Playing
Imagine reading a playful introduction to learning about prime numbers...
Pretend that you are a chemist in a world made of an infinite number of chemical elements. You are trying to figure out the list of elements making up a very large and complicated molecule. You figure that 99% of your molecule’s body doesn't bound with any other molecules, only its surface is reactive, therefore you are left in the dark in regards to the nature of most of your subject. However, you also know that potentially it is made up of a large variety of elements bound together in a twisted maze. You may also find that your molecule is made up of some elements yet unknown to science. Finding a new element would be an amazing discovery! You would have to unspun the tangle starting from its surface, moving inside little by little. Your tools in this process are elements that may or may not split your molecule into smaller, identical structures. How can you tease apart this puzzle? Now, just for fun, lets imagine that your molecule is called 949041. How would you start undoing it? Could it be that 949041 is a single large element (remember you are in a world abounding with an endless variety of them !)? What to do, how to proceed? You would either have to have an unwavering resolve trying to use each known element one by one in several repeated cycles, in the hope that you stumble upon one that unties your knot, but this would be a dull process. Could you stay endlessly motivated? Alternatively, you’d come up with an imaginative solution to this problem. Keep in mind that solutions abound, the only limit is your imagination! Stay within the rules of abstract reality, but stretch them as far as you and they can take it! Play on!
The concept of prime numbers is so simple that playfully introducing them on their own terms is quite impossible; by way of an analogy this introduction becomes easier. The fun involved in primes is not understanding what they are, but perceiving patterns that can be used to find them. The above exercise is meant for high school students who already know what prime numbers are; the exercise asks them to improvise ways of distinguishing primes from composites. The analogy between molecules and composite numbers, elements and primes, chemistry and mathematics, is meant to play on the students' imagination, motivating them to think outside of number theory when dealing with primes. This is a very challenging, open ended activity that calls the students to experiment with any solution technique that they are capable to envision and then put in action. Thus the activity is well suited to the application of Root-Bernstein`s thirteen cognitive tools. This activity doesn't constrain students to specific modes of reasoning, nor to specific representations of their answers that would be foreign to their idiosyncrasies. Relieved of the usual restrictions they can be inventive and playful.
What’s playing? Play involves three basic elements: the player, the object (something to manipulate, virtual or real), rules/constraints. The player is who plays; the player plays with objects; rules/constraints apply to the handling of the objects. Play’s purpose is the player’s amusement. Playing a game either has an objective or it is open ended and doesn’t have any rules and it’s only restricted by physical limitations. For instance, a player can juggle balls and those will fall, this is a rule imposed by gravitation; the player is restricted to remain within these constraints. The player derives joy from playing through manipulating the object to the fullest extent that the rules/constraints and the player’s creativity allow. The more creative the player is the more different configurations/variations/combinations/states of the object he/she can achieve while remaining within the boundaries set by the rules/constraints. Masterfully playing a musical instrument, driving a car to vehicle's and the driver’s limits, or skillfully surfing the ocean's waves, all represent play. In my work on mathematical problem solving primes were the objects of play and rules of arithmetic guided/constrained my play. I was as masterful in perceiving, patterning, abstracting, and building a model as my abilities permitted me to be. Most importantly I had fun during my playful work with primes and I have discovered new ways of looking at primes by playing with them. Working with primes involves discovery, it's a pathless journey. I chose to work on this topic at the first place, because its open endedness presents limitless possibilities for innovation and play.
Nevertheless more strictly defined mathematical problems, those that have stricter conditions and better defined unknowns, can also be solved through play. More on this in the next Module...
Image source: www.nature.com
Visit the next module!
Imagine reading a playful introduction to learning about prime numbers...
Pretend that you are a chemist in a world made of an infinite number of chemical elements. You are trying to figure out the list of elements making up a very large and complicated molecule. You figure that 99% of your molecule’s body doesn't bound with any other molecules, only its surface is reactive, therefore you are left in the dark in regards to the nature of most of your subject. However, you also know that potentially it is made up of a large variety of elements bound together in a twisted maze. You may also find that your molecule is made up of some elements yet unknown to science. Finding a new element would be an amazing discovery! You would have to unspun the tangle starting from its surface, moving inside little by little. Your tools in this process are elements that may or may not split your molecule into smaller, identical structures. How can you tease apart this puzzle? Now, just for fun, lets imagine that your molecule is called 949041. How would you start undoing it? Could it be that 949041 is a single large element (remember you are in a world abounding with an endless variety of them !)? What to do, how to proceed? You would either have to have an unwavering resolve trying to use each known element one by one in several repeated cycles, in the hope that you stumble upon one that unties your knot, but this would be a dull process. Could you stay endlessly motivated? Alternatively, you’d come up with an imaginative solution to this problem. Keep in mind that solutions abound, the only limit is your imagination! Stay within the rules of abstract reality, but stretch them as far as you and they can take it! Play on!
The concept of prime numbers is so simple that playfully introducing them on their own terms is quite impossible; by way of an analogy this introduction becomes easier. The fun involved in primes is not understanding what they are, but perceiving patterns that can be used to find them. The above exercise is meant for high school students who already know what prime numbers are; the exercise asks them to improvise ways of distinguishing primes from composites. The analogy between molecules and composite numbers, elements and primes, chemistry and mathematics, is meant to play on the students' imagination, motivating them to think outside of number theory when dealing with primes. This is a very challenging, open ended activity that calls the students to experiment with any solution technique that they are capable to envision and then put in action. Thus the activity is well suited to the application of Root-Bernstein`s thirteen cognitive tools. This activity doesn't constrain students to specific modes of reasoning, nor to specific representations of their answers that would be foreign to their idiosyncrasies. Relieved of the usual restrictions they can be inventive and playful.
What’s playing? Play involves three basic elements: the player, the object (something to manipulate, virtual or real), rules/constraints. The player is who plays; the player plays with objects; rules/constraints apply to the handling of the objects. Play’s purpose is the player’s amusement. Playing a game either has an objective or it is open ended and doesn’t have any rules and it’s only restricted by physical limitations. For instance, a player can juggle balls and those will fall, this is a rule imposed by gravitation; the player is restricted to remain within these constraints. The player derives joy from playing through manipulating the object to the fullest extent that the rules/constraints and the player’s creativity allow. The more creative the player is the more different configurations/variations/combinations/states of the object he/she can achieve while remaining within the boundaries set by the rules/constraints. Masterfully playing a musical instrument, driving a car to vehicle's and the driver’s limits, or skillfully surfing the ocean's waves, all represent play. In my work on mathematical problem solving primes were the objects of play and rules of arithmetic guided/constrained my play. I was as masterful in perceiving, patterning, abstracting, and building a model as my abilities permitted me to be. Most importantly I had fun during my playful work with primes and I have discovered new ways of looking at primes by playing with them. Working with primes involves discovery, it's a pathless journey. I chose to work on this topic at the first place, because its open endedness presents limitless possibilities for innovation and play.
Nevertheless more strictly defined mathematical problems, those that have stricter conditions and better defined unknowns, can also be solved through play. More on this in the next Module...
Image source: www.nature.com
Visit the next module!