Tuesday, 22 May 2012

Chapter 3 - A message

2 things.

Firstly, I may or may not continue using blogger. I'm considering moving to wordpress. We'll see what happens there.

More importantly, I'm setting myself a bunch of deadlines for completion. I plan to have calculus done (at least as a beta) by the end of june, in time for the start of the coming school holidays. The idea is that I will give access to the material to a bunch of year 12s who are yet to sink their teeth into calculus, to see if this works. I plan to have the rest (again, as a beta copy) done by 23 july, which is when my semester break ends. It's also about a week into the school term.

I have 2 hopes with regards to this. Firstly, that this provides short-term feedback on the calculus (and potentially functions) during my break, so that I can improve these resources.

Secondly, it would be great if some people liked it and used it on some level for the year. But that's a pipedream at this stage. What will definitely be happening though is that most of the stuff will be up, and soon.

:)

END CHAPTER 3

Friday, 18 May 2012

Increasing/decreasing

I want to make sure there's a discussion of increasing functions, strictly increasing functions and why the maximal interval (domain) in which a function is strictly increasing includes its endpoints. It's fairly clear given the definition 'a<b implies f(a)<f(b)', but even still it needs discussion as people often miss this nuance.

Chapter 2 - Chapter 2 - The plan in the beginning of the making of the Towel

I figure I should boil down the MM curriculum into its bare essentials to start, and go from there.

The course outline divides the curriculum into 4 'units' - 'Functions and graphs', 'Algebra', 'Calculus' and 'Probability'. What this blog post will do is outline the main ideas that need to be understood in MM, as I see it. I will miss stuff, but they will be minute details on how these things work which will be dealt with later. The big ideas are what I'll deal with here.

I'll mention quickly before this begins that this is not by any means how I will order the textbook, merely the ideas that need to be in there. Also, I will not focus nearly as much on 'that it works' as much as I will on 'how it works'.

Functions and graphs

Familiarity with a few functions - There are very few different types of graphs to be familiar with - They are:

1) Polynomials
2) x^(m/n) (where m,n are integers)
3) e^x and ln(x)
4) sinx, cosx and tanx
5) |x|

That's basically it. Functions that are lumped together are in the form 'if you understand one, you'll understand the others'.

Transformations - Only dilations, reflections and translations in the x and y directions are dealt with in this course. As such, you need only understand what one generalised transformation means, and the rest should fall out. The generalised transformation is af[b(x-h)]+k.

Inverses - What they are and how to find them.

Combining functions - This only really extends as far as adding functions together ('addition of ordinates'), multiplying them together (in other words, use your cas) or putting functions in functions ('fog', or f(g(x)).

Sketching functions - There are 2 things here. One is applying the generalised transformation mentioned above to sketching, the other is knowing what 'base' form looks like and how to deal with that.

Algebra

This is essentially a revision of previous years' algebra. Very little is taught for the first time. That said, there are specific things that people often need help with.

Solving equations - This basically involves factorising/expanding repeatedly until the function is fully factorised, then applying the 'null factor law'. Other than that, understanding the ideas of inverses, basic algebra of indices/logarithms, a basic understanding of the unit circle and a knowledge of 'equating coefficients' is all that is really needed for this.

Generalisation - As a concept, this is (somewhat surprisingly) often misunderstood, and as such I expect I will focus on this throughout the text.

Matrices - Basic matrix algebra, solving systems of linear equations with them (also using parameters)

For what is supposed to be the first half of the year, that's basically it. In each question set, approximately 1/3 of the questions will be for practising/improving use of algebra or graphing. I will post a blog about the structure of question sets in the future. I will also probably be meshing these two units into one big superunit with 4 or 5 chapters (maybe 'functions in general, methods for solving equations, polynomials and functions of the form x^n, powers and logarithms, trigonometry)

Calculus

This is also mainly a revision of year 11, but this often feels new to people, so it will be treated as being taught from scratch. Also, all of the calculus will most likely be taught in one hit of 2 chapters (differentiation, antidifferentiation).

Differentiation
-This all really boils down to a few things

 - What differentiation actually is
 - Basic knowledge of how to differentiate the functions mentioned in unit 1 (polynomials, e^x etc.)
 - Chain, product and quotient rules.
 - Applying this knowledge to graphs (stationary points, sketching gradient functions)
 - Linear approximation of derivatives
 - Related rates of change (this is merely a direct application of the chain rule)

Antidifferentiation - Again, not much to know

 - What it is, and how it relates to integration
 - How to antidifferentiate a few basic functions: sinx, cosx, x^n (where n is rational) and e^x
 - Some generalised methods of antidifferentiating
 - Definite integrals
 - How all this culminates in the fundamental theorem of calculus

Probability

Admittedly, this part of the course often seems like something tacked on to the end just to keep the statisticians happy. It's not necessarily like that, but it certainly does need a lot of groundwork to get it up to the same level of understanding as other aspects of the course. Luckily enough, there's not that much to learn. There will probably be 3 chapters (Basic tools of probability, discrete random variables, continuous random variables)

Basic tools of probability - Definition of discrete/continuous random variables, expected value, variance, standard deviation, mean/median/mode, conditional probability

Discrete random variables - Bernoulli sequences, markov chains

Continuous random variables - Using integration for continuous probability distributions, normal distributions

I'll mention here that for most cases involving normal distributions, markov chains and bernoulli sequences, very specific CAS-related knowledge and skills which does not come from the previous 3 units are needed, and as such the CAS will be referenced constantly throughout these probability sections.

But that's it! That's the course! Considering that there are about 30 weeks (not including holidays) in which to learn this, I feel that this gives ample time to learn this material. At uni, all this and more is taught to non-methods students (read: little maths background knowledge) in a semester with no problems, so I don't see why it can't be taught to only slightly younger people in significantly more time.

The last thing that needs to be mentioned is that anyone doing methods needs a basic understanding of how to use their CAS. I'm still unsure of the extent to which I will refer to how to actually use the CAS (certainly my hand is forced when probability is being studied). Anyone with an opinion on it, feel free to comment :)

END CHAPTER 2

Thursday, 17 May 2012

Chapter 1 - The introduction to the beginning of the making of the Towel

Victoria's maths curriculum sucks. It really does. Its mid-range year 12 maths course, 'Mathethematical Methods' (MM), also sucks. Big time. I don't mean that in a funny 'I'm referencing the fantastic HHGttG' kind of way. I mean it in an 'it sucks balls' kind of way. It's not that it teaches the wrong material per se - it seems to hit some of the right areas, although there are things (like geometry and linear algebra) which probably deserve more of a mention. It's the focus that sucks - too much emphasis is put on calculation and not enough on deep understanding of ideas.

It's not that this is impossible to test - all of the other subjects manage it, maths at university usually involves a lot of explanations and maths in general is all about expression (more on this later), so it doesn't seem like something that would be that out of place in an exam. It's just that it isn't being tested.

This leads to every single MM textbook (that I am yet to encounter) kowtowing to the needs of this terrible course design - instead of structuring a text to maximise understanding and THEN focussing on exams, most jump straight into 'exam style questions', which leads to problems that I will mention later. Some manage to avoid this particular mire (notably cambridge's text), but fall into other traps like focussing too much on algebraic manipulation over explanation of ideas in their question material. Also problematic. Also will be explained later.

This doesn't mean everything about every text is bad. All it means is that no textbook currently on the market actually has a reasonable structure for the teaching of maths for high school students. Which is ridiculous. There are good reasons for this, which I will undoubtedly outline later, but they are of little importance now.

What is of importance, though, is what's going to change. In response to this lack of quality in the textbook market, I plan to make the Towel. The Towel will be a year 12 MM textbook which aims to avoid these problems. It will be free, electronic, and available to anyone. It will be structured better, and as such it will help people of all ability levels do better. It will give insight into the reasons people actually study maths, and as such it will motivate people to do more maths. It will be awesome.

At least, that's the goal.

Why 'Towel'? For those who haven't read Douglas Adams' 'The Hitchhiker's Guide to the Galaxy, read it. It's fantastic. Anyway, in the book, every good hitchhiker knows where their towel is, with very good reason. Check out a good explanation here (incidentally, Towel Day is coming up on 25/5. Know where your towel is, people!). I reckon that this can and should be seen as a Towel for MM - indispensable for practical and psychological reasons. Or maybe a hitchhiker's guide, with DON'T PANIC on the front. I guess we'll see.

But back to our depressing reality. As a tutor, I am tired of struggling through lessons, having to jump around the textbook in the hope of finding a reasonable (or reasonably placed) diagram, explanation or proof. Most of the time none of these (except maybe a misplaced diagram) are there. I'm tired of the strange question order. I'm tired of a lack of interesting motivating examples. I'm tired of my students being overwhelmed by the vast number of questions without any semblence of a guide for optimal learning given limited time. So I have to remedy this. I have a plan for how exactly a new style of textbook can accomplish this, and I will post detail on that soon.

In the meantime, I want to note the purpose of this blog. From now on, I will be regularly posting what I tutor on here, for all to see. It will therefore be accessible to my students, obviously. The point of the blog is to have my work critiqued. At first the work will probably mainly be theory, but I intend to eventually start putting up questions to prompt further thought and deeper understanding on given bits of material. In essence, I want to make sure the Towel is user-friendly and actually does what it's made to do. But again, more on that later.

So. I will start posting on Saturday, spend a big chunk of my semester break filling in what was missed throughout the first half of this year, and get some semblence of a structure to both this blog and the Towel itself.

I want to have this project complete by the end of the year, for next year's year 12. This is a labour of love - I really see the potential for this to be a useful thing, and as such any help that anyone can give would be really appreciated. Let anyone you know who would be interested in contributing to a better MM text know about this project - tutors and teachers would be great, especially if they have really well-refined ways of teaching this stuff.

One other thing that would be fantastic is a large base of students of all ability and interest levels to critique this as it grows - please let people know about this! The thing is, everyone involved in MM has an interest in being involved in this project - even if you're in year 12 this year, you can benefit in many ways - from simply gaining a better understanding of the ideas through reading the posts to deepening that understanding by figuring out what is missing in my explanations (and of course relaying that on to me so that I can make the explanations more complete).

That's about it for now. Thanks in advanced people! Looking forward to this!

END CHAPTER 1