In addition to the short summaries below, Justin Taylor has (once again) posted a brilliantly organized outline and breakdown of these ideas drawn from a lecture by Bruce Ashford. If you have time, you should read Taylor's whole post (which incorporates a chart and notes created by others from the lecture; the outline below is from Taylor's post). For the best impact, you can listen to the audio from Ashford's lecture. But Taylor's compilation is enough to ground anyone so you can start thinking through how this affects your approach to homeschooling, or public or private education, or Bible studies, or church community.
The third view does something similar, in that it divides Christian scholarship and non-religious studies into two separate, side by side spheres of life. One of the things that attract people to this view is the appearance that non-Christians and Christians alike use the same intellectual tools in secular fields like mathematics, science, and architecture. They believe that what we call general or natural revelation (what God has revealed to all people through nature, without Scripture or faith) and common grace (the blessings and gifts God has given to everyone through nature, regardless of faith, like the ability to think and the ability to use moral reasoning) are completely sufficient for doing all the non-religious work and study in these secular fields. There is no "Christian" math and secular math, for instance. But this view fails to recognize the many ways a Christian mind and Spirit-filled heart transform what we do in mathematics, science, or architecture: the purpose of our work and the methods we use to complete it may be drastically different for a Christian with a redeemed mind.
An atheist may use science to try to prove his belief that there is no God; a Christian can use science to prove that the existence of God is far more likely and plausible than the atheist's belief in a universe made out of nothing but mindless material. An agnostic philosophy professor may spend her whole career teaching students to compare various systems of belief and ideas without ever really giving them an ultimate answer - or finding one herself. A Christian philosophy professor can help students discover how the ideas of philosophy and the greatest thinkers ultimately point the way back to the God who Himself defines reason and logic, revealing how Aristotle or Locke deduced things about nature and humanity that confirm what God reveals of Himself in the Bible. Likewise, the third view often overlooks how the corruption of sin in the mind and heart lead non-Christians away from truth, blinding them to what God reveals of Himself even in philosophy or art. The Christian still uses general revelation and common grace, just like the non-Christian, but the redeemed knowledge and biblical understanding of the Christian allows her to see further and deeper into the meaning of things and connect them to their ultimate, God-intended beauty and perfection. They both start with the same tools, but the Christian is more informed about how and why to use them.
Finally, the two-level approach (first view) and side-by-side approach (third view) can also lead into an inconsistent, unstable dual life where the Christian thinks different rules apply in "Christian" things than in "secular" things. This can become a practice of being "Christian" when at church and in Christian circles, but doing work or politics as if the rules that apply to them are simply the rules the world uses: doing things in the secular world by secular rules. The danger is that we cease to be any different from the world in how we act, and in fact begin to act inconsistent with Christian character and virtue when we're at the office or in the classroom. These same concerns are addressed in H. Richard Niebuhr's Christ and Culture, which I first encountered 15 years ago. I summarize in the next post how Niebuhr compared each of these problems, because his insights are very similar to Ashford's categories yet go much deeper and add more definition. That post also addresses the problems and weaknesses of the second view Ashford described, Grace Against Nature.