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Beliefs and their influence: A mathematical perspective

Matthew Davidson
Curtin University of Technology
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"Maths class is tough!" 'Teen Talk' talking Barbie doll
(introduced February 1992, recalled October 1992).


Background to the project

Mathematics has been the key factor in human advancement. It is fundamental to our way of life and instinctively it has become part of the essence of human nature and used intuitively everyday from intricate to mundane tasks. Yet it brings to mind thoughts of fear, failure and disdain. It is an area which we regard as complex and to be used by only those biologically equipped, that is with superior 'brain power', to wield it at a high intellectual level. How can a so highly crucial discipline become the colloquial 'black sheep' of the learning area family? It is the intention of this paper to identify the effects, if any, of beliefs upon the learning of mathematics. In doing so it uncovers underlying factors, discover solutions and poses further questions on why "maths class is tough!"

The identification of the effects, both positive and negative, will allow for the understanding of how to adjust or formulate teaching approaches in an attempt to improve student learning in mathematics as well as their enjoyment and application of mathematics. Quintessentially the project concentrates on students' beliefs about and towards mathematics and their effects upon students' learning of mathematics.

This research projects focus area is of significance as it helps to identify the beliefs students possess toward mathematics and provides suggestions for educators to help develop positive beliefs and hinder the negative ones. Furthermore, by doing so, teachers will be able to make adjustments to their curriculum to enable it to cater more effectively to the needs of their students. As an extended benefit of this, curriculum writers will be able to produce a more appropriate curriculum to aid its implementers. This can assist the wider community as it will help to remove the stigma associated with mathematics thus enabling more accurate and positive beliefs towards mathematics.

The research question

How do students' beliefs affect their learning in the domain of mathematics? In an attempt to more clearly understand the purpose of the research the subsidiary questions that present themselves as a result of the project focus have been explored further. These expanded formations clarify and provide contextual meaning to individual aspects of the original question.

Literature review

Previous studies have indicated that students have generally positive beliefs about mathematics with a large consensus believing they are competent, or good, at mathematics, with no significant difference between the gender types (Vanayan, White, Yuen, & Teper, 1997; Wilkins & Ma, 2003; Bong, 2004). Specific areas of mathematics had stronger positive responses from students. Areas such as addition and multiplication are associated with the belief that these areas are easier and more purposeful (Vanayan, White, Yuen, & Teper, 1997). A considerable discovery in the research of House (2006) is that self-perceptions of ability, i.e. the belief in one's own ability, as opposed to beliefs about mathematics itself was a greater indicator of success in learning mathematics. This then addresses the root of students' belief - what do students believe makes them successful at mathematics. Carter and Norwood's (1997) research exposed that students believe that doing better than their peers indicates success as opposed to actual achievement, interest and/or enthusiasm. A fascinating revelation by House (2006) showed that students who attributed hard work and study outside of school with success also achieved greater results than their peers. Equally, those who attributed success with natural talent achieved lower mathematics scores than their peers. House's (2006) research went on to show that this applies in cross-cultural settings, demonstrating the research's universal application.

The concept of student beliefs was investigated further by Bong (2004). Here beliefs were looked at as motivational constructs i.e. as causes of related outcomes - positive beliefs equal higher achievement, negative beliefs equal lower achievement. Bong concluded that beliefs as motivational constructs provided a legitimate theory on how beliefs influenced success. Success would motivate students, this motivation would cause students to postulate beliefs about succeeding and thus the beliefs would influence and lead to further achievement. It can be surmised then that student beliefs are more often egocentric in nature, relating to their own abilities, strengths, weakness and personal enjoyment in contrast to beliefs about specific learning areas or education itself.

It can be surmised then that student beliefs do in fact have significant impact upon their learning but have limited impact upon student ability. According to Macnab & Payne (2003), the learning process, i.e. the construction of students' knowledge and understanding, is hindered more prominently by student self-perception of their own abilities than any other factors. House notes in the findings of his studies that Japanese students and United States students have different beliefs in terms of learning but the abilities of the students, when compared, differ almost insignificantly. This correlates with the work of Wilkins and Ma (2003). In their research they came to find that mathematical ability was determined from outside influences, such as that of actual curricula, learning environment, and parental and educator factors. Student belief influenced the way students learned, but evidence from the research of Macnab and Payne (2003) deduced that student ability is due to teachers catering to the beliefs and needs of the students. To synthesise this point more critically it can be said that student ability is not determined by their beliefs directly - their ability is a result of environmental factors and the way in which they learn. The learning process itself is influenced by the beliefs students hold i.e. the confidence in their own abilities, which is built upon students' understanding of the relevance and usefulness of mathematics.

Given this information are we to conclude that students' beliefs in themselves is the determining cause of their success in any academic endeavour? Barlow and Reddish's (2006) research reminds us that there are limitations to this self-perception factor citing the need to identify teaching practices as well as students' concepts of what mathematics actually is, in order to establish the extent to which students' beliefs manipulate their success. Barlow's research also found that student beliefs about mathematics itself were strongly allied with Kogelman and Warren's (1978) twelve mathematical myths. These beliefs appeared to be of social origin and unrelated to actual observable evidences. Many authors acquiesce that these beliefs are detrimental and impede the ability to educate students. The beliefs are hindering the learning, not weakening the ability.

An important commonality of the literature is the reference to student beliefs regarding their perceptions about the value of mathematics in their daily lives. According to Higgins (1997), when students become embittered toward mathematics it is due to their inability to see the relevance and usefulness of mathematics. When mathematical problems are posed as real and authentic problems, invoked from everyday situations and coupled with appropriate language then students begin to appreciate mathematics and acknowledge it as the foundation of all science and technology (Higgins, 1997). House (2006) strengthens this argument with the findings of his research, concluding that the perceived value of mathematics contributes to students belief in their own ability, citing this from an allusion in the research data that this may be due to students future aspirations. They believe they are good at something because they believe that it is a requirement for their supposed future endeavours.

It is reasonable to infer, given the findings of the relevant and previous studies, that beliefs toward mathematics have a formidable influence on students' abilities in mathematics. It has been proposed by some researchers, such as Macnab & Payne (2003), that these beliefs are the singular influence on ability. However this is disputed by Barlow and Reddish (2006) who found that social surroundings, or environmental factors, as well as beliefs, influence ability. It is apparent across all the research that beliefs influence ability, thus this research seeks to uncover what the specific effects of the beliefs are in an effort to find any correlation between specific beliefs and academic effects.

Research methodology

This research took place after a successful application to the Curtin University Human Research Ethics Committee. In conjunction with this, all necessary documents that outline how to conduct an ethical piece of research were referred to and research was carried out using their given guidelines. In addition, the highest level of confidentiality has been used at all times. Pseudonyms have been used to protect the identity of participants and the school involved. No evidence that may indirectly identify individuals has been used in the report. Any personal information has and is only be accessible by the researcher or, if required, by the Academic Supervisor of the researcher.

Participation in this study was completely voluntary. If participants wished to withdraw themselves they were at complete liberty to do so at any time. There were, and remain, no risks to participants in this study.

All participants, and their guardians, were given the researcher's details so that they are able to contact them if they wished. They were also given the contact details of the research supervisor so they were able to contact a neutral party if they had any concerns or felt uncomfortable approaching the researcher.

This project utilised an action research approach. Thus the data has been collected in the following ways:

For the purposes of this research the ability levels were defined as, and based upon the following in relation to mathematics: All participants are early childhood students - Year One. The school is located in a low socio-economic area though this is not reflected in the academic achievement of the school.

The interviews took place individually - the setting being a place removed from their peers to reduce peer influences. "Why do we clap when others clap, eat as others eat, believe what others believe? Frequently it is to avoid rejection or gain social approval" (Myers, 2004). Hence removing the students from their peers. To begin each participant was given a simple explanation of the purpose of the research and thanked in advance for their participation. Each was outlined on the style of questions to be asked and informed that honest responses were vital. The interviews were read back to the participants to confirm the responses had not been misinterpreted and to provide the opportunity to clarify any points.

Each interview underwent qualitative data reduction and thus was analysed independently of any other allowing the any recurring themes to be identified and listed in significance (a point was considered significant if it was a repeated idea during the interview and/or if it was unusual). After this method had been applied to each interview, the results of each were brought together in the hope of isolating the significant and common themes. With this done, several points were identified as recurring and significant to the research.

Findings

It is prudent to return to the research question in order to examine the findings of the research.

How do students' beliefs affect their learning in the domain of mathematics?

Thus, after objective analysis of the data the following themes were identified which were in direct relation to the project focus.

Students' understanding of the purpose and relevance of mathematics: The perception of what mathematics is.

Figure 1

Figure 1: Perception of mathematics

Self-perception of mathematical ability

Enjoyment of mathematics in the classroom and degree of difficulty

Figure 2

Figure 2: Perception of difficulty

Figure 3

Figure 3: Difficulty versus abilty

Table 1: Abbreviated summary of interviews

QuestionParticipant HigherParticipant AverageParticipant Lower
What flavour do you think maths would be?YuckyChocolateStrawberries
Why would it taste like that?"Because maths is bad""Because maths is really lovely but a lot of it makes me sick""I like maths but sometimes it's a bit hard and sometimes I like strawberries but sometimes you get a gross one"
Are you good at maths?YesYesNo
What parts are you good at?Counting and adding (refers to doubling - common in younger in children)Can count to 100 and adding upAdding up
Why are you good at them?Practices at homeIs able to complete maths tasks without parents.Adding is the participant's favourite aspect because they can do it independently.
Is maths hard or easy?EasyEasyHard
What makes it hard/easy?PracticeQuiet surroundings make it easier and practice.It is designed to be hard in order to encourage practice.
Where do you use maths outside of school?Participant stated that maths was only used at school except when she had to practice at home.When playing a sport (referenced cricket)
*Difficult for participant to answer.*
Participant stated that they only used maths at school but others used it in sports and "probably other stuff."
Tell me what you think maths is?"Maths is all about numbers and you do it at school to help you learn your numbers.""Learning about numbers and letters and rhyming."
*Further investigation uncovered 'rhyming' to be a reference to the multiplication tables.
"It's where you write stuff down that the teacher puts on the whiteboard... like how to do it... but I think it's numbers and words."
Do you like maths at school?No
*Stated that she enjoyed it at home.
YesNo
Where does your family use maths at home or in their jobs?Refers to parents cooking.Refers to parents writing cheques and reading recipes.Siblings use maths in homework and parents use it when going to dinner at a restaurant.

Discussion

How do students' beliefs affect their learning in the domain of mathematics? They do so in numerous ways. Outlined below are the three major areas that highlight how the beliefs are influencing the learning process.

Enjoyment

The findings of this study concur with, and support, the findings of House (2006), Vanayan, White, Yuen, & Teper (1997), Wilkins & Ma (2003) and Bong (2004) as the research data indicates students are more likely to enjoy mathematics if they are successful in mathematics. This is supported in the inverse as Participant L, who has had limited success, does not enjoy mathematics. It should be noted that each participant stated they enjoyed the aspects of mathematics that they achieved success in. But is this enjoyment because of their success or do they enjoy it and therefore succeed? Further research into this area is necessary but it can be surmised that, in either case, enjoyment is the counterpart to success. Continuing from this, the research identified that the method in which participants were taught is influential to the extent that they enjoy mathematics - the educators' beliefs are being imparted upon the students and thus become the students' beliefs. Participant H stated that she enjoyed mathematics practice at home but not a school and Participant L referred to mathematics being taught in a traditional rote learning style. This 'traditional' method of teaching has previously been highlighted as influential in creating negative attitudes towards any learning area, but particularly mathematics (House, 2006). All participants indicated through anecdotal conversation that they would enjoy mathematics much more if they were made aware of its purpose, thus we can assume that enjoyment is linked directly to purpose.

Purpose

Interestingly none of the participants could easily provide an example of a use for mathematics outside of school yet all of them indicated during the interviews that mathematical skills were valuable. Furthermore, when asked whether their families used mathematics their responses were limited to references of cooking and recipes which can lead to the conclusion that they are not aware of the broad application of mathematics and the basic and fundamental mathematics that is innate in everyday life. Higgins (1997) observed that students' failure to witness the significance and value of mathematics in everyday life become cynical of it. When discussing mathematics as a taste Participant H referred to it as yucky, while Participants A and L referred to is as something sweet. rationalising that too much is bad for you. It can thus be concluded that as the participants are referring to mathematics negatively they are failing to see its relevance and importance. Yet are they failing to see it because of the manner in which they are taught, their cognitive development, or the curriculum? It is indeed possible that it is one, or a combination, of these or other undefined possibilities. It appears that as these participants could not easily reference any real or authentic everyday mathematical situations that they are not engaged in a curriculum or teaching style that highlights the purpose of mathematics. Though it must be taken into account the age of the students - being of a younger age they are very egocentric in nature - as this may account for the lack of understanding of the purpose of mathematics.

Participant A stated that 'rhyming' was an aspect of mathematics that she found difficult. After this was deciphered to mean the multiplication table (learned in class in a chant style e.g. One 2 is two, two 2s are four, three 2s are six, etc) the participant stated that they did not recognise the relevance of learning numbers in this format. Again it appears that the curriculum and/or the teacher are segregating purpose from curricula.

Self perception

In relation to Participant L's limited success, and thus lack of enjoyment, of mathematics, this child indicated that he did not believe he lacked ability but instead that mathematics had been designed to be difficult. Yet all participants attributed success with practice, a finding supported by the research of Barlow and Reddish (2006). Participants also revealed that they considered themselves to be successful in mathematics if they could work independently of their peers - success was not related to achievement, interest and/or enthusiasm. This correlates with the works of Carter and Norwood (1997) as the ability to work independently outwardly displays success to peers - thus it is a desired ability and sought after more that actual academic success.

The research also lends support to Macnab and Payne's (2003) conclusions that students' ability is significantly influenced by their perception of their abilities. Participant H considered himself extremely capable, Participant A considered himself capable, and Participant L considered himself capable, but not to the extent of his peers - each of which is true. It can be supposed then, given the data, that self-image becomes reality and is a governing factor of ability. The extent to which belief in abilities actually influences abilities is still yet to be determined. Bong (2004) concluded that it accounted for the major part of achieving success and although the number of participants was limited, this research lends credibility to these findings.

To further validate the findings of this research it would be beneficial to undertake it again on a larger scale.

Recommendations

This paper recommends the following implementations and strategies in the following areas.

Students understanding of the purpose and relevance of mathematics: The perception of what mathematics is

  1. Move away from traditional mathematical dogma, the current mathematical initiatives supported and outlined by the documents of the Curriculum Framework (1998), use the terms; explore, investigate, conjecture, solve, justify, represent, formulate, discover, construct, verify, explain, predict, develop, describe, use. These terms show that mathematics should not be 'rigid', but open to exploration.
  2. Ensure that all mathematics which is taught is 'real' mathematics. Real mathematics can be generically stated to be contextual mathematics which is applicable to, and usable in, real life.
  3. Integrate mathematics into other learning areas as opposed to teaching and assessing it on its own. For example, if mathematics was combined with a Society and Environment lesson through map use, the mathematics aspect could be assessed as part of that lesson. This is an example of how to show students that mathematics is applicable to everyday life and situations.

Self-perception of mathematical ability

  1. Not only do students need to be confident in their ability to get the answers right, they need to be confident enough to get an answer wrong and not feel ashamed or embarrassed. As mathematics is a process, they are bound to uncover incorrect answers from time to time, but it does not mean that they are inept in their abilities. Explaining to students that everyone is fallible will help remove negative self-perceptions.
  2. Minimise 'direct competition': as mathematics is a progressive learning area, where students move from stage to stage, some students will move faster than others. Bong's (2004) research indicates that students openly and actively compete with one another. When a student feels they cannot achieve the same level as another student they began to give up and produce negative perceptions of their ability. Competition can be reduced by celebrating all students' achievements and by stopping indirect punishment (i.e. when a student finishes early they are rewarded - the student who requires more time feels they are being punished or missing out, reducing their motivation).

Enjoyment of mathematics in the classroom and degree of difficulty

  1. Positive role model - if the teacher shows enthusiasm for the mathematics learning area it is likely spread to the students who will inturn pour this enthusiasm into their work.
  2. Puzzles and challenges - by setting up problems that arouse the curiosity of students they are likely to become motivated to learn to be able to solve said problems. This becomes intrinsic motivation with a personal sense of achievement.
  3. It is likely that they feel preordained to dislike mathematics (Marsh, 2004), to endure a necessary evil - this is a cultural influence. Thus, to counteract this, all mathematics which students enjoy must be made explicit and obvious to ensure students are aware that they are able to enjoy mathematics.
  4. The primary teacher, when interviewed, noted some common threads that reappear year after year through interviews with parents. Many parents indicate that they believe 'maths is just about numbers', that you must 'have a good memory to be good at maths', maths has a 'mystical quality' to it, and because they had bad experiences their 'children should expect the same.' These beliefs are passed on. It is therefore recommended that parents are invited by educators into the classroom to witness real mathematics.

Recommendations for future studies

Deliberately excluded from the research as they were not within the bounds of the review were students' attitudes, a primary focus on gender beliefs, teaching methodology (though this was identified as a result of the research), areas outside of mathematics, the effect of the home environment, and the differences in age/year level, beliefs and achievement. Therefore future studies would benefit from implementing the recommendations of the paper and assessing their impact, by researching any of the exclusions of this research listed above and by undertaking this same research question across a larger sample across multiple age groups.

References

Barlow, A. T., & Reddish, J. M. (2006). Mathematical myths: Teacher candidates' beliefs and the implications for teacher educators. The Teacher Educator, 145-157.

Bong, M. (2004). Academic motivation in self-efficacy, task value, achievement goal orientations, and attributional beliefs. The Journal of Educational Research, 287-297.

Carter, G., & Norwood, K. S. (1997). The relationship between teacher and student beliefs about mathematics. School Science and Mathematics, 62-67.

Higgins, K. M. (1997). The effect of year-long instruction in mathematical problem solving on middle-school students' attitudes, beliefs, and abilities. The Journal of Experimental Education, 5-28.

House, J. D. (2006). Mathematics beliefs and achievement of elementary school students in Japan and the United States: Results from the third international mathematics and science study. The Journal of Genetic Psychology, 31-45.

Kogelman, S. & Warren, J. (1978). Mind over math. New York: McGraw Hill.

Leedy, G., LaLonde, D., & Runk, K. (2003). Gender equity in mathematics: Beliefs of students, parents, and teachers. School Science and Mathematics, 285-292.

Macnab, D. S., & Payne, F. (2003). Beliefs, attitudes and practices in mathematics teaching: Perceptions of Scottish primary student teachers. Journal of Education for Teaching, 55-68.

Myers, D. G. (2004) Exploring psychology. Worth Publishers.

Vanayan, M., White, N., Yuen, P., & Teper, M. (1997). Beliefs and attitudes toward mathematics among third- and fifth-grade students: A descriptive study. School Science and Mathematics, 345-351.

Wilkins, J. L., & Ma, X. (2003). Modeling change in student attitude toward and beliefs about mathematics. The Journal of Educational Research, 52-63.

Appendix

Action research interview questions

Please cite as: Davidson, M. (2008). Beliefs and their influence: A mathematical perspective. Proceedings Western Australian Institute for Educational Research Forum 2008. http://www.waier.org.au/forums/2008/davidson.html


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