What are the theological approaches to Christian education?

Theological Approaches to Christian Education can vary depending on the institute from which it's being taught. However, there are some common themes that will be taught assuming it is coming from a true perspective of which Christianity stands. Since the Bible is the foundation of Christianity, this true perspective isn't to say there isn't an opinion or two about certain topics that may differ; however, regarless of the institution, all should be teaching the Bible perspective. If not, one has to call into question if the theological approaches of Christianity are really being taught. The most common theological approahces will include: creation, sin (the fall), the crucifixion, resurrection, and redemption. A synopsis of what the teaching may be is as follows.
First, Creation.
Genesis 1:1-2 (NKJV) says, "In the beginning God created the heavens and the earth. The earth was without form, and void; and darkness was on the face of the deep. And the Spirit of God was hovering over the face of the waters."
This tells us that the very first thing in creation was the heavens and earth. There was nothing created before this. The word "void" in Hebrew is "bohu" and means emptiness. There was nothing there.
Next, Sin (the fall). Sin is a term that can be used as an adjective, to describe anything that goes against God's laws or teachings. It can also be used as a verb, as something someone did or does. Once Adam and Eve ate of the fruit that they were not supposed to eat, sin came upon them, as a consequence since they are the first people, sin came upon all generations to come.
Thirdly, the Crucifixion. This is perhaps one of two key theological aspects of Christian theology. The Bible teaches that the crucifixion of Christ, God's only Son, was to bring human kind back to Him. Since sin had fallen on all people, all generations would not be able to have fellowship with Him. He is perfect, He is righteous. Therefore Jesus and sin cannot live in harmony. So, sin cannot be upon people should they want to live in fellowship with Jesus. The Crucifixion, historically the most gruesome way to die, was the act of Jesus Christ taking on the sin of the world, so that it was no longer on anyone who receives it. He did it willingly and for free. He does not ask anything of the people who want to receive what He did for them.
The fourth key point in approaching Christian education is the Resurrection. Equally as important as the third point - the Crucifixion. This is equally important because without the resurrection of Jesus, he is like any other person, a mere mortal. But, by raising from the dead, he became alive. He conquered death. He did not stay dead. It proved his divinity.
John 20:1 tells us, "Now on the first day of the week Mary Magdalene went to the tomb early, while it was still dark, and saw that the stone had been taken away from the tomb." In verse 2, concerned, she runs to some of the disciples and says, "'They have taken away the Lord out of the tomb, and we do not know where they have laid Him.'"
Further in this chapter, they discover He was not stolen or moved, but in fact, raised from the dead.
Finally, the Redemption for all mankind is the result of the previous key points. Because of Christ's crucifixion and resurrection, sin is no longer upon people, should they choose to accept it. This allowed them eternal life with Christ. Mankind has been redeemed. They started out in a perfect world as God intended. But when sin entered the world, it was no longer perfect. So God made a way to bring people back to Him, thus redeeming the human race.
There may be slight differences in the teaching style or methods, however, in Christian education, these five key point would be in the approach.
Sources: www.biblegateway.com
www.biblehub.com (Strong's Hebrew and Greek Concordance)

This is a huge question because there are a lot of different theological components to Christian education. What the question doesn't specify is what type of Christian school it is asking about. For example, is the school a church run school or a parent run school? There is a big difference in how the school is governed and how the curriculum is affected. Additionally, does the school require students to be from Christian homes, or is the school more focused on evangelizing non-churched families that want a private, Christian education because of one reason or another?
I'll answer the question based on schools in the Christian Schools International (CSI) organization. These schools are schools that have been founded from the Christian Reformed worldview. These schools exist to educate students from the Christian Reformed perspective. Students are required to come from Christian homes, and the school believes in a three-part partnership to educate the child. The school works in concert with the home and the church. Each part plays a different role in the overall education of the child. Most CSI schools do have and require students to take Bible classes; however, that is not the extent of the "holistic" Christian education of the child. Each subject is taught from the reformed perspective with an emphasis on educating students about how their faith in God is active and present in all subjects across the curriculum. Theologically, this can be summarized with a "sin, salvation, and service" model or a "creation, fall, and redemption" model. In either case, students are taught how sin has affected creation, society, school subjects, and the like, and that, as God's people, it is their responsibility to proactively work as world reformers, out of thanks for Christ's salvation.
https://www.csionline.org/about