Culture DifferenceThere is a problem of culture that has grown over the years and is transmitted to each new generation of teachers in the training colleges. Physics and Maths teaching have become isolated from each other, with no cross-fertilisation. New styles have been habituated in each subject specialism and they have now become radically different breeds. "Oh, we don't have time for applications!" said one Maths teacher, when questioned.
How can they be so different?Physics teachers are trained and employed in a science context, with a focus on conceptual understanding, measurement, modelling and context. Mathematics teachers have become divorced from applications and have turned inwards. This is not necessarily undesirable, but many of their students (most, if you look at the Maths-Mechanics classes) study Physics and intend to enter Physics related degree courses. The Maths Departments' focus on narrowly defined problems leading to routine processing for a solution encourages students to rely on learnt techniques. This works fine for standard problems, but it is a distraction when dealing with unfamiliar problem types, which require a grasp of fundamental principles.
Student Coping MechanismMany students cope well with the differences, but a few always respond to difficulties badly: home study consists of learning the problem solving technique recipes and cramming for tests. When the unlearned concepts become a cause for declining scores, the response is to do more of the same. The next step is to request extra past papers to hone their technique, but this can lead to frustration as scores fail to improve and hour after hour are consumed chasing the wrong target. It is regular chore telling parents during meetings that their offspring are working terribly hard, but at the wrong things.
Now this is by no means the sole fault of Maths teachers, since this style of learning works well for GCSE Physics, but it is unfortunate that it also works well for A Level Maths. Many have never needed to get to grips with Physics concepts.
Incoherent MathematicsThe strangest difference I have come upon is that algebra is carried out using a bastardised version of quantity calculus. Quantity calculus, or quantity algebra, is the coherent system for dealing with physical quantities mathematically, as specified in the SI. Unit symbols are treated as mathematical entities, and the inclusion of units in workings is invaluable for helping students appreciate the physical basis of calculation, as well as helping them to spot errors when unexpected units appear with the solution.
My Maths teacher colleagues follow the exam board guidance, and claim to use SI units, but the units are all they use. A maths problem will specify, for example, that 'v = velocity in m/s', so the formula presented is unit specific, while in Physics the equations are valid for any coherent set of units, i.e. 'v = velocity'. The weight of a 300 kg mass is labelled as '300g' in a Maths problem, but with g defined as an acceleration, this makes the weight a simple multiple of an acceleration, not a force.
In Physics lessons, I expect my students to write '300 kg x g' to preserve the unit dimension. My colleagues told me that units were omitted because they caused confusion, with grams mixed up with the gravitational g, etc. Of course, there is no actual indication that such mix-ups actually happen. An additional inconsistency, Maths teachers are happy to write '1 mi = 1.6 km', without accepting that this means 'mi/km = 1.6', as this would give the units meaning outside the narrow unit specifications of variables.