It's called Half Kaddish. Did someone cut it in half?
The name makes it sound like a cut-down version. The sources say the opposite. Peninei Halakhah (a modern guide to Jewish law by Rabbi Eliezer Melamed) writes that the Half Kaddish is the core of the Kaddish. It is called "half" only to tell it apart from the other kinds, which have more added on.
You can see this in the Rambam (Maimonides, who died in 1204). When he writes out the words of the Kaddish, he gives only the core, the part said here. Then he turns to the Kaddish the chazan says after the prayer is over, when nothing follows and everyone listens and goes home. To that one, he writes, "the people became accustomed to add" a request that our prayers be accepted, and a prayer for peace.
Abudarham (a commentary on the prayers, Spain, around 1340) describes this point in the morning the same way: the chazan says Kaddish "up to le'ela," through the paragraph that starts with that word. When Kaddish follows a finished prayer, an ending is added. After learning, a prayer for teachers and students is added instead. (The Kaddish DeRabbanan unit covers that one.)
So every longer Kaddish is this one plus something. The core stays the same. What gets added depends on what just happened.
Peninei Halakhah draws a practical rule from this. Answering Amen to the added parts is not as important as answering the core. If you are in the middle of Pesukei DeZimra or the blessings of Shema, you do not stop to answer the additions.
Peninei Halakhah also says when the short form is used: wherever a long break would be out of place. This is such a place. Following the Shulchan Aruch (the main code of Jewish law), Peninei Halakhah rules that you may not stop to talk between Pesukei DeZimra and the blessings of Shema.
Peninei Halakhah, Prayer 23:7 and 16:2,Rambam, Mishneh Torah, Order of Prayer,Abudarham, Kaddish and BarchuSeven Kaddishes a day? Doesn't it turn into background noise?
Fair question. Over a full day of davening with a minyan, you hear Kaddish again and again. The number is not an accident.
The Beit Yosef (Rabbi Yosef Karo's commentary on the Tur, a law code from the 1300s) quotes an older list that explains each Kaddish. The first is this one: "after Pesukei DeZimra, because there is a break of verses there." Next comes one after the Amidah, because it is a mitzvah of its own. Each one closes a piece of davening. The list ends: never fewer than seven a day. The Geonim (heads of the great academies in Babylonia after the Gemara) tied the number to a verse in Tehillim (Psalms): "Seven times a day I praise You."
The Mishnah Berurah (a central commentary on daily Jewish law, around 1900) names the seven. This Half Kaddish is number one. The Arizal (Rabbi Yitzchak Luria, the Kabbalist of Tzfat, 1500s) counted twelve a day, Peninei Halakhah (a modern guide to Jewish law) reports.
But more is not automatically better. The Mishnah Berurah also writes: just as it is good to say fewer blessings, it is good to say fewer Kaddishes. He reports that later authorities protested when ten people gather, read some verses or Mishnah or Gemara, and say several Kaddishes over them.
Not everyone agreed. Peninei Halakhah cites the Eshel Avraham (a commentary on the Shulchan Aruch by the rabbi of Buczacz, in Galicia), who allowed many Kaddishes. His reason: Kaddish does not contain God's name, so an extra one is not said in vain.
What the sources share is the reason for each one. A Kaddish comes where a section of davening has just ended. The Mishnah Berurah's list reads like a map of the day: after the psalms, after the Amidah, after Ashrei at Mincha, and so on. Knowing which section just ended is one way to hear each Kaddish as its own.
Beit Yosef, Orach Chaim 55,Tehillim 119:164,Mishnah Berurah 55:1 and 55:5,Peninei Halakhah, Prayer 23:10Most mornings I daven alone. Am I just missing this part?
Yes, and the sources do not pretend otherwise. Kaddish is said only with ten. (The Kaddish DeRabbanan unit tells where the number ten comes from.) The Shulchan Aruch (the main code of Jewish law) applies the same rule to Barchu. Alone, you go from Yishtabach to the blessings of Shema with neither one.
The rabbis who rule on Jewish law talk about Kaddish as something you hear. Peninei Halakhah (a modern guide to Jewish law by Rabbi Eliezer Melamed) sums it up: every person should hear seven Kaddishes a day. It is built as a call and an answer, so it needs people to answer.
How much did people care about missing it? The Shulchan Aruch describes people who each prayed alone and did not hear Kaddish or Kedushah. Once ten are together, one of them stands and says Kaddish, Barchu and the first blessing before Shema for them. (The Rema notes that the custom became just Kaddish and Barchu.) This is called pores al Shema. If six who missed it cannot be found, it is done even for one person who missed it, and someone who already heard can lead it for him.
You also do not have to be one of the ten to answer. The Tur (a code of Jewish law from the 1300s) rules that when ten are together saying Kaddish, even someone who is not with them may answer. He quotes Rabbi Yehoshua ben Levi, a sage of the Gemara: "Even a wall of iron does not separate Israel from their Father in heaven." The Shulchan Aruch rules the same way.
So when you are alone, this part is skipped, not replaced. The way back in is wide: once ten are together, even one person who missed it can have it said again, and you can answer from the other side of a wall.
Shulchan Aruch, Orach Chaim 55:1, 55:20 and 69:1,Peninei Halakhah, Prayer 23:10,Tur, Orach Chaim 55