Synthesizing
To synthesize means to bring things together, the end result of synthesis is a new product. In our work this new product is the understanding of how the six cognitive skills of perception, patterning, abstracting, embodied thinking, playing, and modeling support one another in all their connections as a single whole. Since these skills involve the use of all of our senses, feelings, intuitions, proprioperception, rational and non-rational reasoning, the synthesis of these can produce a whole new quality of thinking. Synthesis and creativity are similar processes in that they involve making new associations of ideas; the first achieves this through a conscious effort, the second does it through insight. Synthesis brings closure, but it’s not the end of a process.
What is creativity? Creativity is our ability to recombine bits of knowledge gathered through wide ranging personal interests and experiences. Therefore, cultivating diverse interests, and being proactive in seeking out varied life-experiences will build our store of knowledge that serves as the raw material for our creativity. Secondly, we need to understand the processes by which the aforementioned raw materials’ new combinations arise in our minds. The seven cognitive skills explored by CEP 818 foster the mind’s ability to make new associations between formerly disassociated bits of knowledge , leading to new configurations that result in unforeseen insights.
How could creativity be taught? Go ahead, plunge in, be creative; but how? There seems to be a contradiction here: creativity is spontaneous and momentary, if we aren’t in control of our thoughts while creativity “happens”, then how could it be taught? Perhaps it cannot be directly taught, but teaching how to invoke it is quite possible. During this semester in CEP 818 we have seen that the conditions and precursors of creativity can be described, attributes of creative thinking can be understood, and therefore a person’s creative facilities can be systematically developed. By understanding the factors that promote creativity we can methodically develop it in ourselves and in our students. The article Twisting knobs and connecting things: Rethinking Technology & Creativity in the 21st Century argues that creative moments should not be viewed as divine sparks (random and externally imposed), that creativity is not the privilege of gifted people, but a skill everybody can develop in oneself. The apparent contradiction disappears.
Being creative means to think independently and originally as opposed to imitating others’ thought processes for solving our own problems. Reality is ever changing; though two problems/situations may be similar, the details surrounding them are never quite identical. Creativity is essential in tackling new problems, or to solve old ones in innovative ways. Creative approaches in teaching and learning should be promoted because through them students can develop their own personal understanding of concepts and problem solving processes. Being innovative means to know in the context of a process, rather than to know statically by recalling facts. As the Root-Bernstein’s put it “(…) rethinking shifts our educational focus from what to think to how to think in the most productive ways possible.” [1]. The simple knowing of facts compares to the ability of creatively applying knowledge as a list of words compares to eloquently spoken language. Being able to use information, as opposed to just knowing it, is an especially pertinent skill in this information age. The true power of our minds is not in its ability to simply store information, but rather to be able to use stored information to develop a virtually limitless number of new ideas. These two functions, storing information and creating new ideas by the recombination of information, are qualitatively different; creative thinking is a function of the mind that is by far superior to the mind’s mere storage ability .
Having established what creativity is (not conclusively, just in one of the many possible ways) and why it is important, and that multifarious knowledge, interests , and experiences serve as the raw material for creative thoughts, now let’s look back at some of the creativity-promoting cognitive skills explored throughout the semester.
Perceiving
Perception is the observer's transcription of sensory input into his own personal conception of the essence of the object he observes. We may perceive the same old object differently every day, all we need are new eyes to see with. The stages of perceiving are observation, imaging, reinterpretation.
Patterning
Patterning is the process of perceiving order in the arrangement of objects. Based on the perceived order a logical representation of the pattern is created in the form of an image, sound, shape, mathematical formula, etc. An informative pattern either entirely captures the order in the arrangement of the observed objects, or at least creates some sort of analogy between the objects' arrangement and their representation.
Abstracting
To abstract an essence from a source means focusing on a single quality of the source and "pulling it out". This reduced, singled out, quality is then represented in a pure form free of the source's other properties.
Embodied Thinking
Turning outward, engaged in embodied thinking the thinker empathizes with a living or non-living entity, transcending his own mental and physical self, he becomes one with the empathized. Turning inward, the thinker uses proprioperception to examine the state of his own body or a part of it. Through this connection with his physical self the thinker gains an understanding of the state of his state of mind and emotions.
Modeling
In the general sense building a model means to render a scaled down or enlarged replica of a structure. In this sense models are dilations proportional to the objects they model, and thus the modeling process relies on spatial thinking. A model is analogous to the modeled object in appearance but usually it doesn't posses all the functions of the original.
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.
Synthesis (click for a strange loop)
For our purposes to synthesize means to bring things together to form something new. The end result of synthesis is a new product. Synesthesia is a type of synthesis that brings together concepts that are not usually associated. In his autobiography Born on a Blue Day autistic savant Daniel Tammet describes his synesthetic perceptions in detail; a great read on the topic of synesthesia.
How I use CEP 818’s cognitive skills in my work: A case of synthesis
The technical aspect of my approach to mathematics teaching is inspired in large part by George Polya’s guidelines to mathematical problem solving. George Polya was a highly accomplished 20th century mathematician who has made great contributions to mathematics education worldwide; his book How to Solve It is a reference to many mathematics educators over the globe. The power of Polya’s approach to problem solving (including non-mathematical problems) is that it’s based on simple common sense, observation, experimentation, cognitive shortcuts; this approach to problem solving is called heuristics.
Here is a comparison of Polya’s heuristics (lightly simplified) and CEP 818’s cognitive skills promoting creativity. Note that Polya’s approach is usually applied to closed ended problems (as opposed to exploration) while CEP 818’s creative approach applies both to problems that require specific answers, and well as problems that involve discovery:
Polya’s Heuristics CEP 818’s Creative Approach
1. Understand the Problem -------------------- perceiving: observing, imaging
2. Devise a Plan
use symmetry ------------------- patterning, modeling
look for patterns ---------------- observing, patterning
consider special cases --------------- build an analogy, modeling
list things ------------------- perceiving, patterning
draw a picture --------------- perceiving, abstracting
use a simplified problem -------------- build an analogy
write and form an equation ------------- abstracting, modeling
use a formula ---------------- use an existing model
be ingenious ------------------ be creative, playing
3. Carry Out the Plan ---------------- playing
4. Look Back --------------------- synthesizing
Throughout my work in CEP 818 I noticed that Polya’s classical guidelines to problem solving tie in, in some cases match one-to-one, with CEP 818’s cognitive skills promoting creativity. Viewed from this perspective, not unlike the CEP 818’s creators, Polya teaches us to tackle problems of all sorts creatively. I find the revelation of this close correspondence between these two sources on education, removed from each other in time (How to Solve It was published 1945) and other dimensions, a welcome surprise as well as a fresh assurance that teaching mathematics as a creative-heuristic process is not a bad idea.
The steps of Polya’s heuristic and their corresponding counterparts are important in one sense because they help us solve problems. In another sense because they break down barriers created by the habitual pathways of our thinking, they help creating new associations between elements of the “raw material” we have in our minds. By practicing these skills the habit of using them in real applied situations gets instilled in our minds leading to flexible and innovative thinking and can eventually produce more frequent, genuine creative insights. Mathematics is a language that we use to communicate the logical connections in our perceptions about nature. This language is a model itself. Equations, mathematical open sentences, are models of real situations; variables in equations are abstractions of entities that we want to quantitatively understand. Being able to devise a number of different mathematical models for a given real situation brings the element of play into problem solving. In other words, engaging in mathematical thinking without using the creative skills of CEP 818 is unimaginable. Likewise, teaching mathematical thinking only makes sense as a process that involves the same creative cognitive skills.
Synthesizing the ideas explored in this course, and then bringing them together with my existing teaching practices, plus finding many commonalities between the two has been my way of looking back on teaching and learning with creativity.
Let me conclude this final Module with a quote from Polya that I use as my classroom motto, and that in my view is very pertinent to CEP 818's spirit:
"A great discovery solves a great problem, but there is a grain of discovery in the solution of any problem. Your problem may be modest, but if it challenges your curiosity and brings into play your inventive faculties, and if you solve it by your own means, you may experience the tension and enjoy the triumph of discovery."
1. Sparks of Genius: The Thirteen Thinking Tools of the World's Most Creative People, by Robert S. Root-Bernstein, Michele M. Root-Bernstein
Pitching Creativity
Creativity matters! But what is creativity anyway? And, can you develop your creative faculties, or is it talent that some people have and some don’t? Being creative means that you use your knowledge of any kind in an original and meaningful way. Creative people improvise serious ideas as the need arises but remain playful for life! You can also do this by reaching into your mind’s store of knowledge and search for all ideas and experiences relevant, or seemingly irrelevant, to the question you are dealing with. Then, you reimagine those ideas and experiences into something brand new. Creative solutions are as unique as people are distinct individuals, rest assured, you can improve your own creativity! All you need is to be in touch with the wealth of ideas you already possess, and to train your mind to see those old ideas in a new light. Always be observant, and don’t be afraid to play with your perceptions. Look for patterns around you and build analogies between the patterns you find. Don’t be afraid to use your body to think, feel your ideas, trust your intuitions!
For Twitter
Learn about primes while having fun by using your creativity! Perceive patterns, play with abstract models. Expand your mind, have fun!
Click to read The Creative "I" - Part 1.
To synthesize means to bring things together, the end result of synthesis is a new product. In our work this new product is the understanding of how the six cognitive skills of perception, patterning, abstracting, embodied thinking, playing, and modeling support one another in all their connections as a single whole. Since these skills involve the use of all of our senses, feelings, intuitions, proprioperception, rational and non-rational reasoning, the synthesis of these can produce a whole new quality of thinking. Synthesis and creativity are similar processes in that they involve making new associations of ideas; the first achieves this through a conscious effort, the second does it through insight. Synthesis brings closure, but it’s not the end of a process.
What is creativity? Creativity is our ability to recombine bits of knowledge gathered through wide ranging personal interests and experiences. Therefore, cultivating diverse interests, and being proactive in seeking out varied life-experiences will build our store of knowledge that serves as the raw material for our creativity. Secondly, we need to understand the processes by which the aforementioned raw materials’ new combinations arise in our minds. The seven cognitive skills explored by CEP 818 foster the mind’s ability to make new associations between formerly disassociated bits of knowledge , leading to new configurations that result in unforeseen insights.
How could creativity be taught? Go ahead, plunge in, be creative; but how? There seems to be a contradiction here: creativity is spontaneous and momentary, if we aren’t in control of our thoughts while creativity “happens”, then how could it be taught? Perhaps it cannot be directly taught, but teaching how to invoke it is quite possible. During this semester in CEP 818 we have seen that the conditions and precursors of creativity can be described, attributes of creative thinking can be understood, and therefore a person’s creative facilities can be systematically developed. By understanding the factors that promote creativity we can methodically develop it in ourselves and in our students. The article Twisting knobs and connecting things: Rethinking Technology & Creativity in the 21st Century argues that creative moments should not be viewed as divine sparks (random and externally imposed), that creativity is not the privilege of gifted people, but a skill everybody can develop in oneself. The apparent contradiction disappears.
Being creative means to think independently and originally as opposed to imitating others’ thought processes for solving our own problems. Reality is ever changing; though two problems/situations may be similar, the details surrounding them are never quite identical. Creativity is essential in tackling new problems, or to solve old ones in innovative ways. Creative approaches in teaching and learning should be promoted because through them students can develop their own personal understanding of concepts and problem solving processes. Being innovative means to know in the context of a process, rather than to know statically by recalling facts. As the Root-Bernstein’s put it “(…) rethinking shifts our educational focus from what to think to how to think in the most productive ways possible.” [1]. The simple knowing of facts compares to the ability of creatively applying knowledge as a list of words compares to eloquently spoken language. Being able to use information, as opposed to just knowing it, is an especially pertinent skill in this information age. The true power of our minds is not in its ability to simply store information, but rather to be able to use stored information to develop a virtually limitless number of new ideas. These two functions, storing information and creating new ideas by the recombination of information, are qualitatively different; creative thinking is a function of the mind that is by far superior to the mind’s mere storage ability .
Having established what creativity is (not conclusively, just in one of the many possible ways) and why it is important, and that multifarious knowledge, interests , and experiences serve as the raw material for creative thoughts, now let’s look back at some of the creativity-promoting cognitive skills explored throughout the semester.
Perceiving
Perception is the observer's transcription of sensory input into his own personal conception of the essence of the object he observes. We may perceive the same old object differently every day, all we need are new eyes to see with. The stages of perceiving are observation, imaging, reinterpretation.
Patterning
Patterning is the process of perceiving order in the arrangement of objects. Based on the perceived order a logical representation of the pattern is created in the form of an image, sound, shape, mathematical formula, etc. An informative pattern either entirely captures the order in the arrangement of the observed objects, or at least creates some sort of analogy between the objects' arrangement and their representation.
Abstracting
To abstract an essence from a source means focusing on a single quality of the source and "pulling it out". This reduced, singled out, quality is then represented in a pure form free of the source's other properties.
Embodied Thinking
Turning outward, engaged in embodied thinking the thinker empathizes with a living or non-living entity, transcending his own mental and physical self, he becomes one with the empathized. Turning inward, the thinker uses proprioperception to examine the state of his own body or a part of it. Through this connection with his physical self the thinker gains an understanding of the state of his state of mind and emotions.
Modeling
In the general sense building a model means to render a scaled down or enlarged replica of a structure. In this sense models are dilations proportional to the objects they model, and thus the modeling process relies on spatial thinking. A model is analogous to the modeled object in appearance but usually it doesn't posses all the functions of the original.
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.
Synthesis (click for a strange loop)
For our purposes to synthesize means to bring things together to form something new. The end result of synthesis is a new product. Synesthesia is a type of synthesis that brings together concepts that are not usually associated. In his autobiography Born on a Blue Day autistic savant Daniel Tammet describes his synesthetic perceptions in detail; a great read on the topic of synesthesia.
How I use CEP 818’s cognitive skills in my work: A case of synthesis
The technical aspect of my approach to mathematics teaching is inspired in large part by George Polya’s guidelines to mathematical problem solving. George Polya was a highly accomplished 20th century mathematician who has made great contributions to mathematics education worldwide; his book How to Solve It is a reference to many mathematics educators over the globe. The power of Polya’s approach to problem solving (including non-mathematical problems) is that it’s based on simple common sense, observation, experimentation, cognitive shortcuts; this approach to problem solving is called heuristics.
Here is a comparison of Polya’s heuristics (lightly simplified) and CEP 818’s cognitive skills promoting creativity. Note that Polya’s approach is usually applied to closed ended problems (as opposed to exploration) while CEP 818’s creative approach applies both to problems that require specific answers, and well as problems that involve discovery:
Polya’s Heuristics CEP 818’s Creative Approach
1. Understand the Problem -------------------- perceiving: observing, imaging
2. Devise a Plan
use symmetry ------------------- patterning, modeling
look for patterns ---------------- observing, patterning
consider special cases --------------- build an analogy, modeling
list things ------------------- perceiving, patterning
draw a picture --------------- perceiving, abstracting
use a simplified problem -------------- build an analogy
write and form an equation ------------- abstracting, modeling
use a formula ---------------- use an existing model
be ingenious ------------------ be creative, playing
3. Carry Out the Plan ---------------- playing
4. Look Back --------------------- synthesizing
Throughout my work in CEP 818 I noticed that Polya’s classical guidelines to problem solving tie in, in some cases match one-to-one, with CEP 818’s cognitive skills promoting creativity. Viewed from this perspective, not unlike the CEP 818’s creators, Polya teaches us to tackle problems of all sorts creatively. I find the revelation of this close correspondence between these two sources on education, removed from each other in time (How to Solve It was published 1945) and other dimensions, a welcome surprise as well as a fresh assurance that teaching mathematics as a creative-heuristic process is not a bad idea.
The steps of Polya’s heuristic and their corresponding counterparts are important in one sense because they help us solve problems. In another sense because they break down barriers created by the habitual pathways of our thinking, they help creating new associations between elements of the “raw material” we have in our minds. By practicing these skills the habit of using them in real applied situations gets instilled in our minds leading to flexible and innovative thinking and can eventually produce more frequent, genuine creative insights. Mathematics is a language that we use to communicate the logical connections in our perceptions about nature. This language is a model itself. Equations, mathematical open sentences, are models of real situations; variables in equations are abstractions of entities that we want to quantitatively understand. Being able to devise a number of different mathematical models for a given real situation brings the element of play into problem solving. In other words, engaging in mathematical thinking without using the creative skills of CEP 818 is unimaginable. Likewise, teaching mathematical thinking only makes sense as a process that involves the same creative cognitive skills.
Synthesizing the ideas explored in this course, and then bringing them together with my existing teaching practices, plus finding many commonalities between the two has been my way of looking back on teaching and learning with creativity.
Let me conclude this final Module with a quote from Polya that I use as my classroom motto, and that in my view is very pertinent to CEP 818's spirit:
"A great discovery solves a great problem, but there is a grain of discovery in the solution of any problem. Your problem may be modest, but if it challenges your curiosity and brings into play your inventive faculties, and if you solve it by your own means, you may experience the tension and enjoy the triumph of discovery."
1. Sparks of Genius: The Thirteen Thinking Tools of the World's Most Creative People, by Robert S. Root-Bernstein, Michele M. Root-Bernstein
Pitching Creativity
Creativity matters! But what is creativity anyway? And, can you develop your creative faculties, or is it talent that some people have and some don’t? Being creative means that you use your knowledge of any kind in an original and meaningful way. Creative people improvise serious ideas as the need arises but remain playful for life! You can also do this by reaching into your mind’s store of knowledge and search for all ideas and experiences relevant, or seemingly irrelevant, to the question you are dealing with. Then, you reimagine those ideas and experiences into something brand new. Creative solutions are as unique as people are distinct individuals, rest assured, you can improve your own creativity! All you need is to be in touch with the wealth of ideas you already possess, and to train your mind to see those old ideas in a new light. Always be observant, and don’t be afraid to play with your perceptions. Look for patterns around you and build analogies between the patterns you find. Don’t be afraid to use your body to think, feel your ideas, trust your intuitions!
For Twitter
Learn about primes while having fun by using your creativity! Perceive patterns, play with abstract models. Expand your mind, have fun!
Click to read The Creative "I" - Part 1.