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Nonlinear Learning Trajectories from Natural to Decimal Numbers in Zambia Based on Process-Object Dualism

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Title: Nonlinear Learning Trajectories from Natural to Decimal Numbers in Zambia Based on Process-Object Dualism
Language: English
Authors: Yoshitaka Abe
Source: Journal of Education and Learning. 2026 15(2):22-37.
Availability: Canadian Center of Science and Education. 1595 Sixteenth Ave Suite 301, Richmond Hill, Ontario, L4B 3N9 Canada. Tel: 416-642-2606; Fax: 416-642-2608; e-mail: jel@ccsenet.org; Web site: http://www.ccsenet.org/journal/index.php/jel
Peer Reviewed: Y
Page Count: 16
Publication Date: 2026
Document Type: Journal Articles
Reports - Research
Education Level: Elementary Education
Grade 6
Intermediate Grades
Middle Schools
Grade 7
Junior High Schools
Secondary Education
Descriptors: Foreign Countries, Learning Trajectories, Numbers, Arithmetic, Elementary School Students, Elementary School Mathematics, Grade 6, Grade 7, Misconceptions
Geographic Terms: Zambia
ISSN: 1927-5250
1927-5269
Abstract: Understanding the transition from natural to decimal numbers is a central challenge in mathematics education, especially in developing countries. The present study investigates how Zambian primary students conceptualize decimal numbers and which cognitive difficulties they encounter, using Sfard's process-object dualism as a theoretical framework. To capture children's conception, I focused on the sub-item of conception, knowledge. Data were collected over two years from 204 sixth- and seventh-grade students through assessments and interviews. To capture students' cognitive trajectories, a seven-stage learning trajectory was developed. From prior knowledge, the students moved from operational knowledge of natural numbers and decimals to pseudo-structural and structural knowledge. The results revealed that many students demonstrated computational fluency without a deep conceptual grasp and that pseudo-structural knowledge persisted. Typical misconceptions included overgeneralizing natural number rules (e.g., 0.8 ÷ 2 = 4) and misinterpreting place value. Importantly, the results showed that learning does not proceed linearly from one stage to the next. Rather, students often moved back and forth between stages, revisiting earlier forms of reasoning even after demonstrating higher-level understanding. This non-linear and dynamic nature of learning suggests that operational and structural knowledge interact in complex ways. The study proposes a classification system to diagnose students' conceptual stages and provide targeted instructional strategies. By extending process--object dualism to primary-level learning and emphasizing the recursive, interactive nature of concept development, this research offers new insights into improving mathematics instruction and assessment in low-resource educational settings.
Abstractor: As Provided
Entry Date: 2026
Accession Number: EJ1507227
Database: ERIC
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  Data: Nonlinear Learning Trajectories from Natural to Decimal Numbers in Zambia Based on Process-Object Dualism
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  Data: Canadian Center of Science and Education. 1595 Sixteenth Ave Suite 301, Richmond Hill, Ontario, L4B 3N9 Canada. Tel: 416-642-2606; Fax: 416-642-2608; e-mail: jel@ccsenet.org; Web site: http://www.ccsenet.org/journal/index.php/jel
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  Data: Understanding the transition from natural to decimal numbers is a central challenge in mathematics education, especially in developing countries. The present study investigates how Zambian primary students conceptualize decimal numbers and which cognitive difficulties they encounter, using Sfard's process-object dualism as a theoretical framework. To capture children's conception, I focused on the sub-item of conception, knowledge. Data were collected over two years from 204 sixth- and seventh-grade students through assessments and interviews. To capture students' cognitive trajectories, a seven-stage learning trajectory was developed. From prior knowledge, the students moved from operational knowledge of natural numbers and decimals to pseudo-structural and structural knowledge. The results revealed that many students demonstrated computational fluency without a deep conceptual grasp and that pseudo-structural knowledge persisted. Typical misconceptions included overgeneralizing natural number rules (e.g., 0.8 ÷ 2 = 4) and misinterpreting place value. Importantly, the results showed that learning does not proceed linearly from one stage to the next. Rather, students often moved back and forth between stages, revisiting earlier forms of reasoning even after demonstrating higher-level understanding. This non-linear and dynamic nature of learning suggests that operational and structural knowledge interact in complex ways. The study proposes a classification system to diagnose students' conceptual stages and provide targeted instructional strategies. By extending process--object dualism to primary-level learning and emphasizing the recursive, interactive nature of concept development, this research offers new insights into improving mathematics instruction and assessment in low-resource educational settings.
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      – SubjectFull: Learning Trajectories
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      – SubjectFull: Numbers
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      – SubjectFull: Arithmetic
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      – SubjectFull: Elementary School Students
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      – SubjectFull: Misconceptions
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      – SubjectFull: Zambia
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