Thinking & Problem Solving

1. Thinking & Problem Solving

2. Thinking

3. Thought • Thinking—changing and reorganizing the information stored in memory to create new information • Mental processes directed at a goal or purpose. • 5 kinds of units of thought: • Image • Symbol • Concepts • Prototype • rule

4. Image • Image –a visual, mental representation of an object or event. • Imaging works similar to actual visual imaging. • Images are constructed and therefore subject to error. • Ex. Trying to remember your first pet

6. Symbol • Symbol – an abstract unit of thought that represents an object or quantity. • Most common symbols are words • Help us think of things that are not present

7. Concepts • Concept—a label for a class of objects or events that have at least one attribute in common. • Enable us to group chunks of information

8. We develop hierarchies for concepts to organize information in our memory

9. Prototype • Prototype – a representative example of a concept • The closer a new object is to our concept prototype the easier it is to categorize it • If an object has four wheels and doors it probably fits our prototype for…

10. Example • If I said “shoe”, you thought of what type of shoe? • I think of a pump • That is an example of a : • Prototype

11. Rules • Rules – a statement of relation between concepts • All or nothing categorization process • A person cannot be in two places at the same time • Categories for solid, liquid, or gas • A square is…

12. Kinds of Thinking • Convergent Thinking – focused on finding a particular answer to a problem(Directed) • Divergent Thinking – Thinking Creatively to generate as many possible answers to a question or problem as you can.(non-directed) brainstorming • Incubation – Leave the problem for a time, allowing their minds to work on it without conscious effort. (Can led to Insight) • What is the color of snow? • What do cows drink?

13. An old money-lender offered to cancel a merchant’s debt and keep him from going to prison if the merchant would give the money-lender his lovely daughter. Horrified yet desperate, the merchant agreed to let Providence decide. The money-lender said he would put a black pebble and a white pebble in a bag and the girl would draw one. The white pebble would cancel the debt and leave her free. The black one would make her the money- lender’s, although the debt would be canceled. If she refused to pick, her father would go to prison. From the pebble-strewn path they were standing on, the money-lender picked two pebbles and quickly put them in the bag, but the girl saw he had picked two black ones. • What would you have done if you were the girl?

14. Thinking and behavior directed toward attaining a goal that is not readily available. • Must understand the problem correctly to accurately solve it. • Strategies • Subgoals – breaking down the problem into groups • Work backwards– mystery novelists do this • Examine various ways to reach goal – getting to Chicago Problem Solving

15. Jake, Amanda, and Sean are planning to go to a party, and they want to prepare for the party together. The party is across the street from Jake’s home, so the three meet there an hour before the party. When, at the end of the hour, they are ready to go to the party, they discover that is has started to rain heavily. None of them wants to get wet because they are all wearing nice clothing. Unfortunately they have one umbrella for the three of them, and the umbrella is big enough to protect only two people from the rain. How can all three of them get to the party without becoming drenched?

16. Algorithms • A step-by-step procedure for solving a problem • Not always the most efficient method • Use algorithms when we play chess or checkers y + z = r2

17. Example • You are working with a crossword puzzle. • You are trying to fill in a word for which you have all but one letter. • What you have is: CL_FF. • You would try putting every letter of the alphabet starting with A in that blank middle space until you found the letter that formed a word that fit the clue. • This is an example of: • An Algorithm

18. Using an Algorithm • Pick any month of the year. • Look at four dates that form a square in that month and add them together for a total. • Given this total, how can you determine the four dates the person chose using an algorithm? • To find the first date divide the sum by 4 and then subtract 4. • To find the second date just add 1 to the date you got above. • To find the third date add 7 to the first date. • To find the fourth date add 8 to the first date.

19. Heuristics • A rule-of-thumb problem solving strategy that makes a solution more likely and efficient but does not guarantee a solution • Examples: • What is 79 x 10? • Need milk? Go to the dairy sections of store rather than search the entire store. • “I” before “E” except after “C” • They simplify the problem because they let you reduce the number of possible solutions. • These can be handy shortcuts, or they can get us into trouble. (What about words like “weird”)

20. I have flipped a coin 10 times and it has landed on tails every time. What are the odds that it will land on tails when I flip it for the 11th time? • T T T T T T T T T T ?

21. Obstacles to Solving Problems

22. Mental Set • A tendency to approach a problem in a particular way that has worked in the past • A well-established habit of perception or thought • The set may or may not be helpful in solving a new problem

23. Example • There are six eggs in a basket • Six people take one of each. • How is it that one egg can still be left in the basket? • Answer: • The sixth person took the basket as well as the last egg still inside. • This is an example of a • Mental Set

24. Nine dots problem • Without lifting your pencil or re-tracing any line, draw four straight lines that connect all nine dots

25. Nine dots mental set • Most people will not draw lines that extend from the square formed by the nine dots • To solve the problem, you have to break your mental set & the implicit assumption you have to stay within the dots.

26. Functional Fixedness • Type of mental set • Inability to imagine new uses for familiar objects • Have to unlock the door? • Use a credit card. • Tighten a screw without a screwdriver? • Use a coin!

27. The American space agency, NASA wanted to find a way for astronauts to write notes in space. • Problem? In space ink pens don’t always work upside down. • After much research and many research dollars, NASA invented a pen that could be used in space even if the astronaut was upside down. • What did the Russians do? They had their cosmonauts use pencils. • This is an example of: • Functional Fixedness

28. Creativity • Creativity – the capacity to use information and/or abilities in new and original ways • No reasons for why some people are more creative

29. Mounting candle problem • Using only the objects present on the right, attach the candle to the bulletin board in such a way that the candle can be lit and will burn properly

30. Answer to candle problem • Most people do not think of using the box for anything other than its normal use (to hold the tacks) • To solve the problem, you have to overcome functional fixedness

31. Flexibility • Flexibility – the ability to overcome rigidity

32. Example Imagine you are in a room in which two strings are hanging from the ceiling. Your task is to tie the two strings together, but they are so far apart that you cannot reach both of them at the same time. The only other object in the room is a pair of safety scissors. How can you tie the strings together?

35. Recombination • Rearranging the elements of a problem to arrive at an original solution

36. Insight • The apparent sudden realization of the solution to a problem – correct answer suddenly comes to us. • “Unconscious Problem Solving” – You’re not aware of the thought process that led you to an insight.

37. Example • Chimpanzees were placed in an experimental cage with several different objects in the cage. • On the ceiling there are bananas high in the cage so that they were inaccessible to chimps.  • The chimps would holler and jump for the bananas without success, but some of the chimps looked around the cage, saw the various objects, and figured out how to build a scaffold they could climb to reach the bananas. • This is an example of • Insight

39. Can you measure out the amount of water in the right-hand column, using any of the three jars (A, B, and C) with volumes as shown in the middle column?

40. Problem 6 can be solved with a simpler formula (A - C), and so can Problem 7 (A + C). Many people miss these easy solutions because the mental set from the first several problems becomes fixated. Did your thinking stay flexible? Problems 1 through 7 can all be solved by filling Jar B, then pouring off enough water to fill Jar A once and Jar C twice desired volume = B - A - 2C

41. Problem 6 can be solved with a simpler formula (A - C), and so can Problem 7 (A + C). Many people miss these easy solutions because the mental set from the first several problems becomes fixated. Did your thinking stay flexible? Problems 1 through 7 can all be solved by filling Jar B, then pouring off enough water to fill Jar A once and Jar C twice desired volume = B - A - 2C