Problem Solving and Creativity

1. Problem Solving and Creativity

2. Problem solving - definition • Directed thinking towards a goal solution – the task is to choose the best process that will lead to a goal • 1st psychological studies by Wolfgang Kohler (insightful learning)

3. Four Characteristics • 1. problem solving is goal directed • 2. it involves a series of operations • 3. it involves cognitive processes • 4. it involves sub-goal decomposition - reaching overall goal requires reaching sub-goals

4. 3 Steps in Problem Solving • Representing or defining exactly what the problem is to solve • Generating possible ways of solving the problem and choosing the best solution • Evaluating the solution – is it the best solution?

5. Representing the Problem • This is not referring to the surface problem, but the actual problem that has to be solved • Examples: • Mathematical word problems • Bowling pin Problem • Christmas tree problem

6. Word Problems • John and Mary want to buy new plants for their garden. They agree on how many to buy, but not on how many of each to buy. John wants to buy a lot of a few kinds and suggests 10 of each. Mary wants more variety so she wants 4 of each kind. They agree on 5 of each kind. They realize that they have room for 2 more, so they bought 6 of each. How many did they buy?

7. Bowling Pin example • O O O O • O O O • O O • O • O • O O • O O O • O O O O

8. Christmas Tree Example • You have 10 Christmas trees. How would you arrange them in 5 rows of 4 trees each?

9. Generating Solutions • Use of algorithms - a mathematical formula or other procedure that guarantees a correct solution if followed correctly. Can be too time consuming if there is a large number of actions that have to be tried. • Heuristics – choosing a strategy that looks like it will work and trying it – a short cut sometimes called a problem solving protocol

10. Types of Heuristics • 1. Hill climbing heuristic – follow the route that seems to get you closer to the goal – always move toward the goal • 2. Means-ends analysis – compare the current situation to the end goal. Ask what means do I have to get from here to there? Involves the development of sub-goals

11. Types of Heuristics • Working backwards – start at the goal and determine what is the last step to the goal, then the second to the last, etc. • Example 1: Water lilies grow on a lake every summer. It begins with one lily, and each day the amount of lake covered by the lilies doubles. On day number 80 the lake is completely covered. On what day was the lake half covered? • Example 2: You and a friend have each put 50 quarters on a table and take turns removing at least 1, but no more than 5. The person who removes the last quarter keeps all of them. You go 1st. What can you do make sure you get the last one?

12. Use of Analogies • Analogies – identifying the relationship between two concepts or two problems to solve • Recognizing that a new problem has similar beginning situation and end goal as an early problem so you try the same solution

13. Standard Analogies • Tar is to ____ as coal is to ____. • a. roofing: shovel c. black: heat • b. construction; heating d. black; heating • Fur is to bear as ___ is to _____. • a. coat; man c. rug; floor • b. warmth; animal d. wool; sheep • ________ is to torso as branch is to __________. • a. arm; leaf c. fingernails; acorns • b. Leg; twigs d. arm; trunk • Centaur is to horse as ____ is to ____. • a. woman; mermaid c. Mermaid; fish • b. mermaid; woman d. Fish; mermaid • ____ is to ocean as lake is to ______. • a. ship; boat c. Atlantic; Erie • b. island; land d. rough; calm

14. Examples of the use of an analogy to solve problems • Physician has a patient with inoperable tumor in the abdomen. She has radiation equipment that can direct radiation to the tumor and destroy it, but if she makes the radiation strong enough it will kill the surrounding tissue and the patient. If she makes it weak so tissue isn’t destroyed, it won’t destroy the tumor. How does she cure the patient?

15. Examples of the use of an analogy to solve problems • An evil dictator has a fort in the middle of his round country with 10 roads that radiate out from the center to the border. The good general is outside the country and wants to overthrow the dictator. His army has to attack the fort at the same time to win. The dictator has put land mines on the roads. Since he needs to use the roads, the mines are placed so they won’t blow up if a small group uses them, but will if a large army comes down a road they will blow up. How can the general attack the fort and win?

16. Examples of the use of an analogy to solve problems • A high school marching is practicing. They try marching in rows of 12, but they have an extra person. Then they try 8, but still have an extra person. So they try 3, but still have an extra person. They finally try 5 and it works. How many band members are there? (there are more than 45 and less than 200)

17. Types of Heuristics • Drawing a picture: • A man is standing on a bridge 300ft from the near side and 500ft from the far side. A train is coming from the near side. If he runs to the near side, he and the train will reach the end of the bridge at the same time. If runs to the far side, he and the train will also reach the end of the bridge at the same time. He can run 10mph. How fast is the train moving? 500ft 300ft • --------------------------------I-------------------- • I____________________ I______________I

18. Second example • A boat can go upstream 24 miles in the same time it can go downstream 36 miles. In still water the boat’s engines drive the boat 12 miles an hour more than the rate of the current. What is the rate of the current? • ------------------------B--------------- 36 miles 24 miles • --------------- speed of current

19. Second example (cont) • Look at the boat going upstream • 1. it can travel 12mph + speed of the current • 2. going upstream it is going 12mph + the speed of the current – the speed of the current • It has to be going 12 mph for 2 hours and downstream it goes 18mph for 2 hours • Downstream: 12mph + speed of the current + the speed of the current = 18 mph • Or 12 mph + 2 X the speed of the current = 18mph • So the seed of the current is 3 mph

20. 3rd example • Solomon has a 26-volume encyclopedia sitting in alphabetical order on his shelf. Each volume is 3 inches thick (including covers), so the set takes up 78 inches of space. A bookworm is sitting on the front cover of volume “A” and begins to chew his way through the pages directly toward volume “Z”. If he chews at a rate of 6 inches a month how, long will it take him to reach the back cover of “Z”?

21. Problems people have with generating solutions • Water Jar problem: • Problem A B C Amount • 1 29 3 20 • 2. 21 127 3 100 • 3. 14 163 25 99 • 4. 18 43 10 5 • 5. 9 42 6 21 • 6. 20 59 4 31 • 7. 23 49 3 20 • 8. 15 39 3 18 • 9. 18 48 4 22 • 10. 14 36 8 6

22. 2nd Example • Two trains are approaching each other. One is traveling at 20 miles an hour while the other travels at 30 miles and hour. A bird flies back and forth between the two trains at 60 mph. The trains meet in 4 hours. How far did the bird fly? • Called the development of a mental set

23. Problems people have with generating solutions • You are given a box of thumbtacks, a box of matches, and a small candle. Using what you have, how would you mount the candle vertically on a wall so it can be lit? • Two ropes are hanging from the ceiling of a room that contains a short stool and a hammer. If you grab one rope, the other rope is beyond your reach. How can you tie the 2 ropes together? • Functional fixedness

24. Common Errors in Problem Solving • 1. Inaccuracies in reading • Reading material without fully understanding the material • Reading the material too fast • Missing words or misreading words because you are not reading completely • Failing to reread difficult passages

25. Common Errors in Problem Solving • Inaccuracy in thinking • Placing more importance on speed or ease of obtaining an answer rather than accuracy • Not being careful to perform needed operations accurately • Being inconsistent in the way problems are interpreted and solved • Not checking the accuracy of a solution • Drawing a conclusion in the middle of the problem without sufficient thought • Working too quickly results in the above errors

26. Common Errors in Problem Solving • Weakness in problem analysis • Trying to solve the whole problem without breaking it down into sub-goals • Failing to use prior knowledge and experiences • Skipping difficult material or unfamiliar words, etc. • Not properly constructing a representation of the ideas presented in the problem

27. Common Errors in Problem Solving • Lack of perseverance • Making a weak attempt to solve the problem – lack of confidence? • Choosing an answer quickly because it looked or felt right instead of fully analyzing the problem • Jumping to a conclusion halfway through the process

28. Proficient Problem Solvers • Have a positive attitude – confident • Great concern for accuracy • Break problems into parts that can be accomplished • Avoid guessing and jumping to conclusions • More active – do more things and put in more effort

29. Improving Problem Solving Ability • 1. Increase knowledge base – experts have more efficient methods for problem solving • 2. Automate some components through expertise • 3. Follow a systematic plan of attack • 4. Draw inferences from information given and memory • 5. If solution not obvious look at possible heuristics • 6. Reformulate problem – have you defined the problem correctly • 7. Draw a picture

30. Creativity

31. Creativity • Definition – a cognitive activity that results in a new or novel way of viewing a problem or situation • Involves the ability to avoid the traps of problem solving: • Mental sets • Functional fixedness • Misrepresentation of the problem

32. The Process of Creativity • Stage 1: Preparation – knowledge acquisition and developing and testing possible solutions • Stage 2: Incubation – getting away from the problem can lead to creative solutions – doesn’t always occur • Insightful learning • Non-insightful learning • Stage 3: Illumination – sudden discovery of a solution • Stage 4: Verification – evaluation of the insight; does it really solve the problem or is it he best solution

33. Characteristics of a Creative Person • 1. Knowledgeable and intelligent • 2. Personality – self-confident and not afraid to be wrong or look foolish • 3. Motivation – willing to make the cognitive investment and effort to develop a solution

34. Environmental Influences on Creativity • If creativity is valued in the cultural setting, motivation to be creative increases • When practical constraints are put on creative thinking, low creativity • Practical evaluation of creative ideas occurs early – restricts the ideas that can be produced • Thinking about completely impractical ideas can lead to less impractical ideas which can lead to practical innovative ideas • Individual production of ideas produce more creative ideas than group production

35. Measuring Creativity • Most people know who a creative person is, but creativity is difficult to measure • Frequent procedure involves divergent thinking – How many uses for a brick can you think of? • Problem is that evaluating possible solutions just as important as generating