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Linear/Nonlinear Relations of Activity and Fitness with Children’s Academic Achievement


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Medicine & Science in Sports & Exercise: December 2014 - Volume 46 - Issue 12 - p 2279-2285
doi: 10.1249/MSS.0000000000000362
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The health benefits of physical activity (PA) and aerobic fitness for children are well established (20,23,31). An emerging research base suggests that PA and fitness may also have benefits for children’s academic achievement (AA) (22,24). In general, published research indicates a positive, or at worst, a null association of PA and aerobic fitness with AA (1,5,9,10). These positive findings suggest interventions to improve children’s AA by increasing their PA/fitness, and findings from one study provide initial support for this supposition. Donnelly et al. (12) conducted a 3-yr randomized trial with elementary-age students to increase their PA to at least 90 min·wk−1 by integrating PA and academic lessons in intervention classrooms. Secondary results from that study indicated significant improvements in AA for children in the intervention schools compared with that in children in control group schools (12). As researchers consider recommending increasing PA/fitness to improve children’s AA, research needs to address important questions about the efficacy of this approach 1) for children across the spectrum of PA and fitness levels and 2) for achievement across different academic subjects.

The prevailing linear analytic approach in most published research examining the relation of AA with PA and fitness makes the tacit assumption that the effect of PA and fitness on AA will be constant across all levels of children’s PA and fitness. That is, the relation of PA and fitness with AA for children in lower PA/fitness percentiles is presumed the same for children in higher percentiles. Research on health correlates of fitness suggests that the relation of PA/fitness with AA may not follow linear patterns. For example, moving children from the lowest fit quartile to the next fitness quartile seems to have a larger effect on lowering metabolic risk than that in moving children in the higher quartiles up to the next quartile (33). One study that evaluated nonlinear associations of fitness and AA (19) found a significant cubic association between mathematics achievement (but not reading) and aerobic fitness among a population of fifth, seventh, and ninth grade students; unfortunately, the author neither further explored nor interpreted this nonlinear finding.

The research literature indicates that there are important caveats regarding the relation of AA with PA/fitness. First, the relation of PA/fitness with AA may differ by academic subject, with generally a stronger association for mathematics than for other subjects, such as reading (11). Second, overall, the amount of variance in AA explained by PA/fitness tends to be relatively small, approximately 1%–5% depending on the academic subject (19,38), raising concerns about the practical significance of the relations. The omission of relevant constructs (e.g., parental education) (25) could be a contributing factor to the small effect size. In addition, because the relation of AA and socioeconomic factors is well established (14), inclusion of socioeconomic variables is needed to protect against potential confounding factors in the relation of AA with fitness and PA. Finally, the AA–fitness association has generally been more consistent compared with the AA–PA association, although some of this could be due to the use of less robust measures of PA (e.g., self-reported). Overall then, these emerging patterns and limitations of past research suggest correcting for relevant issues (e.g., measurement) to ensure that findings on the relation of PA/fitness with AA are robust.

Extant research findings on the potential of PA and fitness to support children’s AA are encouraging, but an almost exclusive reliance on linear-only analytic models leaves important questions unanswered about the efficacy of increasing PA and fitness to boost children’s AA across subject areas. Thus, the aim of this study was to evaluate whether the relation between PA/fitness and AA in three subject areas was linear or nonlinear across all levels of children’s PA/fitness.


Study Overview

Data reported in this article were collected during the baseline assessment phase of the Academic Achievement and Physical Activity across the Curriculum (A+PAAC) 3-yr intervention study designed to assess the effect of active academic lessons on AA. Further details regarding the A+PAAC study methodology can be found elsewhere (13). Briefly, the goal of A+PAAC is to compare changes in AA between elementary schools that provide academic lessons delivered by classroom teachers using moderate-to-vigorous PA (100 min·wk−1, >3 METs of task) (21). Seventeen elementary schools in northeast Kansas were randomized to receive A+PAAC (n = 9) or serve as controls (n = 8). The study was approved by the human subjects committee at the University of Kansas. The authors declare that there are no conflicts of interest.


All parents/guardians of students in the second and third grades received a flyer describing the study (including exclusion criteria and assessment procedures) and had the opportunity to attend information sessions held at each school. Because of a large response, a random sample of second and third grade students (stratified by grade and gender) in each school was selected from those who provided a parental consent/child assent to complete the outcome assessments used for this study. There were 687 students (age, 7.8 ± 0.6 yr) in baseline assessments. Participant characteristics are provided in Table 1.

Participant demographics.



The Weschsler Individual Achievement Test—Third Edition (WIAT-III) was used to assess AA (36). The WIAT-III is composed of 16 subtests. For this study, five subtests were selected: reading comprehension, oral reading fluency, spelling, mathematics problem solving, and numerical operations. The two reading subtests and two mathematics subtests each form a composite score. The WIAT-III was individually administered by test examiners who were blind to the study’s hypotheses and goals. These test examiners were trained and supervised by a coinvestigator who met the testing company’s required WIAT-III administration qualifications. This coinvestigator monitored the test administration throughout the data collection to ensure that protocol procedures were followed by test examiners. The test administration took approximately 45–50 min per student. A trained member of the research team checked all protocols for accuracy and entered the scores into the WIAT-III computerized scoring assistant, which automatically disallows out-of-range values and computes subtest and composite scores.

The WIAT-III has excellent interscorer reliability (i.e., 0.92–0.99), internal consistency, split-half method (e.g., by age range from 0.83 to 0.98), and test–retest stability (e.g., for children 6–12 yr of age, 0.87–0.96 over 2–32 d) (3). Validity is supported via item reviews of curriculum experts and by correlations with other tests, including the WIAT-II (e.g., 0.62–0.86) and other measures of AA (e.g., 0.60–0.82) (3).

Cardiovascular fitness

The progressive aerobic cardiovascular endurance run (PACER, version 8) was used to assess aerobic fitness. The PACER is based on the 20-m shuttle run (27), with substantial validity and reliability across several age groups (37). Students were instructed to run back and forth between two lines 20 m apart as the time allowed. The pace was initially slow and progressively increased. Students were paced by a beep recorded on a CD (FITNESSGRAM®) to indicate when they should reach each end of the 20-m course. Trained research team members observed the test to ensure that the student traversed the 20-m distance. The test ended for each student when he/she failed to traverse the 20-m distance in the time allotted on two (not necessarily consecutive) occasions. Aerobic fitness was interpreted as the total number of laps completed on the PACER, with a higher number of laps indicating a higher level of aerobic capacity.


To measure PA, children wore an ActiGraph GT3X+ portable accelerometer (ActiGraph LLC, Pensacola, FL) on a belt over the nondominant hip for four consecutive days (including one weekend day) (30). The model GT3X+ accelerometer (3.8 × 3.7 × 1.8 cm, 27 g) contains a solid-state digital accelerometer that measures accelerations by generating an electrical signal proportional to the force acting on it along three axes. The GT3X+ detects accelerations ranging in magnitude from 0.05g to 2.5g and frequency ranging from 0.25 to 2.5 Hz. The acceleration signal is digitized by a 12-bit analog-to-digital converter with a sampling rate set at 30 Hz. The data obtained are called “activity counts” and are stored in the device’s internal memory in 1-s intervals.

Activity counts collected were summarized in 1-min epochs using the ActiGraph software. These activity counts reflect the duration and intensity of activity during a given sampling epoch. Nonwear time was defined by an interval of at least 20 consecutive minutes of activity counts of 0 counts per minute, with an allowance for 1–2 min of counts between 0 and 100. Spurious data time was defined as activity counts ≥16,000 counts per minute, and malfunction time was defined as consecutive identical counts per minute >0 (e.g., 32,767) for >20 min. A valid day was defined as ≥10 h of valid data (35). A minimum of three valid wear days was required to be included in analyses (32). Four hundred and two children (58.5%) met these criteria. Mean counts per minute was calculated by dividing the sum of activity counts for a valid day by the number of minutes of valid data time and averaged across all valid days. Higher mean activity counts indicate a greater amount of PA.


Multilevel regression was used to identify the pattern of relation between PA/aerobic fitness and AA accounting for clustering of students (level 1) within schools (level 2). A relatively small correlation between PA and PACER laps (r = 0.27, P < 0.001) indicated minimal colinearity. Thus, analysis was conducted separately for PA and fitness. Given that PA, fitness, and AA may be associated with sociocultural and economic factors, models included the following covariates: grade, gender, race, ethnicity, mother’s education level, household income, and body mass index (BMI). The covariates and PA/PACER laps (i.e., its degree terms) were all grand mean centered, and their corresponding school-level variables (i.e., centered school means or proportions) and school- and cross-level degree terms were also included in the models to estimate the effects unique at each level. A linear or nonlinear pattern of relation was determined by sequentially comparing model likelihood statistics between two competing models (null model with an nth-degree term vs alternative model with this term and an n + 1th-degree term, where n = 1, …, n) until the statistically significant highest-degree term was identified.

Outliers observed at P < 0.01 in either tail of the distribution were excluded from the modeling (n = 11 for PA; 24 for PACER laps; 3, 7, and 8 for reading, spelling, and mathematics scores, respectively). Normality was confirmed by inspecting histograms and probability plots in each grade and both grades combined. All analyses were conducted using SAS 9.3 (SAS Institute, 2002–2010).


Descriptive statistics of study variables are displayed in Table 2. Intraclass correlations showed nonnegligible variability at the school level (0–0.21 for covariates; 0.09 for PA; 0.06 for PACER laps; and 0.05, 0.07, and 0.06 for reading, spelling, and mathematics scores, respectively), supporting the use of multilevel regression. Multilevel regression results for PA (Table 3) indicated no significant linear or nonlinear association between PA and any of the three AA (all P > 0.05). The final models with PA as the predictor explained 8%, 7%, and 11% of the total student-level variance in reading, spelling, and mathematics scores, respectively.

Study variables by grade and gender.
Regression results for the relation of PA and AA.

Multilevel regression results for aerobic fitness (Table 4) indicated that there was 1) no significant linear or nonlinear association between fitness and reading achievement (P > 0.05) but 2) a significant quadratic association between fitness and both spelling and mathematics achievement (both P < 0.01). No contextual effect (i.e., the difference between the student-level and school-level effects) was significant for the quadratic associations. On average, there was an increase in achievement scores up to 22.7 PACER laps for spelling and 27.1 laps for mathematics (Fig. 1). The final models with fitness as the predictor, along with all covariates, explained 10%, 5%, and 15% of the total student-level variance in reading, spelling, and mathematics scores, respectively. The PACER laps accounted for 35% of the explained student-level variance in spelling scores (or uniquely 2% of total student-level variance) and 37% in mathematics scores (or uniquely 5% of total student-level variance).

Regression results for the relation of fitness and AA.
Predicted achievement scores and trend lines for mathematics and spelling.


This study evaluated whether the relation of PA/aerobic fitness with AA was constant across all levels of PA/fitness and whether the association was consistent across academic subjects. The findings indicated that 1) fitness, but not PA, was significantly correlated with AA, 2) the relation between fitness and AA was nonlinear for spelling and mathematics scores, with no significant relation between AA and reading scores, and 3) the magnitude of the nonlinear association for fitness was stronger for mathematics than that for spelling.

Fitness and AA

This study’s findings suggest that increasing aerobic fitness might have a greater effect on spelling and mathematics achievement for children below a particular fitness threshold than that for those above. The results indicated that approximately 22–28 laps on the PACER measure was the “inflection” point—the point at which the associated positive slope for AA per lap plateaued for spelling and mathematics—suggesting maximal benefits for AA through increasing aerobic fitness among children below a 22- to 28-lap threshold. Compared with aerobic fitness percentiles for children age 8–10 yr (6), the inflection point in this study is associated with the 50%–75% PACER percentiles. Clearly, the findings in this study do not reflect causal relations, given that the study was cross-sectional. However, they suggest that future research focus on evaluating the potential benefits of improving aerobic fitness for AA among children classified in lower fitness percentiles.

The strength of the association between aerobic fitness and both spelling and mathematics achievement in this study is similar in magnitude to these associations in existing research (19,38), which has reported an effect size (i.e., variance explained) in the range of 1%–5%. In this study, aerobic fitness uniquely explained 2% and 5% of the total student-level variance in spelling and mathematics achievement, respectively. As a proportion of the explained student-level variance in AA, however, fitness accounted for just over one-third of the variance, which is considerable in magnitude. The strength of the relation of fitness with spelling and mathematics scores could reflect this study’s measure of fitness; aerobic fitness was assessed as the total number of PACER laps completed, whereas other studies have created a fitness composite score that includes muscular strength, endurance, etc., along with aerobic fitness. Although nonaerobic measures of fitness (e.g., flexibility, strength) were not included in the present study, at least one other study has indicated that, among the components of the FITNESSGRAM®, the PACER has the strongest correlation with AA (7). Thus, the current findings and existing research suggest that among the different measures of fitness, aerobic fitness may be particularly potent for AA.

Although there was an association between fitness and both mathematics and spelling, this study failed to find a significant association between fitness and reading achievement. This overall pattern is consistent with other research (9). The relation of mathematics with fitness could reflect the role of fitness in supporting children’s executive function (2,11,16,17,26). Executive function refers to supervisory control of cognitive functions to achieve a goal and involves allocation of attention and memory, response selection and inhibition, goal setting, self-control, self-monitoring, and skillful and flexible use of strategies (28). Research within the cognitive function literature indicates that children of lower mathematical ability have difficulties performing tasks involving components of executive function, such as inhibition of prepotent information (e.g., Stroop interference) and learned strategies (e.g. Wisconsin Card Sorting Task) and difficulties maintaining information in working memory (4). Reading and spelling have typically been associated with executive function in cases where cognitive dysfunction or a learning disability is present (29). The null findings of this study between reading and fitness are consistent with an executive function explanation, but the significant relation between fitness and spelling is not and warrants further research to determine whether the relation is robust.

PA and AA

The findings on the relation of PA and AA in research literature have been inconsistent, with either a significant positive relation or null relation (1,8,9,12,15,18,34). In this study, PA was not significantly correlated with AA, either linearly or nonlinearly. There could be several reasons for this “null finding.” First, PA may not have a direct relation with AA. However, given the obvious role that moderate-to-vigorous PA over time plays in improving or maintaining aerobic fitness, it seems more likely that this null finding is attributable to other factors. It could be that the unit of measure for PA in this study—aggregate of 3–4 d of accelerometer data with valid data across four consecutive days—may not have adequately represented the full scope of these children’s “typical” activity levels. Valid PA data were also missing for a substantial proportion of children in this study (41.5%), and thus, this smaller subsample may not have adequately represented the relation of PA and AA for the entire sample. The present findings, however, point to the need for longitudinal designs that assess PA over longer periods or on multiple occasions to unmask any potential relation between PA and AA.


This study has particular strengths and limitations that should be noted. Although the cross-sectional design limits any evaluation of how/if PA and aerobic fitness levels relate to AA over time, the findings indicated that many children in this study were below the recommended PA and fitness levels. Barring hereditary and maturational limits on children’s fitness levels, most children should be capable of reaching recommended levels of PA and aerobic fitness (37), and the present findings suggest that increases in these levels might benefit their AA. (Increasing PA is argued as a requisite for increasing fitness.) The use of salient covariates in the analyses, including gender, race, ethnicity, parental education level, household income, and BMI, strengthened the findings by ruling out their potential influence on the relation of PA and aerobic fitness with AA. We suggest that researchers, at minimum, measure and evaluate the influence of these covariates in future studies in which AA is included. It should also be noted that this study’s sample was homogeneous in terms of race (predominantly White) and there was a relatively high proportion of high-income families (21%), which limits generalizability. Lastly, there is limited validity evidence available for the 20-m PACER in 7- to 8-yr-old children and the potential for confounding by motivation to perform or other confounding due to the age of the sample is possible.

Overall, the present study contributes to an emerging literature linking aerobic fitness and PA with AA. This research implies that providing more opportunities to be active and improve fitness could improve AA, at least partially, and schools are uniquely situated to provide such opportunities. With recent financial constraints on educational budgets and increased pressures on schools to meet mandated achievement standards, educators may be tempted to increase their focus on academics, perhaps to the detriment of PA opportunities for children in school. The findings of this study, along with existing research, suggest that this response could ultimately be detrimental to children’s AA. Although further research is needed to better understand the longitudinal effects of increasing aerobic fitness and PA on children’s learning, the present findings suggest that a more proactive approach to increasing fitness is warranted.

We wish to express our appreciation to the schools that participated in this study.

This study was supported by the National Institutes of Health (R01-DK85317). This trial was registered at the US National Institutes of Health Clinical Trials, NCT01699295.

The authors declare that they have no competing interests. No form of payment was given to anyone to produce this manuscript.

The results of the present study do not constitute endorsement by the American College of Sports Medicine.


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