dimensional scaffolding

Concept

The idea of scaffolding is proposed as a didactic strategy for modern physics for several reasons. The term is already used in the theoretical framework of constructivism to reflect the idea of a gradual and sequenced construction of knowledge instead of confronting the student with a complete theoretical edifice. In the case of science teaching, the conceptual scaffolding is facilitated by experimental activities, in the style of open activities with a degree of difficulty and a level of interpretation graduated to the student's ability. It is about structuring the student's thinking in a flexible way, so that they are able to apply it in different situations. When proposing the concept of dimensional scaffolding for the teaching of physics, especially modern physics, we take into account that many of the physical concepts, although they only acquire their full meaning in a three-dimensional space (or in the corresponding four-dimensional space-time) they admit a dimensional reduction while maintaining many of their relevant properties. This can be applied in an inverse sense, that is, presenting first the students with these concepts in spaces of reduced dimensionality. The advantage of this approach, on the one hand, is the ease of visually presenting and interpreting diagrams, figures and graphs in reduced dimensions, On the other hand, once the physical properties in these reduced dimensions are understood, scaling to larger dimensions can be performed. fluently.

Mechanics

From a didactic point of view, it is convenient to find the balance point between the complexity of the concepts and their visualization, taking into account that our ability to perceive geometric relationships varies depending on the dimensions. For one or two dimensions, sensory perception is complete and direct, for three dimensions we have to resort to conventions such as perspective, and for four dimensions (which is the dimensionality of real physical spacetime) our mind lacks resources for direct visual understanding. 

Teaching progression

Thus, for example, focusing on mechanics, in the primary stage there are situations with moving objects (trains, cars, etc.) that are solved by applying the notion of proportionality. In this case, the existence of a one-dimensional space in which the object moves is implicitly assumed. The same notion of proportionality also applies when comparing triangles to establish the size of an object from its shadow, in this case in two-dimensional space. In secondary school the uniform motion (u.r.m) in one dimension is presented first, to which time is added to analyze the corresponding two-dimensional graphs (position/time, speed/time, acceleration/time). It continues in the same dimensionality until the uniformly accelerated movement (u.a.r.m), which has a greater cognitive demand. We also begin to study uniform circular motion (u.c.m), in a two-dimensional space with a fixed one-dimensional path (the circumference). The concept of vector is introduced in high schools, applied in two-dimensional situations (forces in statics, velocities in the plane, tangential and normal accelerations in the u.c.m), reaching the notion of curvilinear motion, both in the parabolic case (horizontal and oblique shots ), as well as in the more general case of the orbits of planets and satellites, to which Kepler's second law is applied from the outset to reduce them to motion in a plane. Also at the end of this stage the notion of relative motion is incorporated (again in a single dimension) to present some relativistic phenomena in a qualitative way. The Lorentz transformation sometimes appears to visually represent these phenomena, in this case adding time to reach the third level of depth, relativistic spacetime, in its simplest version (1,1). In the first university courses, three-dimensional space is introduced to arrive at notions such as Coriolis acceleration, helical or warped trajectories, and the notion of (spatial) curvature is contemplated through metric analysis (algebraic geometry and tensor calculus). The theory of relativity is endowed with a complete four-dimensional formulation (four-vector), to which the tensor calculus is incorporated to arrive at the formulations of general relativity that place us at the current frontiers of modern physics (spacetime curvature, black holes, gravitational waves, etc).

Learning Goals

Depending on the academic trajectory of each individual, the progression through this dimensional scaffolding ends at certain levels. A theoretical physicist would go up to the maximum levels of dimensionality, depth and conceptual complexity, while a scientist of another type may not reach the four-dimensional formulations but does advance to some extent in the three-dimensional spaces, 3-D or (1,2 ) depending on the specialty. A person who finishes science studies at high school without pursuing a scientific career would stay in the two-dimensional space and (perhaps) the simplest space-time (1,1), while following other high school branches -or if he does not study beyond secondary education- the progression in the dimensional scaffolding would stop in previous levels. Physics teachers should take into account, at each stage, the situation point in the dimensional scaffolding in which they are placing students, to adapt their teaching strategies in the sense of facilitating their gradual progression through it.

Teaching sequence

Proposal of a didactic sequence for mechanics and electromagnetism

We have already seen that dimensional scaffolding is implicitly present in normal physics teaching practices. Our intention is twofold: -On the one hand, to make explicit and to provide coherence and theoretical background to the habitual practice in the field of traditional teaching of physics. -On the other hand, and based on pilot experiences already carried out, as well as theoretical reflections on the applicability of physical notions to scenarios of reduced dimensions, we intend to expand and deepen this didactic strategy to adapt it to the teaching and learning of modern physics.

Secondary Education

It would be very convenient to devote some attention to the graphical way in which masses can be compared in a collision using a so-called space-time balance, a concept introduced in our PhD thesis (https://minerva.usc.es/xmlui/handle/10347/3640, in Galician), since it will be the way in which Einstein's equation E = mc2 can be presented in future stages. A summary of that thesis is in English here (https://www.researchgate.net/publication/352167440_A_DIDACTIC_PROPOSAL_FOR_THE_VISUAL_TEACHING_OF_THE_THEORY_OF_RELATIVITY_IN_HIGH_SCHOOL_FIRST_COURSE). At this level, work can begin to be done in two dimensions, starting with circular motion and the composition of forces (static) in the plane, starting with its representation by means of arrows or oriented segments, which we will call vectors, and which admit a double algebraic representation (Cartesian or polar), and operations such as addition, subtraction or product by a scalar.

High School

Once in high school, we will stick to scientific courses, that is, those that have the subject of physics in their curriculum. In the first year of high school students will work with the physical magnitudes in the plane (two-dimensional space), for which the concept of vector is deepened, highlighting the possibility of presenting it in three different and complementary ways: -through its Cartesian components (x, y ), -through its polar components (module and direction), as well as -through a graphical representation (oriented segment or arrow). The operations of addition, subtraction, and product by a scalar (started in the previous course) are now complemented by the product between vectors. 

It is possible to introduce new evidence, such as the invariance of the speed of light, into the graphic analysis of the principle of relativity started in previous courses, and arrive in an elegantly simple way at the Lorentz transformation (a rhombus diagonally inclined and conserving the area) and its physical consequences (temporal dilation, spatial contraction, limiting velocity and equivalence between mass and energy). The geometric deduction of the Lorentz transformation is presented schematically in the “Implications for Teaching” section of our article on learning S.R.T. (https://arxiv.org/abs/2012.15149). We will have thus outlined one of the didactic pitfalls found in the research on the teaching and learning of S.R.T: the convenience of widely using the transformation between I.R.S is shown to be incompatible with the conservation of the speed of light. This derives from the (implicit) use of Galileo's transformation, and is an inevitable consequence of it. By using the Lorentz transformation from the beginning (which according to our research results is within the reach of students at this stage) this important difficulty is radically avoided.

The notions of plane trigonometry (Euclidean) handled in the mathematics subject of previous courses can be complemented with an introduction to the role of bivectors in spherical trigonometry, as well as the concept of Minkowskian trigonometry, which allows the use of hyperbolic functions, which are available in every scientific calculator. It is precisely with the help of these functions that the relativistic gamma factor can be calculated in a simpler way than with the usual algebraic formulation. Since it is a trigonometric function, it has a strict visual interpretation that is easily recognizable in any space-time diagram, which greatly facilitates its integration in a sequential didactic proposal like this one.

In mechanics, in this course planetary motion and orbits of satellites can be approached in a general way by applying the formalism of geometric algebra. Aspects and issues related to the frontiers of physics such as gravitational waves, black holes, Big Bang cosmology, etc. can also be included qualitatively through these diagrams. With the help of these space-time diagrams (1,1), situations such as collisions (elastic or inelastic) can be analyzed as presented in the article “Archimedes meets Einstein” (https://www.researchgate.net/publication/323370346_Archimedes_meets_Einstein_A_millennial_geometric_bridge), in which the Compton effect is interpreted in a one-dimensional version. This is one of the many links between relativistic mechanics and electromagnetism that can be approached at this stage in a preliminary and qualitative way (or semi-quantitative, as they are very simplified versions of real phenomena) to anticipate a subsequent rigorous analysis in case to pursue physics studies. Otherwise, it will be part of the scientific background of these people, helping to found a scientific community with solid foundations in all areas, including modern physics.