A major decision
BeforeCompare a long list of pros and cons.
AfterFirst eliminate options that violate non-negotiable conditions.
A STEP-BY-STEP INTERACTIVE COURSE
What if becoming smarter sometimes means making the problem easier to think with?
We begin with one subtraction problem and add only one idea at a time. By the end, you will understand why the same mind can become more capable when a problem is represented differently—and be ready to choose those forms in Module 2.
15
small lessons
1
new term at a time
0
assumed background
TRY IT
Do the same subtraction in two number systems
Switch the writing system. The problem and your brain stay the same.
M̅ − I
D̅C̅C̅C̅C̅L̅X̅X̅X̅X̅V̅M̅M̅M̅M̅D̅C̅C̅C̅C̅L̅X̅X̅X̅X̅CMXCIX
The value exists, but the writing system gives you no simple column procedure.
The surprise: changing the way the problem is written can change its difficulty dramatically.
You just experienced the entire seed of the idea. Now we will slow it down and name each part.
01
What exactly changed in the subtraction example?
The facts stayed fixed. Their form changed.
A paragraph, a map, a timeline, a spreadsheet, and an equation can contain the same underlying facts while making different patterns easy—or difficult—to notice.
TRY IT
Find when the project went off track
Move the slider. No facts are added or removed.
Paragraph
The brief arrived Monday. Design began Tuesday. The customer changed the goal Friday. Engineering started Thursday. Nobody updated the brief. The launch missed by two weeks.
The turning point is present, but easy to miss.
NOW NAME IT
Representation
Say: “the form the problem takes”
A representation is the form in which a mind or machine encounters information: words, symbols, pictures, spatial layouts, categories, code, or something else.
Do not confuse it with: mere decoration. A useful representation changes what the user can notice or do.HOW THIS FITS
The mind is not working on reality directly. It is working on a particular form of reality.
MindThe thinker
FormHow the problem appears
MovesWhat can be done
PossibilitiesWhat might work
ShortcutsWhat removes work
ChecksWhat catches error
MethodWhat can be reused
CategoriesHow things are grouped
PracticeWhere you use it
02
Where does the thinking actually happen?
Think of the problem form as a workbench.
A good workbench puts the right pieces within reach, arranges related parts together, and makes the next action visible. A bad one can hide useful parts inside a pile.
TRY IT
Arrange the same eight article ingredients
Move one control. Watch what becomes easier to see.
Scattered
Audience
Promise
Proof
Story
Objection
Example
Action
Title
Everything is available, but the next move is unclear.
NOW NAME IT
Cognitive surface
Say: “the thinking workbench”
A cognitive surface is the workspace created by a representation—the visible or manipulable place where reasoning happens.
Do not confuse it with: the physical screen. A notebook page, a card layout, a spoken story, or an internal mental image can also be a cognitive surface.HOW THIS FITS
A representation creates the workbench. The workbench determines which parts are visible and reachable.
MindThe thinker
FormHow the problem appears
MovesWhat can be done
PossibilitiesWhat might work
ShortcutsWhat removes work
ChecksWhat catches error
MethodWhat can be reused
CategoriesHow things are grouped
PracticeWhere you use it
03
What makes one workbench more useful than another?
It gives you better moves.
Modern number notation does more than display quantities. It supports moves such as aligning columns, carrying, borrowing, and multiplying. The available moves change how much work the same mind must do.
TRY IT
Count four apples in several equal groups
Change the number of groups, then switch the available move.
6 groups
ONE GROUP AT A TIMERepeat the same +4 move
step 1
+
step 2
+
step 3
+
step 4
+
step 5
+
step 6
4 + 4 + 4 + 4 + 4 + 4= 24
24 apples6 separate +4 steps
NOW NAME IT
Operator
Say: “a move I can perform”
An operator is an action a representation makes available or cheap: add, group, compare, filter, rotate, connect, split, simulate, or check.
Do not confuse it with: a person in a control room. Here, the word simply means a permitted thinking move.HOW THIS FITS
The form creates a workbench; the workbench makes some thinking moves easy.
MindThe thinker
FormHow the problem appears
MovesWhat can be done
PossibilitiesWhat might work
ShortcutsWhat removes work
ChecksWhat catches error
MethodWhat can be reused
CategoriesHow things are grouped
PracticeWhere you use it
04
What happens when the answer is not immediately visible?
You try possibilities.
You might test routes, explanations, diagnoses, sentences, business ideas, or mathematical steps. Every attempted possibility spends time or compute.
TRY IT
Find which box contains the key
Move from one box checked to twelve. The key is in box eight.
1 tried
1
2
3
4
5
6
7
8
9
10
11
12
You have not yet tested the possibility that contains the answer.
NOW NAME IT
Search
Say: “trying possible answers”
Search is the process of considering or testing candidate answers until one meets the goal.
Do not confuse it with: only typing words into Google. Any systematic exploration of possibilities is search.HOW THIS FITS
Operators are the moves. Search is using those moves to travel from the starting problem toward an answer.
MindThe thinker
FormHow the problem appears
MovesWhat can be done
PossibilitiesWhat might work
ShortcutsWhat removes work
ChecksWhat catches error
MethodWhat can be reused
CategoriesHow things are grouped
PracticeWhere you use it
05
Why can trying possibilities become overwhelming?
Small choices multiply.
If a route has three possible directions at each of four turns, you do not have twelve possible routes. You have 3 × 3 × 3 × 3 = 81.
TRY IT
Watch a route become a branching tree
First add turns. Then change the choices available at every turn. Each complete branch is one route you might have to consider.
1 · LENGTH OF THE ROUTE
2 turns
2 · OPTIONS AT EACH TURN
2 choices each time
2 × 24 complete routes
2 choices repeated across 2 turns create 4 different paths—not 4.
NOW NAME IT
Search space
Say: “everything I might have to try”
A search space is the full collection of candidate answers, paths, or move sequences that could be explored.
Do not confuse it with: physical space. It is a way to count the possibility field.HOW THIS FITS
A representation gives you moves. Repeating those moves creates a field of possible paths.
MindThe thinker
FormHow the problem appears
MovesWhat can be done
PossibilitiesWhat might work
ShortcutsWhat removes work
ChecksWhat catches error
MethodWhat can be reused
CategoriesHow things are grouped
PracticeWhere you use it
06
How can you shrink the possibility field without becoming smarter?
Use rules to remove impossible answers before trying them.
A clue in Wordle, a dietary restriction in a restaurant search, or a budget ceiling in a purchase decision can eliminate entire groups of options.
TRY IT
Find one correct door without opening every door
Each clue rules out half of the remaining doors.
32 doors remain
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
Every door must be considered.
NOW NAME IT
Constraint
Say: “a rule that removes options”
A constraint is a condition an acceptable answer must satisfy. Once visible, it can rule out candidates before they are tested.
Do not confuse it with: always a limitation to overcome. A useful constraint is often a source of speed and clarity.HOW THIS FITS
Search explores possibilities. Constraints make the set of possibilities smaller.
MindThe thinker
FormHow the problem appears
MovesWhat can be done
PossibilitiesWhat might work
ShortcutsWhat removes work
ChecksWhat catches error
MethodWhat can be reused
CategoriesHow things are grouped
PracticeWhere you use it
07
What if the problem is difficult because too many parts are tangled together?
Separate what can be solved separately.
Planning dinner feels like one large problem until you separate menu, shopping, invitations, setup, cooking, and cleanup—and mark the few true dependencies.
TRY IT
Find which dinner tasks actually depend on each other
Reveal the tasks, then keep only the true dependencies. Watch six assumed steps become three parallel workstreams.
One vague job
Host dinner
Choose menuBuy foodCookInvite guestsCleanSet table
Six tasks are present, but their relationships are hidden.
UNKNOWN
coordination burden
“Host dinner” hides the parts and their relationships.
NOW NAME IT
Factorization
Say: “split one knot into smaller problems”
Factorization means dividing a tangled problem into parts that can be solved, checked, or delegated with less interaction.
Do not confuse it with: breaking everything apart. If two parts strongly depend on each other, pretending they are independent creates errors.HOW THIS FITS
Constraints remove invalid paths. Factorization turns one large path-finding problem into several smaller ones.
MindThe thinker
FormHow the problem appears
MovesWhat can be done
PossibilitiesWhat might work
ShortcutsWhat removes work
ChecksWhat catches error
MethodWhat can be reused
CategoriesHow things are grouped
PracticeWhere you use it
08
How can the same amount of information feel smaller?
Treat several details as one meaningful unit.
A familiar area code is three digits, but you remember it as one thing. A chess expert sees a known pattern where a beginner sees many unrelated pieces.
TRY IT
Remember the same phone number
Increase how many digits are treated as one meaningful group.
10 memory units
6035551947
The information is unchanged. Your mind has fewer separate things to hold.
NOW NAME IT
Chunking and compression
Say: “package many details into one useful unit”
Chunking groups details into a meaningful whole. Compression stores or communicates the important structure using fewer separate units.
Do not confuse it with: carelessly deleting information. Good compression keeps what the next task needs.HOW THIS FITS
Factorization makes problems smaller by splitting. Chunking makes them smaller by grouping.
MindThe thinker
FormHow the problem appears
MovesWhat can be done
PossibilitiesWhat might work
ShortcutsWhat removes work
ChecksWhat catches error
MethodWhat can be reused
CategoriesHow things are grouped
PracticeWhere you use it
09
Why is comparison harder when matching facts are far apart?
Put information that must interact next to each other.
A comparison table is powerful because your eyes no longer need to memorize a value, travel to another page, find its match, and hold both in working memory.
TRY IT
Compare three courses
Bring matching facts closer together until your eyes can compare them directly.
Separate pages
Course A$4998 weeks
Course B$2996 weeks
Course C$39912 weeks
Your eyes must remember a value, move, and find its match.
NOW NAME IT
Locality
Say: “bring related facts close together”
Locality means arranging information so things that must be compared or combined are near each other in the representation.
Do not confuse it with: geographic closeness. The closeness may be visual, conceptual, or computational.HOW THIS FITS
Constraints, splitting, grouping, and proximity are four different ways a representation can reduce unnecessary work.
MindThe thinker
FormHow the problem appears
MovesWhat can be done
PossibilitiesWhat might work
ShortcutsWhat removes work
ChecksWhat catches error
MethodWhat can be reused
CategoriesHow things are grouped
PracticeWhere you use it
10
When is a problem ready to work on?
When its important parts are visible and the next move is clear.
“Take a warm trip sometime” becomes workable only after the goal, time window, budget, non-negotiables, and next comparison are separated.
TRY IT
Turn a vague trip idea into something you can decide
Move from a sentence toward a form that suggests the next action.
Raw thought
“Maybe we should take a warm family trip sometime this winter, but not spend too much.”
NOW NAME IT
Structured and operational problem
Say: “the problem has handles”
A structured problem has named parts and relationships. An operational problem goes one step further: its structure makes a concrete next action possible.
Do not confuse it with: a problem that is already solved. It is merely ready to be acted on.HOW THIS FITS
The earlier shortcuts transform a raw situation into a problem with handles.
MindThe thinker
FormHow the problem appears
MovesWhat can be done
PossibilitiesWhat might work
ShortcutsWhat removes work
ChecksWhat catches error
MethodWhat can be reused
CategoriesHow things are grouped
PracticeWhere you use it
11
How does a system learn which attempt is better?
It needs checks that can recognize progress and error.
If you inspect a bridge only after completion, an early mistake contaminates everything built on top of it. Earlier checks make correction cheaper.
TRY IT
Build a five-block bridge with an error in block three
Add earlier checks. Watch how long the hidden error survives.
One final test
Block 1
Block 2
Block 3
Block 4
Block 5
Error discoveredafter the bridge fails
NOW NAME IT
Verifier
Say: “a check that can tell whether it worked”
A verifier is a test, rule, measurement, or judge that can distinguish an acceptable result from an unacceptable one.
Do not confuse it with: guaranteed truth. Every verifier has blind spots and can sometimes be gamed.HOW THIS FITS
A structured problem exposes intermediate pieces. Verifiers can now check those pieces before the final outcome.
MindThe thinker
FormHow the problem appears
MovesWhat can be done
PossibilitiesWhat might work
ShortcutsWhat removes work
ChecksWhat catches error
MethodWhat can be reused
CategoriesHow things are grouped
PracticeWhere you use it
12
What exactly should survive from one solved problem to the next?
Reuse the method—not the old answer.
An apartment decision and a job decision contain different facts. But both can reuse the same comparison rule: put each option in a row and each criterion in a column.
TRY IT
Use the same comparison method on a different decision
Switch from apartments to job offers. The facts change. Then reuse the rule that still works.
CURRENT DECISIONChoose an apartment
Harbor costs $2,100, has a 12-minute commute, and gets 8/10 for natural light.
Oak costs $1,850, has a 28-minute commute, and gets 6/10 for natural light.
All the facts are present, but you must hold matching details in memory to compare them.
NOW NAME IT
Primitive
Say: “a small thinking rule I can reuse”
A primitive is a small reusable move or structure. In this example, the primitive is the row-and-column comparison rule—not the apartment facts or the final choice.
Do not confuse it with: a copied answer or a vague lesson such as ‘be more organized.’ It must be specific enough to perform again.HOW THIS FITS
Verifiers show that a method worked. Saving the specific method lets the next problem begin with useful structure instead of a blank page.
MindThe thinker
FormHow the problem appears
MovesWhat can be done
PossibilitiesWhat might work
ShortcutsWhat removes work
ChecksWhat catches error
MethodWhat can be reused
CategoriesHow things are grouped
PracticeWhere you use it
13
What determines the kinds of building blocks a system can even notice?
The category system decides what counts as the same kind of thing.
Alphabetical pantry shelves make named items easy to retrieve. Meal-based shelves make possible breakfasts easy to see. Neither arrangement is simply “the truth.” Each serves different questions.
TRY IT
Organize the same pantry for a task
Switch the category system. Then ask: “What can I make for breakfast?”
AvocadoA
BaconB
CerealC
LettuceL
OatsO
TomatoT
Easy to find a named item; hard to see a possible meal.
NOW NAME IT
Ontology
Say: “the categories used to divide reality”
An ontology is the set of object types, categories, and relationships a person or system assumes exist.
Do not confuse it with: a mystical claim about ultimate reality. In practical work, it often means the category system underneath a tool.HOW THIS FITS
An ontology decides which primitives exist and which relationships the representation can express.
MindThe thinker
FormHow the problem appears
MovesWhat can be done
PossibilitiesWhat might work
ShortcutsWhat removes work
ChecksWhat catches error
MethodWhat can be reused
CategoriesHow things are grouped
PracticeWhere you use it
14
What should you do differently tomorrow?
Before adding effort, ask whether the problem needs a better form.
The practical habit is simple: identify what you cannot currently see or do, then change how the problem is laid out until that action becomes easier.
Learning something difficult
BeforeRead another explanation in the same format.
AfterTranslate it into an example, diagram, procedure, and test.
A relationship conflict
BeforeArgue harder inside two incompatible stories.
AfterLay out observations, interpretations, needs, and repeating effects separately.
Creative work with AI
BeforeGenerate more drafts.
AfterInvest in better intermediate checks and reusable components.
YOUR TURN
Diagnose the kind of stuckness
Choose the sentence that feels most like your real problem.
USE THE IDEA YOU LEARNED: Use constraints
Write the conditions a good option must satisfy, then remove every option that fails one.
NOW NAME IT
Representation experiment
Say: “try a different problem form and test whether it helps”
A representation experiment holds the goal and facts steady, changes how the problem is encoded, and checks whether a real task becomes easier, faster, or more accurate.
Do not confuse it with: making a prettier diagram. The new form must change performance on a specific task.HOW THIS FITS
You can now change a problem’s form, use the moves it exposes, check the result, and retain a method that transfers. Module 2 teaches the major forms you can choose.
MindThe thinker
FormHow the problem appears
MovesWhat can be done
PossibilitiesWhat might work
ShortcutsWhat removes work
ChecksWhat catches error
MethodWhat can be reused
CategoriesHow things are grouped
PracticeWhere you use it
KEEP THE LANGUAGE
The complete plain-language glossary
Every technical term introduced in the course, in learning order.
Representation
The form in which a mind or machine encounters a problem.
Cognitive surface
The workspace where information is arranged for thinking.
Operator
A move the representation makes easy to perform.
Search
Trying possible answers and checking what happens.
Search space
The full set of answers or paths that might be tried.
Constraint
A rule that removes impossible or unacceptable options.
Factorization
Splitting one problem into parts that can be solved separately.
Chunk / compression
Treating several details as one meaningful unit without losing what matters.
Locality
Putting facts that must interact close enough to compare directly.
Structured problem
A problem whose important parts and relationships are labeled.
Operational problem
A structured problem that makes the next action clear.
Verifier
A check that can tell whether an answer or intermediate step is acceptable.
Primitive
A small reusable thinking move or structure—for example, one option per row and one criterion per column.
Ontology
The categories and relationships a system uses to divide reality.
THE WHOLE IDEA IN ONE SENTENCE
Intelligence is not only the ability to solve a hard problem. It is also the ability to discover a form in which the problem no longer requires as much intelligence.
Same facts
Better form
Easier move
Less search
Earlier check
Reusable method