Intrinsic coordinates are included in the A level Further Mathematics syllabus followed at my school. In the text books produced specifically for our examination board, intrinsic coordinates are introduced via a diagram similar to the one given here from Wikipedia.
Our book then says `From some fixed point of the curve the length of the arc is and the angle often called the gradient angle, is the angle made by the tangent to the curve at with the positive direction of the -axis. It is clear that for every point on the curve, and continuously change and so do and ’.
My immediate questions are the following:
Why is it OK to talk about `the’ angle? Which angle to we mean if the function is decreasing? If we mean the obtuse angle, how can vary continuously as moves over a minimum? If is to vary continuously, it must pass through 0 and become negative, so is it that is defined by = tan ? If we do take this definition for why is there no mention of the fact that can be negative?
I consulted various other books and my colleagues, and looked for something helpful on the web, but made no progress. The diagram and explanation above appear to be the standard ones, but do not help me (or my students) to answer to my questions. The most comforting response I received was from a retired teacher who told me that I was right to be thinking about it, but that the exam questions would avoid any potential difficulties.
I should like to know your thoughts on these questions and about the value of introducing this topic so informally this at this stage in a student’s education. In only a few months these students will be encountering real analysis and abstract algebra, and be expected to construct formal proofs. How does this style of `pure’ maths prepare them for this?