Let me propose a non-standard distinction between two terms (in the context of teaching):

• An application is a problem or a task outside the main scope of the course with a solution presented using the present tools. When a teacher gives an application, the "applied problem" is actually solved in front of the students.
• A motivation is a problem or a task outside the main scope of the course, which can be solved using the tools of the course but such a solution is not presented.

There are, of course, many things between and beyond these two concepts, but let me stick to these to make a point.

A motivation is vague and often plays with heuristic ideas, whereas an application actually shows how to use the course material for something else. Motivations and applications both present topics that are related to the course, but only applications make the connection explicit and clear.

A motivation is easy to give, since there is no need to go to the details and less time is required. The teacher doesn't even have to fully understand the motivation! A motivation is likely to be confusing, as it pours seemingly unrelated stuff into the students and helps them forget what is a definition and what is a heuristic interpretation.

An application actually works (ideally) with the very tools presented in the course, so it both provides a connection to something else and gives an example of the use of the main tools and definitions. Applications take more effort to give, but they give more to the students. A students feels much more empowered if they can, instead of just listing related problems, actually solve a related problem with their newly found tools.

In my opinion one should be very sparing with motivations (in the sense defined above). Convert, if possible, all motivations to applications or at least give the students a chance to do it themselves. If you wish to list several related topics to make the course seem more relevant, try to keep it separate from actual mathematics. When preparing lecture material, make a distinction between vague motivations and well explicated applications, and think which ones are more appropriate. Ask yourself: "Can my students really see how to connect this other thing to this concept from my course? How much do I need to explain to make them see it?" Don't answer too idealistically.

To answer your question (now in the standard meaning of "motivation"): I would say there is too much motivation if its connection to the course material is not made clear. Throwing a bunch of ideas in the air is not very helpful unless you take them (or make the students take them in exercises) and show how to actually work with those ideas and the course material. Rotations and arrows and distorted paintings can be useful in teaching eigenvalues if they are properly connected to the core content of the course.

Let me propose a non-standard distinction between two terms (in the context of teaching):

• An application is a problem or a task outside the main scope of the course with a solution presented using the present tools. When a teacher gives an application, the "applied problem" is actually solved in front of the students.
• A motivation is a problem or a task outside the main scope of the course, which can be solved using the tools of the course but such a solution is not presented.

There are, of course, many things between and beyond these two concepts, but let me stick to these to make a point.

A motivation is vague and often plays with heuristic ideas, whereas an application actually shows how to use the course material for something else. Motivations and applications both present topics that are related to the course, but only applications make the connection explicit and clear.

A motivation is easy to give, since there is no need to go to the details and less time is required. The teacher doesn't even have to fully understand the motivation! A motivation is likely to be confusing, as it pours seemingly unrelated stuff into the students and helps them forget what is a definition and what is a heuristic interpretation.

An application actually works (ideally) with the very tools presented in the course, so it both provides a connection to something else and gives an example of the use of the main tools and definitions. Applications take more effort to give, but they give more to the students. A student feels much more empowered if they can, instead of just listing related problems, actually solve a related problem with their newly found tools.

In my opinion one should be very sparing with motivations (in the sense defined above). Convert, if possible, all motivations to applications or at least give the students a chance to do it themselves. If you wish to list several related topics to make the course seem more relevant, try to keep it separate from actual mathematics. When preparing lecture material, make a distinction between vague motivations and well explicated applications, and think which ones are more appropriate. Ask yourself: "Can my students really see how to connect this other thing to this concept from my course? How much do I need to explain to make them see it?" Don't answer too idealistically.

To answer your question (now in the standard meaning of "motivation"): I would say there is too much motivation if its connection to the course material is not made clear. Throwing a bunch of ideas in the air is not very helpful unless you take them (or make the students take them in exercises) and show how to actually work with those ideas and the course material. Rotations and arrows and distorted paintings can be useful in teaching eigenvalues if they are properly connected to the core content of the course.