Critical Reasoning – The Art of Arguments

As one of the three segments of the GMAT’s verbal section, Critical Reasoning is an important piece of anyone’s pre-test preparation. The issue, however, is that often students have no idea how to prepare for it: CR often seems to fall into that category of things where you just have to magically be smart, at least in many people’s imagination.

Learning to excel on CR, then, means learning what CR is actually trying to assess: your ability to see through BS.

Don’t act so shocked! Seeing through poor arguments, seeing the limitations of a plan and understanding how a conclusion can fail are crucial abilities for anyone in a management role – possibly because there are a lot of BS arguments out there to be wary of! Critical Reasoning, then, is the science of identifying where an argument is weak and seeing what would strengthen or weaken it.

The Argument

According to the GMAT, all arguments are composed of three pieces:

  1. The Evidence
  2. The Assumption
  3. The Conclusion

The Evidence is what they tell you is true. The Conclusion is what they want you to think. The Assumption is the bridge between the tow: what would have to be true for the evidence to prove the conclusion.

That ‘bridge’ is crucial: it has to link the the elements of the evidence to the conclusion. Consider the following argument:

Socrates is human, so Socrates will die one day.

Here, the evidence is ‘Socrates is human’, the conclusion is ‘he will die’ – and already half of you are saying ‘so everyone dies’! But that last part wasn’t in the argument, was it? It was left unsaid – because it’s the underlying assumption. So how did we figure it out so quickly? If we’d never heard the argument before, how would we figure it out at all?

Think about the elements in the evidence: we have ‘Socrates’, obviously, and we have ‘human’. Now look at the conclusion: we have ‘Socrates’ and we have ‘death’. ‘Socrates’ is already there in both – but we have some loose ends with ‘human’ and ‘death’! It’s the assumption that has to tie the two together – ‘everyone dies’.

A Harder Example

Now, that was pretty easy, but the approach pays off on harder questions too! Consider this one:

A recent scientific survey found that home-grown tomatoes contain, on average, 25% more vitamin C than do store-bought tomatoes. So, if you have a cold, you’re better off growing your own tomatoes rather than rushing to the supermarket.

What assumptions are we making here? Well, look at the loose ends: the conclusion talks about store-bought tomatoes and supermarkets – not precisely the same, but certainly close enough not to be the focus of our penetrating analysis! We might start asking hard questions about whether the ‘average’ improvement in vitamin C found in the survey would carry over to our own home-grown tomatoes – but on the other hand, there’s no reason here to think that it wouldn’t!

Both of these are very clever responses that the test-makers can use to construct wrong-answer traps: answers that appear to get at these potential gaps in the argument without actually doing anything of the sort.

Instead, let’s consider the big, honking gap that I suspect many of us missed entirely: the conclusion talks about having a cold, but the evidence doesn’t mention this at all! Which begs the question: who said vitamin C is good for colds in the first place? Even if it were – and it isn’t! – there’s no indication in the argument that this is so! Without some evidence to think that vitamin C helps you fight off a cold, this argument is utterly useless!

The secret is simple – the test-maker is counting on you to make the assumption you should be identifying. It’s the things you take for granted, the things you know are true in the real world that make for the best wrong answer traps and missed assumptions.

Learning to interrogate your own assumptions is central to Critical Reasoning – and as an added bonus, it makes better at spotting BS.

On Maths and Meaning

I’ll confess something – I harbour a lot of unresolved annoyance at my maths education. Growing up in the United States, I was subjected to a the type of maths teaching that was essentially focused on telling what to do and how to do it (in a particular way) – but had no focus on why we would want to do it in the first place!

Consequently, I learned how to find the roots of a quadratic equation, but not what a root actually meant. The result, of course, was that maths were harder to learn and deeply boring. It was only much later, after university, that I decided to reteach myself what I’d learned (in order to prepare for the GRE) and in doing so began to understand maths for the first time.

One big thing to recognise is that maths are simply a version of logic, using numbers and symbols rather than words. Maths aren’t about numbers at all – not really. They’re about relationships – and the more clearly we see the relationships, the easier maths become.

To explain this, I often tell the following story. Like all stories, it’s a lie – but it uses a lie to tell the truth.

Albert Einstein was not the world’s greatest mathematician. By that, I don’t mean to spread the b.s. idea that he was bad at maths – he was fantastically better at it than I will ever be, certainly – but maths themselves were not his forte. After all, he was a physicist! Nonetheless, while working on his theory of relativity, he was confronted by some very difficult maths indeed, maths that he couldn’t do on his own.

So, in the way of all good scientists, he asked for help from others who knew better, corresponding with the greatest mathematical minds of his generation. And with their help, the the theory of relativity was born.

So we might ask ourselves – if Einstein didn’t do it alone, then why does he get all the credit?

I’d argue this: getting the maths right is one thing; knowing what the maths mean is something else again.

When you’re studying for a standardised exam, you need to rewire your brain to think of maths in terms of meaning: the more clearly the maths become a story, the more obvious what it all means, and the more simply you can eliminate answers that are simply insane, or absurd, or just plain wrong.

So – be like (my purely fictional version of) Albert Einstein: focus on the why.

A Quick Language Lesson

So far, we’ve taken a look at maths from a conceptual standpoint but I want to take a small break to mention an interesting linguistic fact: the origin of the word Algebra.

English is famous for having more words than any other language, with the vast majority of them taken (sometimes forcibly! #imperialism) from other languages. Whenever I ask my students which language gave us, the most common answer I hear is ‘Latin!’ It’s understandable – Latin is the source for a ton of our words, especially all the high-status ones – but also totally wrong. After all, the Romans were terrible at theoretical maths – possibly because their numbering system was an utter trainwreck. ‘Greek’ is a better guess, but still incorrect: the Greeks invented Geometry (from ‘geometron’ – or ‘Earth-measuring’), so they got to name it, but it would be another 1000 years or so after Euclid until algebra was invented.

So who gave us Algebra, then? Here’s a hint: the same language that gave us words like alcohol, alchemy and admiral: Arabic! The word ‘algebra’ derives from the title of a treatise written by the great mathematician Muhammad Ibn Musa al-Khwarizmi around the year 820 CE: The Compendious Book of Calculation by Restoration and Balancing. ‘Al-Jabr’ means ‘rejoining’, and it’s central to how we should think about maths: by seeing an equation as equivalent to a balance, we can do the same thing to each side to simplify the work.

So why was al-Khwarizmi able to go where the Roman simply couldn’t? One possible explanation is that he had access to one of the greatest mathematical technologies to have ever been created: the ‘Arabic’ numeral system. But that’s a story for another blog post!

On Average(s)

One of the concepts that standardised tests use to trouble students is averages, ironically enough. Ironic, because calculating averages is one of the few things from maths classes we tend to retain over time – how could averages ever give us trouble?

While calculating averages is easy – simply add everything up and divide by how many things there are – understanding what they are often isn’t so straight forward. In fact, we tend to think of averages as being precisely what they aren’t – and the test is counting on that.

First off: averages aren’t the middle number in the distribution: that’s the median. And they’re also not the most likely number: that’s the mode.

So, what exactly are averages, then?

An average is best thought of as being the balancing point in the data: the place where all the deviations to one side are perfectly balanced by all the deviations to the other! Consider the following distribution:

1 2 3 4 5

It’s pretty obvious that the average here is 3 – and that’s in the middle, right? It is – but only because of where all the other numbers are compared to 3! Let’s look at the distance from ‘3’ to each of the numbers:

-2 -1 0 1 2
1 2 3 4 5

Those distances balance each other out around 3, so 3 is the average. (Also, (1+2+3+4+5)/5 totally equals 3, so the old method works too.) What’s interesting though is that, if we chose different numbers, we would still see the same thing! Consider:

1 5 6 9 19

Here, the average is 8: (1+5+6+9+19 = 40 and 40/5 = 8). So what happens with those distances?

-7 -3 -2 +1 +11
1 5 6 9 19

As you can see, the distances add up to zero! The numbers balance around the average, because that’s what an average does!

So How Does This Help?

By looking at the distances from the average instead of the numbers themselves, it makes it a lot easier to calculate an answer on the types of questions you are likely to get on a standardised exam. Basically – smaller numbers are simpler and easier to deal with, they take less time and you are less likely to make a mistake! Check out this example question:

A group of children had their heights measured for a project. The 18 boys had an average height of 47 inches, while the girls had an average height of 44. If the group as a whole had an average height of 46 inches, then how many girls are in the class?
a) 7
b) 8
c) 9
d) 10
e) 11

Answer and Explanation – The Hard Way

Attacking this question in the traditional way would involve a LOT of annoying calculation. You could try stating that the group’s average height would equal the sum of the kids individual heights divided by the number of kids – so:

46 = (47(18) + 44g)/(18+g)

where g is the number of girls in the group. We could then move that denominator to the other side to get

46(18+g) = 47(18) + 44g
828 + 46g = 846 + 44g
2g = 18
g = 9

All of that is true! And ‘9’ is the right answer! But, seriously – who has time for this? Not you – not on a standardised exam. So, what’s the easier way to do it? Let’s check out that balance.

Answer and Explanation – the Easy Way

First off: how far from the average height are the boys? Well, on average, they’re one inch taller and there are 18 boys – so, that’s 18 inches altogether! To balance that, the girls would have to be 18 inches shorter altogether – and since it’s 2 inches per girl, that gives us 9 girls.

The maths here are literally as simple as multiplying 18 times 1 and then dividing by 2: smaller numbers, easier calculations and it takes almost no time at all.

The Upshot

The point of this all is that maths questions on standardised tests are not simply about memorising formulae and working really quickly: they’re about testing your ability to think critically and use the proper tool for the proper job. Understanding what averages really are allows you to use methods that are simpler, easier and quicker – so you can get more questions right, feel less stress and achieve your high score!

Hercules and the Tortoise

One of the most frustrating maths questions you’re likely to get involves relative speeds. It’s not that the concepts are hard – it’s just that you probably never got taught the concepts at all! And relative speed is one area where conceptualisation makes all the difference.

To see what I mean, consider the ancient Greek philosopher, Zeno. Zeno was different to all the other ancient Greek philosophers, in that his goal wasn’t to show that logic and rationality could lead you to understand the universe – it was to prove that logic and rationality simply aren’t enough!

To show this, he developed some paradoxes – conclusions, logically arrived at, that are totally absurd. One of his most famous involved Hercules and the tortoise. The story goes like this:

Hercules is Big and Strong and Fast. The turtle is … not. So, if Hercules were to chase a turtle, you gotta figure he’s gonna catch him, right?

Hold on, says Zeno. Hercules would have to run up to the place where the turtle was to start with – and during that time, the turtle would have moved a bit. So, Hercules would have to run up to where the turtle is then – during which time, the turtle will have moved a bit. So, Hercules would have to run up to where the turtle is now – during which time, the turtle will have moved a bit. And that will keep happening … forever.

So, logically, Hercules never catches the turtle – even though he’s Bigger and Stronger and Faster!

Now, obviously, Hercules will absolutely catch that turtle in reality – the point is that, proceeding perfectly logically, we’ve ended up believing something crazy.

So, does this mean maths are insane and we should chuck it all in to live on a commune? Not necessarily (unless you’re into that kind of thing, anyway). Instead, recognise this: it doesn’t matter how fast Hercules is – it matter how much faster he is than the turtle!

Let’s say Hercules can run 12 mph, and the turtle can only run 3 mph. Hercules is currently 27 miles behind the turtle. How long will it take to catch him? Instead of trying to create simultaneous equations and thinking about two bodies in motion, let’s just say this: Hercules is 27 miles away from the turtle and he’s catching up at 12 – 3 = 9 miles every hour. So, at that rate, how long would it take Hercules to catch up? 27/9 = 3 hours. Simple as that.

After all – how fast are you moving right now? You might say zero miles per hour – but you are on the Earth and the Earth is rotating, no? And the Earth is circling the Sun, right And the Sun is circling the Galaxy, the Galaxy is circling the Universe and the Universe is accelerating in all directions at once! So how fast are you really going? Well, zero – compared to the Earth!

All speed is relative speed, just as all distance is relative distance. And there’s no reason to privilege Hercules’ speed compared to the Earth, when what matters is the turtle!

Simply conceptualising the problem this way makes an otherwise complicated issue incredibly easy. But then, that’s the GMAT’s way, isn’t it?

Prime Real Estate

One of the most important concepts to learn before taking a standardised exam is how to think about primes – unfortunately, it’s precisely this part of number theory that most students never get a firm grounding in! Getting to grips with primes allows you to find Least Common Multiples and Greatest Common Factors, add and subtract fractions with ease, and even take the roots of non-perfect-squares in your head!

To see why primes are so important, consider this: primes are the only integers divisible by only two different factors – themselves and 1. Which means every other integer can be broken down into factors that are either prime, or not prime – and the not-prime factors can be broken down the same way. What this means is that every integer in this world can be broken down into its prime factors – and that’s what the number really is! 18 is really 2 times 9, which itself is 3 times 3; so really, 18 is just one 2 and two 3s multiplied together! 18 = 2^1 3^2

Here’s where the fun starts. If we want to take the square root of 18, all we have to do it look for pairs of prime factors – in this case, the two 3s. We can take the square root of 3^2, since 3^2 is 9 – and \sqrt{9} is 3! We can’t take the root of 2, though – so we leave it alone. Consequently, the square root of 18 is simply 3\sqrt{2}

Try it yourself! Break down the following numbers and see what roots you can take.

  • 8
  • 27
  • 50
  • 98
  • 54
  • 252
  • 3,000,000,000,000 (for funzies…)