The 'usual' way of growing potatoes is well known to most of us and though good enough it is a compromise based on commercial practices. We gardeners have more time and skill at our disposal than the average farmer and we can use these to look after our potato crops in more careful ways.
Obviously like most crops potatoes do best if good varieties suited to the site are chosen, the soil is well fed with plenty of organic material and most importantly the plants are kept moist at the roots all through the growing season. All the traditional digging, soil breaking and earthing up does give some benefit as potatoes prefer a broken up well aerated soil. But in dry areas more gains in yield come through thorough watering than from any amounts of vigorous digging or added fertiliser. Indeed a thick mulch is often much more use than the usual earthing up.
Potatoes love a cool moist root run and any mulch keeps the soil so longer than with it bare, a mulch also keeps the soil open and loose which potatoes like. Straw is conveniently cheap in the country but best of all is to use grass clippings as mulches. Put on around the haulm in the same way as earthing up repeated thin layers of a grass mulch really improve yields. Not only do they keep the soil moist and open but the clippings rot down feeding the soil life and the potato plants. It's also easier to earth up with clippings than it is to drag up the earth and with a mulch the crop is less digging later to get up!
We can use our cunning to improve our potato crops as fortunately potato plants are very susceptible to manipulation and can be made to respond in all sorts of different manners, they have even had tomato tops grafted on to them which gave rather poor yields of both. But most of all we must guard against inadvertently making their task harder.
Most of us probably chit our potato 'seed'; we start the tubers into early growth in a light frost free place. This is done to force an earlier crop; the stubby shoots produced, although only small, begin the physiological maturation process thus the plant produces it's new tubers sooner once planted than if it had been left unchitted. However as the idea is to make for an earlier crop it is really only worthwhile chitting earlies and second earlies. Indeed chitting maincrop varieties can reduce their yields by shortening their period of real growth in the ground too much - which may occur if they are chitted too long. Luckily if the maincrop seed is kept in the dark it does not mature as fast as being chitted, but may still produce shoots if warm. This is not a problem except that these long soft shoots need care when planting out or they break off.
There is another danger inherent with chitting which is Little or No Potato 'disease'; symptomatic very poor, or no, haulm growth, with early dieback, and only a few if any small new potato tubers formed. This is caused by the chitted seed tuber going from the warmth of the chitting place into too cold a soil. The shock causes it to 'bolt' and instead of making leafy growth it goes into 'autumn' mode and makes new tubers to sit out the winter. These are converted out of the material of the old tuber alone. (The cure is to plant later or pre-warm the soil.)
Indeed I have used this tendency to turn a surplus of the prior year's late stored, and sprouting, tubers into just edible new ones. The surplus are packed in leaf mould, sand or peat and kept nearly dry and in the dark, the old tubers then produce a crop of small new potatoes without adding any water or even seeing the light. These 'freebies' are not of the quality of properly grown ones of course - but they are of more value than a sack full of sprouts and withered tubers!
Another lesser problem with chitting is sprouting the wrong eyes into shoots as apparently not all of them are equally worthwhile. The old boys insisted we stand seed tubers rose-end up whilst chitting - which I could never understand. Then I discovered claims that shoots from eyes at the rose-end crop sooner than those from the eyes from the stem or heel end. Now this will have little effect on early varieties where we want a large number of small new potatoes; all the shoots are left on and the variable speed of cropping becomes almost irrelevant.
However it is different with maincrops where the sprouts are normally thinned before planting out each seed tuber. By reducing the number of shoots left on each tuber we can reduce the number of new tubers produced but with only little effect on the total weight of yield. For long storing maincrops we want fewer larger tubers, and for bakers we desire each plant to make just a couple of huge spuds. Thus we often remove all bar a couple of the shoots on each tuber we plant - but we rarely ensure these are the best ones! If they are from the rose end fine but if from the heel end then apparently they crop late and may fail to catch the growing season.
Of course the time of planting also affects the time of cropping; obviously the earlier the plants are started into growth the earlier they can crop. But too early and they simply suffer in the cold soil and so do miserably yielding poor crops. Thus to ensure success, especially with chitted earlies, it's better to plant later than sooner; these are so quick they can even be planted as late as mid June. On the other hand maincrops must have the longest growing season and need go into the ground as soon as it is warm enough. The old boys summed it up with "Plant your earlies late and your lates early".
Another old claim was that if you saved your own 'seed' that it would crop sooner if it was lifted a week or two early and not left till the plant had matured and the haulm had died down. It certainly pays to carefully select the plant(s) you do save seed from. They must be the healthiest and most normal looking; never ever save seed from odd looking plants and especially avoid any that have died back early leaving a small crop of small tubers - which look ideal for saving as 'seed'. These are the worst to save!
Indeed equally horrendous is saving for seed the small ones left in your sack after the big ones have all been eaten! In only a short time you can inadvertently select for a new very low yielding variety! If just one plant produces a poor crop of small tubers these preponderate amongst those left at the bottom of the sack, selecting these a second year could reduce your crop to next to nothing for no apparent reason. It is less risky to save a few big tubers and cut them up than to plant many small tubers of dubious inheritance.
Cutting up large seed potatoes indiscriminately is risky anyway as again the heel end sprouts ought not be utilised, better to cut the tuber lengthways and to retain only the shoots from eyes at the rose-end. Dividing up a 'seed' tuber encourages rots so should be done well beforehand so the cut surface can heal and not just before planting. Cutting may let the tuber shrivel if it's kept in too dry or too warm a place.
A completely different alternative to cutting up large seed potatoes is to grow on the detached sprouts. Instead of cutting each big tuber up it is allowed to sprout in a light warm place and the shoots pulled off and potted on separately during early spring. Small shoots only an inch or so long can be grown on in a propagator and treated just like tomato plants. If these are then planted out when big enough and after the frosts have finished each shoot can produce as much as a whole 'seed' tuber would have done.
This is also a way of rapidly multiplying a small stock in the first year. One 'seed' tuber produces many shoots, even if you thin them out on planting more tend to appear, and this congestion prevents each shoot growing to it's full potential. By dividing them into separate plants each gets more space and can produce more offspring.
There is a danger with this last though. If you remove all the sprouts from a tuber it will throw a second and third lot from dormant eyes almost too small to see. If these too are all removed then, if the tuber has enough energy, it throws more shoots from under the skin. These may not be genetically the same as the original. For example Golden Wonder reverts to Langworthy, it's non-russetted predecessor from which it was a sport. This process is a possible way of recovering some lost varieties, though we may not always know which they are!
The depth to plant is fraught with confusion. The usual depth recommended seems to be based solely on the ease of commercial cultivation and harvesting. The potato plant makes new tubers on the ends of side shoots coming off the bottom and sides of the main shoots issuing from the eyes and not from the roots as such. These sideshoots may meander through the soil quite some distance but most modern varieties tend to have short ones that cluster the new tubers close in around the plant. Now plant too shallow and you risk all these tubers getting exposed to the light and going green and inedible (though they may be valid for use as next year's seed) as insufficient soil covers them. Plant too deep and the plants may be late emerging and the crop difficult to dig up. The ridge, trench and earth up method of cultivation is neatly designed to balance these opposites.
Leaving that aside, I'm convinced different varieties prefer to be planted at different depths. In particular I like King Edward's which crop poorly on my light dry East Anglian soil and are often rumoured to like deeper planting, so I grew batches planted at different depths. No question of it; King Edward's produced a bigger crop the deeper they were planted. Those planted two feet or more deep were late emerging, stayed alive longest, grew the biggest haulm and gave the biggest yields, but they were an unbelievable task to dig up!