Exploring Students' Understanding of Linear
Independence of Functions with
the Process/Object Pairs Framework
David Plaxco
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,…, fn are linearly independent on I
if 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.
Data Collection
Description
Fix t First
Focus on Scalars
Example
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.
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 a1, a2, a3, 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”
“Can't change degree of t with scalar constants.”
Function Combination
Students seem to attend to linear combinations of
t, t2, and t3 of a variable t that is not constant.
These students did not evaluate the functions for
specific t-values.
Previous Rule
Students relied on rules or generalizations about
vectors in Euclidean spaces to determine linear
(in)dependence of the set of functions.
“No linear combination of t and t2 will ever yield
t3, 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 a1F(t) + a2G(t) + a3H(t) = 0 is ai
= 0, i = 1, 2, 3”
“The set of equations would be L.D. because it
has 3 components in R2.”
“Yes, F(t), G(t), and H(t) are LD b/c there are
three vectors in R2.”
Description
Fix t First
Focus on Scalars
Example
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.
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 a1, a2, a3, 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”
“Can't change degree of t with scalar constants.”
Function Combination
Students seem to attend to linear combinations of
t, t2, and t3 of a variable t that is not constant.
These students did not evaluate the functions for
specific t-values.
Previous Rule
Students relied on rules or generalizations about
vectors in Euclidean spaces to determine linear
(in)dependence of the set of functions.
“No linear combination of t and t2 will ever yield
t3, 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 a1F(t) + a2G(t) + a3H(t) = 0 is ai
= 0, i = 1, 2, 3”
“The set of equations would be L.D. because it
has 3 components in R2.”
“Yes, F(t), G(t), and H(t) are LD b/c there are
three vectors in R2.”
Description
Fix t First
Focus on Scalars
Example
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.
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 a1, a2, a3, 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”
“Can't change degree of t with scalar constants.”
Function Combination
Students seem to attend to linear combinations of
t, t2, and t3 of a variable t that is not constant.
These students did not evaluate the functions for
specific t-values.
Previous Rule
Students relied on rules or generalizations about
vectors in Euclidean spaces to determine linear
(in)dependence of the set of functions.
“No linear combination of t and t2 will ever yield
t3, 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 a1F(t) + a2G(t) + a3H(t) = 0 is ai
= 0, i = 1, 2, 3”
“The set of equations would be L.D. because it
has 3 components in R2.”
“Yes, F(t), G(t), and H(t) are LD b/c there are
three vectors in R2.”
Description
Fix t First
Focus on Scalars
Example
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.
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 a1, a2, a3, 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”
“Can't change degree of t with scalar constants.”
Function Combination
Students seem to attend to linear combinations of
t, t2, and t3 of a variable t that is not constant.
These students did not evaluate the functions for
specific t-values.
Previous Rule
Students relied on rules or generalizations about
vectors in Euclidean spaces to determine linear
(in)dependence of the set of functions.
“No linear combination of t and t2 will ever yield
t3, 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 a1F(t) + a2G(t) + a3H(t) = 0 is ai
= 0, i = 1, 2, 3”
“The set of equations would be L.D. because it
has 3 components in R2.”
“Yes, F(t), G(t), and H(t) are LD b/c there are
three vectors in R2.”
Description
Fix t First
Focus on Scalars
Example
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.
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 a1, a2, a3, 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”
“Can't change degree of t with scalar constants.”
Function Combination
Students seem to attend to linear combinations of
t, t2, and t3 of a variable t that is not constant.
These students did not evaluate the functions for
specific t-values.
Previous Rule
Students relied on rules or generalizations about
vectors in Euclidean spaces to determine linear
(in)dependence of the set of functions.
“No linear combination of t and t2 will ever yield
t3, 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 a1F(t) + a2G(t) + a3H(t) = 0 is ai
= 0, i = 1, 2, 3”
“The set of equations would be L.D. because it
has 3 components in R2.”
“Yes, F(t), G(t), and H(t) are LD b/c there are
three vectors in R2.”
Description
Fix t First
Focus on Scalars
Example
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.
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 a1, a2, a3, 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”
“Can't change degree of t with scalar constants.”
Function Combination
Students seem to attend to linear combinations of
t, t2, and t3 of a variable t that is not constant.
These students did not evaluate the functions for
specific t-values.
Previous Rule
Students relied on rules or generalizations about
vectors in Euclidean spaces to determine linear
(in)dependence of the set of functions.
“No linear combination of t and t2 will ever yield
t3, 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 a1F(t) + a2G(t) + a3H(t) = 0 is ai
= 0, i = 1, 2, 3”
“The set of equations would be L.D. because it
has 3 components in R2.”
“Yes, F(t), G(t), and H(t) are LD b/c there are
three vectors in R2.”
Process/Object Pairs
•
•
•
•
•
Sfard (1991)
Dubinksy (1991)
Gravemeijer (1999)
Zandieh (2000)
Norton (2013)
Process/Object Pairs
•
•
•
•
•
Sfard (1991)
Dubinksy (1991)
Gravemeijer (1999)
Zandieh (2000)
Norton (2013)
Process/Object Pairs
•
•
•
•
•
Sfard (1991)
Dubinksy (1991)
Gravemeijer (1999)
Zandieh (2000)
Norton (2013)
Process/Object Pairs
•
•
•
•
•
Sfard (1991)
Dubinksy (1991)
Gravemeijer (1999)
Zandieh (2000)
Norton (2013)
Process/Object Pairs
•
•
•
•
•
Sfard (1991)
Dubinksy (1991)
Gravemeijer (1999)
Zandieh (2000)
Norton (2013)
Process/Object Pairs
•
•
•
•
•
Sfard (1991)
Dubinksy (1991)
Gravemeijer (1999)
Zandieh (2000)
Norton (2013)
Thurston’s (2006) Various Descriptions of
How One Might Think about Derivative
Infinitesimal
The ratio of the infinitesimal change in the value of a function to
the infinitesimal change in a function.
Symbolic
The derivative of xn is nxn-1, the derivative of sin(x) is cos(x), the
derivative of fog is f′og*g′, etc.
Logical
f′ (x) = d if and only if for every ε there is a δ such that when
0 < ⎥ Δx ⎥ < δ,
Geometric
The derivative is the slope of a line tangent to the graph of the
function, if the graph has a tangent.
Rate
The instantaneous speed of f(t), when t is time.
Approximation
The derivative of a function is the best linear approximation to
the function near a point.
Microscopic
The derivative of a function is the limit of what you get by
looking at it under a microscope of higher and higher power.
Zandieh’s (2000) Framework for the
Concept of Derivative
What’s the Point?
Description
Fix t First
Example
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.
“Can't change degree of t with scalar constants.”
Function Combination
Students seem to attend to linear combinations of
t, t2, and t3 of a variable t that is not constant.
These students did not evaluate the functions for
specific t-values.
“No linear combination of t and t2 will ever yield
t3, 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 a1F(t) + a2G(t) + a3H(t) = 0 is ai
= 0, i = 1, 2, 3”
Previous Context for LI
• Definition of linear independence of vectors:
n are linearly
The vectors v1, v2,…, vn
independent if and only if ai = 0 (i = 1, 2,…, n)
is the only solution to a1v1+ a2v2+…+ an = 0.
What’s the Point?
Description
Fix t First
Example
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.
“Can't change degree of t with scalar constants.”
Function Combination
Students seem to attend to linear combinations of
t, t2, and t3 of a variable t that is not constant.
These students did not evaluate the functions for
specific t-values.
“No linear combination of t and t2 will ever yield
t3, 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 a1F(t) + a2G(t) + a3H(t) = 0 is ai
= 0, i = 1, 2, 3”
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,…, fn are linearly independent on I
if 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.
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,…, fn are linearly independent on I
if and only if ai = 0 (i = 1, 2,…, n) is the only solution
to a1f1(t)+ a2f2(t) +…+ anfn(t) = 0(t).
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).