Your question alludes to a few different domains such as software engineering, education, and philosophy. Each have their own sets of terminology and "problems" they attack, but you may be on to something generalizing a dichotomy that is present in each: a dichotomy in thinking and problem-solving styles.
I made a brief search of the literature from these three fields, using keywords from your question and one additional concept which I think is important: iterative/iteration.
I'll put forth my suggested name and then point out the sources which may support it, or at least raise interesting points for discussion.
"Iterative versus Monolithic thinking"
This phrase describe the two facets of your dichotomy, as I see it. Additionally, mereology is a single word we can use to describe a thinking approach in action (that of a person, an organization, or a project), and its degree or quality of being iterative (bottom-up, part-based) versus monolithic (top-down, whole-based). We might use mereological as an adjective in these cases. Historically, mereology has been primarily used in philosophy and mathematical logic. The Oxford definition:
mereology Philosophy [mass noun] The abstract study of the relations between parts and wholes. (Origin from French, formed irregularly from Greek meros part + -logy.)
Starting from your phrase "you can take a piece of the puzzle and keep failing and changing until it works," I thought of Iterative Development described on the C2 Wiki:
Iterative Development is about planned rework. You create something, review it and then change it (hopefully improving it) based on the feedback.
Iterative methodology is a widespread concept in software engineering. Your question honed in on "rapid feedback", and this phrase has often appeared in computer science literature paired with "iterative". I found several 20th century examples on Google Books, and even more if the search is expanded to the last 16 years.
The pair "iterative versus monolithic" appeared in this 2004 book.
In the comments, aparente001 correlated the computer-based math education you mentioned to drilling (instruction by the means of repeated exercises or practice.) Iteration encompasses this in its simplest meaning, "a repetition". (See etymology.)
I selected the word "monolithic" to represent your phrase "top-down thinking about the problem, fitting the whole system in your head." This was likely from my bias having often read the term in computer science usage. "Holistic" is another possible word for this half of the dichotomy, though it has picked up many fluffy connotations in recent decades and seems a "fad word" to me.
Last, some links to read on mereology as the one-word descriptor of this dichotomy:
From the Stanford Encyclopedia of Philosophy:
Mereology (from the Greek μερος, ‘part’) is the theory of parthood relations: of the relations of part to whole and the relations of part to part within a whole.
Follow to note 1: "In some literature, ‘mereology’ is also used with reference to work in General Systems Theory..." and I would suggest that GST may be another subject to investigate within philosophy, while we wrestle to name this dichotomy.
Also from Standford's Encyclopedia on the subject of "Mechanisms in Science" (a dense read, I should warn):
Darden also emphasized mechanisms as an important framework concept in scientific discovery (Darden 1980, 1982, 1986, 1991). In the discovery of protein synthesis (jointly investigated by molecular biologists and biochemists in the 1950s and 1960s), scientists didn't simply have an “A-ha” moment. Rather, they deployed strategies for revealing how a mechanism works (Darden 2006; Craver and Darden 2013). Darden characterizes the process of mechanism discovery as an “extended, piecemeal process with hypotheses undergoing iterative refinement”
[Emphasis is mine. These "strategies" deployed by scientists are the thinking / problem-solving styles described in your question.]
Following are other appearances in this entry of "the whole" and "monolithic", with my comments added in brackets:
The behavior of the whole [system or body of knowledge] is explained in terms of the activities and interactions among the component parts. These activities and interactions are themselves sustained by underlying activities and interactions among component parts, and so on...
Integrative Pluralism: The mechanistic perspective tends to emphasize integrative pluralism in scientific research (Mitchell 2003, 2009). The goal is not to explain the less fundamental in terms of the more fundamental in a step-wise relating of monolithic theories at one level to monolithic theories at another. Rather, such scientific achievements are collaborative and piecemeal [a recurrence of the "piecemeal" term -- iteration again], adding incremental constraints to an emerging picture of how a mechanism works both at a level and across levels.
Mereology is also briefly mentioned in this article.
My answer grew quite lengthy, and some of the ideas are still not in sharp focus. I welcome all feedback. Your question was a challenging one that deserves further thought and discussion.