tianjara.net | Andrew Harvey's Blog

I've managed to do a couple things all in one here. I've made use of some Geoscience Australia Creative Commons licensed material, in a nice little program with a web API, and I've aggregated some data from the myschool scraper and parser. Putting them all together gives some nice images like this.

The program for generating these images basically takes an SVG template file with placeholder markers and then fills these values based on the CGI parameters. The API is fairly simple so one should be able to work out how to use it from the example in the README file. Here are the files I used to make the graphs (and the svg versions as Wordpress.com won't let me upload them to here).

ps. This gets cut off when viewing it from the default web interface of this blog, use print preview or even better look at the RSS feed to see the cut off parts. Also I tried to ensure the accuracy of the data, but I cannot be 100% sure that there are no bugs, in fact there are discrepancies with the averages I get from my scrape of myschool and the averages provided in the report on the NPLAN website. The numbers I get seem to be consistent (ie. the state rankings seem mostly the same), but nonetheless not exactly the same as those reported in the report. Although I would be very surprised if all the numbers I got were exactly the same as in the report. I mainly did this to use map/graph code I wrote, so if you really care about how certain state averages compare in these tests look at the reports on the NPLAN website.

The lighter the colour the higher the number.

Primary

	2008	2009
Literacy
Numeracy
All

Secondary

	2008	2009
Literacy
Numeracy
All

Following up from my previous post, I have made improvements to the code, and I now have all the NPLAN data too. There are also some data files so you don't need to run the scraper and parser which hopefully this makes the data more usable and to a wider range of people. Now that I have the NPLAN data you can compare schools in terms of their (I assume the numbers are averages) test results. I was going to put in the repository some tables mashing together some of the data in the database, but I've had to research about a silly NSW law first. I'm not exactly sure what I can publish and what the implication of that would be (so best make your own league tables and possibly publish them if you want). The NSW law says,

A person must not, in a newspaper or other document that is publicly available in this State: (a) publish any ranking or other comparison of particular schools according to school results, or, (b) identify a school as being in a percentile of less than 90 per cent in relation to school results.

The folks at the Sydney Morning Herald seem to think that "Published online the same tables infringe no law; printed on these pages they are illegal." This is not what I interpret the law as. Publishing online means that the document is available for access from NSW. However I am confident I can get around this by not hosting anything myself and not hosting in Australia. For this I rely on the great services provided by wordpress.com (Automattic, Inc.) and/or github.com (GitHub, Inc.). Hopefully these US companies wouldn't cave into any threats from the Australian government.

This section of the law carries a maximum of 50 penalty units. Which is currently a fine of $5500, that is a large enough sum for me to take extra care. This is why I'm still not sure if I should put such lists like schools ordered by certain NPLAN results in the github repository.

By the way, this censorship and damaging law raises the same questions and problems (problems for those that wish to avoid criminal or civil charges) about legal jurisdiction over the internet, the classic example is the "yahoo! nazi paraphernalia" debacle.

Footnote: This SQL query should give you an ordered list of schools based on the 2009 year 9 NPLAN results (but I guess if you can load the database dump you can probably write your own queries...).

SELECT s.name, n.score, sub.state
FROM nplan n, school s, (SELECT distinct pcode, state FROM suburb) sub
WHERE n.school = s.myschool_url 
  AND s.postcode = sub.pcode
  AND n.year = 2009 
  AND n.grade = 9 
  AND n.area = 'numeracy' 
ORDER BY n.score DESC;

After overcoming a few problems I managed to write a scraper for the myschool.edu.au data. Unfortunately they choose to put data in HTML, so the scraping process may have led my data to have some unknown errors. I publish (see bottom) the scraped data as I believe that per the IceTV v Nine Network [2009] HCA 14 case, any data that my scraper produces as output from the HTML input is not subject to the copyright of the original HTML content (this also means that I cannot publish the HTML pages) and the Telstra Corporation Limited v Phone Directories Company Pty Ltd [2010] FCA 44 case, that the raw data that is scraped is not subject to copyright.

I wish I could bzip2 up all those HTML pages and give them to you just to save your download, because the myschool.edu.au site doesn't compress their pages when I tell them I accept gzip over HTTP, so it took up almost 2GB of quota to download all the HTML pages, oh well.

Some preliminary statistics from the data.

• There are a total of 9316 (or 9279 after I ran a newer scraper at a later data) schools. Of these,
  • 1538 are Secondary (of which 30% are non-government and 70% are government)
  • 1407 are Combined (of which 68% are non-government and 32% are government)
  • 6054 are Primary (of which 23% are non-government and 77% are government)
  • 317 are Special (of which 15% are non-government and 85% are government)
• So,
  • 6451 are Government (69%),
  • 2865 are Non-government (31%)
• These 9316 schools contain a total of 3 366 351 students of which,
  • 1 745 224 are male (51%)
  • 1 651 127 are female (49%)
• The most schools in 1 postcode is 40, which are all in the postcode 2480.
• The average student attendance rate is 92.007%
  • 91.870% for Government, 92.335% for Non-government
  • 89.205% for Secondary, 92.982% for Primary, 90.675% for Combined, 89.170% for Special.
• There are a total of 265 960 teaching staff (full time equivalent of 241 408) and 124 117 non-teaching staff (full time equivalent of 86 511.9).

I could report a lot of stats like these above, all you need is a basic knowledge of SQL, but as much as I enjoy working out these stats I find graphs and graphics much more intuitive, so that is up next. Because of the vast dimensions to the data you can make all kinds of graphs so what would be best is a system to draw graphics dynamically which allows the user to decide what is graphed, but this takes more work so that is on the todo list.

I've also looked into doing some heatmaps using the geographical location of the schools, I could have used Google Maps, or I could use OpenStreetMap and libchamplain. Both have pros and cons... But for now I used Google Maps because their API is simple and I've always wanted to experiment with it, the downside is I'm not sure about the copyright of their maps and subsequently any derivative works. This image is just a test showing a dot for each school in the system, but its very easy to change the colour, size and opacity of the dots based on features of the school.

Another test (some markers will be missing or in the wrong place, like the ones in NZ!),

[caption id="attachment_1023" align="aligncenter" width="450" caption="Google Earth map showing markers for Australian schools (though not completely accurate). (Copyright notices in image)"][/caption]

Source code? http://github.com/andrewharvey/myschool

Don't want to scrape and parse but want the raw data in a usable form? http://github.com/andrewharvey/myschool/tree/master/data_exports/

Extra thought: Currently the code uses Google's API for geting the geolocation of the school, I could use OpenStreetMap for this also, however it would take more investiagtion to determine what tools exist. At the moment all I know is I have an .osm file of Australia, but schools aren't just one dot, they are a polygon so unless I find some other tools which probably exist, I would need to (probably) just use one of the points in the polygon.

Or I could used the Geographic Names Register for NSW, but that is just for NSW... http://www.gnb.nsw.gov.au/__gnb/gnr.zip

Here is a letter I wrote to the Board of Studies (service@bos.nsw.edu.au).

Hello, I'm not exactly sure who I should address this to, so I hope you can pass it along to the relevant person. I am writing to ask that the Office of the Board of Studies considers licensing their syllabi and examination materials under an open content license (such as Creative Commons, GNU Free Documentation License or another open content license). Currently the Board's course syllabi, HSC and SC examinations and Notes from the Marking Centre are licensed in a way that prevents redistribution and derivative works. The current status of the copyright licenses hiders students and teachers ability to use the syllabi and examination materials for study through sharing and collaboration of content. For example it is to my understanding that students, teachers and anyone else cannot take all the syllabus "dot points" and annotate them with their own content, and republish this for the benefit of others. Similarly the current licence prevents use of syllabus extracts such as "dot points" for collaborative works using modern web tools (such as wiki's). Please note that I have published this letter on the internet (http://andrewharvey4.wordpress.com/2009/03/14/a-letter-to-the-board-of-studies-nsw). If you agree to any reply to this letter to be posted online (with credit of course), please let me know otherwise I will not publish any reply. Thank you for you time, Andrew Harvey (Past HSC student, Currently University Student)

(EDIT: A related post, http://andrewharvey4.wordpress.com/2008/12/24/board-of-studies-nsw-and-their-syllabi-copyright-license/)

This is old news but I've been meaning to write at least something about it.When I did my HSC back in 2007 I found that there were no comprehensive notes for my subjects that suited me. That's no surprise to me, in fact I think most people would find that they to have not found a set of notes that already exists and suits them perfectly. So I wrote my own. I used as many different and variety of sources that I could find and I merged these together into a set of notes that I could understand and reread if I ever forgot.

I made them available here, http://andrew.harvey4.googlepages.com/, or here if you prefer a simple directory listing.

Initially I released them as "all rights reserved", with a disclaimer allowing reproduction for non-commercial use. Since then I've licensed them under a Creative Commons Attribution-Noncommercial-Share Alike 2.5 Australia License.

One thing I did when I was studying and writing my notes was to build upon the works of others. However this was difficult with the current copyright laws and the licensing of most of the material I used. So I had to change it enough to be new works that were not derivatives. Hopefully under this licensing of my notes I will save some others the trouble and allow them to take works that already exist such as my notes and change them or take extracts or add to them to produce and publish a set of study notes that suite their needs, without the need to be breaking the law and risk legal threats.

The license is one thing, but it's still hard to add to my notes to make your own derivative version if I only supplied a PDF version. So to fix this problem I've released the source document (Microsoft Word, sorry but this is what I started it in) so that anyone can easily build upon my work.

Out of all the rights that the Copyright Act 1968 (Cth) grants me, there are at least two that I think all copyright owners should not waiver (most of the time). Those being two moral rights from the Copyright Act 1968 (Cth) (part IX, division 2-3), the right of attribution of authorship, and the right not to have authorship of a work falsely attributed. I agree these should definitely be part of the act I'm glad they were recently added.

In the past week (more like a month now) or so I've had a few requests asking me how I got access to my exam scripts (i.e. my exam responses) and how they (having just completed their HSC) could access theirs. In light of this I thought I would explain why I think exam scripts should be accessible to the student.

About a year ago I made a request for my HSC examination scripts under the Freedom of Information Act 1989 (NSW). The process for submitting a FOI request is documented by the Board here. I was granted copies of these documents[my exam scripts]. In the past people have requested things such as raw marks, I did request those too but that was denied for me. You should note that the Board may or may not grant access to these documents in the future.

Now to why I think students should have access to their scripts, which is mainly because it makes the whole process more transparent (even US President Obama is pressing this with his recent FOIA memo). There should be nothing to hide, students should be able to check what they wrote in the exam. They should be able to publish this along with how their response was marked so that it can be scrutinised and studied by future students. I'm not convinced that this is the best study approach in the long term but that is no excuse for disallowing access to scripts. It would also be great if students could also find out how their questions were marked on a question by question basis.

However I can see reasons why the Board would not want to release exam scripts. It is time and money consuming. Even if the process is automated it still costs money and some time. For this I would accept why the Board would charge a reasonable fee for giving you your scripts.

The Board of Studies is doing the right thing here, they did allow my FOI request so I cannot argue that they are hiding them. Kudos to them for this. I hope two things to happen now, more people become aware that they can get their scripts, and the Board continuing to allow these requests.

In the past month or two I've been watching and listening some of Lawrence Lessig's presentations and I've got his books on my reading list. I could do a lot of blogging on those topics but I wanted to focus on one particular thing. As I was reading Code v2 it lead me to think about a copyright issue that is close to me. It deals with the fact that the Board of Studies NSW, a government organisation copyrights (with a very restrictive license) its syllabi. These syllabi document what students should learn as part of their secondary state education HSC courses. These syllabi are material that students use as part of their study.

For the purposes of review here is the license that the syllabi are provided under,

"© 2002 Copyright Board of Studies NSW for and on behalf of the Crown in right of the State of New South Wales. This document contains Material prepared by the Board of Studies NSW for and on behalf of the State of New South Wales. The Material is protected by Crown copyright. All rights reserved. No part of the Material may be reproduced in Australia or in any other country by any process, electronic or otherwise, in any material form or transmitted to any other person or stored electronically in any form without the prior written permission of the Board of Studies NSW, except as permitted by the Copyright Act 1968. School students in NSW and teachers in schools in NSW may copy reasonable portions of the Material for the purposes of bona fide research or study. Teachers in schools in NSW may make multiple copies, where appropriate, of sections of the HSC papers for classroom use under the provisions of the school's Copyright Agency Limited (CAL) licence. When you access the Material you agree:

• to use the Material for information purposes only
• to reproduce a single copy for personal bona fide study use only and not to reproduce any major extract or the entire Material without the prior permission of the Board of Studies NSW
• to acknowledge that the Material is provided by the Board of Studies NSW
• not to make any charge for providing the Material or any part of the Material to another person or in any way make commercial use of the Material without the prior written consent of the Board of Studies NSW and payment of the appropriate copyright fee
• to include this copyright notice in any copy made
• not to modify the Material or any part of the Material without the express prior written permission of the Board of Studies NSW."

Board of Studies NSW

In my opinion this is absurd! This is depriving students access of material that they require for their studies. This is not a private education institution, this is a government public education system. Students need to know what to study, this document tells students what to study, and as this document is not distributed to students (as in students are not provided a hard copy) the only way they can access it is to copy it, but apparently this is illegal!

The above license does give some rights to school students in NSW ("School students in NSW and teachers in schools in NSW may copy reasonable portions of the Material for the purposes of bona fide research or study."), but why only school students in NSW, what about publishers who are providing material to help students in their studies (for example an annotated copy)? Also why limit the amount students can copy to "reasonable portions"? So basically students cannot make whole copies of this document to aid in their studies!

Also why can't anyone remix the document adding their own annotations or commentary and then publish this? And why only for "personal" use? What if I want to provide a remixed copy to anyone who wants it? If the Board is worried that someone may change the document then republish it and someone mistakes this as an official version, then they should not worry. People are not stupid they know that if they want to ensure the reliability of the document they will go to the source. This is not a valid reason for refusing copying of the document.

These documents should be licensed more freely. They should be in-near public domain allowing anyone to do whatever they want with it. I say "near public domain" because I can understand the Board wanting attribution. But apart from that I don't see any other legal constraint that needs to be placed on these documents. Board of Studies, please consider a license such as the Creative Commons Attribution-Share Alike 2.5 Australia License.

I welcome comments on this matter.

As reported here and here the government will reportedly be issuing NSW years 9-12 students will a free laptop each. If this is true and if the government goes ahead with it there are some issues I think need addressing. As reported in The Age, "Next year NSW would begin rolling out lightweight, three-quarter sized laptops to the 197,000 students in years 9 to 12 with a lightweight." A lot of the issues depend if students will leave the laptops at school or home, or take them to and from every day. However as reported by The Age, Ms Verity Firth the NSW Minister for Education and Training said "That means the kids can bring their laptop in, open it up at their desk and just use it" So lets assume that either students will have to take them to and from school everyday, or can by choice. The first problem I see is if every 9-12 kid in the state has a high chance of having a laptop in their bag then street crime may rise and students may be mugged for their laptop. Unfortunately streets are not yet safe enough. Ms Firth reportedly also said, "It will be connected to the Department of Education network. As it is security coded there will be no impetus to steal one of these, because you won't be able to get access to the network." I can't see how this would technically work, and I very much doubt that it would work. Short of being stolen, what is stopping kids selling the laptop for cash?

Another issue is (more so for year 9-10 rather than 11-12) but I think a lot of students will not look after their laptop. Have you seen a group of year 9 kids, some throw bags around, others get tread on, and on a crammed school bus or crammed corridor some guy will just push everyone like sardines in between a rugby union scrum. Books inside bags can survive this, but laptops stand no chance. Schools will also have to up their security (although this is a good thing) as in the past and probably still now you would be refused to enter a certain classroom unless you left your bag outside in the corridor open to thefts.

They are some of the major concerns I have, but also students will just end up playing games, watching movies, instant messaging, emailing, surfing the web (although not so much due to the locked down nature of the DET provided internet) instead of doing school work or listening to the teachers, though I'm not sure if these things are good or bad. Because these technologies could also be using in a more relevant learning related way. For instance what would be better for the student, reading up the topic on Wikipedia instead of not making sense what the teacher is saying; instant messaging a friend about something the teacher just said instead of talking out loud and disrupting the teacher. I think we should not move so quickly to say this kind of behaviour should be condoned or discouraged rather more discussion about it should occur. This is a whole different argument and whole different matter to what I was originally trying to explore in this post, but although the technology is changing these kinds of things have been happening in the past and will continue to happen in the future. The thing that does change when you introduce technology into the classroom is that instead of the teacher being able to see if you are reading a maths textbook or science fiction novel, the teacher can no longer tell what you are doing on your laptop. These are some of the benefits of laptops in the classroom, however I think more discussion is needed.

Additionally, what will  happen when specific software is needed for a particular lesson. For example AutoCAD for a technical drawing class. You can't remove the classroom PCs yet as they would be the ones with the software installed and licensed on. There may be ways around this though.

The Age reported that "The roll out would cost $2,245 per student". Putting aside network infrastructure $2245 does sound a little too much for a mini-laptop. Currently you can get a HP DV5-1004AX (FK680PA) (CPU: AMD X2 Duo Core 2.0Ghz, RAM: 2GB DDR2, HDD: 320GB SATA, Graphic: ATI Mobile 3450 VGA dedicated 256MB, Screen: 15.4” WXGA (1280 x 800), DVD Burner Lightscribe, Card Reader, WiFi 802.11N, Gigabit LAN...) for $999, minus the cost of Microsoft Vista if you opt for a free OS like Linux. At that is not bulk buying. The $2245 per student may have changed, or may include other costs that may explain a bit more, but it is way too much for just the laptop.

Lastly the internet available at DET schools is mostly useless due to over-blocking sites on the internet. So the use of the laptops would be severely limited by this, although that is another matter entirely.

UPDATE: So it seems I have finally leant something. I should have gone straight to the source, the Premier's Media Release. After reading it I see how the media can change views. For instance "The computers will cost around $500 per unit with running costs bringing the cost per student to $2245.", which is not how I interpreted the media outlets stories.

References: http://www.zdnet.com.au/news/software/soa/NSW-to-censor-student-laptops/0,130061733,339292846,00.htm http://www.smh.com.au/news/national/laptops-in-schools-will-be-antisocial/2008/11/30/1227979845018.html http://www.smh.com.au/news/national/nsw-students-to-get-promised-laptops/2008/11/29/1227491892398.html http://news.theage.com.au/national/nsw-students-to-get-laptop-computers-20081130-6nmm.html

Only a couple of days ago the new Industrial Technology Syllabus to be implemented for the HSC in 2010 was released. It appears they finally weaved out a lot of the bugs making it much clearer and much less ambiguous. You wouldn't think it would take them six years to do this, but turns out it did. The syllabus was not redone, rather just amended.

As for the changes... Well I guess the biggest change is the removal the Building and Construction, and Plastics Industries. I can understand the removal of Building and Construction as there is already a Construction VET course available, it's always a shame to see a subject go so bad news for plastics enthusiasts, although it's understandable when it gathers next to zero candidates each year.

The four sections of the course,

A. Industry study B. Design and management C. Workplace communication D. Industry-specific content and production

have been changed to,

A. Industry Study B. Design, Management and Communication C. Production D. Industry Related Manufacturing Technology.

They have also separated a lot of the preliminary content from the HSC content. This makes a lot of sense previously it appeared that you were supposed to learn the exact same content in both years. Also they have listed "Students learn about" and "Students learn to" dot points for the Major Work.

The most interesting (to me at least) changes were to that of the Graphics Industries specific content (note that they are now called technologies (collectively as focus areas) rather than industries e.g. you would now say the focus area Graphics Technologies). I support many, if not all of these changes although you get the feeling that this is what the original syllabus writers meant to be in the syllabus but simply forgot about and only now noticed that it was missing. I say this because much of the content from the previous HSC exams was based on material and content that was absent from the syllabus but has now been placed in the 2010 one. The order and categorising of this material has been redone and is much cleaner and nicer now.

For instance we now have oblique drawings (with references to cabinet and cavalier) mentioned in the syllabus along with,

• A mention of architectural drawings including plans, elevations, sections, footing details, plumbing, electrical and roofing details, council requirements, site plans, set backs, shadow diagrams, landscape plans and colour palette and material selection. Previously they just said we need to know architectural styles and details without any elaboration.
• axonometric projection
• presentation techniques now include 'fly-thoughs' and prototypes
• and equipment includes, both computer software packages AND mechanical drafting equipment rather than just either, scanners, electronic storage mediums such as external hard drives and flash drives (although they could have mentioned the common practice of storing files centrally on a file server in one place for many people to access, which is the much more common practice in the workplace), display folders, appropriate sized paper and stationary.

The Multimedia Technologies section also is much better now. It now contains the study of different types of fonts, formatting features, page layout elements for publications, features of graphics such as file formats and resolution, methods of obtaining images, image manipulation and editing, audio features such as sampling rate, file formats, analogue vs. digital, video features like frame rate compression, editing, compositing, animation techniques both 2D and 3D with references to motion capture, virtual reality, along with the world wide web, intellectual property, and the list goes on and on... Don't just go by my description here go read the syllabus document, you will be very pleased with the changes or should I say additions.

If I were doing my HSC again, I know for sure I would have a very hard time choosing between multimedia and graphics technologies. They used to be together as one industry back pre 1999, although I must admit it is too much for someone who has done neither before to master both as one 2U subject. I wish you could do both, but they can't allow that because the industry study, design and communication parts would be too common.

As for the common sections (Industry Study, Design and Management and Communication) the improvements here were good too with much more detail. But it's not just the fact that the document is more detailed, but these details are what you would expect. They are in the right direction and are things that should be included. The Design section reinforces that the major project is not just about production of something, but the design aspects that go into it. The only problem I currently have is where is this design meant to be applied. It should be in the most obvious place, but the way the syllabus refers to production makes this slightly unclear. Timber Products and Furniture Technologies would look at the design aspects of the timber products or furniture product that they were producing. But if you were doing Graphics Technologies, your product is a series of drawings and perhaps related media such as flythoughs, etc. Do you look at the design of these drawings, I would say not, rather you should apply design techniques to the thing you are drawing, whether that be a product, building or a mechanical system. I don't think this has been cleared up.

I haven't been up to date with all things related here, so I may have missed some things. But one thing is for sure that I congratulate the Board for their work on this, and I'm sure many HSC students will benefit immensely from this revised syllabus. The syllabus is in much better shape now. As for the content, well I could argue that the material from the stage 5 graphics technology syllabus is more advanced than that of the stage 6 syllabus, and this should not happen. But as long as the stage 5 course is not a prerequisite, and as long as you have less time to cover industry specific content from the stage 6 course than that of the stage 5 course, there is little that can be done.

(PS. As a self advertisment, my 2007 HSC Industrial Technology Graphics Industries Major Work in its entirety can be downloaded from my site here, http://andrew.harvey4.googlepages.com/)

Many of the courses I have studied so far (such as HSC Mathematics and 1st year MATH1131 and MATH1231 [UNSW]) require some form of mathematical proofs to be used. However in these courses they don't really teach us about proof's, rather they tend to say here is the generic proof to use, memorise it and write it out in the exam. They don't always do this, but most of the time they have asked us to do a proof in an exam there is a similar example provided in the notes somewhere.

I have always found doing proof questions very hard and I think the main reason is that they never actually taught us them! Lucky for me I'm doing MATH1081 which has a section all about proof and I must say, I've learnt a lot of very important and interesting things from this course.

A lot of the time when I saw proofs I would think to myself, that is not a proof! Mainly either because it was so obvious that you can just see its true, or because the proof says its true because of such and such reason, with no explanation as to why such and such reason was also true.

However now that I've been taught proofs, things are starting to make sense. Proofs are actually quite simply, they are there just to convince the reader that some result is true based on some agreed upon facts. Proofs just build on things to provide new results based on those results already known. It is quite acceptable to not prove certain things in a proof because they themselves have proofs and if you were to include these proofs every time you find a new result you would have to publish a whole new book with a couple extra pages stuck on the back.

This whole idea of having proof's based on axioms/postulates is very interesting. In Euclid's Elements, Euclid first writes down a series of definitions so that he has something to base his postulates and theorems on. Without first accepting these basic facts nothing can be proved. Some people say that these axioms are obviously true, I tend to disagree. Rather than viewing them are true or false I think it would be more appropriate to not label them as either. They are merely axioms on which you base deductions. I also think you should not just consider things that are based on these axioms. It would of course be very interesting to investigate the mathematics if you have some very weird axioms that most would say are obviously false. Again I would say its not a matter of this fact being true and this being false, rather here is a statement and here is what you can deduce from that statement.

Back to the idea of what is considered a proof, I think my lecturer has explained it well. A proof is merely a argument, something which aims to convince the reader that the statement which you are proving is in fact true based on those agreed upon facts. You simply show how to obtain the result from simpler things which one also accepts to be true.

Another concept which came up was the use of diagrams in proofs. When we studied sets and we had a question which said prove A is a subset of B, I always thought that the easiest way to show this was to draw a Venn diagram and show that A is always inside B. Then again you could always argue that you are just drawing a diagram of the result you are trying to prove!

From an education perspective, I think the very lack of teaching mathematical proof all the way up until math1081 and yet expecting us to understand proofs is completely stupid. If we ever want to have a deeper understanding of the things we learn and know why they work, then we should be taught the fundamentals first!

Now, I'm sure I've said some things which are probably not entirely correct. I accept that, I'm not an expert in maths, so don't expect me to.

Update (18th Oct 2008): I just read this in a maths text book,

If $latex x+3=5,$ then $latex (x+3) + (-3) = 5 + (-3);$ hence $latex x + (3 + (-3)) = 5 - 3 = 2;$ hence $latex x + 0 = 2$; hence $latex x = 2.$

They go on to say,

"Naturally, such elaborate solutions are of interest only until you become convinced that they can always be supplied. In practice, it is usually just a waste of time to solve and equation by indicating so explicitly the reliance on (the laws of elementary algebra)."[1]

I think this is the best example of what my troubles were in understanding the whole concept of proof. I guess the largest challenge that I face now is in examinations how do I know what level of maths I can assume the examiner agrees upon, and how much I actually have to prove.

References: [1] Spivak, M. (1973). Calculus. Addison-Wesley. pp 4-5.