Experiences of transforming the context of the subject matter of an advanced, graduate mathematics course with overly abstract and specific concepts into integrated, real world acoustic context are discussed while still keeping the most important learning objectives of the original advanced mathematical course. The typical student of the course is interested in acoustics, has theoretical and experimental experience of acoustics but is frequently neither skillful nor interested in abstract, advanced mathematics. The only prerequisites are completed courses in mathematics and mechanics at undergraduate level. The learning/teaching spiral of the course typically starts by introducing a concrete example, such as a baby on a swing, and then mathematically trying to model the baby’s motion by the wellknown Newton´s Second Law. Then the equations of motion are slightly generalized and a general solution is derived. Subsequently, home assignments are handed out where the method taught is applied to a slightly new acoustical situation. The course material thus relates to the students’ personal experience and to prior courses, while amplifying the transfer value to new applications, with increasing learning motivation and attention. The outcome of the home assignment frequently shows new insights, such as amplitude dependent period time, a harmonic excitation resulting in multi-harmonic response and other uncommon linear acoustical results. The results are due to the non-linearity of the system analyzed and are normally uncommon from previous modelling point of view but are, nevertheless, very common from practical point of view. Almost all students have experienced the results from real world practices although have never modelled it! Therefore, it is easily for the student to both assimilate the novel knowledge and accommodate it. Then the learning/teaching spiral includes continuously more realistic modelling features, such as damping, friction and viscosity. New insights are subsequently drawn. The course are learned by consecutively shifting between the concrete (relevant acoustical examples, phenomena, applications, hands-on, practical problem solving) and the abstract matter (symbols, principles, fundamental understanding). The students practice problem solving, evaluation and critical thinking skills. However, and most surprising, the students have learned an abstract perturbation method to solve non-linear, ordinary and partial differential equations – an advanced, graduate mathematical solution method, by transforming the overly abstract and specific mathematical context into an integrated, real world and well experienced acoustic context. And yes – we can learn advanced mathematics by acoustics!
