Quantum probability and quantum spin are foundational concepts in quantum physics, not normally taught in pre-university education. In line with recent efforts to modernize quantum science education in schools, this paper presents learning sequences designed to provide an introductory conceptual understanding of these core physical concepts. This is achieved by choosing physical models that provide tangible classical representations that can be contrasted with quantum observations relevant to modern science and technology. This approach allows activity-based learning without the use of algebra. The concept of quantum probability is taught through activities with phasor wheels in which quantum probability is obtained graphically and interpreted using single-photon interference data. Quantum spin and spin vectors are taught by contrasting quantum spin with classical spin, explored through activities with gyroscopes and spinning tops. Quantum concepts are evoked by emphasizing how spin is different and how it manifests in real-world technologies such as magnets and MRI imaging. We describe learning sequences developed following the Model of Educational Reconstruction in which historical discoveries are combined with activities based on classical models and relevant quantum applications. We summarize test results from programs conducted with students aged 11–15. Most students connect the models and activities to quantum concepts and retain knowledge of quantum spin six months after the program. The successful learning outcomes indicate that teachers can effectively introduce quantum probability and quantum spin concepts to middle school using learning sequences such as those presented here.
Quantum probability and quantum spin are foundational concepts in quantum physics, not normally taught in pre-university education. In line with recent efforts to modernize quantum science education in schools, this paper presents learning sequences designed to provide an introductory conceptual understanding of these core physical concepts. This is achieved by choosing physical models that provide tangible classical representations that can be contrasted with quantum observations relevant to modern science and technology. This approach allows activity-based learning without the use of algebra. The concept of quantum probability is taught through activities with phasor wheels in which quantum probability is obtained graphically and interpreted using single-photon interference data. Quantum spin and spin vectors are taught by contrasting quantum spin with classical spin, explored through activities with gyroscopes and spinning tops. Quantum concepts are evoked by emphasizing how spin is different and how it manifests in real-world technologies such as magnets and MRI imaging. We describe learning sequences developed following the Model of Educational Reconstruction in which historical discoveries are combined with activities based on classical models and relevant quantum applications. We summarize test results from programs conducted with students aged 11–15. Most students connect the models and activities to quantum concepts and retain knowledge of quantum spin six months after the program. The successful learning outcomes indicate that teachers can effectively introduce quantum probability and quantum spin concepts to middle school using learning sequences such as those presented here.
Abstract Teaching and learning quantum mechanics is one of the most demanding educational and conceptual challenges, in particular in secondary schools where students do not possess an adequate mathematical background to effectively support the description of quantum behaviour. Educational research shows that traditional approaches, generally based on historical and narrative perspectives, are only partially effective. The reason is that they do not address in depth those basic quantum concepts that radically question the fundamentals of classical physics. A research-based educational program has been proposed to two final-year classes of an Italian scientific high school. In order to build the main concepts of quantum mechanics and their formal basic representation via real and simulated experiments, the program uses the light polarization as a context. A quantum game was then integrated in the educational program, to support students’ learning. Their conceptual paths, monitored by means of tutorials and questionnaires, show significant student learning especially on the concept of state and on appropriating the formalism meaning, whereas students more frequently referred to the geometrical vector representation instead of the algebraic-analytic formula. The quantum game has emerged to support intuition and operative experience in distinguishing the foundational concepts of superposition and entanglement.
Quantum mechanics (QM) has taken on a paradigmatic role in how physics describes and builds new knowledge about the world. It also has a growing role in our society thanks to recent and rapid developments in the field of new quantum technologies. These are becoming increasingly important in our society, in the world of work and which will also decisively influence future socio-economic choices. Also for this reason QM has an increasing role in the education of young people and has been introduced in many different ways also in secondary school in all continents. The change in the theoretical and cultural frame has profound implications that are often not intuitive, which involve the appropriation of a specific way of thinking. Neither macroscopic phenomenology, nor intuition help, because there is no simple evidence of how quantum objects must be thought to interpret that unobservable microscopic world and for which we must enter another interpretative paradigm even if we want to limit ourselves to describe it. PER has carried out several studies on the learning of key concepts, on the learning nodes, the ways of reasoning, the difficulties of the students, their spontaneous ideas, according to two main strands: above all at university level, conceptual difficulties and misunderstandings often linked to a formal approach, which, moreover, underpins the very nature of the entities involved; at secondary school level in relation to how students acquire a quantum view of phenomena on the basis of specific approaches adopted to address QM. In particular, a vast literature, which has increased considerably and especially in relatively recent times, has studied how university students deal with certain concepts such as that of state, superposition, measurement in quantum mechanics rather than how quantum quantities evolve over time. These studies are based on the use of questionnaires that allow to explore different levels: the verbal, formal and representative one, with different representations (eg. wave function, Dirac notation), rather than different representative modes (eg. Graphs of probability amplitudes, or probability distributions). At the secondary school level, the most important studies on student learning were instead conducted evaluating the educational impact and outlining the learning paths of students who had followed formative activities based on differentiated innovative research-based educational paths. In particular, the learning outcomes were compared to those activated by proposals based on the historical reconstruction approach of quantum physics. This approach is privileged in traditional curricula and school texts and consequently in the ways of teachers approaching the topic. It was developed in schools with a narrative or descriptive dimension of solutions based on old physics of quanta for the interpretation of phenomena not explained by classical physics. The innovative approaches instead make use of contexts whose formal treatment is simple (quantum properties described in a two-state space such as polarization or spin) with the aid of simulations that compensate for the problem of students” limited mathematical-formal skills, avoiding any historical reference. In particular, there is a significant difference between the survey tools and the survey contexts in the high school environment compared to surveys conducted with university students. This paper presents a review of the research literature, focusing in particular on the learning processes of secondary school students and outlining how research perspectives have evolved.
This chapter presents teachers and students with the fundamental concepts and principles of quantum computing in a fashion that is both for learners and educators. As quantum technologies are breaking into the lead roles in innovation, integrating quantum literacy into curriculum education is becoming increasingly essential. This chapter makes difficult concepts like qubits, superposition, entanglement, and quantum gates easy to understand and what makes classical and quantum computation distinct in easy-to-understand terms. It offers teachers effective methods for approaching quantum concepts through analogs, simulations, and hands-on devices like IBM’s Quantum Experience. For students, this chapter opens up inquiry by means of interactive and inquiry-based approaches, leading to increased understanding. Interdisciplinary education—physics, math, and computer science—is used as a starting point for the study of quantum mechanics and computing applications. It surveys current trends in quantum curriculum design and offers feasible case studies of effective quantum education initiatives in schools and universities. Problems of the absence of qualified teachers, lack of access to quantum materials, and learning inequalities are observed and accompanied by advice regarding scalable and inclusive methods of pedagogy.
Teaching quantum mechanics at the secondary school level remains a major educational challenge. Traditional historical approaches often present the theory as a sequence of disconnected phenomena, requiring extensive prior knowledge of classical physics before students can appreciate the conceptual structure of quantum physics. In this work, we present the design, implementation, and evaluation of a curriculum based on the two state (qubit first) approach, in which the fundamental postulates of quantum mechanics are introduced through mathematically accessible two level systems and realized with light polarization. The curriculum integrates theoretical instruction, laboratory activities, a coherent progression grounded in the interactive simulations, and inquiry based learning into a coherent pathway that progresses from quantum states and superposition to entanglement and the experimental violation of Bell's inequalities. A distinctive feature of the project is the experimental realization of the violation of Bell's inequalities obtained in a newly established quantum optics laboratory in an Italian high school, where students performed, for the first time in Italy, an authentic Bell test using polarization-entangled photons. The same educational framework was adopted for in service teacher professional development courses. Learning outcomes were evaluated through mixed methods. Preliminary analysis indicates significant improvements in conceptual understanding, confidence in using mathematical formalism, and appreciation of the relationship between theory, experimentation, and the foundations of quantum mechanics. The proposed curriculum shows that two-state formalism and supported by real laboratory experiences enables secondary-school students to engage more meaningfully with both the conceptual foundations of quantum mechanics and their experimental verification.
Quantum computing has emerged as one of the most influential technological developments of the twenty-first century. While considerable attention has been devoted to hardware development, algorithm design, and industrial applications, comparatively less emphasis has been placed on the educational challenges associated with introducing quantum concepts to engineering students. Most available learning resources are designed primarily for physics students and often require prior knowledge of advanced quantum mechanics. Consequently, engineering students frequently encounter significant conceptual barriers when attempting to understand quantum computation. This chapter presents a mathematics-first educational framework aimed at simplifying the learning process for engineering students. Instead of introducing quantum theory through wave mechanics and advanced physical interpretations, the framework begins with familiar mathematical concepts such as vectors, matrices, probability, and linear transformations. The chapter demonstrates how core quantum concepts can be taught using accessible mathematical tools already encountered in undergraduate engineering curricula. A structured educational model is proposed and illustrated through practical examples, matrix operations, simple quantum gates, and engineering-oriented learning activities. The framework is intended to reduce cognitive barriers, improve conceptual understanding, and encourage interdisciplinary learning among future engineers. The chapter further discusses potential applications of quantum mathematics in thermal engineering, computational fluid dynamics, optimization, artificial intelligence, and advanced manufacturing.
Abstract In the last decades, empirical investigations have shown the existence of deep and persistent difficulties with teaching/learning the time independent Schrödinger equation and time evolution. At the same time, research has provided evidence in favor of engaging students in modeling activities to help them improve their understanding. However, there is a lack of model-based curriculum materials for teaching the concepts of energy and dynamics in quantum mechanics. To illustrate how modeling could be included in the learning of these topics, we outline an educational proposal in the context of a spins-first approach. As a starting point to introduce the energy concept for atoms and particles, the learning path leverages student knowledge of the Bohr model. The energy level structure of a hydrogen atom with transitions accompanied by the absorption or emission of a photon provides an intuitive basis for activating the modeling of energy, that is organized around the analysis of contrasting cases on the possible results of spin flip transitions in a magnetic field. In this process, students are guided to build a mathematical representation of energy and to investigate its nature up to a derivation of the energy eigenvalue equation. The learning path allows students to compare and contrast the semi-classical Bohr model with the quantum model, as a support to help them build a consistent understanding of the latter. A different set of contrasting cases on the possible results of spin measurements after different amounts of time in the field is used to activate a qualitative modeling process on time evolution, which is then quantitatively examined by analyzing graphs of the state tomography, guiding students to explore the basic features of quantum dynamics up to a derivation of the Schrödinger equation.