> ajbjb,,6NNZ////L/ZMF0:0000222LLLLLLL,]ORQ|L22@222L?700L?7?7?7200L?7|2L?7?7FG~0a/m5bGGL0MvG\+R?7+R(G?7ZZD/ZZ/The influence of affective variables on students transition to university mathematics
Department of Mathematics and Statistics, University of Limerick, Ireland.
HYPERLINK "mailto:miriam.liston@ul.ie" miriam.liston@ul.ie john.odonoghue@ul.ie
Making the transition to university is a challenging hurdle for most first year students, both personally and academically. This investigation is focussed on the influence of affective variables on students in first year university mathematics in Ireland. The majority of these students are traditional students making the transition from secondary school to university with a small percentage (9%) classified as mature students returning to education from work. Questionnaires were distributed to three groups of students on first year service mathematics programmes (degree courses where mathematics plays a part in the students studies but may not be the main focus) at the University of Limerick at the beginning of the university academic year 06/07. Their attitudes, beliefs, self-concept, conceptions of mathematics and approaches to learning are examined. The impact these concepts, as well as gender and the level of mathematics studied at secondary school, have on performance are also discussed.
introduction
Atkin and Helms (1993) suggest that affective components are as important as the content itself. McLeod (1992) divides affect into three dimensions: attitudes, beliefs and emotions. These affective dimensions along with others such as mathematical self-concept, self-efficacy, confidence, mathematical conceptions and approaches to learning play a major role in students mathematics learning. These affective factors, particularly negative ones, are linked with mathematics. They are becoming so common a factor amongst the least able pupils in secondary schools that we are in danger of assuming that it will inevitably be present (Larcombe, 1985: 6).
There is no question that there is a distinctive gap between secondary and university mathematics and according to Kayander and Lovric (2005: 149), the transition in mathematics is by far the most serious and the most problematic. There are many factors that contribute to this gap but in relation to the conceptual gap, Hoyles et al. (2001: 833) identified three main problem areas between school and university mathematics: lack of mathematical thinking (i.e. the ability to think abstractly or logically and to do proofs), weak calculation competence and the students lack of spirit i.e. lack of motivation and perseverance. Indeed these problems arise at secondary school and are often transferred to university. Ramsden (1992) has reported that studying and learning approaches at university level are influenced by learning and practices at secondary school.
* The majority of the sample (91%) is made up of students making the transition from secondary school mathematics to university mathematics. The rest of the sample (9%) consists of Mature/Adult Learners (over the age of 23) returning to third level education from work.
Research has also shown that students lack basic concepts, knowledge and skills needed at university level (e.g. Kayander and Lovric, 2005) and in Ireland even those with good Leaving Certificate grades (final secondary school examination) struggle with even the more basic aspects of mathematics (National Council for Curriculum and Assessments (NCCA), 2006). These conceptions and approaches, together with the attitudes, beliefs and mathematical self-concept have an onward effect on their university mathematics.
Little or no research has been done in Ireland on the influence of affective variables in this transition to first year university mathematics. In this paper the authors address the issues, briefly outlined above, in an Irish context. The importance of mathematics education in Ireland is reflected in the NCCAs (2006: 6) statement that it is significant in the development of logical thinking and problem-solving skills, as well as its importance as a foundation for other subjects, especially the science and technology subjects. It is of particular importance at senior cycle education given its requirement for admission to third-level courses. The report makes reference to the issue of attitudes to, and beliefs about, mathematics concluding that many students and adults lack confidence in dealing with mathematical issues and processes. As well as the affective domain, the author is concerned with students understanding of mathematical concepts and the approach they adopt when learning. There is due cause for concern in Ireland. Worrying statistics were unveiled after the school Leaving Certificate examination results were announced on Wednesday 15th August 2007. According to the Irish Times, the national paper of record, close to 5,000 students failed mathematics at either Ordinary, Higher or Foundation level, making them ineligible for third level courses. There are three levels of mathematics in the Irish exam system with the highest level referred to as Higher, a lower level referred to as Ordinary and the lowest level that can be taken is called Foundation. The steep decline of numbers taking Higher level mathematics makes for anxious times and will have a knock-on effect on third level education.
role of affective factors in learning
The authors focus on five areas throughout this paper, which they believe through the literature and own research, impact strongly on the transition to university mathematics. Neale (1969: 632) defined attitude to mathematics as an aggregated measure of a liking or disliking of mathematics, a tendency to engage in or avoid mathematical activities, a belief that one is good or bad at mathematics, and a belief that mathematics is useful or useless. There is considerable disagreement among the literature on the correlation between attitudes and achievement. Not all researchers agree on whether the correlation is a strong one or not. For example, Reynolds and Walberg (1992) showed that attitudes towards maths are a predictor of academic performance in that subject, while Fraser and Butts (1982) found little correlation between achievement and attitudes. Despite this lack of agreement, attitudes towards mathematics play an important role in learning and the author believes the students enjoyment and value of the subject strongly impacts on their desire to study it.
Beliefs are one of the three affective domains highlighted by McLeod (1992: 579). He says, Beliefs are central in the development of attitudinal and emotional responses to mathematics. Students beliefs about mathematics differ but Daskalogianni and Simpson (2001: 98) identified two types of beliefs that are relatively constant: 1) systematic where students view mathematics as a methodical subject with exact and definite answers, 2) utilitarian where students view mathematics as being a subject with a direct application to real life and other topics. The author is of the opinion that students with the first type of belief or perception will struggle at university mathematics in comparison to those who can make applications and see the relevance of the mathematics they are studying.
Reyes (1984) defined mathematics self-concept as perceptions of personal ability to learn and perform tasks in mathematics. Bandura (1997) believes a persons self-concept in mathematics is a reflection of their belief in their own ability in mathematics and is formed by past experiences. Gourgey (1982) includes in that definition the persons belief in their ability to perform in situations involving mathematics. The students confidence to do maths can determine if the university mathematics experience will be a positive or negative one.
A factor that often widens the gap in the transition is ones conceptions of mathematics. Conceptions of mathematics investigate students deep understanding of mathematics. Thompson (1992) identifies conceptions of the nature of mathematics as conscious or subconscious beliefs, concepts, meanings, rules, mental images, and preferences concerning the discipline of mathematics. Students approaches to learning are also an important aspect of the transition to university and research suggests there is a link between the two. For example, a study by Crawford et al. (1998) found students conceptions of mathematics influences their approach to learning. Similarly Anthony (2000) says students conceptions of learning have an onward effect on the way they approach their studies and in turn affects the quality of their learning. The type of approach to learning that students adopt is a strong deciding factor on whether students transition to university is successful or not. Marton and Saljos (1976) work focussed on students approaches to learning and they identified two processes, deep-level and surface-level learning. In surface approach learning, the main focus is reproduction of knowledge. Deep-level learning on the other hand aims for comprehension. A correlation between deep approaches to learning and what Crawford et al. (1998) refer to as cohesive conceptions of mathematics as well as a correlation between surface approaches to learning and fragmented conceptions of mathematics is analysed and discussed in this paper. Crawford et al. (1998) describe fragmented statements as focusing on parts rather than wholes while cohesive statements concentrate on the whole picture rather than just constitute parts. The students ability to understand the concepts in mathematics rather than simply carrying out calculations often determines their approach to studying and thus what type of outcome is achieved.
methodology
Development of the Research Instrument
A questionnaire consisting of 78 statements based on attitudinal scales was designed and implemented. Attitudes to maths, beliefs about maths, mathematical self-concept, conceptions of mathematics and approaches to learning were measured. The literature on each of these areas was examined as well as previous studies that employed some of the scales e.g. Miller-Reilly (2005) and Wood and Smith (1993). The following scales were deemed appropriate for the study to investigate the aforementioned areas of concern in an Irish context: Aikens (1974) Two Scales of Attitude Towards Mathematics (Enjoyment and Value)(EM and VM), Schoenfelds (1989) Beliefs about Mathematics (BM), Gourgeys (1982) Mathematical Self-Concept (MSC), Crawford et al.s (1998) Conceptions of Mathematics, which is divided into ten fragmented statements (FCM) and nine cohesive statements (CCM) and Biggs et al.s (2001) Revised two-Factor Study Process Questionnaire, which is also divided into ten deep approach statements (DA) and ten surface approach statements (SA). It should be noted that due to the length of the overall questionnaire a 12-item subset of the original Mathematical Self-Concept 27-item scale was selected, including six positive and six negative statements. A study by Miller-Reilly (2005) found high correlation (0.94) between the scores on the 12-item subset and the full length scale indicating that the subset explained 82% of the variability of the original scale. Thus the author chose to use the smaller subset scale. Similarly studies by both Miller-Reilly (2005) and previously Wood and Smith (1993) found that six items from part of Schoenfelds (1989) instrument investigated students beliefs or perceptions about mathematics. Again the length of the questionnaire was a deciding factor in choosing the six item scale. According to Osborn (2004) there are important issues that occur in comparative research, three of which may be relevant to this study: conceptual equivalence, equivalence of measurement and linguistic equivalence. The author believes the conceptual definitions investigated in this study to have equivalent meaning with the cultures where each of the scales were developed i.e. USA, Australia. English is the first language in these countries as well as in Ireland so no difficulties were envisaged in the concepts in question or indeed in relation to linguistic equivalence. In relation to equivalence of measurement, all scales were measured in accordance with strategies employed by the authors of the scales.
Piloting the Research Instrument
The questionnaire was piloted in May 2005 with six Leaving Certificate students and six secondary school mathematics teachers. The purpose of this was to ensure wording and length were appropriate. The author spoke to each participant following their completion of the questionnaire, discussing the content, wording of questions and the length of the questionnaire. Changes were made to the structure of the questionnaire as due to its length it had appeared off putting to some participants in the pilot study.
Final Research Instrument
The questionnaire was divided into 3 sections. The first section contained demographic information about the student e.g. Leaving Certificate grade and level, etc. In Section A, there are 58 statements examining attitude, beliefs, self-concept and conceptions of maths. Statements were intermingled so students would not notice a pattern developing. Section B consisted of 20 statements looked at students approaches to learning. Students responded to both Section A and Section B in accordance with a Likert scale indicated above the items. For Section A, strong feelings could be indicated on either side of the scale and there was an option for respondents who were unsure of statements (i.e. 1= strongly disagree, 2= disagree, 3=unsure, 4=agree, 5=strongly agree). The Likert Scale for Section B of the instrument consisted of 1 = never or only rarely true of me, 2 = sometimes true of me, 3 = true of me about half the time, 4 = frequently true of me, 5 = always or almost always true of me.
Data Collection
Following the pilot study, the questionnaire was distributed to the three 1st year groups studying service mathematics courses (Science, Engineering and Technological Mathematics 1). The researcher was allocated lecture time with each of the three groups to administer and return the questionnaires as well as to inform students of the purpose and anonymity of the study. Students were reminded that completing the questionnaire was voluntary.
Research Sample
The research sample consisted of students making the transition to university mathematics and who are studying service mathematics courses at the University of Limerick. It included those who may not be making a direct transition to university e.g. mature students (those over the age of 23). Three modules were chosen: Engineering Mathematics 1, Science Mathematics 1 and Technological Maths 1. The authors chose to focus on students from Science, Engineering and Technology (SET) disciplines due to the increasing dependency within Ireland on these areas of the economy. The large sample size and various groups within the sample allowed for much diversity in ability.
Response Rate
Each of these modules is made up of students from a number of different programmes. That is, there are ten different degree programmes within Engineering maths 1, nine within Science maths 1 and 13 within Technological mathematics 1. The response rate was analysed for each of the three service mathematics groups. It was found that Science mathematics 1 had the lowest response rate with 156 out of 289 questionnaires returned (54%). This was followed by Engineering mathematics 1 students where 139 out of a possible 181 responded (77%) and 312 from 390 questionnaires returned (80% response rate) from the largest group Technological mathematics 1.
Data Analysis
The data resulting from the questionnaire at third-level was analysed using the statistical package of SPSS (Version 15). The reliability of each of the scales was analysed using Cronbachs Alpha scores. While the scales used in the questionnaire are those tested in the literature, a confirmatory factor analysis was carried out on all items of each scale and is discussed below. Descriptive statistics revealed the mean and standard deviation for all items. Further, more in depth analysis of the data looked at paired-samples t tests, correlations (Pearsons), and regressions. The results are now outlined.
results
To describe the reliability of each of the scales used in this study, the values of Cronbachs Alpha were calculated. The Cronbach alpha values for the following scales indicated very good reliability (greater than 0.8): EM (.91), MSC (.81), CCM (.83) and DA (.81). Three of the scales had lower Cronbachs alpha scores (0.6-0.8) but are considered acceptable: VM (.76), FCM (.60) and SA (.76). The Beliefs about Mathematics scale showed low reliability (.43) and results were interpreted more cautiously. Confirmatory factor analyses were performed on all items in the questionnaire to see if the items in each scale were loading onto same factor. It confirmed that almost all items, particularly on the scales with strong Cronbachs alpha scores, were loading onto the same factor. The factor analysis on the Beliefs about Mathematics scale confirmed that two items were showing up in other factors thus possibly contributing to the low reliability for the data mentioned above.
Preliminary Analysis of the Scales
Table 1 displays the scales used in the study along with sample items and their mean and standard deviation. As mentioned earlier, respondents were asked to indicate their level of agreement or disagreement with each item: 1= SD, 2= D, 3=U, 4=A, 5=SA. Scoring on negatively worded items was reversed (i.e. 1= SA, 2= A, 3=U, 4= D, 5= SD) on the following scales: Attitude to Mathematics, Beliefs about Mathematics and Mathematical Self-Concept to allow total scores to be calculated. Thus a high score would indicate a favourable attitude, belief or mathematical self-concept. Scoring on the following items was reversed: Items 8-11 on EM scale, items 2 and 8-10 on VM scale, items 1-6 on MSC scale and items 1 and 3-6 on BM scale. For the Conceptions of Mathematics scale and Approaches to Learning scale, scoring on negatively worded items i.e. fragmented conceptions and surface approach statements, was not reversed. This is in accordance with analysis done by the authors, Crawford et al. (1998) and Biggs et al. (2001) respectively.
ScaleNo. of itemsSample ItemMean(SD)Attitude Enjoyment of mathematics (EM)11Item 5: I would like to develop my mathematical skills and study this subject more.3.3(1.12)Attitude Value of Mathematics (VM)10Item 2: Mathematics is not important for the advancement of civilisation and society.4.2(1.01)Mathematical Self-Concept (MSC)12Item 2: If I can understand a maths problem, then it must be an easy one.3.5(1.10)Beliefs about mathematics (BM)6Item 5: To solve maths problems you have to be taught the right procedure, or you cannot do anything.2.9(1.20)Fragmented Conception of mathematics (FCM)10Item 1: For me, mathematics is the study of numbers.2.6(1.11)Cohesive Conception of mathematics (CCM)9Item 8: Mathematics is a logical system which helps explain the things around us.3.7(.86)Deep Approach to Learning (DA)10Item 17: I come to most classes with questions in mind that I want answering.2.6(1.10)Surface Approach to Learning (SA)10Item 20: The way to pass examinations is to try to remember answers to likely questions.2.7(1.22)Table 1 Scales and Sample Items with their mean and standard deviation
Total scores were calculated for all scales with the exception of the conception of mathematics scale where mean scores were used. This is consistent with the analysis conducted by the researchers who created each of the five scales used in this study. The first area of concern is the attitude of students. The mean score obtained was 37.6(SD=8.92). The V scale consists of 10 statements to E scales 11. The mean score on the V scale was 39.3(SD=5.34). The next area of concern was the students beliefs about mathematics using Schoenfelds (1989) scale. The mean score was 20.7(SD=3.27) out of a possible high score of 30. The mean for the MSC scale for the sample was 40.6(SD=7.04).
Crawford et al.s (1998) Conceptions of mathematics scale was incorporated into the questionnaire to determine if students are either fragmented or cohesive learners. It was found that the mean for the CCM scale (3.5(SD= .59)) was slightly higher than the mean for the FCM scale (3.3(SD= .41)). A paired sample t test was carried out to test if the score on the fragmented scale was statistically different from that of the cohesive scale (Ho = there is no difference between the scales, Ha = there is a difference between the two scales). By observing the means discussed above it can be seen that respondents scored more highly on cohesive scale than the fragmented scale. The test was significant (p< 0.01) so we therefore reject the null hypothesis and can confirm that there is a difference between the two scales. In relation to Biggs et al.s (2001) two-factor Study Process Questionnaire, the higher the score on the deep learning scale the better as this indicates a positive response to the deep approach statements and suggests that students favour comprehension rather than reproduction of knowledge. High surface scale scores however would show students were surface learners and aimed for reproduction of knowledge rather than aiming to understand the information. The mean for the DA scale was 29.8(SD=6.77) in comparison to the SA scale mean of 24.3(SD=6.32). A paired samples t test was again significant (p< 0.01) allowing the authors to conclude that the students scored significantly higher in the DA scale than SA scale. When the author examined subscales of the scale however there is evidence of rote-learning and procedural knowledge. For example, students scored a mean of 13.4(SD=3.67) out of possible 25 on the strategy for surface approach learning scale indicating that rote learning is a strategy employed by surface learners in this sample.
Further Analysis (Inferential Statistics)
Correlations between affective variables and final semester exam marks.
The researchers wished to observe if each of the affective variables correlate with mathematical performance. Table 2 below displays the correlations between each of the affective variables and in particular looks at the strength of the correlations between semester one results and the affective factors. The data revealed a positive, although not very strong, statistically significant relationship between EM and final semester one marks (r =.24) and MSC and semester one exam results (r = .22)
SEM 1EMVMMSCCCMFCMDASASEM 11.24**.07.22**.01-.08-.00-.03EM.24**1.57**.73**.42**-.16**.29**-.31**VM.07.57**1.46**.62*-.10*.27**-.23**MSC.22**.73**.46**1.35**-.24**.17**-.19**CCM.01.42**.62**.35**1-.04.32**-.15**FCM-.09-.16**-.10*-.24**-.041-.00.14**DA-.01.29**.27**.18**.32**-.011-.25**SA-.03-.31**-.23**-.19**-.15**.14**-.25**1
** Correlation is significant at the 0.01 level (2-tailed).
* Correlation is significant at the 0.05 level (2-tailed).Table 2: Correlations between affective variables and final semester exam results.
A negative statistically significant relationship (r= -.09) was found between fragmented conceptions of mathematics and semester one results. That is, the higher the fragmented score the poorer the exam result. All other correlations between semester one results and affective variables were found to be non-significant. As expected, each of the affective variables is correlated with one another.
What variables can be used to predict exam results in this sample? (Multiple Regression)
The authors were also interested in how much of the variation in semester one results can be accounted for by the predictor variables. Since the affective variables are correlated they were not combined in the multiple regression. Rather the authors wished to examine the extent to which affective factors predict exam results in conjunction with gender and the level at which each student studied maths to Leaving Certificate level e.g. EM and gender and level. Multiple regressions using stepwise method was used. This creates a regression equation using only the predictor variables that make a significant contribution to the prediction. Results showed that when EM scores and gender and level were put into the equation, level and EM were entered with the other variable, gender, removed. Together EM and Leaving Certificate level account for 13.8% (R= .138) of variance in semester one results. Similar results were found for the multiple regression including MSC scores with gender and level. MSC and level account for 13.6% (R= .136) of the variance in semester one results. Again gender was not included. In the remaining multiple regressions, only the level of Leaving Cert. maths entered into the equation with all other variables removed indicating that they had little or no effect together on exam results at the end of the semester.
The relationship between students conceptions of mathematics and their approaches to learning.
As discussed earlier, the literature predicts a relationship between cohesive conceptions of mathematics and a deep approach to learning and fragmented conceptions of mathematics and a surface approach to learning (Crawford et al., 1998). The authors wished to test if the same was true for this sample. As the conception scale scores are mean scores, the mean scores for the surface and deep approach to learning were calculated and used so as the weighting would be the same on all variables. Pearsons correlation was used. The relationship between cohesive conception of mathematics and deep approach to learning is a statistically significant (p<0.01), positive (r= .32) one. While it is a somewhat weak relationship it is still a positive one suggesting that those with high deep approach scores tend to score highly on the cohesive scale. The correlation between fragmented conceptions and surface approaches to learning was again statistically significant (r = .14, p< 0.01). The relationship was weaker than that of cohesive conceptions and deep approaches to learning but a positive significant one all the same.
discussion and conclusion
Each of the scales discussed above showed good to very good reliability with the exception of the BM scale. Miller-Reilly (2005) in her study found that the item Real maths problems can be solved by commonsense instead of the maths rules you learn in school was difficult to interpret. This item, along with item 5 of the scale did not load onto the same factor as the other items, which may explain its low reliability. In addition, the BM scale used was originally part of a much larger instrument developed by Schoenfeld (1989) and adapted for use by both Miller-Reilly (2005) and Wood and Smith (1993).
Analysis of the questionnaire displays some results that are in agreement with the literature, particularly the relationship between conceptions of learning and approaches to learning. Findings from initial analysis of the scales were quite positive. Aikens (1974) Two Scales of Attitude Toward Mathematics comprises two parts: enjoyment of mathematics (E scale) and value of mathematics (V scale). According to Aiken (1974: 70) the E scale is more highly related to measures of mathematical ability and interest He claims, The V scale is more highly correlated with measures of verbal and general-scholastic ability. Results indicate that while both mean scores were relatively high across the sample, students tend to value mathematics more so than they enjoy it. For BM scale the spread of scores may be attributed to the prior experience the students have had. This often determines how students behave e.g. the amount of time and effort students are willing to invest in a mathematics problem (Schoenfeld, 1989). The statement, to solve maths problems you have to be taught the right procedure, or you cannot do anything, had the lowest mean on this scale (2.9(SD=1.20)). This response would seem to indicate that students consider procedural knowledge to be important, a situation described as problematic by researchers such as Marton and Saljo (1976). The MSC scale revealed positive scores, which were higher than originally anticipated. The Conceptions of Mathematics scale showed that the students tended to lean towards cohesive learners (however, slightly). Similarly on the Approaches to Learning scale, students scored significantly more highly on the DA to learning scale than the SA to learning scale. These are positive findings although somewhat surprising given other findings by Crawford et al. (1998) in Australia and The Chief Examiner Report (2005) from an Irish context where procedural knowledge, rote-learning and general surface approaches to learning are recognised areas for concern.
The statistically significant positive correlation between the affective variables EM, MSC and semester one results indicates that the more highly respondents scored on these scales the better they did in the final exam. A negative significant relationship between the fragmented conception of mathematics and semester one results was not surprising, however one may have expected the relationship to be stronger. It was somewhat unexpected that no relationship exists between the students approaches to learning and their final semester mark. One could speculate that the reason for this is that university exams at do not test deep, conceptual understanding and surface approaches to learning are sufficient for students to pass their mathematics exams. A certain level of procedural knowledge is part of mathematics and as the NCCAs Review of Mathematics in Post-Primary Education Report on the Consultation (2006) explains, some respondents pointed to the benefits of developing a reasonable level of procedural skills, as distinct from rote learning, so that students can call on these skills when solving problems. Perhaps the focus of the teaching and indeed exams must be reconsidered.
The multiple regressions proved interesting and revealed that the level at which students had previously studied mathematics at Leaving Certificate was the most important predictor of exam results. Enjoyment of maths and mathematical self-concept, were the strongest affective predictors of exam results together with the level of maths previously studied. Gender was not a predictor of exam results in any case. There are varying theories in the literature on whether gender predicts performance according to Evans (2000). She reports West el al.s (1986) study found gender not to be a predicator of performance and persistence, whereas Everett and Robins (1991) indicated that females dominated males in performance.
As mentioned earlier, this is a novel study in Ireland. Based on studies carried out by researchers in other countries, and by the authors thus far, it is clear that attitudes, beliefs, emotions, mathematical self-concept, conceptions of mathematics and approaches to learning mathematics are crucial areas in mathematics education and needs attention in an Irish context. While the findings have not been all negative, both literature and particularly studies by Gill (2006) have shown the struggle students endure in Service Mathematics courses. Recent discussion by mathematics education researchers has highlighted the need for a range of methodological instruments to truly interpret students mathematical behaviour and to help build a suitable framework for the study of affect in maths education (Zan et al., 2006). This study is currently being further investigated through the use of qualitative data involving interviews with randomly selected students from each of the three mathematics modules discussed throughout and who have previously completed the questionnaire. Further research aims also to assess the implications of affective variables on pre-service teacher groups in comparison to their peers, which are important if we are to set about changing the perceptions of mathematics teachers in Ireland.
Acknowledgements
This research is funded by the Mathematics Applications Consortium for Science and Industry (MACSI), through Science Foundation Ireland (SFI).
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