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Physics is undergoing rapid developments in quantum technology, and physics students need updated and relevant quantum physics education to equip them for the future. Quantum measurement and superposition are important concepts in quantum technology, but there are few studies on the teaching and learning of these in secondary education. In this paper, we use interview data to study how Norwegian upper secondary students described superposition after having worked with digital learning resources emphasizing qualitative understanding and visualizations, and we compare them to descriptions from university students having completed an introductory quantum mechanics course. Using pedagogical link-making as a lens, we discuss how upper secondary physics can support students’ knowledge building without access to all the building blocks that university students have.
A review by Bouchée et al.1 concluded that students have difficulties with relating the mathematical formalism of quantum mechanics to their own real-world experiences, interpreting nonintuitive quantum physics phenomena and concepts, moving from a deterministic to a probabilistic worldview, and understanding the limitations of language when describing quantum phenomena, concepts, and objects. The latter includes a tendency to interpret classical metaphors and visualizations too literally. Krijtenburg-Lewerissa et al.2 reviewed physics education research on quantum physics and found that two of the least studied topics are quantum measurement and superposition. One paper on superposition found that students were good at calculating probabilities of different outcomes but had trouble identifying the experimental implications of a superposition state.3 Two other studies came from the Norwegian ReleQuant project, which developed digital learning resources for upper secondary quantum physics, emphasizing qualitative understanding and visualizations. Myhrehagen and Bungum4 analyzed secondary students’ reflections on the Schrödinger’s cat thought experiment as part of working with the learning resources, and found that in that context, students often interpreted superposition literally, as quantum objects being in several positions at once. This happened despite the learning resources presenting superposition in terms of quantum states, using two-state visual analogies. Huseby and Bungum5 found that students’ discussions of the electron double-slit experiment revealed a view of measurement as passive observation. They connected this result to a video used in the learning resource, in which Dr. Quantum6 presents the double-slit experiment and illustrates measurement with an eye watching passively.
Here, we present findings from a follow-up study of ReleQuant that uses data from undergraduate university students in addition to upper secondary students, to better understand how building quantum physics knowledge differs on these two educational levels. Our research question is:
How do upper secondary and university students describe superposition, and what can the descriptions tell us about how to support students’ knowledge building about superposition?
Methods
The data comprise 12 semi-structured interviews with 18 upper secondary students and 7 undergraduate university students. The secondary students were in their last year of upper secondary schools in the greater Oslo area and had all used the ReleQuant learning resources including the Dr Quantum video. The university students had just finished an introductory course in quantum physics at the University of Oslo, which emphasized both conceptual understanding and the formalism of quantum mechanics. Most students were interviewed in pairs or groups of three to encourage discussions between students, but for practical reasons, two university students were interviewed alone. The main topics discussed were quantum measurement, superposition, and the double-slit experiment. The first author conducted all interviews, transcribed the audio recordings, and did the analysis. The second author coded parts of the material for validation and took part in discussions and interpretations of data. The coding used was deductive codes from Myhrehagen and Bungum4 as well as inductive codes.
To find out what the descriptions tell us about how to support students’ knowledge building, we draw on pedagogical link-making,7 a framework for teaching and learning scientific conceptual knowledge. The framework takes a constructivist and sociocultural perspective where students build knowledge through making links between new ideas and what they already know, their understanding deepening as more and better links are made. Pedagogical link-making is how teaching supports students in making these links. The framework can be used to better understand students’ learning of conceptual knowledge, and how to support such learning in teaching. We use the framework to understand students’ descriptions of superposition, and to discuss how teaching can support their knowledge building on this topic. The framework describes three types of pedagogical link-making: Link-making to support knowledge building promotes students’ learning by making links between different kinds of knowledge, such as different concepts (e.g., energy and heat) or different forms of representation (e.g., the concept kinetic energy and its mathematical formula). Link-making to promote continuity connects learning experiences separated in time (e.g., referring to last week’s lesson on kinetic energy when introducing heat). Here, we also include connections to different contexts (e.g., linking heat in physics to heat released in a chemical reaction). Link-making to encourage emotional engagement makes students interested or in other ways emotionally engaged with the subject matter (e.g., linking abstract concepts to experiments that can spark students’ interest).
Results and discussion
We identified six categories of descriptions of superposition (Table I). Below we focus on the dominant categories for secondary and university students, respectively: literal interpretations meaning that quantum objects can be in several positions at once and superposition as a mathematical sum of states. We illustrate them with example quotes and interpret and discuss them using our theoretical framework and the research literature.
Categories of descriptions and how many upper secondary (sec) and university (univ) students used them.
| Descriptions of Superposition | Used by | |
|---|---|---|
| Sec (N = 18) | Univ (N = 7) | |
| Literal interpretation, quantum objects can be in several positions at once | 14 | 0 |
| A consequence of lack of human knowledge | 6 | 1 |
| Either-or, the quantum object is in one of several possible states or values | 5 | 0 |
| The quantum object is in all or several possible states or values at once | 5 | 2 |
| Probabilities of events | 5 | 3 |
| A mathematical sum of states | 0 | 6 |
| Descriptions of Superposition | Used by | |
|---|---|---|
| Sec (N = 18) | Univ (N = 7) | |
| Literal interpretation, quantum objects can be in several positions at once | 14 | 0 |
| A consequence of lack of human knowledge | 6 | 1 |
| Either-or, the quantum object is in one of several possible states or values | 5 | 0 |
| The quantum object is in all or several possible states or values at once | 5 | 2 |
| Probabilities of events | 5 | 3 |
| A mathematical sum of states | 0 | 6 |
As many as 14 out of 18 secondary students, but no university students, described superposition in a literal way. One of them said:
An electron can, or, in quantum physics you have superposition, which is when an electron or something can be, like, in several places at once.
Although the learning resources present superposition in terms of states first, the prominence of the literal descriptions suggests that the learning resources have primarily established links between the concept superposition and its meaning in terms of position, not states. This is likely due to the Dr. Quantum video of the double-slit experiment. The film with its visualizations and emphasis on the mystery of the experiment is also likely to elicit emotional engagement, promoting learning.7 One secondary student said:
And that is cool, because I don’t understand it. […] That something can be several places at once and at the same time only in one place.
Moreover, the secondary students expressed that superposition is not intuitive, as students often find.1 The literal interpretation expresses a limited understanding of superposition, linking it to only one physical property/variable (position), and not to any mathematical representation. The literal interpretation is also tough to reconcile with a classical understanding of the concept particle, a common result of overinterpreting analogies1 that we also saw in our data. This is possibly caused by Dr. Quantum saying that “An electron is a tiny, tiny bit of matter, like a tiny marble” and using small balls to illustrate electrons. Some secondary students tried to reconcile this confusion by also using other descriptions of superposition, for example the either-or category as shown here:
I think it is strange [that the electron can be in two different places at once], so I think that maybe it is more that the probability for that it is … that it really is only in one of them.
The university students, however, seemed to avoid confusion by describing superposition with links to mathematics. Six of seven university students did this, for example:
It is a linear combination of different states where each has its own eigenvalue. So that if you do a measurement on a superposition, you have a certain probability of getting one of the eigenvalues that is given by the coefficients in the linear combination.
Here, the student expresses helpful links between superposition and the mathematical concepts linear combination, eigenvalues, probability, and coefficients. The student has also established links between the mathematics and what it means for measurements in physics. In other sections of the interviews, some university students expressed continuity links to superposition in other contexts, such as classical wave physics and electromagnetics. And, importantly, they referred to states in terms of several measurable variables, as in this quote:
… position, spin, angular momentum, speed, and momentum. All of them can be in a superposition.
The descriptions of superposition as a mathematical sum of states that can refer to several physical observables represent a more developed understanding than the literal interpretation. In contrast, it appears that the lack of mathematics and other contexts available to upper secondary students hinders their knowledge building about superposition, especially when they are replaced by classical metaphors and visualizations that are difficult to link meaningfully. Topics like mathematical quantum mechanics or superposition of waves in general are rarely available in secondary school. However, it seems promising to focus on discrete states and superposition for different variables such as energy or spin. Studies where two-state systems were used to teach quantum physics show that students are led away from many common misconceptions, the mathematics can be presented more simply than for the continuous position variable, and the approaches can include fascinating experiments that can boost students’ interest.8,9
Our results also suggest that secondary physics teaching should avoid replacing the mathematics with literal presentations of superposition, and that classical notions of particles as tiny marbles should be handled with care. This means avoiding the Dr. Quantum video and other classical particle visualizations. Moreover, we can promote students’ knowledge building by explicitly discussing different meanings of the concept particle, the classical marble meaning and the quantum object meaning. By making meaningful links between the two and from them to experiments such as the double-slit experiment, we can hopefully help students build a better and less confusing understanding of quantum physics while maintaining their fascination for what it really means about the world at its smallest.
References
Nicoline Berit Campbell Birkeland received her master’s degree in physics from the University of Oslo in 2024. Her field of research was physics education, and her thesis was entitled, “And that is cool, because I don’t understand it. A study of secondary and university students’ descriptions and understanding of measurement and superposition in quantum physics.”
Maria Vetleseter Bøe is an associate professor at the University of Oslo. She has a master’s degree in theoretical physics from the Norwegian University of Science and Technology and a PhD in physics education from the University of Oslo. Her main research interests are quantum physics teaching and learning and motivation and identity in physics education. [email protected]