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.
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Topic: "People Who Understand Math" and "How Math is Taught" (Read 8278 times)