Do teachers want students to add their attempts on solving their own question in an email?

Yes absolutely include it. That's the only way for the instructor to see where the gaps in your knowledge are. If you think it makes the email too long, then put a concise summary at the top and then the details below.

Edit: consider also that in the above post, you attempted to answer your own question. Look how much context that gave to the question. Imagine if you'd posted the title and the bolded statement only. We would not have understood your thought process. You might not have received many useful answers, just comments asking you to clarify what you were asking

In a reasonable grad-level teaching situation, the prof/teacher (should!?!) absolutely want to hear your thoughts/confusions/partial-solutions.

Sure, there is a balance that you/we/they need to find, about balancing your own work (possibly dragging out too long and wasting your own time due to some misapprehension) and "asking questions".

It is generally better to at least have (and express) your own thoughts, even if you've gotten stuck or disheartened. That does also show that you've put some effort in, so are not simply being lazy. :)

It is true that organization and formatting of emails is very important in effective communication. Shorter things are easier to organize, of course, but may be inadequate to communicate your message. Then the inescapable burden of organization itself becomes significant... and problems of clarity (or lack thereof) grow.

But, as most of the time, "being coy" is not a helpful communication scheme.

Conceivably your task really is to "figure it out all on your own", but that's a bit misguided in my opinion (being at least a time-waster), and should not be the default way you think of your professor/teacher.

Put the question up front in a terse form in the subject. This allows to know the topic before opening the mail.

Then repeat the question in a longer single sentence para at the start of your email. This allows to perhaps satisfy you immediately without reading the details. And if it is required to read the details at least the prof knows where the story is headed

Then blather the details after with a header, "Details:" or preliminary efforts, or some topical headers.

And don't really blather in that details section. Write concisely and with a structure. But include details. Your pro con here was good.

Executive communication requires to put the "so what" at the front, not after a buildup. And definitely not after a stream of consciousness.

Oh...and the same applies to SE questions. ;)