Bay 12 Games Forum

Anyway, I find the "that method won't work every time" argument to be pretty much silly. It would be like someone telling you that it's a bad a idea to drive to the grocery store without looking it up up google maps. Because "pull out your driveway, turn left, turn right, turn left" isn't a method that will work to get you to every place. And that's true. But if those directions get you to the grocery store, and you're going to the grocery store, and you know this...there's nothing wrong with doing that rather than using the method that "works in every case" of printing out a map and directions.

In day to day life, you are right. But if you are taking a course on how to navigate using google maps, just because you happen to know one of the routes the teachers are using as examples doesn't mean you shouldn't use them to learn the skill you are being taught.

With many methods you do have to put in the time and effort to know them well enough for future application. If you are using shortcuts to avoid doing that sort of drill then I understand the teachers being annoyed.

More like /ˈfoʊ/.

Which is to say, it's foe with the oe as oa from goat or foal, or the o from go. And now 80% of the board pronounce those three examples completely differently...

So what you're saying is that they're cheap because the government doesn't pay the full price for them, which drive up the price for other commuters. Which in turn you use as a proof that it's an efficient use of money?

Not exactly.

The government pays for the travel taken, at a reasonable discount. The pensioner gets a card that lets them take the bus any time without any fuss. The closest commercial equivalent for the pensioner would be 10x the cost.

Having pensioners able to take public transport honestly brings down the cost. The actual travel taken by pensioners each year would be worth maybe 2-3x what the government actually pays for it, but it seems unlikely anywhere near as much would be taken without the existence of the bus passes. The income from pensioners is not replacing full fare customers but empty seats, so any income for those seats is a net plus for the bus companies.

Still need to play with the Phoenix Freedom Launcher some more in VR, but I kinda like it. Would be nice if it did a bit more damage and need to get used to the controls, but it's about as agile as I'd expect to keep things balanced. Also need to try this.

I just read the article from Richard Littlejohn. I wouldn't say he monstered her, even though he's been targeted more than anyone else. Don't get me wrong, I deeply dislike the man, I just think it's exaggeration and he's a bit of an easy target. He was out of line with his comments in my opinion (which basically consisted of calling her selfish), but he's entitled to them as much as you or I am.

Well if you only read one article then it probably doesn't look like ongoing harassment...

I'd recommend this post.

Yes, when going from point A to B it's useful to know how to go to google maps and print out an satellite map of your trip complete with step by step instructions of every turn along the way. But it's silly to go to all that effort if point A is your front door and point B is your neighbor's.

Except that when you are teaching people to navigate from A to B as a learning exercise, you kinda want them to go through all the steps.

I'll agree that teaching using trivial examples can be a problem, which is why I bothered creating a non-trivial example. Going back to my original post;

Learning the most general methods for classes of problems is the only thing that makes sense, even when shortcuts exist. Those shortcuts usually only apply to a subset of problems, and relying on them too much is setting you up for failure when they aren't around. I'd almost say that teaching the problems where the shortcuts exist is worse than just diving into the deep end, because it creates the illusion that the full methods are overly complex and torturous for those students who recognise the quicker methods.

As it is, I'd let people get away with using the quicker methods when available, but it does hurt learning the methods that most need practice and drilling to actually fix in your mind. There are whole classes of problem I'm sketchy about solving because I didn't run through enough example problems that didn't have neat and cute solutions that let me skip the whole handle turning exercise. The result is I'm not confident enough to turn that handle without constant outside reference.

As for trying to teach people to recognise and take the shortcuts, sure. Except I'm not convinced there is a universal and reliable method for teaching that. So many such shortcuts and tricks involve having well trained intuition at applying more basic mathematical techniques and pattern recognition. Those are things that require drilling and a desire (or at least a willingness) to learn and explore at an early stage. Expecting students who didn't have that to learn those methods is likely futile, especially in any classroom or mandated lesson situation (yes, even one-on-one).



As far as the whole approximation thing, physicist, remember? Pi = 3.

Quick approximation methods and Fermi calculations are my bread and butter. In day to day life I'll use them constantly. But the level of mathematical (and physical) formalism I have had to internalise in order to use them that reliably is significant. And in any serious attempt to solve an actual problem they serve only as a sanity check and calibration to make sure my actual calculations are in the right ballpark.

I don't see anyone suggesting that long forms not be taught. The topic is general non-comprehension of math. In my own personal life, I run into far more people facing problems like "how many third-cups of sugar do I need" and who are unable to do them than I run into cases like yours that are resistant to non-longform methods.

See, I did;

In this case, however, I believe the method being taught by the education system is this horribly complicated mess:

You were explicitly attacking the basic method of solving such problems in favour of shortcut methods.

If you want my real problem with this, it's that not everyone is going to have the basic aptitude for such shortcuts. They rest heavily on intuition and spotting patterns. I've never seen a reliable way to train such skills, and I've seen people have day after day of intense, one-on-one training, at various levels and using different methods. Even more than general maths training it also depends heavily on the commitment of the student and their desire to want to see things that way.

On the other hand, learning the full toolkit works for everyone. I'd actually say it's essential for everyone. Especially the people with strong intuition who don't seem to need it. Those are the people most likely to use mathematics in more advanced situations in the future, so most likely to encounter situations where they genuinely need to apply the full methods. Absolutely everyone's intuition and pattern recognition fails at some point, and you need to have something to fall back on when it does.

The last significant time I had to divide fractions by fractions was calculating probabilities of states in quantum systems; normalising the sum of the square of multiple fractions to equal 1, where the fractions involved were complex numbers involving surds. At this level it's not even a significant step in the mathematics - pretty much just done by observation - but that's because the full method has been solidly internalised and can be applied without really considering it. If I had learned to rely on approximations or computations of such problems I would have had a hell of a harder time of it.


To pull up a completely different example, I never had any basic lessons in using matrices. I kinda had them thrown at me as part of physical systems and had to just read up on the various applicable operations. Most physical systems have significant symmetries or fall into one of a few basic patterns which allowed me to get away without having to remember the full mechanisms you are supposed to learn. I did great on all the problems we had assigned and came out of the course happy, having avoided a pile of extra work by finding shortcuts that had done the job perfectly.

Except that a year or so later I needed to use the same mathematics without the crutch of simple symmetries. And I couldn't, off the top of my head, confidently carry out these basic functions. I may as well have never used matrices in the past for all the good that first term's use did me. I ended up having to go back over the exact same ground I'd covered a year before, only now in addition to a far heavier workload and trying to apply these new skills to far harder problems. At which point there was no chance of me going back and actually doing the sort of basic practice problems that would help fix those methods into my memory permanently.

That's far from the only place where relying on shortcuts and simple patterns rather than trying to learn and understand the basic concepts has cost me understanding, but it's one I still feel today. I've never developed the full intuition for matrices I have for, say, trig functions or calc, always feeling slightly worried I'm misremembering something and going to mess up if I don't check I'm using the right methods constantly.