The barrier of mathematical knowledge is connected inseparably with the development of physics. It is drastically present in physics education. The development of physics proceeds in general through finding an appropriate mathematical apparatus needed to construct a model or theory. It has not always been the case that mathematics was waiting, "ready" to be applied in physics. Many times in history it was physics that was the driving force of the development of an appropriate area of mathematics. This happened with differential and integral calculus. And even if mathematics was ready, the physicist had to discover it anew for himself. This was the case of Einstein who studied non-Euclidean geometry that had been well-known to Hilbert who, however, was lacking in physical knowledge and intuition. Physics is inseparably entangled with mathematics. Still, the common adage that mathematics is the tool of physics is very simplistic - and not only because mathematics has been drawing upon physics for inspiration. The actual relationship is much more involved. It is hard to imagine what would become of mathematics without the empirical input, e.g. geometry. Freudenthal talks about mathematization of reality. And although physics does not usurp the right to embrace all reality, still all of physics is an example of mathematization. Such an interpretation is closer to studying and constructing mathematics along with formulating models of reality, thus the scenario of: "mathematics first, applications later" is obviously not true. Zeldovich develops his lecture on differential and integral calculus for high school students around the example of kinematics. He assumes that intuitive understanding of a physical concept will be helpful while constructing mathematical concepts, such as differentials or integrals. It is the very spirit of a proposal by Freudenthal to introduce the exponent through the example of radioactive decay.
However, in the practice of physics teaching it becomes indispensable to get the pupil familiarized in advance with mathematical concepts. Some craftsman's skill and habits are also vital. Starting with comprehending a simple proportion, the idea of surface area or volume as early as elementary school, up to special functions while studying quantum mechanics at the university level - it is the lack of adequate mathematical background that constitutes an obstacle in understanding new concepts. This obstacle may prove not surmountable. The barrier of mathematical knowledge manifests itself in those areas of physics where using adequate mathematical apparatus is not feasible at school. These areas are mechanics and electrodynamics. Even such an ordinary concept as velocity contains the mathematical notion of a limit, as well as elements from differential and integral calculus. The situation in electrodynamics is even more complex because of differential geometry involved. In order to formulate Maxwell's laws quite advanced concepts are needed, and they are difficult to be introduced in a model demonstration. This is a very serious obstacle. Our students do not usually possess the genius of Faraday who could manage without Maxwell's formalism. These difficulties have been surveyed to a large extent by mathematics educators.
Forgetting this obstacle constitutes a serious didactic error. Studying mathematics, just as much as studying physics, is swarming with cognitive difficulties, the lion's share of which is transferred on to physics. However, with carefully designed curriculum this obstacle may be levelled to a large extent. According to some physicists, e.g. Ginzburg, frequent calculation drills are necessary. Also avoiding mathematical description at all costs, at least in some areas of physics such as e.g. mechanics, is a very short-lived recipe, quite inefficient in long-term teaching.
Certainly, using false mathematical oversimplifications constitutes a major error. Not only do not such methods lead to better understanding, but they also bring a delusion of comprehension.
To conclude: good coordinated with physics, mathematical education is big help. Geometry should be a basic topic.