HOW CONFIDENCE RELATES TO MATHEMATICS
ACHIEVEMENT: A NEW FRAMEWORK
Lesa M Covington Clarkson
Quintin U Love
Forster D Ntow
University of Minnesota
While there is increasing acceptance within mathematics education that confidence
plays a crucial role in students’ mathematics achievement, not much is known about
the nature of this relationship and students’ confidence profiles. This paper uses the
Trends In Mathematics and Science Study (TIMSS) Student Survey data to establish
relationships between students’ level of proficiency and how they describe themselves
in terms of confidence doing mathematics. Based on a set of eighth questions from
the TIMSS survey a theoretical model influenced by the Johari Window was
developed. In this paper, we discuss the development of this model and its
implications to instruction.
INTRODUCTION
Historically, eighth grade students in the U.S. who participated in algebra were
successful white male math students from upper socioeconomic status (Flores, 2007;
Oakes, Ormseth, Bell, & Camp, 1990, Sells, 1973). Recognizing the disproportionate
number of minorities and females in Algebra 1, researchers and educational policy
makers supported the movement to offer “algebra for everyone” (NCTM, 1992). The
“algebra for all” movement began as an initiative designed to increase equity and
diversity amongst the population of students taking Algebra 1 by providing an
opportunity for students from all backgrounds (Moses & Cobb, 2001; Richardson,
Ball, & Moses, 2009). The challenge, however, is that not all eighth grade students
are well prepared for this course leading to persisting achievement gap. The reported
achievement gap necessitates an understanding of the underlying factors rather than
engaging in stereotyping of certain student groups (Gutiérrez & Dixon-Román, 2011;
Martin, 2012).
Recent research focusing on the affective domains suggests that under-preparedness
(prior ability) may not be the only reason and that psychological constructs may also
be related to student achievement in mathematics (Reyes, 1984; Reyes & Stanic,
1988). One of such affective constructs is students’ mathematical confidence which is
believed to serve as a mediator between their motivation and achievement levels.
Morony, Kleitman, Lee& Stankov (2013) in an investigation of the structure and
cross-cultural (in)variance of mathematical self-beliefs between Confucian and
selected European countries reported that confidence is the “single most important
predictor of math accuracy” (p.1). Despite issues related to affected domains such as
confidence being seen as vital in explaining students’ achievement in mathematics,
Op ‘T Eynde, deCorte and Verschaffel (2002) noted that “research on this topic has
not yet resulted in a comprehensive model of, or theory…” where confidence was
taken as part of four characteristics of beliefs (cited in Burton, 2004; p. 358). Morony,
et al., (2013) also called for further research that highlights the extent to which
students’ calibrations of their confidence both reflect and enhance their learning. The
model presented in this paper responds to these calls for further research on students’
confidence and its relationship to their achievement levels. The current study is not
experimental, nor is it attempting to identify causal relationships in mathematics
achievements. Instead, it is a quantitative study that examined the relationship
between students’ mathematics confidence and mathematics achievement which was
followed by the development of a model, CCL Confidence/ Achievement Window that
is related to the Johari Window (Luft, 1969). The following research questions guided
this study: 1) What is the relationship between mathematics confidence and
mathematics achievement? 2) Do higher levels of confidence towards mathematics
lead to higher levels of mathematics achievement? 3) What is significant about the
relationship between confidence and achievement?
LITERATURE REVIEW
Confidence “consists of an individual’s perception of self with respect to achievement
in school” (Reyes 1984, p. 559) and is related to an individual’s self-concept. Pajares
and Miller (1994) define self-efficacy as “a concept-specific assessment of
competence to perform a specific task” (p. 194). Based on these definitions, selfconcept (domain specific) and self-efficacy (item specific) differ at the level of the
construct.
Consistently, both researchers use the term confidence in their definitions. Reyes
states,
Confidence in learning mathematics is a particular component of self-concept
that is specific to mathematics…Confidence in learning mathematics, or selfconcept specific to mathematics, has to do with how sure a person is of being
able to learn new topics in mathematics…. (pp. 560- 561).
Pajares and Miller (1994) report, “It is clear that beliefs regarding confidence are part
of an individual’s self-concept…confidence in learning mathematics [is the]
conceptual forerunner to math self-efficacy” (p. 194). Despite other constructs such
as self-concept and self-efficacy being used interchangeably in the literature, there is
an increasing use of the term confidence as a bridge between these two constructs.
The importance of confidence in explaining students’ motivation and success levels
has led to a number of studies being carried out to determine how well students are
able to calibrate their confidence (e.g. Atherton, 2015; Foster, 2016; Hong, Hwang,
Tai, & Chen, 2014).
Calibration is used to generally refer to how well a student’s confidence and
competence match (Foster, 2016). Empirical studies reveal that there are three ways
in which students calibrate themselves namely: under calibration (high achievement
versus low confidence), over-calibration (low achievement versus high confidence),
and high achievement versus low confidence being well-calibrated (good match
between achievement and confidence) (Foster, 2016). A consistent finding from the
calibration studies is that students tend to be poorly calibrated leading to the
confidence-achievement paradox (e.g. Atherton, 2015; Foster, 2016). For example,
Atherton (2015) reported that females were more likely to be less confident compared
to males. A way of understanding students’ confidence level profile, therefore, is
useful in determining students who know what they know and those who project
over-confidence relative to their actual achievement level. Such a profile, we argue,
will help teachers implement measures that can help students better calibrate their
mathematical confidence and know when to ask for help instead of being overly
confident and obtaining a wrong answer (Fischhoff, Slovic, & Lichtenstein, 1977;
Foster, 2016).
METHODOLOGY
Ladson-Billings (1997) suggests that even when a study is focusing on a specific
group, in this case eighth grade students, it is important “to situate it in the larger
context of mathematics teaching and learning…” (p. 698). For that reason, we begin
our analysis using the U.S. TIMSS data to situate our study in the larger context of
the U.S. We also explored the findings from the TIMSS data analysis by focusing on
journal entries collected from a sample of African American students (although this is
not included in this paper). We chose to focus on a sample of African American
students because data concerning the mathematics achievement of African American
students indicates a prominent achievement gap across various levels and concerns
about instructional quality (Hallett & Venegas, 2011; Horn, 2008; Ladson-Billings,
2006; Palardy, 2015). Hallett and Venegas (2011) in a study of college-bound high
school students concluded that despite increased access to Advanced Placement
courses for students enrolled in low-income urban high schools, students in the study
did not have a sense of being adequately prepared for college. Subsequently, research
is warranted given the disparity among student groups.
Data
The International Association for the Evaluation of Educational Achievement
developed the Trends in International Mathematics and Science Study (TIMSS) to
measure the mathematics and science skills of fourth and eighth grade students
(Williams, Ferraro, Roey, Brenwal, Kastberg, Jocelyn, Smith, & Stearns, 2009). The
United States was one of the 58 participating countries during the TIMSS 2007. The
United States data is available for public use from the National Center for Education
Statistics’ (NCES) Trends in International Mathematics Science Study (TIMSS) 2007
U.S. public use data file (2009).
Sample
The TIMSS 2007 data was collected on a national probability sample of students
nearing the end of their fourth and eighth year of school (Williams, Ferraro, Roey,
Brenwal, Kastberg, Jocelyn, Smith, & Stearns, 2009). This study’s focus is on the
sample of eighth grade students. There are 7,377 eighth grade students sampled from
239 schools across the United States. Of the 7,377 students, 3,656 are males and
3,721 are females. The sampling design of the TIMSS data led to a racially diverse
sample which is relatively proportional to the population of 8th grade students. Of the
7,282 students with valid data, 3,873 are White, 949 are Black, 1,787 are Hispanic,
243 are Asian, 90 are Native American, 58 are Pacific Islanders, and 282 are
Multiracial (2 or more races). Finally, the sample included students with varying
levels of socioeconomic status (measured with Mother’s education level).
Student Questionnaire
The TIMSS 2007 assessment framework documents the importance of contextual
factors on students’ learning; therefore, all participating schools administer
questionnaires at the school, teacher, and student levels (Mullis, Martin, Ruddock,
O’Sullivan, Arora, & Erberber, 2005). The IEA conducts extensive analyses to verify
the reliability and validity of all instruments used during TIMSS administration, and
publishes the information in the TIMSS 2007 Technical Report and User Guide
(Olson, Martin, & Mullis, 2008). We chose to focus only on data collected from the
student questionnaire. The student questionnaires collect information on students’
home background and attitudes towards mathematics and science (Olson, Martin, &
Mullis, 2008). This study examines the relationship between confidence and math
achievement; consequently, only the items of interest are further discussed.
Independent Variables
Eight items from the student questionnaire are used in this study. We theorized that
the following items measure the latent construct of mathematics confidence: I usually
do well in math (BS4MAWEL), I would like to take more math (BS4MAMOR),
Math is more difficult for me (BS4MACLM), I enjoy learning math (BS4MAENJ),
Math is not one of my strengths (BS4MASTR), I learn things quickly in math
(BS4MAQKY), Math is boring (BS4MABOR), and I like math (BS4MALIK; see
Table X). All of the eight items are measured using a four- point Likert scale, where
“Strongly Agree” = 1, “Agree” = 2, “Disagree” = 3, and “Strongly Disagree” = 4.
However, we decided to recode the variables in reverse order, so that larger values
suggest more confidence and interpretation is straightforward. The independent
variable is consistent, scaled -1 to 1 standardized with a mean of 0 and standard
deviation of 1.
Dependent Variables
The dependent variable for this study is mathematics achievement. The TIMSS 2007
data provides five plausible values of mathematics achievement for each student:
BSMMAT01-BSMMAT05. All analyses use the five plausible values as the measure
of students’ mathematics achievement. The use of the plausible values ensures
accuracy of students’ achievement (Williams, Ferraro, Roey, Brenwal, Kastberg,
Jocelyn, Smith, & Stearns, 2009).
Criterion Profile Method
Profile analysis is a statistical procedure used to identify profiles patterns based on
membership of a small subsample. Davison and Davenport (2002) proposed the
Criterion Profile Method (CPM) as an alternative to existing profile analysis
procedures. An advantage of CPM is that the profiles are related to an external
criterion; therefore, the validity of the procedure is greater than other profile analysis
procedures (Culpepper, 2008; Culpepper, Davenport, & Davison, 2008; Davison &
Davenport, 2002). The CPM has been used to study out of school activities that are
related to reading achievement (Culpepper, 2008) and the relationships between race,
academic achievement, and college acceptance (Culpepper, Davenport, & Davison,
2008). CPM is a regression based procedure; consequently, the observed profiles are
related to an external criterion (Dependent Variable). For this reason, the current
study utilizes the CPM model to test the relationship between confidence and math
achievement. Even though profile analysis is a statistical procedure frequently used in
psychology, there is not a significant amount of literature that employs the CPM.
More details of this model and analyses are available in a forthcoming paper.
The Johari Window
The Johari Window, developed by Joseph Luft and Harry Ingram (1969), is a fourpane window that symbolizes information known to ourselves and information
known to others. This interpersonal communication tool has been used in diverse
venues like psychology and leadership to describe the degree and depth of knowledge
individuals have of themselves that is the same or different from the information
others have of the individual. According to Wade and Hammick (1999), one purpose
of the Johari Window is to “help people understand themselves, and to acknowledge
barriers and defences (sic) that may be erecting subconsciously...[and can be used to
help] develop more insight and self awareness (sic)” (p. 10). This, therefore forms the
basis for the connection between the Johari Window and the CCL Confidence
Window.
Known to Self
Not Known to Self
I
II
Area of
Free Activity
(Public)
Known to Others
Blind
Area
III
Not Known
to Others
IV
Avoided or
Hidden Area
Area of
Unknown Activity
Figure 1: The Johari Window
In the diagram (see Figure 1), quadrant one (in the upper left-hand corner) describes
what is known to self and others, the Public area. Quadrant two, upper right-hand
corner, is known as the Blind spot; that is, information known to others but unknown
to self. The Hidden area is in the lower left-hand corner and describes what is known
to self but unknown to others. Quadrant four, lower right-hand pane, represents an
area that is Unknown to self and others.
We combined the concept of the four-panes of the Johari Window with the Cartesian
Coordinate Plane (x- and y-axes) from mathematics to reorganized the information
into quadrants with values greater than or equal to zero or less than or equal to zero
(see Figure 2).
High Achievement
II (c, A)
I (C, A)
Low Confidence
High Confidence
High AchievementHigh Achievement
Low
Confidence
(Blind)
III (c, a)
(Public)
IV (C, a)
High
Confidence
Low Confidence
Low Achievement
(Unknown)
High Confidence
Low Achievement
(Hidden)
Low Achievement
Figure 2: The CCL Confidence Achievement Window (The Johari Window labels
parenthetically)
Graphing dependent and independent variables on this plane disaggregates ordered
pairs into four regions similar to but placed differently from the Johari Window. Our
two characteristics, confidence as a result of the TIMSS confidence items and
mathematics achievement as a result of the TIMSS test, are assigned values between
-1 and 1. Thus, we redefined the quadrants as positive confidence, higher
achievement scores in the upper right-hand corner (+, +); lower confidence and
higher achievement scores in the upper left-hand corner (-, +); lower confidence and
lower achievement in the lower left-hand corner (-, -); and low achievement despite
high confidence in the
lower right-hand quadrant (+, -). While the newly defined window does not match the
corresponding regions of the Johari Window, the characteristics of the premises of the
quadrants’ similarities remain; and our new window closely mimics the mathematical
meaning of the Cartesian coordinate system. The notation that we have selected to
represent each quadrant uses an upper case A for positive achievement, an upper case
C for positive confidence, a lower case a for low achievement and a lower case c for
low confidence. Thus, the notation “(c, A)” represents the second quadrant because of
negative confidence level and high achievement.
The characteristics of the Public area align to the new high confidence-high
achievement quadrant (C,A); here the student and others have confidence in the
student’s ability to perform mathematically. The student’s confidence is intrinsic and
the high achievement is externally positive as well. In other words, both the
individual and those around the student have evidence of success in mathematics. The
Hidden area (unknown to others but known to self) is similar to the new quadrant
where student’s confidence is high but achievement is low (C, a). The student has
confidence in their mathematical performance but has not demonstrated their ability
to others. At this point, it is unclear where the root of the student’s confidence lies.
Negative confidence and high achievement (c, A) is analogous to the Blind area.
Here, students lack confidence in their mathematical ability, yet their performance on
the math assessment is inversely high. The characteristics of this quadrant are also
difficult to explain so early in the research project. The Unknown area is parallel to
the student and the public lacking confidence in the student’s ability to perform
mathematically (c,a). The student does not believe s/he can do math and others are
not able to assume the student is able to perform positively based on past
performances.
RESULTS
The complex sampling design employed by the TIMSS 2007 data requires the use of
software which can properly estimate the standard error of parameter estimates
(Williams, Ferraro, Roey, Brenwal, Kastberg, Jocelyn, Smith, & Stearns, 2009).
Therefore, the analyses for this paper were generated using SAS® software, version
9, and AM version 0.06.03 (American Institute for Research & Cohen, 2005). In
addition, all variables are standardized to have a mean of zero and standard deviation
of one. This was necessary because CPM requires that all variables are on the same
scale (Culpepper, 2008; Davison & Davenport, 2002).
Students’ responses on the eight items of interest are used in the analyses. These
responses only include those from the TIMSS 2007 sample of students. There is great
variation in most of the responses on the items of the student questionnaire. The only
item that does not have a “balanced” distribution of responses is “I usually do well in
math”. Even though only five percent of respondents disagreed a lot with this item, it
was retained for further analyses.
Next, a correlation analysis using the eight items of interest reveals that 26 of the 28
bivariate correlations are above 0.3. This suggests that these items may share some
common variance. As a result, an exploratory factor analysis procedure using
principal axis factoring was performed to test the dimensionality of the eight items
used for the mathematics confidence scale. The results indicate that the items are
measuring a single latent construct. The factor analysis revealed that the dimension
has an Eigenvalue of 3.98 and accounts for 94% of the extracted variance. Finally, a
reliability analysis revealed that the items compose a highly reliable scale (α = .815).
Based on the description of the items, we decided that the latent construct measured
mathematics confidence. Next, we used the items from the mathematics confidence
scale and students’ plausible values on the TIMSS mathematics assessment to test the
relationship of mathematics confidence and mathematics achievement.
A standard multiple regression was performed between the TIMSS math assessment
and the items from the Confidence scale. The Multiple Regression (MR) model was
found to be significant, F (8, 68) = 175.06, p < .001; and accounted for 22.5% of the
variation in the TIMSS mathematics assessment score (R 2 = .225). The results from
the MR model suggest that the item “I would like to take more math” (BS4MAMOR)
does not significantly add to the model; however, we decided to keep it in the model
because of its strong correlation with the scale. Overall, the MR model suggests that
the items on the confidence scale are related to the TIMSS mathematics assessment,
and that they may aid in the prediction of students’ mathematics ability.
Next, a Profile Analysis was performed using the Criterion Profile Methodology
(CPM) model (Davison & Davenport, 2002). The CPM model was significant, F (2,
74) = 709.00, p < .001; and accounted for 22.5% of the variation in the TIMSS math
assessment scores (R2 = .225). The squared multiple correlations of the MR and CPM
are exactly the same, which suggests that the variance from the confidence scale is
partitioned into the level and pattern statistics.
Accordingly, two models were fit to the data, using the level and pattern as the sole
predictors for each model. This allows for the researchers to test the amount of
variation accounted for by the level ( X́ p) and pattern (Covp) statistic individually
(Davison & Davenport, 2002). The results indicate that both the model with level as
the sole predictor, F (1, 75) = 121.16, p < .001, and the model with pattern as the sole
predictor, F (1, 75) = 1401.08, p < .001, are statistically significant. Additionally, the
results indicate that the pattern (R2 = .225) accounts for more variation than the level
(R2 = .027). This suggests that there is a stronger relationship between the pattern of a
student’s confidence profile and mathematics achievement than that of the level of a
student’s confidence profile and mathematics achievement. That is, the results from
the CPM analyses suggest that a highly confident student may not have a high
mathematics achievement scores. Likewise, the results of the analyses suggest that
students doing well in math may not want to take mathematics or enjoy mathematics.
Instead, it is important that students feel that they are doing well and enjoy math
class. These findings have the potential to greatly add to mathematics education
research; therefore, the next step in the analysis is to answer a very important, yet
simple question: WHY?
DISCUSSION
Participation in mathematics tends to be distorted with a one size fits all policy for all
students which assumes that all students need the same fixing without a deep
understanding of the underlying issues surrounding achievement levels (Meaney.,
Trinick., & Fairhall, 2013). Foster (2016) asserts that:
…a person’s knowing whether they know is extremely important, as it
determines whether additional support (from other people, a computer or a
reference source) is required. For example, it is much better to know that you
do not know the value of −2+7, although you think that it might be −5, than it
is to be sure that it is −5 when in fact it is 5. (p.272; emphasis in original).
The CCL Confidence/Achievement Window reported in this paper builds on that of
Foster (2016) by classifying the various ways in which students can calibrate
themselves. The use of the Johari Window, we argue, makes it possible for teachers to
be aware of the unique characteristics students thereby serving as an important step
away from treating all students the same way. By identifying the size of panes like
Hidden or Blind, individuals are able to increase the effectiveness of their interactions
by shrinking these areas in order to open up other panes like Public. Likewise,
identifying the confidence and achievement patterns of mathematics students,
perhaps we as educators will be able to prescribe interventions specific to students
with varying levels of achievement.
REFERENCES
American Institute for Research & Cohen, J. (2005). AM Statistical Software
(Version 0.06.03) [Computer Software]. Retrieved from
http://am.air.org/download.asp
Atherton, M. (2015). Measuring confidence levels of male and female students in
open access enabling courses. Issues in Educational Research, 25(2), 81–98.
Burton, L. (2004). Confidence is everything”–Perspectives of teachers and students
on learning mathematics. Journal of Mathematics Teacher Education, 7(4),
357–381. http://doi.org/10.1007/s10857-004-3355-y
Fischhoff, B., Slovic, P., & Lichtenstein, S. (1977). Knowing with certainty: The
appropriateness of extreme confidence. Journal of Experimental Psychology:
Human Perception and Performance, 3(4), 552–564.
http://doi.org/10.1037//0096-1523.3.4.552
Foster, C. (2016). Confidence and competence with mathematical procedures.
Educational Studies in Mathematics, 91(2), 271–288.
http://doi.org/10.1007/s10649-015-9660-9
Good, C., Aronson, J., & Inzlicht, M. (2003). Improving adolescents’ standardized
test performance: An intervention to reduce the effects of stereotype threat.
Journal of Applied Developmental Psychology, 24(6), 645–662.
http://doi.org/10.1016/j.appdev.2003.09.002
Gutiérrez, R., & Dixon-Román, E. (2011). Beyond gap gazing : How can thinking
about education comprehensively help us (re)envision mathematics education?
In B. Atweh, M. Graven, W. Secada, & P. Valero (Eds.), Mapping Equity and
Quality in Mathematics Education (pp. 21–34). Dordrecht: The Netherlands:
Springer Netherlands.
Hallett, R. E., & Venegas, K. M. (2011). Is increased access enough? Advanced
placement courses, quality, and success in low-income urban schools. Journal
for the Education of the Gifted, 34(3), 468–487.
http://doi.org/10.1177/016235321103400305
Hong, J. C., Hwang, M. Y., Tai, K. H., & Chen, Y. L. (2014). Using calibration to
enhance students’ self-confidence in English vocabulary learning relevant to
their judgment of over-confidence and predicted by smartphone self-efficacy
and English learning anxiety. Computers and Education, 72, 313–322.
http://doi.org/10.1016/j.compedu.2013.11.011
Horn, I. S. (2008). Turnaround students in high school mathematics: Constructing
identities of competence through mathematical worlds. Mathematical Thinking
and Learning, 10(3), 201–239.
Ladson-Billings, G. (2006). From the achievement gap to the education debt:
Understanding achievement in U.S Schools. American Educational Research
Journal, 35(7), 3–12.
Martin, D. B. (2012). Learning Mathematics while Black. Educational Foundations,
26, 47–66.
Meaney, T., Trinick, T., & Fairhall, U. (2013). One size does not fit all: Achieving
equity in Māori mathematics classrooms. Journal for Research in Mathematics
Education, 44(1), 235–263. http://doi.org/10.5951/jresematheduc.44.1.0235
Morony, S., Kleitman, S., Lee, Y. P., & Stankov, L. (2013). Predicting achievement:
Confidence vs self-efficacy, anxiety, and self-concept in Confucian and
European countries. International Journal of Educational Research, 58, 79–96.
http://doi.org/10.1016/j.ijer.2012.11.002
Palardy, G. (2015). Classroom-based inequalities and achievement gaps in first
grade : The role of classroom context and access to qualified and effective
teachers. Teachers College Record, 117(February), 1–48.
National Center for Education Statistics. (2009). Trends in International Mathematics
Science Study (TIMSS) 2007 U.S. public use data file [Data file and codebook].
Retrieved from http://nces.ed.gov/pubsearch/pubsinfo.asp?pubid=2010024
Olson, J.F., Martin, M.O., & Mullis, I.V.S. (2008). TIMSS 2007 Technical Report.
Chestnut Hill, MA: Boston College. Retrieved from
http://timss.bc.edu/TIMSS2007/PDF/TIMSS2007_TechnicalReport.pdf
Pajares, F. & Miller, M.D. (1994). Role of self-efficacy and self-concept beliefs in
mathematical problem solving: A path analysis. Journal of Educational
Psychology, 86(2), 193 – 203.
Palardy, G. (2015). Classroom-based inequalities and achievement gaps in first
grade : The role of classroom context and access to qualified and effective
teachers. Teachers College Record, 117(February), 1–48.
Reyes, L.H. (1984). Affective variables and mathematics education. The Elementary
School Journal, 84(5), 558 – 581.
Williams, T., Ferraro, D., Roey, S., Brenwal, S., Kastberg, D., Jocelyn, L., Smith, C.,
and Stearns, P. (2009). TIMSS 2007 U.S. Technical Report and User Guide
(NCES 2009 – 012). National Center for Education Statistics, Institute of
Education Sciences, U.S. Department of Education. Washington, DC.
Retrieved from http://nces.ed.gov/pubs2009/2009012.pdf