Puzzles and Playing: Power Tools for Mathematical Engagement and Thinking

Transcription

1 Puzzles and Playing: Power Tools for Mathematical Engagement and Thinking Eden Badertscher, Ph.D. SMI 2018 June 25, 2018 This material is based upon work supported by the National Science Foundation under Grant No. DRL Any opinions, findings, and conclusions or recommendations expressed in this material are those of the presenter and do not necessarily reflect the views of the NSF.

2 Typical Role of Puzzles Puzzles and games in mathematics are considered candy or a reward. Why? Puzzles and games are typically provided when other mathematical work has been completed or achieved. Why? What if we thought about puzzles and games as the foundation of work rather than the prize?

3 Perspectives from Mathematical Anthropology and Early Childhood CHALLENGING OUR PERCEPTIONS OF PLAY

4 Play- A Human Cornerstone (Paley) Play, that most ordinary of human functions, as natural as crawling, walking, and running. Without instruction, theses skills flourish. No one is taught to walk or to act out a fantasy [though this can be squashed easily]. The patterns and incentives arise from within. Children pay no attention to the way they walk; they would stumble and fall if they continually observed themselves moving along. The effect would be worse if they watched everyone else as well (Paley, 1990, p. 9).

5 Alan Bishop on Playing (1988, pp ) All cultures play and what is more important, they take their play very seriously. By that I mean that it is essential NOT to treat play as a relatively unimportant aspect of cultural life.

6 Alan Bishop- continued Clearly playing is a form of social activity which is different in character from any other kind of social intercourse playing takes place in the context of a game, and people become players. The real/not real boundary is well established and players can only play with players IF everyone agrees not to behave normally.

7 Alan Bishop- continued Could these characteristics be at the root of hypothetical thinking? Could playing represent the first stage of distancing oneself from reality in order to reflect on and perhaps to imagine modifying that reality? Certainly, Vygotsky (1978) argued that the influence of play on a child s development is enormous (p. 96) in that action and meaning can become separated and abstract thinking can thereby begin. (Bishop, p. 43)

8 Alan Bishop- continued Because play is essentially unserious in its goals, its performance becomes its own reward. So string figures and other play-forms become of interest in themselves and the roots of artistic and aesthetic appreciation can again be seen. The pleasures of aesthetic appreciation must surely account for the popularity and longevity of many games and play activities in all cultural groups.

9 Alan Bishop- continued Once the play-form itself becomes the focus, and a game develops, then the rules, procedures, tasks and criteria become formalized and ritualized. They are also products of playing. Games are often valued by mathematicians because of their rulegoverned behavior which it is said, is like mathematics itself. I think it is not too difficult to imagine how the rule-governed criteria of mathematics have developed from the pleasures and satisfactions of rule-governed behavior in games.

10 Alan Bishop- continued And there is no doubt that people everywhere, adults and children, enjoy participating in rulebound behavior of games, perhaps because they are, unlike reality, social settings where the players all know the rules and agree to play by those rules [and improving is expected].

11 Play, Relevance and Thinking (Paley) Eli and Edward, using fantasy play, are able to visualize a concept; the child finds the natural method for concentration and continuity and satisfies the intuitive belief in hidden meanings. This is why play feels so good. Discovering and using the essence of any part of ourselves is the most euphoric experience of all. It opens blocked passages and establishes new routes. Any approach to language and thought that eliminates dramatic play, and its underlying theme of friendship and safety lost and found, ignores the greatest incentive to the creative process. Play and its necessary core of storytelling are the primary realities in the preschool and kindergarten, and they may well be the prototypes for imaginative endeavors throughout our lives. For younger students, however, it is not too much to claim that play contains the only set of circumstances understandable from beginning to end (Paley, 1990, p. 6).

12 Paley Revisited Play, that most ordinary of human functions, as natural as crawling, walking, and running. Without instruction, theses skills flourish. No one is taught to walk or to act out a fantasy [though this can be squashed easily]. The patterns and incentives arise from within. Children pay no attention to the way they walk; they would stumble and fall if they continually observed themselves moving along. The effect would be worse if they watched everyone else as well (Paley, 1990, p. 9).

13 Themes from Descriptions of Play Instruction unnecessary we are all capable Internal motivation and satisfaction Universal to all cultures Separation from real & idealization/models Connection to aesthetics Agreements for how to interact (discussed explicitly and agreed upon) in ways that deviate from the norm. Appreciation of rule-bound behavior Continuity of understanding from start to finish Creativity, inventing, abstracting

14 TWO ALGEBRAIC PUZZLES: FROGS AND MOBILES

15 Let s consider puzzles From perspectives of: Content Standards for Mathematical Practice Access Connection to Play

16 Leaping Frogs

17 Leaping Frogs

18 Play the Game- 10 minutes Can you implement a strategy so that you are always successful? What about if there are 4 frogs on each side? What about if there are 5 frogs on each side? For any number of frogs is there a solution?

19 Can you find a strategy that is always successful? Keep track of and analyze your solution for patterns:

20 Can you find a strategy that is always successful? Hypothesize a generalization and generate a solution to test:

21 Can you find a strategy that is always successful? Pattern for color of move: Patterns for type of move: ssssssssss + 8 jjjjjjjjjj How many moves would 6 counters on each side require? 10 counters? 100 counters? How can you predict the 42 nd move when using 6 coins on each side?

22 Can you find a strategy that is always successful? Can you prove there will always be a solution to any number of equal counters on each side?

23 Low Threshold High Ceiling It turns out that you can calculate the fewest number of moves for a puzzle with n counters/frogs with the formula n 2 + 2n. Why does a quadratic make sense in this situation? How does the solution change when there is one more counter on one side than the other.

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26 Approaches to Systems Building Substitution = = 4444 SS = 11 Building Linear Combination = = = 4444

27 Play the Game 10 minutes Rules. Each person has their own puzzle to solve, though shapes are equivalent across puzzles in each round. You cannot solve a shape on someone else s puzzle. If you solve a shape, you need to justify your solution and get agreement from the group. You cannot assign a value to a shape on your puzzle until it has received agreement.

28 Why Puzzles? Accessibility Inherent Interest (and some puzzles drive this) Building logic and familiarity with fundamental mathematical ideas without invoking mathematics Expectation of rules and constraints in games allow sense making of constraints of mathematics (e.g. in systems can t solve without adequate clues) in expected ways Can leverage desire to get better at games to build deeper thinking about mathematics

29 Thank you! For questions or comments, please Eden: