An Inquiry-Based Approach to the Distribution Concept in
Methodology
Inquiry Approach to Science Education has been variously defined and several specific examples of scientific inquiry have been supplied. Usually organizations into “stages” of scientific inquiry or patterns of inquiry processes (Wenning, 2007, Banchi & Bell, 2008) are illustrated. These involve a continuous spectrum of inquiry levels that accounts for the gradual shift of the focus of research control from the teacher to the student. A didactic approach based on Guided Inquiry (Wenning, 2007) is intended as an approach where the teacher provides students with research questions and guides students in designing the procedures to test their questions and the resulting explanations. According to this theoretical framework the students are encouraged to plan and carry out their laboratory activities by collecting, formulating and analysing data, for the purpose of gaining a more meaningful understanding of the concepts of physics.
“Inquiry is the scientific process of active exploration by which we use critical, logical and creative thinking skills to raise and engage in question of personal interests. Inquiry helps us to connect our prior understanding to new experiences, modify and accommodate our previously held beliefs and conceptual models, and construct new knowledge” (Liewellyn, 2002). Students are therefore inspired by a scientific investigation procedure, very similar to the one followed by a researcher, a very important aspect of which is the modelling of physical phenomena.
In this case the students work in small teams. They are expected to design and conduct the experiment themselves with little or no guidance by the teacher and only partial pre-lab orientation. The students can use some scientific papers on the subject or on similar problems. They can also use a traditional and microcomputer based laboratory for their measurements and data analysis.
This activity should make able students to trace a path similar to the one some physicists have identified to solve the proposed problem in the past. They are requested to formulate hypotheses, conjectures, explanations and to identify the most appropriate methodology for their research. Students can, then, confirm their assumptions and explicative models on the basis of experimental results. In particular, the idea of an “electron velocity filter” by means of electric fields can be very useful because it naturally introduces students to the concept of a distribution of frequencies.
The different stages of the inquiry procedure are summarized in the diagram reported in Figure 1 (Windschitl, Thompson, Braaten, 2008).
Figure 1. The logic diagram of our teaching sequence.
The didactical experimentation
The teaching experiment was carried out during a 20-hour workshop entitled “An experiment on thermo- ionic emissions: the speed distribution of electrons emitted in a vacuum tube”, carried out at the Faculty of Engineering of the University of Palermo from March to May 2012. The workshop involved 43 students that already completed the curricular mathematical and numerical calculation courses, as well as the general physics courses. In such courses they already faced the problem of statistical distributions, although only in terms of continuous functions and only from a theoretical point of view.
The workshop planning took into account the average level of knowledge of a second/third year university student concerning standard physics (mechanics and electromagnetism), mathematical and numerical analysis, as well as the ability to use software like calculation sheets.
Evaluation of the workshop approach as well as assessment of student learning has been performed through the administration of a questionnaire and a qualitative analysis of videos registered while the experiment was underway.
In this paper, the teaching experiment effectiveness is evaluated by comparing the answers to pre- instruction/post-instruction questionnaires and by taking into account some preliminary results of video- analysis. The administered questionnaire was aimed at investigating the previous student knowledge with respect to the distribution concept, and to obtain information for the gauging of the interventions.
A classification of student answers to the questionnaire items, based on a careful reading of students’
answers within a framework provided by domain-specific expertise, has been performed. In particular, our analysis is focused on the cognitive variables and evaluation criteria that refer to the ability to read and construct a distribution, and the ability to compare different distributions.
A phenomenographic analysis (Marton, 1988, Marton and Pang, 2008) of answers to the pre-instruction questionnaire allowed us to identify clusters of reasoning procedures and classify students in groups.
The workshop was structured into five phases (see Figure 1):
1. Presentation of the inquiry context to the students and definition of the main problem by setting the broad parameter to be investigated.
2. Discussion of the different approaches proposed by students by organising what was known and what was to be known.
3. Generation hypotheses 4. Laboratory activity.
5. Data analysis and pointing out of arguments for appropriate explicative model building (Gilbert, 2002, Greca, 2002).
At the end of the workshop the same questionnaire was administered in order to verify the instruction effectiveness.
The Workshop
At the beginning of the first phase, the problem of identifying methods that can emphasize the distribution of some variables (energy, velocity,….) among particles of a system (atom, molecules, electron) was discussed. Student ideas and proposals have been discussed by the whole class and difficulties in performing experiments making evident the various characteristics have been pointed out. The teacher illustrated several scientific publications in which the problem of experimental determination of the velocity distribution of a system of atoms/molecules has been faced. These systems can be divided into two large categories: a first one, that makes use of mechanical apparatuses; another, including electromagnetic type apparatuses. The first type of apparatuses is usually much more complex and requires complicated and expensive equipment. In fact, Maxwell derived this distribution law in 1860, but Miller and Kusch (Miller and Kush, 1955) performed the first rigorous experimental demonstration almost 100 years later by using a mechanical apparatus
For these reasons, the proposed experimental activity is directed towards the second type of apparatuses and the student interest is focused on ways to obtain a system of electrons. Some historical papers were described (Richardson, 1921, Germer, 1925) and the research was oriented toward the following problem:
“To establish the characteristics of the electrons gas emitted by thermionic effect through the typical variables of statistical mechanics”.
The objective of the second phase was to point out what students knew and what they should know by identifying and making explicit their representations about the problem and defining experimental situations that can represent the problem. The awareness of the necessary additional information allow students to make explicit the research question:
“ How it is possible to realise an electron velocity analyzer able to show how many electrons have velocities in given intervals?”
During the third phase the design problems connected with the design of such electron velocity analyzer have been discussed. Students have been guided in recalling the characteristics of vacuum tubes, like diodes and in analyzing how an electron velocity analyzer can be easily built by using vacuum a appropriately polarized diode. The emitted electrons can be selected within a given velocity range through retarding potentials, and variations in the electron current can be measured.
In the fourth phase students were divided into groups of up to 4. The groups worked independently by designing the measuring apparatus and using the tools provided by the teacher at their request. Each group was responsible of the experimental setting as well as of the choice of the different kinds of measurements.
All the students understood the need to perform a set of measurement maintaining constant the temperature of emitted electrons (an consequently the filament temperature), although many students were not able to explicitly and formally see the relationship between the values of current end the ranges of electron velocities.
Each group performed two sets of experiments by fixing, for each set, the filament temperature and varying the retarding anode potential from zero to values that allow a current in the range of sensitivity of our micro-ammeter. Figure 2 shows the ratio between the anode current I at a given anode potential Va and the value I* (Va = 0) plotted against the anode retarding voltage, Va, for different filament temperatures.
Figure 2. Ratio I/I*plotted against the anode retarding potential Va for different filament temperatures.
Values of I* are in the range 20-500 A
The fifth stage, devoted to data analysis, was preceded by a discussion directed to the formalization of a possible model that would put in relation the current with the electron velocity distribution.
In order to build a model of the electron dynamics, we consider a diode with a regular geometry and a cathode-anode spacing small enough to minimize effects of the space charge. Under this condition, plane parallel geometry may be assumed as a good approximation. In our approximation, a certain retarding potential Vr =V will influence only one component of the electron velocity (say the z-component, vz) and an electron will reach the anode surface if
m V 2eV
Z2 0
− $
(1) where e and m are the electron charge and mass, respectively.
It follows that the electron current, I, in the z direction perpendicular to the vacuum tube surfaces S is ( )
I eS V n v v dvz z z
, min z
=
#
3 (2)where n(vz) indicates the number of electrons having a velocity in the interval vz and vz + dvz, and V 2meV z,min= � If we indicate
( )V mv z 2z2
ε ε= = (3)
we have dε =mv dvz z, and by means a change of variable we express n(vz) as n() by obtaining
I m= e S
#
ev3n^ hε εd (4)where n() is normalized so that I n d n I Sem
0 ε ε 0 0
=
#
3 ^ h = = (5)and n0 is the number of thermionic electrons reaching the anode per unit time and unit surface when V=0 and the electron current is I0�
If the retarding potential V is further increased to (V+V), a further drop in the anode current will take place I me d
S ev n e V dV n d
T = 3 ε ε− 3 ε ε
^ + ^
c
#
h#
^ h h m (6)The second integral on the right-hand side of Equation 6 can be Taylor expanded around the point eV=
and, by taking into consideration only the first two terms of such expansion, in the limit of V->0 we obtain n ddI
ε \− ε
^ h (7)
According to Equation 7, the velocity distribution can be obtained by differentiating the anode current with respect to the anode retarding voltage at each anode voltage (energy) value. Thus, measurement of the anode current as a function of the retarding voltage, coupled with a suitable method of numerical derivation, can lead to the evaluation of the electron velocity distribution.
In order to investigate the shape of the n(vz) distribution function it is need to perform a numerical differentiation of data reported in Figure 2, by applying Equation 7, and to fit the derivative points with a half Maxwellian.
Some students were familiar with the methods of numerical differentiation of experimental data by using the simple method of centred finite difference. By applying such methods to data reported in Figure 2 they obtained data reported in Figure 3.
Figure 3. Calculated derivative points of data reported in Fig. 2 for two different values of filament temperatures. Points are fitted by equation y A= expa−mvz2/2kTek, where A and Te are the fitting parameters. The distribution functions are normalized such that the area under each curve is equal to 1.
By comparing the results of the different groups the limits of the model became explicit, showing that at high temperatures the model and then the Maxwell distribution is no longer in accord with the experimental data. An example is shown in Figure 4.
Figure 4. An example of bad fit. The experimental data don’t agree with the model Data and findings
The pre/post-instruction questionnaire is structured with 4 items that are closely linked to the main problem. Below, we report the questions with the relevant results obtained from the pre/post-instruction comparison.
1. A typical apparatus for the study of properties of a gas is the following:
Figure 5. Experimental apparatus: a Miller and Kush mechanical selector of the silver atoms velocity.
In this apparatus an oven emits atoms of silver, which are collimated on a drum (as a screen) that can rotate at a constant velocity. The surface of the drum is covered with a material that emits light when struck by an atom of silver. In this way the atoms leave a persistent trace on the drum and can therefore be detected.
Explain why the drum is maintained rotating.
The answers are categorised in the following levels:
Level 0: Not answer or only partial answers;
Level 1: Student describes the apparatus but does not give any information about the method of selection;
Level 2: Correctly identifies the method of selection;
Item n. 1 specifically describes the type of particle gas and shows a historic experimental apparatus (the Miller and Kush apparatus), with which it is possible to determine the speed distribution of ion gas emitted by an oven with a vacuum inside it, heated to a high temperature. These ions, which are emitted at different speeds, can leave a mark on a drum rotating at a constant velocity, on which the shape of the distribution is thus determined.
Figure 6. Comparison output-input for the item n. 1, χ =2 30 19. ;p=2 8 10. : −7�
We can confidently say that (See Figure 6) there has been a considerable improvement in the post- instruction answers compared with the pre-instruction ones. Although the proposed apparatus was never described during the activity, after instruction all the students were able to at least describe it correctly. Furthermore, 40% of them managed to identify the selection method, clearly explaining it in their answers. The post-instruction answers made it clear that the identification of the selection method in the description of the apparatus had attained a considerable improvement, compared with the pre- instruction.
2. With respect to item 1, graphically represent the shape that you think will form on the screen.
The answers were categorised in the following levels:
Level 0: Not answer or only partial answers;
Level 1: Student identifies something uniform or a point;
Level 2: Student identifies a distribution;
Item n. 2 is closely related to the previous one.
Figure 7. Comparison output-input for the item n. 2, χ =2 25 42. ;p=1 3 10. # −5�
Figure 7 demonstrates the results of this comparison. In this case the students’ post-instruction answers once again showed a considerable improvement with respect to the pre-instruction one. No students managed to identify the correct distribution in the entry test, but after instruction 9% represent the graph of the Maxwell-Boltzman distribution. Furthermore a large percentage (49%) is able to identify some kind of distribution, partially overcoming the obstacles related to the type of selection method. It should be pointed out that a difference exists between this result and the previous one (aimed at identifying the selection method). In fact, the percentage of students who identify some kind of distribution (49%) is higher than the percentage of students who identify a selection method (40%). This inconsistency of our results may be explained by the fact that some of our students were influenced by their knowledge about the Maxwell-Boltzmann distribution without really understand how the apparatus can make evident such a distribution.
3. Observe the following curves representing three Maxwell-Boltzmann velocity distributions at three different temperatures:
Figure 8. Three normalized Maxwell Distribution for three different temperatures.
State the relation between the temperatures, giving reasons for your answer.
The answers were categorised in the following levels:
Level 0: Student does not identify any information about the distribution.
Level 1: Student identifies some information but not the variance or the connection with temperature.
Level 2: Student identifies the variance as a relevant characteristic of distributions and compares them correctly but does not know the connection with temperature;
Level 3: Student identifies the variances, compares correctly them and knows the connection with temperature;
Figure 9. Comparison output-input for the item n. 3, χ =2 9 16. ;p=0 03. �
As Figure 9 demonstrates, the improvement in post-instruction answers with respect to the pre-instruction ones is considerable, both because the percentage of those who did not manage to identify an answer has decreased, and because 60% managed to correctly identify the variance as a relevant quantity and were able to relate it to temperature. The correct answers were also well explained and complete.
4. If the electrons emitted by the cathode and collected by the anode were all emitted at the same speed, what would be the analytical expression or the graph of the anodic current vs. the tension?
The answers were categorised in the following levels:
Level 0: Not answer;
Level 1: The current does not depend on the velocity of the electrons;
Level 2: The current depends on the velocity but the relation is not known;
Level 3: The shape of the current is constant up to a particular tension value when it becomes zero.
This item partly sets out the same experimental situation in which the students had been operating, but it reverses the logic of the problem. In this case it is assumed that the distribution is known and the question is what the resulting shape of the current vs. tension might be.
In the initial test this question was certainly the most difficult; in fact a very high percentage of the students were unable to answer, as it is shown in Figure 10.
Figure 10. Comparison output-input for the item n. 4, χ =2 56 95. ;p=2 6 10. # −12�
In the post-instruction test questionnaire answers the situation is reversed with respect to the pre- instruction ones. In fact, as Figure 10 shows, more than 60 % of students correctly identified the shape of the current, demonstrating that they have understood the close link between current and velocity distribution and therefore also the experimental evidence that the distribution suggests.
Conclusions
In this paper we describe a 20-hour workshop laboratory, for undergraduate engineering students of University of Palermo, performed by using an Inquiry Approach aimed at understanding the statistical distribution concept in the field of statistical mechanics.
Students used commercial vacuum tubes and easily available measurement devices in order to see that a Maxwellian distribution can be inferred for thermally emitted electrons in a range of temperatures of about 100 K. The students designed the appropriate experimental set up and analysed experimental data by building graphics and using fitting and numerical derivate methods.
Here, we report some partial results of a teaching experiment mainly devoted at the comparison of pre and post instruction questionnaire analysis. These show that students easily understood the use of the method of retarding potential to build distributions and became able to transfer these methods to other contexts. All students showed the awareness of the need for an accurate experimental design and models able to describe the experimental data. Moreover, although a high percentage of students initially showed difficulties especially in the construction of distributions and in identifying the quantity characterizing the distributions, their active construction of distributions aimed at explaining their experimental data stimulated their understanding of general concepts.
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