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CS 501: TA Training Seminar
Neeraj Kumarcs.ucsb.edu/∼leadta
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CS 501: TA Training SeminarTeaching Problem Solving
Neeraj Kumarcs.ucsb.edu/∼leadta
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Plan for Today
Some Tips for E�ective Board-work
Problem Solving Through Problems
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E�ective Board Work
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E�ective Board Work
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E�ective Board Work
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E�ective Board Work
Do you see any problem with this board work?
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E�ective Board Work
Do you see any problem with this board work?
Improper space management, Ordering of text, cannot say what came when..
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E�ective Board Work
Compartmentalize
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E�ective Board Work
Compartmentalize
Plan for Today
– compute sums– . . .
May be even reserve columns for certain sections
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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
![Page 12: CS 501: TA Training Seminarleadta/slides/Week4.pdf · CS 501: TA Training Seminar Teaching Problem Solving Neeraj Kumar cs.ucsb.edu/ leadta](https://reader033.vdocuments.us/reader033/viewer/2022051811/601cd67cc8ca3b40960c7ce3/html5/thumbnails/12.jpg)
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
![Page 13: CS 501: TA Training Seminarleadta/slides/Week4.pdf · CS 501: TA Training Seminar Teaching Problem Solving Neeraj Kumar cs.ucsb.edu/ leadta](https://reader033.vdocuments.us/reader033/viewer/2022051811/601cd67cc8ca3b40960c7ce3/html5/thumbnails/13.jpg)
Teaching Problem Solving
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Teaching Problem Solving
Content of following slides borrowed from Part I. In the classroom of this book
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Problem Solving Through Problems
Teaching Goals
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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
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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”
![Page 18: CS 501: TA Training Seminarleadta/slides/Week4.pdf · CS 501: TA Training Seminar Teaching Problem Solving Neeraj Kumar cs.ucsb.edu/ leadta](https://reader033.vdocuments.us/reader033/viewer/2022051811/601cd67cc8ca3b40960c7ce3/html5/thumbnails/18.jpg)
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..
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Problem Solving Through Problems
Four phases of problem solving
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Problem Solving Through Problems
Four phases of problem solving
Understanding the problem
Making a plan
Carry out our plan
Look back and review solution
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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
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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”
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Understanding the Problem
“Answering a question that you don’t understand is a cardinal sin”
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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
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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
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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
![Page 27: CS 501: TA Training Seminarleadta/slides/Week4.pdf · CS 501: TA Training Seminar Teaching Problem Solving Neeraj Kumar cs.ucsb.edu/ leadta](https://reader033.vdocuments.us/reader033/viewer/2022051811/601cd67cc8ca3b40960c7ce3/html5/thumbnails/27.jpg)
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
![Page 28: CS 501: TA Training Seminarleadta/slides/Week4.pdf · CS 501: TA Training Seminar Teaching Problem Solving Neeraj Kumar cs.ucsb.edu/ leadta](https://reader033.vdocuments.us/reader033/viewer/2022051811/601cd67cc8ca3b40960c7ce3/html5/thumbnails/28.jpg)
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
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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?
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Devising a Plan
Plan ≡ outline of computations or constructions to obtain the unknown
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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
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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?
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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
![Page 34: CS 501: TA Training Seminarleadta/slides/Week4.pdf · CS 501: TA Training Seminar Teaching Problem Solving Neeraj Kumar cs.ucsb.edu/ leadta](https://reader033.vdocuments.us/reader033/viewer/2022051811/601cd67cc8ca3b40960c7ce3/html5/thumbnails/34.jpg)
Carrying out the Plan – Attention to detail
Plan gives an outline. Must examine details to see if they �t the outline
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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
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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
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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.
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Looking back..
No problem is completely exhausted – by looking back one can always improvethe solution or at least our understanding of the solution
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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
![Page 40: CS 501: TA Training Seminarleadta/slides/Week4.pdf · CS 501: TA Training Seminar Teaching Problem Solving Neeraj Kumar cs.ucsb.edu/ leadta](https://reader033.vdocuments.us/reader033/viewer/2022051811/601cd67cc8ca3b40960c7ce3/html5/thumbnails/40.jpg)
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
![Page 41: CS 501: TA Training Seminarleadta/slides/Week4.pdf · CS 501: TA Training Seminar Teaching Problem Solving Neeraj Kumar cs.ucsb.edu/ leadta](https://reader033.vdocuments.us/reader033/viewer/2022051811/601cd67cc8ca3b40960c7ce3/html5/thumbnails/41.jpg)
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?
![Page 42: CS 501: TA Training Seminarleadta/slides/Week4.pdf · CS 501: TA Training Seminar Teaching Problem Solving Neeraj Kumar cs.ucsb.edu/ leadta](https://reader033.vdocuments.us/reader033/viewer/2022051811/601cd67cc8ca3b40960c7ce3/html5/thumbnails/42.jpg)
An Example
“Find the radius of the smallest sphere that completely encloses a unit cube”
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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
![Page 44: CS 501: TA Training Seminarleadta/slides/Week4.pdf · CS 501: TA Training Seminar Teaching Problem Solving Neeraj Kumar cs.ucsb.edu/ leadta](https://reader033.vdocuments.us/reader033/viewer/2022051811/601cd67cc8ca3b40960c7ce3/html5/thumbnails/44.jpg)
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
![Page 45: CS 501: TA Training Seminarleadta/slides/Week4.pdf · CS 501: TA Training Seminar Teaching Problem Solving Neeraj Kumar cs.ucsb.edu/ leadta](https://reader033.vdocuments.us/reader033/viewer/2022051811/601cd67cc8ca3b40960c7ce3/html5/thumbnails/45.jpg)
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?
![Page 46: CS 501: TA Training Seminarleadta/slides/Week4.pdf · CS 501: TA Training Seminar Teaching Problem Solving Neeraj Kumar cs.ucsb.edu/ leadta](https://reader033.vdocuments.us/reader033/viewer/2022051811/601cd67cc8ca3b40960c7ce3/html5/thumbnails/46.jpg)
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
![Page 47: CS 501: TA Training Seminarleadta/slides/Week4.pdf · CS 501: TA Training Seminar Teaching Problem Solving Neeraj Kumar cs.ucsb.edu/ leadta](https://reader033.vdocuments.us/reader033/viewer/2022051811/601cd67cc8ca3b40960c7ce3/html5/thumbnails/47.jpg)
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!
![Page 48: CS 501: TA Training Seminarleadta/slides/Week4.pdf · CS 501: TA Training Seminar Teaching Problem Solving Neeraj Kumar cs.ucsb.edu/ leadta](https://reader033.vdocuments.us/reader033/viewer/2022051811/601cd67cc8ca3b40960c7ce3/html5/thumbnails/48.jpg)
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
![Page 49: CS 501: TA Training Seminarleadta/slides/Week4.pdf · CS 501: TA Training Seminar Teaching Problem Solving Neeraj Kumar cs.ucsb.edu/ leadta](https://reader033.vdocuments.us/reader033/viewer/2022051811/601cd67cc8ca3b40960c7ce3/html5/thumbnails/49.jpg)
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” ;-)
![Page 50: CS 501: TA Training Seminarleadta/slides/Week4.pdf · CS 501: TA Training Seminar Teaching Problem Solving Neeraj Kumar cs.ucsb.edu/ leadta](https://reader033.vdocuments.us/reader033/viewer/2022051811/601cd67cc8ca3b40960c7ce3/html5/thumbnails/50.jpg)
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
![Page 51: CS 501: TA Training Seminarleadta/slides/Week4.pdf · CS 501: TA Training Seminar Teaching Problem Solving Neeraj Kumar cs.ucsb.edu/ leadta](https://reader033.vdocuments.us/reader033/viewer/2022051811/601cd67cc8ca3b40960c7ce3/html5/thumbnails/51.jpg)
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
![Page 52: CS 501: TA Training Seminarleadta/slides/Week4.pdf · CS 501: TA Training Seminar Teaching Problem Solving Neeraj Kumar cs.ucsb.edu/ leadta](https://reader033.vdocuments.us/reader033/viewer/2022051811/601cd67cc8ca3b40960c7ce3/html5/thumbnails/52.jpg)
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!