CS 501: TA Training Seminar

Neeraj Kumarcs.ucsb.edu/∼leadta

CS 501: TA Training SeminarTeaching Problem Solving

Neeraj Kumarcs.ucsb.edu/∼leadta

Plan for Today

Some Tips for E�ective Board-work

Problem Solving Through Problems

E�ective Board Work

E�ective Board Work

E�ective Board Work

E�ective Board Work

Do you see any problem with this board work?

E�ective Board Work

Do you see any problem with this board work?

Improper space management, Ordering of text, cannot say what came when..

E�ective Board Work

Compartmentalize

E�ective Board Work

Compartmentalize

Plan for Today

– compute sums– . . .

May be even reserve columns for certain sections

E�ective Board Work

Compartmentalize

Proceed in-order (add arrows if needed..)

∑ni=1 i = ?

∑(i + 1)2

Plan for Today

– compute sums– . . . = ∑(i2 + 2i + 1)

Rearranging

2∑ i = n2 + n

⇒ . . . . . .

Alternatively, usearithemtic series

May be even reserve columns for certain sections

∑ni=1 i =

n(n+1)2

E�ective Board Work

Compartmentalize

Proceed in-order (add arrows if needed..)

∑ni=1 i = ?

∑(i + 1)2

Plan for Today

– compute sums– . . . = ∑(i2 + 2i + 1)

Rearranging

2∑ i = n2 + n

⇒ . . . . . .

Alternatively, usearithemtic series

May be even reserve columns for certain sections

∑ni=1 i =

n(n+1)2

Erase in LRU (least recently used) order

Teaching Problem Solving

Teaching Problem Solving

Content of following slides borrowed from Part I. In the classroom of this book

Problem Solving Through Problems

Teaching Goals

Problem Solving Through Problems

To help a student solve the problem at hand

Develop student’s ability to solve future problems by himself

Teaching Goals

Problem Solving Through Problems

To help a student solve the problem at hand

Develop student’s ability to solve future problems by himself

Teaching Goals

“Problem solving is a practical skill, like swimming. We acquire any practicalskill by imitation and practice. Trying to solve problems, you have to imitatewhat other people do when solving problems, and �nally doing them by yourown”

Problem Solving Through Problems

To help a student solve the problem at hand

Develop student’s ability to solve future problems by himself

Teaching Goals

“Problem solving is a practical skill, like swimming. We acquire any practicalskill by imitation and practice. Trying to solve problems, you have to imitatewhat other people do when solving problems, and �nally doing them by yourown”

Instill interest for problems and give opportunities for imitation and practice

Dramatize your ideas a little to generate interest, asking questions..

Problem Solving Through Problems

Four phases of problem solving

Problem Solving Through Problems

Four phases of problem solving

Understanding the problem

Making a plan

Carry out our plan

Look back and review solution

Problem Solving Through Problems

Four phases of problem solving

Understanding the problem

Making a plan

Carry out our plan

Look back and review solution

Each of these steps are important!

An instructor must help his students realize importance of each step

Problem Solving Through Problems

Four phases of problem solving

Understanding the problem

Making a plan

Carry out our plan

Look back and review solution

Each of these steps are important!

An instructor must help his students realize importance of each step

Attention to detail is super important, but often overlooked

“The more you sweat in the �eld, the less you bleed in war”

Understanding the Problem

“Answering a question that you don’t understand is a cardinal sin”

Understanding the Problem

“Answering a question that you don’t understand is a cardinal sin”

In most cases, if an interested student cannot understand the problem

Understanding the Problem

“Answering a question that you don’t understand is a cardinal sin”

In most cases, if an interested student cannot understand the problem

– Exposition is not clear or

Understanding the Problem

“Answering a question that you don’t understand is a cardinal sin”

In most cases, if an interested student cannot understand the problem

– Exposition is not clear or

– Problem may not be well chosen or

Understanding the Problem

“Answering a question that you don’t understand is a cardinal sin”

In most cases, if an interested student cannot understand the problem

– Exposition is not clear or

– Problem may not be well chosen or

– Problem is not natural or interesting or

Understanding the Problem

“Answering a question that you don’t understand is a cardinal sin”

In most cases, if an interested student cannot understand the problem

– Exposition is not clear or

– Problem may not be well chosen or

– Problem is not natural or interesting or

– Not enough time was given for getting ‘acquainted’

Questions you should be asking

Understanding the Problem

“Answering a question that you don’t understand is a cardinal sin”

In most cases, if an interested student cannot understand the problem

– Exposition is not clear or

– Problem may not be well chosen or

– Problem is not natural or interesting or

– Not enough time was given for getting ‘acquainted’

Questions you should be asking

What is the unkonown? What are the data? What is the condition? Can you givean example? What will be the solution to that example?

Devising a Plan

Plan ≡ outline of computations or constructions to obtain the unknown

Devising a Plan

Plan ≡ outline of computations or constructions to obtain the unknown

Usually the “bright idea” you need to solve the problem

Questions you should ask your students

Devising a Plan

Plan ≡ outline of computations or constructions to obtain the unknown

Usually the “bright idea” you need to solve the problem

Questions you should ask your students

Do you know a related problem?

Look at the unknown! Is there a problem with a similar unknown?

Could you restate the problem and think of a simpler version? Does this simplerversion give you any useful insights?

Devising a Plan

Plan ≡ outline of computations or constructions to obtain the unknown

Usually the “bright idea” you need to solve the problem

Questions you should ask your students

Do you know a related problem?

Look at the unknown! Is there a problem with a similar unknown?

Could you restate the problem and think of a simpler version? Does this simplerversion give you any useful insights?

At this point you should have ‘guided’ your students to a plan

Carrying out the Plan – Attention to detail

Plan gives an outline. Must examine details to see if they �t the outline

Carrying out the Plan – Attention to detail

Plan gives an outline. Must examine details to see if they �t the outline

Questions you should ask your students

Carrying out the Plan – Attention to detail

Plan gives an outline. Must examine details to see if they �t the outline

Questions you should ask your students

Did you check all the steps?

Can you see that the step is correct?

Can you prove that the step is correct? Intuitively or Formally

Carrying out the Plan – Attention to detail

Plan gives an outline. Must examine details to see if they �t the outline

Questions you should ask your students

Did you check all the steps?

Can you see that the step is correct?

Can you prove that the step is correct? Intuitively or Formally

At this point, students must be honestly convinced that each step is correct.

Looking back..

No problem is completely exhausted – by looking back one can always improvethe solution or at least our understanding of the solution

Looking back..

Questions you should ask your students

No problem is completely exhausted – by looking back one can always improvethe solution or at least our understanding of the solution

Looking back..

Questions you should ask your students

Can you derive the result di�erently?

Do you see a cleaner solution ?

No problem is completely exhausted – by looking back one can always improvethe solution or at least our understanding of the solution

Looking back..

Questions you should ask your students

Can you derive the result di�erently?

Do you see a cleaner solution ?

No problem is completely exhausted – by looking back one can always improvethe solution or at least our understanding of the solution

Can you use the method or the result to solve another problem?

An Example

“Find the radius of the smallest sphere that completely encloses a unit cube”

An Example

“Find the radius of the smallest sphere that completely encloses a unit cube”

1. Understanding the Problem

2. Devising a Plan

3. Executing the plan

4. Looking Back

An Example

“Find the radius of the smallest sphere that completely encloses a unit cube”

1. Understanding the Problem

2. Devising a Plan

3. Executing the plan

4. Looking Back

An Example

“Find the radius of the smallest sphere that completely encloses a unit cube”

1. Understanding the Problem

2. Devising a Plan

3. Executing the plan

4. Looking Back

Ask students to draw pictures? Whatabout problem in 2D? What can yousay about its solution?

An Example

“Find the radius of the smallest sphere that completely encloses a unit cube”

1. Understanding the Problem

2. Devising a Plan

3. Executing the plan

4. Looking Back

An Example

“Find the radius of the smallest sphere that completely encloses a unit cube”

1. Understanding the Problem

2. Devising a Plan

3. Executing the plan

4. Looking Back

Does matching along the longestdimensions su�ce? How to computediagonal of a cube? Great! you have aplan!

An Example

“Find the radius of the smallest sphere that completely encloses a unit cube”

1. Understanding the Problem

2. Devising a Plan

3. Executing the plan

4. Looking Back

An Example

“Find the radius of the smallest sphere that completely encloses a unit cube”

1. Understanding the Problem

2. Devising a Plan

3. Executing the plan

4. Looking Back

Can you prove that longest dimensionssu�ce? Why can you use pythagorustheorem? Can you prove that the angleswill be “alright” ;-)

An Example

“Find the radius of the smallest sphere that completely encloses a unit cube”

1. Understanding the Problem

2. Devising a Plan

3. Executing the plan

4. Looking Back

An Example

“Find the radius of the smallest sphere that completely encloses a unit cube”

1. Understanding the Problem

2. Devising a Plan

3. Executing the plan

4. Looking Back

Can you apply this technique to otherproblems? What about a hypercube in ddimensions? Can you come up withmore general technique?

(1, 1, 1, . . . , 1)√d

An Example

“Find the radius of the smallest sphere that completely encloses a unit cube”

1. Understanding the Problem

2. Devising a Plan

3. Executing the plan

4. Looking Back

Can you apply this technique to otherproblems? What about a hypercube in ddimensions? Can you come up withmore general technique?

(1, 1, 1, . . . , 1)√d

Thanks!