Hmm... ook I kind of get the idea. But I'm still interested... (going a bit off topic now sorry)
Eh, it's on topic as far as I can tell, so.
- Vectors are very intuitively understood, and it takes no mathematical talent to see their myriad applications in physical situations? So it shouldn't be difficult to adapt?
"Intuitively understood"? Because most people I speak to find them conceptually difficult to understand, and in particular, difficult to "picture" a vector. Not to mention that unless you've done physics, a lot of people don't see a lot of the applications of them
- Complex numbers. Agreed, these can be conceptually tricky, but surely there aren't too many difficulties in execution?
I've seen lots of kids struggle to work through stuff - common questions on the boards are questions like, "when do I apply De Moivre's Theorem and when do I let z=x+iy to solve z^2=thing?", despite the fact that either work. A lot of possibilities are opened up, but very little of complex analysis is breached, and so there's very little that can actually be asked, so a massive deal is made out of small things. It creates a false sense of grandeur among a lot of students I've spoken to, and that's intimidating in itself.
- Integration, you're using some given info to find unknown information... whereas differentiation you're applying a definition and hence you'll have standard derivative rules. You can't do this to integrate because there isn't any 'definition' for integration that is comparatively simple as first principles?
And THAT'S what makes it difficult. It's easy to explain why you can't use rules as simply as you can differentiation, but that doesn't make the act of trying to find an integral any easier.
- Conics... well students are introduced to circles in Methods? And conics are just weird variations on a circle?
I will admit I stretched a bit there.
However, it's still a step up for things that people aren't used to, particularly because most kids have to learn the general rules, because they trouble connecting the transformations of functions with the transformations of relationships.
Then again in Methods:
- Function notation, transformations (like why do you replace x with x/a when you dilate by factor a?), matrices (why does matrix transformation for a function rule work?), modulus (what does it do, and how does nesting your function affect the graph? e.g. f(abs(g(x)) or f(g(abs(x)))?
The difference here is for this stuff that you can teach rules that people can (and often do) follow blindly, something that's not easily possible for vectors (particularly come proofs), complex numbers (where multiple methods yield the same answer, due to the nature of doing things in polar or Cartesian form) and integration (which we established there's no "magic formula" for).
- Probability in general, Markov chains, binomial distribution, that stuff. Surely this is trickier both conceptually and practically than Specialist concepts? E.g. for the probability of 2 heads out of 3 tosses, why do I multiply by only 3 and not by 3*2 for 2 'heads' in different order (idk what that even means).
- Why Pr(X=exact value) for a continuous distribution zero? The 'measured values' argument doesn't work... can we explain this by the fact that a binary search for any exact value will never terminate, so its 'impossible' to assign a probability to X=sqrt(2)?
And probability is claimed the most conceptually difficult concept in methods, rivaling the conceptual difficulty of specialist (and usually overpassing it). The difference is, for specialist, it's your whole exam - for methods, it's one question out of 5. Not to mention that, like specialist, due to the conceptual difficulty of probability, VCAA try not to make probability questions that trip you up incredibly. (usually the only tricky stuff they try is conditional probability, or combining PDFs with "integration by recognition" and making something "binomial" even if it wasn't originally so)
- calculation of approximate confidence intervals - you need to use approximation to relate p-hat to p, there's no way you can calculate the exact confidence interval unless you actually know p so we need to use the approximation relation. How does this even still work then? (excuse me if I'm talking rubbish, I tried to study the 2016 SD for tutoring, but I'm leaving it off till later now)
Because estimation - this is stuff that VCE doesn't cover that I'm REALLY pissed off about. Because you don't cover point estimation properly (which would be SUPER easy to include into methods. Like, insanely easy, it's literally just a max/min problem and a bit of conceptual stuff based on sampling distributions [which is already included in the study design])
Basically, as n goes to infinity, p-hat approaches p (that is, p-hat is unbiased), which means that for large enough n, you can use p-hat to approximate p in your confidence interval. It still works because the confidence interval is approximate by nature, so making another approximation is just like rounding off 1.491 to 1.49 and then again on to 1.5. You don't HAVE to take this approach, it's just EASIER to, which is a pretty fundamental part of statistics. Trying to find these tests is usually quite difficult, and we'll never have infinite n, so instead we make approximations to make the maths easier for us and anyone who needs to use it.
For these reasons, I'm inclined to view Methods as somewhat trickier than Specialist.
I can understand why you'd think this, but you've also based 3/4 of your arguments on one area of study in methods, and that area of study also generally has the same treatment from VCAA as specialist does because of that nature.
Besides, at the end of the day, it's all opinionated, so we can't really find a definitive answer as to which is more conceptually difficult.
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