Tilahun Mengesha Afrassa and

John P. Keeves

The Flinders University of South Australia

Over the past 30 years IEA has conducted three international studies of mathematics at the 13-year-old level. In the three studies, in addition to mathematics achievement tests, students were required to respond to an attitude questionnaire concerning their attitudes towards mathematics, the learning of mathematics and school learning and a general information questionnaire about themselves and their mathematics schoolwork. Australia is one of the countries which participated in all the three IEA studies. In this paper path analysis techniques are employed to examine student level factors that influence mathematics achievement of the 1994 Year 8 Australian students (Third IEA Mathematics Study) by using the PLSPATH 3.01 computer program. Conclusions are drawn about the student level factors that influence the mathematics achievement of the 1994 Year 8 Australian students and comparisons are made with the student level factors influencing mathematics achievement at the 13-year-old level in 1964 and 1978.

Previous research studies in Australia have indicated that there are substantial differences between students in their achievement levels in school mathematics, and have identified some of the student level factors that influence mathematics achievement. Keeves (1972), Dungan and Thurlow (1989) and Milne (1992) have argued that students' attitudes towards mathematics were among the student level factors influencing mathematics achievement in Australian schools. Furthermore, these researchers contend that students who express more positive attitudes towards mathematics are likely to achieve at a higher level in mathematics than those students who express less positive attitudes towards mathematics.

Another student level factor that was found to influence the mathematics achievement of Australian students was home background. Keeves (1968, 1972), Rosier (1980) and Ainley et al. (1990) have reported that socioeconomic status of the family influenced the student's school performance. That is, students from higher socioeconomic status homes were likely to perform at a higher level than students from lower socioeconomic status homes.

Keeves (1968, 1972, 1976), Rosier (1980), Bourke (1984, 1986) and Leach and Tunnecliffe (1984) reported that the time allowed and the opportunity given to students to learn mathematics were both factors that influenced the achievement of Australian students in school mathematics.

Research findings (Newman, 1976, 1979, 1983; Morris, 1978; Turner, 1980; Dawe, 1983; Ellerton and Clements, 1990) have found that language is one of the factors that influences achievement in school mathematics.

However, it has been argued by Ellerton and Clements (1989, 1991) that among all school subjects, the least affected subject by linguistic and cultural considerations is mathematics. It can be considered to be a culture-free subject (Ellerton and Clements, 1989, 1991).

The findings of research in Australia have suggested that all students whose first language was not the medium of instruction were educationally disadvantaged. Nevertheless, students who were competent both in their first language and in English which was the medium of instruction had some advantage in learning mathematics over those students who were competent in only one of these languages. Both groups had a clear advantage over students who had lower levels of competence in both languages (Dawe, 1983; Clarkson, 1991, 1992, 1993). Therefore, the level of competence which students had in each language was a vital factor which needed to be considered (Clarkson, 1992).

Causal models of student level factors influencing mathematics achievement of students at the lower secondary school level are required to examine the network of hypothesised interrelationships between variables that are considered important as a result of a theoretical framework and previous research findings. The present study investigates the type and size of interrelationships between specific latent variables and their effects on mathematics achievement and whether these relationships are consistent over time. If the same variables indicated relatively stable effects over time in Australia, they could be taken as evidence for the generality of the model and the coherence of the theoretical framework under investigation. Thus the purpose of this study is to develop a theoretical model of student level factors influencing the mathematics achievements of lower secondary school students in Australia and to examine these hypothesised interrelationships between variables.

From the findings of previous research a model of student level factors influencing mathematics achievement at the the Year 8 level was developed and PLSPATH (Sellin, 1990) was chosen as an appropriate multivariate technique to examine the hypothesised model. The first part of this paper discusses the results obtained when the hypothesised model was tested by employing PLSPATH in the analysis of data from the Third IEA Mathematics Study. The second part of the paper compares the student level factors that influence the mathematics achievement of students on different occasions.

Table 1 shows the Target Populations of the three international studies conducted in Australia under the auspices of the International Association for the Evaluation of Educational Achievement (IEA). In the First IEA Mathematics Study (FIMS), conducted in 1964, two groups of students participated, 13-year-old students in Years 7, 8 and 9 and students in Year 8 of schooling. The total number of students taking part was 4320 (see Table 1).

In the first study only government schools in New South Wales (NSW), Victoria (VIC), Queensland (QLD), Western Australia (WA) and Tasmania (TAS) participated. In the Second IEA Mathematics Study (SIMS), which was administered in 1978, nongovernment schools and the Australian Capital Territory (ACT) and South Australia (SA) were also involved as well as those states that participated in FIMS. Thus in SIMS government and nongovernment school students in six states and one territory were involved. The total number of participants was 5120 students (see Table 1).

Table 1:- Target populations in FIMS, SIMS and TIMS

SRS = Simple-Random-Sample
PPS = Probability-proportional -to-size
R = Restricted to government schools in five states
G = The participant schools were only government schools

In the Third IEA Mathematics Study (TIMS), which was conducted in 1994, government and nongovernment school students in all states and territories including Northern Territory were involved. The total number of students tested in Australia was 7,392 (see Table 1).

In 1964 and 1978 the samples were age samples and included students from Years 7, 8 and 9 in all participating states and territory, while in TIMS the samples were grade samples drawn from Years 7 and 8 or Years 8 and 9. In ACT, NSW, VIC and TAS, Years 7 and 8 students were selected while in QLD, SA, WA and NT samples were drawn from Years 8 and 9.

Therefore, to make the most meaningful possible comparison of mathematics achievement over time by using the 1964, 1978 and 1994 data sets, the following steps were taken. The 1964 students were divided into two groups, 13-year-old students in one group and all Year 8 students including 13-year-old students at that year level as the second group, since in addition to an age sample a grade sample had also been drawn. It is important to observe that 13-year-old students in Year 8 were considered as members of both groups. In the first group, students were chosen for their age and in the second group for their year level. The 1978 students were chosen as an age sample and included students from both government or nongovernment schools.

Meanwhile, in TIMS the only common sample for all states and territories was Year 8 students. Thus only Year 8 students in all states and territories are considered as the TIMS data set in this study.

After excluding schools and the states and territories that did not participate in the 1964 study, two sub-populations of students were identified for comparison between occasions. The two groups were 13-year-old students in FIMSA and SIMS: all were 13-year-old students and were distributed across Years 7, 8 and 9 on both occasions. Hence, these two groups of 13-year-old students were considered to be comparable for the examination of changes in factors influencing achievement over time, between 1964 and 1978. Whereas for the comparison between FIMS and TIMS the other sub-populations consisted of 1964 and 1994 Year 8 students. Students in both groups were at the same year level, although there were differences in the ages between these groups which were tested on the two occasions. Hence, the comparisons in this study are between 13-year-old students in FIMSA and SIMS on the one hand, and FIMSB and TIMS Year 8 students on the other.

In all the three IEA Mathematics Studies students were requested to respond to the mathematics tests and questionnaires. Furthermore, general information about the students' background was collected from those students who participated in the studies using a Student Questionnaire. Three types of information were collected from the students. The first type of information was about themselves, such as their sex, race, age, whether they were born in Australia, how often they spoke English at home and the language their parents mostly spoke at home. The second type of information was about their schools and the learning of mathematics. This information included number of hours spent on different activities in a week, number of hours they gave to mathematics homework, number of hours they devoted to all homework, and about their liking of mathematics as well as their mathematics test results. The third type of information collected from the students was about their home background, such as their father's occupation and education, mother's occupation and education, number of brothers and sisters, number of books at home, father's and mother's country of birth, and languages most often used in their homes.

The purpose of the questionnaires was to obtain background information about each student in order to develop variables that would help to explain differences between students in their achievement in mathematics. Therefore, the three types of information which were collected from the students are considered in the following sections in order to examine which variables might explain differences between students in their mathematics achievement at the 13-year-old and Year 8 levels in Australian lower secondary schools.

In this study the partial least squares path analysis procedure was employed to identify the student level factors that influenced mathematics achievement. Partial least square path (PLSPATH) analysis "is a general technique for estimating path models involving latent constructs indirectly observed by multiple indicators" (Sellin, 1992, p. 398). This procedure is useful in modelling educational and social systems. Thus PLSPATH analysis can be employed as a method of analysing path models which involve latent (indirectly observed) and manifest (directly observed) variables. The PLSPATH model includes an inner model which specifies the hypothesised relationships among the latent variables (LVs) and an outer model that specifies the relationships between the LVs and the manifest variables (MVs) which are their indicators (Sellin, 1992).

PLSPATH identifies the optimal linear relationships between variables and provides estimates of the parameters of the model. Jacobs (1991), Bukowski, Hoza and Boivin, (1993) have argued that PLSPATH is an ideal procedure, because it provides an index of the adequacy of the model, shows the strength of each individual path in the model, and examines the direct and indirect relationships among variables. Kotte (1992) also argued that PLSPATH can be employed efficiently to identify strong and weak relationships between latent and manifest variables. Furthermore, the indices of overall adequacy show whether or not the model produces an accurate representation of relationships among variables in the model. In a causal model certain variables are singled out as causes and others as effects. Consequently, the strengths of the particular paths in the model show how strongly the linked variables are causally related to each other. PLSPATH provides a number of advantages which are most appropriate for this study. It is useful for displaying graphically the pattern of causal relationships among sets of observed and unobserved variables that influence the mathematics achievement level of students. PLSPATH is technically simple, quick and does not require strict distributional assumptions. For these and other reasons PLSPATH is referred to as a soft approach to modelling ( Falk, 1987; Sellin, 1989, 1990).

It is beyond the scope of this study to detail the mathematical and technical aspects of PLSPATH. However, references such as Noonan and Wold (1988), Sellin (1989), Cheung and Keeves, (1990), Falk and Miller (1992) and Sellin and Keeves (1997) are suggested for further reading. Because PLSPATH employs a least square regression procedure in analysis it does not require that variables are normally distributed for the analysis. However, this computer program retreats from significance testing because in so many data sets the assumptions of normality are violated, or the degree of departure from normality can not be determined and because the samples employed can not be considered to be simple random samples.

The starting point in employing PLSPATH is to draw a diagram of the model to be analysed (Falk, 1987). The diagram should include both the outer and the inner models and the hypothesised links between them. In this study theoretical models of student factors that influenced mathematics achievement of Australian students on different occasions were developed prior to analysis. The models specified the variables included in the analyses and their interrelationships were hypothesised. In the path diagram, the MVs or observed variables formed the outer model, while the LVs formed the inner model. The number of MVs and LVs generated by this study for TIMS data set are shown in Figure 1. In total, 25 MVs and 10 LVs were employed in the outer and inner models respectively. The acronyms chosen for the MVs and LVs were intended to reflect their item content (see Table 2). The reader must keep in mind that these acronyms are employed throughout this paper. A list of the MVs employed in this study together with answer categories and coding are given in Appendix 1.

Figure 1 shows the outer model relationships of the hypothesised student level factors thought to influence the achievement in mathematics of students in Australia. There were 25 hypothesised MVs . In developing the outer path model the investigator has the choice between two modes of weight estimation, called outward mode and inward mode (Sellin, 1992). Sellin states that the outward mode indicators assume that the MVs reflect the corresponding latent construct. An example of outward mode is the set of motivation and attitude towards mathematics scales which are assumed to reflect the underlying motivation and attitude of each student. Inward indicators, assume that MVs form or produce a latent construct as presented in Figure 1.

Table 2: Latent and manifest variables employed in the path models for FIMS, SIMS and TIMS data sets

PLSPATH employs a factor analytic procedure for estimating the outward latent constructs. The criterion for the minimum level of a factor loading for the inclusion of a MV in this study was chosen to be 0.30 (Campbell, 1996). However, other researchers have proposed other criteria such as 0.55 (Falk and Miller, 1992), or 0.40 (Harman, 1976; Pedhazur, 1982; Keeves, 1991). MVs with loadings below the predetermined cutting point were dropped from further analysis. PLSPATH uses a regression procedure to calculate the weights of the MVs which form a LV in the inward mode, and weights below 0.07 were removed from further analysis as they indicated that the observed variable did not contribute greatly to the related LVs (Sellin and Keeves, 1997; Sobolewski and Doran, 1996). In order to obtain a robust model, the original model was successively refined to include only significant paths.

In Figure 1 the outward mode is depicted by the arrows pointing from the LVs to their respective MVs, whereas the inward mode is shown by the arrows pointing from the MVs to the corresponding LVs. The figure shows the MVs and the corresponding LVs and the modes specified to estimate the constructs. In order to increase the predictive power of the path model, Sellin (1992) has suggested that researchers should employ the inward mode for exogenous variables and the outward mode for endogenous variables. However, there are situations where this rule would not apply. In accordance with Sellin's (1992) suggestion, the exogenous variables in this study were defined with inward estimation, while the endogenous variables except Time in Learning, which employed the inward mode, all were defined with outward estimation including those constructs that consisted of only one MV. In this latter case the loadings became unity (1.0) (see Figure 1).

The inner model presented in Figure 1 shows the causal links between the LVs. The positions of the LVs in the inner model are based on theoretical considerations. Therefore, in this study of student level factors influencing student achievement in mathematics in Australia, the findings of previous studies were employed as a starting point to hypothesise the causal links between constructs in the inner model (see Figure 1). In determining the final structure of the inner model, it was recommended that direct paths with ß<0.07 should be removed, because such values show an insignificant effect in the estimation of a relationship between two LVs. Hence, the larger the ß value the larger the effect in the path model. This estimation process was repeated successively until all nonsignificant paths had been removed.

The other criterion used to assess the strength of the final path model was the maximum variance explained (R2) of the outcome variable, Mathematics Achievement (Mathachi). The value of R2 gives the maximum variance explained of a construct when the preceding predictor variables are included in the analysis. Thus, the larger R2, the more of the variance is explained.

The modification, trimming or deletion of variables and paths in the path model involves the removing of all paths not contributing to the LVs. The deletion or the removal of paths includes both the outer and inner model. As a result of the trimming procedure some manifest and latent variables were also removed from further analysis.

Tables 2 and 3 show the outer model and the inner model results for the PLSPATH analysis for the TIMS Year 8 students data set. Ten LVs and 25 MVs were included in the model. The results of the PLSPATH analyses are discussed in the following two parts. The first part addresses the results of the outer model and the second part considers the inner model.

In the following discussion the weights and the factor loadings, and the communality, redundancies and the tolerances values of each MV within a construct are discussed with respect to the LV to which that MV contributes. The weights (ßs) are initially considered significant if their values are ß > 0.07, while the factor loadings (ls) are initially significant if l > 0.30. The index employed for measuring the strength of the outer model is the average of the communalities of the MVs (Falk, 1987). Furthermore Falk has argued that the higher the average of the communalities the better the outer model, and an average value of the communalities of 0.30 would generally be considered too low.

Table 2 shows that Homeback is reflected by six MVs, namely Focc, Mocc, Fed, Med, Homebook and Siblings. In the hypothetical model presented in Figure 1, it was assumed to be in an inward mode. However, in the analyses, it was changed to an outward mode. Since this LV had as many as six observed or MVs the outward mode was chosen to avoid problems of multicollinearity. The loadings for five of the six MVs reflecting this antecedent construct, were 0.69, 0.59, 0.77, 0.74 and 0.58 respectively. Siblings was deleted from the model, because the factor loading was -0.06 and this value was below the critical value of 0.30. The important point here is that, in SIMS data sets the MV Siblings contributed to reflect Homeback. However, it did not reflect the same LV in TIMS. This suggested that the drop in the number of children per family and the greater uniformity in family size in 1994, when compared with 1978 in Australia has led to a decline in the importance of this factor. It is of interest to observe that both Fed and Med contribute more to the formation of the LV Homeback than do the other three variables. However, all were highly significant for the development of this construct. The communalities recorded in Table 3 show that the five MVs contributed to this construct.

======================================================================
Variable        Weight/Loading   Communality  Redundancy   Tolerance 
--------------------------------------------------------------------
Homebacko    
	Focc            .69               .48       .00        .25
	Mocc            .59               .35       .00        .21
	Fed             .77               .59       .00        .48
	Med             .74               .54       .00        .47
	Homebook        .58               .33       .00        .09
	Siblings          Deleted
Genderu
	Sex             Deleted by default
Studageu
	Age             Deleted by default
Ethnicityi
	Cntry           Deleted by default
	Fcntry          Deleted by default
	Mcntry          Deleted by default
	Enghome         Deleted by default
Classizeu
	Clssize          1.00              1.00      .02       .00
Viewsu
	Studpart         1.00              1.00      .007      .00
Motivationo
	Motiv1           .75               .57       .02       .35
	Motiv2           .44               .19       .007      .11
	Motiv3           .83               .70       .02       .37
	Motiv4           .79               .63       .02       .24
Timlearni
	Homworkf         .82               .72       .03       .02
	Hourmhw          .27               .19       .009      .03
	Hourmath         .42               .19       .009      .01
Attitudeo
	Mathmrk          .83               .69       .18       .33
	Diffmath         .64               .40       .11       .21
	Likmath          .85               .73       .19       .26
Mathachiu
        Rasch score     1.00              1.00       .34       .00
------------------------------------------------------------------
MEAN COMMUNALITY                    0.57                           

====================================================================


i = inward mode

o = outward mode

u = unity mode

Table 4: TIMS-Year-8 Students- Inner Model Betas
=================================================================
Variable            Beta     Correlation   Tolerance    R2   
------------------------------------------------------------
 
Classize                                               .03
  Homeback          .16         .16             .00   
Views                                                  .01
   Classize         -.08        -.08             .00 
Motivation                                             .04
  Homeback           .11         .11             .0005  
   Views             .15         .15             .0005  
Timlearn                                                .05
   Homeback          .17         .18             .01   
   Views            -.09        -.08             .02  
   Motivation        .09         .10             .03   
Attitude                                                .26
   Homeback          .09         .14             .01   
   Views             .09         .16             .02   
   Motivation        .48         .50             .03   
Mathachi                                                .34
   Homeback          .23         .35             .06   
   Classize          .20         .32             .07   
   Views            -.12        -.11             .04  
   Timlearn          .21         .34             .08   
   Attitude          .30         .36             .06  
-----------------------------------------------------------
MEAN R2                                                 0.12
============================================================

Six LVs were hypothesised as factors that influenced students' Motivation towards mathematics. In Table 5 only three of the hypothesised factors are shown to influence this construct either directly, indirectly, or in both ways. Only Homeback, Classize and Views influenced Motivation (see Table 5 and Figure 2), and the total effects of these variables were 0.11, -0.01 and 0.15 for Homeback, Classize and Views respectively. Homeback influenced Motivation both directly (0.11) and indirectly (-0.002), while, Views (0.15) showed only a direct effect on this construct. However, Classize (-0.01) influenced this construct only indirectly and to a very slight extent. The variance explained for this construct was 0.04. In general, the findings can be summarised as follows:

1. students from higher socioeconomic status backgrounds expressed stronger Motivation towards mathematics than students from lower socioeconomic backgrounds; and
2. students who expressed stronger Views about mathematics also expressed stronger Motivation towards mathematics.

Time in Learning (Timlearn)

Seven LVs were hypothesised to influence this construct. However, the result of the analysis revealed that only Homeback (direct effect = 0.17, indirect effect = 0.01, total effect = 0.18) and Views (direct effect = -0.09, indirect effect =0.01, total effect = -0.07) showed direct and indirect effects while, Motivation (0.09) showed only a direct effect on this construct (see Table 5 and Figure 2). Homeback indicated that students from higher socioeconomic status backgrounds were likely to spend more time in learning mathematics than students from lower socioeconomic status backgrounds. Furthermore, Motivation also influenced this LV directly. Students who indicated stronger Motivation towards mathematics were likely to spend more time on mathematics than students who expressed less Motivation towards mathematics. The effect of Views on Timlearn is negative. This indicates that students who showed less participation in pair or group work in mathematics were likely to spend more time in learning mathematics than those students who participated more in group work. Meanwhile, Classize showed a very small indirect effect (0.006) acting through the mediating variables Views and Motivation (see Table 5). The remaining LVs did not show any influence on this LV. The R2 (0.05) value for this construct was small.

Table 5: TIMS - Year 8 students - Inner Model Statistics
===============================================================================
 Variable         Direct    Indirect     Total    Correlation    Fit     R2
-------------------------------------------------------------------------------
Classize                                                                 .03
  Homeback        .16           -         .16         .16          -
Views                                                                    .01
  Homeback          -         -.01          -.01        -.02        -.01
  Classize      -.08          -         -.08        -.08          
Motivation                                                               .04
  Homeback       .11        -.002        .11         .11          -
  Classize        -         -.01        -.01         .06         .05
  Views          .15          -          .15         .15          -
Timlearn                                                                 .05
  Homeback       .17         .01         .18         .18          -
  Classize        -          .006        .006        .21         .18
  Views         -.09         .01        -.07        -.08          -
  Motivation     .09          -          .09         .10          -
Attitude                                                                 .26
  Homeback       .09         .05         .14         .14          -
  Classize        -         -.01        -.01         .09         .08
  Views          .09         .07         .16         .16          -
  Motivation     .48          -          .48         .50          -
Mathachi                                                                 .34
  Homeback       .23         .11         .35         .35          -
  Classize       .20         .007        .21         .32          -
  Views         -.12         .03        -.08        -.11          -
  Motivation      -          .16         .16         .15        -.04
  Timlearn       .21          -          .21         .34          -
  Attitude       .30          -          .30         .36          -
-------------------------------------------------------------------------
Mean R2                                                                   .12
===========================================================================

Eight LVs were hypothesised to influence students Attitudes towards Mathematics. The result of the PLSPATH analysis demonstrated that four of the hypothesised LVs showed direct and/or indirect effects on this construct. Home Background and Views about Mathematics showed both direct and indirect effects on Attitudes towards Mathematics. The total effects of Homeback and Views were 0.14 and 0.16 respectively (see Table 5 and Figure 2). Classize showed only an indirect effect of -0.01. However, Motivation towards mathematics showed a direct effect (0.48) on Attitudes towards Mathematics and the effect of this LV was much stronger than the other two variables which showed both direct and indirect effects. Thus, students who expressed stronger motivation towards mathematics, students from higher socioeconomic status backgrounds and those students who indicated more positive views about mathematics expressed more positive attitudes towards mathematics than students who expressed weaker motivation, students from lower socioeconomic backgrounds and students who expressed weaker views about mathematics. The important point here was that Gender did not have any effect on Attitudes towards Mathematics when other variables were taken into account. Previous Australian research findings have revealed that boys had more positive attitudes towards mathematics than did girls (Keeves, 1972; Fraser, 1980; Schofield, 1981; Ballenden et al., 1985). However, the findings here did not support these previous findings. In previous analyses in which other variables are taken into account, namely 1964, 13-year-old students, 1964 Year 8 students and 1978 13-year-old students, data analyses involving both 13-year-old in 1964 and 1978 and Year 8 students in 1964 indicated that boys expressed more positive attitudes towards mathematics than girls. However, in 1994, there was no gender effect either direct or indirectly on Attitudes towards Mathematics. Thus the effects of Gender, would appear to have changed over time in respect to attitudes towards the learning of mathematics.

Students' level of mathematics achievement was hypothesised to be influenced by nine LVs. The result of the PLS analysis revealed that three of the nine factors influenced Mathematics Achievement both directly and indirectly, while one other factor influenced Mathematics Achievement indirectly (see Tables 4 and 5 and Figure 2) and two factors showed only a direct effect. The factors that did not show any effect on the outcome variable were Gender, Studage and Ethnicity and since these variables did not enter the model in any way, they do not appear in the final analyses presented. The five factors that had a direct influence on Mathematics Achievement were Homeback, Classize, Views, Timlearn and Attitude.

The five factors that had a direct influence on Mathematics Achievement are discussed in greater detail as follows.

Home Background (Homeback). This LV influenced the mathematics achievement level of Year 8 students directly (0.23) and also with a sizeable indirect effect (0.11). The total effect of this variable on Mathematics Achievement was 0.35. Homeback was the strongest of the factors that influenced this criterion variable. This result indicated that students from higher socioeconomic status backgrounds were likely to achieve at a higher level in mathematics than students from lower socioeconomic status backgrounds within this TIMS Year 8 sample. Previous Australian research findings into the effects of socioeconomic status on mathematics achievement have indicated that students from higher socioeconomic status family backgrounds were likely to achieve at a higher level than their classmates from lower socioeconomic backgrounds (Keeves, 1968; Rosier, 1980; Ainley et al., 1990). A weaker effect was observed in the FIMSB sample which involved Year 8 students as did the TIMS sample, but the FIMSB sample was restricted to students from government schools only, whereas the TIMS sample included all types of schools.

Class Size (Classize). This was a LV that influenced Mathematics Achievement of students directly (0.20) and indirectly (0.007). The total effect was 0.21. This variable indicates that students from larger class groups achieve at a higher level in mathematics than students from smaller class groups. This finding was consistent with the findings in FIMSA, FIMSB and SIMS data sets.

Views about Mathematics (Views). Views also had direct (-0.12) and indirect (0.03) effects on Mathematics Achievement. This variable influenced Mathematics Achievement negatively. Thus, the variable indicated that those students who expressed less participation in working together in pairs or in small groups on different kinds of mathematical activities were likely to achieve at a higher level in mathematics than those who indicated more participation in working together in pairs or in small groups (see Table 5 and Figure 2).

Time in Learning Mathematics (Timlearn). This LV influenced Mathematics Achievement directly. It was the third strongest (ß = 0.21) variable that had a direct effect on Mathematics Achievement (see Tables 4 and 5 and Figure 2). The evidence showed that students who spent more time in learning mathematics were higher achievers in mathematics than students who spent less time in learning mathematics. While this latent variable included time spent in mathematics classes, the most important variate forming this variable was the number of times mathematics homework was assigned in a week. In part the change in composition of this variable could account for its increased contribution in TIMS compared to FIMS and SIMS. However, its greater contribution could arise from greater variability in the 1990s to time given to learning in mathematics classes and on homework.

Attitudes towards Mathematics (Attitude). This LV influenced Mathematics Achievement directly. It was the strongest (ß = 0.30) variable that had a direct effect on Mathematics Achievement (see Tables 4 and 5 and Figure 2). The evidence showed that students who expressed more positive attitudes towards mathematics were higher achievers in mathematics than students who expressed less positive attitudes. This observation was consistent with previous Australian research findings (Keeves, 1972; Schofield, 1981; Milne, 1992).

Motivation towards Mathematics (Motivation). Motivation was the only factor that showed an indirect (0.16) effect on Mathematics Achievement operating through Attitude. The results indicated that highly motivated students towards mathematics were likely to achieve at a higher level in mathematics than less motivated students (see Table 5 and Figure 2). The important point here was that gender did not exhibit any effect on Mathematics Achievement.

A further interesting finding was that the Age of the student did not show any influence on the outcome variable Mathematics Achievement, even though a grade sample was under survey. The background (Ethnicity) of the student also did not show a direct or indirect effect on the outcome variable.

Twenty five MVs and nine LV were originally hypothesised to influence the Mathematics Achievement level of Year 8 students who participated in the Third IEA Mathematics Study in Australia. Among the 25 MVs, seven were removed from further analysis, because the loading of Siblings was below the critical value of 0.30, and the LVs which were formed by the remaining six deleted MVs did not contribute to the inner model and were consequently removed from the analysis.

The results of the analysis revealed that among the nine hypothesised LVs only five of them were identified as student level factors that influenced the mathematics achievements of Year 8 students in Australia in 1994. These student level factors were Homeback, Classize, Views, Timlearn and Attitude. However, Motivation influenced the criterion variable indirectly through Attitude. The mean of the R2 values of the endogenous variables of the inner model was 0.12, which showed that the model was not a strong model. However, 34 per cent of the variance of the criterion variable of Mathematics Achievement was explained by the latent variables in the model.

Table 6 presents the direct and indirect effects of LVs identified as student level factors that influenced mathematics achievement on the different occasions. The first column shows the variables, while the remaining columns show the direct, indirect and total effects of each variable on Mathematics Achievement on each occasion. The direct effects of each variable on the outcome measure Mathematics Achievement were considered to indicate the relative strengths of the factors that influenced Mathematics Achievement on the different occasions.

This construct showed direct and indirect influence on the outcome variable on all occasions except in FIMSA (see Table 6). In FIMSA, it showed only an indirect influence. When FIMSA and SIMS were compared it would appear that the impact of home background had increased markedly over time. In 1964, in FIMSA, the effect was indirect while 14 years later the effect was both direct and indirect. It is important to remember that both groups involved only 13-year-old students. Furthermore, when Year 8 students in FIMSB (0.08) and TIMS (0.23) were compared it would appear that the effect of home background had increased markedly over the last three decades. However, this is almost certainly a consequence not of greater inequity but of a difference in the sample design employed in 1978, and in 1994 when students from nongovernment schools were included in the investigation compared with 1964 when they were not included. Most of the students in non-government schools would be from higher socioeconomic status backgrounds, therefore, there was greater variability in this predictor LV in 1978 and 1994 compared with 1964, and hence stronger effects were detected on the two later occasions compared to the former occasion.

In all Australian groups, namely FIMSA, FIMSB and SIMS Gender had only an indirect effect on Mathematics Achievement, furthermore, it showed neither a direct nor an indirect effect in TIMS.

All students in FIMSA and SIMS were 13-year-olds, therefore Student Age was not considered as a factor in these analyses. However, in FIMSB and TIMS student age was a potential factor, since these groups were all Year 8 students, who were not of the same age group. There was no direct or indirect effect on achievement for either of the Australian data sets. Thus, student age was not found to be a factor that influenced Mathematics Achievement in Australia. The reason might be related to small age differences among Australian students. In FIMSB the age of students ranged from 11 to 16 years and for TIMS ranged from 12.0 to 16.3 years.

The other factor which was considered in all the five groups of students was Classize. In all groups it showed both direct and indirect effects on the outcome variable. The class size effect on Mathematics Achievement between 1964 (0.19) and 1994 (0.20) in Australia was of median strength. The interesting point is that students from larger groups were likely to achieve at a higher level in mathematics in 1964 and the same and perhaps an even greater effect was found after a 30-year period. Thus the finding suggests that to achieve better results, mathematics students should be in larger class groups. This finding was consistent with the results reported by Pidgeon's (1967). From the analysis of the English FIMS data set, he reported that "there is evidence particularly with pupils up to 'O' level that higher mathematics performance is associated with larger classes" (Pidgeon, 1967, p. 140).

Motivation showed both direct and indirect effects on the outcome variable for FIMSA, FIMSB and only an indirect effect for SIMS and TIMS data sets. The direct and indirect effect for FIMSA, FIMSB and the indirect effect for SIMS and TIMS suggest that the effect of Motivation on Mathematics Achievement declined over time. It has been reduced from both direct and indirect effects in 1964 to only an indirect effect in 1978 for 13-year-olds and 1994 for Year 8 students. However, the MVs forming this LV were not identical. In 1964, the manifest variables were Hmwall, which involved the number of hours in a week given by students to all homework and Attitsch, a nine item scale measuring students' attitude towards school and school learning. Whereas in 1994 the MVs involved students' need to do well in mathematics. Therefore, the results might not be comparable, but the MVs were similar for 1964 and 1978 13-year-old students.

Time in learning mathematics was hypothesised as a factor that would influence Mathematics Achievement for all the four groups of students. However, it was found that there was only a direct effect on the outcome variable in the TIMS data set. In Australia in 1994 students who spent more time in learning mathematics were likely to achieve at a higher level than those who spent less time in learning the same subject. The interesting point is that in Australia both in 1964 and 1978 Timlearn did not show either direct or an indirect effects on Mathematics Achievement. However, in 1994 it showed a direct effect (0.21) on Mathematics Achievement. This suggests that the effects of Time in learning mathematics increased between occasions and it became one of the student level factors that had a significant influence on Mathematics Achievement. The MVs which formed this LV in all samples were the time allocated for students to learn mathematics in a week and the time taken by students to do their mathematics homework in a week. These two variables formed the LV for the FIMSA, FIMSB, and SIMS data sets, but for the TIMS data set the frequency of homework given by the teacher in a week was an additional MV that helped to form Timlearn. This MV was also a strong contributor (0.82) in the formation of this construct. Therefore, the increased effect in TIMS might be a consequence of the inclusion of this additional observed variable. However, the changed effect might be due to greater variability among student groups in the time given to learning mathematics on the later occasion than on the earlier occasions. There is also a possible conclusion that the frequency of homework given in a week is more important than the time spent in doing homework to achieve a higher level of performance in mathematics.

This LV was found to influence Mathematics Achievement and it showed a direct effect for all groups of students. In Australia over time, that is between 1964, 1978 and 1994, Attitudes towards Mathematics greatly increased in their effects on Mathematics Achievement. In 1964 the effect was 0.09 while this direct effect over a 30-year period had increased to 0.30. However, in 1964 Aspiration and Futmath were included as variables in the analysis, but in 1978 and in 1994 these two variables could not be introduced into the model. The total effect of Attitude (0.09), Aspiration (0.25) and Futmath (-0.02) for FIMSA data set was 0.32, while for FIMSB the combined effect was 0.25 (see Table 6). If these two variables were excluded from the analyses, the effect of Attitude would probably have been increased. Hence, the increase in the effect of attitude on the outcome variable in 1978 and 1994 might have been due to the removal of the influence of these surrogate variables for Attitude. Moreover, in all samples those students who had positive attitudes towards mathematics were likely to be higher achievers in mathematics than other students. Clearly it is important to find ways and means to improve the attitudes of students towards mathematics.

Students in FIMSA and in SIMS were from Years 7, 8 and 9. Hence, Year level was considered to be a factor that influenced Mathematics Achievement for these two groups of students. In both groups which were age samples, Year Level showed direct and indirect effects on the outcome variable. This indicated that Year Level influenced Mathematics Achievement, that is to say, the higher a student's year level the higher his or her level of mathematics achievement. Year level continued to be a factor from 1964 to 1978. However, it is important to observe that the total effect was markedly stronger for FIMSA than SIMS. This may be consequence of the inclusion of primary school Year 7 students in FIMSA samples for New South Wales, the largest state. By the time the SIMS study was conducted in 1978 New South Wales had reorganised its educational system and Grade 7 was at the secondary school level.

Views was considered to be a factor that would influence Mathematics Achievement for the FIMSA, FIMSB, SIMS and TIMS data sets. However, Views showed only a direct effect for the TIMS data set and very weak indirect effects for the other three Australian data sets. The effect of Views on the outcome variable for the TIMS data set showed that Views was a student level factor that influenced Mathematics Achievement. However, it is important to observe that the influence of Views was very small, specially for the SIMS study. Therefore, the effects of Views on the outcome measure were weak and did not change greatly between 1964 and 1978 in Australia. Furthermore, this factor showed change from an indirect effect in 1964 to a direct effect in 1994. However, the MV that formed the LV in 1964 involved students' views about the methods employed by their mathematics teachers during the teaching-learning process, while in 1994 the MV involved the students' participation in working together in pairs or in small groups on different kinds of mathematical activities. These findings would seem to suggest that such student participatory activities could have a detrimental effect on student learning as measured by the mathematics tests employed in these studies.

Like Views, Values were considered to be a factor that would influence Mathematics Achievement for FIMSA, FIMSB and SIMS data sets. The variable was not considered in the TIMS data set. The variable was formed from two MVs namely, Mathinso which measured the attitude of students towards the place of mathematics in society and Contrenv which measured students' attitudes towards the relationship of man to his environment. This LV showed only indirect effects for the three Australian data sets. It is important to observe that the influence was very small, especially for SIMS where it was only 0.01. Consequently, it was possible to conclude that the effect of Values on the outcome measure did not change greatly over time in Australia.

The indicator used to select similarities and the differences between factors that influenced Mathematics Achievement over time was the direct effects of the same variable for all groups on the outcome variable. Consequently, the first comparison was between FIMSA and SIMS, while the second comparison was between FIMSB and TIMS data sets.

Class Size, Attitudes towards Mathematics and Year Level continued to be factors that influenced Mathematics Achievements between FIMSA and SIMS, that is between 1964 and 1978. Moreover, Gender, Views and Values continued to influence Mathematics Achievements from 1964 to 1978 indirectly (see Table 6). However, Time in Learning did not show any effect either in 1964 or in 1978.

There were differences between 1964 and 1978 with respect to two variables, Home Background and Motivation. Home Background was not directly a factor influencing the outcome variable for FIMSA, but it was found to be a strong factor in 1978. On the other hand Motivation was a direct factor in FIMSA and was not a direct factor in SIMS, since it showed only an indirect effect. This might be as a consequence of a difference in the sample design employed in 1978, in which students from nongovernment schools were included. Most of the students in non-government schools would be from higher socioeconomic status backgrounds, therefore, there would have been greater variability in this predictor variable and the greater likelihood of an effect being detected.

Therefore, Class Size, Attitudes towards Mathematics and Year Level were student level factors that had a stable influence on mathematics achievement over time. Home Background increased from an indirect effect in 1964 to a direct effect in 1978, while Motivation declined from a direct effect in 1964 to an indirect effect in 1978.

Among the nine LVs which were hypothesised to influence Mathematics Achievement, three factors influenced students' Mathematics Achievement on both occasions in 1964 and 1994 at the Year 8 level. These variables were Home Background, Class Size and Attitudes towards Mathematics (see Table 6). All three LVs had effects on the outcome measure which had increased in 1994 when compared to the 1964 data set. These three variables are student level factors influencing mathematics achievement for the last 30 years. Consequently, the findings of this investigation indicate that students from higher socioeconomic status backgrounds, students from larger class groups and students who expressed more positive attitudes towards mathematics are likely to achieve at a higher level in mathematics.

The other similarity between the two Australian groups was that Student Age did not show any effect on the outcome variable.

The differences between the FIMSB and TIMS studies were in Gender, Views, Motivation and Time in Learning. In FIMSB, Gender showed an a trivial indirect effect on the outcome variable, while Gender did not show any effect for the TIMS data set. Motivation showed a direct effect and was considered an important factor in FIMSB, however, its contribution was reduced to an indirect effect for the TIMS data set. Furthermore, Views showed a direct effect in TIMS, but only an indirect effect for the FIMSB data set. The other major difference was for Time in Learning mathematics, this variable did not have any effect in 1964, but it showed a direct effect of 0.21 in 1994. However, it is important to recognise that the MVs which formed Motivation and Views in 1964 and 1994 were different in nature and an additional strong variable (frequency of mathematics homework) was included to form the Time in Learning variable in 1994.

The results of the path analyses for the three different data sets (FIMS, SIMS and TIMS) have revealed that the home background of students, number of students in class and attitudes of students towards mathematics are student level factors influencing achievement in mathematics over the last 30 years. Time in learning mathematics did not show any influence both in 1964 and 1978. However, it influenced the 1994 students' achievement directly, the main reason perhaps being the inclusion of a new and strong manifest variable, frequency of mathematics homework in a week. The findings recorded in their paper are consistent with Carroll's (1963) model of school learning. Carroll has argued that perseverance (motivation, attitude) and time for learning are some of the factors that influence school learning. These factors advanced by Carroll continue to influence the Australian students' mathematics achievement. Thus, teachers, school administrators and curriculum designers need to consider the following points to improve the achievement level of students in mathematics.

3. Curriculum designers should provide enough time for students to learn mathematics and to provide regular homework during a school week, since the number of mathematics homework sessions in a week showed more influence on mathematics achievement than the time the students spent on doing their assignments.