1.
Introduction
Understanding is
not a binary condition but a continuum of cognitive states that determines how
effectively a problem solver can engage with a challenge. While dictionaries
define understanding as knowing meaning, recognizing mechanisms, or
apprehending behavior, problem-solving reveals a far more nuanced hierarchy.
This essay presents understanding as a developmental sequence ranging from
complete misconception to innovative mastery, examining how each level
corresponds to distinct cognitive capacities and pedagogical needs. We also mention
how AI may fit within our hierarchy.
Why connect with
problem-solving? The answer is simple: everything you propose to do can be cast
as a problem. For example, to even discuss what is “happiness,” you first cast it
as a problem, and then focus your available problem-solving tools upon it.
Every question is, in short, a problem.
2. Hierarchy of Understanding
The following levels
describe a progression from ignorance to expertise. Each stage represents a
qualitative shift in how a learner relates to the problem space. In our
experience with education, most of us pass through most of these levels as our
problem-solving skills develop. The depths of problem solving of the later
levels (below) are generally not taught in schools but are substantially developed
through internal reflection.
Level
1: Misunderstanding
At this stage, the problem solver fails to grasp any aspect of the problem.
This may result from unfamiliar terminology, linguistic barriers, or
fundamental categorical errors. The solver cannot distinguish relevant from
irrelevant information and may pursue entirely inappropriate solution paths
based on flawed premises.
Level
2: Term Recognition Without Direction
The solver comprehends the vocabulary and concepts present in the problem
statement but lacks clarity regarding the specific question being asked. This
corresponds to knowing the components without understanding the configuration.
In this situation, the solver simply guesses at solutions, with little guidance
from learning or experience.
Level
3: Comprehension of Terms and Objective
Here, the solver understands both the terminology and the goal but experiences
an executive gap: knowing what is asked yet lacking procedural knowledge for
how to proceed. This is the "tool-less" state of knowing the
destination without the map. Yet, we all pass through this state before our
problem-solving powers are deployed.
Level
4: Procedural Competence
This level marks the threshold of functional understanding is more complex,
where sublevels must be differentiated, subdivided into three gradations:
a) Single-Method
Application: The solver can apply one memorized
procedure or algorithm to the problem. This represents instrumental
understanding (Skemp, 1976), where the solver knows "how" without
necessarily knowing "why." This is usually the level taught in schools.
b)
Method Integration:
The solver can combine two or more procedures and incorporate additional facts
to address the problem. This requires recognizing that complex problems demand
composite strategies. This is more at the college level.
c)
Transformational Thinking:
The solver can see equivalent transformations, reductions, or simplifications
of the problem. This indicates relational understanding (Skemp, 1976), where
the solver perceives the underlying structure and can reformulate the problem
while preserving its essential relationships. This level involves experience,
learning, and innovative thinking. Generally not a school subject, transformational
thinking is beyond many.
Level
5: Divergent Association
The solver can import information from seemingly unrelated domains, applying
analogical reasoning and cross-disciplinary insights. This requires a broad
knowledge base and the capacity to recognize structural similarities across
content areas (Gentner & Markman, 1997). In analogical reasoning, the
solver must not be swayed by its misleading aspects – always present. Analogies
are more like guides, not precise instructions. We are uncertain if this can be
taught at all. It is clearly in the domain of the versatile, inquiring mind.
Level
6: Structural Integration
The solver perceives how the current problem connects to other problems within
the same domain, recognizing it as an instance of broader classes of problems.
This reflects expert pattern recognition (Chi, Feltovich, & Glaser, 1981)
and the presence of well-developed schemata.
Level
7: Innovation and Method Creation
At the highest level, the solver can devise novel solutions or create new
methods. This represents creative expertise, where the solver extends the
domain itself by establishing new connections or developing original frameworks
(Ericsson, 2006). Of course, creating new methods is well beyond normal problem-solving.
Few achieve this. However, microgenius can achieve this on relatively small
problems through people totally immersed in their projects.
3. Knowledge, Understanding, and Wisdom
This hierarchy suggests a
taxonomy of cognitive achievement. Levels 1 through 3 correspond to the domain
of knowledge: the accumulation of information and recognition of facts.
Levels 4 through 5 constitute understanding: the capacity to apply,
analyze, and evaluate using existing frameworks. Levels 6 through 7 approach wisdom:
the ability to synthesize, transcend existing categories, and generate new
knowledge (Alexander, 2016; Sternberg, 2003). On another note, understanding AI
would be more than a simple exercise in testing a given AI engine against these
levels of understanding. Can they reach Levels 6 or 7, the highest of human
levels?
4. Alternative Dimensions of Understanding
Understanding manifests
along additional axes beyond this hierarchical progression. One may possess
thorough or merely partial understanding; one may hold a substantial grasp of
certain aspects while maintaining misconceptions about others. Understanding
functions not merely as a static state but as an active process that integrates
the unknown into the known (Piaget, 1971; Von Glasersfeld, 1989).
As well, even the most
innovative solver may function best with certain types of problems. For
example, a new CEO may be excellent at putting corporate groups together and operating
efficiently, but may fail when those same groups become more and more sophisticated.
The college valedictorian may excel at college-level problems, but when more
serious problems in real-life research arise, they may fail totally. All this
implies that problem-solving can be regarded as a niche skill, supreme at one
level but unable to transcend to the next. The so-called Peter Principle[1]comes to mind, and to an
extent places a soft metric on incompetence.
The Tarot deck offers a
metaphorical parallel to this complexity. Like the branching paths of a tree,
its many directions and viewpoints suggest that understanding is expansive and
multidimensional, comprising fragmented pieces that require integration. While
analytical problems yield to the hierarchical progression described above,
non-analytical problems (those involving interpretation, meaning, or personal
growth) may resist linear categorization and highly overlapping categories. In
such domains, understanding emerges through guided application rather than
algorithmic solution, with the interpreter bringing diverse information to bear
on symbolic representations (Decker, Depaulis, & Dummett, 1996).
5. Conclusion
Effective problem-solving
instruction requires diagnostic assessment of where learners stand within this
hierarchy. A student at Level 2 requires different intervention than one at
Level 4.b, though both may fail to solve the problem correctly. By recognizing
understanding as developmental and multidimensional, educators can target
specific cognitive bottlenecks and guide learners toward the higher levels
where innovation and wisdom reside.
References
1. Alexander,
P. A. (2006). The Oxford handbook of epistemology. Oxford University
Press.
2.
Anderson, L. W., & Krathwohl, D. R.
(Eds.). (2001). A taxonomy for learning, teaching, and assessing: A revision
of Bloom's taxonomy of educational objectives. Longman.
3.
Chi, M. T., Feltovich, P. J., &
Glaser, R. (1981). Categorization and representation of physics problems by
experts and novices. Cognitive Science, 5(2), 121-152.
4.
Decker, R., Depaulis, T., & Dummett,
M. (1996). A wicked pack of cards: The origins of the occult tarot. St.
Martin's Press.
5.
Ericsson, K. A. (2006). The influence of
experience and deliberate practice on the development of superior expert
performance. In K. A. Ericsson, N. Charness, P. J. Feltovich, & R. R.
Hoffman (Eds.), The Cambridge handbook of expertise and expert performance
(pp. 683-703). Cambridge University Press.
6.
Gentner, D., & Markman, A. B. (1997).
Structure mapping in analogy and similarity. American Psychologist, 52(1),
45-56.
7.
Piaget, J. (1971). Science of education
and the psychology of the child. Orion Press.
8.
Skemp, R. R. (1976). Relational
understanding and instrumental understanding. Mathematics Teaching, 77,
20-26.
9.
Sternberg, R. J. (2003). Wisdom,
intelligence, and creativity synthesized. Cambridge University Press.
10.
Von Glasersfeld, E. (1989). Cognition,
construction of knowledge, and teaching. Synthese, 80(1), 121-140.
©2026
[1] The Peter Principle states that
people in a workplace are promoted until they reach a position where they are
no longer good at their job.
Comments
Post a Comment
Please Comment.