COSMO · STUDY + TEACHING DECK

Counting

December 21, 2025

A question worth sharing

Catalan Numbers

One sequence counts balanced parentheses, triangulations, and paths that avoid crossing a boundary.

ThinkCompareExplain

Today’s route

A question becomes an explanation.

Try

Enter through a problem. A partial attempt counts.

Connect

Use Catalan object, first return, bijection to name what you notice.

Convince

Build an argument a partner can check.

The goal is a method someone else can use.

The opener

Start with a question.

Opening problem

How many balanced parenthesis strings use three pairs?

Think · 2 min

Sketch, test a small case, or make a first guess.

Pair · 3 min

Compare methods. Find one thing you agree on and one question.

Be ready to share your route, even if it is unfinished.

Compare before checking

Put two attempts in conversation.

Explain

Tell a partner how you started. Let them finish before responding.

Question

Which step feels certain? Which step still needs evidence?

Build

Combine one useful idea from each attempt into a new route.

Can you explain your partner’s method in your own words?

Board conversation

Notice before naming.

Look for structure

Try a small case and record every step.
Separate what changes from what stays fixed.
State the pattern in a sentence before naming the theorem.

Make a conjecture

We think ___ because ___. A case that might break this is ___.

What changed? What stayed the same?

Useful language

Now the language has a job.

Catalan object

structure counted by $C_{n}$

first return

first time a path returns to a boundary

bijection

reversible structure-preserving match

Use one of these words to explain something you already noticed.

The core idea

Keep the idea. Explain the conditions.

C_0=1,\qquad C_{n+1}=\sum_{i=0}^{n}C_iC_{n-i}

The idea

One sequence counts balanced parentheses, triangulations, and paths that avoid crossing a boundary.

What to remember

A recursive split explains why many different objects share the same count.

What information does this method keep?

Worked example

One route, one decision at a time

The problem

Compute $C_{4}$ from $C_{n}$ = (1/(n+1)) $\binom{2n}{n}$ .

Predict each next step

$\binom{8}{4}$ = 70.
Divide by n+1 = 5.
$\frac{70}{5}$ = 14.
So $C_{4}$ = 14.

Which step needs a reason, not just a calculation?

See the structure

The structure behind catalan numbers

C_0=1,\qquad C_{n+1}=\sum_{i=0}^{n}C_iC_{n-i}

A balanced parenthesis string has one unique split into its first matched pair and its remainder.

Read the representation

One sequence counts balanced parentheses, triangulations, and paths that avoid crossing a boundary.

Look together

Point to the part that explains the relationship. Change one input and predict what changes.

What can you see here that was harder to see in words?

Two approaches

Compare the reasoning.

Structural method

Name the objects and conditions.
Use a reusable relationship.
Explain why the method must work.

Brute force

May help with a small test case.
Can hide the general reason.
Becomes unreliable when the scale grows.

Which route can you justify? What would you test first?

Reasoning clinic

Find the step that needs repair.

Investigate these concerns

May help with a small test case.
Can hide the general reason.
Becomes unreliable when the scale grows.

Repair together

Name the missing condition. Try a small counterexample. Rewrite one step so it is justified.

A useful correction explains why the original step fails.

Work together

Investigate catalan numbers

Your shared task

Groups match parenthesis strings, mountain paths, and polygon triangulations for small n.

Explore

4 min - test cases

Compare

5 min - form a rule

Defend

4 min - defend the rule

Everyone should be able to explain the group’s final claim.

Share → connect

What did the room discover?

Bring your evidence

Share a conjecture, a useful example, or a route that changed.

Connections to investigate

A recursive split explains why many different objects share the same count.
A complete solution names the condition that makes the method valid.
A counterexample is useful when it exposes a missing condition.

Which observation explains the mechanism?

Try → explain → check

Your turn to build the route

The problem

Explain C_(n+1) = sum from i=0 to n of $C_{i}$ C_(n-i), with $C_{0}$ = 1.

Predict each next step

Match the first opening parenthesis with its closing partner.
The inside contains i pairs.
The remainder contains n-i pairs.
Sum over i = 0,...,n; the empty string gives $C_{0}$ = 1.

Which step needs a reason, not just a calculation?

Convince a skeptic

Turn a pattern into a proof.

The claim

Balanced parenthesis strings with n pairs are counted by $C_{n}$ .

Build the argument

The empty string gives one object at n = 0.
The unique decomposition (inside)remainder gives the Catalan recurrence.
This recurrence and its initial value define the Catalan sequence, so the counts equal $C_{n}$ .

What does each step rely on? Where are the conditions used?

Stretch together

Take the idea somewhere new.

Transfer challenge

How many ways can a convex hexagon be triangulated?

Choose a route

Name the objects, the conditions, and a relationship that might help. Start with one test case.

What transfers from the earlier example? What is different?

Peer review

Check the argument together.

Restate

Explain the other person’s route before evaluating it.

Test

Try a small case or boundary case. Does every step still make sense?

Revise

Fix an assumption, add a reason, or find a second method.

Where did your thinking change?

Beyond this problem

Where might this idea help?

A connection

Catalan structures appear in parsing, binary trees, stack algorithms, and geometric subdivisions.

Discuss the model

What objects or quantities are represented?
Which relationship from today is preserved?
What assumption would you check before trusting it?

Invent one more situation with the same structure.

Exit ticket

Leave with an explanation.

Question 1

How many balanced parenthesis strings use three pairs?

Question 2

Explain why this claim is true: Balanced parenthesis strings with n pairs are counted by $C_{n}$ .

Question 3

How many ways can a convex hexagon be triangulated?

Choose one. Write an answer with a reason, then compare with a partner.

Reflect together

Make your thinking visible.

I used to think…

Name a claim or approach that changed.

Now I can explain…

Give the reason, not only the result.

I still wonder…

Write a question worth bringing to the next circle.

Share one sentence. Ask one follow-up question.

The idea to keep

A recursive split explains why many different objects share the same count.

Use the matching practice sets to move from direct examples to proof and transfer.

A good question belongs to everyone in the room.