What do YOU think should be in the HSC exams in the next few years? (2 Viewers)

[study-freak]

ODE's, multi dimensional vectors and matricies and maybe even some PDE's.

Definetly no stats stuff like ANOVA and hypothesis testing, that shit will turn people off maths.

ODEs might be quite reasonable to include in 3/4u, but the rest may be a bit too much. If they include matrices, then they probably can do it only briefly due to time constraints and so aren't likely to be able to teach much of the concepts, e.g. what determinants are useful for/what they mean, eigenvalues, eigenvectors, diagonalisation, etc (at least not thoroughly).
This will in turn put people off linear algebra because most people probably wouldn't get what the hell they are doing those tedious calculations for!

and to learn PDEs, they would have to learn partial derivatives first, before which they should really learn functions of several variables... Sooo much to teach.

Plus the theories behind ANOVA and hypothesis testing are quite interesting, although the procedures might be tedious and boring. But then I reckon these would also be a bit too much to teach in the HSC.

 

[Trebla]

All this talk about chucking this and that into the syllabus...perhaps an equally interesting question is what topics in the current course would you remove to make room for these new topics

 

[mnmaa]

All this talk about chucking this and that into the syllabus...perhaps an equally interesting question is what topics in the current course would you remove to make room for these new topics

all of them

 

[study-freak]

All this talk about chucking this and that into the syllabus...perhaps an equally interesting question is what topics in the current course would you remove to make room for these new topics

Suddenly came across my mind is circle geometry... What is it ever useful for after high school other than that it improves geometric thinking ability? I haven't come across any use of it as of yet.

Edit: oh wait, perhaps in engineering?

 

[1xcv3we]

Suddenly came across my mind is circle geometry... What is it ever useful for after high school other than that it improves geometry skills? I haven't come across any as of yet.

I haven't seen one either. Come to mention it, I haven't seen any of the Euclidean modules used outside of 3/4 unit maths. They should be removed completely IMO and substituted with methods for solving ODEs. Volumes could do with a boot too.

 

[study-freak]

I haven't seen one either. Come to mention it, I haven't seen any of the Euclidean modules used outside of 3/4 unit maths. They should be removed completely IMO and substituted with methods for solving ODEs. Volumes could do with a boot too.

mm, totally agreed.

If they do have space then, I reckon they should also renew the probability section of HSC maths to make it more systematic, employing set notation and perhaps some concepts in discrete maths (combinatorics using function notation (also introduce the terms injective, surjective, and bijective), counting principles) and statistics (equiprobable space, sample space, random variables (e.g. instead of calling it 'binomial probability' or whatever they call it now, introduce the binomial random variable)).

 

[largarithmic]

Suddenly came across my mind is circle geometry... What is it ever useful for after high school other than that it improves geometric thinking ability? I haven't come across any use of it as of yet.

The set of skills you use in Euclidean/Circle geometry problems at school are a far more useful set of skills than learning the specifics of differential equations ever will be at least in any way they'd teach them at school. The reason is this; there's a difference between specific and general skills. Most of school maths is in many ways analogous to cooking. You get taught a recipe (e.g. how to do integration by parts, how to prove various things about conics) and then in the exam you're given slight variations on those recipes which you're meant to carry out. Most of what you learn in say calculus is a very specific skill to do what is admittedly quite an important task - you're learning how to find the derivative of a function; how to follow a pretty significant recipe.

Now the difference with circle geometry is actually profound. In circle geometry you're doing a very inconsequential task (euclidean geometry is barely studied at all any more past high school), but the most important part is that the actual thought process of doing circle geometry is utterly unlike anything else thats in extension 2 maths maybe save some very difficult inequalities questions. In geometry there is no recipe, you have to work out what you're doing for yourself in that question. Which is why lots of otherwise extremely good students find it devilishly hard.

Essentially the mental skill in abstract problem solving and logical deduction is unparalleled in Euclidean geometry as oppose to other areas of maths taught at the school level, and in many ways of a very different type. I'd far prefer that students learn how to use their brain and how to use their brain in a large variety of ways than just learning yet another similar topic a year early; because you can learn a specific skill like solving differential equations at any time, but general thinking takes a much longer time to develop and its far easier to develop it when you're young. And if you ask me, it doesn't really matter someone you pulled of the street can solve a differential equation or not and if it does because they're say an engineer they'd learn it anyway. But I do care whether or not someone can actually think, especially if that person is someone like my doctor. So if you ask me, Id add a hell of a lot more geometry to the syllabus, not take it out.

 

[1xcv3we]

mm, totally agreed.

If they do have space then, I reckon they should also renew the probability section of HSC maths to make it more systematic, employing set notation and perhaps some concepts in discrete maths (combinatorics using function notation (also introduce the terms injective, surjective, and bijective), counting principles) and statistics (equiprobable space, sample space, random variables (e.g. instead of calling it 'binomial probability' or whatever they call it now, introduce the binomial random variable)).

IMO the whole 4 unit course could be scrapped and re-written. I am just reading through it right now and apart from complex numbers (the complex geometry can be scrapped, but keep basic properties, arg mod and de'moivre's thm etc) the rest is basically useless. Alternatively, a better idea again, make 2 separate 4 unit courses: Mathematics Extension 2 Applied strand and Mathematics extension 2 Pure strand. You could have half probability and half based on ODEs. Doing it this way means students who enjoy pure maths are kept happy and those who want something that has a common application in the real world then let them do a course that is similar to that which is in the Victorian VCE (with perhaps a little more depth added).

 

[1xcv3we]

Largarithmic I really like your explaination of Euclidean/Circular geometry and why it should be kept in. I never really thought about it this way. Most of my time in high school and even now doing a Bachelor in Applied Maths/Stats I am always thinking about what is useful to what I am doing. I never really thought of it serving as a brain exercise.

 

[largarithmic]

IMO the whole 4 unit course could be scrapped and re-written. I am just reading through it right now and apart from complex numbers (the complex geometry can be scrapped, but keep basic properties, arg mod and de'moivre's thm etc) the rest is basically useless. Alternatively, a better idea again, make 2 separate 4 unit courses: Mathematics Extension 2 Applied strand and Mathematics extension 2 Pure strand. You could have half probability and half based on ODEs. Doing it this way means students who enjoy pure maths are kept happy and those who want something that has a common application in the real world then let them do a course that is similar to that which is in the Victorian VCE (with perhaps a little more depth added).

Now I understand for most people maths is a tool and not a discipline in its own right, but your conception of what constitutes "use" is incredibly narrow-sighted. We learn huge numbers of things throughout our entire lives and throughout our entire schooling that we won't practically put to any "real world application", but does that make none of them any less worth learning? If you took out things based on incredibly arbitrary notions of what is and is not useful for a students own perception of what they're going to be doing in ten years time, you'd be left with an emaciated skeleton of an education. I find the idea that a fifteen year old, sitting down with their school careers counsellor or parents choosing their year 11 and 12 subject chooses, can have any idea what they are going to be doing in ten years time and then making highly specific judgements on their education because of this absolutely ludicrous.

And that's not considering that perhaps learning things such as circle geometry might possibly have more significant long term use for most people through the general skills acquired and you'd ever get from learning how to do a precise algorithm like integrate a specific type of function.



I should seriously contribute though to how I think the courses should be reformed; I think that maths extension 2 is more or less fine and shouldn't be touched (maybe the exam format should be changed as I've suggested early, but the syllabus is great, and that's not really what this thread is about). But I also think there should be an option 1 unit statistics course which would be available for any student studying mathematics, mx1 or mx2 that would run separately to the other three maths courses as an elective. The focus in this course I think shouldn't be so much on the mathematical details of statistics, but actually teaching numerical and statistical literacy and logic. A sample exam question would present a student with a set of data, the student would do some maths with that data for a few marks, then use that data to make some sort of scientific conclusion.

 

[study-freak]

The set of skills you use in Euclidean/Circle geometry problems at school are a far more useful set of skills than learning the specifics of differential equations ever will be at least in any way they'd teach them at school. The reason is this; there's a difference between specific and general skills. Most of school maths is in many ways analogous to cooking. You get taught a recipe (e.g. how to do integration by parts, how to prove various things about conics) and then in the exam you're given slight variations on those recipes which you're meant to carry out. Most of what you learn in say calculus is a very specific skill to do what is admittedly quite an important task - you're learning how to find the derivative of a function; how to follow a pretty significant recipe.

Now the difference with circle geometry is actually profound. In circle geometry you're doing a very inconsequential task (euclidean geometry is barely studied at all any more past high school), but the most important part is that the actual thought process of doing circle geometry is utterly unlike anything else thats in extension 2 maths maybe save some very difficult inequalities questions. In geometry there is no recipe, you have to work out what you're doing for yourself in that question. Which is why lots of otherwise extremely good students find it devilishly hard.

Essentially the mental skill in abstract problem solving and logical deduction is unparalleled in Euclidean geometry as oppose to other areas of maths taught at the school level, and in many ways of a very different type. I'd far prefer that students learn how to use their brain and how to use their brain in a large variety of ways than just learning yet another similar topic a year early; because you can learn a specific skill like solving differential equations at any time, but general thinking takes a much longer time to develop and its far easier to develop it when you're young. And if you ask me, it doesn't really matter someone you pulled of the street can solve a differential equation or not and if it does because they're say an engineer they'd learn it anyway. But I do care whether or not someone can actually think, especially if that person is someone like my doctor. So if you ask me, Id add a hell of a lot more geometry to the syllabus, not take it out.

I get your point, but I still don't see the whole point of having circle geometry - you can develop those skills with other topics as well. If you want students to think logically, teach them mathematical logic. BoS can also give hard questions in various other topics that can't be done by just using so-called 'recipes'. Practising these questions would improve those skills while also preparing students better for uni than learning circle geo rules and using it.

and doctors and circle geo? Hmm, I doubt it. Circle geo itself is taught based on sets of rules that must be used to get to the answer/prove stuff, though the order of applying those rules may vary. That's still half-recipe taught, don't you think? I don't think it drastically improves abstract and logical thinking.

and I was just thinking, those abstract geometrical thinking can also be taught with introducing vectors to the course? 2 and 3 dimensional. (dot, cross products too)

IMO the whole 4 unit course could be scrapped and re-written. I am just reading through it right now and apart from complex numbers (the complex geometry can be scrapped, but keep basic properties, arg mod and de'moivre's thm etc) the rest is basically useless. Alternatively, a better idea again, make 2 separate 4 unit courses: Mathematics Extension 2 Applied strand and Mathematics extension 2 Pure strand. You could have half probability and half based on ODEs. Doing it this way means students who enjoy pure maths are kept happy and those who want something that has a common application in the real world then let them do a course that is similar to that which is in the Victorian VCE (with perhaps a little more depth added).

Complex geometry is actually somewhat important imo to improve students' understanding of complex numbers, so I don't think it should be scrapped. That's just my opinion.
It's different to the case of circle geo - a topic that's quite independent of all else that's taught in 3u or 4u.

Hmm, perhaps separating pure and applied would cater for various interests of students, but that would also mean there'll be imo too many maths units available. The whole point about having 10 units of subjects compulsory is to make students do at least 4 DIFFERENT subjects, I think. Having that many maths units available would create maths nerds who know little of other subject areas.

 

[Carrotsticks]

Hey carrotsticks can you please give me a link to the MIT online lectures

http://ocw.mit.edu/courses/audio-video-courses/#mathematics

Are the MIT online lectures useful for university maths?

Yep, the concepts are explained well. Good for basics, not so good for advanced.

And uh isn't vector calculus decently hard? like 2nd year uni

Perhaps all I've done so far is basic stuff, but I had a student who was curious about Partial Derivatives and Multiple Integrals. I managed to teach it to them in about 20 minutes. It isn't much more difficult than HSC Calculus.

However, this is excluding further Vector Calculus stuff like Stoke's Theorem.

I have 9th edition of Howard Anton Calculus and I must say his section on limits is not as good (for single variable as well) but he has some great multi-integrals. James Stewart Calculus 6th or 7th edition cover these sections better (Stewart covers MV epsilon deltas too). I have started to read another text book called Calculus One & Several Variables 10th Edition. The latter has a more comprehensive coverage of epsilon deltas. When I did MV epsilon deltas it really comes largely down to trial and error at times unless there is an obvious solution/contradiction or you work out some insane substitution.

I never really found the MIT lectures useful when I did MVC, they have a very different take on it. I found my course notes and Stewart Calculus sufficient. If you want any more details on the above text books just PM me.

I have the same big blue Howard Anton book. I found the limits part to be okay, but I too have heard how great the James Stewart Calculus books are. I am keen to get them, but the pricetag deters me.

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Regarding the Circle Geo debate, I think it is important to have skills where you simply apply everything you know in order to solve the problem. The proofs aren't necessarily machine-like, unlike topics such as Max/Min problems where you can actually give a METHOD to doing them. I found that many geometry problems follow a constant cycle of "What do I need to get to Step A? I need Step B. What do I need to get Step B?" etc etc.

However, I think one other aspect of the HSC is that it's to expose students to the world of Mathematics and how far it can go. This is why it's sad seeing such lack of statistics and number theory, though I am aware that you can only fit so much in one year.

I would prefer to have a HSC that has a bit of a 'taste of everything' versus being highly concentrated on just one thing ie: Calculus.

Also regarding format, I think it would be interesting to have Math 'options' just like in the Sciences. Perhaps one for Mechanics, Pure Math, Statistics, Geometry or Number Theory... so students can just pick whatever they like and just do that whole question.

Of course, this is assuming all the aforementioned topics are already in the current syllabus, and the 'option' questions are just more difficult ones.

 

[Drongoski]

I've 2 copies of James Stewart's Calculus (around 1995 Edn - good enough for me). Got 1 brand new from my friend free (as lecturer at Syd U, he got copies free) and bought 1 second hand for $5.

 

[Carrotsticks]

I've 2 copies of James Stewart's Calculus (around 1995 Edn - good enough for me). Got 1 brand new from my friend and bought 1 second hand for $5.

Can I buy one off you?

 

[ADrew]

Out of interest, Is there a link to the proposed syllabus for HSC mathematics anywhere?

 

[kaz1]

All this talk about chucking this and that into the syllabus...perhaps an equally interesting question is what topics in the current course would you remove to make room for these new topics

Circle Geometry

 

[IamBread]

Circle Geometry

^ I hate circle geometry.

 

[math man]

Even though harder 3u is my fav topic in 4u it should go...it is basically just harder applications of the 3unit course with slight variations.
This to me seems like the board was too ceebs to make another whole topic, which imo they should. Either introduce statistics instead
or matrices or even extend the current topics so complex numbers includes eulers formula and logs, arc length and surface areas in integration,
in curve sketching add more focus on limits (l hopital rule), etc.... Harder 3u is just everywhere and they should really put more focus on just
one topic instead of several.

 

[math man]

^ I hate circle geometry.

circle geo ftw!!! but yeh its place in 4u should be questioned