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ORI GIN AL PA PER
Classroom instructional quality, exposure to mathematicsinstruction and mathematics achievement in fifth grade
Erin R. Ottmar • Lauren E. Decker • Claire E. Cameron •
Timothy W. Curby • Sara E. Rimm-Kaufman
Received: 27 May 2011 / Accepted: 26 May 2012 / Published online: 15 October 2013� Springer Science+Business Media Dordrecht 2013
Abstract This study examined the quality of teacher–child interactions and exposure to
mathematics instruction as predictors of 5th grade student’s mathematics achievement. The
sample was a subset of the children involved in the NICHD–SECC longitudinal study
(N = 657). Results indicate that, even after controlling for student demographic charac-
teristics, more exposure to mathematics instruction was related to increased fifth grade
mathematics achievement for both calculations and applied problems assessments, but
there was no main effect for improved instructional quality. Findings also indicate that, in
classrooms where lower instructional quality was observed, greater exposure to mathe-
matics instruction predicted improved mathematics achievement. Findings are discussed in
terms of differing aspects of mathematics instruction and the possible compensatory role of
exposure to instruction in classrooms of lower quality.
Keywords Exposure � Instructional quality � Mathematics achievement
Introduction
Examining the relation between the instruction that occurs in mathematics classrooms and
student achievement is a particularly timely and relevant topic. American schools are
E. R. Ottmar (&)Department of Psychology, University of Richmond, 28 Westhampton Way, Richmond, VA 23173,USAe-mail: [email protected]
L. E. DeckerEdvance Research, Inc., San Antonio, TX, USA
C. E. Cameron � S. E. Rimm-KaufmanUniversity of Virginia, Charlottesville, VA, USA
T. W. CurbyGeorge Mason University, Fairfax, VA, USA
123
Learning Environ Res (2014) 17:243–262DOI 10.1007/s10984-013-9146-6
experiencing a crisis in mathematics education and mathematics performance in the United
States lags behind other industrial nations at a time when growing dependence on tech-
nology makes mathematics competency essential (American Institutes for Research 2005;
National Science Board 2006).
Past research has suggested that America’s low mathematics performance escalates
during the upper elementary years when students’ positive attitudes towards mathematics,
allocated time spent on teaching and learning mathematics, and achievement all drop
significantly (Porter 1989). Although educational and developmental research suggests that
mathematics achievement trajectories begin to stabilize by third grade (Entwisle and
Alexander 1999; Pianta et al. 2008a, b), meta-analytic work describing the magnitude of
gains in achievement based on a national norming sample show that, on average, children
continue to demonstrate gains during the fifth grade year (Hill et al. 2005). This upper
elementary performance issue has also been found at the national level; the Nation’s
Report Card shows that 18 % of 4th grade students in the United States performs below
basic levels in mathematics (National Center for Educational Statistics 2011). Further, the
fifth grade year warrants attention as it marks the beginning of the transition to middle
school, when mathematics instruction becomes increasingly departmentalised and tracked
(Dauber et al. 1996), children’s motivation often wanes (Anderman et al. 1999) and gains
in mathematics achievement decelerate (Hill et al. 2005).
These findings lead to a line of inquiry into what occurs inside upper elementary
mathematics classrooms that contributes to (or diminishes) children’s mathematics
achievement. There are two common, yet divergent approaches to addressing questions
about what occurs inside mathematics classrooms: the first focuses on the quality of
classroom instructional interactions between the students and the teacher (Roeser et al.
2000; Hamre and Pianta 2001) and the second emphasizes the amount of exposure to
mathematics instruction (Pianta et al. 2008; Sanford and Evertson 1983). Recommenda-
tions from the National Council of Teachers of Mathematics also encompass both the
quality of mathematics instruction (NCTM 2000) and recommendations for the minimum
length of time for which students should experience such instruction (NCTM 2006). While
such recommendations are theoretically sound, there is a need for empirical work to
support or question their evidentiary basis. However, little is known about the individual
and interactional importance of both facets (quality and exposure) of teaching within
classroom instruction.
Taken together, the questions about how quality and exposure matter for children’s
mathematics achievement in the upper elementary school years stand out. Our study
investigated the contribution of the quality and exposure to mathematics within a single
year of upper elementary schooling, which is a developmentally-sensitive period in chil-
dren’s lives.
Quality of mathematics instruction
In our work, we define high-quality instruction in terms of the nature of teacher–child
interactions in the classroom (Pianta and Hamre 2009). By definition, high-quality
instructional interactions include how teachers provide feedback and use language to
promote learning, higher-order thinking and understanding of concepts. These examples of
instructional quality refer to teachers’ efforts to promote an environment of learning and
inquiry (Pianta et al. 2008). Teachers high in instructional quality set the stage for learning
by providing a variety of cognitive tasks, forming connections, scaffolding knowledge and
helping students to develop meta-cognition. For example, teachers who offer sufficient and
244 Learning Environ Res (2014) 17:243–262
123
specific feedback to students, present cognitively demanding tasks, and lead rich discus-
sions can support students’ ability to conduct mathematical computations as well as to
think abstractly about mathematics (Hiebert and Grouws 2007).
Several studies have shown that high instructional quality from teachers is a vital
contributor to students’ mathematics achievement (Matsumura 2002; Meyer et al. 1993;
NICHD 2005; Pianta et al. 2008). Teachers high in instructional quality monitor students’
progress and performance and provide explanations and ideas for higher learning. In
addition, high-quality instruction—requiring concept-focused, rather than answer-focused,
instruction—appears especially important for mathematics, for which learning is incre-
mental and students must draw upon prior knowledge to solve new problems (Kilpatrick
et al. 2001). High-quality instruction appears to play an important role in students’
achievement. For example, Hamre and Pianta (2005) found that first grade students who
were at-risk for school failure, yet experienced positive instructional interactions, achieved
comparably to their counterparts who were not at risk for school failure.
Descriptive analyses have shown that teachers are highly variable in the extent to which
they provide high-quality instruction (Curby et al. 2011; Hanushek et al. 2005). With
regards to mathematics instruction specifically, Nye et al. (2004) found significant varia-
tion between teachers in classroom quality and teacher effectiveness, with up to three times
more variation than reading. While this variability is problematic for students who wind up
having teachers who offer lower-quality interactions, the naturally occurring variation does
enable researchers to better investigate relationships between quality and achievement in
mathematics. However, the quality of teachers’ instruction is unlikely to be the sole
classroom predictor of mathematics achievement, particularly if students spend little time
on mathematics activities (Berliner 1987, 1990). Therefore, it is important to also consider
the amount of time, or exposure, to mathematics instruction that students are afforded.
Exposure to mathematics instruction (quantity)
Classic literature on student learning provides an important starting point for understanding
the impact that exposure to instruction can have on student achievement. Carroll’s (1963)
model of school learning advanced that school learning is a function of opportunity to
learn, measured by allocated time spent teaching or the number of minutes students spent
on schoolwork. Process–product literature (Brophy and Good 1986; Dunkin and Biddle
1974; Purvis and Levine 1975) built on this work by focusing on the impact of time on
learning. These studies established a direct relation between time and learning, showing
that learning and achievement increases when students are engaged in learning activities
for greater amounts of time. For example, students who spend more time in mathematics
instruction show higher levels of achievement than their counterparts who spend less time
in instruction (Caldwell et al. 1982; Gettinger and Siebert 2002).
NCTM recommends that students receive at least 1 h of focused mathematics
instruction per day (NCTM 2006), and many standards-based curricula require teachers to
spend at least 90 min in a single mathematics lesson (Stein et al. 2007). However,
observational work suggests that only 13 % of total instructional time (on average, 45 min
per day) in the United States is spent teaching mathematics content (Hiebert 2003; Phelps
et al. 2012). Similar to quality, however, the amount of time that typical American ele-
mentary teachers spend teaching mathematics varies greatly. One seminal study revealed
that allocated time to mathematics in fifth-grade ranged from 18 to 80 min per day
(Caldwell 1982), suggesting that some students have up to 4 times more opportunity to
learn mathematics compared with other students of the same age. Current research suggests
Learning Environ Res (2014) 17:243–262 245
123
this variability persists (NCES 2011; Phelps et al. 2012). While states and school districts
have begun to require specific amounts of time for each subject, it is common for teachers
to create their own schedules, which can vary significantly from district requirements
(Porter 1989). Moreover, because of transition time and other non-instructional distrac-
tions, researchers have found that only 50–60 % of allocated time is actually spent on
instructional activities, with the other 40–50 % devoted to classroom procedures and non-
instruction, such as transitions (Hollowood et al. 1995).
However, studies have found that more time exposed to mathematics instruction con-
tributes positively to academic growth. Berliner (1990) reported that up to 36 % of vari-
ance in achievement is attributable to time exposed to instruction. Germane to the question
of exposure, Bodovski and Farkas (2007) found that more time spent on mathematics in
classrooms contributed to growth in mathematics achievement for lower-achieving, as well
as higher-achieving students. Similarly, several other studies found that greater exposure to
mathematics instruction resulted in increased mathematics scores (Entwisle and Alexander
1999; Nye et al. 2004; Wang and Goldschmidt 1999). The rationale behind this explanation
is straightforward: students who receive more exposure to a topic are more likely to learn it
than students who receive less exposure (Berliner 1990). Yet, few large observational
studies have directly examined this hypothesis with regard to mathematics learning in
upper-elementary school, particularly while controlling for levels of instructional quality.
It appears that only one study involved how exposure and quality of instruction,
together, contribute to achievement trajectories in elementary school (Pianta et al. 2008a,
b). Using exposures to quality and quantity from 54 months to fifth grade, the authors
found that emotional quality and higher exposure to mathematics activities related to
growth in mathematics achievement trajectories. However, these results were not found for
instructional quality. Taken together, past research suggests that both the quality of, and
exposure to, mathematics instruction are important variables for understanding upper
elementary mathematics achievement; however, little is known about the potential inter-
action between quality and exposure in upper elementary school.
The present study
Building on these past findings, we conceptualised a quadrant that defines the relations
between quality and exposure to mathematics instruction and fifth graders’ mathematics
outcomes (see Table 1). In quadrant I, low instructional quality paired with low exposure
to mathematics instruction is expected to predict the lowest levels of achievement. This is
based on the rationale that, if students are taught little mathematics and the instruction is of
low quality, then they will not learn as much mathematics. In quadrant IV, it is expected
that high-quality instruction coupled with high exposure to mathematics would predict the
highest levels of achievement. Students who are consistently exposed to quantities of high-
quality instruction are likely to perform well on mathematics tasks. However, with regard
to quadrants II and III, little is known about how these combinations of quality and
quantity might be associated with mathematics achievement. Thus, we examined the rel-
ative contributions of instructional quality and exposure to instruction to students’ math-
ematics achievement, as well as the interactive nature of the two.
Two research questions and hypotheses were posited in the present study. First, does
instructional quality and the amount of exposure to mathematics instruction independently
predict mathematics achievement in the fifth grade? We hypothesized that both higher
instructional quality and more exposure to mathematics instruction would each indepen-
dently contribute to students’ higher mathematics achievement, even after controlling for
246 Learning Environ Res (2014) 17:243–262
123
demographic and classroom variables. Second, do varying levels of quality and exposure to
instruction interact to predict mathematics achievement? In other words, we hypothesized
that increased exposure to mathematics instruction might compensate for low-quality
classroom instruction, so that greater exposure to mathematics instruction would be
associated with similar levels of achievement regardless of quality; similarly, higher-
quality instruction could compensate for low exposure to mathematics. The current study
provided an opportunity to consider the extent to which quality and exposure contribute
differentially in predicting students’ mathematics achievement for a sample of children
from 10 of the 50 US states. We tested two models, including the quality and exposure to
instruction for two mathematics outcomes: Calculations (computational ability) and
Applied Problems (conceptual problem-solving ability).
Method
Data for this study were taken from a subsample of participants from the National Institute
for Child Health and Human Development (NICHD) Study of Early Child Care and Youth
Development (SECCYD). Data for the larger study were gathered from 10 sites as part of a
large longitudinal field investigation by the NICHD Early Child Care Research Network
(NICHD-ECCRN). Between January and November 1991, families were recruited through
hospital visits to mothers soon after the birth of their child. Participants were recruited
from hospitals at 10 different geographic locations around the United States. All women
who gave birth were screened for eligibility and for their willingness to be contacted for the
study. From an original sample of 8,986 mothers who gave birth during the time period,
5,416 met eligibility requirements and were willing to be contacted. Mothers were eligible
if they were healthy, over 18 years old, spoke English, were not giving up their child for
adoption, did not have multiple births, lived within an hour of a research site, and did not
plan to move from the area. Mothers (n = 3,015) were selected at random for a 2-week,
follow-up telephone call to determine the health of the baby and to verify any family plans
to move. From this sample, 1,525 continued to meet eligibility criteria and agreed to an
interview. 1,364 of these mothers became study participants when their child was 1 month
old. More specific information about the recruitment and selection of participants from the
NICHD SEECYD study is publicly available online (https://secc.rti.org/). All NICHD
SEECYD participants were then followed from birth to fifth grade, and large amounts of
data were collected at several different times. Data from this larger study have been widely
reported in prior research, although most studies have focused on the early years of
development and schooling (National Institute for Child Health and Human Development
Early Child Care Research Network 2002, 2004, 2005; Pianta et al. 2008a, b).
Participants
Participants in our current study were selected from this larger NICHD sample. Of the
original 1,364 participants, 991 were observed in the fifth grade. Of these 991 participants,
664 students had complete data for the variables of interest. Of the 664 students, 74 were in
the same fifth grade classroom as another child in the sample. However, there were not
enough classrooms with multiple participants to support using a nested structure for this
study. For study students from the same classroom, we randomly selected one child from
each classroom. In addition, seven students had scores on the mathematics assessments that
Learning Environ Res (2014) 17:243–262 247
123
were greater than three standard deviations above or below the mean, and these students
were removed from the sample.
Participants in the present study were 657 fifth graders (325 male, 332 female); 548
(83 %) were white, and 109 (17 %) were of other ethnicities. Their mothers had, on
average, 14.4 years of education (SD = 2.43), ranging from 7 to 21 years, with 178
(27 %) having a high school diploma or less. The mean income-to-needs ratio in children’s
homes was 4.70, based on US census definitions for poverty levels (note that middle-SES
falls within the range 1.85–6 according to Heflin and Pattillo 2002). These 657 students
were not statistically different from the total sample of 991 fifth graders in terms of
ethnicity (v2 = 1.03, p [ 0.05), gender (v2 = \ 0.01, p [ 0.05), or maternal education
(t = 0.11, p [ 0.05). However, a t test revealed that the selected sample had a higher
income-to-needs ratio (M = 4.70) than the full fifth-grade sample (M = 4.10, t = 2.12,
p = 0.03). The relatively low numbers of children living in poverty in this sample have
been reported in past National Institute for Child Health Development papers and should
be kept in mind when interpreting our results (NICHD 2002, 2005).
Students were taught by 657 teachers, with each teacher being identified as a study
child’s primary fifth grade teacher. With regard to preparation, 342 of the teachers
(52.1 %) had a Bachelor’s degree, 311 (47.3 %) had a Master’s degree and 4 (0.6 %) had
completed more than a Master’s degree. The average level of teacher education in this
sample was 3.97 years of college, equivalent to a Bachelor’s degree (SD = 1.01).
Teaching experience averaged 14.2 years, with a range of 0–42 years (SD = 10.72).
Teachers earned a mean annual salary of $36,451, ranging from $15,966 to $105,336
(SD = $16,248).
Procedures
Data were collected from three sources: parents, teachers and trained research assistants.
Parents and teachers completed demographic questionnaires and research assistants con-
ducted classroom observations and assessed students. Prior to data collection, all partici-
pants and their families were contacted and asked to consent to observations and to agree to
complete surveys. Once parental consent was verified, research assistants contacted the
classroom teachers about their schedules and asked the teacher to complete a demographic
questionnaire. Observers then created a schedule that represented a typical school day in
each classroom that also maximized the amount of instructional activities that would be
observed. Trained research assistants conducted a 1-day observation for each teacher–
student pair between January and late April. Each observation lasted around 6 h long and
began at the official start of the school day. The exposure to mathematics variable captured
how often students were exposed to mathematics-related instruction during the time of the
observations.
Measures
Classroom observation system for grade 5 (COS-5)
The Classroom Observation System for Grade 5 (COS-5) (National Institute for Child
Health and Human Development Early Child Care Research Network 2002) was used by
trained and reliable coders to measure both the observed instructional quality of the
classroom and the amount of exposure to mathematics instruction, as well as the emotional
248 Learning Environ Res (2014) 17:243–262
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quality of the classroom. The COS-5 consists of two parts that are described in more detail
below. The first part of the COS-5 was a coding system that was used to measure and
describe classroom quality of teacher–student interactions, along nine dimensions, on a
7-point Likert scale (low: 1, 2; mid: 3, 4, 5; high: 6, 7). The second part of the COS-5
included a time sampling measure to record the amount of exposure to classroom
behaviours and activities. Both parts of the COS assess the classroom, the study children,
and their experiences in the classroom. For this study, teacher level ratings of quality and
time sampling codes of exposure during a 60-min observation window were used as the
variables of interest.
Measuring classroom quality Classrooms were observed for six 10-min intervals, which
occurred throughout the school day. After each 10-min observation, observers gave ratings
of each of the nine classroom-level quality constructs, ranging from 1(low) to 7 (high).
Observers focused on the nature of teacher interactions with the study child in the class-
room. Consistent with how the dimensions were grouped in past NICHD studies using the
COS-5 (Pianta et al. 2008a, b), the nine constructs were then separated into two factor
composites: instructional quality and emotional quality. In this study, instructional quality
was used as the focal variable for quality of instruction, and emotional quality was used as
a control variable. Although emotional support and instructional quality are moderately
correlated, they are distinct factors, which are domain specific with regards to predicting
academic skills (Hamre and Pianta 2007). Instructional quality included ratings on three
dimensions: richness of the instructional methods that teachers used to promote critical
thinking and conceptual development; productive use of classroom instruction; minimi-
sation of distractions and transitions; and extent to which the feedback to students focuses
on learning verses rote memorization. Emotional quality included ratings on five dimen-
sions and aspects of classroom climate: presence of positive features (warm interactions,
evidence of close teacher–student and student–student relationships); absence of negative
features (anger or annoyance on the part of the teacher, threats or poor relationships);
teacher sensitivity to student academic and emotional needs (extent to which the teacher is
in charge of classroom learning); over control (reverse coded); and the degree of chaos in
the classroom (reverse coded).
Measuring exposure to mathematics instruction Exposure to mathematics was measured
as the number of intervals at which students were exposed to mathematics instruction over
six 10-min sampled periods during a typical school day. Coders observed in classrooms for
6 h during which they sampled 10 min of instruction per hour, resulting in a sample of
60 min of classroom time. To gather time sampling data, coders assessed the presence or
absence of mathematics activities during each 1-min interval. Coders observed for 30 s and
then recorded for 30 s throughout each 10-min period. The number of intervals in which
the study child was exposed to mathematics instruction was summed, resulting in potential
values ranging from 0 to 60.
To ensure that the research assistants were properly trained to code observations using
the COS-5, all research assistants from the 10 locations attended a COS-5 training
workshop. During this workshop, they watched videos of classroom instruction and used
the COS-5 standardized coding manual, which identified the different dimensions of
instruction mentioned above, and provided specific anchors and descriptions of each code.
They then conducted six cycles of pilot observations, and were required to attain a 60 %
match with a master coder on time-sampling codes, as well as an 80 % match on the
Learning Environ Res (2014) 17:243–262 249
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quality rating scale (within 1 point) before conducting observations. In addition, observers
conducted paired visits scheduled randomly to estimate live reliability on the measures.
More information about the coding manual, reliability, the training process and the
observation protocol is available at https://secc.rti.org/.
It is important to note that this study sampled quality of and exposure to mathematics
instruction over the entire 6-h school day, rather than the mathematics class only. Most
typically, time exposed to mathematics instruction occurred in the form of a lesson spe-
cifically focused on a mathematical content area (e.g., fractions, decimals, measurement).
However, time exposed to mathematics instruction also referred to time that teachers spent
in teaching mathematical ideas as an incidental part of a lesson in another content area
(e.g., showing how to subtract a higher temperature from a lower temperature in a science
unit, or calculating the number of years between events in a social studies lesson). This
sampling approach represents a strength in the present study as teachers vary in the extent
to which they incorporate mathematics concepts into the full school day (as opposed to
isolating mathematics instruction to only the mathematics period). Gathering data on
mathematics quality and exposure throughout the day provides a more ecologically sen-
sitive approach to understanding the presence of mathematics instruction in children’s
school experiences.
Woodcock-Johnson psycho-educational battery–revised
Two subtests taken from the Revised Woodcock-Johnson Psycho-Educational Achieve-
ment Tests (WJ-R ACH, Woodcock and Johnson 1989) were used to measure student
mathematics achievement: Calculations (students perform basic mathematics calculations)
and Applied Problems (students analyze and solve practical mathematics problems pre-
sented orally). This study used vertically equated W-scores for both Calculations and
Applied Problems scores, which are good for comparing children (i.e. a score of 500 is
average for a 10-year old child) (Mather and Jaffe 2002). For the Calculations subtest, the
test was stopped when participants answered six consecutive questions incorrectly. Sub-
jects were shown pictures, as well as given paper and pencils to help them to solve the
problems. Participants gave their answers to the problems orally, and the research assistant
recorded their answers.
Wechsler Abbreviated Scale of Intelligence (WASI)
Research assistants administered the Wechsler Abbreviated Scale of Intelligence (WASI;
Wechsler 1999) in the fourth grade as a measure of students’ baseline cognitive ability.
The WASI is normed and standardized with a national sample and thus can be used
across schools and all students in the study. Each of the four subtests measures a different
aspect of cognitive functioning including verbal knowledge, nonverbal and verbal rea-
soning, and visual information processing. The entire test took between 30 and 45 min for
students to complete. Subtests included: Block Design (assessing the ability to copy
abstract designs using blocks); Matrix Reasoning (measuring nonverbal reasoning and
visual organization skills); Similarities (assessing the ability to describe similarities
between two concepts); and Vocabulary (measuring the ability to name pictured objects
and define words). A composite cognitive ability score was then calculated from all four
tasks to represent the student’s cognitive ability. This score was treated as a covariate in
subsequent analyses.
250 Learning Environ Res (2014) 17:243–262
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Parent questionnaire
Parents completed a demographic questionnaire, providing information about their family
income (converted into income-to needs ratio) and years of education. They also provided
demographic information about their child (e.g., gender).
Teacher questionnaire
Teachers completed a demographic questionnaire providing information about their edu-
cation, teaching experience and monthly salary.
Approach to analysis
First, descriptive information was examined for all variables of interest. Next, four hier-
archical regression models with four blocks each were used to examine the relative con-
tribution of child, family, teacher and classroom variables in predicting mathematics
achievement. Separate main effect models and interaction models were conducted for two
different achievement outcomes: Calculations and Applied Problems. Child demographic
information, including gender, ethnicity and cognitive ability, as well as family demo-
graphic information (maternal education and income-to-needs ratio; family SES), were
included in the first block. Based on prior research, 4th grade cognitive achievement and
socio-demographic risks of students were controlled in all analyses, including low eco-
nomic resources and low maternal education (Hiebert and Grouws 2007; Natriello et al.
1990; Nye et al. 2004; Shields et al. 2001). Teacher demographic characteristics (teacher
education, certification, experience, and salary), as well as classroom emotional climate,
were entered in the second block. The teacher demographic variables were considered
because such teacher demographic characteristics are conceptually relevant and often
empirically significant and referred to as structural features of quality (Cassidy et al. 1995;
Darling-Hammond 2000). In addition, we controlled for the emotional climate of class-
rooms, based on findings that emotional support contributes to achievement in mathematics
and reading tasks (Connor et al. 2006; Pianta et al. 2008a, b). The inclusion of these
variables allows us to isolate the effects of exposure and instructional quality to deepen
students’ knowledge of mathematics. Observed instructional quality and exposure to
mathematics were entered in the third block. Finally, the interaction between quality and
exposure was entered into the fourth block.
Results: quality and exposure to mathematics instruction predicting mathematicsachievement
We first examined the nature and variation of quality and exposure to mathematics in the
fifth grade. Correlations and descriptive statistics for all variables are reported in Table 1.
Classrooms had a mean instructional quality rating of 3.97 (SD = 0.86) with a range from
1.75 to 6.17. On average, students were exposed to mathematics instruction during 14.46 of
the observed 60 intervals (SD = 6.62), ranging from 0 to 42.75. Correlations revealed that
all child and family demographic variables were significantly related in the expected
direction to both Calculations and Applied Problems, supporting the decision to control for
them in the final regression model. Emotional support was also significantly correlated
with instructional quality (r = 0.56). This correlation provides further justification for
Learning Environ Res (2014) 17:243–262 251
123
controlling for emotional support, and isolating instructional quality as our key variable of
interest. Tests of multicollinearity were conducted to determine the relation between
classroom variables. VIF and tolerance statistics revealed that multicollinearity was not a
concern, with all VIF values less than 2.
Calculations
Table 2 displays all results for the Calculations outcome. Greater exposure to mathematics
instruction contributed to higher achievement on the Calculations subtest, F(11,
656) = 28.11, p \ 0.01, DR2 = 0.01, effect size d = 0.11. These findings were evident
after controlling for child characteristics, F(5, 656) = 56.49, p \ 0.01, and teacher and
classroom characteristics, F(9, 656) = 32.48, p \ 0.01. Instructional quality did not pre-
dict Calculations scores, t = 0.42, p [ 0.05.
In addition to the main effect for exposure to mathematics instruction, the interaction of
quality and exposure was significant in predicting Calculations scores, F(12, 656) = 26.45,
p \ 0.01 (Fig. 1). More specifically, students whose teachers were high in instructional
quality achieved similar Calculation scores, regardless of the amount of exposure to
mathematics instruction. However, students with teachers lower in instructional quality
achieved significantly higher scores for Calculations when they were exposed to more
mathematics instruction. In other words, more exposure to mathematics instruction
appeared to compensate for low-quality instruction in predicting Calculations achievement,
but did not contribute to achievement gains for students who experienced high levels of
instructional quality.
Applied problems
Table 3 displays results for the Applied Problems outcome. Analyses predicting Applied
Problems scores also showed a main effect for exposure to mathematics instruction.
Exposure to mathematics instruction significantly predicted achievement on Applied
Problems, F(11, 656) = 50.69, p \ 0.01, effect size d = 0.06 after controlling for both
child characteristics, F(5, 656) = 109.41, p \ 0.01, and teacher characteristics, F(9,
656) = 60.92, p \ 0.01. No main effects were found for instructional quality.
Similar to calculations, the interaction of instructional quality with exposure to math-
ematics instruction was also significant for Applied Problem scores, F(12, 663) = 49.87,
p \ 0.01 (Fig. 2). Students whose teachers were high in instructional quality achieved
similar Applied Problems scores, regardless of the amount of exposure to mathematics
instruction. However, students with teachers who displayed lower instructional quality
achieved higher scores for Applied Problems when they were exposed to more mathe-
matics instruction.
Discussion
The present study offers two significant findings concerning the contribution of instruc-
tional quality and exposure to mathematics instruction during a typical day in fifth grade.
First, in isolation, higher levels of instructional quality did not predict mathematics
achievement. However, more exposure to mathematics instruction predicted both higher
computational and conceptual problem solving when compared to lower levels of expo-
sure. Second, greater exposure to mathematics instruction appeared to compensate for low
252 Learning Environ Res (2014) 17:243–262
123
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Learning Environ Res (2014) 17:243–262 253
123
instructional quality for both mathematics skills areas. In other words, when more exposure
to mathematics was observed, students in lower-quality instructional settings appeared to
perform more like their counterparts from more instructionally supportive classrooms on
mathematics assessments. It is also interesting to note that the teacher demographic
variables measured—education, experience, certification and salary—did not predict
achievement.
Weak mathematics achievement is a national dilemma that has serious economic and
social implications. However, answers to how we approach problem-solving rest on
establishing a deeper understanding of the contexts in which children develop and learn,
particularly in classrooms. Using a large sample to consider the characteristics of
instruction in a typical day in fifth grade offers a unique lens on the problem of low
mathematics achievement. Further, this study is strengthened by the use of observationally-
based, as opposed to teacher-reported, indicators of instructional quality and exposure to
mathematics. Our study is consistent with other research on mathematics classrooms,
which shows that both overall instructional quality and exposure to mathematics
Table 2 Hierarchical regression analysis summary for variables predicting fifth grade mathematicsachievement: Calculations (N = 657)
Variable Model 1: Main Effects Model 2: Interaction
b SEb b R2 DR2 b SEb b R2 DR2
Step 1: Child and family attributes
Gender (Male = 1) 0.37 0.82 0.02 0.30 0.29 0.82 0.12 0.30
Ethnicity (non-white = 1)
1.42 1.15 0.04 1.53 1.15 0.05
Cognitive ability(WASI)
0.4 0.03 0.45* 0.39 0.03 0.45**
Maternal education(years)
0.70 0.21 0.13** 0.68 0.21 0.13**
Income-to-needs 0.03 0.12 0.01 0.03 0.12 0.01
Step 2: Teacherattributes andemotional classroomclimate
0.31 0.01 0.31 0.01
Teacher education -0.58 0.45 -0.05 -0.51 0.45 -0.04
Teacher experience(years)
0.00 0.05 0.00 0.00 0.05 0.00
Teacher salary 0.00 0.00 0.08 0.00 0.00 0.08
Emotional support 1.24 0.99 0.05 1.24 0.99 0.05
Step 3: Instructionalquality and exposure
0.32 0.01 0.33 0.01
Instructional support 0.25 0.60 0.02 0.25 0.59 0.02
Exposure to mathematicsinstruction
0.21 0.06 0.11** 0.19 0.06 0.10**
Step 4: Interaction
Instructional support xexposure tomathematics
-0.17 0.07 -0.08* 0.33 0.01
*p \ 0.05; **p \ 0.01
254 Learning Environ Res (2014) 17:243–262
123
instruction vary greatly from classroom to classroom (Hamre and Pianta 2005; Pianta et al.
2007; Rivers and Sanders 2002; Schmidt et al. 1999). The findings can be understood in the
context of what appears to be typical in American classrooms; results show that, on
average, moderate levels of instructional quality (average rating of 4 on a 1–7 scale) and
moderate levels of exposure to mathematics-related concepts (i.e., constituting roughly
25 % of the school day). On average, current practice does not reach standards set forth by
NCTM (2000) in relation to quality but does reach standards for exposure, thus providing
context for the focus of future work.
Instructional quality and mathematics achievement
Instructional quality did not contribute to mathematics achievement as a main effect, after
controlling for emotional aspects of teacher quality as well as teacher and child demo-
graphics. However, students in high-quality classrooms performed at comparable levels,
regardless of the amount of exposure to mathematics instruction provided, but this finding
emerged only after considering quantity. The lack of findings for quality of instruction was
counterintuitive, especially given recent research reporting that instructional quality
Table 3 Hierarchical regression analysis summary for variables predicting fifth grade mathematicsachievement: Applied problems (N = 657)
Variable Model 3: Main Effects Model 4: Interaction
b SEb b R2 DR2 b SEb b R2 DR2
Step 1: Child and family attributes
Gender (Male = 1) 1.90 0.67 0.08** 0.46 1.84 0.67 0.08* 0.46
Ethnicity (non-white = 1)
2.60 0.95 0.08** 2.69 0.94 0.09*
Cognitive ability 0.47 0.03 0.59** 0.47 0.03 0.59**
Maternal education(years)
0.34 0.18 0.07 0.33 0.18 0.07
Income-to-needs 0.10 0.10 0.04 0.10 0.10 0.014
Step 2: Teacherattributes andemotional classroomclimate
0.46 0.00 0.46 0.00
Teacher education -0.09 0.37 -0.01 -0.03 0.37 0.00
Teacher experience(years)
-0.02 0.04 -0.02 -0.02 0.04 -0.02
Teacher salary 0.00 0.00 0.08 0.00 0.00 0.04
Emotional support 0.30 0.81 0.01 0.30 0.81 0.01
Step 3: Instructionalquality and exposure
0.46 0.04 0.46 0.01
Instructional support 0.49 0.49 0.04 0.49 0.49 0.04
Exposure to mathematicsinstruction
0.10 0.05 0.06* 0.09 0.05 0.05**
Step 4: Interaction 0.46 0.01
Instructionalsupport 9 Exposure tomathematics
-0.13 0.06 -0.07*
Learning Environ Res (2014) 17:243–262 255
123
predicts achievement using similar measures (Crosnoe et al. 2010; Matsumara 2002). One
explanation could be that the instructional quality measure was based on observations
throughout the entire school day and did not specifically focus on mathematics instruction.
While teachers might receive high-quality ratings on general teaching methods, they might
not necessarily display those same behaviours during mathematics instruction. As a result,
it is uncertain whether teachers who are generally considered high-quality teachers are also
high-quality teachers of mathematics. This problem is partially ameliorated by the fact that
observational data offer more rigour than teacher-report data (Porter 2002) and that the
work focused on teacher–child interactions, not only structural indicators of quality (e.g.,
Fig. 1 Interaction of exposure to mathematics for students experiencing different levels of instructionalsupport (high, mid, low) in predicting achievement in calculations
Fig. 2 Interaction of exposure to mathematics for students experiencing different levels of instructionalsupport (high, mid, low) for achievement in applied problems
256 Learning Environ Res (2014) 17:243–262
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teacher certification, education, experience). Future work, however, could narrow the focus
of the observations only to mathematics instruction.
Exposure to mathematics instruction and achievement
Higher exposure to mathematics instruction (quantity) on a typical day significantly pre-
dicted greater mathematics achievement for both basic skills (i.e., calculations) and more
advanced analytic problem solving (i.e., achievement on applied problems). These findings
support earlier work, suggesting that increasing students’ exposure to mathematics
instruction is associated with students acquiring greater skill proficiency and basic math-
ematics skills (Fisher et al. 1980). These findings are also consistent with the process–
product literature and more current work that suggests that increasing instructional
exposure has been shown to promote achievement (Brophy and Good 1986; Lubienski
2006; Wenglinsky 2004). Although it might seem self-evident that students need to be
provided with and exposed to many mathematical opportunities to learn, few studies have
examined these relations in the fifth grade using a large sample of observationally-based
data prior to the current study.
Our findings show that, within reasonable limits, a classroom that exposes students to
greater time with mathematics can enhance student mathematics achievement. Because
mathematics instruction generally focuses on skills, procedures and problem solving,
increasing the amount of time during which children are exposed to mathematics can
provide students with the necessary time that they need to develop the skills required to
solve basic mathematics problems. Often, complex mathematical concepts cannot be
learned without a long duration of exposure, linking to the developmental and cognitive
premise that students learn what they are exposed. Exposure is required for students to be
able to understand new content and apply it to different situations (Rotherham and Will-
ingham 2009). For example, Klibanoff et al. (2006) reported that the amount of inputs from
the teacher and exposure to mathematics ‘talk’ was related to the amount of mathematics
knowledge that students acquired. For students who struggle in mathematics, increasing the
amount of opportunities that they have to learn and practice mathematics will help them
learn basic skills, which can lead to a stronger understanding. Future work that experi-
mentally manipulates the amount of time spent in mathematics instruction (comparable to
work by Connor et al. 2009 in early reading instruction) might permit researchers to make
causal inferences about exposure to mathematics instruction.
Interactive association between quality and quantity
This study tested hypotheses about how quality and exposure could interact in the class-
room to predict student achievement. The findings showed a moderated effect of quality on
exposure, providing new insight into how quality of instruction might matter more under
some fifth grade conditions (i.e., those offering low amounts of exposure to mathematics
instruction) than others. Specifically, findings suggest that exposing students to mathe-
matics instruction at greater rates could compensate for low-quality instruction in pre-
dicting student achievement (refer to Quadrant III in Table 4, for which students receive
low-quality, but ample doses of, exposure to mathematics).
Two explanations for this finding are plausible. First, becoming proficient in mathe-
matics requires students to master basic skills that build on one another, each with
increasing levels of abstraction and representation (NCTM 2000). Thus, teachers who offer
children more exposure to instruction could provide them with more time to achieve
Learning Environ Res (2014) 17:243–262 257
123
mastery, even if the quality of that instruction is not ideal. Second, more exposure can
contribute to more variety in teachers’ instruction and, as a result, teachers might suc-
cessfully engage and teach more children in their classrooms. Thus, even in conditions of
low quality, more time spent exposing students to mathematics might offer variety and
practice necessary to learn.
Ideally, all teachers strive to provide all students with plentiful, high-quality learning
experiences, as exemplified in Quadrant IV. However, this will not always be possible.
Although it is not ideal to have higher doses of lower-quality instruction, the experiences in
Quadrant III might provide students with enough exposure to mathematics to increase their
achievement in mathematics. Our findings suggest that teachers should devote enough time
to mathematics instruction to ensure that they both expose their students to a coherent,
balanced curriculum and give them time to practice. The lack of emphasis on learning
mathematics outside of school compared to reading adds to the need to ensure that all
students are exposed to ample quality mathematics instruction during the school day. One
approach to achieving this goal is to infuse mathematics concepts and exercises into
lessons focusing on other content areas to increase the amount of exposure to mathematics.
Given the amount of time and training it requires to increase quality, it may be more
practical for teachers and schools to lengthen the amount of time that they spend in
mathematics related instruction. However, it is unwise to make this practical recommen-
dation until studies that permit causal inference have been conducted.
Limitations and future directions
Three limitations require mention. First, the students in the NICHD–SECC study in fifth
grade were more advantaged than the original sample, because of disproportionate rates of
attrition among participants coming from families at-risk for school failure. Others have
reported that the NICHD sample does not nationally represent students who are minorities
or of low socioeconomic status (Hamre et al. 2007; NICHD 2005). Value-added scholars
write that, in privileged communities, where there is greater overlap between what happens
at home and what happens at school, it is more difficult to find schooling effects than in
poor communities (Raudenbush and Williams 1995). Therefore, in a sample with more
socio-demographic diversity, it is possible that quality or exposure would have a different
and perhaps stronger association with student achievement. Second, because of study
design and constraints, our research sampled each classroom on only 1 day during the
school year. Future work would benefit from multiple days of observation, which provide a
better representation of what is typical in school. Third, as discussed earlier, quality was
measured throughout the school day, rather than during mathematics instruction. This
limitation speaks to a broader issue in the study of classrooms, namely, that more general
classroom quality and mathematics instructional quality could be different constructs
(Berliner 2005) and that more data sets that provide observational measures of teacher
quality specific to mathematics are needed (Hill et al. 2007).
Table 4 Four hypothesized quadrants of quality and exposure to mathematics instruction
Level of exposure Level of quality
Low quality High quality
Low exposure I Lowest achievement II Quality compensates for exposure
High exposure III Exposure compensates for quality IV Highest achievement
258 Learning Environ Res (2014) 17:243–262
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With current accountability pressures and the emphasis on student achievement (such as
Race to the Top 2010), teachers and administrators are required to balance their time and
curriculum to ensure that all students achieve at a certain level in multiple subjects. Thus,
teachers already face considerable time constraints. However, research suggests that a
great deal of the day in elementary school is spent in classroom management and transition
(NICHD ECCRN 2005). Strategies and interventions designed to make classrooms run
more smoothly, and to use time more efficiently, can give teachers more time to spend on
content and instruction in mathematics (Rimm-Kaufman et al. 2007). Logically, if there are
fewer classroom management problems, more time can be spent on mathematics and other
core content areas. Further, building a strong classroom community where students are
willing to discuss and participate in activities could also help to increase motivation and
learning and to create a more productive and high-quality learning environment (Ryan and
Patrick 2001). Further educational research that focuses on the impact of quality and
instructional time could strengthen the research base for school decision-making to better
support children’s academic growth.
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