I once taught a youth group class on the Bible to cover for a catechist who was not able to be there. I made reference to a Sunday Mass reading and opened the Bible to give the larger context of the reading. One of the teenagers from the class (who was brought up in the church, attended Sunday Mass consistently, and received all the proper sacraments at the proper time) said, "Wait, you mean that the Sunday readings are from the Bible?!" (not jokingly). He was literally shocked to discover this, and a number of the others in the class said they had never known that either.
Now we are not protestants, so we do not believe that the Scriptures are the only place to find truth (nor do we have the protestant habit of idolizing our own personal interpretation of the Scriptures to keep us reading them regularly), but this is inexcusable for Catholic children to be brought up without this knowledge. I do not have any idea what was being taught to those children before I was assigned as their pastor, but it is clear that they were not exposed to an essential part of our faith: the infallible and inerrant Holy Scriptures. I am not insisting that every Catholic get a doctoral degree in the Bible, but we need to expose our children (and ourselves) to the Scriptures.
Children and adults should be hearing the Scriptures in their original context on a regular basis. It is good for us to include sections of Scripture in the Mass, but that is not the way that they were originally put down on paper. Just reading them methodically has a significant impact on our spiritual well being (in fact, it may be the one thing that helps many protestants to come into full communion with the Church--they hear truth from the Bible on a regular basis). When we read an entire book (Exodus, Hosea, Acts, Romans, Hebrews, etc.) from beginning to end, we learn from it in a new way.
Let me take this one step further: what if your only exposure to the Bible was from the modern lectionary? Meaning, that you never read the Bible as it was originally written out. Do you know what that would mean? That would mean that you never saw the context of any of these passages that we hear in Mass. It would also mean that you never read a number of passages of Scripture. As I have said before, there are large--and important--passages of Scripture that never show up in the Mass readings (in spite of the fact that some passages are repeated multiple times in the standard cycle of readings). These are the very words of God; inspired by the Holy Spirit and given to us to help lead us to salvation. The Church provides them, let us take advantage of them.