This class session provides an example of a “wrap-up” discussion. During a wrap-up discussion, the instructor guides students in thinking together about challenging aspects of a topic that they have just explored through a small group activity.

The example presented here is a “compare and contrast” wrap-up discussion in which the main idea emerges during the discussion rather than during the task that the students have been doing in their small groups. During such discussions, the take home message usually comes in examining the similarities and differences in what the different groups did and found. Thus the function of the wrap-up discussion is to develop new understandings. This contrasts with the typical laboratory session in which the main concept to be learned is illustrated in the task. During such a lab, the function of the wrap-up discussion, if one occurs at all, is to verify that the students obtained the expected result. The wrap-up discussion illustrated here occurs during the second day in a week of instruction known as the ‘Preface.’ The purpose of the Preface is to engage junior-level physics students in reviewing, and in some cases learning for the first time, the linear algebra they will need to use while studying quantum mechanics during the winter term. Day 2 is a double period during which there is time for a small group activity and wrap-up discussion. During the small group activity, Corinne and her teaching assistant moved from group to group and listened in to what the students were saying. As they perceived the need or were questioned, they engaged the small groups in conversation about what the students were doing and thinking. Through such conversations, Corinne could shape the small groups’ progress while becoming aware of the particular difficulties and successes being experienced by these particular students. She then drew on that knowledge in facilitating the wrap-up discussion. During the small group activity and wrap-up discussion, the students were gaining experience in articulating their own ideas while beginning to use the specialized language that they would need to know in learning quantum mechanics

Corinne has designed a set of problems to develop the idea that an eigenvector is an arrow in space whose direction is not changed when the vector is transformed by a matrix. In this activity, the small groups all have the same vectors but different matrices so they are all doing essentially the same calculation, to multiply their matrix times this standard set of vectors to find out what their matrix does, and also to figure out what the determinant is. Thus when each group reports, everybody in the class understands in principal the calculations they did, so the wrap-up discussion can focus on the similarities and differences in the results for the different matrices rather than on the mechanics of the calculations. See the Instructor's Guide for the Linear Transformations activity.

Corinne has carefully chosen the matrices to represent the simplest transformations: rotations in either direction by pi over two, reflections around the x axis or the line y = x, reflections through the origin, multiples of the identity, particularly negative one times the identity, scaling, and then one matrix which is some weird combination of real numbers, not simple. (See Appendix) All are real number matrices. The vectors that she has chosen include the eigenvectors of those matrices, including the eigenvectors of the weird matrix. She has chosen the weird one so that its own entries are real and integer and its eigenvectors also are real and integers.

One of the reasons that Corinne likes this activity is that what the students are expected to do is very straight forward. Every group does the same calculation but with different matrices; then they give a very brief mini-report. She has found over the years that there are lots of opportunities during these mini-reports to have professional-development type discussions, such as pointing out that one should write the determinant of some matrix equals one, not just det equals one, making sure that the students know to report what their matrix is, encouraging them to speak loudly enough, etc. The reports are brief enough that the students are not very intimidated by them. She has found that when she makes some kind of a comment to one group about how to report, presenters for subsequent groups generally do that so there is a chance for them to practice immediately after she has made a suggestion.

Although wrap-up discussions based on the same small group activities have common purposes from year-to-year, the specifics vary considerably depending upon what the students say and do. The following interpretative narrative presents what happened during the January 10, 2006 Day 2 wrap-up discussion of the set of vector transformation problems shown in the Appendix. The purpose of the narrative is to provide an example of a wrap-up discussion with commentary about ways in which the instructor was choosing to guide this process.

The interpretative narrative below is based upon a transcript of the wrap-up discussion during Day 2 on January 10, 2006 and upon reflections recorded by the instructor, Corinne Manogue, and her colleagues while viewing a video of the discussion. In drafting the narrative, Emily van Zee used many of Corinne’s written reflections almost verbatim to try to represent her thinking as closely as possible. In selecting and organizing the reflections, Emily drew upon her own teaching experiences in laboratory-centered physics courses for prospective teachers and her research in the tradition of ethnography of communication (Hymes, 1972; Philipsen & Coutu, 2004; van Zee & Minstrell, 1997a,b). Ethnographers of communication examine cultural practices by interpreting what is said, where, when, by whom, for what purpose, in what way, and in what context. This interpretative narrative presents an example of an instructor welcoming students into the culture of “thinking like a physicist.”

Deciding when to bring small groups together for a whole group discussion can be problematic. Waiting too long means that some groups become restless but starting too soon means that other groups have not yet perceived the pertinent issues and thus will not be ready to absorb what the instructor wants them to learn.

Sometimes Corinne brings the groups back together in the middle of an activity if she finds she is having the same conversation with multiple groups or if several groups are stuck on the same thing so it is more efficient to address them all together. Sometimes she is surprised that the class is interpreting the material differently from the way previous classes have, that there is something different about the way a particular class is interacting with the material, often due to the way that the material has been covered in the previous days leading up to the activity. Other times it is a question that comes up year after year after year.

If Corinne is trying to decide when to start a wrap-up, to signal that the activity is over, typically she either runs out of time in class and does it the next day or she waits until two thirds of the class has an answer and all of the groups understand that there is something tricky to be understood, even if they do not yet see what the trick is. With some tasks she expects that the students will be alerted to a tricky situation when they attempt to do the task she has set them but in other activities, such as this one, the take-home message evolves from the wrap-up discussion, not from the task that the students were doing in their groups. With “compare and contrast” activities like this one, the take-home message usually comes in comparing what the different groups have been doing.

For this activity, the task the students had been set was very straightforward so Corinne waited until essentially all of the groups had completed the task. She gave a second example to work on for groups who finished early. She had instructed the groups to display their work on the whiteboard walls near their tables so that everyone could see their graphs of the transformed vectors and a record of their calculations.

Corinne had some difficulties in getting the students to pay attention when she wanted to start the discussion. In her reflections, Corinne noted that sometimes when she tries to call a class back together, they just keep talking to each other. This is always a signal that there is something going on that they need resolved so she might as well figure out what it is and get it resolved. Often they are still engaged in trying to solve some aspect of the problem that she has set them and they feel like they are making enough progress that they are not interested in what she has to say about it. Sometimes Corinne just gives them more time and sometimes she keeps talking to the couple of groups who are interested and allows those who are still engaged in talking to each other to do that.

In this case, Corinne had stated that the ‘B’ person in each small group would be the one to make the group’s presentation. Amid the chatter that erupted, one of the students told her that they did not know who the “B” person in their group was. So she reviewed the locations in the room by which she had assigned A people, B people, and C people before they had dispersed into the small groups at the beginning of class. In creating groups, sometimes she has people number off and in this case, those numbering to 10 first were to be the A people in ten groups, those numbering to 10 next were to be the B members of their groups and the next 10 the C’s.

Before identifying the first group to present, Corinne reviewed her expectations that the presenter would state what the group’s matrix was, how it had transformed the set of given vectors, and what the determinant of the matrix was. The presenter also was to point out the vectors that were not changed. Then Corinne joked, “And then I get to harass you with questions” to note in a friendly way that there would be discussion of each group’s results as well as the presentation.

In reflecting upon her initiation of this discussion, Corinne indicated a subtle aspect of her instructions to the students. The real purpose of this activity was to get students to recognize conceptually that an eigenvector is a vector whose direction is not changed when multiplied by a matrix but she wanted the students to come to their own realization that that concept is important. So in this set of directions, she indicated that the students would be discussing the determinant and what it means, which could be an interesting and possibly in the long term fruitful distraction, and she stated almost as an aside that she wanted them to say if there were any vectors that were not changed by the transformation.

Corinne concluded her initiation of the discussion by pointing to the board where the given set of vectors had been represented graphically. She asked, “Does everybody agree that these are what the untransformed vectors looked like? We’re going to be comparing to this, in every case.”

The presenter for Group 1 stated, “Our matrix [0,1,-1,0] rotated at ninety degrees, pi over two, clockwise, all the vectors, so none of them stayed the same.” He pointed to the board while saying, “And the determinant was one.”

Then the presenter pointed to the diagonal on the matrix, 0,0, and then to the off diagonal, 1, -1, while saying, “Because it’s zero minus a negative one.”

Corinne responded, “Okay. And all of them rotated. So nothing was unchanged.” In reflecting upon this result, Corinne noted that as far as the students were concerned, there was no a priori reason why this matrix should do the same thing to every vector. She was subtly trying to make that point here when she said “ALL of them rotated.” Some years she makes more of a fuss about that than other years. This is something that should surprise the students but does not. There is this funny operation called matrix multiplication, one can see from the form of it that it takes vectors to other vectors, but why should it do something nice to the set of all vectors? Matrices represent linear transformations and that is a very special property. There might be other kinds of rules that could be written down that might multiply one vector by two and rotate another vector and send another vector to zero but linear transformations treat the set of all vectors in some coherent way. If the students were really curious, they should be surprised by that.

After confirming the result that nothing was unchanged, Corinne continued, “Okay. A little bit of just notational harassment here. You don’t ever say “$\det$ equals one.” An operator acts on something. The determinant of $A$, call it $A$ one, so the determinant of $A$ one is equal to one.” The student added $A_1$ = in front of the matrix and inserted $A_1$ in between $\det$ and $= 1$ to change $\det = 1$ to $\det A_1 = 1$. In reflecting upon this exchange, Corinne commented that every year she has to make this correction, that a determinant is an operator and it has to act on something. She noted that saying “$\det$ equals 1” is the equivalent of saying 'squared equals one' instead of saying “x squared equals one” or is like saying “color equals yellow,” which leaves one wondering, the color of what is yellow?

After correcting the use of language about determinants, Corinne explicitly articulated the difference between this course and many others:

Corinne: What I want us to be thinking about is, I want us to do some physics the way it’s really done. A lot of education is sort of about giving you a really polished presentation, telling you what it is you should expect…but that is not how people do physics.

She then welcomed the students into the culture of physicists by stating how physics is done and how they would be doing physics right now during this discussion:

Corinne: People do a lot of different examples and then they see if they can find something in common from all those examples so what I want you to think about is “Can we figure out anything about what the determinant is telling us about the matrix?…So does the determinant tell us anything? So here’s an example where the determinant is one. Okay? Let’s just remember that for now.”

In reflecting upon this statement, Corinne commented that she wants the students to experience what it is like to deduce a result from looking at many examples–the experience that many professional theoreticians have. This conversation is all about helping set them up for that expectation. She is trying to get the students to experience what it is like to be a theoretician. In most classrooms, students experience theory as they are being told what the theory is but that does not help them understand what it means to come up with new theories. In this activity, her experience has shown that the students often have been taught how to find the determinant but they typically do not have any idea of what the determinant means geometrically. So as they go through the different examples in this activity, she is giving them an opportunity to 'theorize” from the different examples what the geometric meaning might be.

The presenter for Group 1 offered a potential interpretation of a determinant of a matrix equal to one, that in this case, the vectors were not stretched. Corinne concurred that the vectors had not been stretched, only rotated by $\pi/2$.

The presenter for Group 2 began, “Our matrix was $A$ two,” and pointed to it on the board $$A_2 = \pmatrix{0&-1\\1&0\\}$$Next she compared their matrix’s determinant and effect on the given set of vectors to Group 1’s. She stated, “The determinant was one like theirs,” and, as she moved her hand in a counter clockwise direction, continued, “It was a rotation just like theirs but rotated ninety degrees in the opposite direction.” She also offered an interpretation of this difference by comparing the two matrices, “And it’s interesting to note that it’s the transpose of their matrix, so maybe that’s why it’s in the opposite direction.” Corinne responded, “It is the transpose of their matrix so maybe that does have something to do with opposite directions. Is there any other relationship between that matrix and this matrix?”

In reflecting upon this presentation, Corinne noted that the student had volunteered the observation that her matrix was the transpose of the previous group's matrix. However, it turns out that the relevant relationship is that the matrices are inverses of each other. The transpose is flipped on the main diagonal so $$\pmatrix{0&-1\\1&0}$$ becomes $$\pmatrix{0&1\\-1&0}$$ . The inverse is the matrix such that $A$ $A^{-1}$ is the identity (Corinne, is this how you would like to describe inverses? transcript sounded like “A inverses the identity”). For rotation matrices, the transpose is the inverse but that is not true for all matrices. Corinne missed an opportunity to talk to the students about that but chose instead to focus their attention on inverses because the concept of inverse is more fundamental. She wanted them to be thinking about how one can undo a transformation.

Corinne intended to have a conversation about how these matrices were related with each other at some stage but because the student volunteered a relationship, she could 'run with it' rather than trying to start the conversation herself. She finds that student conversations go more smoothly more often when they follow up a student-initiated statement. She could have initiated this conversation with a prompt such as “How is this matrix related to the previous one?” but students might not have known what she was asking for by 'How are these related?” In this case, the student’s statement provided an example and so Corinne’s prompt was to ask for more relationships like that. During such responsive teaching, the students are more likely to understand what it is an instructor means.

Several students offered ways in which the Group 1 and Group 2 matrices were related: “Same determinant” “Both rotations” “Trace is the same.” The presenter for Group 2 responded, “Both Hermitian,” but then changed her mind, “Oh no, they’re not, because they’re real.” Corinne elaborated, “You can take the Hermitian adjoint of a real matrix but it’s just the same as the transpose <right> so yes, they’re Hermitian adjoints.”

In reflecting about this interchange, Corinne noted that the student meant to say that the two matrices were Hermitian adjoints of each other, that you get one by taking the Hermitian adjoint of the other. However, that is not what she said. Then the student said, “they're not because they're real,” so she had two problems, she was thinking that one can not take the Hermitan adjoint of a real matrix and she did not know what the word Hermitian meant. Corinne addressed the second problem but ignored the first one. She used the correct language herself but did not otherwise correct the student’s incorrect use of the word Hermitian because there were too many things going on to stop for that. Corinne noted that students in the classroom say so many wrong things that if one tries to fix all of them when they come up, the presentation becomes hopelessly muddled so one has to make choices. She probably did register that at some stage she would need to address the question of what does the word Hermitian mean. Normally she would do that during Day 5, not Day 2.

Another student offered, “If you multiply them, you get the identity” and Corinne, confirmed, “If you multiply them, you get the identity.” The student continued, “So they’re inverses,” and Corinne again confirmed, “So they’re inverses.”

Corinne reflected that the students were throwing out many different ways in which these matrices were related and that she was confirming each of the relations, supplying a little bit of language help when necessary but not otherwise being evaluative about the students' suggestions. Such confirmations keep the conversation going. Because the students were throwing suggestions out, roughly in the same order that they had been presented during Day 1, it is fairly clear that most of them did not have a good understanding of the geometrical interpretation of any of these operations. With this mention of inverses, Corinne could attempt to pull that out of the student conversation because the inverse of a rotation would be easy to understand geometrically.

Corinne continued the discussion by asking what the fact that these two matrices were inverses had to do with their being rotations. A student responded, “You rotate it one way and you rotate it back the other way, you get the same thing.” Then Corinne concluded consideration of Group 2’s matrix with a mini-lecture. She made a connection to the students’ likely prior knowledge from a linear algebra course, reiterated the use of matrices in physics to transform vectors, summarized the case of rotations, emphasized that one matrix does the same thing to all of the vectors, and reviewed that multiplication by an inverse undoes a transformation.

The presenter for Group 3 started by pointing to their matrix, $$A_3 = \pmatrix{0&1\\1&0}$$ and describing it, “Our’s was this thing that’s sort of backwards from the identity. It’s got the other diagonal as ones. And what it did was it reflected all the vectors along the line $y$ equals $x$.” She moved her hands to illustrate. “ So it’s just a reflection along that line. The blue one was already parallel, was along $y = x$ so it remained the same. And the determinant is minus one.”

After asking the presenter to write the determinant of their matrix on the board, Corinne addressed the class, “Did you understand what she meant about reflection?” She pointed first to the graph of the untransformed vectors on the board and then to Group 3’s graph of the transformed vectors and said, “Here are the untransformed vectors. There are the transformed vectors. So what’s happening?” While moving her hand to model the change, she reviewed the findings, “This purply one is coming down like this. The blue one is staying the same. Okay? Everything is taking the line $y = x$ and just flipping as if the line $y = x$ is a mirror. Is that clear?”

In reflecting on her choice to be so explicit here, Corinne commented that she has been surprised over the years at how many groups have trouble seeing what reflection is, that it is clear they have no experience with that concept. So she made a point of talking about it to give the whole class a chance to think about it for a minute or two because the presenter for Group 3 just said it was a reflection as if everyone knew what that was.

Corinne continued this mini-lecture by reviewing the description of the matrix and its determinant, minus one, and then summarizing their findings so far, “So we’ve got two matrices whose determinants are plus one and one whose determinant is minus one.” Then she asked the student, “ What happened to the lengths of your vectors?’ The student responded, “The same” and Corinne elaborated for emphasis, “The lengths stayed the same.”

The presenter for Group 4 began in a confusing way, “So with our vector we just ran through this with a generic vector and we found that it just reverses the y component. And ah.” Corinne interrupted and coached him by suggesting, “So wait. First tell us what your matrix is.” The student complied with, “Oh. $A$ four, one, zero, zero, one, or negative one, excuse me” $$A_4 = \pmatrix{1&0\\0&-1}$$ . but he then turned to a colleague, questioned the way the vector pointing upward along the y axis had been drawn, erased it, and redrew it correctly pointing downward

Then he described the transformation, “So it just flips all the vectors over the x axis. The one that was lying on the x axis obviously remains unchanged because it has no y component to reverse. The determinant of the thing was one. And as characteristic of all the determinants we have seen to this point, where their magnitude is one, the lengths of the vectors remained the same.” One of his group members corrected him, “Determinant is negative one,” and the presenter agreed, “Negative one, yeah,” and then reiterated “but the magnitude of the determinant is one.”

In reflecting upon this student/student interaction, Corinne noted that although she did not pick up on it at the time, the symbol for the determinant of a matrix is to put the name of the matrix within absolute value signs, $|A_4|$, so this student probably thought one had to take the absolute value of the determinant. She had never consciously noticed that before but this is a misunderstanding that may occur. What she did notice and respond to in his presentation was the use of the word “characteristic” and she prompted him to say the word “hypothesis” by repeating the findings, “The determinant is negative one. The lengths are not changed. All right. And I’m hearing the beginnings of a <hypothesis> a hypothesis, which says that,” and then she prompted him to repeat his hypothesis, “say it again.”

The presenter for Group 4 then restated, “If the magnitude of the determinant is one, it will leave the lengths of the vectors unchanged.” Corinne then prompted the entire class to consider this, “Okay. There’s a hypothesis; so let’s start watching whether or not that happens.” Then she prompted the next step in the kind of thinking she wanted the students to be doing by reviewing that the determinants of some matrices were plus one and some minus one, “Do we see a pattern about the sign of the determinant coming out?”

The presenter for Group 2 offered, “reflection versus rotation.” Corinne asked her to reframe this as a hypothesis, “State it as a hypothesis.” The student then elaborated, “If the determinant is positive one, then there’s a rotation. If the determinant is negative one, there’s a reflection.”

In reflecting upon this development, Corinne noted that in all of the years that she has done this activity, this was the year that the discussion was most framed in terms of making and testing hypotheses. She found it interesting to hear here how that conversation got started. After that year, she has tried to frame the discussion in terms of hypotheses and it has not gone as well. She commented that one can not reproduce a really cool class discussion another year. Each class develops its own best discussion days.

The presenter for Group 5 began by pointing to their matrix, $$A_5 = \pmatrix{-1&0\\0&-1}$$ “So this is our matrix [A5 = (-1, 0, 0 –1)] and the determinant of it was, (writes on board) equals one. Basically it just rotated all the vectors a hundred and eighty degrees leaving their lengths unchanged.” This prompted one of the students to ask a question which began a long discussion about rotations and reflections:

Student: Which way did it rotate?
Corinne: Yes, which way did it rotate?
Presenter: Hundred and eighty degrees
Student ?: Doesn’t matter <laughter>
Corinne: Doesn’t matter! Okay. All right. What kinds of reflections are there in the world?
Student ?: Odd and even
Corinne: Odd and even reflections. Yeah
Student ?: Line and point
Corinne: Line and point. Okay. So we’ve had some examples of reflections in lines. What do you mean by reflection in a point?
Student J: You go through a single point (moves arms). I don’t know how to describe it.You put all the points through that point (moves arm through point made by finger of other hand ) however how far away from the point they are.
Corinne: That was a description
Student J: Okay. It was a muddled one though.
Corinne: Say it again. Say it again with confidence.
Student J: You reflect every point on the line through a point however far away from that point it is.

In reflecting upon this interaction, Corinne commented that geometric things are really hard to describe in words and that this student had done a reasonable job. She had had a running conversation in this class about saying things with confidence, so that was the reason for that prompt. She remarked that this is what distinguishes pompous arrogant physicists from the rest of the population. When the students really do know what they are talking about, she wants them to be able to state their opinions with confidence. She also would like them to recognize when this is being done to them, when someone is stating something with confidence that may not be true. She noted that proof by intimidation is a common scientific phenomenon.

Corinne continued the conversation by asking the presenter whether this was a reflection or a rotation. He responded, “You can call it a reflection by a point at the origin.” Corinne confirmed, “Okay. So if you take the origin as a point of reflection, it’s a reflection. <yeah> It’s also a rotation.” She wanted the students to see that clockwise rotation by pi and a counter clockwise rotation by pi and reflection through a point at the origin are the same transformation. During Day 3, they would be talking about three dimensional transformations and in three dimensions one can distinguish between rotations which leave the $z$ axis fixed and reflections in (or through) the origin, which take the $z$ axis to minus itself.

A student then raised an insightful question, “I wonder what it would do if the, like if it didn’t go through the origin, like if it was offset somehow?” This elicited a mini-lecture about the nature of transformations that might not have been considered otherwise, that linear transformations are only thought about as acting on vectors that start at the origin.

Corinne then returned to the focus on hypotheses:

Corinne: So this is a reflection of a point, or it’s a rotation. Its determinant is plus one. Does that fit our hypotheses? What were our hypotheses?
Student ?: The lengths of the vectors should be the same (?) that one
Corinne: Are the lengths of your vectors the same?
Student ?: (yeah every one)
Corinne: and?
Presenter 2: It doesn’t quite fit our hypotheses, because I said if a determinant was negative one, it would be a reflection.
But the determinant was a positive one and it could be considered a reflection through a point.
Corinne: Okay
Student ?: How do we know if it was a reflection or a rotation?
Presenter 2: It’s both
Student? : It could be either (so that doesn't prove) your hypothesis is wrong
Presenter 2: Maybe this is talking about reflection across a line instead of reflection across a point
Corinne: Okay so we’re going to change our hypothesis a little bit. All right. Physicists do this all the time. You know, like just, if there are exceptional cases, then redefine what you meant. It’s a good strategy.

In reflecting upon this interchange, Corinne noted that the students had started talking to one another. They were doing a nice job of taking responsibility for refining their own ideas. The presenter for Group 2 had said basically that maybe there is a difference between reflections across a line and reflections across a point. Corinne had attempted to affirm that if you can see a difference between two cases then maybe there will be different physical effects for those two cases; there might be two separate rules and that would be ok.

The presenter for Group 6 began, “Our matrix mapped all the lines, all the vectors on to $y$ equals $x$ because the upper row of the matrix is the same as the lower row of the matrix, A6 = [1, 2, 1, 2] $$A_6 = \pmatrix{1&2\\1&2}$$ (See Figure 7). Corinne interrupted him, “Okay. Is that why?”

The student paused and then replied, “Yes,” and added, “That was a confident yes” with a laugh. Corinne replied, “Yes it was. That was a correct and confident yes. And you know what, sometimes you can intimidate people out of asking how do you know (laughter) with a confident yes, but in this case you can’t intimidate me out of asking, so why it is that that the top row being the same as the bottom row matters?”

In reflecting upon this interaction, Corinne noted that she had suspected that the student was remembering something from a linear algebra class but it also might have been something that he had noticed as he made the calculation. She wanted him to expand on why he felt that it was true that their matrix mapped all the vectors onto the line $y = x$ because the upper row of the matrix was the same as the lower row. So she challenged the reason that he gave but rather than expanding on his reason, he chose simply to confirm it. And then he caught himself and engaged in some metacognition. He chose not to engage in giving reasons but at least recognized his behavior for what it was. He commented beautifully that he gave a 'confident yes' harking back to the earlier discussion where Corinne was encouraging students to give their answers with confidence. She had acknowledged the act of metacognition but also clearly indicated that it was not going to get him out of having to explain his reasoning.

Corinne further noted that this whole interaction was about establishing some classroom norms and also about establishing a professional norm. He had not in fact given a confident yes but a querying yes and the students laughed after he said it in what she would interpret as a spirit of camaraderie, we've all been in this boat before, but she gave him credit for striving for a confident yes. She had given the students a sneaky professional tip which they acknowledged with laughter. This was about establishing a classroom norm because it was part of this ongoing conversation about giving confident answers. She wanted the students to give confident answers. However, she was also establishing the classroom norm that doing that was not going to get them out of the expectation of having to explain what they meant. In terms of professional norms, she stated this in the context of on some level helping them possibly pull the wool over someone else's eyes but there was a subtext of somebody may try this on them, giving a confident yes, to keep them from asking why and she wanted them to know that they did not have to be hoodwinked in that way but could respond in the way she did instead.

Corinne also commented upon the laughter, that there was a level of diffusing a threat here. Students are threatened by having to give reasons and discussing that threat on a meta level in this way and making them feel that they have professional resources for handling it at their finger tips to some extent diffuses the threat and that often results in laughter.

The presenter for Group 6 then elaborated with many hand motions, “So the vector on the right side has a top component and a bottom component and in the multiplication the only difference between finding the top component and the bottom component, is that you use the top row instead of the bottom row, right? So if both rows are the same, then both answers are the same.” Corinne responded by reiterating this explanation and then complementing his version, “Does that make sense? I think your explanation was better.”

In reflecting upon this response, Corinne commented upon the use of hand motions both by the student and herself in describing what was happening here. She noted that this student did not like to give reasons, probably because he felt that reasons have to be algebraic, but when he had a chance to use his hands to describe what was happening in the matrix multiplication, he gave a beautiful answer. This is one of the prime effects that she strives for in these first couple of paradigms. She remarked upon how much encouragement it took to get him to attempt this - all of those conversations about answering with confidence, to the class as a whole; it took lots of giving of explanations with hands herself; it took diffusing his terror with humor; and it took insisting that he answer, a couple of times. She also noted that she thought it really important that she told him at the end that his answer was better than her own.

The presenter for Group 6 started to continue, “So the determinant is zero, which is characteristic of the system is not, (?) linearly” LISTEN TO THIS [00:32:36.10] but shaking his hand and moving his arms to indicate a line, muttered, “Linear algebra was a long time ago” to which Corinne responded with humor, “Okay, for some of the students in the class linear algebra hasn’t even happened yet.” The presenter for Group 6 went on to complete his thought, “So the system is linearly - dependent, and so it has a determinant of zero <yes> and the vector that isn’t changed by our matrix is the zero vector.”

In reflecting upon this student’s thinking process, Corinne noted that his uncertainty and pause before saying the word ‘dependent’ is an indication that he, like many students and she herself, has trouble remembering which case is linearly independent and which case is linearly dependent; the words seem backwards from what logic would imply from the way they sound. However, even though the student said linear algebra was a long time ago he was able to produce the right word in the right context.

Rather than digress to discuss the concepts of linear dependence or independence, Corinne chose to focus on the status of the transformed vectors and asked, “Is that the only one that’s not changed? <yeah> Okay. Are there any ones whose direction is not changed?” The presenter for Group 6 responded, “Any vector on the y = x line won't have its direction changed unless it was in the opposite direction<okay> so I’d say some vectors along the line y=x don’t have their direction changed.”

At this point, Corinne shifted from coaching the presenter to engaging the whole group in considering an intriguing issue, the meaning of “change direction” in the context of linear transformations. She asked, “If I take a vector like this (pointing to one on the board) and I multiply it by minus one, have I changed its direction? <no> No. How many people say no? How many people say yes? How many people say it depends on what you mean by ‘change direction’?” <laughter>

Then she acknowledged that the meaning of ‘change direction’ is a question of semantics, and that in the context of multiplying a vector by a scalar, the result is considered to be in the ‘same direction’ even if the scalar is a negative number. She illustrated this visually by pointing to a vector on the board and noting that in multiplying it by minus one “we will choose to call this in the same direction okay, the negative direction being the same direction.” She also made an analogy with the interstate, I-5, with which the students were very familiar, as it runs directly north/south near campus and can be considered to be along the same longitudinal line and in that sense being in the same direction whether north or south. She commented, “so it depends on the context when you want to call these the same direction or when you want to call them different directions. In the context of linear algebra it’s easiest if you call these the same direction.” She returned to the context of Problem 6 by stating. “so your comment that anything on the line y = x doesn’t change directions, none of them change direction, if you’ll let the opposite direction be the same direction.” She also made a joke about the apparent ridiculousness of such a statement.

In reflecting upon this incident, Corinne noted that this is a real sticking point for many students. In casual conversation, turning around and going the opposite way is going the other direction but in all math and physics contexts, multiplication by a negative scalar or even a complex scalar does not change what is meant by the direction of a vector. This is a conversation that she has to have with every class at least once. It had never come up in this example before although it always comes up in Day 2 or Day 4 at some point or she makes it come up. But in this case, the student set up the conversation particularly well by claiming that vectors that reverse their direction were not then going in the same direction. This is an example of fitting content that is part of the intended curriculum into the appropriate moment precipitated by a student.

In reflecting upon the next interaction, Corinne noted that after each group presented or a couple of groups presented, she had returned to the theme of hypotheses about the meaning of the determinant. In this case, she chose first to ask the students about what happened to the lengths of the vectors because the presenter for Group 6 had already referred to that, “All right. So what about our hypotheses? The determinant of the matrix is <zero> zero! And what happened to the lengths of the vectors?” A student responded, “Some of them changed and some of them didn’t change.” Corinne confirmed that and commented, “And the determinant was zero so it’s neither a reflection nor a rotation which is certainly true.” Then a student offered an insight:

Student ? ? There’s no inverse
Corinne: Say more.
Student ? There’s no way to get back out of that once you, back to where you started, because you’ve just erased all information pretty much that the previous vector array had.
Student?: More than one vector can be transformed to the same thing. <yeah>
Student ?: so you don’t know which one

Earlier in the class, Corinne had given the students a geometric definition of inverse, which was that it was the transformation that undid what you previously did. In reflecting upon this student/student interaction, Corinne noted that the students were refining their language and conceptual understanding around this case where one cannot undo what has been done. Noticing that determinant zero matrices have no inverse was an intended outcome of this particular problem. Some of the students, including the one who had just presented, probably knew that as an algebraic statement from a previous linear algebra class. Corinne interpreted this student/student conversation as the students attempt to reconcile two different ideas they had, the algebraic and geometric descriptions of the same concept, and they were working through how those descriptions were actually saying the same thing.

At this point Corinne chose to introduce a new idea, that this transformation is sometimes called a projection, although sometimes the word projection is reserved for something more technical. She provided an analogy, “It’s like shadows. You know. Once you’ve collapsed everything down onto a line, once you’ve collapsed everything down into your shadow, lots of things could have the same shadow.”

The graduate teaching assistant questioned the use of projection here, to which Corinne responded with a question, “So what happened to one, negative one? Several students responded, “It became negative one, negative one,” “It got rotated” One elaborated, “It got rotated as opposed to” “Projection would be zero” cause it was on the line y equals negative x and then, so its projection onto the y equals x line is zero <yes> so it got rotated.”

In reflecting upon this conversation, Corinne noted that the students and the TA rejected the transformation as being a projection because they were thinking of projections geometrically as shadows and there is no possible position for the light source that would make this transformation give the appropriate shadows. She found it interesting that as soon as they rejected the possibility of shadows, that they jumped to the idea that this transformation represented a rotation. Many students are willing to look at how a single vector is transformed and see that as a transformation rather than to look at how the whole set of vectors gets transformed. In this particular example, the rotation interpretation was particularly compelling because the particular vector that they were talking about did not in fact change length and looked to them as if it had been rotated by pi over two. Corinne knew so clearly so it was not a rotation that it did not look like a rotation by pi over 2 to her. She would not apply those words to it. And she found it intriguing that the TA and the students all were compelled to use that language. This was a case of her expert knowledge interfering with an observation. She could not see a natural description because she knew from prior experience that that was not what was actually happening.

What Corinne saw was that everything got smooshed or stretched or collapsed into the line y equals x but she used language that was more like the language of projection than the language of rotation. If one looked at the set of all vectors, they were not all being rotated by the same amount. Students were seeing two snapshots, one before a transformation and one after a transformation and she was asking them the question, what happened? If she saw a star of vectors going to a line, she would say they were smooshed or screnched or she would make up some word to describe some kind of collapsing but the students were willing to look at a single vector and said that it was rotated. That ignored what was happening to all the rest of the vectors.

Corinne knew clearly that the template that was being acted out, was to look at what the transformation did to ALL the vectors all at the same time and furthermore, she knew that since it was a linear transformation there were only four choices: rotation, reflection, projection, and combinations of the above. The students did not know that that was what the template was and so they were busy describing in a natural geometric sense what happened to a single vector. They had correctly described something that was not relevant to the question at hand. Corinne noted that this happens all the time at this middle division level where students are getting their first exposure to a bizillian new templates. Most homework problems involve the application of a new template but students do not know what they are being asked to do. Faculty who know clearly which template they intended to evoke often are startled by how many wrong directions the students can go.

Corinne moved to the board where Group 6 had drawn the transformed vectors and moved her arm to demonstrate transformation of the vector. Then she defined a real projection, in words and in symbols [$AA=A$] but noted, “Sometimes the word projection is loosely used for the things that take a two dimensional space, all the two dimensional vectors and smoosh them down into a one dimensional space; it is a much looser use of this word.” A student then asked an interesting question, “Is there anything that they can be other than identity in that case?” Corinne responded, “yes, I’ll leave that as an exercise. Fine me one that’s not the identity. Okay?”

In reflecting upon this interchange, Corinne noted that this student recognized that the identity satisfied $A$ squared equals $A$ and that was not a projection in the sense she was trying to convey. He could not imagine any other possibilities because this is an exceptional case that does not behave the ways the other cases do. She made the choice to go on rather than talk about this any further. She probably felt that they were running out of time so she put him off by saying it could be an exercise for extra credit.

The presenter for Group 7 read off the components of their matrix, “one, two, nine, four (see Figure 8) and the determinant is fourteen.<negative fourteen> negative fourteen. It kind of looks like it reflected everything around the purple line and then expanded or lengthened the $y$ component a lot more than the $x$ component.” Corinne asked for a repeat, “What did you say about the purple line?” and the student replied, “It’s kind of like it reflected everything around the purple line.”

In a series of exchanges, Corinne and the student worked through what had happened with this matrix acting on the vectors, that they had been “squished together” and the lengths all got longer. Corinne returned to the question of their hypotheses, “So what are we going to do with our hypotheses now?” To which the student responded, “It doesn't violate them.” Corinne asked, “Do we want to make any new hypotheses?” The student offered one, “The square root of the determinant is the magnitude of the scalar for the old vectors,” Corinne responded, “Okay so these lengths were all changed. So that’s something we could check” but noted that that was not quite correct in this case. She also summarized a rule, “matrices with big numbers, are going to stretch things a lot; matrices with little numbers in them are going to scrench things down,” and acknowledged, “So something about the determinant , so there probably is some hypothesis there but we don't quite have it. Okay?” She also queried the student about whether there were any vectors that were not changed by the transformation and agreed, “Purple was, okay so there was one whose direction was unchanged.”

In reflecting upon this interaction, Corinne noted that framing the discussion in terms of hypothesis making was still continuing to work well. As it turns out, there is not a good answer to exactly what the value of the determinant means so she did not push on this aspect of the student’s responses very hard.

The presenter for Group 8 stated, “Ours was one, one, negative one, one. Moving his arm to show clockwise rotation, he added, “All rotated 45 degrees this way and multiplied by square root of two for their lengths.” Pointing to their calculations on the board, he stated, “Determinant was two.” Corinne summarized his report and then asked, “So now, have we got a hypothesis? Several students responded. One suggested, “Um, positive which is what we said makes it rotate, so it did rotate, um magnitude is not one and the lengths changed.” Another student offered, “Seems like the positive determinants, the order of the vectors is still the same, like just going around [moves arm in a circle] the order’s the same but with the negative determinants, the order’s flipflopped.” Corinne confirmed, “That does seem to be happening.”

The presenter for Group 9 stated, “So our transformation just doubled the length of the vectors; it didn’t rotate or reflect or anything because our matrix is just the identity matrix multiplied by two and any vector times the identity matrix is just the same vector back, just scale it by two.” Corinne invited the students to use this example to consider the validity of their hypotheses, “Okay. So there’s another one where the square root of the determinant does tell you how much to scale by. What are we going to do about that? Do you like the square root rule or not?” A student equivocated, “The square root rule seems to be valid when the, well I guess that’s not true, I was going to say it’s valid when the upper right or lower left components are zero because that’s just how some (?) multiplication of the identity matrix (?) [00:44:32.13] (?) And Corinne acknowledged, “We probably don’t have enough examples here.”

In reflecting upon these responses, Corinne noted that she likes this activity because she is asking the students to figure out the role of the determinant. There are some very simple examples with high symmetry like reflections through a line and rotations where the determinant is plus or minus one and the role of the value of the determinant is very clear. However, with an example of a more generic matrix without high symmetry, the determinant can be something like fourteen and there is not a clear rule for what the value of the determinant means. The students were doing some combination of guessing from the examples they were currently seeing and remembering things from courses but since there is not a simple rule they were having a hard time finding one. For examples with high symmetry there is a rule but it just does not apply all the time. If one tries to solve a generic pure math problem, it is really hard to make a clear theorem that is true all the time but with the math problems that are motivated by physical models, the mathematics is often cleaner, which is the story of her research life. She has made a career out of looking at simple mathematical models that have a physical motivation and almost always the mathematics that she has needed has been exactly on the border of what is doable and not doable by a mathematician.

The presenter for Group 10 stated, “The matrix was one, zero, zero, zero. The determinant is zero. And it projected everything onto the x axis. Corinne confirmed, “Okay. So here we've got another zero determinant one.” The presenter for Group 10 continued, “All the lengths became either zero or one.” Corinne confirmed again, “All the lengths became zero or one,” and asked, “Did all the vectors have an x component of one?” The student responded, “Except for one which remained zero.”

With the class coming to close, Corinne initiated a review:

Corrine: … All right. So what do zero determinants do? Student? : Erase information it seems Corinne: Zero determinants seem to erase information. They’re taking two dimensional vectors down to one dimensional vectors. What do determinant plus one’s do? Student ? Rotate Corinne: Rotate Student ?: Rotate counter clockwise Students Corinne: We had two rotations here. Students: Corinne: Determinant plus ones seem to rotate. Determinant minus ones? [00:46:02.10] Student ?: Reflect Corinne: Reflect. What kind of reflection? Student ?: Line Corinne: The line. Because there was a determinant of plus one that was a reflection in a point Student A: (?) CHECK OTHER CAMERA Student: but it did though. It’s opposite what it would be or what it was. It used to be pointing down.

In reflecting upon this series of exchanges, Corinne noted that what was interesting here was that she was trying to get the students to help her summarize by listing the possible cases one by one and asking them to give the conclusion, so for example she was saying determinant one and they were supposed to give a description of which kinds of transformations would have determinant one. For each example, there seemed to be several student/student interactions where they were trying to clarify each other's language and/or correct each other's geometric description of the types of transformations.

Corinne continued summarizing, “And the size of the determinant seems to have something to do with stretching except this matrix over here. Okay. One of the dangers of making hypotheses from matrices which are carefully chosen, simple ones that have only ones and zeros, is that you can miss out on the generic case. The generic case is a, most matrices look like this, right? (See matrix for Group 7). This is just some set of random numbers. Most matrices will look like this. We can start to make rules about the ones that are special but this generic one seems to be blowing some of our rules but not all of them.”

As the period was ending, Corinne provided an overview of the coming sessions Corinne: What we’re going to do over the next couple of days is try to see if we can prove things about various kinds of rules. So does determinant zero really project in the looser sense? Are determinants plus ones always rotations? Are determinant minus ones always reflections? When can we know that the square root of the determinant tells us something about stretching?”

In reflecting upon these comments, Corinne noted that here she was setting up the expectation that over the next few days they would actually try to prove more formally some of the results that they were beginning to suspect were true. Then she went on to change the subject to what she really wanted to be the day’s focus.

Corinne: Now I’ve been asking you as you’ve been going around whether there were any vectors that are not changed by the transformations. Have you been paying attention? Okay? What about rotations? Do they have vectors that are not changed? What about reflections? Do they have vectors that are not changed?
A student offered a technical word learned in a previous course:
Student ?: Eigenvectors
Corinne prompted the student to define this word:
Corinne: What about eigenvectors?
Student ?: They are the eigenvectors, the ones whose directions are unchanged by the matrix She then elaborated on that definition:
Corinne: Who has not seen, never heard of eigenvectors? Only a couple of you. Then we will introduce it for you. For those of you who have not heard of them, eigenvectors of a transformation are just the vectors whose directions is not changed by the transformation. That’s what an eigenvector is geometrically. And that’s the idea of an eigenvector that I want you to hold with you all the way through the winter term. An eigenvector is something that is not changed by the transformation, not changed, it might be stretched, but its direction isn’t changed as long as we count negative directions as being the same direction,that’s why we want to do this [pointing to the line on the board which she had used to talk about the opposite direction being the same direction]. All right?
After emphasizing that an eigenvector’s direction did not change, she asked What is the name for how much the eigenvector is stretched?.
Student ?: Eigenvalue
Corinne: The eigenvalue.
Then she brought the class to a close, by describing an up-coming task:
Corinne: So the job for probably Thursday’s class will be,…You just had a few vectors to check, right?If I give you a matrix, how do you find the vectors whose direction is not changed by the transformation? How do you find the eigenvectors and the eigenvalues for a generic matrix? How do you find them?

She also raised a question that those who had taken a course in linear algebra might have been pondering if they were alert: Corinne: So it looks like rotation matrices have no eigenvectors. Do you believe that? Those of you who have taken linear algebra, a two by two matrix has how many eigenvectors? <two> Two. Always?
Students:
Corinne: Sometimes repeating. Could be zero. There is some rule about how many eigenvectors. Some kinds of two by two matrices always have two eigenvectors. All right. Rotation matrices actually do have two eigenvectors. They’re both zero or <complex> they’re both complex. All right. The eigenvectors of rotation matrices are actually complex.
With mock horror, she indicated a difficulty to be considered: Oh no! Which way do they point?
Students:
Corinne: (holds up hands, shakes head) yeah, which is, if this is the $x$ axis, what is the $i$ component in this direction? [points horizontally] Okay. Can’t do it with physical arrows in space. You can do it in quantum mechanics. Which will be just one of those things you’ll have to say with confidence. Okay? Good. We’ll see you tomorrow.

In reflecting upon this last issue, Corinne noted that the geometric interpretation of a complex eigenvector is a well known problem and related to her traditional research, so it is a problem that she has thought about a lot. Her research is all about three component octonionic columns. During the example Day 4 wrap-up discussion on January 10, 2008, she asked a student to draw the column of a complex vector, typically called a spinner in quantum mechanics, In this context it would just be the vector. She thinks there is a pedagogical difference between asking the students which way does it point and asking them to draw it. One can make a column with i's, write it down algebraically, and it has a physical meaning in quantum mechanics but not one that can be drawn. It takes certain kind of huspa to ask the students to do something that is not possible but it is very effective, if one can get them to not be resentful about it. The six small groups had worked on different problems and were presenting their solutions to the whole group during the wrap-up discussion. During compare and contrast discussions, the take home message usually comes in comparing what the different groups do. In this example, the students were considering similarities and differences in the ways in which the various matrices transformed a given set of vectors.

She gave the six small groups the same set of vectors but different matrices. The task was to use the group’s matrix to transform the vectors and then to plot the transformed vectors graphically. The set of given vectors had included eigenvectors for many of the matrices so that the students could see visually any whose direction was not changed by being operated on by a matrix. None of the vectors included complex components. In addition to plotting graphs of the transformed vectors, the students were to calculate and report the determinant of their matrix.

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