Recently, Professor Li Xiaodong and Dr. Jiang Ronghuan of the School of Psychology published an empirical study entitled “From Additive to Multiplicative Thinking: Individual Cognitive Patterns Revealed Through Bayesian Classification” in the internationally leading journal Journal of Educational Psychology. Founded in 1910, Journal of Educational Psychology is a top-tier journal of the American Psychological Association dedicated to empirical research in educational psychology. It enjoys broad academic influence and is widely recognized as one of the leading journals in the field. This publication is the first paper from Shenzhen University’s psychology program to appear in Journal of Educational Psychology. Dr. Jiang Ronghuan of the School of Psychology, Shenzhen University, is the first author, Professor Li Xiaodong is the corresponding author, and Shenzhen University is the first corresponding institution. This research was supported by the Youth Fund for Humanities and Social Sciences Research of the Ministry of Education and the Shenzhen Talent Program Start-up Fund for Doctoral Researchers.
The study focuses on a key and widespread challenge in the development of students’ mathematical cognition: the transition from additive thinking to multiplicative thinking. Using a hypothesis-driven Bayesian classification approach, the researchers systematically identified for the first time eight cognitive patterns that emerge during this transitional stage. The findings not only confirm the advantage of multiplicative thinking at the group level, but also reveal, at the individual level, the diversity and regularities of cognitive development. In doing so, the study offers a new perspective for understanding the complex mechanisms underlying the development of mathematical thinking and provides solid empirical support for personalized mathematics education interventions based on cognitive diagnosis.
The “Inertia” of Mathematical Thinking: From the Blind Use of Addition to the Blind Use of Multiplication
When solving mathematical problems, students often fall into a kind of cognitive set: they incorrectly use addition in problems that require multiplication, and vice versa. For example, consider the following problem: “Ann and David are making dolls. They work at the same speed, but David started earlier than Ann. When Ann had made 4 dolls, David had made 12. When Ann has made 20 dolls, how many dolls has David made?” The correct answer is 20 + (12 − 4) = 28, which reflects an additive relationship. Yet many students mistakenly apply multiplication and answer 20 × (12/4) = 60. In another scenario—“Mary and Sue are washing dishes. They start at the same time, but Sue washes faster than Mary. When Mary has washed 4 dishes, Sue has washed 12. When Mary has washed 20 dishes, how many dishes has Sue washed?”—the correct answer is 20 × (12/4) = 60, reflecting a multiplicative relationship, but students may instead incorrectly apply addition. This phenomenon of “overgeneralization,” that is, the overuse of either addition or multiplication, has long been recognized as a classic challenge in mathematics education.
Previous research has largely examined this issue at the group-average level, often overlooking the substantial differences among individual students. The present research team argues that this developmental transition is far from uniform and that considerable heterogeneity may exist within the student population.
A Theory- and Hypothesis-Driven Bayesian Classification Approach: Constructing a “Cognitive Profile” for Each Student
To examine individual differences in greater depth, this study moved beyond both traditional group-comparison approaches and purely data-driven analyses of individual variation by adopting a hypothesis-driven Bayesian classification method. Drawing on dual-process theories of heuristic and algorithmic processing, overlapping waves theory, and developmental models of mental structure, the study theoretically derived in advance eight hypothetical cognitive patterns that students might exhibit when solving additive and multiplicative problems, along with the corresponding behavioral response profiles associated with each pattern—namely, accuracy and response time across the two types of problems (see Figure 1 for the hypothesized response patterns).
For example, a student with an Add-dominant cognitive pattern would respond both quickly and accurately to additive problems, and would also respond quickly to multiplicative problems, but with low accuracy, because the student fails to recognize that the additive strategy being applied is not appropriate for multiplicative problems. By contrast, a student with a Mul-inhibition cognitive pattern would show relatively high accuracy on both types of problems, but a longer response time on additive problems, because additional cognitive resources are required to inhibit interference from multiplicative thinking. The study also proposed a cognitive pattern that had not been identified in previous research—namely, the Intuition type. Students of this type respond both quickly and accurately to both kinds of problems, suggesting that they have formed an effective “mental structure” for problem solving.
Figure 1. Hypothesized behavioral performance (accuracy and response time) under different cognitive patterns.
Note: Add = addition; Mul = multiplication. The horizontal and vertical axes in the figure are used only to illustrate response accuracy and relative speed, and do not represent actual behavioral values.
A total of 229 students, ranging from Grade 3 to Grade 8 as well as university students, completed a conflict matching task (Figure 2), during which both response accuracy and response time were recorded. The researchers then applied a naïve Bayesian classification approach to calculate, on the basis of each student’s response pattern across all items, the posterior probability that the student belonged to each hypothetical pattern. In this way, each student could be assigned the pattern label that best matched his or her cognitive characteristics. This method makes it possible to draw probabilistic inferences about individual cognitive processing mechanisms on the basis of a limited set of response data.
Figure 2. Experimental procedure of the match–conflict task.
Key Findings: Three Major Cognitive Patterns and a Clear Developmental Trajectory
Group-level analyses showed that, across all grade levels, students performed better on multiplicative problems than on additive problems, with higher accuracy and faster response times for the former (see Figure 3). This finding is consistent with the research team’s previous work, which suggested that students generally exhibit a multiplicative-thinking heuristic and that the activation of multiplicative thinking tends to be more automatic.
Figure 3. Overall performance of students at different grade levels on the two types of problems.
The Bayesian analyses, however, revealed that students could be classified into eight distinct cognitive patterns (see Figure 4), among which three emerged as the major patterns. The first was the Add-inhibition pattern, in which students tended to rely on an additive heuristic but were able to successfully inhibit this tendency when solving multiplicative problems, arriving at the correct answer at the cost of additional processing time. The second was the Mul-inhibition pattern, in which students tended to rely on a multiplicative heuristic but could successfully inhibit this tendency when solving additive problems, again achieving accurate performance by investing more time. The third was the Intuition pattern, in which students demonstrated a high degree of proficiency and automaticity in both addition and multiplication, solving both types of problems quickly and accurately with little need for conscious inhibitory effort.
The study further identified a clear developmental trend: as grade level increased, the proportion of students showing the Add-inhibition pattern declined significantly, whereas the proportions exhibiting the Mul-inhibition and Intuition patterns rose steadily. This suggests that the development of mathematical thinking is not merely a matter of acquiring knowledge, but also a process in which inhibitory control continues to develop and cognitive processing becomes increasingly automatized. The Intuition pattern appears to represent a more advanced stage of cognitive development, in which formal mathematical rules have been internalized as automated “mental programs.”
Figure 4. Posterior probabilities of each cognitive pattern across grade levels after Bayesian classification.
Perhaps the most important contribution of this study lies in its integration of cutting-edge cognitive psychology theory with educational practice. Through a highly interpretable method such as Bayesian classification, teachers and educational assessment systems may in the future be able to rapidly diagnose students’ cognitive patterns. Based on such diagnostic results, personalized intervention may become possible. For students showing the Mul-inhibition pattern, who need to suppress an excessive tendency toward multiplicative reasoning, executive function training could be designed to strengthen their ability to shift thinking appropriately in additive contexts. For students with the Add-inhibition pattern, who need to inhibit an overreliance on additive reasoning, metacognitive strategies such as “pause and think” may be encouraged, alongside contrastive exercises featuring similar surface features but different underlying relations, to help build bridges toward the application of multiplicative thinking. For students who have already reached the Intuition pattern, more challenging and non-routine problems should be provided to further deepen conceptual understanding and strategic flexibility while preventing cognitive rigidity.
This study marks an important step in the field of arithmetic reasoning development by showing that the integration of individual-centered Bayesian methods with traditional group-level analyses can provide a more comprehensive and fine-grained account of students’ cognitive patterns. In doing so, it offers important scientific evidence and practical insight for advancing student-centered educational practice.
Citation:
Jiang, R., Li, X., & Cai, M. (2026). From additive to multiplicative thinking: Individual cognitive patterns revealed through Bayesian classification. Journal of Educational Psychology. Advance online publication. https://dx.doi.org/10.1037/edu0001005
