Download Context Factors and Mental Models – Examples in E&M

Context Factors and Mental Models – Examples in E&M
Topic 1:
Particle trajectories in electric fields
(1) Take a look at the picture on the right. The two charges
(positive and negative) are at rest. Some of the electric field
lines are also shown as broken
lines. A small positive test charge (green dot) is then placed
at the position shown.
From the pictures given below, which one correspond to the
most likely path (solid line) that the test charge (green dot)
would take?
A
B
C
D
E
F
Workshop questions:
1.
Identify the possible context factors in the above diagrams.
2.
What are the possible mental models corresponding to the different
choices?
Topic 2:
Charged particle in an electric field
1. A negative charge (red dot) is placed at rest in an electric field region (depicted by
arrows) as shown below.
Which way will the charge move?
A.
It will not move at all.
B.
It will move up.
C.
It will move down.
D.
It will move to the right.
E.
It will move to the left.
2.
A negative charge is placed at rest in an electric field region. The electric field points
from left to right.
Which way will the charge move?
A.
It will not move at all.
B.
It will move up.
C.
It will move down.
D.
It will move to the right.
E.
It will move to the left.
F.
Cannot say for sure.
3. A negative charge (red dot) is placed at rest in an electric field region (depicted by
arrows) as shown below.
Which way will the charge move?
A.
It will not move at all.
B.
It will move up.
C.
It will move down.
D.
It will move to the right.
E.
It will move to the left.
4. A negative charge (red dot) is placed at rest in an electric field region (depicted by
arrows) as shown below.
Which way will the charge move?
A.
It will not move at all.
B.
It will move up.
C.
It will move down.
D.
It will move to the right.
E.
It will move to the left.
Workshop questions:
1.
Identify the possible context factors in the above questions.
2.
What are the possible mental models associated with the choices?
3.
What are the possible mental models that are common to topic 1 and
topic 2?
Topic 3:
Induced currents
The following situations involve a CONDUCTING rectangular wire loop in a magnetic
field. The magnetic field has CONSTANT magnitude and direction in the region shown
by a cross X. The magnetic field is into the paper.
1.
In the three pictures shown below the wire loop is moving at a constant velocity
(i.e. at a constant speed in the fixed direction as shown by the arrow).
In picture I the loop is just moving into the magnetic field region.
In picture II the loop is moving in the field.
In picture III the loop is moving out of the field region.
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1.
Which situations correspond to an induced current in the loop?
A.
I
B.
II
C.
III
D.
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E.
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F.
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G.
All
H.
None
2.
Consider the same three pictures WITHOUT THE LOOP HAVING ANY
SPEED. That is, the loop is at rest at the respective positions. Which
situations correspond to an induced current in the loop?
A.
I
B.
II
C.
III
D.
I and II
E.
II and III
F.
I and III
G.
All
H.
None
3. The following situations involve a CONDUCTING circular wire loop in a magnetic field. The
magnetic field has CONSTANT magnitude and direction in the region shown by a cross X. The
magnetic field is into the paper.
Consider the following three pictures:
In picture IV the loop is at REST in the magnetic field region.
In picture V the loop is MOVING AT CONSTANT VELOCITY in the direction shown.
In picture VI the loop is ROTATING clockwise AT CONSTANT ANGULAR VELOCITY about
an axis through its center.
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Which situations correspond to an induced current in the loop?
A.
IV
B.
V
C.
VI
D.
IV and V
E.
V and VI
F.
IV and VI
G.
All
H.
None
Workshop questions:
1.
Identify the possible context factors in the above
diagrams and questions.
2.
What are the possible mental models?
Particle trajectories in electric fields - observations
Majority agreed with A. Here, the drawn field lines trigger students to
bringing up the (incorrect) notion that charges ‘follow’ the field lines. One major
reason for this reasoning is the belief that the drawn field lines are the only
ones that exist (this fact is investigated more in our later questions). The
ordinary usage of the language such as ‘opposite charges attract each other’
also leads to this type of incorrect reasoning – thus they can be thought of as
the linguistic context. The few students who answered the correct choice E
based their judgment on the fact that the test charge should start tangentially
to the field line at the point of concern. Another context issue with the pictures
is that students often think that the field does not exist outside the region of the
lines that are already shown. So, the test charge crossing the lines (C) and
moving ‘beyond the field region’ (E) are taken as unacceptable. (Note: There
can be some disagreement about E and F as to the nature of whether the
depiction of the test charge colliding with the negative charge is correct. These
were simply given for students to correctly realize that the test charge start off
tangentially to the field line rather than worry about the end result since all
other pictures depict the same end scenario. It is understood that this a threebody problem which lacks any analytical solution and the subsequent motion
depends very sensitively on the initial conditions.)
Charged particle in an electric field - observations
The question was based on earlier surveys and interviews. About ~50%
answered the questions consistently and correctly. About ~40% fluctuated
depending on the context factors.
Q1 was given based on the belief that the arrows pointing straight at the
negative charge would trigger responses similar to that were given for the
positive nuclei and the electrons scenario (see slide 3 and 4). Our
assumption was proved correct among several students but they selected
answer D to Q1 for more elaborate reasons. Students in all of these problems
seems to be visualizing a set of positive charges at the blunt end and a set of
negative charges at the pointed end of the arrows. This is in conformity with
the popular exposition that ‘electric field lines point from positive to negative
charges’. Thus, a set of negative charges were imagined at the arrow tips
which would repel the negative test charge to the right.
Some students considered the electric field region to exist only in the area
where arrows were depicted and not as a sample representation of a broad
region. In Q2 ~10% selected F as the choice. In that reasoning ‘a picture was
essential for the correct evaluation since the subsequent motion of the charge
depends on where the charge is been placed’.
In Q3 when the charge was placed ‘in between’ the field lines ‘nothing
happened to the charge’. Here as was discussed earlier students tend to
model the region not as a continuum of field lines but rather as a discrete set.
However, an opposite model also emerged in the sense that ‘the charge is
placed IN the field’ which gave rise to the dynamics. It also gave rise to the
wrong dynamics because of the ‘asymmetry’. This manifested through the
fact that as shown, there were two field lines below the charge as opposed to
one above it which tend to ‘drag the charge’ downwards.
In Q4 since the charge was placed ‘on a field line’ dynamics emerged.
However, the opposite model also emerged, this time through ‘symmetry’.
Since it seems that the charge is placed in the middle of the picture, this led
some students to abandon the meaning of the arrows and argue that the topbottom, left-right look symmetrical thereby holding the charge in place without
any movement.
The negative charge is mentioned instead of a positive charge in order not
to ‘trivialize’ the problem and to get the students a bit more involved with the
questions. About 50% of the studentswere consistent in answering all the
questions through a physicist’s conception. The rest fluctuated depending on
the contextual elements. Simply put, the diagrammatic representations were
interpreted as they meet the eye rather than abstract and idealized
representations.
About 18% selected choice A to Q1. The charge been placed at the end of
the arrows in Q1 prompted the issue of the ‘end of the electric field region’ (“it
will not move since it is not ‘in’ the electric field”). The ‘end of the field’ also
gave rise to the choice D. One argument goes as follows: “The electric field
points from positive charge to the negative charge. The arrows stop just before
the negative charge so I am assuming the field region borders the apex of the
arrows and indicates the location of the negative charge causing the electric
field. Like charges repel. The left side is theoretically closest to the negative
charge. [The test charge] Moving right is due to the negative charges toward
the left”. Here the student has taken the phrase ‘electric field lines go from
positive to the negative charges’ to imagine a set of negative charges at the tip
of the arrows which would in turn repel the negative test charge.
In Q2 the absence of a picture gave way to the answer F which 10% of the
students selected (“it depends on the position of the charge”). We also
observe a 10% increase (not necessarily the same students as in the previous
case) in choice A for Q3 where the charge is considered not within the
influence of the electric field lines (“no field lines appear to be contacting the
charge”). However, this particular representation has also given rise to
arguments like “[the charge] is actually in the field…will be attracted to the
positive side, which is to the left”.
The distribution of answers to Q1 and Q4 did not differ much. It is surprising
that there is only ~3% drop from Q1 to Q4 with regard to choice A. On careful
examination of student reasoning it seems that ‘symmetry’ played a part in Q4.
Due to the position of charge in Q4 the left-right and up-down regions look
symmetrical in the picture with the charge at the center. This leads to a field
surrounding the charge homogeneously and isotropically thus ‘balancing out
the influence of the field on the charge’ (“it [the charge] will not move because
a constant field is around it and therefore canceling out any tendencies to
move”). The same symmetric arguments gave rise to an increase of choice C
by ~6% in Q3 as compared to the others. Since there are two field lines below
the charge as opposed to one line above it, the charge would be more
influenced by the lower region ‘dragging’ the charge down (“It [the charge] will
move down. The field is stronger beneath the charge”). In both cases the
direction of the field lines and its meaning is abandoned.
Induced currents - observations
The questions were based on the previous interviews and surveys. It is
particularly designed to make the identifications that were difficult to make with
‘similar’ kind of questions in CSEM. An attempt has been made to make the
questions clean and without any difficulties in visualizing the scenarios given. The
major fact that we try to isolate through these questions is the model of motion
that students give precedence over the rate of change of flux in inducing a
current.
The majority selected G as the choice for Q1 whereas in Q2 it was H. This
confirms that translations seems to be the predominant model students have. This
is possibly because of the emphasize in such question in the classroom and in
textbooks. Do look at this situation more carefully Q3 is developed with a circular
loop. Majority selected B followed by E which is about a half compared to those
answered B. This helps us identify further, that among translational and rotational
motion, translational motion is indeed the predominant model. Unlike in the
electric field context discussed above, the depicted magnetic field region does not
give rise particular context issues in these questions. The students’ attention is
mainly fixed on movement than on the field. Also, the question is phrased clearly
(e.g., ‘…loop is just moving into the magnetic field region.’) for the students to get
an idea of where the boundaries of the region are.
Student comments:
They all have a velocity (Q1: G)
They do not have any velocity (Q2: H)
They have a speed (Q3: E)
Because in all cases the loop is moving (Q1: G)
Because the loops are not moving (Q2: H)
Because its the only loop moving a distance (Q3: B) – Translatory motion
References
R. Warnakulasooriya and L. Bao, “Preliminary studies on students’ understanding of
electricity and magnetism for the development of a model-based diagnostic instrument”,
Proceedings of the 2001 Physics Education Research Conference, pp 127-130.
R. Warnakulasooriya and L. Bao, “Towards a model-based diagnostic instrument in
electricity and magnetism - an example” , Proceedings of the 2002 Physics Education
Research Conference, (submitted).