Concrete Understanding, Problem-Bank Coverage, and the Construction of an Abstract Layer
Why do people struggle to learn mathematics? The usual explanations are familiar: they are not intelligent enough, they do not work hard enough, they have not done enough exercises, or their foundations are weak. These explanations are sometimes correct, but they cannot account for several other common situations. Some people complete a great many exercises and still perform inconsistently on exams. Others earn excellent grades and may be regarded as good at mathematics for years, yet forget the material soon after the exam. When they move from primary school to secondary school and the courses become harder, their previous advantage suddenly disappears.
I believe these two groups may share the same problem. During their mathematical education, they acquired a concrete, metaphor-based understanding and learned correspondences between problem types and solution procedures, but they never built an abstract layer belonging to the mathematical concepts themselves.
Here, “not learning well” does not refer only to low exam scores. A person who can pass an exam but cannot retain, transfer, or rederive the knowledge has not truly learned it well either. An exam is a measurement made at one moment. It does not automatically prove that abstract ability has formed.
Three Layers in Mathematical Learning
For the purpose of this discussion, I divide mathematical learning into three layers: the concrete layer, the problem-type layer, and the abstract layer.
The concrete layer corresponds broadly to arithmetic. People begin with quantities and specific problems drawn from daily life, then use operators such as addition, subtraction, multiplication, and division to deal with them. The learner knows how to calculate and can use these calculations to solve concrete problems. Even though numbers and operation signs are already present, the numbers and operations remain attached to objects that can be counted, divided, or compared. This is the main content of the concrete layer.
This way of learning does not automatically disappear at more advanced stages. A teacher may describe a function as a “machine,” draw a vector as an arrow, explain a derivative as the rate at which something changes, or use area and accumulation to introduce an integral. In each case, arithmetic experience, images, and everyday concepts provide a concrete entrance to a new mathematical object.
These entrances are useful. A learner will struggle to approach a completely unfamiliar concept if it offers no point of entry at all. The problem is that a metaphor provides an entrance but is not the mathematical concept itself. A function is not a machine, a vector is not an arrow, and a derivative is not merely a matter of “how fast something changes.” A metaphor preserves part of a concept while omitting many of its conditions and relations. If the learner stops there, what has been acquired is a feeling of similarity, not an understanding in the formal sense.
The problem-type layer is built through repeated practice. When learners see a certain form of question, they know which formula to call upon. When they see a particular condition, they know what transformation should come next. As the volume of practice grows, they can recognize familiar problem types more quickly and may eventually respond almost automatically.
This is certainly an ability, but it is still not the abstract layer. The problem-type layer asks, “Which problem I have seen before does this one resemble?” The abstract layer asks, “What structure composes this problem, which conditions matter, and what can I derive from the definition?” The two processes may produce the same answer, but their internal workings are entirely different.
The abstract layer does not depend on any single concrete example. It consists of definitions, relations among objects, permitted operations, and structures that remain invariant across different representations. Once learners enter this layer, they can change the symbols, diagrams, problem settings, or even the natural language being used while the concept itself remains stable.
That stability is the sign that a mathematical concept has genuinely been established.
What Exactly Is the Abstract Layer?
The abstract layer described above can be defined more precisely. In this essay, the abstract layer is not a more complicated or more technical explanation in natural language. It is a formal system that can operate as independently of natural language as possible. Natural language can help someone enter the system, explain it from the outside, and communicate its conclusions, but it does not constitute the abstract layer itself. What operates inside the abstract layer is a system of objects, symbols, relations, operations, and rules.
For this discussion, the abstract layer of mathematics can be understood through three foundational directions: algebra, geometry, and analysis.
The abstract layer of algebra consists of variables, equations, matrices, operators, and the rules connecting them. In f(x) = y, for example, x and y do not have to correspond to everyday objects. They can simply occupy positions within a structure. The letters can be replaced while the relation itself remains. The same is true of a matrix. A matrix is not merely the image of a table filled with numbers. It is a formal object that participates in transformations and operations according to defined rules.
The abstract layer of analysis also relies mainly on symbols. Functions, limits, derivatives, integrals, and various operators are organized by strictly defined relations. Words such as “change,” “approach,” or “accumulation” can give a beginner an entrance, but once one enters analysis, what matters is the system of limit conditions, functional relations, and operational rules. Images supplied by natural language cannot replace those definitions.
Geometry may appear closer to the concrete layer because it uses visible figures, but a geometric figure is not an everyday object. A point in a diagram has no size, and a line is not a thin physical rod. Geometric figures express position, direction, distance, adjacency, symmetry, and transformation. They form a graphical symbol system. The fact that a figure can be seen does not mean it remains at the concrete layer.
The abstract layer is therefore not the same as “what cannot be seen,” nor is it restricted to formulas. Equations, matrices, and geometric figures can all carry abstraction. They share one defining feature: the symbols remain as semantically neutral, or even as semantically blank, as possible. Their meaning does not come from an inherited everyday meaning in natural language. It comes from their definitions, positions, and relations within the formal system.
This is why natural language should enter the abstract layer as little as possible. It cannot be eliminated altogether because learning, communication, and proof all require language. Within the internal structure of a concept, however, natural language should remain in the role of an external explanation. It can tell us how symbols are defined, but it should not replace the symbol system itself. Otherwise, learners can easily mistake the image produced by a natural-language term for the mathematical object.
Whether the abstract layer has been built can be tested by asking whether learners can leave natural-language metaphors and work directly inside the symbol system. Can they continue a derivation from relations in an equation or matrix? Can they read invariants from a geometric figure? Can they move the same structure from one formal representation to another without depending on an everyday story to keep it intelligible? If a concept stops operating as soon as natural language is removed, it still belongs mainly to the concrete or metaphorical layer.
How Metaphor-Based Understanding Creates a False Sense of Understanding
By “metaphor-based understanding,” I do not mean metaphor as a rhetorical device. I mean a way of knowing a mathematical concept. Faced with an unfamiliar mathematical object, a learner first calls upon a familiar object, concrete experience, or image, then connects the similar features of the two and uses the familiar object as a substitute for understanding the new concept directly.
The process can be written as follows:
Familiar object or image → selection of similar features → use of those features to understand the mathematical object
For example, understanding a function as a machine maps the feature “an input produces an output” onto a function. Understanding a vector as an arrow uses direction and length to approach the concept of a vector. Understanding a derivative as velocity uses everyday experience of changing speed to approach the derivative. Learners do receive some information from these metaphors, but what they receive is the resemblance between a mathematical object and a familiar thing, not the complete definition of the mathematical object.
Metaphor-based understanding mainly answers, “What is this concept like?” Abstract understanding must answer, “Which conditions define this concept, what relations exist among its objects, and what reasoning is permitted?” If someone cannot explain a function without the machine, discuss vectors without arrows, or understand derivatives without velocity, then the mathematical concept remains attached to the metaphor. It has not yet become an abstract object capable of operating independently.
The way to determine whether a learner is stuck at the metaphorical level is not to ask whether a metaphor has been used. The question is what remains after the metaphor is removed. If the learner can still state the definition, identify the conditions under which it holds, and continue reasoning, then the metaphor was only an entrance. If the concept disappears when the metaphor is removed, then the learner mainly understood the metaphor itself.
The most important side effect of metaphor-based understanding is not simply premature familiarity. It is the creation of a false sense of understanding. The learner really does understand the metaphor and the similar features shared by the familiar and mathematical objects. The error comes in the next step: the learner mistakes an understanding of the metaphor for an understanding of the mathematical concept itself.
This false sense is difficult to detect precisely because the subjective feeling of “I understand” is not baseless. The learner has genuinely acquired some content. The problem is that this content comes from an old everyday concept and the resemblance between the two objects, rather than from the definition, conditions, and relations of the mathematical object. The learner has not understood nothing. The learner has understood the wrong object, or only the small part of the target object covered by the metaphor.
When a new concept is explained as something already familiar from daily life, it is therefore easy to feel that it has been understood. But understanding the metaphor and understanding the mathematical object are not the same thing. The former finds a similar location in an old conceptual network. The latter requires a new system of definitions and relations to be built.
If that construction never happens, learners will keep returning to the original image when they encounter later problems. They are not operating on the mathematical object. They are operating on their own understanding of the image. As soon as a problem exceeds the range covered by the metaphor, their understanding fails.
Treating a function as a machine, for instance, can help explain inputs and outputs. But someone who remains dependent on the machine image may struggle with functions that involve no obvious process of calculation yet still satisfy the definition. The same is true of treating vectors as arrows. An arrow can express direction and magnitude, but it does not by itself provide the structures of vector spaces, linear combinations, or changes of basis.
Metaphors can therefore be retained, but they must be recognized as scaffolding. At some point, the learner must leave the metaphor and enter the definition.
The Value of Practice Depends on What It Trains
Drilling problems is usually regarded as the most direct way to improve mathematical performance. From the standpoint of exams, it does have some effect. The more problems learners have encountered, the more familiar they become with standard types. When an exam falls within the range they have already practiced, they are more likely to find the corresponding procedure. But this effectiveness is first of all effectiveness for an exam. It does not allow us to infer that an abstract layer has been built.
Whether practice helps learners enter the abstract layer cannot be answered in a single way. It depends on their approach to learning and on where they direct their attention while solving problems.
If learners have no concept of an abstract layer and simply repeat problems, each problem is mainly stored as a type and a sequence of steps. They can quickly assign a new problem to an old category, then reproduce a rehearsed solution. Under this approach, more practice increases coverage of the problem bank and may improve exam performance, but abstract ability does not appear automatically.
This path can be written as:
Practice problems → memorize types and procedures → expand problem-bank coverage
Another kind of practice begins with awareness of the abstract layer. Learners first try to understand definitions, relations, and structures, then use problems as material for testing that understanding. Different problems reveal whether the abstract concept they have built is stable. They expose missing conditions, invalid reasoning, and boundaries the concept does not yet cover. The learner then revises the concept and tests it again with new problems.
This path is:
Build a preliminary abstract concept → test it with problems → discover errors or boundaries → revise the concept → test again
Here, practice does not simply add items to memory. It repeatedly tests and iterates the abstract layer that has already begun to form. Problems provide feedback, and learners use that feedback to adjust their understanding. This kind of practice has value because what it improves is not problem-bank coverage but the abstract concept itself.
Two people may complete the same number of exercises and arrive at completely different results. One is trying to increase the number of problems that can be classified as “I have seen this before.” The other is testing whether an understanding of the mathematical structure is actually valid. Practice alone therefore neither guarantees that an abstract layer will form nor deserves to be dismissed. The real question is whether the exercises are expanding a problem bank or revising and strengthening abstract concepts.
Ordinary exams have difficulty distinguishing these two learning processes because they may produce identical answers. If exam questions closely resemble the training set, problem-bank coverage may affect the score more directly than abstract ability does.
A simple conceptual model is:
Exam score ≈ α × problem-bank coverage + β × abstract ability
Here, α and β are not fixed constants obtained from statistics. They represent how much a particular exam depends on each of the two abilities. The more standardized the questions, the larger α becomes. The more unfamiliar the questions, and the more they require a student to redefine the problem, organize the conditions, and complete a derivation, the larger β becomes.
The same person may therefore score highly when α is large and rapidly lose that advantage in an environment where β is large. Familiar question types, fixed exam ranges, and intensive training can conceal the absence of an abstract layer. The problem becomes visible only after a learner enters a new course and the form of the questions changes so that the old problem bank no longer provides sufficient coverage.
Two Apparently Opposite Ways of “Not Learning Well”
The first is immediately visible: poor performance in the current course.
These learners have mastered some problem types, but their coverage is insufficient. As soon as a question changes its wording, recombines its conditions, or moves slightly beyond the practiced range, they cannot decide which method to use. Because the abstract layer has not been built, they also find it difficult to return to the definition and analyze the problem from the beginning.
What appears externally is an inability to solve problems. More precisely, it is an inability to solve problems they have not seen before.
The second is less visible: exam performance is good or even excellent, but the knowledge cannot be retained or transferred.
These learners may have practiced enough problems for the exam range to lie almost entirely within their problem bank. They can recognize the types, call upon formulas, and reproduce procedures fluently, so their grades are not poor. Once the exam is over, however, those separately stored types and procedures begin to disappear. Because they were never organized into an abstract structure, they are also difficult to rederive after they have been forgotten.
This produces a strange phenomenon. A person may once have earned a high grade in a mathematics course, yet some time later be almost unable to explain what was learned. The person remembers having solved many problems but cannot reorganize those problems into knowledge. When faced with a new field, the old knowledge is also difficult to transfer.
This is still a failure to learn well. The exam score merely conceals it for a time.
What Does an Exam Measure?
Existing exams can certainly measure many things, but a single exam has difficulty separating problem-bank coverage from abstract ability. As long as the questions remain within the training distribution, the two abilities may produce similar external performance.
If drilling is treated as training, a large number of standardized questions can make learners highly adapted to that range. They respond quickly to problems inside it, but speed does not mean they can handle problems outside it. An exam score describes performance within a distribution of questions. On its own, it cannot answer several more important questions: Can the learner still identify the structure after the wording changes? Can a forgotten formula be rederived? When facing a problem never seen before, can the learner organize the reasoning from the definitions onward?
Highly standardized exams do not measure these abilities easily. As a result, low scores expose some learners' genuine difficulty early, while high scores temporarily protect others from seeing the same difficulty. The first group knows that it does not know. The second may continue believing that it already understands.
The First Step Is to Recognize That the Layers Are Different
The first step toward solving this problem is to recognize that the concrete layer, the problem-type layer, and the abstract layer are not the same thing.
Whenever learners encounter a new concept, they can ask themselves several questions:
- Do I currently understand an example, or the definition itself?
- If the metaphor is removed, can I still explain the concept?
- If all the symbols are replaced, can I still recognize the same structure?
- If I forget the formula, can I derive it again from the definition?
- If the setting of the problem changes completely, do I know which conditions remain valid?
- Can I explain why two apparently different problems belong to the same mathematical structure?
These questions do not reject intuition. They test whether intuition has continued to develop toward the abstract layer.
A more deliberate learning process can be written as:
Concrete example → preliminary intuition → formal definition → change of representation → extraction of invariants → derivation from the definition → transfer to an unfamiliar problem
When studying functions, a learner may begin with the “machine,” but must later return to the mapping relation and the conditions under which it holds. When studying derivatives, one may begin with rates of change, but must later understand limits and local linear relations. When studying vectors, one may first draw arrows, but must eventually be able to discuss operations and structure without them.
After solving a problem, learners should not merely check the answer. They can change one condition and examine the point at which the original solution fails. They can try to recover a remembered formula from the definition. They can also compare several problems that look different and identify what remains unchanged. Only through this process does practice shift from accumulating a problem bank to training abstraction.
The Second Step Is to Study Directly in English
For students from a Chinese educational background, the second effective direction is specific: study mathematics directly in English.
There is no need to leave this open as a choice among any number of foreign languages. For most students from a Chinese background, English is the second language they can realistically use, and it also occupies the dominant position in contemporary academic communication. Whether the concern is access to learning materials, the terminology of a field, or eventually entering professional research, English is the most practical choice.
Words in a Chinese speaker's native language already belong to a dense semantic network. A word carries more than its dictionary meaning. It automatically connects with daily experience, concrete images, emotions, and situations in which it has been used before. When a learner encounters a mathematical concept named in Chinese, the word itself can easily create a certain kind of understanding. That understanding may help, but it may also pull the concept back toward a familiar concrete layer too soon.
English technical vocabulary usually has fewer dense everyday connections for a native Chinese speaker. Even when someone speaks English well, many mathematical terms still function as relatively independent new labels. Because these labels are not surrounded by as much lived experience, they can serve as comparatively neutral conceptual containers.
English letters, mathematical terminology, and formal definitions together create a new system of representation. The learner's first response to a term is no longer the inherited meaning or visual form of a Chinese word. Instead, a direct connection must be built between the English term and the mathematical definition.
The more desirable path is:
English term ↔ formal definition ↔ mathematical symbol ↔ relations among objects
Rather than:
English term → Chinese translation → literal Chinese meaning → everyday image
If every English term is immediately translated back into the most familiar Chinese meaning, this semantic distance quickly disappears. The value of studying in English is not the acquisition of a second vocabulary list. It is the temporary prevention of an old semantic network from taking control of a new concept.
Learners can read definitions directly in English, use mathematical symbols to record relations, and then explain the conditions in their own words. Once the abstract structure of a concept has become stable, they can return to expressing it in Chinese. At that point, Chinese becomes another representation of the concept rather than the source from which the concept itself is derived.
How to Tell Whether the Abstract Layer Has Been Built
The abstract layer is not a mystical subjective state, nor is it merely the feeling that one has “figured it out.” It can be observed through concrete changes.
Before the concept is established, the learner depends on the surface features of a problem. Afterward, the learner can more easily identify common structures across problems that look different. Before, forgetting a formula brings the work to a halt. Afterward, the formula can be rederived from definitions and known relations. Before, the learner needs a fixed metaphor. Afterward, the learner can move directly among equations, matrices, geometric figures, and other formal representations while recognizing that natural language only explains these structures from the outside.
There is another direct test: replace the familiar terminology, examples, and problem setting. If the learner can still preserve the definitions, relations, and reasoning, then the concept has begun to detach from its concrete carrier. If the understanding disappears as soon as the representation changes, what has been mastered is probably still an image or a problem type.
Effective learning therefore does not completely exclude concrete understanding, nor does it require the learner to stop solving problems. It requires learners to know which layer they currently occupy and to complete the transitions consciously. The concrete layer begins with everyday quantitative relations and handles specific problems. The problem-type layer supplies operational experience. The abstract layer organizes these scattered experiences into a structure that supports derivation and transfer.
Many people struggle with mathematics not because they have never understood anything. They understood the examples, understood the metaphors, and remembered how many problems were solved. The problem is that the learning process stopped there.
If learners can recognize that stopping point and deliberately train themselves to move from examples to definitions, from procedures to relations, and temporarily away from the familiar semantics of their native language, then mathematical learning can cease to be the coverage of an existing problem bank and begin to form an abstract ability that can be retained, used for derivation, and transferred to new situations.