Why is problem solving important?

1. Why is problem solving important? • Cognitively Guided Instruction (CGI) • Achievement increases when teachers know more about the problem solving strategies of their students

2. What is problem solving? • Using heuristics? Memorized rules of thumb, and procedures • Using algorithms? A complete procedure. • Gestalt: Reasoning about the whole? • What you do when you don’t know what to do? • Representing the problem until you can solve it. (Polya, Mayer) • “Coming up with the answer” (4th grader)

3. Types of problems • Well defined- “What is four times six?” • Small problem space • Clear operation • Clear result • Ill defined- “What could we do to improve our quality of life?” • Vast problem space • Arguable solution

5. Story Problem Floor tiles are sold in squares 30 centimeters on each side. How much would it cost to tile a rectangular room 7.2 meters long and 5.4 meters wide if the tiles cost $0.72 each?

6. A Problem Solving Model(Mayer, 1987) Problem Representation Problem Solution

7. Problem Representation Translation • Linguistic: knowledge of language used in the problem. For example… • “Floor tiles” and “tiles” refer to the same objects. • The room is a rectangle. • Semantic: facts about the world. One meter equals 100 centimeters.

8. Problem Representation (Cont.) Translation • Integration: • Finding the right problem schemas to use on this item, for example: this is an area problem so… • Area= Length x Width, and then you figure you can… • Multiply the two dimensions of the room to get its area, • Multiply sides of a tile to get its area, • Divide the total room area by the area of one tile to find number of tiles that will fit the room.

9. Translation Problem Representation Integration Planning and Monitoring • Strategic information: Knowledge of how • To plan a solution once a representation has • been created for the problem. • Solving an equation • Evaluating your progress Problem Solution Execution

10. Translation Problem Representation Integration Planning & monitoring Execution Problem Solution Procedural: Carry out the steps 7.2 x 5.4= 38.88 .3 x .3= .09 tile area in meters 38.88/.09= 432 total tiles 432 x .72= $311.04

11. The Answer is….. $311.04

12. Can I correctly interpret the words in the problem? (linguistic & semantic) Translation Problem Representation Integration Can I put some structure to the givens? (schematic) Plan & Monitor Problem Solution Execution Can I work with the problem now that I have represented it? (strategic) Can I carry out the remaining steps my plan calls for to get an answer? (procedural)

13. What Hinders Problem Solving? • Functional fixedness: failing to see a tool in a new or unorthodox way. • Response set: Problems bias you toward a particular type of strategy. • What percent is 4?

14. What helps Problem Solving? • Draw a picture • Use multiple representations of the content • Breaking the problem down to identify subgoals. • Working backwards and forwards • Remaining calm and relaxed: Regulated adrenaline levels.

15. What kind of problem solving do we emphasize? • Most textbooks emphasize the problem solution phase of problem solving. • Our problem solving is influenced by our representations. • Placing values on a number line • Asking students to draw a picture of a problem • Research has indicated that we can improve our representation skills (Brenner, et al, 1999)

16. Discovery and Guided Discovery • Discovery: Open problem solving with little teacher involvement. • Guided Discovery: Teacher Structures the problem solving task and provides guidance at key points

17. Direct or Expository Instruction • Highly organized presentation: Teacher organizes material and sequences it • Presentation of procedures first • Emphasizes deductive logic

18. Guided Discovery v.s. Direct or Expository Instruction • Should we give concepts or procedures first? • Can you think of a way that discovery has worked or not worked for you? • How would you use discovery learning in a lesson?

19. Outline • What is transfer? • Why is studying transfer important? • A brief transfer task. • What are the components of transfer? • Are there different types of transfer? • What do we know about how transfer works? • What kind of new research is being done?

20. What is Transfer? • Transfer is taking knowledge acquired from one set of tasks and applying that knowledge to a new task. • Fundamental question: “Are people able to see commonalties in related problems and make use of them in their attempts to find solutions?”

21. Importance of Transfer I • Transfer has become an important part of the definition of conceptual understanding. • Part of NCTM and NSES curriculum standards. • A means of testing or demonstrating understanding.

22. Importance of Transfer II • Crucial difference between experts and novices. • Ex. Feynman: O-rings in cold temp. • Teachers and schools aspire to be teaching critical thinking skills. • Do you transfer knowledge between problems?

23. The Problem... • In a far away land there was a fortress with an evil dictator who mistreated the local villagers. • At that time there was also a general who commanded a rebel army, and he wanted to seize control of the fortress and free the villagers. • There were lots of roads to the dictator’s castle but the dictator had them booby-trapped with bombs so that only a small number of persons could travel safely on each road at a time. • If the general had sent sufficient forces down one of the roads to the castle, the explosives would have blown-up his troops.

24. The Solution….. • The general decided to spread out his forces and send small numbers of troops down each of the roads to the fortress. • The general’s troops were then able to avoid the bombs and converged on the fortress at the same time, capturing it from the dictator, and liberating the villagers. -Adapted from Gick & Holyoak, 1980.

25. Consider a New Problem... Given a human being with an inoperable stomach tumor, and rays which destroy organic tissue at sufficient intensity, by what procedure can one free the patient of the tumor by using these rays and at the same time avoid destroying the healthy tissue which surrounds it?

26. The Dictator Problem...

27. The Tumor Problem...

28. The Two Problems are Analogous

29. Important Points About This Transfer Task • This problem is one of the classic problems in cognitive psychology. • The problem is a good example of how the cognitive study of transfer began. • Today, the definition of transfer has widened: Research now studies the application of different areas of knowledge to new problems.

30. What are the Components of Transfer? • Surface cues. • Problem structure.

31. An Example from School Math... • Billy has three marbles and Alley has five marbles. How many do they have altogether? • Surface cues: Marbles, Billy and Alley. • Structure: 3 + 5= 8. • How do we help students to look past the surface features to the problem structure?

32. Different Types of Transfer I • Positive Transfer: Previous learning facilitates performance. • Moving from one column addition to two column addition. • Negative Transfer: Previous learning hinders performance. • Drawing familiar objects.

33. Different Types of Transfer II • Near Transfer- The problems share many of the same features. • Example: One column addition to two column addition. • Far Transfer - Different surface features, but the problems share some underlying similarities. • Example: Using addition to take an inventory.

35. Tumor Problem • An example of far positive transfer. • The two problems are analogous. • Both use the divide and conquer strategy. • Without hints 30% success rate on second problem. • 10% success rate working only the second problem.

36. How Does Transfer Work? • Schema theory. • Reasoning by analogy. • Schemata at work: Blind Date

37. Schema Theory • Schemata are organized networks of knowledge that you use to make sense of the world. • Smallest unit of meaning in thinking: Analogous to a concept. • Give coherence to information by influencing your: • Ability to pick useful strategies. • Ability to remember relevant facts that go with the current problem. dog

38. A Schema Fit for a Dog..

39. Activated Dog Schema

41. The Schema of an Expert • Larger schemas: More knowledge. • More integrated Knowledge: More connections within and between schemata. • Perhaps learning that is integrated with prior knowledge transfers more frequently.

42. Thinking by Analogy • Schema activation: “It’s like a puppy but it never gets any bigger?” • “It goes yip, yip, and never shuts up.” • One schema of knowledge (puppy) is mapped on to Chihuahua schema.

43. Problem features: Fortress Mined roads Streams of attacking troops that vary in size. Solution: Small groups of troops move along each road to converge on the fortress Problem features: Tumor Surrounding tissue Rays that can vary in intensity Solution: Weak rays are all focused on the tumor Dictator (base) Tumor (target) Analogous Problem Structures

44. Positive Transfer Remains Elusive • Studies finding low rates of transfer. • In everyday life it appears that we transfer knowledge all the time. • Typical experiments may be a difficult way to study successful transfer.

45. Transfer in the Classroom... • Recent research in classrooms has shown some evidence of transfer (Brenner, et al., 1997). • Connecting tables, graphs, and equations: multiple representations of content.

46. Can you write a rule that explains the shipping errors? Y=2x+1

47. A Classroom Based Study of Transfer. • The study involved two groups of students: • A treatment group receiving our experimental unit. • A comparison group that received their regular instruction. • The students were local middle school students taking their first pre-algebra class. • After 20 days of instruction the two groups were tested.

48. Transfer Results... • Students were able to transfer their understandings of tables, graphs and equations to a function word problem more frequently than the comparison group. • Interviews with the students over a year later indicated that they still remembered content from the lessons.

49. Conclusion: Will this Lecture Transfer? • You now... • Know the definition of transfer. • Know that it is recognized an important part of the national math and science standards. • Have worked some problems designed to demonstrate transfer. • Have considered new ideas about how problems might be related. (near / far)

50. Conclusion (cont...) • Can think about whether previous learning ultimately helps or hurts your efforts to solve a new problem. (positive/negative) • Have an understanding about the basic cognitive principles that underlie transfer. (schemata, analogical reasoning) • Have seen an example of some of the latest research in transfer.