It is human nature to put all data into context in order to tell a story, re-create the world by this means, and thereby be able to predict outcomes (causality).
ART OF MEASUREMENT: How Playing Sudoku Can Teach Math and Bridge the Digital Divide
Most of the incarcerated operate under a warped narrative that they are unable to change on their own.
Civilization advances by extending the number of important operations that we can perform without thinking about them. All things that we “know” no longer demand our attention. To know something is to do it automatically, without thinking, to categorize it at a glance, or to ignore it entirely. The nervous system is “designed” to eliminate predictability from consideration, and to focus limited analytical resources where focus would produce useful results.
It is human nature to put all data into context in order to tell a story, re-create the world by this means, and thereby be able to predict outcomes (causality). Most of the incarcerated operate under a warped narrative that they are unable to change on their own.[1]
In prison, a common consequence of this narrative is a math phobia.[2] Math phobia is merely a habit. Habits often begin as a complex positive feedback loop.[3] Next, the very thought of this fearful event occurring again triggers anxiety. The person notices their racing heart, and this triggers more anxiety — now you have a positive feedback loop.[4]
We can break the positive feedback loop just like with a concert’s speakers. To change the decision-making process by playing Sudoku is analogous to moving the microphone away from the speakers at a concert and thus breaking the positive feedback loop. It works because the conscious mind has extended the decision-making process and therefore it has time to actually work on the problem.[5]
The Prison University Project (“PUP”) provides an environment to change the education narrative to one that includes developing critical thinking skills, including math.
I. THESIS
The science of learning has always included a creative mix to memorize and classify data and their interpretation by theory. Traditional learning does not address either the failing cultural narrative that supports a math-phobia or the use of active learning tools necessary to develop critical thinking essential to succeed in math.
A framing mechanism to develop analytic ability is playing SUDOKU as inspired by NARRATIVE (FORTUNA vs. SAPIENTIA). Students can gain mental rigor from playing Sudoku and learn to differentiate between uncertainty, quantifiable risk and certainty. Math no longer requires guesswork. They can better understand the digital world and develop the practical active learning concepts of thinking critically, thinking creatively, communicating effectively and interacting effectively.
II. THE SCIENCE OF LEARNING
Education seeks to employ a creative mix to memorize and classify data and their interpretation by theory, i.e., to develop reasoning. Traditional teaching methods of lecture, rote memorization and lots of homework are ill suited to resolve the common learning impediments of PUP students[6] and result from a failure to understanding that these individuals need a full-spectrum teaching methodology to develop basic reasoning.[7]
The development of basic reasoning was a problem solved by mainstream society, therefore, no longer a focus of the education system.[8] After the solution was implemented, even the fact that such problems had ever existed disappeared from view.[9] Teaching focuses on information transmission, learning is about knowledge acquisition. Prison education often uses traditional teaching methodology for those that have demonstrated they do not learn that way. Overall, 52 percent of incarcerated adults are below the basic competency of Level 2 in numeracy with 65 percent of black men below the basic competency level.[10]
The purpose of lecture and learning is to transfer information from dynamic memory (only exist as long a consciously maintained) to structural memory (that can persist even when they are not being actively considered). The concept of active learning has made significant inroads to enhance student learning over the past decade by re-incorporating basic reasoning.[11]
This structural memory acquisition process has a physical manifestation in the brain. It causes structural changes in the brain that will endure over time.[12] There are various means to establish this critical-thinking ability, including narrative and awareness training.
III. SUDOKU — A Mechanism to Develop Critical Thinking
Sudoku can serve as a cornerstone of active learning. It will allow the brain to develop the neural pathways necessary to extend the decision-making process and allow mathematical concepts to rise to consciousness thereby students can solve problems.
At its core, the failure to teach these skills young means it will only develop via practice — lots of practice. A student playing Sudoku develops procedural pathways in the brain that aid in learning both math and accessing information in whole and in its context. For those intimidated by math and concerned about their ability to interact in the digital world, learning Sudoku accomplishes three goals.
· Students can learn even if they do not intend to do so. They can learn simply because of using information — PLAY THE GAME.
· With practice, they can transition from doing something consciously to doing it automatically — CREATING A TOOL.
· Once the procedural process becomes familiar and transitions to structural memory, Students can apply it in other relevant contexts — USE THE TOOL.
A. PLAYING THE GAME
“Su” means number in Japanese. “Doku” refers to a single place on the puzzle board what each number can fit into.”[13]
As with solving math, Sudoku players “rely solely on pure deductions, and they spontaneously acquire various deductive tactics, which differ in their difficulty….” Sudoku puzzles are “of pure deduction, and their solution depends ultimately on the ability to make valid deductive inferences. To draw conclusions that must be true given the truth of their premises.”[14] This is the same method used to solve math problems.
The more complex the Sudoku puzzle is, the more stages and tactics a player must use to solve the puzzle. The kicker is that “individuals were not inclined to make guesses” rather to transition from simple tactics to “advanced tactics embedded in an advanced strategy…from their discovery that they can make no further progress …relying solely on simple tactics. In other words, players progress from System 1 reasoning that exploits automatic processes to System 2 that uses logic and calculation.”[15]
The theory in “The Psychological Puzzle of Sudoku” postulates seven distinct simple exclusion/inclusion tactics that individuals acquire, but emphasizes they are not forced to use any of these tactics to solve the puzzle.[16]
B. CREATING A TOOL
Sudoku as an activity can develop procedural pathways that draw on a set of cognitive processes that aid in developing mental rigor necessary for math and accessing/evaluating information stored as different mental procedures (The timeless battle between Fortuna and Sepientia).[17]
Sudoku is problem solving that encompasses a set of conative processes that employs reasoning to make inferences from knowledge and draw conclusions. The player must draw upon memory, attention, and perception to identify and solve the puzzle.
C. USING THE TOOL
By accomplishing these three goals, PUP students will apply and deploy the decision-making and practical active learning concepts:
· Thinking critically
· Thinking creatively
· Communicating effectively
· Interacting effectively
Playing Sudoku allows to develop and apply a new cognitive face to the math problem can completely defuse a math phobia.[18]
IV. THE NEED FOR THE KNIFE (aka Critical Thinking)
Critical thinking, or the Art of Measurement, employs a metaphoric rhetorical Knife as a tool to divide between sense and nonsense. This analytic knife is the ability to name and split the whole world into parts of your own choosing, split the parts and split the fragments of the parts, finer and finer and finer until you have reduced it to what you can use.
The art of using the Knife takes place not as an inherent, natural trait, but as a learned trait, usually gained through pre-memory-like observation of parents.
A. Sudoku: Framing Tool for Education Pathway
If not gained in infancy, its acquisition requires education. Most schooling presumes a competency in mental rigor necessary to develop critical thinking, and so ignores further development of the skill. Students frequently have no conceptual framework in which to develop and deploy critical thinking to make a reasoned choice.
We need a framing tool to develop and practice basic reasoning necessary to learn math. One active learning tool that can foster the development of reasoning and make math accessible to PUP students is playing the game, Sudoku.
By playing Sudoku, any student can pass every required math class at PUP and gain a crucial skill necessary to operate in our modern digital world. Sudoku aids the development of attention, perception, and memory as well as an environment to practice formal logic and reasoning.
B. Sudoku Relies on Pure Deductions
If you have never played Sudoku, it is devilishly easy. Most players quickly learn several simple tactics used in a basic strategy.
C. Competition for Social Status
Evolution designs humans, especially those in prison, to compete, but the winners are those that compete for social status. Society developed what we call reason, as commonly deployed, to help us spin, not to help us learn. PUP students’ fear of failure blocks them from learning the skills necessary to do math and forcing all too many to avoid further education.
Critical thinking will not emerge on its own from regular coursework. Humans rarely abstract general principles from examples, therefore, we should not assume students would identify and formulate the common strategies and methods that underlie critical thinking.
In addition, even if studies can recognize a key principle in a class, they rarely apply those principles outside of the learning domain where they learned them.
D. Consciousness with the KNIFE
Consciousness has been characterized as “a global neural workspace” or blackboard where many different parts of the brain have access to information. “Consciousness emerges when incoming sensory information, inscribed onto such a blackboard, is broadcast globally to multiple cognitive systems.”[19]
The Knife allows us to recognize faulty arguments built into stories and evaluate whether a chain of reasoning leads to a valid conclusion. As we gain info-literacy, we can recognize there are hierarchies in source quality, pseudo-facts often masquerade as facts, biases can distort, and framing does matter in the Art of Measurement.
The goal is to use the Knife to divide facts into manageable pieces, make a measurement, and then incorporate these into our narrative.
Once we define the data with the Knife, we use this data to develop a model that maximizes the likelihood of the outcome shown from the data P(D|M).[20] This process of generating a model is the art of measurement. With probabilities, we can quantify future events essential to any rational decision.
V. NARRATIVE: FORTUNA AND SAPIENTIA
Pedagogically embedding the training mechanism of the Knife within narrative encourages the patience to practice the development of diligence and the ability to restart processes without losing focus — all essential to succeed in math.
Critical thinking will not emerge on its own. Humans rarely abstract general principles from examples, therefore, we use STORY to help identify and formulate the common strategies and methods that underlie critical thinking.
A. The epic narrative: the eternal struggle between Fortuna and Sapientia, or guessing and knowing
A significant aid to learning uses a narrative environment rich in retrieval cues, i.e., making use of associations. We call this narrative construct, story. Humans build stories on a series of interlocking causes and effects. We call this plot. By using concepts embedded in story, we can have a host of associations — cues — to learn and deploy the TOOL with clear divisions between types of unknowns.
The narrative STORY enhances the vital ability to differentiate an unmeasured risk from probability or certainty: Certainty is known; uncertainty is unknown; and probability is a fixed range of uncertainty. The goal will be to bring into conscious action these concepts and then, through practice, allow them to settle into semi-conscious habit.
**Two magnificently dressed young women sit upright on their chairs, calmly facing each other. Yet neither takes notice of the other.
B. FORTUNA
Fortuna (Lady Luck), the fickle, wheel-toting goddess of chance, sits blindfolded on the left while human figures desperately climb, cling to, or tumble off the wheel in her hand. The wheel has a square pedestal, a metaphor for the sharp corners of unknowing choice just beyond the corner. Can she make the correct choice?
C. SAPIENTIA
Sapientia (Wisdom), as reason personified, sits opposite Fortuna, facing her.
The goddess of quantifiable reason, Sapientia the calculating and vain deity of science, gazes into a hand-mirror, lost in admiration of herself. Also clad in opulent raiment, she stares into an oval mirror of reason with a circular pedestal, lost in admiration of her self-brilliance. An allegory of the danger of omniscience, do her choices have inherent correctness? With knowledge, nothing is hidden on an oval…
SAPIENTIA’s most exceptional power and greatest threat is her own capacity for self-recognition and self-admiration — endless capacity for pride — presuming omniscience. That is why she is looking into the mirror.
SAPIENTIA has good reason for self-admiration. With science’s development of Probability Theory, SAPIENTIA has shrunk FORTUNA’s domain.
D. Narrative Construct the Story
We seek to use THE KNIFE (Tool) in the ART OF MEASUREMENT (Critical thinking) to battle with FORTUNA and SAPIENTIA.
O SAPIENTIA
Wisdom is radiant and unfading,
And she is easily discerned by those who love her,
And is found by those who seek her.
To fix one’s thought on her is perfect understanding,
And he who is vigilant on her account will
Soon be free from care,
Though she is but one, she can do all things….
Wisdom 7:7–15 excerpts….
O FORTUNA
Like the moon you are changeable
Ever waxing and waning…
Poverty and power
You melt them like ice
Fate-monstrous and empty
You whirling wheel, you are malevolent
Well-being is vain and always fades (to nothing)
Shadowed and veiled, you plague me too….
Roman marching song adapted in CARMINA BURANA by Carl Orff
E. Who Will Win — Fortuna or Sapientia
Critical thinking is our dance with these two ladies, Fortuna/Risk and Sapientia/Wisdom.[21]
If “Consciousness is the virtual world composed by the scenarios”, (Wilson 120) we create based on narrative, by creating a narrative where math is controlled and numbers predictable -where answers ALWAYS can be found, we now have the frame to perform math.[22]
My goal is to use the art of the narrative to chain the fear of math.
VI. CHALLENGE: FAILING CULTURAL NARRATIVE
Many of the incarcerated are deadly afraid of math; therefore, they avoid all education. Potential PUP students that fear math, have a cultural blind spot. “Can’t they SEE just how easy it is?”
What blocks the learning process (it is not a question of IQ)?
PRISON is like a hospital: a negative error culture. To make errors of any kind, good or bad, is always dangerous. In this at-risk community, a mistake is considered weakness, and everything must be done to hide it. Such a culture has little chance to learn from errors and discover new opportunities.[23]
To encourage a Positive Error Culture, PUP must educate about the keys of Risk Literacy and the ability to recognize and deal with both risk and uncertainty where the sense of false certainty is our greatest risk.
None of this is easy. If reason conflicts with a strong emotion, do not try to argue. Enlist a conflicting and stronger emotion i.e., FORTUNA v. SAPIENTIA narrative.
To over-come this learning impediment, prison educational institutions must acknowledge that humans’ function as physical, psychological and socio-cultural individuals. The key is how to develop cross-level coherence among these facets. Active learning requires a student to turn off the fixation on self (and fear of math) and operate on multiple levels. Only then can learning happen.
The student must learn the fundamental skill of effective decision-making first. Students must first learn to assess risk (quantifiable) and uncertainty (not quantifiable).
By acknowledging the student environment, an educator may tailor a solution to a math phobia. The average PUP student has faced radical uncertainty in the urban-poor environment (“Urban Poor”). Accordingly, many PUP students had to make adult choices without first learning to quantify risk. If they are unable to set a probability range because of uncertainty and the unknown, they cope with this uncertainty by default to using a common social narrative — I can’t do math.[24]
PUP seeks to create an environment where its students will be able to generate the mental space to make independent choices (instead of defaulting to the failing narrative).[25]
My participation in more than a dozen courses has revealed to me that PUP students need a critical thinking tool — THE KNIFE — to learn Plato’s “the art of measurement” aka rational decision-making in our modern world. Given the incarcerateds’ heightened level of vigilance and competitiveness that can interfere with school learning, developing rational decision-making is not easy.
This modern cultural narrative assumes an inherent ability of dis-interest not commonly found among PUP students. They live in an environment where there is no space/time in the decision-making loop to either conceive of most common abstractions or avoid impulse reactions demanded by the misinterpretation of their environment as demanding either domination or submission.
The PUP student operates in an all-encompassing audio-field of protection against a dangerous world that requires hyper-vigilance and quick reaction to survive. There is no school-to-prison pipeline, rather, a cradle-to-prison pipeline as participants are primed for actions society defines as criminal. The Urban Poor learn hyper-vigilance at home but practice it as a defense mechanism that has the unintended consequence of inhibiting learning by PUP students.[26]
The brain is a sense-making machine and relies on intuition when it has no alternative narrative. Sudoku is an alternative as a play-based means to teach Plato’s “the art of measurement” aka rational decision-making in our modern world.
This paper will use the game of Sudoku to set out the constituent elements of modern world decision-making and practical active learning concepts:
· Thinking critically
· Thinking creatively
· Communicating effectively
· Interacting effectively
VII. SUDOKU AS A FRAME
By creating a frame, i.e., focusing the PUP students’ attention upon a game, educators can introduce multiple concepts aimed at solving > winning. PUP students can use many different strategies to solve the puzzle. The PUP student now has access to these same problem-solving strategies to solve math or other real-life problems.[27]
A. The Game
Sudoku is a pure logic game where you can begin anywhere (a minimum of 17 squares are pre-determined) but one mistake means you lose. As with math, there is always a right answer.[28]
In truth, it is impossible to make only one mistake! For if, you make a mistake, the impact cascades across the game (sort of like a JINGA tower falling if you pull out the wrong block). Sudoku is a game without competition (no fear of losing to others and can be played alone), always the single right answer and a given block can transition from unknown, probability, to certainty (necessary transitions in understanding).[29]
B. Game Benefits
By using a single-player puzzle, we can avoid the need for competition and speed demanded by social imperatives of competition. In addition, PUP students’ diligent practice develops their FOCUS on a single topic/issue over time; therefore, enhancing concentration.[30]
By characterizing practical knowledge in terms of production systems consisting of a condition/action pair, Sudoku improves active learning abilities. A PUP student by satisfying a condition can perform an action. (“When all other numbers are placed in the row except 5, therefore, the empty spot must be 5’) “IF…, THEN”[31]
The purpose of Sudoku is to develop habits of the mind “a cognitive skill that with practice come to be triggered automatically… without requiring conscious deliberation.”[32]
C. Deploy Mental Systems
Another means of defusing a math phobia is the application of different mental systems. As there are two distinct kinds of awareness, the sequential and simultaneous, commonly considered the left and right hemispheres of the brain, according to Ian McGilchrist’s characterization. Evolutionary speaking, there is a great advantage in being able to bring these two distinct types of attention to bear on a problem at the same time. “The very fabric of our experience emerges from the interaction and integration of each hemisphere’s separate means of perceiving,” according to Nick Sousanis in UNFLATTENING. [33]
D. Problem-solving Heuristics
The framing and focusing mechanism of Sudoku will allow for the regular application of the scientific method[34] or commonly used problem-solving heuristics including random search, hill climbing and means — ends analysis.[35]
Sudoku provide a context in which students can recognize underlying principles and have practice applying them in numerous different situations.
Playing Sudoku enhances the vital ability to differentiate an unmeasured risk from probability or certainty: Certainty is known; uncertainty is unknown; and probability is a fixed range of uncertainty. For example, a player can use risk application theory to fill in a box (limit a square to only two possible numbers); this “probability lock” can be in one (box, row, column, number) dimension or multiple.
The goal will be to bring into conscious action these concepts and then through practice allow them to settle into semi-conscious habit.
E. Sample Script
A sample script presenting this mechanism:
We are going to work on developing your math muscle. It will require dedication and mental sweat. Then we will add a series of exercises to allow for the use of THE KNIFE. Using THE KNIFE can develop the skill known as the ART OF MEASUREMENT. All knowledge is accessible through the ART OF MEASUREMENT.
We will develop your MATH MUSCLE so that you may learn to wield THE KNIFE with such skill that the ART OF MEASUREMENT becomes available. All knowledge is accessible through the ART OF MEASUREMENT. You will become SUDOKU WARRIORS.[36]
VIII. ESTABLISHING DIFFERENT CONTEXTS: DEPLOYING THE KNIFE
The analytic knife is the ability to name and split the whole world into parts of your own choosing, split the parts and split the fragments of the parts, finer and finer and finer until you have reduced it to what you can use. You must cut deep to get at the root of a thing.
You have the illusion that parts are named, as they exist. However, you can name them quite differently and organize them quite differently depending on how the knife moves. Once you have arrived at workable pieces, it is time to measure. In our modern world, we have a precise guide to clear thinking the art of measurement. It always works.[37]
One must be extremely careful and rigidly logical when dealing with the world: one logical slip and an entire scientific edifice comes tumbling down. One false deduction about the world can hang you up indefinitely. We can regularly experience this by mistakes in Sudoku.
To break a positive feedback loop, we need to practice using the blackboard, and expose the math problem to multiple mental subsystems as once. Most beginning students erroneously believe there is only one way to solve a math problem. Rarely is this true.
IX. ASSOCIATIVE LEARNING: ADDITIONAL BENEFITS
This constitutes my aspirational attempt to address the needs of this at-risk community as it prepares to transition back to the WORLD.
Using the game of Sudoku can operate as a powerful tool of deployed reason. By recasting the game of Sudoku as a metaphor for the modern world, we can now deploy this tool in new ways to access the digital world.
This skill allows you to look at any content in several ways at once, just as Sudoku allows you to consider a number as part of a row, column or box. You can fill any square in any order — the whole puzzle is available at once, like a digital cloud.
A. Digital Cloud
This digital cloud is just another way of describing content that is stored and accessible on the Internet. The cloud makes it possible to binge watch an entire season at once or view an old episode of The Vampire Diaries, Sleepy Hollow or The Simpsons. It allows people to download and watch shows like Game of Thrones or EMPIRE at their convenience or stream and watch them live.
A member of the Digital Age (as a unified field expert) would be completely comfortable with the boxes-within-a-box structure. They will also be comfortable with looking at the puzzle as one unified field, not as 81 separate problems. The ability to look at each square in multiple dimensions — its membership in a row, column and 3X3 box — allows the player to fill ANY SQUARE with confirmation across these multiple dimensions. This cross-referencing also speeds up the process.
B. Effective Decision Making
Sudoku as a secondary tool teaches how to understand risk and deal with uncertainty and most important, how not to confuse the two. The process of assigning an objective probability to the hypothesis (“I think 4 goes here.”) and then modify the model in light of new data (Since there is already a 4 in this row, it can’t take another 4.”)
RISK: If risks are known, good decisions require logic and statistical thinking (math works).
UNCERTAINTY: If some risks are unknown, good decisions also require intuition and smart rules of thumb heuristics (or Bayesian probability theory).
We must play the game as a group/team to show how to learning by making mistakes and inoculate students from the scourge of defensive decision making: A person or group ranks option A as the best for the situation but chooses an inferior option B to protect itself in case something goes wrong.
If we play Sudoku as a group, risk aversion is normal. “The emotional fabric of defensive decisions differs from that of risk aversion… [As] part of the crowd behavior” you can take excessive risk. “The problem is not simply risk aversion, but lack of a positive error culture.”[38]
If you want to understand the defining power of modern society, play Sudoku…
X. FURTHER STUDIES
By understanding the activation of different brain networks, we may be able to enhance learning. “The findings of Fugelsang and Dunbar (2005) suggest that during inductive reasoning the human brain may be specifically turned to require learning mechanisms when evaluating data that are consistent with preexisting hypotheses, and to recruit error detection mechanisms when evaluating date that are not consistent with hypotheses….by understanding the underlying brain networks that are involved in various complex tasks, we can begin to understand how the subcomponents of inductive reasons (e.g., attention, error processing, conflict monitoring and working memory) interact.”[39]
XI. CONCLUSION
People need to think. They need to simulate the world and plan how to act in it. True thinking is complex and demanding. It involves uncertainty. They must learn to tolerate uncertainty. You must modify your premises and adjust your thoughts. In consequence, thinking is emotionally painful as well as physiologically demanding; more than anything else — except not thinking — to paraphrase Peterson. Memory is not a description of the objective past. Memory is a tool. Memory is the past’s guide to the future. That is the purpose of memory. It is not ‘to remember the past,’ but to avoid mistakes.
In conclusion, playing Sudoku puzzles can yield human competence in stark contrast to many psychological theories. Any student can acquire the ability to make deductions about abstract concepts and do math.
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SOURCES:
Building the Intentional University: Minerva and the Future of Higher Education edited by Stephen M. Kosslyn and Ben Nelson 2017 MIT Press, Cambridge
Risk Savvy: How to Make Good Decisions by Gerd Gigerenzer 2014 of the Max Planck Institute for Human Development, Penguin, New York
Consilience: The Unity of Knowledge by Edward O. Wilson 1999, Vantage, New York
Maps of Meaning: The Architecture of Belief by Jordan B. Peterson 1999, Rutledge, New York
The Guttenberg’s Galaxy by Marshall McLuhan 1962, University of Toronto, Canada
Thinking Fast and Slow by Daniel Khaneman and Patrick Egan 2011, Farrar, Straus, and Giroux, New York
Protagoras by Plato Sections 356d — 357a
UNFLATTENING by Nick Sousanis 2015, Harvard University Press, Cambridge
Cognitive Psychology: Mind and Brain by Edward E. Smith and Stepen M. Kosslyn 2014, Harlow, Essex, Pearson
12 Rules For Life: An Antidote to Chaos by Jordan B. Peterson 2018, Random House, Canada
Mathematical Mindsets: Unleashing Students’ Potential Through Creative Math, Inspiring Messages and Innovative Teaching by Jo Boaler 2015, John Wiley & Sons, New Jersey
Effects of Two Types of Sudoku Puzzles on Students’ Logical Thinking by Youngyun Baek, Bokyeong Kim of Korea National University of Education, South Korea and Department of Instructional Technology, University of Virginia, USA, 2006
The Psychological Puzzle of Sudoku in THINKING & REASONING 14(4). 342–364 by N.Y. Louis Lee, Geoffrey P. Goodwin, P.N. Jo
[1] “Much more of our sanity than we commonly realize is a consequence of our fortunate immersion in a social community….If a child has not been taught to behave properly by the age of four, it will forever be difficult for him or her to [socialize]…They fall further and further behind, as the other children continue to progress… “ Peterson 135
[2] In studies by Carol Dweck, “about 40 % …held a damaging fixed mindset….” Pg. 5
“Students with a fixed mindset are more likely to give up easily, whereas students with a growth mindset keep going even when work is hard and are persistent; displaying what Angela Duckworth has termed “grit” (Duckworth & Quinn, 2009).” Pg. 6 Development and validation of the short grit scale Journal of Personality Assessment, 91(2), 166–174
[3] “A positive feedback loop requires an input detector, an amplifier, and some form of output…. the trouble starts when the input detector detects that output, and runs it through the system again, amplifying and emitting it again.” Peterson 19
[4] “There are many systems of interaction between brain, body and social world that can get caught in positive feedback loops.” Peterson 22
“The fixed mindset thinking that is so damaging — a mindset in which students believe they either are smart or are not — cuts across the achievement spectrum….” Pg. 7
[5] “The new evidence from brain research tells us that everyone, with the right teaching and messages, can be successful in math, and everyone can achieve at the highest levels in school.” Boaler 2
“Your nervous system responds in an entirely different manner when you face the demands of life voluntarily.” Peterson 27
[6] Educated Western elites have predictable routines and patterns of behavior that foster critical thinking. They are mostly implicit, as “our knowledge has been shaped by our interaction with others.” Peterson 163
[7] What so often counts most in schools is the important but incomplete cognitive resource of knowledge. Fixed knowledge and algorithms are easier to teach, learn, and test than the tangled web of processes that make up problem solving. Martinez, M.E. (1998) What is problem solving? Phi Delta Kappan, 79(8), 605–609 quoted in Kosslyn & Nelson (2017)
[8] Consciousness itself might be considered as that organ which specializes in the analysis and classification of unpredictable events. Peterson 150
[9] As Plato’s Phaedrus said in rejecting the technology of the printed word, that over-reliance today on teaching as to learning is analogous to the written word in ancient Greece. “By telling them of many things without teaching them, you will make them seem to know much, while for the most part they know NOTHING as men filled, not with wisdom, but with the conceit of wisdom, they will be a burden to their fellows.”
DANGER: Education may seem the best way to pull a people out of this ancient evil, however, the modern iteration of education does not create the necessary cultural foundation (similar to the challenge Plato faced with the Poets). Our educational system does not stand a chance of curing this ancient evil that holds a people in THRALL.
[10] Overall, 52 percent of incarcerated adults are below the basic competency of Level 2 in numeracy with 65 percent of black men below the basic competency level. On a 500 scale, U.S. prison inmates average 220 with black men averaging 206.
U.S. Program for the International Assessment of Adult Competencies (PIAAC) of incarcerated adults, http://nces.ed.gov/pubsearch
[11] “Students may be unready for some mathematics because they still need to learn some foundational, prerequisite mathematics they have not yet learned, but not because their brain cannot develop the connection because of their age or maturity. When students need new connections, they can learn them.” Boaler 8
[12] “If you learn something deeply, the synaptic activity will create lasting connections in your brain, forming structural pathways, but if you visit an idea only once or in a superficial way, the synaptic connections can ‘wash away’ like pathways made in the sand.” Boaler Pg. 1
[13] “Effects of Two Types of Sudoku Puzzles on Students’ Logical Thinking” by Youngyun Baek, Bokyeong Kim of Korea National University of Education, South Korea and Department of Instructional Technology, University of Virginia, USA
[14] As with solving math, individuals playing Sudoku “rely solely on pure deductions, and they spontaneously acquire various deductive tactics, which differ in their difficulty….” P343 Sudoku puzzles are “of pure deduction, and their solution depends ultimately on the ability to make valid deductive inferences. To draw conclusions that must be true given the truth of their premises.” Just as with solving math problems.
pg. 343 “The Psychological Puzzle of Sudoku” in THINKING & REASONING, 2008. 14(4). 342–364 by N.Y. Louis Lee, Geoffrey P. Goodwin, P.N. Johnson-Laird
[15] The theory postulates seven distinct simple exclusion/inclusion tactics that individuals acquire, but emphasizes they are not forced to use any of these tactics to solve the puzzle. Pg. 346–347 & Pg. 355 of “The Psychological Puzzle of Sudoku” in THINKING & REASONING, 2008. 14(4). 342–364 by N.Y. Louis Lee, Geoffrey P. Goodwin, P.N. Johnson-Laird
[16] “The Psychological Puzzle of Sudoku” Pg. 346–347
[17] “In sum, the puzzles establish that logically naïve individuals have the competency to make deductions about abstract matters, and that they enjoy exercising this ability, contrary to all the theories above that impugn it.” Pg. 360
[18] “In conclusion, the solution to the puzzle of Sudoku yields an insight into human competence that is in stark contrast to many psychological theories: reasoners readily acquire the ability to make deductions about abstract contents….” Pg. 362
[19] “What is Consciousness?” by Christof Koch in Scientific American, June 2018
[20] “The logical thinking of players in the vectorial stages of cognitive development.” “In the vectorial strategy…apply abstract reasoning and understanding …symbolic representation.” Pg. 21 of “Effects of Two Types….”
[21] Plato said only knowledge can provide salvation “specifically some kind of measurement” and that “the art of measurement in contras, would make the appearances lose their power and showing us the truth…[is] nothing other than arithmetic….”
[22] Science explains feeling, while art transmits it. Art …transmits feeling among persons of the same capacity.” (Wilson 127) Art is the means by which people of similar cognition reach out to others in order to transmit feeling.
[23] Risk Savvy: How to Make Good Decisions by Gerd Gigerenzer 2014 of the Max Planck Institute for Human Development pg.
[24] According to Kahneman, we can now understand human irrationality in a systemic way. We can thereby design and deploy algorithms that correct for these consequential flaws in human thought processes.
“Consciousness is the virtual world composed by the scenarios” we create based on narrative. Wilson 120
[25] People need to think. They need to simulate the world and plan how to act in it. True thinking is complex and demanding. It involves uncertainty. So, you have to tolerate uncertainty. You have to modify your premises and adjust your thoughts. In consequence, thinking is emotionally painful as well as physiologically demanding; more than anything else — except not thinking — to paraphrase Peterson. Peterson 241
[26] McLuhan provides the best exposition of how the modern Print Culture allows for the creation of a point-of-view necessary to break the quick-trigger of violence systemic to traditional cultures comparable to the Urban Poor.
[27] This is more than a game; it is an idea. “An idea that grips a person is alive…It wants to express itself, to live in the world.” Peterson 195
[28] Sudoku is a puzzle consisting of a 9X9-square grid subdivided into nine 3X3 boxes. Some of the squares contain numbers. The object is to fill in the remaining squares so that every row, column and 3X3 box contains each of the numbers 1 to 9 exactly one. A solitary individual usually plays it.
[29] To a person from the pre-digital culture, rational means uniform, continuous, and sequential “you start at the beginning, step one, and go to the end.” Sudoku, however, is unsettling — begin anywhere? — No apparent rhyme or reason. This can seem disorienting and confusing to a pre-digital culture expert because of the mis-definition of reason with a linear sequential methodology. McLuhan defined this condition as “our typographical cultural bias.”
[30] People are rational. We forget “The capacity of the rational mind to deceive, manipulate, scheme, trick, falsify, minimize mislead, betray, prevaricate, deny, omit, rationalize, bias, exaggerate and obscure is …endless….” Peterson 217
Sudoku Myth is a safe place to acknowledge your capacity to rationalize over critical thinking.
[31] “Memory is not a description of the objective past. Memory is a tool. Memory is the past’s guide to the future. If you remember that something bad happened, and you can figure out why, then you can try to avoid that bad thinking happening again. That’s the purpose of memory. It’s not ‘to remember the past.’ It’s to stop the same damn thinking from happening over and over.” Peterson 239
[32] Building the Intentional University: Minerva and the Future of Higher Education edited by Stephen M. Kosslyn and Ben Nelson 2017 pg
[33] The left breaks down and isolates information into segmented parts and the right addresses the whole in its context.
[34] The scientific method is a cycle of observing, theorizing, formulating hypotheses, collecting date to test hypotheses, revising the theory and then generative and testing new hypotheses.
[35] Random search is a fallbacks heuristic that we use when other heuristics either do not work or are cognitively too demanding.
Hill climbing is where the player looks one move ahead and chooses the move that resembles the goal.
Means — ends analysis is a more demanding, but more successful, strategy in which the problem is broken down into sub problems. Smith 430–3
[36] Sudoku also allows for group-play. The need to collaborate and develop the skills of practical active learning: Thinking critically, Thinking creatively, Communicating effectively, Interacting effectively.
The team members must accomplish this. The internally generated ‘rules of thumbs’ are the key — they can do it themselves… If XX, Then YY! Set probabilities! Eliminate alternatives — It can’t be seven!
[37] “SCIENCE is neither a philosophy not a belief system. It is a combination of mental operations that has become increasingly the habit of educated peoples as the most effective way of learning about the real world.” Wilson 49
[38] Risk Savvy: How to Make Good Decisions by Gerd Gigerenzer 2014 of the Max Planck Institute for Human Development pg. 156
[39] Smith pg. 448
