First learn to solve any problem, then learn to solve it faster

School education is often structured as if the student's main task is to memorize as many correct solution methods as possible.

For every type of object, their own names are introduced. For every combination of properties, separate signs. For every common case, a special formula.

The student memorizes:

• how to find the area of a rectangle;
• how to solve a quadratic equation;
• how to calculate uniform motion;
• how to determine current;
• how to balance a chemical reaction of a certain type.

As long as the problem resembles an example from the textbook, the system works. But it is enough to slightly change the conditions, remove the familiar name, or add extra data — and the student no longer understands which formula needs to be applied.

They know many solutions, but do not possess a general method for constructing a solution.

Knowledge of special cases creates the illusion of universality

Suppose a child knows the formulas for the area of a rectangle, triangle, parallelogram, rhombus, and trapezoid.

This creates a feeling that they know how to find the area of polygons. But if an irregular heptagon appears before them, the familiar system stops working.

The figure does not resemble any studied template. It has no suitable name. No special formula is found in memory.

Although the area can still be determined by universal methods:

• break the figure into triangles;
• set the coordinates of the vertices and apply the shoelace formula;
• break the region into known parts, adding and subtracting their areas.

Special formulas were not useless. They allowed solving individual problems quickly. But the student was given optimizations before being given the foundation that these optimizations shorten.

As a result, they can quickly solve familiar problems and cannot slowly solve unfamiliar ones.

First, you need a method that works almost always

Effective learning should start with a minimal universal toolkit.

Such a toolkit may not be the fastest. It may require more actions, calculations, and time. But it must allow the student to independently approach the widest possible class of problems.

For polygons, such a core could be:

1. Representing the figure through a sequence of vertices.
2. Measuring distances between points.
3. Determining angles and directions.
4. Checking for equality, parallelism, and perpendicularity.
5. Breaking a complex figure into simple parts.
6. Calculating the perimeter as the sum of side lengths.
7. Calculating the area of an arbitrary contour.
8. Checking the result by another available method.

Possessing this set, the child will be able to explore a triangle, quadrilateral, or twenty-gon, even if they have never encountered its name before.

Perhaps their solution will turn out to be long and inefficient. But it will exist.

This is fundamentally more important than the ability to instantly solve only the problem whose form the textbook has anticipated in advance.

Optimization should shorten an already understood solution

Once the universal method is mastered, the student begins to notice repeating properties.

For example, the area of any rectangle can be found by a universal method — splitting it into triangles or applying the coordinate formula. But if the sides of the rectangle are perpendicular and have lengths a and b, all calculations can be shortened to:

S = ab

Now this formula does not look like an independent rule that needs to be memorized.

The student understands:

• for which set of properties it works;
• from which general method it is derived;
• which calculations it shortens;
• why it cannot be applied to an arbitrary quadrilateral without verification;
• what to do if they forget the formula.

The short formula becomes not the basis of understanding, but an optimization of an already known solution.

Such an order is more stable:

universal method → repeating properties → regularity → short formula

Instead of the usual:

object name → ready-made formula → typical problem

Names are compressed sets of conditions

Terms play a similar role.

The word "rectangle" does not create a special geometric object. It compactly communicates a set of constraints:

• before us is a quadrilateral;
• all its angles are right angles.

From these conditions, other properties can be derived: parallelism and equality of opposite sides, equality of diagonals, and methods for calculating area.

At the same time, a rectangle may possess additional properties. If all its sides are equal, it is simultaneously a square, a rhombus, a parallelogram, and a quadrilateral.

The name does not necessarily describe the object completely. It communicates only those guarantees that are needed for a certain level of reasoning.

In this sense, a term is similar to a compressed record:

quadrilateral + four right angles = rectangle

But school education often flips this dependency.

First, the child is shown a characteristic picture and told its name. Then they are asked to memorize the properties related to it. Because of this, a false impression arises as if the figure is a rectangle, rhombus, or trapezoid because it looks accordingly.

A more natural path looks different:

1. Obtain data about the object.
2. Measure or calculate its properties.
3. Check the fulfillment of certain conditions.
4. Draw conclusions from the discovered properties.
5. Find the name of the corresponding class if necessary.

Then the term becomes the result of research and a convenient way of transmitting information, rather than a password to access a memorized formula.

Not all useful knowledge needs to be stored in memory

Curricula often treat all knowledge almost equally. If a term or formula is included in the textbook, it is assumed that the student must study and memorize them.

But different knowledge performs different functions.

It is useful to divide them into at least three levels:

Level	Purpose
Universal core	Allows exploring the widest possible class of objects under known conditions of applicability
Optimizations	Accelerates the solution of frequent or structured cases
Reference book	Stores rare names, features, formulas, and specialized methods

The universal core must be mandatory. Without it, a person remains dependent on pre-familiar templates.

Optimizations are worth studying after a clear need for them arises. Then the student understands not only the formula itself, but also the reason for its existence.

Reference material does not have to be constantly kept in memory. It is enough to know:

• that such a method exists;
• what problem it solves;
• where to find it;
• what its limitations are;
• how to check the correctness of application.

The ability to quickly restore or find rare knowledge is often more useful than the ability to store it for years without application.

Every mandatory knowledge must justify its cost

Any new term or method takes time.

It is necessary to:

• understand it for the first time;
• connect it with already known ideas;
• learn to apply it correctly;
• distinguish it from similar methods;
• maintain it in memory;
• periodically repeat it.

Therefore, it is not enough to say that some knowledge is true or interesting in itself. To include it in the mandatory core of learning, one needs to understand what function it performs.

A new term, formula, or method must provide at least one of the following:

• expand the class of solvable problems;
• reduce the number of calculations;
• reduce the probability of error;
• increase accuracy;
• compactly transmit essential properties;
• allow proving correctness;
• help understand or control the tool being used.

If the material does not give any of this, it does not have to be completely excluded. But perhaps its rightful place is not in mandatory memory, but in the reference layer.

The usefulness of knowledge must be compared with the cost of its acquisition.

A rare term that is easy to find and that does not open up new methods may have low educational value. And a simple universal principle applicable in dozens of different areas — very high.

This principle applies not only to geometry

In algebra, the universal core can be:

• concepts of variable and equality;
• allowable transformations;
• system of constraints;
• functions;
• checking the solution by substitution.

The formula for the roots of a quadratic equation then becomes a specialized algorithm for expressions of a certain structure, rather than a magical sequence of symbols.

In physics, the basis can be:

• quantities and units of measurement;
• state of the object;
• change;
• interaction;
• conservation laws;
• reference frame;
• error.

The formula s = vt in such an approach is immediately considered as a shorthand for a special case in which speed is constant, rather than as a universal law of any motion.

In computer science, the core consists of:

• data;
• state;
• conditions;
• sequences;
• loops;
• functions;
• algorithms;
• correctness;
• computational complexity.

Specific languages, libraries, and application programming interfaces become replaceable tools on top of this foundation.

The same architecture repeats in different disciplines:

universal model → general method → conditions of applicability → private optimizations → automation → verification

Computers and AI make this approach even more important

Previously, the need to remember many special formulas could be justified by the fact that a person had to perform calculations independently.

Today, a significant part of calculations can be entrusted to programs and AI. An intellectual system is capable of recognizing an object, selecting an algorithm, performing thousands of operations, and checking several solution options.

This does not make understanding unnecessary. But it changes its purpose.

A person is less and less required to compete with a computer in the speed of manual calculations. It is much more important to understand:

• what exactly is being calculated right now;
• what data was used;
• what assumptions were made;
• whether the chosen method is applicable;
• how accurate the result is;
• whether it contradicts the initial problem.

The universal core allows asking these questions. Specific formulas without general understanding do not.

If a person knows only a ready-made template, they depend on the coincidence of the new problem with the old example. If they understand the general model, they can explore an unfamiliar situation independently or meaningfully use an external tool.

A student must be able to be slow

One of the main criteria for good learning can be formulated as follows:

First, a person must learn to solve an unfamiliar problem slowly. Only after that does it make sense to teach them to solve familiar problems quickly.

A slow universal method provides independence from templates.

The student may forget the name of the figure, the formula, or a special feature, but is still capable of:

1. Representing the object through known elements.
2. Determining available data.
3. Applying basic operations.
4. Obtaining necessary properties.
5. Finding a solution.
6. Checking the result.

When such a foundation exists, special knowledge becomes truly useful. They speed up work, but do not replace the ability to reason.

Without this foundation, education produces a person who knows the correct answers to many pre-prepared questions, but does not know how to act when the question is formulated differently.

The goal of learning is not the maximum amount of knowledge

The effectiveness of education cannot be measured only by the number of studied terms, formulas, and topics.

More material does not necessarily create more opportunities. Sometimes it only increases the load on memory and hides a small number of general regularities under a multitude of special names.

A more sensible goal of learning is to form a system in which a person knows how to:

• explore an unfamiliar object;
• build its model;
• apply universal methods;
• notice repeating properties;
• derive shorter solutions from them;
• find rare knowledge as needed;
• transfer calculations to suitable tools;
• check the obtained result.

Then knowledge ceases to be a collection of ready-made answers.

It becomes an extensible system:

understanding → solution → regularity → optimization → new level of understanding

It is precisely this system that allows a person not only to reproduce what has already been studied, but also to master what no one has taught them in advance.