How to Teach Puzzle-based Learning

1. How to Teach Puzzle-based Learning Zbigniew Michalewicz 1

2. Outline of the talk • Getting started • Icebreakers • Effective teaching approaches • Understanding the problem • Some techniques (e.g. simplification, performing a Gedanken: What if? So what?) • Summary

3. A book The talk is based on a new book (Springer, August 2014)

4. Puzzle-based Learning Puzzle-based Learning is an approach to develop thinking skills, mental stamina and perseverance at solving problems. The approach is based on unstructured, generally context-free (i.e. does not require domain knowledge) and usually entertaining problems, better known as puzzles…

5. General perspective The ultimate goal of Puzzle-based Learning is to lay a foundation for students to be effective problem solvers in the real world. At the highest level, problem solving in the real world calls into play three categories of skills:  (1) dealing with the vagaries of uncertain and changing conditions; (2) harnessing domain specific knowledge and methods; (3) critical thinking and applying general problem solving strategies.  These three skill categories are captured in the three forms of learning – project, problem, puzzle-based: 

6. General perspective

7. Outline of the talk • Getting started • Icebreakers • Effective teaching approaches • Understanding the problem • Some techniques (e.g. simplification, performing a Gedanken: What if? So what?) • Summary

8. Getting started Two key components: • The instructor • Motivated students

9. Getting started Two key components: • The instructor • Motivated students

10. Logicians Annual International Conference on Logic: 31 participants one master it is necessary to test, whether all participants are logicians each participant gets from a master a dot of some colour on his forehead the master assures one participant (who expressed a concern) that everyone will be able to guess the colour of his/her dot at some stage (when a bell rings…)

11. Logicians This is what happened. When the bell ring: 1st time: 4 people left 2nd time: all with red dots left 3rd time: no one left 4th time: at least one left shortly afterwards: a participant, who expressed concern left together with his sister; both had dots of different colour… At that time there were still some people left… How many times did the bell ring?

12. Getting started Two key components: • The instructor • Motivated students – Real-world connections – A few examples on ‘thinking’ in wrong directions…

13. Connection with “real-world” When technology is changing beneath your feet daily, there is not much point in hiring for a specific, soon-to-be-obsolete set of skills. You have to try to hire for general problem-solving capacity, however difficult that may be. An interview puzzle is a filter to prevent bad hires.

14. Connection with “real-world” Both the solver of a puzzle and a technical innovator must be able to identify essential elements in a situation that is initially ill-defined. It is rarely clear what type of reasoning is required or what the precise limits of the problem are.

15. A warm up Below is a schematic of a printed circuit board (PCB). The problem is to either connect each number to its prime (1 to 1’; 2 to 2’ etc) by a continuous line or to provide an argument that it is impossible. 1 2 3 4 3’ 4’ 1’ 2’

16. A warm up Below is a schematic of a printed circuit board (PCB). The problem is to either connect each number to its prime (1 to 1’; 2 to 2’ etc) by a continuous line or to provide an argument that it is impossible. 3 4 1 How about if we renumber as below? 2 1’ 2’ 3’ 4’

17. A warm up The key is to first connect … 1 2 3 4 3’ 4’ 1’ 2’

18. A warm up … and the rest is easy…. 1 2 3 4 3’ 4’ 1’ 2’

19. Unusual puzzle…

20. Unusual puzzle…

21. Unusual puzzle…

22. Unusual puzzle… After cutting off the right-bottom sub-square, divide the remaining shape into 4 identical pieces…

23. Unusual puzzle…

24. Unusual puzzle… Divide the bottom-right sub-square into 5 identical pieces…

25. Unusual puzzle…

26. Pattern recognition Our ability to recognize patterns is of utmost importance. If we can identify a pattern, then we can build a model to find a solution (e.g. to find the next occurrence of a symbol, number, action, or event). Marilyn Burns, in her book I Hate Mathematics, wrote: “The password of mathematics is pattern.” Indeed, in many branches of mathematics we search for patterns which allow some generalizations. But we search for patterns everywhere; we even recognize patterns in words: 26

27. Pattern recognition Aoccdrnig to a rscheearch at Cmabrigde Uinervtisy, it deosn’t mttaer in waht oredr the ltteers in a wrod are, the olny iprmoetnt tihng is taht the frist and lsat ltteer be at the rghit pclae. The rset can be a toatl mses and you can sitll raed it wouthit porbelm. Tihs is bcuseae the huamn mnid deos not raed ervey lteter by istlef, but the wrod as a wlohe. 27

28. Pattern recognition Try to memorize the following coding of numbers 1 – 9: 1 2 3 4 5 6 7 8 9

29. Pattern recognition 1 2 3 4 5 6 7 8 9 What is the coded version of the number 3875 ? 29

30. 2 3 • 5 6 7 8 9 Pattern recognition The “pattern”: 1 2 3 4 5 6 7 8 9

31. Pattern recognition 1 2 3 4 5 6 7 8 9 What is the coded version of the number 3875 ?

32. Pattern recognition • the first symbol is “different” as the only one with red border • the second symbol is “different” as the only one without green shade • the third symbol is “different” as the only one which is not square • the fourth symbol is “different” as the only one with red dot • the fifth symbol is most different as the only one which is not unique with respect to some property…

33. An observation For the following 23 design activities, which do you view as the six most important?

34. An observation For the following 23 design activities, which do you view as the six most important?

35. An observation For the following 23 design activities, which do you view as the six most important?

36. Outline of the talk • Getting started • Icebreakers • Effective teaching approaches • Understanding the problem • Some techniques (e.g. simplification, performing a Gedanken: What if? So what?) • Summary

37. Lottery Take your business card (or piece of paper with your name) and write one natural number (i.e. 1, 2, 3, …). The winner is … … the person who writes the smallest number… …which is unique… Good luck!

38. A quiz Here is a quiz that can be used on the first day of class. Consider handling it out before any instructions. As soon as the students are in their seats, your first words to them can be: “Here is the first quiz; you have 5 minutes…” Here it is:

39. A quiz

40. A game There are n dots on a plane(flat surface). There are two players, A and B, who move alternatively; A moves first. The rules of the game are the same for both players: at each move they can connect two points, but they cannot connect points which were already directly connected to each other or connect a point with itself. 40

41. A game The winner is the one who makes the dots connected (i.e. there is a path between any two dots, however, not every two dots have to be connected directly). What is the winning strategy for player A, if such exists? 41

42. Outline of the talk • Getting started • Icebreakers • Effective teaching approaches • Understanding the problem • Some techniques (e.g. simplification, performing a Gedanken: What if? So what?) • Summary

43. Effective teaching approaches Key questions: What knowledge and skills should students learn? How can I facilitate their learning? How do I determine how well they have learned via formative and summative feedback?

44. Effective teaching approaches • show the student how you solve a problem (think aloud exercise) • get the student to articulate his/her thought process • present a variety of puzzle types • give frequent homework assignments • try “flipping the classroom” approach • match the tone and your language to the puzzle • select carefully puzzles for use in the classroom (purpose, groups, hints, class discussion) • consider online activities

45. Assignments What is this assignment supposed to achieve in terms of learning outcomes? What level of knowledge do your students have in this area? What level of knowledge do you wish your students to have after this assignment? How much time do you have for this? Have your students seen a similar or related puzzle before? Have you demonstrated this in class? Do the students have a reason to do this assignment? Do the students understand the value of this assignment? Do you understand how to do this assignment? Have you tested it? Does the assignment require additional equipment or resources? Do you need to provide a guide? Do you need to provide a rubric? Do you have a list of hints? What is the impact if a student cannot solve any of the problems?

46. Effective teaching approaches • assessment • grading (A, B vs. 8, 7) • uniform rubric • rewarding effort and outcome • feedback is essential • increasing and maintaining confidence • peer teaching • final exam questions (note the difference between in-class puzzles, homework, exams, great-challenges)

47. Outline of the talk • Getting started • Icebreakers • Effective teaching approaches • Understanding the problem • Some techniques (e.g. simplification, performing a Gedanken: What if? So what?) • Summary

48. Understanding the problem A farmer has: 20 pigs, 40 cows, and 60 horses. How many horses does he have, if he calls the cows horses?

49. Understanding the problem A farmer has: 20 pigs, 40 cows, and 60 horses. How many horses does he have, if he calls the cows horses? Answer: The farmer has 60 horses…

50. Critical Thinking “Arguments for banning guns are mostly myths, and what we need now is not more laws, but more law enforcement. One myth is that most murderers are ordinary, law-abiding citizens who kill a relative or acquaintance in a moment of anger only because a gun was available. In fact, every study of homicide shows the overwhelming majority of murderers are career criminals, people with lifelong histories of violence. The typical murderer has a prior criminal history averaging at least six years, with four major felony arrests.”