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voltage will be across capacitor,” “since they are
in series the voltage should be the same across
all of the bulbs”), and admitted to making some
guesses. The responses of students who received
invariants prompts and instruction revealed they
more often attempted to revise and correct their
misconceptions:
•
“when you go from parallel to series, the
intensity will decrease since the voltage is
split across both bulb 1 and bulb 2.”
•
“because the lowere resistance will need
more power.” “the lower resistance will
result in more power being dissapated be-
cause p=v^2/R”
Example 1
It was clear that after receiving feedback on
invariants, most students attempted to revise and
correct their misconceptions.
•
1
st
explanation: “Since frequency and cur-
rent are related linearly, and increase in
frequency will increase current.”
Invariant Selection Analysis
•
2
nd
explanation: “Actually, current and fre-
quency have no relation so current does not
change with frequency.”
For each circuit problem, our experts selected and
agreed upon the invariants that were most relevant
and helpful for solving the problem. They also
identified which invariants were clearly irrelevant.
The impedance of an inductor, for example, is
irrelevant for a circuit with only capacitors. The
remaining invariants were placed in a third cat-
egory. These invariants were technically involved
in the circuit's behavior, but were not necessarily
useful for solving the problem asked about the
circuit. Ohm's law, for example, is involved in
many of the circuits used in our tests, but is not
always an important one for answering a particular
question about the circuit.
We needed a way to quantify how well stu-
dents selected invariants for a particular problem.
In this case a simple percent correct measure is
insufficient for characterizing students' use of
invariants, and would reward students who select
more invariants regardless of their importance
to the problem. To control for such response
biases, we utilized a nonparametric discrimina-
tion measure known as Yule's Q. Nelson (1984)
contrasted simple percentage correct measures,
d'
measures and Yule's Q, and advocated Yule's Q
over
d'
on the basis that it was thought to make
weaker assumptions about the data and required
fewer observations. For our purposes this measure
rewards the selection of invariants our experts
agreed were appropriate for a problem, while
Example 2
•
1
st
explanation: “More current flows
through the bulbs when they are in series.
this makes a brighter light.”
•
2
nd
explanation: “More voltage through
the bulbs in parallel will make for brighter
bulbs.”
•
3
rd
explanation: “Power is what determines
the light intensity. More power makes for
more intensity.”
The students also articulated some of the invariant
principles they learned:
•
“There is only one path for the current to
flow through and both bulbs lie on this
path.”
•
“With low frequencies most of the voltage
is across the capacitor since its impedance
is high when frequency is low.”
•
“because the lower resistance in Peter's
circuit will result in more power consump-
tion. lower resistance = more power”
•
“since it is in series connection, the same
resistances and current will produce the
same power for all the bulbs. p=i^2R”
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