A circle measures 360 degrees. If this circle were marked with numbers like a clock, every number would represent 30 degrees farther from 0 and closer to 360 degrees.

To find the angle formed when an angle marker at 10:00 is rotated to the position normally marked for 5:00 in a counterclockwise direction, we can first identify the positions of these hours on a circle.

- The position for 10:00 is represented by \(10 \times 30 = 300\) degrees from the 12:00 position.
- The position for 5:00 is represented by \(5 \times 30 = 150\) degrees from the 12:00 position.

Now, to find the angle formed by moving counterclockwise from 10:00 to 5:00, we calculate the difference in their degree measures:

1. Starting at 10:00 (300 degrees), we want to move to 5:00 (150 degrees).
2. The angle going counterclockwise can be calculated as:
\[
\text{angle} = 300^\circ - 150^\circ = 150^\circ
\]

Thus, the degree measure of the angle formed by moving counterclockwise from the position of 10:00 to the position of 5:00 is \( \boxed{150} \) degrees.