A Key Predictor for Children’s Success in Math

A thorough understanding of addition and subtraction requires knowledge of the inverse relation between these operations.

According to Piaget (1952), any inverse relation poses a significant challenge for children between the ages of 5 and 8 because they lack the ability to engage in "reversible" thought processes. They fail to recognize that if 4 + 8 = 12, then 12 - 8 = 4 because they do not realize that addition is the inverse of subtraction. However, Piaget never tested this claim directly.

Recent research showed that there is considerable variability in the understanding of the inversion principle among children of the same age (Baroody & Lai, 2007; Bryant et al., 1999; Rasmussen et al., 2003; Sherman & Bisanz, 2007). Some evidence suggests that children’s initial understanding of inversion may stem from their interactions with tangible materials (Bryant et al., 1999; Canobi, 2005; Ching, 2023; Ching & Wu, 2019; Gilmore & Papadatou-Pastou, 2009).

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The knowledge of inversion can potentially enhance children's ability to perform calculations (Bisanz et al., 2009; Greer, 2012). If children have a good understanding of the inversion principle, they may be able to utilize their knowledge of number facts in a more flexible manner, leading to success in a broader range of calculation problems compared to those who lack such flexibility. For instance, if children know that 9 + 7 = 16 and understand inversion, they can apply this knowledge to solve additional questions: "16 - 9 = ?" and "16 - 7 = ?" By leveraging their understanding of inversion, they can deduce the missing values in these subtraction problems.

Similarly, the use of "indirect addition" to solve challenging subtraction problems relies on recognizing and utilizing the inverse relationship between addition and subtraction. To illustrate, in a subtraction problem like 42 - 39, understanding inversion allows a child to convert it into an addition problem. They can count up from 39 to 42, recognizing that the difference is 3, and thus conclude that 42 - 39 = 3. Reasoning in this manner would be challenging without a solid grasp of the inverse relation between addition and subtraction.

Research by Gilmore and Bryant (2006) suggests that understanding the inverse relation may be important for children to learn to add and subtract proficiently. In this study, they used cluster analysis on samples of 6- to 8-year-old children who were presented with inversion and control problems (where calculation was required for the control problems), and identified three distinct groups of children from the analysis.

The first group displayed a clear understanding of inversion and exhibited strong calculation skills. These children performed better in the inversion problems compared to the control problems, but their scores in the control problems were also relatively high. The second group comprised children who demonstrated a limited understanding of inversion and weak calculation skills. The final group consisted of children who exhibited a good understanding of inversion but had weak calculation skills. In other words, these children performed better in the inversion problems than in the control problems, but their scores in the control problems were low.

Notably, the discrepancy between understanding inversion and calculation skills was observed in one direction but not the other. Gilmore identified a group of children who could effectively employ the inversion principle despite struggling with calculations. However, she found no evidence of a group of children who excelled at calculations but were unable to utilize the inversion principle. Thus, children do not necessarily need to possess strong addition and subtraction skills in order to grasp the relationship between these two operations. On the contrary, understanding the inverse relation may be a prerequisite for learning to add and subtract proficiently.

Nunes et al. (2007), who employed a combination of longitudinal and intervention methods, have provided additional evidence for the association between understanding reasoning principles and mathematics learning. The researchers' longitudinal study involved testing children in their first year of school. Their mathematics achievement was assessed in the second year (14 months apart) using standardized tests. A reasoning test included an assessment of children's understanding of the inverse relation between addition and subtraction, as well as additive composition and correspondence. The findings showed that the inverse relation task remained a significant predictor of mathematics achievement even with rigorous controls in place, explaining an additional 12 percent of the variance.

The study also included an intervention component. Children who initially showed underperformance in the logical assessment were assigned to either a control group or an intervention group. The control group received no additional intervention, while the intervention group received specialized instruction on reasoning principles for one hour per week over a 12-week period, during the time their peers were engaged in regular mathematics lessons. Results showed that the intervention group significantly outperformed the control group, with the intervention group's mean score surpassing the 50th percentile, indicating performance above the average.

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The intervention study did not isolate the effect of inversion, as the children received instruction on multiple reasoning principles deemed essential for their learning. While separate studies examining the impact of each reasoning principle could be conducted, the authors argue that each principle is central to children's mathematics learning and should be addressed collectively.

While experimental studies provide evidence of what can be achieved under controlled conditions, it is necessary to determine if similar outcomes can be obtained in authentic educational environments. The transition from the laboratory to the classroom warrants careful consideration. Overall, children’s understanding of inversion is a good predictor of their mathematical success, and improving this understanding has the result of improving children’s mathematical knowledge in general.

Currently, some children spend a significant amount of time practicing number facts at home and in school, often attempting to memorize them as isolated pieces of information. However, combining the learning of number facts with mathematical principles that help children connect and relate one number fact to another, such as inversion, could provide them with more flexible knowledge and engender more engaging learning experiences.