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Nonsymbolic ratio processing longitudinally predicts fraction knowledge in earlier grades

Fractions are among the most important mathematical concepts children learn about in school: a strong understanding of fractions supports later success in algebra and other advanced mathematics. However, many students continue to struggle with fractions well into adolescence and adulthood. Identifying the foundations of fraction knowledge is an important step into understanding why fractions are so hard to many people. In a recent longitudinal study, recently published in the Developmental Psychology Journal, our team investigated how children's ability to process numerical magnitudes predicts the development of fraction knowledge across elementary school years. 

Before children receive formal instruction on fractions, they can already make judgments about quantities and proportions. For example, they may be able to determine which of two mixtures contains more juice than water, even without using numerical symbols. Researchers refer to this ability as nonsymbolic ratio magnitude processing.  Children also develop the ability to compare and understand the magnitudes represented by symbolic fractions, such as determining whether 3/4 is larger than 2/5. This skill is known as symbolic fraction magnitude processing.  Previous studies have shown that both nonsymbolic ratio processing and symbolic fraction processing are related to fraction achievement. However, most research has relied on measurements taken at a single point in time, making it difficult to understand how these skills contribute to fraction learning across development.  

To better understand how nonsymbolic ratio magnitude processing and symbolic fraction processing relate to fraction knowledge time, we followed two groups of students for approximately one year:  A younger cohort followed from 2nd to 3rd grade, and an older cohort followed from 5th to 6th grade  Students completed a nonsymbolic ratio comparison task, a symbolic fraction comparison  task, and a mixed (nonsymbolic ratio vs. symbolic fraction) comparison  involving both nonsymbolic ratios and symbolic fractions in their first year in the study. They also completed the Fraction Knowledge Assessment in both year so the study. This design allowed us to examine which skills predicted future growth in fraction knowledge.  

We found that students in both cohorts improved their fraction knowledge over time. However, growth was substantially larger in the younger cohort, suggesting that the most dramatic gains in fraction understanding occur during the earlier stages of fraction learning.  Our results also revealed an interesting developmental pattern.  For younger children, the ability to compare nonsymbolic ratios predicted later fraction knowledge, even after accounting for domain-general cognitive factors. In other words, children who were better at understanding proportions without symbols tended to develop stronger fraction knowledge one year later.  However, this association was not observed in the older cohort. By 5th and 6th grade, nonsymbolic ratio processing no longer uniquely predicted fraction knowledge.  In contrast, symbolic fraction magnitude processing predicted fraction knowledge in both age groups.  These findings suggest that nonsymbolic ratio understanding may provide an important foundation for the initial acquisition of fraction concepts, but that symbolic fraction understanding becomes increasingly important as children gain experience with formal mathematics.  

Understanding how fraction knowledge develops has important implications for education.  Our findings suggest that early intuitions about proportions and ratios may support children's first steps toward understanding fractions. As students progress through school, however, successful fraction learning appears to depend increasingly on their ability to understand and manipulate symbolic fraction magnitudes.  These results contribute to a growing body of evidence showing that mathematical learning is built upon both informal numerical intuitions and formal symbolic knowledge. By understanding how these systems interact across development, researchers and educators can design more effective approaches to supporting fraction learning and identifying students who may need additional support.  Ultimately, helping children develop a strong understanding of fractions is critical because fractions serve as a gateway to more advanced mathematical concepts. Understanding the cognitive foundations of fraction learning brings us one step closer to helping all students succeed in mathematics.

Novel tool to measure fraction skills: the Fraction Knowledge Assessment (FKA) 

To reach this goal, our team developed a series of grade-appropriate versions of a novel test: the Fraction Knowledge Assessment (FKA).  In a research article we recently published in the Journal of Experimental Child Psychology, we report how effective the assessment was.

The original Fraction Knowledge Assessment was developed by Matthews and colleagues in 2016 to measure adults' understanding of fractions. Building on this foundational work, we adapted the assessment to evaluate fraction knowledge across elementary school grades and to support longitudinal investigations of fraction learning and development.  Our team developed FKA versions for students in 2nd, 3rd, 5th, 6th, and 8th grades. The goal was to create assessments that were developmentally appropriate while maintaining sufficient overlap to allow us to track growth over time. Unfortunately, data collection for the 8th-grade version was disrupted by the COVID-19 pandemic, so our study focused on the other grades. 

The FKAs included items from previous research studies, and national and international math assessments. The items measured children's conceptual fraction knowledge (knowing what), such as knowing that 1/3 is greater than 2/9 and that there are infinite fractions between two fractions. The assessment also included items measuring procedural fraction knowledge (knowing how), such as adding two fractions with unlike denominators or dividing a fraction by another fraction.

We followed two cohorts of students. One cohort was followed from 2nd to 3rd grade, while a second cohort was followed from 5th to 6th grade. To minimize potential practice and learning effects associated with repeated testing, we created distinct versions of the FKAs for each grade level. These versions included some unique items and some items that remained the same across all versions. Then, we used a series of statistical analyses to examine the properties of FKA and track fraction knowledge growth.

First, we used a statistical technique known as known as Confirmatory Factor Analysis to investigate if FKA worked better as a general metric of fraction knowledge (one-factor model), or as a measure of conceptual and procedural fraction knowledge separately (two-factor model). Results showed that conceptual and procedural fraction knowledge were not consistently separated in the FKA, suggesting it works as a general measure of fraction knowledge.  

Next, we evaluated how difficult each FKA item was and also how well they could distinguish children with stronger fraction knowledge from those with weaker fraction knowledge, using a specific type of statistics known as Item Response Theory (IRT). We created an ability score for each child and, using IRT and anchor items, linked the assessments across grades so that scores from different grade levels could be compared on a common scale. This means that we were able to investigate how much children's fraction knowledge improved from 2nd to 3rd grade, and from 5th to 6th grade, even though the tasks were slightly different.  These procedures allowed us to measure developmental growth while accounting for differences in item properties across grade levels, making our analyses more sensitive than only looking at raw scores.

Finally, we analyzed fraction knowledge change across grades. We found that participants' fraction knowledge improved across the year, with steeper growth from the 2nd to the 3rd grade than from the 5th to the 6th grade. Also, children who started the study with better fraction knowledge (in 2nd or 5th grade) also had better fraction knowledge one year later (in 3rd or 6th grade). We also found that children with better working memory (the ability to temporarily hold and manipulate information in our minds) and basic calculation skills also had better fraction knowledge in both cohorts. 

You can learn more about the study here. Also, we are sharing the FKA and the statistical analysis scrips (R code), so that other researchers and educators interested in measuring fraction knowledge can use them.