A student opens a physics problem.
There are numbers in the text. There is probably a formula somewhere in the chapter. The obvious strategy seems simple:
find the formula → insert the numbers → calculate → write the answer.
Sometimes this works.
That is precisely why the habit is so difficult to break.
A student can use an incomplete method and still obtain the correct numerical answer. The danger appears later, when the wording changes, an irrelevant number is added, the familiar variable disappears, the same physical principle is presented through a graph, or the problem requires two models instead of one.
Suddenly, the formula that seemed to represent knowledge no longer tells the student what to do.
The underlying problem is not necessarily a lack of formulas.
It is often a failure to distinguish between a physical situation, a model of that situation, and the mathematical representation of the model.
A more complete architecture of physics problem solving looks like this:
situation → system → phenomenon → model → assumptions → quantities → relationships → equations → mathematical transformation → calculation → units → physical interpretation → validation → transfer
The calculation is only one part of that chain.
And in many difficult physics problems, it is not the hardest part.
A Physics Problem Begins Before the Equation
Consider a familiar problem:
A car travels at 60 km/h for two hours. How far does it travel?
The calculation is immediate:
s = vt
so
s = 60 × 2 = 120 km.
But the apparent simplicity hides several decisions.
We have interpreted 60 km/h as the relevant speed.
We have treated that speed as constant.
We have interpreted two hours as the duration of motion.
We have decided that the required quantity is distance travelled.
We have ignored changes in direction because they do not matter for the question as posed.
We have accepted a simplified model of motion in which the relationship
s = vt
is appropriate.
The equation did not choose itself.
A model made the equation possible.
This distinction becomes increasingly important as physics becomes more advanced.
Reality Does Not Arrive with Equations Attached
A falling object does not carry a label saying:
v² = v₀² + 2aΔx
A resistor does not display:
V = IR
A moving object does not announce:
p = mv
And an accelerating system does not tell us which version of Newton’s laws will be most useful.
Reality gives us events:
objects move;
materials deform;
temperatures change;
waves propagate;
charges interact;
fluids flow;
light refracts;
bodies collide.
Physics requires us to construct a representation of those events.
The intellectual transition is therefore not:
numbers → formula
but:
reality → relevant physical structure → model → mathematical representation.
That is a fundamentally different activity.
The First Question Is Not “Which Formula?”
Suppose a ball is released from a height of 20 metres.
A student who has learned to search for numerical clues may see:
20 m
and immediately start scanning memory for formulas containing height.
A physicist—or a student learning to think physically—asks different questions.
What is the system?
What motion is being considered?
Is the initial velocity zero?
Is gravitational acceleration approximately constant?
Are we near Earth’s surface?
Is air resistance negligible for the required accuracy?
What quantity are we trying to determine?
The number 20 becomes useful only after we know what it represents.
20 metres is not merely a number. It is a measurement of a physical quantity inside a particular model.
This is why two problems containing exactly the same numbers can require completely different reasoning.
Numbers Do Not Determine the Physics
Imagine three problems containing the values:
10, 20, 5.
In one problem they may represent speed, distance and time.
In another they may represent resistance, voltage and current.
In another they may represent mass, height and energy.
The numerical pattern tells us almost nothing.
Physics is not the art of asking:
Which formula has the same number of variables as the problem has numbers?
The relevant equation is determined by the physical relationship, not by the appearance of the data.
That distinction explains why memorising increasingly large formula collections eventually produces diminishing returns.
The student may know fifty equations and still face the harder question:
Which one represents this situation?
A Formula Is Compressed Knowledge
Consider one of the most famous expressions in mechanics:
F = ma
It occupies only a few characters.
But those characters sit inside a much larger conceptual structure involving force, mass, acceleration, interactions, reference frames, vectors and the dynamics of a system.
The compactness of an equation is one of the great powers of mathematics.
It is also an educational trap.
Because the representation is short, students can mistake the representation for the explanation.
It is not.
An equation represents a relationship. It does not automatically explain why that relationship applies, what assumptions support it, or where its usefulness ends.
This is a general principle of scientific thinking:
representation ≠ phenomenon ≠ mechanism.
A graph is not the event.
A diagram is not the object.
An equation is not nature.
They are tools that preserve selected relationships while deliberately leaving other information outside the representation.
Every Useful Model Leaves Something Out
Physics would be almost impossible if every school problem had to reproduce reality in complete detail.
Consider the instruction:
Neglect air resistance.
Air has not disappeared.
The statement means that, for the purpose and precision of this model, its effect is being excluded.
Likewise, we may treat:
a body as a point mass;
a surface as perfectly horizontal;
a string as massless;
a battery as ideal;
a gravitational field as uniform;
friction as negligible;
a gas as ideal.
These are not necessarily mistakes.
They are modelling decisions.
A useful model does not contain everything that exists.
It contains what is necessary to answer the question at the required level of accuracy.
That is why the same real-world system may require different models for different questions.
Air resistance may be irrelevant when estimating the fall of a small dense object over a short distance.
It becomes essential when modelling a parachutist.
The world has not changed.
The question has.
And therefore the appropriate model has changed.
“Given” Is a Translation Process
Many students are taught to begin a physics solution with a section called Given.
That can be useful—but only if we understand what the operation is doing.
It is not clerical copying.
It is translation.
Suppose we read:
A car increases its speed from 10 m/s to 25 m/s in 5 seconds.
We encode the situation as:
v₀ = 10 m/s
v = 25 m/s
t = 5 s
If acceleration is required:
a = ?
Something intellectually important has already happened.
We have moved from natural-language description to a structured system of physical quantities.
Only then can we ask what relationship connects them.
For uniformly accelerated motion:
v = v₀ + at
and therefore:
a = (v − v₀)/t
The equation was not selected because the problem contained three numbers.
It was selected because the model of the motion justified the relationship.
Drawings Are Part of the Reasoning
Suppose an object rests on a horizontal surface while an external force acts on it.
A student can immediately start searching for equations.
But a free-body diagram may reveal more than another minute of formula hunting.
Once the forces are represented, we can distinguish:
gravity;
the normal force;
friction, if relevant;
the applied force;
other interactions, if present.
A good diagram externalises relationships that working memory would otherwise have to maintain internally.
That makes diagrams cognitive tools, not decorations.
But the same warning applies.
The diagram is still a representation.
A real object is not a rectangle.
A force is not literally an arrow.
The diagram works because it removes most details and preserves the relationships needed for the analysis.
A strong physics student therefore learns to move in both directions:
physical situation → diagram
and
diagram → physical meaning.
Units Are Part of the Physics
Suppose a speed is given as:
72 km/h
while the rest of the problem uses metres and seconds.
Inserting the number 72 directly into an SI-based calculation can produce a completely wrong result.
Why?
Because 72 is not the physical quantity.
72 km/h is.
A numerical value without its unit is an incomplete representation of a measured physical quantity.
This is why unit analysis should not be treated as an administrative step performed after the “real physics” is finished.
It is part of the reasoning itself.
If
v = s/t
then the dimensional structure must be consistent with:
m/s.
If a calculation that supposedly produces velocity ends with kilograms per metre, something is wrong regardless of how elegant the arithmetic looks.
Dimensional Analysis Can Reject an Equation Before We Calculate Anything
Suppose someone proposes:
s = vt²
as an equation for distance, where v has dimensions of velocity and t is time.
Then:
[v] = L/T
and
[t²] = T²
so:
[vt²] = LT.
But distance must have dimensions:
[s] = L.
The proposed equation is dimensionally inconsistent.
No numerical calculation was required.
This is an important shift in thinking.
Instead of treating units as labels attached after calculation, we use dimensions as structural evidence about whether the equation itself can make physical sense.
A Correct Calculation Can Produce a Physically Absurd Answer
Imagine that a student calculates a person’s mass as:
8,500 kg.
Perhaps every arithmetic operation was performed correctly.
That does not rescue the answer.
Physics is not satisfied by internal mathematical consistency alone.
The result must return to the physical world.
We must ask:
Is this plausible?
Order-of-magnitude reasoning is one of the simplest and most powerful ways to detect errors.
A car travelling for roughly one hour at roughly 100 km/h will not cover 10,000 km.
An ordinary room will not have an area of one million square metres.
A human body will not normally have a mass of several tonnes.
Approximation is therefore not the enemy of precision.
It is often the first defence against meaningless precision.
Physics Requires Thinking in Two Directions
Problem solving involves at least two transformations.
The first is:
reality → mathematics
We identify the system, construct a model and represent relationships mathematically.
The second is:
mathematics → reality
We interpret the result and decide what it means for the physical system.
A weak solution often performs only the middle operation:
formula → number.
A complete solution closes the loop.
Not Every Physics Problem Needs Numbers
Consider:
If the mass of an object doubles while its speed remains unchanged, what happens to its kinetic energy?
We know:
Eₖ = ½mv²
If m doubles while v remains constant, kinetic energy doubles.
No calculator is necessary.
Now suppose speed doubles:
(2v)² = 4v²
so the kinetic energy becomes four times as large.
This kind of qualitative reasoning exposes whether the student sees the equation as a relationship or merely as a container into which numbers are inserted.
A powerful learning habit is therefore:
predict first, calculate second.
Before touching the calculator, ask:
Should the answer increase or decrease?
Should it be positive or negative?
Should it double, quadruple or remain unchanged?
What approximate magnitude should we expect?
The numerical result can then be tested against an existing physical expectation.
Graphs, Equations, Diagrams and Words Are Different Languages of the Same Model
A physical process may be represented through:
a verbal description;
an equation;
a table;
a diagram;
a graph;
a simulation.
Strong understanding includes the ability to move between these representations.
For example, a velocity-time graph is not merely a line to be visually inspected.
Its slope may encode acceleration.
Its area over a time interval may encode displacement.
Its shape can reveal changing motion.
Its intersections and extrema may have physical interpretations.
So understanding motion means more than knowing a motion equation.
It means being able to move through a network such as:
situation → description → diagram → equation → graph → interpretation.
If knowledge works only when presented in one familiar format, it remains fragile.
“I Understand It When the Teacher Explains It” Is Not the Same as Being Able to Solve It
This is one of the most important distinctions in learning physics.
When an experienced teacher demonstrates a solution, many difficult decisions have already been made.
The teacher has probably:
identified the relevant quantities;
ignored irrelevant information;
selected the system;
chosen the model;
recognised the governing principle;
constructed the equation;
selected an efficient sequence of steps.
The student sees the finished route.
And the route may make perfect sense.
But recognising a good route after someone else has constructed it is not equivalent to constructing that route independently.
There is a difference between:
recognition
and
retrieval + selection + construction.
That gap explains the familiar experience:
“I understood everything in class, but when I opened the homework I didn’t know where to start.”
The problem may not be comprehension of the teacher’s solution.
The missing skill may be generating the solution architecture without the teacher.
Transfer Is a Better Test Than Repetition
Suppose a teacher solves a problem about a car moving with constant acceleration.
If the student’s next problem changes only “car” to “bus” while preserving every structural feature, success tells us relatively little.
The student may simply have copied the visible pattern.
A stronger test changes something that forces a new decision.
The same applies to Ohm’s law.
Knowing
I = V/R
is useful.
But can the student also:
solve for resistance?
solve for voltage?
predict what happens to current when resistance increases?
interpret the relationship graphically?
identify a situation where the simple relationship is insufficient?
connect the equation to the behaviour of a real circuit?
Knowledge becomes robust when it survives changes of representation, wording and context.
Sometimes a Physics Problem Is Actually a Mathematics Problem
A student may understand the physical model perfectly but struggle to solve the equation.
The bottleneck may involve:
fractions;
algebraic rearrangement;
powers;
proportions;
graphs;
trigonometry;
unit conversion.
In that case, explaining the physical principle five more times may achieve very little.
The physical understanding is not the problem.
The mathematical operation is.
This is one reason why physics learned through another language can become especially interesting educationally: subject knowledge, mathematical operations and language can interact, but they should not be confused with one another.
A learner working with physics in English, German or Spanish may encounter several simultaneous demands:
physical modelling + mathematics + academic language + terminology + explanation.
The correct educational response is not to collapse all of these into “the student doesn’t understand physics.”
It is to identify the actual bottleneck.
Sometimes the Opposite Happens
A student may be excellent at algebra.
Give them an equation and they can rearrange it immediately.
Yet they may still be unable to determine which equation represents the situation.
Then the problem lies earlier in the chain:
situation → model.
This is not a mathematical failure.
It is a modelling failure.
And giving the student more algebra practice will not solve it.
One Wrong Answer Can Hide Many Different Failures
Suppose a student gets the wrong answer.
That visible outcome could have been caused by very different mechanisms.
Perhaps the student:
misread the wording;
misidentified the physical phenomenon;
selected the wrong system;
chose an inappropriate model;
ignored an important interaction;
used a law outside its valid conditions;
confused two quantities;
failed to convert units;
made an algebraic error;
made an arithmetic error;
interpreted a vector as a scalar;
failed to check the final result.
The visible error is one thing.
Its cause is another.
That is why the statement:
“You are bad at physics”
has almost no diagnostic value.
A useful diagnosis might instead look like:
situation ✓
model ✓
quantities ✓
physical relationship ✓
algebra ✗
Now we know what to train.
Or perhaps:
situation ✓
model ✗
In that case, twenty additional substitution exercises may reinforce the wrong skill.
Good Teaching Does Not Remove Every Decision
There is a subtle problem with help.
Help can solve the student’s difficulty—or solve the student’s problem for them.
If a teacher immediately says:
“Use conservation of energy here,”
one of the central intellectual decisions has already been removed.
A better prompt might be:
What changes between the initial and final states?
or:
Which forms of energy are present?
or:
Is there a quantity that should remain constant under the assumptions of this model?
The student receives support without losing ownership of the reasoning.
This leads to a useful principle:
The best hint is not the hint that makes the answer easiest to obtain. It is the smallest hint that restores productive thinking.
Support Should Eventually Disappear
At the beginning, a teacher may demonstrate the complete process.
Later, the teacher may provide questions rather than solutions.
Then the student may be asked to justify the model.
Then to select the governing principle independently.
Eventually, the student should be able to construct the entire route.
The progression can be represented as:
demonstration → guided reasoning → partial independence → independent solution → transfer to unfamiliar problems.
The purpose of instruction is not to make students excellent spectators of expert solutions.
It is to make the reasoning available when the expert is absent.
Solve Symbolically Before Substituting Numbers
This is one of the most useful habits in physics.
Suppose:
v = v₀ + at
and acceleration is required.
First derive:
a = (v − v₀)/t
Only then substitute:
a = (25 − 10)/5.
Why?
Because symbolic reasoning preserves structure.
Numbers can hide it.
When numbers enter too early, students may lose sight of which quantities are being related and why.
A symbolic expression also makes dimensional analysis easier and often reveals simplifications before arithmetic begins.
Numbers Can Create an Illusion of Progress
Calculation feels productive.
Keys are being pressed.
Lines are being written.
Values are changing.
But a student can calculate an incorrect model with perfect arithmetic.
That is why thirty seconds spent deciding what the system is can be more valuable than three minutes spent accurately calculating the wrong equation.
In physics, speed of calculation cannot compensate for an error made at the level of modelling.
The Answer Must Return to the Phenomenon
Suppose we calculate:
a = 3 m/s².
What does that mean?
Within the model, velocity changes by 3 m/s every second.
Suppose:
P = 200 W.
What does that say about the rate of energy transfer?
Suppose:
η = 0.85.
What does an efficiency of 85% mean for the system being analysed?
A complete answer has at least two layers:
mathematical result
and
physical interpretation.
Without the second, physics can deteriorate into algebra with physical symbols.
Change the Problem After You Solve It
One of the best tests of understanding is to alter one condition.
Ask:
What if the mass doubles?
What if the time is halved?
What if friction disappears?
What if resistance increases?
What if the direction of the applied force changes?
What if the initial velocity is no longer zero?
What if air resistance can no longer be neglected?
This turns one solved exercise into a family of models.
The student begins to see not only what the answer is, but what controls the answer.
That is much closer to scientific reasoning.
A Physics Problem Is a Small Exercise in Scientific Modelling
Even a conventional school problem can contain a simplified version of a scientific reasoning cycle:
observation or described situation
↓
definition of the system
↓
model
↓
assumptions
↓
relationships
↓
mathematical representation
↓
prediction or calculation
↓
interpretation
↓
validation
Real scientific research is obviously much more complex.
But the intellectual skeleton is already visible.
That is one reason physics problem solving matters beyond examinations.
It trains the ability to convert a messy situation into a structured model without confusing the model with reality.
What Does It Mean to “Understand” a Formula?
Knowing an equation by heart is not enough.
For an important equation, a student should gradually be able to answer questions such as:
What does every quantity represent?
What units does each quantity have?
Which relationship is expressed?
Under what assumptions does it apply?
What happens if one variable increases?
What happens if a variable approaches zero?
Can the equation be rearranged?
What would its graph look like?
What real physical process does it represent?
When would this equation become insufficient?
Which alternative model would then be needed?
At that point, the formula is no longer an isolated memory object.
It has become part of a conceptual system.
Should Physics Formulas Be Memorised?
Sometimes, yes.
Fluent access to fundamental relationships reduces unnecessary cognitive load.
A student who has to rediscover every basic equation from scratch cannot devote full attention to higher-level reasoning.
The problem is not memory.
The problem is confusing:
remembering with understanding
or:
remembering with application.
A more useful sequence is:
knowledge → understanding → selection → application → validation → transfer.
Memory supports this system.
It should not replace it.
Learn the Boundary of the Model
Suppose a student knows:
s = vt.
That is useful.
But an equally important question is:
When is this relationship no longer sufficient?
If velocity changes significantly over the interval, a different description is required.
So genuine knowledge includes both:
where a model works
and
where the model stops being adequate.
This is a deeper form of expertise than formula recognition.
It teaches students that scientific knowledge is conditional, structured and connected to assumptions.
There May Be More Than One Good Solution
Some physics problems can be approached through:
kinematics;
Newton’s laws;
energy;
momentum;
geometry;
graphs;
symmetry;
conservation principles.
Two valid methods do not indicate a defect in the problem.
Comparing them can reveal its structure.
One method may be shorter.
Another may expose the mechanism more clearly.
A third may generalise more easily.
The shortest solution is therefore not automatically the best solution for learning.
Experts compress reasoning because the underlying structure is already available to them.
Beginners often need to build that structure before compression becomes useful.
First build the model. Then learn to compress it.
Why Worked Solutions Can Produce False Confidence
A student reads a complete solution.
Every line makes sense.
The student closes the book and opens a new problem.
Nothing happens.
This is not paradoxical.
While reading a worked solution, the architecture has already been constructed by someone else.
The student is evaluating decisions, not necessarily generating them.
Worked solutions are valuable—but they should be used carefully.
One strategy is to reveal only the first step.
Then continue independently.
Or reveal the governing principle but construct the equation yourself.
Or compare your model with the solution before looking at the calculation.
External support should gradually decrease.
A Correct Answer Is Evidence, Not Proof of Understanding
Two students can both obtain:
20 m/s.
The first may have recognised a superficial pattern and guessed the appropriate equation.
The second may have:
identified the phenomenon;
defined the system;
made the assumptions explicit;
constructed the model;
selected the relationship;
checked dimensions;
calculated correctly;
interpreted the result;
tested its plausibility;
predicted what would happen if the conditions changed.
The number is identical.
The educational outcome is not.
This is why a correct answer is a visible marker of performance, not a complete description of the mechanism that produced it.
Sometimes an incorrect answer accompanied by transparent reasoning is more educationally useful than a correct answer produced by an invisible guess.
A Practical Physics Problem-Solving Architecture
When you open your next physics problem, delay the calculator.
Use the first minute to construct the problem.
Step 1. Describe what is happening
Explain the situation in your own words.
Do not merely repeat the wording.
Step 2. Define the system
What object, collection of objects or process are you analysing?
Where are the boundaries?
Step 3. Identify the physical phenomenon
Is this primarily about motion, forces, energy, momentum, electricity, waves, heat, fluids or another domain?
Step 4. Draw a representation if useful
Sketch the geometry, motion, forces, circuit or states.
The drawing does not need to be beautiful.
It needs to expose relationships.
Step 5. Identify known and unknown quantities
Write them with units.
Distinguish data from assumptions.
Step 6. State the assumptions
What is constant?
What is negligible?
What idealisation is being used?
Step 7. Predict qualitatively
Should the unknown be large or small?
Positive or negative?
Increasing or decreasing?
What approximate scale seems plausible?
Step 8. Choose the physical principle
Do not begin with a memorised formula.
Begin with the governing relationship or principle.
Step 9. Build the equation symbolically
Keep the physical structure visible before inserting numbers.
Step 10. Check dimensions
Does the equation have the correct dimensional structure?
Step 11. Substitute and calculate
Now the arithmetic has a justified role.
Step 12. Interpret the result
What does the answer actually tell us about the system?
Step 13. Test plausibility
Check units, sign, scale and limiting cases.
Step 14. Change one condition
Can you predict how the result would change?
If you can, you are no longer solving only one exercise.
You are working with the model.
When a Student Keeps Making Mistakes in Physics
The immediate response should not automatically be:
More exercises.
More practice can help.
But practice is effective only when it targets the mechanism that needs improvement.
A student may need help with:
reading the problem;
distinguishing relevant from irrelevant information;
building diagrams;
identifying forces;
selecting models;
understanding graphs;
algebra;
units;
proportional reasoning;
independent method selection;
checking answers;
or working with fewer hints.
Ten additional exercises aimed at the wrong bottleneck may simply make the wrong habit more automatic.
Diagnosis should come before volume.
When the Student Knows the Formulas but Still Cannot Solve Problems
Do not immediately add more formulas.
Take one familiar equation and interrogate it.
What does each quantity mean?
What assumptions support the equation?
How are the variables related?
What happens when one doubles?
What happens near zero?
What graph represents the relationship?
What units should appear?
Can the equation be rearranged?
Where does the model stop working?
What real situation does it describe?
One equation explored deeply can sometimes teach more physics than ten equations memorised superficially.
When the Student Understands the Teacher but Cannot Work Independently
The next intervention may not be another explanation.
It may be the gradual removal of support.
Remove:
the ready-made equation;
the name of the law;
the completed diagram;
the prepared “Given” section;
the prescribed order of operations.
Then observe where independent reasoning stops.
That point is educationally valuable.
It tells us what the student can already generate and what still depends on external support.
Physics Is Not a Collection of Formulas
Formulas matter enormously.
But physics begins before them and continues after them.
It begins with:
What is happening?
Then:
What system am I analysing?
Then:
What model is appropriate?
Then:
What assumptions am I making?
Then:
Which quantities and principles describe the model?
Only then does the question become:
Which mathematical relationship represents it?
And after the calculation:
What does the result mean?
Then:
Does it make physical sense?
And finally:
Can the same reasoning survive a change in the situation?
That is a much richer architecture than:
given → formula → answer.
Learning Physics at Levitin Language School
At Levitin Language School, academic subjects are treated as an independent educational layer rather than as an accessory to language lessons.
The educational architecture includes three distinct possibilities:
Languages
Academic Subjects
Language + Subject
Physics can therefore be studied as physics: through phenomena, models, mathematical relationships, problem solving, interpretation and transfer.
Where it serves the learner’s actual objective, physics can also become an environment for academic language development. Our dedicated Physics category develops the subject as a field of reasoning rather than a collection of isolated formulas.
Students who need to work across disciplines and languages can also explore Learn Physics in English, German or Spanish.
This does not mean replacing physics with vocabulary lists.
If physics is studied through English, the student still needs to understand the physics.
If it is studied through German, the physical model does not become secondary to terminology.
If it is studied through Spanish, the subject should not collapse into translation practice.
The layers can interact without becoming indistinguishable.
For learners who also need systematic language development, Levitin Language School provides dedicated routes for learning English, learning German and learning Spanish.
The objective determines the architecture.
Sometimes the learner needs physics.
Sometimes language.
Sometimes both.
The route should reflect the real task.
The Real Goal of Physics Problem Solving
In a textbook, someone tells you that the next exercise belongs to a particular chapter.
In reality, the world does not label situations:
Newton’s second law problem.
Conservation of momentum problem.
Ohm’s law problem.
First there is a situation.
Then someone has to decide what matters.
That is why the deepest goal of learning to solve physics problems is not:
I recognise this exercise because I have seen one like it before.
It is:
I can construct a defensible model of a situation I have not seen before, use mathematics to reason about it, and decide whether the result makes physical sense.
At that point, formulas stop being answers waiting to be selected.
They become what they were supposed to be all along:
tools for thinking about the physical world.
About the Author
Tymur Levitin
Founder and Director of Levitin Language School and Language Learnings
Language teacher, certified translator, author
Languages • Academic Subjects • Language + Subject
© Tymur Levitin. All rights reserved.
