One observation: People understandably hesitate telling a half-truth. When you teach a heuristic picture to someone, you also need to teach them about how fuzzy it is and when it starts to break down. A more calculational proof has the virtue of being self-contained and robustly transmissible. This is even more important when writing a textbook.

Then there are other cases where I'm puzzled why certain heuristic means of understanding and organizing knowledge don't seem to be usually taught. Take the concept of normal subgroups. In one of his books, V.I. Arnold says that a subgroup is normal when it is relativistically invariant, and he doesn't develop that line of thought any deeper. That statement is a good example of a heuristic analogy that is specific in its detail but general in its spirit. However you phrase it, certainly you should give your students the idea that a normal subgroup is something whose structure is invariant with respect to the parent group's symmetries. As a litmus test, your students should be able to tell whether these subgroups are normal at a glance, without calculation:

Let $E^2$ be the Euclidean group of the plane and let $O^2$ be the subgroup fixing some point.

Subgroups of $E^2$:

• Translations along some particular direction.
• Translations along every direction.
• Translations and glides along every direction.
• Reflections in every line.
• Rotations around some particular point.
• Symmetries of a tessellation.

Subgroups of $O^2$:

• Symmetries of a regular polygon.
• Reflections in a line.

The way I think about the non-normal cases is that there is something something non-isotropic about them, some structure that the subgroup preserves that is not preserved by the parent group. For example:

• Translations along some particular direction: Rotations don't preserve the direction.
• Translations along every direction: No special directions, so it's normal.
• Translations and glides along every direction: Ditto.
• Reflections in every line: Ditto. (This combines the two previous cases.)
• Rotations around some particular point: Translations don't preserve the point.

One observation: People understandably hesitate telling a half-truth. When you teach a heuristic picture to someone, you also need to teach them about how fuzzy it is and when it starts to break down. A more calculational proof has the virtue of being self-contained and robustly transmissible. This is even more important when writing a textbook.

Then there are other cases where I'm puzzled why certain heuristic means of understanding and organizing knowledge don't seem to be usually taught. Take the concept of normal subgroups. In one of his books, V.I. Arnold says that a subgroup is normal when it is relativistically invariant, and he doesn't develop that line of thought any deeper. That statement is a good example of a heuristic analogy that is specific in its detail but general in its spirit. However you phrase it, certainly you should give your students the idea that a normal subgroup is something whose structure is invariant with respect to the parent group's symmetries. As a litmus test, your students should be able to tell whether these subgroups are normal at a glance, without calculation:

Let $E^2$ be the Euclidean group of the plane and let $O^2$ be the subgroup fixing some point.

Subgroups of $E^2$:

• Translations along some particular direction.
• Translations along every direction.
• Translations and glides along every direction.
• Reflections in every line.
• Rotations around some particular point.
• Symmetries of a tessellation.

Subgroups of $O^2$:

• Symmetries of a regular polygon.
• Reflections in a line.

The way I think about the non-normal cases is that there is something non-isotropic about them, some structure that the subgroup preserves that is not preserved by the parent group. For example:

• Translations along some particular direction: Rotations don't preserve the direction.
• Translations along every direction: No special directions, so it's normal.
• Translations and glides along every direction: Ditto.
• Reflections in every line: Ditto. (This combines the two previous cases.)
• Rotations around some particular point: Translations don't preserve the point.