Introduction

In Victoria, Australia, secondary mathematics students are expected to use technology and pen-and-paper (P&P) for mathematics (Victorian Curriculum and Assessment Authority (VCAA), 2015). One school subject incorporating technology is Year 11 Mathematical Methods (MM). MM contains four areas of study, (i) Functions and Graphs, (ii) Algebra, (iii) Probability and Statistics and (iv) Calculus. Content from the Algebra area of study is incorporated within the other areas of study as required; hence, students learn and use algebra throughout Year 11. The use of ‘numerical, graphical, geometric, symbolic and statistical functionalities of technology’ (VCAA, 2015, p. 36) must be incorporated within each area of study as applicable. Three outcomes are assessed, broadly related to concepts and skills; problem solving, modelling and investigations; and technology. The latter requires students to ‘Select an appropriate functionality of technology in a range of mathematical contexts and provide a rationale for these selections’ (p. 36); hence, part of learning in Year 11 MM involves learning to make appropriate choices about when and how to use technology. CAS is well established within MM, having been incorporated since 2001 (Leigh-Lancaster, 2010) and remaining the approved technology for high-stakes Year 12 examinations (VCAA, 2018); hence, for the students reported here, technology (i.e. CAS) was an integral part of mathematics.

Calculators incorporating a CAS allow symbolic manipulation of expressions and equations; these symbolic functionalities of CAS can automate many mathematical procedures (see Fig. 1). Using CAS in mathematics presents opportunities for teaching and learning by enabling, for example, access to multiple representations and real-world problem-solving (Pierce & Stacey, 2010). CAS can also develop student understanding (Bawatneh, 2012; Heid, 1988) and reasoning (Granberg & Olsson, 2015; Kieran & Saldanha, 2005). Students need to make effective use of CAS to benefit from the learning opportunities presented by CAS (Pierce, 2001). A key element of effective use of CAS is the capacity to make judicious use of CAS, which involves the ‘use of CAS in a strategic manner’, being ‘discriminate in functional use of CAS’ and undertaking ‘pedagogical use of CAS’ (p. 55). Pierce and Stacey (2004b) argue that affective factors (e.g. beliefs, emotions and attitude) impact both how students use CAS and the extent to which they attempt to overcome technological obstacles, so a student’s attitude may impact their capacity to make judicious use of CAS. Although previous studies have explored the link between attitude and experience with CAS (e.g. Orellana, 2016; Schmidt, 2010), and attitude has been shown to influence CAS use (e.g. Orellana, 2016; Pierce et al., 2007), the literature review did not identify any studies that have explored the stability of students’ beliefs (a component of their attitude) as they gain experience with CAS. This article builds upon existing literature that has explored students’ beliefs about CAS by using a repeated measures research design (Kraska, 2010) and collecting data from the same (rather than separate, e.g. Orellana, 2016; Schmidt, 2010) cohort of students, to determine the stability of students’ beliefs about CAS as they gain experience with CAS. Given that attitude has been identified as a requisite for the effective use of CAS (Pierce & Stacey, 2004b) and beliefs can influence behaviour (Ajzen, 2005), developing an understanding of the stability, or otherwise, of students’ beliefs as they gain experience with CAS may provide insight into where pedagogical interventions could be targeted to positively support effective use CAS.

Fig. 1
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Sample use of CAS for algebra in Year 11 MM

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Theoretical framework

Rationale for focus on beliefs

The focus on beliefs for this article results from using a bidimensional definition of attitude in the first-named author’s PhD study. While attitude can be broadly defined as ‘a predisposition to respond favourably or unfavourably to an object, person, institution or event’ (Ajzen, 2005, p. 3), Di Martino and Zan (2010) found three predominant definitions of attitude within mathematics education:

  • Simple definitions, where attitude is synonymous with affect and often refers to emotions

  • Bidimensional definitions consist of beliefs and emotions

  • Tripartite definitions consist of beliefs, emotions and behaviour.

Each definition presents advantages and disadvantages for a given study (Di Martino & Zan, 2011). Simple definitions limit discussion of attitude to emotions and hence would exclude research on beliefs about CAS. Tripartite definitions are problematic because incorporating behaviour can result in circular discussions whereby behaviour describes, and is used to infer, an attitude (Di Martino, 2016). This contrasts to simple and bidimensional definitions where behaviour may be a result of an attitude (Di Martino & Zan, 2011). Utilising a bidimensional definition (see Fig. 2) allowed the PhD study to build upon existing literature and avoid issues with the inclusion of behaviour. While emotions are a key part of a student’s attitude towards CAS, they are not reported in this article.

Fig. 2
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Bidimensional definition for ‘attitude towards CAS’

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Defining beliefs

The investigation of beliefs was informed by Phillip (2007) definition of beliefs as ‘lenses through which one looks in interpreting the world’, which are ‘psychologically held understandings, premises, or propositions that are thought to be true’ (p. 259). There is no universally accepted definition of beliefs (Zhang & Morselli, 2016); however, Phillip’s definition features in the Encyclopedia of Mathematics Education (Liljedahl & Oesterle, 2020) and shares similarities with others in mathematics education (e.g. Leder et al., 2002; Spector, 2012), thus indicating some agreement with this definition. The investigation of beliefs is important as individuals learn to favour behaviours that align with their beliefs (Ajzen, 2005), hence suggesting beliefs about CAS would influence a student’s CAS use and their capacity to benefit from the learning opportunities presented by CAS.

In defining beliefs as a component of the affective domain, McLeod (1992) described beliefs as being ‘generally stable’ (p. 578) and unlikely to change once formed. This stability is because, for a belief to change, an individual must reject a view previously held to be true and accept a new view that potentially opposes the existing belief (Grootenboer & Marshman, 2016). Although McLeod (1992) described beliefs as stable, Furinghetti and Pehkonen (2002) reported that the idea of the stability of beliefs was contentious among the experts in their study because of their different understandings and definitions of beliefs. Liljedahl et al. (2012) conducted a review of 92 journal articles to identify the validity of the claim that ‘stability is an inherent and definable characteristic of beliefs’ (p. 101) and concluded that use of the word stability to describe beliefs has a ‘varied and disparate use … from difficult to change, to slow to change, to resistant to change’ (p. 112). Liljedahl et al. (2012) reported that a range of definitions for beliefs were used in literature, which aligns with Furinghetti and Pehkonen’s finding that different researchers have different understandings and definitions of beliefs. Liljedahl et al. (2012) concluded that beliefs can change; thus, the mathematics education community should cease using stability as a defining characteristic of beliefs. Rather than considering beliefs to be inherently stable, Liljedahl et al. (2012) argue that researchers should consider the stability of beliefs in the context of whether a belief is held or not held over time as determined through empirical evidence. We were unable to identify studies that had examined the stability of students’ beliefs about CAS or technology; hence, one aim of this study is to determine the stability of students’ beliefs about CAS as they gain experience in using CAS for Year 11 mathematics.

McLeod (1992) explained that beliefs develop over a long period. Students in this study were new to learning mathematics with CAS, so we expected their beliefs would form throughout the year. Leder et al. (2002) categorise beliefs as descriptive, inferential and informational according to how the beliefs were formed. Descriptive beliefs are formed through an individual’s experiences and reflect what they think to be true (e.g. I believe that CAS is useful for learning mathematics). Ajzen (2005) explained that beliefs about a stimulus (e.g. CAS) form through associating the stimulus with either positive or negative attributes, such as the outcome of a behaviour (e.g. the success or otherwise of completing a problem with CAS). Inferential beliefs are formed through inference based on logical deductions. Such beliefs may ‘go beyond directly observable events and could be based on prior descriptive beliefs’ (Leder et al., 2002, p. 179). For example, a student who developed the descriptive belief that CAS is useful for checking answers may subsequently form an inferential belief that CAS is useful for learning mathematics.

Informational beliefs are those based on the actions and information provided by external sources (Spector, 2012). An example may be a student who forms a belief that CAS is useful for mathematics based on their teacher’s explanations and demonstrations of CAS for doing mathematics. In this case, the social nature of beliefs is apparent as students’ beliefs are impacted by external sources such as a teacher or peer (Op’t Eynde et al., 2002). Shuman and Scherer (2014) argued that beliefs could also be formed from an individual’s perception of a stimulus, not just their experiences. For example, a student may believe that CAS (i.e. a stimulus) is hard to use without having used one. This belief would be inferential or informational, depending on whether their belief was formed from beliefs about other technologies being hard to use (i.e. inferential) or being told by another student that CAS is hard to use (i.e. informational). Overall, these three types of beliefs are formed by a student’s direct or indirect experiences with CAS. Consequently, it could be expected that we would observe changes in students’ beliefs about CAS as they gain experience with CAS.

Students’ beliefs about CAS

Beliefs cannot be observed or measured but are inferred (Ajzen, 2005). Beliefs discussed below are inferred from student comments, questionnaire items or findings from studies that investigated the use of, or attitudes towards, CAS.

Beliefs about CAS in mathematics relate to the role that CAS can play in mathematics. Pierce and Stacey (2001) reported that undergraduate calculus students’ beliefs about mathematics, including whether using CAS was ‘real mathematics’ (p. 43), impacted CAS use. Over half of the 41 Year 9 students in Ng’s (2003) study viewed CAS as useful for mathematics, consistent with Weigand and Bichler (2010), who found that almost two-thirds of 412 Grade 10 and 11 students viewed CAS as beneficial when learning mathematics. These findings suggest that students can hold beliefs that using CAS supports mathematics learning. Schmidt (2010) investigated over 2500 Grade 11 and 12 students’ attitudes towards CAS and found that most believed using CAS supported problem-solving, avoiding mistakes and provided opportunities to check answers. When learning to use CAS, approximately 85% of students in Ng’s (2003) study viewed CAS syntax as easy to remember and understand, but only 39% viewed CAS (TI-92) as easy to use. So, students may believe it is hard to use CAS for mathematics.

Beliefs about speed relate to perceptions of how quickly problems can be completed with either P&P or CAS and may reflect perceptions of the amount of P&P working. Mohammad (2019) reported Grade 13 students who preferred using CAS for problems deemed time-consuming, with CAS viewed as quicker when problems required extensive P&P working. Ball (2015) reported teachers encouraging students to choose between CAS and P&P based on speed. A survey of 522 senior secondary students found that CAS was primarily used to complete problems quickly (Kissane et al., 2015), so beliefs about speed, which may result from teacher advice, could contribute to this use. In Schmidt’s (2010) study, the item ‘I can work faster when using the TI-89’ (p. 106) had the greatest number of positive responses of any item; hence, many believed working with CAS (TI-89) was faster than P&P. In contrast, Ball and Stacey (2005) reported Year 12 students choosing to use P&P instead of CAS for speed, hence suggesting they held a belief that P&P is faster than CAS. Consequently, students may hold beliefs about the speed of either CAS and/or P&P.

Students view CAS as useful in various contexts, including in mathematics lessons, homework and examinations (Ng, 2003). A subsequent study of 32 junior college students found CAS useful for solving equations, simplifying expressions and checking solutions (Ng et al., 2005). Students also find CAS useful for experimenting with mathematical ideas and relationships (Kissane et al., 2015). In contrast, students value the ability to identify and track errors in P&P working (Ball & Stacey, 2005), so they may also hold beliefs about the usefulness of P&P.

Of the above, only Ng (2003), Ng et al. (2005) and Weigand and Bichler (2010) used a longitudinal research design with the same cohort of students. Hence, little is known about how beliefs, including beliefs about speed and the role of CAS in mathematics, change as students gain experience with CAS. This article addresses this gap.

Research question

This article reports the beliefs about CAS held by the students of one Year 11 MM class as they learnt to use CAS and whether these beliefs were stable over time. In this article, the term ‘CAS’ refers only to symbolic features of CAS as this was the context in which students were using CAS and hence the focus of questionnaire items. The research question was ‘How did the beliefs about CAS that were held by the students of one Year 11 MM class change as they gained experience with CAS?’.

Students’ attitudes towards CAS tend to become more positive as they gain experience with CAS (Orellana, 2016), and beliefs (a component of attitude) tend to form over a long period of time (Leder et al., 2002; McLeod, 1992). Students in this study were new to working with CAS, and had little prior experience with CAS; hence, it was anticipated that their beliefs about CAS would change across the school year as they gained experience with it and developed an appreciation of the benefits (or not) of using CAS for mathematics. Beliefs can impact behaviour (Ajzen, 2005), so beliefs about CAS may positively or negatively impact CAS use. This article does not assume that beliefs are stable but will determine stability from empirical data as Liljedahl et al. (2012) suggested. In this article, a stable belief is defined as one held (or not held) at both the start and end of the study. It is important to consider the stability of beliefs that form soon after students commence working with CAS as, if the beliefs are stable, they may have lasting impacts on their CAS use. Consequently, there may be a need to support students who are new to learning mathematics with CAS to form beliefs that would positively influence their CAS use, or to change existing beliefs that would negatively influence CAS use. This article contributes to literature by investigating whether stability is a characteristic of students’ beliefs about CAS. The understanding of the beliefs held by students new to working with CAS, and the stability (or otherwise) of these beliefs, may support teachers and researchers to develop teaching interventions that positively support CAS use.

Research design

Participants were the students of one Year 11 MM classFootnote 1 in a co-educational government school in Victoria, Australia. A criterion for school selection was that students should have little experience with CAS, so they developed experience with CAS during the study. Eight students had not used CAS before Year 11, and four reported limited use; all were considered novice CAS users. Although discussion of how students used CAS is outside the scope of this study, students were working in a classroom where the use of technology (in this case, CAS) was expected to be embedded into teaching, learning and assessment (VCAA, 2015). A key outcome of MM relates to the use of technology to ‘develop mathematical ideas, produce results and carry out analysis’ (p. 36), which suggests the students were working in a classroom where the teacher would have supported the functional and pedagogical use of technology. The classroom teacher was an experienced teacher with over 15 years of experience, including the years in which CAS was embedded into the curriculum, so we expect they had a range of strategies to support students to learn to use CAS.

A questionnaire was used because they enable the identification of beliefs held by an individual through their agreement or disagreement with individual statements (Ajzen, 2005). A literature review was conducted to identify suitable questionnaires (e.g. Ng et al., 2005; Pierce et al., 2007; Schmidt, 2010) for the larger study, but none met all selection criteria.Footnote 2 Consequently, a new questionnaire was developed with four sections; the section on beliefs is reported here and the items are provided in Table 8. Students completed the questionnaire in April (after studying Linear, Quadratic and Cubic functions) and November (after studying Calculus) of one school year. When completing the questionnaire, students were instructed to consider their use of symbolic functionalities of CAS for completing the algebra required in these topics.

Questionnaire development

Questionnaire items were developed following Cohen et al.’s (2018) method. First, a literature review was conducted, described above, to identify statements (i.e. questionnaire items or student comments) where a belief was stated or could be inferred. Items 19, 20 and 21 were generated from a student’s comment that ‘I will trust the calculator more than I trust myself doing it by hand’ (Ball & Stacey, 2005, p. 125). The idea of trust reflects a belief about whether answers are correct using CAS (Item 20). To contrast this belief with P&P, items 19 and 21 were developed.

A further example is the development of items 8 and 11 from Schmidt’s (2010) questionnaire item ‘I can work faster when using the TI-89’ (p. 106), where students responded on a Likert scale. Although Schmidt’s item does not refer to P&P explicitly, the term ‘faster’ implies a comparison between the speed of CAS and P&P. Those agreeing with Schmidt’s item believe CAS is faster than P&P, while those disagreeing believe the opposite; hence, two beliefs about speed were developed (see Table 1). Beliefs refer to algebra as this was the context of the larger study. The first-named author then rephrased each belief as a new questionnaire item. Some beliefs refer to P&P (e.g. answers are correct when using P&P; item 19) because of the expectation that beliefs about P&P could influence CAS use (e.g. choosing P&P over CAS to track errors; Ball & Stacey, 2005).

Table 1 Beliefs and items inspired by ‘I can work faster when using the TI-89’ (Schmidt, 2010)
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Cohen et al. (2018) explained two processes for piloting questionnaires. A statistical analysis was not possible due to the small sample. Instead, feedback on content and format was collected from subject matter experts. Co-authors reviewed all items, focusing on rewording items for clarity and removing unnecessary items. Several mathematics education colleagues piloted the questionnaire.

The 22 beliefs and items generated through this process are provided in Table 8.

A 5-point Likert scale was used to determine whether a belief was held. Although 7-point Likert scales are more sensitive and reliable than 5- or 3-point scales (Cohen et al., 2018), data were collected using a 5-point scale (Strongly Disagree, Disagree, Unsure, Agree, Strongly Agree) because we grouped responses of Agree or Strongly Agree as both responses indicate a belief was held. The remaining response options indicated the belief was not held.

Questionnaire analysis

The analysis focussed on beliefs at the start and end of the study (hereafter ‘start’ and ‘end’). Table 2 illustrates the analysis of Belief A. Each student either holds (H) or does not hold (H′) a belief at the start and end. Five students held Belief A at the start and end (response pattern HH), while one did not hold this belief at either time (H′H′). Belief A is ‘stable’ for these six students as their response did not change. In contrast, Belief A is ‘unstable’ for the remaining six students as they moved from holding to not holding the belief (HH′) or vice versa (H′H). Each belief was categorised according to stability using the criteria defined in Table 3.

Table 2 Example of following tables
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Table 3 Belief categories
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Limitations

The small sample size limits the findings of this study. The 12 students reported here were new to learning mathematics with CAS, so we expected they were forming beliefs about CAS use. Hence, the study contributes to an understanding of changes in students’ beliefs when learning with CAS and provides implications of these changes for teachers as they support student CAS use. We acknowledge that statistical validation and refinement of the questionnaire would strengthen the reliability and validity of this instrument, and this should be considered before future use.

Results and discussion

The study aimed to investigate the stability of students’ beliefs about CAS as they gain experience with CAS. Where a belief reflects a positive view of CAS, it is seen to contribute to a positive attitude towards CAS (e.g. CAS is useful for checking answers, item 9). In contrast, beliefs reflecting a negative view towards CAS would contribute to a negative attitude (e.g. It is hard to learn how to solve problems with CAS, item 27). Beliefs referring to P&P only, and that do not involve a comparison between CAS or P&P, do not contribute to a students’ attitude towards CAS (e.g. It is hard to learn how to solve problems with P&P, item 25).

The results are summarised in Table 4 which provides the number of beliefs in each of the categories defined in Table 3. A total of 16 (of 22) beliefs were categorised as ‘Mostly stable’ or ‘Very stable’; none as ‘Mostly unstable’ or ‘Very unstable’; and six as inconclusive. For this sample of students, stability (rather than instability) is a feature of students’ beliefs about CAS.

Table 4 Categorisation of 22 beliefs according to stability
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Students tended to hold beliefs contributing to a positive attitude towards CAS with a greater degree of stability than those that would contribute to a negative attitude towards CAS (5 of 11 cf. 1 of 8 for ‘very stable’). Given that negative attitudes can negatively influence CAS use, a lesser degree of stability for these beliefs may be desirable if fewer students held beliefs that would negatively influence their CAS use at the end of the study than at the start. It is not possible to determine this trend from Table 4, but changes in students’ beliefs are described in the following sections.

‘Very stable’ beliefs

Table 5 shows the seven beliefs categorised as being very stable (i.e. stable for 10 or more students). The modal response pattern for six of these beliefs was HH, so most students were stable in holding these beliefs. In contrast, the modal response pattern for the belief that There is a need to choose between CAS or P&P before solving a problem (item 23) was H′H′, so most students were stable in not holding this belief.

Table 5 Response patterns for beliefs categorised as ‘Very stable’
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Five of the seven beliefs would contribute to a positive attitude towards CAS, so overall, these results indicate that many students viewed CAS favourably. For example, items 7 and 9 both relate to usefulness. Given the high number who responded HH to these items, students had recognised the potential to use CAS to solve problems quickly and check answers. As all 12 students held the belief that CAS is useful for checking answers (item 9), and 10 held the belief that CAS is useful for solving problems quickly (item 7), it could be argued that using CAS to check answers was an accepted practice, either developed through teacher advice, peers or positive experience of CAS. These results provide a contrast to Ng et al. (2005), who reported that perceptions of the usefulness of CAS increased with experience as students in this study appeared to quickly develop their perception of the usefulness of CAS.

The beliefs It is important to become good at solving problems with P&P in MM (item 26) and It is important to become good at solving problems with CAS in MM (item 28) are reflective of the expectations of the development of both technology and P&P skills in the curriculum (VCAA, 2015). Both beliefs were widely held by students, which suggests an awareness of this expectation. Ng (2003) reported students who were reluctant to use CAS where CAS was permitted in class but not in assessments. The pervasiveness of the belief that It is important to become good at solving problems with CAS in MM (item 28) contrasts with Ng (2003) and may reflect the curricular requirements to incorporate technology into teaching, learning and assessment (VCAA, 2015). These beliefs may be inferential or informational beliefs formed based on the expectation for technology and P&P in the curriculum, or teacher advice about the importance of learning to use CAS or P&P. Beliefs are generally described as taking a long time to form (McLeod, 1992) yet holding these beliefs at the start of the study suggests that students formed these beliefs prior to completing the questionnaire in week 8 of term 1. In this case, it appears students’ beliefs formed quickly upon, or before, commencing Year 11 MM.

The remaining three beliefs relate to whether students felt they could choose between CAS and P&P. These beliefs are important as CAS is used more frequently when access to CAS is not restricted (Kendal & Stacey, 2001). The pervasiveness and stability of the belief that CAS can be used whenever students like (item 24; HH for nine students) suggest students worked in a classroom where teachers did not restrict CAS use. Ten students also responded HH for P&P needs to be used before CAS (item 16), so although the teacher did not restrict CAS use, students believed problems should be attempted with P&P before CAS. All who responded H′H′ to There is a need to choose between CAS or P&P before solving a problem (item 23) responded HH to item 16; these students may have felt they did not need to choose because P&P should be used before CAS. Students who hold this belief may prefer P&P and thus may not benefit from potential affordances of CAS (e.g. automation of routine procedures benefits students as they are not constrained by their P&P facility and can complete problems outside their P&P facility).

Overall, these very stable beliefs demonstrate that students generally viewed CAS favourably and recognised the importance of learning to use CAS, and potential uses for CAS, in mathematics. However, beliefs that P&P needs to be used CAS suggest a preference for working with P&P rather than CAS.

‘Mostly stable’ beliefs

Table 6 provides the nine beliefs categorised as ‘mostly stable’. The modal response pattern for five beliefs was HH (items 10, 12, 13, 14 and 15) and H′H′ for the remaining four (items 11, 20, 21 and 25). Overall, this indicates that many students were stable in their beliefs. However, it is important to note that the modal response pattern is less than half the class for items 13, 15 and 20, so there was a greater range in students’ beliefs for these items. For these items, the variation resulted from differences in whether students were stable in holding (or not holding) the belief and does not indicate that these beliefs were less likely to be stable than the other beliefs in this category.

Table 6 Response patterns for beliefs categorised as ‘mostly stable’
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The categorisation of the belief that CAS is useful for answering problems correctly (item 10) as ‘mostly stable’ is a contrast to the other two beliefs about usefulness (items 7 and 9), which were categorised as ‘very stable’. This difference may be explained by students who did not believe that Answers are correct when using CAS (item 20), or I can get more answers correct using CAS than P&P (item 21). It could be that either (i) students who do not believe they can obtain correct answers do not believe that CAS is useful for checking answers or (ii) students who are more assured that they can obtain correct answers with P&P than CAS do not feel that they need to check their answers using CAS. Ng et al. (2005) reported that perceptions of the usefulness of CAS increased with experience. Consequently, we expected some students who did not find CAS useful at the start would change their belief by the end (H′H). For item 10, we observed the opposite change, with four students holding the belief at the start but not the end. We anticipate this change is a result of increased P&P facility; hence, students had a reduced reliance on CAS for solving problems correctly.

Beliefs about mathematics, including whether CAS is considered ‘real mathematics’, can impact CAS use (Pierce & Stacey, 2001). The idea of CAS being ‘real mathematics’ was investigated through items related to what should be recorded as working out and whether CAS use is acceptable in MM. Items 11, 12, 13, 14 and 15 relate to this idea. Responses of HH for items 12, 13 and 14 indicate a perceived need to show P&P working. Using CAS requires changes to written solutions (Ball & Stacey, 2003) as CAS can ‘gobble up’ (Flynn & Asp, 2002) intermediate steps of working. Students holding these beliefs may not have developed ways to record working when CAS has been used. Students responding H′H′ or HH′ for items 13 or 14 may have developed strategies to record working when CAS has been used. Together, these beliefs may explain the prevalence of the ‘very stable’ belief that P&P needs to be used before CAS (item 16) as students may have felt that P&P should be used before CAS because P&P working needs to be recorded. However, it is noteworthy that changes in beliefs about the need to show P&P working (i.e. those who responded H′H or HH′) did not result in changes to students’ belief that P&P needs to be used before CAS; there may be other beliefs or norms that are underpinning students’ belief that P&P should be used before CAS.

Overall, these ‘mostly stable’ beliefs largely relate to the role of CAS and P&P in the classroom. The stability of such beliefs is not surprising, given that students have many years of experience working in a classroom where P&P mathematics is expected.

‘Inconclusive’ beliefs

Table 7 shows the six beliefs categorised as inconclusive; i.e., the number of students with stable and unstable beliefs was equal. In contrast to beliefs in the previous categories, the modal response patterns for four of the six beliefs indicate instability. Students’ beliefs were more varied than in other groups, which is a result of half the response patterns indicating instability for these items. This variation contrasts with beliefs in the ‘mostly stable’ category, where variation could be attributed to differences in whether students were stable in holding (or not holding) the beliefs.

Table 7 Response patterns for beliefs categorised as ‘inconclusive’
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The categorisation of the belief Mathematics is not being done properly if CAS is used (item 18) as ‘inconclusive’ contrasts with other beliefs about the role of CAS in mathematics, which were categorised as ‘mostly stable’. Holding this belief at the start could reflect students’ prior experience of mathematics where there was an expectation of P&P use; hence, CAS was not seen as a valid option. Responses of HH′ indicate six students came to incorporate CAS into their mathematical routines, as we would expect when learning to use CAS. It is possible that coming to view CAS as a valid option for mathematics accounts for some of the changes observed in beliefs about the need to show P&P working (e.g. Item 12). One student did not view CAS as a valid option (item 18; HH); this student responded HH for all items related to P&P working, so their view that CAS is not valid mathematics corresponds to their beliefs about P&P working. We assume students who responded HH for the belief There is no need to use CAS for a problem that can be solved with P&P (item 17) have strong P&P skills, so there is no need to use CAS to check answers. Students not holding the belief may use CAS to check answers after using P&P. This belief was unstable for six students; HH′ may reflect an increased understanding of potential uses for CAS and its affordances. Those responding H′H may have developed a preference for P&P or became more discerning about CAS use as they developed further understanding of the affordances of CAS or P&P for solving problems.

The categorisation of the belief Most algebra problems can be solved faster with CAS than P&P (item 8) as ‘inconclusive’ contrasts with the belief Most algebra problems can be solved faster with P&P than CAS (11) which was ‘mostly stable’. These items involved a comparison of CAS to P&P, so a reversed pair was used to determine which belief was held. Holding either belief indicates that students had made some determination about the relative speed of CAS or P&P for solving problems; hence, we would not expect a student to hold both beliefs concurrently. Two of the three who responded HH for Most algebra problems can be solved faster with CAS than P&P (item 8) also responded H′H′ for item 11, but one responded H′H. In contrast, of the seven who responded H′H′ for item 11, only two gave the expected HH to item 8. Another two responded H′H′ to both items, so they did not judge the relative speed of CAS or P&P. These students may have recognised that the speed of CAS or P&P depends on the problem or procedure, so neither is faster most of the time. Responding HH′ for Most algebra problems can be solved faster with CAS than P&P (item 8) suggests students moved from believing CAS was faster than P&P to not, potentially because they recognised the relative speed of CAS or P&P is context-dependent. Development of P&P facility may also impact this belief, as in the end, students may have increased their ability to perform P&P procedures quickly.

Categorisation of the beliefs It is easier to solve problems with CAS than P&P (item 22) and It is hard to learn how to solve problems with CAS (item 27) as ‘inconclusive’ suggests students have different perceptions about how easy it is to learn how to solve problems with CAS. Beliefs about ease of use were included as some students find CAS hard to use (Ng et al., 2003), which impacts their ability to use CAS. In addition to expectations around technology use, MM students are expected to have strong P&P skills (VCAA, 2015), so students need to learn to solve problems with and without technology. Consequently, it was unsurprising that many did not hold the belief It is hard to learn how to solve problems with P&P (item 25), which was categorised as ‘mostly stable’. Learning to use CAS can take up to 1 year (Weigand & Bichler, 2010); as students were new to using CAS, we expected responses of HH′ for It is hard to learn how to solve problems with CAS (item 27). This change occurred for five students, but another four still held this belief at the end. One obstacle that students may encounter when using CAS is the inability to recognise when and how CAS can be used (Drijvers, 2002). Students encountering this obstacle might find it hard to learn how to solve problems with CAS because they do not have a sufficient understanding of how CAS can be used; hence, developing an understanding of when and how to use CAS would support students’ capacity to use CAS.

Overall, it is not possible to determine whether stability is a feature of these beliefs as equal numbers of students provided stable or unstable response patterns. Where changes did occur, it is anticipated that they resulted from increased CAS facility and an increased understanding that CAS is valid for mathematics. Further research could explore beliefs over a longer period of time to determine whether these beliefs trend towards stability or instability as students gain further experience with CAS.

Conclusions and implications

We expected students to form beliefs about CAS across the year as they were new to learning mathematics with CAS. In contrast to our expectation, the tendency for students to stably hold beliefs across the study indicated that beliefs were formed prior to, or within 8 weeks of, commencing learning mathematics with CAS. This finding contrasts with existing literature which describes beliefs as being formed over a long period (e.g. McLeod, 1992). If students’ beliefs were formed prior to commencing Year 11 MM, the beliefs may be inferential in nature (Leder et al., 2002) with their experiences with other technologies (e.g. computer-based mathematical software, smartphone apps, scientific or graphical calculators) or pen-and-paper informing their beliefs about CAS. Consequently, it would be important for teachers to monitor and support students to develop positive beliefs about technology throughout their study of mathematics, so they commence Year 11 MM holding beliefs that would positively influence their CAS use. Alternatively, if beliefs form within weeks of commencing working with CAS, it would be important for teachers to encourage students to view CAS positively when introducing CAS. For example, beliefs about usefulness of CAS would positively influence CAS use, so demonstrating CAS features and discussing affordances may support the development of these beliefs.

Within one Year 11 class, we might expect students to hold similar beliefs due to the teacher’s influence and shared classroom experience (Leder et al., 2002); hence, the diversity in beliefs was unexpected. This finding may indicate that many of the beliefs reported in this study are descriptive (Leder et al., 2002) and formed through students’ experiences rather than advice from their teachers or peers. The literature review showed little is known about students’ beliefs about CAS and how they change, even though attitude has been found to become more positive as students gain experience with CAS (e.g. Orellana, 2016). Identifying the stability of students’ beliefs about CAS builds on literature highlighting the importance of supporting a positive attitude towards CAS (Pierce & Stacey, 2004a) by providing a new level of specificity with which teachers can consider, and support, attitudes towards CAS.

The categorisation of beliefs as ‘very stable’, ‘mostly stable’ or ‘inconclusive’ is a new contribution to the literature on students’ beliefs about CAS. The finding that 16 of the 22 beliefs were ‘very stable’ or ‘mostly stable’, with none being ‘mostly unstable’ or ‘very unstable’, shows that stability (rather than instability) is a feature of students’ beliefs about CAS. However, as noted by Liljedahl et al. (2012), this does not that students’ beliefs cannot change as there were several instances of students holding a belief at the start, but not the end, or the reverse. Attitude has been shown to become more positive as students gain experience with CAS (Orellana, 2016; Pierce & Stacey, 2001), so the stability reported here may suggest that changes in students’ attitudes are driven by emotions, or emotions and behaviour, depending on the definition of attitude being considered.

The identification of ‘very stable’ and ‘mostly stable’ beliefs provides a contribution that will support teachers and researchers in considering how to best support CAS use. For example, ‘very stable’ beliefs, such as CAS can be used whenever students like (item 24), would positively support students’ CAS use as students make greater use of CAS in classrooms where CAS use is not restricted (Kendal & Stacey, 2001). These beliefs may have formed because of the classroom norms institutionalised by the teacher. Future research could identify classroom experiences or teaching practices that supported students to hold such beliefs, so they may be replicated in other settings. In contrast, the very stable belief P&P needs to be used before CAS (item 16) could negatively influence CAS due to preferencing of P&P (Pierce & Stacey, 2004a). For students who are learning in a system where technology is expected to be used, there is an imperative for teaching to incorporate technology and for students to learn to work with technology. Consequently, this belief may be a candidate for a targeted intervention to change the belief.

‘Mostly stable’ and ‘inconclusive’ beliefs did change for some students, which is consistent with Liljedahl et al. (2012) conclusion that all beliefs, including those described as stable, can change. Given some of these changes may positively support CAS use, for example, students who held the belief Answers are correct when using CAS (item 20) at the end but not the start, further research could identify teaching practices that contributed to this change. Such research could then enable teachers to develop targeted interventions to support more students in changing their beliefs. Related to this is the possible relationship between the specific beliefs reported here. For example, beliefs about CAS is proper mathematics may impact beliefs about usefulness of CAS as students who do not consider CAS a valid option may not find it useful. Such understanding could facilitate teaching interventions that support beliefs and emotions that positively influence CAS use.