There is strong evidence from research conducted in the United States that fraction magnitude understanding supports mathematics achievement. Unfortunately, there has been little research that examines if this relation is present across educational contexts with different approaches to teaching fractions. The current study compared fourth and sixth grade students from two countries which differ in their approach to teaching fractions: Australia and the United States. We gathered data on fraction and decimal magnitude understanding, proportional reasoning, and a standardized mathematics achievement test on whole number computation. Across both countries, reasoning about rational magnitude (either fraction or decimal) was predictive of whole number computation, supporting the central role of rational number learning. However, the precise relation varied, indicating that cross-national differences in rational number instruction can influence the nature of the relation between understanding fraction and decimal magnitude and mathematics achievement. The relation between proportional reasoning and whole number computation was fully mediated by rational magnitude understanding, suggesting that a key mechanism for how reasoning about rational magnitude supports mathematics achievement: proportional reasoning supports the development of an accurate spatial representation of magnitude that can be flexibly and proportionally scaled, which in turn supports children’s mathematics learning. Together, these findings support using measurement models and spatial scaling strategies when teaching fractions and decimals.
Being able to reason about fraction and decimal magnitudes is essential for progress in mathematics (
Unfortunately, most research on the predictive role of magnitude understanding has taken place in the United States, limiting generalizability to other cultures and educational contexts (with exception of
In the United States, there is growing evidence of a strong, consistent, and predictive relation between understanding fraction magnitude and mathematics achievement. Cross-lagged panel models demonstrate longitudinal stability of this relation across fourth through seventh grade (
Extending to decimal magnitude,
To our knowledge, there are no studies that have examined the role of decimal understanding supporting mathematics achievement outside of the United States. However, two studies have examined the role of fraction magnitude understanding (
International comparisons of mathematics education can characterize differences in state or national standards, teacher’s content knowledge, preparation, and instructional practices, and their relation to student outcomes.
The current study focuses on two countries that are more similar: Australia and the United States. Both countries are primarily Western and English speaking. Primary school educators from Australia and the United States have similar rates of completing a mathematics degree (13%), completing at least a bachelor’s degree (93-100%), years of experience (average 13-15 years), and having undertaken professional learning in mathematics content and pedagogy in the past two years (
At the time of this study, the sequence and progression of fraction and decimal instruction described in state and national standards is also similar between Australia and the United States. The bulk of fraction instruction in both countries occurs over third through fifth grade, with foundational fraction concepts (e.g., partitioning and understanding “half”) beginning in first grade (
Importantly, Australia and the United States differ on their use of measurement models to teach fractions. The United States emphasizes, almost exclusively, a part-whole interpretation (
Measurement models have been identified as a key component in intervention studies for developing an accurate sense of fraction magnitude (e.g.,
Understood within overlapping waves theory (
It may be the case that a linear representation of magnitude supports mathematics achievement because it involves the ability to flexibly and proportionally scale mathematics information. Indeed,
To date, we are not aware of any studies examining how magnitude understanding may mediate the relation between proportional reasoning and mathematics achievement. In
The present study examines the relation between fraction magnitude understanding and mathematics achievement in two countries – Australia and the United States – that differ on their use of measurement models in fraction education. This extends the work by
A sample of 156 students from Australia and the U.S. participated in this study. Information sheets and consent forms describing the study were sent home to all families with children in fourth or sixth grade (10 different classes) at three participating schools. The participating schools were private schools matched for demographics of the population served: religious affiliation, size, coed, tuition, and similar indexes of community socioeconomic advantage (e.g., the schools reported serving predominately white, English-speaking, middle to higher socioeconomic families). While each school used their own teacher-designed curriculum, all curricula were explicitly mapped onto their respective national education standards. The United States educators reported exclusively using part-whole models to teach fraction magnitude in fourth grade, in contrast with the Australian fourth grade educators reporting the inclusion of measurement models. In both countries the average age of students in fourth grade is nine to ten years old and in sixth grade is 11-12 years old. The Australian sample was comprised of 41 students in fourth grade and 34 students in sixth grade (45% female, 55% male), and the U.S. sample was comprised of 35 students in fourth grade and 46 students in sixth grade (48% female, 52% male).
The same number line estimation tasks (0-1, 0-5) as
A decimal comparison task was developed based on
The “Goldilocks” proportional equivalence task developed by
The mathematics subsection of the Wide Range Achievement Test (WRAT;
The researcher administered the assessments in a group setting at the classroom level. Students completed the assessments in the following fixed order: fraction number line estimation, proportional reasoning, decimal comparison, and then whole number computation.
An a priori power analysis using G*Power (
Separate two (country) by two (grade) ANOVAs were conducted to examine differences in 0-1 and 0-5 number line estimation. There were significant main effects of country: Australian students were more accurate than U.S. students on the 0-1 number line estimation task,
A two (country) by two (grade) ANOVA was conducted to examine differences in decimal comparison. There was no significant effect of country,
A two (country) by two (grade) by three (version type) ANOVA was conducted to examine differences in proportional reasoning. There was a significant main effect of country: with Australian students scoring higher than U.S. students,
A two (country) by two (grade) ANOVA was conducted to examine differences in performance on the whole number computation. There was a significant main effect of country: Australian students scored higher than U.S. students,
Australian students significantly outperformed U.S. students on fraction number line estimation, proportional reasoning, and whole number computation. There were no significant country level differences on decimal comparison. See
To examine cross-national differences in the relative contributions of fraction number line estimation, decimal comparison, and proportional reasoning to whole number computation, we conducted hierarchical linear regression analyses. When
See
Task | Decimal Comparison Accuracy | Proportional Reasoning Accuracy | Whole Number Computation |
---|---|---|---|
Australia | |||
0-1 Fraction Number line PAE | .44** | .48** | .41** |
0-5 Fraction Number line PAE | .41* | .05^{ns} | .19^{ns} |
Composite Fraction Number Line PAE | .52** | .37* | .41** |
Decimal Comparison Accuracy | – | .40* | .61** |
Proportional Reasoning Accuracy | – | .31* | |
Whole Number Computation | – | ||
United States | |||
0-1 Fraction Number line PAE | .35* | .55** | .39* |
0-5 Fraction Number line PAE | .71** | .50** | .57** |
Composite Fraction Number Line PAE | .57** | .60** | .54** |
Decimal Comparison Accuracy | – | .42* | .39* |
Proportional Reasoning Accuracy | – | .25^{ns} | |
Whole Number Computation | – |
*
Multiple regression analyses with enter method was used to predict whole number computation. Regression assumes homogeneity of slopes, which is assessed through the inclusion of interaction terms (
Predictor | ||||
---|---|---|---|---|
Model 1 (Including nonsignificant and significant interaction terms) | ||||
Country | .71 | 5.05 | 1.40 | .17 |
Fraction number line estimation (FNLE) | .45 | .29 | 1.54 | .13 |
Decimal comparison | .30 | .28 | 1.08 | .29 |
Proportional reasoning | .38 | .27 | 1.38 | .17 |
FNLE*country | .16 | .20 | .84 | .41 |
Decimal comparison*country | .38 | .19 | 1.96 | .05 |
Proportional reasoning*country | .31 | .18 | 1.77 | .08 |
Model 2 (Including only significant interaction terms) | ||||
Country | .72 | 2.28 | .32 | .75 |
Fraction number line estimation (FNLE) | .14 | .07 | 1.97 | .05 |
Decimal comparison | .59 | .41 | 1.42 | .16 |
Proportional reasoning | .08 | .08 | .99 | .32 |
Decimal comparison*country | .71 | .28 | 2.58 | .01 |
Because the interaction terms between country and fraction number line estimation and between country and proportional reasoning were nonsignificant, as recommended, we removed these variables and reran the analysis. This final model included the following predictor variables: country, fraction number line estimation, decimal comparison, and proportional reasoning, as well as an interaction term for decimal comparison by country. The results explained 51% of the variance in whole number computation,
See
Task | Decimal Comparison Accuracy | Proportional Reasoning Accuracy | Whole Number Computation |
---|---|---|---|
Australia | |||
0-1 Fraction Number line PAE | .201^{ns} | .384* | .221^{ns} |
0-5 Fraction Number line PAE | .398* | .318^{ns} | .480* |
Composite Fraction Number Line PAE | .419* | .417* | .458* |
Decimal Comparison Accuracy | – | .094^{ns} | .316^{ns} |
Proportional Reasoning Accuracy | – | .457* | |
Whole Number Computation | – | ||
United States | |||
0-1 Fraction Number line PAE | .410** | .599** | .506** |
0-5 Fraction Number line PAE | .465** | .464** | .501** |
Composite Fraction Number Line PAE | .498** | .589** | .571** |
Decimal Comparison Accuracy | – | .348* | .348* |
Proportional Reasoning Accuracy | – | .288^{ns} | |
Whole Number Computation | – |
*
A multiple regression was conducted for the sixth-grade sample following the same parameters as the fourth-grade sample. With the inclusion of all interaction terms, the overall model explained 62% of the variance,
Predictor | ||||
---|---|---|---|---|
Model 3 (Including nonsignificant and significant interaction terms) | ||||
Country | .00 | 6.29 | .00 | 1.00 |
Fraction number line estimation (FNLE) | .14 | .23 | .60 | .55 |
Decimal comparison | .13 | .66 | .20 | .85 |
Proportional reasoning | .43 | .24 | 1.84 | .07 |
FNLE*country | .12 | .17 | .73 | .47 |
Decimal comparison*country | .008 | .40 | .02 | .99 |
Proportional reasoning*country | .39 | .17 | 2.29 | .03 |
Model 4 (Including only significant interaction terms) | ||||
Country | 2.18 | 3.39 | .64 | .52 |
Fraction number line estimation (FNLE) | .29 | .07 | 3.91 | .001 |
Decimal comparison | .16 | .19 | .81 | .42 |
Proportional reasoning | .50 | .22 | 2.33 | .02 |
Proportional reasoning*country | .43 | .16 | 2.76 | .01 |
Because the interaction terms between country and fraction number line estimation and between country and decimal comparison were nonsignificant, as recommended, we removed these variables and reran the analysis. This final model included the following predictor variables: country, fraction number line estimation, decimal comparison, and proportional reasoning, as well as an interaction term for proportional reasoning by country. The results explained 63% of the variance in whole number computation,
To examine if the relation between proportional reasoning and whole number computation is mediated by fraction number line estimation or decimal magnitude comparison, we conducted a parallel mediation analysis using the PROCESS macro (version 3.5) on SPSS (
Although a growing body of work has found that the ability to reason about rational magnitude supports mathematics learning (e.g.,
In the current study, Australian students outperformed students from the United States on fraction number line estimation. This finding is aligned with
Consistent with existing literature (e.g.,
One reason decimal magnitude understanding may be more important in the Australian fourth grade context, is that decimals play a large role in understanding the metric system, which is Australia’s official measurement standard and used across all industries. While the United States national standards includes learning metric units, they also include learning customary units (e.g., feet and inches;
In contrast, Australian sixth grade students, after receiving the bulk of fraction instruction that includes emphasis on a measurement model, may be better equipped to utilize proportional reasoning and spatial scaling strategies when completing mathematics problems. Measurement model instruction may change the spatial representations (more accurate linear representation) and/or strategy (spatial scaling strategies) underlying both proportional reasoning and rational number problem-solving. The stronger relations among these measures are consistent with a common underlying change in the Australian students spatial and numerical thinking (though further research is needed to discriminate between this explanation and alternatives). In contrast, in early United States fraction instruction, fourth grade students tend to use whole number properties as a strategy to reason about fraction magnitude (
Finally, the findings from the current study suggest that a key mechanism for how reasoning about magnitude (including rational numbers) supports mathematic achievement is proportional reasoning. The relation between proportional reasoning and whole number computation is fully mediated by rational magnitude understanding, suggesting that proportional reasoning may support the development of an accurate spatial representation of magnitude that can be flexibly and proportionally scaled, which in turn supports children’s mathematics learning. Indeed, using a holistic analog representation to reason about magnitude requires imaging the internal proportion shrinking or growing. Experimental studies, however, are required to determine causal effects.
Although the current study contrasted two countries that are similar on many important factors, it is impossible to align all national, cultural, and education contexts. Subsequently, there may be other factors that influence fraction and decimal magnitude learning and their relation to mathematics achievement. For example, Australian students outperformed the United States students on mathematics achievement, which contrasts with the TIMSS (
Differences between the Australian and US students on the WRAT could also be explained by this sample of Australian students happening to have superior math skills. We do not believe this to be the case because we matched the samples based on a range of demographic, cultural, and educational factors. Nevertheless, even if the samples had different overall mathematics skills, that does not change our finding that reasoning about rational magnitude and proportional reasoning supports whole number computation cross-nationally. Further, country is not predictive of whole number computation in the final regression model, emphasizing the relative importance of rational magnitude and proportional reasoning over any country-level differences that may exist. It seems likely that higher WRAT scores may be a consequence of having higher fraction and decimal magnitude understanding and proportional reasoning skills (
It may also be the case that fraction instruction varies within each country, including aspects not identified within this study (e.g., how much time is spent on rational number instruction). Future work should consider a wider range of cultural and educational settings in exploring how and when reasoning about different kinds of magnitude support mathematics achievement. In addition, future research should also consider the role of other individual differences (e.g., SES, working memory, gender, other spatial skills).
Another consideration concerns the proportional reasoning measure, which was presented in three versions: continuous, discretized, and discrete. We originally chose to assign participants to a single version, because completing one type of proportional reasoning task can influence strategies on subsequent tasks (
Notably, a larger sample size would have provided our study with more power to observe potentially present smaller effects. Indeed, the inclusion of interaction terms in our models further reduced our power. However, we observed large effect sizes in both countries indicating that rational magnitude understanding is important cross-nationally, but the specific relation varying between country, which was accounted for by proportional reasoning when put together in a single model. While our sample size may not have been sufficient to detect if a small effect were present, having no effect versus a small effect in this instance would not change the interpretation – there is a differential relation between math and rational number in Australia compared to the USA.
A key finding from the current study is that rational magnitude understanding was predictive of mathematics achievement across Australia and the United States, even though mathematics achievement was measured using whole number computation problems. This supports the integrated theory of numerical development (
The authors have no funding to report.
The authors have declared that no competing interests exist.
The authors have no additional (i.e., non-financial) support to report.