This document describes a study that investigated students' understanding and knowledge structure of electromagnetism after completing a traditional high school course. The study found deficiencies in three areas: 1) Students did not recognize Maxwell's equations as central ideas. 2) Students lacked conceptual understanding of relationships like the connection between electric fields and their sources. 3) Students struggled to apply concepts in problem solving. To address these issues, the researchers proposed a model integrating problem solving, conceptual learning, and building a hierarchical concept map with Maxwell's equations at the top. An experiment found this approach helped students more than isolated drilling or conceptual instruction alone.

1. From problem solving to a knowledge structure: An example
from the domain of electromagnetism
Esther Bagno and Bat-Sheva Eylon
Department of Science Teaching, The Weizmann Institute of Science, Rehovot, 76100, Israel
͑Received 6 September 1995; accepted 29 January 1997͒
An investigation of students’ knowledge after a traditional advanced high-school course in
electromagnetism shows deficiencies of their knowledge in three major areas: ͑1͒ the structure of
knowledge—e.g., realizing the importance of central ideas, such as Maxwell’s equations ͑expressed
qualitatively͒; ͑2͒ conceptual understanding—e.g., understanding the relationships between the
electric field and its sources; and ͑3͒ application of central relationships in problem solving. To
remedy these deficiencies we propose an instructional model which integrates problem solving,
conceptual understanding and the construction of the knowledge structure. The central activity of
the students is a gradual construction of a hierarchical concept map organized around Maxwell’s
equations as central ideas of the domain. The students construct the map in five stages: ͑1͒
SOLVE—they solve a set of problems that highlight the central ideas in the domain; ͑2͒
REFLECT—they reflect on the conceptual basis of their solutions; ͑3͒ CONCEPTUALIZE—they
perform activities that deal with relevant conceptual difficulties; ͑4͒ APPLY—they carry out
complex applications; ͑5͒ LINK—they link their activities to the evolving concept map. This
integrative model ͑experimental treatment͒ was compared to an isolated treatment of drill and
practice or treatment of conceptual difficulties without linkage to the proposed knowledge structure.
The comparison shows that students in the experimental treatment performed better than the other
students on measures of recall, conceptual knowledge and problem solving. Students in the
experimental treatment were also able to transfer and extract central ideas in a domain different than
physics. © 1997 American Association of Physics Teachers.
I. INTRODUCTION
The focus of advanced courses of electromagnetism at the
high-school level, or introductory courses at the college
level, is the acquisition and application of abstract concepts
and relationships. Such courses usually involve a mathemati-
cal treatment of central relationships and sophisticated
problem-solving tasks. These characteristics of the subject
matter in electromagnetism present a difficult challenge to
the students. The concepts are removed from students’ expe-
rience, and the mathematical aspects add another obstacle to
those who are unsophisticated mathematically. It is plausible
to assume that students would not form an adequate repre-
sentation of knowledge in this domain and as a result will
have difficulties in effectively using and retaining the knowl-
edge.
Based on previous research1,2
this investigation focused on
three central aspects related to knowledge representation: ͑1͒
the structure of knowledge; ͑2͒ the understanding of central
ideas; and ͑3͒ the relationship between conceptual under-
standing and its application in problem solving.
(1) The structure of knowledge: Research3
has shown that
in studying new information, people extract a hierarchy of
ideas, where the ‘‘important’’ information ͑as grasped by the
learner͒ is placed at the top of the hierarchy. This kind of
organization enables effective retrieval and facilitates
performance.4
As time passes, lower levels of the hierarchy
will be forgotten and people will remember the top level
‘‘important’’ information. A comparison of novice and ex-
pert physicists in different domains of physics consistently
shows that experts’ hierarchical organization of knowledge is
based on the physical principles of the domain while novices
often represent surface features of situations.5
These findings
suggest that it is important for students to know what are the
central ideas in a domain, since these are the ideas that they
are likely to remember and use in the future. In the domain
of electromagnetism these ideas should include some repre-
sentation of Maxwell’s equations and the Lorentz force.
Qualitative reasoning is an important goal at all levels of
science instruction, hence another desired feature of the
knowledge structure in physics is the qualitative representa-
tion of relationships in addition to their mathematical repre-
sentation.
(2) Conceptual understanding: Many studies have docu-
mented students’ conceptual difficulties in the domain
of electromagnetism. For instance, in dealing with electric
circuits students often confuse related concepts such as
current, voltage, energy and power; misinterpret schematic
representations2
and fail to grasp the electric circuit as a
system.6
They don’t relate macro and micro relationships in
electric circuits7
and are unable to link electrostatics and
electrodynamics.8
These conceptual difficulties are espe-
cially pronounced in performing qualitative tasks.
(3) The relationship between conceptual understanding
and problem solving: The fact that students can solve prob-
lems does not indicate that they understand the fundamental
ideas that form the basis for solving these problems. For
example, finding currents and voltages in a specific electric
circuit by using Kirchhoff’s laws does not guarantee under-
standing the concept of voltage. Conversely, understanding
of fundamental ideas does not guarantee success in solving
problems, since mastery of strategies and skills are necessary
as well.
This study investigated whether in the course of conven-
tional instruction of electromagnetism students form an ad-
equate representation of knowledge, in terms of the previous
726 726Am. J. Phys. 65 ͑8͒, August 1997 © 1997 American Association of Physics Teachers

2. three aspects, and whether one can encourage the develop-
ment of such representations by appropriate instructional
means.
The present paper consists of three parts: ͑1͒ The first part
describes a diagnostic study that looked into students’
knowledge representation in the domain of electromagne-
tism. The results of the study highlight some deficiencies in
students’ knowledge. ͑2͒ Following an analysis of the
sources of the observed deficiencies, the second part suggests
a model of learning and instruction to remedy students’
learning difficulties in the domain. The model includes a
proposed knowledge structure and a didactic approach. The
approach treats conceptual knowledge and problem solving
in an integrative manner and relates them to the proposed
knowledge structure which is formed actively by the stu-
dents. ͑3͒ The third part describes an instructional study that
evaluates the efficacy of the integrative model of instruction
and compares it to an approach which treats the learning
difficulties without linking conceptual knowledge and prob-
lem solving to the proposed knowledge structure.
II. STUDENTS’ REPRESENTATION AND
UNDERSTANDING OF KEY RELATIONSHIPS IN
ELECTROMAGNETISM
In the diagnostic study, we investigated some aspects of
students’ knowledge in electromagnetism after they had
completed a standard course on the topic. The study was
conducted among high-school students who majored in phys-
ics. The sample consisted of nine 12th grade classes ͑about
250 students, ages 17–18͒. These students had completed
their course in electricity and magnetism and were preparing
for their physics matriculation examination. The level re-
quired in such classes is roughly that of first year college in
the US or the A-level course in the UK. All classes were in
good high schools and were taught by experienced physics
teachers.
The following sections describe investigations that exam-
ined three questions:
͑1͒ Which ideas do students view as central in electromag-
netism? Are the key relationships, summarized by Max-
well’s equations, included among these main ideas?
͑2͒ In what form do students represent the main ideas? Do
they represent ideas qualitatively or only in mathemati-
cal, symbolic representation?
͑3͒ How well do students understand the key relationships in
electromagnetism? How well do they apply the relation-
ships in solving problems?
A. Students’ view of main ideas in electromagnetism
Table I lists seven key relationships which summarize
qualitatively principles associated with Maxwell’s equations
and the Lorentz force. These key relationships were used as
reference for analyzing students’ answers.
1. The tasks
A written questionnaire was administered to classes of the
sample during one class period ͑of approximately 25 min͒.
The questionnaire included several retrieval tasks represent-
ing retrieval cues that students are likely to use. Each task
defined a different criterion for selecting the information
͑e.g., retrieval by importance, retrieval by label͒. By compar-
ing the answers on the different tasks, we hoped to get a
comprehensive picture of students’ representation.
The tasks were given in the following order:
(a) Free recall: The task was presented as follows: ‘‘Sum-
marize in a few sentences the main ideas of electromagne-
tism according to the order of their importance. Don’t use
formulae!’’ Students’ answers can shed light on the relative
importance they attach to different ideas in the domain. It is
possible to identify omissions of ideas that are considered to
be important by physicists, or overemphasis of unimportant
ideas. Furthermore, this kind of summary can suggest what
information is likely to be remembered by the students after
a long period of time.
(b) Cued recall: The cues were labels, intended to facili-
tate access. The task was phrased in the following manner:
‘‘Next to each of the following concepts, write as many re-
lationships as possible that include the concept: (i) Electric
Field (ii) Magnetic Field.’’ Performance on cued recall is
usually superior to performance on free recall, since the cue
facilitates search in memory.4
Thus the answers on this task
can reveal some additional information that students know
but have difficulty in retrieving.
(c) Contextual recall: The task was phrased as follows:
‘‘Suggest as many ways as possible to produce: (i) an Elec-
tric Field (ii) a Magnetic Field.’’ In this task we attempted to
highlight dynamical aspects of the knowledge by requiring
students to use it more selectively. The task provides a con-
text in which the concepts are used.
2. Results
Figure 1 shows the results for the recall tasks.
(a) Free recall: Less than 45% of the students mentioned
the relationships. Several of the figures are quite striking,
including the following.
Table I. Key relationships of E.M. used to analyze student’s representation
of knowledge.
Equation Key relationship
Symbolic
representation
1. ͶE–dSϭ
⌺q
⑀0
A charged particle produces
an electric field
q→E
2. FϭqE An electric force is exerted
on a charged particle in the
presence of an electric field
E→F(q)
3.
dq
dt
ϭI Moving charges are current q→I
4. ͶB–drϭ␮0⌺I Current produces a magnetic
field
I→B
5. Fϭqv؋B A magnetic force is exerted
on a current in the presence
of a magnetic field
B→F(I)
6. ͶE–drϭϪ
d␾B
dt
A change in a magnetic field
produces an electric field
⌬B
⌬t
→E
7. ͶB–drϭ⑀0␮0
d␾E
dt A change in an electric field
produces a magnetic field
⌬E
⌬t
→B
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3. ͑1͒ A high proportion of students considered Ohm’s law to
be one of the most important ideas of electromagnetism,
consistent with previous findings9
labeled humorously:
‘‘The three principles of electromagnetism: VϭiR; i
ϭ V/R; R ϭ V/i.’’
͑2͒ A comparison of the three recall tasks shows that the
high proportion of students who mentioned the sixth re-
lationship ͑a change in magnetic field produces an elec-
tric field͒ is an artifact. The overestimation resulted from
the counting of the labels ‘‘Lenz’s law’’ or ‘‘induced
emf’’ as evidence for a relationship between the mag-
netic and the electric fields.
͑3͒ The symmetry which exists between the electric and the
magnetic fields is not reflected in students’ summaries.
Less than 5% of the students in the sample mentioned
the production of a magnetic field by a changing electric
field ͑the seventh relationship͒.
(b) Cued Recall: We assumed that if the key relationships
exist in students’ cognitive structure, then provision of key
concepts would facilitate their retrieval. Figure 1 shows that
the recall patterns in the free and cued recall tasks are quite
similar and suggests that the summaries reflect quite accu-
rately the content of stored information, and that the missing
information is not due to retrieval difficulties.
(c) Contextual Recall: We expected that a context in the
form of a specific goal ͑suggest possible ways of producing
magnetic and electric field͒, would facilitate the retrieval of
information relative to the free and cued recall tasks. We
expected four relationships—two ways for producing electric
fields, and two ways for producing magnetic fields.
On the basis of Fig. 1, it can be concluded that the provi-
sion of context aided students in the recall of most relation-
ships. However only 10% of the students in the sample
claimed that a change in magnetic field is accompanied by an
electric field. ͑This finding provides further support for the
suspicion that students do not relate the labels Lenz’s law or
induced emf to the production of an electric field.͒ Less than
10% mentioned production of a magnetic field by a changing
electric field.
These results suggests that these relationships were prob-
ably not internalized and deserve some treatment.
B. Form of representation for key relationships
The form of each statement was categorized into one of
the following:
͑1͒ A qualitative verbal statement about a relationship or a
property of a concept. For example: ‘‘An electric charge
produces an electric field.’’
͑2͒ A verbal translation of a formula. For example: ‘‘Cur-
rent equals charge over time.’’
͑3͒ A mathematical formula. For example: Fϭqv؋B.
͑4͒ A label. For example: ‘‘Gauss’s law,’’ ‘‘electric field.’’
The performance of each student was computed relative to
the total number of statements given by that student.
Table II summarizes the results for each of the categories.
All students followed the instructions and did not write
down equations. However, some of the students adhered to
the instruction by writing down a literal translation of an
equation into words. Although the results suggest that stu-
dents have the skill to phrase central ideas of electromagne-
tism in a verbal form, these results should be considered
cautiously. The fact that students write down a qualitative
statement when explicitly required to do so, does not imply
that they would use this form of representation by them-
selves when appropriate.
Fig. 1. Average percentage of recall for the key relationships ͑see Table I͒ in the recall tasks. For contextual recall the four relevant relationships are included.
Table II. Average performance of the various categories of form in the
diagnostic study (Nϭ250).
Form
Percent out of total
number of statements
͑a͒ qualitative 45%
͑b͒ ‘‘verbal’’ formula 20%
͑c͒ formula 0%
͑d͒ label 18%
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4. C. Conceptual understanding
Based on previous experience and research we designed a
questionnaire with five ideas in electromagnetism that stu-
dents find difficult. The ideas were presented by verbal state-
ments ͑see Table III͒ and students had to indicate whether
the statements were true or false and give a detailed expla-
nation. In addition to analyzing the questionnaires, we inter-
viewed several students to further investigate the sources of
difficulty.
1. The first statement (application of ⌬B/⌬t˜E)
Students have difficulty in determining the direction of the
induced magnetic field. The major source of difficulty has to
do with fuzzy encoding. An examination of the relevant text-
books suggests that sentences like ‘‘The induced current re-
sists its cause’’ are too vague. Students interpret these sen-
tences incorrectly. For example, opposes the change is
interpreted as being in the opposite direction.
2. The second statement (application of q˜E)
Students’ explanations indicate that many consider the
electric field to have a static nature. Namely, the field exists
in space and applies forces on charges, and it does not
change even when a new charged particle enters the region.
The purpose of this statement was to examine whether stu-
dents can apply it in dynamic situations.
Consider for example, the following dialogue between a
teacher (T) and a student (S).
T. Consider the following: ‘‘In a certain region there is an
electric field. If an electric charge is removed from this re-
gion, the electric field will change.’’
S. No, the field will not change.
T. Why?
S. Because the field is a property of every point in the re-
gion as a result of other electric charges in the vicinity. If
you remove a charge it does not matter, because the field is
Kq/r2
.
T. Here is another statement: ‘‘Between the plates of a ca-
pacitor, there is a certain electric field (an illustration was
provided). When an electric charge enters the region be-
tween the plates the electric field will change.’’
S. In this case the field will not change.
T. Why?
S. Because the field is created by the two charged plates, if
you bring into the region between the plates an electric
charge, it will move or be affected by the field, but it will not
change the field. (Here follows a long dialogue about the
creation of an electric field).
S. I am sure that the field would not change. It is the force
that acts on a charge.
T. Is there any way to change this field?
S. Yes, by changing the distance between the plates, or by
changing the potential.
As in the other cases most presentations in textbooks sup-
port this perception of the student, since the electric field, a
difficult and nonintuitive concept, is presented merely as a
force applier. Also, the problems usually deal with static
situations such as: ‘‘Four charges are fixed in the four cor-
ners of a rectangle; find the resultant electric field,’’ and do
not illuminate the dynamic nature of the electric field. Even
in problems dealing with charged particles entering a region
with a constant electric field, students are never asked to find
the new field. They are usually asked about the path of the
particle, its velocity, energy, etc. An attempt to develop a
dynamic conception of electric fields is included in the recent
instructional materials by Chabay and Sherwood.7
Table III. Distribution of the various categories of the conceptual under-
standing task.
Statements
%
incorrect
%
incorrect
in
category
1. As you know, induced current is produced
by a changing magnetic field. This induced
current may produce a magnetic field in the
direction of the magnetic field which produced
it.
72
a. Fuzzy encoding: ‘‘induced current resists its
cause, i.e., is in the opposite direction’’
42
b. Incorrect application of principle: violates
conservation of energy ⇒ should decrease
energy ⇒ must be in the opposite direction.
18
c. Confusion of variables: i and B must be
always perpendicular.
7
d. Over generalization: like law of inertia ⇒
induced field is always in the opposite direction.
7
e. No reason 26
2. A charged particle enters a region with a
constant electric field. The field in this area
changes because of the new charge.
40
a. The electric field is a ‘‘property’’ of the
region—its task is to apply force on a charge in
it.
82
b. No reason 18
3. At the point where the electric field is zero,
the electric potential is also zero.
62
a. Incorrect use of formulas: 70
Confusing voltage (V) with potential (P):
Eϭ 0⇒V ͑voltage͒ϭ ͐E drϭ 0⇒P
͑potential͒ϭ0.
Confusing energy (Ep) with field ͑E͒:
P ͑potential͒ϭ Ep /q ϭ E/q⇒ if Eϭ 0 then
Pϭ0.
b. Over-use of parallelism between field and
potential: Field and potential are caused by
charges. ⇒ no field, no potential
27
c. No reason 3
4. A constant magnetic field never changes the
speed „magnitude of velocity… of a charged
particle which moves in it.
46
a. Recitation of formula: field applies force,
and force causes acceleration according to
Newton’s second law of motion
40
b. Blind substitution into formula 37
c. No reason 23
5. The velocity of a charged particle moving in
a magnetic field is always perpendicular to the
direction of the field.
37
a. Recitation of formula: v, B and F are
always perpendicular according to left hand or
right screw law
81
b. No reason 19
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5. 3. The third statement (relationship between E and V)
Many students do not understand the relationship between
the concepts of electric field and electric potential. The first
category in Table III ͑incorrect use of formulas͒, deserves
special attention. It is composed of two kinds of errors: ͑1͒
no differentiation between the concepts of potential and po-
tential difference, and ͑2͒ erroneous interpretation of a sym-
bol.
Concerning the second category, examination of presenta-
tions in textbooks suggests the possibility that the proximity
of introducing the electric field and the electric potential, as
well as the similarity of the formulas for their calculation,
may lead to the confusion of the terms. The exercises which
accompany the text lead to the same impression.
The following dialogue highlights some conceptions of
students about electric field and electric potential:
T. (reads the original statement)
S. This is not correct.
T. Why?
S. Because the potential is created by something which is
further apart and the field is not. Also field is a vector and
potential a scalar.
T. Explain.
S. For example, if you have a positive and negative charge,
in the middle Eϭ0, but not P.
T. Have you heard about reference point?
S. Yes, for P the reference point is infinity. The electric
field is measured relative to the distance from the charge
that creates it but the potential does not depend on that
distance.
T. Distance from what?
S. No, it must depend on the distance, since the equation is
Kq/r. So it depends whether one is close to infinity or not.
This kind of dialogue was quite common. When students
become confused, they recite equations which do not mean
much to them. In particular, the meaning of the symbols
͑e.g., in Kq/r͒ and the relation to the choice of a reference
point is not understood.
4. The fourth and fifth statements (The Lorentz force)
While students seemed able to determine the direction of a
magnetic force on a moving charged particle by using the
right hand rule, we suspected that a qualitative understanding
of this relationship was missing.
The first category in statement 4 shows that misconcep-
tions in one domain may cause difficulties in another: 40%
of the students who gave incorrect answers, attached accel-
eration only to a change in the magnitude of velocity and not
in its direction—a well documented misconception in me-
chanics. The second category includes all those students who
tried to find out how change in the magnetic field, one of the
variables of the formula, may cause change in velocity, an-
other variable in the formula.
We suspect that the difficulty with statement 5 is caused
by the fact that so many problems in electromagnetism deal
with charged particles whose initial direction is perpendicu-
lar to the direction of the magnetic field. This may lead stu-
dents to the incorrect generalization, that the path of a
charged particle in a magnetic field is always circular. This
was confirmed in the second stage of the study, when we
asked students to judge the following related statement:
‘‘The path of a charged particle moving in a magnetic field
is circular’’ and 60% of the students considered it to be
correct. Dialogues with students confirm this interpretation.
D. Relating representation deficiencies to instruction
In order to find the relationship between the ‘‘what’’ and
the ‘‘why,’’ we have conducted a survey which examined:
͑1͒ the presentation of theory in many popular physics text-
books, ͑2͒ the exercises included in these textbooks, and ͑3͒
the Israeli matriculation examinations in the last 10 years.
Our conclusions are as follows:
͑1͒ Although some of the textbooks attempt to locally orga-
nize the information ͑e.g., within single chapter͒ by giv-
ing a summary or a table, there are no comprehensive
attempts to organize the information at a global level.
This can explain the difficulty of students in producing a
global view of the information by themselves since there
is no structure to support retrieval. The low level of re-
call is not surprising.
͑2͒ In each of the textbooks examined, Ohm’s law is central,
either in the presentation of theory or in the exercises. In
each of the matriculation examinations, one-third of the
problems in electricity require the application of Ohm’s
law. It is not surprising that Ohm’s law is considered by
students to be one of the most central laws of electro-
magnetism.
͑3͒ Neither the textbooks nor the matriculation examinations
emphasize the idea that a change in magnetic field is
related to the production of an electric field, while the
idea of an induced emf is emphasized in the presentation,
the examples, and the exercises. This may explain the
results of the cued and contextual recall regarding this
relationship.
͑4͒ In the examined textbooks, the induced magnetic field is
presented as an element of a complicated integral, while
Lenz’s law is usually presented in a vivid and concrete
way. The only excuse for the ‘‘existence’’ of this mag-
netic field seems to be the need for completeness in
Maxwell’s equations. There are very few relevant exer-
cises in the textbooks, and even fewer in the matricula-
tion examinations.
͑5͒ Regarding the form of the relationships, the examination
of the textbooks shows that there is no emphasis on
qualitative analysis and statement of relationships. In ad-
dition, problems requiring such phrasing do not exist.
The lack of a global view in the textbook may also be
the cause of students’ difficulties in providing qualitative
statements at a high level of generality such as the state-
ments in Table I.
In conclusion, this part of the diagnostic study identified
some inaccurate conceptions of students in the domain of
electromagnetism that should be treated carefully in the de-
sign of instruction. It should be noted that we investigated
only a sample of important conceptions. Additional studies
are necessary to provide a comprehensive survey of concep-
tual understanding in this domain.
III. AN INTEGRATIVE MODEL OF INSTRUCTION
The results of the diagnostic study suggest that even
though students spend considerable time solving problems
that are based on the central ideas in the domain of electro-
magnetism, they do not automatically extract a knowledge
structure which includes the central ideas. Students may also
730 730Am. J. Phys., Vol. 65, No. 8, August 1997 E. Bagno and B.-S. Eylon

6. experience many conceptual difficulties. Hence there is a
need either to redesign existing courses of electromagnetism
or to design auxiliary instructional materials that would en-
able students to:
͑1͒ Construct an overall structure of knowledge in this do-
main. In particular, organization should highlight the re-
lationships summarized by Maxwell’s equations and the
Lorentz force.
͑2͒ Develop an understanding of difficult concepts and rela-
tionships in electromagnetism.
͑3͒ Relate aspects ͑1͒ and ͑2͒ to problem solving.
͑4͒ Form qualitative representations as well as mathematical
representation of Maxwell’s equations.
We hypothesize that the old saying, ‘‘The whole is larger
that the sum of its parts,’’ holds also in these cases, and a
treatment which integrates problem solving, conceptual un-
derstanding and the construction of a knowledge structure
may lead to better learning than an isolated treatment of each
individual aspect.
A. A useful representation of knowledge
Ausubel’s learning theory10
suggests that hierarchical
structures should be useful in promoting understanding and
recall. Novak and co-workers11
have developed the idea of
‘‘concept maps’’ as an exemplary learning/teaching strategy.
Many other studies have also shown the utility of such maps
in diagnosis and promoting meaningful learning. On the ba-
sis of previous instructional research, Eylon and Reif12
sug-
gest that a useful representation of knowledge ͑1͒ should
include the central information in the domain ͑principles as-
sociated with Maxwell’s equations͒, ͑2͒ has to highlight im-
portant features ͑e.g., parallelism between electric and mag-
netic fields in vacuum͒, ͑3͒ has to be hierarchical ͑from the
general to the specific͒ and ͑4͒ has to be economic ͑e.g.,
presentation by concept maps͒. Eylon and Reif emphasize
that it is not sufficient for students to construct a structure of
knowledge, it is also essential to actively develop methods
for using it.
In accordance with the previous discussions we propose a
hierarchical knowledge structure in electromagnetism. The
proposed structure has, in principle, properties that can fa-
cilitate the linkage of conceptual and procedural aspects.
This linkage may be obtained through an appropriate didac-
tic approach in which students actively develop the concept
map by themselves through a problem-solving approach.
B. A proposed knowledge structure
The seven previously described key relationships ͑Table I͒
summarize principles and definitions associated with Max-
well’s equations and Lorentz force. These relationships were
represented in a hierarchical structure that includes several
interconnected layers. The different layers constitute differ-
ent levels of the hierarchy. The first layer ͑see Fig. 2͒ pre-
sents a skeleton of the domain at the most general level and
consists of a two-dimensional map with four key concepts
͑electric charge, electric current, electric field, and magnetic
field͒ and relationships among them represented by arrows.
For example, q→E represents the following relationship: ‘‘A
charged particle produces an electric field’’ ͑Gauss’s law͒.
Additional layers represent progressively more specific in-
formation about the concepts and the relationships, while the
global first layer states only that there is some relationship
between an electric charge and the electric field which it
produces. The next levels specify this relationship in greater
detail including accurate formulas, characteristic examples of
how this formula is being derived and used, etc.
C. The didactic approach
Students construct the structure through active problem-
solving. As a result, the concepts and the relationships were
directly linked to characteristic tasks that students encounter
in the study of electromagnetism. Figure 3 shows a represen-
tative sequence of learning events in which the understand-
ing of two concepts is improved while they are also at the
same time related to the central structure. The explicit link-
age of the problems to the structure and their use in dealing
with the relevant concepts and relationships in the develop-
ment of the map links procedural and conceptual aspects.
The learning sequence consists of several stages ͑see Fig.
3͒:
Fig. 2. A concept map for electromagnetism.
Fig. 3. A representative sequence of learning events in which the relation-
ship between two concepts is constructed.
731 731Am. J. Phys., Vol. 65, No. 8, August 1997 E. Bagno and B.-S. Eylon

7. Stage 1: SOLVE—The student solves a problem ͑or prob-
lems͒ in which the relevant relationship between A and B
plays a central role. These problems can be selected from
standard problems that are used in regular instruction.
Stage 2: REFLECT—The student identifies the relation-
ship, compares it to other relevant relationships, recognizes
differences and similarities and finally formulates the rela-
tionship verbally, symbolically and visually. For example, a
bidirectional arrow between the electric field and potential
difference is used (E↔V) in order to emphasize that the first
concept can be defined by the second one and vice versa. The
written materials provide feedback on important issues asso-
ciated with this stage.
Stage 3: CONCEPTUALIZE—The student develops and
elaborates the concepts. This is the stage in which common
misconceptions are illuminated and important differences be-
tween concepts such as potential and potential difference are
clarified.
Stage 4: APPLY—At this stage the following means are
used to help students apply their knowledge and create an
improved knowledge structure:
͑1͒ Concrete examples including non-routine situations il-
lustrate the relationship.
͑2͒ Students are asked to apply the already defined relation-
ships in non-familiar problem solving. For example,
from a graph of the electric potential versus the distance,
the graph of the electric field versus the distance has to
be derived.
͑3͒ Students are asked to use the concept map to describe
various physical processes. For example: ‘‘Use the con-
cept map to describe the charging of a capacitor con-
nected in series with a resistor, and a battery.’’ Special
attention is given to misconceptions. Non-routine prob-
lems which create conflicts are used in each chapter in
order to highlight inconsistencies.
Stage 5: LINK—The written materials provide compact
tables to facilitate retention and retrieval. The student links
the new part of the concept map including A and B and the
relevant relationship to the previously existing concept map.
The proposed structure and didactic approach were imple-
mented in the instructional unit in electromagnetism,13
and in
a set of seven interdomain organizational units, MAOF14
͑overview in Hebrew͒, in mechanics, electricity and magne-
tism. The design of the approach is described in greater de-
tail in Bagno15
and an upcoming article.
IV. THE INSTRUCTIONAL STUDY
The integrative approach, described in the last section, is
characterized by two important features:
͑1͒ Formation of an explicit relationship between problem-
solving and a knowledge structure.
͑2͒ Treatment of conceptual difficulties in relation to a
knowledge structure.
We hypothesized that these features contribute signifi-
cantly to the learning process. We expected that students
studying with the integrative approach would have better
learning outcomes both in problem solving and in conceptual
understanding than students who carry out the same activi-
ties without the knowledge structure.
A. Method
We investigated this hypothesis by comparing the effects
of three treatments:
͑1͒ Treatment E consisted of studying, in addition to regular
instruction, the integrative unit in electromagnetism that
was designed according to the didactic approach de-
scribed previously.
͑2͒ Treatment C1 consisted of studying an alternative in-
structional unit that included all the exercises and prob-
lems given to E, together with the treatment of concep-
tual difficulties and the same feedback. It did not include
the active development of the concept map and thus
problems and concepts were not related explicitly to a
knowledge structure.
͑3͒ Treatment C2 served as a comparison and students re-
ceived only the regular instruction of the teacher includ-
ing preparation for matriculation examination.
E and C1 were administered as self-instructional units at
the end of regular instruction of the topics, allowing its use
with any textbook and instructional approach used by the
teacher.
The proposed design allowed us to evaluate the following:
͑1͒ Comparison of E with C1: Comparing the effect of an
integrated treatment of problem solving, conceptual un-
derstanding and construction of a knowledge structure
with an isolated treatment of the above.
͑2͒ Comparison of E with C2: Comparing the effect of the
integrative treatment to that of regular instruction.
͑3͒ Comparison of C1 with C2: Comparing a systematic re-
view of a topic which includes a careful choice of prob-
lems that deal with all concepts and relationships, with a
standard review in which the choice of problems is usu-
ally less systematic and less comprehensive.
The sample consisted of 190 students, who majored in
physics. These students had completed their course in elec-
tricity and magnetism and were preparing for their physics
matriculation examination. All classes were in good high
schools and were taught by experienced physics teachers.
All students ͑including C2͒ were given a pretest in class
after they had finished their regular course of electromagne-
tism but before they started studying the self-instructional
units, and a post-test about a month after its completion. The
pretest and the post-test were administered in the classrooms
and each lasted about 45 min. The tests examined four as-
pects: ͑1͒ content and form of knowledge representation, ͑2͒
conceptual understanding, and ͑3͒ application and ͑4͒ trans-
fer.
B. Results
1. Content and form
Students were asked ‘‘to summarize in a few sentences the
main ideas in electromagnetism in order of their impor-
tance.’’
Figure 4 shows the occurrence of the key relationships in
the summaries. Since there were no differences between C1
and C2 they were lumped together ͑C͒. Also the average
across the sample ͑S͒ for the pretest is given. T tests com-
paring E and C show that all the differences are significant at
least at the 0.01 level, except for I→B and B→F(I) ͑rela-
732 732Am. J. Phys., Vol. 65, No. 8, August 1997 E. Bagno and B.-S. Eylon

8. tionships 4 and 5͒. For all groups these items were among
the dominant ones in the pretest and remained so in the post-
test.
It was found that in the post-test, students in E, as com-
pared to students in C1 and C2, not only recalled more key
relationships, but the number of correct statements in their
summaries was larger, and there were more verbal state-
ments ͑versus a list of equations͒.
It is plausible to assume that the superiority of the E group
resulted from a hierarchical internal organization of the do-
main, that was developed by the students, where the highest
level of the hierarchy included the key relationships.
Similar results were found on a cued recall task although
the differences were smaller. This is not surprising, since
cues help retrieve relevant information and thus the structure
is less important.
2. Conceptual understanding
The treatment of selected conceptual difficulties was given
both to E and to C1. Thus it was expected that these groups
would perform better than C2. The conceptual understanding
test included the first four items of Table III. For each state-
ment the student had to judge whether it is correct and to
explain the choice. Only correct judgements accompanied by
correct explanations were accepted, thus the score for each
item was 1 or 0. There were two matched versions of this
test, one for the pretest and one for the post-test.
Since there were differences among the groups on the
items of the pretest, the mean score in the pretest served as a
covariate for an ANCOVA that was performed on the mean
score of the post-test. A comparison of EϩC1 with C2
yielded a significant difference (pϽ0.005). The adjusted
mean scores were 64.9 for EϩC1 and 52.1 for C2. A priori t
tests, comparing pretest and post-test scores for the two
groups, show significant improvement for EϩC1 ͑tϭ2.61,
pϽ0.01͒, but not for C2. Figure 5 shows the average per-
centage of students who failed in the pretest and succeeded
in the post-test in EϩC1 and C2.
These results imply that a judicious choice of exercises
and problems that focus on common conceptual difficulties
leads to improved performance over a systematic review. A
closer examination of the test shows that in some of the
items the success rate increased considerably in E but only
moderately in C1 or C2. These items were confusing and a
proper solution required a good understanding of the rela-
tionships. For example, a common source of confusion of
students concerning Lenz’s law was that in magnetic induc-
tion, the induced field is opposite in direction to the field
which induces it. In fact it is opposite in direction to the
change in the inducing field, and might be directed, at a
given instant of time, along the same direction as the induc-
ing field. Student’s confusion probably arises from mislead-
ing wording of the law in many textbooks or misguided in-
terpretation of the negative sign in the equation. A
hierarchical representation of this relationship which in-
cludes a higher level interpretation of Lenz’s law and which
involves energy considerations can help students to avoid
such confusion. ͑The ‘‘conceptualize’’ stage of the didactic
approach͒. Thus an organizing, hierarchical structure in ad-
dition to exercises can aid students in cases that require
higher-level qualitative reasoning.
3. Application
The application part consisted of two tasks: A standard
and a non-standard problem ͑see Figs. 6 and 7͒.
(a) Standard problem: Two scores were given to each stu-
dent: one for the direction ͑correct/incorrect͒ and one for the
explanation. In particular we were interested in seeing
whether students would use a qualitative ‘‘energy-based’’
consideration ͑prohibition of infinite buildup of energy, etc.͒.
Since students’ background could affect their success in
problem solving, an ANCOVA ͑with background score in
the pretest as covariate͒ was performed on the proportion of
students who correctly predicted the correct direction and on
the proportion of students who used an ‘‘energy-based’’ ex-
planation. A statistically significant difference was found
both for the ‘‘direction’’ measure (pϽ0.0001), and for the
‘‘energy-based’’ consideration, measure (pϽ0.0005). Dun-
Fig. 4. Average percentage of the key relationships ͑see Table I͒ in the free
recall summary task: in the pretest for the whole sample ͑S͒ and in the
post-test for treatments CϭC1ϩC2 and E. Fig. 5. Average percentage of students in C2 and EЈϭEϩC1 who failed in
the pretest and succeeded in the post-test in the conceptual understanding
task.
733 733Am. J. Phys., Vol. 65, No. 8, August 1997 E. Bagno and B.-S. Eylon

9. can’s multiple range test shows that E outperformed both
C1 and C2 (pϽ0.03) in both measures. There was no differ-
ence between C1 and C2.
It is interesting to note that the additional drill for C1 did
not help beyond regular instruction in this task. This is not
surprising, since in review sessions teachers solve standard
problems with their students. More detailed analysis of stu-
dents’ answers shows that students in the E group employed
more accurate considerations. They were also able to debug
their solutions by using an alternative method.
(b) Non-standard problem: The second problem described
a complicated unfamiliar physical system ͑see Fig. 7͒. This
problem required a comprehensive search of the whole do-
main and a selection of relevant information. The problem
was quite difficult, and we did not expect students to provide
a completely correct analysis. The following is an example
of an acceptable analysis:
When you close the switch, current starts to flow through
the circuit. The current produces a magnetic field in the
right-hand coil. This changing magnetic field produces an
electric field in the left-hand coil which causes current in the
left-hand coil. Energy-based considerations can be used to
find the direction of the induced current. After a relatively
long period of time there is no current in the left-hand coil
because the magnetic field in the right-hand coil doesn’t
change any more. When the switch is opened the current in
the right-hand coil decays and there is a transient current in
the left-hand coil.
We were interested in the following aspects:
͑1͒ How ‘‘rich’’ is their analysis of the situation? ͑How
many correct statements can they give about the situa-
tion?͒
͑2͒ What kind of considerations do they employ? ͑For ex-
ample, do they use an ‘‘energy-based’’ consideration?͒
An ANCOVA ͑with background serving as covariate͒ was
performed on these scores. A significant effect of the treat-
ment was found for each of the measures (pϽ0.002). Dun-
can’s multiple range test shows that E outperformed both
C1 and C2 (pϽ0.05), but there was no difference between
C1 and C2.
It is plausible to assume that the organizing structure
helped students to better retrieve the information necessary
for analyzing complex situations. The experimental unit
stressed the use of the concept map in solving problems. In
particular, it guided students on how to retrieve relevant in-
formation in analyzing unfamiliar physical situations.
Figure 8 presents the adjusted mean scores of the two
measures for the standard and nonstandard problems.
4. Transfer
Students were presented with a paragraph taken from a
textbook in immunology, an unfamiliar topic to these stu-
dents, and were asked to write down the main concepts and
relationships in it. Since only group E dealt with the structure
of knowledge, groups C1 and C2 were lumped together in the
analysis. Two measures were obtained for each student: ͑1͒
the number of main concepts and ͑2͒ the number of main
relationships.
The judgement was based on an a priori list of main con-
cepts and relationships that we had prepared.
Table IV describes means for these measures and the re-
sults of an ANOVA comparing E with C1ϩC2. A significant
difference was found between the groups.
V. CONCLUSIONS AND IMPLICATIONS
The diagnostic study focused on three aspects of students’
representation of knowledge following a standard course in
electromagnetism: What is represented? in what form it is
represented? and how accurate is the representation?
The results suggest that students’ knowledge representa-
tion is deficient in several respects.
͑1͒ Often, it does not include central relationships ͑e.g.,
Maxwell’s equations͒ in any form, neither mathematical
nor qualitative.
͑2͒ There is an overemphasis of subsidiary information at
the expense of more central relationships. For instance,
many students consider Ohm’s law to be of central im-
portance and completely disregard electromagnetic in-
duction.
͑3͒ It seems that students lack a coherent organization of
concepts and relationships in this domain to facilitate the
process of retrieval. Thus, in tasks requiring a compre-
Fig. 6. The standard problem.
Fig. 7. The nonstandard problem.
734 734Am. J. Phys., Vol. 65, No. 8, August 1997 E. Bagno and B.-S. Eylon

10. hensive search of information, they have difficulty re-
trieving even the partial information that they store.
͑4͒ Most students seem to represent the relationships only in
mathematical form and do not have access to more quali-
tative representations that are important in experts’ rea-
soning. Furthermore, even students who are capable of
providing such a qualitative description of the knowl-
edge do not do so spontaneously.
͑5͒ As in other scientific domains, students hold many inac-
curate ideas in electromagnetism and erroneously inter-
pret the central relationships. More specifically, this
study highlights some difficulties students have in under-
standing the relationship of an electric field to its
sources, motion of charges in a magnetic field and inter-
pretations of electromagnetic induction.
To remedy the situation we propose an integrative instruc-
tional approach that is centered around the construction of a
hierarchical concept map by the students. The map is con-
structed by students in five stages ͑SOLVE, REFLECT,
CONCEPTUALIZE, APPLY, LINK͒. Students solve prob-
lems and add the concepts and relationships that are used in
the problems to the map. As a result a well-organized linkage
is formed between conceptual knowledge and how it is used
in problem solving. When new concepts are added to the
map, the relevant conceptual issues and difficulties are
treated and thus conceptual knowledge is naturally linked to
the structure. The hierarchical design of the map at different
levels of detail is helpful for recall and problem solving:
higher level information helps retrieve more detailed infor-
mation.
The performance of students learning according to the in-
tegrative approach ͑E͒ was compared with that of students in
two comparison groups: C1—an isolated treatment of con-
ceptual difficulties and C2—a standard review of electromag-
netism.
The results show clearly an overall advantage of students
in E over students in C1 and C2 in all aspects: recall, concep-
tual understanding and problem solving. There was also a
transfer effect: Students in E learned how to identify impor-
tant ideas and relationships in the presentation of an unfamil-
iar topic. It is plausible to assume that these learning out-
comes result from a useful knowledge representation formed
by students in E. The effect of an isolated treatment of con-
ceptual difficulties like in C1 seems to be limited to the par-
ticular aspects that are treated and has limited effect on recall
and problem solving. The results suggest that a deliberate
effort is necessary to connect the new understanding of con-
cepts to an overall structure and to procedural knowledge.
Several questions can be raised considering the proposed
instructional approach:
– Would it be useful to integrate such a treatment as part of
the regular teaching of the course?
– Would it be useful to allow students to design their own
representation of the domain?
– What is the long-term effect of the treatment?
These questions require further investigation.
In practice, the proposed approach has several advantages
͑1͒ It can be administered after students have finished a
regular course in the domain. It does not make any as-
sumptions about the didactic approach used in the regu-
lar course. Thus the same unit can be used with different
courses ͑as was the case in the present study͒ as long as
the syllabus is similar.
͑2͒ It is designed as a self-study unit that takes a relatively
short time ͑an average of about 4 h͒.
Fig. 8. Distribution of correct answers and ‘‘energy-based’’ considerations in the standard and non-standard problems in the post-test.
Table IV. Average number of main concepts and main relationships in the
transfer test for E and CϭC1ϩC2.
Average
͑SD͒
E
Nϭ69
C
Nϭ111
F
͑p͒
No. of concepts 2.59 1.89 22.18
͑1.15͒ ͑0.84͒ ͑0.0001͒
No. of 3.36 2.07 40.77
relationships ͑1.53͒ ͑1.18͒ ͑0.0001͒
735 735Am. J. Phys., Vol. 65, No. 8, August 1997 E. Bagno and B.-S. Eylon

11. ͑3͒ In the process of creating the map, the students exercise
in a systematic manner problems in the whole domain
and get an overview of all the material that they have
learned. Thus the unit can also serve as part of a review
that teachers normally perform at the end of a course.
Considering the fact that the time spent by students in the
experimental and the comparison groups was about the same,
it is recommended to adopt the integrative approach that can
lead to considerable gains in learning with relatively little
investment of time.
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736 736Am. J. Phys., Vol. 65, No. 8, August 1997 E. Bagno and B.-S. Eylon