Syllabus development (2 Viewers)

[kony]

noooo the beloved conics.

 

[buchanan]

The most significant bit is as follows.

Note: In the information presented below, proposed course topics shown in italics are new topics in the calculus-based courses or are new in the particular course described.

Appropriate arrangement of the proposed course topics into Preliminary/HSC courses will be undertaken in the next phase (Syllabus Development Phase) of the syllabus development process.

Mathematics Advanced

(For each topic, the necessary assumed knowledge, skills and understanding are to be
identified for review.)

Proposed course topics:

• Counting techniques
• Probability (simple and counting methods)
• Real functions and their graphs
• Trigonometry
– Right-angle triangle trigonometry
– Introduction to trigonometric functions (using radians)
– Calculus of trigonometric functions (including applications)
• Differential calculus
– Introduction
– Geometrical applications of differentiation
• Data analysis
– Types of variables, measures of centre and variability, graphical representations of
data
– Simple discrete and continuous probability models, expected value and the Normal
distribution
• Sequences and series
• Integral calculus
– Introduction
– Areas and volumes
• Logarithmic and exponential functions (including applications of calculus)
• Mathematical modelling
– Applications of calculus

Mathematics Extension 1 (includes Preliminary Mathematics Extension)

Proposed course topics:

• Circle geometry
• Further algebra (including sum and product of roots of quadratic equations, quadratic
identities)
• Transformations of graphs
• Other inequalities
• Polynomials
– Polynomial equations, graphs
– Multiple roots of polynomials
• Elementary difference equations and the discrete logistic growth model
• Mathematical induction (series and divisibility only)
• The binomial theorem, binomial identities and the binomial probability distribution
(including expected value)
• Further trigonometry (sums and differences, general solutions, auxiliary angle, and angle
between two lines)
• Methods of integration (including substitution, the primitive of sin²x and cos²x)
• Inverse functions (including inverse trigonometric functions)
• Further applications of calculus involving mathematical modelling (including motion, modified growth and decay, and Newton’s method)

Mathematics Extension 2

Proposed course topics:

• Further inequalities
– induction with inequalities
– proof using graphs and calculus
• Complex numbers and polynomials over the complex field
– geometric representation
– vectors
– powers and roots
– curves and regions
• Graphs
– sketching basic curves
– addition, subtraction, multiplication, division and reflection
– general approach to curve sketching
• Integration techniques
– t-formulae
– partial fractions
• Volumes
– slicing
– cylindrical shells
• First and second order ordinary differential equations and modelling
– including aspects of mechanics
– simple harmonic motion

 

[Trebla]

Omg, there's a bit of statistics in there...
Quite a lot more of some first year university material seems to appear in this one. I guess this gives better preparation for university science degrees....

Where is the 'Harder 3 Unit' in the Mathematics Extension 2 course, which virtually allows any random challenging quesion to be examinable? Is it gone? If it's gone, then the most defining/famous feature of Mathematics Extension 2 is gone because that's what makes it a reputable course on a national standard.

 

[buchanan]

Harder 3 unit is gone. As is conics.

They have also deleted central limit theorem and confidence intervals from the writing brief.

As for component A/B - well that has yet to be decided. It has to be decided in phase 3 (syllabus development) which has only just started.

 

[pyrrha]

No harder Ext1 in the Ext2 course - but that was the best part!
Second-order differential equations in mechanics in Ext 2
SHM and induction with inequalities are gone from Ext 1
Emphasis in statistics in 2U, not just General - good for uni
Conics gone ... no, wait.

Mixed feelings here.

Hey, is anyone else concerned about the disappearance of recurrence relations from integration?


EDIT: Just saw the discrete logistic growth model in Ext1. Awesome stuff.

 

[pyrrha]

It does actually relate to the recurrence relations in the induction component - but more importantly, it's actually interesting. Most of the syllabus is very bland.

EDIT: I guess you could also argue that it ties in with exponential growth and decay.

 

[buchanan]

it tends to be very boring.

The first thing in the syllabus should be a statement like:

Make Maths Interesting!

Everything after that should be to assist teachers and students to achieve that goal - including examinations.

It's a pity that the current syllabus and the one they are replacing it with are boring.

The 2006 and 2007 4 unit papers were also boring.

We should stop this boringness and start doing something more interesting.

Examples:

Terry Tao's blog: http://terrytao.wordpress.com

Valentin Vornicu's Mathlinks forum: http://www.mathlinks.ro/Forum/forums.html

Landon Clay's Claymath: http://www.claymath.org

MAA's new books on Euler:

EULER AND MODERN SCIENCE
N. N. Bogolyubov, G. K. Mikhailov, & A. P. Yushkevich, Editors
Translated by Robert Burns

EULER AT 300: AN APPRECIATION
Robert E. Bradley, Lawrence A. D'Antonio, C. Edward Sandifer, Editors

HOW EULER DID IT
C. Edward Sandifer

THE EARLY MATH OF LEONHARD EULER
Edward Sandifer
Volume 1: The MAA Tercentary Euler Celebration

THE GENIUS OF EULER: REFLECTIONS
William Dunham, Editor
Volume 2-The MAA Tercentenary Euler Celebration

(available at http://www.amazon.com)

 

[pyrrha]

The problem is that all the interesting stuff will only be taken seriously by students who honestly love maths. I'm all for making maths more interesting - I've written projectile motion questions with cannonballs from pirate ships and Newton throwing objects at greater and greater speeds from a tall tower, but all my students just sit there and yawn. The only one who was interested in maths outside the syllabus spent all his lessons studying Cosmology for the HSC.

 

[buchanan]

There are lots of ways to make it interesting. If one way fails, try another way.

 

[buchanan]

Fighting against philistines in the teaching profession often makes maths interesting.

Philistine teachers make excuses for not doing interesting maths - which goes something like this:

- it's not in the syllabus

or

- you'll have to wait till uni to do it

For example for many years they said you can't prove irrationality of e till uni and it's not in the syllabus. Then we saw it in the <a href="http://www.geocities.com/ext2papers/4u_1990-1994.pdf">1993 4 unit exam</a>!

Then they started saying you have to wait till you do 4 unit before you can prove it - until I put a <a href="http://www.angelfire.com/ab7/fourunit/eisirrational.pdf">2 unit proof</a> on my website!

I find this stuff rather amusing.

And so it is with Wallis' product. The same philistines said it can't be done in high school. Then a <a href="http://community.boredofstudies.org/showpost.php?p=2911043&postcount=59">calculus method</a> appeared in the 1995 4 unit exam. Then they started saying you have to wait till you do 4 unit. This month an elementary proof not requiring any calculus appeared in the American Mathematical Monthly which I have attached. An older preprint version however can be found <a href="http://www.ep.liu.se/ea/lsm/2005/002/lsm05002.pdf">here</a>.

So if any philistine teacher says to you that you have to wait till uni (or you can only do it in 4 unit) - just educate your teacher with this.

 

[Yip]

buchanan, the "2u proof" of the irrationality of e on your website requires knowledge of the taylor series expansion of e, which is not even in the 4u syllabus. The reasoning used in that proof seems more in the level of harder 3u imo. I also think you go a bit far in criticizing these "philistine teachers". Of course, one should mention that there may exist better ways to do a particular problem that is not strictly in the syllabus, rather than saying that the way the syllabus says to do it is the only way. However, going too deep into these alternate methods can be counterproductive and confusing to the average student. A teacher of a class who has high overall ability in math may be able to illustrate these alternate methods. However, most classes, especially in non-selective public schools, do not have the ability required in order to benefit from it. I don't think the teacher should be called "phillistine" just because of the caliber of the class.
That is my two cents anyway.

 

[buchanan]

Are you referring to e=&sum;1/n!? I agree that this can be done with Taylors series. However, it can also be done without it - again within 2 unit. I prefer however not to dwell on this point in the 2 unit proof of the irrationality of e.

I also prefer not to make excuses for mediocrity ..........

 

[buchanan]

If you ask mathematicians what they do, you always get the same answer; they think. They are trying to solve difficult and novel problems. They never think about ordinary problems.

- M. Evgrafov, Literaturnaya Gazeta, no. 49 (1979) 12.

So the BOS should not only keep Harder 3 Unit and Component B, but also introduce Harder 4 Unit (as some schools have already done in their teaching programs).

 

[kony]

If only they introduced 5 unit maths.

 

[buchanan]

It has previously been suggested to the BOS that they bring back the 2-year Level 1 course, and that they should include Number Theory if they did that. It seems they won't be doing that. Instead their writing brief seems to be obsessed with the utility of maths and pays scant regard to the unity of maths. It is a widely held view that the final syllabus will be substandard if they continue down this path.

"The theory of Numbers has always been regarded as one of the most obviously useless branches of Pure Mathematics. The accusation is one against which there is no valid defence; and it is never more just than when directed against the parts of the theory which are more particularly concerned with primes. A science is said to be useful if its development tends to accentuate the existing inequalities in the distribution of wealth, or more directly promotes the destruction of human life. The theory of prime numbers satisfies no such criteria. Those who pursue it will, if they are wise, make no attempt to justify their interest in a subject so trivial and so remote, and will console themselves with the thought that the greatest mathematicians of all ages have found in it a mysterious attraction impossible to resist."

G. H. Hardy, Prime numbers, Brit. Assn. Rep., 1915, pp. 350-354

 

[darkliight]

The utility thing you mention is not just a BOS phenomenon. Many universities (in Australia) already just focus on applied maths, with pure maths subjects there for support only. Although it has been mostly pure for a long time, Newcastle is in the process of moving to applied now too.

It seems applied maths attracts students and pays dividends, so you can't blame the universities. In turn, you can't blame the BOS for preparing students to take it on (or related subjects like engineering).

With our governments focus on trades, tafe, utility, etc, I think pure maths is the least of their concerns too. It seems its a case of get used to it, or head to the US or Europe.

 

[buchanan]

It seems its a case of get used to it

It's a case of the uni's, governments and the BOS having to get used to being publicly exposed for the amateurs that they are - and a case of teachers ignoring them and doing it properly instead.

Many schools advertise themselves as striving for excellence. Can they do this with a substandard syllabus? The answer is yes, but only if teachers ignore the BOS and do it properly. The problem is that only a minority of teachers would have the guts to stand up to the BOS.

 

[darkliight]

It's a case of the uni's, governments and the BOS having to get used to being publicly exposed for the amateurs that they are

Are you sure you have the right definition of amateur? How are the universities the amateurs here?

and a case of teachers ignoring them and doing it properly instead.

I've seen a lot more bad teachers than good teachers. I'd rather the bad ones point to a text book and tell the kids to keep themselves entertained for 50mins (like I've seen some do) then take it upon themselves to choose what to teach. You can't have teachers teach what they want instead of what the kids are going to be assessed on either.

As for your excellence bit, that does not depend on the subjects taught. You're not excellent if you know or teach some number theory.

It is a shame the pure maths pool here seems to be shrinking away, but as I said, while ever we have a "skills" focused government that has "knowledge for the sake of knowledge" way down on its list of priorities, it is going to stay that way.

 

[buchanan]

You seem to be saying that teachers should just accept a substandard syllabus and if good students want to learn pure maths they should give up on Australia and go to the US or Europe. Some schools won't accept this. We already have several schools teaching the IB for example.

As for uni's, the criticism probably can't be applied equally to all of them. Some are still running good pure maths courses, eg., at UNSW: http://www.maths.unsw.edu.au/pure/purehome.html

So we don't need to give up on Australia just yet.

And the final syllabus hasn't been written yet, so we needn't give up on the HSC either (at least not yet).

I'm not the only one complaining about the unnecessary and inappropriate focus on utility. A keynote presentation in the 2007 MANSW conference also highlighted it (a conference convened by the Senior Project Officer of the Syllabus Committee).

 

[darkliight]

I'm saying, teachers should teach the syllabus and I don't think the syllabus would be substandard if it didn't include number theory. I'm also saying that if the government is obsessed with trades and skills, then you can expect to see a shift refelecting that in the maths syllabus. In fact, it goes beyond high schools and into the universities - another thing I said.

It's simple supply and demand - so long as the government and industry are demanding applied maths, it will be met with supply from schools and universities.

If good students want to learn more pure maths then they can at a university that offers it, and again, supply and demand will kick in, else, there is the US and Europe to further your studies (though that comment was more aimed at undergrad+). Just because one place does not offer what you want doesn't mean that it's a substandard institution, be it a high school, university or otherwise.

I don't think anyone should give up either, quite the opposite in fact, we shouldn't resign ourselves to the fact that a maths syllabus without number theory is going to be substandard.