listen man I follow you from Iraq, I know your channel not to much views but your authentic and original content is worth watching, so please keep going.
This is unintentional on your part but the videos have a very particular vibe and "aesthetic" similar to that of ones found during the 2000s to early 2010s. Grainy footage, always done in one area, partly a personal blog and partly an educational resource on a particular subject. It definitely makes the channel more appealing compared to the mainstream of saturated slop which is only meant for zoomer retention.
I appreciate that feedback! Honestly, I don't even realize it! I guess I am indeed a creature of my particular era so it's sort of the format I am the most used to. I could not ever imagine myself hopping from place to place with so many visual distractions. Thank you again for your comment, because I take my commenters very seriously and feel a great sense of respect for you all.
That sounds like you don't know what mainstream math content looks like. One of the biggest maths channels on RUclips is 3Blue1Brown, and while they certainly don't have "grainy footage" or a "personal blog" (at least not on the main channel), they do have excellent maths content. If you're looking for someone who puts more value on rigorous definitions rather than ways to intuitively understand the underlying concepts, you can also take a look at videos like those of @MathMajor, which basically give a comprehensive explanation in the form that you would expect from a proper maths course on the topics he presents. The point in either case is that you shouldn't be demeaning towards people for no good reason.
@kappasphere - I worry about what I do, and have no time to look at what others do. I answer for what I do, and have no idea who all those people you mention are. If you are so keen and enamored with whoever those people are, you are more than welcome to spend your time on their pages or wherever it is they go. I am not "demeaning", I am honest. There's a difference, and it sounds like you cannot handle the truth, which is exactly what I mentioned in my video if you had bothered to watch it. I don't need to look at anyone else to do my job, thank you. Like I said, if you are are so into whatever other places you have mentioned, maybe spend time there rather than doing propaganda for them for free on MY channel. Don't let me keep you from enjoying your trendy non-grainy heroes!
Thank you! This happened on Friday evening. The channel went over the 8K mark. I feel very blessed to have the subscribers I do, and the engagement has been amazing! I just can't keep up always, but I'll do my best.
Not directly math related, but on the topic of "making your student feel bad" - personal story from my time with BNSF Railway during my Conductor Trainee employment phase - on one of the first trains I Conducted (under the supervision of a fully licensed Conductor of course) I was with an engineer by the name of Christopherson. He was good, but he was tough; put me on the spot several times and didn't mince words if I fell short, which I did several times. Grilled me on radio procedure, train position, manifest, HazMat, and timetables (basically the Railroad's version of a road map, but with much higher stakes). The more he rattled me, the more I fell short. Yes, it was embarrassing, and yes, you could say my feelings got hurt. But after finally yarding our train and going to the hotel, I poured over all those procedures and timetables until I fell asleep and after callup the next morning I piled into the crew van with my Conductor and Christopherson to go to our next train. Christopherson didn't even say "Good morning", he went right back to asking me about the situations we'd encountered the day before, to which I had concrete answers and could explain where I went wrong and how I would handle things differently with hindsight. He just smiled and said, "You're welcome." He wasn't trying to be mean, he'd just been with enough trainees to know how to get us out of our comfort zone and MAKE US THINK! It worked too, because I still remember the details of that trip to this day, and it brought me a lot closer to earning my place in that seat 👍
Your ideas on truth resonate deeply. I want to expand on your statement about how neutrality comes from cowardice/fear. I would say anything that tries to conform to being "neutral" comes from fear. In fact, I could even say any negative emotion-sadness, depression, anxiety, anger, frustration, shame, disappointment, guilt, disgust, loneliness-all come from fear. Fear of being different; fear of not being accepted; fear of being out of your comfort zone; fear that things won't work out; fear of not being enough; etc. I am still sorting out all these fears, but I can acknowledge that they exist. My, what a better place the world would be if people could actually admit that. There is a book called "The Course in Miracles" that delves very deeply into this. I can see that you hold to the faith of Christianity; coincidentally, "The Course in Miracles" was 'written' by Jesus Christ. It wasn't truly written by Jesus, but a woman claimed to have had a voice claiming to be Jesus who told her to take notes. I'm not sure of the validity of that, but the information in that book rings true and hits a lot of the ideas that you mentioned and even expands upon them. Truth is truth; it does not matter where or how it came. Connecting to my previous remark, "The Course in Miracles" claims anything that is not love is fear. There are more similar ideas mentioned in "The Course in Miracles," and if you haven't read it yet, I recommend checking it out. I think you would find the book interesting.
My goodness, what a powerful message and so very true. The I don't want to get involved comment with regard to the defence of someones honour is terrible and definitely a sign of character or should I say lack of it.
That is a great summary of the necessary math symbols. I'll need to remind myself of the Weierstrass p. I believe that something like 2^aleph-naught would be an example.
Hello I am a 25 year old viewer . I have watched one of your previous videos where you mentioned the books that impacted your math journey. I have read and worked through the exercises of those books. I was wondering if you would have time to make another video as a sort of continuation to the previous one. I didn’t go for a science degree at university but I find myself more interested in mathematics lately, I might just go back to school and get a math degree who knows. Thank You for your insightful content.
No color! Great choice. Professor Cromwell, perhaps it would be better to write a letter, but you mentioned your morning devotions in this video. I think that a good morning routine is the foundation of success. Can you tell a bit more about how your day starts?
Absolutely! Ever since I was in high school and truly got serious about my faith in God, I made it a point to begin the day with morning devotions. Usually, I will simply pick a random scripture without looking at the Bible and wherever my finger ends up, that's what I read. This morning it happened to be the book of Judges. When I'm at my devotions, I believe it grounds me in a spiritual connection with God. Our great Puritan ancestors had this habit, and many Christians after the 20th century lost the habit of devotionals (much like, unfortunately, evening services have been lost in many churches and vespers). Aside from that, I always have a very light breakfast. Usually, my first drink of the day is a cup of coffee (made with a 19th century percolator because I like percolated coffee rather than the Keurigs and drip makers you see these days!) If not, I will use a 1930s vacuum coffee pot. Then, after about 1 hour or so, I will have shredded wheat on some days, with some honey, and a 2nd cup of coffee with cream. If I don't have shredded wheat, I will simply have a slice of toast with marmalade (naturally made, no artificial ingredients). I make my own bread at home as well, for toast, since most store bought bread has terrible ingredients. It is also very important that personal hygiene be up to standard, whether it be a morning tub, or evening tub, I wouldn't go so far as to say cleanliness is next to godliness but it sure follows godliness! A man who is sloppy and doesn't take care to keep his body clean/washed has poor character and lack of respect for others.
@citytutoring I don't believe that it is always true that someone with poor hygiene has no respect for others, but it doesn't make the best impression. I remember going on a bus to a job fair at college and there was an old man not well dressed at all. My suit probably did not make the best impression to HIM.
Hello Professor, could you make a video giving advice on how to solve math proofs when you are stuck and don't know what to do? I find that the logical flow of an argument is easy to grasp, but how it was come up with I am unsure about. I would best explain it as, once I have the correct idea, the rest of the proof follows easily. However, when I have no idea at all I feel uneasy about where to begin, even though I recount certain definitions and theorems that may apply to the given problem, it just doesn't seem to lead anywhere.
I enjoy your videos, Sir. I had other questions in regards to book recommendations. Is Selby's Geometry and Trigonometry for Calculus worth purchasing? If not, what is a better alternative? Thank you.
Hi professor Cromwell , nice video , I only had one question at the end you included the element 0 in the set of natural numbers N . I thought N was the same as Z+ the set of positive integers and didn't have zero as a member ?
Great question! Zero is an integer, and (for some) zero is NOT a Natural Number, but it is not a settled question. I consider it a natural number, but there is a lot of debate around that.
The way I remember natural numbers is that counting numbers come "naturally" and therefore start at 1. It used to be that 1 was considered a prime number.
I think it depends on where you use them. Places where you include 0: If you have an addition structure (it is nice to have a neutral element), if you define them in formal set theory (Neumann ordinals for example), if you use them for "repeated action" formalism for other algebraic structures for example. Places where we ussually don't include 0: If we use it as an index set (0 can also be an index, but we specify it if we intend to use it), if we have a filter/ filterbase structure on it (cannot tell you why... we just do it), if we use it as infinite summation index (yes... most of the time we specify bounds), if we use it for enumerative/ countability arguments. Hope I could help.
Thanks for the video Professor Cromwell. Since you have mentioned both morning devotions and mathematical symbols, I was curious about what sources on philosophy and theology you enjoy and recommend. As a side question, how many hours of daily study of mathematics do you recommend?
Unfortunately, in places like Jamaica. The students in day care vs. The students in elementary school vs. The kids in middle school vs. High school are not the same kids and really not everyone makes it. Even if you gave any attention to the kids in day care, it would mean the world to them. It's really sad.
@@citytutoring No, I was only there for a semester. Logistically it was actually quite terrible, and I'm glad I got out when I could. Starting next semester at UMD in less than a week. In fact I took no math classes last semester because they had strung a bunch of unrelated math classes together as pre requisites, so I could only take one at a time essentially. Then the only available class was full.
The Art of Problem Solving series is good for Math olympics, but I would not use them as textbooks in my classes. They do not, for example, teach you anything about Set Theory, and there are not enough computational fluency problems in their books. Most of their problems are very interesting, no doubt, but not adequate for a standard Math course.
@@citytutoring Well, the truth is that I’m interested in Calculus and want to learn it with the Spivak book, with enough Mathematical Maturity developed, I want to be able to understand the rigorous language, but I don’t know what textbooks to read beforehand to get familiar with the rigor, hence why I asked you about the Art of Problem solving books which I have heard are rigorous in nature. I’m currently reading a Mathematical Logic book, then I was planning to read the Art of Problem solving Number Theory book, then I want to delve into some rigorous algebra or precalculus books that will prepare me for Spivak. Essentially, I seek to build my mathematical maturity to rigorous textbooks. And textbooks are indeed my preferred method of study. Oh, and should I even read a logic or number theory book? I only started the logic book given that you’ve been talking about it lately.
It may have been me who mentioned term logic yesterday. The reason being is because in Philadelphia, there is a researcher of Artificial General Intelligence known as Pei Wang. I am in accordance with him though he never taught me and I never worked under his paradigm (yet).
I find the Book of Judges the most difficult to read. It's an unbroken chain of tragedy but what's really depressing is that it seems like all that suffering, all those mistakes result in no lessons learned whatsoever. I've spoken with ministers who try to put lipstick on the pig, who claim that it's not as bad as it looks. For example, they will say that the saga of Jephthah's daughter is not what it seems, that the poor girl was not slaughtered. Unfortunately, neither my reading comprehension nor ability to reason are impaired and I read the Bible often enough to know what I'm looking at.
You can look at the book of Judges as how God repeatedly raises up deliverers, demonstrating His sovereign grace and unmerited favor. These judges, though flawed, point to the ultimate Judge and Deliverer, Jesus Christ. The saga of Jephthah, while tragic, reflects how God's purposes can still be accomplished through human brokenness. The fate of Jephthah’s daughter is debated, but for me it reveals the tragic consequences of sin and the importance of careful vows before God. Whether she was sacrificed or dedicated to lifelong virginity, her story reflects the need for humility and reverence before God, aligning with Ecclesiastes 5:2: “Be not rash with your mouth, nor let your heart be hasty to utter a word before God.” Re: Jephthah's daughter. Although Jephthah had professed faith in the true God, he was imitating the wickedness and pagan practices of the Canaanites (that is how he fell into the sin mentioned in Deut. 12:31). At that time, the Israelites were very influenced by that Canaanite culture. The bleakness of Judges makes the coming of Christ in the New Testament shine even brighter. Judges 2:16-19 encapsulates the repetitive failure of human judges, pointing us to the perfect and eternal Judge.
@@citytutoring Thank you for your thoughts. My problem is not so much with the ancient story as with the fact that our culture follows a similar path as Israel. In Galatians 6:7 the apostle Paul said, "God is not mocked..." which in my mind does not bode well for our civilization. In reading Judges I see the fate of this generation although I'm old enough that I'll probably not see the conclusion of things; but my children might.
Actually, they are the same. My two favorite classes were always Algebra (Advanced Algebra, and also Abstract Algebra). They remain my favorite classes to teach to this day.
@citytutoring Abstract Algebra is the same as Modern Algebra right? I'll be 53 in several weeks but I decided to return to get my mathematics degree. I'm going to bone up on things and hopefully enter my community college for College Algebra and run through the Calculus program. I just want to get to Differential Equations and then decide what next. I was a truck driver for awhile. It was a nightmare all the time and I got a few dogs now and just want to settle. I love your channel
@@markkromer1238 Yes sir. Abstract Algebra is (unless otherwise specified) Modern Algebra usually in college course descriptions. However, "College Algebra" is not Abstract Algebra, usually...it usually means a course very similar to High School level Algebra II/Pre-Calc at best. Thank you for your support! You have the best company! Dogs are always better company!
I'm also a serious learner, I want to become a mathematician, to make my dad's wish come true. He named me David, because he wanted me to become the next David Hilbert.
@jeffwads: Common Core Math has had disastrous results in this country...for both teachers, and students, in different ways. The Common Core, as an idea, is not "bad", however the teaching methods associated with it are not always appropriate for certain ages, and we also have a problem of teachers who do NOT know how to teach on those standards. I see nothing wrong with, for example, teaching different ways to multiply numbers, but many people are simply not trained well enough to teach something like that. In addition, many teachers at the elementary level are not good at Math themselves: www.jstor.org/stable/40248135 Aside from these points, I oppose federal Math standards as I believe it is unconstitutional federal overreach. "No Child Left Behind" is an utter failure. It has worsened educational outcomes! Nothing the federal government has touched in this country has resulted in anything good for the American people. Everything the feds touch, just turns to ash.
In accounting its different. Rounding pennies when reconciling monthly bank statements can come in handy. Better to eat the five cents than spend hours of expensive company time finding the discrepancy. That's where I love the concept of materiality. But in mathematics, I definitely love exact numbers, with square roots and pi and all that mess left in without rounding. It's much more artsy, meaningful, and beautiful.
Respectfully, it is your platform and you do whatever you want on it, but to be fair, I write whatever I want here as well. Well this whatver is going to be my opinion. I would personally feel ashamed to wash math and morality together. Though both are based on philosophy, math describes reality, morality prescribes our position in reality. Though you make sort of good arguments, it ruins the video for me about teaching/learning math and honestly is kinda out of place. If I were you, I would make these into 2 different videos from now on, so the people who wanna watch you talk about politics/morality can just do that and the ones like me, who are not American and don't give a damn about the American political situation, just the method and perspective of teaching Math can just watch that.
@xxateszxx: It really isn't hard to fast-forward the video as I indicated in the beginning. I don't think the lessons I discuss in this video are exclusive to Americans either. Morality is universal. Sorry you don't feel that way. Thank you for your comment.
@@citytutoring Well you are right on the larger part of it, it was a sort of less thought through comment from me, but I don't think that these subjects fit together, though my tutoring mostly included uni students same age as me, or a few years younger than myself, so I never had to talk about respect or being attentive since these people are basically adults as much as I am and usually are thinking just as logically as I do, just have a lack of understanding of a smaller topic.
listen man I follow you from Iraq, I know your channel not to much views but your authentic and original content is worth watching, so please keep going.
Thank you! I send you my greetings and well wishes for you. Iraq has a fascinating and rich history!
Immaculate linguistic segment as always professor, you must be popularized for your exquisite educational role. Looking forward to more!
This is unintentional on your part but the videos have a very particular vibe and "aesthetic" similar to that of ones found during the 2000s to early 2010s. Grainy footage, always done in one area, partly a personal blog and partly an educational resource on a particular subject. It definitely makes the channel more appealing compared to the mainstream of saturated slop which is only meant for zoomer retention.
I appreciate that feedback! Honestly, I don't even realize it! I guess I am indeed a creature of my particular era so it's sort of the format I am the most used to. I could not ever imagine myself hopping from place to place with so many visual distractions. Thank you again for your comment, because I take my commenters very seriously and feel a great sense of respect for you all.
Now that's the kind of awareness the young people of today need.
Very true!😊
That sounds like you don't know what mainstream math content looks like. One of the biggest maths channels on RUclips is 3Blue1Brown, and while they certainly don't have "grainy footage" or a "personal blog" (at least not on the main channel), they do have excellent maths content.
If you're looking for someone who puts more value on rigorous definitions rather than ways to intuitively understand the underlying concepts, you can also take a look at videos like those of @MathMajor, which basically give a comprehensive explanation in the form that you would expect from a proper maths course on the topics he presents.
The point in either case is that you shouldn't be demeaning towards people for no good reason.
@kappasphere - I worry about what I do, and have no time to look at what others do. I answer for what I do, and have no idea who all those people you mention are. If you are so keen and enamored with whoever those people are, you are more than welcome to spend your time on their pages or wherever it is they go. I am not "demeaning", I am honest. There's a difference, and it sounds like you cannot handle the truth, which is exactly what I mentioned in my video if you had bothered to watch it. I don't need to look at anyone else to do my job, thank you. Like I said, if you are are so into whatever other places you have mentioned, maybe spend time there rather than doing propaganda for them for free on MY channel. Don't let me keep you from enjoying your trendy non-grainy heroes!
I like the way you describe each symbol so poetically
Another great video, also congratulations on 8k subs, I'm not sure how long ago you hit that but you're already at 8.2k almost!
Thank you! This happened on Friday evening. The channel went over the 8K mark. I feel very blessed to have the subscribers I do, and the engagement has been amazing! I just can't keep up always, but I'll do my best.
Thanks for the math logic lessons!I will become stronger again!
Thank you for this video. The definitions were clean, memorable, and workable. I look forward to the problems!
Not directly math related, but on the topic of "making your student feel bad" - personal story from my time with BNSF Railway during my Conductor Trainee employment phase - on one of the first trains I Conducted (under the supervision of a fully licensed Conductor of course) I was with an engineer by the name of Christopherson. He was good, but he was tough; put me on the spot several times and didn't mince words if I fell short, which I did several times. Grilled me on radio procedure, train position, manifest, HazMat, and timetables (basically the Railroad's version of a road map, but with much higher stakes). The more he rattled me, the more I fell short. Yes, it was embarrassing, and yes, you could say my feelings got hurt. But after finally yarding our train and going to the hotel, I poured over all those procedures and timetables until I fell asleep and after callup the next morning I piled into the crew van with my Conductor and Christopherson to go to our next train. Christopherson didn't even say "Good morning", he went right back to asking me about the situations we'd encountered the day before, to which I had concrete answers and could explain where I went wrong and how I would handle things differently with hindsight. He just smiled and said, "You're welcome." He wasn't trying to be mean, he'd just been with enough trainees to know how to get us out of our comfort zone and MAKE US THINK! It worked too, because I still remember the details of that trip to this day, and it brought me a lot closer to earning my place in that seat 👍
Your ideas on truth resonate deeply. I want to expand on your statement about how neutrality comes from cowardice/fear. I would say anything that tries to conform to being "neutral" comes from fear. In fact, I could even say any negative emotion-sadness, depression, anxiety, anger, frustration, shame, disappointment, guilt, disgust, loneliness-all come from fear. Fear of being different; fear of not being accepted; fear of being out of your comfort zone; fear that things won't work out; fear of not being enough; etc. I am still sorting out all these fears, but I can acknowledge that they exist. My, what a better place the world would be if people could actually admit that.
There is a book called "The Course in Miracles" that delves very deeply into this. I can see that you hold to the faith of Christianity; coincidentally, "The Course in Miracles" was 'written' by Jesus Christ. It wasn't truly written by Jesus, but a woman claimed to have had a voice claiming to be Jesus who told her to take notes. I'm not sure of the validity of that, but the information in that book rings true and hits a lot of the ideas that you mentioned and even expands upon them. Truth is truth; it does not matter where or how it came. Connecting to my previous remark, "The Course in Miracles" claims anything that is not love is fear. There are more similar ideas mentioned in "The Course in Miracles," and if you haven't read it yet, I recommend checking it out. I think you would find the book interesting.
My goodness, what a powerful message and so very true. The I don't want to get involved comment with regard to the defence of someones honour is terrible and definitely a sign of character or should I say lack of it.
That is a great summary of the necessary math symbols. I'll need to remind myself of the Weierstrass p. I believe that something like 2^aleph-naught would be an example.
Hello I am a 25 year old viewer . I have watched one of your previous videos where you mentioned the books that impacted your math journey. I have read and worked through the exercises of those books. I was wondering if you would have time to make another video as a sort of continuation to the previous one. I didn’t go for a science degree at university but I find myself more interested in mathematics lately, I might just go back to school and get a math degree who knows. Thank You for your insightful content.
No color! Great choice. Professor Cromwell, perhaps it would be better to write a letter, but you mentioned your morning devotions in this video. I think that a good morning routine is the foundation of success. Can you tell a bit more about how your day starts?
Absolutely! Ever since I was in high school and truly got serious about my faith in God, I made it a point to begin the day with morning devotions. Usually, I will simply pick a random scripture without looking at the Bible and wherever my finger ends up, that's what I read. This morning it happened to be the book of Judges. When I'm at my devotions, I believe it grounds me in a spiritual connection with God. Our great Puritan ancestors had this habit, and many Christians after the 20th century lost the habit of devotionals (much like, unfortunately, evening services have been lost in many churches and vespers).
Aside from that, I always have a very light breakfast. Usually, my first drink of the day is a cup of coffee (made with a 19th century percolator because I like percolated coffee rather than the Keurigs and drip makers you see these days!) If not, I will use a 1930s vacuum coffee pot. Then, after about 1 hour or so, I will have shredded wheat on some days, with some honey, and a 2nd cup of coffee with cream. If I don't have shredded wheat, I will simply have a slice of toast with marmalade (naturally made, no artificial ingredients). I make my own bread at home as well, for toast, since most store bought bread has terrible ingredients.
It is also very important that personal hygiene be up to standard, whether it be a morning tub, or evening tub, I wouldn't go so far as to say cleanliness is next to godliness but it sure follows godliness! A man who is sloppy and doesn't take care to keep his body clean/washed has poor character and lack of respect for others.
@citytutoring I don't believe that it is always true that someone with poor hygiene has no respect for others, but it doesn't make the best impression. I remember going on a bus to a job fair at college and there was an old man not well dressed at all. My suit probably did not make the best impression to HIM.
I thought this was going to be a math video?! 😂 Subscribed!
I probably mentioned earlier, but the iff statement is used in math sometimes as well for if and only if.
Hello Professor, could you make a video giving advice on how to solve math proofs when you are stuck and don't know what to do? I find that the logical flow of an argument is easy to grasp, but how it was come up with I am unsure about. I would best explain it as, once I have the correct idea, the rest of the proof follows easily. However, when I have no idea at all I feel uneasy about where to begin, even though I recount certain definitions and theorems that may apply to the given problem, it just doesn't seem to lead anywhere.
I enjoy your videos, Sir. I had other questions in regards to book recommendations. Is Selby's Geometry and Trigonometry for Calculus worth purchasing? If not, what is a better alternative? Thank you.
Hi professor Cromwell , nice video , I only had one question at the end you included the element 0 in the set of natural numbers N . I thought N was the same as Z+ the set of positive integers and didn't have zero as a member ?
Great question! Zero is an integer, and (for some) zero is NOT a Natural Number, but it is not a settled question. I consider it a natural number, but there is a lot of debate around that.
The way I remember natural numbers is that counting numbers come "naturally" and therefore start at 1. It used to be that 1 was considered a prime number.
I think it depends on where you use them. Places where you include 0: If you have an addition structure (it is nice to have a neutral element), if you define them in formal set theory (Neumann ordinals for example), if you use them for "repeated action" formalism for other algebraic structures for example. Places where we ussually don't include 0: If we use it as an index set (0 can also be an index, but we specify it if we intend to use it), if we have a filter/ filterbase structure on it (cannot tell you why... we just do it), if we use it as infinite summation index (yes... most of the time we specify bounds), if we use it for enumerative/ countability arguments. Hope I could help.
Thanks for the video Professor Cromwell.
Since you have mentioned both morning devotions and mathematical symbols, I was curious about what sources on philosophy and theology you enjoy and recommend.
As a side question, how many hours of daily study of mathematics do you recommend?
Unfortunately, in places like Jamaica. The students in day care vs. The students in elementary school vs. The kids in middle school vs. High school are not the same kids and really not everyone makes it. Even if you gave any attention to the kids in day care, it would mean the world to them. It's really sad.
Not sure I understand?
15:22 wow I just transferred out of UF! Doing pure mathematics at UMD college park now! Trying to graduate early, I just want to do math all day!
Interesting! Did you know the Professor I link in the description?
@@citytutoring No, I was only there for a semester. Logistically it was actually quite terrible, and I'm glad I got out when I could. Starting next semester at UMD in less than a week. In fact I took no math classes last semester because they had strung a bunch of unrelated math classes together as pre requisites, so I could only take one at a time essentially. Then the only available class was full.
Sir what do you think about the Art of Problem solving books? Specifically the Algebra books and Number Theory books they have?
The Art of Problem Solving series is good for Math olympics, but I would not use them as textbooks in my classes. They do not, for example, teach you anything about Set Theory, and there are not enough computational fluency problems in their books. Most of their problems are very interesting, no doubt, but not adequate for a standard Math course.
@@citytutoring Well, the truth is that I’m interested in Calculus and want to learn it with the Spivak book, with enough Mathematical Maturity developed, I want to be able to understand the rigorous language, but I don’t know what textbooks to read beforehand to get familiar with the rigor, hence why I asked you about the Art of Problem solving books which I have heard are rigorous in nature. I’m currently reading a Mathematical Logic book, then I was planning to read the Art of Problem solving Number Theory book, then I want to delve into some rigorous algebra or precalculus books that will prepare me for Spivak. Essentially, I seek to build my mathematical maturity to rigorous textbooks. And textbooks are indeed my preferred method of study. Oh, and should I even read a logic or number theory book? I only started the logic book given that you’ve been talking about it lately.
It may have been me who mentioned term logic yesterday. The reason being is because in Philadelphia, there is a researcher of Artificial General Intelligence known as Pei Wang. I am in accordance with him though he never taught me and I never worked under his paradigm (yet).
Were you discussing Aristotelian logic?
I find the Book of Judges the most difficult to read. It's an unbroken chain of tragedy but what's really depressing is that it seems like all that suffering, all those mistakes result in no lessons learned whatsoever. I've spoken with ministers who try to put lipstick on the pig, who claim that it's not as bad as it looks. For example, they will say that the saga of Jephthah's daughter is not what it seems, that the poor girl was not slaughtered. Unfortunately, neither my reading comprehension nor ability to reason are impaired and I read the Bible often enough to know what I'm looking at.
You can look at the book of Judges as how God repeatedly raises up deliverers, demonstrating His sovereign grace and unmerited favor. These judges, though flawed, point to the ultimate Judge and Deliverer, Jesus Christ. The saga of Jephthah, while tragic, reflects how God's purposes can still be accomplished through human brokenness. The fate of Jephthah’s daughter is debated, but for me it reveals the tragic consequences of sin and the importance of careful vows before God. Whether she was sacrificed or dedicated to lifelong virginity, her story reflects the need for humility and reverence before God, aligning with Ecclesiastes 5:2: “Be not rash with your mouth, nor let your heart be hasty to utter a word before God.”
Re: Jephthah's daughter. Although Jephthah had professed faith in the true God, he was imitating the wickedness and pagan practices of the Canaanites (that is how he fell into the sin mentioned in Deut. 12:31). At that time, the Israelites were very influenced by that Canaanite culture.
The bleakness of Judges makes the coming of Christ in the New Testament shine even brighter.
Judges 2:16-19 encapsulates the repetitive failure of human judges, pointing us to the perfect and eternal Judge.
@@citytutoring Thank you for your thoughts. My problem is not so much with the ancient story as with the fact that our culture follows a similar path as Israel. In Galatians 6:7 the apostle Paul said, "God is not mocked..." which in my mind does not bode well for our civilization. In reading Judges I see the fate of this generation although I'm old enough that I'll probably not see the conclusion of things; but my children might.
Judges is a cool book.
Do you have a favorite class to teach? And is that class different from your favorite math class you took as a student?
Actually, they are the same. My two favorite classes were always Algebra (Advanced Algebra, and also Abstract Algebra). They remain my favorite classes to teach to this day.
@citytutoring Abstract Algebra is the same as Modern Algebra right? I'll be 53 in several weeks but I decided to return to get my mathematics degree. I'm going to bone up on things and hopefully enter my community college for College Algebra and run through the Calculus program. I just want to get to Differential Equations and then decide what next. I was a truck driver for awhile. It was a nightmare all the time and I got a few dogs now and just want to settle. I love your channel
@@markkromer1238 Yes sir. Abstract Algebra is (unless otherwise specified) Modern Algebra usually in college course descriptions. However, "College Algebra" is not Abstract Algebra, usually...it usually means a course very similar to High School level Algebra II/Pre-Calc at best. Thank you for your support! You have the best company! Dogs are always better company!
I'm also a serious learner, I want to become a mathematician, to make my dad's wish come true. He named me David, because he wanted me to become the next David Hilbert.
my dad named me after himself and i think i'm becoming to be like him
You probably answered this somewhere else, but, what do you think of that Common Core math? I can't imagine you are too impressed.
@jeffwads: Common Core Math has had disastrous results in this country...for both teachers, and students, in different ways. The Common Core, as an idea, is not "bad", however the teaching methods associated with it are not always appropriate for certain ages, and we also have a problem of teachers who do NOT know how to teach on those standards. I see nothing wrong with, for example, teaching different ways to multiply numbers, but many people are simply not trained well enough to teach something like that.
In addition, many teachers at the elementary level are not good at Math themselves: www.jstor.org/stable/40248135
Aside from these points, I oppose federal Math standards as I believe it is unconstitutional federal overreach.
"No Child Left Behind" is an utter failure. It has worsened educational outcomes!
Nothing the federal government has touched in this country has resulted in anything good for the American people. Everything the feds touch, just turns to ash.
22:28 LOL! And physicists too, they aren't as bad as engineers but it unironically irks me when they round numbers.
In accounting its different. Rounding pennies when reconciling monthly bank statements can come in handy. Better to eat the five cents than spend hours of expensive company time finding the discrepancy. That's where I love the concept of materiality. But in mathematics, I definitely love exact numbers, with square roots and pi and all that mess left in without rounding. It's much more artsy, meaningful, and beautiful.
Why is the first half of your video colorless?
Because my message is colorful enough.
You look like if Xavier Milei was a sane person and british.
Why the video is in black and white?
Because my message is colorful enough.
Respectfully, it is your platform and you do whatever you want on it, but to be fair, I write whatever I want here as well. Well this whatver is going to be my opinion. I would personally feel ashamed to wash math and morality together. Though both are based on philosophy, math describes reality, morality prescribes our position in reality. Though you make sort of good arguments, it ruins the video for me about teaching/learning math and honestly is kinda out of place. If I were you, I would make these into 2 different videos from now on, so the people who wanna watch you talk about politics/morality can just do that and the ones like me, who are not American and don't give a damn about the American political situation, just the method and perspective of teaching Math can just watch that.
@xxateszxx: It really isn't hard to fast-forward the video as I indicated in the beginning. I don't think the lessons I discuss in this video are exclusive to Americans either. Morality is universal. Sorry you don't feel that way. Thank you for your comment.
@@citytutoring Well you are right on the larger part of it, it was a sort of less thought through comment from me, but I don't think that these subjects fit together, though my tutoring mostly included uni students same age as me, or a few years younger than myself, so I never had to talk about respect or being attentive since these people are basically adults as much as I am and usually are thinking just as logically as I do, just have a lack of understanding of a smaller topic.