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## Example of Small Group Conversation about Geometric and Algebraic Representations of a Physical Quantity

## Physics 422, Friday October 26, 2007, Day 10

**by Emily van Zee, Corinne Manogue, and Len Cerny**

This video clip starts at 54:42 and ends at 58:17 in the video 071026Ph422Grp6.mov

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.

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 algebraic expression they have written for the current density $J$ is: $J = I \delta(z) \delta(R) (Q/2πR) (2πR/T)$

Len has been watching the videos to identify places where students are not understanding one another's arguments and found this to be an intriguing example. He has been looking for instances of epistemic framing (Bing, 2008). Bing's four epistemic frames are: calculation, physical mapping, mathematical coherence, and authority. Len has been wondering whether there is a difference in framing from thinking of the physical object compared to thinking of a geometric representation of that physical object. Physical mapping is a huge category and situations such as this might indicate the need for subdivisions of that category or additional categories.

In summarizing the context for this conversation, Len noted that Jack (lower right) was leading the group, trying to take the $d\phi$ and $J$ and leave those in the equation, to retain them, whereas other groups had all switched to $I$'s $\lambda$ and $d\phi$'s. Len commented that these group members are seeing that they need to look at the $z$, $r$ and $\theta$ pieces. They realize that the $z$ and $r$ are going to be delta functions, reduced points in those dimensions, and therefore they need to now look at what they consider the theta piece to be and this is where they see the interesting parts of $J$ to occur. They have thrown in this $I$ at the start and as far as he can tell they do not realize that this $I$ and the expression they are developing here in the bracket are the same thing.

Jack (lower right) has pointed to their charge density expression $Q/2\pi R$ with the comment “But this has to be with respect to $d(\phi)$, right? Charge over length is,…this has to go,…we need this related to $d(\phi)$, so for,…we've got small sections.”

In watching the video, Corinne noted that this looks like the first place where someone is realizing that they have to chop something up. Jack is talking about small sections, he gestures a small distance, and then he draws a little $d\phi$ wedge with radius $R$ and an arc, and labels these (what does he write?) Corinne noted that $dQ$ is the amount of charge on an arc and the density is $\lambda$ so $dQ = \lambda R d\phi$.

Seth (upper right) seems to check his understanding of what this wedge represents, “$R d(\phi)$ equals $Q$?”

In watching the video, Len noted that Jack seems to mumble “$R d(\phi) = Q$” in reply and it is unclear whether he is just mumbling back what Seth has just said or whether he has considered the question and is confirming that interpretation. Corinne agreed that it is unclear what Jack intends and what Seth hears.

This is the first voicing of Seth’s apparent equating of an expression for the length of the arc with an expression for the charge. Len described the subsequent conversation between Jack and Seth as two people talking past each other the entire time. Seth never seems to understand Jack's argument and Jack never seems to understand Seth's interpretation.

Jack's argument seems to be that he is looking at the charge density, $\lambda$, as $dQ$ over $R d\phi$ (the initial use of $Q$ above shifts to $dQ$ in subsequent statements). Seth is using the geometric diagram to conclude that $R d\phi = dQ$. Len’s view is that one student is using the diagram less literally in representing $dQ$ as an arc whereas the other student is interpreting that arc as a length and that $dQ$ is an alternate description for that physical length. Therefore this conversation seems to be an example of the contrast between thinking about the physical situation versus thinking about a partial model of that physical situation.

At this point in the video, Peter (on the left) directs attention away from the drawing. He points to the long expression for $J$ and asks, “This equals $J$, huh?” Seth seems to repeat the phrase, “This equals $J$, huh?” and Peter recasts the question, “Is this true?”

Jack responds to Peter’s question, “Not quite.” Peter presses, “What are we missing?” and Jack replies, “We need a $d(\phi)$.”

Having drawn a small angle $d\phi$ with radius $R$ and an arc, Jack seems to be trying to connect that visual representation to the algebraic expression that they have written on the board.

Seth elaborates, “We need somehow to, like, incorporate, like, an $R d(\phi)$, right?” and Jack confirms, “Yeah.”

Seth then points to Jack’s wedge drawing and the $Rd\phi$ expression and reviews the situation: “So, there's your…so we got our $dz$, our $dr$, we need $R d(\phi)$. $R d(\phi)$ is $dQ$” (pause) “right? “ He looks at Jack. Jack seems to nod affirmatively and Seth continues, “for example” as an apparent hedge and then points to the end of the long algebraic expression, “So, how can we make this look like a $dq$?”

This is the second statement by Seth, now expressed in terms of $dQ$ rather than $Q$, that the arc length $Rd\phi$ equals a charge. Seth is thinking like a physicist in that he seems to be actively trying to figure things out by stating aloud what he is understanding and drawing attention to a piece of a complex algebraic expression. Jack is sitting oriented toward Seth and seems to be attending to what Seth is saying. They are engaged in a physics conversation, with Jack drawing a visual representation of what he is thinking and Seth trying to connect what he thinks is needed to make progress in developing an algebraic expression with what his partner has offered in his drawing.

Seth continues thinking aloud while pointing to the $Q$ in $Q/2\pi R$, “”So this is the charge,…total charge divided by…” and Jack contributes “the length.” Seth confirms, “the total length, that makes sense” while moving his hand around in a small circle as if he were mimicking the form of the ring.

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.

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, “$r d (\phi)$.”

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 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: $\frac{dQ}{Rd\phi}$ 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.”

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 $Rd\phi$ 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.

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 $Rd\phi$?” 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.

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 $\lambda Rd\phi$. 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.

T. J. Bing, Ph.D. thesis, University of Maryland, 2008, http://www.physics.umd.edu/perg/dissertations/Bing/