How to Solve a Problem You've Never Seen Before | Tymur Levitin
How to Solve a Problem You Have Never Seen Before: The Architecture of Independent Thinking
“A familiar exercise tests whether you can reproduce a path. A real problem begins when the path is no longer given.”
— Tymur Levitin
Many students can solve a problem immediately after watching someone else solve one like it.
Then one detail changes.
The wording is different.
The numbers look unfamiliar.
The problem combines two topics instead of one.
Or nobody tells the student which method to use.
Suddenly:
“I don't know how to do this.”
This moment is often interpreted as failure.
It may actually be one of the most educationally valuable moments available to us.
Because now we can finally observe what happens when the learner does not already possess a ready-made path.
Can they investigate?
Represent?
Connect?
Test?
Reject?
Revise?
Verify?
In other words:
Can they solve a problem — rather than reproduce a solution?
An Exercise and a Problem Are Not Quite the Same Thing
The distinction is not absolute, but it is useful.
An exercise often trains something already identified.
The chapter is called:
Linear Equations
and the instruction says:
Solve the following equations.
Much of the intellectual decision has already been made.
The learner knows approximately which toolbox to open.
A problem contains more uncertainty.
The learner must determine:
What kind of situation is this?
What matters?
What do I know?
What is missing?
Which ideas might apply?
How can I test them?
The difference is not necessarily difficulty.
A complicated exercise may require a great deal of work.
A simple problem may require only one insight.
The difference is the amount of the path that the learner must construct independently.
Problem Solving Begins Before the Solution
Students often focus immediately on:
What operation should I do?
But operations belong somewhere in the middle of the process.
Before acting, we need to understand what we are acting on.
A useful problem-solving architecture is:
Orient → Represent → Connect → Generate → Test → Evaluate → Revise → Transfer
I call this the Independent Problem-Solving Cycle.
It is not a rigid algorithm.
Real thinking moves backward and forward.
But the model helps us see what successful problem solving actually requires.
1. ORIENT
What is actually happening here?
Before solving, orient yourself inside the problem.
Ask:
What is being asked?
What information is given?
What is unknown?
What constraints exist?
What can be ignored?
This sounds elementary.
It is not.
Many errors occur before any calculation or formal reasoning begins.
The learner solves the wrong problem correctly.
Or starts manipulating information without understanding the objective.
In language learning, something similar happens when a learner produces a grammatically correct sentence that does not fit the communicative situation.
Technical correctness cannot rescue a misunderstood task.
2. REPRESENT
Can I express the problem in another form?
Representation changes thinking.
A mathematical word problem can become:
an equation;
a diagram;
a table;
a graph.
A programming task can become:
pseudocode;
a flowchart;
smaller functions.
A historical question can become:
a timeline;
a causal map;
a comparison table.
A language problem can become:
a contrast between meanings;
a set of contextual examples;
a comparison across languages.
Changing representation is not decoration.
Sometimes the original form hides the relationship we need.
A different representation makes it visible.
This is one reason representation occupies an important place in Understanding Mathematics: How Mathematical Thinking Develops.
3. CONNECT
What do I already know that resembles this?
New problems are rarely completely new.
They usually contain relationships we have encountered somewhere else.
The challenge is recognizing them.
Ask:
Have I seen a similar structure?
Which concept might be relevant?
What previous problem shares something with this one?
What happens in a simpler version?
Can another subject help me understand it?
This is where knowledge becomes useful.
Not because the learner remembers the exact answer.
Because existing knowledge becomes a network from which new reasoning can begin.
4. GENERATE
What could I try?
A learner who expects every problem to have one immediately obvious method can freeze when certainty disappears.
Problem solving requires hypothesis generation.
Possible moves include:
try a simpler case;
work backward;
draw something;
estimate;
look for a pattern;
divide the problem into parts;
change representation;
test an extreme case;
use analogy;
construct an example;
construct a counterexample.
At this stage we do not need certainty.
We need plausible directions.
This is an important intellectual shift:
from
“I don't know the solution”
to
“I don't know the solution yet, but I have something I can test.”
5. TEST
What happens if I follow this idea?
Now reasoning meets evidence.
In mathematics, calculate.
In programming, run the code.
In science, compare a prediction with observation or data.
In language, use the formulation and examine whether it produces the intended meaning.
In historical reasoning, compare the claim with available evidence.
A hypothesis that fails is not automatically wasted effort.
It tells us something about the structure of the problem.
The learner needs to become comfortable with the idea that an unsuccessful attempt can still reduce uncertainty.
6. EVALUATE
Does the result make sense?
Getting an answer is not the same as finishing.
Ask:
Does it satisfy the original conditions?
Is the magnitude plausible?
Does the argument contradict known information?
Did the code solve the intended problem or merely run without errors?
Does the sentence communicate the meaning I intended?
Can I verify the result another way?
Evaluation turns the learner from executor into supervisor of their own thinking.
7. REVISE
If it failed, what exactly failed?
This is where productive problem solving separates itself from random trial and error.
Do not merely try something else.
Diagnose the attempt.
Was the problem misunderstood?
Was the representation poor?
Was the relevant concept wrong?
Was the strategy reasonable but execution inaccurate?
Did an assumption fail?
Did new information appear?
Revision should use information generated by the failed attempt.
Then failure becomes part of the reasoning process rather than its opposite.
8. TRANSFER
What have I learned that belongs beyond this one problem?
The final answer can be correct and still produce little future value.
After solving, ask:
What was the key idea?
Where else could it work?
What made the problem difficult?
Which clue did I initially miss?
Could I solve it another way?
What would change if one condition changed?
This is how a solved problem becomes new intellectual infrastructure.
The learner does not merely leave with an answer.
They leave with a stronger system for the next unknown situation.
The Independent Problem-Solving Cycle
The complete model is:
ORIENT → REPRESENT → CONNECT → GENERATE → TEST → EVALUATE → REVISE → TRANSFER
Or, in questions:
What is the problem?
→ How can I represent it?
→ What do I already know?
→ What could I try?
→ What happens?
→ Does it make sense?
→ What should I change?
→ What can I carry forward?
These questions can be taught.
Eventually they can become internal.
That is one route toward independent thinking.
Why Giving the Method Too Early Can Remove the Problem
Suppose a learner receives a difficult task.
After ten seconds, the teacher says:
Use this formula.
The student may now complete the calculation successfully.
But one important part of the task has disappeared:
method selection.
The learner has practised execution.
They have not practised deciding what to execute.
This does not mean teachers should never give hints.
Hints are essential.
The question is what the hint removes.
A useful hint reduces difficulty while preserving as much productive thinking as possible.
A Hint Has a Size
Compare:
“The answer is 12.”
with:
“Try using this formula.”
with:
“Which quantities seem related?”
with:
“Could you represent the situation visually?”
All are forms of help.
But they remove different amounts of intellectual work.
A good teacher chooses the smallest useful intervention.
Enough to restart thinking.
Not so much that the teacher takes over the thinking.
Why Students Become Dependent on Examples
Worked examples are powerful.
They show structure efficiently.
But there is a danger.
If every new problem arrives immediately after an almost identical example, the learner can develop a strategy:
Find the matching example and imitate it.
This can produce excellent-looking work.
Until the match disappears.
The solution is not eliminating examples.
It is gradually increasing the distance between:
example
and
new task.
Change the representation.
Mix problem types.
Remove labels.
Combine concepts.
Ask the learner to explain why an example works.
Eventually ask them to generate the method before seeing one.
“I Understand It When You Explain It”
This sentence is extremely informative.
It often means exactly what the learner says.
They do understand the explanation.
But following a path and generating a path are different tasks.
The broader distinction is part of our Four-Level Learning Model:
Knowledge → Understanding → Ability → Independence
Understanding is not fake simply because independent performance has not yet developed.
It is one stage of the system.
The next task is to convert supported understanding into increasingly independent ability.
Why Productive Struggle Matters
There is a fashionable temptation to say:
Struggle is good.
Not necessarily.
Confusion without useful structure can simply waste time.
A learner who lacks essential prerequisite knowledge may struggle indefinitely.
Difficulty becomes productive when the learner has enough resources to investigate and receives enough support to continue.
The objective is not suffering.
It is to preserve the part of the task where thinking needs to develop.
The Teacher as a Regulator of Cognitive Responsibility
At the beginning, the teacher may carry much of the responsibility.
The teacher:
selects the representation;
identifies the concept;
suggests the strategy;
checks the answer.
As learning develops, responsibility shifts.
The student begins doing those things.
This gives us another progression:
Teacher identifies → learner recognizes
Teacher chooses → learner chooses
Teacher checks → learner checks
Teacher diagnoses → learner diagnoses
This transfer of cognitive responsibility is one of the central goals of good individual education.
It is also why Online Academic Tutoring Should Not Be Homework Help: How Individual Subject Learning Actually Works places independence above simple task completion.
Problem Solving in Mathematics
Mathematics makes the process highly visible.
A learner must often:
translate a situation into mathematical form;
identify relationships;
select a strategy;
perform operations;
verify the result;
generalize.
The correct answer matters.
But the reasoning architecture matters more for future problems.
That is why our mathematical framework distinguishes concept, representation, relationship, strategy, operation, verification, explanation and transfer.
The full model is available in Understanding Mathematics: How Mathematical Thinking Develops.
Problem Solving in Physics
Physics adds another challenge.
The learner must decide how a real or imagined physical situation maps onto a model.
Which forces matter?
Which can be ignored?
What assumptions are reasonable?
Which variables describe the system?
A formula is useful only after the situation has been represented appropriately.
This is why knowing equations is not identical to being able to solve physics problems.
Problem Solving in Programming
Programming makes revision unavoidable.
Code fails.
Unexpected behaviour appears.
The learner must:
locate the problem;
form a hypothesis;
test;
inspect output;
revise;
test again.
Debugging is problem solving made visible.
An error message is not simply a sign of failure.
It is information about the system.
Problem Solving in Language
Language may seem different because everyday communication is not normally presented as a “problem.”
But spontaneous communication contains constant micro-problems.
You do not know a word.
What do you do?
The other person misunderstood.
How do you repair the interaction?
Two grammatical constructions seem possible.
Which meaning do you need?
You cannot formulate the perfect sentence quickly enough.
Can you express the idea another way?
Fluent communication depends partly on the ability to continue when the ideal linguistic resource is unavailable.
This is why language learning involves decision-making, not only memorization.
The broader mechanism is described in How Language Learning Actually Works: From Words and Rules to Thinking, Decisions and Communication.
Problem Solving Through Another Language
Now combine the systems.
A student solves mathematics in German.
Studies biology in English.
Learns programming through English documentation.
The learner may face two simultaneous problems:
the intellectual problem
and
the linguistic problem.
If we confuse them, diagnosis becomes difficult.
Does the learner not understand the mathematical relationship?
Or do they not understand the German wording?
Can they solve the physics problem but not explain the reasoning in English?
This distinction is central to Learn a Subject Through Another Language: When Language Becomes a Tool for Knowledge.
Language can carry problem solving.
But language can also become one of its bottlenecks.
When AI Can Solve the Problem
AI changes the educational environment.
A learner can paste a problem into a system and receive a solution.
That can be useful.
But now we need to distinguish two objectives.
Objective 1
Obtain the solution.
Objective 2
Develop the learner's ability to solve future problems.
Sometimes we need the first.
Education primarily concerns the second.
A powerful way to use AI educationally is not merely:
Solve this.
But:
Give me one hint without solving it.
Check my reasoning.
Show me where my first incorrect step occurs.
Give me a similar problem.
Challenge my explanation.
Ask me questions that help me find the method.
The tool then supports the problem-solving cycle instead of replacing it.
The Goal Is Not to Make Every Problem Familiar
It is impossible.
New situations will always appear.
Education that depends entirely on familiarity eventually reaches a boundary.
A more robust objective is to develop learners who can enter unfamiliar territory with a method for thinking.
They may not know the answer.
They may initially choose the wrong approach.
They may need help.
But they possess a way to begin.
That changes the meaning of:
“I don't know.”
It no longer means:
“I cannot proceed.”
It can mean:
“I have not solved it yet.”
A Practical Protocol for an Unfamiliar Problem
When you do not know what to do, resist the immediate search for the solution.
Run the cycle.
1. ORIENT
Write what is known, unknown and required.
2. REPRESENT
Change the form of the problem.
3. CONNECT
List relevant concepts or similar structures.
4. GENERATE
Produce at least one plausible approach.
5. TEST
Follow it far enough to obtain information.
6. EVALUATE
Check the result against the original problem.
7. REVISE
If it fails, identify why before changing direction.
8. TRANSFER
After solving, extract the principle.
At first this protocol can be explicit.
Eventually much of it becomes internal.
That is exactly what we want.
Independence Is Not Never Needing Help
Independent thinkers use resources.
Teachers.
Books.
Search engines.
AI.
Colleagues.
Experts.
The difference is that help becomes part of their strategy rather than a replacement for strategy.
They can formulate a better question.
Evaluate an answer.
Compare alternatives.
Return to the problem with new information.
Knowing how to use assistance is itself part of independent problem solving.
From Solving Tasks to Building Thinkers
A lesson can finish with every exercise completed and still leave the learner dependent.
Another lesson may finish with fewer completed problems but a stronger ability to approach the next one.
Quantity matters.
Practice matters.
Correctness matters.
But education also needs to ask:
Who performed the important thinking?
If the teacher repeatedly identifies, selects, explains, checks and corrects everything, the student may become very successful at following.
If responsibility gradually moves toward the learner, something else develops.
Judgment.
Strategy.
Self-correction.
Transfer.
Independence.
“The purpose of teaching is not to remove every unknown from the learner's path. It is to make the learner increasingly capable of entering the unknown.”
— Tymur Levitin
Continue Learning
The broader educational model behind this page is Knowing vs Understanding: The Four Levels of Real Learning.
For academic learning and the development of independence, read Online Academic Tutoring Should Not Be Homework Help: How Individual Subject Learning Actually Works.
For mathematical reasoning specifically, continue with Understanding Mathematics: How Mathematical Thinking Develops.
For learning academic content through another language, explore Learn a Subject Through Another Language: When Language Becomes a Tool for Knowledge.
Individual Online Education
Levitin Language School provides individual online education for children, teenagers, university students and adults internationally.
Our educational ecosystem develops three connected areas:
Languages · Academic Subjects · Language + Subject
Depending on the learner's needs, lessons can focus on language development, academic knowledge, problem solving, or learning and using subject knowledge through another language.
International and U.S.-focused resources are also available through Language Learnings.
Contact Levitin Language School
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About the Author
Tymur Levitin
Founder & Director, Levitin Language School
Educator and author working across language learning, academic education, problem solving, comparative thinking and integrated Language + Subject learning.
His educational framework focuses on understanding systems, diagnosing bottlenecks, developing decision-making and gradually transferring cognitive responsibility from teacher to learner.
Levitin Language School: https://levitintymur.com/
Language Learnings — USA: https://languagelearnings.com/
Language Thinking Laboratory: https://languagethinkinglab.blogspot.com/
Author contact: tymurlevitin@levitintymur.com
© Tymur Levitin — Founder & Director, Levitin Language School. All rights reserved.
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