- Understand the question. Understand all parts of the question, are all terms familiar. What does a solution look like?
- Understand the problem. See structure, relationships. You fully understand the problem when you have the solution.
- What does it feel like to understand? People typically don't know that they don't understand or that there is something to understand.
- How do I know that I understand? Indicators:
- Ability to simply and concisely express yourself (ie in english).
- See connections.
- Situation seems simple (have an aesthetic sense, don't be satisfied with complexities, things tend to be simple once you understand). Strip away all unnecessary data, connections, facts etc.
- Ability to explain "WHY", justify your position. I see the reason behind everything.
- I am
**certain**that I am correct (since I sufficiently argued all points with myself). - Marking Scheme
Mark What you did 5 **Reason, apply, problem solve, present:**Correctly applies ideas to reach a clear, concise, correct, cleanly presented solution. Can reason about ideas presented. Can apply them to different settings. etc. Minor technical flaws allowed.4 As in 5 but lacks some clarity, not as concise as possible, some minor flaw, not a clean presentation. 3 **Semantics:**Demonstrates correct understanding of basic definitions, ability to correctly apply basic techniques, theorem statements etc.2 Some major flaw in the understanding of basic definitions, techniques, theorem statements. Examples are: Applies theorems to wrong settings. Does not understand the goal (ie tries to prove the wrong result). Solves the wrong problem. 1 **Syntax:**Ability to restate given definitions, theorems, techniques.

- Develop an aesthetic sense. A simple solution is best. A concise explanation indicates understanding.

- You are part of the problem.
- Goal: Teach yourself enough about the problem so that the solution is
**simple, clear and obvious**. - I am human!!
- Price to pay in order to understand.
- We only learn simple things.
- Keep your EGO out of it. No "I'm stupid" etc.

- How do I teach myself? Know what you know, know what you don't and how to cover the difference.
- Be introspective
- Be willing to say "I don't understand."
- Take action
**NOW**(put the problem in your brain) - Try something AND LEARN FROM IT.
- Know what you are doing and why.
- Doing something simple
- No matter what, there is always some evidence for...
- You are always making progress if you LEARN.
- Move from firm ground to firm ground.

- Wait for answer to pop into your head.
- Don't get started early. Put problem in your head, allow your brain time to connect and understand.
- Not recognizing the cost (effort required) to obtain the solution.
- Not addressing the abilities you are lacking in order to solve the problem.
- Not checking yourself. Ask yourself "Is it correct?"
- Managing yourself poorly (getting down on yourself).
- Fear of others knowing that you are stupid. Not asking questions in class. 'stupid' is irrelevant. The winner is the one that comes out of class understanding the most.
- Not learning from your efforts.
- Not simplifying, breaking down problem.
- Not committing yourself to understanding UGLY points.
- Not knowing that you don't know.

- Learning a new programming language
- How many 0s are at the end of 999! (Num0s.java Num0s.pl)
- Find the last non-zero digit in 999!
- Locker Room problem (LockerRoom.java)
A locker room has lockers numbered 0,1,2,...,1023. All lockers are initially closed.

The first person walks in and flips every locker The 2nd person flips every other locker (starting at 0) The 3rd person flips every third locker (starting at 0) . . . The 1023 person flips every 1023 locker (starting at 0) At the end, which lockers are open and which are closed? - Card swapping problem: 25 people in a circle, a deck of 50 cards
with numbers 1,1,2,2,...,25,25 on them. Each person gets 2
randomly selected cards. At each step, each person looks at
their two cards and passes a smallest card left.
Prove that eventually some person will be holding two cards with the same number on them.

- Compress.java
- The Jumble Problem
- 15 puzzle (example of a simple solution, exercises include: have students write this, have them comment this)