The Ramah writes in the Shulchan Aruch on Hilchot Purim:

"There are those who say that one must give, before Purim, half of the coin which is currently in use in his place (country), in memory of the half Shekel which they (Israel) gave in Adar."
 

Chazal give several reasons for the bringing of the half Shekel:

1. Because they sinned (with the golden calf) at the half of the day (midday) they will give half a Shekel.

1. "Raish Lakish said: it is manifest and known to He who said and the world was created, that in the future Haman will weigh (pay) Shekalim against Israel, therefore He preceded their Shekalim to his Shekalim. And so it is written (Megillah 13b): On the first of Adar they announce regarding the (bringing of) Shekalim, and regarding the kilayim (clearing the fields of growths of forbidden mixtures)."

1. Because they sold the firstborn of Rachel for twenty pieces of silver (Yerushalmi).
    

And we must ask: what is the connection between these reasons?
 

The Ohr HaChayim HaKadosh (Shemot 19:2) says that the verse: "And they journeyed from Refidim and they came to Midbar Sinai and they camped in the desert, and Israel camped there across from the mountain" alludes to three things which are necessary for Talmud Torah:
 

"The first is vigor and vitality in engaging in the Torah, since laziness is a 'weed' which undermines attaining it… as attaining it cannot be achieved except by great effort, and regarding this the scripture said 'and they journeyed from Refidim' meaning that they had to journey (distance themselves) from 'rifyon yadayim' (weakness of the hands, implying indifference). In other words, we must serve HaShem with enthusiasm.

 
And the second issue is lowliness and humility, as the words of Torah do not endure except in one who lowers himself and regards himself akin to a desert, and regarding this it says 'and they camped in the desert' indicating lowliness and humility like a desert, which anyone can tread upon.

And the third issue is in the spirit of convening of sages, by their joining together full-heartedly and wholly – not being בד בבד (each one alone), a situation about which Scripture says (Yirmiyahu 50:36) חרב על הבדים (and the Talmud Brachot 63b interprets this as: a sword upon those who study Torah in isolation). But rather, they must convene together and sharpen each other and display good will to each other, and this is implied in: 'and Israel camped there across from the mountain' in the singular form of 'camped' (ויחן) – as Israel all became one, as one man, and now they are worthy of receiving the Torah. In short: love and camaraderie of friends."

In the same spirit, the Kli Yakar says: "And they journeyed from Refidim, meaning: away from the place where they were involved in quarreling and dispute, since the place was called masa umeriva (testing and fighting) and רפידים (Refidim) is composed of the same letters as פרידים (divided) denoting the discord which was between them. Like כשב and כבש (are the same word – lamb). And Chazal interpreted this (Sanhedrin 106a) in terms of raf yadayim (weakening of hands [grip]), since one depends on the other – because of the disunity which was between them, their grasp upon the Torah was weakened."
 

The Midrash Yalkut Shimoni says: "The Torah was given through three things: fire, water and the desert. Fire, as it says: and all of Mount Sinai was in smoke. Water, as it says: HaShem, when You went out of Se'ir…the clouds also dripped water. The desert, as it says: In the Sinai desert."
 

And indeed, these three things which accompanied the giving of the Torah parallel the three foundations which we mentioned above:

Fire – alludes to dedication and enthusiasm in Avodat HaShem.
 

Water – hints at the characteristic of humility: "Why are divrei Torah likened to water? Just as water flows from a high place and goes to a low place, so divrei Torah do not endure except in one who regards himself as lowly (Ta'anit 7a)."
 

The desert – this indicates dibbuk chaverim (camaraderie of friends): the desert is a place which is hefker (free) for everyone, similarly the Rav must teach Torah to all for free. Torah learning for free results in dibbuk chaverim. Therefore a Torah scholar is called chaver (friend or companion), a word with the same root as chibbur (connection).
 

In the dispute between Rabbi Eliezer and Chachamim (Babba Metziah 59b), Rabbi Eliezer brings several proofs that the correct Halacha is according to his opinion: "He said to them: if the Halacha is as I say, this carob (tree) will prove it. The carob uprooted from its place (and moved the distance of) a hundred cubits, and some say: four hundred cubits. They said to him: A (Halachic) proof cannot be brought from a carob. He said: If the Halacha is as I say, amat hamayim (the water conduit) will prove it. The water reversed its course (and flowed upwards). They said: a proof cannot be brought from amat hamayim. He said: if I am correct, the walls of the Beit Midrash will prove it. The walls of the Beit Midrash leaned and threatened to fall.
 

These three things as well correspond to the aforementioned elements:
 

The carob corresponds to fire, since it expresses sufficing with little and relinquishing worldly pleasures for the sake of learning Torah – that is, dedication to study. The strength to devote oneself to Torah study can be acquired though enthusiasm and desire for study.
 

Amat HaMayim, naturally, corresponds to water.
 

Beit HaMidrash corresponds to dibbuk chaverim: Torah is learned in the Beit Midrash, where all gather together to study, and not each person going to learn by himself.
 

The Holiday of Purim is the completion of Matan Torah, as Chazal said "They (re)affirmed their acceptance of it in the days of Achashverosh" (Shabbat 88a). Here lies the significance of the similarity between Kippurim – Ki Purim (Yom Kippur - like Purim): in both of them Israel received the oral Torah - only then (at Sinai) by coercion, and now by choice.
 

The half Shekel as well embodies these three foundations:
 

Enthusiasm and dedication: He (HaShem) showed him (Moshe) a coin of fire, and its weight was half a Shekel, and He said to him: like this they will give. (Rashi on Shemot 30:13) The coin of fire demonstrates enthusiasm and devotion to the Torah.  

Humility: The bringing of half a Shekel comes to teach us that every person should have a broken and unassuming heart: he should consider himself lowly.
 

Unity: The bringing of specifically half a Shekel shows that no person is complete without his fellow man. The half Shekels were used for casting the adanim (sockets) for the Mishkan. We are familiar with Rashi's words, that the Mishkan comes to atone for the sin of the calf. The sin of the golden calf resulted from several factors: deficiency in Ahavat HaShem, lack of humility and disunity (according to the Kli Yakar on parshat Shemini).  
 

The three reasons which Chazal give for bringing the half Shekel parallel the three foundations we opened with:

1. The remedy for Haman's decree: the decree came about because of lack of enthusiasm for Avodat HaShem, as the Midrash Yalkut Shimoni says "'Yeshno (there is) one nation'… yashnu min haMitzvot (they are tired of doing the Mitzvot)." [According to this, we can understand Rashi on Megilla 16a, where he writes that Haman knew that he will fail when he saw Mordechai teaching Israel the Mitzvah of the Omer, because Sefirat HaOmer demonstrates Israel's desire and enthusiasm to receive the Torah].


2. Atonement for the sin of the calf: As we explained above, this sin was the result of haughtiness. And the rectification for this is the characteristic of humility.


3. Atonement for the sale of Yosef: Yosef was sold because of disharmony, and the remedy for this lies in the power of unity.