Research Matters - to the Science
Teaching Conceptual Understanding to Promote
Students' Ability to do Transfer Problems
by William C. Robertson
Consider the following steps in a basic algebra problem:
Solve for x:
x + 3 = 5
x = 5 - 3
x = 2
Now suppose two different students have learned how to solve
problems such as the one above and they now encounter a new
Solve for x:
4x = 16
Let us further suppose that our students have never seen a problem
involving multiplication of algebraic variables. The students think
out loud as they try to solve this problem. Here is what they might
Student 1: "I want to solve for x, so I
need to get x by itself. In order to maintain the equality as before,
I must do the same thing to both sides of the equation. I can isolate
x if I multiply 4x by 1/4, so that's what I'll do to both sides of
(1/4)(4x) = (1/4)(16)
x = (1/4)(16)
x = 4
Student 2: "I need to get x by itself
again. In the previous problems, I took the number to the other side
and made it negative, so I'll do that again."
x = 16 - 4
x = 12
Which of the two students would you say understands the concepts
associated with solving linear equations? Which student has memorized
a set of procedures or algorithms? If you said that student 1
understands the concepts, you are in agreement with most cognitive
psychologists who study how people solve problems. Student 1 is able
to successfully apply the concepts in a novel situation, which is an
indication that the student understands the concepts. Unfamiliar
problems that require previously-encountered concepts for their
solution are "transfer problems."
Cognitive Structures Associated With
Conceptual understanding is superior to memorized algorithms for
solving transfer problems (Katona, 1940; Mayer, 1974; Mayer, Stiehl
& Greeno, 1975). Conceptual understanding is a worthwhile goal of
science teaching; but what is conceptual understanding and how is it
taught? In this article we shall take a look at the models of human
memory and knowledge (cognitive structures) that are associated with
conceptual understanding and the ability to do transfer problems.
These models will then be used to recommend specific teaching
Studies in complex domains such as solving science problems
(Bromage & Mayer, 1981; Heller & Reif, 1984; Robertson, 1986)
have suggested that conceptual understanding is associated with
connections -- connections between science concepts and everyday life
and connections among the different science concepts in a discipline.
Someone who is good at solving transfer problems does not randomly
connect concepts (which might occur when using memorized algorithms
to solve problems) but rather integrates the concepts into a
well-structured knowledge base. Broad, organizing concepts are
situated at the top of the hierarchy and useful ancillary knowledge
is contained in lower levels, as Figure 1 shows for selected physics
concepts. The concept map illustrated in Figure 1 is not complete,
but it does clearly show the relative importance and appropriate
connections among some physics concepts. For instance, the map shows
that friction and electrical forces are not major problem-solving
principles but merely types of forces that one might consider when
using the principle of Newton's Second Law.
Major principles such as Conservation of Energy are at the top of
the hierarchy, and physical characteristics of systems (e.g. whether
a spring is present) are at the bottom, which lets the student know
the relative importance of these ideas.
What to Do?
For the purpose of solving transfer problems, this well-structured
knowledge base appears to be more important than the utilization of
strategies such as setting goals and subgoals and working backwards
from the goal. These strategies may be helpful, but without utilizing
an accompanying "connectedness" of concepts specific to the
discipline, one cannot be good at solving problems in science and
other complex domains.
If one agrees that conceptual understanding in a discipline is
desirable, then what can a teacher do about it? The following are
appropriate teaching strategies suggested by research:
1) Help your students to see the structure of your discipline. Show
them the "big picture" -- how concepts connect with one another and
with everyday experiences. Concept Mapping (Novak & Gowin, 1984)
shown in Figure 1, is an excellent tool for illustrating how concepts
are related. Explicate the "ancillary knowledge" associated with
formulas and principles -- this is knowledge that students use to
"make sense" of a formula rather than just memorize it.
2) The most important thing you are presenting to students
is the big picture, so allow them to concentrate on the big picture
by making sure that they can use certain skills almost automatically.
For example, one is not free to acquire a conceptual understanding of
an equation such as F = ma if the use of an algebraic symbols in an
equation is not second nature. Similarly, one cannot begin to
concentrate on the meaning of words if one doesn't know the alphabet
well. This is not to say that skills should be memorized; you should
teach them in a meaningful way, just as you should teach higher-level
concepts in a meaningful way. However, students should then practice
the skills until they no longer present a hindrance to concentrating
on more important matters.
3) Install in your students the desire to make sense of the subject
matter. Encourage them to look for the connections among concepts and
to structure these concepts in a hierarchy. Encourage your students
to be dissatisfied with explanations that are to be memorized rather
than understood. Although a well structured knowledge base in one
discipline will not transfer to another discipline, perhaps the
ability and desire to look for the appropriate conceptual structure
in the new discipline is transferable.
4) Test your students for their ability to solve transfer problems.
Testing students on problems that are exactly like ones they have
done in their homework is a sure way to promote memorization of
problem types. If, however, students know that the test problems will
be unfamiliar, they are more likely to try to acquire the conceptual
understanding necessary to do them. If you are able to help your
students truly understand concepts, perhaps at some point during the
year they will stop referring to the transfer problems as "trick
5) Finally, allow your students the time necessary to acquire
conceptual understanding. People need time to establish connections
and see how concepts fit together. Reduce the number of topics you
cover in your science course. Teach fewer (important) concepts in
greater depth. Allow more time in laboratory explorations that are
meaningful rather than an exercise in following recipes. It is better
for your students to understand a limited number of science concepts
than to memorize many concepts that they are unable to apply in novel
William C. Robertson is a staff associate
at the Biological Sciences Curriculum Study. He is a member of NARST, an
organization dedicated to improving science teaching through research.
Bromage, B. K., & Mayer, R. E. (1981). Relationship between
what is remembered and creative problem solving performance in
science learning. Journal of Educational Psychology,
Chi, M. T. H., Feltovich, P., & Glaser, R. (1981).
Categorization and representation of physics problems by experts and
novices. Cognitive Science, 5, 121-152.
Heller, J. I. , & Reif, F. (1984). Prescribing effective human
problem-solving processes: Problem description in physics.
Cognition and Instruction, I, 177-216.
Katona, G. (1940). Organizing and memorizing.
New York: Columbia University Press.
Mayer, R. E. (1974). Acquisition processes and resilience under
varying testing conditions for structurally different problem solving
procedures. Journal of Educational Psychology,
Mayer, R. E., Stiehl, C. C., & Greeno, J. G. (1975).
Acquisition of understanding and skill in relation to subjects'
preparation and meaningfulness of instruction. Journal of
Educational Psychology, 67, 331-350.
Novak, J. D., & Gowin, D. B. (1984). Learning how to
learn. New York: Cambridge University Press.
Robertson, W. C. (1986). Measurement of conceptual understanding
in physics: Predicting performance on transfer problems involving
Newton's second law. Doctoral dissertation, University of
Simon, D. P., & Simon, H. A. (1978). Individual differences in
solving physics problems. In Siegler, R. (Ed.), Children's
thinking: What develops? Hillsdale, NJ:
Research Matters - to the Science
is a publication of NARST