There are many types of explanations, but there are extreme forms along a continuium from completely procedural, stating only what you have done, to highly sophisticated and rich with implications, stating the necessary principles at play but not saying how those principles underly what you did. We will call the first extreme a Procedural explanation (all the steps but no insight). We will call the second extreme a Conceptual explanation (all insight without directions). We will aim for Reasoned explanations -- less extreme than a purely conceptual explanation but certainly in the conceptual neighborhood. A reasoned explanation is one that is rich with imagery, motives, and implications AND spells out their connections with the task at hand.
Example
Problem:
Given angle BAC and point D internal to BAC, construct a circle that is tangent to rays AB and AC and which contains point D.
Procedural Explanation:
- Construct ray k, the bisector of angle BAC.
- Construct ray AD.
- Let point E be a point on k.
- Construct line m, the line perpendicular to AB through R.
- Construct point S, the intersection of side AB and m.
- Construct the circle centered at E and passing though S (circle c1).
- Construct point G, the intersection of ray AD and c1.
- Construct segment ET.
- Construct line j, passing through point D parallel to segment EG.
- Construct point U, the intersection of line j and ray k.
- Construct the circle centered at U passing through D. This circle will be tangent to rays AB and AC.
Reasoned explanation.
Suppose we have another circle, c1 that is tangent to rays AB and AC.
Then:
- the target circle c (the one that will pass through D) will be a scaled version of c1, so anything true of c1 will have a proportional counterpart in c.
- both circles will be centered on the angle bisector of the angle, because all points equidistant from the sides of an angle lie on the angle's bisector.
- If we let point G be the intersection of ray AD and circle c1, then the segment connecting G and c1's center (E) will be parallel to the segment connecting D and c's center (H).
- Segment HD will be c's radius.
- SO ... to get the target circle:
- Construct the bisector k of angle BAC.
- Let E be a point on the bisector
- Construct a circle c1 centered at E that is tangent to the angle's sides (make a perpendicular segment to line AB from E; this will be c1's radius).
- Construct ray AD and find the intersection G between it and circle c1.
- Connect G and E. Segment GE in c1 will be parallel to a corresonding radius in c.
- Construct line j parallel to GE through D
- Find the intersection H of j and the angle bisector. H is the center of the circle that passes through D and is tangent to the angle's sides.