Basics of Strategy From a Third Grade Class

11 10 2009
 
Last week my students and I were reviewing the strategy of the game of Nim in my third grade class.

Nim is a very simple game that can introduce big ideas of strategy.

The object of the game is to force your opponent to take the last marker. That is, the loser removes the last marker. Of course, the game could be played that the winner is the one who takes the last marker.

Construct three rows of markers: the top row has three markers, the center row has five markers and the bottom row has seven markers. The number of markers for each row and the number of rows is open to options. But, 3-5-7 Nim provides for a short game and yet complex enough for variety and analysis.

Players take turns removing as many markers as they like in one row only. Each player must remove at least one marker per turn. The word Nim probably comes from the Shakespearean word meaning “to take away” or “steal”.

This game can be played anywhere because one can use toothpicks, little rocks or play animals,marks on a chalkboard, or on a foggy window.

Nim is a game that has been played, in various forms, on at least four continents for at least four centuries. Like tic tac toe, it is a challenging game until one realizes that there is a correct way to play. In the case of tic tac toe, there is a correct way for both players and, if both players make the correct moves, the game must always end in a tie.

In the case of Nim, when one player makes the correct moves, he will always win. (Whether this is the player who goes first or the player who goes second depends on the variation of nim being played.)

Where the perfect strategy for Tic Tac Toe is discoverable by a bright child, discovery of the correct nim strategy takes a mathematical intuition of the highest order for one without mathematical experience.

After two months of playing one of my third grade students studied with his parents for a week to discover a winning strategy. He came to class and confidently won me in a game. He has now become a partner teacher in strategy for the class.

For some websites on Nim try:
http://www.eserc.stonybrook.edu/wise/HSfall2000/Nim.html
http://www.archimedes-lab.org/game_nim/nim.html
http://www.2020tech.com/fruit/

This gave a good introduction to the basics of strategy. Here I laid the foundation to three basic ways students approach games:

1. Superstitious Plans
According to the writer Raymond Lamont Brown: “Superstition is a belief, or system of beliefs, by which almost religious veneration is attached to things mostly secular; a parody of religious faith in which there is belief in an occult or magic connection.”

Another way to put it is that superstition is an irrational or nonscientific belief in the existence of certain powers operant in the world, with positive or ill (usually ill) effects. These are rituals or patterns of behavior that are believed to have some power to influence the outcome of the game.
What are some examples you have seen as we played the game?

2. Psychological Ploys
The art or practice of using tactical maneuvers to further one’s aims or better one’s position. The use in a sport or game of aggressive, often dubious tactics, such as psychological intimidation or disruption of concentration, to gain an advantage over one’s opponent. Here the concepts of Gamesmanship versus sportsmanship are introduced. Also mentioned are ways players try to psych-out your opponent. Psychological Ploys are the use of dubious (although not technically illegal) methods to win a game
What are some examples you have seen as we played the game?

3. Strategic Play
Strategy is a careful plan or method. Victory is completely dependent on your reasoning and pattern recognition skills, and completely independent of luck
What are some examples you have seen as we played the game?

But it is amazing how such a simple game can introduce the ways people approach life.





Performance Pictures of Bob Bishop

4 10 2009

These are pictures of Bob Bishop performing

in Taipei, Taiwan.

Teipei 103

Teipei 053

2008 010

This picture is Bob Bishop performing at the Special Olympics Banquet in Idaho.

tvmc4





The Joy and Art of Problem Solving

24 09 2009

Since……

 

Problems are inevitable and unavoidable.

 

They are the means by which we grow.  They are not necessarily “bad.”

 

There is no such thing as a problem without a gift in it.

 

Problem solving is one of the critical and central activities in one’s life.

 

Problems come in all shapes, sizes, varieties, and levels of difficulty.

 

Problems grow more complex each year.

 

Problem solving can be easier, more effective, and more fun if you have a flexible system for solving problems.

 

There is no substitute for experience.  If you want to become a better problem solver, you must practice, practice, practice.  Hence, the more problem solving you do, the better problem solver you become.

 

 

Some tools for solving problems…..

PROBLEM SOLVING:

A Student’s Guide

Rule 1 

If at all possible, avoid reading the problem.  Reading the problem only consumes time and causes confusion.

Rule 2

Extract the numbers from the problem in the order in which they appear. Pay no attention for numbers written in words.

Rule 3

If rule 2 yields three or more numbers, the best bet for getting the answer is adding them together.

Rule 4

If there are only two numbers which are approximately the same size, then subtraction should give the best results.

Rule 5

If there are only two numbers in the problem and one is much smaller than the other, then divide if it goes evenly-otherwise, multiply.

Rule 6

If the problem seems like it calls for a formula, pick a formula that has enough letters to use all the numbers given in the problem.

Rule 7

Never, never spend too much time solving problems.

This set of rules will get you through even the longest assignment in the minimum time with little or no thinking.

 

 

Tools That May Really Help

Problem Solving Tools You May Use 

1. Rephrasing:

Often a problem seems complex or hard to understand simply because the words used are complicated, vague, or confusing.  By rephrasing the problem in your own words, you can get it organized in your mind.  Put the problem in your own words until you feel comfortable with your understanding of the problem.

Try stating the goal in your own words and as completely as you possibly can.

 

2. Possibility listing:

One of the easiest and most effective ways to get control of a confused situation is simply to itemize the variables and possibilities involved.  This involves making a list of the key factors involved. In this case the further analysis of the puzzle can be transformed into a list of factors that make the puzzle a problem.

Try listing the variables and factors of the problem.

 

3. Identify sub goals:

When a problem is complex, breaking it down into sub problems and solving each part is helpful.  By analyzing the problem carefully and not being distracted by the first thing that comes to mind, you may be able to discover the one key factor that lies at the heart.

Try simplifying the problem or the puzzle by breaking it down into sub-problems and then solving the parts.

 4. Trial and error:

This is the weakest and often the most inefficient method.  It is randomly trying one possibility, then another, and then another. This method is also called guess and check. The correct solution is discovered by chance. This method is testing all the possibilities at random. (It is very probable you will use other methods instead of making a completely exhaustive search)

Try guessing and checking your solution

 

5.     Estimate, predict or project

            Get an idea what the solution would be close to. Predict the range of where the answer might be.

          Try estimating what the answer would be close to

 

  6.  Best first analysis:

 This searching strategy involves testing the most probable or most desirable (or promising) possibility first. This method can also be used on sub goals.  If the first method attempted fails to produce a solution the second best choice is tried.

Try the most desirable choice first.

 

 

7. Worst first analysis:

This searching strategy involves testing the least probable or desirable (or promising) possibility first. This method can also be used on sub goals.  If the first method attempted fails to produce a solution the second least desirable choice is tried.

Try the least desirable choice first.

 

 

8. Process of Elimination:

This method is organizing the possibilities by eliminating what does not work. This process may be used to solve sub goals and categorizing trial and error testing.

Try eliminating the possibilities that do not work.

 

 

9. Jump the Track:

Often problem solvers get stuck in a mental rut and do the same process over and over.  Stopping to reconsider the whole course of your attack on the problem may help.  Start again with a completely different approach or a different point of view. Enlarge the range of options to include unusual ones.

Try a totally different approach.

 

 

10. Look for patterns:

By examining the puzzle carefully, a pattern for arranging the pieces or in the solution may be observed. This may be patterns in shapes, color, size, process of steps or a hidden code.

Try looking for a hidden pattern.


 

11. Draw or use a diagram, table, or model:

Problems are often approached by sketching out the process on paper.  Often Athinking with a pencil@ helps clarify the thinking process.

Try looking using a pencil to sketch or keep track of your thinking process.

 

12. Work backwards:

When the goal is clear, you can begin there and work backwards.  Taking a completed puzzle apart piece by piece, or working a maze backwards or completing describing the finished puzzle may help in the process.

Try working backwards by understanding what the solved problem must look like.       

 

13. Simplify

Do a simpler problem of the same kind to understand the method.  Apply that method to the present problem.

          Try doing a simpler problem of the same kind and apply that method.

 

14. Logic

          When there are steps that depend on each other, decide which step goes first. After that, decide the steps that follow in a reasonable order.  Discover how the steps fit together with phrases such as: If I do this, then this will happen.

          Try breaking the steps of problem into a reasonable order.

 

15. Act it Out

                                                                                                                                                                                        

          Often it helps to play act the problem by demonstrating the situation physically.                                                             

Try play acting the problem by demonstrating the situation.

 

16.  Create an equation                                                

Practice some algebra by using letters as variables to represent unknown quantities. Solving the equation leads to the solution of the problem

Try using algebra as a mathematical “shortcut”.





Math Magician helps Students to Want to Study Numbers USA

17 09 2009

By Courtney Cobb – Journal Writer

math

POCATELLO, IDAHO – A new spin has been put on mathematics as Tendoy Elementary students use some magic to study various math concepts.

Bob Bishop, the Math Magician, has delighted students in kindergarten through sixth grade and teachers with his magic skills and math abilities over the past week.

“Math is so necessary in life,” he said. “It’s not just making math fun, but it’s also trying to attach some sense of understanding for students.”

Fifth grade teacher Vicki Reeder’s class had the opportunity to spend some time with Bishop while working on problem solving skills.

Students worked with calculators, the box of magic, learned how to do multiplication tables with their fingers, played a game called fast and loose and other activities.

During a game of fast and loose, Bishop produced a single chain and proceeded to fold it into a series of loops.

Students were asked to pick a loop and place their finger inside it. If they had guessed correctly the loop would stay around their finger. However, if they guessed incorrectly, the loop would slip away.

“You will win if you know mathematics, but you’ll lose if you don’t,” Bishop said.

Students learned how to follow the loops and determine the correct place to put their fingers.

Bishop has been performing for students and other audiences for 10 years and says he continually teaches students and teachers how math can be fun.

He said many students work with arithmetic but don’t fully understand problem solving skills.

With the help of a little magic, students are forced to observe the environment around them for any changes and think about possible outcomes.

“Generally students don’t really care to do math because it’s not fun,” Bishop said. “By making it interesting and proving to them they can do it, it helps to raise their self-esteem and interest level in math.”

Bishop will perform along with Tendoy Elementary students at 6:30 p.m. today for a Math Night.

Fifth grade student Quinci Shelley is acting as Bishop’s assistant during the show and said she can’t wait to perform for other students.

“I think it’s cool and it’s a good opportunity for us,” she said. “Some people don’t like math, but when they see this show it sparks their interest.”

Fifth grade student Brant Leo will lead the audience in applause, but said working with Bishop has been great because he’s learned new things.

“He’s helping students to improve their math by using cool tricks,” he said.

Bishop also worked with teachers after school and gave them various activities they can do with students in their classrooms.

“By making math fun, students will learn to enjoy it more and it will give them a sense of pride as they figure out difficult problems,” he said.