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===== Example of Small Group Conversation about Geometric and Algebraic Representations of a Physical Quantity ===== ===== Example of Small Group Conversation about Geometric and Algebraic Representations of a Physical Quantity =====
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This video clip starts at 54:42 and ends at 58:17 in the video 071026Ph422Grp6.mov This video clip starts at 54:42 and ends at 58:17 in the video 071026Ph422Grp6.mov
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 +//This interpretative narrative is based upon a video of the class session and discussions with the instructor and the director of the Physics Paradigms Program, Corinne Manogue, and Len Cerny, a doctoral student.  A postdoc, Elizabeth Gire, interacts with a small group in the video. In writing the narrative, Emily van Zee drew upon her research in the tradition of ethnography of communication (Hymes, 1972; Philipsen & Coutu, 2004; van Zee & Minstrell, 1997a,b), a discipline that studies cultures through the language phenomena observed.  This interpretative narrative presents an example of students growing into participants in the culture of “thinking like a physicist.”//
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This narrative presents an example of a small group of three students working together on a large whiteboard on which they have written a complex algebraic expression.  One of them draws a geometric representation of a relevant physical quantity and the others try to understand what this drawing means and how it relates to the algebraic expression they have written and the physical quantity it represents. This narrative presents an example of a small group of three students working together on a large whiteboard on which they have written a complex algebraic expression.  One of them draws a geometric representation of a relevant physical quantity and the others try to understand what this drawing means and how it relates to the algebraic expression they have written and the physical quantity it represents.
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The students are thinking about how to write a formula for the current density, $J$, of a ring of charge $Q$ and radius $R$ that is spinning with period $T$.  The students are thinking about how to write a formula for the current density, $J$, of a ring of charge $Q$ and radius $R$ that is spinning with period $T$. 
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The algebraic expression they have written for the current density $J$ is: The algebraic expression they have written for the current density $J$ is:
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Seth points again to the same term in the algebraic expression and comments, “That looks,...this to me looks like a $dq$, right?" Seth points again to the same term in the algebraic expression and comments, “That looks,...this to me looks like a $dq$, right?"
-Jack seems to agree, “Oh, OK, yeah,” and then says “$dQ$ over” while writing $dQ$ and drawing a line under the $dQ$ to indicate division.  He then points to his diagram while saying, “then our partial length” and finishes with “is going to be $r d(\phi)$” while writing $rd\phi$ under the $dQ$ to form the expression: $\frac{dQ}{r d\phi}$   Jack seems to be expressing orally a clear connection between the visual representation of his drawing and the algebraic expression he is naming as he writes.+Jack seems to agree, “Oh, OK, yeah,” and then says “$dQ$ over” while writing $dQ$ and drawing a line under the $dQ$ to indicate division.  He then points to his diagram while saying, “then our partial length” and finishes with “is going to be $r d(\phi)$” while writing $rd\phi$ under the $dQ$ to form the expression: $\frac{dQ}{r d\phi}$Jack seems to be expressing orally a clear connection between the visual representation of his drawing and the algebraic expression he is naming as he writes.
-Apparently associating rd, the length of the arc, with the small charge dQ that would be located there, Seth questions Jack’s algebraic expression, “"So dQ over dQ?"+Apparently associating $rd\phi$, the length of the arc, with the small charge $dQ$ that would be located there, Seth questions Jack’s algebraic expression, “"So $dQ$ over $dQ$?"
-Jack points to the denominator of his expression and names it, “r d (phi).”+Jack points to the denominator of his expression and names it, “$r d (\phi)$.”
-Seth reiterates his understanding that the arc, labeled rd, IS the little piece of charge located there, “But then it...but, like, r d(phi) is dQ, so, like, that'd be dQ over dQ."  This is the third iteration by Seth of his understanding that the arc length rd is equal to a charge, dQ, and he uses that understanding to infer an expression that seems to puzzle him, dQ/dQ, which would equal 1 and therefore not be useful.+Seth reiterates his understanding that the arc, labeled $rd\phi$, IS the little piece of charge located there, “But then it...but, like, $r d(\phi)$ is $dQ$, so, like, that'd be $dQ$ over $dQ$."  This is the third iteration by Seth of his understanding that the arc length $rd\phi$ is equal to a charge, $dQ$, and he uses that understanding to infer an expression that seems to puzzle him, $dQ/dQ$, which would equal $1$ and therefore not be useful.
-Jack starts to respond, “wait”  but is interrupted by Peter, “Uh, that’d be a big R by the way, just (?)..”  Although Peter has been restless, reaching for and sipping his drink, and just generally fidgeting, he has apparently been following the discussion enough to notice Jack’s shift from writing capital R to lower case r.+Jack starts to respond, “wait”  but is interrupted by Peter, “Uh, that’d be a big $R$ by the way, just (?)..”  Although Peter has been restless, reaching for and sipping his drink, and just generally fidgeting, he has apparently been following the discussion enough to notice Jack’s shift from writing capital $R$ to lower case $r$.
-Jack at first defends his choice to shift to a little r, “"Um, well, no, really it has to be a little r, because it's changing” but then reconsiders, “No, wait, no, it's not, it's got to be a big R,...” and changes the lower script r to a capital R in his expression: dQ_ +Jack at first defends his choice to shift to a little $r$, “"Um, well, no, really it has to be a little $r$, because it's changing” but then reconsiders, “No, wait, no, it's not, it's got to be a big $R$,...” and changes the lower script $r$ to a capital $R$ in his expression: $\frac{dQ}{Rd\phi}$
-         Rd+
Peter confirms this, “yes” and Jack states, “..because it's not changing" recognizing that the radius of the spinning ring is fixed. Peter confirms this, “yes” and Jack states, “..because it's not changing" recognizing that the radius of the spinning ring is fixed.
The group laughs.  Seth commends the exchange, "That was really good intuitive..." and Jack seems about to elaborate when Liz comes by with an open, “How's it going over here?"  Jack evaluates their progress, "Not good." The group laughs.  Seth commends the exchange, "That was really good intuitive..." and Jack seems about to elaborate when Liz comes by with an open, “How's it going over here?"  Jack evaluates their progress, "Not good."
-As Liz moves to the other side of the table where she can read the writing right side up, Seth begins to articulate what they have done so far, “"So we've got, for, J = I times, here's our z component…here's our R component…And we still need our Rd(phi), so we decided that Rd(phi) equals dQ, so we..."+As Liz moves to the other side of the table where she can read the writing right side up, Seth begins to articulate what they have done so far, “"So we've got, for, $J = I$ times, here's our $z$ component…here's our $R$ component…And we still need our $Rd(\phi)$, so we decided that $Rd(\phi)$ equals $dQ$, so we..."
-This is the fourth iteration by Seth of his understanding that the arc Rd equals a charge, dQ, and with the “we decided” he now attributes this understanding to the entire group.  Liz interrupts with a “Wait, wait, wait. I am confused…” but does not address this statement.  She seems to be reacting to what she is seeing on the whiteboard, the entire algebraic expression that they have written there, rather than what she is hearing in the details of Seth’s explanation.+This is the fourth iteration by Seth of his understanding that the arc $Rd\phi$ equals a charge, $dQ$, and with the “we decided” he now attributes this understanding to the entire group.  Liz interrupts with a “Wait, wait, wait. I am confused…” but does not address this statement.  She seems to be reacting to what she is seeing on the whiteboard, the entire algebraic expression that they have written there, rather than what she is hearing in the details of Seth’s explanation.
-In watching the video, Len noted that Seth seemed to be still seeing the arc length Rd as being equal to the little charge, dQ, located there.  Corinne noted that the issue was not addressed even though he had now made this statement several times within the group and in front of an instructor.+In watching the video, Len noted that Seth seemed to be still seeing the arc length Rd as being equal to the little charge, $dQ$, located there.  Corinne noted that the issue was not addressed even though he had now made this statement several times within the group and in front of an instructor.
-Len wondered if Jack is imagining a ghost lambda, is he picturing a Rd?  Jack appears to differentiate between the dQ and Rd but his communications back to Seth seem to confirm Seth's equating the two.  Corinne commented that this is typical of communications; people hear that part of what someone is saying that confirms what they are thinking and do not necessarily hear nuances that are unrelated to what they are thinking.  One of the issues about having students working together in groups is that they are not precise with the language so it is much easier for miscommunications to happen.  +Len wondered if Jack is imagining a ghost lambda, is he picturing a $\lambda Rd\phi$?  Jack appears to differentiate between the $dQ$ and $Rd\phi$ but his communications back to Seth seem to confirm Seth's equating the two.  Corinne commented that this is typical of communications; people hear that part of what someone is saying that confirms what they are thinking and do not necessarily hear nuances that are unrelated to what they are thinking.  One of the issues about having students working together in groups is that they are not precise with the language so it is much easier for miscommunications to happen. 
-The implications of this are that the students are only beginning to learn how to distinguish between the actual physical thing that they are talking about and the representations of the thing that they are talking about that they are drawing and the algebraic expressions that they are writing down.  So “is this physical thing a dQ, which is telling me something about charge?  Or is this thing the geometric length Rd?”  The students are just not picking up on those nuances very carefully and so even Jack, who drew the original picture and knew that he meant something with a Q, is agreeing to a statement that is just the geometric quantity Rd and Seth is never seeing the difference between them. It is typical of students at this stage, to not distinguish between different physical phenomena that can share the same geometric representation. +The implications of this are that the students are only beginning to learn how to distinguish between the actual physical thing that they are talking about and the representations of the thing that they are talking about that they are drawing and the algebraic expressions that they are writing down.  So “is this physical thing a $dQ$, which is telling me something about charge?  Or is this thing the geometric length $Rd\phi$?”  The students are just not picking up on those nuances very carefully and so even Jack, who drew the original picture and knew that he meant something with a $Q$, is agreeing to a statement that is just the geometric quantity $Rd\phi$ and Seth is never seeing the difference between them. It is typical of students at this stage, to not distinguish between different physical phenomena that can share the same geometric representation.
-Corinne commented that her extended research group has been having a running discussion about whether or not it is a good idea pedagogically to use the symbol dQ or dM to represent a small amount of charge or a small amount of mass.  One argument goes that if you have an extended charge distribution and you want to know what the total charge is you should write down for the students Q equals the integral of dQ so that the total charge is the sum of a bunch of little charges.  Some people in the group believe that writing that down would help students understand that they have to chop up something large and then sum up the pieces.   +Corinne commented that her extended research group has been having a running discussion about whether or not it is a good idea pedagogically to use the symbol $dQ$ or $dM$ to represent a small amount of charge or a small amount of mass.  One argument goes that if you have an extended charge distribution and you want to know what the total charge is you should write down for the students $Q$ equals the integral of $dQ$ so that the total charge is the sum of a bunch of little charges.  Some people in the group believe that writing that down would help students understand that they have to chop up something large and then sum up the pieces. 
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-Other people in the group believe that the symbol d should be reserved for things that are differentials in the precise mathematical sense. dQ is not the differential of anything and so they do not want that written down.  This particularly rears its ugly head with the switch over to thermodynamics and the students need to distinguish between things that are exact differentials like dU for the internal energy versus things that are not exact differentials like d(slash)Q or d(slash)W or the heat or the work.  This distinction is so important that the d(slash) symbol has been developed.  Her personal view in these slightly lower level courses has been to emphasize that what one is chopping up is always physical space and that one then adds up some physical quantity on that little chopped up piece so that in this case she would always write Q = integral of Rd.  She has been on the fence about using dQ explicitly because she has always felt like it would help some people and make it worse for other people.  So this is a video clip where one student is spontaneously using dQ probably because it has been used by either his high school or intro course teachers, it is a common symbol, and it totally confuses one of the other students in this group.  Now we have some actual evidence about what happens to students around this question.+
 +Other people in the group believe that the symbol d should be reserved for things that are differentials in the precise mathematical sense. $dQ$ is not the differential of anything and so they do not want that written down.  This particularly rears its ugly head with the switch over to thermodynamics and the students need to distinguish between things that are exact differentials like $dU$ for the internal energy versus things that are not exact differentials like FIXME d(slash)Q or d(slash)W or the heat or the work.  This distinction is so important that the d(slash) symbol has been developed.  Her personal view in these slightly lower level courses has been to emphasize that what one is chopping up is always physical space and that one then adds up some physical quantity on that little chopped up piece so that in this case she would always write $Q$ = integral of $\lambda Rd\phi$.  She has been on the fence about using $dQ$ explicitly because she has always felt like it would help some people and make it worse for other people.  So this is a video clip where one student is spontaneously using $dQ$ probably because it has been used by either his high school or intro course teachers, it is a common symbol, and it totally confuses one of the other students in this group.  Now we have some actual evidence about what happens to students around this question.
 +**References**
T. J. Bing, Ph.D. thesis, University of Maryland, 2008, http://www.physics.umd.edu/perg/dissertations/Bing/ T. J. Bing, Ph.D. thesis, University of Maryland, 2008, http://www.physics.umd.edu/perg/dissertations/Bing/
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 +Hymes, D. (1972).  Models for the interaction of language and social life.  In J. Gumperz & D. Hymes (Eds.), Directions in sociolinguistics: The ethnography of communication  (pp. 35-71).  New York: Holt, Rinehart & Winston.
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 +Philipsen, G. & Coutu, L. (2004). The Ethnography of Speaking. In K. L. Fitch & R. E. Sanders (Eds.), Handbook of language and social interaction (pp.l 355-380.  Mahwah, NJ: Lawrence Erlbaum.
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 +van Zee, E. H. & Minstrell, J. (1997a).  Reflective discourse: Developing shared understandings in a high school physics classroom.  International Journal of Science Education, 19, 209-228.
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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.
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