[update] High School visit

Today I visited my old high school. Before this, I visited a few times during first and second year Biomedicine. I don't think I've been there since then, and I don't know many students there anymore.

There were some teachers which have left since they taught me, but some were still there, and I talked to them today. Some of them had quite a few things to say, especially since our school merged with two other schools, and there was a lot of change happening after I left.

Most of the teachers I talked to asked what I was doing now. I'm not someone who hides that I'm doing medicine, and I told them that I finished Biomedicine last year and I'm doing Medicine now. I met some of my past science, math teachers and coordinators, including the physics teachers who helped teach me some of the physics I would later need for the GAMSAT.

A few teachers commented that they still talked about me, even though I haven't been studying at the school for almost four years now. It's somewhat nice to be recognized in some sense, but on the other hand, it is somewhat sad that for some subjects I did, there haven't been students getting 40+ study scores (out of 50; 40 is top 9%) for some of the past years, which caused me to stand out further in the minds of some teachers.

That contrasts with some other schools which have been able to almost consistently get quite a few almost each year. However, my high school was one with students from a lower SES on average, so it wouldn't be expected to do that well. That said, this is a problem with the education system in Australia. In some other countries, the achievement gap isn't as great between low and high socioeconomic status students. The government seems to be doing something about it though, and I can only hope that what they're doing works properly.

One of the teachers also mentioned that more people were leaving the school to go to selective schools. This isn't really a surprise, given that new selective schools have gone up in our side of Melbourne (Sir John Monash Science School and Nossal). However, I can't help to think that some of the reason that people leave is because the Accelerated Program people do not get to finish a year early anymore like I did, which I've discussed earlier this year in this blog.

I certainly would have considered leaving the school for a selective school more seriously given the current situation if I was in the accelerated program now, being unable to finish a year early. Talking to teachers, there also doesn't appear to be any indication of those accelerated people staying another year translating into much better students at Year 12 level, so I still don't agree with what they've done to the program there. Even if those accelerated students were measurably better (which doesn't seem to be the case), it doesn't seem right to me to keep them back another year if they could enter university anyway without staying another year in high school.

[update] GAMSAT and VCE tutoring

I've decided that I should offer my services in tutoring people for the GAMSAT and VCE Chemistry, Physics, Math Methods, and Specialist Mathematics. I've been tutoring on and off for various subjects at VCE, high school, and primary school level since 2009, and I currently do some voluntary tutoring every week.

As you would know from reading my blog, I received a GAMSAT overall score of 83 (100th percentile), including a Section III score of 100 (maximum):

GAMSAT 2011 Percentile Curve:  http://gamsatau.acer.edu.au/results/normInfo.php?yr=2011

See the GAMSAT Science 100 + VCE tutor tab at the top of the page for more information. I have exams coming up next month so I may not be able to tutor much (especially closer to the exam date) until they finish (27 November). After that, I will be away from Melbourne from December 5-22, 2012. Apart from that, I should be available on the summer holidays.

Number tricks: multiplying and dividing by 10

The GAMSAT this year happened yesterday. If you did the GAMSAT yesterday, I hope that you went well.

While I was helping someone with some GAMSAT questions for practice this year, without a calculator, I was reminded of some number tricks which can sometimes help. When calculators are available they aren't really required, but without calculators, the tricks can be helpful. You won't always be able to use tricks in calculations, but when you can, they can save time.

The trick I'll be discussing is multiplying and dividing by multiples of 10. Essentially, this simply involves moving the decimal point. It is sometimes useful if you are dealing with decimal multiplications or divisions.

Example 1:

300*0.04

Just divide the 300 by 100 (ie move the decimal point 2 to the left for 300) and multiply the 0.04 by 100 (ie move the decimal point 2 to the right for 0.04). Then the product becomes easier.

=3*4=12

Example 2:

0.0075/0.00015

Multiply the numerator and denominator by 100000 (ie move the decimal point for both 5 spaces to the right).

=750/15=50


Tricks like these can sometimes cut out time for decimal calculations.

No more calculators in GAMSAT...

It seems like there's no more calculators allowed in the GAMSAT anymore! That's quite a change. http://gamsat.acer.edu.au/sit/prohibited-items/

This brings the GAMSAT more in line with the MCAT; MCAT doesn't allow calculators. It also means that what I've said in my blog earlier about figuring out stuff on the calculator for logs can't be done; they either have to give you easy numbers, or they have to give you options in multiple choice which are spaced apart enough so that only one of them is plausible.

(ie, from my previous example, if you can use non-calculator methods to narrow down to a certain range, even though you can't figure out things to a few decimal points without tables or memorizing values. For instance, we can still say that 1/10 of a substance is left somewhere between 3 half lives (1/2^3=1/8) and before 4 half lives (1/2^4=1/16) because 1/8>1/10>1/16 and work from there, without knowing what the decimals are after the 3).

It also means that the whole e^loge type decay formula which I don't memorize is pretty useless now, without a calculator or log tables, because the only advantage it used to confer is not needing to memorize as many log laws (apart from the most fundamental one; a^x=y means logay=x. One of my friends said his tutor called this the "sock rule" because if you draw a bubble looking like a sock around a and x in the log form, the orientation is just like a^x with x as the power, and a as the base. Then y, which is outside of the sock, is on the other side of the = sign). And now you can't be expected to calculate things accurately in decay anyway, unless it's a "good number" (ie, whole number of half lives).

I think actually they already space the numbers far apart enough in multiple choice for these types of questions so you don't need to figure it out accurately, but I preferred to figure them out exactly anyway for these types of problems sometimes.

I wonder how much the format of questions will change because of this ban on calculators. Maybe there doesn't actually need to be any change, but it will increase constraints on time. In any case, I do think it'd be good for med students to be able to do more basic sums and multiplication without calculators.

sin(x+y) does not equal sin(x)+sin(y)

Following on from my physics talk about vector quantities, it reminded me of what my Real Analysis (second year math subject) lecturer told our class during a lecture. I already knew it, but it does seem to be a mistake some people make. I paraphrase slightly:

f(x+y) does not always equal f(x)+f(y). An example is, sin(x+y) does not equal sin(x)+sin(y). Some people sometimes write it on exams, and it's not correct. Actually, sin(x+y)=sin(x)cos(y)+sin(y)cos(x).

[note: the angle expansion formula is not required for the GAMSAT, so don't worry about memorizing it if you aren't doing math subjects or subjects which require significant math, but you should know that sin(x)+sin(y) doesn't equal sin(x+y), in case you have to solve some algebra and think you've found a neat shortcut which doesn't exist]

I think some people get confused because a(b+c)=ab+ac, so they think sin(x+y)=sin(x)+sin(y), but it's not like that. a(b+c) means a times (b+c), but sin(x+y) doesn't mean sin times (x+y).

How does this relate to the GAMSAT? I think sometimes you might have to manipulate logs in certain questions. No, I don't mean logs you find from trees, even if I say "natural log". I mean logarithms. In radioactive decay or other types of decay with half-lives, you might need logarithms to find out stuff. The "log" also pops up in the decibel formula. There's a whole set of log laws, which can be thought of as the reverse of exponent laws.

In particular, loga(x)+loga(y)=loga(x*y), provided that x and y are positive, and a is positive (which is required for the logarithm to exist). Don't think that loga(x)+loga(y)=loga(x+y), because it's not.

Also, loga(x^n)=n*loga(x) is a law you might need.

Anyway, perhaps I'll finish with an example. Simple one, but one with bad numbers, so you can't guess whole numbers easily.


"A radioactive compound initially is 1000 grams. After 60 hours, there is 100 g of the original compound remaining. What is the half life?"

Actually, before I'll begin, I'll mention that there is some weird formula for calculating this type of stuff with a e^(logex) type thing in it which I haven't bothered to memorize. The advantage is that you don't have to remember your log laws when using it, but the disadvantage is that it's not that easy to remember. Personally, I like to remember the formula more logically, even if it means I have to remember the log laws to solve something on the calculator.

Basically, after every half life, the amount of original compound remaining is halved. So after one half life there's half remaining, and after the second half life there's 1/2 * 1/2=(1/2)^2=1/4 remaining; after the next one it's 1/4*1/2=(1/2)^3 remaining. See the pattern? But in the question, we have 1/10 remaining, and 10 is not a multiple of 2, so we can't exactly just work it out the easy way.

Anyway, from the discussion above (there's more rigorous ways to prove it though, but for our purposes an illustration is OK), the proportion remaining of the original compound is (1/2)^n, where n=number of half lives.

Then to convert that into an actual amount, multiply by the original amount.

So, N=N0(1/2)^n, where N=amount remaining, N0=initial amount of compound. To me, knowing this formula is simpler than memorizing a whole formula with e^loge etc. However, to use it backwards you need to know your log laws, unlike the other way where you don't need to know so much.

Anyway, back to the question.

N=1000, N0=100.

100=1000(1/2)^n
1/10=(1/2)^n

now if you use this formula, this is where we get logs into play. Take the log of both sides (it doesn't matter which base, but your calculator has two logs; base 10 (the button "log") or base e (the button "ln"), so it has to be one of these. It would be simpler if there was a log base 2 for this particular question, but it doesn't exist on your calculator. So if we don't have a log base 2 on the calculator, we need a work-around.

log (1/10)=log (1/2)^n

Now use the log law: loga(x)^n=n*loga(x) on the RHS of the equation:

log (1/10)=n*log (1/2)
n=log(1/10)/log(1/2) (put this onto your calculator)
=3.3219

Now this is the number of half lives. We wanted to know the half life. Now, we know from the question that the time taken was 60 hours, so:

60 hours=3.3219 half lives

What's one half life? Divide 60 by 3.3219.

Answer is 18.06

So half life is 18 hours. That is the answer.

As I said before, there's another formula which is more complex to remember that some people use for it, but which means you don't need to know your log laws. Personally, because I do remember my log laws, I find this way simpler because I don't really have to remember a more complex formula for exponential decay involving half lives.

Hope this helps. I don't really plan to turn this blog into a GAMSAT blog, but I might still mention a few things here and there.

Physics review: Vector quantities

OK... I decided to write up something for any potential GAMSAT sitters reading this site.

Quite commonly when someone begins learning physics, and doing problems, they sub numbers into formulas to get the answer. Yes, that's what you're supposed to do for numerical answers to get the solution, although sometimes there may be a few steps involved rather than just one step.

However, a common elementary mistake is to forget about directions. When we are dealing with vector quantities, we must keep in mind not only the magnitude, but also the direction. Otherwise, your answers will come out weird. Just as an illustration, if I'm facing south and initially travelling south at 3 m/s then turn around and travelling north at 6 m/s, it's different from if I was initially travelling north at 3 m/s and just sped up to 6 m/s. In the first instance, my change in velocity was 9 m/s north (i.e. 6-(-3), or 6+3), but in the second instance it was only 3 m/s north (i.e. 6-3).

In two or three dimensions, there are a few ways to approach vector problems. You can draw vectors and add them "head to tail", etc, and/or separate into orthogonal components and deal with each component separately. For one dimension, the main thing required is just to assign one direction as positive and label the signs of all variables accordingly (ie, opposite direction means that the variable's value is negative).

I'll just finish with an example now. Perhaps I'll just use the same example as I started off with.

"A person was travelling south at 3 m/s initially. He slows down, then turns around and then runs north, reaching 6 m/s. He does all this in 3 seconds. What is his average acceleration?"

Well, in this case, I'll assign north to be positive.

His initial velocity is south 3 m/s, and since north is positive, south is the opposite direction and is negative. So u=-3 m/s. (u=initial velocity)

He is finally travelling north, which was assigned as positive. So v=+6 m/s.

"Time" is 3 seconds; t=3 s (time is actually a scalar quantity, so usually it should be positive)

Anyway, so a=(v-u)/t=(+6-(-3))/3
=9/3
=3 m/s²

Since this is positive, and we assigned north as positive, this means that the acceleration's direction is north. Technically the acceleration needs both a magnitude and a direction; so just saying it's 3 m/s² isn't enough. Anyway, so from that, acceleration is 3 m/s² north.


Now, actually it still works if we assigned south as positive; you don't have to worry about which direction to assign as positive in two directions too much, although it's good to choose one where you don't have too many negatives, since it makes things easier.

Anyway, if I assigned south as positive instead (just to be different),

initial velocity=u=3 m/s south=+3 m/s (since south is positive)
final velocity=v=6 m/s north=-6 m/s (since north is opposite to south, where south is positive)

time=3 s

a=(v-u)/t
=(-6-3)/3
=-3 m/s²

However, since south is positive, and the answer is negative, that means acceleration is 3 m/s² north (ie, since south is positive, north is negative in this instance; and we have a negative answer). Same as before, 3 m/s² north, as long as you interpret the sign correctly by the constraints you gave initially.

A common mistake might be to just use magnitudes; forgetting that directions are important. Doing this:
u=3 m/s
v=6 m/s
t=3 s

a=(v-u)/t=(6-3)/3=1 m/s²; different from the correct answer of 3 m/s² north!

Plugging in magnitudes without thinking about directions only gives the correct answer usually when the direction is unchanged; ie if the person was initially going 3 m/s north and accelerated to 6 m/s north, or initially 3 m/s south and accelerated to 6 m/s south in 3 seconds, then the acceleration's magnitude would be 1 m/s². Not for the question given above though!

So... when doing physics and plugging numbers into formulas, remember to take into account directions! In 1D, this means assigning a positive direction, and figuring out whether a value takes a positive or negative sign, before solving the algebra for the answer. Otherwise, if you've got a few quantities in different directions, your answer is probably going to turn out wrong.

.97 c

You know, every time I see a shop say something is on sale for .97 c, I feel tempted to check out 2 of those items for free since 1.94 c should round down to 0 as it is less than 2.5 c. And if they don't let me, to then give the shop attendant a lesson on decimal notation. Seriously, do some people who make the signs not know the difference between .97 c and 97 c? The first value is $0.0097 and the second is $0.97.

I once saw this as a price for one of Woolworths' reusable bags by the way.


On another note, it's good that the public transport fare issue has made its way to The Age. http://www.theage.com.au/national/education/end-of-the-line-on-cheap-fares-for-melbourne-uni-pioneers-20110301-1bd9d.html

Standardized testing results in education: good or overemphasized?

I recently saw an article in The Age which has criticized the alleged overemphasis of the NAPLAN, the national standardized test for math and English over Australia for years 3, 5, 7 and 9 [Official quits at school's test results spike]. This prompted me to write this essay.



There is no doubt that Australia could do with an increase in literacy and numeracy standards of school pupils. One only has to look at the results of TIMSS to find that Australians do not perform close to the top nations in these areas. Standardized tests are used widely around the world to measure performance in areas. They are a benchmark, with results of people able to be easily compared with one another. As such, they can provide meaningful comparisons between different people who do the same test.

Since standardized tests enable meaningful comparisons, they should be used in education. In the case of Dallas Primary School, they saw an increase in scores for the NAPLAN. The most obvious conclusion, therefore, is that the quality of the students' math and literacy skills has increased. The use of the standardized test has given us a relatively objective measure of this.

However, detractors may argue that the test has been overemphasized. Particularly, that students are trained "just to do well in the test". But this should not be an issue, if the test is written well. If the test covers good math and literacy skills in great detail and depth, then an increase in scores indicates an increase in student ability. Therefore, virtually all the training for students "just to do well in the test" would have the effect of increasing math and literacy skills; the wanted outcome, with the remaining small amount devoted to exam strategy which will become important later on as students move into higher education.

How would you explain a spike in data though? Well, to me, there is no good reason why a real increase in math and literacy skills of the students isn't a good explanation. Two years is a long time, and the students can learn a lot in this time. Especially considering how low Australian math standards are, there is no good reason why students cannot catch up in two years. Of course, there could be other effects too though; more than 30% of people did not sit the test and they could have been low performers. However, a lot of these were excluded due to arriving in Australia for only a short time, and the departmental investigation found no artificial inflation from exclusion from the school. Then again, it's also possible that this is a freak result. We'll find out after the results come out this year. But that doesn't invalidate the test.

As for the "world's best practice" alluded to in the article? Well, just look at the TIMSS results. It seems like the "world's best practice" of numeracy and literacy standards is found in East Asian nations and countries like Singapore. And our Australian performance is not even close! It's not rocket science here, it's not as if we're at the top of the game. If you want "world's best practice", then learn from them. That's what we should be doing.

Standardized testing and aiming to improve upon benchmarks is a great way to increase our literacy and math standards. It is perhaps the only objective way to note improvement. If the test is written well, then most increases in scores would mean increases in real math and literacy skills. However, if the test is not sound, then it should be improved. But they should not be dispensed with completely.

Australia needs to improve its pre-university math and science education

The current state of primary and secondary math and science education is far from ideal. Learning mathematics and science is very slow until the last two years of high school, then the pace picks up enormously in the important last two years before university, and then again in university. As such, the extremely slow pace before year 11 hardly prepares aspiring science university students well for the next 5 or more years of their study.

This can be shown in international benchmark tests such as TIMSS (Trends in International Mathematics and Science Study). According to the TIMSS 2007 Math Report (page 35, 38; 47, 50th in pdf), at year 8 level, Australians perform statistically significantly worse than Taiwan, Korea, Singapore, Hong Kong, Japan, Hungary, England, Russia, the US and Lithuania. Also, a 95th percentile Australian would not even reach the 75th percentile mark in Singapore, Korea or Taiwan; while an average Australian would fall within the bottom quartile in Taiwan, Korea, Singapore, Hong Kong and Japan. In other words, our supposedly gifted students, those within the top 5%, can not even compare with the top quarter of the top math nations. This is a definite cause for concern.

It is also concerning that Australia is below the United States, despite some of their states having a very poor "fuzzy math" curriculum. Such curriculum had very little emphasis on simple basic techniques such as manipulating fractions or even multiplication or division without a calculator. Such reforms had come because the old regime was not perfect. However, this demonstrates what can happen when reforms go wrong.

It could be said that these East Asian nations have more emphasis on the students doing well in math by parents, as a matter of culture. However, even without this emphasis for some students, the top Australian students, especially those who find the current curriculum up to year 10 unchallenging, could do very well simply with a much stronger curriculum than currently.

Australia is currently in a process of implementing a national curriculum. While a reform of math standards is definitely needed, I find it doubtful that standards will reach those seen in Taiwan, Korea or Singapore. But I do hope so, it is definitely needed here.