Four partial meanings of the definite integral articulated as a global meaning¶

Theory · All theories
Evidence · 3 claims (3 for) · 1 study (1 theoretical), q2 · 0 of 1 report an effect size · 3 claims rest on one study

Description¶

The article proposes that instruction should distinguish four partial meanings of the definite integral: as a quantity of magnitude bounded between convergent sequences, as the limit of Riemann sums, as a cumulative function, and as the incremental difference of the cumulative function. "From the point of view of the processes of mathematical instruction at High School and University levels, we consider useful to distinguish four types of meanings, the integral as:" Each partial meaning contributes relevant aspects of the global meaning, understood as an articulated network of partial meanings.

Design Implications¶

Context¶

Requirements¶

• Curricular planning must articulate the partial meanings rather than present any single meaning in isolation

Constraints¶

• Meaning 1 alone is not sufficient for robust understanding; the Riemann-limit calculation is complex compared with antiderivative-based meanings 3 and 4

Target Learners¶

• high school and university calculus students

Target Learning Objectives¶

• understanding the definite integral and its applications

Claims¶

• Abusive generalization processes in the intuitive presentation are a source of potential semiotic conflicts [+W]
• Students can apply procedural knowledge to solve integration problems but show limited understanding of basic integration concepts [+W]
• Understanding Stewart's formal definition of the definite integral demands mobilizing a system of prior knowledge across a network of semiotic functions [+W]

Related Theories¶

Examples¶

• Plan integral instruction around the articulation of partial meanings, since the intuitive meaning alone is insufficient

Key Sources¶

• Burgos, M., Bueno, S., Pérez, O., & Godino, J. D. (2021). Onto-semiotic complexity of the Definite Integral. Implications for teaching and learning Calculus. REDIMAT – Journal of Research in Mathematics Education, 10(1), 4-40. https://doi.org/10.17583/redimat.2021.6778