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.





What We Can Learn From Games

11 10 2009

by Ben Bishop

I came across an interesting article in the Idaho Statesman two days ago. It was a great article on teaching math to students and one of the final points was to play games that emphasized math like Monopoly or Risk. I am glad that any game can be used to teach concepts; however that seems to be an oversimplification of the potential of a game. If that is all a game is (a conceptual teaching tool) then the higher levels of Bloom’s famous taxonomy are not being reached at all. I can see games like Snakes and Ladders being used for this purpose (after all that is basically a counting game) but doing this is a higher grade class like 6th grade would turn me off of math and games in general.Games are expressions of the struggles we face, a miniature version of reality without the painful loss. If they are used like a set of flash cards then the purpose is lost. Yes, I’ll learn about probability when I roll the dice, I’ll learn about trivia when I play a question game, etc… It’s the participation of play that teaches me not just the memorization of facts and figures.Here’s a link to an article that says about the same thing but in a more logical manner:
http://www.thegamesjournal.com/articles/Ethics5.shtml

Till next time,
Ben Bishop





Lessons from Strategy Games

11 10 2009

I was just interviewed for a local newspaper concerning what we can learn from games. I was quoted saying that students may not naturally learn from games. We often say how helpful games are in teaching life lessons, math, strategy or whatever. As a teacher, I have realized that unless a teacher is explicit, students do not transfer their knowledge to other areas. Here are some ideas about drawing life lessons from games.

Here are some guidelines for guidelines:

They must be easily repeatable, quotable, meaningful, and memorable

They must be a generalization to all or a specified group

They are commands or they implies what we should do

They are short and concisely worded

Examples of Life Lessons written by my students….

-When you don’t understand the rules, you cannot play the game of life successfully.

-Be willing to learn new things so you are more equipped to make better choices and decisions.

-Commit to paying attention and reflecting upon the actions and behaviors of those around you.

-Your actions determine your outcomes.

-Your life experience is made up of the choices you make and the outcomes that accompany them each and every day.

-If you hope to have a winning life strategy you have to be honest about where your life is right now.

-Life rewards action.

-You must realize that your plans will alter and sometimes change along the way. Winners adapt to these new developments.

-A strategy requires courage, commitment and energy in order to succeed.

-When you know your goals, you will recognize which choices support them and which do not.

-Study and dissect your mistakes so you can avoid repeating them.

-Study and analyze your successes so you can repeat the behavior that has brought you positive results.

-Losers just make it up as they go along

Here is a quote that sums up my thoughts………

There are one-story intellects,
two-story intellects,
and three-story intellects with skylights.
All fact collectors who have
no aim beyond their facts
are one-story men.
Two-story men compare, reason, generalize, using the labor
of fact collectors as their own.
Three-story men idealize,
imagine, predict-
their best illumination comes
from above the skylight.
–Oliver Wendell Holmes





Life Lessons from Chess

4 10 2009
 The following is an essay by Benjamin Franklin on chess. 

THE GAME OF CHESS is not merely an idle amusement; several very valuable qualities of the mind, useful in the course of human life, are to be acquired and strengthened by it, so as to become habits ready on all occasions; for life is a kind of Chess, in which we have often points to gain, and competitors or adversaries to contend with, and in which there is a vast variety of good and ill events, that are, in some degree, the effect of prudence, or the want of it. By playing at Chess then, we may learn:

 

1st, Foresight, which looks a little into futurity, and considers the consequences that may attend an action; for it is continually occurring to the player, “If I move this Piece, what will be the advantage or disadvantage of my new situation? What use can my adversary make of it to annoy me? What other moves can I make to support it, and to defend myself from his attacks?”

 

2d, Circumspection, which surveys the whole Chess-board, or scene of action: – the relation of the several Pieces, and their situations; the dangers they are repeatedly exposed to; the several possibilities of their aiding each other; the probabilities that the adversary may make this or that move, and attack this or that Piece; and what different means can be used to avoid his stroke, or turn its consequences against him.

 

 

3d, Caution, not to make our moves too hastily. This habit is best acquired by observing strictly the laws of the game; such as, if you touch a Piece, you must move it somewhere; if you set it down, you must let it stand.

 

 

Therefore, it would be the better way to observe these rules, as the game becomes thereby more the image of human life, and particularly of war; in which if you have incautiously put yourself into a bad and dangerous position, you cannot obtain your enemy’s leave to withdraw your troops, and place them more securely, but you must abide by all the consequences of your rashness.

 

 

And lastly, we learn by Chess the habit of not being discouraged by present bad appearances in the state of our affairs; the habit of hoping for a favorable chance, and that of persevering in the search of resources. The game is so full of events, there is such a variety of turns in it, the fortune of it is so subject to vicissitudes, and one so frequently, after contemplation, discovers the means of extricating one’s self from a supposed insurmountable difficulty, that one is encouraged to continue the contest to the last, in hopes of victory from our skill; or, at least, from the negligence of our adversary: and whoever considers, what in Chess he often sees instances of, that success is apt to produce presumption and its consequent inattention, by which more is afterwards lost than was gained by the preceding advantage, while misfortunes produce more care and attention, by which the loss may be recovered, will learn not to be too much discouraged by any present successes of his adversary, nor to despair of final good fortune upon every little check he receives in the pursuit of it.

That we may therefore, be induced more frequently to choose this beneficial amusement in preference to others, which are not attended with the same advantages, every circumstance that may increase the pleasure of it should be regarded; and every action or word that is unfair, disrespectful, or that in any way may give uneasiness, should be avoided, as contrary to the immediate intention of both the parties, which is, to pass the time agreeable.

1st, Therefore, if it is agreed to play according to the strict rules, then those rules are to be strictly observed by both parties; and should not be insisted upon for one side, while deviated from by the other: for this is not equitable.

2d, If it is agreed not to observe the rules exactly, but one party demands indulgences, he should then be as willing to allow them to the other.

3d, No false move should ever be made to extricate yourself out of a difficulty, or to gain an advantage; for there can be no pleasure in playing with a man once detected in such unfair practice.

4th, If your adversary is long in playing, you ought not to hurry him, or express any uneasiness at his delay; not even by looking at your watch, or taking up a book to read: you should not sing, nor whistle, nor make a tapping with your feet on the floor, or with your fingers on the table, nor do anything that may distract his attention: for all these displease, and they do not prove your skill in playing, but your craftiness and your rudeness.

5th, You ought not to endeavor to amuse and deceive your adversary by pretending to have made bad moves; and saying you have now lost the game, in order to make him secure and careless, and inattentive to your schemes; for this is fraud and deceit, not skill in the game of Chess.

6th, You must not, when you have gained a victory, use any triumphing or insulting expressions, nor show too much of the pleasure you feel; but endeavor to console your adversary, and make him less dissatisfied with himself by every kind and civil expression that may be used with truth; such as, you understand the game better than I, but you are a little inattentive, or, you play too fast; or, you had the best of the game, but something happened to divert your thoughts, and that turned it in my favor.

7th, If you are a spectator, while others play, observe the most perfect silence: for if you give advice, you offend both the parties: him against whom you give it, because it may cause him to lose the game: him in whose favor you give it, because, though it be good, and he follow it, he loses the pleasure he might have had, if you had permitted him to think till it occurred to himself. Even after a move or moves, you must not, by replacing the Pieces, show how they might have been placed better; for that displeases, and might occasion disputes or doubts about their true situation.

 

All talking to the players lessens or diverts their attention; and is, therefore, unpleasing; nor should you give the least hint to either party, by any kind of noise or motion; if you do, you are unworthy to be a spectator.

If you desire to exercise or show your judgment, do it in playing your own game, when you have an opportunity, not in criticizing or meddling with, or counseling the play of others.

 

Lastly, If the game is not to be played rigorously, according to the rules before mentioned, then moderate your desire of victory over your adversary, and be pleased with one over yourself.

 

 

Snatch not eagerly at every advantage offered by his unskilfulness or inattention; but point out to him kindly, that by such a move he places or leaves a Piece en prise unsupported; that by another, he will put his King into a dangerous situation, &c.

 

By this general civility (so opposite to the unfairness before forbidden) you may happen indeed to lose the game; but you will win what is better, his esteem, his respect, and his affection; together with the silent approbation and the good will of the spectators.

When a vanquished player is guilty of an untruth to cover his disgrace, as “I have not played so long, – his method of opening the game confused me, – the men were of an unusual size,” &c all such apologies, (to call them no worse) must lower him in a wise person’s eyes, both as a man and a Chess-player; and who will not suspect that he who shelters himself under such untruths in trifling matters, is no very sturdy moralist in things of greater consequence, where his fame and honor are at stake? A man of proper pride would scorn to account for his being beaten by one of these excuses, even were it true; because they have all so much the appearance, at the moment, of being untrue.

 

 Author: Benjamin Franklin




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”.





The Social and Emotional Needs of Gifted Students

19 09 2009

Often teachers think that gifted students are those who just need more intellectual stimulation. The phase “they can take care of themselves” is often heard. There are other needs that must be addressed to serve the needs of the gifted students.  Here are just a few…………….

Perfectionism: The ability to see how one might ideally perform, combined with emotional intensity leads many gifted children to unrealistically high expectations of themselves. 

 

Underachievement: This is the discrepancy between potential and performance or ability and achievement. When a gifted student is not working up to his or her potential this is called underachievement.

 

Avoidance of risk taking: In the same way the gifted see the possibilities, they also see potential problems in undertaking those activities.  Avoidance of potential problems can mean avoidance of risk-taking, and may result in underachievement.

 

Uneven development: Motor skills, especially fine-motor, often lag behind cognitive conceptual abilities.  These children may see in their “minds eye” what they want to do, construct, or draw. However, motor skills do not allow them to achieve the goal.  Intense frustration and emotional outbursts may result.

 

Multi-potentiality: Gifted children often have advance capabilities and may be involved in diverse activities to an almost frantic degree.  Though seldom a problem for the child, this may create problems for the family, as well as quandaries when decisions must be about career selection.

 

Peer relationships:  Gifted students find that they often need both social and intellectual peers and they need to develop relationships with both.

 

Excessive self-criticism: The ability to see possibilities and alternatives may imply that youngsters see idealistic images of what they might be, and simultaneously berate themselves because they see how they are falling short of an ideal.

 

Emotional intensity and stress: Because of the areas stated above and the uneven coping abilities, gifted students may feel deeper and may experience intense stress.

 

Social Skills: Often gifted students put much emphasis on the advanced thought process to the neglect of social skills that seem to come naturally to others.  These areas could be listening skills, communication skills, and friendship skills.





Motivation Quotes from Leonardo Da Vinci

19 09 2009

As a well-spent day brings happy sleep, so life well used brings happy death.

Iron rusts from disuse; water loses its purity from stagnation and in cold weather becomes frozen; even so does inaction sap the vigor of the mind.

Patience serves as a protection against wrongs as clothes do against cold. For if you put on more clothes as the cold increases, it will have no power to hurt you. So in like manner you must grow in patience when you meet with great wrongs, and they will then be powerless to vex your mind.

You do ill if you praise, but worse if you criticize, what you do not understand.

Simplicity is the ultimate sophistication.

Obstacles cannot crush me. Every obstacle yields to stern resolve. He who is fixed to a star does not change his mind.

Every now and then go away, have a little relaxation, for when you come back to your work your judgment will be surer since to remain constantly at work will cause you to lose power of judgment. Go some distance away because then the work appears smaller, and more of it can be taken in at a glance, and a lack of harmony or portion is more readily seen.

For once you have tasted flight you will walk the earth with your eyes turned skywards, for there you have been and there you will long to return.

He who loves practice without theory is like the sailor who boards ship without a rudder and compass and never knows where he may cast.

I have been impressed with the urgency of doing. Knowing is not enough; we must apply. Being willing is not enough; we must do.

I have offended God and mankind because my work didn’t reach the quality it should have.

I love those who can smile in trouble, who can gather strength from distress, and grow brave by reflection. ‘Tis the business of little minds to shrink, but they whose heart is firm, and whose conscience approves their conduct, will pursue their principles unto death.

It had long since come to my attention that people of accomplishment rarely sat back and let things happen to them. They went out and happened to things.

Life is pretty simple: You do some stuff. Most fails. Some works. You do more of what works. If it works big, others quickly copy it. Then you do something else. The trick is the doing something else.

The noblest pleasure is the joy of understanding.

Where the spirit does not work with the hand, there is no art.

Who sows virtue reaps honor.

Anyone who conducts an argument by appealing to authority is not using his intelligence; he is just using his memory.   

Study the science of art and the art of science.

Learn how to see and remember that everything is connected to everything else.

Good students naturally thirst after knowledge.

There are three classes of people.  Those who see: those who see when they are shown: those who do not see.

Poor is the pupil who does not surpass his teacher.