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===== Example Wrap-up Discussion #2===== ===== Example Wrap-up Discussion #2=====
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  * Day 1 had been a 50-minute interactive lecture that introduced basic matrix manipulations such as matrix multiplication and determinants.   * Day 1 had been a 50-minute interactive lecture that introduced basic matrix manipulations such as matrix multiplication and determinants.
-  * Day 2 had been a double period.  First the students worked in small groups.  Each group used a given matrix to transform a set of given vectors and then plotted the transformed vectors graphically.  The set of given vectors had included eigenvectors, those whose direction would not be changed by being operated upon by one of the matrices.  During the wrap-up discussion, the students had considered similarities and differences in how the various matrices had transformed the same given vectors in various ways (SEE <html> <a href="http://www.physics.oregonstate.edu/portfolioswiki/doku.php?id=activities:main&file=lineartrans">Instructor's Guide for Linear Transformations Activity</a> </html> and [[.:lineartranslong|Narrative for Linear Transformations Activity]]).  Corinne had designed Day 2’s problems to develop the idea that an eigenvector is an arrow in space whose direction is not changed by a transformation. +  * Day 2 had been a double period.  First the students worked in small groups.  Each group used a given matrix to transform a set of given vectors and then plotted the transformed vectors graphically.  The set of given vectors had included eigenvectors, those whose direction would not be changed by being operated upon by one of the matrices.  During the wrap-up discussion, the students had considered similarities and differences in how the various matrices had transformed the same given vectors in various ways (SEE <html> <a href="http://www.physics.oregonstate.edu/portfolioswiki/doku.php?id=activities:main&file=prlineartrans">Instructor's Guide for Linear Transformations Activity</a> </html> and [[.:lineartranslong|Narrative for Linear Transformations Activity]]).  Corinne had designed Day 2’s problems to develop the idea that an eigenvector is an arrow in space whose direction is not changed by a transformation.
  * Day 3 had been a 50-minute interactive lecture that presented properties of rotation matrices in two and three dimensions.   * Day 3 had been a 50-minute interactive lecture that presented properties of rotation matrices in two and three dimensions.
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  * Day 5 would complete the Preface with a 50-minute interactive lecture that presented the abstract properties of vector spaces and linear transformations.   * Day 5 would complete the Preface with a 50-minute interactive lecture that presented the abstract properties of vector spaces and linear transformations.
-The goals of Day 4’s small group activity and wrap-up discussion were for the students to learn an algebraic procedure for finding eigenvalues and eigenvectors for a matrix and to see that the vectors that they were getting from their algebraic calculations were the same ones that they had seen to be unchanged when plotted graphically on Day 2.  (See the <html> <a href="http://www.physics.oregonstate.edu/portfolioswiki/doku.php?id=activities:main&file=eigenvectors">Instructor's Guide</a> </html> for the Eigenvectors and Eigenvalues activity.)  By comparing and contrasting the solutions presented during the wrap-discussion, the students could gain a deeper understanding of the relations between geometric and algebraic representations of eigenvectors.  During their study of quantum mechanics in the weeks ahead, Corinne wanted the students to understand that finding eigenfunctions for the Hamiltonian would mean looking for the functions that are unchanged or just scaled by a constant.+The goals of Day 4’s small group activity and wrap-up discussion were for the students to learn an algebraic procedure for finding eigenvalues and eigenvectors for a matrix and to see that the vectors that they were getting from their algebraic calculations were the same ones that they had seen to be unchanged when plotted graphically on Day 2.  (See the <html> <a href="http://www.physics.oregonstate.edu/portfolioswiki/doku.php?id=activities:main&file=preigenvectors">Instructor's Guide</a> </html> for the Eigenvectors and Eigenvalues activity.)  By comparing and contrasting the solutions presented during the wrap-discussion, the students could gain a deeper understanding of the relations between geometric and algebraic representations of eigenvectors.  During their study of quantum mechanics in the weeks ahead, Corinne wanted the students to understand that finding eigenfunctions for the Hamiltonian would mean looking for the functions that are unchanged or just scaled by a constant.
The six small groups of students had worked on large whiteboards at their tables as shown in Figure 1.  The six small groups of students had worked on large whiteboards at their tables as shown in Figure 1. 
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When Corinne called the small groups together for the wrap-up discussion, she was quite explicit in stating her intentions, that she had demonstrated an easy generic case for them and then had engaged the small groups in cases “that have something special about them…that you need to notice.”  All of the examples that the students had been working on in their small groups were ones that had either stumped her or her students in the past.  Her intent in the wrap-up discussion was to bring out the small number but frequently encountered ways of being stumped in solving for a matrix’s eigenvalues and eigenvectors.  When Corinne called the small groups together for the wrap-up discussion, she was quite explicit in stating her intentions, that she had demonstrated an easy generic case for them and then had engaged the small groups in cases “that have something special about them…that you need to notice.”  All of the examples that the students had been working on in their small groups were ones that had either stumped her or her students in the past.  Her intent in the wrap-up discussion was to bring out the small number but frequently encountered ways of being stumped in solving for a matrix’s eigenvalues and eigenvectors. 
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-In acknowledging the challenge of the work the students had been doing, Corinne also affirmed their image of themselves as future physicists, “if you're going to have problems, I wanted you to have them here in class and not two years from now when you're trying to solve one of these by yourself in a job somewhere.”  Then she issued a direct order  “So I want you to pay attention to the presentations, particularly today, pay attention to the presentations of all of the groups because all of these have some funny little nuance to them.” 
{{whitepapers:narratives:080110day2hour2classroomlayout.png?400 |Figure 2: Classroom Layout During Group Presentations.}} {{whitepapers:narratives:080110day2hour2classroomlayout.png?400 |Figure 2: Classroom Layout During Group Presentations.}}
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 +In acknowledging the challenge of the work the students had been doing, Corinne also affirmed their image of themselves as future physicists, “if you're going to have problems, I wanted you to have them here in class and not two years from now when you're trying to solve one of these by yourself in a job somewhere.”  Then she issued a direct order  “So I want you to pay attention to the presentations, particularly today, pay attention to the presentations of all of the groups because all of these have some funny little nuance to them.”
As shown in Figure 2, each group’s presenter placed the group’s white board on the ledge of the blackboard in the front of the room so that all could see their work.  The other students remained in small groups at their tables.  Corinne sat at a table near the back of the room.  By placing herself within the class, she ensured that in talking to her, the presenter also would be addressing the entire class.  From this position, she also could engage the entire class in conversation as needed. As shown in Figure 2, each group’s presenter placed the group’s white board on the ledge of the blackboard in the front of the room so that all could see their work.  The other students remained in small groups at their tables.  Corinne sat at a table near the back of the room.  By placing herself within the class, she ensured that in talking to her, the presenter also would be addressing the entire class.  From this position, she also could engage the entire class in conversation as needed.
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The first group to present had produced the solution shown in Figure 3 [00:24:27.26] for the two dimensional matrix $A_1$. The group’s presenter began by identifying the matrix they had been given, “So this is, $A$ one is our matrix: zero, minus one, one, zero.” The first group to present had produced the solution shown in Figure 3 [00:24:27.26] for the two dimensional matrix $A_1$. The group’s presenter began by identifying the matrix they had been given, “So this is, $A$ one is our matrix: zero, minus one, one, zero.”
-{{ whitepapers:narratives:080110hour2group1.png?400|Figure 3: Whiteboard Presented by Group 1.}}+{{ whitepapers:narratives:080110day2hour2group1.png?400|Figure 3: Whiteboard Presented by Group 1.}}
Corinne interrupted to make a connection to the problems they had worked on earlier in the week, “Before you go on, this is one of the one's from Tuesday's activity; what group had this transformation?”  She wanted the students to think back to what they had done in the previous activity in order to bring the resources that they had developed there to this discussion.  By making such connections explicit, she was modeling ways of thinking that she wanted the students to adopt, seeking prior knowledge that can be brought to bear on the current focus of attention. Corinne interrupted to make a connection to the problems they had worked on earlier in the week, “Before you go on, this is one of the one's from Tuesday's activity; what group had this transformation?”  She wanted the students to think back to what they had done in the previous activity in order to bring the resources that they had developed there to this discussion.  By making such connections explicit, she was modeling ways of thinking that she wanted the students to adopt, seeking prior knowledge that can be brought to bear on the current focus of attention.
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van Zee, E. H. & Minstrell, J. (1997b).  Using questioning to guide student thinking.  The Journal of the Learning Sciences, 6, 229-271. van Zee, E. H. & Minstrell, J. (1997b).  Using questioning to guide student thinking.  The Journal of the Learning Sciences, 6, 229-271.
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