13

Most studies of education and learning in relation to electric circuits are done with students studying in primary or secondary school. Few studies deal with learning and understanding of electric circuits at the university level. Although courses for engineering students deals with topics typically not covered in school these studies are relevant for the university level. As will be shown below university students have problems with the same topics and concepts in electric circuits as younger students has.
terms of conceptual maps and try to make the connections explicit that the students establish among diverse concepts such as current, voltage, energy and resistance. The studies of Driver (1985) and Shipstone et al. (1988) about students conceptions of electrical circuits, carried out in several European countries, points at similar ideas for different communities: diminishing intensity along a circuit, distribution of voltage in the branch, influence of the form of presentation of the circuit, etc. It is possible to see similarities in the frequency of appearance of the different conceptions on basic
Hesse in Germany, England).
In other studies about reasoning in electrical circuits Duit (1984), Caillot (1983) and Joshua (1984), it was shown that the students tend to use topological strategies
(diagrams) more than to reasoning with the concepts taught about current.
According to the research of Osborne (1980) and Shipstone and Gunstone (1984) the students use the following models to explain the movement of the electricity in simple circuits:
1. Only a conductor is necessary to take the current to a bulb.
2. There are two types of current, the positive one and negative that flow from opposite poles of the battery.
15

16 2 .
Study 1
3. The current flows trough the whole circuit but it is partially consumed in the bulb, for that, there is less current for the rest of the circuit.
The laws of Ohm and Joule that establish the quantitative relations in a circuit, it represents for the students two types of difficulties, from the first type proportional calculation and from the second type the used magnitudes. On the other hand, the majority of the results obtained with children and students can be interpreted in terms of linear causality. This form of reasoning can have different forms depending on the situation.
According to studies of Hierrezuelo and Montero (1991); Driver et al. (1985, 1994) students have problems with:
• Difficulties to distinguish between, and to use, terms like: potential difference, voltage, current, energy, power, etc.
• The current is thought of as a material fluid.
• They do not see the necessity to for a circuit to be closed for there to be an electrical current.
• They tend to make local analysis and think that voltages and currents are only affected there the change is made not else were in the circuit.
• They tend to use sequential reasoning and for example think that the current become weakened or exhausted as it pass trough a circuit or think there be only charges after the circuit element changed.
• Tend to interpret the voltage as a property of the current, instead of considering the current to be a consequence of the difference in potential between two points of a conductor.
• The students have difficulties to interpret the graphical representations of the circuits. They are not able of associating the real circuits with their graphical representations, though it is a question of simple assemblies.
Another area of difficulties in the quantitative understanding of electrical circuits is the nature of the implied concepts. To understand the physical theory on the electrical circuits it is not enough to understand the relations of proportionality between the different magnitudes is necessary besides to integrate the meaning of the different magnitudes inside these relations.
In the understanding of a problem of circuit he will construct a net of the bulb of nodes, the battery, the voltage, the current, etc. Together with the specific relations between these nodes.
In the research of Fredette and Lochhead (1980) come to the conclusion that the majority of the students who come to the University do not have a clear idea of what a circuit is.

2.2
Method and Sample 17
The Conceptions of the Students on Current, Voltage and Resistance The meaning current in the daily language about electricity and electrical applications differ from the formal language of electricity in physics. For example, the physical terms such as electricity, the current, the voltage and resistance are also used in the daily language, but their meaning is different.
The misunderstandings in lessons of physics, is therefore, probably because the teacher is not conscious of these differences between his way of speaking and the way in which the students speak about electrical phenomena.
Linear Causal Effect Between Batteries and Bulbs Tiberghien and Delacˆ
(1976) analyzed how the children handle batteries and bulbs and what explanations they give in relation with their actions. The results of this study are that the children use very general explanations of the functionality of a simple electric circuit. In general, they establish a causal connection between the battery and the bulb and make clear that there is an agent who moves between the battery and the bulb.
They are seeing to the electrical current or the electricity as an agent. The electricity or current are stored in the battery and can “rest” in conductors. The agent is consumed in the bulb. There is no idea of conservation of electricity in these children.
The linear causal effect between the battery and bulb does not hence imply a closed circuit.
The Local Reasoning Local reasoning describes the fact that the students concentrate their attention in a point in the circuit and do not consider what happens in another parts of the circuit. An example of local reasoning is that many students look at the battery as a source of constant current and not as a source of constant voltage.
The battery as a source of constant current delivers the current, independent from the circuit that is connected to the battery.
The Voltage in Closed Circuits One of the most difficult concepts in the basic electricity is the concept of voltage or differential potential. Before the voltage was related to “the force of a battery” or “the intensity or the force of the current”. Later the students use the concept of voltage as having the same properties that the concept current.
The Sequential Reasoning Establishes that the students analyze a circuit in conditions of “before” and “after” when the current “goes through” a specific place. A change in “the beginning” of the circuit influences the element later, where the change
“in the end” does not influence elements placed before. The information of change is transmitted by the electrical current.
Also the students in the level of university they use the sequential reasoning in other situations Closset (1983).
The particular aim of this research was to find information about difficulties of engineering students to understand concepts of electricity and solve electric circuits problems.

18 2 .
Study 1
And as technique to collect data were applied two questionnaires with close and open questions about:
• Basic concepts of electricity.
• Solve electric circuits problems.
• Topics involved in the analysis of electric circuits.
The strategy of analysis used was quantitative and qualitative, making categorizations, comparing information by tables.
A first questionnaire was applied (see Appedix A.1) to a group of 22 engineering stu-
They were given information about the meaning of the test and they were kindly available to answer it and they received instructions on how to answer it.
The questionnaire was divided in two parts. The first part was objective, evaluating the knowledge of the students. The second part was more subjective asking about the personal opinions of the students concerning electric circuits.

2.3
Questionnaire 1
Question 9
19
9: In the lab work if the measure of a resistance is infinite, it means:
Table 2.1: Answers from students to question 9
Experiences from the teaching electric circuits and in previous research mentioned before shows that students present confusion between the concept of open circuit and short circuit. With this question we want to know the answer from students (considering they have done this experience in the lab) relating the measure of the resistor with the concept in question.
The expected answer was “Open Circuit”. The table show that most of the students choose the correct answer. About one third of the students choose “Short Circuit”.
Nobody choose the rest of the options presented.

20
Question 10
2 .
Study 1
10: If the voltage in the end of a resistor is constant, the current in the resistor is ... to the resistance in its ohm’s value
Table 2.2: Answers from students to question 10
In an electric circuit of direct current and alternating current, the resistance is the property that is acting as the opposition to the flow of current. When applying an alternating voltage to a resistor, an alternating current is flowing through the resistor.
The magnitude of the current in any instant is directly proportional to the magnitude of the voltage in this instant and inverse proportional to the resistor value.
According to Ohm’s law the students should answer: Inverse proportional.

2.3
Questionnaire 1
Questions 11 and 12
11: The total value (ohm) “R” of some resistors connected in “series” is ...
Than any else resistor consider it individually.
12: In a circuit connected in series the current ... In all the points.
21
Table 2.3: Answers from students to question 11

22 2 .
Study 1
Table 2.4: Answers from students to question 12
For question 11 the answer expected is: HIGER. The “potency” is the speed that charge works. When many charges are connected in series in a circuit, all of them consumes some potency. The total potency consumed in the circuit is the sum of the potency consumed in each charge. The flow of the current depends of the total resistance of the circuit and for a circuit connected in series the total resistance is the sum of resistances of individual charges. For question 12 the expected answer is: IS
THE SAME.
According with the statistic result all students answer the question expected, just one student in question 12 answered the opposite one.
Both questions 11 and 12 concern the students’ understanding of connection series in electric circuits. According the researche of McDermott (1993):
“A failure to think holistically in dealing with compound systems is one kind of reasoning difficulty that may be hard to disentangle from conceptual confusion. For example, in predicting bulb brightness, students often considered only the order of a bulb in an array. Many claimed that the first bulb in a series network was the brightest. This error is consistent with the misconception that current is ’used up’ and also with improper use of local sequential reasoning”
Closset (1983) concluded that the models used by the students depend on the particular situation and how the question is written. On the other hand, the majority of the results obtained with children and students can be interpreted in terms of linear causality. This form of reasoning can have different forms depending on what situation.
Another area of difficulties in the quantitative comprehension of electrical circuits is the nature of the concepts implied. The current flowing in a circuit depends on the voltage source and the total resistance in the circuit. When many charges are used in series, the total resistance of the circuit is the sum of the individual resistances.

2.3
Questionnaire 1
Question 16
16: The current exist only in a closed circuit? (True/False)
23
Table 2.5: Answers from students to question 16
The circuit has to be closed or complete for current to flow. If the conductor is opened in any point, then in the negative part, the electrons are accumulated, while in the positive side the electrons are attracted. The movement of electrons is stopped and it implies the flow of current too. For this reason the answer expected is TRUE.
And just one student answered FALSE.
“An example of a common difficulty that research has shown to be especially persistent is the apparently intuitive belief that current is ‘used up’ in a circuit.” McDermott (1993)
If a negative charge is applied to the end of a conductor, this charge repels free electrons to the other end of the conductor. The current flows only for an instant, until enough electrons has accumulated in the other end of the conductor to produce an equal negative charge, so that it will not allow more electrons. This is static electricity but to have electric current, the free electrons have to be in movement. When a source of energy is used, to apply opposite charges to the two ends of the conductor, the negative charge repels the electrons in the entire conductor. In the positive side of the energy source electrons are consumed and move from the conductor to the energy source. For each electron that leaves the conductor, at the positive side of the the energy source, a new is supplied at the negative side of the energy source. This is a complete circuit or closed.

24
Questions 14 and 15
2 .
Study 1
14: The tension between the ends of every branch of a net in parallel is not equal (True/False)
15: Resistances of 22Ω , 33Ω and 44Ω are connected in parallel. Their total resistance will be less than 22 Ω (True/False)
Table 2.6: Answers from students to question 14
Table 2.7: Answers from students to question 15
A parallel circuit is a circuit that has one or more points where the current is divided and follows different trajectories. When elements are connected in parallel, every element has a different potential. Instead of having the differential potential of the source in every charge, as in a circuit in series, it has the total differential potential of

2.3
Questionnaire 1 25 the source in itself (because all charges connected in parallel are also connected directly to the terminals of the source of energy). For this reason the sentence of the question
14 is FALSE. The question 15 has the intention to show a sequence of quantitative summarize of the resistors. In the way that it is presented the question is easy to suppose that the total value of the resistors in the circuit should be major than the each one, as in the case of resistors connected in series; but, according with the proprieties of the electric circuit for any number of resistors in parallel, the reciprocal of the equivalent resistance equals the sum of the reciprocals of their individual resistances, so it means that the equivalent resistance is always less than any individual resistance.
A circuit connected in parallel has different electric behavior than connected in serie.
In consequence the correct answer for this question is TRUE, and it is possible to probe it with this mathematical model:
1
22
1
+
1
33
+
1
44
1
=
0 .
106060 . . .
= 9 .
428571 . . .
9 .
43Ω < 22Ω
(2.1)
(2.2)
However, as we can observe in the quantitative result that most students answered wrong. But comparing this result with the result from question 14 we found inconsistence and also for questions 11 and 12 about circuits connected in series, we observe a difference: although all students answered right in question 11 and just one student answered wrong in question 12, the inconsistence.
“In treating the parallel branches as independent, the students were not recognizing the difference between parallel branches connected across a battery and parallel branches connected elsewhere. Instead of using qualitative reasoning to check that their predictions were consistent with what they knew about current and potential difference, the students relied on a rule that they had incorrectly memorized.” McDermott (1993)
The following questions were applied with the intention to know, from the students’ point of view, about the group of concepts and elements involved in the analysis of circuits as a subject and to be more close the side of the students not just as object of study.

26
Question 1
1: What is your opinion about the subject Electric Circuits? Explain why.
• Iteresting • Important
• Boring
• Incomprihensible
• Not important
•
•
•
Easy
Complicated
Necessary
2 .
Study 1
Table 2.8: Answers from students to question 1
Students’ comments on their answers The students were asked to motivate why they choose a particular option; the following are the students comments and motivation for their answers.
Interesting
• Because it gives tools to analyze electric circuits
• Because it makes more easy to get data of circuits
Important

2.3
Questionnaire 1
• Because it is the base to understand other subjects
Easy
• Because it make more easy analysis of electric circuits
Complicated and Boring
• Because it became just a requirement
Necessary
• Because it is the base of technique knowledge for electronic career.
• To continue my career.
• To simplify calculus
• To understand electric circuits
• To make more easy the analysis of electric circuits
Question 5
27
5: Is it necessary to be skillful in mathematics to understand the analysis of electric circuits? Explain why.
• Yes • No
Students’ comments on their answers The students were asked to motivate why they choose a particular option; the following are the students comments and motivation for their answers.
Yes
• Because is more easy to understand mathematical operations
• It helps, because principally are mathematic calculus
• Is necessary to have it because appearing mathematical operations and tools like the Laplace transform inverse that are more complicated than the system of equations.
• To make more easy the calculus
No
• The mathematical calculus are easy
• If you know interpret the unknown quantities the mathematical operations used are basic.

28 2 .
Study 1
• All the concepts have been studied previously.
• Although is fundamental to have any skillful to understand
• I think is more useful to be skillful to solve electric circuits than in mathematics
• Although, I think, is it necessary to have clear different aspects of mathematics, for example: the Laplace transform .
Although the question don’t specify the sentence: “skillful in mathematics”, the students related it with to “make easy to understand...” and to “do calculus...” and it is interesting to observe that the topic of the Laplace transform appear in both options
(yes/no) as complicated topic.
Question 8
8: Which do you identify with when solving electric circuits?
A
B
C
D
E
F
I understand the problem and sometimes I get the right answer without mathematical methods or theorems
I don’t understand the physical phenomena
Sometimes I fail, but only in mathematical errors
There are many mathematical methods, and
I don’t know which one to use
Without understanding much of the physical phenomena I get the right result by applying mathematical methods
Other

2.3
Questionnaire 1 29
Table 2.9: Answers from students to question 8
“Those who learn successfully from lectures, textbooks and problemsolving do so because they constantly question their own comprehension, confront their difficulties and persist in trying to resolve them. Most students taking introductory physics do not bring this degree of intellectual independence to their study of the subject.” McDermott (1993)
Questions 3 and 4
The intention of question 3 was to know which topic the students considered as the most difficult to learn when they studied the subject Electric Circuits.
The purpose of question 4 was to know which topic the students considered as the most relevant to the students when they solve electric circuits. Important topic in the sense that the student remember this topic as something that, to the student, seemed important to the course. This is not necessarily the topic that they understood the best.
3: In the subject of Electric Circuit, do you consider any topic difficult to understand? And why?
4: In the subject of Electric circuit, which topic do you remember more? And why?

30 2 .
Study 1
Table 2.10 is a Table of Contigency , where it is possible to observe the intersection
between the topic of Laplace inverse (vertical) and Laplace transform (horizontal) and correspond to the answers about the topic (of electric circuits) that students consider more difficult to understand (Question 3) and the topic (of electric circuits) that they remember more, that is, they had to work more on it (Question 4).
Table 2.10 also shows that over all the topics related with electric circuits, the topic
of Laplace transform (that it is a mathematical tool solve electric circuits problems) was mentioned many times from the students as a difficult topic.

2.3
Questionnaire 1 31
Table 2.10: Table of contigency for questions 3 and 4

32 2 .
Study 1
A second questionnaire was applied (Appendix A.2) to the same group of engineering where they take lectures.
An interesting point observed was that after the test some students made questions to know about their mistake and they expressed some important points that were considerate.
The strategy that we used in the second questionnaire was to change the conditions for a problem that the students are used to solve. In our study the students were asked to solve a problem considered as a basic exercise they solve during classes, with the exception that it was to be solved using direct current instead of alternating current.
This created a conceptual conflict that the students had to resolve. The motivation for doing this was to see how the students interpret the physical phenomena and what kind of mathematical models they use related to this phenomena.
1: Find the value of the current trough the coil and justify your answer

2.4
Questionnaire 2
The solution we expected The most common solution given by the students
33
10 = 4 [ I ( s ) + i x
] + ( s + 6) I ( s )
10 = 4 [ I ( s ) + i x
] +
4 s i x

34
Figure 2.1: The power source is direct current
Figure 2.2: The capacitor functions as an inifinite resistance
We expected the following answer:
The most common answer from the students was as following:
Figure 2.3: The coil functions as a conductor
2 .
Study 1

2.4
Questionnaire 2 35
Figure 2.4: The current through the inductor is the same as the total current in the circuit
Figure 2.5: The power source was considered by the students as alternating current
Figure 2.6: The students applied theorems to analyze the circuit and simplify it
Figure 2.7: The students obtained a reduced circuit, and from it they got a system of equations to solve

36 2 .
Study 1
Figure 2.8: The students ends up with the following equations
10 = 4 [ I ( s ) + i x
] + ( s + 6) I ( s )
10 = 4 [ I ( s ) + i x
] +
4 s i x
(2.3)
(2.4)
When they applied the Laplace transform formulas they realized that they needed more information to find the solution
Only one student in the group asked how to solve the problem using direct current.
While other students tried to solve it by applying the Laplace models.
Reviewing their answers, raised the question of how the students interpret this process that they expressed in the circuit.
This is the question that is pushing for more research.
After the questionnaire some students where interested in to know the right solution and they manifested particular (but important) comments about some aspect that they consider important to analyze electric circuits:

2.4
Questionnaire 2
Student Comments
Student
S1
S1
S2
S3
S4
S5
Comment
Antes de llegar a la asignatura de
Translation
Before coming to the subject of Analysis of Circuits I, in first courses, the Laplace transform might be explain more
Laplace.
natura estar´ıa bien realizar tareas extra, como ejercicios puntuales hechos en casa.
En las clases de Teor´ıa de Circuitos/ de corriente continua y corriente alterna
(diferencias) y poner ejemplos de lo que estamos haciendo y que sucedan en la
Me parece totalmente correcta la asignatura y el temario es suficiente...sin emasignatura es un poco aburrida.
creo que es necesario que se le de mas importancia. al funcionamiento en s´ı del circuito (que hace ese circuito) en vez de buscar tensiones o corrientes en puntos que a simple vista parecen poco significativos.
Otra cosa que veo importorio tendr´ıan que ser mas abundantes.
solo de forma presencial e intentar acmuy importante tener una idea amplia y clara de lo que es la electricidad en sus distintos comportamientos de los elecomportamiento y la naturaleza misma de la electricidad. Si se comprende mejor atica y t´ Eso es lo que complejo y dif´ıcil de explicar ya que es como mas “invisible” e intuitivo que
To improve the comprehension of the subject; it would be nice to realize extra tasks, as punctual exercises done at home
In lectures of Circuits Theory /Analysis of Circuits I, lacking explanation about what is direct current and alternating current (also differences) and to see examples of what we are doing and that happen in real life/daily
The subject seems to me totally correct and the agenda is sufficient ... although,
I have to say that the subject is a little bit boring too
In the subject of Analysis of Circuits
I believe that it is necessary to give more importance to the functioning circuit (what does this circuit do?) instead of looking for tensions or currents in points that seem to be slightly significant. Another thing that I consider important is that the practices in the laboratory would have to be more abundant
In my opinion I believe that they should explain fewer mathematics and theorems, just as presence form and they should try to close the phenomenon over more in itself to the student (the physical phenomenon) since I believe that it is very important to have a wide and clear idea of what is the electricity in it basic form to be able to understand better the mathematical equations and it different behaviors of electronic elements; why they are designed on this way and not of another form? And it is in the behavior and the nature itself of the electricity. If the phenomenon is understood from the beginning, then I believe that it is easier to assault the mathematical and technical theory. This is what I don’t find in my studies. For other one I understand that it is a complex phenomenon and difficult to explain because it is like ”invisibly” and intuitively that other physical phenomena
37

38 2 .
Study 1
(continued)
Student Comment
S6 muy mal organizado, en las asignaturas mas especializadas de la carrera teor´ıa de circuitos,...) se dispone de apenas dos horas a la semana de teor´ıa; dir´ıa que es poco tiempo para coger una buena base, porque viniendo del bachillerato la unica base es matem´ Yo, los cambios los plantear´ıa no a nivel de
Translation
I believe that the current study plan is badly organized, in the more specialized subjects of the career (as basic electronics, digital circuits, theory of circuits, etc.) have only two hours per week of theory; I would say that it is little time to get a good base, because coming from high school the technological background is practically void, the only base is mathematics.
I would propose changes not only to the level of analysis of circuits, but also to a much more wide level
S7
S8
Yo sugiero aplicar en las horas de labolo que se explica en teor´ıa y no hacer tanque hacemos en clase de teor´ıa.
I suggest to apply in the hours of lab more practical checking than is explained theoretically and not to do so many calculations that of all forms already it is what we do in class of theory
I think that it should relate more the subject to practical real cases

2.5
Conclusions
(continued)
Student Comment
S9 de Circuitos I), pero creo que si pro-
Translation
I believe that the subject Circuit Analysis
I is one of the most profitable subjects that we have studied, but I believe that if they penetrate deeper into the topics of condensers and coils it would be better bobinas ser´ıa mejor.
Table 2.11: Comments made by some students after Questionnaire 2
39
According with results we consider the next conclusions:
• It is important to note that the engineering level education needs a special field of study. One of the reasons for this is that the concepts managed in this level are complex, comparing with the concepts in lower level education. The engineering students need to be able to combine and apply new and previous knowledge to solve real situations.
• It is important to observe that while almost all students gives the right answer in the first questionnaire about topics implicated in electric circuit, they showed difficulty to solve the electric circuit problem in the second questionnaire. The topics covered in both tests are the same but the way the questions are asked is different. While the first questionnaire asks direct questions where the students have to choose the right answer, the second is an application of more advanced concepts, that imply a higher level of knowledge, where it is necessary to know, not only all the concepts involved, but also their behaviour.
“Predicting the effect of a change in a circuit requires a more sophisticated level of holistic reasoning.” (McDermott, 1993)
We do not intend to affirm the statement above, only with the results from this “classic” test because it was formulated like an exam, and could therefore have influenced the answers from the students. However, we do not discard the possibility that the solving electric circuits implies handling threshold concepts.
• In both questionnaires we observed some topics that for the students are not easy to manage when they solve electric circuits. A constant topic that appears is the
Laplace transform and we consider to put more attention in this topic for the next study to know is it local or general difficulty.
“Students specialising in electrical engineering or engineering or engineering physics typically need to study not only AC-circuits but methods for handling more complex circuits and are usually request to learn to apply various transform methods (phasor, Fourier, Laplace)

40 2 .
Study 1 and Fourier-series in circuit analysis.” (Bernhard and Carstensen,
2002)
Most students say that it is easy to apply the mathematical theorems or methods without understanding the physical phenomena and get correct answer. It implies that students are able to develop “their system” to solve problems with specific conditions.
By just applying the right mathematical method is possible to get right answer. But, when the conditions for the problem are changed “their system” will not work and some of them do not know how to handle the situation. This is worrying in engineering education because the students lose the principal aim (that is learn) and just becomes machines, very skilful at using computers and calculators, producing good answers but without understanding the meaning of that calculus.
It is important to notice the difference between to analyze and to calculate.

To be able to analyze electrical circuits it is necessary to have knowledge of the laws, theorems and fundamentals of the theory of electric circuits, trying to become skilled in the use of the concepts and technologies of differential calculation and diverse variables. It is also necessary to have an interest in having a good “mathematical base”, such as algebra, complex numbers, differential equations and the Laplace transform.
Further, it is necessary to study the concepts of static and dynamic models of the basic semiconductor devices, as well as to study and to characterize basic circuits and subcircuits functionality. The integrated-analogical circuits used in the subjects of basic electronics allows the student to have a wider vision of the electronic applications.
The analysis of electric circuits among engineering students are of special interest to study, because engineering students are required to combine both previous knowledge and develop new knowledge in the process of analyzing electric circuits. Few researches has been made on understanding of electric circuits in higher level education:
“Research on students learning and understanding electric circuit theory is still in its infancy. Student’s conceptions in circuit theory and electricity are not as well investigated as those in mechanics.” Bernhard and
Carstensen (2002)
This study is a survey of the answers from 109 engineering students from three dif-
UPC, Catalonia.
The method used to collect data was a questionnaire (see Appendix A) to electric engineering students with open questions about the use of mathematical models,
41

42 3 .
Study 2 physical concepts and procedures.
The data analysis strategy for each stage was quantitative and qualitative analysis:
Categorizing, comparing and summarizing information.
Presenting and explaining results and their interrelation and conclusions.
Sweden Mexico Catalonia
Figure 3.2: Students from three different contexts
Engineering students learn to solve electric circuits in different ways according with the plan of studies of every country. And our approach is not to compare curriculum or programs or knowledge to establish a score to say which one is better. We are interested to know if the result gotten in Study 1 is just local situation and also to know how the students interpret the mathematical models that they use to solve electric circuits problems
Results from the survey show the different arguments from engineering students explaining the meaning of complex concepts and formulas used to solve the electric circuits.
The questionnaire had to be adapted not only with the language for each country, but also with the different technical words used.
The first part consisted in to answer true or false to each statement concerning electrical behaviour made in the question. Some of the statements had the same meaning, but expressed in different ways.
The second part consisted in to explain an already solved problem.
The Tables 3.1, 3.2 and 3.3 show the results of the answers for each question in Ques-
tionnaire 3 in Appendix A.

3.3
Results 43
Table 3.1: All answers to question 8 in Questionnaire 3 (Appendix A ) from students in
Mexico

44 3 .
Study 2
Table 3.2: All answers to question 3 in Questionnaire 3 (Appendix A ) from students in
Sweden

3.3
Results 45
Table 3.3: All answers to question 3 in Questionnaire 3 (Appendix A ) from students in
Catalonia

46 3 .
Study 2
Dependence between Potential Difference and Current
Figures 3.3, 3.4 and 3.5 show the analysis to the answers of the questions (a), (p),
(b) and (q), that express the relation of dependence between potential difference and current, but in different ways. We can see that the three countries are consistent in their answers to questions (a) and (p), but at the same time all of them are inconsistent in their answers to questions (b) and (q).
Figure 3.3: Relation between answers to questions (a), (p), (b) and (q) from mexican students
Figure 3.4: Relation between answers to questions (a), (p), (b) and (q) from swedish students
Figure 3.5: Relation between answers to questions (a), (p), (b) and (q) from catalonian students
The following tables show the relations between different answers from the students, where the results were the same from the three countries.

3.3
Results
(m): Current is an electric flow
(n): Voltage is an electric flow
47 m \ n
True 21 43 0 0
False 1 0 0 0
True & False 0 0 0 0
No Answer 0 0 0 1
22 43 0 1
m \ n
True 0 17 0 0
False 4 2 0 0
True & False 0 0 0 0
No Answer 0 0 0 2
4 19 0 2
17
6
0
2
25
64
1
0
1
66
Relation between questions (m) and (n), where the following was the highest frequency in each country
1. Mexico
(m): True and (n): False
2. Sweden
(m): True and (n): False
3. Catalonia
(m): True and (n): False m \ n
True
False
True & False
No Answer
2
0
0
0
2
13
0
0
0
13
0
0
0
0
0
1
0
0
2
3
16
0
0
2
18
Table 3.4: Relations between questions (m) and (n)

48
(i): You can feel Voltage by touching a wire
(j): You can feel Current by touching a wire
3 .
Study 2 i \ j
True 12 9 0 0
False 40 3 0 0
True & False 0 0 0 0
No Answer 0 0 0 2
52 12 0 2
i \ j
True 1 9 0 0
False 12 2 0 0
True & False 0 0 0 0
No Answer 0 0 0 1
13 11 0 1
10
14
0
1
25
21
43
0
2
66
Relation between questions (i) and (j), where the following was the highest frequency in each country
1. Mexico
(i): False and (j): True
2. Sweden
(i): False and (j): True
3. Catalonia
(i): False and (j): True i \ j
True
False
True & False
No Answer
1
8
0
1
10
1
5
0
0
6
0
1
0
0
1
0
0
0
1
1
2
14
0
2
18
Table 3.5: Relations between questions (i) and (j)

3.3
Results
(b): Potential Difference is necessary to get a Current
(g): The Voltage is the cause of electric Current
49 b \ g
True 28 17 0 1
False 3 5 0 1
True & False 0 0 0 0
No Answer 3 5 0 3
34 27 0 5
b \ g
True 12 1 0 2
False 4 4 0 1
True & False 0 0 0 0
No Answer 0 0 0 1
16 5 0 4
15
9
0
1
25
46
9
0
11
66
Relation between questions (b) and (g), where the following was the highest frequency in each country
1. Mexico
(b): True and (g): True
2. Sweden
(b): True and (g): True
3. Catalonia
(b): True and (g): True b \ g
True
False
True & False
No Answer
7
1
0
0
8
5
4
0
0
9
0
0
0
0
0
1
0
0
1
1
13
5
0
1
18
Table 3.6: Relations between questions (b) and (g)

50
(b): Potential Difference is necessary to get a Current
(q): Current can occur without Voltage
3 .
Study 2 b \ q
True 9 37 0 0
False 6 3 0 0
True & False 0 0 0 0
No Answer 3 5 0 3
18 45 0 3
b \ q
True 2 12 0 1
False 2 7 0 0
True & False 0 0 0 0
No Answer 0 1 0 0
4 20 0 1
15
9
0
1
25
46
9
0
11
66
Relation between questions (b) and (q), where the following was the highest frequency in each country
1. Mexico
(b): True and (q): False
2. Sweden
(b): True and (q): False
3. Catalonia
(b): True and (q): False b \ q
True
False
True & False
No Answer
1
2
0
0
3
11
3
0
0
14
0
0
0
0
0
1
0
0
0
1
13
5
0
0
18
Table 3.7: Relations between questions (b) and (q)

3.3
Results
(c): A Voltage impulse will cause a Current
(d): The Voltage is the force driving Current
51 c \ d
True 19 15 0 2
False 14 7 0 0
True & False 0 0 0 0
No Answer 0 3 0 6
33 25 0 8
c \ d
True 15 2 0 1
False 4 3 0 0
True & False 0 0 0 0
No Answer 0 0 0 0
19 5 0 1
18
7
0
0
25
36
21
0
9
66
Relation between questions (c) and (d), where the following was the highest frequency in each country
1. Mexico
(c): True and (d): True
2. Sweden
(c): True and (d): True
3. Catalonia
(c): True and (d): True and also
(c): No Answer and (d): No
Answer c \ d
True
False
True & False
No Answer
3
2
0
0
5
2
6
0
0
8
0
0
0
0
0
1
1
0
3
5
6
9
0
3
18
Table 3.8: Relations between questions (c) and (d)

52 3 .
Study 2
The task for the students was to give an explanation, or interpretation, to the electric
phenomena that occured in the circuit in Figure 3.6
Figure 3.6: Circuit from the questionnaire
The circuit in Figure 3.6 has two solutions; first condition and second condition. To
the first case; when the circuit has been connected to the source of 12V for a long time, the following phenomena occurs:
1. The coil acts as a ”short circuit” (like a conductor)
2. The capacitor as a ”open circuit”
Figure 3.7: The circuit in the first case
In this case only the resistor acts, and the original circuit is reduced to the circuit in
Figure 3.8
Figure 3.8: The reduced circuit
The intention to ask the students what happends in the circuit, with the elements like the coil and the capacitor, is to know how they interpret the electric phenomena and to compare the answers from the different countries.

3.3
Results 53
Examples of answers from the students
In this part of the questionnaire we explained to the students the aim of it and we try that they do not take it as an ordinary test were their knowledge is evaluated, the problem is solved and they just have to explain the meaning of the mathematical model. This kind of activity is not common for the students in neither Mexican nor
Catalan context. This is because in the examinations they use to solve and give the result, but usually they do not give an explanation or interpretation of their solution.
Figure 3.9: Part of the questionnaire answered by a swedish student

54 3 .
Study 2
Figure 3.10: Part of the questionnaire answered by a mexican student
Answers from the engineering students, explaining the meaning of the values in the circuit.
Table 3.9 shows the answers from the students that coincide with the expected
answer. And we can appreciate that in three countries although the answer of “Ohm law” is superficial, it was found in the three countries.

3.3
Results
Answers from Students
55
Mexico Sweden Catalonia
Inductor is like short circuit.
Capacitor is like open circuit.
Resistor keeps equal
3 7 4
Ohm law 2 2 1
Table 3.9: Expected Answers; The same arguments found in all the three different contexts
In Table 3.10 shows the misunderstandings of concepts in students when they ex-
plain certain parts of the problem.
Answers from Students
Mexico
The coil opens and the capacitor closes
The inductor is like a short circuit and it doesn’t flow current .
The capacitor is like open circuit and doesn’t flow current
It doesn’t exist variation
Something but I don’t know
I don’t interpret x x x
In coil what ever?
And a current which blocked in the resistance
The capacitor is broken and this imply coil is broken and current goes throws by resistor
Table 3.10: Misunderstandings
Sweden x x x
Catalonia x x

56 3 .
Study 2
We can apreciate in Table 3.10 not incidence of answers in the the different countries.
Answers from Students
Mexico Sweden Catalonia
By the capacitor.
By drop voltage and differential potential.
Total current pass trough in the short circuit and the voltage in capacitor is the same than voltage source.
V (0) is in parallel to voltage source.
I (0) applying Ohm law in a closed circuit.
All current pass trough the inductor, for this reason it doesn’t exist voltage, in the capacitor doesn’t not pass current, in consequence it doesn’t exist current and voltage either.
The values you got when you started the measure.
Capacitor doesn’t carry any current at low frequency and may there for be removed from the circuit which becomes open at the point.
All the current goes trough the resistor
Direct current
In C arises a potential ( V ) x x x x x x x x x x
I (0) is the intensity that coil let go x
V in parallel is the same in all the networks
The inductor is charged and let flow current by conductor
Table 3.11: Explanations of the meaning of values in the electric circuit x x

3.4
Conclusions 57
Table 3.12 shows other clasification of the students answers that are diverse and do
not have incidence.
Answers from Students
Mexico Sweden Catalonia
When the current flows in coil through the time; the coil presents a impedance and we find a voltage.
By “integral- differential” equation
L di dt coil is the voltage who pass by the
I have forgotten x x x x
Everything that comes “in” most comes “out”
As Kirchoff ’s law said x x
The sum of entire “in” potentials is x equal to the sum of “out” potentials
Table 3.12: Explanations of the mathematical meaning of values in the electric circuit
We can see that there are inconcistencies in the majority of the answers to the questions that relate to either potential difference being the cause of current or current being the cause of potential difference. It can be originated from the way the questions were written (in every country the expressions used sometimes change), from misunderstandings, or because there is not a concencus among the teacher regarding this aspect.
In the results from the analysis in Appendix B.
1. The students from Catalonia showed five times as the most frequent answer two combinations of answers, this did not occur among the students from either
Mexico or Sweden.
2. We observed that some of the answers from the Catalonian and Swedish students agree, while for the same questions, the Mexican students answer the complete opposite.

58 3 .
Study 2
3. In four combinations of answers the students from the three countries agreed completely:
(a) That current is an electric flow, and voltage is not an electric flow.
(b) That you cannot feel voltage touching a wire, but you can feel current touching a wire.
(c) That potential difference is necessary to get a current, and that the voltage is causing electric current.
(d) Potential difference is necessary to get a current, but current does not occur without a voltage.
The students almost agreed (only a small difference in the answes from the students in Catalonia) that voltage impulse cause a current. And that the potential difference is the force driving the current.
According to the results from the students in Table 3.10 we found that the students
are able to manage the concepts without understanding the meaning of them. We can relate these results with the difficulties that Meyer and Land suggest exists the learners may be left in a state of liminality (Latin ‘limen’ – a theshold). Liminality refer to an individual or a group – a suspended state in which understanding approximates to a kind of mimicry. The transition is problematic. troubling and often humbling, and students often mimic the new status without understanding the meaning of what they are doing.
Baillie (2005)