1. Teacher discussion of crucial aspects, cardinal concepts and elements peculiar to Quantum Mechanics starting from an educational proposal
Lorenzo Santi, Alberto Stefanel
Physics Department of the University of Udine, Italy

2. In the wide panorama of proposals about the teaching of QM
Am. J. Phys. 2002, Special Issues 70 (3)
Phys Educ. 2000, Special Issues 35 (6)
two approaches result mainly familiar to the teachers:
the historical approach, that is of some use in the school as teaching method of quantum physics;
 the approach to QM through the wave function, rarely used in school until PC-era, however often known from university education.

3. The teacher formation on innovative proposals, aiming at the construction of key concepts of quantum theory, shows the problems of
overcoming the conviction that the previous two approaches are actually the only ones possible in didactics
overcoming well known learning problems and, in particular in the case of teachers graduated in mathematics, lack in basic concepts of the theory (Zollmann 1999)

4. To face these problems, the research in teacher education pointed out the importance of integrating the content knowledge education with the pedagogical content knowledge (Schulman 1986, Michelini 2004, Duit 2005)
In this perspective, a module for the in-service physics teacher education focused on an innovative didactical proposal developing an approach to QM that follows the Dirac formulation, was designed.
(Ghirardi G. C., Grassi R., Michelini M., 1995, 1996, 1997
Michelini, Ragazzon , Santi, Stefanel 2000, 2001, 2004

5. This proposal aims at introducing the concept of state and the superposition principle as key points of the theory and the basic elements of formal representation of states and operators in vector spaces. (Michelini, Stefanel 2006, In Opatija Girep Sel. Papers: MICHELINI 2008, M. MICHELINI, L. SANTI, A. STEFANEL 2008)
The module was proposed to a group of teachers selected all over Italy , in the context of the Master II - IDIFO.

6. In this contribution:
- the structure of the educational module is discussed, and in particular the disciplinary and didactical knots on which the discussion was developed on the web.
-the teacher learning path about the content knowledge (CK) and about the pedagogical content knowledge (PCK).
-Two level of analysis were considered:
the collective process level, developed in the web and integrated with in-person (presence) meetings,
the level of the individual path of some teachers as examples of case studies.
Results/conclusions

7. Research questions
The research questions that we planned are the following:
RQ1 - Which are the most problematic knots regarding QM in a group of high level student-teachers?
RQ2 - Which difficulties one faces in a teacher education based on a new proposal of QM teaching?
RQ3 - Which learning paths results more effective for a real improvement of the student-teacher PCK?
RQ4 - How do student-teachers modify the proposal of reference when asked to design a didactical path?

8. The Sample A group of 22 teachers (17 attended the entire path)
16 out of 22 graduated in physics and 6 in mathematics.
All of them had a long teaching experience and often collaborated with the research groups of the involved universities, except for one teacher, graduated in physics and working for a optical materials and instruments company.

9. The educational module
The educational module is subdivded into three main steps:
Step A – WebCourse A Pre-questionnaire Course focus: the presentation and discussion in web forum of the knots of the reference proposal + work-sheets
Step B – In-person meeting for discussing on the rationale of the proposal itself and the unsolved knots
Step C – WebCourse B constituted by a web didactical laboratory, aimed at designing a micro module focused on the reference proposal analyzed in the previous steps (in some cases implemented in class)

10. The forum discussion lines, that constituted the core of Step A, were:
F0 – discussion about the knots raised by the pre-questionnaire.
F1 – Polarization as property of light: phenomenological approach to light polarization, based on simple experiments with light interacting with Polaroid and birefringent crystals.
F2 – Probabilistic interpretation of quantum measurement processes: single photon phenomena and probabilistic re-interpretation of the Malus law.
F3 – Dynamical properties of a quantum system: polarization as dynamical property of photons and the quantum state.
F4 – Interpretative hypotheses (mixture of pure state, coexisting properties, superposition): exploration of hypotheses about the description of the 45° polarization photons state.
F5 – Impossibility to attribute a trajectory to a quantum system: analysis of the interaction of photons with two aligned inverse calcite crystals.
F6 – Quantum states, vector of state and the formal representation of the superposition principle: from the Malus law expressed using the scalar product of unit vectors of state, to the formal representation of the superposition in the case of 2 state system.
F7 – Observables and linear operators: Polaroid and projectors, polarization as quantum observable and the linear operator representing it.
F8 – Generalization of the formalism: from photon polarization to an arbitrary (discrete) quantum observable.
F9 – Non-locality: EPR like experiments, entanglement and non-locality.

11. In Step B, which is in-presence, the rationale of the reference proposal was discussed on three levels: 1) exemplifying how to face the most critical points of the reference proposal in-class; 2) discussing about the most difficult disciplinary knots and about the main learning difficulties of the quantum concepts learning; 3) comparing the approach followed in the reference proposal with the other proposals discussed in other courses of the Master program and briefly delineated in the WS paper by Cazzaniga and Giliberti.
Step C was the most important one for the development and the verification of the ability in managing with an innovative proposal and building on it the plan of a coherent educational path to be proposed in-class.

12. Analysis methods and instruments
Several are the instruments used for monitoring the educational path of student-teachers:
A pre-questionnaire (15 open and multi-choice questions - concepts of QM and related PCK) filled during home-work.
Two types of maps, conceptual – organizationalthe maps were designed both at the beginning of the module (initial maps) and at the end (final maps).
Statements, questions posted in the web forum.
Statements video recorded in Step B.
The didactical projects designed in Step C.
ones in which one correlates the concepts considered relevant on the theme from the disciplinary point of view, and organizational in which one correlates the disciplinary contents with the contexts chosen to face them in the didactical proposal (these maps have been proposed as individual and group activities);

13. Selected tools and criteria:
- Questionnaire – frequencies of the answers given by the teachers in the multi-choice questions; frequencies of categories a priori and re-thought a posteriori knowing the motivation of the answers of the multi-choice and open questions.
- Statements in web forum: from the quantitative point of view (frequencies of statements for each teachers in the web forum and statements of the tutor); from the content point of view (faced disciplinary knots and underlined didactical aspects).
- Didactical designing – Maps – Here are considered only the final maps that were the basis for the didactical designing (which is the conclusion of Step C). The concepts emerged in the maps were first categorized and then their frequencies were analyzed and compared with those of the pre-questionnaire.

14. Data Analysis over the entire sample
Here a global analysis in order to identify the general aspects that characterize the sample is proposed.

15. Answers topics - first two questions of the pre-questionnaire
Q1 – List 3 themes regarding QM, that you believe to be of particular interest for high school students and tell the reasons of your choice.
Q2 – List 2 elements that in your opinion characterize quantum mechanics with respect to classic mechanics and tell the reasons of your choice.

16. The themes that student-teachers believe to be important to face at school ( Q1) are mainly about the old physics of quanta (26) and are different (in quality and quantity) from the ensemble of aspects that characterize QM with respect to classical physics – Q2

17. Problematic aspects: quantization=discrete spectra
Q12): “Discuss the following assertion: in QM, in a system, all physics observables can have discrete (i.e. non-continuous) values only”.
R1. “It is one of the axioms of QM. During the measurement process of an observable A, this observable has a well-defined value a, i.e. more measurements of the same observable in the same system will result with certitude in the same value a.
R2. “This is true only under particular conditions, i.e. in the borderline cases in which classical mechanics apply.”
R3. “In QM each physical observable is associated to a linear operator and thanks to the linear operators it is possible to quantize a physical system: the possible outcomes of a measurement coincide with the operator eigenvalues, while the eigenvectors are the possible states in which the system can be found”

18. Web discussion
The frequencies of the contributions of the 22 student-teachers (Nc) and of the course tutors (Nt) to the several subjects of discussion in which the Step A web forum.
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19. Web discussion
The frequencies of the contributions of the 22 student-teachers (Nc) and of the course tutors (Nt) to the several subjects of discussion in which the Step A web forum.

20. WebForum F2
The dynamics activated in F2 results particularly complex both for the several sub-subjects introduced by the student-teachers (at least 5 can be reconstructed) and for the length of the several proposed statements, often about more than one knot at the same time.

21. WebForum F2 5 Sub-subjects:
-from high intensity experiments to single photon ones
“Inductive “ (experiment based) vs “deductive” (axiom based) strategies in teaching QM
Disciplinary knots on measurement theory
Peculiarity of QM theory and role of mathematics
Scientific theory and rationality
Sometime, sub-subjects were developed with rich exchange of ideas between two student-teachers alternated and often concluded by other student-teachers, which suggested the analysis of other aspects of the discussion.

22. 1st - sub-subject (20 statements) : from high to low intensities experiments and probabilistic/quantum description
Student-Teacher A:
“Why is it necessary to talk about a single photon? Is it only to exclude the fact that the effect is due to an interaction between photons? In order to let emerge the probabilistic interpretation, isn’t it sufficient to show that I is proportional to the number of photons?”
Example of Student-Teachers B-C…aswer:
“It is clear that it is really the single photon experiment that forces the physicist to move to a quantum interpretation of the polarization. Otherwise it would not be possible to conciliate the experimental observation with the experimental postulate of the photon indivisibility”.

23. Other problems linked to the previous one about the “existence” of the photon:
“Is it possible not to talk about the photon? “
“Is it postulated [in the educational path]?
“Is the photon given plausibility by means of movies like PSSC?” “…and about the way to characterize the photon in terms of its properties (Describing them? Introducing them phenomeno-logically? Is it sufficient for students?)”

24. 2nd - sub-subject (6 statements): supposed shift between the strategy to explore the polarization phenomenology and the moving to the analysis of ideal single photon experiments.
Student Teacher D: observe a shift from an approach defined as “inductive” to a “deductive” one. At this point he asks if it is preferable to adopt an “inductive” approach to QM (plausibility of a theoretical description, starting from the experiments) or one defined as “deductive” (starting from the rules – the postulates – and then see how they work).

25. 2nd - sub-subject (6 statements): supposed shift strategy
To this question, 4 statements answer directly-4 indirectly.
Example: Teacher E statement:
“If I have understood, the steps to make with the students are the following:
[.. ]
1) lets examine the polarization phenomenology by using intense beams and lets obtain some results.
2) lets decrease the beam intensity until we are ideally operating with single photons.
3) lets understand that the interaction with the Polaroid is stochastic.
4) lets search for the answers by reformulating the law in terms of probability of transmission of single photons.
Thus, the initial phenomenological part has the fundamental goal to introduce to the probabilistic interpretation, which, in the path logic, must emerge to the students’ eyes as the interpretative answer introduced to explain the indeterminism that emerges operatively from the simulated experiments of single photon.”

26. 3rd - sub-subject (11 statements) disciplinary knots about measurement theory:
- intrinsic stochasticity of microscopic processes, non-local nature, identity and indeterminism, wave-particle dualism, complementarity and causality in QM (8 statements)
alternative QM theories and opportunities of discussing with the students (3 statements).

27. 4th - sub-subject (8 statements) peculiar nature of the theory and role played by mathematics in it:
“In the microscopic phenomena explanation, QM has a different intrinsic (this time it is needed) nature from those that are the basis of any other physics theory. But also different from any other theory that we would formulate to explain any possible new phenomenon that we observe around us.” And in addition, “the approach (here followed) that has its rationale only in the mathematics used and not in its purely physics explanation”.
5th sub-subject (7 statements) From the role of the formalism in QM, a discussion started about the rationale of the scientific theories and of QM in particular.

28. WebForum F3
F3 is about the “modalities by which a property is operatively attributed to photons and about the sense and the meaning of this attribution”, as it was specified in a web statement.
The discussion dynamics results more focused on the proposed theme, but also less rich of statements, of real interaction in the web discussion and of proposed ideas.
The main sub-subjects of discussion are three.
1st - formulation of the uncertainty principle
2nd - advantages of the reference proposal, importance of introducing concepts within a context and then generalize
3rd - human perception indeterminacy as example of superposition

29. 1st sub-subject (17 contributions):
uncertanty principle in the case of position and impulse variables and in the case of polarization.
It emerges also the missing distinction between the (vectorial) property of the system and the vector that describes the state in which the system is: “The indetermination principle is constructed as a consequence of the existence of (observable) incompatible properties. The vertical polarization property and the one at 45° are considered distinct observable quantities. But, are they not distinct values that a single polarization observable can assume? In our case, have we not a single observable physical quantity (the polarization)?”.
Other statements are linked to this issue: the indeterminism, the mutual exclusion, the formulation of the hypothesis of the statistic mixture and that one of the coexisting properties and the comparison between the experimental outcomes and foreseen results on the basis of the hypothesis.

30. 2nd sub-subject (3 statements) : advantages of the reference proposal
“At the beginning it was hard for me. But once one gets in use with the context (which I think is needed in order to allow students to propose their ideas and their interpretative hypothesis), he starts to appreciate its potentialities, i.e. the fact that it opens to key concepts of the theory likewise the existence of incompatible properties, the indeterminism, the influence of the measurement process, the superposition principle”.
The importance of introducing concepts within a context, and them generalize in an other one.

31. 3rd Sub-subject - it is possible to find an analogy between superposition states and situation of visual perceptual ambiguity?

32. Student-teacher A:
“The polarization experiments showed that in the quantum formalism superposition states exist with respect to the possible values that an observable can assume.
[…]
Imagine, while we are walking, that our sight is captured by a human figure in a shop window and that we are not able to distinguish if we are looking at a human being or at a mannequin. While we are approaching the shop, both exclusive possibilities are still there for a certain amount of time, until the distance that we need to cover to reach the shop window is enough decreased that we are able to doubtlessly distinguish the mannequin. Is this a good example of superposition state of two possibilities [..] as it happens in the polarization case?”

33. Student-teacher B: “But in this example, the two properties (of being a human being and of being a mannequin) are mutually exclusive and not incompatible. I can perceive the two “properties” as equally probable but I am allowed to think that the “system” has just one of the two, independently from the fact that I am approaching the window to verify one or the other property.”

34. Student-teacher C: “I agree with student-teacher B
Student-teacher C: “I agree with student-teacher B. The mannequin possesses the property of being a mannequin”
Student-teacher D: “I agree that this example is dangerous, it does not solve the knot of the non epistemic indeterminism of QM and continues to root the idea that the superposition state is bound to the impossibility of the observer to distinguish the two states”.

35. The basic maps for the didactical designing
Summary of the contents of the maps’ concepts that the 17 teachers (those who completed the whole proposed module) included in their didactical designing of a learning path to be proposed to high school students based on the reference proposal.
From the comparison of Figure bbbb with zzz, the radical change in contents considered important for the developing of the didactical proposal is clear: in figure bbbb, that refers to the initial surveying, the predominant category refers to aspects of quantum physics but few of these are characterizing elements of QM as a theory; in figure zzz, the basic aspects of the theory are shown. What can be clearly seen is the pre-eminence of the polarization context, underlining that for 12 teachers the reference context stayed the one of the polarization, for other 5 the context of diffraction and spin (not present in the diagram because its frequency is 2) are also introduced.

38. Uncertainty principle
Decrease
Measurement theory, incompatibility concept, concept of state:
More attention

39. Uncertainty principle Decrease
Web-forum F2
Uncertainty principle
Decrease
Measurement theory, incompatibility concept, concept of state:
More attention
With respect to the reference proposal it emerges a smaller attention to:
quantum particles and trajectories
identity of quantum particles
role of quantum interference in showing the QM peculiarities with respect to classical physics.

40. Case studies
Here we present in details a case of low-level student-teacher and a high level student-teacher [low/high level from what emerged in the pre-questionnaire]
Student-Teachers F (low level):
MQ Aspects to be proposed to high school :
the photoelectric effect,
the Compton effect,
the two-slits experiment
Aspects characterizing QM vs CM : indetermination principle and the quantization (discrete value)
QM includes CM as limit case and the passing from the micro to the macro world consists in the passing of quantities assuming discrete values to quantities assuming continuous values.

41. Measurement of an observable
 F believes that the interaction with the measurement apparatus produces a perturbation on the system, while “the probabilistic aspect is present even in thermodynamics and the measurement indetermination is always present in CM, even if it can be reduced” and that “the dependence of the measured quantity from the measurement apparatus has very interesting philosophical consequences that could provide stimulus to our high school students often de-motivated”.
F. believes that this aspect could be proposed in-class considering the two-slits experiment, which can not be completely explained just using the wave –like behavior, nor the only corpuscular aspect of light.
F. correlates the uncertainty principle to the existence of incompatible observables, and the quantum indeterminism to the necessity of giving a definition of state different from the classic one.
F. believes that, even in principle, it is impossible to associate a trajectory to a particle (aspect which is correlated to the position-impulse indetermination).

42. Criticalities:
the different use of probability in the case of throwing a coin and in the case of the photon-Polaroid interaction (“in the case of throwing a coin there are only two possible final states which are equi-probable, while in the photon case there are infinite possible final states”, and F. says that he would not be able to explain this second case in-class)
the superposition principle (he does not answer about its physical meaning nor the possible didactical treatment);
Meaning of the vectorial description of a state (identification of vector of state and vectorial formalization of a measured physical observable):
 he says that the state vector of a particle having an impulse at 45° with respect to two orthogonal axes x and y is represented by a linear combination of vector a and b that represent the state of the particle having the impulse defined along the x and y axis respectively.

43. The QM education proposal of F is focused on the indetermination principle, that is the trait d’union between the historical roots of quantum ideas and part of the innovative concepts with respect to the classical vision (The superposition principle, the particle indistinguishability, the impossibility to associate a trajectory to quantum particles).

44. The teacher student H – High level
Final conceptual map (base for educational project)
Initial Conceptual Map

45. Student-teacher G –High level
The maps does not change noticeably. But, it is of quite importance the change he proposed (with respect to the reference one) by using a sequence of steps analogous to those in the reference proposal integrating the polarization context with the spin one, which was studied by means of simulation of Stern and Gerlach apparatus acting on atomic beams.

46. Student-teacher G –High level
The maps does not change noticeably. But, it is of quite importance the change he proposed (with respect to the reference one) by using a sequence of steps analogous to those in the reference proposal integrating the polarization context with the spin one, which was studied by means of simulation of Stern and Gerlach apparatus acting on atomic beams.

47. Conclusions The formative educational module on QM:
E phases proposed to 22 selected student-teachers
The core of this work:
analysis and discussion of the proposed learning/teaching module, and its tutorials,
focused on the construction of basic concepts of the theory and on the recognition of the conceptual role of formalism.

48. RQ1 - even well prepared teachers have a vision of the teaching of QM oriented to the physics of quanta
It also emerged that the uncertainty principle is considered a key one in QM,
likewise, for many other student-teachers, the quantization of physical observables (discrete spectrum).

49. RQ2 - main difficulties encountered on the learning path are about leaving context usually explored with high school students (e.g. the uncertainty principle known only for position and impulse, but not correlated to incompatible observables in general; the context of free propagation, rather that contexts of two-state systems which are simpler) and the known approaches.

50. As far as RQ3 is concerned, what produced the main changes in PCK:
The didactical proposal focused on basic concepts of QM and on the detailed analysis of the instructional materials;
The rich exchange by peers and engagement on the web about the different basic concepts of QM followed in the reference didactical path;
The direct involvement in the construction of educational path and tutorials
The main changes are the passing
from physics of quanta that has as final goal the indetermination principle,
to a vision in which the superposition principle has an important role in the organization of the didactical proposal of the quantum theory.

51. In the designing of personal didactical paths, superposition principle but attention focused on the measurement theory and on the concept of state (the focuses of the web-discussion)
In the specific case of high level competence student-teachers, the main integrations are about new phenomenological contexts with a similar approach to the one proposed, like the one of the spin and the diffraction phenomenology (RQ4).