Greg,
I think that Karl and kyo have given us some good suggestions. I hope my intuition from experience with many different grammars may be helpful. 

The first question is whether "studying" is best analyzed as a gerund (a verbal noun) or as a participle (a verbal adjective).  
The participle is the preferred analysis when its subject is being modified.  
  "I am studying math" makes me busy in this activity.  
  "Studying math I am ready to solve that problem." qualifies me for some action.
The gerund focuses on the activity of the verbal.  
  "I like studying math" makes it the object of my preferring.  
  "Studying math helps me get ahead" makes the activity the subject. 
As object of a preposition it must be the gerund.  
  "I am busy in studying math" tells about the activity that makes me busy.
  "I spent the whole day in studying math" tells about the activity that caused me to spend the day. 
The preposition "in" relates the prepositional phrase as an adverbial.
This seems to make the gerund analysis the closest related pattern and seems to be right at home with other adverbials.  
It does not exclude other adverbials from accumulating along with it:
  "I spent the whole day with the children in the park playing football." ("the whole day" is a direct object)
  "I am busy all day with the children in the park playing football." ("all day" is an adverbial noun phrase)
This accumulation of adverbials in a preferred order is called its cartology. Adjectivals do the same sort of thing in front of the noun phrase they modify. 

--- [log in to unmask] wrote:

From:         Greg Campbell <[log in to unmask]>
To:           [log in to unmask]
Subject: Sentence Pattern Question
Date:         Sat, 12 Oct 2019 09:31:19 -0400

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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