By 'falange', do you mean 'phalanx', the Greek spearwall formation? I wrote a bit here about the continued use of the spearwall in the early medieval period, and how it remained a central part of English strategy into the twelfth century. The spearwall had proved its worth against mounted knights at Hastings in 1066 and again at the Battle of the Standard, but was a large and unwieldy formation, especially against far-more-mobile cavalry and even more so if you were not on a battlefield of your choosing. The English spearwall had performed so well at Hastings because it had been able to seize a high ridgeline and dominate the battlefield, but it failed when it was lured out of formation and ambushed by Norman mounted knights. The greater mobility and strategic diversity afforded by a heavy cavalry force meant that while spear walls remained important, they were slowly relegated to the role of a holding force or protection for archers, rather than the battle-winning role that they had once enjoyed.

I recommend Strickland's Anglo-Norman Warfare for more.

Since u/BRIStoneman has discussed the traditional hoplite phalanx which used large shields and one-handed spears, I will talk a bit about the Macedonian Phalanx used by Alexander the great and his successors which was armed with full-sized, two-handed pikes up to 20 feet long.

The pike did see a sudden resurgence in popularity at the tail end of the middle ages, as did the study of classical warfare. Yet the concept of using both a shield and a pike at the same time apparently continued to baffle a number of early modern military writers. Machiavelli, in The Art of War came to the conclusion that classical authors must have been mistaken on the subject.

. . . the Macedonian phalanx, which was armed with spears more than 20 feet long, called sarissae, with which they broke in upon the enemy, and yet kept good order in their ranks. Although some authors say they had shields too, yet I cannot see, for the above mentioned reasons, how they could manage them and the spears at the same time. Besides, in the battle between Aemilius Paullus and Perseus, king of Macedonia, I do not remember any memtion made of shields, but only of the sarissae, which were very troublesome to the Romans; so that I imagine the Macedonian phalanx was like the Swiss regiments of today, whose strength lies wholly in their pikes.

Though Machiavelli wasn't an experienced soldier himself, Raymond de Fourquevaux was and in his 1548 "Instructions for the Wars" similarly didn't think that the idea made much sense:

. . . but that should seeme impossible, for as much as they would have been troubled enough to have used one of those weapons alone: and as for to suse them both at once, would have been but little for their ease, for the Pike alone requireth both handes. On the other part, the shield served but only to cover them, because it was not very maniable. . .

Keep in mind that an 18-20 foot pike is a pretty heavy, and when the point is lowered nearly all that weight has to be supported by the soldier's left arm. This could get really tiring fast, and strapping a shield to the left arm would have only made it more so. Second, one of the main benefits of a shield in combat is that it provides an active defense, i.e it can be moved around to protect a wider area as well as to intercept or deflect enemy blows, or even used agressively. With a two handed pike however, the left arm must remain close to the soldier's body to better support the weight of the weapon and any movement with the left hand, for instance to protect the face, is going to throw the point of his pike out of line and create a gap in the wall of spears. A round shield attached to the pikeman's left arm then is generally going to remain motionless during the battle, only covering the pikeman's center of mass, which is already protected by his corslet.

This doesn't mean that shields were never used by pikemen during the 16th century or were considered completely useless. In fact Fourquevaux pointed out that shields wouldn't have been nearly so cumbersome if the Macedonians wore them on their backs instead, and only slung them forward if their pike broke or for close combat. He also recommended that French pikemen should wear shields on their back as well. Quite a few other writers agreed with this and pikemen with shields on their back do occasionally show up in artwork but the practice never became that widespread. William Garrard, whose The Arte of VVarre was published posthumously in 1591 was another proponent of giving pikemen lightweight leather targets to use in close combat, however he also mentions that at the time, the practice was primarily limited to

. . . amongst some nations for want of brest plates of Iron, vse tand lether, paper, platecoates, iackets, &c. For a gorget, thicke folded kerchefes a∣bout their neck, a scull of Iron for a head péece, and a Uenetian or lether Shéeld and Target at their backes, to vse with their short Swordes at the close of a battaile, and in a throng.

This suggests that most soldiers who did have access to armored plates perhaps felt it was less necessary to carry a shield as well. Or perhaps even a light shield seemed like too much of an extra hassle during marches and long campaigns. During this period armor in general was becoming less popular among most soldiers due to its weight. At the beginning of the 17th century, english soldiers in ireland were reportedly leaving behind even breastplates and helmets "almost buried in the soil at every quarter."

One notable example of pikemen fighting with shields at the same time comes from the Battle of Pinkie Cleugh in 1547, where William Patten described that the scots fought ". . . holdynge their pykes in both handes, and thearwith in their left their bucklers. . .". He describes the shields in a bit more detail further on:

And they wear, nue boordes endes cut of, being about a foot in breadth and half a yarde in leangth: ha∣uyng on the insyde, handels made very cunnyngly of .ii.[two] cordes endes

Since Patten also mentions that the scottish troops were only wearing jacks and helmets otherwise for protection, this fits with Garrard's observation that small targets were mainly limited to troops that were less well armored. Alternatively, it might be that the scottish pikemen were carrying these shields to help protect against the large number of archers being fielded by the English, but as Patten himself pointed out, they didn't help against artillery.