Thanks for building the MIST Academy community! I interview all new students nowadays, and it definitely helps. I am also running Alabama ARML practices. Unfortunately I don't have as big a reach as you did in the Birmingham area, but I hope my online students from around the states are able to gain what I gained from attending classes at MIST. Most of my students are from outside AL now. Hopefully I am continuing your legacy of teaching a bit in my own way.
Congratulations on seeking out new methods in education and for capturing your wisdom for posterity. And of course many thanks.
It’s clear the ancient techniques are serving society and civilization poorly. The industrialization of education (and medicine) have been massive errors (or, more charitably, necessary phases of imperfection in the evolution of both). A more personal approach to both is sorely needed. No doubt it takes huge amounts of energy and enthusiasm.
I would be interested in hearing about any of your students who decided to become teachers themselves using your methods. Did you have more advanced students teaching the less in your schools?
As a homeschool mama, I can attest that this sort of instruction absolutely requires high amounts of energy and enthusiasm. But, this dialogic method of learning and teaching is soooo important and serves far better than simply plopping a child in front of a textbook and workbook.
I agree with you re the industrialization of education and medicine. We’ve realized it is not good for our food supply. Now we need to realize it’s not good for much of anything else. Except perhaps making cars.
Listened to the whole video on my walk today. It was great!
I felt a deep sense of gratitude for the countless hours my dad spent educating me as a teenager after school hours and on weekends. At the time I hated it, and he certainly was not as patient as you, nevertheless he took countless hours out of his own schedule to educate me. That 1-on-1 instruction is invaluable.
I like the fact you did no shy away from starting with a hard question for each new problem and then broke it down into easier scenarios depending on their comprehension, yet always managed to eventually help the student answer the original hard question on their own. (State the goal, establish the competency level, then figure out the best way to get them to the goal whilst remaining extremely attentive to how they respond at each step.)
Loved her accent and the way she could speak perfect English and Spanish seamlessly.
Thank you for taking the time to put this together and providing a guide for others to follow. Much appreciated.
------
I'm amazed at how sharp young brains are. Given the right conditions and instruction they can absorb so much information. This gives me so much hope for the future. Thank you.
Starting with a problem that requires unraveling like a puzzle room, or by at least chopping it up into steps or examining the simpler versions/cases is crucial to helping students along a path that dodges some of the build-up of the feeling of intimidation of "hard problems". Sure, every discipline involves hard stuff. Sure, people learn that hard stuff all the time. There are a few basic formulas, and practice breeds confidence in applying them in a dark maze.
She also had another ingredient, which is a positive attitude. That's a credit to her parents.
If you can spare the time, I'd love to know the most straightforward math behind solving the Fibonacci question you mentioned (you can keep it super brief, just satisfy my curiosity). I read it through several times and felt compelled to mess with it on the legal pad next to my keyboard but honestly didn't know how to do anything besides brute-forcing it, which told me that there's a fundamental formula somewhere that I simply never learned- I never really had any advanced math beyond trig.
My father- a public school teacher at a terrible, extremely violent, highly dysfunctional public school where it was not expected that any real teaching would be going on, a fact which left him bitter, depressed, and drunk much of the time- also did this with me, mainly with experimental science. I think one of the first truly expensive "toys" I got as a child was a microscope. But it ran the whole gamut- looking at water from the bird bath under the aforementioned microscope and comparing it to tap water, trying (TRYING) to build model rockets, dissecting dead animals found in the park, etc.
I regret that I never became any kind of research scientist, but that incredibly focused intellectual attention was a) a big reason I have any happy memories of childhood at all and b) a big part of why I can bring a high degree of intellectual rigor and regard for the integrity of experimental methods to the various work I HAVE done over the years.
It's not something every single human could hope to receive outside of a Star Trek utopia, but it is a tremendous gift to a child.
I appreciate the way you and Iris developed the rules for each more complicated counting problem. More impressed with your patience (wonderful) and Iris perseverance even when for a time she seem stuck. She never gave up.
I homeschooled my children. When I looked at math education and textbooks for 6-8th grade I became appalled with the fact that almost no new concepts were taught 3 yrs in a row. When it came time to teach algebra, I decided to use a college algebra and trig text. The text was out of print but was very good with great example problems. My oldest graduated with a chemical engineering degree and my youngest graduated cosmetology school.
The best video I have seen on education is:
2+2=5 | Two & Two - Nominated as Best Short Film, Bafta Film Awards, 2012
"I felt the problem was even broader—that everyone is at a dramatic disadvantage anywhere there is not a healthy culture of learning math."
But math is hard. "The easy path is always mined."
Similarly, a solid background in chemistry and physics helps one to understand systematic error and error analysis.
NB Cool physics factoid: A substance that absorbs a particular frequency of EM radiation will always reflect that EM radiation, but the reverse isn't always true. Some things reflect, but don't absorb. (e.g., mirrors) The reason is because the index of refraction (IOR) is complex. The real part has to do with reflection and the imaginary part has to do with absorption. Also, the IOR varies with EM frequency for substances. My physics research had to do with measuring reflectivity for a straight chain alcohol and comparing it with a straight chain carboxylic acid containing the same number of carbons. I uploaded the reflectivity data to a computer, which ran a Fast Fourier Transform on the data to generate absorption values which I then compared to a library of absorption values for the alcohol and carboxylic acid components. I found absorption for a C=O in one set of data, but not in the other. QED. Nothing new theoretically, but there was practical application in space exploration and in medicine.
Great teaching Mathew! It may be easier to teach the factorial method of counting prior to the choose method? (i.e. first, keep the number of items/people to choose positions equal to the number of positions available). Great approach to teaching all these concepts. I love it.
I love this approach for persons of a ages. You are doing a great service for all of us. If you have books (math stuff in particular) that you can recommend please do.
Part of what I did as a consultant was tutor professors at universities like Stanford, or CEOs of tech corporations. I had graduate students at major universities like Harvard from time to time.
But mostly I focused on creating a guided path whereby students interested in further explorations could diverge from school curriculum around the end of elementary school to essentially what would be a complete undergraduate curriculum by age 17. Some of my students won medals at the International Mathematical Olympiad or events such as the Intel Science Fair. During the nine years I ran schools, I was invited by China's Minister of Education to build one in Beijing.
You said you worked with 2000 students or more, not counting time at AoPS. But how much time with each student, and did you really achieve a ratio of greater than 8:1 students to teacher?
I literally have your AoPS number theory book within arms-reach right now. I'm about to begin teaching it to my student this semester (I homeschool.) But I cannot imagine Socratic methods scaling to every day instruction except in the hands of the most gifted teacher and equivalently gifted students.
We have brick and mortar AoPS in our town; they're grinds. They aren't taught socratically even though they're problem based. Online AoPS grew but didn't scale; quality has suffered, and now their goals have shifted. When ed schools became enamored with constructivism,they invent TERC and Everyday Math which became fuzzy math appreciation. Modern schoolteachers can't possibly use the Socratic method because they don't know a good idea from a bad. They wouldn't recognize a contradiction.
Moreover it is exhausting to do this and to do it for every subject every day. There's another reason Socrates did it with adults, too. There were premises (even if false) on which to hang an argument. And of course, when he did succeed with the youngerish, they killed him. We should proceed cautiously.
You're asking questions that would take a few hours to answer. As much as I would like to, I'm trying to keep a schedule lately whereby I sleep. But I do write Education articles and make videos from time to time.
Excellent! My grandfather was a Math teacher and would sit down with me and do math problems when he came to visit and I think that is what fueled my interest in the subject. I had some good and bad experiences in school, but I always loved math competitions. I did not enjoy my college statistics class, but now I'm having to go back and re-learn everything just to keep up with everything that is going on. This lesson made crystal clear what had always been kind of a fuzzy concept.
I saw a Math teacher with a different approach. Works for a class, where you have different levels of students.The main idea is that you have different answers to math questions. You have an easy answer, and some more complex answers.
That way students can feel satisfied that they can answer the question. And then they can explore from interest the more complex answers that are possible. The system works well in magic squares, number patterns, fill in the +/-/*/div etc. Because you always get an answer with little effort, it invites to continue and stay focused.
I think this same system can be applied to statistics & probabilities, graphs, geometries, etc. Always have an easy answer, and let the smart-ass people explore more complex ones.
What you describe in the video: "How can you make this simpler?" Seems a good idea too. It is hard to make it simpler in a correct way. So you may need to explore those variants too.
In my experience, 5 girls never can choose a seat.
Warning: Don't flip the coin imaginary times. Then you may end existence.
I got A's in Calculus (thanks to a diligent TA who showed me how to recognize cases where certain manipulations could successfully solve the problems. I've always found combinatorics quite difficult to grasp because combinations become highly complex. Do you think the way you and Iris approached this is better than recognizing a rule based approach?
I get that concept, however, in order for Iris to become comfortable with probability and statistics, she is eventually going to be dealing with pattern recognition. Using your concept:
really fast analysis. She expressed this in the beginning saying,"I don't understand the question."
I'm not sure what your concern is about. The purpose of the interview is to reach a point of lack of comprehension on some level, and keep talking about it while letting the student figure out as much as they can. That's closer to the way the real world works, not to mention it helps students gain experience that gives them confidence not to give up when they first read a sentence or paragraph in a math book that they don't understand.
Thanks for building the MIST Academy community! I interview all new students nowadays, and it definitely helps. I am also running Alabama ARML practices. Unfortunately I don't have as big a reach as you did in the Birmingham area, but I hope my online students from around the states are able to gain what I gained from attending classes at MIST. Most of my students are from outside AL now. Hopefully I am continuing your legacy of teaching a bit in my own way.
Thank you for sharing your time with all of us.
You did very well.
Iris. You were under a lot of pressure there, and you performed wonderfully. Congratulations.
Congratulations on seeking out new methods in education and for capturing your wisdom for posterity. And of course many thanks.
It’s clear the ancient techniques are serving society and civilization poorly. The industrialization of education (and medicine) have been massive errors (or, more charitably, necessary phases of imperfection in the evolution of both). A more personal approach to both is sorely needed. No doubt it takes huge amounts of energy and enthusiasm.
I would be interested in hearing about any of your students who decided to become teachers themselves using your methods. Did you have more advanced students teaching the less in your schools?
As a homeschool mama, I can attest that this sort of instruction absolutely requires high amounts of energy and enthusiasm. But, this dialogic method of learning and teaching is soooo important and serves far better than simply plopping a child in front of a textbook and workbook.
I agree with you re the industrialization of education and medicine. We’ve realized it is not good for our food supply. Now we need to realize it’s not good for much of anything else. Except perhaps making cars.
Listened to the whole video on my walk today. It was great!
I felt a deep sense of gratitude for the countless hours my dad spent educating me as a teenager after school hours and on weekends. At the time I hated it, and he certainly was not as patient as you, nevertheless he took countless hours out of his own schedule to educate me. That 1-on-1 instruction is invaluable.
I like the fact you did no shy away from starting with a hard question for each new problem and then broke it down into easier scenarios depending on their comprehension, yet always managed to eventually help the student answer the original hard question on their own. (State the goal, establish the competency level, then figure out the best way to get them to the goal whilst remaining extremely attentive to how they respond at each step.)
Loved her accent and the way she could speak perfect English and Spanish seamlessly.
Thank you for taking the time to put this together and providing a guide for others to follow. Much appreciated.
------
I'm amazed at how sharp young brains are. Given the right conditions and instruction they can absorb so much information. This gives me so much hope for the future. Thank you.
Starting with a problem that requires unraveling like a puzzle room, or by at least chopping it up into steps or examining the simpler versions/cases is crucial to helping students along a path that dodges some of the build-up of the feeling of intimidation of "hard problems". Sure, every discipline involves hard stuff. Sure, people learn that hard stuff all the time. There are a few basic formulas, and practice breeds confidence in applying them in a dark maze.
She also had another ingredient, which is a positive attitude. That's a credit to her parents.
If you can spare the time, I'd love to know the most straightforward math behind solving the Fibonacci question you mentioned (you can keep it super brief, just satisfy my curiosity). I read it through several times and felt compelled to mess with it on the legal pad next to my keyboard but honestly didn't know how to do anything besides brute-forcing it, which told me that there's a fundamental formula somewhere that I simply never learned- I never really had any advanced math beyond trig.
1. Start simple. Solve for sequences of lengths 1, 2, 3, 4...
My father- a public school teacher at a terrible, extremely violent, highly dysfunctional public school where it was not expected that any real teaching would be going on, a fact which left him bitter, depressed, and drunk much of the time- also did this with me, mainly with experimental science. I think one of the first truly expensive "toys" I got as a child was a microscope. But it ran the whole gamut- looking at water from the bird bath under the aforementioned microscope and comparing it to tap water, trying (TRYING) to build model rockets, dissecting dead animals found in the park, etc.
I regret that I never became any kind of research scientist, but that incredibly focused intellectual attention was a) a big reason I have any happy memories of childhood at all and b) a big part of why I can bring a high degree of intellectual rigor and regard for the integrity of experimental methods to the various work I HAVE done over the years.
It's not something every single human could hope to receive outside of a Star Trek utopia, but it is a tremendous gift to a child.
I appreciate the way you and Iris developed the rules for each more complicated counting problem. More impressed with your patience (wonderful) and Iris perseverance even when for a time she seem stuck. She never gave up.
Thank you both
I homeschooled my children. When I looked at math education and textbooks for 6-8th grade I became appalled with the fact that almost no new concepts were taught 3 yrs in a row. When it came time to teach algebra, I decided to use a college algebra and trig text. The text was out of print but was very good with great example problems. My oldest graduated with a chemical engineering degree and my youngest graduated cosmetology school.
The best video I have seen on education is:
2+2=5 | Two & Two - Nominated as Best Short Film, Bafta Film Awards, 2012
https://www.youtube.com/watch?v=EHAuGA7gqFU
You will have to read the subtitles because I believe the language is Persian (Farsi)
Was at school a lot of years ago and hated math. When’s the next lesson? can’t wait!
Great stuff, thank you. All of the kids in my family gets a math focus because it falls under the category of "Things other people can't do".
You can only make money 5 ways. Steal it, inherit/win it, marry it, do something other people don't want to do, or do something other people can't do.
There is no doubting the utility of math. It's easier, too, when it can be learned in an enjoyable environment.
https://www.maa.org/external_archive/devlin/LockhartsLament.pdf
They all know math. One is a plumber and makes more than an attorney per hour. LOL
Math is foundational. It is about discourse, reason, logic proof and beauty.
Ah you quoted Lockhart. Hated and despised by nearly all mathematics professors.
His imprecision angers them. his playfulness enrages them.
"I felt the problem was even broader—that everyone is at a dramatic disadvantage anywhere there is not a healthy culture of learning math."
But math is hard. "The easy path is always mined."
Similarly, a solid background in chemistry and physics helps one to understand systematic error and error analysis.
NB Cool physics factoid: A substance that absorbs a particular frequency of EM radiation will always reflect that EM radiation, but the reverse isn't always true. Some things reflect, but don't absorb. (e.g., mirrors) The reason is because the index of refraction (IOR) is complex. The real part has to do with reflection and the imaginary part has to do with absorption. Also, the IOR varies with EM frequency for substances. My physics research had to do with measuring reflectivity for a straight chain alcohol and comparing it with a straight chain carboxylic acid containing the same number of carbons. I uploaded the reflectivity data to a computer, which ran a Fast Fourier Transform on the data to generate absorption values which I then compared to a library of absorption values for the alcohol and carboxylic acid components. I found absorption for a C=O in one set of data, but not in the other. QED. Nothing new theoretically, but there was practical application in space exploration and in medicine.
Great teaching Mathew! It may be easier to teach the factorial method of counting prior to the choose method? (i.e. first, keep the number of items/people to choose positions equal to the number of positions available). Great approach to teaching all these concepts. I love it.
I love this approach for persons of a ages. You are doing a great service for all of us. If you have books (math stuff in particular) that you can recommend please do.
I may put together a recommendation list sometime.
It sounds more like you taught an 8 th grade or thereabouts math class rather then " opened " and " closed a school " .
Nice smear.
What "sounds like" what?
Part of what I did as a consultant was tutor professors at universities like Stanford, or CEOs of tech corporations. I had graduate students at major universities like Harvard from time to time.
But mostly I focused on creating a guided path whereby students interested in further explorations could diverge from school curriculum around the end of elementary school to essentially what would be a complete undergraduate curriculum by age 17. Some of my students won medals at the International Mathematical Olympiad or events such as the Intel Science Fair. During the nine years I ran schools, I was invited by China's Minister of Education to build one in Beijing.
You said you worked with 2000 students or more, not counting time at AoPS. But how much time with each student, and did you really achieve a ratio of greater than 8:1 students to teacher?
I literally have your AoPS number theory book within arms-reach right now. I'm about to begin teaching it to my student this semester (I homeschool.) But I cannot imagine Socratic methods scaling to every day instruction except in the hands of the most gifted teacher and equivalently gifted students.
We have brick and mortar AoPS in our town; they're grinds. They aren't taught socratically even though they're problem based. Online AoPS grew but didn't scale; quality has suffered, and now their goals have shifted. When ed schools became enamored with constructivism,they invent TERC and Everyday Math which became fuzzy math appreciation. Modern schoolteachers can't possibly use the Socratic method because they don't know a good idea from a bad. They wouldn't recognize a contradiction.
Moreover it is exhausting to do this and to do it for every subject every day. There's another reason Socrates did it with adults, too. There were premises (even if false) on which to hang an argument. And of course, when he did succeed with the youngerish, they killed him. We should proceed cautiously.
You're asking questions that would take a few hours to answer. As much as I would like to, I'm trying to keep a schedule lately whereby I sleep. But I do write Education articles and make videos from time to time.
https://www.campfire.wiki/doku.php?id=rounding_the_earth:the_education_wars
Excellent! My grandfather was a Math teacher and would sit down with me and do math problems when he came to visit and I think that is what fueled my interest in the subject. I had some good and bad experiences in school, but I always loved math competitions. I did not enjoy my college statistics class, but now I'm having to go back and re-learn everything just to keep up with everything that is going on. This lesson made crystal clear what had always been kind of a fuzzy concept.
I saw a Math teacher with a different approach. Works for a class, where you have different levels of students.The main idea is that you have different answers to math questions. You have an easy answer, and some more complex answers.
That way students can feel satisfied that they can answer the question. And then they can explore from interest the more complex answers that are possible. The system works well in magic squares, number patterns, fill in the +/-/*/div etc. Because you always get an answer with little effort, it invites to continue and stay focused.
I think this same system can be applied to statistics & probabilities, graphs, geometries, etc. Always have an easy answer, and let the smart-ass people explore more complex ones.
What you describe in the video: "How can you make this simpler?" Seems a good idea too. It is hard to make it simpler in a correct way. So you may need to explore those variants too.
In my experience, 5 girls never can choose a seat.
Warning: Don't flip the coin imaginary times. Then you may end existence.
I got A's in Calculus (thanks to a diligent TA who showed me how to recognize cases where certain manipulations could successfully solve the problems. I've always found combinatorics quite difficult to grasp because combinations become highly complex. Do you think the way you and Iris approached this is better than recognizing a rule based approach?
Understanding is more important than rules, though rules can still help pattern recognition and intuition.
I get that concept, however, in order for Iris to become comfortable with probability and statistics, she is eventually going to be dealing with pattern recognition. Using your concept:
really fast analysis. She expressed this in the beginning saying,"I don't understand the question."
I'm not sure what your concern is about. The purpose of the interview is to reach a point of lack of comprehension on some level, and keep talking about it while letting the student figure out as much as they can. That's closer to the way the real world works, not to mention it helps students gain experience that gives them confidence not to give up when they first read a sentence or paragraph in a math book that they don't understand.