Abstract
Mathematical connections are widely considered an important aspect of learning linear algebra, particularly at the introductory level. One effective strategy for teaching mathematical connections in introductory linear algebra is through inquiry-based learning (IBL). The demands of IBL instruction can make it difficult to implement such strategies in courses in which the instructor faces various constraints. The findings presented here are the product of an action research study in which IBL instructional materials were designed for a large-enrolled introductory linear algebra course with limited class time. This resulted in IBL being presented in a limited capacity alongside traditional lecture in what will be described as adapted inquiry. Specifically, these IBL materials were designed as vehicles through which students could form mathematical connections. This study was conducted with the goal of determining what mathematical connections students appear to be able to exhibit within the context of an adapted inquiry approach to IBL instruction.
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Appendix
Appendix
Codes Used in Analysis of Student Logical Implication Connections.
Code | Statements from the Span Theorem |
---|---|
A1 | The equation \(A\mathbf{x}=\mathbf{b}\) has at least one solution for every \(\mathbf{b}\) in \({\mathbb{R}}^{m}\) |
A2 | There exists a vector \(\mathbf{b}\)in \({\mathbb{R}}^{m}\) such that \(A\mathbf{x}=\mathbf{b}\) does not have a solution |
B1 | \(A\) has a pivot position in every row |
B2 | \(A\) does not have a pivot position in every row |
C1 | Every vector \(\mathbf{b}\) in \({\mathbb{R}}^{m}\) is a linear combination of the columns of \(A\) |
C2 | There exists a vector \(\mathbf{b}\) in \({\mathbb{R}}^{m}\) that is not a linear combination of the columns of\(A\) (not every vector in \({\mathbb{R}}^{m}\) is a linear combination of the columns of \(A\)) |
D1 | The columns of \(A\) span \({\mathbb{R}}^{m}\) |
D2 | The columns of \(A\) do not span \({\mathbb{R}}^{m}\) |
E1 | The linear transformation \(\mathbf{x}\to A\mathbf{x}\) maps \({\mathbb{R}}^{n}\) onto \({\mathbb{R}}^{m}\) |
E2 | The linear transformation \(\mathbf{x}\to A\mathbf{x}\) does not map \({\mathbb{R}}^{n}\) onto \({\mathbb{R}}^{m}\) |
Code | Statements from the Linear Independence Theorem |
---|---|
F1 | The equation \(A\mathbf{x}=\mathbf{b}\) has at most one solution for every \(\mathbf{b}\) in \({\mathbb{R}}^{m}\) |
F2 | There exists a vector \(\mathbf{b}\) in \({\mathbb{R}}^{m}\) such that \(A\mathbf{x}=\mathbf{b}\) has infinitely many solutions |
G1 | The corresponding linear system only has basic variables (has no free variables) |
G2 | The corresponding linear system has a free variable |
H1 | \(A\) has a pivot position in every column |
H2 | \(A\) does not have a pivot position in every column |
I1 | \(A\mathbf{x}=\mathbf{0}\) has only the trivial solution |
I2 | \(A\mathbf{x}=\mathbf{0}\) has (infinitely many) nontrivial solutions |
J1 | No column of \(A\) is a linear combination of the others |
J2 | At least one column of \(A\) is a linear combination of the others |
K1 | The columns of \(A\) are linearly independent |
K2 | The columns of \(A\) are linearly dependent (not linearly independent) |
L1 | The linear transformation \(\mathbf{x}\to A\mathbf{x}\) is one-to-one |
L2 | The linear transformation \(\mathbf{x}\to A\mathbf{x}\) is not one-to-one |
Code | Statements exclusive to the Invertible Matrix Theorem |
---|---|
M1 | The equation \(A\mathbf{x}=\mathbf{b}\) has exactly one solution for every \(\mathbf{b}\) in \({\mathbb{R}}^{n}\) |
M2 | There exists a vector \(\mathbf{b}\) in \({\mathbb{R}}^{n}\) such that \(A\mathbf{x}=\mathbf{b}\) does not have exactly one solution |
N1 | \(A\) has a pivot position in every row and in every column |
N2 | \(A\) does not have a pivot position in every row or \(A\) does not have a pivot position in every column |
O1 | \(A\) is row equivalent to the \(n\times n\) identity |
O2 | \(A\) is not row equivalent to the \(n\times n\) identity |
P1 | \(A\) is invertible |
P2 | \(A\) is not invertible |
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Payton, S. Fostering mathematical connections in introductory linear algebra through adapted inquiry. ZDM Mathematics Education 51, 1239–1252 (2019). https://doi.org/10.1007/s11858-019-01029-9
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DOI: https://doi.org/10.1007/s11858-019-01029-9