traditional teaching pervades schools, these categories are still relevant and can be used to code transcriptions of learner work whilst modelling. Goldin (2007, p. 289) states that affective competencies refer to a person’s ability to use affect positively during a mathematical activity, and that future research should be devoted to studying how learners change global affective structures regarding mathematics. Goldin’s concept of a global affective structure was also included as this concept extends the discussion on beliefs. Global affect refers to complicated affective structures which include emotions, beliefs, values, stories, memories and one’s history. Goldin (2007, p. 289) further set out three essential affective structures which were also used to code data:

Beliefs seem to form the underlying core of these affective structures, although these structures are more holistic than learner beliefs. These affective structures wereused for coding the data to trace affective competency development. Both Schoenfeld’s and Goldin’s categories could be used to analyse learner beliefs whilst they are engaged in a modelling task. It may be valuable to use existing concepts and categories to code data on modelling: Schoenfeld’s categories focus specifically on typical learner beliefs, whilst Goldin’s categories were used to focus on more widespread affective facets of learning.

From the above, it is apparent that modelling is very different to traditional mathematics tasks. Modelling requires a larger domain of cognitive, meta-cognitive and affective competencies than traditional tasks. The process that learners go through when modelling is displayed in a normative modelling cycle in Figure 1.

Modelling is more complex than traditional tasks (Chamberlin & Moon, 2005; Lesh & Doerr, 2003a). In a traditional classroom learners remain at a single node of the cycle – working mathematically, whilst the teacher or textbook is responsible for the rest of the cycle. From this experience learners could develop negative beliefsabout what mathematics is, how it is learnt and how it is relevant to real life.

Once learners are placed in a directorship role of the entire cycle, their beliefs about mathematics should start to change. Initially their negative beliefs leave themfrustrated and they try to solicit a traditional ‘didactical contract’ (Brousseau, 1997, p. 39) by asking the teacher to show them the method or solution. Blomhoj and Jensen (2007) describe a key to modelling as learning to cope with feelings of ‘perplexity due to too many roads to take and no compass given’ (p. 49). This type of perplexity is not easily overcome. As learners gain experience of modelling, they are able to develop their modelling competencies. By modelling, learners are responsible for wider aspects of problem-solving, which changes their experiences, and this may result in a change in affective competencies. Modelling tasks also provide learners with evidence that mathematics is useful and that they can use mathematics to solve problems. With modelling tasks, groups of learners negotiatemeaning with each other and create a model for the situation in their own way without any predetermined, memorised methods.

FIGURE 1: A modelling cycle.

Competence is taken from Weinert’s definition (Henning & Keune, 2007, p. 225), as the sum of available abilities and skills and the willingness of a learner to solve aproblem and to act responsibly concerning the solution. Modelling competencies encompass all the abilities and skills a learner uses in solving a modelling problem. A learner employs these abilities and skills together with certain beliefs, and these beliefs affect the way in which the learner may successfully use available abilities and skills. McLeod (1992, p. 578) also identified beliefs as a concept in the affective field that is more stable than emotions or attitudes. This was an indication thatbeliefs could be considered a competency. Beliefs were included as a competency so that an awareness or development/change in learner beliefs could be monitored together with the other competencies that were identified. Since the instrument did not use Likert-type scales but a qualitative open coding, the development of learner beliefs as a natural progressing entity could be documented before and after a modelling programme.

Beliefs as an element of affect play an important role in learning and are a consequence of learning experiences. The relationship between teaching approaches and beliefs has been established (Spangler, 1992). Since this cycle is not easily broken, the focus of research should then be on using the teaching approach-beliefs cycle andnot working at breaking the cycle. If the relationship between teaching and beliefs about mathematics exists, then changing instruction to a modelling approach may lead to more positive beliefs about what mathematics is, how it is learnt and how it is relevant to real life. Beliefs are a stable, affective construct developed over a long period of time. However, Harel and Lesh (2003) show that during 90 minutes of modelling, the stages oflearner thinking are strikingly similar to those described by van Hiele (1986, cited in Harel & Lesh, 2003) or Piaget (1958, cited in Harel & Lesh, 2003). These ‘compact versions of developmental sequences’ (Harel & Lesh, 2003, p. 381) assisted in hypothesising that learner beliefs may also develop over a relatively short period of time whilst they are involved with modelling tasks.

Learner ideas about what constitutes a mathematical task form part of their beliefs. Group 2 had the following discussion after a heated argument in Task 3, that can be related to beliefs about mathematics and the nature of mathematical tasks (in the following protocols, letters denote the names of different children):

Learner S: ‘It’s incorrect!’
Learner T: ‘Nothing is incorrect with this, ok.’
Learner S: ‘I am really good at maths.’
Learner T: ‘Yes, but with these problems ...’ (Biccard, 2010, p. 124)

Here Learner T is moving towards changing beliefs about mathematics problems, but Learner S remains unchanged, holding that ‘I am really good at maths’. This shows that Learner T may have been becoming more aware of a difference between these problems and traditional tasks.

In learners’ written reports after each session, they had to write down where they needed assistance. Group 3 wrote this after the first session of Task 1:

This shows that the learners expected to apply some mathematics that they knew, without really understanding what they were doing (S3).

By Task 3, they were discussing how to find a ratio, with the focus on their own understanding of the task (development of S3):

Learner G: ‘We need to find something related to that.’ [Pattern measurement and real measurement.]
Learner E: ‘We need to measure this which is 27, 4 … now we need to find something that will give us ... something that is …’
Learner M: ‘Related … We are trying to see our answer.’

Learners were informally interviewed during Task 2, when they were asked by the researcher about their experience of working in groups as opposed to working individually (S4) and both groups responses are given in Box 2.

By the end of Task 3, the following comments were made by Group 3, showing a gradual change in their beliefs about the nature of mathematical tasks (Development of S5):

The thing about the beginning of a task is that we are always so confused. [Learner G]

That’s impossible. Look, I don’t think we are going to get a direct answer at the very beginning of a task. [Learner G]

No, we’ll never do that, unless we have a time machine. [Learner W] (Biccard, 2010, p. 125)

The problems involved a large amount of reading, which is not common in traditional classrooms. Since real problems are set in a context that needs to be understood, often through reading, learner response to reading the supporting material for each task allowed S6 to be gauged. During Task 1, the learners were unsure what all the reading material was about, as one indicated by saying ‘I don’t think the teacher meant for us to read this stuff’.

This shows that they may not have believed that mathematics problems involved much reading. However, by Task 3 they did not question the supporting material for each task, but simply referred to it when they needed to:

Learner S: ‘This problem is a little more serious than the last. Oh well, what the heck, let’s give it a try. How do we start the problem A?’
Learner A: ‘Ok. Read this.’
Learner J: ‘Do you think we have to make this outside also?’
Learner A: ‘No, that is a photo frame.’
Learner J: ‘Read what we are asked to do. Just read it again …’

Evidence was found, which includes elements from S1-S6, that learners believe that mathematics must be done neatly and that whole number answers aremore ‘correct’ than messy decimals. An incident showing that learners viewed mathematics as a neat, systematic endeavour is noted at a later stage, and is also mentioned as it directly affected Group 3’s modelling competencies. It was noticed that Group 3 spent a lot of time rewriting a list of items from a catalogue for Task 2 (Catalogue Problem), and lost valuable time doing this. The next week, they were questioned about their decision:

Researcher: ‘Why did you write the prices over? When they were here?’ [on the instruction sheet provided] ‘Was there a reason?’
Learner M: [A little embarrassed] ‘We didn’t want to mess the page.’ (Biccard, 2010, p. 124)

During Task 2, Group 2 noticed that they had reached an answer of a whole number. This led them to think that they must be on the right track. They still assumed that neat, round numbers were more correct than decimal numbers, and Learner J exclaimed ‘[b]ut ours and yours adds up to R200 [exactly], there is a connection!’ (Biccard, 2010, p. 124).

By Task 3, this group worked with a decimal of up to 5 digits without thinking that it was incorrect.

When looking at Goldin’s essential affective structures, it was found that modelling elicits the affective structures he describes. Mathematical integrity as an affective structure includes an awareness of one’s limitations in understanding and acknowledging them and working towards removing them (Goldin, 2007). This structure is strongly affective and confirms learner beliefs in mathematics and their experiences of it. It was found that learners were unafraid to acknowledge theirconfusion or frustration with these tasks – which is largely kept silent during traditional instruction. If it is evident, the teacher feels responsible for clearing up the confusion due to the didactical contract (Brousseau, 1997) that exists. With modelling tasks learners had to support and provide scaffolding for each other. The following condensed excerpt illustrates the need for mathematical integrity (G1) that modelling elicits (Task 1):

Learner S: ‘We are really stuck.’
Learner J: ‘I don’t get it.’
Learner J: ‘What about the adding 20 method?’
Learner S: ‘That was your method. No, that was a random thing.’
Researcher: ‘Can you explain that to me?’
Learner J: ‘No, I can’t back it up.’
Learner J: ‘We are trying to see the significance between it.’ [foot size and height]
Learner S: ‘You mean a relationship?’
Learner J: ‘Maybe it can be a pattern?’

The input from learners corresponded with mathematics self-identity (G2) are listed in Box 3.

Learners were slowly starting to place themselves, their thoughts and values into the mathematical arena, which resulted in a much fuller mathematical experience.

The following highlights mathematical intimacy (G3), which includes a ‘vulnerable interaction involving one’s sense of self during mathematical activity’ (Goldin,2007, p. 291); the final question on the questionnaire asked for any comments the learners wished to offer, and a number of learners stated: ‘When are we going to do this again?’

The following comment by one learner fits into Csikszentmihalyi’s concept of flow: ‘I did enjoy the problems because you got to go way out of the ordinary and when you’re done you feel so fantastic’.

Affective structures and beliefs are brought to light whilst learners are involved in modelling tasks, which allow one to view learner beliefs and affective structures formed over the entire span of learner experiences with mathematics. Modelling therefore fulfills two different but mutually important roles: it serves as a mirror in which learners reflect not only their mathematical knowledge and skills but also their affective orientations, and also suitably fills the role of generating anenvironment within which to build more positive beliefs about mathematics.

McLeod’s (1992) view that what is needed is a change in the curriculum that fosters learners’ negative beliefs is also accepted. Modelling allows for such a change or modification to both the curriculum and teaching strategies. Possibly, as suggested by Wilkins and Ma (2003), recognition of negative trends is a central first step for teachers in attempting to reverse negative learner beliefs. As such, these results are concurrent with the findings of Wilkins and Ma (2003), in that exposing learners to challenging mathematics may curb the development of negative beliefs about mathematics.

Returning to Walberg’s (1984) model of productivity (p. 21), modelling affects at least one factor in each of the three main areas. With regard to learners’ personal factors, modelling increases learner motivation. Modelling also makes significant changes to the teaching factors by improving the quality and amount of time spent on the learning experience, and affects environmental factors by making changes to the traditional classroom environment and interaction.

Modelling allows Mandler’s affective aspects (McLeod, 1992) to come into view. Within the modelling sessions learners did hold certain beliefs about mathematics, and some of these beliefs showed signs of positive change towards the end of the programme. Learner blockages were evident – and so were their routes through these blockages; it was their use of informal methods through these blockages that built more positive beliefs about mathematics. In order to maintain or consolidate these positive attitudes, repetition of these tasks is necessary.

Modelling as a teaching approach in mathematics classrooms will lead to learners using their own (often invented) methods and allow learners to mathematise situations in ways that are meaningful to them. Learners are not expected to follow a rehearsed method or a set procedure (that they often do not understand). The model that results from engaging in a well-formulated modelling task is constructed by learners and shows their understanding of the problem situation.

In Perrenet and Taconis’ (2009) study, that showed significant shifts in the beliefs of university students, the students were asked to explain their own shift in beliefs to point out differences between their secondary school and university mathematics experiences. According to these students, the difference was in the nature of the problems that they were given to solve. They said that school problems were standard closed problems, whilst those at university were more challenging and complex. Goldin (2007, p. 294) affirms that teacher-guided modelling produces long-term effects that have cognitive and affective consequences.

Modelling as an environmental factor allows learners to work in groups, collaborating with each other. From a socio-constructivist view, affective processes are considered an essential part of problem-solving and learning (Op’t Eynde, De Corte & Verschaffel, 2006) so they form a key part of a modelling environment and are also developed through this environment. From a social-interactionist lens, individuals construct meaning through social interaction but, more importantly, social interaction is the process that forms learner conduct and is not simply a setting for this conduct to take place (Yackel, 2000). Social interaction therefore forms learner beliefs about mathematics. This too was evident in the study.

The input and suggestions of the reviewers are acknowledged and the authors would like to thank them for their contribution.