Student and Teacher Characteristics on
Student Math Achievement
, Eui Kyung Kim
, Sun Ah Lim
Using data from the Programme for International Student Assessment (PISA), this study implements two statistical
analyses to investigate the effects of student and teacher characteristics on students’ mathematical achievement. First,
the authors conduct an exploratory factor analysis to explore the factor structure for the various student and teacher
variables of interest in this study. Second, they perform hierarchical linear modeling to analyze students’ and teachers’
multilevel structure in a school. The results suggest that student characteristics such as mathematics interest, instrument
motivation, mathematics self-efficacy, mathematics anxiety, mathematics self-concept, and out-of-school study time
predicted 39.9% of mathematical achievement variance. The results also suggest that mathematics self-efficacy had
the largest effect on mathematical achievement. Teacher characteristics such as teacher-directed instruction, cognitive
activation, teacher support, classroom management, and student–teacher relations predicted 34.9% of mathematical
achievement variance. This study’s results have implications for educators in fostering a positive learning environment to
increase students’ mathematics interest and self-efficacy, and focus on specific teacher characteristics to increase
students’ mathematical achievement.
mathematical achievement, multilevel, PISA data, student characteristics, teacher characteristics
Received 25 August 2018; accepted 20 November 2020
Increasing students’ mathematical achievement is of
national concern. Results from international compar-
ison studies such as the Programme for International
Student Assessment (PISA) suggest that 15-year-old
U.S. students continually score lower on standard-
ized mathematical achievement tests than 15-year-
old students in other countries (OECD, 2012).
Moreover, within the United States, the nation’s
report card revealed that 42% of fourth-graders,
35% of eighth-graders, and 26% of twelfth-graders
were placed at or above the proficient levels for their
respective grades, where proficient indicates that stu-
dents demonstrated competency with challenging
subject matter (Kena et al., 2015). Given these
statistics, there is a critical need to understand the
factors influencing students’ mathematical
achievement and address this achievement gap
between U.S. students and their international
There has been a plethora of empirical research
focusing on the student characteristics and teacher
characteristics that influence students’ mathematical
achievement scores. In terms of student characteristics,
the extant literature suggests a number of factors,
including students’ mathematics interests, motivation,
mathematics self-efficacy, mathematics anxiety, mathe-
matics self-concept, and out-of-school study time, influ-
ence academic achievement (Beaton et al., 1996;
College of Education, Hankuk University of Foreign Studies
Graduate School of Education, University of California, Riverside
Department of Education, Chonbuk National University
Department of Education, University of California, Santa Barbara
Sukkyung You, College of Education, Hankuk University of Foreign
Studies, Imun-dong, Dongdaemun-gu, Seoul 130-791, South Korea.
Journal of Pacific Rim Psychology
Volume 15: 1–13
!The Author(s) 2021
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Else-Quest et al., 2013; Fast et al., 2010; Lee, 2009;
Marsh & Martin, 2011; Pajares & Miller, 1994;
Stevens et al., 2004). In terms of teacher characteristics,
there has been research suggesting that teacher-directed
instruction, cognitive action, teacher support, classroom
management, and student–teacher relations influence
students’ mathematical achievement (Baumert et al.,
2010; Dever & Karabenick, 2011; Hochweber et al.,
2014; Hughes et al., 2011; Levpuscek & Zupancic,
2009). These student- and school-level factors have
been demonstrated to increase students’ college and
career opportunities, as well as their future income pros-
pects (National Mathematics Advisory Panel, 2008).
Students’ mathematical achievement scores have been
widely used as indicators of students’ academic perfor-
mance in school. However, there is much debate as to
which factors influence students’ mathematical
achievement and how they vary across countries.
Previous studies have provided empirical evidence for
the effects of student- and school-level factors on U.S.
students’ mathematical achievement. However, most
studies have examined only one or some of these char-
acteristics as predictors of mathematical achievement,
but not yet explored various student and teacher char-
acteristics simultaneously. Thus, based on the existing
theoretical and empirical research, this study utilizes a
conceptual framework that student academic outcome
(i.e., mathematical achievement in this study) is
explained by various student- and school-level factors.
The conceptual model is depicted in Figure 1.
The previous literature has identified several student-
level variables that explain mathematical achievement.
Specifically, these student characteristics include math-
ematics self-efficacy (i.e., perceived confidence in math-
ematics; Fast et al., 2010; Lee, 2009; Pajares &
Kranzler, 1995; Pajares & Miller, 1994, 1995;
Stevens et al., 2004), mathematics self-concept (i.e.,
a self-evaluation of one’s own general ability in math-
ematics; Lee, 2009; Marsh & Martin, 2011; Pajares &
Miller, 1994), mathematics anxiety (i.e., a feeling of
worry or discomfort toward mathematics that impedes
performance; Bandalos et al., 1995; Lee, 2009; Ma,
1999; Schulz, 2005), mathematics interest (i.e., interest
in learning and achievement in mathematics; Else-
Quest et al., 2013), mathematics motivation (i.e., the
extent to which individuals embrace challenges and
are motivated to perform well in mathematics;
Gottfried et al., 2013; Stevens et al., 2004), and out-
of-school study time (i.e., the amount of time spent
studying outside school; Beaton et al., 1996). These
characteristics may act as primary determinants of stu-
dents’ mathematical achievement.
Although many existing studies have supported the
positive impacts of selected student characteristics,
some studies have provided mixed results. For instance,
Pajares and Miller (1994, 1995) found that mathemat-
ics self-efficacy and mathematics self-concept were pos-
itively related to mathematical achievement among U.
S. undergraduate students. However, Lee’s (2009)
study used the PISA 2003 data to study how mathe-
matics self-efficacy, mathematics self-concept, and
mathematics anxiety related to students’ mathematical
performance and found different results across coun-
tries. Specifically, students in Asian countries, such as
Korea and Japan, tended to have low mathematics
self-efficacy, low mathematics self-concept, and high
mathematics anxiety, yet had high mathematical
achievement. Conversely, students in western countries
such as Finland, the Netherlands, and Liechtenstein,
tended to have high mathematics self-efficacy and
mathematics self-concept and low mathematics anxi-
ety, yet had high mathematical performance.
Mathematics anxiety has often been negatively associ-
ated with mathematical achievement (e.g., Tomasetto,
2020), but some studies have found a positive associa-
tion between them (e.g., Hunt et al., 2017). These
results suggest that mathematics self-efficacy and self-
concept may be differentially related to mathematical
achievement across countries.
Figure 1. Conceptual Framework.
2Journal of Pacific Rim Psychology
Many studies have reported positive relations
between mathematics interest and mathematical
achievement (e.g., Aunola et al., 2006; Fisher et al.,
2012). Interest in a particular subject can be accompa-
nied by attention and concentration (Hidi, 2006).
However, mixed results have been reported on the
effect of mathematics interest on achievement. For
example, some researchers have found a significant
positive relationship between mathematics interest
and achievement (Liu, 2009; Singh et al., 2002), while
others have reported insignificant or inconsistent
oller et al., 2001; Marsh et al., 2005). There
have also been some mixed reports on the effects of
out-of-school study time on mathematical achieve-
ment. Cheema and Sheridan (2015) found that time
spent on homework outside of school had a significant
impact on mathematical achievement. However,
Beaton et al.’s (1996) study, which used Trends in
International Mathematics and Science Study
(TIMSS) data, suggested that the highest mathematical
achievement was associated with students who studied
a moderate amount outside of school (i.e., 1 to 3 hours
per day). More study time outside of school did not
necessarily relate to increases in mathematical achieve-
ment scores. Students who have academic difficulties
may need additional time to study outside of school to
These mixed results on the associations between stu-
dent characteristics and mathematical achievement
indicate that student characteristics may not play a
consistent role in mathematical achievement for all stu-
dents, suggesting the need to examine their relations
among U.S. students specifically. Furthermore, math-
ematics motivation has been consistently positively
associated with mathematical achievement (e.g.,
Areepattamannil et al., 2011; Zhu & Leung, 2011),
but there are limited studies that have examined math-
ematics motivation with other various student charac-
teristics simultaneously as achievement predictors.
Thus, this study will use selected student-level variables
to understand their effects on U.S. students’ mathemat-
ical achievement. The study also aims to include
teacher-level variables to understand how student-
and teacher-level variables relate to students’ mathe-
Based on an extensive literature review, several teacher
characteristics were identified as primary school-level
variables contributing to mathematical achievement.
These teacher characteristics include teachers’ support
(i.e., perceived psychological and practical support
from teachers; Levpuscek & Zupancic, 2009), classroom
management (i.e., methods and strategies a teacher uses
to maintain a positive classroom environment;
Hochweber et al., 2014), student–teacher relationships
(i.e., students’ perception of the closeness with teachers;
Hughes et al., 2011), cognitive activation (i.e., encour-
aging students to think more deeply to find solutions
and reach the answer, rather than focusing on the
answer itself; Baumert et al., 2010), and teacher-direct-
ed instruction (i.e., instruction in which teachers are
primarily communicating mathematics to students;
National Mathematics Advisory Panel, 2008).
Perceived teacher support and positive student–
teacher relationships facilitate student learning as
teachers are the primary organizers of classroom activ-
ities (Patrick et al., 2007; Simons-Morton & Chen,
2009). They help students to develop academic capac-
ities within their classroom (Bedeck, 2015) and to con-
centrate on academic tasks better (Ryan & Patrick,
2001). The existing literature provided mixed empirical
evidence of the impacts of teacher support and stu-
dent–teacher relationships on mathematical achieve-
ment. For example, some studies have found positive
relations between perceived teacher support and stu-
dent–teacher relationships and mathematical achieve-
ment scores (e.g., Klem & Connell, 2004; Paulo et al.,
2007), but others have reported inconsistent or no sig-
nificant relations (e.g., Rueger et al., 2010; Valiente
et al., 2019).
Teachers’ use of cognitive activation as a teaching
strategy has also been linked to higher mathematical
achievement (e.g., Burge et al., 2015). Examples of cog-
nitive activation tasks include having classroom discus-
sions, encouraging students to explain and validate
their solutions, and prompting students to discover
multiple solutions to a problem (Baumert et al.,
2010). Cognitive activation has been found to mediate
teachers’ pedagogical content knowledge and
students’ progress in mathematical performance, indi-
cating the importance of cognitive activation on the
path to mathematical achievement (Baumert et al.,
2010). Additionally, classroom management, defined
as teachers’ ability to create a functioning learning
environment (Hochweber et al., 2014), helps students
stay on task and enables teachers to better monitor
students’ progress (Blair & Dennis, 2010). Children in
well-managed classrooms have shown higher mathe-
matical achievement than those in ineffectively man-
aged classrooms (Stronge et al., 2011). However,
Blazar (2015) found no significant classroom manage-
ment impact on mathematical achievement when other
classroom variables were examined simultaneously,
suggesting the need to further examine various class-
room effects on mathematical achievement.
Although there is no doubt that teachers play a sig-
nificant role in students’ academic achievement, there
has been an ongoing debate on which instructional
You et al. 3
style is effective for students’ mathematical achieve-
ment (National Mathematics Advisory Panel, 2008).
More specifically, there have been mixed results as to
whether a teacher-directed instructional approach
effectively improves students’ mathematical perfor-
mance. Teacher-directed instruction is focused on the
teacher and ranges from highly scripted direct instruction
to interactive lecture styles. Similar to other teacher var-
iables, there have been mixed results about the impacts
of teacher-directed instruction on academic achievement.
Morgan et al. (2015) reported that teacher-directed
instruction was significantly associated with the mathe-
matical achievement of students with mathematics diffi-
culties. Other studies, however, have indicated that too
much teacher-directed instruction might slow the devel-
opment of students’ conceptual understanding (Hiebert,
1999; Woodward & Howard, 1994).
Since the legislation of No Child Left Behind, a large
body of research has been conducted to identify factors
related to students’ academic achievement. Previous
studies have attempted to divide variables into school
and student levels to extract school effects and student
effects (Kang et al., 2005; Lee & Chung, 2011; Mu~
& Chang, 2007). We conducted a multilevel analysis to
address the data’s nested structure, where students are
nested within schools. Statistically, mixing school-level
variance with student-level variance can generate mis-
leading results. Therefore, the research design needs to
be multilayered, with student and teacher characteris-
tics in the same school. This will reveal whether stu-
dents’ academic achievement varies depending on the
teachers or their teaching abilities. Thus, the present
study aims to investigate the effects of student and
teacher characteristics on students’ academic achieve-
ments through multilevel analysis.
Data and Sample
This study used the fifth PISA survey, which tested 15-
year-olds on reading, mathematics, and science in 2012.
Specifically, this study investigated the relation of these
student-level variables and school-level variables to stu-
dents’ mathematical achievement using a U.S. sample
drawn from the PISA 2012 data. The U.S. sample con-
tains 4,978 students and 162 schools.
Outcome Variable. The outcome variable measuring
mathematical achievement was based on a total of 85
mathematics test items distributed over 13 booklets.
The balanced incomplete block design was used to max-
imize subject-matter coverage, which meant that no
student had the opportunity to attempt all mathematics
items during the 2-hour assessment. The scaled achieve-
ment scores took the form of five plausible values for
each student, where each was generated as a random
draw from an estimated ability distribution of students
with similar item response patterns and backgrounds
(OECD, 2012). For all measures, we used Cronbach’s
ato assess the reliability of a set of scale or test items.
The reliability coefficient for the mathematical assess-
ment scores (i.e., five plausible values) was .985.
Mathematics Interest. Mathematics interest measures
the degree to which students think about their views on
mathematics, such as enjoying reading about mathe-
matics, looking forward to mathematics lessons, enjoy-
ing mathematics, and being interested in learning
in mathematics (Organisation for Economic Co-
operation and Development, 2013). The mathematics
interest variable included four items, where students
responded to the items on a 4-point Likert scale that
ranged from 1 ¼strongly agree to 4 ¼strongly disagree.
These responses were reverse-coded so that higher
scores indicate a higher interest related to mathematics.
The reliability coefficient for mathematics interest was
Instrumental Motivation. Instrumental motivation
measures the degree to which students are motivated
to learn mathematics because they perceive mathemat-
ics as being useful to them and to their future studies
and careers (Organisation for Economic Co-operation
and Development, 2013). The instrumental motivation
variable was measured by four items, including stu-
dents’ perception of mathematics as worthwhile for
work, worthwhile for their career chances, important
for future study, and helping them to get a job.
Students had the option to respond on a 4-point
Likert scale that ranged from 1 ¼strongly agree to
4¼strongly disagree. These responses were reverse-
coded so that higher scores indicate higher instrumen-
tal motivation. The reliability coefficient for instrumen-
tal motivation was .910.
Mathematics Self-Efficacy. Mathematics self-efficacy
measures the degree to which a student believes in
their ability to successfully perform or accomplish a
particular mathematical task or problem (Hackett &
Betz, 1989). Mathematics self-efficacy was measured
using eight items with response options on a 4-point
Likert scale ranging from 1 ¼very confident to 4 ¼not
at all confident. These responses were reverse-coded so
that higher scores indicate higher mathematics self-
efficacy. The reliability coefficient for mathematics
self-efficacy was .852.
4Journal of Pacific Rim Psychology
Mathematics Anxiety. Mathematics anxiety measures
the degree to which a student feels helpless or emotion-
al stress when dealing with mathematics (Schulz, 2005).
The mathematics anxiety items were measured using
five items with response options on a 4-point Likert
scale ranging from 1 ¼strongly agree to 4 ¼strongly
disagree. These responses were reverse-coded so that
higher scores indicate higher anxiety about mathemat-
ics. The reliability coefficient for mathematics anxiety
Mathematics Self-Concept. Mathematics self-concept
measures the degree to which a student evaluates their
own self-worth related to mathematics (Pajares &
Schunk, 2001). Mathematics self-concept was mea-
sured using five items on a 4-point Likert scale ranging
from 1 ¼strongly agree to 4 ¼strongly disagree. These
responses were reverse-coded so that higher scores indi-
cate a higher self-concept of mathematics. The reliabil-
ity coefficient for the mathematics self-concept variable
Out-of-School Study Time. The out-of-school study
time variable indicates the number of hours per week
that a student spends on out-of-school study time for
all school subjects. Out-of-school study time was mea-
sured using six items: homework, guided homework,
personal tutor, classes organized by a commercial com-
pany, study with a parent or other family member, and
online lessons. Students had the option to freely indi-
cate how many hours per week they spent on each out-
of-school study activity. The reliability coefficient for
out-of-school study time was .618.
Teacher-Directed Instruction. The teacher-directed
instruction variable measures students’ perceptions of
their teacher’s directed mathematics instruction. There
were five items used to measure students’ perceptions
of teacher-directed instruction, including the extent to
which the teacher sets clear goals; encourages thinking
and reasoning; checks understanding; summarizes pre-
vious lessons; and informs about learning goals.
Students had the option to respond on a 4-point
Likert scale: 1 ¼every lesson,2¼most lessons,3¼some
lessons, and 4 ¼never or hardly ever. The values were
reverse-coded so that higher scores indicate the higher
frequency of a student’s perceptions of teacher-directed
instruction. The reliability coefficient for teacher-
directed instruction was .763.
Cognitive Activation. Cognitive activation measures
the degree to which students perceive how often their
mathematics teacher incorporates cognitive activation
tasks in class, such as having classroom discussions,
encouraging students to explain and validate their
solutions, and prompting students to discover multiple
solutions to a problem (Baumert et al., 2010).
Cognitive activation was measured using nine items,
including the extent to which a student perceives that
their mathematics teacher encourages them to reflect
on a problem; gives problems that require students to
think for an extended period of time; asks students to
use their own procedures; presents problems with no
obvious solutions; presents problems in different con-
texts; helps students learn from their mistakes; asks for
explanations; presents problems that require students
to apply what they have learned; and gives problems
with multiple solutions. Students had the option to
respond using a 4-point Likert scale ranging from
1¼always or almost always to 4 ¼never or rarely.
These values were reverse-coded so that higher scores
indicate higher cognitive activation by mathematics
teachers in the classroom. The reliability coefficient
for mathematics cognitive activation was .869.
Teacher Support. Teacher support measures the
degree to which students perceive that their mathemat-
ics teacher provides support. Four teacher support
items measured students’ perception of the extent to
which their mathematics teacher lets students know
they have to work hard; provides extra help when
needed; helps students with learning; and gives students
opportunities to express opinions. Students responded
to these items on a 4-point Likert scale ranging from
1¼strongly agree to 4 ¼strongly disagree. The values
were reverse-coded so that higher scores indicate a
more positive student perception of teacher support.
The reliability coefficient for the mathematics teacher
support variable was .840.
Classroom Management. Classroom management
measures the degree to which students perceive how
their mathematics teacher manages the classroom.
The classroom management variable was measured
using four items, including the extent to which students
perceive their teacher gets students to listen; keeps the
class orderly; starts lessons on time; and has to wait a
long time for students to quieten down. Students
responded on a 4-point Likert scale ranging from
1¼strongly agree to 4 ¼strongly disagree. All of the
items except for one (i.e., the teacher has to wait a
long time for students to quieten down) were reverse-
coded. The reliability coefficient for classroom man-
agement was .746.
Student–Teacher Relationship. Student–teacher rela-
tionship measures the degree to which a student per-
ceives their relations with teachers at their school. Five
student–teacher relationship items measured the extent
You et al. 5
to which students perceive their teacher gets along with
most students; is interested in students’ well-being; lis-
tens to students; helps students; and treats students
fairly. Students responded to these items on a 4-point
Likert scale ranging from 1 ¼strongly agree to
4¼strongly disagree. The items were reverse-coded so
that higher scores indicate students’ perception of more
positive relationships with teachers. The reliability
coefficient for student–teacher relationship was .833.
Two statistical analyses were carried out: exploratory
factor analysis and multilevel analysis. For the data han-
dling and descriptive analysis, SPSS 16.0 was used.
Correlations among the variables were examined to
verify the descriptive statistics and model. Multilevel
analysis was conducted using hierarchical linear model-
ing. We conducted hierarchical linear modeling to
address the nested structure of the data, where students
are nested within schools. Statistically, mixing school-
level variance with student-level variance can generate
misleading results. Student and teacher characteristics
from the same school are not independent of one another
and therefore their responses do not meet the indepen-
dence assumption for regression analysis. The estimates
thus reduce the statistical variance and yield more liberal
results in significant tests (Bryk & Raudenbush, 1992;
Salvucci & Weng, 1995).
We built three models, starting with a one-way anal-
ysis of variance model, which allowed for partitioning
of the total variance in overall mathematical achieve-
ment into within-school and between-school variances.
The second model—regression with the means-as-
outcomes model—incorporated the student-level
characteristics. The third model incorporated the
school-level characteristics. Our results are presented
in the following section.
We implemented a correlation analysis to assess the
relations between student characteristics, teacher char-
acteristics, and students’ mathematical achievement.
As shown in Table 1, there are significant correlations
between student and school characteristics with varying
levels of significance, except for the relations of out-of-
school study time to mathematical anxiety, mathemat-
ics self-concept, and classroom management. The
mathematical achievement score was significantly asso-
ciated with all student and teacher characteristics.
Exploratory Factor Analysis
A robust weighted least squares estimation was used
with a promax rotation. Empirical approaches, such as
examining a scree test and patterns of the factor loading,
were considered within a theoretical framework and the
extant literature to confirm that the final factor selection
was interpretable and substantively plausible. Through
this process, an 11-factor model emerged as the most
meaningful and parsimonious model. The 5-factor solu-
tion yielded a root mean square error of approximation
of .06 and standardized root mean square residual of
.04, which is considered acceptable (Hu & Bentler,
1999). Table 2 displays the factor loadings of the 11
different factors obtained from the exploratory factor
analysis, and the Cronbach’s aand intercorrelations
for these factors.
We conducted hierarchical linear modeling to identify
mathematical achievement predictors by examining
student and teacher characteristics in their respective
schools (see Table 3). First, we implemented a null
Table 1. Intercorrelation Matrix for Study Variables.
1 2 3 4 5 67891011
1. Mathematics score
2. Mathematics interest .140***
3. Instrumental motivation .153*** .607***
4. Mathematics self-efficacy .569*** .407*** .361***
5. Mathematics anxiety .418*** .440*** .305*** .466***
6. Mathematics self-concept .414*** .636*** .452*** .548*** .776***
7. Out-of-school study time .085*** .132*** .110*** .139*** .001 .040
8. Teacher-directed instruction .041* .332*** .309*** .270*** .187*** .233*** .091**
9. Cognitive activation .053** .279*** .275*** .276*** .137*** .217*** .109*** .617***
10. Teacher support .057** .278*** .295*** .219*** .182*** .208*** .065* .548*** .593***
11. Classroom management .214*** .231*** .252*** .287*** .229*** .233*** .035 .412*** .414*** .569***
12. Student–teacher relationship .150*** .283*** .316*** .274*** .213*** .243*** .129*** .483*** .411*** .460*** .391***
*p<.05. **p<.01. ***p<.001.
6Journal of Pacific Rim Psychology
model (Model 1), which provided information regard-
ing the partitioned total variance in overall mathemat-
ical achievement scores into within-school and
between-school variances. The intra-class correlation
was .24, indicating that 24% of the variance in overall
mathematical achievement scores is between school-
Second, Model 2 included student characteristics to
examine their effects on mathematical achievement.
The results from Model 2 show that the test language
conducted in English (b¼11.926, p<.001) positively
affected mathematical achievement scores. Students’
gender, parental education, and economic wealth did
not significantly affect mathematical achievement
scores. Mathematics interest (b¼6.351, p <.001) and
mathematics self-efficacy (b¼10.118, p <.001) had a
statistically significant positive effect on mathematical
achievement, whereas mathematics anxiety
(b¼7.180, p<.001) had a significant negative effect
on mathematical achievement.
Third, Model 3 added teacher characteristics to
Model 2 to assess the school-level effects on student
Table 2. Factor Loadings of Items Used for This Study.
Mathematics interest (Cronbach’s a¼.910)
1. Enjoy reading .840
2. Look forward to lessons .899
3. Enjoy mathematics .913
4. Interested .894
Percentage of variance explained 78.677
Instrumental motivation (Cronbach’s a¼.906)
1. Worthwhile for work .871
2. Worthwhile for career chances .886
3. Important for future study .893
4. Helps to get a job .884
Percentage of variance explained 78.051
Mathematics self-efficacy (Cronbach’s a¼.852)
1. Using a <train timetable>.682
2. Calculating TV discount .750
3. Calculating square meters of tiles .775
4. Understanding graphs in newspapers .730
5. Solving Equation 1 .609
6. Distance to scale .717
7. Solving Equation 2 .645
8. Calculate petrol consumption rate .700
Percentage of variance explained 49.419
Mathematics anxiety (Cronbach’s a¼.877)
1. Worry that it will be difficult .826
2. Get very tense .841
3. Get very nervous .831
4. Feel helpless .803
5. Worry about getting poor <grades>.805
Percentage of variance explained 67.456
Mathematics self-concept (Cronbach’s a¼.898)
1. Not good at mathematics
2. Get good <grades>.805
3. Learn quickly .885
4. One of best subjects .863
5. Understand difficult work .837
Percentage of variance explained 71.483
Out-of-school study time (Cronbach’s a¼.618)
1. Homework .437
2. Guided homework .729
3. Personal tutor .685
4. Commercial company .598
5. With parent .727
6. Computer .644
Percentage of variance explained 41.538
Teacher-directed instruction (Cronbach’s a¼.763)
1. Sets clear goals .768
2. Encourages thinking and reasoning .714
3. Checks understanding .754
4. Summarizes previous lessons .664
5. Informs about learning goals .698
Percentage of variance explained 51.928
Cognitive activation (Cronbach’s a¼.869)
1. Teacher encourages to reflect on problems .747
2. Gives problems that required to think .714
Table 2. Continued.
3. Asks to use own procedures .663
4. Presents problems with no obvious solutions .575
5. Presents problems in different contexts .752
6. Helps learn from mistakes .732
7. Asks for explanations .691
8. Apply what we learned .733
9. Problems with multiple solutions .697
Percentage of variance explained 49.337
Teacher support (Cronbach’s a¼.840)
1. Lets us know we have to work hard .772
2. Provides extra help when needed .862
3. Helps students with learning .877
4. Gives opportunity to express opinions .789
Percentage of variance explained 68.255
Classroom management (Cronbach’s a¼.746)
1. Students listen .850
2. Teacher keeps class orderly .886
3. Teacher starts on time .777
4. Wait long to <quiet down>
Percentage of variance explained 59.627
Student–teacher relationship (Cronbach’s a¼.833)
1. Get along with teachers .686
2. Teachers are interested .797
3. Teachers listen to students .817
4. Teachers help students .782
5. Teachers treat students fairly .787
Percentage of variance explained 60.113
Standardized factor loadings from exploratory factor analysis (factor
A reversed item.
You et al. 7
mathematical achievement. The results from Model 3
show that among all of the school-level variables,
teacher-directed instruction (b¼12.931, p <.05) and
teacher support (b¼36.615, p<.01) negatively pre-
dicted mathematical achievement scores. On the other
hand, cognitive activation, classroom management, and
student–teacher relationship were significant positive
predictors of student mathematical achievement scores.
Student–teacher relationship (b¼39.319, p <.001) was
the strongest predictor of the achievement outcome, fol-
lowed by classroom management (b¼28.447, p<.01)
and cognitive activation (b¼8.540, p<.05).
The effect of the conditional model is shown in the
accumulated explained variance (R
). The student-level
model (Model 2) explained 39.9% of the variance in
mathematical achievement among classrooms and the
school-level model (Model 3) explained 34.9% of the
variance in mathematical achievement among schools.
The accumulated variance explained by all the
predicting variables included in this analysis was 12%
of the variance.
The current study aimed to investigate factors that pre-
dict student mathematical achievement by evaluating
student- and school-level factors. The hierarchical
linear modeling analysis found that student variables
predicted 39.9% of students’ mathematics achievement
and the teacher variables predicted 34.9%. Specifically,
among the student variables, mathematics interest and
mathematics self-efficacy had positive impacts on
mathematical achievement, whereas mathematics anx-
iety had a negative effect. Among the teacher variables,
teacher-directed instruction and teacher academic sup-
port had negative effects on students’ mathematical
achievement, while cognitive activation, classroom
Table 3. Multilevel Analysis Results for Mathematics Achievement.
Model 1: Null model Model 2: Student level Model 3: School level
Intercept 481.012*** (3.746) 460.217*** (7.057) 461.703*** (6.842)
Gender (Female ¼1) 1.533 (2.134) 1.431 (2.131)
Mother’s school 0.205 (1.538) 0.236 (1.537)
Father’s school 2.200 (1.396) 2.054 (1.393)
Wealth 1.388 (1.061) 1.206 (1.071)
Language (English ¼1) 11.926*** (3.347) 11.859** (3.350)
Effect of student characteristics
Mathematics interest 6.351*** (0.617) 6.368*** (0.616)
Instrument motivation 0.341 (0.554) 0.339 (0.554)
Mathematics self-efficacy 10.118*** (0.438) 10.128*** (0.438)
Mathematics anxiety 7.180*** (0.590) 7.187*** (0.590)
Mathematics self-concept 0.735 (0.541) 0.736 (0.541)
Out-of-school study time 0.087 (0.140) 0.091 (0.141)
Effect of teacher characteristics
Teacher-directed instruction 12.931* (5.820)
Cognitive activation 8.540* (4.230)
Teacher support 36.615** (10.612)
Classroom management 28.447** (7.913)
Student–teacher relationship 39.319*** (6.310)
School level 1938.130 1894.976 1233.035
Student level 6087.792 3656.817 3658.189
Total 8025.922 5551.793 4891.224
Intra-class correlation 0.241 0.341 0.252
Accumulated variance explained (R
School level 0.022 0.349
Student level 0.399 0.000
Total 0.308 0.119
*p<.05. **p<.01. ***p<.001.
8Journal of Pacific Rim Psychology
management, and student–teacher relationship had
Student Characteristics and Mathematical
First, mathematics interest significantly influenced
mathematical achievement, which is consistent with
previous studies (e.g., Fisher et al., 2012). For example,
Singh et al. (2002) provided empirical evidence for the
impact of mathematics interest on mathematical
achievement among 24,599 middle school and high
school students collected by the National Center for
Education Statistics. The current study also reports
that mathematics interest positively predicted academic
achievement. These results suggest that students who
are interested in mathematics are more immersed and
invest more time in studying mathematics, leading to
Second, although mixed results exist, many previous
studies have reported the positive effect of mathematics
self-efficacy on mathematical achievement. For exam-
ple, self-efficacy has been shown to be more strongly
associated with mathematical performance than
English and writing (e.g., Pajares, 1996). Another
study, conducted by Shores and Shannon (2007), also
found that self-efficacy was a significant variable that
increased mathematical achievement among 761 fifth-
and sixth-graders. Similarly, Yum and Park (2011)
studied the relation between mathematics self-efficacy
and mathematical achievement among seventh- to
ninth-graders through a 3-year latent growth model.
They reported a continuous positive effect of mathe-
matics self-efficacy on mathematical achievement. The
current study supports these previous results by show-
ing that mathematics self-efficacy had the largest
impact on mathematical achievement among all the
student variables, suggesting the importance of improv-
ing students’ perceived confidence in mathematics to
improve their mathematical performance.
Third, this study’s findings support mathematics
anxiety’s adverse impact on mathematical achievement.
Previous studies have reported inconsistent results on
the relation between mathematics anxiety and mathe-
matical achievement. For instance, Ma’s (1999) meta-
analysis of 26 studies indicated a significant negative
correlation between anxiety and academic achievement.
Similarly, Shim’s (2000) study of 219 high school stu-
dents showed that mathematics anxiety was negatively
correlated with mathematical achievement. Shim (2000)
argued that negative emotions, such as anxiety, interfere
with learning, and negatively affect achievement. The
present study reveals comparable findings, where math-
ematics anxiety had a statistically significant negative
effect on mathematical achievement. Given that
mathematics anxiety may interfere with manipulating
numbers and solving mathematical problems
(Richardson & Suinn, 1972), mathematics interventions
could incorporate the means to reduce students’ mathe-
matics anxiety to improve their mathematical
Teacher Characteristics and Mathematical
The student–teacher relationship was the strongest pre-
dictor of students’ mathematical achievement at the
school level, followed by teacher support, classroom
management, teacher-directed instruction, and cognitive
activation. Specifically, the student–teacher relationship,
classroom management, and cognitive activation posi-
tively predicted mathematical achievement, whereas
teacher support and teacher-directed instruction nega-
tively predicted mathematical achievement.
First, our results provide additional empirical sup-
port to the existing literature for student–teacher rela-
tionships’ positive influence on students’ mathematical
achievement (e.g., Croninger & Lee, 2001). The quality
of the student–teacher relationship plays a primary role
in students’ cognition and academic achievement (Kim
& Lee, 2015). The current study suggests that the stu-
dent–teacher relationship has the most considerable
effect on mathematical achievement—more than
other student or teacher characteristics—emphasizing
the critical role of a positive student–teacher relation-
ship in improving students’ mathematical achievement.
Second, teacher support had a negative relation to
mathematical achievement. It is important to note that
this does not necessarily imply that more academic sup-
port from teachers results in decreased mathematical
achievement scores. Instead, we interpret this finding
as students who had lower mathematical achievement
needed more support from teachers. Given that previ-
ous studies have reported mixed results on teacher sup-
port and mathematical achievement (e.g., Klem &
Connell, 2004; Rueger et al., 2010), more research is
needed to understand this relationship.
Third, this study shows that teachers’ classroom
management positively predicted students’ mathemati-
cal achievement. Our finding provides additional sup-
port for the positive impacts of classroom management
on mathematical achievement (Kim, 2000; Stronge
et al., 2011), suggesting that better organized and man-
aged classrooms enable students to perform to the best
of their ability and promote academic achievement.
Fourth, this study shows the negative effect of
teacher-centered classes on students’ mathematical
achievement. This result contributes to the existing lit-
erature by supporting the findings of previous studies
(e.g., Hiebert, 1999; Woodward & Howard, 1994) that
You et al. 9
teacher-centered classes, or teacher-directed instruction,
might hinder students’ voluntary learning and thus neg-
atively influence mathematical achievement. However,
Morgan et al. (2015) noted that students who are strug-
gling with mathematics benefit from teacher-directed
instruction more than student-centered instruction.
Thus, more research is needed to understand which
type of instruction works better for students depending
on their mathematical performance level. Lastly, cogni-
tive activation positively influenced mathematical
achievement, supporting previous findings (Burge
et al., 2015; Lampert, 2001; Sizmur et al., 2015) that
classes emphasizing students’ cognitive activities could
help students construct their knowledge and show
improved mathematical performance.
This study’s results contribute to the existing litera-
ture by using a large nationally representative data set
of 15-year-old students across the United States to
examine student and teacher factors related to student
mathematical achievement. The study emphasizes the
importance of student mathematics interest, student
mathematics self-efficacy, the student–teacher relation-
ship, teacher cognitive activation, and classroom
management, providing implications for educators.
Specifically, interventions designed to improve stu-
dents’ mathematics interest and mathematics self-
efficacy may promote students’ mathematical achieve-
ment. We have found that teacher characteristics had
the most effect on students’ mathematical achievement,
especially the student–teacher relationship. It is widely
acknowledged that teachers are an essential component
in education (Driel et al., 2001) and determine the class
(Borich, 2000). Building positive relationships with stu-
dents, using cognitive activation, and more student-
centered instruction are suggested to improve mathe-
matical performance. This can come from inquiry-
based activities that are student-centered and challenge
students’ mathematical thinking while listening to and
encouraging students (Savery, 2006). Fostering these
student and teacher characteristics can be the first
step in improving U.S. students’ mathematical achieve-
ment and, ultimately, rankings in international mathe-
matical achievement comparisons.
Limitations and Future Directions
The findings of this study should be interpreted with
consideration of the study’s limitations. The study
examined the influence of teacher and student charac-
teristics on student mathematical achievement.
Although the student characteristics mostly focused
on mathematics-related factors, out-of-school study
time was the amount of time spent studying any sub-
ject. Out-of-school study time specific to mathematics
may have more substantial impacts on mathematical
achievement. In future studies, researchers need to
investigate other factors that might influence mathe-
matical achievement, such as cognition, metacognition,
Future studies could also expand on the current
findings by using various achievement scores (i.e., read-
ing and science) and including family and peer charac-
teristics. Additionally, the present study relied on
students’ self-reported data. Although it is arguably a
unique source of information regarding student and
teacher characteristics, it is possible that variables relat-
ed to student perceptions of teacher characteristics may
be underreported or overreported by the students. The
present study was also limited to only teacher-related
factors at the school level and did not examine other
contextual variables such as peer factors. Despite these
limitations, however, this study’s results contribute to
our understanding of various student and teacher char-
acteristics and their impacts on student mathematical
Declaration of Conflicting Interests
The authors declared no potential conflicts of interest with
respect to the research, authorship, and/or publication of this
The author(s) disclosed receipt of the following financial sup-
port for the research, authorship, and/or publication of this
article: This work was partially supported by the Hankuk
University of Foreign Studies research fund granted to
Sukkyung You https://orcid.org/0000-0002-7779-7965
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