Hi, I'm Daniel, a teacher and independent curriculum writer. I teach, make videos, and write resources for teachers, parents, and students pursuing education rooted in goodness, truth, and beauty.
I am not so sure this approach would work very well in public schools, only in private schools with highly selective admissions criteria and highly trained and educated teachers. So 99 percent of American children will not and cannot receive it. They are simply not ready or able to begin with. Not to mention the consensus against "Eurocentric" learning based on the work of "Dead White European Males" because of the radically different demographics of the country. Nowadays the overwhelming trend is toward "multiculturalism", "diversity," "equity", and "decolonization" of the curriculum. Even many expensive private schools are going in this direction. And to top it off, the classroom of today is dominated by kids who have their minds (and eyes) focussed on what is on their cellphones, or literally have buds in their ears, or head phones over their ears, listening to their favorite music. I suppose the speaker has never witnessed or heard of this happening in schools, but it does. And, btw, there are still beatings taking place in schools. It is just the teachers and staff members who are receving the beatings. All of this is to say that the speaker has the best of intentions, quite noble actually, but he does not seem grounded in, well, what is happening on the ground. But more power to him, resisting the forces of social and cultural dissolution and chaos. I wish him well.
im not a fan of this approach, learn maths rigorously yeah, but historically? we're better at maths now than we were like noone should learn Galois theory as Galois wrote it. The modern treatments are far superior in generality and application to actual maths
Utter poppycock. Does this guy know any math? Has he ever taught anything? Aside from badgering people who do and who have? One great thing about "math" is that it becomes progressively clearer what it's actually about. That's why it makes literally no sense to go back looking for some lost essence.
It is just false to maintain that, since we speak English, what happened in 16th-century England has had the decisive effect on U.S. education. It is false because our educational system is primarily modeled on 19th-century German education. From Kinder-garten through the research university, our educational system of the period following the Civil War (the one in the United states, not the one in England) was shaped by educational authorities who consciously modeled, with variations, what the Germans were doing. Next, the entire claim that classical education was reformed by focussing on Latin in 16th-century England is ridiculous. ALL European education was based on Latin, going back to the Roman Empire. The Bible was in Latin. Church schools were the only schools. Man, this whole video stinks. When the Italian Renaissance revivified Platonism, intellectuals across Europe engaged more seriously in mathematics, since Plato taught that the divine was approached by means of mathematics -- a doctrine articulated eloquently by Galileo. Nothing to do with English educational reform. Finally, the most ridiculous aspect of this hopeless, clueless video is the failure to understand that mathematics is involved in SOLVING PROBLEMS. If you can solve a set of challenging problems well, you're a trained mathematician. Sigh.
Mathematics as a field underwent a revolution in the late 19th century, and the material from before around 1880 is completely trivial compared to what came later. While it is important to read the historical material as you suggest, it is very important to do so only after learning 20th century methods, because they are so much more powerful than the earlier methods. For example, Euclid's approach to the Pythagorean theorem is relatively dated, although it needs to be learned. It is important to learn the modern formulations of the theorem first, and then examine the classics. This is difficult to do. Also, your emphasis on the classical literature is not good, it is a type of traditionalism, which undercuts progress.
Please do a quick TLDR of what the video is going to be about. You may not realize this, but there are a billion videos on youtube and not a lot of time, so if you do get suggested by the algorithm, please don't expect people to stick around for a detailed treatment of things you are passionate about. I mean 14 minuts to get to what classical maths is, in a video that is titled "What is classical mathematics" ...
I was hoping to find a personal way of reviewing my self in geometry... But this was how we were taught geometry... So, goodbye... I'm not even hoping to be a teacher... Education is not for me... I am for math. I am for engineering. Not education... Because I finished a bachelor's degree in engineering, not education.
I love this! Looking back, it wasn’t until I got to grad school that I started learning math in a way similar to this classical approach you mentioned. I wonder how much more deeply I could understand things had I started learning this way sooner.
That's a good question. Doing problems is an excellent way to learn the skill aspect of math, but math is not merely a computational skill. There is also quite a bit of theoretical understanding one must have, and the best way to achieve that is through a combination of problem solving and note-taking. The point of the note-taking is to enable you to make connections between ideas and better retain the information you've studied--which in turn will enable you to make better connections. To put it another way, problem solving is a way of looking at the individual trees in the forest, but the reflection inherent in note-taking provides you with a broader perspective, ideally to see large sections of the forest.
Well said Daniel. To carry on with the idea of engagement I also use my notes to agree or disagree with the author, why an approach is brilliant or silly, and general free flowing thoughts on the topic. It somewhat ties back with your video on writing your own textbook but I'm just modifying the given text/lesson with my reactions to it. (+1 for the Blackwing Box.)
I was educated at a Jesuit School. I had five years of Latin and three years of ancient Greek. I also have a doctorate in Mathematics with a 30 year academic and industrial career in my field. While I enjoyed learning classical languages, they had NOTHING to do with learning math. You’re teaching the history of mathematics- not mathematics. Mathematics evolves in part by simplifying and generalizing older theorems and proofs. Forcing students to read the works of Greek mathematicians handicaps their progress in mathematics because the original proofs no longer provide the best explanations. In fact it hobbled the development of the most significant mathematician of the twentieth century, Alan Turing. Evangelicals lost all credibility in mathematics when Van Til published his epistemology.
Thanks for your comment. While I understand your points, I respectfully disagree that students are handicapped by being exposed to historical math texts. I don't advocate teaching solely historic Greek mathematics and never have. What I'm attempting to do is re-humanize high school mathematics at a time when textbooks are soulless and ultra-pragmatic, completely failing to address the wonder, complexity, creativity, and beauty of mathematics in favor of bare computational competency. What I'm working toward is a way of teaching high school mathematics in a way that simultaneously 1) produces greater competency (by emphasizing the interconnectness of mathematical ideas, which standard high school textbooks do badly), and 2) gives students a glimpse into the beauty and creativity of mathematics by exposing them to its historic development (including pointing out how old prejudices get in the way of progress--e.g. your example or Viete's algebra being crippled by the law of homogeneity). In any case, you might disagree with my methods, which is fine, but those are my goals, and I think historical texts are key in achieving them. There is no discipline that does not benefit from studying its own history. Thanks again for your thoughtful input! I appreciate the feedback.
Agree with what you say in its entirety - "Memory is a byproduct of thought" - i.e. if you do not think about it, you will not remember it. I used to encourage students to do work at home by explaining how this helps to make memories more permanent. The more pathways that exist to a memory, the stronger the reinforcement, so by associating the learning with different environments / people / emotional states /etc. more pathways to the memory are formed. The biggest issue most young people face when trying to learn, I think, comes from poor teaching in High School, this is caused by multiple factors but one thing that always helps is the understanding that we can only build strongly on the things we already know (our schema) so 1) we must identify students' existing schemata and 2) Ask when planning for every lesson you teach "What is the one core concept every pupil must add to / have in their schema to succeed in the learning outcome of the lesson". Linking key concepts to pupils' underlaying schema is a very powerful tool, as a well developed schema allows pupils to work things out for themselves more easily and completely. Teaching is not the imparting of knowledge, it is guiding pupils to learn whilst being a facilitator, dispeller of misconceptions, motivator, solid point in reality (always the same, ready to repair relationships, reacting the same way to every event, every time) and to impart a love of learning. Keep up the great work, the world needs more people like yourself, helping the next generation to succeed and grow.
Great points! Thanks for your comment. I'll admit that I disagree that teaching isn't the imparting of knowledge. There must necessarily be *some* content that is passed from teacher to student; you can't teach students to "think for themselves" in a vacuum. But--and I think this is your point--we do our students a disservice when teaching becomes *merely* an imparting of knowledge. It's a balancing act, like nearly everything else in life, and it's all too easy to err by extremes.
Skip to 13:48 to get the point in the video. He want to expose students to great articles in the history of math. (Similar to the way that we read the great authors when we study litterature)
It is very difficult to read the classics of mathematics without a foundation in modern textbook mathematics. Although it is important to read the classics, it must always be done after you learn the modern methods, because it cuts down the time to read the classics by a factor of 10, sometimes a factor of 100.
I actually think that the emphasis on logic and proofs is very important to transport the beauty of mathematics. I like the attitude of mathematics that you actually need to prove something before you believe it and I think it's important to prove that to students. It does probably help many to tell stories about mathematicians who found those insights, but I personally don't need that. Rather than how other people got to insights, I care about how I can get to those insights. I also don't need to hear stories of Columbus to be curious about America, I need to learn about its impact on our lives and our impact on America. Learning is a lot about personal relevance, and mathematics have a lot more personal relevance than just beauty that is not transported sufficiently.
I care more about ''visuals', Can you picture it and explain it simply? Can you develop an intuition of it that a child can understand and use it as a metaphor to help explain your ordinary world? Sure.. Study Euclid, but it's more important to develop intuitions about say Logarithms and Calculus can describe and predict your own development and experiences. How Trigonometry can explain cyclic behavior. Chaos Theory predict future recourse. Can Mathematics enhance one's reading of Literature and Movies?
The best math classes I have attended had prepared note sheets. These were Calculus II and III. Each class would have multiple exercises where we would work out the problems and fill in the sheets. The problems were very relevant to the tests and quizzes. The notes were also graded. That was just one way to get students to engage more with the material and have some structured examples that they could study before tests.
That sounds like a good idea as well. Were these note sheets presenting problems analogous to the examples presented in class? Or were they identical to the examples given by the professor?
Alternatives: shoving the content into flashcards or into patchwork of hypertext documents (like private Wiki), instead of assemblying a textbook out of it.
Those are good alternatives. I like the flashcard idea better than the wiki idea; I steer students away from digital solutions whenever possible for various reasons. The main downside of the flashcards is that it becomes difficult to treat them hierarchically without introducing cataloging methods or something like a Zettelkasten system. If a student is already experienced with such methods, however, index cards would be a pretty cool alternative.
Hi why avoiding digital solutions? I use my tabled Samsung Ultra for reading/annotating the textbook and for doing all the exercises. All my notes and worksheets are clean and not messy, such as when I used the paper. I can erase erros and use simple coloring schemes and highlighting. Also all material is nicely organized in folder and/or labelled. The screen is huge so that I can put the textbook and the notes on each half of it. Writing with the digital pen is such a pleasure. I still use paper occasionally, if I have not the tablet with me or if I don't plan to keep the stuff I'm writing.
England?? What about Italy? I get it's about education and not 'research', but none of the English people you mentioned at 3:36 was a great mathematician.
Precisely. I'm not discussing mathematicians here, but rather the theorists who gave us classical education. Later in the video, I situate mathematics within the classical framework. Thanks for your comment!
Very insightful video! One big question I have coming out the other side pertains to your first point about moving too quickly through material. I definitely agree that we shouldn't be leaving students who still have confusions behind by proceeding to new content too hastily, however I wonder about the opposite problem: What is to be done for the exceptional student who may feel bored having to go through all the repetition necessary to ensure everyone is on the same page? The reason I'm curious about this is I suspect the exceptional student could become disengaged with the subject in much the same way as the struggling student, but for opposite reasons. Instead of feeling like they're left behind they may feel like they're never truly moving forward (perhaps to put it another way they may feel like they're being held back from "getting to the good bits", whatever that may mean to them). I find it difficult to reconcile the needs of these two hypothetical students (struggling and exceptional), let alone accommodating the whole spectrum of needs we may find in between. I'd love to get your thoughts on this since you seem to have dedicated a great deal to thinking about these kinds of issues. P.S. A bit of background about myself: I am not a teacher, I am currently an undergraduate physics student. I don't know if I'll go into teaching after, but I certainly think about education quite frequently. This is why I've made plenty of use of the speculative "may" above since I haven't personally found myself in quite this scenario yet. The only experience I have in teaching mathematics so far is in the extremely informal setting of helping my peers where I can when they get lost. There's definitely a sense of gratification I get in the moment where clarity manifests itself in their expression, it helps me feel like I really do know what I'm talking about. Who knows, maybe this sense of gratification I get will inevitably push me to pursue teaching sometime down the line!
This is a great question. Unfortunately, there's no easy answer for how to balance the pace to account for both struggling and exceptional students. Practically speaking, if the environment allows for it, perhaps one good thing to do would be to give the exceptional students opportunities to present/teach key aspects of the lesson to the class. This would have to be done carefully to avoid creating an obvious cultural split between "good" and "bad" students in the classroom. However, keeping gifted students engaged by allowing them to share their insights is usually beneficial to all. Thanks for your comment!
I was always using the gym analogy too. It’s great to see someone more experienced and qualified thinking in the same lens. Thanks for the amazing video.
Would you happen to have any recommendations for a returning student with a decent understanding of pre-algebra but has ambitions to understand math up to and beyond calculus and linear algebra but from a classical perspective?
@andso7068, great question. TRACK 1) If you wanted to trace the birth of algebra in the most thorough way, I'd begin by reading Diophantus alongside my guidebook _Diophantine Algebra_ (polymathclassical.com/curriculum-diophantine-algebra/). You could then proceed to read through Viete's _Preliminary Notes On Symbolic Logistic_ and the first 2 or 3 books of Zetetica (all contained in this book: www.amazon.com/Analytic-Art-Francois-Vi%C3%A8te/dp/0486453480). From there, things get tricky. Descartes should be one of the next stops on your trip, but you can't really appreciate what he's trying to do in his _Geometry_ without first reading bits of Apollonius' _Conics_ . That said, the first book of Descartes' _Geometry_ is relatively approachable, even without knowing anything about Apollonius. At this point, it becomes increasingly difficult to tackle the primary sources without first having mastered the mathematics in question. TRACK 2) If you wanted to do things an easier way, I'd suggest beginning with _Mathematics for the Nonmathematician_ by Morris Kline and _Journey through Genius_ by William Dunham. From there, I'd pick and choose a few interesting primary sources (based on what caught my attention in those two books), and read them alongside a standard math textbook (to fill in any gaps). For Calculus, it would be interesting to read Newton alongside a standard calculus text--or, for a more humanities-based approach to calculus, you might also look into Mitch Stoke's _Calculus for Everyone_ . Again, yours is a great question, and it's the same question I've been trying to answer for years, and no simple solution exists yet. But I hope what I've laid out above is helpful.
Hello, what’s your plan for the channel? Hoping more videos! This is the best mathematics channel on RU-vid as it focuses more on building the world of math for yourself in order to develop deep and thorough understanding. Thank you!
Thank you! I do have plans to continue making videos. I've got two more videos planned out; I just need to find time to record them. I teach full-time during the year and am writing a Ph.D. dissertation, so finding time to make videos can be a bit of a challenge--but I definitely plan to stick around. Thanks for your comment!
I learned a lot from your video. I’m curious on your thoughts about the negative consequences of using only European examples of math thinkers and doers. For example, with Algebra and our number system being rooted in Persia, are you really providing examples of creativity if you present the European re-presentation of Persian constructions? Also, I’m not very familiar with classical education. When you describe literature and Latin instruction, is it accurate to say that you de-emphasize students’ opportunities to write and create their own work? Is it not a detriment to not provide opportunities for students to be problem solvers themselves? Or to construct their own theorems or proofs? I know you said math should not be considered a mental exercise. However, when would students have the opportunity to become critical thinkers and problem solvers and creative with mathematics? Maybe I’m misunderstanding. I would appreciate some clarity here.
Those are great questions. Rather than respond here, I'm hoping to make a video soon to address them. I'll let you know once I do. Thanks for your comment!
@KristynSartin, I've just published a video addressing your questions. Thanks again for asking some great questions and giving me the chance to engage with them!
Renaissance humanism was a worldview centered on the nature and importance of humanity, that emerged from the study of Classical antiquity. Bing search
Or, just teach the actual classical mathematical arts of the Quadrivium from the actual classical texts that were standard in Late Antiquity and the Middle Ages (all of which have readily available English translations). Arithmetic • Nicomachus of Gerasa's _Introduction to Arithmetic_ **or** • St. Boethius' _De arithmetica_ (his Latin translation and edition of Nicomachus' earlier Greek text) Geometry • Euclid's _Elements_ Music • Nicomachus of Gerasa's _Manual of Harmonics_ **or** • St. Boethius' _De institutione musica_ (again, his Latin translation and edition of Nicomachus' earlier Greek text) And Astronomy • Claudius Ptolemy's _Almagest_
That said, I appreciate the other thoughts presented in the video. Like exposing them to various examples of different discoveries of mathematics and the beauty of it.
That's a great idea! The problem I'm working on at present is how to present high school mathematics _as students need it today_ using classical texts as a framework. It's not so hard to do with Geometry, since Euclid's _Elements_ is a single text and pretty thorough. It's much more difficult to do with Algebra, since Algebra was developed in bits and pieces across various treatises over many centuries. Ideally, students should take Calculus at some point, so the curriculum has to be arranged so as to make that possible. Essentially, this makes places us in the difficult position of having to balance the teaching of primary texts with decontextualized math instruction in order to make 1) the primary texts intelligible and 2) give students the math they need for college, etc. I wish we lived in a world where I could do exactly what you outline. That would be spectacular. I'm just not sure how to make that work for high school, given the expectations and requirements of high school math. Your outline would be great for a supplementary course of study, or as part of an undergraduate program _a la_ St. John's College, etc. Thanks for your comment!
@@polymathclassical William Michael, of the Classical Liberal Arts Academy, is actually transitioning his school's curriculum for Algebra to be based on and work through Leonhard Euler's _Elements of Algebra_ (1765). I have a copy of it myself (the Cambridge Library Collection edition), but just looking at it makes my brain go fuzzy, so best wishes to him in his endeavors. 😂 He might be on to something, though. Any way to make such a fundamental text for modern mathematics more accessible is appreciated, and having two great teachers both coming up with their own ways to do so benefits everyone. So there's a thought. I'm at a loss for calculus, though, as I never took it formally and it still frightens me (as all math does on a primal level, if I'm being honest; I'm much more a history and theology person).
@@HickoryDickory86 I've also been considering using _Euler's Elements of Algebra_ as a core text, so that's good to know that I'm not the only one with that idea. I might consider that further, then. I haven't thought ahead to Calculus much, but I've heard great things about Mitch Stokes' book _Calculus for Everyone_ . I actually own a copy, but haven't had a chance to read through it yet. Thanks for your valuable input!
Thanks for the comment. That's the one downside to getting one's information primarily out of books--there's no one around to tell you how names are pronounced!
