“Education must also train one for quick, resolute and effective thinking. To think incisively and to think for one’s self is very difficult. We are prone to let our mental life become invaded by legions of half truths, prejudices, and propaganda. At this point, I often wonder whether or not education is fulfilling its purpose. A great majority of the so-called educated people do not think logically and scientifically. Even the press, the classroom, the platform, and the pulpit in many instances do not give us objective and unbiased truths. To save man from the morass of propaganda, in my opinion, is one of the chief aims of education. Education must enable one to sift and weigh evidence, to discern the true from the false, the real from the unreal, and the facts from the fiction.
“The function of education, therefore, is to teach one to think intensively and to think critically. But education which stops with efficiency may prove the greatest menace to society. The most dangerous criminal may be the man gifted with reason, but with no morals.
“We must remember that intelligence is not enough. Intelligence plus character—that is the goal of true education. The complete education gives one not only power of concentration, but worthy objectives upon which to concentrate. The broad education will, therefore, transmit to one not only the accumulated knowledge of the [human] race but also the accumulated experience of social living.” —Martin Luther King, Jr., “The Purpose of Education,” published in 1947 in the Morehouse College campus newspaper The Maroon Tiger
“The vision of the teacher should be at once precise like that of the scientist, and spiritual like that of the saint. The preparation for science and the preparation for sanctity should form a new soul, for the attitude of the teacher should be at once positive, scientific and spiritual.” —Dr. Maria Montessori, p. 107, The Advanced Montessori Method Volume I
“Michael provokes and encourages logical, rational thinking that you will use beyond your time with him. He helped me develop critical thinking that immensely benefited me. He will teach you how to deconstruct complex problems and come to a decision yourself.” —Jacob L, ex-high school student
“Michael has been a great tutor that I’ve known for more than 5 years; not only has he helped me in my education but also in my health and body. Not only has he taught me fitness techniques but he has also shown me how to think about how the techniques help my body and how to keep it in top shape for the long run, like keeping my back straight so that it won’t compromise my spine when I’m much older. He has helped me prepare for my Air Force physical exam, showing me ways to preserve my energy to be able to run longer. I’m sure you can learn more than a thing or two just by hearing him out. I thought exercising was hard, but really, it’s not hard at all after you get used to it. Everything he’s taught made me actually know that I can change and become the man I’ve always wanted to be.” —Tam W, ex-high school student (mathematics) and fitness client
Introduction and Background
My teaching philosophy has changed through the decades. At the start, when I was a young adult and out of college, what I had in place of a philosophy was a hodgepodge of habit, ignorance, good ideas, wrong ideas, confusion, all mostly isolated and unconnected and unverified. I did not, fully and on principle, know how to know: I did not have an understanding of logical standards and conceptual practice. After decades of struggle, decades of immersion in philosophy, decades of immersion in conceptual thought, decades of teaching, I have now an explicit, evidenced, proven, developed philosophy of teaching — and understand that I have more to learn. This is an early draft, however. In the future, I will have to edit, add to, subtract from, develop, and improve it. What’s more, as Blaise Pascal first said, but quoting Ben Franklin: “I have already made this paper too long, for which I must crave pardon, not having now time to make it shorter.”
Metaphysics: Who and Where We Are
We are rational animals living in a natural world, a world causal, lawful, and intelligible. As humans, we are physical, social, independent, emotional, rational, perceptual, experiential, biological, ecological, historical. We can perceive, think, reason, imagine, dream.
We develop. We do not emerge, like Athena from Zeus’s head, a ready-made adult. We are not ready-born Gods of Wisdom and War. To gain mental and physical prowess that we need for the good life and the moral life takes work. We have to go through stages of development, as humans have known for millennia but as we’ve learned more and more through those millennia.
We gain more and more independence — varying for each individual depending on numerous and varied factors — as we go from helpless newborn to toddler to child to adolescent to young adult to adult. We start out having no control and needing total assistance, and, typically, develop into self-sovereign social animals who can take care of self and contribute to the tribe.
We do this in a world where much is out of our control, some is in our control.
Our heart beats automatically, breathing is semi-autonomous, our bodies grow automatically, stars are out of our control, planetary orbits and quantum entanglement are out of our control, the world does what it does regardless of us, perception is a strict cause-effect relationship — but we can each control the conceptual content of our own mind.
As the Serenity Prayer says — God or some other God; God or no God —
“God, grant me the serenity
To accept the things I cannot change;
Courage to change the things I can;
and wisdom to know the difference.”
Many animals train, or “educate,” their young. The young learn survival skills, they learn the ways of the world, they learn how to find food, they learn what is food, they learn what to seek and what to avoid. As long as they are in a “species-appropriate” environment, the environment helps make them functional and healthy. As wildlife tracker, survivalist, and naturalist Tom Brown III says “Animals are instruments played by their environment.” And it’s something scientists are learning more and more about as they delve into and develop the science of epigenetics.
So also with humans. We are animals. We are biological. We are part of nature, not separate from it.
But we know from our own experience, from the world around us, and from human history, that in some things we are correct, in some we are in error, and in some we lie. We learn the differences at a young age. Some people also go on to identify that it’s in thought, not perception, that those things can happen. And some learn, at a higher level of abstraction, that it’s in reason that those differences happen, reason being an aspect of human consciousness, not a part or separate faculty on its own.
One type of human error is sometimes found in fad or convention. We are social animals, so we inevitably, by nature, seek to belong to some tribe and to conform to it — albeit in varying degrees and measures for each individual person.
As people have learned through history, in slow, hard-earned steps, conceptual thought is at the foundation of much that we do and of all that is distinctively human. It is critical to knowing the truth. It affects us in all sorts of small ways, in our success at survival, and in life-or-death ways. Farming. Building. Planning. Hunting. Gathering. Tool-making. Technology. Ethics. Aesthetics. Politics. Policing. Safety. Security. Friendship. Judgement. Law. Mathematics. Science. Technology. Health. Wellness. Fitness. Philosophy. Natural history. Ecology.
So, to get to the root of what we need training in to survive, we need training in reasoning: its content and its method. While we can and should get training in other things, the one crucial, vital thing we need in education is training in reasoning.
In human training or education, we need to learn ideas that will help us survive and thrive, and we need to learn techniques and methods of how reason works. We need to learn to use reason independently so we can check the truth of our own conclusions (and those of others) so we can live better on our own and can help the tribe stick to the truth. We live best when we commit to the truth and when fad and convention are informed by truth.
So my focus is on reason as a tool of survival, a tool of surviving and thriving physically, independently, socially, emotionally, rationally, perceptually, experientially, biologically, ecologically, historically. (Mine is not a Platonic or Kantian view of reason, nor that of irrationalists or metaphysical materialists. Et. al. )
We need understanding of the world and an understanding of our understanding so we can each make our way through the unknown that is our future.
It is for all people everywhere all through time.
“Under the sky, under the heavens, there is but one family.” —Bruce Lee
“I look to a day when people will not be judged by the color of their skin, but by the content of their character.” —Martin Luther King, Jr.
Epistemology: How We Know
“From the statements made by the noble Shaykh, it is clear that he believes in Ptolemy’s words in everything he says, without relying on a demonstration or calling on a proof, but by pure imitation (taqlid); that is how experts in the prophetic tradition have faith in Prophets, may the blessing of God be upon them. But it is not the way that mathematicians have faith in specialists in the demonstrative sciences.” —Ibn al-Haytham (Abū ʿAlī al-Ḥasan ibn al-Haytham, c. 965 – c. 1040.)
“Nothing is in the intellect that was not first in the senses.“ —Thomas Aquinas, De veritate, q. 2, art. 3, arg. 19
“Galileo’s radical renewal sprang, nevertheless, from the Aristotelian mind set, as it was taught at the Jesuits’ Collegio Romano: human reason has a basic ability to recognize and understand the objects registered by the senses. The objects are real. They have properties that can be perceived, and then ‘further processed’ according to logical rules. These logical concepts are also real (if not in exactly the same way as the physical objects).” —Atle Naess, Galileo Galilei: When the World Stood Still
Perception is the fundamental on which all conceptual knowledge is built. It is the cognitive and epistemological given. It is valid knowledge from which we can form valid concepts and generalizations. Hume, Kant, Popper, Kuhn, et. al. were in error.
All animals move through the natural environment, relying on direct perception, as J.J. Gibson and others have identified and discussed. Perception is not Cartesian but Ecological. (The work of John Boyd, his OODA Loop, is also relevant in this context.) From there, humans, having a rational consciousness, can develop more and more abstract concepts and propositions.
But we have to develop cognitively as we grow from newborn to adult, a development Piaget and others have studied.
“Jean Piaget’s theory of cognitive development suggests that children move through four different stages of mental development. His theory focuses not only on understanding how children acquire knowledge, but also on understanding the nature of intelligence. Piaget’s stages are:
Sensorimotor stage: birth to 2 years
Preoperational stage: ages 2 to 7
Concrete operational stage: ages 7 to 11
Formal operational stage: ages 12 and up
“Piaget believed that children take an active role in the learning process, acting much like little scientists as they perform experiments, make observations, and learn about the world. As kids interact with the world around them, they continually add new knowledge, build upon existing knowledge, and adapt previously held ideas to accommodate new information.” —Kendra Cherry, “The 4 Stages of Cognitive Development: Background and Key Concepts of Piaget’s Theory” (verywellmind.com, 31 March 2020)
Reasoning is volitional. We can make mistakes in forming concepts, engaging in abstraction, identifying causally relevant similarities or differences, engaging in selective focus, identifying essences, identifying causes, forming cognitive connections, deducing, inducing, theorizing, etc. To do it right, we need to obey rules of reasoning and rules of reality.
Abstract knowledge is hard-earned and is not obvious. So we should be patient and should respect different opinions — besides, each of us is wrong on some things, too. No one is omniscient.
We learn different things in different ways.
In physical fitness, we learn by doing. We need to perceive, experience, and do. When teaching, I have to do some conceptual communication: what we are doing, why, how it helps, how one thing is connected to another. Some things we get only on the conceptual level; we cannot grasp principles in perception or by doing. But fitness is heavily experiential.
In academic subjects, I work on helping students develop conceptual understanding. I work to make sure their “learning” is not merely memorizing words, imitation, or habituating social behavior. I strongly disagree with the philosophies of education of Sparta, Nazi Germany, Communist countries, etc. We can learn something still, of course, from some of them like Vygotsky, and we should study some of their thought, but it is fundamentally wrong.
We still need to learn words, remember them, and associate them with concepts and propositions. We need basic learning here. But for conceptual learning we need more.
Students construct their own content, but, living in a complex world and having a complex consciousness, need guidance through reality and thought so they can properly form concepts and propositions logically and truthfully. They need full content, the logic of the conceptual level, and connection to other causally relevant ideas. The teacher provides the wisdom to do so fully, accurately, logically, and efficiently.
Education should not be indoctrination and should not be training by force, social pressure, or intimidation. Education should be free — but not free as in “unschooling.” We need guidance and direction in our education. We need the oversight of wisdom. We need someone who has a grasp of a subject, its methods, its connection to other human knowledge, its connection to human survival, and its use of logic — things we can never know when swamped in learning specific concepts of a subject, things we don’t know, and things we don’t even know we don’t know. All this will vary by student; we have to take into account the nature of each student in determining what to do and when.
I use my understanding of logic to help students form concepts and use them. It informs what I know, how I know, what I teach, how I teach, how I answer questions, how I help students with difficulties, how I structure knowledge, scope, and sequence, and more.
Concepts of Logic.
- Concept-formation: the cognitive process of forming a mental unit from a number of things based on their similarity(s) and against a background of contrasting things, all in a broader cognitive context.
- Classification: the cognitive process of relating a number of concepts based on some rules of comparison and on the concepts’ relative abstraction and generality.
- Definition: a statement that identifies the (epistemological) essence of a concept; it identifies the nature of a thing.
- Argument analysis: the cognitive process of assessing the truth and validity of a chain of reasoning.
- Deduction: the cognitive process of applying a generalization to a concrete individual or to another generalization no more general than the first.
- Induction: the cognitive process of forming a generalization.
- Reduction: the cognitive process of retracing a generalization from top down; it is like “reverse engineering” the induction.
- Abstraction: the cognitive process of focusing on similar aspects of a number of real, concrete individuals or of a number of already-formed abstractions, amidst a range of differences and in a context.
Principles of epistemology.
- Foundation. All conceptual knowledge is an identification of and is abstracted from the evidence of the senses.
- Context. All conceptual knowledge is conditioned by other cognitive elements.
- Hierarchy. All conceptual knowledge has a structure of abstraction based on perceptual experience; some abstractions come from and build on other abstractions.
- Integration. All conceptual knowledge is interrelated. Integration is involved in all aspects of conceptual knowledge, bottom to top, depth to breadth.
- Independence. It is the individual who has conceptual knowledge.
“ádántsiíkees” (i.e., “think for yourself”) —part of the Navajo HézhodJif Kehgo Na’nitin (the Blessing Way (Positive) Teaching)
“Each truth you learn will be, for you, as new as if it had never been written.” —Ancient African proverb
“Trust yourself. Think for yourself. Act for yourself. Speak for yourself. Be yourself. Imitation is suicide.” —Marva Collins
“I fully agree with you about the significance and educational value of methodology as well as history and philosophy of science. So many people today — and even professional scientists — seem to me like someone who has seen thousands of trees but has never seen a forest. A knowledge of the historic and philosophical background gives that kind of independence from prejudices of his generation from which most scientists are suffering. This independence created by philosophical insight is — in my opinion — the mark of distinction between a mere artisan or specialist and a real seeker after truth.” —Albert Einstein (Letter to Robert A. Thorton, Physics Professor at University of Puerto Rico (7 December 1944)
“One of the advantages of a multidisciplinary educational approach is that you gain a more comprehensive grasp of the world. A multidisciplinary approach combines aspects of each department into the study plans of the other, rather than focusing on individual divisions and their subject topics individually.
“Multidisciplinary learning isn’t simply a theory of education; it’s a practical manner of seeing the world. Why should school be any different? When you graduate, you’ll be interacting with individuals from all walks of life, so why should school be any different? It will be easier for you to perform in the working world if you can compile and collaborate with individuals from many walks of life.
“There is yet another breakthrough with this approach. Those who experience this process of learning will gain a strong and deeper understanding of the subject. When that happens, a student has a clear vision of what he likes, where his aptitude lies and what he wants to pursue.
“This is completely missing in a subject-based curriculum approach. And why is that the case? It is because when a student learns subjects in isolation, he does not get the kind of exposure to knowledge that is needed for him to take a clear call on whether this subject works for him or not?” —Shilpa in ”Multi-Disciplinary Approach that Boosts Learning Outcome to the Next Level” (11 March 2022)
Ethics: What is Right and Helps Us Live
“Peace will always rule the day where reason rules the mind.” —Marva Collins, Ordinary Children, Extraordinary Teachers
“An error means a child needs help, not a reprimand or ridicule for doing something wrong.” —Marva Collins
As introspection, everyday life, and history show, we are volitional, rational animals. What is good is based on both choice and nature. We do best when we choose a value, when we understand what is good and why, and when the value actually promotes human life. So students need guidance and advice, not oppression, intimidation, or compulsion.
Besides, to respect their metaphysical nature is to respect their choice — in context of their stage of development and cognitive context. We understand that we do not let a 2-year old run into a racing highway even if the 2-year old wants to. We prevent it; we apply to their action some adult wisdom until they have it on their own. However, we would rarely prevent a 32-year old from most actions we think wrong; they are self-sovereign, and need to learn on their own and to take the consequences of their decisions and actions.
In teaching, I help students develop the knowledge and skills to make good decisions. I do not force-feed them, and I do not mindlessly accept anything and everything they do. I believe in neither authoritarianism nor apathy.
I’ll help them put an issue in a broader context, help them see and think through the implications and consequences of an idea, give them some ideas to think about, discuss Daniel Dennett’s rules of discussion (aka rules of “criticizing with kindness” or “arguing with kindness and care”), discuss The Five Whys (i.e., a root-cause analysis), discuss reasoning, and discuss how to use question-and-answer skills to think through something on one’s own. I’ll discuss real people from history, fictional characters in literature, and historic events.
And, in teaching math, science, SAT/ACT prep, and more, I am giving students cognitive tools and confidence to make good decisions and to know the truth.
As long as students are not out of generally accepted ethical bounds, I help them work through issues on their own. I do not program, “brain-wash,” deceive, intimidate, etc. I respect their conscience and their beliefs.
“…the function of man is to live a certain kind of life, and this activity implies a rational principle, and the function of a good man is the good and noble performance of these, and if any action is well performed it is performed in accord with the appropriate excellence: if this is the case, then happiness turns out to be an activity of the soul in accordance with virtue.” —Aristotle, Nicomachean Ethics, 1098a13
“He is happy who lives in accordance with complete virtue and is sufficiently equipped with external goods, not for some chance period but throughout a complete life.” —Aristotle, Nicomachean Ethics, 1101a10
Content: What We Need To Know
We need to understand history, science, mathematics, literature, and grammar/language. History, because it is the story of humanity and human nature, and more broadly the story of all that has happened in the world to this point, all that has conditioned and shaped us and made us possible. Science, because it is the organized study of the external world and natural law. It teaches method and content we need to be true to nature and human nature. Especially important is biology: the science of life, nutrition, diet, sleep, exercise, farming, ecology, environment, and the world we need in order to thrive as the rational animals we are. Mathematics, because it is the science of measurement, teaching some tools of science and providing clues to the essence of deduction, induction, and the application of knowledge. Literature, because it is the form of art that uses concepts as its medium; it is the predominantly conceptual art. Grammar and language, because they are important tools for conceptual thought.
Physical education should be focused on training and teaching students to understand fitness, health, and wellness and to have biomechanically correct practices. It should teach technique. It should teach how to breath, sleep, and eat, and why. And it should do so by teaching what humans have done through history to be fit so we can better sift out right from wrong in what inevitably comes up in human society: convention, fad, error, etc. And, as relevant, we should teach what other animals do. Some things, after all, are principles. And sometimes, seeing the effects of some practice on “the other” helps us see it in ourselves.
Physical education should not be exclusively sport or “fun:” sports, or any human movement, done without good technique is detrimental to our health. It merely strengthens dysfunction. We need to learn appropriate technique and biomechanics so we strengthen optimal function, can practice it through life, and have both conceptual and experiential knowledge. Physical education should be about “leading out” good practices, not just “doing stuff.”
Other content in education is secondary or tertiary.
In all things, we should in treating seek to develop conceptual awareness of a subject and to embed it in human survival.
A teacher is one who helps train youth to survive and thrive in this world, with all its complexities. A teacher’s role is to help youth become self-sovereign social animals living in a natural world. What we need to survive is identified in history, ecology, and evolution.
I love the identity in all our infinite variety, and I love the infinite variety in our identity.
A teacher is one who understands a survival skill and helps train youth to master it. The survival skill that is especially important and fundamental is human reasoning. A teacher should understand how an idea or skill is used across many areas of life and many interests. A teacher should not know an idea or skill without knowing anything about how it helps us survive.
All conceptual knowledge being based in the evidence of the senses, all humans having perceptual access to reality, and all humans having a consciousness rational in nature, knowledge of truth and the good is “democratic:” it is accessible to all. No person or group has some special insight like Platonic Philosopher Kings. We each can err, we each can lie, but we each can also know truth and good on our own.
Assessment should hit a variety of measures, should be appropriate to the learning being assessed, and should be appropriate to the level of the student. It should include fact recall, analysis, synthesis, multiple choice, free response, essay, discussion, conversation, decision-making, wisdom, practice, and lifestyle (does something become part of a student’s life and thought).
When teaching, I constantly ask students questions, and constantly ask why and how they got an answer, even before telling them whether the answer is right or wrong. I seek to get them to focus on chains of reasoning rather than isolated ideas, and seek to have them put answers or ideas in a broader context. Some things I teach I know that the student will not grasp or understand for years or decades; they will grasp the meaning and value of these things, if they get them, when they are in their 20s or 40s. So, some assessment is very long-term indeed.
In fitness, I assess students by their technique at a task, by measures of volume, intensity, and complexity at that task, by their performance in an “obstacle course” of various skills, by their skill at moving in natural environments, and by their practice of skills and good technique in life outside the “gym” or “training session.”
In fitness, learning is sculpting motion in an environment, which takes time, practice, and feedback. Learning is habituating good fitness technique appropriate for some context. Learning in fitness happens by doing skill-based training that helps us refine technique, build strength, and increase our level of skill. We learn by interacting with our environment, as JJ Gibson and others have identified in their ecological approach to learning, perception, and movement.
For fitness, no one needs to know anatomy and physiology; no animal on earth or through history suffers from not knowing those subjects — they do not even know that they don’t know. (Those subjects are, of course, important for physical therapy and medical care.) Animals get fit and strong by moving through their environment to survive. So do we — given a species-appropriate environment and lifestyle. We need to keep the ancient in the modern and couch the modern in the ancient. We need ancient tradition and modern science.
In academics, learning is conceptualizing things in the perceptual world we are immersed in and connecting those concepts into a total awareness of reality. Students need to identify similarities and differences in things to form a mental unity: a concept. They need to find naturally, causally, cognitively relevant similarities in a kind of thing, which they learn in part by contrasting or differentiating it with other things in a context. They need to construct a mental unit of, say, “dog” or “adverb” or “projectile” or “differential,” and make a habit of associating it and a word. Then they need to construct connections between this concept and others. They need to form propositions, and to analyze, synthesize, classify, define, investigate, and process. Going deep and broad takes more time and effort than does memorizing words or going wide and shallow, but the former is what we need for a life as a human: a rational, self-sovereign social animal living in an ecosystem.
A teacher is one who helps youth understand the good and the ethical, who encourages them to choose it, but who leaves them free to choose it (in varying measure depending on the student’s age and cognitive function). Also, students have to care about what they learn — hence teachers need to motivate them by demonstrating how an idea is important for human survival or for the student’s own personal interests. Sometimes we have to motivate by joke, game, drama, acting, etc. We have to involve them, we have to show them we care, we have to show them we’ll be an adult who will help them be safe and secure and who they can trust.
I believe that, for good classroom management, a teacher needs:
To be someone who deserves respect
To earn respect
To have some wisdom, the more the better
To be committed to experience and the evidence of the senses
To be alert and to be a quick thinker
To adapt to circumstance
To see circumstance in context of principle and wisdom
To be professional
To be friendly, polite, personable, and benevolent
To have a sense of humor
To have good character
To be a good person
To have good ethics
A teacher needs to know about both individual psychology and group dynamics.
A teacher needs to consider the individual, the class, the school, the groups the individual belongs to, the family, the locale, the society.
What matters infinitely more than “having experience” is being a good person and having good relations with others. After all, we all know parents, police, professionals, and teachers who have done their thing for a long time, but still yet are not good at it — and some of whom abuse their responsibilities. And what makes for a good parent is not “having experience,” but having good character and wisdom and being able to learn and adapt.
I respect students and aim to earn their trust. I don’t treat them like dictation machines or helpless children, but as competent humans making their way on this earth, immersed in their own activities and interests — interests personal, social, and natural — and working to become adults.
I strive to be knowledgeable, be wise, be clear, be relevant, and be interesting — and so to win their attention and earn their interest. I strive to bring wisdom and the whole human story to bear on their own story, and to help them see their story benefits from knowing the whole human story.
I aim to have good classroom management skills by demonstrating that I care, that I can help, that I recognize students as competent, good human individuals, that I want to help them be the best they can be, and that we are in this together. We are, at least temporarily in life, in a cooperative tribe.
So, for example, in an SAT prep class on a Saturday, taught to two girls who were mutual friends, one said after one class: “Today was fun.” And the other agreed. Through the whole 5-week prep course, they were motivated, they learned, they had fun.
During the COVID crisis of 2020, in online classes via Outschool (which uses Zoom), students were glad to attend my classes, and parents would report that they never had to remind their child to attend. In many cases, I know students were present, interested, and involved — they did not turn off their camera then go do something else — because they answered every question in the chat, and their answers were relevant even if not correct.
In some two-week long summer classes, I’d get private chat notes, day after day, from more than one student, saying “I love this class.”
“For the past 2 years, my son has taken a different math class over the summer with Michael Gold just to sharpen up before the upcoming school year, and the payoff has been tremendous. He loves this teacher’s style, is surprisingly riveted during the class, always on time, and absorbs the equivalent of an entire years worth of work in 2 weeks, helping him to breeze through the academic school year with great grades and a real sense of satisfaction. You’re kids might roll their eyes at you at first, but they will thank you if you are lucky enough to grab a spot in one of Michael Gold’s classes. He is fast, clear, to the point and gives real life application examples to make math make sense. 10 stars if I could.” —Lia J., parent, “Summer Algebra 2 Review/Preview, Session 1,” 10 Jul 2021
“Never before was my son as motivated and interested in a course. It was a delight to see and fun to hear him share about the lessons without prodding.” —Paula L., parent, “High School Geometry, Semester 1,” 2 Jan 2022
Students I tutor and teach often tell me some unfortunate things some of their teachers do. They are glad that I do not do the same. I do not get bent out of shape, and lose my temper and senses, if I am not called “sir.“ Or if the student does not understand something. Or if they question me. Or if they want a different explanation. I sometimes use the opportunities (discussing an angry outburst of a teacher) to help students learn how to analyze a situation, think about it in a broader context, understand people, understand themselves, and be wise.
I learn and adapt quickly. In Feb 2023, I did a 20-25 minute fitness presentation and activity with about 20 boys aged 6-9. Some friends, who have their boys in Trail Life, asked me to be part of one regular hour-long Monday evening meeting. So, I thought up ahead of time what I could do with these boys — in an age range that I’ve never worked with — that’d help them in fitness and on their camping and hiking trips. I went over some basic technique of walking, running, breathing, lifting, and crawling. At the start, I was working with all 20 boys, but saw that they had too much downtime and some were getting distracted. So I adjusted on the fly, and, after demonstrating good technique in a skill, broke the boys into three groups and asked some parents if they would take care one group and other parents if they would take care of another. We’d practice, come together for a new skill, then break apart again to practice. Everyone loved the evening, and some parents said it was one of the best meetings they’ve had. Awesome. I love to help. I also followed up with my friends, giving them some links to more fitness information that they could (and did) share with other parents in the group.
Through the decades, I have had classes of 10-20 in public schools, I have had individuals or small groups in in-person tutoring, I have had individuals, small groups, and classes of 10-15 on Zoom, I have had 8-hour-long fitness seminars with a few adults who I’ve kept interested, involved, and learning the whole time.
Years ago, when I taught in public high schools, some of my classes would have surprise birthday parties for me. Or they would have lunches where all the students brought some dish in. That was earned even when I was young, unwise, and seriously sleep-deprived. Now I am older, wiser, more experienced, more personable, and more respected.
And I’ve had some ex-students, when older and married with kids, connect on Facebook. They appreciate my teaching them. One of them even said that I inspired her to get a degree in mathematics!
A teacher trains youth in a subject in order to teach reasoning in order to teach human survival skills in this natural world. Content should be primarily what youth need to understand in order to live well.
Electronics are great tools, but get more and more user-friendly as the months go by. They are easy to use, and require no understanding of the underlying physics. As part of modern life, they should be incorporated into education. But, even though we are fascinated by them and sometimes find them “sources of magic,” they are not and should not be the focus of education, which is and always remains training youth to survive and thrive as rational, self-sovereign social animals in this natural world.
Electronics are tools, and tools are no more useful than what our knowledge and creativity can do with them. And they are our servants; we are not theirs. The liberal arts always are and remain the arts of free, independent peoples. I believe that, in education, we should use calculators, computers, the Internet, smart phones, and other electronic technologies, but only as aids to or concretizations of thought (“concretizations being, for example, marking lines on a parchment or in the sand to help us tally). They should not be a substitute for thought or independent awareness. Technology yes, but cognitive method and epistemology is far more important and fundamental.
“The more monies we spend, the less children learn; because the more machines we have there, the more gadgets, the more gimmicks, the less children have to really think — the less they have to use their innate abilities, their curiosity, their brains.” —Marva Collins
Some techniques of learning I use are what the Learning Scientists (https://www.learningscientists.org/ ) call spaced practice, retrieval practice, elaboration, interleaving, concrete examples, dual coding. The latter is the idea that everyone needs every “learning style:” reality demands it. We need to be holistic and multidisciplinary.
As Robert Heinlein said: “A human being should be able to change a diaper, plan an invasion, butcher a hog, conn a ship, design a building, write a sonnet, balance accounts, build a wall, set a bone, comfort the dying, take orders, give orders, cooperate, act alone, solve equations, analyze a new problem, pitch manure, program a computer, cook a tasty meal, fight efficiently, die gallantly. Specialization is for insects.”
Cognitive Integration, Cognitive Hierarchy, Life, and Logic.
While some examples of how math is used in science or technology are good, as are abstract, abstruse applications, what students need are things they can get in the here and now. They need examples meaningful in their practical life as it is now. They need things they could dig into deeply and broadly now, not only in some future life or in someone else’s life. And they need to see me taking it seriously, and walking the walk. Here are some examples, sketched out but not developed in the depth and breadth I can.
In teaching the Ideal Gas Law, PV = nRT, I dig into the science, the scientific development, the logic, and the practice. I ask why scientists did V = kT and P = k/V first. “Why did they not get it? Why didn’t they just get that PV = nRT??” Students do not know why, of course: they do not know the history of science. So what scientists did can seem silly or look like stupidity to them, as might the different theories of acids and bases or as might the different theories of our solar system. But understanding what happened in the history of science will help students themselves develop knowledge in their own futures, especially in their careers.
So I explain how Boyle was studying the natural world before any accurate, useful thermometer had even been invented. While someone can get a qualitative idea of how temperature affects pressure or volume, no one can identify a quantitative, mathematical relationship without a quantitative, mathematical thermometer. No thermometer, no V = kT or P = kT. Then, after the invention of the thermometer and the scales of Fahrenheit and Celsius, scientists could discover that V = V0(1 + T/273) and so also P is a function of T.
It’s not obvious that the individual laws can be combined, or how or why to do so. No one can get the idea that the laws can be put together into PV = kT, because the temperature scales they had, Fahrenheit and Celsius, had zero at the freezing point of water and had the potential to go indefinitely into the negative. At this point, they need an absolute temperature scale: we cannot really have e.g. V = kT if volume does not go toward zero when temperature goes toward zero. And we cannot just arbitrarily make it so because we feel like it; we need empirical and theoretical reasons to do so.
Eventually, William Thomson, Lord Kelvin, got things together. So scientists could grasp that PV = kT. But not until the atomic theory was developed, the concept of “mole” was formed, and moles could be measured, could scientists go further and grasp that PV = knT. And not until the kinetic theory of heat (and the atomic theory) were developed could scientists grasp that PV = nRT.
I give students a “hierarchy diagram” of this to help them grasp and retain it.
It is important, I think, to let them know that PV = nRT is not a dead-end or final conclusion — as I thought when I was young. PV = nRT has context, and is embedded in broader contexts. It does not apply when work is done. It took the study of heat and gasses for scientists to grasp that PV can be equivalent to work, which idea was one thing that led to the broader idea of the conservation of energy. The further study of physics led them to Van der Waals’ equation: (P + a/V^2)(V + b) = RT.
But I let them know some ways it matters in everyday life. In car racing, race teams have to know that P = kT and have to apply the idea. Racing cars must start out a race with underinflated tires, or they will have blowouts. The idea is also important for us: underinflated tires compromise how our cars drive. Underinflated tires decrease our control over a car, decrease handling, affect tire traction, etc. Underinflation causes the sidewalls to wear and fail more quickly because of the lack of mechanical support. P = kT matters for our safety and wellbeing. It is not theoretical or something affecting others. It’s real and it’s in our “backyard.”
In all this, I can share with students scientific articles, scientists’ original research, scientific history, etc. I can dig into observation, concept-formation, induction/generalization, definition, classification, the logic of experiment, the use of mathematics, theory-formation, and more.
One simple idea; so much to learn. Students learn logic and how knowledge develops, and they learn how knowledge and science are practical. They learn ideas and methods they can use in all areas of life, wherever they turn their minds to think.
Years and decades ago, I knew little about the significance and deeper value of geometry and geometric proof. But, though clueless, deep inside, I wanted to know. I wanted to know for myself, and I wanted to know to better serve my students.
After I had already started teaching, I lucked into my hands a copy of Plane Geometry by Welchons and Krickenberger. In it, they said: “It is said that Abraham Lincoln studied geometry in order to make better arguments in court.” That was intriguing — but very empty and unsatisfactory. How did it help? In what ways exactly? What is the detail? What was the process? What were the logical principles involved?
And so I began investigating. Curiosity, learning, and understanding helps me, which in turn helps my students. They benefit. But I cannot help them any more than I’ve helped myself. The understanding and wisdom I can teach and communicate is limited by the understanding and wisdom that I possess.
Through the years, I found out a little from sites such as the National Park Service’s Lincoln Memorial site, where they discuss Lincoln’s education and early years. I found out much more in Drew R. McCoy’s scholarly article “An ‘Old-Fashioned’ Nationalism: Lincoln, Jefferson, and the Classical Tradition.” Then, around 2020, during the COVID crisis, I dug deeper by reading (most of) the 464-page book Abraham Lincoln and the Structure of Reason by David Hirsch and Dan Van Haften. All of which I share with students, so they can pursue it if they are interested.
Decades ago, I’d discuss some of what Lincoln got out of geometric proof, but could little develop the idea.
Now, given what I’ve learned about Lincoln — and in context of my deep, decades-long study of logic and reason — I can develop the importance of geometric proof to all areas of thought, in depth and in breadth. Physics. Chemistry. Scientific method. Law. Politics. Ethics. Judging other people. Understanding other people. Getting along with others. Being fair and being understood. Writing. Logic. Epistemology. And more.
I find it valuable, given human history, to discuss how geometric proof relates to judging people well, understanding the other person, committing to the truth. Geometric proof helps us better understand Dennett’s rules of discussion:
“How to compose a successful critical commentary:
- You should attempt to re-express your target’s position so clearly, vividly, and fairly that your target says, “Thanks, I wish I’d thought of putting it that way.”
- You should list any points of agreement (especially if they are not matters of general or widespread agreement).
- You should mention anything you have learned from your target.
- Only then are you permitted to say so much as a word of rebuttal or criticism.”
—Daniel C. Dennett, Intuition Pumps And Other Tools for Thinking
Such a process is extremely, immeasurably valuable. I love introducing it to students and discussing it with them. It is a process to use when possible, but is also a framework to adjust based on situation, context, and time-pressure: sometimes we can work through it fully, but sometimes we have to adjust it to do the best we can with an immediate, 30-second response.
And it can lead us to The Five Whys, aka Root-cause Analysis, another important technique for students to know, one that we can use wherever we turn our minds to think, all through life.
I strive to help students be, in their thinking, like a lawyer, who “must study the evidence presented during the trial and choose that which has the greatest bearing on the case; he must search the court records for past decisions on similar cases; and then he must arrange his argument in a logical order, proving each statement by giving an acceptable reason for it.” —Welchons and Krickenberger, Plane Geometry
And to be like Abraham Lincoln, who was “well-known for the logical precision of his political speeches, in which he characteristically stated propositions and proceeded to prove them, both by adducing documentary or empirical evidence and by deducing them from axioms or self-evident truths.” —Drew R. McCoy, “An ‘Old-Fashioned’ Nationalism: Lincoln, Jefferson, and the Classical Tradition.”
One simple idea; so much to learn. Students learn logic and how knowledge develops, and they learn how knowledge and science are practical. They learn ideas and methods they can use in all areas of life, wherever they turn their minds to think.
I have gone through a similar development with area and volume. I have a deeper, broader understating of the concepts: I have connected them to numerous and varied areas of human thought and action. In the past decades, in digging into interests — e.g., nature, ecology, fitness — I have learned things I never knew before, and things I did not even know there were to know.
For example, the ratio of area to volume is important for thermoregulation and thermodynamics in animals. Allen’s Rule states that “the body surface-area-to-volume ratio for homeothermic animals varies with the average temperature of the habitat to which they are adapted (i.e. the ratio is low in cold climates and high in hot climates.)” (Wikipedia entry “Allen’s rule.”) So, depending on time, student interest, and student cognitive level, we can delve into why Polar Bears are bigger than Grizzly Bears are bigger than Black Bears. We can delve into a reason why some Horse breeds are bigger than others. We can think about Mice vs. Elephants — and why Elephants have big ears. We can discuss scaling, fractals, metabolism, body temperature, ecology, evolution, and more.
And now, being a certified fitness trainer who trains in nature and the wilderness, area and volume are real for me in ways they are not for many others.
I lift, carry, and throw logs and pieces of concrete. They do not have numbers on them as does stuff at gyms. To know what to do on light, medium, and heavy days, to know my progress, to know the meaning of what I am doing — and to know if I can even lift something at all — I must calculate and estimate the weight of things. No way around it. I cannot bring a scale out all over the woods and weigh whatever I want. I have to math it out. I have to measure the dimensions of a thing, calculate the volume of a triangular prim or cylinder or conic frustum, find the thing’s approximate density on the internet, then calculate the weight.
Many students get the reality and practicality of this. It has meaning. They say it is “cool.” And I agree.
From volume and weight in this context, we can travel all sorts of paths and investigate all sorts of contexts: levers, lever arm, leverage in the deadlift, biomechanics, rotational dynamics, strength training, adaptation, diet, human history, human fitness, logical context, classification, causality, animals in an ecology, epigenetic, and more.
Math texts have some applications of conic sections that are good, and I have used those for decades — but back at the beginning of my teaching career I did not understand the science or the historical development of those applications. After studying epistemology and studying the history and philosophy of science, I get conics on a whole different level, in depth and breadth. Now I know exactly how and why they were used by Copernicus, Kepler, Galileo, and Newton, and can teach it to students, depending on time, student interest, and student cognitive level. And I can teach other applications such as satellite dishes, car headlights, road construction, LORAN, etc.
I can go into some detail how Kepler used Mars to determine that ellipses are needed to understand planetary orbits and how Galileo used certain experiments to determine that parabolas are needed to understand (local) projectile motion — and how Galileo knew that, under similar conditions but without the earth present, a projectile would spiral into the center.
So that students get that this is about human survival and is not some “ivory tower” thing, I will put it all into the human story of using stars to know where we were in the year, when seasons would change, when we’d have more or less rain, when flooding would occur, when to plant crops, when animals would be migrating, etc. It develops into navigation, which helps people not only survive and thrive and get things done, it helps people avoid ship wreck, loss of wealth, and bankruptcy — and, more importantly, injury and death. Human life is at stake.
Conics are of general human importance, but they are also of personal importance.
I have used them in biology and natural history and in fitness.
Since 2017 or before, I’ve been learning and studying wildlife tracks. I love it. I’ve learned to identify lots of species — and love that I have still so much more to learn about it all.
In May 2020, I found some tracks I was not sure about. Pure, awesome serendipity. My thought was Western Rat Snake, given the area and my context. I took lots of pictures, and posted them on a Facebook group dedicated to studying and sharing wildlife tracks from an educational perspective. Someone said they were American Alligator tracks. Wow! That was a first for me! A “lifer.”
So I looked up pictures and videos of tracks of Western Rat Snake, Snapping Turtle, Alligator Snapping Turtle, and American Alligator, and came to the conclusion that, yes, the tracks were from American Alligator.
Having some pictures with a tape measure and ruler in them already, but wanting more, I went back out the next day and got some good pictures of the size of the tail-drag tracks. I found they had an elliptical shape, and so I could measure two perpendicular axes to characterize the tracks.
In contrast, some snake tracks I found around that time had some (semi) circular sections and some elliptical sections, of different measure than the Alligator.
Now that is a fascinating story students love. It connects math, biology, animals, nature, and wonder — and leads into physics, biomechanics, ecology, and evolution. I have pictures; I share them. It is real.
I love to discuss how the tail tracks were formed. What was the physics? What was the biomechanics? I take students through thinking about how Alligators use contralateral motion, just as we do, and how that motion causes their body to deform and the tail to respond. How is it different than the motion of Snakes? And Turtles? We open the door on all sorts of questions, inquiry, and paths of study.
Developing the discussion into tracking leads to all sorts of observations and questions about track shape, depth, pressure, substrate, animal foot, animal locomotion, “pressure-release” methods of tracking, ecology, animals being “shaped” by their environment, epigenetics, human health, life, knowledge, understanding, appreciation, and being immersed in the world with curiosity.
I’ve also found logs for exercise that deform from elliptical near the bottom to circular near the top — something interesting, given conic sections are mathematically related, but something important for calculating the volume and then the weight of the log! I have pictures; I share them. It is real.
On understanding HRV using ellipses, I can go on a whole long story, as well. The size of an ellipse on an HRV Poincare plot tells us information about our health and our status to exercise. But all this takes some discussion of HRV — something students can grasp, can do something with using apps, can use to learn about their own fitness and health, and can dig into on their own. HRV makes math and science real and relevant.
And I love discussing how we use hyperbolas as error bounds for lines of best fit. We can use different hyperbolas for different estimations in that regard. In fact, I have given students equations to graph in their calculators to see a line of best fit and hyperbolas around it, along with context they came from: a problem out of the classic Snedecor and Cochran book Statistical Methods.
In posting about my local Creek after a rain, I showed some data and graphs about rainfall measures in our area and about the stream height at various places along the Creek. Someone else posted about the stream flow rate, and said he preferred flow rate to height. Maybe in part he is more interested in flow rate because he kayaks and canoes — if I remember the person right. So, off I go looking to find the data he had — which gets me also into flow rate and how it is determined, which takes me on a joy ride into its physics, math, statistics, measurement, calculus, Riemann sections, and more — which all I share with my nature/ecology group on Facebook and post on my nature/ecology website, and some of which I share with students, one of whom had recently done Riemann sums in calculus. She loved it.
The students got to see some real examples of math, stats, and science, examples they could grasp the meaning of and which was part of their lives, not the life of some scientist off somewhere in a lab or university.
The whole science of calculating flow rate gives students a glimpse of the depth and breadth of knowledge involved in calculating something seemingly so simple as flow rate, a glimpse of all the work and thought that goes into it, and a glimpse of all sorts of avenues that could be pursued in this context. The science of stream flow is a wealth of ideas and applications I can tap far into the future. And it is not an isolated concrete; it is not sui generis. It is an example of the kind of thing all humans need to do and of what students will need to do in their futures.
In all this, I can share with students scientific articles, scientists’ original research, scientific history, etc. I can dig into observation, concept-formation, induction/generalization, definition, classification, the logic of experiment, the use of mathematics, theory-formation, and more.
This situation is like any other: there is no dead-end. All areas have infinite avenues and paths a student could explore, some paths being a “road less traveled.”
“Two roads diverged in a wood, and I—
I took the one less traveled by,
And that has made all the difference.”
—Robert Frost, “The Road Not Taken”
Curiosity and investigation always open up new vistas and lead to new connections.
In theory and in practice, I strive to help each student become a rational, self-sovereign social animal at home on this wonderful earth. Rational. Ethical. Wise.
Results: Student and Parent Insights
1. “Mr. Gold has been an outstanding calculus teacher for our son. He was genuinely invested in the students’ ability to both master the math and become more independent and philosophical thinkers. He always provided detailed, helpful feedback and consistently went the extra mile, giving plenty of his own time in overtime to ensure the material was understood. He definitely has our recommendation. Thank you!!” —Michele, parent, “Calculus, Semester 2,” 26 Apr 2022
2. “Never before was my son as motivated and interested in a course. It was a delight to see and fun to hear him share about the lessons without prodding.” —Paula L., parent, “High School Geometry, Semester 1,” 2 Jan 2022
3. “Prior to working with Michael, I had very little hope of actually pursuing my dream to work in the aerospace industry as an engineer. Yet here I am, finishing an engineering degree at A&M this Fall and working at Bell Helicopter as an engineer. I wasn’t good at math. I didn’t like math (in fact, I avoided it). With Michael’s help, I came to understand the importance of not only mathematics, but also reason, rationality, and a constant pursuit of knowledge. His emphasis on critical thinking and questioning changes the fundamental way student’s think about and approach problems, regardless of the specific nature of the problem. In other words, don’t expect him to do your homework – expect to come out asking for seconds. You will take away a lot more than you ever expected, and it will stick with you for a long time. I honestly wish more teachers were like Michael – maybe they should take a lesson or two! I cannot recommend Michael enough for any student of any subject, this guy is simply the best.” —Drew T, ex-high school student
4. ”Gold’s tutoring helped me achieve A’s in my classes and great scores on standardized tests, but this was just the ‘tip of the iceberg.’ With each lesson I also learned how to better think logically, carefully, and reasonably. These skills are essential to life and have helped me immensely — I still use the learning/study methods I developed through Gold’s lessons to earn top scores in many college courses and to do well in my work and hobbies. I will forever be grateful to Gold Academy, and highly recommend the service to any student who is looking to start learning, thinking and living.” —Rajiv R, ex-high school student
5. “This class was excellent! My daughter said ‘He is the best teacher ever! He actually explains why things work so you understand what’s going on instead of just memorizing it.’ She took Calc 1 a year ago and wanted to review. She feels she understands it so much better now. She said they covered an entire semester in two weeks more in depth than her original college-level class. She can’t wait to take the next level.” —Danielle L., parent, “Summer Calculus Review/Preview, Session 1,” 7 Aug 2021
6. ”I really did learn more than geometry and math taking classes with you. I learned to reason and use logic to start solving things in everyday life. It’s pretty useful at work when I use reasoning to find root causes, and to find and fix problems. You teach people how to THINK!” —Dylan T, ex-high school
7. “For the past 2 years, my son has taken a different math class over the summer with Michael Gold just to sharpen up before the upcoming school year, and the payoff has been tremendous. He loves this teacher’s style, is surprisingly riveted during the class, always on time, and absorbs the equivalent of an entire years worth of work in 2 weeks, helping him to breeze through the academic school year with great grades and a real sense of satisfaction. You’re kids might roll their eyes at you at first, but they will thank you if you are lucky enough to grab a spot in one of Michael Gold’s classes. He is fast, clear, to the point and gives real life application examples to make math make sense. 10 stars if I could.” —Lia J., parent, “Summer Algebra 2 Review/Preview, Session 1,” 10 Jul 2021
8. ”Michael expounds all that is science in such an interesting and exciting way! He always keeps my kids engaged and thinking that science is fun! He presents science in a logical, rational, integrated way — making it easy and possible to take everything they are learning and apply it in ‘the real world’… not just a list of useless facts, but tangible, knowable, valuable life lessons. He builds a great foundation with biology first and all the rest of the sciences following and integrating from there. Who knew that biology, physics, physical fitness, and chemistry are all related and integrated? Just take a few science classes with Michael, and you’ll see for yourself! Whether you’re homeschooling and want to seriously upgrade your science curriculum or just want to supplement your child’s deplorably lacking public school science education.. Michael will make science fun, interesting, and relevant to your kids’ lives – and yours!” —Melanie K, mother of three to whom I taught science
9. “Michael Gold is an awesome trainer. His workouts are designed to improve all elements of fitness: coordination, strength, stamina, mobility, just to name a few. Back in college I did a year of CrossFit and later on went through weight lifting programs with trainers at my gym. Michael has definitely been #1 when it came to proper form and joint/ligament strengthening. I trained with Michael for about 7 months and I saw huge improvements in my endurance and fitness. I felt better overall and even had some surprising benefits. I now effortlessly make baskets when throwing paper into my garbage bin, whereas, before working with Michael, I would always miss. And I am now much more aware of my environment, which especially helps when driving. He is big on safety and makes sure he trains to match people’s fitness levels whatever they may be. I would definitely recommend him to anyone trying to improve their fitness and well-being in whatever capacity.” —Joseph S, business professional
10. “Total Human Fitness is an experience unlike anything I was expecting. Coming from a background of physical contact sports such as hockey and football, I wanted to get myself back into form, without a florescent lighting tan, without aggravating previous injuries from the glory days. Being outside was the greatest reward, and the challenge I was looking for. Michael was with me every step of the way, building routines for with me and for me, helping me to build my body correctly and avoid injuries. This is not a boot camp for those who need someone mindlessly yelling in your face. This was an in depth structured experience, for me to be my own drill sergeant, because that is where true success lies, in becoming the best version of yourself. Starting off simply, then constantly challenging yourself to master simplicity, until your skills are your foundation. This a custom tailored experience, and I recommend Michael’s expertise and logical approach. Be prepared to learn with Michael, he’s a great teacher and an invaluable resource of knowledge. Be prepared to be immersed in nature. Be prepared to bring your mind, body, diet, and thinking to new levels. I recommend the Total Human Fitness experience for all those looking to build the best self version they achieve. Get out of the box and get involved!” —Matt H, entrepreneur
11. “Michael Gold’s ‘Summer Geometry Preview/Review’ was a in-depth yet efficient way (8-classes over a two-week period @ 1.5 hours per class) to get a preview of what to expect in the fall. My daughter wants to stay sharp over the summer (around the fun and relaxation that all kids need) and the structure of a class that meets live is the only option for us. Also, the class was comprised of like-minded students and a nice group of kids.
“We first came to know Mr. Gold through his ‘Logic Essentials: How to Think Well’ class. I highly recommend this class because logic and deductive reasoning are the cornerstones of thinking well. We need our kids to flag poor arguments and discern the truth. Unless you are attending an elite prep school, I don’t know where you would find exposure to a class like this. I am so grateful that Mr. Gold makes this class offering available. So when I shared Mr. Gold’s Geometry offering with my daughter, she wanted to sign up knowing he is a wonderful teacher.
“What was most telling – my daughter never forgot about the class date or time. I know Mr. Gold is engaging because if I walked past her room, she was never distracted. He connected with kids on all fronts using lecture, illustrating problems on the board and showing YouTube videos. He has a warm personality, inflection in his voice and good sense humor. After each class ended, my daughter wanted to show me her notes and explain a problem set she learned. Also, I received a detailed email from Mr. Gold outlining everything that was covered in each class, along with links to videos, books, geometry worksheets and more. I also appreciate that homework was optional – your child can leisurely gain exposure or go all out on mastering the subject matter – there’s no pressure.
“Michael Gold has high standards for teaching and truly cares about his students. My family joins his long list of five star reviews because he’s that good. My daughter and I highly recommend Mr. Gold’s ‘Summer Geometry Preview/Review’ and ‘Logic Essentials: How to Think Well’. We enthusiastically encourage you to sign up for any class he teaches!” —Mary B, parent, “Summer Geometry Preview/Review, Session 1,” 19 Aug 2020
1. Private Tutoring: https://goldams.com/about/testimonials/testimonials-academics-education/
2. Teaching on Outschool: https://goldams.com/about/testimonials/testimonials-academic-classes-on-outschool/
Also on Outschool itself: https://outschool.com/teachers/Michael-Gold
3. Logic and Thinking Skills: https://goldams.com/about/testimonials/testimonials-logic-and-thinking-skills/
4. Fitness Training: https://goldams.com/about/testimonials/testimonials-fitness-training/
Influential Thoughts & Thinkers
1. “He who learns but does not think is lost! He who thinks but does not learn is in great danger.” —Confucius
2. “We are what our thinking makes us.” —Nigerian Proverb
3. “When we are no longer able to change a situation, we are challenged to change ourselves.” —Viktor Frankl
4. “If I am walking with two other men, each of them will serve as my teacher. I will pick out the good points of the one and imitate them, and the bad points of the other and correct them in myself.” —Confucius
5. “A little learning is a dang’rous thing;
Drink deep, or taste not the Pierian spring:
There shallow draughts intoxicate the brain,
And drinking largely sobers us again.“
—An Essay on Criticism by Alexander Pope
6. “An answer brings no illumination unless the question has matured to a point where it gives rise to this answer, which thus becomes its fruit. Therefore learn how to put a question.” —Ancient African proverb
7. “A pupil may show you by his own efforts how much he deserves to learn from you.” —Ancient African proverb
8. “I should even think that in making the celestial material alterable, I contradict the doctrine of Aristotle much less than do those people who still want to keep the sky inalterable; for I am sure that he never took its inalterability to be as certain as the fact that all human reasoning must be placed second to direct experience.” —From the Second Letter of Galileo Galilei to Mark Welser on Sunspots, p. 118 of Discoveries and Opinions of Galileo, translated by Stillman Drake, (c) 1957 by Stillman Drake, published by Doubleday Anchor Books, Doubleday & Co., Garden City, New York.
8. “Sense represents the first stage. Next comes memory in the case of the animals which have it. In man memory leads up to experience; many memories of the same thing occurring come in the end to have the force of a single experience.” —Aristotle, Chapter 1, Book A, the Metaphysics
10. ”There is no human experience that can be termed true science unless it can be mathematically demonstrated. And if thou sayest that the sciences which begin and end in the mind are true, this cannot be conceded, but must be denied for many reasons, and firstly because in such mental discourses experience is eliminated, and without experience there can be no certainty.” —Leonardo Da Vinci, Thoughts on Art and Life
11. “The seeker after truth is not one who studies the writings of the ancients and, following his natural disposition, puts his trust in them, but rather the one who suspects his faith in them and questions what he gathers from them, the one who submits to arguments and demonstration and not the sayings of human beings whose nature is fraught with all kinds of imperfection and deficiency. Thus the duty of the man who investigates the writings of scientists, if learning the truth is his goal, is to make himself an enemy of all that he reads, and, applying his mind to the core and margins of its content, attack it from every side. He should also suspect himself as he performs his critical examination of it, so that he may avoid falling into either prejudice or leniency.” —Ibn al-Haytham (Abū ʿAlī al-Ḥasan ibn al-Haytham, c. 965 – c. 1040.) Source: quote at the beginning of Chapter 11, “The Physicist,” in the book The House of Wisdom by Jim Al-Khalili
12. “Ecological psychology focuses on the organism-environment coupling and not on what happens at the subpersonal, neural level. However, this does not mean that this level of description is not important for enabling the whole process of perception. As Travieso and Jacobs (2009, p. 403) claimed: ‘Note, though, that one cannot reject the validity of studies with smaller or larger levels of analysis; that would be analogous, we think, to rejecting current neurobiology because subatomic particles are not taken into account.’ In this sense, J. J. Gibson understood that at a neural level, the systems do not compute, transform, or enrich information, but that they “resonate” to ecological information (J. J. Gibson, 1966, p. 5), an idea that resembles current dynamic approaches to neuroscience (Freeman, 2000; Raja, 2018).” —Lorena Lobo, Manuel Heras-Escribano, and David Travieso, “The History and Philosophy of Ecological Psychology” (Front. Psychol., 27 November 2018, Volume 9 – 2018
13. “Ecological Psychology is an embodied, situated, and non-representational approach pioneered by J. J. Gibson and E. J. Gibson. This theory aims to offer a third way beyond cognitivism and behaviorism for understanding cognition. The theory started with the rejection of the premise of the poverty of the stimulus, the physicalist conception of the stimulus, and the passive character of the perceiver of mainstream theories of perception. On the contrary, the main principles of ecological psychology are the continuity of perception and action, the organism-environment system as unit of analysis, the study of affordances as the objects of perception, combined with an emphasis on perceptual learning and development.” —Lorena Lobo, Manuel Heras-Escribano, and David Travieso, “The History and Philosophy of Ecological Psychology” (Front. Psychol., 27 November 2018, Volume 9 – 2018
14. “An important aspect of specificity is that it allows for the lawful character of ecological information. In this sense, ‘[e]cological information is lawful not in the Newtonian sense of being universal in space and time, but in an ecological sense of being regular within an ecological context of constraint’ (Warren, 2005, pp. 342–343). When the agent explores the environment and encounters this ecological information, the agent uses that information for guiding its intentional behavior (Reed, 1983, p. 90). What follows from the detection or picking up is that there is no need to process and enrich the ecological information, since it is sufficiently informative of the affordances of the environment as for guiding the behavior of the agent by itself. In this sense, non-representationalism and the rejection of cognitive processing is a consequence of both the specificity and the picking up of ecological information. For this reason, the detection of ecological information implies the direct perception of affordances.” —Lorena Lobo, Manuel Heras-Escribano, and David Travieso, “The History and Philosophy of Ecological Psychology” (Front. Psychol., 27 November 2018, Volume 9 – 2018
15. ”As you begin the study of geometry, you no doubt wish to know why the subject is taught and what benefit you will receive from its study.
“Primarily, geometry teaches you how to reason. The habit of correct thinking acquired in its study is beneficial to all. By its study you will be able to converse more logically and read with a greater understanding.
“It is said that Abraham Lincoln studied geometry in order to make better arguments in court. Before a lawyer makes his argument to a jury, he must study the evidence presented during the trial and choose that which has the greatest bearing on the case; he must search the court records for past decisions on similar cases; and then he must arrange his argument in a logical order, proving each statement by giving an acceptable reason for it.” —Welchons and Krickenberger, p. 1, Plane Geometry, Revised Edition, (c) 1943
16. “[Marva Collins] insisted on frequent reading and writing, through which children learn about their world and how what may seem separate matters when taught in subject silos, readily connect in thinking minds.” —From: https://www.thewestsideschools.ca/blog/marva-collins (A note on their website, when I looked on 25Mar2023, says they are now closed.)
17. “The holistic programme we offer in our classrooms allows the children to build a comprehensive picture of all the aspects of the world. It is meant to show the child that everything is interconnected and interrelated. Thanks to this, it becomes possible for him to make the necessary connections, to sort out useful from useless information, to gain an understanding of the world from different angles, and to develop a clear vision of this interconnected web. The hope is that, with time, the child becomes able to base his opinions, decisions and solutions to problems on an increasingly clearer vision.” (p. 19)
“Reaching this comprehensive or more so, global view is a long-term process that begins with an extensive period of exploration. Indeed, the level of abstraction that is required to make rational decisions can only grow from a very concrete experience of the world.” (p. 18) —Sandra Van Cauter, Directress Primary Group 1, at International Montessori Tervuren, the November 2015 newsletter Montessori Motion of International Montessori Schools (the International Montessori Schools, Child Development Centres and International Baccalaureate World School).
18. “I’ll say that the single most surprising study in the book, to me, was conducted at the U.S. Air Force Academy: The Academy provided a unique environment for studying the impact of teaching quality on learning, because students have to take the same sequence of courses and the same tests, and they are randomized to professors, and then re-randomized for each subsequent course, so you can truly track the impact of teaching. Basically, the study found that teachers who are the best at helping students do well in their own class now also systematically undermine the future development of those same students, who then go on to underperform in future classes. That’s a deeply counterintuitive finding, but it’s also a theme in the book—that behavior which causes the best performance right now can damage longer-term development.
“I think something that applies at every level of athletes is that they should introduce variety into whatever they’re doing. A good symbol to keep in mind is Cirque du Soleil, which, by the way, includes plenty of former Olympians. They experimented with having performers learn the basics of disciplines outside of those they actually perform and found that, among other benefits, it dramatically reduced their injury rates.” —David Epstein in Brad Stulberg’s “Going Broad—Not Narrow—is the Best Route to Lasting Success: An interview with David Epstein”
19. ”Logical thinking is correct and productive thinking. Building a logical system of thought means correctly combining ideas in such a way that they produce other ideas. … [T}he process of building a logical system of thought is important, also to all of you regardless of what field of work you enter. Studying how to make your thinking about geometry logical can help to make all of your thinking logical. This is important, for much of your success and happiness will depend upon your ability to think logically.” —Welchons and Krickenberger, p. 5, Plane Geometry, (c) 1958
20. “Many people make an unfortunate and invalid separation between heart and mind, or feeling and intellect. In some contemporary traditions, abetted by attitudes stereotypically centered on Southern California, feelings are exalted as more ‘real’ and the only proper basis for action – if it feels good, do it – while intellect gets short shrift as a hang-up of outmoded elitism. Statistics, in this absurd dichotomy, often become the symbol of the enemy. As Hilaire Belloc wrote, ‘Statistics are the triumph of the quantitative method, and the quantitative method is the victory of sterility and death.’
“This is a personal story of statistics, properly interpreted, as profoundly nurturant and life-giving. It declares holy war on the downgrading of intellect by telling a small story about the utility of dry, academic knowledge about science. Heart and head are focal points of one body, one personality.
“But match people with the same cancer for age, class, health, socioeconomic status, and, in general, those with positive attitudes, with a strong will and purpose for living, with commitment to struggle, with an active response to aiding their own treatment and not just a passive acceptance of anything doctors say, tend to live longer.
“We still carry the historical baggage of a Platonic heritage that seeks sharp essences and definite boundaries. (Thus we hope to find an unambiguous ‘beginning of life’ or ‘definition of death,’ although nature often comes to us as irreducible continua.) This Platonic heritage, with its emphasis in clear distinctions and separated immutable entities, leads us to view statistical measures of central tendency wrongly, indeed opposite to the appropriate interpretation in our actual world of variation, shadings, and continua. In short, we view means and medians as the hard ‘realities,’ and the variation that permits their calculation as a set of transient and imperfect measurements of this hidden essence.
“But all evolutionary biologists know that variation itself is nature’s only irreducible essence. Variation is the hard reality, not a set of imperfect measures for a central tendency. Means and medians are the abstractions.” —Stephen Jay Gould, “The Median Isn’t the Message”
(c) Michael Gold