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Fostering mathematical connections in introductory linear algebra through adapted inquiry

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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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Correspondence to Spencer Payton.

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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