One of the reasons I really like the MEI OCR A-Level Mathematics is the section 6 in the Core 1 unit entitled "the Language of Mathematics". I am a massive believer in mathematics being a tool we use to communicate abstract ideas and to translate real-life situations into mathematical models that can be logically analysed. I love working on proofs with pupils and using proper maths notation and symbology, including notation for different subsets of the complex numbers - including N for the natural numbers.
It was a few years into my teaching career that it suddenly dawned upon me the links between the set notation I had studied in my degree, and the algebra I was using with my pupils - x as a member of R, n s a member of N, or z as a member of C; and in particular how I could apply this to my teaching. I started off by making it clear to my upper set GCSE and A-Level pupils why certain letters were used in certain contexts (i.e. n is used in sequences because it refers to only whole numbers, whilst the assumption is that we use x when values could be fractions or irrational numbers) but more recently I have started using the idea with lower attaining pupils, particularly with sequences. I am writing about it today because I introduced it with Year 8 set 4 (of 5) with fabulous results. Starting with the natural numbers being n, we looked at defining the different times tables (2n, 3n etc) as well as sequences like the square numbers, all starting from the natural numbers. By the end of the lesson I had several people being able to find the first 5 terms of n3 + n2 + n (remember, set 4 of 5 in Year 8!) and even some being able to explain why 154 was not in the sequence 3n - 1. The idea of the natural numbers, and building other sequences from them for me is much more powerful than some of the other approaches to position to term rule (ghost number is one that seemed to grow in popularity). Just be clear the idea is relatively straight forward and looks like this:
"Write the first 5 terms of the sequence 4n - 3"
"4n is the 4 times table, so it goes 4, 8, 12, 16, 20, ...
So the sequence 4n - 3 is each number in the 4 times table subtract 3, i.e. 4-3, 8-3, 12-3, 16-3, 20-3, ...
So 4n - 3 = 1, 5, 9, 13, 17, ...."
So if you are teaching about ghost numbers, or another approach, consider this as an alternative that should hopefully lead to increased conceptual understanding.