Problem
Based Learning in Basic Physics - I
School Science 49 (3) Sept 2011
A. K . Mody
V. E.S. College of Arts,
Science and Commerce, Sindhi Society
Chembur, Mumbai – 400 071
H. C. Pradhan
HBCSE, TIFR, V. N. Purav
Marg, Mankhurd
Mumbai – 400 088
In this article- first in the series of articles we
present a problem based learning course that we have used successfully to build
capacity of physics students. We present the learning objectives in different
area of basic physics and what each problem tries to achieve with its solution.
There is a criticism that at every stage
of learning, success in examinations in present education scenario relied
heavily on reproduction of material that the students had learned. No
importance is given to development of necessary skills that can make students
think like professionals. If carefully chosen as a part of learning process,
problems can encourage the cultivation of a group of skills, which can be
important constituent of the expertise of a professional. Problem based
learning is found to be a convenient method to teach subject/s (Baden 2000) and
many experiments are being tried all over the world with positive outcome. Problem solving brings to bear essentially
reasoning about the subject. Problem solving is scaffolding/building up higher
objectives of learning. As per Bloom (1980), these objectives are,
comprehension, application, analysis and synthesis. Doing science itself in a
way is problem solving.
At a level of basic physics all problems chosen have to be
well defined and as Baden has described, in the subject of science, formal
teaching has to precede problem based learning. Apart from being used for
testing, problems can become good instruments to help students construct their
knowledge. This will also be in tune with NCF –2005 guide lines of ‘promoting
problem-skills, problem solving abilities and applications of physics
concepts/content, useful in real life situations, for making physics learning
more relevant, meaningful and interesting.
We have tried selecting our special problems for the course designed and following Reddish(1994) termed them as touchstone
problems. Although we have used them in different sense than Redish.
By touchstone problem we mean a problem which satisfies
more than one of the following criteria.:
(i)
A problem which incorporates basic principle/s
(ii)
A problem which is attractive enough or is rich in
context
(iii) The
problem should be sufficiently difficult but not too difficult to put students
off.
(iv) should require steps that are not mechanical
but involve some decision making
(v)
The problem should have a reasonable goal
(vi) The
problem should guide students to comprehend the topic and/or application.
We have conducted such a problem
solving course covering topics from basic physics with similar problems. Most
of the problems required one or more of the above-mentioned strategies to be
used. The problems were of the level of standard textbook ‘Fundamentals of
Physics’ by Halliday, Resnik and Walker (2004) and Young (2004). Most problems
were chosen from the textbooks mentioned, competitive exams like JEE (Joint
Entrance Exam for admission to Indian Institute of Technologies) and Physics
Olympiads or equivalent. Source of some problem is unknown as authors have lost
track of them over the period of time but found them to be indispensable. Some
problems were specially designed as dictated by need.
The
strategy we used to make students solve the problems can be called as
constructivist, features of which are as follows.
(i)
Instructors should play a role of facilitator and help
learner to get his or her own understanding of the concepts & let the
learner play an active role in the learning process.
(ii)
The learning
environment should be designed to support (by making books available) and
challenge (through problems: touchstones & auxiliary (additional/smaller)
and counter questions as well as guided intervention) the learner’s thinking.
This way learning becomes an active process where learner learns to discover
principles, concepts and facts themselves.
(iii) Requires
instructor has culture, values and background to become an essential part of
the interplay between learners and tasks in the shaping of meaning.
(iv) Students
learn by building upon knowledge they already possess themselves and guided
interventions correct errors, which creeps in their understanding.
(v)
There should be enculturation. Students should be
introduced to culture of the subject. In our case students were introduced to
culture of doing physics by solving problems.
(vi) Most
importantly, there should be effective scaffolding. That is, students are not
given answers to any questions, but have to be guided (using interventions like
auxiliary problems, counter questions, cognitive conflicts) to converge to the
answer themselves.
Based on how students work
through, it can also be used for formative assessment (Mody 2011). This way,
students also get immediate feedback of their thinking and construction of
knowledge as teacher uses his/her constructivist scaffolding. This is very much
on par with the vision of National Focus group on Examination Reforms (NCERT 2006a), which has noted, ‘In
the long term (about a decade) we envision a vastly different system built upon
entirely new foundations. This system would actually make the teacher the
primary evaluator of her students. This system would not be one-shot but
continuous, would extend beyond the cognitive domain and beyond pen and paper;
and, hopefully, be seen by all not as a burden but as a tool for further
learning. In this system the primary role of boards would change radically –
from direct testing at present to rigorous validation of school-based,
teacher-based assessment. If any direct testing by boards were still to be
needed it would be of a very different type – optional, open-book and
on-demand’. In this scheme emphasis is on students construction of
knowledge and teacher can very much use it to assess students.
The problems on motion that we
dealt with are given as follows with the learning objectives and what each
problem is expected to achieve along with solution. This problems can be used
by teachers to teach finer aspects of concepts. Teacher may use any of the
method they prefer: (i) It could be done on board in the class (ii) may be used
as tutorials (iii) may be given as home assignment (iv) may be used as a tool to
help students construct their own knowledge. We have used these problems as
class room work (as in (iv)) where teacher uses constructivist method as
discussed above and described by Pradhan (2009a). As a result of such a course we found
students capacity increased and were ready to face challenges, which earlier
they never thought of.
As per Downey(1967) , ‘The core of
good thinking is the ability to solve problems. The essence of problem solving
is the ability to learn in puzzling situations. Thus, in the school of these
particular dreams, learning how to learn pervades what is taught, how it
is taught, and the kind of place in which it is taught. The students gather
around learning problems and study how they think and make conscious efforts to
learn to think more effectively. (as quoted in Joyce and Weil 2005). Thus
we can achieve aim of education as per
NCF-2005, which is to learn how to learn and process of construction of
knowledge through such problem based method.
The methodology is discussed in
detail with auxiliary problems by Pradhan (2009a). Results of our course were
encouraging (Pradhan 2009b). Teachers have here responsibility to chose
appropriate auxiliary problems, counter questions, etc… as per their strategy.
We are presenting our course based on problems as a series of articles starting
with Mechanics-motion, with solution.
Mechanics
Learning Objectives:
1.
Understand motion without worrying about origin of motion,
especially force.
2.
Motion under constant acceleration, in one and two dimensions.
Motion is always relative to an observer and hence how does state of an
observer changes description as we switch from one observer to another. In two
dimensions there is a special class, which is circular motion.
3.
Understanding motion: linear and rotational (or equilibrium)
in the light of Newton’s Laws of motion.
4.
Understanding some of the interactions in the light of
conservation principles (like momentum, energy etc…). For example, the latest
experiment in LHC (large hadron collider) also has these principles involved of
course in much detail beyond the scope of this course. We have tried
incorporating at a basic physics level.
5.
To become familiar with mathematical structure of dealing
with what is covered in above-mentioned points.
1. Kinematics
in 1-D
An elevator ascends with an upward acceleration of 1.2
m/s2 . At the instant its upward speed is 2.4 m/s, a loose bolt
drops from the ceiling of the elevator 2.75m from the floor. Calculate
(a)
The time of flight of the bolt from the ceiling to the floor of the
elevator.
(b)
The displacement and the distance covered by the bolt during the free
fall relative to the elevator shaft. (Irodov 1988)
Tasks involved in this problem are:
1.
To identify the reference
frame.
In this
case students can work with either of two different frames: (1) elevator
and (2) ground based (what problem
specifies as elevator shaft).
2.
To specify value of velocity,
acceleration and displacement using proper sign convention in each frame.
3.
To recognize that time is same (Galilean
invariant) in both the reference frames.
4.
To be able to understand the
difference between distance travelled and displacement.
This problem
gives a thorough picture of use of Kinematical equations that are to be
used for motion with constant accelerations.
The
following problem can be used as an auxiliary problem to illustrate use of
kinematical equations, sign convention and unit conversion.
A car
moving on a straight highway with speed of 126 kmh-1 is bought to a
stop within a distance of 200 m. What is
the retardation of the car (assumed uniform), and how long does it take for the
car to stop? [NCERT 2006b]
2. Projectile Motion
A ball starts
falling with zero initial velocity on a smooth inclined plane forming an angle a with
the horizontal. Having fallen the distance h, the ball rebounds elastically off
the inclined plane. At what distance from the impact point will the ball
rebound second time? (Irodov 1988)
Tasks involved in this
problem are:
- To use energy conservation principle to find speed at the impact.
- To use momentum conservation principle and geometry to find direction of motion after impact.
- To understand meaning of elastic collision.
- To either break motion into two 1-D motion or treat it as a simple projectile motion problem.
- To make a decision about point of impact so that it can be incorporated into equations of projectile.
Smaller problems needed to do this problem need students to understand
either how equations of projectile are used or how motion can be broken into
two different 1-D motions.
The following auxiliary problem can be used
to make students familiar with the use of equations that can be obtain for
motion of a projectile.
A soccer player kicks a ball at an angle of 37o from the
horizontal with an initial speed of 20 m/sec. Assuming that the ball moves in a
vertical plane,
a)
Find time (time of ascent) at which the ball reaches
the highest point in its trajectory.
b)
How high does the ball go (maximum height reached)?
c)
At what instant the ball hits the ground (time of
flight)?
d)
What is the horizontal range of the ball ?
e)
What is the velocity of the ball as it strikes the
ground? [Take g = 10 m/s2]
3. Relative
Velocity
A motor boat,
with its engine on in a running river and blown over by a horizontal wind is
observed to travel at 20 km/hr in a direction 53o East to North. The
velocity of the boat with its engine on in still water and blown over by the
horizontal wind is 4 km/hr Eastward and the velocity of the boat with its
engine on over the running river, in the absence of wind is 8 km/hr due South.
Find
a.
The velocity of the boat in magnitude and direction,
over still water in the absence of wind
b.
The velocity of the wind in magnitude and direction.
[Source: Unknown]
The following problem can be used as an auxiliary problem
to illustrate use of Cartesian vectors to represent velocity in two dimensions
and find relative velocity.
A girl riding a bicycle with a speed of 5 m/s towards north
direction, observes rain falling vertically down. If she increases her speed to
10 m/s, rain appears to meet her at 45o to the vertical. What is the
speed of the rain? In what direction does rain fall as observed by a ground
based observer? [NCERT 2009]
Tasks involved in this problem are:
1.
To find relative velocity and
2.
Rectangular resolution of vectors.
This
problem was chosen as it involves basic idea of relative velocity and technique
of using Cartesian representation of vectors.
Solutions to Touchstone Problems:
1.
Kinematics in 1-D:
Inside the elevator
:
Initial speed u = 0,
Distance through which bolt
falls h =2.75 m
Acceleration of the bolt towards
the floor = acceleration of the
elevator
+ acceleration due to gravity = a
+ g
=
11.0 m/s2
h = ut + ½ (a + g) t2 gives time of flight to the elevator floor
= 
= 0.707 sec
= 0.707 sec
Outside the elevator :
u = 2.4 m/s and a = g
. Time of flight being Galilean invariant remains same.
hence bolt’s displacement in time
t is s = ut + ½ gt2 = - 0.753
m (negative sign indicates downward displacement).
As seen from out side, bolt moves up and comes down (due to initial
upward speed) crossing the starting point till it hits the floor. This upward
motion upto the point when its speed becomes zero. Thus v2 = u2 +
2as gives H = 0.294 m.
Thus total distance travelled = 2H
+ s = 1.34 m
2.
Projectile Motion:
Conservation of energy gives speed of impact u = 
.
.
Since plane is inclined at an angle a to the horizontal and collision is perfectly
elastic, the ball will be like a projectile launched at speed u and at an angle
90 - 2a
to the horizontal. Taking point of impact to be origin, trajectory of the ball
is given by 
, where x is
horizontal distance and y is vertical distance. It hits the plane whose
equation in the plane of the trajectory can be taken as y = - x tana The point of intersection (second impact if
is at a distance L from first impact) on
the plane (Lcosa , - Lsina)
substituted in equation of trajectory yields L = 8h sina
.
, where x is
horizontal distance and y is vertical distance. It hits the plane whose
equation in the plane of the trajectory can be taken as y = - x tana The point of intersection (second impact if
is at a distance L from first impact) on
the plane (Lcosa , - Lsina)
substituted in equation of trajectory yields L = 8h sina
.
3. Relative
Velocity:
Taking west to east as x-direction, south to north as y-direction
and taking vb as boat speed w.r.t. still water, vr
river speed and vw as wind speed, we have
vb + vr
+ vw = 20 cos 37 i + 20 sin 37 j
vb + vw = 4 I
vb + vr
= - 8 j
gives vb = -12 i –20 j :
23.33 km/hr at 59o2’ south of west
and vw = 16 i + 20 j : 25.61 km/hr at 51o21’
north of east.
References:
1.
Baden Maggi Savin: ‘Problem-based Learning in
Higher Education: Untold Stories’, The Society for Research into Higher
Education & Open University Press (2000)
2.
Benjamin S. Bloom, Ed., Max D. Engelhart, Edward J.
Furst, Walker H. Hill, David R. Krathwohl,
‘Taxonomy of Educational Objectives, Vol. I, Longman Inc. (1980)
3.
Halliday, Resnick and Walker, Fundamentals of Physics
by 6th Ed., John Wiley & Sons (2005)
4.
Irodov I. E., ‘Problems in General Physics’,
MIR Publication (1988)
5.
Bruce Joyce and Marsha Weil in Models of Teaching,
PHI (2005)
6.
Atul Mody., ‘On New System of Grading for Students’
Learning of Physics’, Epistme-4, HBCSE, Mumbai (2011)
7.
National Curriculum Framework – 2005, NCERT, New Delhi
(2005).
8.
National
Focus group on Examination Reforms : A Position Paper; NCERT, New Delhi (2006a)
9.
Physics
Textbook for class XI part-I, NCERT New Delhi (2006b)
10. Physics Exemplar problems class XI, NCERT New
Delhi (2009)
11. Pradhan
H.C. & Mody A. K., ‘Constructivism
applied to physics teaching for capacity
building of undergraduate students’, University News, 47 (21) 4-10, (2009a)
12. Pradhan
H.C. & Mody A. K., ‘Supplementary Programme for Capacity Building
of Physics Undergraduate Students’,
Physics Education, 26 (2) 93-98, (2009b)
13. Edward F.
Redish, “Implications of cognitive studies for teaching Physics,” Am. J.
Phys. 62 (9), 796 - 803 (1994)
14. Alan H.
Schoenfeld : Mathematical Problem Solving, Academic Press INC (1985)
15. Young and
Freedman , “Sears and Zeemansky’s University Physics,” 11th Ed.,
Pearson Education (2004)
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