Concept Learning in Unified Signals and Systems

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Concept Learning in Unified Signals and
Systems Steven R. HallDepartment of Aeronautics
and Astronautics, MIT Reem NasrSchool of
Education, Boston University Wallenberg CDIO
Meeting Vadstena, Sweden February 2426, 2003
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Overview

• Misconceptions and active learning
• Analytical vs. conceptual understanding in
  Unified Signals and Systems
• Free response concept tests
• Research study using oral problems
• Conclusion

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Role of Misconceptions in Learning

• Learners construct, before or during formal
  instruction, misconceptions that impede further
  learning
• There are a limited number of common
  misconceptions for a given scientific domain
• Scholarly research on common misconceptions
  exists for only a few subject areas (e.g.,
  Newtonian mechanics)
• Learning process can be enhanced significantly if
  the instructor is aware of misconceptions

Active learning techniques can significantly
improve conceptual understanding
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Active Learning In Unified Engineering

• Cooperative activities
• Opening organizing discussion
• Concept Tests / Free response concept tests
• Turn-to-your-partner discussions
• In class demo
• Individual activities
• Cold calling
• Reading quiz at the beginning of the lecture
• Pop quiz on material
• Muddiest part of the lecture
• Oral problems

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Example Concept Test

• A black box is sitting over a hole in a table.
  It is isolated in every way from its surroundings
  with the exception of a very thin thread which is
  connected to a weight.
• You observe the weight slowly moving upwards
  towards the box.
• 1) This situation violates the First Law of
  Thermodynamics
• 2) Heat must be transferred down the thread
• 3) The First Law is satisfied, the energy in the
  box is increasing
• 4) The First Law is satisfied, the energy in the
  box is decreasing
• 5) The First Law is satisfied, the energy in the
  box is constant

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Analytical vs. Conceptual Understanding

• In order to assess conceptual learning in Unified
  Signals and Systems, the quizzes include both
  analytical and conceptual questions
• By including analytical and conceptual questions
  on a single topic, can compare the two types of
  learning
• On Fall 2002 quiz, had two pairs of analytical /
  conceptual questions
• Resistive networks
• Inductive networks

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Passive Circuit Problem (Analytical)

• For the circuit above, calculate
• The current in the 4 ? resistor. Make sure that
  you specify the direction of the current.
• The potential difference between points P and Q.
  Make sure that you specify the sense of the
  potential difference.

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Passive Circuit Problem (Conceptual)

• A circuit consists of three identical light
  bulbs, two batteries, and a switch, as shown
  above. When the switch S is closed, do the
  following increase, decrease, or stay the same?
• The intensities of bulbs B and C
• The intensity of bulb A
• The current flow in the 24 V battery
• The power supplied by the 24 V battery to the
  circuit
• The power supplied by the 12 V battery to the
  circuit

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Conceptual vs. Analytical Understanding
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Resistor / Inductor Circuit (Analytical)

• For the circuit above, write differential
  equations in state-space form that describe the
  circuit dynamics. Note that the component values
  are not given, so your answer should be in
  symbolic form. You do not need to solve the
  equations.

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Resistor / Inductor Circuit (Conceptual)

• Consider the circuit above with three identical
  light bulbs, a battery, an inductor, and a
  switch. Initially, the switch is open, and has
  been open for a long time. The switch is then
  closed.
• Immediately after the switch is closed, the
  intensity of bulb B (increases, decreases, stays
  the same).
• Immediately after the switch is closed, the
  intensity of bulb C is (less than, greater than,
  the same as) the intensity of bulb A.
• After the switch has closed for a long time, the
  intensity of bulb B is (less than, greater than,
  the same as) the intensity of bulb A.
• After the switch has been closed for a long time,
  the switch is then opened. Immediately after the
  switch is opened, the intensity of bulb A
  (increases, decreases, stays the same).
• Immediately after the switch is opened, the
  intensity of bulb B (increases, decreases, stays
  the same).

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Conceptual vs. Analytical Understanding
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Why the Difference in Performance?

• Performance was about the same for the two
  analytical problems
• Performance was significantly better for the
  inductive conceptual problem
• In class, students struggled with a tricky
  inductive circuit concept test
• Resistive circuit concept test was not as
  challenging
• Resistive circuit concept test failed to make
  students confront their misconceptions

The instructional technique used in a single
lecture can cause a significant difference in the
educational outcome
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Free Response Concept Tests

• Generation of good concept tests depends on
  understanding misconceptions
• Mud cards generate vague misconceptions, which
  are difficult to translate into good concept
  test
• Current supply thing
• Constitutive relations for inductors and
  capacitors
• Free response concept tests can be used to
  surface misconceptions

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Free Response Concept Tests

• Typical free response concept test
• A pulse, u(t), is the input to a smoother, with
  impulse response g(t). Sketch the output, y(t)
  g(t)u(t).

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Free Response Concept Tests Responses
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Free Response Concept Tests Responses
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Fundamental Research Question

• Can generate some misconceptions from active
  learning techniques in class
• In other fields (physics), many studies required
  to determine typical misconceptions
• Research question Can misconceptions be
  determined only from in-class, active learning
  activities?

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Conceptual Understanding Research Study

• Want to determine whether we have surfaced all
  (or most) misconceptions using active methods
• Clinical Interviews are generally regarded as the
  best way to assess student understanding
• In Fall term 2002, began clinical study of
  student understanding of Signals and Systems
  material in Unified Engineering
• Study will continue in Spring term 2003

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Study Methodology

• Used oral problem solving sessions / interviews
  to assess student misconceptions
• Oral problems are part of the course (all
  students must participate)
• Students who volunteer are taped (audio video of
  blackboard)
• 54/70 students volunteered in Fall 2002
• Student has 1/2 hour to read problem, work alone,
  then interviewed for 1/2 hour
• Interview emphasizes conceptual understanding
• 70 students were split into 4 cohorts of about
  18, and interviewed over the course of 4 weeks
• Students in each cohort answered the same
  question
• Taped interviews were transcribed, analyzed, and
  coded
• Representative interviews used to generate
  quotations that illustrate typical misconceptions

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Oral Problem 1

• Problem
• Consider the circuit above, where
• V1 3 V, R2 3W, R3 6 W, R4 2 W, V5
  10 V
• Find the voltage across each element, and the
  current through each element.
• Which elements dissipate or absorb power? Which
  elements supply power? Explain.
• Interview questions
• Explain your general approach to the problem.
• Explain the node equation
• How does the behavior of the circuit change when
  a 12V voltage supply and a 1W resistor are
  connected in series across the 10V (V5) voltage
  supply?

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Concept Sign Convention

• Misconception The voltage across an element
  labeled with / signs must be positive.
• S56 Well, its 4 volts because you cant have a
  negative voltagea negative voltage It was
  negative because I was doing it with relation to
  the node.
• P But for example, if this were not just a
  homework problem, but an exam, and I said write
  in the space below v4 and you would pass the
  course if do this right and you fail the course
  if you get it wrong, what would you put there?
• S56 Then I would work it out on some scratch
  paper and then I would put 4 volts there.
• P Even with the plus and minus sign here
• S56 Yes.

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Concept Potential / Voltage

• Misconception Potential accumulates
• P So what I'm really getting at is that let's
  say that I have a real physical circuit, I have a
  3-volt battery in the circuit and I have a
  10-volt battery in the circuit. Is it plausible
  that you get 12 volts somewhere in the circuit?
• S49 It seems like you could It seems like the
  voltages could sum. I guess that's what I think.
  UmmI guess the maximum would be 13 volts.

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Concept Current Source (Oral Problem 2)

• Misconception There is no voltage drop across a
  current source
• S Theres no voltage across a current source
  its just a current source
• Misconception A current source is an ohmic
  element
• P Whats the voltage drop across the current
  source?
• S Well thats an unknown. Thats in terms of I,
  so I start to think well V IR, I always write
  that. And were looking for V, and we know I,
  well we have a variable for I, so maybe we should
  write I as V/R.

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Oral Problem 3

• Problem
• Find a set of differential equations that
  describes the node voltages as a function of
  time.
• Find the characteristic values of the system.
• Find the characteristic vectors of the system.
• If the initial conditions are v1(0) 4 V,
  v2(0) 0 V, v3(0) 0 Vwhat is e1 (t)?
• Interview questions
• Explain your general approach for solving the
  problem.
• Explain what physically happens in the circuit as
  it evolves from its transient state to its steady
  state.
• Plot the graph of voltage versus time for each
  capacitor.

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Concept Voltage

• Misconception Voltage is conserved
• S35 The voltages should be equal across, at
  least. So that means youre going to take 4
  volts from C1 and disperse it across three
  different capacitors equally. So at steady
  state, they should be at four-thirds volts for
  each one.

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Concept Voltage

• Misconception Voltage always dissipates
• S58 I know that this C1 is going to dissipate
  the voltage because there is no voltage source
  and there are two resistors, but first it should
  initially charge up C2 and C3, I think And then
  eventually over time it should all be dissipated
  in the resistors Im not sure I understand how
  they the capacitors can all stay at 1 volts.
• P Ok, now I open the switch and so what youve
  told me is that immediately after I open the
  switch its still at V, but what happens longer
  term, you know.
• S64 Its going to dissipate back down to 0.

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Concept Capacitance

• Misconception Voltage is proportional to
  capacitance
• S35 That means the capacitor would have twice as
  much twice the voltage as capacitors one and
  three.
• S75 All right, okay. C2 and C3 are going to
  increase. I guess that for e3 will go to the same
  point as e1 and I would guess that e2 was going
  to climb up as well, but its going to reach a
  higher potential.
• S94 Eventually it reaches some sort of balance.
  C1 and C3 are going to wind up with the same
  voltage, and C2 would have twice as much.

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Conclusion

• In our studt, we surfaced numerous misconceptions
  that we had not anticipated
• Preliminary conclusion is that it is not possible
  to construct a complete set of student
  misconceptions on the cheap, without a formal
  study
• Strong evidence that appropriate active learning
  enhances conceptual understanding
• Where do we go from here?
• Continue study
• Research to inform development of teaching
  materials
• Consortium of institutions to spread load

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• Questions? Discussion?