Len's dissertation work has uncovered an interesting snippet from a small group:

One student labels part of the arc of a circle as $dQ$. The other students initially don't understand the difference between this and $r d\phi$. One of these eventually understands, the other never does.

Summer 2011: Jeff is making a videoclip of this. Original Video File: 071026Ph422Grp6.wmv

Here are Len's initial transcript and observations:

## Transcript excerpt for dQ

[00:54:42.01] Jack, POINTS to velocity expression 2πR/T , “But this has to be with respect to dφ right? Charge over length is,…(GESTURES a small distance with thumb ring finger)…this has to go, (WRITES a circle around Q/2πR)…we need this related to dφ, so for,…we've got small sections.” (GESTURES a small distance with thumb and pointer finger, DRAWS a small wedge and WRITES labels on the end of the wedge as dQ and the angle as dφ)

[00:55:08.16] Seth, “So Rdφ equals Q?”

[00:55:15.07] Jack (mumbles), ” Rdφ equals Q.”

{pause}

[00:55:29.03] Peter, POINTS to J = I δ(z)δ(R) Q/2πR 2πR/T and says, “This equals J, huh?”

[00:55:31.11] Seth, “This equals J, huh?”

[00:55:32.09] Peter, “Is this true?”

[00:55:33.11] Jack, “Not quite”

[00:55:36.01] Peter, “What are we missing?”

[00:55:38.04] Jack, “We need a dφ”

[00:55:45.16] Seth, “We need somehow to, like, incorporate, like, an Rdφ right?”, (WRITES “Rdφ” on board next to ring drawing)

[00:55:50.15] Jack, “Yeah.”

[00:55:51.27] Seth, POINTS at Jack's wedge drawing and the Rdφ expression, “So, there's your…so we got our dz, our dr, we need Rdφ. Rdφ is dq, right? For example. So, how can we make this look like a dq?”

[00:56:16.11] Seth, POINTS to Q in Q/2πR part of expression, “So this is the charge,…total charge divided by…”

[00:56:22.16] Jack, “The length.”

[00:56:23.29] Seth GESTURES around ring and says, “The total length, that makes sense.”

[00:56:26.03] Jack, “So…”

[00:56:26.03] Seth, “That looks,…this to me looks like a dQ, right?”

[00:56:28.10] Jack, “Oh, OK, so, yeah, dQ over (POINTS at drawing of wedge),…then our partial length is going to be rdφ right?”, (WRITES dQ/ rdφ)

[00:56:37.00] Seth, “So dQ over dQ?”

[00:56:38.24] Jack, POINTS at dφ in dQ/ rdφ expression, “Uh, dφ”

[00:56:40.28] Seth, “But then it…but, like, rdφ is dQ, so, like, that'd be dQ over dQ.” (POINTS at dQ/ rdφ expression)

[00:56:44.27] Jack, “Wait.”

[00:56:47.21] Peter, “Uh, that'd be a big R, by the way, just …[?]…”

[00:56:52.13] Seth (laughs)

[00:56:54.15] Jack, “Um, well, no, really it has to be a little r, because it's changing. No, wait, no, it's not, it's got to be a big R,…” (WRITES a capital R into expression to get dQ/ Rdφ)

[00:57:04.03] Peter, “Yes.”

[00:57:04.14] Jack, ”…because it's not changing.”

[00:57:06.01] Group laughs

[00:57:07.03] Seth, “That was really good intuitive…”

[00:57:09.20] Jack, “Um…”

LIZ ARRIVES

[00:57:11.17] Liz, “How's it going over here?”

[00:57:13.01] Jack, “Not good.”

[00:57:13.25] Liz, “Not good? OK.”

[00:57:15.09] Evan from Group 5 says, “We have a question.”

[00:57:16.12] Liz (to Evan), “I'll be right there.”

[00:57:17.11] Evan from Group 5 says, “Oh, OK, I thought you were leaving them.”

[00:57:18.04] Liz (to Evan), “Nope.”

[00:57:18.21] Liz, “I've come in so that I can read this side.”

[00:57:21.06] Seth (POINTS at Jack's J = I δ(z)δ(R) Q/2πR 2πR/T equation), “So we've got, for, J = I times, here's our z component.” (POINTS at δ(z))

[00:57:27.08] Liz, “Mm-hm.”

[00:57:28.19] Seth POINTS at δ(R), “Here's our R component.” [00:57:30.05] Liz, “OK”

[00:57:30.20] Seth POINTS at Rdφ written near ring, “And we still need our Rdφ so we decided that Rdφ equals dQ, so we…”

[00:57:37.23] Liz, “Wa, wa, wa, wait. I am confused. I think you're convolving some things.” (Gesticulates) “So first is” (Gestures by using both hands to make brackets around J = I δ(z)δ(R) Q/2πR 2πR/T equation) “J you're saying is this.”

Jack’s drawing labels dQ next to a dl arc. Thus, if the drawing is accepted at face value, a logical conclusion based on the drawing would be that Rdφ = dQ, or Rφ = Q. Seth seems troubled by this, but seems to accept it without any analysis in terms of units or physical meaning. Jack, although making a mumbled repeat of Seth’s statement, appears to actually understand that his dQ label does not represent the dl arc. Jack appears to distinctly separate the dQ and dl concepts a little over a minute later at [00:56:28.10] when he says, “Oh, OK, so, yeah, dQ over (POINTS at drawing of wedge),…then our partial length is going to be rdφ right?”, (WRITES dQ/ rdφ). Meanwhile Seth repeatedly claims that Rdφ = dQ. Seth never really resolves this issue, because he brings it up with Liz and Liz focuses on other issues instead.

Jack is interpreting the arc of the wedge to represent a small bit of charge dQ, while Seth is interpreting this same arc of the wedge to be a representation of a small arc length dl which in this case is equivalent to dQ. Bing blurs geometric to symbolic and physical thinking to symbolic representation. However, in this case Jack is thinking physically and using the geometric drawing to reflect his thoughts. Seth is taking the geometric drawing at face value and missing the shorthand that Jack is attempting to use. This produces an extended mismatch of understanding. The wedge drawing was Seth’s object of discussion, whereas for Jack, the physical ring was still the focus, and the drawing was only a shorthand representation of this physical reality. This leads to an extended geometric mapping versus physical mapping misunderstanding.

This is reminiscent of the studies by McDermott (or is it someone else) where students would make a graph of velocity that looked like the ramp down which the ball was rolling. The are cases throughout Ed research (but I can’t bring specifics to mind) where students will start analyzing a graph of something as representing the physical something, like taking a graph of height versus time for a ball thrown in the air and having students thinking the ball took a parabolic path.

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