Seeing that the class of 2016 is taking over the forums, I thought I'd summarise what I've learned in terms of studying for VCE Mathematics subjects - I'm trying not to repeat what's been said before, but here a few things that really helped me throughout year 12 (and the rest of what I've learned in recent years really) which I think I'd like to share. My personal experience has been in Methods and Spesh, although I guess this would be applicable to Further (it's still maths, albeit easier)
Conceptual understandingI cannot stress enough the importance of conceptually understanding every detail when studying mathematics. The basis of topics in maths, unlike the other sciences, are largely accessible in their entirety to the VCE student. This is in contrast to chemistry and to some degree physics, where you need to see with 'only one eye open' (or else do uni extension subject haha).
Conceptually understanding a formula, rule or interpretation in maths for me often involved understanding the
proof. Proofs are a very important part of mathematics, and sadly neglected in much of VCE maths. For instance, why is the derivative of sin(x) = cos(x)? What is the geometric meaning of a vector derivative? How is it derived? What about a definite vector integral? What does a limit actually mean? Why does Euler's formula work, and what factors may affect its accuracy? And how exactly does a Markov chain work? In terms of the derivative expression, why does a maximum correspond to a minimum on a reciprocal graph?
These are the questions which you need to ask yourself when studying mathematics - it is
NOT SIMPLY A FORMULA! (exception: probability - this can be very tricky, and the 'logical', conceptual conclusion may be different from the correct conclusion)
So in general, look for a geometric and algebraic interpretation for each topic you learn, be it functions and transformations, or differential equations. Geometric interpretations can be key to solving complex extended-response questions at VCAA level, such as the interpretation of a vector derivative (which is in a way a tangent of a space curve)- or the geometric parallels between complex numbers and vectors. It would be beneficial to read broadly (Paul's Maths Notes, and Khan Academy to a certain extent, but I definitely would recommend David Guichard - Single Variable Calculus (lyryx) (a free open e-textbook, google it)
The VCE level textbooks tend to be dry and uninteresting, and Essentials is probably the best. However, the above (Guichard, Single Variable Calculus) covers most requirements of Methods and Specialist comprehensively, with interesting extras - you may even be able to substitute reading it instead of your textbook! These higher-level textbooks in general give far more insight than say the Heinemann books (or maybe MathQuest)
That said, however, don't go off on a massive tangent unless you have time (e.g. during holidays). Your Year 12 SACs and assessments obviously take priority. I did nevertheless go off on a wild mathematical adventure during Term 1 holidays and learn about separable, first order homogeneous, first order linear, second order linear homogeneous and second order linear inhomogeneous differential equations - the terminology was a mouthful already!.
Write down what you learnI never felt comfortable learning stuff unless I had A4 loose leaf lined paper and a binder to write it down on. And I didn't make random unstructured notes - I organised my binder from A-Z and grouped topics that I covered. Personally, I need to write down stuff I learn - and it assists with learning and conceptual understanding too.
I'd include worked examples on my own notes, less for future reference (I rarely forgot stuff I learned this way) but more for getting a hang of how to solve the problem/use the formula/use the concept. I would attempt the textbook's worked solutions first without reading their solutions (or with reading) and then compare mine with theirs (or just check the answer at times). This was like 'training wheels' before launching into doing actual questions w/o solutions.
I've actually made a scanned copy of my notes binder - it contains everything I've learnt about maths (except from Uni maths) and is over 200 pages. (for your reference, I hope it helps). And I'd strongly encourage you to work on your own.
https://onedrive.live.com/redir?resid=2692D5EC8060E581!6573&authkey=!AO6WU14SZcjoQeQ&ithint=file%2cpdf
WORK HARD and START EARLYEdit: yes, as Eulerfan pointed out, this may not work for everybody. My post is reflective of what worked for me, so take it as a suggestion of one thing which worked for one guy. It is definitely not the only way, but it is one. The above (conceptual understanding) however is probably the only way though
There is no substitute for hard work. I personally started studying for my VCE subjects (all of them but englang
) in November 2014. For Methods and Spesh, I spent roughly 1.5 hours a day each, sometimes 2 each. For Methods, I believe I covered most if not all of Unit 3 material before starting term (can't really remember, I think I was up to differentiation). However, note that I had covered most of this stuff in random places before (before VCE), so it was mainly review. Specialist much the same, was up to Chapter 6 of Essentials by February. Starting this early proved to be a massive advantage (I stayed months if not terms ahead of my class throughout most of the year, although stuff caught up to some extent towards the end).
For Specialist and Methods, I finished the course roughly in late July/early August, and yes, I did 95% of the textbook exercises (all except for the really stupid ones.
Might add some more stuff to this later on, but I hope this gives some guidance as to how to study for VCE maths (and maths/stuff in general, not just VCE)
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