# Moving Away from Lecture in Undergraduate Mathematics: Managing Tensions within a Coordinated Inquiry-Based Linear Algebra Course

## Abstract

This study describes how nine university professors managed their teaching when a shift from lecturing to inquiry-based learning was mandated in a large enrollment course, linear algebra for math majors. We describe the tensions that emerged and how they were resolved, in part via the production of worksheets that were used in teaching. We describe the way in which professional obligations towards the discipline and towards the institution shaped the tensions and the ways in which they were resolved by this group of faculty. We offer a conceptualization of the enactment of instruction that was prompted by the study and implications for continued investigation of change strategies in undergraduate mathematics.

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1. 1.

The notion of double bind was first proposed by Bateson (1973) to describe conflicting messages in communication patterns between Bali mothers and children. Mellin-Olsen (1987, 1991) introduced the notion in mathematics education, also in the sense of conflicting messages tied to mathematical situations, specifically regarding how explicit is the control over aspects of problem solving activity, specifically its goals and the availability of resources. In this investigation we follow Herbst’s (2006) sense of double-bind.

2. 2.

By a coordinated course we mean a course with more than one section for which there are mechanisms put in place to guarantee that all students are taught “consistent core material” (Rasmussen and Ellis 2015). This notion does not imply that all sections of a course will be the same, as there are many elements to contribute to variations across sections of a course.

3. 3.

1. Linear Equations, 2. Linear transformations, 3. Subspaces of Rn and their dimensions, 4. Linear Spaces, 5. Orthogonality and Least Squares, 6. Determinants, 7. Eigenvalues and eigenvectors, 8. Symmetric Matrices and Quadratic forms, and 9. Linear Differential Equations.

4. 4.

This number is difficult to establish, as some problems may have several parts; so individually a 4-problem worksheet may have 17 different sub-questions (e.g., Lewis’s worksheet for sections 1.2 and 1.3).

5. 5.

Names are pseudonyms.

6. 6.

We use “IBL” when participants are discussing their perceptions of inquiry-based learning.

7. 7.

The analysis of the textbook in Appendix 2 expands on the textbook sections that were problematic to the instructors.

8. 8.

Ulrich, the post-doc who did not participate of the interview, also created his own worksheets.

9. 9.

The Moore Method is named after R. L. Moore, a mathematician who believed that students should create mathematics by working ideas on their own starting with a few key definitions, proposing conjectures, and exploring them in ways that would allow further discoveries of mathematics (Jones 1977; Mahavier 1999). In a class using Moore’s method, the instructor writes a definition on the board and lets the students ponder it and come up with propositions based on that definition without direct instruction.

10. 10.

By non-normative we mean that mathematicians would not recognize the term. A search in Wolfram alpha gives the dictionary (non-mathematical) definition of redundant as “repetition of same sense in different words.”

11. 11.

Bretscher uses the term linear spaces (as opposed to vector spaces) here.

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Correspondence to Vilma Mesa.

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This work has been conducted with support from the Educational Advancement Foundation and the University of Michigan, through a Transforming Learning for Third Century (TLTC) Quick Wins/Discovery award.

## Appendices

### Part 1: Background, Linear Algebra, Inquiry-Based Learning

1. 1.

2. 2.

Prior to coming to this campus what did you know about I-B-L? What experience did you have teaching with I-B-L?

3. 3.

I-B-L can have different meanings to different people. What does inquiry-based learning mean to you?

4. 4.

In your opinion what should be the goals and objectives of any Linear Algebra course? How are they different from the goals and objectives of the Linear Algebra course in this campus?

5. 5.

Fall 2015: Tell us your impression of how well students mastered the goals for the course. (Winter 2016: Tell us your impression of how well students have mastered the goals for the course so far)

6. 6.

Who did/do you collaborate with most during the Fall semester/this semester? What was the nature of the collaboration?

7. 7.

Please describe the role of <your role as> the course coordinator. What kinds of support did you receive from the coordination?

8. 8.

What was most useful to you to learn about I-B-L?

9. 9.

Please describe the supports that were useful to you to learn to teach with I-B-L. How were they helpful?

<there were probes for the various elements that were designed to support the course, including textbook weekly meetings, discussion with colleagues, and observations>.

### Part 2: Worksheets and Quiz

1. 10.

Please tell us what are key features you seek to ensure your worksheets have (e.g., variety, order, level of inquiry, types of problems, etc.)

2. 11.

1. (a)

How did you come up with this (WL, WD, Q)?

2. (b)

Why do you like it?

3. (c)

What objectives/goals does this (WL, WD, Q) target?

4. (d)

What resources did you use in creating it?

5. (e)

How did the students work on the worksheet?

6. (f)

What changes, if any, would you make to it?

1. 12.

Bi-weekly log question: What was your strategy for adapting the worksheets and at the same time managing your time constraints?

2. 13.

For instructors teaching for a second semester:

1. (a)

What is different about teaching this course for a second semester?

2. (b)

What did you learn from last semester?

3. (c)

What are you using now to create the worksheets for your class?

### Appendix 2: Content Contrast of Two Textbooks

We provide a contrast between two textbooks, Bretscher (2013) and Friedberg et al. (2002), hereafter FIS, in order to provide a context for the discussions about content that framed much of the tensions we describe in this study.

### Bretscher’s Presentation of Content

According to Henry, Bretscher was chosen as the textbook for the course because it provided students with a resource that helped them develop intuition about linear algebra concepts and procedures. Henry chose the textbook after reviewing various options and, after having given it “a try,” found it sufficiently readable and usable by the students, one from which “students could learn best.” His perception was that the problems were “decent” and sufficient for helping students develop the needed intuitive understanding of linear algebra ideas, despite its shortcomings regarding the definitions used for linear independence and linear transformations, and the particular ordering of the chapters. During the first semester in which all the sections used IBL with Bretscher, the faculty expressed significant concerns with these shortcomings during the weekly course planning meetings.

#### Linear Independence

In Section 1.5, FIS presents the following definitions:

Definition. A subset S of a vector space V is called linearly dependent if there exists a finite number of distinct vectors u1, …, un in S and scalars a1, …, an, not all zero, such that

$${a}_1{u}_1+{a}_2{u}_2+\cdots +{a}_{\mathrm{n}}{u}_{\mathrm{n}}=0$$

In this case we also say that the vectors of S are linearly dependent.

Definition. A subset S of a vector space V that is not linearly dependent is called linearly independent. As before, we also say that the vectors of S are linearly independent. (Friedberg et al. 2002, p. 36-37)

In contrast, Bretscher defines linear dependence in a logically equivalent way but using the non-normativeFootnote 10 term redundant:

Redundant vectors;4linear independence; basis

Consider vectors $${\overrightarrow{v}}_1,\dots, {\overrightarrow{v}}_m$$in n.

1. a.

We say that a vector $${\overrightarrow{v}}_i$$ in the list $${\overrightarrow{v}}_1,\dots, {\overrightarrow{v}}_m$$ is redundant if $${\overrightarrow{v}}_i$$ is a linear combination of the preceding vectors $${\overrightarrow{v}}_1,\dots, {\overrightarrow{v}}_{i-1}{.}^5$$

2. b.

The vectors $${\overrightarrow{v}}_1,\dots, {\overrightarrow{v}}_m$$ are called linearly independent if none of them is redundant. Otherwise, the vectors are called linearly dependent (meaning that at least one of them is redundant).6

3. c.

We say that vectors $${\overrightarrow{v}}_1,\dots, {\overrightarrow{v}}_m$$ in a subspace V of n form a basis of V if they span V and are linearly independent.7 (p. 125)

In the call for the fourth footnote in the title, “Redundant vectors;” Bretscher states: “The notion of a redundant vector is not part of the established vocabulary of linear algebra. However, we will find this concept quite useful in discussing linear independence” (p. 125). With this statement, Bretscher explicitly recognizes that that redundancy is not a normative definition in linear algebra.

Bretscher justifies the inclusion of a different definition because it is “useful.” Exactly what he meant by this is not explicitly stated. One affordance of this definition, which he might be referring to, is that it helps demonstrate how each vector of a linearly independent set cannot be formed by linear combinations of the other vectors in the set. The normative definition of having no nontrivial solution to an equation does not reveal this relationship. Bretscher’s definition makes the relationship between the vectors in the set more intuitive. One could imagine that once a first vector is pictured in a vector space, the next vector in the linearly independent set cannot be formed by transforming the first vector. A third vector that needs to be linearly independent in turn cannot be formed by transforming or combining the first two vectors, and so on for the m vectors in the linearly independent set. This presentation is consistent with Bretscher’s intent for his textbook, described in the preface as emphasizing visualization.

Laura summed up the definition’s main affordance and limitation saying: “it is basically a formalization of our intuition of linear independence but it is hard to use in proofs.” Consider the problem of showing that vectors in a given set are linearly dependent. With FIS’s normative definition one needs to find the scalars that satisfy the equation $${a}_1{u}_1+{a}_2{u}_2+\cdots +{a}_{\mathrm{n}}{u}_{\mathrm{n}}=0$$ (see Friedberg et al. 2002, p. 36–37). But in order to use this equation in proof-writing, Bretscher needs to do additional work. First, he defines an intermediary definition, linear relation, as follows:

Definition 3.2.6: Consider the vectors $${\overrightarrow{v}}_1,\dots, {\overrightarrow{v}}_m$$ in n. An equation of the form $${c}_1{\overrightarrow{v}}_1+\cdots +{c}_m{\overrightarrow{v}}_m=\overrightarrow{0}$$ is called a (linear) relation among the vectors $${\overrightarrow{v}}_1,\dots, {\overrightarrow{v}}_m$$” (p. 127).

Next he introduces a theorem that relates this definition his own definition of linear dependence:

Theorem 3.2.7… The vectors $${\overrightarrow{v}}_1,\dots, {\overrightarrow{v}}_m$$ in n are linearly dependent if (and only if) there are nontrivial relations among them (p. 128).

The inclusion of the extra definition 3.2.6 and theorem 3.2.7, which resemble the normative FIS definition, show that Bretscher presents both the normative and non-normative definitions to students rather than only one, on account of facilitating students’ “imagin[ing] the computations” (p. ix).

#### Linear Transformations

FIS defines a linear transformation as follows:

Definition. Let V and W be vector spaces (over [a field] F). We call a function T : V → W a linear transformation fromVtoW if, for all x, y ∈ V and c ∈ F, we have

T(x + y) = T(x) + T(y) (and)

T(cx) = cT(x) (FIS, 2002, p. 65)

Bretscher, in contrast, proposes two definitions for linear transformations; one in Chapter 2 and one in Chapter 4. In Chapter 2, he defines linear transformations as follows:

Linear transformation s 2

A function T from m to n is called a linear transformation if there exists an nxm matrix A such that $$T\left(\overrightarrow{x}\right)=A\overrightarrow{x}$$,

For all $$\overrightarrow{x}$$in the vector space m (p. 45).

and states Theorem 2.1.3 as follows:

Theorem 2.1.3: Linear transformations

A transformation T from m to n is linear if (and only if)

1. a.

$$T\left(\overrightarrow{v}+\overrightarrow{w}\right)=T\left(\overrightarrow{v}\right)+T\left(\overrightarrow{w}\right),\mathrm{for}\ \mathrm{all}\ \mathrm{vectors}\ \overrightarrow{v}\ \mathrm{and}\ \overrightarrow{w}\ \mathrm{in}\ {\mathbb{R}}^m,$$ and

2. b.

$$T\left(k\overrightarrow{v}\right)= kT\left(\overrightarrow{v}\right),\mathrm{for}\ \mathrm{all}\ \mathrm{vectors}\ \overrightarrow{v}\ \mathrm{in}\ {\mathbb{R}}^m\ \mathrm{and}\ \mathrm{all}\ \mathrm{scalars}\ k.$$(p. 45)

which corresponds to the normative definition of linear transformations. In the proof for Theorem 2.1.3 Bretscher constructs the needed matrix, justifying his non-normative definition by arguing that “correct” definitions do not exist. In the call for footnote 2 Bretscher writes,

This is one of several possible definitions of a linear transformation; we could just as well have chosen the statement of theorem 2.1.3 as the definition (as many texts do). This will be a recurring theme in this text: Most of the central concepts of linear algebra can be characterized in two or more ways. Each of these characterizations can serve as a possible definition; the other characterizations will then be stated as theorems, since we need to prove that they are equivalent to the chosen definition. Among these multiple characterizations, there is no “correct” definition (although mathematicians may have their favorite). Each characterization will be best suited for certain purposes and problems, while it is inadequate for others (p. 45).

This is of course correct. It might be a matter of preference, and in a course in which students are expected to understand how mathematics is generated, such work might be useful, even if the definitions proposed do not conform to normative, accepted practices. An advantage of Bretscher’s definition is that it is useful for the purpose it is being used for. The first three chapters of the textbook focused on matrices in Euclidean spaces (i.e., n-dimensional spaces of real numbers). Proposing a definition of linear transformation that requires a matrix furthers his goal of helping students gain intuition about what the linear transformations do. Though it is slightly more limited than the definition in FIS (it reduces vector spaces to those in n), it gives students a context for manipulation and recognition of linear independence. By defining linear transformations as those involving matrices, Bretscher delays the use of typical linear transformations such as differentiation, integration rotations, reflections, or projections until Chapter 4.

In Chapter 4, Bretscher introduces the normative definition of a linear transformation:

Definition 4.2.1. Linear transformations, image, kernel, rank, nullity

Consider two linear spacesFootnote 11V and W. A function T from V to W is called a linear transformation if

$$T\left(f+g\right)=T(f)+T(g)\ \mathrm{and}\ T(kf)= kT(f)$$

for all elements f and g of V and for all scalars k. These two rules are referred to as the sum rule and the constant-multiple rule, respectively. (…) (p. 178).

By waiting until Chapter 4 to introduce this definition, Bretscher delays the use of typical linear transformations such as differentiation, integration rotations, reflections, or projections. This aligns with Bretscher’s interest in “keep[ing] abstract exposition to a minimum… The examples always precede the theorems in this book” (2013, p. x). The tension between contextualization and abstraction is not new (Tall 2004). Choosing the contextualization can be rationally justified on grounds of accessibility, while the normative definition (e.g., FIS’s) is useful for proof-writing, but will not give students the contextualization needed to make the material more accessible.

#### Order of Material

The order in which content is presented in Bretscher differs from the order in FIS. FIS introduces linear independence before linear transformations. Bretscher presents linear transformations before linear independence, and as mentioned earlier, Bretscher introduces the two definitions a second time when he defines general vector spaces in Chapter 4. The first three chapters in Bretscher are dedicated to showing students “the language of linear algebra in n” (2013, p. 166). The two definitions are presented again in terms of general vector spaces in the fourth chapter, and then in the same order as in FIS. In FIS and in Chapter 4 of Bretscher, the definition of linear transformations is introduced after linear independence because transformations are about preserving the structure of the vector spaces, which we conjecture is motivated by wanting to preserve the relationships between the vectors in those spaces. Emphasizing the preservation of linearity with linear independence in the early chapters of Bretscher would not have been appropriate, because the reader would not yet know why there was anything important to preserve.

The decision to introduce linear transformations before linear independence also fits with Bretscher’s goals for intuition and visualization of the subject. Chapter 1, “Linear Equations” (p. vii), includes topics such as linear systems of equations and matrix algebra, so defining linear transformation functions gives a formal label to the operations Bretscher wants the student to become familiar with. In Section 2.2, immediately after the section that gave the definition of a linear transformation, Bretscher shows linear transformations visually with scaling, projections, reflections, rotations, and sheers. In doing so, Bretscher prioritizes giving students a concrete, graphic intuition of these transformations before stepping into the more abstract concept of linear independence, where visualization beyond three dimensions is more difficult.

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