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Exploring Students' Understanding of Linear Independence of Functions through the Process/Object Pairs Framework

This study investigates student comprehension of linear independence in vector-valued functions using the Process/Object Pairs framework developed by key educational theorists such as Sfard, Dubinsky, and Gravemeijer. We define linear independence as the condition where the only solution to a linear combination of functions equaling zero is when all coefficients are zero. By employing observational data and theoretical frameworks, we aim to enhance instructional approaches to teaching this fundamental concept in linear algebra.

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Exploring Students' Understanding of Linear Independence of Functions through the Process/Object Pairs Framework

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  1. Exploring Students' Understanding of Linear Independence of Functions with the Process/Object Pairs Framework David Plaxco

  2. Linear Independence of Functions • Definition of linear independence of vector-valued functions: Let fi: I = (a,b) → n, I = 1, 2,…, n. The functions f1,f2,…,fnare linearly independent on Iif and only if ai= 0 (i = 1, 2,…, n) is the only solution to a1f1(t)+ a2f2(t) +…+anfn(t) = 0 for all t I.

  3. Data Collection

  4. Process/Object Pairs • Sfard (1991) • Dubinksy (1991) • Gravemeijer (1999) • Zandieh (2000) • Norton (2013)

  5. Process/Object Pairs • Sfard (1991) • Dubinksy (1991) • Gravemeijer (1999) • Zandieh (2000) • Norton (2013)

  6. Process/Object Pairs • Sfard (1991) • Dubinksy (1991) • Gravemeijer (1999) • Zandieh (2000) • Norton (2013)

  7. Process/Object Pairs • Sfard (1991) • Dubinksy (1991) • Gravemeijer (1999) • Zandieh (2000) • Norton (2013)

  8. Process/Object Pairs • Sfard (1991) • Dubinksy (1991) • Gravemeijer (1999) • Zandieh (2000) • Norton (2013)

  9. Process/Object Pairs • Sfard (1991) • Dubinksy (1991) • Gravemeijer (1999) • Zandieh (2000) • Norton (2013)

  10. Thurston’s (2006) Various Descriptions of How One Might Think about Derivative

  11. Zandieh’s (2000) Framework for the Concept of Derivative

  12. What’s the Point?

  13. Previous Context for LI • Definition of linear independence of vectors: The vectors v1,v2,…,vnnare linearly independent if and only if ai= 0 (i = 1, 2,…, n) is the only solution to a1v1+ a2v2+…+an= 0.

  14. What’s the Point?

  15. Linear Independence of Functions • Definition of linear independence of vector-valued functions: Let fi: I = (a,b) → n, I = 1, 2,…, n. The functions f1,f2,…,fnare linearly independent on Iif and only if ai= 0 (i = 1, 2,…, n) is the only solution to a1f1(t)+ a2f2(t) +…+anfn(t) = 0 for all t I.

  16. Revisiting the Definition of LI of Functions • Definition of linear independence of vector-valued functions: Let fi: I = (a,b) → n, I = 1, 2,…, n. The functions f1,f2,…,fnare linearly independent on Iif and only if ai= 0 (i = 1, 2,…, n) is the only solution to a1f1(t)+ a2f2(t) +…+anfn(t) = 0(t).

  17. References Dubinsky, E. (1991). Reflective abstraction in advanced mathematical thinking. In Advanced mathematical thinking (pp. 95-126). Springer Netherlands. Gravemeijer, K. (1999). Emergent models may foster the constitution of formal mathematics. Mathematical Thinking and Learning, 1(2), 155-177. Norton, A. (2013). The wonderful gift of mathematics. Sfard, A. (1991). On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin. Educational Studies in Mathematics, 22, 1–36. Thurston, W. (1995). On proof and progress in mathematics. For the Learning of Mathematics, 15(1), 29–37. Zandieh, M. (2000). A theoretical framework for analyzing student understanding of the concept of derivative. Research in Collegiate Mathematics Education, IV (Vol. 8, pp. 103-127).

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