Name: Meridith Who is asking: Student Level: Secondary
Question: Hi Meridith As a teacher at multiple levels, I would say the main use for studying some math 'beyond what you will be teaching' is perspective. There are choices you make in teaching at prior levels which are formed by where it will take the students. Let me give a few examples. Integration and Algetiles. There is a basic 'metaphor' which connects the geometric measurements where one divides a region into two pieces and says that the 'area' of the whole is the sum of the areas of the pieces' (in the plane). A similar process works for volume in 3space, and length on the line. This metaphor is used in algetiles, when one compares areas for formulae such as A(B+C) = AB + AC, etc. It is also used as the basic metaphor for patterns called 'proofs without words' in which one uses a visual shape (actually areas of pieces and the whole) to 'see' patterns such as: 1 + 3 + 5 + ... (2n1) = n^{ 2}. This metaphor is also at the heart of integration. (By the way, integration for area and volume predates differentiation by over a thousand years  Archimedes effectively used integration, as did Pappus). In my experience, students come out of high school without having thought about the basic pattern  a pattern which is at the heart of senior undergrad, grad courses on 'measure theory'. Similarly, in geometry, it really helps to understand what current higher geoemetry is about. It is about transformations, symmetry, etc. Ideas like 'congruence', SAS, etc. are at their core, about transforamtions. What can be moved onto what  and what properties are unchanged (invariant) under the effects of such transformations. If one has not looked more deeply into geometry (as it is applied today in diverse areas like CAD, Robotics, protein modeling ... ) then it is common to get wrapped up in 'logic games' around proofs, and forms of proofs. Logic has its place, but it does not actually teach geometric thinking of the types that will be needed by students if they go on into computer science (computational geometry); chemistry (stereochemistry and 3D symmetry); engineering; ... In combinatorics  counting problems, I have seen the difference it makes to have a larger overview. At one level there are many small changes in wording of counting problems that turn out to make a big difference in how one calculates. If you have a larger vision of how these variants work, some higher understanding (again related to symmetry), it seems like a mess of small differences and memorized pieces. [I once gave my class an assignment, then had a replacement teacher take it up the next day. He gave different answers than I would  he did not have the larger understanding, coming from something called Polya's counting theorem.] I hope these examples give some idea when it is a real advantage to a teacher if they know more of what 'comes next' in the progression of mathematics. There are connections. These connections make a difference in being able to talk with students about 'why are we doing this?'. Walter WhiteleyMathematics and Statistics York University
