Learning Aids for
study Kinematics in Algebra and Calculus based Physics
Valentin Voroshilov, Ph.D.
Physics Department,
617-918-3656
valbu@bu.edu
The main tool Physics teachers use to measure students’ understanding of Physics is Physics problems1. At the same time the hardest work a Physics teacher experiences while teaching Physic is teaching how to solve Physics problems.
There is a major contradiction a Physics teacher has to overcome: the number of problems students have to study how to solve is far more the number of problems the solutions of which a teacher can demonstrate during the course. One of the possible ways to solve this contradiction is to find the difficulties which an average student has to overcome when solving problems and which are common for any Physics problem and to give students a hand with learning the general ways of overcoming those difficulties. In other words on the basis of teaching how to solve a specific Physics problem a teacher may teach students to the general way of problem solving in Physics.
This approach was formulated first in Mathematics Education by Gorge Polya in his outstanding book “How to Solve it” in 1945.2 Even today the book is as exiting and useful as sixty years ago. Many of the ideas and technical steps from Polya’s book can be used in teaching Physics; however, problem solving in Physics has some aspects which have to be covered in a specific way3.
The most common recommendations a student meets in a
textbook or gets from a teacher usually are “draw a picture as detailed as
possible” and “follow the given algorithm” (see, for example, 4 and
the bibliography given in it).
In this article I am presenting six learning aids
which I have been using for many years of teaching an algebra/calculus based
Physics to high school and college students. Presented
learning aids are aiming to help students to become a problem solver in
Kinematics.
Because the first serious acknowledgement of Physics begins with Kinematics it is very important to get students the impression that study Physics is not just for celebrities; that they are capable of study Physics and that study Physics is not just an intuitive process but there is logic behind this process and everybody can acquire this logic. When during the studying of Kinematics students realize a specificity of a physical thinking this gives a positive effect for a speed of further studying and the depth of understanding of all the subsequent sections of Physics.
All the learning aids help, students having difficulties
with creating the solution of a problem, to find a lead in the right direction
on the way to the solution. There is no ideal way for using the aids because of
many differences in students’ backgrounds and behavioral habits, but a teacher
usually can use an aid when seeing a student got stuck and cannot figure out
the step should be performed to move the solution ahead. These aids are
especially effective when a student is just starting to look for the solution
of a given problem and the one does not know what the first step can be done.
In the situation like this a teacher can use a learning aid as the tool for
initiating the student’s work (“You could try this aid” I tell usually to the
student, or ask “did you try that aid?).
Let us now describe the learning aids which can be used when teaching Kinematics.
One of the difficulties students meet on the way of
the solving a problem is misunderstanding the problem (see, for example4,5). The common difference between a student and a
physicist is the student does not know how to interpret the text of the problem
in terms of Physics; the student cannot translate the problem from an everyday
life language into the language of Physics.
In this situation as a learning aid a teacher can use a
Dictionary connecting the everyday lexicon and Physics terminology.
Let us consider the following problem as an example.
An airplane
needs to get the speed of 100 m/s for the takeoff. The engines produce total
acceleration of 8.33 m/s2. Find the time it takes for the plain to
reach the speed.
When a Physicist
read this problem the one does translate it immediately and intuitively
into the following (or similar) text.
A body moves
from rest with a given constant acceleration and at unknown instant of time has
a given speed.
For a Physicist an airplane, a rocket, a stone, a rock, an arrow, a car, a particle, the Sun, a train, etc. are just a body or an object. And when a Physicist reads such words as “flying, dropping, starting, shooting, thrown, driving” etc the one reads actually just “moving”. The word “rest” translates as “zero velocity”. The words “starting” or “stopping” translates as “velocity is changing”. Hence, one of the first learning aids a teacher can give out to students is the Dictionary Physicists use for the translation. It can look like this, for example (see the Table 1).
Table 1
|
An empirical term (everyday
word) |
A theoretical term, category |
Physical quantities describing the category (and the common notations) |
|
A car, a stone, an arrow, … |
A body, An
abject |
A mass (m),
coordinates (x, y, z), a volume (V), etc |
|
Goes, drops, rolling, pulling, flies, … |
Moving, At a motion |
Displacement
(S), distance (L), velocity (v), acceleration (a), time taken for the motion
(t), etc |
|
Getting at rest, moving from rest, making a turn, … |
Changing the velocity, Accelerating |
Displacement
(S), distance (L), average velocity (vav),
initial velocity (vi), final/terminal velocity (vf), , time taken for the motion (t),
acceleration (a), etc |
|
Lies, hangs, sits, … |
At rest |
The speed is 0,
v = 0 |
The first column lists the everyday
words students used to use when describing the events around and can meet while
reading the text of a problem; the second column gives the translation from an
everyday language into Physics language; and the third column lists the usual
physical quantities which can be used in order to describe the corresponded
object or process.
The
other widespread obstacle on the way of creating the solution of a problem is an
inability to identify the physical model which has to be used in order to
describe the physical situation a student is having in the problem4,5.
There
are several learning aids we can offer to students in order to help them to
identify the necessary physical model.
I find it
is very helpful to give in the beginning of the course the brief description of
the Physics as a whole6 and to show that the problem solving process
starts from locating the part of Physics containing the description of
phenomena happening in the situation given in the problem.
Even
when students just start to study Classical Mechanics they have some ideas
about electricity and magnetism and optics from everyday life and a teacher can
use it. A teacher can summarise this information in a table showing the context
of main possible situations students can meet when study Physics (see Table 2).
When discussing a problem a teacher can ask what part of Physics for the Table 2
should be used to describe the situation given in the problem.
Table 2
|
Indicators
of a situation |
Section
of physics (phenomena studied) |
|
Objects
change positions |
KINEMATICS
(describing of motion) |
|
Objects act against each other, have an obviously observed influence on each other (a body in a liquid; moving
springs; two surfaces at a contact; one body presses or pulls the other; two
bodies are attracting or repelling the each other) |
DYNAMICS
(forces between objects) |
|
An oscillating body (on a spring, on a thread,
about a pivot point) |
OSCILLATIONS
(moving periodically back and force) |
|
The motion of many molecules has to be considered |
KINETIC
THEORY OF MATTER |
|
Processes on a gas (usually change in volume,
pressure or temperature has to be considered) |
THE GAS LAWS
|
|
Bodies are heated or cooled up and it is important
that their internal energy varies |
THERMODYNAMICS |
|
Charged objects (without a motion) |
ELECTROSTATICS |
|
Moving charged particles (usually along with the
consideration of wires, EMF or generators) |
DIRECT CURRENT or ALTERNATE CURRENT |
|
Wires with a current (linear or in loops) and/or a
number of magnetic arrows |
MAGNETISM |
|
Light is transferring or reflecting or refracting
(there are bulbs, mirrors, prisms etc.) |
OPTICS |
|
Very fast moving objects, processes with atoms and
nucleuses, photons and other unusual words |
NONCLASSICAL
PHYSICS |
Another learning aid is
helping to identify (i.e. to recognize) the physical model matching to the problem.
The idea of the aid is that within the specific subsection of Physics we can
analyse the specific parameters to find the appropriate physical model.
For example, in
Kinematics, in many cases we can analyze the value of two main parameters:
1. The form of a
trajectory;
2. The behaviour of a
speed.
In the beginning of the
study Kinematics in “99 cases from 100” we deal with the following values of these parameters:
The form of a trajectory –
a) A STRAIGHT LINE; b) A CIRCLE.
The behaviour of a speed –
a) DOES NOT VARY
(constant); b) VARIES (changing).
Corresponding to the
values of the parameters, four main kinematics models can be found (see the Table
3):
Table 3
|
The
behaviour of a
speed |
A STRAIGHT LINE |
|
|
DOES NOT VARY |
A linier motion with a constant speed |
A uniform
circular motion |
VARIES
|
A linier motion with a constant acceleration
(remember, it is not exact definition, but for 99 % of problems it is true,
and it is always worth to check it out) |
A uniformly accelerated circular motion (true for
almost 99 % of a typical problems) |
Of
course, we cannot use this table to solve any
problem in Kinematics, but we don’t need it! We use this table just to get
students a main idea of how to identify a
physical model.
After the correct
identification of the model we can make next steps, which are fixing the
physical quantities and choosing the correct equations we need to use to
describe the physical situation we have met in the problem.
To do that two following
learning aids can be useful (see the Tables 4 and 5).
The table of the
correspondents between a kinematical model and the usual physical quantities
needed to give the quantitative description of the model (the Table 4).
Table 4
|
MODEL |
MAIN PHYSICAL QUANTITIES |
|
A linier motion with a constant speed |
Displacement
(initial point, final point), distance, trajectory, velocity, speed, time
taken |
|
A linier motion with a constant acceleration |
Displacement (initial point, final point),
distance, trajectory, time taken, initial velocity, final/terminal velocity,
(initial instant, final instant), acceleration. |
|
A uniform circular motion |
Displacement (initial point, final point),
distance, velocity, time, angle, angular displacement, amount of revolutions,
frequency, angular velocity, period, centripetal acceleration, the radius of
the circle. |
|
A uniformly accelerated circular motion |
Displacement (initial point, final point),
distance, velocity, time, angle, angular displacement, angular velocity, angular
acceleration, centripetal acceleration, tangential acceleration, the radius
of the circle. |
|
The mixed model |
Concepts of parent models; interval of motion,
average velocity, average speed; average acceleration. |
The table of the correspondence between a kinematical model and formulae needed to give the quantitative description of the model (Table 5).
Table
5
|
Model |
Formulas |
|
A linier motion with a constant speed |
v = s/t; s = x – xo, a
= 0 |
|
A linier motion with a constant acceleration |
v = vo
+ at; s = x – xo s = vot
+ at2/2 |
|
A uniform circular motion |
w = j/t; wT = 2p; n = N/t;
v = wR n = 1/T; ac = v2/R; j = s/R |
|
A uniformly accelerated circular motion |
j = s/R; w = wo + εt; j = wot + ε t2/2 v = wR; ac = v2/R; at = ε R |
One
of the most effective learning aids that gives students the understanding of
how does a physicist work while working on a problem is a System of Operationally
Connected Categories (SOCC).
We
can link SOCC with a knowledge system, a schemata, a mental model or concept
map3,7. The main idea for this aid is that a
real knowledge is always a network of concepts; it is a complex of
interconnected terms and senses. Moreover, connections in fact make a knowledge
being knowledge. We know in Physics;
even the smallest interaction can drastically change the property of a system. Same
is true for a system of knowledge. Without connections all the words in our
memory are just the names for some objects we can point at: “The desk”, “The chair”.
Without connections a generalisation does not exist; there is no “Furniture”.
Connections make the combination of words being knowledgeable.
In
Physics all our work actually is looking for connections, even on such a simple
level as solving high school or undergraduate physics problems.
To build an example of
SOCC first we have to choose the most important terms, categories, names of
physical quantities within a specific part of the subject. While doing that, we
have to make sure that each of the terms has a direct logical connection with
at least one of the others. In Physics any direct connection usually means
there is a formula (at least one) which both quantities go into (a direct
logical connection can be also a verbal statement; for example, a definition of
straight line connects the terms “an object”, “dimension”, “infinitely”).
Now we can make a simple
graph.
Each important concept/term/category/name we can indicate by a circle with a number (or name) inside. For example, the circle with number 33 inside it -
- indicates the
term «an angular speed» on my SOCC.
Each important direct logical connection between two
terms can be indicated by a line that connects them.
The statement or the formula must include BOTH the terms linked by the corresponded line.
For example, the line
connecting circles 33 and 30,
represents the formula, relating a radius of a circle (R,
number 30) with an angular speed (w, number 33).
After finishing our work
we have the set of vertexes connected by the set of lines (see Fig. 1).
Fig. 1
Each vertex answers a
question “What kind of physical quantity can be used to describe the situation within
the given part of a subject”.
Each line answers question
“Is there a direct connection between two given physical quantities”; or we can
say in other words “Does the value of this quantity affect the value of that
one”?
In addition, we can give students an actual
formulation of each link. For example, the line 33 – 30 represents the formula v
= wR (the same formula
is represented also by the lines 33 – 32 and 30 – 32, where the vertex 32
represents the linear speed of a body under a circular motion).
The schematics above (Fig.
1) visualises the main quantities/terms, and relations/connections/dependencies
between physical quantities within the given part of Physics.
When necessary, we can
provide a brief explanation of main vertexes and links. For example:
1. An object is a body (a
car, a spacecraft, a stone etc.).
2. A
point mass is a body, which sizes are not essential for the given problem.
Usually it is right for small objects, particles, which are placed in the
vicinity of large objects (a stone falls down to the surface of Earth; a train
goes from one city to another etc.).
5. Circular motion is a
motion when all points of a body lie on circles, which centres belong to one
line (axis), and the planes of the circles are mutually parallel.
7. Distance is the length
of a trajectory. To find a distance it is necessary to put a thread along a
trajectory, then to stretch this thread into a strait line and to apply it to a
ruler and read the number.
8. Displacement is a
vector connecting an initial and final position of a body.
…
28. The displacement of
the body undergoing a linear motion with a constant acceleration is s = vot
+ ½at2 The given formula is represented by the lines 16 - 28
and 23 - 28.
…
33. An angular speed is a
physical quantity describing the rate of changing of angular displacement
during the time. The angular speed is defined by the formula w = j/t, which is represented by lines 16 - 33 and 31 -
33. The angular speed also is connected to the speed by the formula v = w R, which is represented by a line 19 - 33.
There
is no need to show here the full list of the quantities and equations because it
is the matter of a teacher’s choice; there is no ideal graph; the structure
depends on teacher’s preferences.
The
small version of the SOCC can be drown quickly for any given problem a teacher
is going to explain to students or the teacher can ask student to draw the SOCC
as a specific assignment helping to understand connections between quantities.
The visualisation of logical connections between quantities helps very much to
get students the understanding of the general way of how our brain works while
working on the solution of a problem. But more important, it gives students the
understanding that any problem solving is always just a finding the necessary
connections between necessary quantities.
Let us emphasise that a teacher doesn’t have to use every aid every time
when students are having troubles with solving a problem. There is no fixed
rule when it is appropriate to use an aid and when it isn’t. Usually, when we see
a student got stuck or when a student is asking a question, we can ask first if
he or she tried to use any aid already, or ask “show me the SOCC for this problem”
or “give me the list of the quantities involved in the situation”, etc. Of
course, in a class of 30 people during one hour discussion section it is no
easy to balance a teacher’s work and students’ work, but on the other hand this
is what a teacher for.
I believe
when solving a problem writing down the necessary equations is the final step
of analysis; Physics is done after that, Math is starting. The main cause for
misunderstanding Physics and for disability to solve Physics problems is the lack
of experience of making the analysis
which leads to the necessary equations. This is the focus, the main goal
and the most valuable result of Physics education.
References:
1. Erich Mazur, “Peer Instruction”, Prentice Hall, Inc., 1997).
2. George Polya, “How to Solve It: a new aspect of mathematical method”, (Princeton University Press, Expanded Princeton Science Library Edition, 2004).
3. Mark Vondracek,
“Improving Student Comprehension by Thinking about a Topic in
Valentin Voroshilov, “An Universal Algorithm for Solving
School Problems in Physics”, (in the book “Problems in Applied Mathematics and
Mechanics”, Perm,
Valentin Voroshilov, “Quantitative Indicators
of the Learning Difficulty of Physics Problems”, (in the book “Problems of
Education, Scientific and Technical Development and Economy of Ural Region”,
4.
Donald Scarl, “How to Solve Problems: For Success in Freshman Physics, Engineering and Beyond”, (Dosoris Press, 6th Edition, 2003).
5. David Hestenes, “Modeling Methodology for Physics Teachers”, Proceedings of the International Conference on Undergraduate Physics Education (College Park, August 1996) (located on www.modeling.asu.edu).
6. Richard P. Feynman, “Six Easy Pieces”, Helix Books, 1994).
7. F.K.L. Chit Hlaing (F. K. Lehman), “Cultural models (and Schemata) and Generative Knowledge Domains: How are they related?”, Paper for the panel on Cultural Models and Schema Theory, American Anthropological Association Annual Meeting, October, 2000, San Francisco (located on real.anthropology.ac.uk/AAA2000SF).
Valentin Voroshilov, “Application of the System of Operationally-Interconnect
Categories for Diagnosing the Level of Students' Understanding of Physics”, (in
the book “Artificial Intelligence in Education”, part 1. -