Students in search of extension work in maths, particularly as far as Sixth Term Entrance Papers (STEP) are concerned, should peruse Carr’s Synopsis of results in Elementary Pure Mathematics.
This book is famous for being the volume used by the autodidact Ramanujan to teach himself mathematics. The book is merely a list of results and student how works through them, proving each one, will develop a strong ability in the subject. This life line into higher mathematics could not be produced in the same form in today’s educational and publishing context but, perhaps, Carr deserves a place as one of the subject’s great educators.
Higher mathematics is about generalisation and abstraction. Because of this, school mathematics students can find university text books unreadable. In such a context how can harder, higher level questions be set using the frame work of sixth form mathematics? The answer to this question is to wind the clock back to the 1880s and see how the subject was constructed then. Carr does this for us; many useful stages in school mathematics are listed here with extensions and generalisation written in generally sixth form recognisable form.
Take question 1 from this year’s (2016) STEP I exam. Pupils will only have to be familiar with the first six results to tackle this quickly.
The question play is all about the divisibility of,
which result 6, above, tells us is
Question game play leads us to factorise large numbers (without use of calculator):
in a difference of two squares (result 1) construction.
This preamble from Euclid’s Elements is where mathematics education still goes wrong, even after 2000 years or so.
Sound notions of what equality is are required as a bedrock of algebra. Dynamic approaches of teaching equation solving, in which terms and numbers move and change sign, seem to work at first but lack the simplicity of mathematical logic and create all sorts of problems when randomly, but apparently sensibly, applied.
Linking core maths 3 functional transformations to Step algebra and solutions of equations.
Abstract algebra, the un-codified implications of which need to be fully understood at school to gain good grades in exams. School algebra and numeracy is all about persuading pupils that the numbers we have work by these rules, and not by the various coping patterns they learned earlier in their education.
First six results in Carr’s Synopsis outline the gentle art of algebraic division.
Added as complement to last post.
This 7th century visualisation of quadratic algebra, thanks to Islamic scholars, is a great place to start the academic year in mathematics.
Rules of Indices provide an excellent first look at a developing mathematical structure. You start with a definition which provides a nice rule. You then generalise, trying to preserve the rule. What you get is a marvellous structure which is all a consequence of the first definition.
Euclid Book II proposition 9 and 10 proved by area decomposition marries algebra and geometry with a recipe for making Tartans.