Traditional grammar books limit their discussion of elements required 
after a verb to direct/indirect objects and subject/object complements. 
There are, however, actually more patterns than that, especially with 
nonfinite phrases/clauses, which cannot always be forced into those 
categories. This is a good example of one of them. Exactly what to call 
it depends on the framework you're using, and since the terminology 
overlaps in potentially confusing ways, we need to be careful how we 
define things.

"studying math" is a complement of the verb (but not a subject or object 
complement), meaning that the verb is governed by "spent." That 
particular choice of verb sets the patterns allowed and requires we put 
a limited range of constituents here:

I spent the whole day studying math.
I spent the whole day in a panic.
?I spent the whole day. (only valid with prior context)
*I spent the whole day to study math.

For your students, I would just call this another type of complement. If 
you want a specific term, the Cambridge Grammar of the English Language 
calls this type of complement "catenative."

Traditional grammar would call "studying math" a participial phrase, and 
since participles have the same complement patterns as ordinary verbs, 
they can have direct objects. So yes, "math" is the D.O. of "studying." 
And its function would, indeed, be called "adverbial" if you're using a 
framework that uses analogies with word categories to describe functions.

More modern accounts would call this a nonfinite clause, since they 
reject the idea that you need an overt subject to call something a 
clause. Among other benefits of this approach, once we think of 
"studying math" as having an implied subject, we can also note the 
relationship of that implied subject to the matrix. In the most common 
case, the implied subject of "studying" would be the object of the matrix:

I saw Jane studying math. (= Jane studies math)

In your example, the implied subject is the subject of "spent." This 
sort of verb is often called "subject-raising," from the idea (based on 
the visual metaphor of a tree diagram) that the subject of the nonfinite 
clause is lifted up to the subject of the matrix. Verbs like "see" are 
called "object-raising."


On 10/12/2019 6:31 AM, Greg Campbell wrote:
> Can anyone lend some input to what we have in the following sentence:
>
> I spent the whole day studying math.
>
> A student submitted it wanting "studying" to be the object complement, which it definitely isn't. One colleague is seeing "studying math" as potentially functioning adverbially (how you spent the day), and another sees "studying" as a second verb (lacking the helper verb) taking "math" as a direct object. It strikes all of us as a pretty common sentence pattern, yet we can't seem to pin it down. I know we're potentially missing something obvious, but any input is appreciated!
>
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