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

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Linear Independence of Functions Definition of linear independence of vector-valued functions: Let f i : I = (a,b) → n, I = 1, 2,…, n. The functions f 1, f 2,…, f n are linearly independent on I if and only if a i = 0 (i = 1, 2,…, n) is the only solution to a 1 f 1 (t)+ a 2 f 2 (t) +…+ a n f n (t) = 0 for all t I.

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Data Collection

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DescriptionExample Fix t First Students evaluated the functions at one or more fixed value of t. Most then evaluated if the resulting real-valued vectors were LI or LD. Focus on Scalars Students carried out algebraic manipulations on the homogeneous vector equation (or related system of equations) without evaluating for specific t-values. They typically interpreted the results of their algebraic work by fixing the set of parameters and considering which/how many values of t satisfied the equation. “It is LI at some values of t for a given a set of values a, b, and c but not at others” “No, because for any a 1, a 2, a 3, there exists a time t when the vectors are linearly independent, except for the zero vector.” “There are plenty of t's where aF + bG + cH ≠ 0 with the required values of a and b” Function Combination Students seem to attend to linear combinations of t, t 2, and t 3 of a variable t that is not constant. These students did not evaluate the functions for specific t-values. “Can't change degree of t with scalar constants.” “No linear combination of t and t 2 will ever yield t 3, so they are LI.” “No because there is no way you can get one of the other vectors from a l.c. of the others. The only solution to a 1 F(t) + a 2 G(t) + a 3 H(t) = 0 is a i = 0, i = 1, 2, 3” Previous Rule Students relied on rules or generalizations about vectors in Euclidean spaces to determine linear (in)dependence of the set of functions. “The set of equations would be L.D. because it has 3 components in R 2.” “Yes, F(t), G(t), and H(t) are LD b/c there are three vectors in R 2.”

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DescriptionExample Fix t First Students evaluated the functions at one or more fixed value of t. Most then evaluated if the resulting real-valued vectors were LI or LD. Focus on Scalars Students carried out algebraic manipulations on the homogeneous vector equation (or related system of equations) without evaluating for specific t-values. They typically interpreted the results of their algebraic work by fixing the set of parameters and considering which/how many values of t satisfied the equation. “It is LI at some values of t for a given a set of values a, b, and c but not at others” “No, because for any a 1, a 2, a 3, there exists a time t when the vectors are linearly independent, except for the zero vector.” “There are plenty of t's where aF + bG + cH ≠ 0 with the required values of a and b” Function Combination Students seem to attend to linear combinations of t, t 2, and t 3 of a variable t that is not constant. These students did not evaluate the functions for specific t-values. “Can't change degree of t with scalar constants.” “No linear combination of t and t 2 will ever yield t 3, so they are LI.” “No because there is no way you can get one of the other vectors from a l.c. of the others. The only solution to a 1 F(t) + a 2 G(t) + a 3 H(t) = 0 is a i = 0, i = 1, 2, 3” Previous Rule Students relied on rules or generalizations about vectors in Euclidean spaces to determine linear (in)dependence of the set of functions. “The set of equations would be L.D. because it has 3 components in R 2.” “Yes, F(t), G(t), and H(t) are LD b/c there are three vectors in R 2.”

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DescriptionExample Fix t First Students evaluated the functions at one or more fixed value of t. Most then evaluated if the resulting real-valued vectors were LI or LD. Focus on Scalars Students carried out algebraic manipulations on the homogeneous vector equation (or related system of equations) without evaluating for specific t-values. They typically interpreted the results of their algebraic work by fixing the set of parameters and considering which/how many values of t satisfied the equation. “It is LI at some values of t for a given a set of values a, b, and c but not at others” “No, because for any a 1, a 2, a 3, there exists a time t when the vectors are linearly independent, except for the zero vector.” “There are plenty of t's where aF + bG + cH ≠ 0 with the required values of a and b” Function Combination Students seem to attend to linear combinations of t, t 2, and t 3 of a variable t that is not constant. These students did not evaluate the functions for specific t-values. “Can't change degree of t with scalar constants.” “No linear combination of t and t 2 will ever yield t 3, so they are LI.” “No because there is no way you can get one of the other vectors from a l.c. of the others. The only solution to a 1 F(t) + a 2 G(t) + a 3 H(t) = 0 is a i = 0, i = 1, 2, 3” Previous Rule Students relied on rules or generalizations about vectors in Euclidean spaces to determine linear (in)dependence of the set of functions. “The set of equations would be L.D. because it has 3 components in R 2.” “Yes, F(t), G(t), and H(t) are LD b/c there are three vectors in R 2.”

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DescriptionExample Fix t First Students evaluated the functions at one or more fixed value of t. Most then evaluated if the resulting real-valued vectors were LI or LD. Focus on Scalars Students carried out algebraic manipulations on the homogeneous vector equation (or related system of equations) without evaluating for specific t-values. They typically interpreted the results of their algebraic work by fixing the set of parameters and considering which/how many values of t satisfied the equation. “It is LI at some values of t for a given a set of values a, b, and c but not at others” “No, because for any a 1, a 2, a 3, there exists a time t when the vectors are linearly independent, except for the zero vector.” “There are plenty of t's where aF + bG + cH ≠ 0 with the required values of a and b” Function Combination Students seem to attend to linear combinations of t, t 2, and t 3 of a variable t that is not constant. These students did not evaluate the functions for specific t-values. “Can't change degree of t with scalar constants.” “No linear combination of t and t 2 will ever yield t 3, so they are LI.” “No because there is no way you can get one of the other vectors from a l.c. of the others. The only solution to a 1 F(t) + a 2 G(t) + a 3 H(t) = 0 is a i = 0, i = 1, 2, 3” Previous Rule Students relied on rules or generalizations about vectors in Euclidean spaces to determine linear (in)dependence of the set of functions. “The set of equations would be L.D. because it has 3 components in R 2.” “Yes, F(t), G(t), and H(t) are LD b/c there are three vectors in R 2.”

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DescriptionExample Fix t First Students evaluated the functions at one or more fixed value of t. Most then evaluated if the resulting real-valued vectors were LI or LD. Focus on Scalars Students carried out algebraic manipulations on the homogeneous vector equation (or related system of equations) without evaluating for specific t-values. They typically interpreted the results of their algebraic work by fixing the set of parameters and considering which/how many values of t satisfied the equation. “It is LI at some values of t for a given a set of values a, b, and c but not at others” “No, because for any a 1, a 2, a 3, there exists a time t when the vectors are linearly independent, except for the zero vector.” “There are plenty of t's where aF + bG + cH ≠ 0 with the required values of a and b” Function Combination Students seem to attend to linear combinations of t, t 2, and t 3 of a variable t that is not constant. These students did not evaluate the functions for specific t-values. “Can't change degree of t with scalar constants.” “No linear combination of t and t 2 will ever yield t 3, so they are LI.” “No because there is no way you can get one of the other vectors from a l.c. of the others. The only solution to a 1 F(t) + a 2 G(t) + a 3 H(t) = 0 is a i = 0, i = 1, 2, 3” Previous Rule Students relied on rules or generalizations about vectors in Euclidean spaces to determine linear (in)dependence of the set of functions. “The set of equations would be L.D. because it has 3 components in R 2.” “Yes, F(t), G(t), and H(t) are LD b/c there are three vectors in R 2.”

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DescriptionExample Fix t First Students evaluated the functions at one or more fixed value of t. Most then evaluated if the resulting real-valued vectors were LI or LD. Focus on Scalars Students carried out algebraic manipulations on the homogeneous vector equation (or related system of equations) without evaluating for specific t-values. They typically interpreted the results of their algebraic work by fixing the set of parameters and considering which/how many values of t satisfied the equation. “It is LI at some values of t for a given a set of values a, b, and c but not at others” “No, because for any a 1, a 2, a 3, there exists a time t when the vectors are linearly independent, except for the zero vector.” “There are plenty of t's where aF + bG + cH ≠ 0 with the required values of a and b” Function Combination Students seem to attend to linear combinations of t, t 2, and t 3 of a variable t that is not constant. These students did not evaluate the functions for specific t-values. “Can't change degree of t with scalar constants.” “No linear combination of t and t 2 will ever yield t 3, so they are LI.” “No because there is no way you can get one of the other vectors from a l.c. of the others. The only solution to a 1 F(t) + a 2 G(t) + a 3 H(t) = 0 is a i = 0, i = 1, 2, 3” Previous Rule Students relied on rules or generalizations about vectors in Euclidean spaces to determine linear (in)dependence of the set of functions. “The set of equations would be L.D. because it has 3 components in R 2.” “Yes, F(t), G(t), and H(t) are LD b/c there are three vectors in R 2.”

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

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

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

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

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

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

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InfinitesimalThe ratio of the infinitesimal change in the value of a function to the infinitesimal change in a function. SymbolicThe derivative of x n is nx n-1, the derivative of sin(x) is cos(x), the derivative of f o g is f′ o g*g′, etc. Logicalf′ (x) = d if and only if for every ε there is a δ such that when 0 < Δx < δ, GeometricThe derivative is the slope of a line tangent to the graph of the function, if the graph has a tangent. RateThe instantaneous speed of f(t), when t is time. ApproximationThe derivative of a function is the best linear approximation to the function near a point. MicroscopicThe derivative of a function is the limit of what you get by looking at it under a microscope of higher and higher power. Thurston’s (2006) Various Descriptions of How One Might Think about Derivative

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

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What’s the Point? DescriptionExample Fix t First Students evaluated the functions at one or more fixed value of t. Most then evaluated if the resulting real-valued vectors were LI or LD. Function Combination Students seem to attend to linear combinations of t, t 2, and t 3 of a variable t that is not constant. These students did not evaluate the functions for specific t-values. “Can't change degree of t with scalar constants.” “No linear combination of t and t 2 will ever yield t 3, so they are LI.” “No because there is no way you can get one of the other vectors from a l.c. of the others. The only solution to a 1 F(t) + a 2 G(t) + a 3 H(t) = 0 is a i = 0, i = 1, 2, 3”

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Definition of linear independence of vectors: The vectors v 1, v 2,…, v n n are linearly independent if and only if a i = 0 (i = 1, 2,…, n) is the only solution to a 1 v 1 + a 2 v 2 +…+ a n = 0. Previous Context for LI

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What’s the Point? DescriptionExample Fix t First Students evaluated the functions at one or more fixed value of t. Most then evaluated if the resulting real-valued vectors were LI or LD. Function Combination Students seem to attend to linear combinations of t, t 2, and t 3 of a variable t that is not constant. These students did not evaluate the functions for specific t-values. “Can't change degree of t with scalar constants.” “No linear combination of t and t 2 will ever yield t 3, so they are LI.” “No because there is no way you can get one of the other vectors from a l.c. of the others. The only solution to a 1 F(t) + a 2 G(t) + a 3 H(t) = 0 is a i = 0, i = 1, 2, 3”

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Linear Independence of Functions Definition of linear independence of vector-valued functions: Let f i : I = (a,b) → n, I = 1, 2,…, n. The functions f 1, f 2,…, f n are linearly independent on I if and only if a i = 0 (i = 1, 2,…, n) is the only solution to a 1 f 1 (t)+ a 2 f 2 (t) +…+ a n f n (t) = 0 for all t I.

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Revisiting the Definition of LI of Functions Definition of linear independence of vector-valued functions: Let f i : I = (a,b) → n, I = 1, 2,…, n. The functions f 1, f 2,…, f n are linearly independent on I if and only if a i = 0 (i = 1, 2,…, n) is the only solution to a 1 f 1 (t)+ a 2 f 2 (t) +…+ a n f n (t) = 0(t).

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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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