I cannot teach anybody anything, I can only make them think.” ~ Socrates
From my 17+ years of schooling, and even today, schools systems teach facts and figures, and then have the students regurgitate them for a grade. In my opinion, the educational system, especially in this country, is deficient in the way the schools approach education. Like Socrates, schools should teach students how to learn. That is not to say that teachers are bad, there are many that are excellent in their profession. The system they work in, and the sometimes the way they are forced to teach, is the problem.
I couldn’t see how anyone could be educated by this self-propagating system in which people pass exams, and teach others to pass exams, but nobody knows anything.
Richard Feynman
When a person graduates from high school or college, they need to know how to learn as they enter the world of employment. Once employed, they must “learn” how to do the work assisgned them. Yes, they may have someone who “teaches” or “guides” them on their tasks, but they must make the effort to learn. We are human beings, not robots.
To learn, a person must first need to be willing to take the time to study and meditate on the subject, to do the necessary research, and to ask the appropriate questions. This leads to knowledge. Next, they need to see how that knowledge fits in with what we already know, discerning its benefit to us personally. That results in understanding. Finally, they should use what they have learned, putting it into action in our lives. This is wisdom.
What is the difference between knowledge, understanding, and wisdom? Let’s illustrate the difference this way: Imagine you are standing in the middle of a road and a bus is coming toward you. First, you recognize that it is a bus—that is knowledge. Next, you realize that if you remain standing there, you will be hit by the bus—that is understanding! So you jump out of the way of the bus—that is wisdom!
Wisdom is not a product of schooling but a lifelong attempt to acquire it.
Albert Einstein
What then is the best way to learn? Each individual is unique and may learn in a way different from others. Begin your quest on learning and find the something that was not taught in school, or while you are in school, that would help you understand a mathematical concept. I missed many “somethings” and have placed them in this website. I hope what I have placed here will help you to have an aha moment in understanding mathematics!
I have listed some links below that will assist you in your quest to learn.
In Praise of Teachers
I have had many outstanding teachers (e.g., Mr. Jack Bowen and Mrs. Marion Kinzele of Perry High School, Massillon, OH), and have read many teachers blogs (e.g., Math Giraffe and iteachalgebra), use websites (e.g., CUEMATH and Math Is Fun), and learn teaching techniques from various websites (e.g., The Socratic Method: What it is and How to Use it in the Classroom and Asking Questions: Six Types). Many teachers are trying hard to make their students think on their own. It must be hard to work inside an educational system that wants teachers to teach how to take standardized test rather than how to think and learn. For all you teachers out there trying hard to improve the lives of your student: Take a bow! You deserve it!
Additional Reading
Canen, Cathy. 2018. “Using The Socratic Method With High Schoolers | Paradise Praises”. Paradise Praises. https://paradisepraises.com/socratic-method/.
⭐ Cheney, Chloe. “Pros And Cons Of Formal & Informal Learning”. 2022. Medium. https://chloecheney44.medium.com/pros-and-cons-of-formal-informal-learning-abc8e59f22c.
Education is the process of gaining or imparting knowledge and being successful later. This whole process demands learning, and it is done in many ways. And to acquire knowledge, you have to undergo at least each process of learning. ‘Learning’ gives us the idea of classes, extracurricular activities, syllabus, structured rooms, and the entire academic process that our kids or we experience. Even though you learn daily from incidents you face, people you meet, or the experiences, the list goes on. Today, you will learn about the learning types an education sector or the workplace follows: formal and informal learning.
“Live The #Mathlife With Lessons, Activities, Videos, & Much More”. 2021. iteachalgebra. https://iteachalgebra.com/.
“The Feynman Technique: A Beginner’s Guide To Learning Fast”. 2019. Develop Good Habits. https://www.developgoodhabits.com/feynman-technique/.
Using the Feynman Technique allows you to apply the concepts that you learn to real-world problems because you are able to grasp the concepts and processes of complex ideas.
“Richard Feynman On Teaching Math To Kids And The Lessons Of Knowledge”. 2016. Farnam Street. https://fs.blog/2016/07/richard-feynman-teaching-math-kids/.
“The Least Efficient Way To Teach Math”. 2021. Medium. https://medium.com/math-simplified/the-least-efficient-way-to-teach-math-abdc7177217.
The US ranks 30th in the world for math1. I’m not surprised as I teach a remedial math class to people in the Army (soldiers need to pass a timed competency test to determine what jobs are available to them. The class I teach is a test prep class.) It’s not the students who are at fault here. 95% of the kids I teach (yes, 20 somethings are kids. I’m old.) are bright and hard-working. So what went wrong?
Movies
“Dead Poets Society – Wikipedia”. 2021. en.wikipedia.org. https://en.wikipedia.org/wiki/Dead_Poets_Society.
Maverick teacher John Keating uses poetry to embolden his boarding school students to new heights of self-expression.
“Hidden Figures – Wikipedia”. 2021. en.wikipedia.org. https://en.wikipedia.org/wiki/Hidden_Figures.
The story of a team of female African-American mathematicians who served a vital role in NASA during the early years of the U.S. space program.
“Mr. Holland’s Opus – Wikipedia”. 2020. en.wikipedia.org. https://en.wikipedia.org/wiki/Mr._Holland%27s_Opus.
In the fall of 1964, 30-year-old Glenn Holland is a successful and talented musician and composer from Portland, Oregon. He retires and takes a position as a music teacher at John F. Kennedy High School so that he can spend more time with his young wife Iris and work on his symphony. The film covers his 30-year teaching career, set amongst the changes in American society.
“Music Of The Heart – Wikipedia”. 2021. en.wikipedia.org. https://en.wikipedia.org/wiki/Music_of_the_Heart.
In 1988 New York City, Roberta Guaspari, a recently divorced violinist, lives with her two sons, Alexi and Nicholas Tzavaras, and her mother, Assunta Guaspari. With Assunta’s encouragement, Guaspari attempts to rebuild her life and is recommended to the head teacher of East Harlem’s Central Park East School. Despite having little experience in actual music teaching, she accepts a substitute violin teaching position at Central Park East. With a combination of her toughness and determination, she inspires a group of children, and their initially skeptical parents. The program slowly develops and attracts publicity, eventually expanding to Central Park East II and River East Schools.
“Stand And Deliver – Wikipedia”. 2021. en.wikipedia.org. https://en.wikipedia.org/wiki/Stand_and_Deliver.
In the early 1980s, Jaime Escalante becomes a mathematics teacher at James A. Garfield High School in East Los Angeles. The school is full of Latino students from working-class families who are far below their grade level in terms of academic skills and also have a lot of social problems. Escalante seeks to change the school culture to help the students excel in academics. He soon realizes the untapped potential of his class and sets a goal of having the students take AP Calculus by their senior year. Escalante instructs his class under the philosophy of ganas, roughly translating to “desire” or “motivation”.
TV
“Numb3rs”. 2021. imdb.com. https://www.imdb.com/title/tt0433309/.
Working for the F.B.I., a mathematician uses equations to help solve various crimes.
Books
The Thread That Runs So True: A Mountain School Teacher Tells His Story. New York: Simon & Shuster, 1949.
First published in 1949, Jesse Stuart’s now classic personal account of his twenty years of teaching in the mountain region of Kentucky has enchanted and inspired generations of students and teachers. With eloquence and wit, Stuart traces his twenty-year career in education, which began, when he was only seventeen years old, with teaching grades one through eight in a one-room schoolhouse. Before long Stuart was on a path that made him principal and finally superintendent of city and county schools. The road was not smooth, however, and Stuart faced many challenges, from students who were considerably older—and bigger—than he to well-meaning but distrustful parents, uncooperative administrators and, most daunting, his own fear of failure. Through it all, Stuart never lost his abiding faith in the power of education. A graceful ode to what he considered the greatest profession there is, Jesse Stuart’s The Thread That Runs So True is timeless proof that “good teaching is forever and the teacher is immortal.”
⭐ I suggest that you read the entire reference. Other references can be read in their entirety but I leave that up to you.
I update the content on this website periodically to include more information on a that will help the reader understand a mathematical concept, or adding additional pages to cover mathematical topics that may be difficult to understand. Happy reading!
Mathematical Mysteries 