The Physics of
Learning: Do Students Obey the Laws of Physics?
David LaBrecque 3/12/98 Send
questions or comments to: DavidLab@maine.edu
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Artificial Intelligence
Extending a physics
analogy to teaching and learning creates a rich set of terms and relationships
that are already familiar to scientists. This paper defines those terms,
suggests methods for quantifying those terms and thus provides teachers with a
method for optimizing student learning. The author’s role in developing a 1000
student per year, computer networked chemistry lab provides a unique opportunity
to analyze these terms and relationships.
As teachers and learners,
we use terms applied to our physical world to help us explain and discuss our
teaching and learning experiences. For example: "Students should pass in
their work on time. Teachers develop optimum
learning paths. I ran out of energy studying last
night. We had a heated discussion. I’m going to force
myself to learn this." Can we learn more about teaching and learning by
carrying this analogy further? An interesting exercise is to utilize the rich
resource of terms and mathematical relationships in physics. Extending the
analogy this way may help science teachers define pedagogical parameters and
more fully understand the relationships between those parameters.
The knowledge that a scientist
or a student has is a very complex network of facts and relationships. Still,
presenting the idea of learning space, the subject matter teachers want to
teach, in a very simple way will help us define terms and relationships in the
physics analogy. Later on we can use these terms and relationships to help us
more fully understand learning space. See addendum at end of paper.
We begin by mapping out
learning paths on a multi-dimensional grid. For example the top of the hill
shown below could represent a general concept like Newton's second law: F=ma.
Height above the grid is a measure of complexity. Concepts like F=ma are
complex because there are a vast network of relationships and facts needed to
derive the concept and to apply it. A learning path involves following a path
made up of certain facts and relationships. The learner actually updates their
understanding, their own network of facts and relationships, as they learn the
path. Such learning, a modification or enhancement of one's own network of facts
and relationships, requires the learner to exert some effort and time. It
requires energy.
The topography of learning
space is composed of a network of meaningful learning paths. Complete
understanding of a general concept involves being familiar with all of the
paths that make up the hill. Incomplete understanding may involve following
only a few paths or perhaps following a shortened path to the summit by
starting further up the hill and just accepting as fact the supporting
relationships that appear lower down the hill.
Science teachers often
complain that their students either easily forget or can’t apply the general
concepts they have "learned" in class. Unlike the teachers, students
have not established a network of learning paths that would help them remember
and apply those general concepts.
Students must force
themselves to learn. The amount of force depends on how steep the learning path
is. The amount of energy required is a function of the force and the distance.
Students may waste a lot
more energy following a learning path than is needed. They may travel in
circles or they may encounter friction. We’ll define a high friction path as
one filled with distractions and small meaningless exercises. Movement on a
high friction learning path generates heat, which we can define as frustration,
dissatisfaction or anger. As in the real world, there is always some friction
along learning paths. This friction decreases significantly each time a student
repeats a learning path.
A closer examination of
learning space provides insight into the value of inductive learning versus
deductive learning. Inductive learning is where the learner does not know what
or where the desired knowledge is. The teacher provides them with numerous
facts along the base of the hill and hopes they can induce their way to the top
of the hill. The learner does not know what concept is at the top of the hill.
Deductive learning involves showing the learner one or more paths to the peak.
The learner is then asked to follow the same paths down the hill with homework
problems. Ultimately the learner may be asked to solve a problem that requires
them to blaze their own path up or down the hill. As the student learns, the
paths become smoother with less friction and the student requires less energy
to follow them.
Other physics parameters
can be used to define a student's abilities to travel in learning space. For
example, students bring a certain amount of potential energy to a class.
A high power student can concentrate and focus on a subject well.
Mass is inversely related to a persons
motivation to learn. It is a person’s personal philosophy about learning. A
closed mind, a person who does not believe there is anything more to learn or
that learning has no value, is a very massive object. Students with deep-rooted
misconceptions are high-mass objects. A huge amount of force is required
to force the student along a learning path. The desire to learn, the inverse of
mass, may come from various sources. It may be instinctive or it may be an
altruistic desire or it may be seen as just a means to get a better grade or
job.
Extending this analogy can
help science teachers label, define and analyze what good experienced teachers
already know about teaching and learning: Teachers should attempt to guide
students along numerous learning paths. Teachers should utilize the energy a
student brings to class. Encourage students to bring more energy to class.
Don’t allow students to waste their energy. Observe and maintain learning
momentum. Students naturally follow the paths of lowest energy. The teacher
must ensure those paths do involve learning the material. Each student has a
certain amount of learning power, a measure of the students ability to focus
and concentrate. Each student has a unique desire to learn. Teachers should
encourage this desire.
Quantifying Terms and
Optimizing Student Learning
Specific mathematical
relationships also arise from the physics analogy. These common sense
relationships can be used to create a mathematical model of a classroom. If we
can quantify certain parameters as a student’s concentration power, a student's
"mass" and the time a student spends on a learning path we may be
able to calculate the energy required to follow the learning path from E = ò P · dt. If we can quantify a student’s desire to learn then we
could map out the topography of learning space, i.e., the material in a course,
from E = mgh. The following table summarizes terms and relationships that may
be useful.
Physics Term Equivalent Educational Term Symbol, Relationship
Path Learning Path r
Space Knowledge = f(x,y),
Learning Space x,y, or more
dimensions
Height Knowledge Complexity h
Force Learning
effort F
= dE / ds
Work Work W
= ò F · ds
Power Ability to
concentrate, focus P =
dE / dt
Energy Energy E
= ò P · dt
Potential Well E(r) F = dE(r) / ds
Velocity Learning rate v
Mass Inverse of desire to learn. m
Momentum Learning momentum p = m v
Friction Distractions, roughness
in learning path k
Heat Frustrations,
Dissatisfaction, Anger Q
Gravity Desire to not work g
Potential Energy Available amount of energy P.E. = mgh
Kinetic Energy Current
energy being used K.E.
= p /2m
By examining data from a
large group of students, we can generate a topographical map of learning space
for a given topic or course. We can evaluate paths quantitatively and thus
select the best paths for each individual.
Quantifying Terms
Quantifying a student’s
potential energy, concentration power and mass may be accomplished with a
pre-class self-evaluation form as shown below. Another approach would be to let
a student proceed with an endless exercise until they exhaust themselves. The
time they spend on the exercise and the progress they make is related to their
energy and concentration power.
Example
Questionnaire
Estimate the amount of
energy you bring to this class as compared to other classes you’ve taken on a
scale of 1-5. 5 is the highest energy.
Estimate the amount of
energy you bring to this class as compared to other students in this class.
Estimate the total amount
of time in hours per week you expect to spend on this including class time.
Estimate how highly motivated
you are to learn this material.
You have no calculator.
Calculate the square root of 2 to as many places as possible until you want to
leave. Write down your answer and the amount of time you spent on that answer.
Make sure your answer is correct.
We now have a means to
evaluate the learning paths we create for students. Are the learning paths too
steep? Are the steps in the learning path too broad? Has the student followed
enough progressively steeper paths to a general concept to effectively follow
new deduction paths on their own? A series of exercises can be generated to
test how much the student has learned:
1. Exact repetition of
learning path followed in class.
2. Replacement of numerical
values in this learning path.
3. Redefine one or more variables
in the learning path.
4. Unique application of
general concept.
Collecting this data,
mapping out learning space and tracking student energy levels and momentum is
greatly simplified with a networked computer system. Students can perform
computerized exercises that are designed to collect and track this information.
The network can time students. Fortunately such a network already exists and is
being used to teach over 1000 students per year general chemistry.
Addendum: Learning Space
and Networks
If we think of a person’s
knowledge and skills as somehow imprinted on a vast network of neurons in the
brain, then we can realize that this network has certain features. The most
general knowledge and skills or general concepts must have more connections.
For example, Newton’s second law, F=ma, might have a pyramidal or mountain like
shape of network connections.
The mountain continues down
with network connections to the person’s more basic background experiences.
The complexity of a concept
depends on the number of network connections it has or the height of the
mountain. These connections give rise to a unique topography for a given
concept.
When we talk of teaching
and learning, we are talking about the transfer of the teacher’s (or textbook author’s)
network of knowledge to the student’s network of knowledge. We can think of the
teacher’s knowledge as the learning space the student must traverse in a
course. We can think of learning as a change of network states, from the
student’s current state to the desired learning space state.
Implanting a network of
knowledge, a change of states, in the student requires energy. It also depends
on the student’s background of knowledge.
State changes also depend
on the student’s ability to efficiently think of new states and test them to
see if they match with the desired state in learning space. This ability
depends on student energy, momentum, concentration and mass. These are
quantifiable parameters.
Momentum: How fast a
student creates new states and compares them to the desired learning space
state.
Energy: This is a function
of the student’s momentum and the time the student can work.
Concentration: How many
network connections the student can work with at a given moment.
.