Percentages (Procedures vs Understanding)

I can’t remember if I’ve ranted about this before somewhere, but here it is for posterity anyway.

Have you ever noticed how at school you’re taught this:

percent

Basic percentages. Divide your numbers, multiply by 100, and you get your percentage. Easy, simple, procedural, and easily rattled off in an exam.

Except, no. No. Not at all. You don’t multiply by 100. That gets you the number, it doesn’t get you the percentage. “100*(14/20)” gets you “70”, not “70%”.

This is because by its own definition, a percentage is a fraction, where by one full unit is normalised to “100”. So 70%, as a digital number, is 0.7.* As a fraction, 70% is 7/10, or 14/20 or 70/100. It is not equivalent to 70. Although it’s implied to get you that number (the equivalent implicit multiplication by 1,000 gets you the rarely used “per-mille” unit, ‰), at no point does multiplying by 100 actually get that actual percentage. It’s effectively included in the definition already.

* It could be anything, of course. 70% of 2 is 1.4 – but you get that by multiplying 2 by 0.7, not by 1.4 or by 70. In any case, the whole thing itself, no matter what it is, is normalised to 100, the equivalent decimal is normalised to 1 the same way.

Have you ever looked at the “%” and “‰” symbols?

In fact, the word “per” usually translates to “divided by” (kilometres per hour means kilometres travelled divided by the time taken) and “cent” means 100. So “percent” means “divided by 100”. The symbol “%” is quite literally a unit, and the unit conveys meaning just as much as km/hr or m/s or Js or Kgm-2s-2. And sticking numbers in there and having a “per hundred” or “per 250” or something like that isn’t unheard of, and pops up whenever it’s convenient to rescale your units to sensible numbers. That’s already what we do when we talk about “kilometers per hour” because the SI unit is the metre, so kph is actually “1,000 meters per hour” – or “1,000 meters per 3,600 seconds” since we may as well go all out on this.

What you have really written when you’ve formally put “x100” in your expression/equation is the following:

percent_2

“(14/20) * 100 = 70%” implies that “7,000 = 70”. Which is absurd.

If you take out that “x100” you get 14/20 = 70/100 = 70%, which is arithmetically correct.

So far, so obnoxiously trivial.

But I think from a pedagogical point of view this might, actually, be quite important. Not just in a narrow, pedantic sense about a bit of numeracy, but in a wider sense about how we (by which, I mean “schools”) teach things as procedures to be followed, rather than as concepts to be applied and understood. The “x100” bit is certainly implied, and it gets you the right number, but it’s not a formal part of getting you the percentage. Sticking it there as a formality strips out understanding percentages, and changes it into a set of steps to be triggered one after each other, without stopping to think about exactly what is happening.

The trouble with procedural steps is that they then only get applied to one situation and one situation only. Thinking about “%” as a unit that means “per 100” is, in fact, incredibly powerful, as looking at units and letting them guide you will let you blag your way through physics, mechanics, thermodynamics, kinetics and near-enough all times that arithmetic rears its head in science. But no, every school kid out there is left just thinking that when they want a “%”, they need to divide and multiply by 100. It’s nothing but sticking a bit of trivia into a drop down menu to be used in a few narrow situations.

And not to mention that procedural steps put together are notoriously difficult to recall. For anyone with even a mild gift at numbers the percent thing might look too simple, so to illustrate this let’s jump to an example from chemistry:

That’s a rotary evaporator, a common piece of laboratory equipment for evaporating solvents – whenever you see a generic scientist on the TV and they’re not using a Gilson pipette, they’ll probably be using a rotovap. The thing just screams “lab” at you.

The main aim is to use a water bath to heat a sample and evaporate solvent. It also uses a reduced pressure so that you don’t need as much heat to do it – the vacuum does the hard work for you. It’s a fairly simple piece of kit under all that mess, and only a handful of components are involved. Yet the first time an undergraduate chemist sees one they practically shit themselves.

So what’s the first thing a student will look for? Of course, the instructions – usually a point-by-point procedure on how to go about doing it.

And they read the procedure.

And they follow it.

And they do it.

And, hell, they successfully complete the task without getting parts of their body stuck in a lettuce and screaming “my god, the blood, it’s everywhere!”.

And then they promptly forget how to do it less than ten minutes later.

I’m not kidding, the recall on using these things is fucking appalling if all students are given is the step-by-step instructions.

It’s not because the equipment is particularly complicated. It’s just that when written out formally the procedure is about a dozen steps long, and it induces a sudden panic about doing things in the right order. “Do I do this before that? Do I turn this valve first or press this button first, and… oh gods, when do I turn this dial and when do I stop it… and…”

Well, pretty soon you’re dealing with a supposedly grown adult freaking the fuck out.

But it doesn’t have to be that way.

If you know why the bloody thing works in the first place, the procedure pretty much writes itself.

“Do I lower the flask into the water first, or turn the vacuum on first?” Well, if you know that the vacuum lowers the boiling point of the solvent, then you’ll know that heating it up first, and then turning the vacuum on risks flash-boiling the entire thing as you lower the boiling point to below the water bath’s temperature. If you turn the vacuum on first, then the pressure lowers, the solvent evaporates, adiabatic expansion cools it down, then you warm it up by lowering the flask into the water bath.

The same thing applies to gas lines, where instructions tend to be along the lines of “Open tap 1, now close tap 5, after that close tap 6 and open tap 2 slowly, break the seal on tap 4 and close tap 1 again…” Even I glaze over reading those things andknow what the hell I’m doing with that kit! Yet if you ask “now, what do you need to expose to the vacuum pump right now?” and let them figure out which tap to open, they can usually do it. You might have to stop, flick them on the nose, and actually prompt the question, but it’s absolutely not beyond the capabilities of someone to figure it out on their own. The procedure writes itself.

It’s a bit more information to take in at first, and it might be quite a bit of effort to actually teach it compared to writing down the procedural list. But you can’t get the procedure wrong once you’ve learned the actual inner workings of the equipment: because the wrong procedure makes no sense at all.

And that can apply back to percentages, too. Someone just taught it might ask “I can’t remember, do I divide by 100, or multiply by 100 to get the percentage?” Don’t laugh there, anyone just taught to rote-memorise the procedure can seriously fall into that trap. But when you actually know what “%” means, that question literally answers itself.

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